Optical Metrology for Precision Engineering 9783110542363, 9783110541090

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Wei Gao, Yuki Shimizu Optical Metrology for Precision Engineering

Also of Interest Optical Electronics. An Introduction Jixiang Yan,  ISBN ----, e-ISBN (PDF) ---- e-ISBN (EPUB) ---- Metrology of Automated Tests. Static and Dynamic Characteristics Viacheslav Karmalita,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Optical Measurement Mechanics Kaifu Wang,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----

Frontiers in Optics and Photonics Federico Capasso, Dennis Couwenberg (Eds.),  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----

Metrology and Theory of Measurement Valery A. Slaev, Anna G. Chunovkina, Leonid A. Mironovsky,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----

Wei Gao, Yuki Shimizu

Optical Metrology for Precision Engineering

Authors Prof. Wei Gao Department of Finemechanics Graduate School of Engineering Tohoku University 6-6-01 Aramaki Aza aoba Sendai 980-8579 Japan [email protected] Prof. Yuki Shimizu Department of Finemechanics Graduate School of Engineering Tohoku University 6-6-01 Aramaki Aza Aoba Sendai 980-8579 Japan [email protected]

ISBN 978-3-11-054109-0 e-ISBN (PDF) 978-3-11-054236-3 e-ISBN (EPUB) 978-3-11-054127-4 Library of Congress Control Number: 2021943619 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Cover image: Wei Gao Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Precision engineering is a discipline to develop and apply principles of design, manufacture, control and measurement for precision machines and precision manufacturing. Precision metrology, often called dimensional metrology, is one of the cornerstones to support precision engineering. It is the science and technology of measurement for quantifying physical quantities derived from the SI base quantities of length and time, including geometrical properties of an object such as length, angle and surface topography, as well as kinematic properties such as displacement, velocity and acceleration. A high degree of precision and a low degree of uncertainty are required for such a measurement. Precision metrology is particularly important for modern manufacturing industries based on mass production of interchangeable parts by using various machine tools. The concept of interchangeability has been the greatest achievement of the twentieth century, which has substantially changed not only the ways of manufacturing but also the lifestyles of human beings. In interchangeable manufacturing, the parts are required to be made to the designed geometrical specifications so that the parts made in different places by different operators can be exchangeable with each other for efficient assembly of new products and easy repair of existing products. This can only be made possible when the necessary level of quality control is established in the manufacturing processes based on precision measurements of the geometrical specifications of the manufactured parts. Precision measurement of the kinematic properties of machine tools is also essential for supporting the interchangeable manufacturing. On the other hand, light has long been utilized in measurement activities of human history. Optical measurement technologies have evolved to an independent branch of metrology called optical metrology. Optical metrology can be applied for measurement of almost all kinds of physical quantities, including the SI base quantities of time, length, mass, temperature, luminous intensity and their derived quantities. Noncontact, fast and sensitive are some of the important features of optical metrology. Optical measurement technologies are playing an increasingly important role in precision metrology where accurate measurements are required to be made in a short time and with minimal influence on the measurement target. This book presents the state-of-the-art optical technologies of dimensional metrology for precision engineering. This book is composed of 12 chapters. The manuscripts of Chapters 1–5, 7–9 and 11 were mainly written by Wei Gao and the rest were written by Yuki Shimizu. Yuki Shimizu also prepared most of the figures and made the final edition of the entire book. The first half of the book, from Chapters 1 to 6, presents optical sensors and systems with conventional continuous-wave laser source. Optical sensors for the measurement of angular and translational displacements, which are fundamental quantities in precision metrology, are presented in Chapters 1–4. Chapter 1 describes optical angle sensors based on laser autocollimation, which can make nanoradian angle measurement while maintaining a compact https://doi.org/10.1515/9783110542363-202

VI

Preface

sensor size. The angle sensor is then expanded for three-axis measurement by using a refraction grating as the target reflector in Chapter 2. In Chapters 3 and 4, surface encoders are presented for XY, XYZ or Z–θZ multi-axis motion measurement by using grating scales. A concept of mosaic surface encoder is also presented for expansion of the measurement range. Chapter 5 presents a number of unique optical scanning profilers for surface topographic measurement under in-line and/or onmachine conditions. In Chapter 6, fabrication of grating scales by interference lithography is presented together with fast and accurate calibration technologies. The second half of the book, from Chapters 7 to 12, presents the next-generation optical measurement technologies based on ultrashort pulse laser and optical frequency comb. Ultrashort pulse laser angle sensing methods are presented in Chapter 7 by taking use of the unique characteristics of ultrashort pulses, including second-harmonic generation. Chapter 8 provides a mathematical analysis on the optical frequency comb, where the theorem of Fourier transform, including Fourier integral and Fourier series, is applied to analyze the relationship between a train of ultrashort pulses and the optical frequency comb as well as to reveal the key features of optical frequency comb. Chapters 9 to 12 present optical sensors based on optical frequency comb; an angle comb for absolute angle measurement, a chromatic confocal comb for absolute displacement measurement, a position comb for absolute position measurement, and applied comb metrology including a comb surface profiler. I wish that this book can provide the newest information on the advancement in optical dimensional metrology to engineers in areas of precision engineering, optical engineering, mechanical engineering and nanotechnology, as well as the postgraduate and undergraduate students learning in these fields. I sincerely wish that this book could inspire research fellows and postgraduate students to create new ideas in dimensional metrology, especially in the next-generation measurement technologies based on optical frequency comb and ultrashort optical pulses. This book is the record of an important part of the research work that the authors have been involved in for over the past decade. I would like to thank my colleagues and many students in the Precision Nanometrology Laboratory for their significant contributions to the development of the technologies presented in this book. Hiraku Matsukuma, So Ito, Yuanliu Chen, Xiuguo Chen, Akihide kimura, Yusuke Saito, Takemi Asai, WooJae Kim, SungHo Jang, Xinghui Li, Zengyuan Niu, Xin Xiong, Chong Chen, Lue Quan, Wijayanti Dwi Astuti, Dong Wook Shin, Kentaro Uehara, Shinji Sato, Haemin Choi, Kouji Hosono, Hiroshi Muto, Shinichi Osawa, Takeshi Ito, Dai Murata, Tatsuya Ishikawa, Yukitoshi Kudo, Ryo Aihara, Taiji Maruyama, Masaya Furuta, Jun Tamada, Taku Nakamura, Shaoqing Yang, Zongwei Ren, Shuhe Madokoro, Kazuki Mano, Ryo Ishizuka, Yuri Kanda, Ryo Sato and Shota Takazono are some of them. Mr. Ryo Sato carefully checked and revised Chapters 8 and 11. Prof. Hiraku Matsukuma also made comments to these two chapters. I would like to thank Ms. Karolina Sobańska, Ms. Cruz-Kaciak Aneta, Mr. Leonardo Milla, Ms. Chao Yang, Mr. Joachim Katzmarek and other staffs of De Gruyter for their dedicated efforts that have

Preface

VII

made this book possible. The financial support from the Japan Society for Promotion of Science is also appreciated. Special thanks to Dr. Hajime Inaba, Head of Optical Frequency Measurement Group, National Metrology Institute of Japan, AIST, for his invaluable advice on the development of the fiber-based frequency comb system in Chapter 11. The invaluable advice from Prof. Lijiang Zeng, Tsinghua University on grating fabrication is also appreciated. I wish to express my thanks to my wife, Hong Shen, for her warm support. The same appreciation also goes to Mrs. Yayoi Shimizu, Prof. Yuki Shimizu’s wife, for her support to her husband. This book was written in a difficult situation of COVID-19. It would never have been completed without the patience, sacrifice, encouragement and support from the two families. Finally, I would like to thank my co-author, Prof. Yuki Shimizu, for his hardworking on this book as well as his great contribution to the development of the measurement technologies presented in this book. Wei Gao Sendai, Japan March, 2021

Contents Preface

V

Chapter 1 Laser autocollimator 1 1.1 Introduction 1 1.2 Nanoradian laser autocollimator 3 1.3 Rangefinder autocollimator 22 1.4 PD-edge method associated with laser autocollimation 1.5 Summary 42 References 43 Chapter 2 Three-axis angle sensor 47 2.1 Introduction 47 2.2 Three-axis angle sensor 50 2.3 Three-axis inclination sensor 2.4 Summary 98 References 99

72

Chapter 3 Surface encoder 103 3.1 Introduction 103 3.2 Three-axis surface encoder 105 3.3 Six-degree-of-freedom surface encoder 112 3.4 Reduction of cross-talk error in surface encoder 3.5 Linear-rotary surface encoder 145 3.6 Summary 159 References 160 Chapter 4 Mosaic encoder 165 4.1 Introduction 165 4.2 Mosaic linear encoder 165 4.3 Mosaic surface encoder 169 4.4 Four-probe three-axis mosaic surface encoder 4.5 Summary 198 References 199

128

174

32

X

Contents

Chapter 5 In-line and on-machine surface profiler 201 5.1 Introduction 201 5.2 Flexspline gear profiler 202 5.3 Rotary die cutter profiler 210 5.4 Microstructure slope profiler 219 5.5 Diamond tool profiler 228 5.6 Microdrill bit profiler 237 5.7 Mixed-probe profiler 247 5.8 Summary 256 References 257 Chapter 6 Fabrication and calibration of scale grating 261 6.1 Introduction 261 6.2 Orthogonal two-axis Lloyd’s mirror interferometer 262 6.3 Non-orthogonal two-axis Lloyd’s mirror interferometer 277 6.4 High-resolution calibration of linear grating scale 313 6.5 Fast calibration of planar scale grating 322 6.6 Summary 331 References 332 Chapter 7 Ultrashort-pulse angle sensor 335 7.1 Introduction 335 7.2 Ultrashort-pulse laser angle sensor 7.3 Chromatic dispersion angle sensor 7.4 Second harmonic wave angle sensor 7.5 Summary 375 References 376

340 348 356

Chapter 8 Optical frequency comb 379 8.1 Introduction 379 8.2 Mathematical model of optical frequency comb 8.3 Effects of dispersion on optical frequency comb 8.3.1 Effect of phase velocity dispersion 389 8.3.2 Effect of group velocity dispersion 399 8.4 Stabilization of optical frequency comb 406 8.4.1 Stabilization of repetition frequency 406

380 389

Contents

8.4.2 8.4.3 8.5

Stabilization of carrier-envelope frequency Compensation of group velocity dispersion Summary 422 References 422

Chapter 9 Angle comb 425 9.1 Introduction 425 9.2 Intensity-domain angle comb 426 9.3 Optical frequency-domain angle comb 9.4 Summary 477 References 478

408 416

451

Chapter 10 Chromatic confocal comb 481 10.1 Introduction 481 10.2 Optical comb chromatic confocal probe 483 10.3 Range-expanded chromatic confocal comb 494 10.4 Stabilized chromatic confocal comb 505 10.5 Improved chromatic confocal comb 522 10.6 Summary 537 References 537 Chapter 11 Position comb 541 11.1 Introduction 541 11.2 Real-time monitoring of repetition frequency 11.3 Fiber frequency comb 554 11.4 Position comb 581 11.5 Summary 596 References 597 Chapter 12 Optical comb applied metrology 601 12.1 Introduction 601 12.2 Multi-beam form measurement 602 12.3 Micro-prism angle measurement 607 12.4 Diffraction grating calibration 628 12.5 Summary 636 References 638 Index

641

543

XI

Chapter 1 Laser autocollimator 1.1 Introduction A precision linear motion slide is an important element in precision engineering. It is used to generate precision linear motions in machine tools and scanning-type measuring instruments [1, 2]. A linear motion slide, which is often called a linear slide or a linear stage, is composed of a moving table and a straight guideway. A moving table is sometimes called a carriage or a saddle, and a straight guideway is often referred to as a rail. In a linear slide, the carriage is moved to a command position along a straight line. The line is referred to as the axis of motion of the carriage (X-axis). The closeness of the command position to the actual position of the carriage is called the positioning accuracy. On the other hand, the closeness of the actual path of the carriage movement to the X-axis is referred to as the straightness motion accuracy. Many precision linear slides are required to have sub-micrometric positioning accuracy and straightness motion accuracy [3]. For slide precision positioning, it is necessary to make closed-loop feedback control of the slide carriage by using the result of position measurement with an accurate position sensor, which is often a linear encoder [4]. With the advances in sensor technologies, sub-micrometric or even nanometric precision positioning of the slide carried has been realized [5]. On the other hand, the straightness motion accuracy is mainly determined by the form accuracy of slide guideways. A pair of parallel guideways is typically employed in a linear slide to provide physical constraints to the slide carriage in the directions vertical to the axis of motion of the carriage [6]. Any flatness errors in the surface form of each guideway will cause corresponding out-ofstraightness motion errors of the slide carriage [7]. The flatness error of the guideway surface can also generate tilt motion errors of the slide carriage. For obtaining submicrometric straightness motion accuracy, it is necessary to machine the guideways with sub-micrometric flatness accuracy. In addition, it is also necessary to identify the tilt motion accuracy after the carriage is assembled with the guideways. As shown in Fig. 1.1, not only the tilt motion accuracy of the slide carriage but also the flatness of the slide guideway can be measured by an autocollimator. In the machining process, the slide guideway is first ground by a grinding machine to a flatness on the order of several micrometers [8, 9]. To further reduce the flatness error to sub-micrometer level, a hand scraping operation is carried out. Although such an operation is time-consuming and highly dependent on the skill of the operator, it is effective to improve the guideway flatness beyond the machining accuracy of a machine tool [9]. It is thus an essential operation for production of precision linear slides used for high-end machine tools and measuring instruments. Before the scraping operation, a measurement operation is carried out to measure the https://doi.org/10.1515/9783110542363-001

2

Chapter 1 Laser autocollimator

actual flatness of the guideway surface by using an autocollimator [6]. Based on the measured flatness, the excess areas on the guideway surface are roughly scraped away by using a scraper. Then, the autocollimator is used again to identify the residual flatness error of the scraped guideway surface. Such a pair of measurement and scraping operations is repeated until a sub-micrometric flatness is achieved.

Fig. 1.1: Measurement of slide guideway flatness and carriage title error motion by using an autocollimator.

It is also necessary to link the measured tilt motion error of the moving carriage as well as the measured flatness of the slide guideways with the X-directional positions of the measurement points with a millimeter order accuracy. In the case of the hand scraping operation, the information of the X-direction positions is employed for identifying the positions of the excess areas to be scraped on the guideway surfaces. It has been a problem for a conventional autocollimator since it does have the function for position measurement. It is also a problem for the conventional autocollimator when it is used to measure the tilt motion error of the slide carriage during the assembly process where the linear encoder for closed-loop feedback positioning of the slide table is not available. This makes the error correction process and the assembly process more timeconsuming and inefficient. A laser interferometer has a nanometric resolution and a long-range measurement of the displacement of a reflector [10]. A multi-axis laser interferometer with multiple laser beams [11] or a surface encoder based on grating interferometry [12, 13] can also measure the tilt angle of the reflector. However, most of the commercial interferometers are incremental types and can only measure relative displacement instead of absolute position [14]. The output of a laser interferometer will also be lost if the optical path of the laser beam is blocked. The high cost of a laser interferometer is another shortcoming and the high-resolution capability is also not necessary for the millimetric position measurement in the scraping process and the assembly process.

1.2 Nanoradian laser autocollimator

3

Laser autocollimators, which are composed of a laser light source and an autocollimation unit with a collimator objective (CO) and a light position detector (LPD), have the advantages of fast measurement speed and compact size [15–18]. This chapter presents a laser rangefinder autocollimator, which is a laser autocollimator combining with a low-cost laser rangefinder based on the time-of-flight method [19], for use in the scraping and the assembly processes of a linear slide. A nanoradian laser autocollimator, which can be used for evaluation of an ultrahigh precision linear slide, is also presented [20] together with a PD-edge method associated with laser autocollimation [21, 22].

1.2 Nanoradian laser autocollimator With the advances in nanotechnology, the resolution of an angle sensor for measurement of an ultraprecision linear slide (linear stage) with a nanometric positioning accuracy is required to be higher than 0.001 arc-second (5 nanoradians). The resolution of a laser interferometer system is typically lower than 0.04 arc-second [23]. An autocollimator employing a CO with a long focal length can achieve a resolution of 0.005 arc-second with a charge-coupled device [24] or 0.002 arc-second with an analog silicon-based sensor [25]. However, such autocollimators tend to be bulky; for example, the total length of the sensor head for the autocollimator with a resolution of 0.002 arc-second is more than 270 mm. It is sometimes difficult to employ such a large autocollimator in the production line of ultraprecision linear stages where the space is restricted. It is thus more desirable to improve the resolution of laser autocollimation [26] that has the advantage of compactness. As shown in the previous section, a bi-cell photodiode (BPD) or a quadrant photodiode (QPD) is typically employed as a LPD to achieve a high response speed. For such a laser autocollimator, the measurement sensitivity does not depend on the focal length of the CO and a resolution of 0.01 arc-second has been achieved with a short focal length where the focused light spot on the QPD has a small diameter comparable to the QPD cell gap [27]. However, it is difficult to further improve the resolution since most of the energy in the focused light spot with a further smaller diameter will be lost at the insensitive areas (gaps) of QPD, resulting in a degradation of a signal to noise ratio (S/N) of the autocollimator output [28]. As a solution to this problem, a new optical configuration with single-cell photodiodes (SPDs) is presented in this section for angle measurement with a nanoradian-order resolution. The optical configuration of a two-directional (2D)-type nanoradian laser autocollimator is shown in Fig. 1.2 [18, 20, 22]. A laser diode (LD) and SPDs (SPD1, SPD2 and SPD3) are employed as the light source and LPDs, respectively. The light beam from LD is formed to be a collimated measurement beam by a collimating lens (CL). For simplicity, the divergence of the collimated measurement beam is not considered in the following discussions. After being reflected from the reflector surface,

4

Chapter 1 Laser autocollimator

the beam is divided into two parts by using a beam splitter (BS1). One part is made incident to SPD3 for monitoring the intensity fluctuation of LD. The other part is further divided into two sub-beams again by using another beam splitter (BS2). The two sub-beams are made to focus on the SPD1 and SPD2 planes through the COs CO1 and CO2, respectively. The combinations of SPD1/CO1 and SPD2/CO2 are used as the autocollimation units of θY and θZ, respectively.

Fig. 1.2: Optical configuration for a 2D-type nanoradian laser autocollimator with single-cell photodiodes (SPDs).

In the θY unit, the focused light spot is positioned on the edge of the PD cell of SPD1 to detect the light spot displacement Δu(v) due to the tilt motion ΔθY. Similar to that shown in eq. (1.1), the relationship between ΔθY and Δu(v) can be described by the following equation: ΔuðvÞ = f · tan 2θYðZÞ ≈ 2f ΔθYðZÞ

(1:1)

where f is the focal length of CO1. Assume the center of the focused light spot is positioned on the edge of the PD cell on SPD1 when ΔθY = 0, and the intensity distribution of the light spot on the plane of SPD1 is denoted by I1(u,v). The output of the θY-unit uSPD_out [%] can be calculated by the following equation: ÐÐ I1 ðu, vÞdudv × 100 (1:2) uSPD out = ÐÐ S1 S1 total I1 ðu, vÞdudv where S1 means the area of the light spot received by the PD cell of SPD1. S1_total is the total area of the light spot on the plane of SPD1. The numerator in eq. (1.2) can be estimated from the magnitude of the photocurrent output of SPD1 and the denominator can be estimated from half of the magnitude of the photocurrent output of SPD3. Similarly, the output of the θZ-unit vSPD_out [%] can be calculated by

1.2 Nanoradian laser autocollimator

ÐÐ vSPD

out

= ÐÐ

S2 I2 ðu, vÞdudv

S2 total I2 ðu, vÞdudv

× 100

5

(1:3)

where I2(u,v) is the intensity distribution of the light spot on the plane of SPD2. S2 is the area of the light spot received by the PD cell of SPD2. S2_total is the total area of the light spot on the plane of SPD2. The numerator and denominator in eq. (1.3) can be estimated from the magnitudes of the photocurrent outputs of SPD1 and SPD3, respectively. If I1(u,v) and I2(u,v) are uniform distribution, eqs. (1.2) and (1.3) can be simplified as follows: 1

uSPD

2 out =

vSPD

2 out =

1

πðd1 =2Þ2 + d1 · Δu πðd1 =2Þ2

πðd2 =2Þ2 + d2 · Δv πðd2 =2Þ2

× 100 = 50 +

8f ΔθY × 100 πd1

(1:4)

× 100 = 50 +

8f ΔθZ × 100 πd2

(1:5)

where Ds1 and Ds2 are the diameters of the focused light spot on the planes of SPD1 and SPD2, respectively. Differing from the case of using QPD, there is no restriction on the acceptable focused light spot diameters Ds1 and Ds2 when an SPD is employed for light position measurement. The measurement sensitivity can therefore be maximized by minimizing the focused light spot diameters Ds1 and Ds2 to the light diffraction limit. As a result, the optical configuration shown in Fig. 1.2 is expected to achieve a higher measurement resolution compared with that achieved using QPDs. When the CO is free from aberrations, the focused light spot diameter ddiff on the focal plane of the CO will be determined by the diffraction limit as shown in Fig. 1.3. Assume that the light beam of diameter D0 received by the CO has a uniform intensity distribution. ddiff can be expressed by the following equation [29]: ddiff =

2.44f λ D0

(1:6)

where λ and D0 are the wavelength and the diameter of the collimated beam from CL, respectively. The focal plane of an aberration-free system is often referred to as a paraxial focal plane [29]. Meanwhile, in practice, the CO is not free from the aberrations. As a result, the focused light spot diameter at the paraxial focal plane will be affected by the aberration. For the laser autocollimator, the spherical aberration is one of the factors dominating the diameter. As can be seen in Fig. 1.4, when D0 becomes large, a marginal ray strikes the position with a distance of dTA from the optical axis on the paraxial focal plane due to the influence of the spherical aberration. dTA can be determined by the Gaussian optics as follows [30]:

6

Chapter 1 Laser autocollimator

dTA =

D0 3 32f 2 ðn − 1Þ2

"

 2 # R2 n+2 R2 n − ð2n + 1Þ + R2 − R1 R2 − R1 n 2

(1:7)

where n, R1 and R2 are the refractive index and surface radii of the CO, respectively. From eq. (1.7), it can be seen that the influence of the spherical aberration increases with the increase of D0. According to eqs. (1.6) and (1.7), an increase in D0 will cause a decrease of ddiff but an increase of dTA, which has opponent effects on the measurement resolution of light spot position by SPD. Therefore, both the influences of light diffraction and spherical aberration need to be taken into consideration in the determination of D0 for the nanoradian laser autocollimator. It should be noted that the spherical aberration not only causes an increase in the diameter of the focused light spot, but also causes a disturbance in its intensity distribution, which influences the sensitivity of the laser autocollimator. In the following, computer simulation is carried out based on wave optics to investigate the influences of the spherical aberration.

Fig. 1.3: Focused light spot without the influence of spherical aberration.

Figure 1.5 shows an optical model for simulating the autocollimation unit. In the optical model, a plano-convex lens is employed as the CO, while a pupil plane and the SPD plane are placed at its front focal position and back focal position, respectively. The pupil plane having an aperture diameter of D is included in the optical model so that the corresponding measurement beam diameter D0 can be adjusted. The intensity distribution of the focused light spot I1(u,v) on the SPD plane can be described by the following equation [31]:

1.2 Nanoradian laser autocollimator

7

Fig. 1.4: Focused light spot with the influence of spherical aberration.

I1 ðu, vÞ =

ðð 2    1  2π  ð x, y ÞPðx + u, y + vÞ exp − j U ð xu + yv Þ dxdy l  2 2 λf λf

(1:8)

where P(x,y) and Ul(x,y) are a pupil function and a complex field across the pupil plane, respectively. The pupil function P(x,y) is defined as follows: ( 1 ðwhen ρðx, yÞ < 1Þ Pðx, yÞ = (1:9) 0 ðwhen 1 < ρðx, yÞÞ where ρ(x,y) is the normalized radial coordinate, and can be described as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 ρðx, yÞ = (1:10) D=2 Once I(u,v) is calculated from eq. (1.8), the autocollimator outputs can be calculated from eqs. (1.4) and (1.5). In the optical model, the pupil plane is assumed to be illuminated by a plane wave with an amplitude of A uniformly across the surface. The influence of the spherical aberration of the CO can be expressed as a wavefront error, and can be implemented in the calculation by using the complex field Ul(x,y). In the following calculations, Ul(x,y) represented by the following equations is applied to eq. (1.8) [30, 31]:

8

Chapter 1 Laser autocollimator

8 >
 : 4 2 A · exp j 2π λ Sðρ ðx, yÞ − ρ ðx, yÞÞ ðwith aberration, at best focusÞ

(1:11)

where S is referred to as the peak aberration coefficient. In the case of a planoconvex lens, S can be described by [30, 32] h i − n3 + ðn + 2Þ + ð3n + 2Þ ðn − 1Þ2 − 4ðn2 − 1Þ · D4 S= (1:12) 32nðn − 1Þ2 f 3

Fig. 1.5: An optical configuration for computer simulation based on wave optics.

Parameters employed in the following calculations are summarized in Tab. 1.1. The values of the parameters are determined to be consistent with the following experiments. Figures 1.6, 1.7 and 1.8 show examples of the Ul(x,y) calculated by

1.2 Nanoradian laser autocollimator

9

using the parameters in Tab. 1.1. The wavefront errors are represented in a wave number. As shown in Fig. 1.7, the influence of the spherical aberration becomes significant near the edge of the projected aperture diameter. Tab. 1.1: Parameters used in the computer simulation. Parameter

Value Unit

Wavelength (λ)

 [nm]

Focal length of the collimator objective (f)

 [mm]

Refractive index of the collimator objective (n)

. –

Aperture diameter on the pupil plane (corresponding to the measurement laser beam diameter) (D)

– [mm]

Fig. 1.6: Wavefront errors due to the spherical aberration of the collimator objective applied to the simulation model (D0 = 14 mm) without the spherical aberration.

At first, the relationship between the intensity distribution I1(u,v) and the beam diameter D0 is calculated. Figures 1.9 and 1.10 show the calculated I1(u,v) for D0 of 2 and 14 mm, respectively. The cross sections of the results are shown in Figs. 1.11 and 1.12, respectively. In the figures, I1(u,v) is normalized by the maximum intensity calculated with the aberration-free system In the case of small D0 (=2 mm) shown in Fig. 1.9, the influence of the spherical aberration on I1(u,v) at the paraxial focal plane is negligibly small. However, the influence becomes large with the increase of D0, as shown in Fig. 1.10. Meanwhile, even under the influence of the spherical aberration, I1(u,v) at the best focal plane is found to be almost identical to that calculated with the aberration-free system. The 1/e2 diameter of the focused light spot on the SPD plane is then calculated with respect to D0 based on the results of Figs. 1.11 and 1.12. As shown in Fig. 1.13, the 1/e2 diameter of the focused light spot at the best focus continuously decreases

10

Chapter 1 Laser autocollimator

Fig. 1.7: Wavefront errors due to the spherical aberration of the collimator objective applied to the simulation model (D0 = 14 mm) without the spherical aberration with the spherical aberration, at paraxial focal plane.

Fig. 1.8: Wavefront errors due to the spherical aberration of the collimator objective applied to the simulation model (D0 = 14 mm) without the spherical aberration with the spherical aberration, at the best focal plane.

with the increase of D0. On the other hand, the diameter at the paraxial focus decreases and increases with the increase of D0 when D0 is smaller and larger than 11 mm, respectively. By using the results of Fig. 1.9, the output uSPD_out of laser autocollimator is calculated. Figure 1.14 shows the calculation procedure. At first, the total power of the focused light spot on the whole SPD plane is calculated. After that, the relationship between the displacement of the focused light spot Δui and the tilt angle ΔθYi of the reflector is evaluated. The sum of the total power of the focused light spot on the SPD active area is then calculated. Finally, uSPD_out at each ΔθYi is evaluated. Figures 1.15 and 1.16 show the results of simulation for D0 of 2 mm and 14 mm, respectively. The curves of the autocollimator output uSPD_out versus the tilt motion

1.2 Nanoradian laser autocollimator

11

Fig. 1.9: Calculated intensity distribution of the focused light spot on the SPD plane. D = 2 mm.

Fig. 1.10: Calculated intensity distribution of the focused light spot on the SPD plane. D = 14 mm.

of the reflector Δθ(uSPD_out–Δθ curves) are plotted in the figures. The sensitivities of the autocollimator outputs are also calculated from the data in the uSPD_out–Δθ curves for Δθ of −5 arc-seconds to 5 arc-seconds. The results are plotted in Fig. 1.17. As shown in Fig. 1.15, the uSPD_out–Δθ curves are almost the same regardless of the existence of the spherical aberration when D0 is 2 mm. When D0 is 14 mm, however, the sensitivity of the autocollimator decreases significantly at the paraxial focal plane as shown in Fig. 1.16. It should be noted that the sensitivity clearly decreases

12

Chapter 1 Laser autocollimator

Fig. 1.11: Cross sections of the intensity distributions in Fig. 1.9.

Fig. 1.12: Cross sections of the intensity distributions in Fig. 1.10.

Fig. 1.13: Variation of the focused light spot diameter on the SPD plane.

at the best focal plane under the influence of the spherical aberration, even though the laser beam diameter is almost the same as that of the aberration-free system, as shown in Fig. 1.17. From these results, it is revealed that a slight disturbance in the intensity distribution can cause a decrease in the sensitivity of the autocollimator output.

1.2 Nanoradian laser autocollimator

Fig. 1.14: Calculation procedure for calculation of the laser autocollimator output.

Fig. 1.15: Calculated uSPD_out–Δθ curves. D = 2 mm.

13

14

Chapter 1 Laser autocollimator

Fig. 1.16: Calculated uSPD_out–Δθ curves. D = 14 mm.

Fig. 1.17: Variation of sensitivity as a function of D0.

The results of the computer simulation were compared with those of experiments. At first, the influence of the spherical aberration on the focused light spot diameter was investigated experimentally. Figure 1.18 shows a schematic of the experimental setup. A laser beam from a 685 nm wavelength LD was collimated by using an aspherical lens. The collimated laser beam was then expanded by using a beam expander, and the expanded laser beam was made to pass through a diameter-variable iris. The beam expander was employed not only for expanding the laser beam diameter but also for making the intensity distribution of the laser beam to passing through the iris as uniform as possible. The laser beam that had passed through the iris with an aperture diameter of D0 was then made to go through a plano-convex CO with a focal length f of 100 mm. The focused light spot was monitored by a commercial beam profiler (BeamScan XYS/LL/5 μm, Photon Inc.). At the beginning of the experiment, the position of the beam profiler along the optical axis was adjusted with a narrow laser beam with a diameter D0 of 2 mm. The beam profiler was kept stationary during the experiment at a position almost identical to the paraxial focus. The variation of the 1/e2 diameter of the focused light spot was measured

1.2 Nanoradian laser autocollimator

15

Fig. 1.18: Experimental setup for measurement of the focused light spot.

when D0 was increased from 2 to 14 mm, Fig. 1.19 shows the results. In the figure, the focused light spot diameter calculated in the simulation is also plotted. A good agreement can be found between the results of the simulation and the experiment.

Fig. 1.19: Measured focused light spot diameter.

A laser autocollimation unit was constructed for investigating the influence of the spherical aberration on the sensitivity of the autocollimation unit output. Figure 1.20 shows the optical configuration of the autocollimator unit. The reflector in Fig. 1.18 was

16

Chapter 1 Laser autocollimator

mounted on a piezoelectric (PZT) tilt stage, which had been calibrated by using a commercial autocollimator (Möller–Wedel Optical GmbH, Elcomat 3,000). The beam profiler in Fig. 1.18 was replaced with the autocollimation unit, which consists of a BS and two SPDs: SPD1 and SPD2. Both SPD1 and SPD2 were placed on the focal plane of the CO. SPD1 was aligned on the focal plane in such a way that the center of the focused light spot was made to coincide with the edge of the PD cell. Meanwhile, the PD cell of SPD2 was aligned to capture the whole energy of the focused light spot. It should be noted that the alignments of the SPDs were carried out by setting the iris aperture diameter D0 to be 2 mm. The PD cell of SPD1 was located at a position almost identical to the paraxial focus. Bi-cell Si PIN photodiodes (Hamamatsu S4204) were employed for the SPDs. As shown in Fig. 1.21, one of the PD cells of the photodiode was employed as the SPD1.

Fig. 1.20: A schematic of the laser autocollimation unit for evaluation experiment of sensitivity.

In the experiments, a continuous sinusoidal tilt motion was applied to the reflector via the PZT tilt stage. Photocurrent outputs from SPD1 and SPD2 were converted into voltage outputs by transimpedance amplifiers. The voltage outputs were recorded simultaneously by a digital oscilloscope. Figures 1.22, 1.23 and 1.24 show the

1.2 Nanoradian laser autocollimator

17

Fig. 1.21: Pictures of the SPD1.

Fig. 1.22: Measured uSPD_out–Δθ curves. D0 = 2 mm.

Fig. 1.23: Measured uSPD_out–Δθ curves. D0 = 10 mm.

measurement results for different D0. In the figure, the calculated uSPD_out–Δθ curves by simulation are also plotted. The sensitivities of the autocollimation unit outputs, which were calculated from the data in the measured uSPD_out–Δθ curves ranging from Δθ of −5 to 5 arc-seconds, are shown in Fig. 1.25. As can be seen in the figure, it

18

Chapter 1 Laser autocollimator

is effective to improve the sensitivity by increasing D0 when D0 was smaller than 10 mm. On the other hand, the sensitivity decreased with the increase of D0 when D0 was larger than 10 mm, which was due to the influence of spherical aberration of the CO. The results of the experiments were consistent with those of simulation.

Fig. 1.24: Measured uSPD_out–Δθ curves. D0 = 14 mm.

Fig. 1.25: Comparison of sensitivities by experiments and simulation.

A nanoradian laser autocollimator with a compact size was then designed and constructed based on the results of simulation and experiments shown above. Figure 1.26 shows the optical layout of the nanoradian laser autocollimator. It is a one-directional (1D) type that is simplified from the 2D type shown in Fig. 1.20. The autocollimator was designed in a size of 100 mm (X) × 150 mm (Y). The same LD with a wavelength of 685 nm in Figs. 1.18 and 1.20 was employed as the light source. The transimpedance amplifier and the digital oscilloscope in Figs. 1.18 and 1.20 were also employed. However, the beam expander in Figs. 1.18 and 1.20 was not employed due to the limitation in sensor size. A CL, which was provided by an optics manufacturing company, was selected for generating a measurement laser beam that had a smooth intensity distribution profile with reduced interference and speckle noises. The diameter of the measurement laser beam was 5 mm, which was determined by the CL. For this reason, investigation of the effect of a beam diameter D0 on the angular resolution was limited within 5 mm in the following experiment. An optical isolator, which was a combination of a polarizing

1.2 Nanoradian laser autocollimator

19

beam splitter (PBS) and a quarter-wave plate was employed to avoid the influence of stray light rays on the laser source. An achromatic doublet lens with reduced spherical aberration was employed as the CO. The focal length of the CO was set to be 100 mm. The same photodiodes shown in Fig. 1.21 were utilized as the SPDs. A picture of the nanoradian laser autocollimator is shown in Fig. 1.27, which was built on a vibration isolation table. A flat mirror reflector was mounted on the PZT tilt stage.

Fig. 1.26: Optical layout of the nanoradian laser autocollimator.

The short-term stability of the nanoradian laser autocollimator was tested, from which the angular resolution could be estimated. The center of the focused light spot was aligned to be on the edge of the PD cell of SPD1 by using the PZT tilt stage for the test. During the test, the tilt stage was kept stationary and the outputs from SPD1 and SPD2 were captured. The variation of the voltage outputs of SPD1 over a term of 5 s is shown in Fig. 1.28. The standard deviation (σ) of the voltage outputs from both the SPD1 and SPD2 was confirmed to be approximately 0.3 mV, corresponding to a fluctuation of the sensor output of approximately 0.00323%. The uSPD_out–Δθ curves were measured by applying a periodic sinusoidal tilt motion about the Y-axis was applied to the reflector, from which the sensitivity of the

20

Chapter 1 Laser autocollimator

Fig. 1.27: Picture of the nanoradian laser autocollimator.

Fig. 1.28: A typical waveform of the SPD output through the transimpedance amplifier.

nanoradian laser autocollimator output could be evaluated. Figure 1.29 shows the measured uSPD_out–Δθ curves. In the measurement, D0 was adjusted by inserting an aperture between the CL and the PBS as shown in Fig. 1.26. The results by simulation are also shown in Fig. 1.29 for comparison. The sensitivities of the nanoradian laser autocollimator outputs were calculated by using the data in the measured uSPD_out–Δθ curves ranging from Δθ of −5 to 5 arc-seconds. As expected, the sensitivity was improved in proportion to D0. A fairly good correlation can be found between the results of the computer simulation and experiments. When D0 was set to be 5 mm, the sensitivity was 4.57%/arc-second. Regarding the fluctuation of the output shown in Fig. 1.30 (0.00323%), a resolution of approximately 0.0007 arc-second (3.4 nrad) is expected to be achieved by the nanoradian laser autocollimator. Dynamic tilt motion measurements were carried out by using the nanoradian laser autocollimator. The outputs of the laser autocollimator were acquired when a periodic sinusoidal tilt motion with an amplitude and a frequency of 0.001 arcsecond (4.8 nrad) and 1 Hz, respectively was applied to the reflector by the PZT tilt stage. The results are shown in Fig. 1.31. In the experiment, D0 was set to be 5 mm. It

1.2 Nanoradian laser autocollimator

Fig. 1.29: The uSPD_out–Δθ curves of the nanoradian laser autocollimator by experiment and simulation.

Fig. 1.30: Sensitivities of the nanoradian laser autocollimator.

Fig. 1.31: Measurement result of a nanoradian angular motion.

21

22

Chapter 1 Laser autocollimator

can be seen in the figure that the sinusoidal tilt motion was clearly detected, verifying the capability of the laser autocollimator for nanoradian angle measurement.

1.3 Rangefinder autocollimator As shown in Fig. 1.1, an autocollimator is an optical instrument used with a reflector for measurement of the tilt angle of the reflector. The autocollimator has the advantages of easy use, high measurement resolution and long working range [27]. During the hand scraping process as well as the assembly process, the optical path of the autocollimator may be intermittently blocked by the operator. The measurement can be re-started from the blocked position without losing the tilt angle information measured by the autocollimator. This is an important feature for practical use in the hand scraping and assembly processes. When the autocollimator is employed for surface flatness measurement in the hand scraping process, the reflecting mirror is fixed on a two-footed base. The base is moved on the surface of the guideway along the X-axis step by step over the length of the surface to be measured in such a way that the forward foot of the base is positioned at the point where the rear foot previously rested. At each of the measurement points, multiplying the reading of the autocollimator and the distance between the two feet of the base gives the height difference between the two feet. The flatness of the guideway surface can thus be obtained from the height difference data [33]. Assuming that the distance between the two feet of the base is 100 mm, a 0.2 arc-second reading of the autocollimator corresponds to a height difference of approximately 0.1 μm, which is good enough for the sub-micrometer flatness measurement of the guideway surface. In the assembly process, the autocollimator is employed to measure the tilt motion errors (pitch and yaw) of the slide carriage when it is mounted on the guideways. A tilt motion error of the slide carriage is one of the sources for the positioning error of the slide. A linear slide is used to move a cutting tool in a machine tool or a measuring probe in a measuring instrument. In most cases, the cutting position or the measuring position has a certain distance from the slide axis of motion, which is called the Abbe offset. Even a small tilt motion error can generate a sub-micrometric error in the cutting or measuring position (positioning error), which is referred to as the Abbe error [17, 34]. For example, a 0.2 arc-second tilt motion error will cause a positioning error of approximately 0.1 μm for a 100 mm Abbe offset. If the measured tilt motion error is too large, it is necessary to return to the hand scraping process for further correction of the flatness errors of the slide guideways. Figure 1.32 shows a schematic of a laser rangefinder-autocollimator that can measure both the tilt angle and the position of a reflector by combining an autocollimation unit and a rangefinder unit [19, 35]. The measurement resolutions/ranges for tilt angle and position are set to be 0.2/±40 arc-seconds, 1 mm/5 m, respectively. In the sensor, a LD is employed as the light source. The laser light with a wavelength λ from

1.3 Rangefinder autocollimator

23

Fig. 1.32: A schematic of the laser rangefinder-autocollimator.

the LD is collimated by using a CL before it is projected onto the target reflector. Denoting the diameter of the output beam at the CL by D0 and the beam divergence angle by βbeam, the beam diameter Dbeam at a position x can be written by Dbeam ðxÞ = D0 + βbeam · x

(1:13)

It should be noted that the beam divergence is not shown in the figure for the sake of clarity. As shown in the figure, the specular reflection light from the reflector is bent by the BS1 before it enters the autocollimation unit. The autocollimation unit consists of a CO and a LPD placed at the focal position of the objective. Denote the tilt angle of the reflector at position x by Δθ(x), the focal length of the objective by f and the displacement of the focused light spot on the detector by ΔdLPD(x), respectively. For a small Δθ(x), it can be expressed by the following equation [36]: ΔθðxÞ = arctan

ΔdLPD ðxÞ ΔdLPD ðxÞ ≈ 2f 2f

(1:14)

Meanwhile, in the rangefinder unit for position measurement of the reflector, a collecting lens (CL) is employed to collect a part of the diffuse reflection light rays. If the distance of the reflector from the LD is long enough, the optical path from the LD to the reflector and that from the reflector to the light intensity detector 1 (LID 1) of the rangefinder unit can be regarded as the same. Under this condition, the position x of the reflector can be obtained based on the principle of time of flight from the following equation [37]: c x = Δt 2

(1:15)

24

Chapter 1 Laser autocollimator

where c is the speed of light and Δt is the time for a pulsed light to travel from the LD to the detector via the reflector.

Fig. 1.33: A schematic of the laser rangefinder unit with phase modulation.

As can be seen in eq. (1.15), the resolution of the simple time-of-flight principle is determined by time measurement. For a millimeter position measurement resolution, a pico-second order time measurement resolution is required, which is difficult to achieve. A phase modulation technique [38] shown in Fig. 1.33 is typically employed to improve the resolution of the rangefinder for position measurement. BS2 is added to the optical path of the rangefinder to generate a reference beam. The reference beam is received by LID 2. Assume that the light intensity of the pulsed laser light from the LD is modulated with a frequency fref. By measuring the phase difference Δϕ between the pulse train in the reference laser beam and that in the measurement laser beam, the position x of the reflector can be obtained from the following equation [38]: x=

c Δ’ 4πfref

(1:16)

Table 1.2 shows the specifications of a commercial laser rangefinder unit that is selected for the laser rangefinder-autocollimator [39]. As can be seen in the table, a modulation frequency of 66.67 MHz is adopted for a position measurement resolution of 1 mm over a range of 30 m [39]. A low price is also an advantage of the laser rangefinder. An autocollimation unit is then designed for integration with the rangefinder unit. At first, the diameter DC of the CO and the width wBS of BS1 are determined based on Fig. 1.34. It can be seen in the figure that the specular reflection light from the reflector is received by the collimator lens at position P when the tilt angle of the reflector at the position of LR reaches its maximum (ΔθR). The distance from P to the center O of the CO is denoted by tR and the laser beam diameter at the CO is denoted by Dbeam (2LR). The following equations can be obtained: DC = 2tR + Dbeam ð2LR Þ

(1:17)

25

1.3 Rangefinder autocollimator

Tab. 1.2: Specifications of the commercial laser rangefinder [39]. Item Wavelength of laser diode Resolution Range Modulation frequency (fref) Measurement time Beam diameter (D) Beam divergence angle (bbeam) Working temperature Size

Value Unit . [μm]  [mm]  [m] . [MHz] . [s]  [mm] . [rad] −‒ [ºC]  ×  ×  [mm]

Fig. 1.34: Determination of the diameter of CO and the size of BS1 for the autocollimation unit.

tR = LR tanð2ΔθR Þ ≈ 2LR ΔθR

(1:18)

Substituting eqs. (1.13) and (1.18) in eq. (1.17) gives

 DC = D0 + 2LR 2ΔθR + βbeam

(1:19)

Based on D0 (=3 mm), LR (=5 m), ΔθR (=40 arc-seconds) and βbeam (=0.16 mrad), DC is calculated to be 8.48 mm. A CO with a diameter of 10 mm was selected based on the result. Similarly, the width of WBS of BS1 is determined to be 10 mm based on the calculated DC.

26

Chapter 1 Laser autocollimator

A BPD or a QPD can be chosen as the LPD for 1D or 2D angle measurement [16, 26, 40]. Figure 1.35 shows the light spot that is focused on the LPD with a gap of gLPD (=10 μm) by the CO. The diameter Ds of the light spot is expressed by DS =

4f λ + f βbeam πD0

(1:20)

The first term in the right side of eq. (1.20) is determined by light diffraction and the second term is caused by the beam divergence [29].

Fig. 1.35: Determination of the focal length of CO for the autocollimation unit.

Assume that the light spot is located at the center of the LPD when the tilt angle of the reflector is zero and the width of the LPD cell is much larger than the diameter of the light spot. The light spot moves a distance of ΔdR to its leftmost or rightmost positions when the tilt angle of the reflector reaches its maximum value of ΔθR. The relationship between gLPD, Ds and ΔdR can be expressed by DS ≥ gLPD + 2ΔdR = gLPD + 4f ΔθR

(1:21)

The focal length of the CO can then be obtained as follows by combining eqs. (1.20) and (1.21): f≥

4λ πD0

+

gLPD βbeam − 4ΔθR 4

(1:22)

f is calculated to be larger than 63 mm from eq. (1.22) and a CL with a focal length of 80 mm is selected for assurance of a large enough measurement range of tilt angle. Figure 1.36 shows the layout of the laser rangefinder-autocollimator [19]. A QPD is employed as the light position photodetector in the autocollimation unit for measurement of 2D tilt angles (pitch ΔθY and yaw ΔθZ). As can be seen in the figures, the autocollimation unit is placed on the same side as the CL of the rangefinder unit for a compact structure. The size of the laser rangefinder-autocollimator is

1.3 Rangefinder autocollimator

27

Fig. 1.36: Layout of the laser rangefinder-autocollimator.

Fig. 1.37: Picture of the laser rangefinder-autocollimator.

250 mm × 205 mm × 63 mm. It is compact enough for use as a standalone measuring instrument in the hand scraping and assembly processes of a precision linear slide. Figure 1.37 shows a picture of the laser rangefinder-autocollimator. The laser beam from the rangefinder unit works is a high-frequency modulated pulsed laser beam. However, constant laser power is required for the laser autocollimation unit. It is necessary to avoid the influence of the pulsed laser power on the measurement of the autocollimation unit. A sequence shown in Fig. 1.38 is then employed to control the laser power. In each measurement cycle, the laser power is first controlled to be constant in the term T1 for the measurement of tilt angle by the

28

Chapter 1 Laser autocollimator

autocollimation unit. Then, the laser is switched to output a beam with pulsed laser power in the term T2 for the measurement of position by the rangefinder unit. The start of each measurement cycle shown in the figure as well as the switching from T1 to T2 are made by pushing the measurement button of the rangefinder manually. This is acceptable for the hand scraping and assembly processes since only static measurements are required.

Fig. 1.38: Sequence of laser power control.

Fig. 1.39: Setup for testing the basic performances of the autocollimation unit.

The basic performances including the range, linearity and stability of the laser rangefinder-autocollimator were tested. Since the performance of the commercial rangefinder unit for position measurement has been reported by the manufacturer, the test was focused on the autocollimation unit for tilt angle measurement. Figure 1.39 shows the setup for the test. A PZT-driven 2D tilt motion stage was employed to generate the yaw (ΔθZ) and pitch (ΔθY) tilt angles. A mirror was mounted on the tilt motion stage as the reflector for the autocollimation unit. The tilt angles of the stage were closed-loop controlled with strain gauges integrated into the stage. The initial tilt angles of the target mirror were adjusted to locate the center of the specular reflection beam from the reflector to that of the QPD. Figures 1.40 and 1.41 show the 2D outputs of the autocollimation unit when tilt angles were applied to the mirror in the ΔθZ and ΔθY directions, respectively. The distance x of the reflector was set to be 242 mm. It can be seen that a range of ±50 arc-seconds was achieved in the two directions. The linearity error of the output was within ±4.2 arc-seconds. The main reason for the nonlinearity was the non-uniform intensity

1.3 Rangefinder autocollimator

29

distribution of the laser beam. The results of the short-term stability test are shown in Figs. 1.42 and 1.43, from which the resolution of the autocollimation unit can be indirectly evaluated. During the test, the reflector was kept stationary. The test term was set to 5 s. The stabilities were within 0.15 arc-second for ΔθZ and 0.18 arc-second for ΔθY, which implies the resolution of the autocollimation unit satisfied the requirements use in the hand scraping and assembly processes of precision linear slides.

Fig. 1.40: Two-dimensional outputs of the autocollimation unit when tilt angles were applied to the reflector (L = 242 mm).

Fig. 1.41: Two-dimensional outputs of the autocollimation unit when tilt angles were applied to the reflector (L = 242 mm).

Figures 1.44 and 1.45 show the results of testing the cross-talk errors between the measurements of ΔθY and ΔθZ. A sinusoidal tilt angle motion was applied to the reflector by the PZT tilt motion stage along the direction of ΔθZ. The frequency and amplitude of the periodic tilt motion were 1 Hz and 50 arc-seconds, respectively. The outputs of the autocollimation unit in the two directions under this condition are shown in Fig. 1.44 where the cross-talk error in the direction of ΔθY was evaluated to

30

Chapter 1 Laser autocollimator

Fig. 1.42: Short-term stability of the autocollimation unit θY.

Fig. 1.43: Short-term stability of the autocollimation unit θZ.

Fig. 1.44: Cross-talk errors of the autocollimation unit in ΔθY.

1.3 Rangefinder autocollimator

31

Fig. 1.45: Cross-talk errors of the autocollimation unit in ΔθZ.

be 1.8 arc-seconds. Then the sinusoidal tilt motion was applied in the direction of ΔθY. The corresponding outputs are shown in Fig. 1.45 where the cross-talk error in the direction of ΔθZ was evaluated to be 1.4 arc-seconds. The cross-talk errors were caused by the misalignment of the axes of the QPD. The laser rangefinder was employed to measure a linear guide. The setup is shown in Fig. 1.46. A reflector was mounted on the stage plate of the linear guide. The axis of motion of the linear guide was set to be the X-axis. The measured pitch (ΔθY) and yaw (ΔθZ) error motions are shown in Figs. 1.47 and 1.48, respectively. The position of the reflector at each measurement point was detected by the rangefinder unit. It can be seen from the figure that the yaw and pitch error motions were measured to be approximately 25 arc-seconds and 50 arc-seconds, respectively, over a movement distance of 110 mm along the X-direction.

Fig. 1.46: Setup for measurement of a linear guide.

32

Chapter 1 Laser autocollimator

Fig. 1.47: Measured tilt error motions of a linear guide (yaw error ΔθY).

Fig. 1.48: Measured tilt error motions of a linear guide (pitch error ΔθZ).

1.4 PD-edge method associated with laser autocollimation As presented in the previous sections, the sensitivity of laser autocollimation for angle measurement is significantly influenced by the diameter of the laser beam focused on the PD cell. In addition, with the increasing use of lasers in research laboratories and machine shops for measurement and materials processing, evaluation of the diameter of a small focused laser beam is getting more important for the assurance of product quality and/or fabrication resolution. A simple and cost-effective method for diameter measurement of a focused laser beam is thus required. The knife-edge method [41–43] is the most well used method for beam diameter measurement. Figure 1.49 shows a schematic of the knife-edge method. In this method, a knife-edge is made to scan across the laser beam. The beam diameter is evaluated by reconstructing the intensity distribution of the beam by measuring the optical powers passed through the knife-edge during the scanning. On the other hand, the influence of light diffraction occurring at the knife-edge becomes more

1.4 PD-edge method associated with laser autocollimation

33

significant as the decrease of the laser beam diameter, which cannot be neglected where the photodetector is placed at a certain distance from the knife-edge. As shown in Fig. 1.50, this problem can be overcome by placing a thin-film structure, known as a knife pad [44–47], right above the PD cell in a photodetector. However, a precise knife pad is required for this method, which is not easy for ordinary research laboratories and machine shops. In addition, the measurement of a focused laser beam can be influenced by the thickness of the knife pad. The measurement is also influenced by the positioning errors and motion errors of the linear scanning mechanism for moving the knife in the knife-edge method or the knife pad and the photodetector in the knife-pad method.

Fig. 1.49: The knife-edge method for diameter measurement of a focused laser beam.

Fig. 1.50: The knife-pad method.

34

Chapter 1 Laser autocollimator

In this section, a new method referred to as the PD-edge method is presented. The method associated with laser autocollimation can make diameter measurement of a focused laser beam diameter in a simple and cost-effective manner [21]. Assume the center axis of the focused laser beam with a diameter of d is aligned to coincide with the Z-axis. The light intensity distribution I(x,y) in the focal plane (XY-plane) of the focusing lens can be expressed by a Gaussian function as follows [29]: ! 2ðx2 + y2 Þ (1:23) I ðx, yÞ = I0 exp − ðd=2Þ2 where (x,y) is the coordinates in the focal plane and I0 is the light intensity at the center axis of the laser beam. d is the diameter where I(x,y) decreases to 1/e2 of I0. Denoting the X-position of the knife-edge in Fig. 1.50 by x′, the total power of the laser beam P(x′) passing through the knife edge can be calculated as follows: ð ðx′ ∞

 P x′ =

Iðx, yÞdydx

(1:24)

−∞ −∞

Meanwhile, the total power of the whole laser beam P0 can be calculated by: ∞ ð

∞ ð

P0 =

Iðx, yÞdydx = −∞ −∞

πI0 d2 8

(1:25)

Pðx′Þ, which corresponds to P(x') normalized by P0 can therefore be obtained as:

 ðx′ ∞ ð

 P x′ 8 ′ P x = = Iðx, yÞdydx P0 πI0 d2

(1:26)

−∞ −∞

By making the knife edge to scan over the laser beam to obtain Pðx′Þ and taking the derivative of Pðx′Þ with respect to x′, I(x,y) can be reconstructed [18, 41, 42], from which d can be evaluated. Figure 1.51 shows a schematic of the PD-edge method for diameter measurement of a focused laser beam. The edge of a PD is employed as the knife edge and the PD itself is employed as the photodetector. The beam diameter can be measured by moving the PD along the X-direction so that the PD-edge can scan across the focused light spot. Since the knife-edge and the detector surface are one component and on the same plane, the problems inherent in the conventional knife-edge method and the knife-pad method can be solved. In addition, taking into consideration the PD is located at the focal plane of the focusing lens, the combination of the PD and the lens can be regarded as a laser autocollimation unit where the focusing lens functions as a CO. Based on the principle of laser autocollimation, a θY-directional tilt motion of the beam incident to the lens will be converted into a linear motion of the focused light spot along the X-direction. Therefore, the linear scan motion necessary for the

1.4 PD-edge method associated with laser autocollimation

35

diameter measurement of the focused light spot can be easily generated by a rotary or a tilt stage in a cost-effective way as shown in Fig. 1.51.

Fig. 1.51: A setup for the PD-edge method.

Assume that the PD–edge is located at the center of the focused light spot when the angular displacement ΔθY of the mirror reflector is zero. The change in the total power of the irradiated laser beam ΔP with respect to ΔθY (=Δϕ/2) can be expressed by f tan ð2Δθ ∞ ð

ΔP =

ð0

∞ ð

Iðx, yÞdydx − −∞

−∞

∞ ð

≈ ðf tan 2ΔθÞ · rffiffiffi π = I0 df Δθ 2

−∞



Iðx, yÞdydx −∞ −∞

∞    ð − 8y2 − 8y2 dy ≈ 2f Δθ dy I0 exp I0 exp d2 d2

(1:27)

−∞

where f is the focal length of the focusing lens. From eqs. (1.25) and (1.27), ΔP, which corresponds to ΔP normalized by the total power of the whole laser beam P0, can be obtained as follows: , rffiffiffi rffiffiffi π πI0 d2 2 4f ΔP = = I0 df Δθ Δθ (1:28) 8 2 πd By denoting the gradient of ΔP with respect to ΔθY as S (=ΔP Δθ), eq. (1.28) can be modified as follows: pffiffiffi 4 2f (1:29) d ≈ pffiffiffi πS

36

Chapter 1 Laser autocollimator

where the relationship between d and S is obtained. As can be seen in the equation, since f is a known parameter in the setup, d can be estimated by S. Figure 1.52 shows typical photodetector outputs with respect to the linear scan positions in the conventional knife-edge/knife-pad methods. For the conventional methods, the photodetector output, which is a function of the displacement of a knife-edge/knife-pad, is required to be monitored over a scan range across most of the focused light spot for the evaluation of d. Figure 1.53 shows the outputs of the PDedge method with respect to the angular position of the reflector. Since d is evaluated from the gradient S of the output curve in a small angular range Δθ, the tilt scan range of the reflector can be very small, which is an advantage of the PD-edge method.

Fig. 1.52: Outputs of the conventional methods for beam diameter measurement.

Fig. 1.53: Output of the PD-edge method for beam diameter measurement.

An experimental setup shown in Figure 1.54 was built to test the characteristics of a commercial PD, specifically the sensitivity to input optical power around its edge area. A picture of the setup is shown in Fig. 1.55. A LD with a wavelength λ of 685 nm and a maximum laser power of 70 mW was employed as the light source. The light rays emitted from the LD were collimated by an aspheric lens to form a collimated laser beam. The beam was then expanded to a large beam with a diameter of approximately 26 mm by using a beam expander. The expanded collimated laser beam was made to focus on the edge of one of two cells of the bi-cell PD (S4202, Hamamatsu

1.4 PD-edge method associated with laser autocollimation

37

photonics). An optical microscopic image of the bi-cell PD is shown in Fig. 1.56. The PD was mounted on an X-directional linear slide whose displacement was measured by a length gauge. As shown in Fig. 1.56, three edges (A, B, C) of the right-side PD were tested. Edge-A is the edge adjacent to the neighboring the left-side PD and EdgeC is the conjugating edge between the active cell and the electrode. Edge-B is an independent edge. The length of each of the edges was 1 mm.

Fig. 1.54: An experimental setup for testing the characteristics of PD-edges.

In testing of Edge-A and Edge-C, the PD was moved along the X-direction in a step of 10 μm to scan across the focused laser beam. The focused beam was kept stationary and its diameter d was approximately 130 μm. During the scan, a photocurrent from the PD at each X-position was recorded. The photocurrent was converted into a voltage output, referred to as the PD output, by using a transimpedance circuit. For testing Edge-B, it was necessary to rotate the PD holder 90°. Figure 1.57 shows the PD output curves in the testing of the edge. As can be seen in the figure, Edge-C was most sensitive to the PD displacement, indicating that Edge-C was most proper for use in the PDedge method as well as the knife-pad method of beam diameter measurement. Edge-C of the PD was first applied to the conventional knife-pad method in the setup shown in Figs. 1.54 and 1.55. During the measurement, the PD was moved along the X-direction with a constant velocity of 5 μm/s. The PD output and the length gauge output were acquired simultaneously during the scan. The measurement results are shown in Fig. 1.58. By taking the derivative of the PD output with

38

Chapter 1 Laser autocollimator

Fig. 1.55: Picture of the setup in Fig. 1.43.

Fig. 1.56: A microscope image of the PD and the edges for test.

respect to the X-position of the PD, the intensity distribution of the focused laser beam was reconstructed in the figure, from which the diameter of the focused laser beam, which was denoted by dknife, could be evaluated based. As shown in Figs. 1.54 and 1.55, a diameter-variable iris was placed between the beam expander

1.4 PD-edge method associated with laser autocollimation

39

Fig. 1.57: PD output curves in the testing of the edges.

Fig. 1.58: Measurement results by the knife-pad method with Edge-C.

and the objective lens for adjusting the diameter D0 of the collimated laser beam before it was made incident to the objective lens. Edge-C of the PD was then applied to the PD-edge method in the setup shown in Fig. 1.59. A picture of the setup is shown in Fig. 1.60. The optical components including the laser source, the beam expander and the diameter-variable iris were the same as

40

Chapter 1 Laser autocollimator

those in Fig. 1.55. A reflector mounted on a PZT tilt stage was employed to tilt the collimated laser beam with a diameter D0 about the Y-axis. The laser beam from the reflector was focused on the PD for the diameter measurement. The PD was mounted on a manual three-axis positioning stage. The collimated laser beam was divided into two beams by using a BS. One beam was received by a reference PD for monitoring the light intensity deviation of the laser source. The other beam was employed for the diameter measurement where the center of the focused laser beam was positioned on Edge-C of the PD when the PZT tilt stage was at the initial angular position.

Fig. 1.59: Experimental setup for the PD-edge method of beam diameter measurement.

A small angular displacement Δθ of ±1 arc-second was applied to the reflector by using the PZT tilt stage while the PD output was acquired, from which the gradient S of the PD output can be evaluated. The ±1 arc-second angular displacement of the reflector corresponded to a ±0.17 μm linear displacement of the focused laser beam on the PD. D0 was changed by using the diameter-variable iris from 1 to 10 mm. The evaluated S at each D0 is shown in Fig. 1.61. As can be seen in the figure, S increased with the increase of D0. As shown in eq. (1.29), an increase in S corresponded to a decrease in the measured diameter of the focused laser beam by the PD-edge method, which was referred to as dedge. The evaluated dedge based on eq. (1.29) at each D0 was also plotted in Fig. 1.61. dedge by the PD-edge method and dknife by the knife-pad method are compared in Fig. 1.62. A good agreement can be found between the results by the two different methods. This verified advantage in the PDmethod, that is, the gradient S acquired within a narrow range is effective enough for the accurate evaluation of the focused laser beam diameter.

1.4 PD-edge method associated with laser autocollimation

Fig. 1.60: A picture of the setup shown in Fig. 1.48.

Fig. 1.61: Measured S and dedge by the PD-edge method.

41

42

Chapter 1 Laser autocollimator

Fig. 1.62: Comparison between the measured beam diameters by the PD-edge method and the Knife-pad method.

1.5 Summary A laser rangefinder-autocollimator has been presented for the hand scraping process and the assembly process of slide stages. The 2D tilt angle components (pitch and yaw) of a reflector were mounted on the guideways or those of the moving carriage of the stage while providing the position information of the reflection for the necessary operations in the processes. The angle measurement and the position measurement are based on the laser autocollimation method and the time-of-flight method, respectively. A commercial rangefinder with a resolution of 1 mm and a range of 30 m has been chosen as the rangefinder unit. A laser autocollimation unit has been designed to integrate with the rangefinder unit. Geometrical models have been established for designing the diameter and focal length of the CO as well as the size of the BS used in the autocollimation unit. The laser rangefinder-autocollimator had a size of 250 mm (X) × 205 mm (Y) × 63 mm (Z). The performances of the autocollimation unit, including sensitivity, stability, linearity, measurement range as well as cross-talk error have been investigated. It has been verified that the constructed laser rangefinderautocollimator could satisfy the design goals of measurement resolutions and ranges. A nanoradian laser autocollimator with a compact size of 100 mm (X) × 150 mm (Y) has then been presented for ultrasensitive tilt angle measurement of precision linear stages. Influences of the spherical aberration of CO on the sensitivity of laser autocollimation have been investigated by computer simulation based on wave optics. The simulation results have revealed that the increase of the measurement laser beam diameter D0 is effective in improving the autocollimator sensitivity when D0 is smaller than 10 mm where the influence of the spherical aberration is insignificant. In the measurement experiment, the measured focused light spot diameter and the measured autocollimator sensitivity had a good agreement with the simulation results. The designed and

References

43

constructed nanoradian laser autocollimator has reached a resolution of 0.001 arcsecond (4.8 nrad) with a measurement laser beam diameter D0 of 5 mm. A new PD-edge method associated with laser autocollimation has been presented for diameter measurement of a focused laser beam diameter. In this method, the lens for focusing the laser beam is treated as the CO in an autocollimation unit. The focused laser beam is aligned on the PD cell edge of a photodiode, which is located at the focal position of the focusing lens. A small angular displacement Δϕ is given to a collimated laser beam before being made incident to the lens. The angular displacement of the collimated laser beam will generate a linear scan motion of the focused laser beam across the PD-edge. The output of the photodiode during the scan is acquired to obtain the gradient S of the PD output, from which the diameter of the focused laser beam diameter d can be evaluated. It has been verified that a commercial photodiode can be utilized for the PD-edge method. Experimental results have demonstrated the feasibility of the PD-edge method for diameter measurement of micrometric focused laser beams.

References [1]

Gao W, Haitjema H, Fang FZ, Leach RK, Cheung CF, Savio E, Linares JM. On-machine and inprocess surface metrology for precision manufacturing. CIRP Ann 2019, 68, 2, 843–866. [2] Buice ES, Otten D, Yang RH, Smith ST, Hocken RJ, Trumper DL. Design evaluation of a singleaxis precision controlled positioning stage. Precis Eng 2009, 33, 4, 418–424. [3] Gao W, Arai Y, Shibuya A, Kiyono S, Park CH. Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage. Precis Eng 2006, 30, 1, 96–103. [4] Oiwa T, Katsuki M, Karita M, Gao W, Makinouchi S, Sato KOY. Report of questionnaire survey on ultra-precision positioning; technical committee of ultra-precision positioning. Japan Soc Precis Eng Tokyo, Japan 2016, 81, 10, 904–910. [5] Gao W, Kim SW, Bosse H, Haitjema H, Chen YL, Lu XD, Knapp W, Weckenmann A, Estler WT, Kunzmann H. Measurement technologies for precision positioning. CIRP Ann – Manuf Technol 2015, 64, 2, 773–796. [6] Moore WR, Special M, Co T. Foundations of Mechanical Accuracy. 1970. [7] Hwang J, Park C-H, Gao W, Kim S-W. A three-probe system for measuring the parallelism and straightness of a pair of rails for ultra-precision guideways. Int J Mach Tools Manuf 2007, 47, 7–8, 1053–1058. [8] Fan K-C, Torng J, Jywe W, Chou R-C, Ye J-K. 3-D measurement and evaluation of surface texture produced by scraping process. Measurement 2012, 45, 3, 384–392. [9] Seiki M. Topics about “Scraping.” (Accessed March 16, 2021, at http://www.mitsuiseiki.co. jp/machine/tabid/192/Default.aspx) [10] Kunzmann H, Pfeifer T, Flügge J. Scales vs. Laser Interferometers Performance and Comparison of Two Measuring Systems. CIRP Ann – Manuf Technol 1993, 42, 2, 753–767. [11] Zhang JH, Menq CH. A linear/angular interferometer capable of measuring large angular motion. Meas Sci Technol 1999, 10, 12, 1247–1253. [12] Kimura A, Gao W, Kim W, Hosono K, Shimizu Y, Shi L, Zeng L. A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement. Precis Eng 2012, 36, 4.

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[13] Li X, Gao W, Muto H, Shimizu Y, Ito S, Dian S. A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage. Precis Eng 2013, 37, 3, 771–781. [14] Jang Y-S, Kim S-W. Distance measurements using mode-locked lasers: A review. Nanomanufacturing Metrol 2018, 1, 3, 131–147. [15] Ennos AE, Virdee MS. High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Precis Eng 1982, 4, 1, 5–8. [16] Saito Y, Gao W, Kiyono S. A single lens micro-angle sensor. Int J Precis Eng Manuf 2007, 8, 2, 14–18. [17] Gao W, Saito Y, Muto H, Arai Y, Shimizu Y. A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann – Manuf Technol 2011, 60, 1, 515–518. [18] Shimizu Y, Tan SL, Murata D, Maruyama T, Ito S, Chen Y-L, Gao W. Ultra-sensitive angle sensor based on laser autocollimation for measurement of stage tilt motions. Opt Express 2016, 24, 3, 2788. [19] Tan SL, Shimizu Y, Meguro T, Ito S, Gao W. Design of a laser autocollimator -based optical sensor with a rangefinder for error correction of precision slide guideways. Int J Precis Eng Manuf 2015, 16, 3, 423–431. [20] Murata D An ultra-sensitive optical angle sensor. Tohoku University, Master thesis, 2014. [21] Shimizu Y, Maruyama T, Nakagawa S, Chen Y-LY-L-L, Matsukuma H, Gao W. A PD-edge method associated with the laser autocollimation for measurement of a focused laser beam diameter. Meas Sci Technol 2018, 29, 7, 0–6. [22] Maruyama T An ultra-high sensitive optical angle sensor. Tohoku University, Master thesis, 2016. [23] Keysight Technologies. Keysight Technologies 55280B Linear Measurement Kit Data Sheet Optical Setup for Linear Measurements Basic Equipment. (Accessed November 18, 2020, at www.keysight.com/find/5530) [24] MÖLLER-WEDEL OPTICAL. Electroninc autocollimator. (Published 2007 Accessed March 14, 2021, at www.moeller-wedel-optical.com) [25] Newport. CONEX-LDS Electronic Autocollimator. (Accessed November 18, 2020, at https:// www.newport.com/f/conex-lds-electronic-autocollimator) [26] Saito Y, Arai Y, Gao W. Investigation of an optical sensor for small tilt angle detection of a precision linear stage. Meas Sci Technol 2010, 21, 5, 054006. [27] Gao W. Precision Nanometrology. London, Springer London, 2010. [28] Bennett SJ, Gates JWC. The design of detector arrays for laser alignment systems. J Phys E 1970, 3, 1, 65–68. [29] Hecht E. Optics. 5th Pearson, 2017. [30] Smith JW. Modern Optical Engineering. McGraw Hill, 2007. [31] Goodman JW. Introduction to Fourier Optics. Roberts & Company, 2004. [32] Mahajan VN Optical imaging and aberrations. 1000 20th Street, Bellingham, WA 98227-0010 USA, SPIE, 1998. [33] Otsuka J, Ichikawa S, Masuda T, Suzuki K. Development of a small ultraprecision positioning device with 5 nm resolution. Meas Sci Technol 2005, 16, 11, 2186–2192. [34] Bryan JB. The Abbé principle revisited: An updated interpretation. Precis Eng 1979, 1, 3, 129–132. [35] Meguro T, Shimizu Y, Gao W An angle sensor with a laser rangefinder. Proceedings of International Conference on Leading Edge Manufacturing in twenty-first Century: LEM21, The Japan Society of Mechanical Engineers, 2011, 1–4. [36] Ennos AE, Virdee MS. High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Precis Eng 1982, 4, 1, 5–8.

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[37] Gueuning F, Varlan M, Eugene C, Dupuis P Accurate distance measurement by an autonomous ultrasonic system combining time-of-flight and phase-shift methods. Conference Record – IEEE Instrumentation and Measurement Technology Conference, Vol 1. IEEE, 1996, 399–404. [38] Amann M-C, Bosch T, Marc L, Risto M, Marc R. Laser ranging: a critical review of usual techniques for distance measurement. Opt Eng 2001, 40, 1, 10. [39] Leica Geosystems. DISTO Lite5 User Manual. (Accessed November 18, 2020, at https://shop. leica-geosystems.com/sites/default/files/2019-03/leica_disto_lite5_en.pdf) [40] Saito Y, Arai Y, Gao W. Detection of three-axis angles by an optical sensor. Sensors Actuators, A Phys 2009, 150, 2, 175–183. [41] Arnaud JA, Hubbard WM, Mandeville GD, De La Clavière B, Franke EA, Franke JM. Technique for Fast Measurement of Gaussian Laser Beam Parameters. Appl Opt 1971, 10, 12, 2775. [42] Suzaki Y, Tachibana A. Measurement of the μm sized radius of Gaussian laser beam using the scanning knife-edge. Appl Opt 1975, 14, 12, 2809. [43] Firester AH, Heller ME, Sheng P. Knife-edge scanning measurements of subwavelength focused light beams. Appl Opt 1977, 16, 7, 1971. [44] Dorn R, Quabis S, Leuchs G. Sharper focus for a radially polarized light beam. Phys Rev Lett 2003, 91, 23, 233901. [45] Huber C, Orlov S, Banzer P, Leuchs G. Corrections to the knife-edge based reconstruction scheme of tightly focused light beams. Opt Express 2013, 21, 21, 25069. [46] Huber C, Orlov S, Banzer P, Leuchs G. Influence of the substrate material on the knife-edge based profiling of tightly focused light beams. Opt Express 2016, 24, 8, 8214. [47] Marchenko P, Orlov S, Huber C, Banzer P, Quabis S, Peschel U, Leuchs G. Interaction of highly focused vector beams with a metal knife-edge. Opt Express 2011, 19, 8, 7244.

Chapter 2 Three-axis angle sensor 2.1 Introduction Precision positioning of a machining tool or a measuring probe is a common operation in precision engineering, in which a linear stage is employed for one-axis applications or a combination of multiple linear stages is employed for two- or three-axis applications [1–3]. As shown in Fig. 2.1, such a precision linear stage is typically equipped with a position sensor, often a linear encoder or a laser interferometer, for closed-loop control of the stage along its axis of motion. It is necessary to align the measuring axis of the position sensor coaxially with the axis of motion of the stage to avoid the Abbe error caused by angular error motions of the stage [4–6]. The positioning systems for ultraprecision metrological applications, such as the line scale comparator and the metrological scanning probe microscope, are basically designed to obey the Abbe principle [7, 8]. Many other positioning systems, such as those for machine tools and coordinate measuring machines (CMMs), however, do not satisfy the Abbe principle [9]. In such a system, the position of the machining tool or the measuring probe has a certain distance from the axis of the position sensor, which is called the Abbe offset.

Fig. 2.1: Closed-loop control of a linear stage.

Even a small angular motion of the linear stage could cause a large amount of Abbe error if the Abbe offset is large. Figure 2.2 shows a commercial ultraprecision linear stage driven by a linear motor. A linear encoder with a resolution of 0.14 nm is employed as the position sensor for closed-loop control. In this example, the Abbe offset is approximately 100 mm. As shown in Fig. 2.3, in this case, an angular motion with an amplitude of 1 arc-second will generate an Abbe error with an amplitude of 0.48 μm. Taking into consideration the amplitude of the angular error motion of a precision stage is typically larger than 1 arc-second [10] and most of the precision https://doi.org/10.1515/9783110542363-002

48

Chapter 2 Three-axis angle sensor

Fig. 2.2: Abbe offset in a commercial ultraprecision air-bearing linear stage.

stages are required to have a positioning precision better than 0.1 μm, the Abbe error is a large error factor for the precision stage. Detection of the angular error motion of the stage is thus important for evaluation and compensation of the Abbe error.

Fig. 2.3: Influence of the Abbe error on positioning accuracy.

2.1 Introduction

49

Fig. 2.4: Pitch and yaw two-axis measurement by a conventional autocollimator.

Assuming that the stage travels along the Z-axis, the pitch, yaw and roll errors are the components of the angular error motion about the X-, Y- and Z-axes, respectively. As shown in Fig. 2.4, a conventional autocollimator [11] can be used to detect the pitch error (Δθx) and yaw error (Δθy) of the linear stage [10, 12]. Figure 2.5 shows a schematic of the autocollimator. In the detection, a light beam from the autocollimator is projected onto a flat mirror reflector along the surface normal of the mirror, which is mounted on the stage moving table. The reflected beam from the flat mirror reflector is received by the autocollimation unit, which consists of a collimator objective and a two-directional light position-sensing detector placed at the focal plane of the collimator objective [13, 14]. As presented in the previous chapter, based on the principle of autocollimation, the pitch and yaw errors will be converted into Y- and X-directional linear displacements of the light spot on the detector (Δy, Δx), respectively, from which the pitch and yaw errors can be simultaneously detected. Commercial autocollimators can achieve a high resolution of 0.01 arc-second. However, the roll error (Δθz) cannot be detected by conventional autocollimators because the reflected beam from the flat mirror reflector does not respond to the rotational motion about the surface normal of the flat mirror reflector. This chapter presents a three-axis angle sensor that can simultaneously detect the three-axis components of the angular error motion of a precision linear stage (Fig. 2.6). In such a three-axis angle sensor, the flat mirror used in the conventional two-axis autocollimator is replaced with a refractive grating reflector. The diffracted

50

Chapter 2 Three-axis angle sensor

beams reflected from the grating reflector are detected by a sensor head for the three-axis angle measurement [15–22]. Since the working distance between the sensor head and the reflector/moving table along the Z-direction varies with the movement of the moving table, the sensor is referred to as the variable working distance type. A constant working distance type sensor head is then presented for extending the detectable moving stroke of the linear stage. A three-axis inclination sensor [23, 24] is also presented.

Fig. 2.5: Schematic of a conventional two-axis autocollimator.

2.2 Three-axis angle sensor Figure 2.7 shows the concept for three-axis angle measurement. Compared with a conventional two-axis autocollimator shown in Fig. 2.5, a diffractive grating reflector is employed to replace the flat mirror reflector. In addition to the zeroth-order diffracted beam, the positive and negative first-order diffracted beams from the grating reflector are also received by the collimator objective. Three detectors are employed

2.2 Three-axis angle sensor

51

Fig. 2.6: Simultaneous measurement of pitch, yaw and roll errors by a three-axis angle sensor.

to detect the displacements of the focused diffracted beams on the focal plane of the collimator objective associated with the pitch, yaw and roll angular motions. Figure 2.8 shows the displacements of diffraction light spots focused on the detectors located at the focal plane of the collimator objective. As shown in the figure, if a pitch angle Δθx is applied to the grating reflector, it will generate the same amount of linear displacement on the three detectors along the Y-direction, which is the same as the conventional two-axis autocollimator. Similarly, if a yaw angle Δθy is applied to the grating reflector, it will generate the same amount of linear displacement on the detectors along the X-direction, which is also the same as the two-axis autocollimator. On the other hand, if a roll angle Δθz is applied to the grating reflector, the two first-order diffracted light spots will rotate about the zeroth-order diffracted light spot, which can be utilized for the detection of Δθz. Based on the results in Fig. 2.8, the three-axis angles Δθx, Δθy and Δθz can be measured simultaneously by using the setup shown in Fig. 2.9 or that in Fig. 2.10. In Fig. 2.9, detectors A and B are used for detecting the displacements of the zeroth-order and one of the first-order diffracted beams. The two first-order diffracted beams are detected in Fig. 2.10 for the three-axis angle measurement. Figure 2.11 shows a three-axis angle sensor based on the principle shown in Fig. 2.9 for detection of the angular error motion of a precision linear stage moving along the Z-axis. The magnitudes of the angular error motions are assumed to be small. A grating reflector is mounted on the stage moving table. The s-polarized light from a laser diode (LD) is collimated by a collimating lens (CL) and bent by a

52

Chapter 2 Three-axis angle sensor

Fig. 2.7: Concept of the three-axis angle detection.

Fig. 2.8: Displacements of diffraction light spots on detectors for three-axis angle measurement.

polarizing beam splitter (PBS). The collimated light beam is then circularly polarized by a quarter-wave plate (QWP) before it is projected onto the grating reflector along the normal direction of the grating surface. The reflected zeroth-order and positive first-order diffracted beams from the grating reflector are converted into p-polarized beams by the QWP so that they can pass through the PBS to reach the

2.2 Three-axis angle sensor

53

Fig. 2.9: Three-axis angle measurement by detecting zeroth-order and positive first-order diffracted beams.

autocollimation unit, which consists of a collimator objective and two-position sensing detectors. The two-position sensing detectors are located at the focal plane of the collimator objective to detect the X- and Y-directional displacements of the diffracted beam spots focused on detectors A and B. Since the distance between the sensor head and the grating reflector, which is called the working distance, varies with the position of the stage moving table, the three-axis angle sensor shown in Fig. 2.11 is referred to as the variable working distance type. The displacements of the diffracted beam spots caused by the angular motion of the stage are shown in Fig. 2.12. A0 and A1 are the initial positions of the zeroth-order and the first-order diffracted beam spots, respectively. If the distance between A0 and A1 is denoted by L, L can be expressed by [13] L = f tan α1

(2:1)

where f is the focal length of the collimator objective. a1 is the diffraction angle of the first-order diffracted beam, which is given by [14] α1 = arcsin

λ g

(2:2)

where λ is the wavelength of the LD and g is the grating period of the grating reflector. Assume that the first stage has a yaw error Δθy, then a pitch error Δθx, finally a roll error Δθz in sequence. In response to Δθy, the zeroth-order and first-order

54

Chapter 2 Three-axis angle sensor

Fig. 2.10: Three-axis angle measurement by detecting positive and negative first-order diffracted beams.

diffracted beams will have the same angular deviations of Δθy, which are converted into X-directional linear displacements of beam spots on the detectors by the collimator objective. As shown in Fig. 2.12, the zeroth-order and first-order diffracted beam spots move to B0 and B1 along the X-axis, respectively. Let the displacements of the beam spots be ΔxA and ΔxB, which are the X-directional outputs of Detector A and Detector B, respectively. ΔxA is equal to ΔxB. Because Δθy is small, it can be obtained by [13]: Δθy =

ΔxB Sθy

Sθy = 2f

(2:3) (2:4)

where Sθy is the conversion ratio of the angular motion Δθy to the linear displacement ΔxB with a unit of mm/rad. Similarly, Δθx makes the zeroth-order diffracted beam spot move from B0 to C0 and the first-order diffracted beam spot from B1 to C1. In Fig. 2.12, ΔyA and ΔyB1 are the displacements generated by Δθx along the Y-axis, which are the Y-directional outputs of detectors A and B, respectively. Δθx can be obtained by Δθy =

ΔyA Sθx

Sθx = 2f

(2:5) (2:6)

where Sθx is the conversion ratio of the angular motion Δθx to the linear displacement ΔyA with a unit of mm/rad.

2.2 Three-axis angle sensor

55

Fig. 2.11: A three-axis angle sensor of variable working distance type for linear stage measurement.

Fig. 2.12: Displacements of the diffracted light spots on Detectors A and B.

Because the roll error Δθz is a rotational motion about the Z-axis, which is consistent with the axis of the zeroth-order diffracted beam, the zeroth-order diffracted beam spot keeps stationary at point C0 when Δθz occurs. On the other hand, the roll error generates an angular deviation of Δθz to the first-order diffracted beam. As shown in Fig. 2.10, the first-order diffracted beam spot will rotate about C0 with a radius of L from C1 to D1 in responding to the angular deviation. For a small Δθz, the Y-directional displacement ΔyB2 can be written as

56

Chapter 2 Three-axis angle sensor

ΔyB2 = LΔθZ

(2:7)

Δθz can thus be obtained by ΔθZ =

ΔyB2 − ΔyB1 SθZ

(2:8)

where ΔyB2 is the total Y-directional output of detector B. Substituting eqs. (2.1) and (2.2) into eq. (2.8) gives ΔθZ =

ΔyB2 − ΔyA SθZ

SθZ = f tanðarcsin α1 Þ

(2:9) (2:10)

where Sθz is the corresponding conversion ratio of angular motion to linear displacement with a unit of mm/rad. Consequently, the pitch error Δθx and the yaw error Δθy can be obtained from the Y- and X-directional outputs of detector A. The roll error Δθz can be obtained from the difference between the Y-direction outputs of detectors B and A. Figure 2.13 shows the optical layout of a three-axis angle sensor designed for the variable working distance type. Two identical autocollimation units are employed to receive the zeroth-order and the first-order diffracted beams, respectively, for easier optical alignment and reduction of the influence of lens aberrations. A p-polarized light with a wavelength (λ) of 685 nm is output from the LD, which was located in the center of the sensor for a balanced layout. The focal length of the CL was 11 mm. The reflected zeroth-order diffracted beam was converted into an s-polarized light by the QWP and bent at the PBS before it was received by autocollimation unit A. The reflected firstorder diffracted beam was received by Autocollimation unit B without passing through the QWP. Two mirrors (M1 and M2) were added to bend the light beams so that the sensor can be made compact. The collimator objectives (CO1 and CO2) were identical and the focal length (f) was 25.4 mm. Quadrant photodiodes (QPDs) were employed as the position sensing detectors in the autocollimation units. The grating reflector had a grating period (g) of 1.7 μm. The conversion ratios of angular motion to linear displacement Sθy, Sθx and Sθz were evaluated to be 50.8, 50.8 and 10.2 mm/rad based on eqs. (2.4), (2.6) and (2.10), respectively. It can be seen that Sθz was smaller than Sθx and Sθy, which means that the three-axis autocollimator is less sensitive to the roll error than the pitch error and the yaw error. To increase the sensitivity in detection of the roll error, an optical magnifier is added to the optical path of the first-order diffracted beam. As shown in Fig. 2.13, two lenses (L1 and L2) were arranged in such a way that the secondary focal point F1 was overlapped with the primary focal point F2 of L2. The angular magnification M can be expressed by

2.2 Three-axis angle sensor

57

Fig. 2.13: Optical layout of a three-axis angle sensor designed for the variable working distance type.

M=

f1 f2

(2:11)

where f1 and f2 were the focal lengths of L1 and L2, respectively. M is determined to be 4.1 by choosing f1 and f2 to be 25.4 mm and 6.24 mm, respectively. The diameter of L1 is 15 mm. Figure 2.14 shows a picture of the three-axis autocollimator constructed based on the design in Fig. 2.13. The size of the sensor head is 85 mm (L) × 85 mm (W) × 47 mm (H). The basic performance of the sensor was tested by using the setup shown in Fig. 2.15 where a commercial two-axis autocollimator is employed as the reference. The grating reflector was mounted on a three-axis PZT (piezoelectric) tilt stage to generate the Δθx, Δθy and Δθz tilt motions. At first, the two-axis autocollimator was placed at Position I

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Chapter 2 Three-axis angle sensor

Fig. 2.14: Picture of the three-axis autocollimator constructed based on the design in Fig. 2.13.

with its optical axis along the Z-axis so that it could detect the Δθx and Δθy motions. The same Δθx and Δθy motions were also detected by the three-axis angle sensor with its optical axis along the Z-axis at the same time. The two-axis autocollimator was then moved to Position II with its optical axis along the X-axis while the three-axis angle sensor was kept at its original position. Under this configuration, the Δθz tilt motion could be detected simultaneously by the two sensors. Figure 2.16 shows the investigated non-linear components of the three-axis angle sensor over a range of 40 arc-seconds. The nonlinear error components were identified to be approximately 2.68, 2.06 and 2.11 arc-seconds in the detection of Δθx, Δθy and Δθz, respectively. The nonlinear error components can be compensated based on the results in the figure. Figure 2.17 shows the result for testing the resolutions of the three-axis angle sensor. In the test, a sinusoidal tilt motion with an amplitude of 0.01 arc-second and a frequency of 1 Hz was applied by the PZT stage in the Δθx, Δθy and Δθz directions, respectively. It can be seen that the sinusoidal tilt motions were well detected, demonstrating that the resolutions of the three-axis angle sensor were better than 0.01 arc-second in all three directions. It should be noted that the first-order diffracted beam will move across the collimator objective when the stage moves in the Z-direction. It is necessary to determine the diameter of the collimator objective based on the required range of motion along the Z-axis. Figure 2.18 shows an optical layout of a three-axis sensor demonstrating the relationship between the range of motion of the grating reflector Rz and the radius

2.2 Three-axis angle sensor

Fig. 2.15: Setup for investigating the basic performance of three-axis angle sensor.

Fig. 2.16: Results of testing non-linear error components of the three-axis angle sensor.

59

60

Chapter 2 Three-axis angle sensor

of the collimator objective rc. As shown in the figure, the relationship between Rz and rc can be expressed by rc = RZ tan α1

(2:12)

Substituting eq. (2.2) into eq. (2.12) gives   λ rc = RZ tan arcsin g

(2:13)

Based on eq. (2.13), for a three-axis angle senor with a laser wavelength (λ) of 685 nm and a grating period (g) of 1.7 μm, if the range of motion along the Z-axis Rz is required to be longer than 100 mm, the radius of the collimator objective rc should be larger than 41.35 mm. Figure 2.19 shows the required diameters of the collimator objective (2rc) for different ranges of motion of the stage. It can be seen that a large diameter of approximately 900 mm is required for the collimator objective for a 1 m stage range of motion, which is a significant challenge for the design and construction of the sensor.

Fig. 2.17: Results of testing resolutions of the three-axis angle sensor.

2.2 Three-axis angle sensor

61

Fig. 2.18: Relationship between the range of motion of the grating reflector Rz and the radius of the collimator objective rc in a three-axis angle sensor of variable working distance type.

Fig. 2.19: Required diameter of collimator objective for different stage ranges of motion (λ: 685 nm, g: 1.7 μm).

62

Chapter 2 Three-axis angle sensor

As shown in Fig. 2.20, a precision linear encoder is often equipped with a linear encoder, which is composed of a linear scale and a read head. The linear scale is attached to the stage moving table with its graduations aligned along with the stage motion of axis (Z-axis) and the read head is fixed on the stage base. When a threeaxis angle sensor of variable working distance type is employed for measurement of the angular motions of the stage, the sensor head is fixed on the stage base with its optical axis aligned along the Z-axis. The grating reflector for the angle sensor is attached to the stage moving table and the surface normal of the grating reflector is aligned along the Z-axis. As presented in the previous section, the collimator objective of the angle sensor is required to have a large diameter for a long stage moving range. In addition to this drawback, since the working distance of the angle sensor changes with the movement of the stage table along the Z-axis as shown in Fig. 2.21, the angle sensor also suffers from the influence of environmental fluctuations, especially when the stage table moves away from the sensor head. On the other hand, the linear encoder works with a constant working distance where the distance between the read head and the scale is kept constant. Since the scale-read head distance is typically small, the linear encoder is robust to environmental fluctuations.

Fig. 2.20: A linear encoder and a three-axis angle sensor of variable working distance type in a precision linear stage.

To solve the above problems, a setup is proposed in Fig. 2.22 for three-axis angular motion measurement of a precision linear stage where the angle sensor works at a constant working distance. The concept is based on the fact that the linear scale of the encoder, which is a diffractive grating, can be employed as the grating reflector

2.2 Three-axis angle sensor

63

Fig. 2.21: Working distances for the sensors in Fig. 2.20.

Fig. 2.22: Concept of three-axis angular motion measurement in a constant working distance.

for the three-axis angle sensor. The angle sensor head can be placed side by side with the read head of the linear encoder so that the two heads can read the linear scale simultaneously for measurement of position and angular motions at the same time. Taking into consideration of the physical sizes of the encoder read head and the angle sensor head, it is necessary to use a longer linear scale for covering the full moving stroke of the stage as shown in Fig. 2.23. From this point of view, it is also desired to design a three-axis angle sensor head with a compact size. In the following, the angle sensor is referred to as the CWD (constant working distance) three-axis angle sensor for clarity.

64

Chapter 2 Three-axis angle sensor

Fig. 2.23: Constant working distances in the measurement.

Fig. 2.24: Pictures of a pair of read head and scale of a commercial linear encoder.

Figure 2.24 shows the pictures of a pair of the reading head (LIP 401) and scale (D-8225) of a commercial linear encoder made by Dr. Johannes HEIDENHAIN GmbH. The scale has a grating period (g) of 4 μm over a scale length of 80 mm. The width of the grating graduations is 3 mm. The size of the read head is 60 mm × 30 mm × 30 mm. A three-axis angle sensor head was designed and constructed by using the linear scale as the grating reflector. Figure 2.25 shows a schematic of the sensor head. The optical layout is basically the same as that in Fig 2.9 where the zeroth-order and the first-order diffracted beams are employed for the three-axis angle measurement. The working distance is designed to be 10 mm. Each diffracted beam is received by an individual collimator lens. The diffracted beam is aligned to pass through the center area of the lens so that the influence of lens aberration can be reduced. Table 2.1 shows the specifications of the collimator objectives. A LD with a small size of ϕ5.6 mm × 3.5 mm is employed as the light source for the sensor. Table 2.2 shows the specifications of the LD. The wavelength of the laser beam is 685 nm. The maximum optical power that can be output from the LD is 35 mW and the output power can be adjusted by the operating current. The laser light rays from the LD are made to be a collimated beam with a diameter of D0 by

2.2 Three-axis angle sensor

65

using a CL. The diffraction angle of the first-order diffracted beam α1 is calculated to be 9.86° based on the wavelength of the laser beam and the grating period of the scale. Quadrant photodiodes (QPDs) are selected as position sensing detectors. Table 2.3 shows the specifications of the QPD. Since the gap between the PD cells is 10 μm, the dimeter of the focused laser light spots Ds should be larger than 14 μm as shown Fig. 2.26. For making the angle sensor that has enough measurement range, Ds is set to be 20 μm. Based on the following equation where fco and λ are 20 mm and 685 nm, respectively, the diameter of the collimated beam D0 is calculated to be smaller than 0.872 mm: D0 =

4fCO λ πDS

(2:14)

Fig. 2.25: Schematic of the CWD three-axis angle sensor.

Tab. 2.1: Specification of the collimator objective. Item

Value

Manufacturer

Edmund optics

Type

TS Achromatic lens

Model number

#-

Diameter

. mm

Effective diameter

. mm

Focal length fco

 mm

66

Chapter 2 Three-axis angle sensor

Tab. 2.2: Specification of the laser diode (LD). Item

Value

Manufacturer

Mitsubishi Electric

Model number

MLR-

Size

ϕ. mm × . mm

Beam divergence angle

Horizontal direction βH

°

Vertical direction βV

°

Wavelength λ

 nm

Optical power (max.)

 mW

Operating current (max.)

 mA

Operating voltage

. V

Monitor current

. mA

Tab. 2.3: Specification of the quadrant photodiode (QPD). Item

Value

Manufacturer

Hamamatsu Photonics es

Type

Quadrant Si photodiode

Model number

S-

Size

ϕ . mm × . mm

Gap between PD cells

 μm

Sensing area

. mm × . mm

Spectral response range

 nm ~ , nm

Peak sensitivity wavelength

 nm

As shown in Fig. 2.27, D0 is determined by the beam divergence angle of the LD and the focal length of the CL fcl. By taking the beam divergence angle in the vertical direction in Tab. 2.2, fcl is calculated to be 1.97 mm based on the following equation. Table 2.4 shows the specifications of the CL selected for the three-axis angle sensor. It should be noted that since the LD has different beam divergence angles in the vertical and horizontal directions, the collimated laser beam has an elliptical sectional shape with diameters of 0.887 mm × 0.420 mm. The focused laser spot on the QPD also has an elliptical sectional shape. The diameters are calculated to be 19.7 μm × 41.55 μm based on the following equation:

2.2 Three-axis angle sensor

67

Fig. 2.26: Focused diffraction laser spot on QPD.

Fig. 2.27: Determination of the focal length of collimating lens.

fcl =

D0 2 tan

βV 2

(2:15)

The distance WD between linear scale and the assembly of the PBS and QWP, which is the working distance of the angle sensor, is calculated based on the geometrical relationship shown in Fig. 2.28. The specifications of the PBS-QWP assembly are shown in Tab. 2.5. WD is calculated to be 18.7 mm from the following equation:

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Chapter 2 Three-axis angle sensor

Tab. 2.4: Specification of the collimating lens. Item

Value

Manufacturer

Edmund optics

Type

LIGHTPATH Aspherical lens

Model number

#-

Diameter

 mm

Effective diameter

 mm

Focal length fcl

 mm

WD =

5 mm 5 mm − 10.5 mm = − 10.5 mm = 18.7 mm sin α1 sin 9.86

It can be seen that the required working distance of 10 mm can be satisfied.

Fig. 2.28: Determination of the working distance.

Tab. 2.5: Specifications of the PBS-QWP assembly. Item

Value

Manufacturer

Edmund optics

Model number

CM

Size

 mm × . mm ×  mm

Center wavelength

 nm

(2:16)

2.2 Three-axis angle sensor

69

Fig. 2.29: Picture of the CWD three-axis angle sensor.

Figure 2.29 shows a picture of the constructed CWD three-axis angle sensor. The size of the sensor head is 50 mm × 30 mm × 30 mm. As shown in Fig. 2.30, the CWD three-axis angle sensor is more compact than the read head of the linear encoder. In experiments, the sensitivity of the established three-axis angle sensor was initially evaluated. In the tests, while the scale grating was held stationary in the setup, the angular displacement of 50 arc-seconds about the roll direction was given to the scale grating. The variations of the readings of QPD1 and QPD2 during the rotational motion of the scale grating are shown in Figs. 2.31 and 2.32, respectively. For both QPD1 and QPD2, the focused laser beams on the photodetector plane may be shifted in accordance with the applied roll angle, as shown in Fig. 2.31. The roll angle measurement sensitivity was evaluated to be 0.489%/arc-second and 0.373%/arc-second for QPD1 and QPD2, respectively. The sensitivity of the pitch angle measurement was tested in the same way. Variations of the reading of QPD1 and QPD2 during the change in the pitch angle of the scale grating are shown in Figs. 2.33 and 2.34, respectively. The developed three-axis angle sensor was confirmed to have a sensitivity of 0.0185%/arc-second for pitch angle measurement, from Fig. 2.34. As explained in theory, the sensitivity for pitch angle calculation is lower than for roll and yaw angle measurement. Meanwhile, due to the measurement principle, the focused laser beam on the photodetector plane of QPD1 does not move in accordance with the grating pitch motion. A slight change in

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Chapter 2 Three-axis angle sensor

Fig. 2.30: Comparison between the two sensor heads.

the reading of QPD1 can be observed in Fig. 2.33. Moreover, as shown in Fig. 2.34, the cyclic deviation was observed in the reading of QPD2. Potential causes for these phenomena include the distortion of the focused laser beam, light diffraction due to the circular aperture of the CL and misalignments in the experimental setup. The divergences of the readings of active cells in QPD1 and QPD2, while the PZT tilt stage was held stationary, are shown in Figs. 2.35 and 2.36. From these data, the standard deviation of the reading of each PD cell output was evaluated to be approximately 0.4 mV (1σ). By using the value, standard deviations of the readings of QPD1 and QPD2 were evaluated to be 3.23 × 10−3% and 3.29 × 10−3%, respectively, corresponding to resolutions of 0.0066 arc-second and 0.18 arc-second, respectively, for roll and pitch angle measurement. To further validate the measurement resolution, sinusoidal tilt motions applied to the scale grating through the PZT tilt stage were then measured by the established three-axis angle sensor. A commercial signal generator was employed in the experiment to supply the control signal for the PZT tilt stage. The PZT tilt stage generated sinusoidal roll and pitch motions with an amplitude of approximately 0.01 and 0.20 arc-seconds, respectively. In advance of these tests, the motions of the PZT

2.2 Three-axis angle sensor

71

Fig. 2.31: Variation of the voltage output from QPD1 due to the applied roll angle.

Fig. 2.32: Variation of the voltage output from QPD2 due to the applied roll angle.

Fig. 2.33: Variation of the voltage output from each QPD due to the applied pitch angle (QPD1).

tilt stage were calibrated using a commercially available laser autocollimator. The roll and pitch motions observed by the three-axis angle sensor are shown in Figs. 2.37 and 2.38, respectively. The control signals from the function generator are also plotted in the figures. The developed three-axis angle sensor was confirmed to have a resolution of better than 0.01 arc-second for measuring roll motion. For pitch motion estimation, the sensor was verified to have a resolution of approximately 0.2

72

Chapter 2 Three-axis angle sensor

Fig. 2.34: Variation of the voltage output from each QPD due to the applied pitch angle (QPD2).

Fig. 2.35: Noise level of the output signal from each PD cell (QPD1).

arc-second. The resolutions are supposed to increase with applications with a laser beam with a shorter wavelength or/and a scale grating with a shorter grating pitch, according to the theory. Electrical noise reduction in the performance of QPD is also efficient in achieving improved resolution of the measurement.

2.3 Three-axis inclination sensor In precision machine tools and precision measurement instruments, a precision linear slide is one of the most critical elements [25]. A precision linear slide can achieve highly precise and accurate positioning with the help of precision positioning sensors such as laser interferometers or linear encoders [26]. Meanwhile, quantitatively

2.3 Three-axis inclination sensor

73

Fig. 2.36: Noise level of the output signal from each PD cell (QPD2).

Fig. 2.37: Roll motion of the PZT tilt stage measured by the developed three-axis angle sensor.

testing the angular error motion of a linear slide has become more critical in order to obtain even higher positioning accuracy in recent years [27]. Electronic levels and autocollimators are major optical measuring instruments to be used for the evaluation of tilt angles of a precision linear slide [4, 28]. Simultaneous two-degree-offreedom measurements of the tilt angles of a linear slide can be carried out by using two-degree-of-freedom location detectors such as charge coupled devices [29] or

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Chapter 2 Three-axis angle sensor

Fig. 2.38: Pitch motion of the PZT tilt stage measured by the developed three-axis angle sensor.

multi-cell photodiodes [30]. Only by positioning a reference reflector on a slide table and changing the optical axis of an autocollimator with respect to the target, angle measurement can be carried out in a simple way. When a two-axis autocollimator is positioned such that its optical axis coincides with a linear slide’s main motion axis, it is possible to measure both the pitch and yaw angles of the linear slide simultaneously. Meanwhile, it is not possible to measure a roll angle that could greatly impact the positioning precision of a linear slide when the measurement axis of a displacement sensor is located offset from the motion axis of the linear slide [26]. With the employment of an additional autocollimator, the roll angle can be measured. However, this also induces another problem since a long reference mirror that can cover the long travel range of a linear slide should be used for measurement of the slider roll angle. Instead of an autocollimator, a laser interferometer system may be used to measure the roll angle of a linear slide having a long travel range [10, 31]. In the meanwhile, in the case where the optical path of a measurement laser beam is blocked, the measurement operation has to be started again from its beginning due to the lack of position information. In the case of the slide development process, this may be a fatal concern, where the measurement would be constantly carried out along with a hand scraping procedure to correct the shape errors of the slide guideways [32]. It is therefore preferred to develop angle sensors/inclination sensors that can be placed on a slide table and can perform extremely precise simultaneous tilt angle measurements on the X-, Y- and Z-axes. Some other measuring instruments such as fiber optic gyroscopes [33], ring laser gyroscopes [34] and vibrating structure gyroscopes [35] can also satisfy the above requirements [36]. The simultaneous measurement of three-axis tilt angles can be realized by using a gyro

2.3 Three-axis inclination sensor

75

for each measurement axis. Especially, ring laser gyroscopes [34] have a fairly high measuring resolution to be applied to the tilt angle evaluation of a linear precision slide. However, such a high precision ring laser gyroscope could have a bias instability of 0.01°/h due to the gyroscope theory under which the tilt angle is acquired by integrating the detected angular velocity [37]. For the evaluation of the angular error motion of a precision linear slide, which is usually in the order of several arcseconds, this cannot be ignored. Also, the high cost of the ring laser gyro may also be a deterrent to using it in a machine shop for testing linear slides. The three-axis laser autocollimator [15] is expected to address the above issues. In the meanwhile, the accuracy of yaw motion measurement could be degraded due to external disturbances such as temperature deviation or mechanical vibrations in a long optical path of the measurement laser beam when measuring a linear slide with a long travel range. A new design of a three-axis inclination sensor is thus introduced in this study to achieve simultaneous measurement of three-axis tilt angles of a precision linear slide. The principle of the fluid-based-type clinometer [38] is expanded by incorporating the three-axis laser autocollimator principle into the concept [39]. The optical sensor head of the three-axis laser autocollimator and the scale grating mounted on a reference float can be used for simultaneous measurement of three-axis tilt angles. In the setup, the motion of the scale grating can be measured by the optical head of the laser autocollimator fixed on the casing. A theoretical analysis of the angular motion of the reference float is carried out as the first step of research on the basis of a simplified kinematic model. After that, for the verification of the proposed concept, experiments are carried out through developing a prototype three-axis inclination sensor. Figure 2.39 displays a diagram of the proposed design of the three-axis inclination sensor. The sensor consists of an optical sensor head and a reference float floating in a casing. A scale grating with line pattern structures and a reference float on which the scale grating is placed are combined as the reference float. Related ideas can be found in the literature using liquids as a reference [40]. In the meanwhile, simultaneous measurement of rotational motion about the three axes with the enhancement of three-axis laser autocollimation is expected by the suggested threeaxis inclination sensor [15]. The tip of a tiny needle supports the middle point on the bottom surface of the float. The needle supports are attached rigidly to the bottom of the case so that the reference float can be kept in the sensor device stationary along with the X-, Y- and Z-directions. This setup allows the reference float to have a relative angular motion about three-axes with respect to the casing. Three-axis inclination measurements are done by installing the inclination sensor in a precision linear slide. The reference float floating on the liquid surface can be treated as the datum for tilt measurement about the X- and Y-axes, since the natural position of the liquid surface in the casing is always aligned with the gravity orientation. The tilt angles of the measurement target about the X- and Y-axes can be determined by detecting the relative tilt angles of the optical sensor head in

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Chapter 2 Three-axis angle sensor

Fig. 2.39: Liquid-surface-based inclination sensor based on the three-axis laser autocollimator.

relation to the reference float. It is also possible to use the reference float as a datum for tilt angle measurement about the Z-axis. Now the grating is floating, and therefore shearing force between the float surface and the liquid will generate an external torque to the reference float. The float may also be influenced by the friction force between the bottom surface of the reference float and the tip of the supporting needle. Therefore, by employing liquid having low viscosity resistance, the motion of the float about the Z-axis is expected to be suppressed under the condition where the friction force by the supporting needle is enough small; with the increase in the inertial force of the reference float, the angular position of the reference float along the Z-axis with respect to a fixed table is expected to be kept stationary. By employing another reference three-axis inclination sensor mounted on a table where the target slide is mounted, the angular displacement of the slide can be evaluated. The drift motion of the reference float about the Z-axis (yaw drift) degrades the accuracy of the yaw angle measurement in the proposed three-axis inclination sensor. This can be induced by the liquid flow around the reference float. Therefore, it is necessary to design the reference float while paying attention to its response to the liquid flow about the Z-axis. The motion equation of the reference float of the Zaxis can be represented by the second-order differential equation as follows: Iz

d2 θYaw ðtÞ = MExt ðtÞ dt2

(2:17)

In the above equation, IZ is the moment of inertia about the Z-axis of the reference float, θYaw is the angular displacement of the float about the Z-axis. The external

77

2.3 Three-axis inclination sensor

torque about the Z-axis applied to the float is denoted by MExt. The parameter t denotes time. We now discretize the equation as follows by introducing the discretized time ti (i = 1, 2, . . ., N): ΔθYaw ðti Þ Δt Iz

−

ΔθYaw ðti − 1 Þ Δt = MExt ðti Þ

Δt

(2:18)

In the above equation, ΔθYaw(ti) can be defined as θYaw(ti) − θYaw(ti −1). The following equation can be obtained by solving this equation concerning ΔθYaw(ti): ΔθYaw ðti Þ = ΔθYaw ðti − 1 Þ +

MExt ðti Þ 2 Δt Iz

(2:19)

The above equation means that the large IZ and the decrease of MExt lead to the reduction of the yaw drift. In this study, the model of the external torque applied to the reference float is treated as the sum of MBottom(t) and MSide(t), the shear torque by the liquid flows beneath the float and that at the side of the float. On the basis of a model shown in Fig. 2.40, MBottom(t) is firstly estimated. We now presume that it is possible to handle the liquid in the casing as a Newtonian fluid. We also assume that the liquid only flows about the Z-axis. Under this condition, τ, the shear stress can be represented by the following equation [41]: τ=μ

dv dz

(2:20)

In the above equation, the shear viscosity of the liquid is denoted by µ. The parameters v and z denote the velocity and the Z-directional displacement, respectively.

Fig. 2.40: Shear torque model between the bottom surface of the casing and the reference float.

Now the fluid velocities vf and vc at the bottoms of the float and the casing, respectively, can be obtained by vf = rωf(t) and vc = rωc(t), where ωf(t) and ωc(t) are the angular velocities of the float and the casing, respectively. Therefore, the shear stress dτ to be added to the region dS can be calculated by the following equation:

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Chapter 2 Three-axis angle sensor

dτ = μ

vc − vf ωc ðtÞ − ωf ðtÞ = μr h h

(2:21)

In the polar coordinate system, by the radial position r and angular position θ, dS can be obtained by dS = rdrdθ. Thus, the following equation expressing dMBottom(t), the shear torque at dS, can be obtained as follows: dMBottom ðtÞ = rdτdS = μr3

ωc ðtÞ − ωf ðtÞ drdθ h

(2:22)

MBottom(t), the total shear torque to the bottom of the float, can then be obtained through the integration operation of eq. (2.22): 2ðπ ð a

MBottom ðtÞ =

μr3

ωc ðtÞ − ωf ðtÞ πμa4 ðωc ðtÞ − ωf ðtÞÞ drdθ = 2h h

(2:23)

0 0

It should be noted that the radius of the float is denoted by a in the above equation. MSide(t) is also estimated for the modeling of the yaw motion of the reference float. A schematic of the shear torque model between the sidewall of the casing and that of the float is shown in Fig. 2.41. To explain the motion of the reference float, a cylindrical coordinate system is used in the model. The shear stress τrθ by the liquid viscosity can be expressed by the following equation [41]:   ∂ vθ  1 ∂vr (2:24) + τrθ = μ r ∂r r r ∂θ In the above equation, vr and vθ denote the radial and circumferential components of fluid velocity, respectively.

Fig. 2.41: Shear torque model between the sidewalls of the casing and the reference float.

For the sake of simplicity, now we assume vr = 0. Under this condition, eq. (2.24) can be simplified as follows:

2.3 Three-axis inclination sensor

τrθ = μr

∂ vθ  ∂r r

79

(2:25)

By deriving Navier–Stokes equations, the circumferential liquid velocity component vθ can be obtained. Now we express the body forces along the Z-, circumferential and radial directions as FZ, Fθ and Fr, respectively. Incompressible momentum Navier–Stokes equations can be expressed in the cylindrical coordinate system as follows, by denoting the Z-directional liquid velocity component as vZ [42]: ∂vr ∂vr vθ ∂vr vθ 2 ∂vr + vr + − + vz ∂t ∂r r ∂θ r ∂z  2  1 ∂p ∂ vr 1 ∂vr vr 1 ∂2 vr 2 ∂vθ ∂2 vr + Fr + + − =− − + +ν ∂r2 ∂z2 ρ ∂r r ∂r r2 r2 ∂θ2 r2 ∂θ

(2:26)

∂vθ ∂vθ vθ ∂vθ vr vθ ∂vθ + vr + − + vz ∂t ∂r r ∂θ r ∂z  2  1 ∂p ∂ vθ 1 ∂vθ vθ 1 ∂2 vθ 2 ∂vr ∂2 vθ + Fθ + + + =− − + +ν ∂r2 r2 r2 ∂θ2 r2 ∂θ ∂z2 ρ r∂θ r ∂r

(2:27)

∂vz ∂vz vθ ∂vz ∂vz + vr + + vz ∂t ∂r r ∂θ ∂z  2  1 ∂p ∂ vz 1 ∂vz 1 ∂2 vz ∂2 vz + Fz + + =− + +ν ∂r2 ∂z2 ρ ∂z r ∂r r2 ∂θ2

(2:28)

In the above equations, the density, pressure and dynamic viscosity of the fluid in the casing are denoted by ρ, p and ν, respectively. It should be noted that these equations cannot be analytically solved. Therefore, to obtain vθ, the following boundary conditions are applied to convert them into linear differential equations [42]: (a) The flow is a steady one, and is not a function of time (∂/∂t = 0). (b) ρ can be treated as a constant; namely, the flow is incompressible. (c) vr = vz = 0. Here, we assume the continuity equation expressed in the cylindrical coordinate system as follows: 1 ∂ðvr rÞ 1 ∂vθ ∂vz + =0 + ∂z r ∂r r ∂θ

(2:29)

By applying the condition (c) to the above equation, ∂νθ/∂θ = 0 can be obtained; this implies that the component of the circumferential velocity of the flow is constant with respect to θ. In respect to this, we now presume the following boundary conditions as well: (d) Along the Z-direction, the circumferential velocity component is uniform (∂νθ/∂z = 0)

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Chapter 2 Three-axis angle sensor

(e) Along the circumferential direction, the pressure distribution can be treated as uniform (∂p/∂θ = 0) The following linear differential equations can be obtained by applying these boundary conditions to eqs. (2.26–2.28):

0=

vθ 2 1 ∂p =− r ρ ∂r

(2:30)

∂2 vθ 1 ∂vθ vθ + − 2 ∂r2 r r ∂r

(2:31)

0= −

1 ∂p ρ ∂z

(2:32)

Deriving eq. (2.30) gives the following equation: vθ = C1 r + C2

1 r

(2:33)

The parameters C1 and C2 are constants in the above equation. At the sidewall of the float, the fluid velocity component vθ can be obtained as vθ = aωf(t). In the same way, at the sidewall of the casing, the fluid velocity component vθ can be obtained as vθ = Rωf(t). In the equation, the radius of the casing is denoted by R. By using these equations, C1 and C2 can be derived as follows: C1 =

 1 2 a ωf ðtÞ − R2 ωc ðtÞ a2 + R 2

(2:34)

 a2 R 2

ωc ðtÞ − ωf ðtÞ 2 2 a +R

(2:35)

C2 =

From eqs. (2.33) to (2.35), the following equation can be obtained:  

2 

1 1 2 2 2 a ωf ðtÞ − R ωc ðtÞ r + a R − ωf ðtÞ + ωc ðtÞ vθ = 2 a − R2 r

(2:36)

The shear stress τrθ by the fluid viscosity can thus be obtained as follows from eqs. (2.26) and (2.24):   

2 

 1 d 1 2 2 2 a ω ðtÞ − R ω ðtÞ + a R − ω ðtÞ + ω ðtÞ τrθ = μr c c f f dr a2 − R2 r2 (2:37) 2μa2 R2 ðωc ðtÞ − ωf ðtÞÞ = ðR2 − a2 Þr2 By the above equation, the shear stress on the sidewall of the float can thus be obtained as follows:

2.3 Three-axis inclination sensor

  2μR2 ðωc ðtÞ − ωf ðtÞÞ τrθ r = a = R 2 − a2

81

(2:38)

Beneath the liquid surface, the area of the float sidewall Sside can be expressed as Sside = 2πaT, where T is the draft of the float. Therefore, MSide(t), the shear torque applied to the float sidewall can be derived as follows: Mside ðtÞ = τrθ · Sside · a =

4πμa2 R2 Tðωc ðtÞ − ωf ðtÞÞ R 2 − a2

(2:39)

Finally, the following equation can be obtained as follows by applying MBottom(t) and Mside(t) to eq. (2.17):  4   d2 θYaw ðtÞ a 4R2 a2 T

ωc ðtÞ − ωf ðtÞ = πμ + Iz 2 2 2 2h R − a dt (2:40)

 = πμK ða, h, R, T Þ ωc ðtÞ − ωf ðtÞ In the above equations, a factor K including the parameters of the casing and the reference float is implemented. The factor K is desired to be minimized as much as possible in order to minimize the yaw motion of the reference float caused by the angular motion of the casing, as can be seen in the equation. A design analysis is then conducted to analyze how K is influenced by the design parameters. It should be noted that with regard to the developed prototype sensor mentioned in the following section of this chapter, the draft float T is set to be 8.8 mm. The variance of K as a function of the reference float radius a under the condition of R = 140 mm is shown in Fig. 2.42. Variations of K with respect to h ranging from 1 to 128 mm are plotted in the figure. K was found to decrease with the decrease of a. This finding suggests that the reduction of a tends to decrease the yaw motion of the reference float caused by the casing’s angular motion. K, meanwhile, has been found to increase significantly with an increase of a over 135 mm; This finding means that the increase in a decreases the clearance between the float and the casing side surfaces. As a result, the transmitting torque between them increases due to the fluid viscosity. Furthermore, the increase in h is also observed to contribute to the reduction of K. The theoretical studies mentioned above have shown that the following designs are required to minimize the reference float’s yaw drift: (a) Maximize R, the radius of the casing, with respect to a, the radius of reference float. (b) Minimize a. (c) Maximize h, the distance between the casing and the float

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Chapter 2 Three-axis angle sensor

Fig. 2.42: Variation of K as a function of the float radius a (R = 140 mm and T = 8.8 mm).

In the meantime, the minimum value of a needs to be calculated by taking into account not only the buoyancy to be produced by the reference float, but also the gravitational force of the float including masses of fixing screws, optical window and the scale grating. Furthermore, the allowable sizes of R and h will be restricted by the design of the three-axis inclination sensor. It should be noted that, for the sake of simplicity, the effect of a on Iz is not considered. The yaw drift of the reference float is simulated by using the design parameters. With regard to the angular velocity of float ωf(t), eq. (2.40) can be updated as follows:   dωf ðtÞ πμ a4 4R2 a2 T ðωc ðtÞ − ωf ðtÞÞ (2:41) + 2 = Iz 2h R − a2 dt By the Runge–Kutta method [43], the angular velocity of the reference float f(t) is numerically solved. A step input with a casing angular velocity ωc of 1 arc-second/s is assumed in the numerical simulation. A time step employed for the calculation is 10 ms. Table 2.6 summarizes the parameters used in the simulation. The result of a computational simulation is shown in Fig. 2.43, indicating the difference in the angular velocity of the float relative to time t. An angular displacement of the float θc can be acquired as shown in Fig. 2.44 by integrating the numerically simulated ωc(t) shown in Fig. 2.43. The sum of yaw drift increases with the increase of time, as shown in the figure. On the other hand, a yaw drift of less than 1 arc-second is predicted after 100 s; this value is comparable to or better than that by the conventional gyros [37]. To further analyze the properties of the reference float, the pitch and roll motions of the reference float are also modeled, following the modeling of yaw motion as described above. The reference float used in this research is constructed to have a disk-like shape. Both the roll and pitch motions of the reference float can thus be

2.3 Three-axis inclination sensor

83

Tab. 2.6: Parameters employed for the calculations. Component

Parameters

Symbol

Value

Float

Radius

a

 mm

Draft

T

 mm

Moment of inertia about the Z-axis

Iz

. × − kg m

Radius

R

 mm

Water level

h

 mm

Type

–

Water

Viscosity

μ

. × − Pa s

Density

ρ

. ×  kg/m

Casing

Fluid

Fig. 2.43: Simulated angular velocity of the reference float about the Z-axis due to the yawing given to the casing.

Fig. 2.44: Simulated angular displacement of the reference float about the Z-axis due to the yawing given to the casing.

84

Chapter 2 Three-axis angle sensor

treated by the same model. In the following, as shown in Fig. 2.45, attention is paid to the modeling of the roll motion of the reference float.

Fig. 2.45: Analytical model of the reference float about its roll motion.

The following equation describes the angular motion of the reference float: IX

d2 θRoll dθRoll +c =M 2 dt dt

(2:42)

In the above equation, the moment of inertia of the reference float about the X-axis at the gravitational center is denoted by IX. The parameters M, c and θRoll are a torque applied to the reference float, the viscosity coefficient of the fluid and the roll angle of reference float. Now Fb, the buoyancy to be generated by the float can be expressed as follows:  Fb = t 0 , ρgVf sin θRoll , ρgVf cos θRoll (2:43) In the above equation, Vf, g and ρ denote the volume of reference float beneath the liquid surface, gravitational acceleration and fluid density. A center of the volume beneath the liquid surface coincides with the point of application of buoyancy (yf, zf) that can be obtained by the following equations: R R xdV zdV yf = R , zf = R (2:44) dV dV The point of application of buoyancy (yf, zf) in this analysis is estimated using a feature in the commercial CAD (computer aided design) program. Table 2.7 summarizes the parameters applied to the simulation. The variation of (yf, zf) according to the change in θRoll in the case of T = 10 mm and a = 30 mm is shown in Fig. 2.42.

2.3 Three-axis inclination sensor

85

Tab. 2.7: Design parameters of the reference float. Parameters

Symbol

Value

Moment of inertiaof the floatabout the X-axis

IX

. × − kg m

Sensitivity coefficient

α

−. m/rad

Gravity

g

. m/s

Fig. 2.46: Point of application of buoyancy.

With respect to θRoll, yf can be expressed by a linear function, while zf can be described by a second-order polynomial function as shown in Fig. 2.46. For the sake of simplicity, zf is assumed to be zero because it is small enough to be neglected compared with yf. It is assessed that the gradient of the estimated yf is −0.1091 mm/ arc-second under the condition with T and a of 10 and 30 mm, respectively. A positional vector rres from the center of buoyancy to (yg, zg), the gravitational center of the reference float can be expressed as follows:  rres = rf − rg = t 0 , yf − yg , zf − zg (2:45) By using the vector, the rotational moment associated with the buoyancy applied to the float can be expressed by the following equation: Mf = rres × Fb   = t 0 , yf − yg , zf − zg × t 0 , ρgVf sin θRoll , ρgVf cos θRoll  = t ðyf − yg ÞρgVf cosθRoll − ðzf − zg ÞρgVf sin θRoll , 0, 0  = t ρgVf ðyf cos θRoll + zg sin θRoll Þ, 0, 0

(2:46)

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Chapter 2 Three-axis angle sensor

In the above equation, since the float has a symmetrical profile, yg is treated to be zero, as well. In addition, the measurement range of the inclination sensor is limited to be less than ±50 arc-seconds. Therefore, under the approximations of sinθRoll ≈ θRoll and cosθRoll ≈ 1, Mf can further be simplified as follows:  Mf = t ρgVf ðα + zg ÞθRoll , 0, 0 (2:47) The rotational moment about the X-axis could be generated in Fig. 2.45, according to the above equation. Applying Mf in eq. (2.47) to eq. (2.42) gives the following equation of the roll motion of the float: IX

d2 θRoll dθRoll +c = ρgVf ðα + zg ÞθRoll dt2 dt

(2:48)

The same discussion can be applied to the pitch motion of the float. It should be noted that, from the above equation, the natural frequencies of the roll and pitch motions of the float fr and fp, respectively, can be derived as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi

ffi − ρgVf α + zg − ρgVf α + zg 1 1 fr = , fp = (2:49) IX IY 2π 2π It should be noted that fp can exist in the region of zg < − α since α < 0. When designing the reference float, care should be taken to ensure that the frequency bandwidth of the three-axis inclination sensor does not interfere with the natural frequency fp. An optical head of the angle sensor on the basis of the three-axis laser autocollimation [15] is employed in the proposed three-axis inclination sensor, while a scale grating with linear pattern structures with a constant period of p is used as a reflector and is mounted on the float. The geometric relationship between the scale grating and the optical head is shown in Fig. 2.47. The zeroth-order diffracted beam and firstorder diffracted beams in positive and negative directions will be produced by a measuring laser beam projected onto the scale grating, as shown in the figure. For the sake of clarity, the negative first-order diffracted beam is not indicated in the figure. Based on two-axis laser autocollimation [13], the two-dimensional (2D) tilt angles of the scale grating along the X- and Y-axes can be determined. The following equations explain the relationship between the angular displacements of the scale grating ΔθRoll and ΔθPitch and the spot displacements Δv0 and Δh0 on the detector plane, respectively [44]: 1 Δh0 ΔθRoll = arctan , f 2

1 Δv0 ΔθPitch = arctan f 2

(2:50)

In the above equation, the focal length of the collimator objective is denoted by f. On the contrary, QPD0 in Fig. 2.43 cannot detect the angular displacement of the scale grating about the Z-axis. However, QPD1, which captures the first-order diffracted beam from the grating reflector, can sense the scale angular displacement

2.3 Three-axis inclination sensor

87

Fig. 2.47: A schematic of the measurement principle of the three-axis laser autocollimation.

about the Z-axis. According to the grating equation [14], the angle of diffraction ϕ1st of the first-order diffracted beam having a wavelength λ can be expressed by the following equation: ’1st = arcsin ðλ=pÞ

(2:51)

In the above equation, p is the pitch of the grating. Now we denote the propagating directions of the first-order diffracted beam in the case with the yaw angle of zero and ΔθYaw as r and r’, respectively. From the geometric relationship, by denoting the propagating direction of the zeroth-order diffracted beam as n, the following equation can be obtained [15]: r′ = r cos ΔθYaw + nðn · rÞ½1 − cos ΔθYaw  + ðn × rÞ sin ΔθYaw

(2:52)

Assuming that ΔθYaw is small, eq. (2.52) can further be modified as follows: dr = r′ − r ffi ðn × rÞΔθYaw

(2:53)

By using r and dr, the angle between r and r′ (ΔψZ) can be expressed by the following equation: ΔψZ ffi

kdrk = ΔθYaw sin ’1st krk

(2:54)

ΔψZ in the above equation corresponds to the angle of incidence of the first-order diffracted beam, which is defined with respect to the optical axis of the collimator objective 1. The relationship between Δv1_Yawing, the displacement of the beam spot on QPD1 along the V1-axis and ΔθYaw can be described by the following equation, on the basis of the principle of the laser autocollimation:

88

Chapter 2 Three-axis angle sensor

Δv1

Yawing

= f tan ΔψZ ffi f ΔψZ =

f λΔθYaw p

(2:55)

Modification of the above equation with respect to ΔθYaw can give the following equation: ΔθYaw =

p Δv1 fλ

(2:56)

Yawing

In the meanwhile, the focused spot on QPD1 can be affected by ΔθPitch, as well. Therefore, the deviation of the yaw angle can be obtained through the arithmetic operation that can be expressed as follows: ΔθYaw =

p Δv1 fλ

Yawing

=

p

Δv1 − Δv1 fλ

 Pitching

=

p ðΔv1 − Δv0 Þ fλ

(2:57)

The optical configuration of the optical sensor head for the three-axis inclination sensor is shown in Fig. 2.48. Table 2.8 summarizes the design parameter of the optical sensor head. A LD with a wavelength of 685 nm was employed as the light source for the optical sensor head. As the photodetectors for the optical sensor head, two types of QPD were employed; one is having an insensitive gap among the active cells of 5 µm (S6795, Hamamatsu Photonics), while the other is having that of 10 µm (Hamamatsu Photonics S1651-03). The former and latter were employed to capture the diffracted beams in the zeroth and the first orders, respectively. In the optical sensor head, all the optical components were aligned in a size of 100 mm × 100 mm. It should be noted that, in the case of evaluating a stage system with a long travel range, the influence of the earth curvature should be corrected for further better measurement accuracy [45]. A diagram of the designed reference float, consisting of a float, a scale grid, an optical window, a cover and screws, is shown in Fig. 2.49. For the reduction of the yaw drift, the reference float was designed in a compact size regarding the results of the theoretical investigation described above. In order to minimize the overall weight of the reference float, attention was also paid to the use of lightweight materials for each part. Table 2.9 summarizes the design parameters of the reference float, which were mainly dominated by the scale grating size. Natural frequencies of the reference float about the X- and Y-directions were estimated to be approximately 6 Hz based on eq. (2.49). With the decrease of IX and IY through the design modification of the reference float, higher natural frequencies are expected to be achieved; on the contrary, this modification could induce the increase of IZ, and could affect the float yaw drift. The developed reference float with a scale grating is shown in Fig. 2.50. It is important to keep the reference float steady along with the X-, Y- and Z-directions while facilitating its rotational motions. Therefore, in the system, a supporting needle is implemented. As shown in Fig. 2.51, the reference float is mounted on a jig, which has a dent at the middle of its bottom. The needle

2.3 Three-axis inclination sensor

89

Fig. 2.48: Optical sensor head.

Tab. 2.8: Design parameters of the optical head for the three-axis angle sensor. Parameters Light wavelength Aperture diameter

Value  nm  mm

Grating period

. μm

Focal length of the collimator objective

. mm

is fixed at the center of the casing surface so that the needle tip contacts with the reference float. This configuration makes it possible to suppress the three-translational motions of the float, while allowing the rotational motions about three axes. In the developed three-axis inclination sensor, the sensor head and the reference float are incorporated, the schematic of which is seen in Fig. 2.52. The setup also contains an alignment mechanism for the sensor head and the casing in an

90

Chapter 2 Three-axis angle sensor

Fig. 2.49: Reference float.

Fig. 2.50: A photograph of the reference float.

Fig. 2.51: Reference float and a supporting needle.

2.3 Three-axis inclination sensor

91

Tab. 2.9: Specification of the developed reference float. Parameters

Symbol

Value

Radius

a

 mm  mm

Thickness Draft

T

. mm . × − kg

Mass Moment of inertia about the Z-axis

Iz

. × − kg m

Moment of inertia about the X/Y-axes

IX/IY

. × − kg m

Material

–

Polyoxymethylene

Density

ρ

 kg/m

inner diameter of 240 mm. Through poles and positioning systems, the casing is rigidly attached to the optical sensor head. The reference float is made to float on the water, while a needle fixed on the bottom surface of the casing point supported the float. With a diameter of approximately 300 mm and a height of 360 mm, the whole sensor device has been constructed.

Fig. 2.52: Three-axis inclination sensor.

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Chapter 2 Three-axis angle sensor

The basic characteristics of the proposed system were verified in experiments by the use of the established prototype three-axis inclination sensor. At first, the performance of the optical sensor head mounted on the system was tested. A photograph of the experimental setup is shown in Fig. 2.53. The reference float was placed on a stage system fixed on the bottom surface of the casing under which the water was drained to prevent the effect of the instability of the reference float being floating on the fluid surface. A two-axis precision PZT tilt stage was employed as the stage method to add tilt angles to the reference float. A commercial two-axis laser autocollimator (Elcomat 3,000, Möller-Wedel, Germany) was also used to assess the tilt angles of the reference float along the Y- and Z-axes. As a reflector for the commercial laser autocollimator, a mirror was mounted on the side facet of the PZT tilt stage. It should be noted that the roll angle (tilt angle about the X-axis) cannot be determined by the commercial autocollimator in the configuration shown in Fig. 2.53. This could however be determined by positioning the commercial autocollimator through re-mounting the reflector on the other side of the PZT tilt stage while aligning the measurement axis of the commercial autocollimator to the Y-axis. In the tests, the PZT tilt stage added tilt angles to the reference float. The angular displacement of the reference float was measured by the reference commercial laser autocollimator and the developed optical sensor head. The measured angular displacements about the X-, Y- and Z-axes are shown in Figs. 2.54–2.56, respectively. The horizontal and the vertical axes indicate the angular displacement measure by the reference commercial laser autocollimator and that by the developed optical sensor head, respectively. In the figure, nonlinear errors are also plotted. The outputs from the established sensor and those from the commercial laser autocollimator showed close correlations as shown in the figure. In an angular displacement range of 50 arc-seconds, the non-linear error of the pitch, roll and yaw measurements were evaluated to be ±1.05, ±0.93 and ±1.20 arc-seconds, respectively. These values are close to the results published in the previous work [15]. Meanwhile, in contrast with those for pitch and roll angle measurement, the sensitivity of yaw angle measurement was evaluated to be low; as can be seen in eqs. (2.51) and (2.57), this is primarily due to its different theory of angle detection. The pitch and roll measurement of the established three-axis inclination sensor can be calibrated and by the technique mentioned above. For the calibration of the yaw measurement, the three-axis inclination sensor can be placed on a precision spindle installed with a precision rotary encoder. The stability of the three-axis inclination sensor was tested in experiments after verification of the static characteristics of the formed optical sensor head. A setup used in the experiments is shown in Fig. 2.57. Initially, the point-support effect of the needle was confirmed. The reference float was made to float without the needle support on the water surface in the casing and the differences of the three-axis outputs from the optical sensor head were tested. Results are shown in Fig. 2.58. The continuation of the experiment was difficult due to the instability of the reference float on the water surface, as shown in the figure. The experiment was again conducted

2.3 Three-axis inclination sensor

Fig. 2.53: Experimental setup with the three-axis inclination sensor.

Fig. 2.54: Reading of the developed inclination sensor (roll).

Fig. 2.55: Reading of the developed inclination sensor (pitch).

93

94

Chapter 2 Three-axis angle sensor

Fig. 2.56: Reading of the developed inclination sensor (yaw).

after the reference float was point supported by a needle fixed on the float’s bottom surface. Results are shown in Fig. 2.59. By the enhancement of point supported by the needle, reliable measurement of the pitch and roll angle was realized, as shown in the figure. The point supported by the needle also enabled the yaw angle measurement, which was difficult under the condition where the reference float was free from the point support. Throughout the tests, the water level in the casing was adjusted to be 28 mm by trial and error in order to change the contact force between the reference float and the needle to improve sensor stability. Standard deviations of 3.4, 3.9 and 32.9 arc-seconds were obtained for the stabilities of pitch, roll and yaw angle output, respectively, in a period of 60 s. A comparatively significant variation in the performance of the yaw angle was found to be largely due to the lower sensitivity and volatility of yaw measurement caused by standing waves in water. Due to the instability of the reference float, the results obtained by the produced prototype were not as good as expected from the theory of the three-axis laser autocollimation [15]. In some applications, allowing the low-frequency bandwidth of the three-axis inclination sensor, signal processing techniques such as low-pass filtering may minimize the resolution to the level of 1 arc-second. To suppress the effects of standing waves, a ring-shaped cover plate was inserted into the three-axis inclination sensor since all the external disruptions that contribute to the appearance of standing waves in water are difficult to remove. A 5 mm thickness cover plate was made to float on the water surface. Outer and inner diameters of which were designed to be 90 and 230 mm, respectively, so as to avoid collision with the reference float. The stability of the reading of the three-axis inclination sensor with the cover plate evaluated in experiments is shown in Fig. 2.60. Standard deviations of 1.8, 2.9 and 18.6 arc-seconds were observed for the reading of pitch, roll and yaw angles, respectively, in a period of 60 s. As shown in the figure, in particular, a low-frequency component in the performance of the yaw angle was decreased. The cover plate added was supposed to act as a damper to suppress the standing waves on the water surface, and decreased the yaw motion. Another key concern of the developed three-axis inclination sensor, yaw drift caused by the liquid flow in the casing, was treated in experiments. A schematic of

2.3 Three-axis inclination sensor

95

the experimental setup is shown in Figs. 2.61 and 2.62. On a rotary stage, which could be freely rotated about the Z-axis, the three-axis inclination sensor was mounted. Meanwhile, the rotary stage could not spin the three-axis inclination sensor with its actuator because of its heavyweight. To give the angular displacement about the Z-axis to the inclination sensor, an external linear slide was used. Beneath the casing, the external linear slide was placed so that it can push a rod that connects the case and the head of the optical sensor. For holding stationary the three-axis inclination sensor after the angular displacement about the Z-axis is applied to the sensor, a magnet was attached to the top plate of the linear slide. In the setup shown in Fig. 2.61, as a reference sensor, a commercial autocollimator was used to validate the reading of the three-axis inclination sensor.

Fig. 2.57: The reference float supported by the needle.

Fig. 2.58: Stability of the readings of the three-axis inclination sensor output.

Figure 2.63 shows the result without the cover plate. Both the readings of the commercial autocollimator and the inclination sensor of the three axes were plotted in the figure. The provided yaw motion was successfully detected by the established three-axis

96

Chapter 2 Three-axis angle sensor

Fig. 2.59: Stability of the readings of the three-axis inclination sensor output with the support by a needle.

Fig. 2.60: Stability of the readings of the three-axis inclination sensor output.

Fig. 2.61: Experimental setup for the evaluation of drift about the Z-axis.

2.3 Three-axis inclination sensor

97

Fig. 2.62: A photograph of the Experimental setup for the evaluation of drift about the Z-axis.

inclination sensor. An angular displacement of about 1,200 arc-seconds was added to the three-axis inclination sensor according to the output from the commercial autocollimator. The output from the inclination sensor, meanwhile, was approximately 300 arc-seconds. This outcome means that the yaw drift was approximately 900 arcseconds due to the angular motion produced by the linear slide, which was greater than predicted from the simulation. The frictional force between the reference float and the tip of the needle is a potential root cause of these effects, which were different from those calculated in the theoretical analysis. The difference in fluid behavior between the reference float and the casing may also cause the shear effect. On the other hand, after providing the step input to the float, the variations in the inclination sensor outputs were found to be small. The sensor reading was also found to be quite stable, as well. This finding shows the feasibility of applying this sensor to assess the accuracy of a linear slide with a slight angular error motion about each axis. The result obtained by using the cover plate is shown in Fig. 2.64. The presence of the cover plate was checked to have no effect on the yaw drift of the reference float. In this study, attention has been paid to checking the possibility of the suggested three-axis inclination sensor. To demonstrate the viability of the built prototype sensor, only simple tests were carried out. A further detailed investigation is required for the assessment of the performances of the built sensor such as the static and dynamic characteristics of the sensor reading, signal processing technique for the improvement of the resolution. Design optimization of the supporting needle and the reference float is also required.

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Fig. 2.63: Reading of the developed inclination sensor when measured the yawing motion of the reference float (without the cover plate).

Fig. 2.64: Reading of the developed inclination sensor when measured the yawing motion of the reference float (with the cover plate).

2.4 Summary In this chapter, a compact three-axis angle sensor for measurement of three-axis tilt motions (roll, yaw and pitch) of a precision linear stage has been introduced. Instead of a flat mirror reflector employed in the conventional two-axis laser autocollimator, a 2D scale grating is employed as the reflector for measurement. By employing the zeroth- and first-order diffracted beams, angular error motions such as the pitch, roll and yaw can be detected simultaneously. A prototype optical setup has been designed and constructed for the verification of the feasibility of the proposed concept of the three-axis angle sensor, while employing detector units composed of a collimator objective and a QPD. It should be noted that the principle of angle measurement about the laser beam axis (Z-axis) is different from those for the X- and Y-axes, and the sensitivity becomes different. To improve the sensitivity of angle measurement about the Z-axis, an optical magnifier has been integrated into the prototype. Experimental results have demonstrated that the developed three-axis autocollimator can

References

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achieve a resolution of better than 0.01 arc-second, and can distinguish sinusoidal angular motions in all three axes. In addition, the concept of the three-axis angle sensor has been extended to the three-axis inclination sensor, in which a reference float with a 2D scale grating is employed as the reference for measurement.

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Buice ES, Otten D, Yang RH, Smith ST, Hocken RJ, Trumper DL Design evaluation of a singleaxis precision controlled positioning stage. Precis Eng 2009, 33, 4, 418–424. Weck M, Wahner U Linear magnetic bearing and levitation system for machine tools. CIRP Ann – Manuf Technol 1998, 47, 1, 311–314. Erkorkmaz K, Gorniak JM, Gordon DJ Precision machine tool X-Y stage utilizing a planar air bearing arrangement. CIRP Ann – Manuf Technol 2010, 59, 1, 425–428. Bryan JB The Abbé principle revisited: An updated interpretation. Precis Eng 1979, 1, 3, 129–132. Kunzmann H, Pfeifer T, Flügge J Scales vs. laser interferometers performance and comparison of two measuring systems. CIRP Ann – Manuf Technol 1993, 42, 2, 753–767. Steinmetz CR Sub-micron position measurement and control on precision machine tools with laser interferometry. Precis Eng 1990, 12, 1, 12–24. Sawabe M, Maeda F, Yamaryo Y, Simomura T, Saruki Y, Kubo T, Sakai H, Aoyagi S A new vacuum interferometric comparator for calibrating the fine linear encoders and scales. Precis Eng 2004, 28, 3, 320–328. Weckenmann A, Hoffmann J Long range 3 D scanning tunnelling microscopy. CIRP Ann – Manuf Technol 2007, 56, 1, 525–528. Knapp W Measurement uncertainty and machine tool testing. CIRP Ann – Manuf Technol 2002, 51, 1, 459–462. Gao W, Arai Y, Shibuya A, Kiyono S, Park CH Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage. Precis Eng 2006, 30, 1, 96–103. Estler WT, Queen YH Angle Metrology of Dispersion Prisms. CIRP Ann 2000, 49, 1, 415–418. Elster C, Weingärtner I, Schulz M Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors. Precis Eng 2006, 30, 1, 32–38. Ennos AE, Virdee MS High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Precis Eng 1982, 4, 1, 5–8. Hecht E Optics, 5th Pearson, 2017. Gao W, Saito Y, Muto H, Arai Y, Shimizu Y A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann – Manuf Technol 2011, 60, 1, 515–518. Saito Y Sensitive angle sensors based on laser autocollimation. Tohoku University, Doctoral thesis, 2010. Murata D An Ultra-sensitive Optical Angle Sensor. Tohoku University, Master thesis, 2014. Maruyama T An ultra-high sensitive optical angle sensor. Tohoku University, Master thesis, 2016. Muto H, Saito Y, Arai Y, Gao W An optical sensor for measurement of angular motions of a planar motion stage. The Proceedings of Conference of Tohoku Branch, The Japan Society of Mechanical Engineers, 2010, 176–177.

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[20] Saito Y, Arai Y, Gao W Development of a sensitive 3-axis angle sensor based on laser autocollimation. Proceedings of the 23rd Annual Meeting of the American Society for Precision Engineering, ASPE 2008 and the 12th ICPE, American Society for Precision Engineering, 2008. [21] Saito Y, Arai Y, Gao W Multi-axis angle sensor based on laser autocollimation. Proceedings of the 10th Anniversary International Conference of the European Society for Precision Engineering and Nanotechnology, EUSPEN 2008, euspen, 2008, 333–337. [22] Saito Y, Arai Y, Gao W Development of a three-axis angle sensor. J Japan Soc Precis Eng 2008, 74, 9, 997–1001. [23] Shimizu Y, Kataoka S, Ishikawa T, Chen Y-L, Chen X, Matsukuma H, Gao W, Liquid-SurfaceBased Three-Axis A Inclination sensor for measurement of stage tilt motions. Sensors 2018, 18, 2, 398. [24] Ishikawa T Development of a precision attitude sensor. Tohoku University, Master thesis, 2015. [25] Schwenke H, Knapp W, Haitjema H, Weckenmann A, Schmitt R, Delbressine F Geometric error measurement and compensation of machines-An update. CIRP Ann – Manuf Technol 2008, 57, 2, 660–675. [26] Gao W, Kim SW, Bosse H, Haitjema H, Chen YL, Lu XD, Knapp W, Weckenmann A, Estler WT, Kunzmann H Measurement technologies for precision positioning. CIRP Ann – Manuf Technol 2015, 64, 2, 773–796. [27] Gao W Precision Nanometrology, London, Springer London, 2010. [28] ISO – ISO 230-1:2012 – Test code for machine tools – Part 1: Geometric accuracy of machines operating under no-load or quasi-static conditions. (Accessed November 19, 2020, at https:// www.iso.org/standard/46449.html) [29] MÖLLER-WEDEL OPTICAL. Electronic autocollimators. (Published 2007Accessed March 14, 2021, at www.moeller-wedel-optical.com) [30] Gao W, Ohnuma T, Satoh H, Shimizu H, Kiyono S, Makino H A precision angle sensor using a multi-cell photodiode array. CIRP Ann – Manuf Technol 2004, 53, 1, 425–428. [31] Zhang JH, Menq CH A linear/angular interferometer capable of measuring large angular motion. Meas Sci Tech 1999, 10, 12, 1247–1253. [32] Tan SL, Shimizu Y, Meguro T, Ito S, Gao W Design of a laser autocollimator-based optical sensor with a rangefinder for error correction of precision slide guideways. Int J Precis Eng Manuf 2015, 16, 3, 423–431. [33] Vali V, Shorthill RW Fiber ring interferometer. Applied optics 1976, 15, 5, 1099. [34] Lawrence A The Ring Laser Gyro, New York, NY, Springer, 1998, 208–224. [35] Yazdi N, Ayazi F, Najafi K Micromachined inertial sensors. Proc IEEE 1998, 86, 8, 1640–1658. [36] Titterton D, Weston J Strapdown Inertial Navigation Technology, Institution of Engineering and Technology, 2004. [37] Barbour NM Inertial Navigation Sensors. (Accessed November 19, 2020, at https://apps.dtic. mil/sti/pdfs/ADA581016.pdf) [38] Shimizu Y, Kataoka S, Gao W High Resolution Clinometers for Measurement of Roll Error Motion of a Precision Linear Slide. Chinese J Mech Eng 2018, 31, 1, 92. [39] Shimizu Y, Kataoka S, Ishikawa T, Chen Y-LY-L, Chen X, Matsukuma H, Gao W A liquid-surfacebased three-axis inclination sensor for measurement of stage tilt motions. Sensors 2018, 18, 2, 398. [40] Debenham M, Dew GD Determining gravitationl deformation of an optical flat by liquid surface interferometry. Precis Eng 1980, 2, 2, 93–97.

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Batchelor GK An Introduction to Fluid Dynamics, Cambridge University Press, 2000. Temam R Navier-Stokes Equations, North Holland, North Holland, 1979. Süli E, Mayers DF An Introduction to Numerical Analysis, Cambridge University Press, 2003. Saito Y, Arai Y, Gao W Investigation of an optical sensor for small tilt angle detection of a precision linear stage. Meas Sci Tech 2010, 21, 5, 054006. [45] Evaluation of surface plate flatness measurements – Eindhoven University of Technology research portal. (Accessed November 19, 2020, at https://research.tue.nl/en/publications/ evaluation-of-surface-plate-flatness-measurements)

Chapter 3 Surface encoder 3.1 Introduction Translation stages used for precision engineering can be classified into single-axis X stages, two-axis XY stages and three-axis XYZ stages in terms of the number of axes [1, 2]. Measurement of the X-, Y- and Z-directional positions of a three-axis XYZ stage is essential for evaluation of the positioning error in each axis, which is defined as the difference between the command position and the actual position of the stage. It is also necessary to carry out three-axis displacement measurement for a two-axis XY stage or a single-axis X stage. In the case of an X stage, the moving element of the stage moves along the linear guideway of the stage in the X-axis. The measurement result of the X-directional position of the moving element is employed for evaluation of the positioning error of the stage over the travel range of the stage, typically on the order of 10 mm to 1 m [3–5]. On the other hand, although the moving element of the stage is physically restricted to the X-directional motion by the linear guideway, the manufacturing errors of the linear guideway will cause the moving element to have Y- and Z-directional error motions, which are called out-ofstraightness error motions. Although the error motions have small amounts of displacements, typically on the order of 100 nm to 10 μm [6–9], measurement of such error motions is important for precision stages because they often directly influence the manufacturing/measurement accuracy. Similarly, in the case of an XY stage, three-axis displacement measurement is carried out for evaluation of the X- and Y-directional position errors and the Z-directional out-of-flatness error motion. Meanwhile, the three-axis motion accuracies of translation stages used in ultraprecision applications such as semiconductor manufacturing and ultra-precision machining [10] are required to be evaluated with sub-nanometric resolutions. Most of the conventional three-axis displacement measuring systems, including the threeaxis laser interferometer systems [11] and the laser tracking systems [12], are based on single-axis laser interferometers. A single-axis laser interferometer, in which the wavelength of the laser is treated as the graduation pitch, can measure the displacement along the optical axis of the laser beam with a high-resolution and fast speed. In a three-axis laser interferometer system, three single-axis laser interferometers are arranged in the X-, Y- and Z-axes for measurement of the displacement in each of the axes, respectively. In a laser tracker, the three-axis displacement is obtained from the relative distance of the target retroreflector measured by a single-axis laser interferometer and the azimuth and elevation of a beam-steering mirror measured by rotary encoders. Such a system can measure the three-axis displacement of a stage moving in a large spatial volume. The flexibility in the optical path arrangement also makes

https://doi.org/10.1515/9783110542363-003

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the system that may have small Abbe errors [13]. However, because the length of the optical path is proportional to the measurement range of displacement, the stability of a laser interferometer-based three-axis measuring system is easily influenced by the instabilities in air temperature, air humidity and air refractive index over the optical path. This is a serious problem for measurement of precision stages used in industries since most of such stages are operated in the air. An optical encoder is another well-used sensor for measurement of stage displacement [14–16]. In an optical linear encoder, an optical sensor head is employed to read the physical graduations of a linear scale grating, from which the X-directional displacement of the scale grating with respect to the optical sensor head can be measured. Because the length of the optical path in an optical encoder, which is basically twice the gap between the optical sensor head and the scale surface, is small and constant, the optical encoder is more robust to environmental variations compared with the laser interferometer. This is an important advantage for industrial applications. The linear encoder has been expanded for two-axis measurement of the X-directional position and Y- or Z-directional error motion by using linear scale grating [17, 18]. Two-axis XY planar encoders have also been developed for measurement of the X- and Y-directional positions by using an XY planar scale grating with X- and Y-directional grating structures [19–22]. In an XY planar encoder, a laser beam from the optical sensor head is projected onto the moving scale grating. The X-directional positive and negative first-order diffracted beams from the scale grating interference with each other to generate interference signals, from which the X-directional displacement can be obtained. Similarly, the Y-directional displacement can be obtained from the interference between the Y-directional positive and negative first-order diffracted beams from the scale grating. The XY planar encoder has been successfully used for two-axis XY position measurement of CNC (computerized numerical control) machine tools [23] and photolithography scanners [24]. This chapter presents a three-axis optical encoder, which is called the threeaxis surface encoder, for measurement of displacements in the X-, Y- and Z-directions by using planar gratings [25–32]. The surface encoder can be employed for three-axis measurement of a planar motion stage [33–35], which has large moving ranges in the X- and Y-directions on the order of 100 mm and a small moving range in the Z-direction on the order of 100 μm. It can also be employed for measurement of the X-directional position and Y- and Z-directional out-of-straightness error motion of a linear stage. Differing from a conventional two-axis planar encoder, the three-axis surface encoder employs a pair of planar gratings with identical grating structures, one as the stationary reference grating and the other as the moving scale grating. The X- and Y-directional first-order diffracted beams from the scale-grating interference with the corresponding diffracted beams from the reference grating, from which the X-, Y- and Z-directional displacements of the scale grating can be obtained simultaneously. Application of the surface encoder for measurement of a linear-rotary stage [36, 37] is also presented.

3.2 Three-axis surface encoder

105

3.2 Three-axis surface encoder Figure 3.1 shows the phase shifts in diffraction beams associated with in-plane displacement Δd of a grating with a grating pitch g. As shown in Fig. 3.1(a) and 3.1(b), the positive first-order (+1st-order) diffracted beam is generated by in-phase rays from the slits of the grating. For simplicity, only two slits are shown in the figure for the sake of clarity. Figure 3.1(c) shows the initial phases of the diffraction beams when the phase of the zeroth-order (0-order) beam is let to be zero. Figure 3.1(d) and 3.1(e) shows the phase shifts Δφ+1 and Δφ0 generated in the +1st-order and 0-order beams, respectively, when the in-plane displacement Δd is applied to the grating. The phase shifts for the +1st-order and ̶1st-order beams summarized in Fig. 3.1(f) are employed in the three-axis surface encoder. Figure 3.2 shows the fundamental structure of the three-axis surface encoder, which is composed of an optical sensor head and a scale XY planar grating [26]. The scale XY planar grating has periodic grating structures with a pitch of g in the X- and Y-axes. The components of the optical sensor head are a laser source, a nonpolarizing beam splitter (BS), a detector unit and a reference XY planar grating. The reference XY planar grating is identical to the scale XY planar grating, except for the plate size. The light beam from the laser source with a wavelength λ is divided into two beams by the BS. When the divided beams are projected onto the scale and reference XY planar gratings, the X- and Y-directional ±1st-order diffracted beams are generated with a diffraction angle of α1. Assume that the wavefront functions of the X- and Y-directional ±1st-order diffracted beams from the scale XY planar grating are Usα (α = X + 1, X – 1, Y + 1, Y – 1), and those from the reference XY planar grating are Urα (α = X + 1, X – 1, Y + 1, Y – 1). Usα and Urα are superimposed by the BS to interference with each other. The interference waves Um (m = X + 1, X – 1, Y + 1, Y – 1) are expressed as follows:     2π 2π UX+1 = USX+1 + UrX+1 = U0 exp i Δx · exp i ð1 + cos α1 ÞΔz + U0 (3:1) g λ     2π 2π (3:2) UX−1 = USX−1 + UrX−1 = U0 exp − i Δx · exp i ð1 + cos α1 ÞΔz + U0 g λ     2π 2π (3:3) UY+1 = USY+1 + UrY+1 = U0 exp i Δy · exp i ð1 + cos α1 ÞΔz + U0 g λ     2π 2π (3:4) UY−1 = USY−1 + UrY−1 = U0 exp − i Δy · exp i ð1 + cos α1 ÞΔz + U0 g λ where Δx, Δy and Δz are the three-axis displacements of the scale XY planar grating with respect to the sensor head. U0 represents the amplitude of ±1st-order diffracted beams, which is proportional to the intensity of the light source.

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Fig. 3.1: Phase shifts in diffraction beams associated with in-plane displacement Δd of grating.

The X- and Y-directional ±1st-order interference signals Iα are obtained as the conjugated complex multiplication of Um:    2π 2π IX+1 = UX+1 · UX+1 = 2U02 1 + cos ð1 + cos α1 ÞΔz (3:5) Δx + g λ    2π 2π ð1 + cos α1 ÞΔz (3:6) IX−1 = UX−1 · UX−1 = 2U02 1 + cos − Δx + g λ    2π 2π ð1 + cos α1 ÞΔz (3:7) IY+1 = UY+1 · UY+1 = 2U02 1 + cos Δy + g λ

3.2 Three-axis surface encoder

107

Fig. 3.2: Principle of three-axis surface encoder.

 IY−1 = UY−1 · UY−1 = 2U02

  2π 2π 1 + cos − ð1 + cos α1 ÞΔz Δy + g λ

(3:8)

The XYZ-directional displacements can be calculated from the outputs of interference signals, which are functions of Δx, Δy and Δz. The interference signals have signal periods of g in the X- and Y-directions and λ/(1 + cos α1) in the Z-direction. The measurement ranges in X- and Y-directions are determined by the area of the scale XY planar grating encoder and that in the Z-direction is influenced by the coherent length of the laser source. On the other hand, the measurement resolution in the Z-direction is determined by subdivision of the laser wavelength, which is similar to a laser interferometer. The measurement resolutions in the X- and Y-directions are determined by subdivision of the grating pitch, which is similar to a linear encoder or a planar grating encoder. To obtain balanced resolutions in all three axes, it is desired to design the grating pitch g to be close to the laser wavelength λ. g and λ have the following relationship [38]: g=

λ sin α1

(3:9)

where α1 is the diffraction angle of the first-order diffracted beam. As shown in eq. (3.9), a decrease of g will cause an increase of α1. When g gets close to λ, α1 will get close to 90°, which makes the design and construction of the

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optical sensor head of the surface encoder difficult. Taking into consideration that a laser diode (LD) with a wavelength of 685 nm is employed in the optical sensor head, the grating pitch of the XY planar gratings is determined to be 1 μm. The corresponding diffraction angle α1 is 43.28°. Simple rectangular structures, which are formed by a square hole array, are chosen as the grating structures for the XY planar gratings. The pitch g of the square hole array in the X-direction or the Y-direction is corresponding to the grating pitch of the XY planar gratings, which is 1 μm. The depth and the width of each square hole are determined on the basis of the diffraction efficiencies of the first-order diffracted beams, which are employed in the surface encoder for measurement. The depth and width of the square hole were determined to be 0.17 and 0.75 μm, respectively, which can offer the maximum efficiency of 16% for the {1st(X), 0th(Y)}-order diffracted beam based on a Fourier model method [39]. The scale and reference XY planar gratings with identical grating structures were made by laser interference lithography [25]. The grating area of the reference XY planar grating was 10 mm × 10 mm and that of the scale XY planar grating was 100 mm × 100 mm in maximum. The thickness of the reference XY planar grating plate was 5 mm and that of the scale XY planar grating plate was 10 mm. Figure 3.3 shows a schematic of the optical sensor head designed for the 1 μm pitch XY planar gratings. The size of the optical sensor head was 50 mm(X) × 70 mm (Y) × 40 mm(Z). The working distance, which was the gap between the scale grating and the optical sensor head along the Z-axis, was set to be 4 mm. The LD-unit consisted of an LD with a wavelength of 685 nm, a collimating lens (CL) and an aperture having a diameter of 1.5 mm. The light beam from the LD was collimated by the CL and the diameter of the collimated light beam was made to be 1.5 mm after passing through the aperture. Transparent one-axis gratings with a grating pitch of 1 μm were employed to bend the optical paths of the XY-directional ±1st-order diffracted beams with a diffraction angle of 43.28°. Figure 3.4 shows the dimension of the transparent one-axis grating. The transparent one-axis gratings were also fabricated by using the same laser interference lithography equipment used to fabricate the scale and reference XY planar gratings. However, coating of chromium and aluminum layers was not applied to the transparent one-axis gratings, which was different from the scale and reference XY planar gratings. A picture of the fabricated transparent one-axis gratings is shown in Fig. 3.5. Figure 3.6 shows a picture of the constructed three-axis surface encoder. The scale XY grating has an area of 100 mm (X) × 100 mm (Y). The specifications of the optical components used in the three-axis surface encoder are shown in Table 3.1. To remove the influence of the variation in the light intensity of the laser beam and to distinguish the direction of displacement, each of the X- and Y-directional ±1storder interference signals shown in eqs. (3.5–3.8) was divided into four sets of signals with a phase interval of 90° by using polarizing beam splitters (PBSs) and

3.2 Three-axis surface encoder

109

Fig. 3.3: Optical layout of three-axis surface encoder.

quarter-wave plates (QWPs), which are denoted by Iα (0°), Iα (90°), Iα (180°), Iα (270°) (α = X + 1, X – 1, Y + 1, Y – 1), respectively. The three-axis displacements Δx, Δy and Δz of the scale XY planar grating with respect to the optical sensor head can thus be obtained from the X-, Y- and Z- directional sensor outputs shown in the following equations. The signal periods of interference signals were 1 μm (=g) in the X- and Y-directions and 396 nm (=λ/(1 + cos α1)): Δx =

1 g ðarctan TX+1 − arctan TX−1 Þ · 2 2π

(3:10)

Δy =

1 g ðarctan TY+1 − arctan TY−1 Þ · 2 2π

(3:11)

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Chapter 3 Surface encoder

Fig. 3.4: Dimension of the transparent one-axis grating.

Fig. 3.5: The fabricated transparent one-axis gratings.

3.2 Three-axis surface encoder

111

Fig. 3.6: The constructed three-axis surface encoder. Tab. 3.1: Specification of the optical components in the three-axis surface encoder. Manufacturer/model/specifications Laser diode

Mitsubishi Electric Corporation/MLR-/output:  mW, wavelength:  nm

Collimating lens Edmond optics/,-J/focal length: . mm ° prism mirror

Size:  mm ×  mm ×  mm

PBS + QWP × 

Edmond optics/size:  mm ×  mm ×  mm

BS

Edmond optics/,-H/size:  mm ×  mm ×  mm

λ/ plate

Edmond optics/,-K/film-type wave plate (attached to BS)

PBS + QWP

Edmond optics/size:  mm ×  mm ×  mm

Detector unit

Hamamatsu Photonics K.K./Quadrant photodetector, Detective area:  mm ×  mm

Δz =

1 1 λ · ðarctan TX+1 + arctan TX−1 + arctan TY+1 + arctan TY−1 Þ (3:12) · 4 1 + cos α1 2π

where    IX+1 ð0 Þ − IX+1 ð180 Þ Δx ð1 + cos α1 Þ = tan 2π + Δz IX+1 ð90 Þ − IX+1 ð270 Þ g λ    IX−1 ð0 Þ − IX−1 ð180 Þ Δx ð1 + cos α1 Þ = tan 2π − + Δz TX−1 = IX−1 ð90 Þ − IX−1 ð270 Þ g λ

TX+1 =

(3:13) (3:14)

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   IY+1 ð0 Þ − IY+1 ð180 Þ Δy ð1 + cos α1 Þ TY+1 = = tan 2π + Δz IY+1 ð90 Þ − IY+1 ð270 Þ g λ    IY−1 ð0 Þ − IY−1 ð180 Þ Δy ð1 + cos α1 Þ = tan 2π − + Δz TY−1 = IY−1 ð90 Þ − IY−1 ð270 Þ g λ

(3:15) (3:16)

The basic performances of the developed surface encoder were tested by using the experimental setup shown in Fig. 3.7. A small 1 μm pitch XY planar grating with a grating area of 10 mm × 10 mm was mounted on a three-axis PZT stage with its surface normal direction along the Z-axis as the scale grating for the optical sensor head of the surface encoder. For comparison, three commercial laser interferometers were employed to detect the displacement of the three-axis PZT stage along the X-, Y- and Z-directions, respectively. Three plane mirrors were mounted on the same PZT stage as target mirrors for the laser interferometers. The optical sensor head was mounted on a manual stage unit consisting of a manual translation station with its axis of motion along the Z-axis and three manual tilt stages that can apply inclination angles to the optical sensor head. The alignment of the optical sensor head with respect to the scale XY planar grating was carried out by adjusting the manual stages. At first, the resolutions of the optical sensor head in the X-, Yand Z-directions were tested by detecting the three-dimensional (3D) vibrations of the three-axis PZT stage. The PZT stage was kept stationary during the test. Figure 3.8 shows the X-, Y- and Z-directional outputs of the surface encoder and the laser interferometers, respectively. It can be seen that the developed sensor could detect the XYZ-directional displacements with a resolution of better than 1 nm. Figures 3.9–3.11 show the quadrature interference signals and the corresponding nonlinear components of the measurement errors when X-, Y- and Z-directional displacements were applied to the scale XY grating, respectively. The displacements of the PZT stage measured by the commercial laser interferometers were used as the reference values for evaluation of the interpolation errors. It is observed that the XY- and Z- directional interpolation errors synchronizing with the quadrature interference signals were within ±10 and ±3 nm, respectively. In other words, the XYZ-directional interpolation errors were within ±1% of the signal period of the quadrature interference signals. As shown in Figs. 3.10 and 3.11, the Y- or X-directional quadrature interference signals changed though only X- or Y-directional displacement was applied to the scale XY grating, respectively. These changes were caused by the inclination angle of the scale grating [26].

3.3 Six-degree-of-freedom surface encoder A six-degree-of-freedom (six-DOF) surface encoder can be constructed by incorporating the three-axis surface encoder with the three-axis angle sensor demonstrated

3.3 Six-degree-of-freedom surface encoder

Fig. 3.7: Setup for testing the performance of three-axis.

Fig. 3.8: Results of resolution test of the three-axis surface encoder.

113

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Fig. 3.9: Quadrature interference signals and corresponding nonlinear components of the measurement errors when an X-directional displacement was applied.

in the previous chapter. Figure 3.12 shows the principle for realizing six-DOF measurement. A displacement assembly and an angle assembly are employed for the measurement. The displacement assembly, which is based on the principle of a three-axis surface encoder, is for measurement of the translational displacement motions Δx, Δy and Δz. The angle assembly, which is based on the principle of a three-axis autocollimator, is for measurement of the angular motions θx, θy and θz. The displacement assembly and the angle assembly are combined with each other to form a simple sensor configuration through sharing the same laser source.

3.3 Six-degree-of-freedom surface encoder

115

Fig. 3.10: Quadrature interference signals and corresponding nonlinear components of the measurement errors when a Y-directional displacement was applied.

Figure 3.13 shows a schematic of the six-DOF surface encoder. A collimated beam is output from a laser source consisting of an LD with a wavelength of λ and a collimating lens. After passing through the beam splitter 1 (BS1), the beam is divided into two beams by the beam splitter 2 (BS2), which are projected onto the scale and the reference gratings, respectively. The scale and the reference gratings have an identical grating pitch g in the X- and Y-directions. The X- and Y-directional first-order diffraction beams are generated at each of the grating surfaces with a diffraction angle of β. The diffraction beams from the scale grating are bent at BS2 and those from the reference grating pass through BS2. These two groups of diffraction beams interfere with each

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Fig. 3.11: Quadrature interference signals and corresponding nonlinear components of the measurement errors when a Z-directional displacement was applied.

other and the superimposed interference signals (X + 1, X – 1, Y + 1, Y – 1) are detected by four separate photodiodes (PDs) (AX + 1, AX – 1, AY + 1, AY – 1). The photoelectric current outputs IAX+1, IAX–1 from PDs AX + 1 and AX – 1 are cosine functions of the combinations of the primary displacement motion Δx and the secondary translational displacement motion Δz. Similarly, the photoelectric current outputs IAY+1, IAY–1 from PDs AY + 1 and AY – 1 are cosine functions of the combinations of Δy and Δz. The amplitudes of the translational displacement motions Δx, Δy and Δz can be evaluated from the interference signals detected by the PDs. The interference signals as well as the PD outputs have signal periods of g in the X and Y-directions and λ/(1 + cos α1) in the Z-

3.3 Six-degree-of-freedom surface encoder

Fig. 3.12: Principle for six-DOF measurement (XZ-view).

Fig. 3.13: A schematic of the six-DOF surface encoder.

117

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Fig. 3.14: XZ-view of the optical design for the six-DOF surface encoder.

3.3 Six-degree-of-freedom surface encoder

119

direction. Meanwhile, a part of the zeroth-order and the X-directional negative firstorder diffracted beams, which are indicated by 0th and −1st, from the scale grating passes through BS2 and gets into the angle assembly. The beams bent at BS1 are received by two autocollimation units. Each of the autocollimation units is composed of a quadrant photodiode (QPD) and a collimator objective lens with a focal length f. The QPD is located at the focal plane of the collimator objective to detect the linear displacement of the diffracted beam spot focused on the QPD. Assuming that the linear displacements of the beam spots on the QPDs are denoted by ΔhC, ΔvC, ΔhD, ΔvD, which can be obtained from the photoelectric current outputs of the QPD cells (ICk, IDk, k = 1, 2, 3, 4), the three secondary angular motions θx, θy and θz can be calculated as follows based on the principle of three-axis angle detection shown in the previous chapter: θx =

Δhc 2f

(3:17)

θy =

Δvc 2f

(3:18)

g ΔhD fλ

(3:19)

θz =

Because the QPD outputs are linear functions of the angular motions, both the amplitude and the moving directions of the angular motions can be obtained. Figure 3.14 shows the optical layout of the sensor head with additions of polarization components of PBSs, QWPs and polarizers to the displacement assembly of Fig. 3.12 for identification of the moving directions of the translational displacement motions. A new design of using two sets of PD units in the displacement assembly is employed to simplify the design of using four sets of PD units in the three-axis displacement sensor for measurement of Δx, Δy and Δz shown in the previous section. Figure 3.15 shows a schematic of the displacement assembly with two sets of PD units. The optical design is made in such a way that the photoelectric current outputs of the corresponding PD cells in QPD A and B, which detect the interference signals of the diffraction beams from the scale grating and the reference grating, have a 90° phase difference with each other. The quadrant outputs of the QPDs make it possible for the surface encoder to recognize the moving directions of Δx, Δy and Δz. Assuming that the AC components of the photoelectric current outputs of the PD cells in QPD A and QPD B are denoted by IAm and IBm (m = X + 1, X – 1, Y + 1, Y – 1) [25], respectively, Δx, Δy and Δz can be evaluated by      g IBX+1 IBX−1 − arctan (3:20) Δx = arctan IAX+1 IAX−1 4π

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Fig. 3.15: A schematic of the displacement assembly with two sets of PD units.

     g IBY+1 IBY−1 − arctan arctan IAY+1 IAY−1 4π      λ 1 IBX+1 IBX−1 + arctan Δz = · · arctan IAX+1 IAX−1 8π 1 + cos ϕ

Δy =

(3:21)

(3:22)

    IBY+1 IBY−1 + arctan + arctan IAY+1 IAY−1 On the other hand, the transmitted beams at BS1 in Fig. 3.14 propagate to the modified angle unit. The diffraction beams from the reference grating are blocked by a polarizer (polarizer 1) and only the diffraction beams from the scale grating, which carry the information of the three secondary angular error motions θx, θy and θz for measurement, are received by the autocollimation units of the angle assembly. The X-directional positive first-order diffraction beam from the scale grating was blocked mechanically by a black plate placed behind Polarizer 1. To avoid the physical conflict between the two autocollimation units, a mirror (Mirror 1) is added to

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121

separate the zeroth-order and the X-directional negative first-order diffracted beams. Figure 3.16 shows a compact six-DOF surface encoder that can be can fit into a surface motor-driven motor stage shown in Fig. 3.17 [40, 41]. The surface motor stage with its primary axes of motion along X- and Y-axes, which has been improved from its original design [42], is composed of a platen (moving element) and a stage base. The moving element is suspended by four aerostatic bearings. An assembly of a permanent magnet and an electromagnet is installed in each of the air-bearings to apply a controllable attractive force to the moving element made of stainless steel as a preload. The gap between the moving element and each of the aerostatic bearing can be independently adjusted in a small range of 20 µm by changing the force of the electromagnet in the aerostatic bearing, which is proportional to the current applied to the electromagnet. The moving element can be moved in the X- and Ydirections by two X-linear motors (X-motors) and two Y-linear motors (Y-motors), respectively. The X-linear motor and the Y-linear motor have the same configuration, which is composed of a paired permanent magnet array and a stator (coils). The stator is mounted on the stage base and the permanent magnet array is mounted on the moving element. The moving range is 40 mm in both the X- and Y-directions. In addition to the two primary motions Δx and Δy, the stage can also generate four secondary motions, which are the translational displacement motion Δz along the Z-axis and the angular motions θx, θy and θz about X-, Y- and Z-axes, respectively. The secondary motions are basically generated for compensation of the error motions of the stage moving element when it is moving along the X- and Y-directional primary axes of motion. The translational motion Δz and the angular motions θx, θy of the moving element can be adjusted in a range of 20 μm and 20 arc-seconds by controlling the gaps between the moving element and the air bearings. The range of the angular motion θz, which can be adjusted by controlling the forces generated by the two pairs of coils and permanent magnet arrays of the X-motors or the Y-motors, is set to be 20 arc-seconds for matching those in the directions of θx and θy. The XY surface motor stage is expected to carry out precision positioning in the two primary axes of motions (Δx, Δy) and precision compensation of error motions in the four secondary axes of motion (Δz, θx, θy, θz) with sub-micrometer/sub-arc-second resolutions. The six-DOF surface encoder is designed for measurement and control of the stage motions in the two primary and four secondary axes. The scale grating of the surface encoder is mounted on the back of the moving element, and the sensor head is mounted on the stage base. For the surface encoder, it is the highest priority to have a compact sensor head so that the sensor head can be set within the limited space of the stage base, which is among the four coils of the X- and Y-motors. The maximum space for mounting the sensor head is 100 mm (X) × 100 mm (Y) × 30 mm (Z) and that for mounting the scale grating is 80 mm (X) × 80 mm (Y) × 20 mm (Z). In addition to the sensor size, the measurement resolutions and ranges are set as the priorities for the prototype surface encoder as the first stage of the research. The

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design goals for resolutions of the surface encoder are determined to be 10 nm for the translational displacement motions and 0.1 arc-seconds for the angular motions. The measurement ranges of the surface encoder should be larger than the moving ranges of the XY surface motor stage. As shown in Fig. 3.16, the optical components of the sensor head fit into a space of 60 mm (X) × 70 mm (Y) × 15 mm (Z) and the size of the sensor head, including the electric cables and the base plate for mounting the optical components is 95 mm (X) × 90 mm (Y) × 25 mm (Z), which is slightly smaller than the mounting space shown in Fig. 3.17. In the sensor head, taking into consideration that the height of the space for the sensor head along the Z-direction is limited within 30 mm, a mirror (mirror 2) is employed to change the main plane of the sensor head from XZ to XY, which is parallel to that of the scale grating. Collimator objectives with a short focal length of 15 mm are employed for a compact size of the angle assembly. The wavelength (λ) of the LD and the grating pitch (g) of the gratings are set to be 405 nm and 0.57 μm, respectively. The diffraction angle α1 is calculated to be 45.3° based on eq. (3.9). The planar gratings were fabricated by interference lithography based on a dual-beam interferometer [18]. The scale grating and the reference grating had an identical grating structure, which was composed of a square hole array. A Fourier model method was employed to design the width and depth of the square hole for getting a high diffraction efficiency of the first-order diffraction beams used in the surface encoder [39]. The depth and width of the hole were determined to be 0.38 and 0.18 μm, respectively, which generated a diffraction efficiency of 15% for the first-order diffracted beams. Optical flats were employed as the grating substrates. The grating structure was fabricated by exposing the layer of photoresist coated on the grating substrate with the interference fringe pattern generated by the dual-beam interferometer. A grating pitch (0.57 μm) was set to be shorter than that for the three-axis surface encoder (1 μm) shown in the previous section. The resist-based grating structure was then evaporated with aluminum so that a reflective-type XY planar grating could be obtained. The scale grating had a square shape with a side length of 60 mm, which determined the X- and Ydirectional measuring ranges of the prototype surface encoder. The reference grating has a round shape with a diameter of 10 mm. The thickness of the scale grating and that of the reference grating were 10 and 5 mm, respectively. The performance of the six-DOF surface encoder was tested with the setup shown in Fig. 3.18. A 0.57 μm pitched XY planar grating with a grating area of 10 mm in diameter was employed as the target scale grating for the experiment. The target grating was mounted on a five-DOF stage assembly, which was composed of a PZT XYZ-stage and a two-axis PZT tilt stage for generating the three-axis translational displacement motions (Δx, Δy, Δz) and the two-axis angular motions (θx, θz). The translational displacement motions and angular motions of the PZT stages were employed as the references for testing the performances of the surface encoder. Each axis of the PZT XYZ-stage was feedback controlled by using a capacitive displacement sensor over a maximum moving range of 25 μm. The closed-loop linearity

3.3 Six-degree-of-freedom surface encoder

Fig. 3.16: A designed and constructed six-DOF surface encoder.

123

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Chapter 3 Surface encoder

Fig. 3.17: The six-DOF surface encoder in a surface motor stage.

3.3 Six-degree-of-freedom surface encoder

125

Fig. 3.18: Setup for testing the six-DOF surface encoder.

of the PZT motion was 0.02% of the travel range. The open-loop controlled PZT tilt stage, which had a range of 200 arc-seconds in each axis, was calibrated by using a commercial photoelectric autocollimator with a resolution of 0.01 arc-second and linearity of 0.1%. In the calibration, the relationship between the driving voltages applied to the PZT tilt stage with respect to the tilt angle of the stage measured by the autocollimator was investigated and recorded. Based on the calibration result, the actual tilt motion of the stage could be estimated from the applied driving voltage to the stage in the experiment of testing the surface encoder. For the generation of the angular motion θy, the PZT tilt stage was rotated about the X-axis by 90°. The sensor head was mounted on a manual alignment stage assembly, which was employed to set the initial position of the sensor head with respect to the target scale grating. The alignment stage assembly was composed of a Z-axis manual stage and a three-axis manual tilt stage. Figure 3.19 shows the results of the resolution test in measurement of the threeaxis translational motions Δx, Δy and Δz. The target grating was moved by applying a command step motion of 2 nm to the PZT XYZ-stage in the target grating stage assembly along each of the translational axes. The step motions of the target grating were measured by the surface encoder. The outputs of the surface encoder were compared with those of the capacitive sensors in the PZT XYZ-stages for feedback control of the PZT translational motions. As shown in Fig. 3.19, the step motions in

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Chapter 3 Surface encoder

Fig. 3.19: Resolutions of the six-DOF surface encoder for displacement measurement.

the X-, Y- and Z- axes were well distinguished by the surface encoder. It can also be observed that the resolutions of the capacitive sensors, which had large noise components in their outputs, were worse than those of the surface encoder. Similarly, the resolutions of the surface encoder in the detection of angular motions of θx, θy and θz were tested through applying step tilt motions to the target grating by the PZT tilt stages in the target grating stage assembly. Figure 3.20 shows the results. It can be seen that the surface encoder could distinguish step tilt motions of 0.1 arc-second, 0.1 arc-second and 0.3 arc-second in the θx-, θy- and θz-directions, respectively. Figure 3.21 shows the results of investigating the nonlinear errors in the displacement outputs of the surface encoder. When the test was carried out in the Xand Y-directions, the target scale grating was moved by the PZT XYZ-stage over five periods of the grating period g along the corresponding direction for investigation of the non-linear error components caused by the interpolation errors of the surface encoder, which were associated with the subdivision of the interference signals within the signal period. Periodic error components, which corresponded to the interpolation errors, can be identified in the Δx and Δy outputs of the surface encoder.

3.3 Six-degree-of-freedom surface encoder

127

Fig. 3.20: Resolutions of the six-DOF surface encoder for angle measurement.

The peak-to-valley (PV) amplitudes of the interpolation errors were about ±6 nm and ±7 nm in the two directions, respectively, which were about 1% of the grating period. When the testing was carried out in the Z-direction, the target scale grating was moved by the PZT XYZ-stage over 1.2 μm along the Z-direction, which corresponded to approximately five signal periods of the Δz output of the surface encoder for investigation of the interpolation error in this direction. It can be seen that the amplitude of the interpolation error was approximately ±6 nm. It can be seen that the amplitudes of the interpolation errors were comparable to those of commercial laser interferometers and linear encoders associated with similar interpolation errors. Figure 3.22 shows the nonlinear error components in measurement of θx, θy and θz. In the test, the target scale grating was tilted by the PZT tilt stage about the X-, Y- and Z-axes, respectively. As can be seen in the figure, the nonlinear error components over a range of 60 arc-seconds were approximately 2.2 arc-seconds, 1.4 arc-seconds and 2.5 arc-seconds for θx, θy and θz, respectively.

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Chapter 3 Surface encoder

Fig. 3.21: Nonlinear components of the six-DOF surface encoder for displacement measurement.

3.4 Reduction of cross-talk error in surface encoder For multi-axis sensors, it is important to avoid cross-talk errors in and between the sensor outputs along different axes [30, 43, 44]. In this section, the cross-talk error components in the angular outputs of the six-DOF surface encoder are investigated, based on which the initial design of the optical layout of the sensor head shown in the previous section is improved by changing the position of the angle assembly in the sensor head for the cross-talk error reduction [45]. The improved design is then optimized to further reduce the remained cross-talk error components by replacing a cube-type BS with a plate-type BS.

3.4 Reduction of cross-talk error in surface encoder

129

Fig. 3.22: Nonlinear components of the six-DOF surface encoder for angle measurement.

Since only the scale diffracted beams are necessary in the angle assembly for the angular motion measurement, optical isolation in the initial design of the previous section had been adopted by using polarization components to prevent the entry of the reference diffracted beams into the angle assembly. As shown in Fig. 3.23, a cube PBS is employed for dividing the light from the LD into a P-polarized beam and an S-polarized beam. The divided beams are projected onto the scale and reference gratings after passing through QWP1 and QWP2, respectively. The diffracted beams reflected back from the scale grating are changed to S-polarization by QWP1 so that the scale beams can be reflected at the hypotenuse surface of the PBS toward BS2. The reference beams, which are changed to P-polarization by QWP2, transmit through the PBS toward BS2. Half of the combined scale and reference beams are reflected at the BS and then received by the displacement assembly. The other half of the combined scale and reference beams pass through BS2 toward the angle assembly. Polarizer 1 is added in front of the angle assembly to block the reference beams. The transmission

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Chapter 3 Surface encoder

Fig. 3.23: Optical isolation in the initial design for preventing the entry of the reference diffracted beams into the angle assembly.

axis of Polarizer 1 is adjusted to be parallel with the polarization direction of the scale beams and orthogonal to that of the reference beams so that only the scale beams can be received by the angle assembly. Experiments were carried out with the setup shown in Fig. 3.18 to confirm crosstalk errors in the outputs of the angular assembly of the six-DOF surface encoder of the initial design shown in Fig. 3.23. Figure 3.24 shows the measured outputs of the angular assembly of the previous surface encoder when only a Z-directional translational displacement was applied to the scale grating by the PZT stage while the reading head was kept stationary. The angular error motions of the PZT stage were confirmed to be sufficiently small. The output signals were smoothed by the simple moving average method. The number of averaged points was 100. Periodical cross-talk error components can be observed in the θX, θY and θZ-outputs. In the ideal case, the angular outputs are expected to be constant, that is, the output curves should be straight lines parallel to the horizontal axis of the figure when an X, Y or Z-directional translational displacement is applied to the scale grating. In the actual case, however, deviations of the angular outputs will be associated with the translational displacement, which are error components of the angular outputs. The deviation of each angular output is defined as the cross-talk error. As shown in Fig. 3.24, cross-talk error components can be observed in the θX, θY and θZ output curves. The PV amplitude of the deviation of each angular output curve,

3.4 Reduction of cross-talk error in surface encoder

131

Fig. 3.24: Measured cross-talk errors in the outputs of the angle assembly of the initial design when only a Z-directional translational displacement was applied to the scale grating.

that is, the cross-talk error curve, in Fig. 3.24 was calculated to make a quantitative evaluation of the error. The cross-talk errors were thus evaluated to be 0.38, 0.48 and 3.24 arc-seconds in the θX, θY and θZ outputs, respectively. As can be seen in the figure, the cross-talk error curves are dominated by periodic components with specific periods PZ–θX, PZ–θY and PZ–θZ along the Z-axis. Based on an analysis of the spatial wavelengths of the dominant periodic components in the spatial spectrum obtained by discrete Fourier transform (DFT), PZ–θX, PZ–θY and PZ–θZ were evaluated to be approximately 200, 200 and 250 nm, respectively. The results when only an Xdirectional translational displacement was applied to the scale grating by the PZT stage are shown in Fig. 3.25. From the spatial wavelengths of the dominant periodic component obtained by DFT, P X–θZ was evaluated to be approximately 500 nm.

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Fig. 3.25: Measured cross-talk errors in the outputs of the angle assembly of the initial design when only an X-directional translational displacement was applied to the scale grating.

Based on the experimental results shown above, the reason for the periodical cross-talk errors is analyzed. In the ideal situation, all the reference beams will be blocked by Polarizer 1 as shown in Fig. 3.23. However, in a practical situation shown in Fig. 3.26, due to the imperfection of the PBS and QWP2, the zeroth-order and the first-order reference beams arriving at Polarizer 1 will include not only the expected P-polarization component but also the unexpected S-polarization component. The S-polarization component will pass through the polarizer and enter the angle assembly, which is referred to the leakage beam in the figure. Figure 3.27 shows the light spots on QPD1 for detecting θX and θY. Similar light spots can be observed on QPD2 for detecting θZ. In the ideal situation without leakage beam, only the zeroth-order reference beam is received by the angle assembly. The beam is focused to be a small light spot on the QPD by the collimator objective, which is referred to as the scale beam spot in the figure. The centroid point of the light spot

3.4 Reduction of cross-talk error in surface encoder

133

Fig. 3.26: Beam leakage at the angle assembly.

will be the center of the spot. Based on the principle of laser autocollimation, a θX motion of the scale grating will cause a linear displacement of the centroid on the QPD along the V-direction. Similarly, a θY motion will cause a linear displacement of the centroid along the W-direction. The 2D displacements of the centroid are detected by the QPD, from which the θX and θY can be evaluated. On the other hand, in a practical situation with the leakage beam, an additional light spot will be formed on the QPD, which is referred to as the reference leakage beam spot in Fig. 3.27. Since the two light spots have the same polarization direction, interference fringes will be generated on the QPD due to the interference of the two light spots. The number of fringes is determined by the alignment of the optical components in the sensor head. If the optical components are perfectly aligned, the reference beam and the scale beam will be parallel and completely overlap with each other, resulting in a zero interference fringe. In an actual case, there will be a certain number of fringes due to the imperfect alignment. In either case, the centroid point is now the weighted center of the scale beam spot, the reference leakage beam spot and the interference fringes, which is denoted as C p in the figure. Cp will move on the QPD in responding to the θX and θY motions of the scale grating. However, additional displacements of Cp will also be caused by the variations of the interference fringes, that is, the variations of the phase in the interference signal of the zeroth-order scale beam and the zeroth-order leakage reference beam.

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Fig. 3.27: The light spots on QPD under the actual situation with leakage beam.

Fig. 3.28: Phase shifts in the zeroth-order scale beam caused by Δz of the scale grating.

Here, we assume that the wavefronts of the reflected zeroth-order and positive firstorder diffracted beams from the reference grating are denoted by Er0 and Er1, respectively. Without considering the initial phases, Er0 and Er1 can be expressed as Er0 = E0

(3:23)

Er1 = E0

(3:24)

where E0 is the amplitude of the wave. The diffraction efficiencies for the zerothorder beam and the first-order beam are assumed to be equal to each other. Since the reference grating is kept stationary, Er0 and Er1 will not change when the scale grating is moved.

3.4 Reduction of cross-talk error in surface encoder

135

Fig. 3.29: Phase shifts in the first-order scale beam caused by Δz of the scale grating.

Fig. 3.30: Phase shifts in the zeroth-order scale beam caused by Δx (or Δy) of the scale grating.

Let the wavefronts of the reflected zeroth-order and positive first-order diffracted beams from the scale grating at the initial position of the scale grating to be denoted by Ei-s0 and Ei-s1, respectively. Without considering the initial phases at this position, Ei-s0 and Ei-s1 can be expressed as

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Fig. 3.31: Phase shifts in the first-order scale beam caused by Δx (and/or Δy) of the scale grating.

Ei−s0 = E0

(3:25)

Ei−s1 = E0

(3:26)

When the scale grating is moved with a translational motion Δz along the Zdirections, optical path differences shown in Figs. 3.28 and 3.29 will be generated, resulting in the following phase shifts ϕZ-s0 in the zeroth-order diffracted beam and ϕZ-s1 in the positive first-order diffracted beams: 2π 2Δz λ

(3:27)

2π ð1 + cos α1 ÞΔz λ

(3:28)

ϕZ−s0 = ϕZ−s1 =

where λ is the wavelength of the LD and α1 is the diffraction angle shown in eq. (3.9). If the scale grating is moved with translational motions Δx and Δy along the Xand Y-directions, the optical path differences shown in Figs. 3.30 and 3.31 (Δy is not shown in the figure for clarity) will generate the following phase shifts in the scale zeroth-order and the positive first-order diffracted beams: ϕXY−s0 = 0

(3:29)

3.4 Reduction of cross-talk error in surface encoder

ϕXY−s1 =

2π ðΔx + ΔyÞ g

137

(3:30)

The wavefront functions of the scale zeroth-order and positive first-order diffracted beams can then be written as Es0 = E0 eiðϕz−s0 + ϕxy−s0 Þ = E0 ei λ 2Δz

(3:31)

Es1 = E0 eiðϕz−s1 + ϕxy−s1 Þ = E0 ei g ðΔx + ΔyÞ ei λ ð1 + cos α1 ÞΔz

(3:32)

2π

2π

2π

Assuming that the zeroth-order and the positive first-order reference beams have the same leakage rate of k, the leakage reference beams can thus be expressed by El0 = kE0

(3:33)

El1 = kE0

(3:34)

The wavefront function of a point of the interference fringes generated on QPD1 by El0 and Es0 can be written as Els0 = El0 + Es0

(3:35)

Similarly, the wavefront function of a point of the interference fringes generated QPD2 by El +1 and Es +1 can be written as Els1 = El1 + Es0

(3:36)

The intensities of the interference waves on QPD1 (Ils0) and QPD2 (Ils+1) can thus be obtained as   2π 2 2 Ils0 = Els0 · Els0 = E0 1 + k + 2k cos 2Δz (3:37) λ    2π 2π ð1 + cos α1 ÞΔz (3:38) ðΔx + ΔyÞ + Ils1 = Els1 · Els1 = E02 1 + k2 + 2k cos g λ When a Z-directional translational motion Δz, which is referred to as the out-ofplane motion, is applied to the scale grating, the centroid point on QPD1 will oscillate with a period of λ/2 as can be seen in eq. (3.37). The periods for the variations in the displacement outputs of QPD1, that is, the outputs of θX and θY (the cross-talk errors in θX and θY) can be obtained as PZ−θx = PZ−θy =

λ 2

(3:39)

Since a Blu-ray LD with a λ of 405 nm was employed in the sensor head, PZ-θX and PZ-θY are thus calculated to be 202.5 nm. These are in a good correspondence with the values of PθX and PθY in Fig. 3.24.

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In the presence of Δz, the period for the oscillation of the centroid point on QPD2 will be λ/(1 + cos α1) as can be seen in eq. (3.38). The periods for the variations in the displacement output of QPD2, that is, the output of θZ (the cross-talk error in θZ), can thus be obtained as PZ−θz =

λ λ

 = 1 + cos α1 1 + cos arcsin gλ

(3:40)

For λ of 405 nm and g of 0.57 μm, PθZ is calculated to be approximately 238 nm. This value also well agrees with that of PZ − θZ and Fig. 3.24. On the other hand, it is observed in Fig. 3.24 that the magnitude of the periodic cross-talk error component in θZ is larger than those in θX and θY. This can be caused by the difference in the diffraction efficiencies of the zeroth-order and the first-order diffracted beams. The alignment errors of the optical components in the sensor head are other possible reasons. When an in-plane motion (Δx or Δy) is applied to the scale grating, it can be seen from eq. (3.37), there will be no change in the intensities of the interference waves on QPD1, which means that the centroid point will not oscillate and the displacement outputs of QPD1, that is, the outputs of θX and θY will not change periodically. This corresponds to the results shown in Fig. 3.25. However, on the other hand, the in-plane Δx or Δy motion will cause an oscillation of the centroid point on QPD2, resulting in a periodic change in the displacement output of QPD2, that is, the output of θZ (the cross-talk error in θZ) as can be seen in eq. (3.38). The period of the cross-talk error in θZ caused by Δx or Δy can be expressed by PXY−θz = g

(3:41)

The calculated value of PXY − θZ (0.57 μm) is in good correspondence with the experimental result shown in Fig. 3.25. Consequently, the cross-talk errors in the outputs have been successfully identified to be caused by the interference between the scale beams and the leakage reference beams to the angle assembly based on the above analysis. To reduce the cross-talk errors, it is thus necessary to completely prevent the entry of the reference beams into the angle assembly. An improved design of the sensor head is then presented for preventing the entry of the reference beams into the angle assembly based on the fact that the imperfection of the polarization components (the PBS, QWPs, polarizer) is unavoidable, which had been the reason for causing the leakage of the reference beams in the previous sensor head. Figure 3.32 shows a schematic of the improved design. A BS is added between the scale grating and QWP1 to split the diffracted beams from the scale grating. The reflected beams by the BS are received by the angle assembly for measurement of the angular motions of the scale grating. It can be seen that the angle assembly is out of the optical path of the reference beams and there is no

3.4 Reduction of cross-talk error in surface encoder

139

Fig. 3.32: A schematic of the improved design of the sensor head.

Fig. 3.33: Optical layout of the improved sensor head.

need to employ polarization components for preventing the entry of the reference diffracted beams into the angle assembly. As a result, the cross-talk errors caused by the leakage reference beams in the previous sensor head are expected to be avoided.

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Fig. 3.34: 3D view of the improved six-DOF surface encoder.

A detailed optical layout of the improved sensor head is shown in Fig. 3.33. The angle assembly of the improved design is modified from that of the initial surface encoder by employing not only the zeroth-order and positive first-order diffracted beams but also the negative first-order diffracted beam for improving the performance in θZ measurement [46]. In the displacement assembly, the diffracted laser beams are divided into two groups (groups A and B) by a beam splitter (BS1). One group is directly detected by PD unit A, while the other is detected by the PD unit B. It should be noted that a 90-degree phase difference is given to each of the beams in the group B with respect to each of them in the group A, respectively. This makes it possible for the surface encoder to recognize the moving direction in the same manner as a conventional optical linear encoder [47]. For compactness of the optical system, two mirrors are located between the PBS and the reference grating in order to keep the optical path difference within the coherence length. Figure 3.34 shows the mechanical structure of the improved six-DOF surface encoder. The size of this improved surface encoder is 98 mm (X) × 95 mm (Y) × 28 mm (Z). The size of this improved design is slightly different from the initial one (95 mm (X) × 90 mm (Y) × 25 mm (Z)) shown in the previous section.

3.4 Reduction of cross-talk error in surface encoder

141

Fig. 3.35: Measured cross-talk errors in the outputs of the angle assembly of the improved surface encoder for Δx.

Experiments were carried out to investigate the cross-talk errors in the angular outputs of the improved sensor head. Figure 3.35 shows the measured cross-talk errors in the outputs of the angle assembly of the improved six-DOF surface encoder, when only Δx was applied to the scale grating. It can be seen that the cross-talk errors were almost completely reduced from those in the initial design of the previous section, demonstrating the effectiveness of the improved design of the sensor head. Figure 3.36 shows the measured cross-talk errors in the outputs of the angle assembly of the improved six-DOF surface encoder, when only Δz was applied to the scale grating. Although the cross-talk errors were significantly reduced compared with those in the initial design, periodic error components are still observed in Fig. 3.36. After an analysis of the possible reasons, it is identified that the residual periodic cross-talk error components are due to the internal reflection phenomenon of

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Fig. 3.36: Measured cross-talk errors in the outputs of the angle assembly of the improved surface encoder for Δz.

light at the cube surface of BS1. As shown in Fig. 3.37, the light from LD is divided into two parts. The transmitted part (Beam 1) is projected onto the scale grating and the reflected part (Beam 2) propagates toward the cube surface T of BS2. Here, we assume that Beams 1 and 2 have an identical wavefront function of Ein as follows: Ein = Ein0

(3:42)

where Ein0 is the amplitude of the wave. It is known that a certain percentage (typically 4%) of the light incident normally on an air-glass interface will be reflected back [38]. This will happen at the cube surface T of BS2. Therefore, a part of Beam2, which is denoted as Beam2T, will enter the angle assembly. In a practical case, Beam 2T can be received by both QPD C for measuring the θX- and θY-motions, and QPD D (or QPD E) for measuring

3.4 Reduction of cross-talk error in surface encoder

143

Fig. 3.37: Internal reflection at cube surface T in BS1.

the θZ-motion. Beam 2T will then interfere with the scale zeroth-order and firstorder beams to generate interference fringes on QPD C and QPD D (or QPD E). The wavefront function E2T0 of the part of Beam 2T received by QPD C and the wavefront function E2T1 of the part of Beam2T received by QPD D (or QPD E) can be expressed by E2T0 = m0 Ein = m0 Ein0

(3:43)

E2T1 = m1 Ein = m1 Ein0

(3:44)

where m0 and m1 are constants determined by the reflectance at the cube surfaceT and the reflection/transmission ratio (R/T) of BS2, as well as the alignment of the components in the sensor head. Taking into consideration the phase shifts in the scale diffracted beams by ΔZ of the scale grating, the wavefront functions of the scale zeroth-order and first-order diffracted beams can be written as 2π

Es0 = n0 Ein0 ei λ 2Δz 2π

Es1 = n1 Ein0 ei λ ð1 + cos α1 ÞΔz

(3:45) (3:46)

where n0 and n1 are constants determined by the diffraction efficiency of the scale grating and R/T of BS2. α1 is the diffraction angle. The wavefront function of a point of the interference fringes generated on QPD C by E2T0 and Es0 can be written as Eqs0 = E2T0 = Es0

(3:47)

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Similarly, the wavefront function of a point of the interference fringes generated on QPD D (or QPD E) by E2T1 and Es1 can be written as Eqs1 = E2T1 = Es1

(3:48)

The intensity of the interference waves on QPD C Iqs0 and the intensity of the interference waves on QPD D (or QPD E) Iqs1 can thus be obtained as   2π 2 Iqs0 = Eqs0 · Eqs0 = Ein0 n20 + m20 + 2m0 n0 cos 2Δz (3:49) λ    2π 2 n20 + m20 + 2m0 n0 cos ð1 + cos α1 ÞΔz (3:50) Iqs1 = Eqs1 · Eqs1 = Ein0 λ When Δz is applied to the scale grating, the centroid point on QPD C will oscillate with a period of λ/2 as can be seen in eq. (3.49), which will be the period for the periodic cross-talk error components in ΔθX, ΔθY. On the other hand, when Δz is applied to the scale grating, the centroid point on QPD D (or QPD E) will oscillate with a period of λ/(1 + cos α1) as can be seen in eq. (3.50). This will be the period for the cross-talk errors in ΔθZ. The analysis results are well consistent with the results shown in Fig. 3.36. It should be noted that the differences between m0, n0 and m1, n1 will make a difference between the amplitude of the periodic cross-talk error component in ΔθX, ΔθY and that in ΔθZ. Based on the spatial wavelengths of the dominant periodic components evaluated by DFT, PZ–θX, PZ–θY and PZ–θZ were evaluated to be around 200 nm. The residual periodic cross-talk error components caused by Δx and Δy of the scale grating were 0.074 and 0.046 arc-second, respectively, which achieved the targeted 0.1 arc-second. However, the residual periodic cross-talk error component caused by Δz of the scale grating was identified to be 0.38 arc-second, which was larger than the targeted goal. Therefore, an optimized design shown in the following is made to reduce this error. In the optimized design, the BS2 shown in Figs. 3.32 and 3.33 are replaced with a plate-type beam splitter (plate-type BS2) for avoiding the influence of the internal reflection shown in the previous section. As shown in Fig. 3.38, when the light from LD reaches Surface 1 of the plate-type BS2, the light is divided into two parts. The transmitted part is projected on the scale grating after being refracted at Surface 1 and Surface 2 of the plate-type BS2. The reflected part at Surface 1 propagates toward the outside of the sensor head and will not be received by the angle assembly. Therefore, the interference between Beam 2T and the scale diffracted beams, which caused the periodic cross-talk error components in the improved design, can be avoided. Surface 1 of the plate-type BS2 transmitted 70% of the incident laser beam, while Surface 2 of the plate-type BS2 was anti-reflective coated at the wavelength of the laser beam. Figure 3.39 shows the outputs of the angle assembly of the optimized six-DOF surface encoder when only a translational motion along Z-directions was applied to

3.5 Linear-rotary surface encoder

145

the scale grating. Almost no periodic cross-talk error components are observed in the figure. It can be seen that the cross-talk errors have been reduced to be much less than 0.1 arc-second, which demonstrates the feasibility of the optimized design where the targeted goal of 0.1 arc-second in all three directions has been achieved. Table 3.1 summarizes the results of the initial design, the improved design and the optimized design.

Fig. 3.38: Employment of a plate-type beam splitter in the optimized design of the sensor head.

3.5 Linear-rotary surface encoder Precision linear-rotary (Z–θZ) stages for moving light specimens along and about the Z-axis with high-resolution, high-speed, wide moving stroke are desired in various areas of nanotechnology, such as manipulation of cells and micro-parts. Linear stages and rotary stages are fundamental components of precision positioning systems [48, 49]. Many such single-axis stages, which have been well developed, are commercially available. A Z–θZ stage can also be constructed by superimposing a linear stage and a rotary stage. However, the stacked structure of the Z–θZ stage, in which one stage is moving on the other, tends to reduce the positioning accuracy and speed. Efforts have been made to develop mechanisms that can generate both the Z-motion and θZ-motion to the same moving element by using electromagnetic motors or ultrasonic motors [50]. However, the electromagnetic mechanisms are too complex to make the compact design of a stage system. It is also difficult to tune the resonator and adjust the pre-load of the ultrasonic motor. More importantly, precision positioning by such systems is difficult because there

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Chapter 3 Surface encoder

Fig. 3.39: Measured cross-talk errors in the outputs of the angle assembly of the optimized surface encoder for Δz. Tab. 3.2: Cross-talk errors with various optical designs of the six-DOF surface encoder for Δz. θX arc-second

θY arc-second

θZ arc-second

Initial design

.

.

.

Improved design

.

.

.

Optimized design

.

.

.

are no proper linear-rotary position sensors. In this section, a Z–θZ surface encoder is presented for servo control of a precision Z–θZ-axis stage based on friction drive for moving light specimens with PZTs as the actuators. Magnetic force generated by a permanent magnet is utilized to stabilize the contact condition between the moving element and the friction component for the friction drive.

3.5 Linear-rotary surface encoder

147

Fig. 3.40: Schematic of the Z–θZ stage based on friction drive.

Figure 3.40 shows a schematic of the Z–θZ stage. The stage consists of two driving units and a moving element. The driving units, which are kept stationary, are placed with an angular distance of 90°. Each of the driving units is composed of two PZTs and a friction component. The friction component, the PZTs and the PZT base are glued with each other. The friction components are made by permanent magnets and the moving element is a hollow-type steel cylinder so that the moving element can be attached to the driving units by the magnetic force. The Z-PZTs and the θ-PZTs are aligned along the axial direction (Z-direction) and tangential direction of the moving element, respectively. It should be noted that the pitching and yawing error motions are associated with the Z-directional movement of the moving element because the moving element is only supported by the driving units aligned at the same Z-directional position. The error motions can be reduced by using an additional driving unit at different Z-position. Adding a shaft in the hollow-type

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Chapter 3 Surface encoder

moving cylinder as a guideway is another solution for this problem. For the sake of simplicity, the design shown in Fig. 3.40 is employed in the experiment for feasibility test of the driving mechanism of the Z–θZ stage based on the fact the Z-directional stroke of the stage is short and the moving element is light. Figure 3.41 shows a photograph of the driving unit, which consists of the PZT base, the PZTs and the friction component. The PZT base is an important element to determine the performance of the driving unit, especially the frequency response. The PZT-base has a monolithic structure made by stainless steel so that the PZT-base is rigid for assurance of the frequency response of the driving unit. The surfaces of mounting the two PZTs are machined to be flat and orthogonal to each other for assurance of the squareness of the PZT axes as well as the strength of gluing the PZTs. The size of the PZT-base is 20 mm (L) × 20 mm (W) × 10 mm (H). Small-sized PZTs and permanent magnets are employed to obtain a high bandwidth of the driving units. The size of the PZT is 3 mm (L) × 2 mm (W) × 3 mm(H). The stroke of the PZT is approximately 1.4 μm with an applied voltage of 120 V. The material of the permanent magnet is neodymium and the size is 4 mm (L) × 4 mm (W) × 3 mm (H).

Fig. 3.41: A picture of the PZT unit.

To generate the θZ-motion as shown in STEP 1 of Fig. 3.42, the θ-PZTs are first expanded slowly so that the moving element can be moved with the PZT by the friction force between the friction component and the moving element. Figure 3.43 shows the voltage applied to the PZTs for generating the Z- and/or the θZ-motions. The friction force here is mainly determined by the magnetic force Fmagnet generated

3.5 Linear-rotary surface encoder

149

Fig. 3.42: Generation of the θ-motion.

by the friction component and the static coefficient of friction between the friction component and the moving element. Under this condition, the moving element will follow the displacement ΔdPZT of the Z-PZTs to rotate about the Z-axis by an angle Δθ expressed by Δθ =

ΔdPZT R

(3:51)

where R is the outer radius of the moving element. In Step 2, the θ-PZTs are extracted rapidly so that the moving element cannot be moved by the friction force to follow the fast movement of the PZT. As a result, the moving element will slip to remain on site. The friction force here is mainly determined by the magnetic force Fmagnet and the dynamic coefficient of friction between the friction component and the moving element. Consequently, the moving element rotates an angle Δθ after Steps 1 and 2. By repeating this operation, the moving element can be driven continuously about the Z-axis. Similarly, the moving element can be moved along the Z-axis as shown in Fig. 3.44. It should be noted that the Z-PZT B is not shown in Fig. 3.44 for clarity.

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Chapter 3 Surface encoder

Fig. 3.43: Waveform of the voltage applied to the PZT.

Fig. 3.44: Generation of the Z-motion.

The Z–θZ stage has a simple and compact structure. It can generate stable motions because the contact force between the friction component and the moving element can be kept constant in a simple manner by using the magnetic force. The force of holding the moving element, which is also generated by the magnetic force, can be

3.5 Linear-rotary surface encoder

151

kept stable. On the other hand, although the motion of the stage is discontinuous in each of the expansion and extraction cycles of the PZT, which is common for all friction driving mechanisms, smooth and fast motions can be generated by raising the PZT driving frequency. Fine motions can be realized by reducing the PZT driving amplitude. Figure 3.45 shows a photograph of the Z–θZ stage. The driving units are mounted on the base of the stage with a size of 45 mm (L) × 45 mm (W) × 35 mm (H). The cylindrical moving element consists of two parts. Part 1 is made by steel, which is used for the friction drive. Part 1 is attached to the driving units by the magnetic force. Part 2 is made by aluminum, which is used for the Z–θZ surface encoder. Lengths of Part 1 and Part 2 along the Z-direction are 12 and 15 mm, respectively. The two parts are bonded coaxially as one moving element. It should be noted that the form error of Part 1 as well as the coaxiality error between Part 1 and Part 2 will introduce error motions along the radial directions of the moving element. The outer diameter and the inner diameter of the moving element are 21.36 and 20 mm, respectively. Weights of the PZT, the friction component and the moving element are 0.15, 0.35 and 10 g, respectively. The moving range of the stage along the Z-direction, which is determined by the length of Part 1 and the size of the permanent magnet, is 12 mm. There is no range limitation for the stage to rotate in the θZ-direction. The magnetic force of the friction component against the moving element is 5 N and a specimen with a maximum mass of 50 g can be mounted on the moving element.

Fig. 3.45: A picture of the Z–θZ stage.

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Chapter 3 Surface encoder

Fig. 3.46: A schematic of the Z–θZ surface encoder.

Because the Z–θZ stage does not have separate guideways for the motions along the Z-direction and the θZ direction, cross-talk errors will occur in the open-loop drive of the stage. To solve this problem, a Z–θZ surface encoder is developed for closed-loop servo control of the stage. Figure 3.46 shows a schematic of the Z–θZ surface encoder. It is composed of a sensor head and a sinusoidal grating scale [51– 53]. As shown in Fig. 3.47, the scale is generated on the Part 2 surface of the moving element by diamond turning [54]. The grating surface is composed of sine waves along the axial direction (Z-direction) and the circumferential direction of the moving element. The sine waves are used as scales for measurement of the Z- and θZmotions of the moving element. The pitch and amplitude of the sine wave are 100 μm and 100 nm, respectively. An optical read head is designed to read the sinusoidal scales. As shown in Fig. 3.46, the collimated laser light with a diameter of 5 mm from an LD is divided into multi-beams with pitches of 100 and 141 μm along the Z- and Y-directions, respectively. The multi-beams are projected onto the surface of the moving element after passing through a cylindrical lens, which is used to match the pitch of the multi-beams formed on the surface of the moving element with that of the sinusoidal grating. The center of the moving element is aligned at the focal position of the cylindrical lens. The focal distance of the cylindrical lens is 15 mm. The reflected multi-beams carrying the information of Z- and θZ-motions of the moving element are received by a QPD that is placed at the focal position of a collimator objective with a focal distance of 30 mm. The Z- and θZ-displacements can be obtained from

3.5 Linear-rotary surface encoder

153

Fig. 3.47: The sinusoidal scale grating of the Z–θZ surface encoder.

the two-dimensional outputs of the QPD, respectively. Figure 3.48 shows a picture of the Z–θZ surface encoder. The size of the optical read head is 58.5 mm (L) × 44 mm (W) × 40 mm (H). The sensor has a resolution of 15 nm and 0.27 arc-seconds in the Z- and θZ-directions, respectively. The measurement range of the sensor along the Z-direction is determined by the length of Part 2 of the moving element, which is 15 mm. There is no limitation in the measurement range of θZ-direction. The bandwidth of the sensor is 587 Hz. Saw-toothed voltages shown in Fig. 3.43 are applied to the PZTs to test the driving performances of the Z–θZ stage along the Z-direction and the θZ-direction, respectively. It is confirmed experimentally that the driving frequency of the PZT voltage can be set within wide bandwidths up to 31 kHz for the Z-PZTs, and up to 26 kHz for the θ-PZTs, respectively. This is a major advantage over ultrasonic motors in which the driving frequency has to be set exactly at the resonant frequency. Figures 3.49 and 3.50 show the relationship between the PZT driving amplitude and the mean velocity of the stage. The driving frequencies are set to be 23 and 20 kHz for the Z-PZTs and θ-PZTs, respectively, at which the stage reaches the highest velocities. The duty

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Chapter 3 Surface encoder

Fig. 3.48: The sensor head of the Z–θZ surface encoder.

ratios are 70% for the two sets of PZTs. The stage velocity is measured by the Z–θZ sensor. It can be seen from the figures that the stage velocity increases with the increase of PZT amplitude in both the Z- and θZ-directions when the PZT displacement is larger than certain threshold values, which are approximately 0.13 μm in the Z-direction and 0.16 μm in the θZ-direction, respectively. The threshold value is mainly determined by the friction force between the surfaces of the moving element and the friction component. The nonlinear components in the results are caused by the nonlinear behaviors of the driving units. The maximum velocities are 16 mm/s and 10.5 rpm in the two directions, respectively. Ideally, the maximum velocity is the same as that of the PZT during the period of slow expansion shown in Fig. 3.43. Practically, there are two main factors reducing the maximum velocity. One is the limited driving frequency of the PZT, which is determined by the frequency response of the driving unit associated with the mechanical structure. The other is the backward motion caused by the inertia force, which is generated during the fast contraction motion of the PZT shown in Fig. 3.43. This is also related to the friction force between the surfaces of the moving element and the friction component. Figures 3.51 and 3.52 show the relationship between the duty ratio of the PZT driving frequency and the mean velocity of the stage. The driving frequencies are the same as those in Figs. 3.49 and 3.50. The PZT driving amplitude is set to be 60 V. It can be seen that the stage velocity is approximately zero, which means that the stage keeps stationary, when the duty ratio is 50%. The stage moves to the positive direction when the duty ratio becomes larger than 50% and reaches the maximum velocity

3.5 Linear-rotary surface encoder

Fig. 3.49: Relationship between the amplitude of PZT voltage and stage velocity (θ).

Fig. 3.50: Relationship between the amplitude of PZT voltage and stage velocity (Z).

155

156

Chapter 3 Surface encoder

Fig. 3.51: Relationship between duty ratio of PZT voltage and stage velocity (θ).

Fig. 3.52: Relationship between duty ratio of PZT voltage and stage velocity (Z).

3.5 Linear-rotary surface encoder

157

when the duty ratio is 70%. On the other hand, the stage moves to the opposite direction when the duty ratio is less than 50% and reaches the maximum velocity in this direction when the duty ratio is 35%. This result can be utilized for control of the moving direction of the stage.

Fig. 3.53: Precision positioning by friction drive (θ).

Closed-loop positioning of the stage was carried out by using the Z–θZ sensor as the feedback sensor. The output of the Z–θZ sensor, which is the actual Z and θZdirectional position information of the stage, is input to a personal computer (PC) via an analog-to-digital (A/D) converter. The difference between the command position information and the actual position information is output to a saw-toothed wave generator via a digital-to-analog (D/A) converter after passing through a PID controller implemented in the PC. The output of the saw-toothed wave generator is applied to the PZTs via PZT amplifiers. During the closed-loop positioning, the frequency of the saw-toothed wave output of the generator is kept stationary and the PZT displacement is controlled by the amplitude of the saw-toothed wave output, which is proportional to the output of the PID controller. The direction of the moving element is controlled by changing the duty ratio of the saw-toothed wave output. Figure 3.54 shows the results of closed-loop positioning. The frequency of the sawtoothed wave output is set to be 20 kHz. The duty ratios are set to be 35% and 65% for the forward and backward directions, respectively. The stage is moved by the friction drive, which can drive the stage over a range longer than the PZT stroke. In this

158

Chapter 3 Surface encoder

Fig. 3.54: Precision positioning by friction drive.

driving mode, the stage can move with micro-steps of 0.5 μm and 10 arc-seconds in the Z- and θZ-directions, respectively. The positioning resolution in this drive mode is mainly limited by the driving performance of the stage and the controller. As shown in Figs. 3.52 and 3.53, the sensor output in the θZ-direction is less stable than that in the Z-direction. This is because there is no rotating axis inside the hollowtype moving cylinder. If the stage only needs to move in a small range within the PZT stroke, the stage can be directly driven by the PZT with a higher resolution. The input of the PZT amplifier is switched from the output of the saw-toothed wave generator used in the friction drive mode to the output of the D/A converter of the PC. During the manual switching operation, the moving element is kept stationary by the magnetic force of the driving units. Figure 3.55 shows the result of micro-step positioning by the direct drive of PZT along the Z-direction. The motion of the moving element is also measured by an optical fiber displacement sensor for reference. It can be seen that a 10 nm resolution can be reached along the Z-direction, which corresponds to a resolution of 0.2 arc-seconds along the θZ-direction. The resolution in this driving mode is mainly limited by the performance of the sensor.

3.6 Summary

159

Fig. 3.55: Micro-step positioning by direct drive of PZT.

3.6 Summary A three-axis surface encoder, which consists of a scale XY planar grating and an optical sensor head, has been designed and constructed for sub-nanometric displacement measurement along the X-, Y- and Z-directions simultaneously. To achieve sub-nanometric resolutions not only in the Z-direction, in which the wavelength of the laser source in the optical sensor acts as the graduation period, but also in the X- and Y-directions, in which the grating period acts as the graduation period, XY planar gratings with a rectangular hole array with a short period of 1 μm have been designed and fabricated. The maximum area of the scale XY planar grating, which determines the measurement range of the surface encoder in X- and Y-directions, was 100 mm × 100 mm. An optical sensor head has been designed and constructed for detecting the three-axis displacement of the scale XY planar grating. Interference signals were generated by superimposing the corresponding first-order diffracted beams reflected from the scale XY planar grating and the reference XY grating. The optical arrangement of the sensor head has been designed in such a way that four sets of interference signals with a phase interval of 90° are generated for removing the influence of the variation of the light intensity and for distinguishing the moving directions. The optical sensor head was constructed in a dimension of 50 mm (X) × 70 mm (Y) × 40 mm (Z).

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Chapter 3 Surface encoder

The three-axis surface encoder has then expanded to a six-DOF version. The six-DOF measurement has been realized by combining a three-axis displacement sensor and a three-axis autocollimator in a simple manner through sharing the same laser source. Efforts have been made to design and construct the sensor head with a compact size of 95 mm (X) × 90 mm (Y) × 25 mm (Z) so that it can fit into an XY surface motor stage with two primary axes of motions (Δx, Δy) for precision positioning and four secondary axes of motion (Δz, θx, θy, θz) for compensation of error motions with sub-micrometer/sub-arc-second resolutions. The grating period and grating area of the planar grating were designed and fabricated to be 0.57 μm and 60 mm (X) × 60 mm (Y), which dominate the resolution and measurement range in the X- and Y-directions, respectively. Reduction of the cross-talk errors in the sixDOF surface encoder has then been presented. The cross-talk errors in the outputs of the previous prototype of the six-DOF surface encoder have been successfully identified based on theoretical analysis, based on which the sensor head of the surface encoder was optimized. As a result, the cross-talk errors have been successfully reduced. The concept of the surface encoder has also been applied to the measurement of Z–θZ motion. An optical Z–θZ sensor has also been designed and constructed to measure a PZT-driven Z–θZ stage. An aluminum part, on which a sinusoidal grating is fabricated for the Z–θZ sensor, is connected coaxially to the steel part of the moving element used for the Z–θZ stage. Experimental results for precision positioning of the Z–θZ stage by using the Z–θZ sensor as the feedback sensor have been presented.

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[10] Robonano catalogue. (Accessed March 13, 2021, at https://www.fanuc.co.jp/en/product/robo nano/alphanmia.html) [11] Keysight Technologies: Santa Rosa, CA, USA 2017. Optics and Laser Heads for LaserInterferometer Positioning Systems. [12] Leica Absolute Tracker AT401. (, at https://w3.leica-geosystems.com/downloads123/m1/me trology/at401/brochures/leicaabsolutetrackerat401_en.pdf) [13] Bryan JB. The Abbé principle revisited: An updated interpretation. Precis Eng 1979, 1, 3, 129–132. [14] Kunzmann H, Pfeifer T, Flügge J. Scales vs Laser Interferometers Performance and Comparison of Two Measuring Systems CIRP. Ann – Manuf Technol 1993, 42, 2, 753–767. [15] Teimel A. Technology and applications of grating interferometers in high-precision measurement. Precis Eng 1992, 14, 3, 147–154. [16] Kimura A, Arai Y, Gao W. A three-axis displacement sensor based on interference of diffracted beams. J Japan Soc Precis Eng 2008, 74, 9, 976–980. [17] Kimura A, Gao W, Arai Y, Lijiang Z. Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness. Precis Eng 2010, 34, 1, 145–155. [18] Kimura A, Gao W, Lijiang Z. Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder. Meas Sci Technol, 2010, 21, 5. [19] Heidenhain. Encoders – For machine tool inspection and acceptance testing. (Published 2017, at www.heidenhain.de) [20] Kao C-F, Lu S-H, Lu M-H. High resolution planar encoder by retro-reflection. Rev Sci Instrum 2005, 76, 8, 085110. [21] Watanabe Y, Gao W, Shimizu H, Kiyono S. Optical design of a double pass surface encoder. Proceedings of SPIE – The International Society for Optical Engineering 2005, 6052. [22] Shimizu Y, Osaw S, Meguro T, Lu W, Gao W Surface characterization of a diamond turned XY sinusoidal grating. Procedia Engineering, Vol 19. 1st CIRP Conference on Surface Integrity (CSI), 2011, 337–342. [23] Guo S, Jiang G, Zhang D Research on Error Modelling and Identification of 3 Axis NC Machine Tools Based on Cross Grid Encoder Measurement Related content Position-independent geometric error identification and global sensitivity analysis for the rotary axes of five-axis machine. J Phys Conf Ser Published online 2006. [24] Hocken R, Schattenburg M Needs and advances in metrology for precision motion control in mechatronics. 18th IMEKO World Congr 2006 Metrol a Sustain Dev 2006, 3, 1–5. [25] Kimura A, Gao W, Kim W, Hosono K, Shimizu Y, Shi L, Zeng L. A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement. Precis Eng 2012, 36, 4, 576–585. [26] Kimura A Multi-Axis Displacement Sensors Based on Interference of Diffracted Beams. Tohoku University, Doctoral thesis, 2010. [27] Xinghui L. Fabrication of Scale Gratings and Application to Surface Encoders, Tohoku University, 2014. [28] Li X, Gao W, Muto H, Shimizu Y, Ito S, Dian S. A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage. Precis Eng 2013, 37, 3, 771–781. [29] Matsukuma H, Ishizuka R, Furuta M, Li X, Shimizu Y, Gao W. Reduction in cross-talk errors in a six-degree-of-freedom surface encoder. Nanomanufacturing Metrol 2019, 2, 2, 111–123. [30] Furuta M. High-Precision Multi-Axis Surface Encoder, Tohoku University, 2017.

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[31] Shimizu Y, Furuta M, Chen Y-L, Chen X, Matsukuma H, Gao W. Uncertainty analysis of a sixdegree-of-freedom surface encoder for a planar motion stage. Procedia CIRP 2018. 75, 355–360. [32] Muto H An optical sensor for measurement and control of a planer motion stage. Tohoku University, Master thesis, 2012. [33] Kramar JA. Nanometre resolution metrology with the Molecular Measuring Machine. Meas Sci Technol 2005, 16, 11, 2121–2128. [34] Holmes M, Hocken R, Trumper D. The long-range scanning stage: A novel platform for scanned-probe microscopy. Precis Eng 2000, 24, 3, 191–209. [35] Gao W. An XYθZ Planar motion stage system driven by a surface motor for precision positioning. Smart Devices and Machines for Advanced Manufacturing, London, Springer London, 257–281. [36] Gao W, Sato S, Arai Y. A linear-rotary stage for precision positioning. Precis Eng 2010, 34, 2, 301–306. [37] Sato S Precision positioning of a theta-Z actuator. Tohoku University, Master thesis, 2006. [38] Hecht E. Optics, 5th Pearson, 2017. [39] Li L. New formulation of the Fourier modal method for crossed surface-relief gratings. J Opt Soc Am A 1997, 14, 10, 2758. [40] Dejima S, Gao W, Yanai H, Kiyono S. Precision positioning of a surface motor-driven multiaxis stage using a surface encoder. (2nd report). experiment of dynamic micro-positioning. J Japan Soc Precis Eng 2003, 69, 3, 433–437. [41] Gao W, Nakada T, Dejima S, Kiyono S. Precision positioning of a surface motor-driven multiaxis stage using a surface encoder. J Japan Soc Precis Eng 2001, 67, 12, 1981–1985. [42] Gao W. Precision Nanometrology, London, Springer London, 2010. [43] Matsukuma H, Ishizuka R, Furuta M, Li X, Shimizu Y, Gao W. Reduction in cross-talk errors in a six-degree-of-freedom surface encoder. Nanomanufacturing Metrol 2019, 2, 2, 111–123. [44] Ishizuka R Two-axis Absolute Encoder Using Optical Frequency Comb. Tohoku University, Master thesis, 2020. [45] Ishizuka R, Matsukuma H, Shimizu Y, Gao W Cross-talk error analysis of a multi-degree-offreedom surface encoder for a planar motion stage. 2018 IEEE International Conference on Advanced Manufacturing (ICAM), IEEE, 2018, 36–39. [46] Saito Y, Arai Y, Gao W. Detection of three-axis angles by an optical sensor. Sensors Actuators, A Phys 2009, 150, 2, 175–183. [47] Gao W, Kim SW, Bosse H, Haitjema H, Chen YL, Lu XD, Knapp W, Weckenmann A, Estler WT, Kunzmann H. Measurement technologies for precision positioning. CIRP Ann – Manuf Technol 2015, 64, 2, 773–796. [48] Van Brussel H, Van De Vijver W, De Voider M, Devos S, Reynaerts D. A fast, high-stiffness and high-resolution piezoelectric motor with integrated bearing and driving functionality. CIRP Ann – Manuf Technol 2006, 55, 1, 373–376. [49] Brecher C, Spachtholz G, Paepenmüller F. Developments for high performance machine tool spindles. CIRP Ann – Manuf Technol 2007, 56, 1, 395–399. [50] Isobe H, Kyusojin A. Frequency characteristics of non-contact ultrasonic motor with motion error correction. Precis Eng 2007, 31, 4, 351–357. [51] Hirabayashi R, Noh YJ, Arai Y, Gao W. Hot embossing of a sine-grid for a surface encoder. Nanotechnol Precis Eng 2007, 5, 102–106. [52] Oka T, Ohmura Y, Nakashima T, Gao W, Hane K, Kiyono S. Displacement Metrology Based on Grating Imaging With a Cylindrical Lens Array and a Phase Grating, Ottevaere H, DeWolf P, Wiersma DS, eds., 2005, 58580T.

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Chapter 4 Mosaic encoder 4.1 Introduction As presented in Chapter 3, a surface encoder, either a two-axis type (XZ) or a threeaxis type (XYZ), can measure the in-plane primary motions along the X- and/or Y-axis as well as the out-of-plane secondary motions along the Z-axis with the employment of a scale grating (a linear scale grating in the two-axis type, or an XY planar scale grating in the three-axis type). A sub-nanometric resolution can be achieved by using a scale grating with a short grating pitch in the order of 1 μm. On the other hand, the measurement range of the surface encoder for the primary motions along the X- and/ or Y-axes is determined by the length/size of the scale grating. Recently, the positioning range of a precision stage required for the state-of-the-art instruments is becoming larger and larger. For example, the maximum diameter of the wafer used in semiconductor manufacturing is now shifting from 300 to 450 mm [1]. It is therefore necessary for the surface encoder to have a large enough range along the X- and/or Yaxes for measurement of the primary motions. For this purpose, it is basically required to use a large-sized scale grating. This can be achieved by developing a largescale scale manufacturing system to fabricate the scale gratings. However, such a manufacturing system is extremely expensive [2]. More critically, the deformation of a large-sized scale grating, which is proportional to its weight, will cause considerable measurement errors in long stroke measurements. In this chapter, a concept of the mosaic surface encoder with a mosaic scale grating is presented [3–10]. The mosaic scale grating consists of multiple scale gratings arranged along the X-direction for a two-axis-type surface encoder [11] or in a matrix in the XY-plane for a three-axis surface encoder [12, 13]. It is expected to solve the above problems as an effective large-scaled scale grating. By using a twoaxis XZ optical sensor head, the feasibility of the mosaic linear encoder is presented in Section 4.2. The sensor head is then expanded to a three-axis version in Section 4.3, followed by a three-axis mosaic surface encoder with a four-probe unit presented in Section 4.4.

4.2 Mosaic linear encoder The expansion of the measurement range of an optical linear encoder can be achieved by employing a mosaic linear scale, which is composed of multiple scale gratings aligned in a line with a certain gap as shown in Fig. 4.1 [11]. Meanwhile, two major issues are needed to be addressed; the first major issue is the misalignment between the neighboring scale gratings. Now we consider the two neighboring https://doi.org/10.1515/9783110542363-004

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linear scales (Scales I and II), whose normal and line pattern structures are aligned to be parallel with the Z- and Y-axes, respectively. Regarding the concept of the mosaic linear scale, attention should be paid to the alignment of Scale II with respect to Scale I. However, there should exists a translational misalignment along the Z-axis (Δu in the figure) and angular misalignments about the X-, Y- and Z-axes as shown in Fig. 4.1. Although the precise positioning of a linear scale can reduce these misalignments, it is not a realistic way to prepare for a precision multi-axis positioning system for each of the scales in a mosaic linear encoder. It is thus necessary to develop a method capable of employing a mosaic linear scale without complicated alignment mechanisms for each of the scale gratings.

Fig. 4.1: Mosaic linear scale grating.

The second major issue is the physical gap between the neighboring two scale gratings along the X-axis (Δs shown in Fig. 4.1). When a measurement laser beam is projected onto the gap, the light intensity of the diffracted beams will dramatically be reduced, resulting in the poor signal-to-noise ratio of the reading out of the optical head in the mosaic linear encoder. These issues can be overcome by employing an optical sensor head having multiple measurement probes. Figure 4.2 shows a schematic of the displacement measurement in a mosaic linear encoder having two measurement probes. The distance between the two measurement probes is set to be narrower than the length of a linear scale; this configuration enables one measurement probe to be projected onto a scale grating while the other is projected onto the gap between the two scales. By

4.2 Mosaic linear encoder

167

Fig. 4.2: The approach of using two sensor heads for reading the mosaic scale.

using the multi-axis encoder (the details of which are described in Chapter 3) as each of the measurement probes and stitching the obtained reading outputs, the issues of the misalignments and the physical gap of the neighboring two scales can be addressed. Figure 4.3 shows an example of the configuration of an optical sensor head based on the concept of the multi-axis surface encoder (in Chapter 3) having two measurement probes. The main feature of the configuration is that the pair of parallel measurement probes is generated from a single laser beam by using two circular apertures. As shown in Fig. 4.3, the two measurement probes share the same reference grating, the polarizing beam splitters (PBSs) and quarter-wave plates (QWPs). Moreover, quadrant photodiodes are employed in the detector unit of the optical sensor head; these ideas contribute to preparing for the optical sensor head in a compact size. A photograph of the optical sensor head fabricated based on the configuration shown in Fig. 4.3 is shown in Fig. 4.4. Experiments were carried out by using the developed optical sensor head. Figure 4.5 shows the setup for testing the encoder outputs when the mosaic scale grating was moved along the X-direction. As can be seen in the figure, a pair of linear scales having a length and an identical grating pitch of 12.7 mm and 1.67 μm,

168

Chapter 4 Mosaic encoder

Fig. 4.3: Optical design of a two-axis mosaic encoder with a sensor head of two probes.

respectively, was mounted on a precision air-bearing linear slide whose primary axis is set to be parallel with the X-axis. In the experiments, the displacement of the mosaic linear scale along the X- and Z-axes was detected by the developed optical sensor head during the scanning along the X-direction. Regarding the distance of two measurement laser beam along the X-direction (3 mm), the gap between the two neighboring scales was set to be 2 mm. It should be noted that experimental results [11] demonstrated that each of the probes had a resolution and nonlinear errors comparable to the multi-axis surface encoder described in Section 4.3. Figure 4.6 shows the X- and Z-directional readings of the two probes (probes A and B) in the optical sensor head of the mosaic linear encoder obtained when the pair of scales were made to travel along the X-direction. As can be seen in the figure, at least one probe could successfully detect the X- and Z-displacements even when the other probe was projected onto the gap between the two scales. A stitching operation was carried out based on the X-directional readings of the probes A and B shown in Fig. 4.6. Figure 4.7 shows the stitched X-directional readings from Fig. 4.6, and Fig. 4.8 shows the nonlinear component extracted from the stitched X-directional readings shown in Fig. 4.7. As can be seen in the figure, the

4.3 Mosaic surface encoder

169

Fig. 4.4: Picture of the two-axis mosaic encoder.

stitching operation was successfully carried out without any gaps or discrepancies at the stitched point. These experimental results demonstrated the feasibility of the mosaic linear encoder with a multi-probe optical reading head.

4.3 Mosaic surface encoder The two-axis mosaic encoder is expanded to a three-axis version by using an optical multi-probe generator shown in Fig. 4.9 [4]. The optical multi-probe generator is composed of a triangle prism, two PBSs, a QWP and a mirror. The light beam emitted from an laser diode (LD) is made to be a collimated beam by passing through a collimating lens and an aperture. The polarization of the transmitted beam is then modified by the PBSs as shown in the figure. Multi-beams from the multi-probe generator have a stronger intensity compared to the simple-aperture design shown in the previous section. Quadrant photodiodes (QPDs) shown in Fig. 4.10 are applied to detect the X- and Y-directional interference signals generated from the probes (Probe I, Probe II). The influence of the gap between two gratings can be avoided by switching the interference signals of the probes.

170

Chapter 4 Mosaic encoder

Fig. 4.5: Setup for testing the encoder outputs when the mosaic scale grating was moved along the X-direction.

Figure 4.11 shows the optical layout of the three-axis mosaic grating encoder with the multi-probe generator. The X, Y and Z-directional positions of the optical head with respect to the mosaic scale grating can be obtained from the XYdirectional positive and negative first-order quadrature interference signals from the multi-probes, with the algorithm shown in the previous section. Figures 4.12 and 4.13 show the configuration and a photograph of the mosaic surface encoder for measuring the three-axis displacements of a one-axis linear stage, respectively. The components of the three-axis mosaic surface encoder are listed in Tab 4.1. The size of the optical sensor head is 110 mm (X) × 110 mm (Y) × 45 mm (Z). The XY grating has grooves of 600 lines/1 mm, which is equal to a grating period of approximately 1 μm. The working distance between the scale grating and the optical sensor head is designed to be 7 mm. The LD unit consists of an LD with a wavelength

4.3 Mosaic surface encoder

171

Fig. 4.6: Readings of the probes in the optical sensor head.

of 685 nm, a collimating lens (CL) and a hole aperture of 0.5 mm in diameter. The arrangement of two transparent one-axis diffraction gratings allows only the X- and Y-directional ±1st-order diffracted beams to be received by the detector units. The resolutions of the three-axis mosaic surface encoder were investigated in the experimental setup shown in Figs. 4.14–4.16. An XY grating was installed on an XYZ-axis PZT stage. The output from a commercial laser interferometer with a resolution of 0.08 nm was compared with that of the mosaic surface encoder. The sampling frequency was 10 kHz. Figures 4.17–4.19 show the outputs of the mosaic encoder and the laser interferometer detecting the PZT displacements. It can be

172

Chapter 4 Mosaic encoder

Fig. 4.7: Stitched output of the two-axis mosaic encoder in the X-axis.

Fig. 4.8: Stitched output of the two-axis mosaic encoder in the X-axis (at the stitching point).

Fig. 4.9: Optical layout of the multi-probe generator.

4.3 Mosaic surface encoder

173

Fig. 4.10: QPDs for detecting the interference signals.

seen that the mosaic surface encoder had a sub-nanometer resolution in all the XYZ-axes. Figure 4.20 shows the experimental setup for evaluating the mosaic performance of outputs from the multi-probes of the three-axis mosaic surface encoder. Two XY gratings with a grating period of 1 μm were mounted on a linear air-bearing slide to construct the mosaic scale grating. The length of the XY grating along the Xdirection was 30 mm. The gap between the gratings was set to be 0.5 mm, which was smaller than the interval between the probes. Figure 4.21 shows a photograph of the setup. The mosaic scale grating was moved by the linear air-bearing slide along the X-direction with a speed of 1 mm/s. The sampling frequency of the probes was 10 kHz. Figure 4.22 shows the outputs of Probe I and Probe II of the three-axis mosaic surface encoder when an X-directional displacement was applied to the mosaic scale grating. When Probe II came to the gap in the mosaic scale grating, the amplitude of the output from Probe II declined as shown in Fig. 4.22. At the same moment, the amplitude of the output of Probe I was kept constant. Similarly, when Probe I came to the gap, the amplitude of the output from Probe I declined and that from Probe II was kept constant. Figure 4.23 shows the X-directional stitching results of the outputs from Probes I and II. The results in the Y-direction are shown in Figs. 4.24 and 4.25, and those in the Z-direction are shown in Figs. 4.26 and 4.27. It can be seen that the influences of the gap between gratings were successfully removed in the stitching outputs along X, Y and Z-directions.

174

Chapter 4 Mosaic encoder

Fig. 4.11: Optical layout of the three-axis mosaic surface encoder with the multi-probe generator.

4.4 Four-probe three-axis mosaic surface encoder The two-probe optical head shown in the previous sections is improved to a fourprobe design shown in Fig. 4.28 [13, 14]. As shown in the figure, the four probes are arranged in a square on the mosaic scale grating with certain gaps in the X- and Y-directions among the probes so that at least one probe can be on the XY-axis scale grating. Since each probe can measure the XYZ-axes translational motions based on the principle described above, by choosing the appropriate probe in each case, the three-axis surface encoder with the four probes can measure

4.4 Four-probe three-axis mosaic surface encoder

Fig. 4.12: 3D view of the three-axis mosaic surface encoder.

Fig. 4.13: A picture of the three-axis mosaic surface encoder.

Tab. 4.1: Optical components of the optical sensor head. Item

Manufacturer

Model

Specifications

Laser diode

Mitsubishi electric corp.

MLR-

Wavelength:  nm Power:  mW

Collimating lens

Edmund optics

NT-

Focal length: . mm NA: .

Prism

Sigma Koki Co. Ltd.

RPB--

Size:  mm ×  mm ×  mm

175

176

Chapter 4 Mosaic encoder

Tab. 4.1 (continued) Item

Manufacturer

Model

Specifications

PBS + QWP

Edmund optics

(Special order)

Size:  mm ×  mm ×  mm

Prism

Sigma Koki Co. Ltd.

RPB--

Size:  mm ×  mm ×  mm

PBS

Sigma Koki Co. Ltd.

PBS-

Size:  mm ×  mm ×  mm

λ/ plate

Sigma Koki Co. Ltd.

WPM--P

Size:  mm ×  mm

Objective lens

Sigma Koki Co. Ltd.

Slb--pm

Focal length:  mm

BS

Sigma Koki Co. Ltd.

NPCH--

Size:  mm ×  mm ×  mm

QPD

Hamamatsu Photonics K.K.

S

Active area:  mm ×  mm

Scale grating

–

Fig. 4.14: Experimental setup for resolution investigation (X).

Fig. 4.15: Experimental setup for resolution investigation (Y).

Pitch:  μm

4.4 Four-probe three-axis mosaic surface encoder

Fig. 4.16: Experimental setup for resolution investigation (Z).

Fig. 4.17: Measurement results of the resolution test (X).

Fig. 4.18: Measurement results of the resolution test (Y).

177

178

Chapter 4 Mosaic encoder

Fig. 4.19: Measurement results of the resolution test (Z).

Fig. 4.20: Experimental setup for evaluating the mosaic performances of the surface encoder.

Fig. 4.21: Experimental setup for resolution test.

4.4 Four-probe three-axis mosaic surface encoder

179

Fig. 4.22: Outputs from by Probe I and Probe II in X-direction.

Fig. 4.23: Stitching result in X-direction.

XYZ-directional translational motions of the mosaic scale grating without the influence of its gaps. Figure 4.29 shows an XZ-view of the optical layout of the four-probe three-axis surface encoder. The basic optical configuration of the three-axis surface encoder with a single beam is extended to have four probes and to capture the diffracted beams from the XY-axis scale gratings. Four detector units, each of which is composed of four QPDs, are employed to detect the X- and Y-directional ±1st-order diffracted beams from the four-probe (probes A, B, C and D). The employment of the QPDs enables the optical configuration of the optical sensor head to be simple and compact for the integration into the planar stage system [15]. One beam splitter (BS), three PBSs, five QWPs and four detector units (detector 0°, detector 90°, detector 180° and detector 270°) are installed. The fast axes of QWPs (QWP1, 2, 3 and 5) are set to be 45° with respect to the axis of the p-polarization of the light source. The

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Chapter 4 Mosaic encoder

Fig. 4.24: Outputs from Probe I and Probe II in Y-direction.

Fig. 4.25: Stitching result in Y-direction.

fast axis of QWP4 is set to be 0°. The four-probe with approximately the same intensities from the four-probe unit are separated into s-polarized and p-polarized beams by PBS1. The four probes pass through QWPs 1 and 2 and reach the scale and reference gratings, respectively. The polarization states of the four probes are changed from linear to circular after the diffraction beams pass through QWPs 1 and 2. The diffraction beams from each XY-axis scale grating pass through QWPs 1 and 2 again. Consequentially, the diffraction beams diffracted on the XY-axis scale and reference gratings become the p- and s-polarized beams, respectively. These beams were divided by BS. The beams passed through BS will be divided by PBS2 after passing

4.4 Four-probe three-axis mosaic surface encoder

181

Fig. 4.26: Outputs from by Probe I and Probe II in Z-direction.

Fig. 4.27: Stitching result in Z-direction.

through QWP4, which fast axis is set to be 45° so that the interference signals can be generated from the p-polarized beams and s-polarized beams incident to QWP4. The p-polarized components of the interference beams will be detected by the detector 0°, while the s-polarized ones will be detected by the detector 180°. In the same manner, the beams bent by the BS will be divided by PBS3 and their polarization states and propagating direction will be controlled by QWP5 and PBS3. As a result, the p-polarized beams and the s-polarized beams will be detected by the detector 90° and 270°, respectively.

182

Chapter 4 Mosaic encoder

Fig. 4.28: A schematic of the four-probe three-axis mosaic surface encoder.

Fig. 4.29: Two-dimensional view of the optical layout of the four-probe three-axis mosaic surface encoder.

4.4 Four-probe three-axis mosaic surface encoder

183

Figure 4.30 shows the relationship between the positions of the four probes on the mosaic scale grating and the interference spots on each cell of the QPDs. As shown in the figure, even if three probes out of four probes are on the gaps, the XYZ-directional translational motions of the mosaic scale grating can be measured by using the interference signal from the QPD cell, on which the interference spot is not affected by the gaps. It should be noted that angular misalignments of each XY-axis scale grating in the mosaic scale grating about the X-, Y- and Z-axes will affect the measurement accuracy of the three-axis surface encoder. However, in the previous study by the authors, it was confirmed that the angular misalignments between adjacent two XY-axis scale gratings could be compensated by carrying out a matrix calculation with the data acquired with some simple operations [13, 16].

Fig. 4.30: Interference spot on the detector.

It should be noted that three probes are enough in principle as the multi-probe of the three-axis mosaic surface encoder to avoid the influence of the gaps in the mosaic scale grating. On the other hand, from the viewpoint of the multi-probe generation, the details of which are described in the following section of this chapter, four probes in a matrix are easy to control the properties of each probe such as light intensity, probe diameter or measurement sensitivity. In addition, due to the symmetric layout of the XY-axis scale grating in the mosaic scale grating, the four probes are considered to be sufficient as the multi-probe of the optical sensor head for the three-axis surface encoder. The four probes were therefore employed as the multiprobe in this chapter. In order to generate the four probes for the optical sensor head of the three-axis surface encoder, a new optical module is designed. Figure 4.31 shows a basic concept of a four-probe unit. Two cube-type BSs are employed to divide a collimated single laser beam from the LD into four parallel laser beams; the first BS (1st BS) divides the incident laser beam along the X-direction into two parallel laser beams, while the second BS (2nd BS) divides each of the two incident laser beams along the

184

Chapter 4 Mosaic encoder

Y-direction into two parallel laser beams. Figure 4.32 shows the XZ-view of the 1st BS for dividing the incident beam into two beams along the Z-direction. A laser beam from the LD passes through a collimating lens and a circular aperture, and then enters the 1st BS with an incident angle of 45° with respect to the normal of the BS surface. The beam will be bent as it enters the BS due to the difference of the refractive index of the air (n1) and that of the beam splitter (n2).

Fig. 4.31: Optical setup for generating multi-probe with a pair of cube beam splitters.

Fig. 4.32: Geometric configuration of the multi-probe generator.

According to Snell’s law, the angle of the refracted beam θ1 in the BS can be expressed as follows:   n1 (4:1) θ1 = arcsin sin 45 × n2

4.4 Four-probe three-axis mosaic surface encoder

185

A BS consists of two triangular glass prisms glued together at their bases. The refracted beam will be divided into two beams at the gap between the triangular glass prisms since some of the beams will be transmitted to the gap, while the other will be reflected at the gap. After that, both the beam will come out from the BS. Due to the difference in the refractive index, the path of each beam will also be bent as it comes out from the BS. The refracted angle of the two beams θ3 can be expressed as follows:       n1 n1 = arcsin sin arcsin sin 45 × (4:2) θ3 = arcsin sin θ1 × n2 n2 As can be seen in eq. (4.2), the two beams will come out from the BS with the refraction angle of 45° with respect to the normal of the BS surface. It should be noted that the generated two beams are parallel to each other. In the same manner, the 2nd BS will divide each incident laser beam into two beams. As a result, totally, the four parallel laser beams can be generated from the single laser beam by using two cube BSs. As the design of the four-probe unit for the optical sensor head, the distance between each beam should be set so that each interference spot generated by the diffracted beams from the gratings can be detected by each corresponding cell on QPDs. Assume that the incident beam enters the BS with the incident angle of 45° and an offset of H from the gap between the two triangular glass prisms. According to the geometric relationship, the distance L between the divided laser beams that come out from the BS can be calculated as follows: H+

pffiffiffi L + d1 + d2 = 2h 2

(4:3)

H tanð45 − θ1 Þ

(4:4)

L 2 tanð45 − θ1 Þ

(4:5)

d1 = d2 =

where h denotes the side length of the cube BS. According to eqs. (4.3), (4.4) and (4.5), the distance L in the four-probe can be adjusted by changing H. Figures 4.33 and 4.34 show a developed 3D view and a photograph of the multi-probe generating unit. The distance between the two probes was designed to be 4 mm. Assume that the refractive indices of the air (n1) and BS (n2) were 1.00000 and 1.51347, respectively, the offset of the incident beam H with respect to the gap between the two triangular glass prisms was set to be 2.7 mm from eq. (4.3). The intensity deviation among the four probes was verified to be within 10%.

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Chapter 4 Mosaic encoder

Fig. 4.33: A schematic of the multi-probe generating unit.

Fig. 4.34: A photograph of the multi-probe generating unit.

Figures 4.35–4.37 show examples of possible optical misalignments in the fourprobe unit and corresponding distortions on the generated four probes. Not only the misalignments of incident angles of the laser beam with respect to the BS surface, but also the angular misalignments between the 1st BS and the 2nd BS would cause the non-parallelism of each laser-beam probe in the four probes. The nonparallelism of each probe would affect the measurement sensitivity of each probe for translational motions along with X-, Y- and Z-axes. Due to those misalignments in the four-probe unit, each probe would have inclination angles with respect to the

4.4 Four-probe three-axis mosaic surface encoder

187

normal of the XY-axis scale grating. Assume that the Probe α (α = A, B, C and D) has inclination angles of Δθα and ΔΨα about the Y-axis and the X-axis, respectively, with respect to the normal of the grating surface. Due to these inclination angles, pseudo-grating pitch gX’ and gY’ along the X- and Y-axes, respectively, would be shorter than the ideal grating pitch g. In addition, measurement of the translational motion along the Z-axis could also be affected by the inclination angles since the optical path difference between the measurement beam and the reference beam could become longer than that without the inclination angles. According to equations of the surface encoder in Chapter 3 and the geometric relationship between the XY grating and the probe α, when the translational motion Δx, Δy and Δz along the X-, Y- and Z-axes, respectively, are given to the scale grating, the translational motion Δxα, Δyα and Δzα to be detected by the Probe α can be expressed by the following equation: 0 1 0 1 0 10 1 cos Δθα Δx Δx 0 tan Δθα Δxα B C B C B CB C cos ΔΨα − tan ΔΨα @ Δyα A = A@ Δy A = @ 0 A@ Δy A (4:6) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δz Δz Δzα 0 0 1 + tan2 Δθα + tan2 ΔΨα

Fig. 4.35: Multi-probe distortion due to the incident angle misalignment about Y-axis.

According to eq. (4.6), the measurement sensitivity of each probe will be affected by the non-parallelism of the four probes. Meanwhile, the influences of the nonparallelism of the four probes on the displacement measurements can be compensated based on eq. (4.6). In practice, the measurement sensitivity of each probe can be compensated by using a probe arbitrarily chosen from the four probes as a reference probe, and carry out matrix calculations based on eq. (4.6). Now, we consider the sensitivity compensation between probes A and B. The relationship between the measurement results of the three-dimensional translational grating motions measured by the probe A [ΔxA, ΔyA, ΔzA]T and the probe B [ΔxB, ΔyB, ΔzB]T can be described as follows:

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Fig. 4.36: Multi-probe distortion due to the incident angle misalignment about Z-axis.

Fig. 4.37: Multi-probe distortion due to the angular misalignment of the 2nd beam splitter with respect to the 1st beam splitter about X-axis.

0

ΔxB

1

0

ΔxA

1

B C B C B ΔyB C = Q· B ΔyA C @ @ A A ΔzB 0

ΔzA

cosΔθB B cos ΔθA

B B B =B B B B @

0

0

cos ΔψB cos ΔψA

0

0

θA cosθB 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi − tancosθ + tanΔθB 2 2 A

1

C0 1 C  C ΔxA C ψA cosψB CB 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi tancosψ − tanΔψB CB ΔyA C A A C@ 1 +tan2 ΔθA + tan2 ΔψA C C Δz rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A A 1 +tan2 Δθ + tan2 Δψ 1+ tan ΔθA +tan ΔψA

(4:7)

B

B

1+ tan2 ΔθA + tan2 ΔψA

It is difficult to get to know the matrix Q from the inclination angles of probes A and B with respect to the normal of the XY grating. However, the matrix Q can be calculated by using data acquired from experiments. In the experiments, three different sets of XYZ-directional translational motions [Δxi, Δyi, Δzi]T (i: experiment number) would be given to the XY grating, while detecting the grating motions by the surface

4.4 Four-probe three-axis mosaic surface encoder

189

encoder. From the data [ΔxAi, ΔyAi, ΔzAi]T and [ΔxBi, ΔyBi, ΔzBi]T measured by probes A and B, the following equation can be acquired: 2 3 2 3 ΔxB1 ΔxB2 ΔxB3 ΔxA1 ΔxA2 ΔxA3 6 7 6 7 (4:8) 4 ΔyA1 ΔyA2 ΔyA3 5 = Q · 4 ΔyB1 ΔyB2 ΔyB3 5 ΔzA1

ΔzA2

ΔzA3

ΔzB1

ΔzB2

ΔzB3

From eq. (4.8), the matrix Q can be calculated, and the measurement sensitivity of the probe B can be compensated with respect to that of Probe A. In the same manner, the measurement sensitivity of probes C and D can also be compensated with respect to that of Probe A. To confirm the feasibility of the proposed optical sensor head with a four-probe unit, experiments were carried out. At first, the measurement sensitivity of each probe in the four probes was investigated. Figure 4.38 shows the schematic of the experiments. The optical sensor head was placed against the XY-axis scale grating mounted on a precision linear stage, whose driving axis was set to be X-axis. In the following experiments, small XY-axis scale gratings with a size of 30 mm (X) × 35 mm (Y) were used in the mosaic scale grating to verify its feasibility. The translational motion of the mosaic scale grating along the X-axis was measured by using the four probes, while all the four probes from the optical sensor head were made incident to the same XY-axis scale grating. In the same manner, translational motions of the XY-axis scale grating along Y- and Z-axes were also measured. Figure 4.39 shows the X-, Y- and Z-directional translational motions of the XY-axis scale grating measured by each laser-beam probe. The XY-axis scale grating was moved along each direction with a constant velocity of 0.25 mm/s by the linear stage. The outputs from the four probes were captured simultaneously. The X-, Y- and Z-directional translational motions of the XY-axis scale grating were successfully measured by using all four probes. It should be noted that the gradients of each plot in Fig. 4.39 are slightly different from each other. Figure 4.40 shows deviations of the translational motions measured by probes B, C and D, with respect to those of probe A. The deviations are proportional to the stage displacements. These results imply that the measurement sensitivity of each probe was different from the others, mainly due to the non-parallelism of the four probes as predicted eq. (4.7). The sensitivity of each probe can be compensated by carrying out a calculation with experimental data as described in the previous section. In this chapter, three different types of XYZ-directional translational motions (Δx, Δy, Δz) = (70, 70, 70), (70, −70, 70) and (70, 70, −70) were applied to the XY-axis scale grating. The following matrix QAB is an example for the compensation of the sensitivity of the probe B with respect to that of Probe A: 2 3 0:999980277 0:00000522759 0:0000019673 6 7 (4:9) 0:998617286 − 0:0000479895 5 QAB = 4 0:0000113679 0:0000203644

− 0:000021749

1:000006515

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By using the acquired matrixes, the X-, Y- and Z-directional translational motions measured by each probe shown in Fig. 4.40 were compensated as shown in Fig. 4.41. It was verified that the first-order components of the deviations were reduced by the compensations.

Fig. 4.38: Schematic of the experiments.

Fig. 4.39: Comparison of the displacement measured by each probe in the multi-probe.

4.4 Four-probe three-axis mosaic surface encoder

191

Fig. 4.40: Deviations of the translational motions measured by probes B, C and D, with respect to those of Probe A.

By using the similar experimental configuration shown in Fig. 4.38, the measurement resolution of each probe in the four probes was investigated. An XYZdirectional PZT stage was employed instead of the linear stage, and the XY-axis scale grating was mounted on the PZT stage. Although the PZT stage is controlled in a closed loop with a high-sensitive capacitance displacement sensor, it is vibrating in each direction with its natural frequency. The mechanical vibration of the PZT stage in each direction was measured by the developed three-axis surface encoder with the four-probe unit. As a reference, a commercial laser interferometer with a measurement resolution of 0.08 nm was employed to verify the stage vibration. The outputs of each probe in the four probes of the three-axis surface encoder and the laser interferometer were captured simultaneously by the digital oscilloscope with a sampling frequency of 10 kHz. Figures 4.42, 4.43 and 4.44 show the outputs of each probe in the three-axis surface encoder and the laser interferometer. In the figure, waveforms in a randomly chosen period are plotted. Good correlations were observed among the acquired output waveforms. From these results, it was verified that each probe in the four probes has sub-nanometric measurement resolution.

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Fig. 4.41: Deviations of the translational motions measured by probes B, C and D, with respect to those of probe A (with the compensation based on the matrix).

Fig. 4.42: Outputs of each probe in the three-axis surface encoder and the laser interferometer (X).

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193

Fig. 4.43: Outputs of each probe in the three-axis surface encoder and the laser interferometer (Y).

Fig. 4.44: Outputs of each probe in the three-axis surface encoder and the laser interferometer (Z).

Interpolation errors of each probe for the X-, Y- and Z-directional displacements were also investigated. The same setup was employed to investigate the nonlinear error components of the surface encoder output over a short-range translational motion along each axis, from which the interpolation error of the surface encoder associated with the subdivision of the interference signals within a signal period could be identified. The X-, Y- and Z-directional translational motions were given to the XY-axis scale grating by the three-axis PZT stage. The deviations of the outputs of four probes in the three-axis surface encoder with respect to the output of the laser interferometers are shown in Figs. 4.45–4.47. Periodic error components, which correspond to the interpolation errors, can be observed in each probe output. The periods of the main components of the interpolation errors were 0.5, 0.5 and 0.4 μm in the X-, Y- and Z-directions, respectively. These periods observed in the

194

Chapter 4 Mosaic encoder

experiments well agreed with the periods that can be predicted from the equations of the surface encoder in Chapter 3. The interpolation errors were approximately 5 nm (95% confidence) in the X-, Y- and Z-directions, respectively, which were within 1% of the signal periods.

Fig. 4.45: Deviations of the outputs of four probes in the three-axis surface encoder with respect to the output of the laser interferometers (X).

Fig. 4.46: Deviations of the outputs of four probes in the three-axis surface encoder with respect to the output of the laser interferometers (Y).

After confirming the measurement sensitivity and resolution of each probe in the four probes of the developed three-axis surface encoder, a long-range translational motion was measured by using the mosaic scale grating. Figures 4.48 and 4.49 show the schematic of the experimental setup. A pair of the XY-axis scale gratings with a grating period of 1 μm was aligned along X-direction with a gap of about

4.4 Four-probe three-axis mosaic surface encoder

195

Fig. 4.47: Deviations of the outputs of four probes in the three-axis surface encoder with respect to the output of the laser interferometers (Z).

1 mm, and was employed as the mosaic scale grating. The relative angles between the two XY-axis scale gratings about the X-, Y- and Z-axes were carefully adjusted to maximize the amplitudes of the interference signals from each XY-axis scale grating. The mosaic scale grating was mounted on a linear stage driven by a DC servo-motor so that a translational motion along the X-axis could be given to the mosaic scale grating. The XYZ-directional motion of the mosaic scale grating was measured by the proposed four probes. At first, all four probes were projected onto the same XY-axis scale grating. The translational motion along the X-direction was given to the mosaic scale grating with a velocity of 0.5 mm/s, while the outputs from the four probes were being captured by the oscilloscope with a sampling frequency of 5 kHz. The motion of the mosaic grating was verified by the laser interferometer. In the experiment, the developed four probes can detect not only the X-directional translational motions but also the Y- and Z-directional out-of-straightness components of the stage motion. Figures 4.50–4.55 show the X-, Y- and Z-directional motions of the mosaic scale grating measured by each probe in the four probes when a long-range translational motion along the X-axis was given to the mosaic scale grating, respectively. When the probes B and D arrived at the gap in the mosaic scale grating, the amplitudes of the interference signals of probes B and D started to decrease, resulting in the incorrect sensor outputs from those probes. However, at the same moment, the probes A and C kept on staying on the grating surface, and could measure the stage displacement. Similarly, when the probes A and C came to the gap between two XY-axis scale gratings, these probes could not measure the stage displacement, while the other probes (probes B and D) were on the grating and could measure the X-directional stage displacement. Therefore, by stitching the outputs from probe A and probe B (or probe C and probe D), the long-range X-directional displacement could successfully be measured.

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Chapter 4 Mosaic encoder

Fig. 4.48: A photograph of the experimental setup.

Fig. 4.49: A schematic of the experimental setup.

Fig. 4.50: Stitched measurement result of the translational motions of the mosaic scale grating.

Stitching of the grating motions measured by each probe in the four probes was carried out. At first, the relative tilt of the probe output (probe A in Fig. 4.38) was adjusted to get an exact match in the overlapped region of an adjacent measurement (probe B in Fig. 4.38). After that, the offset was adjusted between the two probe outputs so that the average value of the overlapped region of each probe output could be matched [17, 18]. Figure 4.50 shows the outputs of probes A and B stitched at the overlapped region. Figure 4.51 shows the close-up of the result shown in Fig. 4.50.

4.4 Four-probe three-axis mosaic surface encoder

197

Fig. 4.51: Stitched measurement result of the translational motions of the mosaic scale grating (at the stitching point).

Fig. 4.52: Stitched measurement result of the Y-directional error motion of the mosaic scale grating.

Fig. 4.53: Stitched measurement result of the Y-directional error motion of the mosaic scale grating (at the stitching point).

The amplitude of the residue of the stitching error was within ±50 nm. In the same manner, the Y- and Z-directional out-of-straightness components of the stage motions were also successfully stitched, the results of which were shown in Figs. 4.52 and 4.54, respectively. Periodic deviations were found in the stitched Y- and Z-directional out-of-straightness components of the stage motions. The period of the deviation was approximately 400 μm, which corresponded to the pitch of the ball screw employed in the linear stage. A long-range translational motion along the Y-axis and translational motions along the X- and Z-axes associated with the long-range Y-directional

198

Chapter 4 Mosaic encoder

Fig. 4.54: Stitched measurement result of the Z-directional error motion of the mosaic scale grating.

Fig. 4.55: Stitched measurement result of the Z-directional error motion of the mosaic scale grating (at the stitching point).

motion were also successfully measured. From these results, the feasibility of the developed four-probe surface encoder on measurements of the primary X- and Ydirectional motions and secondary Z-directional motion was verified. A disadvantage of stitching is that errors tend to get amplified when multiple measurements are stitched together. Due to this error propagation, the measurement uncertainty may increase in long-range translational motion measurement. Investigation on the measurement uncertainty of the three-axis surface encoder with the four-probe unit and the mosaic scale grating will be carried out in the future work when the mosaic scale grating with multiple XY-axis scale gratings achieving the measurement range of 300 mm (X) × 300 mm (Y) is prepared.

4.5 Summary In this chapter, the concept of a mosaic surface encoder is presented. To expand the measurement range along the in-plane XY directions, a mosaic XY scale grating, which is constructed by placing multiple XY gratings in the XY-plane along the Xand Y-directions, is applied. The feasibility of the mosaic encoder was experimentally verified by using an XZ two-axis mosaic linear encoder with two optical probes. A mosaic XY grating with a grating period of 1 μm was constructed by using two XY

References

199

gratings with a gap of 1 mm. It was confirmed that by switching the signals from the two probes, the displacement of the mosaic grating could be measured without the influence of the gap between the XY gratings. A two-probe-type optical head with an improved multi-probe generator was then designed and constructed for expanding the XZ two-axis mosaic linear encoder to a three-axis XYZ mosaic surface encoder. Experimental results have verified the capability of the mosaic surface encoder for measurement of three-axis displacements. Finally, a four-probe unit, which generates four probes being aligned in a matrix with a gap of 4 mm, has been designed and constructed for the three-axis mosaic surface encoder. The light intensity of each probe in the four probes has been confirmed to be approximately the same as each other. Experimental results of a short-range stage displacement measurement revealed that each probe in the four probes can measure the X-, Y- and Z-directional displacements of the XY-axis scale grating independently, with measurement resolution of better than 1 nm and nonlinearity error of approximately 5 nm (95% confidence). The experimental results of a long-range stage displacement measurement verified that the developed optical sensor head with the four-probe unit could be used for the extension of its measurement range both along the X- and Y-directions.

References International Roadmap for Devices and Systems (IRDSTM) 2018 Edition – IEEE International Roadmap for Devices and SystemsTM. IEEE Press. (Published 2019 Accessed July 15, 2020, at https://irds.ieee.org/editions/2018) [2] Jitsuno T, Motokoshi S, Okamoto T, Mikami T, Smith D, Schattenburg ML, Kitamura H, Matsuo H, Kawasaki T, Kondo K, Shiraga H, Nakata Y, Habara H, Tsubakimoto K, Kodama R, Tanaka KA, Miyanaga N, Mima K. Development of 91 cm size gratings and mirrors for LEFX laser system. J Phys Conf Ser 112, Institute of Physics Publishing 2008, 032002. [3] Kimura A Multi-Axis Displacement Sensors Based on Interference of Diffracted Beams. Tohoku University, Doctoral thesis, 2010. [4] WooJae K Self-Calibration and Applications of Diffraction Gratings for Precision Nanometrology. Tohoku University, Doctoral thesis, 2013. [5] Hosono K, Kim WJ, Kimura A, Shimizu Y, Gao W. Surface encoders for a mosaic scale grating. Int J Autom Technol 2011, 5, 2, 91–96. [6] Li X, Shimizu Y, Ito T, Cai Y, Ito SS, Gao W. Measurement of six-degree-of-freedom planar motions by using a multiprobe surface encoder. Opt Eng 2014, 53, 12, 122405. [7] Shimizu Y, Ito T, Li X, Kim W, Gao W. Design and testing of a four-probe optical sensor head for three-axis surface encoder with a mosaic scale grating. Meas Sci Technol 2014, 25, 9, 094002, 15. [8] Hosono K A Surface Encoder for a Mosaic Grating. Tohoku University, Master thesis, 2011. [9] Ito T Multi-probe Surface Encoder for a Mosaic XY Grating. Tohoku University, Master thesis, 2014. [10] Hosono K, Kimura A, Kim W-J, Gao W, Zeng L A multi-probe surface encoder for mosaic XY grating. Proceedings of the 10th International Symposium on Measurement and Quality Control 2010, ISMQC 2010, IMEKO TC14, 2010, 282–285. [1]

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[14] [15]

[16]

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Kimura A, Hosono K, Kim WJ, Shimizu Y, Gao W, Zeng L. A two-degree-of-freedom linear encoder with a mosaic scale grating. Int J Nanomanuf 2011, 7, 1, 73–91. Kim WJ, Shimizu Y, Hosono K, So I, Gao W. Design and test of a three-axis mosaic surface encoder. Key Eng Mater 2012, 523–524, 919–924. Shimizu Y, Ito T, Li X, Kim W, Gao W. Design and testing of a four-probe optical sensor head for three-axis surface encoder with a mosaic scale grating. Meas Sci Technol 2014, 25, 9, 094002, 15pp. Gao W, Saito Y, Muto H, Arai Y, Shimizu Y. A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann – Manuf Technol 2011, 60, 1, 515–518. Kimura A, Gao W, Kim W, Hosono K, Shimizu Y, Shi L, Zeng L. A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement. Precis Eng 2012, 36, 4, 576–585. Ito T, Shimizu Y, Kim W, Hosono K, Ito S, Gao W. Design and testing of a four-probe sensor head for a mosaic grating surface encoder. Proceedings of International Conference on Leading Edge Manufacturing in twenty-first Century: LEM21, The Japan Society of Mechanical Engineers, 2013, 381–384. Jansen MJ, Haitjema H, Schellekens PH. Scanning wafer thickness and flatness interferometer. Geyl R, Rimmer D, Wang L, eds. Opt Fabr Test Metrol 5252, SPIE 2004, 334. Jansen M, Schellekens P, Haitjema H. Development of a double sided stitching interferometer for wafer characterization. CIRP Ann – Manuf Technol 2006, 55, 1, 555–558.

Chapter 5 In-line and on-machine surface profiler 5.1 Introduction Optical profilers have long been employed for precision measurement of surface topography [1]. Compared with a contact surface profiler [2], an optical profiler is advantageous in high-speed and nondestructive measurement due to its noncontact nature. Full-field systems, represented by optical microscopes and form interferometers, are the most widely used surface profiles. In such a full-field profiler, a light beam is projected onto an area of the specimen surface, which is referred to as the field of view of the system, for capturing the two-dimensional (2D) and/or threedimensional (3D) topographic information over the area [3]. In an optical microscope, an objective lens is employed to collect the surface texture image within a small field of view. The surface texture image is then magnified to a larger image for detection in an image sensor so that the surface texture can be imaged with a necessary lateral resolution determined by the size of the field of view and the number of the camera pixels or by the diffraction limit. For making the measurement along the vertical direction in a 3D optical microscope, the objective lens of the microscope is mechanically moved along the vertical direction by using an actuator, or the wavelength of the light source of the microscope is electronically swept, for the image sensor to take a series of images of the surface texture. In surface form interferometry, an interferogram, that is, a stationary interference fringe field, is generated based on two-beam interference. The interferogram is corresponding to the distribution of optical path difference (OPD) between the specimen surface and a reference surface. An intensity image of the interferogram is captured by an image sensor, from which the height information of the specimen surface is obtained in relation to light wavelengths. The phase-shifting technique is employed to analyze a set of data frames of interferogram images taken through stepping the reference surface by an actuator with a sub-wavelength step where a high vertical resolution can be achieved. Although the surface topographic information over a certain area can be obtained in a relatively short time, full-field optical profilers also have some disadvantages, especially in 3D measurement and under in-line and/or on-machine conditions [3]. For example, it is necessary to scan the objective or the reference surface in the vertical direction for a set of images to obtain the surface height information, any vibrations during the scan will cause errors in the reconstructed 3D surface topography. Due to the limitation of the image sensor, it is also a trade-off between the lateral resolution and field of view of a full-field profiler. It is difficult for a fullfield profiler to measure a wide area with a high lateral resolution. It is also difficult for a full-field profiler to measure complicated 3D shapes. High cost and large instrument size are other disadvantages of full-field profilers. https://doi.org/10.1515/9783110542363-005

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Chapter 5 In-line and on-machine surface profiler

In addition to full-field systems, optical scanning profilers, also referred to as probe-scan systems, are also widely employed in surface metrology. In an optical scanning profiler, an optical probe is scanned over the specimen surface by scanning stages in a point-by-point or a continuous-path mode for 3D measurement of surface topography. The lateral resolution of such a measurement is determined by the light spot size of the optical probe and/or the spacing between the sampling points on the surface. The lateral resolution is basically determined by that of the optical probe. The lateral and vertical measurement ranges are determined by the stroke of the scanning stages. Surface texture measurement and surface form measurement can be carried out by selecting a proper combination of probe and scanning stages. For surface texture measurement, an optical stylus with a small light spot and scanning stages with high resolutions are employed. For surface form measurement, the spot size of optical probe can be more flexible, and the scanning stages are required to have large enough strokes. Compared with full-field optical profilers, optical scanning profilers have better flexibility in workpiece shapes and measurement areas although a longer measurement time is needed. An optical scanning profiler is typically cheaper and smaller, which is easier to be applied in in-line and/or on-machine surface metrology [3]. In this chapter, a number of unique optical scanning profilers for in-line and/or on-machine measurement are presented [4–15]. In Section 5.2, a flexspline (FS) gear profiler is presented for measuring the pitch error of a FS gear machined by a hobbing machine [7]. In Section 5.3, a rotary die cutter profiler is presented for measuring the cutting edge height of a rotary die cutter machined by a grinding machine. Laser triangulation displacement sensors are employed in the systems of the two sections [6, 8, 15]. Section 5.4 presents an on-machine slope profiler for a microstructured surface on a diamond turning machine by using an optical two-axis slope sensor with a multi-spot light beam [9, 10]. Section 5.5 presents a diamond tool profiler in which a focused laser spot is employed to scan the cutting edge of a diamond tool for measurement of the contour profile of the edge [5, 12, 13]. In Section 5.6, a microdrill bit profiler is presented for run-out measurement of microdrill bits with an optical micrometer [14, 16]. Section 5.7 presents a multi-probe profiler based on an error-separation method called the mixed method [4, 11, 17–19], in which the outputs from two displacement probes and a slope probe are employed for separating the surface profile from the motion errors of the scanning stage.

5.2 Flexspline gear profiler A harmonic speed reducer has a high-speed reduction ratio with near-zero backlash [20]. It also features excellent positioning accuracy and repeatability, high torque capacity, high efficiency, minimal wear and long life, etc. [21]. As shown in Fig. 5.1, a harmonic speed reducer is composed of three components: 1) the wave generator,

5.2 Flexspline gear profiler

203

which is a thin-raced ball bearing fitted onto an elliptical plug; 2) the FS, which is a flexible cylinder with external teeth; 3) the circular spline which is a solid ring with internal teeth [21]. Since its invention in 1955 by Walter Musser [22], harmonic speed reducers have been used in a wide range of precision products, such as industrial robots, machine tools, medical equipment, and spacecraft [23]. All the components of the harmonic speed reducer are made of steel and are fabricated in various mechanical cutting machines. The dimensional tolerances required for the harmonic speed reducer components have been getting increasingly tight. To assure the quality of the fabricated components, it is necessary to measure the cutting errors accurately and feed the information back to the machines. Of the harmonic speed reducer components, quality control of the FS gear is one of the most difficult and important tasks. The required cutting accuracy of the FS gear, which greatly affects the performance of the harmonic speed reducer, is on the order of several microns.

Fig. 5.1: Components of a harmonic speed reducer.

Conventionally, the cutting error measurement is accomplished through measuring the fabricated FS gear by a combination of Van Keuren wires and a micrometer [24]. As shown in Fig. 5.2, the wire is placed between two gear teeth, and the measurement over wires is obtained by the micrometer. When the tooth is overcut, the tooth thickness decreases and the space interval increases, causing the contact points of the wire to move toward the gear center. Consequently, the cutting error can be evaluated from the measurement over wires. However, the measurements are very difficult because the module of the gear is small, and the gear is elastic. An optical

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Fig. 5.2: Contact measurement method of using Van Keuren wires.

system shown in Fig. 5.3, namely, the FS gear profiler, is then developed for the cutting error measurements [7]. The measurement is accomplished through comparing the results with a reference FS gear whose cutting error is sufficiently small. As shown in Fig. 5.3 where the reference gear is measured, the two laser probes, which are kept stationary with a certain spacing D, are used to detect the surfaces of two different teeth. The detection position on each tooth surface is set close to the pitch circle of the gear where the profile of the tooth surface can be approximated to be a straight line within a small area around the detection point. The diameter of the laser spot on the tooth surface is approximately 20 μm. The gear is mounted on a rotary table and can be rotated about the Z-axis. The outputs of the displacement probes with respect to the rotational angle are shown in Fig. 5.3, where θ is the rotational angle of the gear. As shown in the figure, the outputs of the two probes, m10(θ) and m20(θ), vary in opposite manners and intersect at a rotational position θA where the output equals mA. It should be noted that, since the measurement range of the probe is on the order of several hundreds of microns, only a small area of the tooth surface, which can be approximated to be a straight line, is scanned by the probe. In the second step, the FS gear under test is mounted on the rotary table for measurement after the removal of the reference gear. As shown in Fig. 5.3, the cutting error of the gear causes an offset Δα in the output of each probe. The outputs of the two probes, m1(θ) and m2(θ), also intersect at an angular position θB, which θB is different from θA because of the uncertainty of the initial orientation of the gear along the circumference direction when mounting the gear. Here, the eccentric error of the gear relative to the axis of the rotary table and the radial error motion of the table is not considered. It can be understood from the figure that taking the difference of the two probe outputs at θ = θi, where m2(θi) = mA, yields 2Δα without the influence of the uncertainty of the initial orientation of the gear along the circumference direction.

5.2 Flexspline gear profiler

205

Fig. 5.3: (a) Step 1: measurement of the reference flexspline gear without cutting error. (b) Step 2: measurement of the flexspline gear with cutting error.

Fig. 5.4: Geometric model in the presence of eccentric and rotational errors.

Figure 5.4 shows the measurement model, in which the eccentricity error of alignment and rotational error of the gear stage are taken into consideration. To reduce the influence of these errors, the measurement is carried out over one rotation of the gear. We denote the eccentricity error by eC, and the X-, and Y-directional

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components of the rotation error by eX(θi), eY(θi), respectively, where θi corresponds to the angle shown in Fig. 5.3. The probe outputs m1(θi) and m2(θi) can then be expressed as follows: m2 ðθi Þ = mA + feC cos θi + eX ðθi Þg tan φ2 ðθi Þ + eC sin θi + eY ðθi Þ

(5:1)

m1 ðθi Þ = mA + 2Δα + feC cos θi + eX ðθi Þg tan φ1 ðθi Þ + eC sin θi + eY ðθi Þ ði = 1, 2, . . . , N Þ

(5:2)

where N is the number of the gear, and ϕ1(θ) and ϕ2(θ) are the angles of the tooth surfaces of the gear. Subtracting eq. (5.2) from eq. (5.1), the differential output Δm(θi) becomes Δmðθi Þ = 2Δα − feC cos θi + eX ðθi Þgftan φ1 ðθi Þ − tan φ2 ðθi Þg

(5:3)

It can be seen that the Y-directional components of the eccentricity and rotation errors are removed. To reduce the influence of the X-directional error components, an average of the data over one rotation is carried out. The averaged result Δmave can be expressed as follows: Δmave =

N X Δmðθi Þ i=1

= 2Δα −

N N X feC cos θi + eX ðθi Þgftan φ1 ðθi Þ − tan φ2 ðθi Þg N i=1

= 2Δα − ΔEX

(5:4)

where ΔEX =

N X feC cos θi + eX ðθi Þgftan φ1 ðθi Þ − tan φ2 ðθi Þg N i=1

(5:5)

The residual error term, ΔEX, which is caused by nonsynchronous rotation components of the X-directional rotational error, is expected to be reduced greatly by the averaging operation. Averaging of data can also be carried out over multiple rotations to reduce ΔEX further. As a result, the cutting error, which is a function of Δα, can be evaluated from Δmave accurately. Figure 5.5 shows a schematic of the experimental system, which consisted of two laser probes and a rotary stage on which the gear is mounted. The resolution of the laser probe was approximately 10 nm [19]. The spacing D between probes was approximately 35 mm. The probe outputs were taken into a personal computer through an analog-to-digital (A/D) board. The rotary stage was driven by a stepping motor. The radial error motion of the stage was approximately 5 μm. A hydrostatic cylindrical fixture, on which the FS gear was mounted, was installed on the rotary stage.

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The eccentric error of the axis of the fixture as well as that of the FS gear relative to the axis of the rotary stage was measured to be approximately 5 μm. The time for the data acquisition of one rotation was approximately 60 s.

Fig. 5.5: Photograph of the experimental system.

Three FS samples with cutting errors of 0, −5 and +5 μm were used as specimens in the experiment. The samples are referred to as the +0 μm FS, the –5 μm FS and the +5 μm FS, respectively. The cutting errors were generated by adjusting the depth of cut while machining the samples on the hobbing machine. Measurements were made after the laser probes and the rotary stage had been powered for an hour. The temperature variation of the measurement room was approximately 1 °C. Figure 5.6 shows the results of a stability test of the probes in the experimental system. The test duration was 30 min. As shown in the figure, the probes showed good stability of approximately 0.1 μm.

Fig. 5.6: Results of probe stability tests.

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Tests were carried out to investigate the influence of the cutting oil attached on the FS surface. Figure 5.7 shows the output of one of the probes acquired while the FS was rotated by the rotary table. The figure shows the results both before and after the layer of cutting oil on the FS surface was removed by compressed air. The horizontal axis shows the rotary angle of the rotary stage and the vertical axis shows the probe output. It can be seen that the laser probe did not function well in the case of the sample with oil on its surface. However, the laser probe was able to measure the sample after the surface was cleaned by compressed air. This indicates that blowing off cutting oil from the sample surface with compressed air is an efficient method for cleaning the sample surface in practical use.

Fig. 5.7: Laser probe outputs before and after removal of the cutting oil from the surface using compressed air.

Figure 5.8 shows the measured differential output Δm(θi) defined in eq. (5.15) for different FS samples. The gear surfaces were cleaned with compressed air before the measurement. The data shown in the figure are from one rotation. The horizontal axis shows the rotary angle of the stage and the vertical axis shows the differential output Δm(θ i ). It can be seen that influences of the eccentric error and the rotational error are included in Δm(θ i ). The eccentricity errors differed for each FS sample since the samples were mounted on the stage at different times. Figure 5.9 shows Δmave defined in eq. (5.16), which is the average of the differential output Δm(θi) over one rotation. The horizontal axis of the figure represents the applied FS cutting error, while the vertical axis represents Δmave. The results of repeated measurements for each FS are shown in the figure. It can be seen that the cutting errors have a linear relationship with Δmave, which indicates the feasibility of

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Fig. 5.8: Differential output Δm(θi).

the measurement method developed. The measurement uncertainties of Δmave were approximately ± 1.8 μm (3σ), which meet the production line requirement.

Fig. 5.9: Measurement results of cutting errors.

Measurements were also performed with the FS sample mounted on the rotary table at different orientations. The 0 μm FS sample was used in the test. Table 5.1 shows the measured Δmave as a function of the orientation angle. As shown in the table, the standard deviation of the measured Δmave was approximately 0.37 μm, which was in the same order as the results shown in Fig. 5.9.

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Tab. 5.1: Measured Δmave as a function of the orientation angle. Initial orientation of FS, degree

Measured value of Δmave, μm



.



.



.



.



.

Standard deviation

.

5.3 Rotary die cutter profiler As shown in Fig. 5.10, a rotary die cutter is an important cutting tool for use in the roll-to-roll cutting process, in which sheet material is inset in between a pair of rotating rolls, one of which is the rotary die cutter with a cutting edge generated over its outer surface. Sheet products such as diapers, solar cells and electrode sheets for secondary batteries can be manufactured by such a process at a high speed. The cutting edge of a rotary die cutter is made of cemented carbide with high wear resistance. Precision grinding is typically employed for machining the cutting edge. However, it is difficult to machine the cutting edge with the designed accuracy because cemented carbide is a hard-to-machine material. Therefore, it is necessary to measure the machined cutting edge and feedback the data to the compensation grinding process. To improve the measurement efficiency and measurement accuracy, it is desired to make measurement on the grinding machine, that is, onmachine measurement. Figure 5.11 shows the schematic of an optical system, namely the rotary die cutter profiler, which has been developed for this purpose [6, 15, 25]. The rotary die cutter is supported by two V-blocks mounted on an X scanning stage, which is driven by a servo-motor with a rotary encoder. The maximum stroke of the X scanning stage is 1 m. The moving distance of the X scanning stage can be measured by the rotary encoder. The rotation axis of the rotary die cutter is coincident with the Xaxis. A commercial laser triangulation displacement sensor is mounted on the stage base for scanning the cutting edge. Figure 5.12 shows a sectional profile of the rotary die cutter. The tool has two identical cutting edges with the same height with respect to the side ring surfaces, which act as the reference surfaces. The width of the

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Fig. 5.10: The schematic of a rotary die cutting used in roll to roll.

ring surfaces is 20 mm, which is much larger than the diameter of the laser spot. On the other hand, the width of the cutting edges, which is approximately 30 μm, is smaller than the diameter of the laser spot. This makes it difficult for the laser displacement sensor for measuring the edge part. A new algorithm is then proposed to solve this problem.

Fig. 5.11: Schematics of the measurement system for a rotary die cutter.

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Fig. 5.12: Schematic of the cross-sectional profile of the rotary die cutter A–A′.

Figure 5.13 shows a schematic of the commercial laser triangulation displacement sensor used in the rotary die cutter profiler. The sensor consists of a laser diode, a pair of lenses and a position-sensitive detector (PSD) [26]. The laser beam which illuminates from the laser diode is focused on an object surface by Lens 1. The incident light is reflected on the surface. Then the reflected light is collected by Lens 2, and detected by the PSD. The distance variation ΔZ of the object surface along the Z-axis can be measured by calculating the displacement ΔP of the reflected light spot on the PSD according to the triangulation principle as follows [18]: ΔP = ΔZ × sin α × f =d

(5:6)

where ΔZ is the distance variation of the object surface, ΔP is the position change of the light spot on the PSD, f is the focal distance of Lens 2, d is the distance from the focal point of Lens 2 to the PSD and α is the incident angle of the laser beam projected onto the object surface.

Fig. 5.13: Principle of a laser triangulation displacement sensor.

The laser displacement sensor is suitable for high-speed measurement because the laser sensor has a wide bandwidth and can make the measurement in a noncontact condition. Figures 5.14 and 5.15 show the schematic diagrams of the laser beam scanning on the reference surface and the cutting edge surface of the cutting tool,

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respectively. As shown in Fig. 5.14, the width of the reference surface is much larger than the diameter of the laser spot. When the laser displacement sensor scans across the reference surface, all of the light spot is reflected by the surface and the Z-directional position of the reference surface can be determined from the displacement output of the laser displacement sensor, which is defined in eq. (5.18). On the other hand, as shown in Fig. 5.15, the width of the cutting edge is typically smaller than the diameter of the laser spot. As the laser displacement sensor scans across the cutting edge, only a part of the laser spot is reflected from the top surface of the cutting edge. This could make the displacement output of the sensor to be different from the actual Z-position of the top surface of the cutting edge as shown in Fig. 5.15(a). This is the reason the height information of the cutting edge cannot be obtained directly from the sensor displacement output. A new method is thus proposed to solve this problem. In this method, the light intensity output of the laser displacement sensor, which represents the light intensity of the laser light reflected back from the surface, is also acquired in parallel with the displacement output. As shown in Fig. 5.15(b), when the center of laser spot is located at the center of the cutting edge top surface, the displacement output of the sensor would coincide with the actual Z-position of the surface. At that time, the light intensity output of the laser displacement sensor reaches to the maximum. Thus, the height of the cutting edge can be determined from the displacement output when the light intensity output reaches to the maximum.

Fig. 5.14: The laser beam scans across the reference surface.

Figure 5.16 shows a simulation result, in which the intensity of the light spot is assumed to have a Gaussian distribution. As can be seen in the result, the displacement output of the sensor decreases linearly with respect to the displacement of the X-axis stage, which is different from the case when the width of the cutting edge is larger than the diameter of the laser light spot. On the other hand, the light intensity output of the sensor shows a parabolic change. The Z-directional position of the cutting edge top, which corresponds to the height of the cutting edge with respect to

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Fig. 5.15: The laser beam scans across the cutting edge top surface. (a) The right part of the laser spot is reflected by a cutting edge surface. (b) The center of the laser spot coincides with that of the cutting edge surface.

the reference surface, is determined from the displacement output of the sensor when the light intensity output of the sensor reached the maximum.

Fig. 5.16: The simulation result of a test piece scanning.

Measurement experiments were then carried out to confirm the characteristics of the displacement output and the light intensity output of the laser displacement sensor when the sensor scanned across a test piece, on which a cutting edge was formed, was employed as the specimen. The geometry of the cutting edge was similar

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to that of an actual rotary die cutter. Figure 5.17 shows a photograph of the experimental setup for measurement of the test piece. The width of the cutting edge was 20 μm and the included edge angle was 60° as shown in Fig. 5.17(b). The test piece was mounted on the X-axis stage shown in Fig. 5.17. The displacement sensor was kept stationary. The diameter of the laser spot was measured by the knife-edge method [27]. The diameter of the laser spot was measured to be 54 μm, which was larger than the width of the edge of the test piece.

Fig. 5.17: The measurement setup for a test piece.

The test piece was moved by the X-axis stage to scan the edge of the test piece with the laser displacement sensor. The scanning speed of the stage was 0.1 mm/s. Figure 5.18 shows the outputs of the laser displacement sensor during the scanning. It can be seen that, similar to the results shown in Fig. 5.16, the displacement output kept decreasing, while the light intensity output showed a parabolic change. In comparison with the simulation result shown in Fig. 5.16, the measurement result indicates a similar tendency. The ratios of the sensor displacement output to the displacement applied by the X-axis stage were −0.76 and −0.72 for the experimental result and the simulation result, respectively. These results have verified the feasibility of the proposed method for measurement of the Z-directional position of the top of the edge surface, which has a width smaller than the diameter of the light spot of the laser displacement sensor. The agreement of the simulation results with the experimental results indicated the feasibility of the proposed method. When the width of the cutting edge is smaller than the size of the laser spot of the displacement sensor, the displacement output of the laser displacement sensor continuously decreases when the laser spot scans over the cutting edge. The height of the

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Fig. 5.18: Outputs of the laser displacement sensor when scanning across the test piece.

top of the cutting edge with respect to the reference surface can be obtained from the output of the displacement sensor at the moment when the light intensity output reaches the maximum. The same measurements were made when the laser displacement sensor was set at different Z-positions with respect to the test piece. As shown in Fig. 5.17, the laser displacement sensor was mounted on a manual Z-stage. The Z-position of the displacement sensor was changed by the Z-stage. The displacement of the Z-stage was measured by a contact-type displacement sensor with a resolution of 0.1 μm. As shown in Fig. 5.19, at each of the Z-positions, the displacement sensor was moved by the X stage along the X-axis to scan across the cutting edge. The corresponding Z-position of the cutting edge top was determined from the displacement output of the laser displacement sensor when the light intensity output reached its maximum. Figure 5.20 shows the measured results. The horizontal axis shows the output of the contact-type displacement sensor and the vertical axis shows the measured height position of the cutting edge by the proposed method. The residual errors of the result in Fig. 5.20 from a linear fitting were smaller than 0.3 μm. The cutting edge height of a rotary die cutter was measured. The rotary die cutter was moved by the X stage so that the reference area and the cutting edge of the tool could be scanned by the laser displacement sensor. The scanning speed of the stage was 3 mm/s. The sampling interval was 0.3 μm. The straightness error motion of the X stage was compensated by the reversal method [28, 29]. Figures 5.21 and 5.22 show the displacement outputs and the light intensity outputs of the laser displacement sensor during the scanning from the left-side ring to the right-side ring of the rotary die cutter, respectively. As shown in Fig. 5.21(a), the displacement outputs of the laser displacement sensor were almost constant on the two side rings. Figure 5.21(b) and 5.21(c) shows the displacement outputs at the two cutting edge areas. As can be seen from the figures that the displacement output decreased linearly, which was consistent with the analysis result and the experimental results. In

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Fig. 5.19: Schematics of the measurement system for linearity test.

Fig. 5.20: The result of the linearity test.

Fig. 5.22(a), the light intensity outputs show small changes in the areas of the side rings, which were caused by the variations of reflections on the surfaces. Figure 5.22 (b) and 5.22(c) shows the light intensity outputs at the cutting edge areas. It can be seen that the light intensities showed parabolic changes, which was consistent with the analysis and simulation results shown above. The height of the left cutting edge was calculated from the measurement results to demonstrate the feasibility of the proposed method. The displacement outputs of the two side rings were employed to eliminate the effect of the imperfect alignment of the axis of the rotary die cutter and that of the X stage. The outputs of the laser displacement sensors at the cutting edge area were fitted by the least square method for removing the influence of electronic noises in the outputs. The height of the cutting edge was obtained from the displacement output when the light intensity output reached the maximum. The edge height was calculated to be 2.71 μm.

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Fig. 5.21: The result of the rotary die cutter measurement by using the laser displacement sensor. (a) The displacement output from the left-side ring to the right-side ring, (b) from the left cutting edge and (c) from the right cutting edge.

The rotary die cutter was also measured by using a contact-type displacement sensor, that is, a dial indicator, in comparison with the proposed method. The laser displacement sensor was replaced by the contact-type displacement sensor. The rotary die cutter was moved by the X stage so that the rotary die cutter could be scanned by the contact type displacement sensor. The scanning speed of the X stage was 1 mm/s. The range of the displacement measurement was 0.8 mm. The sensor bandwidth was 20 Hz. Figures 5.23 and 5.24 show the experiment results obtained as different scanning speeds with the contact type displacement sensor and the laser displacement sensor, respectively. In the experiment, the rotary die cutter was scanned at four different scanning speeds. The scanning speeds were 1 mm/s, 2 mm/s, 3 mm/s and 4 mm/s, respectively. The scanning by each of the sensors was repeated ten times at

5.4 Microstructure slope profiler

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Fig. 5.22: The result of the rotary die cutter measurement by using the laser displacement sensor. (a) The light intensity output from the left-side ring to the right-side ring, (b) from the left cutting edge and (c) from the right cutting edge.

each scanning speed. The measurement repeatability errors by the two sensors at each of the scanning speed were compared. As shown in Fig. 5.23, the repeatability errors by the contact-type displacement sensor were 0.4 μm at 1 mm/s and 1.3 μm at 4 mm/s. The repeatability error at 4 mm/s was three times bigger than that at 1 mm/s. On the other hand, in Fig. 5.24, the repeatability errors by the laser displacement sensor were almost the same for all the scanning speeds, indicating the advantage of the noncontact measurement by the laser displacement sensor.

5.4 Microstructure slope profiler A sinusoidal microstructured surface, called angle grid, has been employed for a surface encoder [30–32] to measure planar motions of precision XY stages [33–36].

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Fig. 5.23: Relationship between the stage speed and the edge height measured by the contact dial indicator.

Fig. 5.24: Relationship between the stage speed and the edge height measured by the laser displacement sensor.

The surface of angle grid is a superposition of sinusoidal waves in the X- and Ydirections. The sinusoidal profile of the angle grid surface is designed to have spatial wavelengths of 100 μm and amplitudes of 100 nm in both the X-direction and the Y-direction. The corresponding amplitude of the slope profile of the sinusoidal surface is 2π mrad. Taking into consideration the moving stokes of most precision XY stages, the area of the grid surface, which determines the measurement range, is designed to be from several tens of millimeters to several hundreds of millimeters in diameter. Since the angle grid surface is used as the reference of position measurement, it is of a high priority to fabricate the sinusoidal surface accurately over a large area with a low fabrication cost. Based on the specifications of the angle grid surface, the diamond turning with a fast tool servo (FTS) [37–40] is chosen for the fabrication of the angle grid surface. Single-point diamond turning has the advantage of producing difficult geometries with high form accuracy and good surface finish [41, 42]. It can be used to the sinusoidal profile on a plane surface over an area of

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221

150 mm in diameter [43]. However, on the other hand, the diamond turning process is quite time-consuming for fabricating microstructured surfaces over a large area, resulting in a high fabrication cost. The maximum area of the fabricated surface is also limited by the size of the spindle and the stroke of the slide of the diamond turning machine. One way to overcome these shortcomings is the employment of the replication technique [44, 45]. Replication is widely used for low-cost and mass production of various optical elements. For replication of the angle grid surface over a large area, it is necessary to accurately fabricate the sinusoidal profile on a cylinder used as the replication master. An on-machine measurement system, namely microstructure slope profiler, for the cylindrical master grid is presented. Figure 5.25 shows a schematic of the microstructure slope sensor constructed on a diamond turning machine. The FTS is mounted on the tool post of a diamond turning machine to cut the surface of a cylindrical workpiece. The workpiece is mounted on the spindle with its axis along the Z-direction. The spindle is moved along the Z-axis by the carriage of the diamond turning machine. The carriage movement and the spindle rotation are synchronized so that the Z–θ coordinates of the cutting point can be determined by the rotary encoder output of the spindle.

Fig. 5.25: Schematic of the fabrication and measurement system for the cylindrical master grid.

Figure 5.26 shows a schematic of the FTS unit developed for fabricating sinusoidal microstructures on cylindrical surfaces. Figure 5.27 shows a photograph of the FTS unit. Since the size of the FTS unit in the Z-direction will limit the fabrication area of the cylindrical workpiece on the spindle side as shown in Fig. 5.25, a compact

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design of the FTS unit is carried out for a larger fabrication area. The compact size is also helpful for the improvement of the FTS unit in thermal and dynamic characteristics, which are important for the long-time fabrication of large-area microstructured surfaces.

Fig. 5.26: Schematic of the FTS unit.

Fig. 5.27: Photograph of the FTS unit.

As shown in Figs. 5.26 and 5.27, the FTS unit consists of a hollow-type piezoelectric (PZT) ring actuator and a capacitance displacement probe. The PZT actuator with an internally pre-stressed stainless casing has characteristics necessary for the FTS unit, including compact size (35 mm diameter × 34 mm length), high stiffness (450 N/μm),

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large bandwidth (30 kHz) and high displacement resolution (0.24 nm). A small capacitance probe (4 mm diameter × 34 mm length) is mounted in the hollow of the PZT actuator with a diameter of 14 mm so that the FTS unit has almost the same size as the actuator, which is half of the first-generation FTS unit used for the fabrication of sinusoidal micro-structures on plane surfaces [43]. The sub-nanometer resolution and 20 kHz bandwidth of the capacitance sensor also match those of the actuator. Figures 5.28 and 5.29 show the static response and the dynamic response of the FTC-unit, respectively. The bandwidth of the closed-loop controlled FTS unit is approximately 3 kHz (−3 dB).

Fig. 5.28: Static response of the FTS unit.

Fig. 5.29: Dynamic response of the FTS unit.

The slope profile of the sinusoidal microstructures generated on the cylinder surface is measured by a two-axis optical slope sensor. The following equations express the slope profiles of the sine waves along the circumferential direction (θZdirection) and the axial direction (Z-direction), respectively.

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    1 ∂f ðθ, zÞ 2π 2πr 2πr αðθÞ = = A cos θ = AS cos θ r ∂θ λ λ λ     ∂f ðθ, zÞ 2π 2π 2π = A cos z = AS cos z βð z Þ = ∂z λ λ λ

(5:7) (5:8)

where A and λ are the 100 nm amplitude and a 100 μm wavelength of the sinusoidal structures. AS is the corresponding amplitude of the slope profile, which is calculated to be 2π mrad. The two-axis slope sensor is mounted on the tool post of the diamond turning machine. Figure 5.30 shows a schematic of the sensor, which is based on the principle of laser autocollimation [19, 46–50]. The p-polarization laser light from a laser diode is collimated to a beam with a diameter of 6 mm. The beam then passes through an aperture plate, on which a 2D array of 30 × 30 micro-rectangular apertures are generated by using the photolithography process, to form a bundle of beams. The size of each micro-apertures is 50 μm (Z) × 74 μm (Y). The pitch spacing is 100 μm in the Z-direction, which is the same as the wavelength of the sinusoidal structure. The pitch spacing in the Y-direction is 148 μm. Differing from the slope sensor used in a surface encoder [32, 43], a cylindrical lens is used to converge the multiple beams to the center of the cylindrical workpiece so that the measurement of the slope profile of the sinusoidal microstructures will not be influenced by the curvature of the cylinder surface on which the microstructures are generated. The focal length of the cylindrical lens is chosen in such a way that the multiple beams are projected onto the workpiece surface at the same phase positions in different periods of the periodic sinusoidal waves, which satisfies the same condition when the surface encoder is used. The averaging effect of the multiple beams is utilized to reduce the influence of high-frequency error components in the surface profile. The output of the slope sensor thus corresponds to an averaged value over 30 periods of the since wave. The multiple beams reflected from the surface are bent at the polarization beam splitter (PBS) and then received by the autocollimation unit consisting of an autocollimation lens and 2D linear PSD located at the focal plane of the lens. The linearity of the detector is 0.1% of the measurement range. Fabrication and on-machine measurement experiments were carried out to test the developed system. An aluminum workpiece with a diameter of 54.5 mm and a length of 100 mm was vacuum chucked on the spindle. The cutting area was on the free-end side of the workpiece. A length of the cutting area along the axial direction of the workpiece was 50 mm. The sinusoidal microstructures with 100 μm wavelength and 100 nm amplitude were generated on the cylindrical workpiece by the FTS unit. Positioning resolutions of the spindle and the Z-carriage of the diamond turning machine were 0.001° and 10 nm, respectively. The rotating speed and feed speed of the spindle and the Z-carriage were set to be 20 rpm and 5 μm/rev. The focal length of the cylindrical lens in Fig. 5.30 was 42 mm. A single-crystal diamond

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Fig. 5.30: Schematic of the slope sensor.

cutting tool with a round nose geometry was employed for fabrication. A rake angle was 0°, and a clearance angle was 7°. The nominal radius of the round nose (in the XZ-plane) was 1 mm. It has been shown in a previous paper of the authors [43], the round nose geometry will cause a periodic error component along the axial direction (Z-direction) of the workpiece with a wavelength of 50 μm, which is half of the wavelength of the desired sinusoidal structures. The error component can be compensated for based on the model shown in Fig. 5.31 on the condition of knowing the local radius of the portion of the tool tip, which actually cuts the surface. In the fabrication, the 1 mm normal radius was first employed for compensation. It was not necessary to carry out such error compensation along the circumferential direction because the radius of the tool edge (in the XY-plane), on the order of 50 nm, was small enough compared with the curvature radius of the desired sinusoidal structures. The slope profile of the fabricated sinusoidal structures was measured by the slope sensor without removing the workpiece from the spindle. The surface was cleaned by ethanol before measurement. The sensor was kept stationary during the measurement. The workpiece was rotated by the spindle with a speed of 6 rpm and moved by the Zcarriage with a speed of 250 mm/min for sectional slope profile measurements along the conferential direction and the axial direction, respectively. Figures 5.32 and 5.33 show the measured slope profiles along the two directions, respectively. Only the data over a scanning length of 1 mm, which is corresponding to 10 periods of the sinusoidal structures, are shown in the figures for clarity. As shown in the figures, sinusoidal signals corresponding to slopes of the sinusoidal structures were output by the sensor.

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Fig. 5.31: Model for compensation of the tool geometry.

Fig. 5.32: Measured sectional slope profile along the circumferential direction of the workpiece.

Fig. 5.33: Measured sectional slope profile along the axial direction of the workpiece.

Figures 5.34 and 5.35 show spectrum distributions along the two directions, respectively. As shown in Fig. 5.34, the sinusoidal structures with the 100 μm wavelength have been accurately generated in the circumferential direction. In the axial direction shown in Fig. 5.35, however, there is a large error component with a slope amplitude of 0.632 mrad (corresponding to a height amplitude of 10.1 nm) at a frequency of 0.02 μm−1 (50 μm wavelength). The error component, which was caused by the round nose geometry of the tool, indicating that the 1 mm radius of the tool used in the

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compensation was not accurate. Because the portion of tool tip actually cutting the sinusoidal structures was very small, on the order of 10 μm along the tool circumference, the local radius of the tool could be much different from the nominal radius. To reduce the error component, fabrications were repeated with different radius values for compensation. The fabricated surface was measured by the slope sensor at each time to get the spectrum distribution shown in Fig. 5.35. As a result, it was confirmed that the compensation with the radius value of 1.5 mm could reach the best result. As shown in Fig. 5.36, the error component at the frequency of 0.02 μm−1 was reduced to 0.112 mrad, which was corresponding to a height amplitude of 1.8 nm.

Fig. 5.34: Spectrum distribution along the circumferential direction.

Fig. 5.35: Spectrum distribution along the axial direction.

Fig. 5.36: Improved spectrum distribution along the axial direction.

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5.5 Diamond tool profiler Single-point diamond cutting tools are often used with a nose radius of several 10 nm [51] for ultra-precision machining. One of the crucial factors that will impact the efficiency of the machined surfaces is the shape error of the tool cutting edge [52] in the manufacture of precision products such as optics and micropatterned functional surfaces. Therefore to ensure machining precision, the shape of the cutting edge of the instrument should be tested periodically [53, 54]. Furthermore, since the demounting and remounting operation of a cutting tool from a machine tool could induce the tool misalignment, the tool cutting edge is desired to be measured on-machine. Even if such an installation fault is slight, in the case of ultra-precision machining, it should be taken seriously [52, 55]. Scanning electron microscopes (SEMs) [56, 57] is one of the standard ways of determining the outline of a cutting tool. However, using the acquired SEM images, which are qualitative findings, to quantitatively evaluate the tool type, is a costly and time-consuming task [58]. In addition, the SEM observation requires a vacuum condition, which prevents the SEM to be applied to the on-machine evaluation of the tool cutting edge. Another appropriate method for measurement of a tool cutting edge is atomic force microscopes (AFMs) [59, 60]. There are some benefits of AFMs, such as three-dimensional (3D) imaging, sub-nanometric precision of measurement and low-measuring force. To determine the shape of the tool cutting edge [61], an AFM-based measuring instrument that is ideal for onmachine measurement has been proposed. Instead of using optical microscopes on commercial AFMs, an optical alignment device was used to coordinate the AFM probe with the cutting edge [62] instrument in the proposed instrument. Onmachine quantitative measurement of the cutting edge tool [63] has been carried out by the proposed instrument. However, owing to the restricted measuring range of the AFM method, it takes a long time for a large-scale cutting tool to measure its shape. In this section, it is presented to use the optical alignment device built in the AFM-based measuring instrument as a micro-optical probe for fast and quantitative measurement of large-scale multi-millimeter cutting edge tools. A micro-optical probe with a focused beam spot, namely a diamond tool profiler, is employed for the evaluation of the tool edge contours. The deviation of the light intensity, which is part of the micro-optical probe that passes through the edge of the instrument, will be translated into the contour of the tool edge. A schematic of the proposed micro-optical probe is shown in Fig. 5.37(a). The axis of the microoptical probe is aligned to have a right angle with respect to the rake face of a diamond cutting tool. The optical probe consists of a focused laser beam generated by an objective lens and a photodetector (PD). The light intensity of the laser beam passed through the edge of the tool contour is captured by the PD. A photoelectric

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current output from the PD is converted into the voltage output by a trans-impedance amplifier. The voltage output is treated as the PD output in the following. Figure 5.37(b) shows a schematic of the relationship between the Y-position of the optical probe and the PD output. It should be noted that the Y-position is defined with respect to the tool edge. The PD output can be translated to the relative location of the probe with respect to the instrument edge contour by referencing the relationship curve in Fig. 5.37(b). The edge contour will then be acquired from the divergence of the PD output and the information on the absolute probe position while information on the absolute probe position is available from the tool positioning system.

Fig. 5.37: A micro-optical probe. (a) Optical setup (b) PD output and the Y-position of the optical probe with respect to a tool edge.

A schematic of how the PD output can be transformed with respect to the tool edge to the relative location of the probe is shown in Fig. 5.38. The relative direction of the middle point of the optical probe with respect to the tool edge is denoted by h(x); the scanning direction of the micro-optical probe is denoted by g(x). The following equation can thus indicate the tool edge contour f(x): f ðxÞ = gðxÞ + hðxÞ

(5:9)

At each X-position (xi, i is sampling number), both PD output V(xi) and probe position g(xi) are recorded simultaneously, while the optical probe is made to trace g(x), the scanning path. Then by referencing the curve of the relationship seen in Fig. 5.37(b), it is possible to transform the PD output V(xi) to the relative position.

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Fig. 5.38: A schematic of converting the PD output into the Y-position of the probe with respect to the tool edge.

A schematic of the positioning of the tool edge with respect to the optical probe is shown in Fig. 5.39. It is necessary to position the tool edge on the focal plane of the objective lens in the optical stylus. Furthermore, the tool edge needs to be positioned in the focal plane so that the beam waist of the optical stylus would overlap the tool edge, as shown in the figure. Figure 5.40 indicates the variance of the PD output with respect to the tool displacement in the Z-direction. The PD output varies as the sectional size of the beam increases, and when the tool is located at the beam waist of the optical probe, it has a minimum value. The Z-directional alignment can thus be carried out in the proposed method by tracking the PD output in the XYplane prior to the probe scanning. In the proposed method, the scanning motion accuracy, the spot size of the optical probe, as well as the signal quality of the PD output, are major contributors that mainly affect the resolution. It should be remembered that in terms of the tool nose radius, the beam position is small enough to be viewed as a “point” on the cutting edge. A measuring device with an optical system for a micro-optical probe has been developed for the measurement of form errors and radii of tool edge contours. A schematic of the built system is shown in Fig. 5.41. A laser diode (LD), a beam

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Fig. 5.39: Alignment procedure of the tool edge to the optical probe.

Fig. 5.40: PD output and the Z-position of the tool in the optical probe.

splitter (BS), three lenses and two PDs are included in the optical probe. On a diamond turning machine equipped with a precision three-axis positioning system, the optical system is mounted for on-machine measurement. A collimated laser beam with a wavelength of 405 nm propagating along the Z-direction is generated by using the LD and the collimating lens. The BS is then used to split the collimated laser beam into two beams: a reference beam and a measurement beam. The PD for reference (PDR) tracks the deviation of the light intensity of the reference beam in order to compensate for the effect of the LD power drift and some noises due to the electric field around the LD. The measuring beam is focused by an objective lens on the edge of the tool. Aspheric

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achromatic lenses having a numerical aperture of 0.25 were employed as the collimating lens and the objective lens. At the beam waist, the laser beam spot has a diameter of 2 μm (full width at half maximum, FWHM). According to the diffraction theory, the spot r0 radius can be expressed by the following equation [64]: r0 ∝

λ NA

(5:10)

A smaller laser beam spot that can reach a better resolution is provided by an objective lens with a greater numerical aperture (NA), according to the above equation. However, when applying the objective lens with a high NA, attention should be paid to the light-collecting portion of the micro-optical probe since the light passing through the tool edge will spread easily.

Fig. 5.41: A schematic of the on-machine diamond tool profiler.

During the scanning of the micro-optical microscope, a part of the direct beam passing through the instrument edge is collected by the PD for measuring (PDM). As a measuring signal, the PDM output is used. In this chapter, on the same electrical circuit board the I–V circuits for both PDM and PDR are arranged so that effects of circuit power instability and noises related to the electric field can be reduced, which would impact the signal quality of the PDs. A diamond tool to be evaluated was mounted on the XYZ carriage slides of the turning machine, while the optical probe was mounted on the table the spindle in the diamond turning machine. In both the X- and Y-directions, the carriage slides have a nanometer-order positioning resolution with repeatability of less than ±50 nm; the precision positioning and scanning of the optical probe can be realized by the carriage slides. During the tracing of the optical probe along the tool tip, both the XYZ-directional orientation of the optical probe and the PD outputs were recorded concurrently.

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The edge contours obtained from the PD output of the sensor are shown in Fig. 5.42. It was found that the measurement outcomes agreed with each other, as shown in the figure. A large variation over the entire measurement spectrum of 400 nm from the straight form was found. The measurement repeatability was found to be below 10 nm at the smooth edge contour regions. The effect of the system’s mechanical vibration, whose frequency was approximately 1 Hz, was observed. At the steeply deviated edge contour field, the measurement repeatability became worse. The main explanation for this is that the X-directional positioning of the micro-optical probe differs slightly for each measurement. The straight cutting edge with submicron shape precision was effectively tested over the measurement range of 3.5 mm by the established measurement method, as shown in Fig. 5.42.

Fig. 5.42: The edge contours measured by the system.

Another diamond tool with a circular cutting edge was also tested using the onmachine measuring method that was developed. Figure 5.43 displays the schematics of two types of the method with different optical probe scanning paths used in this study. The first type is referred to as the “raster-scan method” in which every scan line has been set perpendicular to the contour of the tool edge as shown in Fig. 5.43. To define the scanning path, a priori knowledge of the shape of the edge contour will be needed in the process. “A “pseudo” edge contour was thus formed in this method by detecting multiple points on the edge of the tool and fitting a circle on them, as seen in Fig. 5.43.

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The second type is referred to as the “direct-scan method,” in which the tool edge contour is traced by the one-pass scanning of the probe, as shown in Fig. 5.44. In the direct-scan method, the pseudo-edge contour measured in the raster-scan method was also employed. One of the advantages of the direct-scan method is the reduction of measurement time. Meanwhile, one of the drawbacks of the direct-scan method is that the method cannot accept the deviation of the tool edge contour from the pseudo-scanning path due to the limited diameter of the measuring probe.

Fig. 5.43: Schematics of optical probe-scanning paths in the raster-scan.

Fig. 5.44: Schematics of optical probe-scanning paths in the direct-scan.

By using the scanning methods of both types, the cutting edge of the tool with a nose radius of 0.9 mm was evaluated. As seen in Fig. 5.45, the relationship between the PD output and the relative position δ was acquired in advance of the evaluation of the tool edge contour. Initially, using the raster-scan process, the tool was measured. The PD output obtained during each line scan is seen in Fig. 5.46. The length and the number of sampling points were set to be 20 μm and 320 points, respectively, for each scan line. A scanning interval was set to be 2.5 μm along the pseudo-edge contour.

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Fig. 5.45: The relationship between the PD output V and the relative position δ was acquired in experiments.

Fig. 5.46: The PD output obtained by the raster-scan method.

The estimated edge contour, determined from the matrix data of the PD output shown in Fig. 5.46, is shown in Fig. 5.47. According to the geometric relationship of the micro-optical probe having a diameter of 2 mm, with respect to the tool edge whose nose radius is 0.9 mm, the intensity of the optical probe I of 50.04% is expected when the axis of the micro-optical probe is aligned on the tool edge. Therefore, using that value, the tool edge contour was subtracted from the matrix data in Fig. 5.46. The repeatability was evaluated to be 0.20 μm; that was greater than that observed in the case of measuring straight cutting edge. The lengthy measurement time is one of the main reasons for these results. By the raster-scan method, a measurement period of approximately seven minutes was required. The contribution from the thermal drift was thus regarded as the main cause of the degradation of the measurement repeatability. Using the direct-scan process, the edge contour was also measured. The edge contour determined by the direct-scan technique is seen in Fig. 5.48. The scanning path of the laser beam g(x), tool edge contour f(x) calculated based on eq. (5.42), the relative position of the laser beam δ(x) from the scanning path are plotted in the

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Fig. 5.47: Estimated edge contour determined from the matrix data of the PD output in Fig. 5.46.

figure. Furthermore, the out-of-roundness obtained from f(x) in Fig. 5.48 is plotted in Fig. 5.49. The tool edge contour evaluated in the two repetitive measurements was also found to align well with each other. A difference between the two repetitive trials was evaluated to be smaller than 80 nm. A far shorter measuring period (6 s) required for the direct-scanning method was considered to contribute to improving the repeatability. The comparison of the out-of-roundness obtained by the two types of the scanning method is seen in Fig. 5.50. In the figure, it was found that the

Fig. 5.48: The edge contour determined by the direct-scan technique.

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results obtained by the methods well agreed with each other in the region of “A,” while a lack of high-frequency component was observed in the region “B” obtained by the direct-scanning method. The degradation of the measurement sensitivity as a result of the large deviation of the probe-scanning path with respect to the tool edge contour is considered as the main root cause of the above results.

Fig. 5.49: Out-of-roundness obtained by the direct-scan technique.

Fig. 5.50: A comparison of the tool out-of-roundness obtained by the two types of the scanning method.

5.6 Microdrill bit profiler With the development of high-density printed circuit boards (PCBs), the demands of sub-micrometer hole drilling (0.5‒0.1 mm) are increasing [65, 66]. Compared to laser drilling and electrochemical drilling, mechanical drilling is the most useful

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Fig. 5.51: Definition of the run-out of a microdrill bit.

machining method for deep through-hole drilling in practice [67, 68]. To satisfy the required dimensions of holes on PCBs, micro drill bits are required. Similar to a large-scale drill bit, a micro drill bit is composed of three parts, which are the point part, the body part and the shank part. The run-out is caused by the axial straightness error between the shank part and the body part as shown in Fig. 5.51. Different distances are caused by the axial straightness error from the axis of the shank part to the two margins shown in the figure, where the distance from the axis of the shank part to each margin is the rotation radius of the margin. When the microdrill bit is rotated, the different rotation radii of the two margins cause different rotation paths. The run-out of the micro drill bit is defined as the difference between the rotation paths of the two margins. Compared with large-scale drill bits, it is more difficult to measure the run-out of micro drill bit. In this section, a microdrill bit profiler composed of a laser scan micrometer (LSM) and a concentricity gauge. Although LSM-based measurement system is already being used in industries for measurement of the run-out of micro drill bits, the spindle error motion of the concentricity gauge is relatively large, which significantly influence the measurement accuracy of run-out. The influence of the spindle error motion of the concentricity gauge is thus treated in the microdrill bit profiler to improve the measurement accuracy of run-out.

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Fig. 5.52: Schematic of the microdrill bit profiler.

A schematic of the microdrill bit profiler is shown in Fig. 5.52 [69]. The shank part of the microdrill bit is supported by a pair of main rollers and a top roller of the concentricity gauge. Two timing belts connect the shaft of the drive handle and the shafts of the two main rollers. By manually rotating the drive handle, the two main rollers can be rotated by the two timing belts to drive the micro drill bit by the friction between the surfaces of the shank part and the main rollers. Since the diameters of the shaft of the drive handle and the shafts of the main rollers are the same,

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the drive handle and the main rollers are synchronized rotated. It should be noted that one revolution of the micro drill bit corresponds to a fraction of one revolution of the mail rollers when the diameters of the shank part of the micro drill bit and the main rollers are different. The motion of the micro drill bit is monitored by the LSM for the measurement of the run-out, in which the LSM includes a controller and a measurement part. Figure 5.53 illustrates the measurement part of the LSM [70]. A polygonal mirror is employed to scan the laser beam in the measuring area. In the LSM, the existence of the microdrill bit will separate the measurement laser beam into three segments: top edge, bottom edge and the diameter. The top edge segment indicates the distance from the upper surface of the microdrill bit to the top scanning position. The bottom edge segment indicates the distance from the lower surface of the microdrill bit to the bottom scanning position. The diameter segment corresponds to the distance between the upper and lower surfaces of the microdrill bit; namely, the microdrill bit diameter. Within the top edge segment and the bottom edge segment, the laser beams are collected by a condenser lens, and are then obtained by a photoelectric element, while the microdrill bit blocks the laser beam within the diameter segment. Using the known LSM scan velocity, the outputs of the top edge segment and the diameter segment, as well as the bottom edge segment in each scanning cycle, can be obtained after converting the received signal of the photoelectric element to the time domain.

Fig. 5.53: Measurement principle of laser scan micrometer.

As shown in Fig. 5.51, the run-out of the microdrill bit is defined by the difference between the rotational path of the two margins. The rotation radius of each margin is defined with respect to the shank axis. The top edge segment of the LSM dented by TEdge(θ) is employed to measure the two margins for the evaluation of the run-out, where the angular position of the microdrill bit is represented as θ. Figure 5.54 shows the measurement principle. The minimum measurement output of the top edge segment in the first half of the rotation of the microdrill bit is denoted by TEdge(θj) in the measurement output on Margin 1. The subscript j denotes the circumferential index

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Fig. 5.54: Measurement principle of the run-out.

of the data sampling. In the same manner, measurement output on Margin 2 is denoted by TEdge(θj + π). The run-out can be calculated as the absolute value of the difference in the two margins, as follows:   Run − out = TEdge ðθj Þ − TEdge ðθj + πÞ

(5:11)

Meanwhile, the Y-position of the microdrill bit can be changed by the spindle error motion of the concentricity gauge, as shown in Fig. 5.55. Denoting the spindle error motion as eSpindle_1,2, the run-out under the influence of eSpindle_1,2 can be rewritten as follows:   Run − out = TEdge ðθj Þ − TEdge ðθj + πÞ + espindle 1, 2  (5:12) As can be seen in the above equation, eSpindle_1,2 could directly affect the measurement of the run-out. The spindle error motion of the concentricity gauge is thus required to be isolated to guarantee adequate measurement accuracy. Meanwhile, as described above, under the condition of fine positioning of the concentricity gauge with the minimized error motion, the microdrill bit can be measured with enough accuracy. For the fine positioning of the concentricity gauge, it is thus necessary to evaluate the spindle error motion of the concentricity gauge. The reversal method is a well-known method [29, 71–74] employed for the evaluation of error motions of slides or spindles in machine tools. Two displacement probes and a measuring artifact are typically used to evaluate the spindle error motion, in which the two displacement probes are arranged on two opposite sides of the artifact. A large-diameter artifact is often used in operation for the measurement of the spindle error motion to address the measurement error induced by the misalignment error of the two displacement probes. However, due to the limited gripping capability of the concentricity gauge, it is not possible to employ a large-diameter

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Fig. 5.55: Spindle error motion influencing the run-out measurement.

artifact in the proposed method. In this study, for the achievement of the accurate assessment of the spindle error motion in the concentricity gauge, the top edge segment and the bottom edge segment of the LSM are employed instead of using two additional displacement probes to detect the two opposing sides of a measurement artifact in a small diameter. The misalignment error of the top edge segment and the bottom edge segment is negligibly small on the level of 10 nm, due to the wellcalibrated linearity of the LSM over the whole measuring range; this contributes to the case of employing an artifact with a small diameter. Figures 5.56 and 5.57 show a schematic of the measurement of spindle error motion of the concentricity gauge by the LSM. In the figure, eSpindle(θM) denotes the spindle error motion where θM is the angular position of the main rollers. As the measurement artefact, a pin gauge in a small diameter is employed. The artefact is mounted on the concentricity gauge. The upper surface and the lower surface of the pin gauge are determined by the top and lower edge segments of the LSM at the same time. The top edge segment is denoted by TEdge, while the bottom edge segment is denoted by BEdge. As shown in Fig. 5.58, before and after a 180° reversal operation of the pin gauge, the spindle error motion of the concentricity gauge is measured. TEdge_Before and BEdge_Before before the reversal operation can be represented as follows: TEdge

Before ðθM Þ = eForm ðθp Þ + eSpindle ðθM Þ

(5:13)

TEdge

Before ðθM Þ = eForm ðθp

(5:14)

+ πÞ − eSpindle ðθM Þ

In the above equation, the form error of the pin gauge including the influences of out-of-roundness error as well as the straightness error is denoted by eForm(θp), where the angular position of the pin gauge is represented as θp. In the same

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manner, the measurement outputs of the two edge segments after 180 rotation of the pin gauge can be expressed as follows: TEdge

After ðθM Þ = eForm ðθp

+ πÞ + eSpindle ðθd Þ

TEdge

Afger ðθM Þ = eForm ðθp Þ − eSpindle ðθd Þ

(5:15) (5:16)

From the above equations, eSpindle(θM) can then be obtained as follows while eliminating the influence of eForm(θp): eSpindle ðθM Þ =

1 TEdge 4



Before ðθM Þ − BEdge Before ðθM Þ + TEdge After ðθM Þ − BEdge Before ðθM Þ

(5:17)

Fig. 5.56: A schematic of the evaluation of the spindle error motion of a concentricity gauge. (in 3D view)

Fig. 5.57: A schematic of the evaluation of the spindle error motion of a concentricity gauge. (in XY view)

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Fig. 5.58: A principle of the evaluation of spindle error motion.

The evaluations of the run-out of the microdrill bit and the spindle error motion of the concentricity gauge were carried out based on the method shown in Fig. 5.58. The concentricity gauge (JSLP-10C, Universal Punch Corp.) had two main rollers with a diameter of and a top roller with a diameter of 6.3 mm. A commercial laser scanning micrometer (LSM-902/6900, Mitutoyo Corp.) with a measurement range of 32 mm was employed in the setup. The measurement throughput of the LSM was 800 cycles/s. Through the serial interface connection, the measurement data of the LSM was transferred to a 64-bit personal computer. Photographs of the pin gauge and the microdrill bit employed in the following experiments are shown in Figs. 5.59 and 5.60. In order to measure the spindle error motion of the concentricity gauge, the pin gauge with a diameter and a length of 3 and 50 mm, respectively, was used as the measurement artifact. The length and the diameter of the shank part of the microdrill bit were 26 and 3 mm, respectively, while those of the microdrill bit were 24 and 0.75 mm, respectively. The spindle error motion eSpindle(θM) of the concentricity gauge was evaluated. During the experiment, the concentricity gauge was operated manually, while the angular position θM was monitored by using the scale on the manual handle on the concentricity gauge. eSpindle(θM) was evaluated in a constant angular interval of 5.625°. Three repetitive trials were made for the evaluation of eSpindle(θM) over one rotation. Figures 5.61 and 5.62 show TEdge and BEdge obtained before and after the reversal operation. Figure 5.63 shows the e Spindle(θM) obtained through the arithmetic operation based on eq. (5.17). A mean peak-to-valley (PV) of eSpindle(θM)

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Fig. 5.59: A pin gauge.

Fig. 5.60: A microdrill bit.

was evaluated to be 7.10 μm with a repeatability of 2.54 μm; this result contained the influence of the manual operation of the concentricity gauge. In addition, the non-repeatable runout of the two main rollers could affect the measurement results.

Fig. 5.61: Sensor readings before the reversing operation of the pin gauge.

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Fig. 5.62: Sensor readings after the reversing operation of the pin gauge.

Fig. 5.63: Measurement results of the spindle error motion.

As can be seen in the results, a minimum value of 1.86 μm (PV) was observed at the angular positions from 70° to 110°. This angular position range corresponds to the approximately one-tenth revolution of the concentricity gauge; this is comparable to the angular region required for the evaluation of the whole circumference of the microdrill bit having a diameter and a length of 3 and 22 mm, respectively. The microdrill bit was calibrated at the defined fine location of the concentricity gauge in order to distinguish eSpindle(θM) from the concentricity gauge to achieve the necessary measurement accuracy. For measurement of the two margins by the top segment of the LSM, the microdrill bit was mounted on the concentricity gauge in such a way that it can be rotated over one revolution by the fine angular position of the rollers in the concentricity gauge with a small error motion. After that, for the second test, the microdrill bit was rotated back to the beginning of the finer location. Ten repetitive trials were made. Figures 5.64 and 5.65 summarize the results.

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Repeatability of measurement for Margins 1 and 2 were evaluated to be 0.51 and 0.96 μm, respectively. The mean of the run-out of the microdrill bit was evaluated to be 13.57 μm on the basis of eq. (5.11). Figure 5.65 shows the run-out obtained at each trial. A PV and a standard deviation of the observed run-out were evaluated to be 1.16 and 0.40 μm, respectively, through ten repetitive trials. Uneven rotational speed, the thermal drift of the scanning beam in the LSM during the measurement could be potential contributors to the observed repeatability.

Fig. 5.64: Measured two margins.

Fig. 5.65: Measured run-out at the fine position in the concentricity gauge.

5.7 Mixed-probe profiler In an optical scanning profiler, an optical probe is scanned over the specimen surface by using a scanning stage. A reading error in the probe output will be caused by the motion errors (scan errors) of the scanning stage along the probe axis [2]. The multi-probe method, based on the concept of error-separation is an effective way to compensate for both repeatable and non-repeatable scanning errors, including

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thermal errors and vibration [3]. In a multi-probe method, a number of probes are employed to scan the specimen surface simultaneously, from which a set of simultaneous equations are established using the outputs of the probes. Each probe output contains a surface profile term and a scan error term. Since the probes detect different points on the specimen surface, the surface profile term has a different phase in each probe output. The scan error can be removed in a differential output based on the equations, from which the surface profile can be reconstructed. The two-displacementprobe method [75, 76], two-slope-probe method [77], three-displacement-probe method [28, 78] are some of the conventional multi-probe methods. The measurable spatial wavelength by a multi-probe system is limited by its probe arrangement. It is typically lower than that restricted by the probe size, which is the case in a single-probe system. This is because that certain harmonic components of the specimen surface profile will be lost in the profile reconstruction process, which is referred to as the harmonic loss problem. This is one of the significant shortcomings of conventional multi-probe methods. This section presents a mixed-probe profiler based on the mixed method [17–19], in which two displacement probes and a slope probe are employed to completely solve the harmonic loss problem in straightness measurement.

Fig. 5.66: Principle of the conventional two-displacement-probe method.

At first, the principle of the mixed-probe method is demonstrated in comparison with the conventional multi-probe methods. Figure 5.66 gives the principle of the conventional two-displacement-probe method [75, 76]. Two displacement probes are set on a scanning stage that can scan a surface when the stage moves along the X-direction. The profile of the surface is described by function h(x). Let P0 be the representative point of the stage, and let d1 and d3 be the distances from point P0 to the probes, respectively. The height positions of the specimen surface at points A

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Fig. 5.67: Principle of the conventional two-slope-probe method.

and C are detected by probes 1 and 3, respectively. The outputs of the displacement probes can be expressed as follows: m1 ðxÞ = hðx + d1 Þ + eZ ðθÞ + d1 eP ðθÞ

(5:18)

m3 ðxÞ = hðx + d3 Þ + eZ ðθÞ + d3 eP ðθÞ

(5:19)

where eZ(x) the translational error of the scanning stage along the Z-axis, and eP(x) is the pitching error of the stage about the Y-axis. The differential output m2D(x) of the two-displacement-probe method can be denoted as m2D ðxÞ = m1 ðxÞ − m3 ðxÞ = hðx + d1 Þ − hðx + d3 Þ + ðd1 − d3 ÞeP ðxÞ

(5:20)

where the scan error component eZ(x) is canceled. However, the scan error component eP(x) remains in the differential output. Assuming that eP(x) is sufficiently small gives m2D ðxÞ ≈ hðx + d1 Þ − hðx + d3 Þ

(5:21)

Figure 5.67 shows the principle of the conventional two-slope-probe method [77]. Two-slope probes are set on a scanning stage to detect the local slopes of the specimen surface at points A and C, respectively. The outputs of the slope probes can be expressed as follows: μ1 ðxÞ = h′ðx + d1 Þ + eP ðθÞ

(5:22)

μ3 ðxÞ = h′ðx + d3 Þ + eP ðθÞ

(5:23)

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The differential output m2S(x) of the two-slope-probe method can be denoted as follows: m2S ðxÞ = μ1 ðxÞ − μ3 ðxÞ = h′ðx + d1 Þ − h′ðx + d3 Þ

(5:24)

where both eZ(x) and eP(x) are canceled.

Fig. 5.68: Principle of the conventional three-displacement-probe method.

Figure 5.68 shows the principle of the conventional three-displacement-probe method [28, 78]. Compared with the two-displacement-probe method shown in Fig. 5.66, one displacement probe (probe 2) is added to detect the height position of the specimen surface at Point B. The probe output can be expressed as follows: m2 ðxÞ = hðx + d2 Þ + eZ ðθÞ + d2 eP ðθÞ

(5:25)

The differential output m3D(x) of the three-displacement-probe method can be obtained as follows from eqs. (5.18), (5.19) and (5.25): m3D ðxÞ = m1 ðxÞ − m3 ðxÞ − ðd1 − d3 Þ

m3 ð x Þ − m 2 ð x Þ d3 − d2

hðx + d3 Þ − hðx + d2 Þ = hðx + d1 Þ − hðx + d3 Þ − ðd1 − d3 Þ d3 − d2

(5:26)

where the scan errors eZ(x) and eP(x) are canceled. To reconstruct the surface profile h(x) from the differential output of each multiprobe method, the equation of the differential output can be treated as a system, in which h(x) is treated as the input and the differential output as the output of the

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system. For example, for the differential output of the two-displacement-probe method in eq. (5.21), h(x) and m2D(x) as the input and output, respectively. The relationship between the input h(x) and the output m2D(x) can be defined by the following transfer function of the two-displacement-probe method: G2D ð f Þ =

M2D ð f Þ Hð f Þ

(5:27)

where f is the spatial frequency. M2D(f) and H(f) are the Fourier transforms of m2D(x) and h(x), respectively. It should be noted that the transfer function shows the sensitivity of the system to the surface profile in the spatial frequency domain. Since H(f) can be obtained from M2D(f) and G2D(f) from eq. (5.27), h(x) can then be evaluated by the inverse fast Fourier transform (IFFT) of H(f). To identify the transfer function G2D(f) of the two-displacement-probe method, h(x) and H(f) are assumed as follows: hðxÞ = H ð f Þ = ej2πfx

(5:28)

Substituting eq. (5.28) into eq. (5.21) gives M2D ð f Þ = m2D ðxÞ = ej2πf ðx + d1 Þ − ej2πf ðx + d3 Þ = ej2πfx ðej2πfd1 − ej2πfd3 Þ = H ð f Þðej2πfd1 − ej2πfd3 Þ (5:29) For simplicity, assume the representative P0 is at the center of the two probes in Fig. 5.66. G2D(ω) can be evaluated as follows: G2D ð f Þ = j2 sin πfD

(5:30)

where D is the distance between the two probes. Similarly, the transfer function of the two-slope-probe method where P0 is set at the center of the two probes in Fig. 5.67 can be evaluated as follows: G2S ð f Þ = − 4πf sin πfD

(5:31)

The transfer function G3D(ω) of the three-displacement-probe method is obtained in eq. (5.32). In this case, P0 is set at the position of probe 2, which is at the center of probes 1 and 3 in Fig. 5.68, which is referred to as the symmetric probe arrangement G3D ð f Þ = 2ðcos πfD − 1Þ

(5:32)

The amplitude of G3D(ω) is shown in Fig. 5.69. It can be seen that the amplitude of G3D(ω) reaches zero at spatial frequencies of 1/2nD, where n is an integer. At such frequencies, which are referred to as the zero points, the output of the system, that is, the differential output of the three-displacement-probe method becomes zero. This means that the corresponding harmonic components cannot be measured by the three-displacement-probe method, which is referred to as the harmonic loss problem. Such a problem is also inherent in the two-displacement-probe method

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Fig. 5.69: The transfer function of the three-displacement-probe method for symmetric probe arrangement.

and the two-slope-probe method, as well as any other types of multi-probe systems where the same types of probes (displacement probe or slope probe) are employed. Figure 5.70 shows the principle of the mixed-probe method. One of the displacement probes in the three-displacement-probe method (probe 2 in Fig. 5.68) is replaced with a slope probe. The mixed-probe method is unique from the conventional multiprobe method from the point of view of using both displacement probe and slope probe. Let the output of the slope probe be μ2(x), then

Fig. 5.70: Principle of the mixed-probe method.

5.7 Mixed-probe profiler

μ2 ðxÞ = h′ðx + d2 Þ + eP ðθÞ

253

(5:33)

The differential output mm(x) of the mixed-probe method, in which the scan errors eZ(x) and eP(x) are canceled, can be denoted as mM ðθÞ = m1 ðxÞ − m3 ðxÞ − ðd1 − d3 Þμ2 ðxÞ = hðx + d1 Þ − hðx + d3 Þ − ðd1 − d3 Þh′ðx + d2 Þ

(5:34)

(d1, d2, d3) = (D, 0.5D, 0)

Fig. 5.71: The transfer function of three-displacement-probe method at symmetric probe arrangement.

The transfer function GM(f) of the mixed method can be expressed as GM ðf Þ =

MM ðf Þ = ej2πf ðx + d1 Þ − ej2πf ðx + d3 Þ − j2πf ðd1 − d3 Þej2πf ðx + d2 Þ Hðf Þ

(5:35)

When probe 2 is located at the center of probes 1 and 3 in Fig. 5.70, that is, in the symmetric probe arrangement, GM(f) is shown in Fig. 5.71. It can be seen that the amplitude of the GM(f), which is the sensitivity of the mixed-probe method to the surface profile in the frequency domain, increases with the increase of frequency f, and there are no frequencies where the sensitivity becomes zero. This indicates that the mixed-probe method can theoretically measure all the frequency components of the surface profile without losing any harmonics, which is the most significant improvement from the conventional multi-probe method. Figure 5.72 shows the amplitude of the GM(f) in an asymmetric probe arrangement where the position of the slope

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Chapter 5 In-line and on-machine surface profiler

probe is overlapped with one of the displacement probes. It can be seen that the sensitivities of the mixed-probe method in the low-frequency range are much higher than those of the symmetric probe arrangement shown in Fig. 5.71, while the sensitivities in the higher frequency range are kept the same, which means that the asymmetric probe arrangement is better than the symmetric probe arrangement. Figure 5.73 shows a schematic of an optical probe that detects the height and local slope at the same point on the specimen surface [19]. Figure 5.74 shows a 3D drawing of a sensor for a mixed-probe profiler that combines two of the probes shown in Fig. 5.73. The sensor can detect the heights and local slopes at two points. Figure 5.75 shows a picture of the constructed mixed-probe profiler. The sensor has been applied to onmachine measurement of mirror profiles [18, 19]. (d1, d2, d3) = (D, 0, 0)

Fig. 5.72: The transfer function of the three-displacement-probe method at asymmetric probe arrangement.

5.7 Mixed-probe profiler

Fig. 5.73: Schematic of an optical probe that can detect the height and local slope at the same point.

Fig. 5.74: Three-dimensional drawing of the mixed-probe profiler.

255

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Chapter 5 In-line and on-machine surface profiler

Fig. 5.75: Picture of the mixed-probe profiler.

5.8 Summary A number of optical scanning profilers have been presented for the measurement of specimen surfaces in in-line and on-machine conditions. At first, a methodology was presented for the measurement of cutting errors in FS gears of harmonic speed reducers through the use of two laser displacement probes instead of Van Keuren wires and a micrometer. An averaging operation of the data over one rotation of the gear stage was found to remove the influence of eccentricity and rotation errors. The use of compressed air proved effective in removing the oil from the gear surface, enabling the laser probes to retain their functionality during practical use. A number of optical scanning profiles were then presented for measurement of small parts. A method was presented to measure the height of the cutting edge with respect to a reference surface on the tool by using a laser displacement sensor. In this method, the use of the displacement output when the light intensity output reaches the maximum enables to measure the height of cutting edge whose width is smaller than that of the laser spot. The measurement of a test piece confirmed the basic performance of the proposed method. The experimental results of a rotary die cutter have demonstrated the applicability of the proposed method for measurement of actual cutting tools. It has been verified that the repeatability of measurements by the proposed method was less than 0.5 μm for different scanning speeds ranging from 1 to 4 mm/s. An FTS-based fabrication system and an optical slope sensor were then presented for generation and measurement of sinusoidal microstructures on a cylindrical workpiece, which is used as the master grid of a surface encoder for replication. The FTS unit has a compact size (35 mm diameter × 80 mm

References

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length) and a high bandwidth (3 kHz). A cylindrical lens has been integrated into the slope sensor so that slopes of the sinusoidal structures can be detected without the influence of curvature of the cylindrical workpiece. Meanwhile, an optical method for edge contour measurement by using a micro-optical probe was presented to evaluate large-scale tool cutting edge over the measurement range of several millimeters in a short time. An optical setup for the optical micro probe was developed, and its feasibility on the edge contour measurement was investigated in experiments. Finally, an error-separation method has been proposed to improve the measurement accuracy of the run-out of microdrill bit by using a commercial LSM measurement system. The LSM and a small-diameter pin gauge were employed to measure the spindle error motion of the concentricity gauge. Based on the measurement result of the spindle error motion, a fine position of the concentricity gauge, which had the smallest spindle error motion, has been determined. The measurement of the microdrill bit has been carried out at the fine position. In the last part of this chapter, the mixed-probe surface profiler has been presented for removing the scan errors of the scanning stage based on the concept of errorseparation. Differing from the conventional multi-probe methods using the same type of probes (displacement probes or slope probes), the mixed-probe method, which employs two displacement probes and one slope probe, can completely solve the harmonic loss problem inherent in conventional multi-probe methods so that all the necessary spatial frequency components of the specimen surface profile can be measured.

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[57] Asai S, Taguchi Y, Horio K, Kasai T, Kobayashi A. Measuring the very small cutting-edge radius for a diamond tool using a new kind of sem having two detectors. CIRP Ann – Manuf Technol 1990, 39, 1, 85–88. [58] Lane BM, Dow TA, Scattergood R. Thermo-chemical wear model and worn tool shapes for single-crystal diamond tools cutting steel. Wear 2013, 300, 1–2, 216–224. [59] Lucca DA, Seo YW, Komanduri R. Effect of tool edge geometry on energy dissipation in ultraprecision machining. CIRP Ann – Manuf Technol 1993, 42, 1, 83–86. [60] Born DK, Goodman WA. An empirical survey on the influence of machining parameters on tool wear in diamond turning of large single-crystal silicon optics. Precis Eng 2001, 25, 4, 247–257. [61] Gao W, Motoki T, Kiyono S. Nanometer edge profile measurement of diamond cutting tools by atomic force microscope with optical alignment sensor. Precis Eng 2006, 30, 4, 396–405. [62] Jang S, Asai T, Shimizu Y, Gao W. Optical analysis of an optical probe for three-dimensional position detection of micro-objects. Int J Autom Technol 2011, 5, 6, 862–865. [63] Gao W, Asai T, Arai Y. Precision and fast measurement of 3D cutting edge profiles of single point diamond micro-tools. CIRP Ann – Manuf Technol 2009, 58, 1, 451–454. [64] Grosjean T, Courjon D. Smallest focal spots. Opt Commun 2007, 272, 2, 314–319. [65] LaDou J. Printed circuit board industry. Int J Hyg Environ Health 2006, 209, 3, 211–219. [66] Nguyen NT, Huang X. Miniature valveless pumps based on printed circuit board technique. Sen Actuat A Phys 2001, 88, 2, 104–111. [67] Sen M, Shan HS. A review of electrochemical macro- to micro-hole drilling processes. Int J Mach Tools Manuf 2005, 45, 2, 137–152. [68] Rajurkar KP, Sundaram MM, Malshe AP Review of electrochemical and electrodischarge machining. Procedia CIRP, Vol 6. Elsevier B.V., 2013, 13–26. [69] Niu Z, Chen Y-L, Shimizu Y, Matsukuma H, Gao W. Error separation method for precision measurement of the run-out of a microdrill bit by using a laser scan micrometer measurement system. J Manuf Mater Proc 2018, 2, 1, 4. [70] Combines High-Rate Scanning with Highly Accurate Measurement. (Accessed November 29, 2020, at www.mitutoyo.jp) [71] Gao W, Lee JC, Arai Y, Noh YJ, Hwang JH, Park CH. Measurement of slide error of an ultraprecision diamond turning machine by using a rotating cylinder workpiece. Int J Mach Tools Manuf 2010, 50, 4, 404–410. [72] Lee J, Gao W, Shimizu Y, Hwang J, Oh JSJS, Park CHCH. Spindle error motion measurement of a large precision roll lathe. Int J Precis Eng Manuf 2012, 13, 6, 861–867. [73] Niu Z, Chen YL, Matsuura D, Lee JC, Kobayashi R, Shimizu Y, Ito S, Gao W, Oh JS, Park CH. Precision measurement of Z-slide vertical error motion of an ultra-precision lathe by using three-probe method. Int J Precis Eng Manuf 2017, 18, 5, 651–660. [74] Chen Y-L, Niu Z, Matsuura D, Lee JC, Shimizu Y, Gao W, Oh JS, Park CH. Implementation and verification of a four-probe motion error measurement system for a large-scale roll lathe used in hybrid manufacturing. Meas Sci Technol 2017, 28, 10, 105004. [75] Tanaka H, Tozawa K, Sato H, O-hori M, Sekiguchi H, Taniguchi N. Application of a new straightness measurement method to large machine tool. CIRP Ann – Manuf Technol 1981, 30, 1, 455–459. [76] Kiyono S, Gao W. Profile measurement of machined surface with a new differential method. Precis Eng 1994, 16, 3, 212–218. [77] Kiyono S, Asakawa Y, Inamoto M, Kamada O. A differential laser autocollimation probe for on-machine measurement. Precis Eng 1993, 15, 2, 68–76. [78] Aoki Y, Ozono S. On a new method of roundness measurement based on the three-point method. J Japan Soc Precis Eng 1966, 32, 383, 831–836.

Chapter 6 Fabrication and calibration of scale grating 6.1 Introduction One of the primary components in modern optical measuring instruments, such as spectrometers and/or optical encoders, is a diffraction grating [1]. Interference signals produced by superimposing diffracted light rays from the scale are utilized for the position sensing [2–4]. By using the pattern structures on the scale grating as scale graduations, measurements can be carried out. The measurement resolution of the state-of -the-art linear encoder can achieve several tens of pm with the use of the signal interpolation technique. The measurement range of a linear encoder, mainly determined by the length of the grating, can reach several tens of meters [5]. In addition, for simultaneous measurement of multi-degree-of-freedom displacement, multi-axis encoders using two-dimensional (2D) grating structures as a scale have also been developed [6– 9]. The 2D gratings to be used as their scales are needed to have the following characteristics in order to realize a high measuring resolution and accuracy of the multi-axis encoders: First, 2D grating systems that have standardized grating intervals need to be fabricated. Second, for better signal-to-noise ratios of the interference signals, diffraction efficiencies of both the negative and positive diffraction beams need to be consistent. Finally, in order to obtain more improved measurement resolution, the grating period is required to be as short as possible. Laser interference lithography (LIL) is a promising approach for the fabrication of 2D grating structures [8, 10, 11], among many fabrication methods for pattern structures. A laser beam is separated into two beams by amplitude division or wavefront division in LIL. The two coherent beams will be superimposed to produce interference fringes that can be used for pattern exposure. While several optical configurations have been developed so far, due to its simple optical configuration [9], the Lloyd’s mirror interferometer is a good candidate for manufacturing grating structures. One-dimensional grating structures can be created by the traditional one-axis Lloyd’s mirror interferometer, which has a mirror oriented perpendicularly with respect to a substrate. In addition, 2D grating structures can be fabricated with a second exposure phase after 90° rotation of the grating substrate. However, in the second exposure, the grating structures produced in the first exposure will be affected by the background light, and the depths of the grating structures in the Xand Y-directions will be different. As a consequence, both the negative and positive diffraction beams would not be compatible with diffraction efficiencies. As it specifically impacts the signal-to-noise ratio of the interference signals, this is a fatal concern for the use in an encoder system. Several optical configurations that can produce 2D grating structures at a single exposure have been developed to solve the problem induced by the two-step exposure [11, 12, 21–24, 13–20]. Among them, https://doi.org/10.1515/9783110542363-006

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orthogonal 2D pattern structures with a symmetric profile can be generated by an orthogonal two-axis Lloyd’s mirror interferometer [13–15, 23, 24], and a non-orthogonal two-axis Lloyd’s mirror interferometer [11, 12, 19, 20]. Sections 6.2 and 6.3 treat these interferometers for the fabrication of 2D grating pattern structures. Meanwhile, the evaluation/calibration of a scale grating is another important task to assure the measurement accuracy of an encoder system, since the accuracy of the grating pitch directly affects the measurement accuracy of the linear encoders [2]. Many attempts have thus been made to measure the pitch variation of a scale grating [25–27] so far. For the determination of the pitch of a small area scale grating with several tens of picometers uncertainty, a critical-dimension scanning electron microscope (CD-SEM) and a critical-dimension atomic force microscope (CD-AFM) can be used [28]. Especially, the three-dimensional (3D) profile data of the pattern structures of the grating can be obtained by CD-AFMs. However, owing to their small measurement area and low scanning speed, it is not a practical way for these measuring instruments to measure the entire length of a linear scale [29]. On the other hand, the whole length of a linear scale can be measured by a linear scale comparator with high throughput and a sub-nanometric resolution [30–34], although the optical method-based edge detection technique prevents the linear scale comparator from measuring a linear scale with a grating pitch smaller than 4 μm. Another disadvantage of the linear scale comparator is a complex system that needs a highly controlled measurement environment. Therefore, it is beneficial to develop a system capable of measuring the pitch variation of a linear scale whose grating pitch is in the order of a micrometer or sub-micrometer over its entire length. In responding to the background described above, several methods employing the wavefronts of the first-order diffracted beams from the grating under evaluation have been proposed [16, 25, 26, 35– 38]. In Section 6.4, a method developed on the basis of the laser autocollimator for robust measurement of the grating pitch of a linear scale is explained. In Section 6.5, a method capable of evaluating the pitch deviation of a 2D scale grating with a commercial Fizeau form interferometer is introduced.

6.2 Orthogonal two-axis Lloyd’s mirror interferometer The conventional one-axis Lloyd’s mirror interferometer [39], which can be employed for the fabrication of one-dimensional grating structures, is the basis of the optical configuration of the orthogonal two-axis Lloyd’s mirror interferometer. Schematics of the conventional one-axis Lloyd’s mirror interferometer and the orthogonal two-axis Lloyd’s mirror interferometer are indicated in Figs. 6.1 and 6.2. Another mirror is added to the conventional one-axis Lloyd’s mirror interferometer to construct the optical setup for the orthogonal two-axis Lloyd’s mirror interferometer. The normal of the additional mirror, which is referred to as the Y-mirror in this section, is set to be perpendicular to both the normal of the substrate and the X-mirror. A collimated

6.2 Orthogonal two-axis Lloyd’s mirror interferometer

263

laser light can be split into the following five beams by using these two mirrors to generate interference fringe fields: Beam1: Projected onto the substructure directly Beam2: Projected onto the substrate after being reflected by the X-mirror Beam3: Projected onto the substrate after being reflected by the Y-mirror Beam4: Projected onto the substrate after being reflected by the X- and Y-mirrors Beam5 (4’): Projected onto the substrate after being reflected by the Y- and X-mirrors

Fig. 6.1: A schematic of the conventional one-axis Lloyd’s mirror interferometer.

Fig. 6.2: A schematic of the orthogonal two-axis Lloyd’s mirror interferometer.

It is possible to quantify the 2D fringe patterns produced by the interference between the five beams by superimposing the beams’ electric fields. The electric field of a laser beam can be represented by the following equation [13]: ~ ei expðj~ ki ·~ r + γi Þ Ei = Ei · ~

ði = 1, 2, 3, 4, 5Þ

(6:1)

! ei , γi and ki are the position vector, the real electric In the above equation, ! r , Ei, ! field amplitude, a unit vector in the polarization direction of the laser beam, the initial phase and the wave vector, respectively. Figure 6.3 shows the geometric relationship between the incident beam and the interferometer. Wave vectors of the

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Chapter 6 Fabrication and calibration of scale grating

beams in the orthogonal two-axis Lloyd’s mirror interferometer can be denoted by the following equation with the use of the angle between the direction of the incident beam and the substrate surface θi and the azimuthal angle of the laser light with respect to the X-axis ϕ [40]: 0 1 0 1 − cos θ1 cos ϕ cos θ2 cos ϕ 2π B 2π B C C ~ k2 = k1 = @ − cos θ1 sin ϕ A , ~ @ − cos θ2 sin ϕ A , λ λ − sin θ1 − sin θ2 0 1 0 1 0 1 − cos θ3 cos ϕ cos θ4 cos ϕ cos θ5 cos ϕ 2π 2π 2π ~ @ cos θ3 sin ϕ A , ~ @ cos θ4 sin ϕ A, ~ @ cos θ5 sin ϕ A k3 = k4 = k5 = λ λ λ − sin θ3 − sin θ4 − sin θ5 (6:2) In the above equations, the wavelength of the laser light is denoted by λ. I(r), the interference fringe field, can thus be expressed by the following equation: I ð~ rÞ =

5 X ,=1

E,2 + 2

5 X X

Em En~ en cos em · ~

m=2 n

nm

Mode field diameter

. ± . (@, nm)

mm

Focal length of the collimator objective

. (for λ =  nm)

mm

Resolution of detection frequency



GHz

Fig. 7.22: Relationship between the Z-directional displacement of the fiber edge and the power of each optical frequency component in the captured laser beam.

measurement method. In the experiment, a small angular displacement Δθ in a step of 3.3 arc-seconds was applied to the target mirror by using the PZT tilt stage, and variation of the spectral range from 185 to 200 THz was measured by the optical spectrum analyzer at each angular position of the mirror reflector. In the experiments, the frequency resolution bandwidth of the optical spectrum analyzer was set to be 200 GHz. According to the proposed principle, the light spots generated by the optical frequency comb are dispersed corresponding to the optical frequency by the chromatic aberration of the collimator objective. Therefore, the larger light spot interval was suitable for verification of the proposed principle. It is obvious from eq. (7.27) that the light spot interval on the detector becomes larger by increasing the initial incident angle θ0. Therefore, in this experiment, the amount of θ0 was set to be as large as possible in order to acquire a large spot interval hi. It should be noted that the angle α between the axis of the incident femtosecond laser beam and the optical axis of collimator objective and the initial incident angle θ0 were treated

354

Chapter 7 Ultrashort-pulse angle sensor

as known design parameters. Meanwhile, α will not affect the measurement as long as the measurand is the relative angular displacement of a target of interest. In the following experiments, θ0 was treated to be approximately 5.0°. Figure 7.23 shows the relationship between the angular displacement Δθ of the mirror reflector and the changes in measured light power of the four optical frequency components. Figure 7.24 shows a close-up view of the central part of Fig. 7.23. In the same manner as the result shown in Fig. 7.22, the changes in light power are normalized in Figs. 7.23 and 7.24. As shown in the figures, the angular positions giving maximum light power were found to be different for each of the optical frequency components. This was caused by the chromatic dispersion of the collimator objective. These experimental results demonstrated the feasibility of the proposed angle measurement method combining the optical frequency comb with the chromatic dispersion of the collimator objective.

Fig. 7.23: The change in power of each optical mode captured by the optical spectrum analyzer due to the angular displacement given to the mirror reflector.

Fig. 7.24: Close-up view of the central part in Fig. 7.23.

7.3 Chromatic dispersion angle sensor

355

Further detailed analyses on the data obtained in the above-mentioned experiment were also carried out to estimate a resolution of the angle measurement by the proposed method. Figure 7.25 shows the angular displacement Δθ giving maximum light power in each frequency component. In the figure, the angular displacement giving maximum light power was determined by fitting a quadratic function to the change in normalized light power of each of the frequency components. As shown in Fig. 7.25, the angle detection sensitivity in the experiment setup was evaluated to be 0.88 THz/arc-second from the slope of the linear approximation; namely, when an angular displacement of 1 arc-second is given to the mirror reflector, the change in peak optical frequency of 0.88 THz can be observed in the optical spectrum captured by the optical spectrum analyzer. From the obtained sensitivity, under the condition of a frequency resolution bandwidth of the optical spectrum analyzer of 200 GHz, the measurement resolution of the developed angle sensor was estimated to be approximately 0.23 arc-second. Although the resolution obtained by the developed prototype optical setup is lower than that expected by the conventional laser autocollimators, further optimization of the optical setup including the employment of a collimator objective having a further larger effect of chromatic dispersion is expected to improve the resolution. It should be noted that, in this chapter, the initial incident angle θ0 has been treated as a known parameter in the optical setup. In addition, the experiments have been done in a limited angular displacement due to the limitation on the developed experimental setup employing the PZT tilt stage having a small stroke for giving an angular displacement to the mirror reflector.

Fig. 7.25: Estimation of the measurement sensitivity of angular displacement by the proposed method.

356

Chapter 7 Ultrashort-pulse angle sensor

7.4 Second harmonic wave angle sensor In this section, the possibility of making use of the unique characteristic of a femtosecond laser, that is, the high peak power and high-intensity electric field, is explored for angle measurement. This characteristic is conventionally employed to generate second harmonic waves based on the nonlinear optical phenomenon [23, 24], which is well known as frequency doubling or SHG. SHG has been usually applied as a wavelength conversion method of lasers, where maximization of the output power of the second harmonic wave has attracted for various applications [25, 26]. Meanwhile, it is also known that the output power of the second harmonic wave correlates with an angle between wave vectors and a crystal optic axis [14, 23, 26, 27], which is expected to be employed for angle measurement. In the following, the principle of angle measurement based on frequency doubling with an ultrashort pulse laser is presented [28].

Fig. 7.26: Linear polarization generated by a weak input light wave.

When a light wave enters a dielectric medium, the internal charge distribution of the dielectric will be distorted by the electric field of the light wave to generate an oscillating electric dipole [23]. A dipole is composed of two charges, one plus and one minus. The plus and minus charges are separated by the input electric field E(t) of the light wave. Since E(t) is a time-dependent harmonic wave (sinusoidal wave), the two charges vibrate along a straight line in a corresponding time-dependent harmonic motion. This results in a time-dependent harmonic dipole moment. As shown in Fig. 7.26, when E(t) is small, the dipole moment per unit volume, which is called the electric polarization P(t), is proportional to E(t) as follows [23]: PðtÞ = ε0 χð1Þ EðtÞ

(7:28)

7.4 Second harmonic wave angle sensor

357

where ε0 is the vacuum permittivity and χ(1) is a dimensionless constant called the electric susceptibility. It should be noted that the position of the positive charge is assumed to be stationary in Fig. 7.26 for clarity.

Fig. 7.27: Nonlinear linear polarization generated by an input light wave with extremely high intensity.

Based on eq. (7.28), P(t) will increase linearly with the increase of E(t). However, when the medium is subjected to a very strong input light wave such as an ultrashort pulse wave that has an extremely high light intensity, typically on the order of 108 V/m, P(t) will be saturated due to the nonlinear properties of the medium and it can also be asymmetric due to the anisotropic molecular structure of the medium as shown in Fig. 7.27. Consequently, nonlinear components will be included in P(t) as follows: PðtÞ = ε0 χð1Þ EðtÞ + ε0 χð2Þ E2 ðtÞ + ε0 χð3Þ E3 ðtÞ +   

(7:29)

where χ(n) (n = 2, 3, . . .) is the nth-order nonlinear coefficient of the electric susceptibility. The nonlinear components of P(t) can then generate nonlinear optics phenomena such as the optical Kerr effect, self-phase modulation and SHG. Here the second harmonic wave generated by the second-order nonlinear coefficient χ(2) of the electric susceptibility is employed for angle measurement.

358

Chapter 7 Ultrashort-pulse angle sensor

Fig. 7.28: The fundamental and second harmonic waves propagating in a dielectric medium.

Assume the electric field E(t) of the input light wave, which is referred to as the fundamental wave, has an angular frequency ω1 and an amplitude A. E(t) can be expressed by EðtÞ = A cos ω1 t

(7:30)

The second-order nonlinear polarization can then be written as 1 1 Pð2Þ ðtÞ = ε0 χð2Þ E2 ðtÞ = ε0 χð2Þ A2 cosð2ω1 tÞ + ε0 χð2Þ A2 2 2

(7:31)

It can be seen that a second harmonic wave with an angular frequency of 2ω1 is generated in P(2)(t). The fundamental wave and the second harmonic wave interact with each other when they propagate through the medium as shown in Fig. 7.28 [29]. Figure 7.29 shows a schematic of the angle measurement method using the second harmonic waves generated by an ultrashort pulse laser. The optical setup is mainly composed of an ultrashort pulse laser source, a photodetector and a nonlinear crystal mounted on a measurement target that rotates about the Y-axis in the figure. The optic axis of the nonlinear crystal is aligned to be parallel to the XZ-plane in the figure. A collimated beam from a femtosecond laser with a beam diameter of ϕ as the fundamental wave from the laser source, which is also aligned to be parallel to the XZ-plane, is made incident to the nonlinear crystal to generate second harmonic waves. Since

7.4 Second harmonic wave angle sensor

359

Fig. 7.29: Schematic of angle measurement using the second harmonic wave generated by an ultrashort pulse laser.

the intensity of the generated second harmonic wave I2 depends on the angle θ between the optic axis of the nonlinear crystal and the axis of the incident laser beam, θ can be obtained by detecting I2 with the photodetector. Assume that the ordinary and extraordinary axes of the nonlinear crystal are aligned to be parallel to the Y- and X-axes, respectively, while the incident fundamental wave and the second harmonic wave are polarized in the Y- and X-directions, respectively. Denoting the intensity of the incident fundamental wave as I1, the relationship between I2 and θ can be expressed by the following equation [27]: I2 =

8π2 deff 2 2

no ðλ1 Þ ne ðθ, λ2 Þε0 cλ1

I2 2 1

L2 ΔkðθÞL sinc2 S 2

(7:32)

where deff is the effective nonlinear coefficient [27], ne and no are refractive indices of a negative uniaxial crystal (ne < no) for extraordinary and ordinary rays, respectively, λ1 and λ2 are wavelengths of the fundamental wave and the second harmonic wave, respectively, c is the speed of light in vacuum, S is the cross-sectional area of the collimated laser beam and L is the crystal length. Δk is a phase mismatching that can be represented as follows:

360

Chapter 7 Ultrashort-pulse angle sensor

ΔkðθÞ = 2k1 − k2 =

4π ½no ðλ1 Þ − ne ðθ, λ2 Þ λ1

(7:33)

where k1 and k2 are the magnitude of the fundamental wave vector and the second harmonic wave vector, respectively. As can be seen in eq. (7.31), I2 is proportional to sinc2[Δk(θ)L/2] and becomes maximum when Δk(θ) is zero. In the case with a uniaxial birefringent crystal such as beta-barium-borate (BBO), regarding the refractive index ellipse shown in Fig. 7.31, the angle θm referred to as the matching angle that satisfies no(λ1) = ne(θm, λ2), as well as Δk(θm) = 0, can be found; this procedure is known as the index matching [30].

Fig. 7.30: Second harmonic waves with matched phases in a negative uniaxial crystal.

As can be seen in eq. (7.32), I2, which is the sum of the second harmonic waves generated in the nonlinear crystal, is proportional to the square of I1. At a position where the incident fundamental wave generates the second harmonic wave, the two are coherent with each other. The fundamental wave continues to generate additional contributions of the second harmonic wave while propagating through the nonlinear crystal. In the case where the phase is mismatched (no(λ1) ≠ ne(θm, λ2)) as shown in Fig. 7.30, effective SHG cannot be accomplished due to the different propagating speeds of the fundamental and second harmonic waves in the crystal. Meanwhile, in the case where the phase is matched (no(λ1) = ne(θm, λ2)) as shown in Fig. 7.31, all the second harmonic waves are combined totally constructively, and I2 can be maximized. As described above, efficient SHG can be accomplished

7.4 Second harmonic wave angle sensor

361

Fig. 7.31: Second harmonic waves with mismatched phases in a negative uniaxial crystal.

by index matching. Although the angle-dependence of SHG is a well-known phenomenon, in this chapter, an attempt is made to utilize the phenomenon for detecting small angular displacement of the nonlinear crystal. Ideally, a collimated laser beam shown is expected in angle measurement. This can be realized in an SHG-based angle sensor when a femtosecond laser source with a high output power is available. Since it is difficult to make a collimated laser beam with a small beam diameter, which is typically an order of 1 mm, when the output power of a femtosecond laser is not enough, the intensity of a collimated femtosecond laser beam cannot reach the threshold intensity for SHG. In this case, it is effective to focus the laser beam in a nonlinear crystal to generate the second harmonic wave, although the applications of such an angle sensor with a focused laser beam are limited. As the second step of theoretical calculations, the response curve of the second harmonic wave power with respect to the angle under the condition of such a focused beam is investigated in the following analysis. Figure 7.32 shows a schematic of the focused laser beam inside a nonlinear crystal. The laser spot diameter 2w0 at the beam waist is expressed as follows: 2w0 =

4f λ1 πϕ

(7:34)

where ϕ is the diameter of a collimated laser beam made incident to an objective lens, f is the focal length of the objective lens and λ1 is the light wavelength of the laser beam. The Rayleigh length b/2, which is known as the distance along the propagation direction of the beam from the beam waist to the place where the area of the cross-section is doubled, can be expressed as follows:

362

Chapter 7 Ultrashort-pulse angle sensor

b=

8f 2 λ1 πϕ2

(7:35)

For the case of focusing a fundamental wave into a nonlinear crystal, the length of nonlinear crystal as L, the angle between the optic axis of the nonlinear crystal, and the laser beam axis as θ, eq. (7.32) can be modified as follows [31, 32]: I2 = KI1 2 Lk1 hðσ, ξÞ

(7:36)

where K=

8πdeff 2 n1 2 n2 ε0 cλ1 2

(7:37)

2π λ1

(7:38)

k1 =

1 σ = bΔkðθÞ 2 ξ= hðσ, ξÞ =

2π 2=3 ξ

∞ ð

−∞

(7:39)

L , b

jHðσ, ξÞj2 e − 4s2 ds =

(7:40) π2 jHðσ, ξÞj2 ξ

(7:41)

ðξ

Hðσ, ξÞ =

1 cos στ + τ sin στ 2ðz − f Þ dτ and τ = π 1 + τ2 b

(7:42)

0

The above equations are valid under the condition that the influences of the walkoff effect and the light absorption in the crystal are negligibly small [31, 32]. As can be seen in eq. (7.36), the power of the second harmonic wave depends not only on the phase mismatching Δk(θ) but also on a double of the Rayleigh length b, which is a function of three variables: λ, f and ϕ. To estimate the sensitivity of angle measurement by the proposed method based on SHG, theoretical calculations are carried out based on eq. (7.33). As the first step of theoretical calculations, an attempt is made to select a nonlinear crystal appropriate for the proposed optical angle measurement method with a specific mode-locked femtosecond laser source. The refractive index of a medium ne for an extraordinary ray is a function of two variables θ and λ, and can be expressed by the following equation:

7.4 Second harmonic wave angle sensor

363

Fig. 7.32: A schematic of second harmonic generation and focusing parameters.



sin2 θ cos2 θ + ne ðθ, λÞ = Ne 2 ðλÞ no 2 ðλÞ

 − 1= 2 (7:43)

where Ne(λ) = ne(90°, λ) and no(λ) = ne(0°, λ). It is known that the refractive index Ne(λ) or no(λ) of a transparent medium can be expressed by the following empirical equation [27]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B − Dλ2 (7:44) Ne ðλÞ, no ðλÞ = A + 2 λ −C where coefficients A, B, C and D are medium-specific constants. These parameters for the refractive indices no and Ne for BBO, lithium-iodate (LiIO3) and lithium niobate (LiNbO3) whose phase-matching angles have relatively smaller dependences on the wavelength around 1,560 nm are summarized in Tab. 7.3 [27]. According to eqs. (7.33) and (7.43), the phase-matching angle θm of a medium can be obtained as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ½no ðλ1 Þ − 2 − ½no ðλ2 Þ − 2 (7:45) θm = arcsin ½Ne ðλ1 Þ − 2 − ½no ðλ2 Þ − 2 By using the parameters summarized in Tab. 7.3, phase matching angles of BBO, LiIO3 and LiNbO3 are calculated based on eq. (7.45). Figure 7.33 shows the results. To investigate the wavelength dependence of the phase-matching angles, calculations are carried out in a wavelength range of 1,500 nm to 1,620 nm, which corresponds to the spectral width of the erbium-doped fiber laser employed in the following experiments. As can be seen in the figure, the dispersion of the phase-matching angle of BBO is found to be smaller than those of LiIO3 and LiNbO3 in the wavelength range; this wavelength-independent characteristic of the phase-matching angle of BBO is ideal for the proposed optical angle measurement method based on the SHG.

364

Chapter 7 Ultrashort-pulse angle sensor

By using the calculated phase-matching angles shown in Fig. 7.33, sinc2[Δk(θ)L/2] term in eq. (7.32) for each crystal is also calculated. Figure 7.34 shows the results, which are referred to as the response curves showing the relationship between the power of the second harmonic wave P2 and the angle θ (eq. (7.32)). In the figure, sinc2 [Δk(θ)L/2] term is calculated for the cases with wavelengths of 1,500, 1,560 and 1,620 nm, respectively. As can be seen in the figure, peaks of the curves for several wavelengths are almost overlapped with each other for the case with BBO crystal, while the peaks of the curves for the cases with LiIO 3 and LiNbO3 are not. In the proposed angle measurement method, the photodetector detects the power of second harmonic waves regardless of the wavelength. The integrated values of sinc2[Δk(θ)L/2] in the spectral range are therefore calculated to simulate the photodetector output. Figure 7.35 shows the result. The peak of the response curve for BBO crystal is several times higher than those for LiIO3 and LiNbO3. In addition, the variation of the photodetector output with BBO crystal is similar to the sinc 2[Δk(θ)L/2] curve shown in Fig. 7.34. Meanwhile, those for LiIO3 and LiNbO3 are significantly different from the sinc2[Δk(θ)L/2] curves. Here, the sensitivity of the angle measurement system is defined as the slope of second harmonic wave power to angular displacement. The relative sensitivities of the second harmonic wave to angular displacement in the range between 45 and 55% of the peaks at the left shoulders of the curves in Fig. 7.35 are shown in Tab. 7.4. BBO crystal is therefore a suitable for the optical setup with the erbium-doped fiber laser employed in the following experiments. Tab. 7.3: Parameters for the empirical equation of refractive indices. Crystal

A (–)

B (–)

C (μm)

D (μm−)

LiNbO

for no (θ = °) for Ne (θ = °)

. .

. .

. .

. .

LiIO

for no (θ = °) for Ne (θ = °)

. .

. .

. .

. .

BBO

for no (θ = °) for Ne (θ = °)

. .

. .

. .

. .

Based on the theoretical equations, calculations are also carried out to obtain the response curves for several f. The parameters used in the calculations are summarized in Tab. 7.5. Figure 7.36 shows the results. It can be seen that the response curves showed a similar trend to the result of BBO shown in Fig. 7.34 where a collimated beam with a large power is employed, indicating the possibility of angle measurement based on SHG with a focused laser beam. On the other hand, the full width at half maximum (FWHM) of the response curve in Fig. 7.6 is wider than that of BBO in Fig. 7.34). This means the sensitivity of angle measurement based on SHG with a focused laser is lower than that with a collimated beam. In Fig. 7.36, the second

7.4 Second harmonic wave angle sensor

Fig. 7.33: Wavelength dependence of phase-matching angles.

Fig. 7.34: Calculated sinc2[Δk(θ)L/2] with respect to angular displacement for a collimated femtosecond laser beam.

365

366

Chapter 7 Ultrashort-pulse angle sensor

Fig. 7.35: The integrated value of sinc2[Δk(θ)L/2] with respect to a wavelength between 1,500 and 1,620 nm. Tab. 7.4: Relative sensitivity of simulation results. Crystal

Relative sensitivity (arbitrary units)

LiIO

.

LiNbO

.

BBO

.

harmonic wave power is found to increase as the decrease of f. This was mainly due to the smaller laser spot diameter with smaller f, since the irradiance of the second harmonic wave is proportional to the square of irradiance of the fundamental wave. This can be utilized to improve the sensitivity of angle measurement based on SHG with a focused laser beam. Tab. 7.5: Parameters used in the simulation of SHG. K (W−)

P (W)

λ (nm)

L (mm)

ϕ(mm)

. × –

,

,

.

.

f (mm) , , or 

To demonstrate the feasibility of the proposed angle measurement method based on SHG, experiments were carried out by using the developed prototype optical setup, a schematic of which is shown in Fig. 7.37. It is noted that, in addition to mode-locked ultrashort pulse lasers, other pulsed laser sources such as Q-switched Nd:YAG lasers can also generate second harmonic waves. However, mode-locked ultrashort pulse lasers are more stable in output powers, and are therefore more suitable for angle measurement using second harmonic waves than other pulsed laser sources since

7.4 Second harmonic wave angle sensor

367

Fig. 7.36: Calculated second harmonic wave power with respect to angular displacement for different focal lengths of focusing lenses.

Fig. 7.37: Experimental setup for observing spectra of fundamental and second harmonic waves.

SHG is a nonlinear process associated with light intensity. The commercial modelocked femtosecond fiber laser (C-fiber, Menlo Systems) used in the previous sections was employed in Fig. 7.37. It had a central wavelength of 1,560 nm with a 15 mW output power. The pulse repetition rate was approximately 100 MHz, while the pulse duration was approximately 150 fs. The resultant peak power of the laser pulse was approximately 1,000 W. The femtosecond laser emitted from an edge of a singlemode optical fiber was collimated to be a collimated beam with a beam diameter of ϕ by a collimating lens. ϕ is 3.6 mm. Since the power of the femtosecond laser source was low and it was difficult to make a collimated beam with a small diameter, the intensity of the collimated beam is too small to make SHG. The technique of using a focused laser beam for SHG shown in Fig. 5 was employed in the experiment. For this purpose, the collimated laser beam, whose polarization direction was aligned to be parallel to the Y-axis in Fig. 7.37, was focused into a nonlinear crystal mounted on a rotary stage by an achromatic focusing lens having a focal

368

Chapter 7 Ultrashort-pulse angle sensor

length of 40 mm. A type I BBO crystal, in which the angle between its optic axis and the crystal surface was designed to be its matching angle, was employed as the nonlinear crystal. By using an objective lens, both the generated second harmonic wave and the unconverted fundamental wave were coupled into a multimode fiber having a core diameter of 50 μm. These coupled light waves were observed by an optical spectrum analyzer (AQ6370C-20, Yokogawa Electric) having a wavelength resolution of 0.02 nm. First, the spectrum of the second harmonic wave was verified. The BBO crystal was rotated by the rotary stage in a step of 48 arc-seconds, which corresponds to 500 steps of the rotary stage, to investigate the angle-dependence of the spectrum of the second harmonic wave. Figures 7.38 and 7.39 show the observed spectra of the fundamental wave and the second harmonic wave, respectively. In each figure, the integrated laser power over the observed spectrum was also plotted. As can be seen in the figure, second harmonic waves were successfully generated. In Fig. 7.39, a strong dependence of the power of the generated second harmonic wave on the angular displacement of the BBO crystal was observed indicating the possibility of angle measurement based on SHG with a focused beam for a femtosecond laser source with small power. The maximum conversion efficiency of the fundamental wave to the second harmonic wave was evaluated to be approximately 4%. The angle in which the intensity of the second harmonic wave reaches the maximum was considered to be that the incidence angle of the fundamental wave into the BBO crystal was a matching angle. It should be noted that the observed second harmonic spectra were Gaussian-like ones, while the spectrum of the fundamental wave was flat; the root cause of this difference is mainly due to the chromatic aberration of the focused fundamental light wave, the details of which are explained in the following of this section. For the proposed angle measurement method, the power of the second harmonic wave I2 is preferred to be as high as possible. Regarding eq. (7.39), a higher power I1 of the fundamental wave contributes to increasing I2. It is therefore required for the optical setup to optimize the beam focusing of the fundamental light in the BBO crystal. According to eqs. (7.37) and (7.38), the focused spot diameter of the fundamental wave, which is one of the important parameters to describe the condition of beam focusing, can be changed by the focal length f of the focusing lens. The influence of focused laser beam diameter was therefore evaluated in experiments by using several focusing lenses having different focal lengths. Figure 7.40 shows the optical setup for the proposed angle measurement method based on SHG. In the setup, the BBO crystal mounted on a precision rotary table was placed at the focal plane of the focusing lens. To reduce the influence of chromatic aberration, achromatic doublet lenses were employed as the focusing lens. A polarizer was placed just behind the BBO crystal in such a way that the transmission axis of the polarizer was set to be parallel to the X-axis so that the second harmonic light wave could transmit the polarizer, while the unconverted

7.4 Second harmonic wave angle sensor

369

Fig. 7.38: Observed fundamental wave spectra and the integrated power of the spectra intensity.

Fig. 7.39: Observed second harmonic spectra and the integrated power of the spectra intensity.

fundamental light wave could be absorbed. The transmitted second harmonic light wave was detected by a photodiode with a cutoff frequency of 30 MHz. The photocurrent signal from the photodiode was converted into the corresponding voltage signal by a trans-impedance amplifier and was captured by an oscilloscope. The angular displacement was given to the BBO crystal by the rotary table. A flat mirror

370

Chapter 7 Ultrashort-pulse angle sensor

Fig. 7.40: A schematic of the experimental setup.

Fig. 7.41: Experimental results of the second harmonic generation with angular displacements.

Fig. 7.42: PD output stability for detection of SHG.

7.4 Second harmonic wave angle sensor

371

was also mounted on the rotary table so that the angular displacement given to the BBO crystal could be verified by a commercial autocollimator employed as a reference. Experiments were carried out to evaluate the sensitivity of the developed optical setup. Figure 7.41 shows the variations of signal from the photodiode due to the angular displacement given to the BBO crystal. In the figures, the photodiode signals obtained by the setups with different focal lenses having different focal lengths were plotted. As predicted in the theoretical calculation results, the power of the second harmonic wave was found to be higher with the decrease of focal length f. In addition, the observed powers of the second harmonic waves showed similar angular dependence compared with the simulation results. Meanwhile, the measurable angle range, in which the variation of the output signal from the PD can be observed, was found to be wider by several times compared with the range predicted in the simulation results. The sensitivity was evaluated as the steepest region of these curves. The results are summarized in Tab. 7.6. Figure 7.42 shows a typical noise component in the output signal from the PD in the case with the focusing lens having a focal length f of 75 mm. The results of the noise level (2σ) are also summarized in Tab. 7.6. From the obtained sensitivities and noise components, in each case, a resolution of the developed angle measurement method was evaluated as the ratio of the noise component (2σ) to the sensitivity. From the above results, a resolution of approximately 0.4 arc-second was estimated to be achieved by the developed setup.

Fig. 7.43: Output of the angle measurement system versus the applied angular displacement.

Experiments were further extended to verify the resolution of the developed setup. The BBO crystal was rotated by the rotary table in a step of 0.4 arc-second, while the output signal from the PD was monitored. Figure 7.43 shows the variation of PD output observed during the experiment. In the figure, the output signal from the laser autocollimator is also plotted. As can be seen in the figure, it was verified that the developed angle measurement system could distinguish the given angular displacement. Meanwhile, as shown in Figs. 7.36 and 7.41, the sensitivity of the developed angle measurement system observed in the above experiments was lower than

372

Chapter 7 Ultrashort-pulse angle sensor

Fig. 7.44: The system for ray tracing.

Tab. 7.6: Specification of the presented angle measurement system. Focal length of the focusing lens(mm)

Noise level(mV)

Sensitivity (mV/arc-second)

Resolution (arc-second)

Resolution (arc-second)



.

.

.

.



.

.

.

.



.

.

.

.

Tab. 7.7: Specification of the lens and the position of the BBO. Focal length ofthe focusing lens(mm)

t(mm)

t(mm)

R(mm)

R(mm)



.

.

.

.

.

.



.

.

.

.

.

.



.

.

.

.

R(mm)

.

d(mm)

.

that predicted in theoretical calculation results. One of the reasons for this difference is considered to be due to the chromatic aberration of the focusing lens. To investigate the influence of chromatic aberration, ray tracing is carried out for the developed optical setup. The optical model used for the ray tracing is shown in Fig. 7.44. In the model, an achromatic doublet lens composed of an N-LAK22 substrate and an N-SF6 substrate is employed as the focusing lens, where the data of ref. [33] is used for Sellmeier equations of these glasses. The origin of the ray tracing is set at the center of the surface S1. The lens parameters are summarized in Tab. 7.7. A ray

7.4 Second harmonic wave angle sensor

373

Fig. 7.45: The results of ray tracing for Δθ = 0°.

Fig. 7.46: The results of ray tracing for Δθ = 1°.

Fig. 7.47: The results of ray tracing without the BBO crystal.

whose incidence axis is x0 from Z-axis is refracted at Surfaces 1, 2 and 3 in the figure, and is then is refracted at Surface 4 of the nonlinear crystal. The result of the ray tracing for the cases with a lens focal length of 40 mm is shown in Fig. 7.45 for the case with Δθ = 0°. In the figure, the results with wavelengths of 1,500, 1,560 and 1,620 nm

374

Chapter 7 Ultrashort-pulse angle sensor

are plotted. Ray tracing is also carried out for the case with Δθ = 1°. The result is shown in Fig. 7.46. To clarify the influences of refraction at Surface 4, ray tracing is also carried out for the case without BBO crystal. The result is shown in Fig. 7.47. To evaluate the influences of chromatic aberration of the focusing lens, a parameter ΔZ, which is the Z-directional distance between the focal positions of the rays with wavelengths of 1,500 and 1,620 nm, is introduced. Table 7.8 summarizes ΔZ for the cases of f = 40, 75 and 150 mm. The difference between the sensitivities observed in the experiments and those predicted in theoretical calculation results can be explained by the influences of the chromatic aberration summarized in the table: in the experiments, the second harmonic wave was effectively generated around the beam waist position of the light with a wavelength of 1,560 nm (Z1560 nm), since the spectrum of the second harmonic wave had its central peak around 780 nm. This was because the irradiance of the laser around Z1560 nm was larger than those in other areas due to the chromatic aberration, and the resultant SHG occurred effectively around Z1560 nm; in other words, the chromatic aberration made the effective crystal length Leff shorter. The full width at half maximum of PD output voltage dependence on angular displacement in Fig. 7.41 is summarized in Tab. 7.9. FWHM for shorter f is found to be wider than that for longer f. The ratio of ΔZ to b is also summarized in the table. The ratio of ΔZ to b for shorter f is also found to be larger than that for longer f. Therefore, the effective crystal length Leff is shorter for shorter f due to the stronger influence of the chromatic aberration. Thus, the resolution using f = 75 mm is higher than that using f = 40 mm. For shorter Leff, FWHM of sinc2[Δk(θ)L/2] becomes wider. FWHMs of sinc2[Δk(θ)L/2] calculated for various L are plotted in Fig. 7.48. The experimental results are found to take closer value for L = 0.25, 0.5 and 0.8 mm for f = 40, 75 and 150 mm, respectively. These Leff are several times longer than ΔZ; this result indicates that SHG is localized around Z1560 nm within Leff. Therefore, the SHG spectra in Fig. 7.39 are not similar to the fundamental wave spectra. Surface 4 of the BBO crystal makes the chromatic aberration ΔZs 1.6 times longer than those without the BBO crystal as shown in Tab. 7.8. Tab. 7.8: Specification of the lens and the position of the BBO. ΔZ without BBO crystal(mm)

Focal length of the focusing lens(mm)

Angle(degree)

ΔZ with BBO crystal(mm)

.



 

. .

.



 

. .

.



 

. .

7.5 Summary

375

Tab. 7.9: FWHM of second harmonic power dependence on angular displacement and ΔZ/b. FWHM(arc-second)

ΔZ/b(–)



,

.



,

.



,

.

Focal length of the focusing lens (mm)

Fig. 7.48: FWHM of SHG power dependence on angular displacement for various crystal lengths.

7.5 Summary The possibility of applying mode-locked ultrashort pulse fiber lasers to angle measurement has been explored. At first, taking into consideration most of the ultrashort pulse fiber lasers have a center wavelength of 1,560 nm and the light from such a fiber laser is invisible, a unique alignment technique employing a retroreflector has been presented to determine the zero position of the optical setup of an angle sensor using an ultrashort pulse fiber laser. An ultrashort-pulse sensor based on laser autocollimation with the zero-alignment method has been developed. The sensitivity and the resolution of the angle sensor have been confirmed to be 0.0062 V/ arc-second and 2.0 arc-seconds, respectively. The feasibility of the zero-alignment method has been verified by experiments. Then an ultrashort pulse chromatic dispersion angle sensor is presented by utilizing the wide spectral of an ultrashort pulse laser. The sensor has been designed in such a way that each optical mode in a reflected ultrashort pulse laser beam is separated by chromatic dispersion of a collimator objective to generate a group of focused laser beams that can be utilized as the scale for measurement of an angular displacement by detecting the change in the peak frequency over the optical spectra

376

Chapter 7 Ultrashort-pulse angle sensor

obtained by a photodetector. An experiment has been carried out to investigate the effect of chromatic dispersion by the collimator objective employed in the angle sensor. The experimental result has demonstrated that a focal length change of approximately 15 μm can be generated by using an ultrashort-pulse fiber laser whose optical spectrum is ranging from 185 to 200 THz. Results of the extended experiments have also demonstrated that the resolution of 0.23 arc-second can be expected by the developed angle sensor. A new optical angle measurement method has also been presented by making use of the unique characteristic of high peak power and a high-intensity electric field of ultrashort pulses, which can generate second harmonic waves based on the nonlinear optical phenomenon. Results of the theoretical analysis have clarified that BBO crystal is suitable for the proposed angle measurement method when the ultrashort pulse fiber laser source is employed. It has been demonstrated by theoretical analysis and experiment that a focused ultrashort pulse laser spot is effective to realize SHG-based angle measurement for an ultrashort pulse laser source with a small power where the intensity of a collimated beam from such a laser source is too small to make SHG. Experimental results with the developed measurement system have demonstrated the feasibility of the proposed angular measurement. To clarify the reason for the discrepancy between the results of theoretical calculation and those of the experiments, investigations have been carried out based on the ray tracing. It has been clarified that chromatic aberration has been one of the main root causes of these problems, taking the results of ray tracing and observed second harmonic spectra into consideration. It has also been clarified that the shorter focal length of the focusing lens has made the influence of chromatic aberration stronger, and has made the effective crystal length Leff shorter, resulting in the degradation of the sensitivity.

References [1] [2] [3] [4]

[5]

[6]

Udem T, Reichert J, Holzwarth R, Hänsch TW. Accurate measurement of large optical frequency differences with a mode-locked laser. Opt Lett 1999, 24, 13, 881. Nelson LE, Jones DJ, Tamura K, Haus HA, Ippen EP. Ultrashort-pulse fiber ring lasers. Appl Phys B Lasers Opt 1997, 65, 2, 277–294. Udem T, Reichert J, Holzwarth R, Hänsch TW. Absolute optical frequency measurement of the cesium d1 line with a mode-locked laser. Phys Rev Lett 1999, 82, 18, 3568–3571. Gao W, Kim SW, Bosse H, Haitjema H, Chen YL, Lu XD, Knapp W, Weckenmann A, Estler WT, Kunzmann H. Measurement technologies for precision positioning. CIRP Ann – Manuf Technol 2015, 64, 2, 773–796. Jin J, Kim YJ, Kim Y, Kang CS, Kim SW Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser. Proc 6th Int Conf Eur Soc Precis Eng Nanotechnology, EUSPEN 2006 2006, 1, 13, 414–417. Salvadé Y, Schuhler N, Lévêque S, Le FS. High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source. Appl Opt 2008, 47, 14, 2715–2720.

References

[7] [8] [9] [10] [11]

[12]

[13]

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27] [28]

[29] [30]

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Jang Y-S, Kim S-W. Distance measurements using mode-locked lasers: A review. Nanomanufacturing Metrol 2018, 1, 3, 131–147. Udem T, Holzwarth R, Hänsch TW. Optical frequency metrology. Nature 2002, 416, 6877, 233–237. Tamada J Angle Sensor Using Optical Frequency Comb. Tohoku University, Master Thesis, 2017. Madokoro S Angle Sensor Using Femtosecond Laser. Tohoku University, Master thesis, 2019. Matsukuma H, Madokoro S, Nakao M, Shimizu Y, Gao W An optical angle sensor based on second harmonic generation of a modelocked laser. Proceedings of SPIE – The International Society for Optical Engineering, SPIE, 2018. Madokoro S, Shimizu Y, Chen Y-L, Gao W. Ultra-precision angle sensor with a mode-locked laser source. Proceedings of International Conference on Leading Edge Manufacturing in twenty-first Century: LEM21, Japan Society of Mechanical Engineers, 2017, 1–4. Tamada J, Kudo Y, Chen Y-L, Shimizu Y, Ito S, Gao W Determination of the origin position for an angle sensor with a femtosecond laser. Proceedings of International Conference on Leading Edge Manufacturing in twenty-first Century: LEM21, Japan Society of Mechanical Engineers, 2015, 1–4. Yariv A, Yeh P. Photonics, Oxford University Press, 2006. Femtosecond Erbium Laser | Menlo Systems. (Accessed January 13, 2021, at https://www.men losystems.com/products/femtosecond-lasers-and-amplifiers/c-fiber/) Ennos AE, Virdee MS. High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Precis Eng 1982, 4, 1, 5–8. Srivastava DK, Wintner E, Agarwal AK. Effect of focal size on the laser ignition of compressed natural gas-air mixture. Opt Lasers Eng 2014, 58, 67–79. Yuki S, Siew Leng T, Dai M, Taiji M, So I, Yuan-Liu C, Wei G. Ultra-sensitive angle sensor based on laser autocollimation for measurement of stage tilt motions. Opt Express 2016, 24, 3. Saito Y, Gao W, Kiyono S. A single lens micro-angle sensor. Int J Precis Eng Manuf 2007, 8, 2, 14–18. Ishizuka R Two-axis Absolute Encoder Using Optical Frequency Comb. Tohoku University, Master thesis, 2020. Tamada J, Kudo Y, Chen Y-LY-L, Shimizu Y, Gao W. Determination of the zero-position for an optical angle sensor. J Adv Mech Des Syst Manuf 2016, 10, 5, 0072. Shimizu Y, Madokoro S, Matsukuma H, Gao W. An optical angle sensor based on chromatic dispersion with a mode-locked laser source. J Adv Mech Des Syst Manuf 2018, 12, 5, 1–10. Hecht E. Optics, 5th Edition, Pearson, 2017. Franken PA, Hill AE, Peters CW, Weinreich G. Generation of optical harmonics. Phys Rev Lett 1961, 7, 4, 118–119. Seka W, Jacobs SD, Rizzo JE, Boni R, Craxton RS. Demonstration of high efficiency third harmonic conversion of high power Nd-glass laser radiation. Opt Commun 1980, 34, 3, 469–473. Solid-State Laser Engineering, New York, Springer, 2006. Dmitriev VG, Gurzadyan GG, Nikogosyan DN Handbook of Nonlinear Optical Crystals, Vol. 64, Berlin, Heidelberg, Springer Berlin Heidelberg, 1999. Matsukuma H, Madokoro S, Dwi W, Yuki A, Wei S, New Optical A. Angle measurement method based on second harmonic generation with a mode ‑ locked femtosecond laser. Nanomanufacturing Metrol 2019, 2, 4, 187–198. Kazuo K. Nonlinear Optical Frequency Conversion of Ultrashort Light Pulses. (Accessed February 12, 2021, at http://qopt.iis.u-tokyo.ac.jp/lecture/pdf/NLOtutorial2.pdf) Koechner W. Solid-State Laser Engineering, Vol. 1, New York, NY, Springer New York, 2006.

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[31] Boyd GD, Kleinman DA. Parametric interaction of focused Gaussian light beams. J Appl Phys 1968, 39, 8, 3597–3639. [32] Kleinman DA, Ashkin A, Boyd GD. Second-harmonic generation of light by focused laser beams. Phys Rev 1966, 145, 1, 338–379. [33] Schott AG Technical Information [TIE-19] Temperature Coeffcient of the Refractive Index., 2016. (at https://www.schott.com/d/advanced_optics/3794eded-edd2-461d-aec5 -0a1d2dc9c523/1.1/schott_tie-19_temperature_coefficient_of_refractive_index_eng.pdf)

Chapter 8 Optical frequency comb 8.1 Introduction At the end of the 1970s, Eckstein et al. demonstrated that a train of picosecond light pulses can be described as a superposition of phase-locked modes that are equally spaced in frequency domain [1], which is referred to as optical frequency comb. As demonstrated in Fig. 8.1 an optical frequency comb consists of a series of discrete, equally spaced modes of extremely narrow bandwidth that can be accurately identified by two radio frequency (RF) components; the mode spacing (repetition frequency) ωrep and the carrier-envelope offset ωceo. Through ωrep and ωceo, the optical frequency comb can be directly linked to RF time standards, which had led to a revolutionary simplification in optical frequency metrology at the end of the 1990s. In 1999, Udem et al., at Max-Planck Institute, Germany, made absolute optical frequency measurement of the Cesium D1 line by using the modes from a mode-locked femtosecond laser with a repetition frequency of 75 MHz as the ruler in frequency space [2]. Although the experiments were carried out by phase-locking the center mode of the frequency comb by using a He-Ne frequency standard, the possibility of realizing absolute frequency measurement from measurement of the interval between a frequency ω and its second harmonic 2ω was demonstrated [3]. This socalled self-referencing technique, which employs modes of ω and 2ω to identify the carrier-envelope offset ωceo, was then presented by Jones et al., at University Colorado and National Institute of Standards and Technology, USA, in 2000 [4], based on which the optical frequency comb can be fully stabilized with a direct linkage to a stable microwave clock. Since then, the optical frequency comb has become an essential and effective tool for optical frequency metrology. In addition to optical frequency metrology, optical metrology has also been applied to distance measurement. In 2000, Minoshima et al., at National Metrology Institute of Japan, demonstrated high-accuracy measurement of 240-m distance in an optical tunnel by using an optical frequency comb where a series of beat frequencies between the comb modes over a wide spectrum range were utilized for extending the range of absolute measurement of distance with high resolution and good stability [2, 4, 5]. Kim et al., at KAIST, Korea, have also developed a number of combbased distance measurement technologies [5]. The optical frequency comb is expected to play a more important role in not only distance measurement but also in a wide range of precision dimensional metrology, that is, precision metrology. For this purpose, it is important to make an insight about the mathematics behind optical frequency comb and the physical phenomena associated with optical frequency comb such as the influence of dispersion.

https://doi.org/10.1515/9783110542363-008

380

Chapter 8 Optical frequency comb

Fig. 8.1: Ultrashort pulse and optical frequency comb.

This chapter, therefore, presents fundamentals of optical frequency comb from point of view of mathematics. A mathematical model of the optical frequency comb is presented in Section 8.2, based on which the effects of dispersion on optical frequency comb are presented in Section 8.3, followed by stabilization and compensation of optical frequency comb in Section 8.4.

8.2 Mathematical model of optical frequency comb Figure 8.2 shows a train of ultrashort pulses observed at a fixed point on the X-axis along which the light propagates. The pulse is formed by mode-locking of the cavity longitudinal modes with angular frequencies ωm. The dispersion, that is, the dependence of the refraction index of the medium in the resonant cavity on wavelength or frequency, is not considered. The pulse makes round-trips within the resonant cavity with a round-trip cavity length Lcav, and the pulse passes through the fixed point once a round-trip. Assume that the pulse envelope g(t) is repeated with a time interval of τrep0 to form a train of pulse envelopes denoted by a function gP0(t), which is referred to as the pulse envelope train function. The reciprocal of τrep0 is referred to as the repetition frequency frep0 of the pulse envelope. The pulse electric field EP0(t) of the train of pulses is the product of gP0(t) and a carrier wave Ec(t) with an angular frequency of ωc or 2πfc. Mathematically, EP0(t) can be written as

8.2 Mathematical model of optical frequency comb

381

EP0 ðtÞ = gP0 ðtÞEc ðtÞ

(8:1)

Ec ðtÞ = ejωc t

(8:2)

In the following, a mathematical model is established to analyze the pulse electric field of the pulse train and its frequency spectrum, which is referred to as the optical frequency comb. The pulse envelope g(t) and its Fourier transform G(ω) are related by the following Fourier integrals [6]: ∞ ð

GðωÞ =

gðtÞ =

1 2π

gðtÞe − jωt dt

(8:3)

GðωÞejωt dω

(8:4)

−∞ ∞ ð

−∞

It should be noted that G(ω) is a continuous function and can be referred to as a continuous Fourier transform of g(t).

Fig. 8.2: Pulse train observed at a fixed point in a resonant cavity without dispersion.

Fig. 8.3: Pulse envelope g(t) and its Fourier transform G(ω).

382

Chapter 8 Optical frequency comb

As illustrated in Fig. 8.3, if g(t) is a Gaussian function, G(ω) will also be a Gaussian function, which can be expressed by gðtÞ = g0 e GðωÞ = G0 e

− 1 2 t2

−

2σt

(8:5)

1 ω2 2σ2ω

(8:6)

where σt and σω are RMS (root-mean-square) widths of the Gaussian functions g(t) and G(ω), respectively. g0 and G0 are the amplitudes of the Gaussian functions at t = 0. The relationship between σt and σω, and that between g0 and G0 are written by σt =

1 σω

(8:7)

σω g0 = pffiffiffiffiffi G0 2π

(8:8)

Let s(t) be an infinite sequence of impulse functions with a time interval of τrep0. The Fourier transform S(ω) of s(t) is also an infinite sequence of impulse functions with an angular frequency interval of ωrep0, which the repetition angular frequency of the pulse envelope in a resonant cavity without dispersion. ωrep0 is expressed by ωrep0 = 2πfrep0 =

2π τrep0

(8:9)

s(t) and S(ω), which are illustrated in Fig. 8.4, can be expressed as follows [6]: 1 sðtÞ = 2π ∞ ð

SðωÞ = −∞

∞ ð

SðωÞejωt dω =

X

−∞

gðtÞe − jωt dt = ωrep0

δðt − nτrep0 Þ,

n = 0, ± 1, ± 2, . . .

(8:10)

n

X

δðω − mωrep0 Þ,

m = 0, ± 1, ± 2, . . .

m

Fig. 8.4: An infinite sequence of impulse time functions (s(t)) and its Fourier transform S(ω).

(8:11)

8.2 Mathematical model of optical frequency comb

383

The periodic function gp0(t) in eq. (8.1), which is the series of pulse functions g(t) with a time period τrep0, can be mathematically produced from the convolution of g(t) and s(t) as: gP0 ðtÞ = gðtÞ*sðtÞ

(8:12)

Substituting eq. (8.10) into eq. (8.12) gives: X X δðt − nτrep0 Þ = gðt − nτrep0 Þ, gP0 ðtÞ = gðtÞ* n

n = 0, ± 1, ± 2, . . .

(8:13)

n

Correspondingly, the Fourier transform GP0(ω) of gP0(t) is the product of G(ω) and S(ω) as GP0 ðωÞ = GðωÞSðωÞ

(8:14)

If G(ω) is continuous at ω = mωrep0, then GðωÞδðω − mωrep0 Þ = Gðmωrep0 Þδðω − mωrep0 Þ,

m = 0, ± 1, ± 2, . . .

(8:15)

Combining eqs. (8.11), (8.14) and (8.15) gives X X GP0 ðωÞ = GðωÞωrep0 δðω − mωrep0 Þ = ðωrep0 Gðmωrep0 Þδðω − mωrep0 ÞÞ, m

m

m = 0, ± 1, ± 2, . . .

(8:16)

gP0(t) and GP0(ω) are illustrated in Fig. 8.5. It can be seen that S(ω) works as a sampling function to sample the continuous Fourier transform G(ω) with angular frequency interval ωrep0, which is referred to as the repetition frequency of the pulse envelope g(t). As shown in eq. (8.9), ωrep0 is determined by the time interval τrep0, which is referred to as the repetition time of the pulse envelope g(t). The sampled G(ω), that is, GP0(ω) is a discrete function composed of a sequence of equidistant impulses. The mth impulse of GP0(ω) with an angular frequency mωrep0 is referred to as the mth mode. Since the amplitude of each impulse in the sampling function S(ω) is ωrep0, the amplitude am of the mth mode in GP0(ω) can be expressed by am = GP0 ðmωrep0 Þ = ωrep0 Gðmωrep0 Þ

(8:17)

Here denote the product of the pulse envelope g(t) and the carrier wave Ec(t) by Eg(t), which is referred to as the single pulse wave. Eg(t) and its Fourier transform Hg(ω) can be related to the following equations. ∞ ð

Hg ðωÞ =

Eg ðtÞe −∞

Eg ðtÞ = gðtÞEc ðtÞ =

1 2π

− jωt ∞ ð

∞ ð

dt =

gðtÞejωc t e − jωt dt = Gðω − ωc Þ

−∞

Hg ðωÞejωt dω = −∞

1 2π

(8:18)

∞ ð

Gðω − ωc Þejωt dω −∞

(8:19)

384

Chapter 8 Optical frequency comb

Fig. 8.5: The periodic function gP0(t) and its Fourier transform GP0(ω).

Fig. 8.6: The pulse electric field Eg(t) of its Fourier transform Hg(ω).

Fig. 8.7: The pulse electric field E P0(t) of the train of pulses and its Fourier transform HP0(ω) (the optical frequency comb in a resonant cavity without dispersion).

8.2 Mathematical model of optical frequency comb

385

Eg(t) and Hg(ω) are illustrated in Fig. 8.6. It can be seen that the Fourier transform G(ω) in Fig. 8.3 is shifted along the axis of frequency by an amount of ωc, which is referred to as the frequency shifting effect as one of the Fourier transform properties [6]. Similarly, since EP0(t) in eq. (8.1) is the product of gP0(t) and the carrier wave Ec(t), the Fourier transform HP0(ω) of EP0(t) can be written as: ∞ ð

gP0 ðtÞejωc t e − jωt dt = GP0 ðω − ωc Þ

HP0 ðωÞ =

(8:20)

−∞

Substituting eq. (8.16) into eq. (8.20) gives X δðω − ωc − mωrep0 Þ HP0 ðωÞ = Gðω − ωc Þωrep0 = ωrep0 =

X

X

m

ðGðmωrep0 Þδðω − ωc − mωrep0 ÞÞ

(8:21)

m

ðωrep0 Gðmωrep0 Þδðω − ω0m ÞÞ

m

EP0(t) and HP0(ω) are illustrated in Fig. 8.7. HP0(ω) is referred to as the optical frequency comb of the train of pulses in a resonant cavity without dispersion. ω0m is the angular frequency of the mth impulse mode. It can be seen that the angular frequency of the mth impulse mode in HP0(ω) has a frequency shift of ωc from that of the mth impulse mode in GP0(ω) while the amplitude is kept as the same am of eq. (8.17). The optical frequency comb is composed of discrete comb modes with a comb envelope of G(ω-ωc). The position of each comb mode on the frequency axis, that is, the mode position, is determined by that of the center mode ωc and the comb spacing ωrep0, that is, the repetition frequency of pulse envelope. It should be noted that ωrep0 is determined by the pulse envelope repetition time τrep0 as in eq. (8.9). Assuming there are K periods of the carrier wave during the time period of τrep0 (or 2π/ωrep0) in Fig. 8.2, the relationship between ωc and ωrep0 can be expressed in the following equation: ωc = Kωrep0

(8:22)

The angular frequency ω0m of the mth impulse mode in Hp(ω) can thus be expressed by: ω0m = ωc + mωrep0 = ðK + mÞωrep0 ,

m = 0, ± 1, ± 2, . . .

(8:23)

where m and K are integers. On the other hand, the train of pulses is physically formed by mode-locking of the cavity longitudinal modes shown in Fig. 8.8. The electric field of the mth cavity mode at time t is denoted by Em(t) as follow:

386

Chapter 8 Optical frequency comb

Fig. 8.8: The gain curve in a resonant cavity without dispersion.

Em ðtÞ = bm ejωm t ,

m = 0, ± 1, ± 2, . . .

(8:24)

where bm is the amplitude determined by the lasing spectrum Glasing(ω) of the lasering medium in the cavity (Fig. 8.8). ωm is the angular frequency of the mth cavity mode shown in eq. (8.25). ωm = ωcen + mωrep , m = 0, ± 1, ± 2, . . .

(8:25)

Here ωcen is the central frequency of the gain curve. The pulse electric field EP(t) of the train of pulses can be obtained from the sum of the Em(t) of the cavity modes at time t generates as follows: X X Em ðtÞ = bm ejωm t , m = 0, ± 1, ± 2, . . . (8:26) Ep ðtÞ = m

m

Substituting eq. (8.25) into eq. (8.26) gives: Ep ðtÞ = ejωcen t gP ðtÞ

(8:27)

bm ejmωrep t , m = 0, ± 1, ± 2, . . .

(8:28)

where gp ðtÞ =

X m

Since gP(t) is a periodic function with a time period of τrep0, eq. (8.28) is the Fourier series expressed in the exponential form [6]. Based on the theory of Fourier series, bm can be written as τrep0 =2

bm =

1 τrep0

ð

− τrep0 =2

gP ðtÞe − jmωrep0 t dt

(8:29)

8.2 Mathematical model of optical frequency comb

387

gP(t) and g(t) have the following relationship within the limits of integration of eq. (8.29), from –τrep0/2 to τrep0/2. gP ðtÞ = gðtÞ, − τrep 0 =2 < t < τrep 0 =2

(8:30)

Assuming g(t) is zero outside the limits of integration of eq. (8.29) gives τrep0 =2

bm =

1 τrep0

ð

gðtÞe

− jmωrep0 t

ωrep0 dt = 2π

− τrep0 =2

−ð∞

gðtÞe − jmωrep0 t dt

(8:31)

−∞

Rewriting eq. (8.17) based on eq. (8.3) gives ∞ ð

am = ωrep0 Gðmωrep0 Þ = ωrep0

gðtÞe − jmωrep0 t dt

(8:32)

−∞

By comparing eqs. (8.1) and (8.27), (8.31) and (8.32), respectively, it can be seen that ωc = ωcen

(8:33)

am 2π

(8:34)

bm =

This indicates the mathematical model HP0(ω) of the optical frequency comb in Fig. 8.7 is corresponding to the lasing spectrum Glasing(ω) of the lasing medium in the resonant cavity in Fig. 8.8 where the central frequency ωcen of the gain curve is corresponding to the carrier frequency ωc. Then we consider the traveling of the pulse wave gc(t) in a resonant cavity without dispersion along the X-axis. The phase of a harmonic light wave with angular frequency ω can be written as: φðωÞ = ωt − kðωÞx

(8:35)

where k(ω) is the wave number determined by wavelength λ(ω) as 2π λðωÞ

(8:36)

λðωÞ =

2π c ω nðωÞ

(8:37)

kðωÞ =

ω nðωÞ c

(8:38)

kðωÞ = Since

k(ω) can then be rewritten as

where n(ω) is the index of refraction of the lasing medium in the resonant cavity.

388

Chapter 8 Optical frequency comb

The phase velocity of the harmonic light wave is then defined as vw ðωÞ = −

ð∂φ=∂tÞx ω c = = , ð∂φ=∂xÞt kðωÞ nðωÞ

m = 0, ± 1, ± 2, . . .

(8:39)

Letting k(ω), n(ω) and vw(ω) at ω = ωc be kc, nc and vc, respectively, kc and vc can be expressed by kc = kðωc Þ =

ωc nc c

(8:40)

c nc

(8:41)

vc = vw ðωc Þ =

In a resonant cavity without dispersion, n(ω) will be a constant. Assuming n(ω) equals to nc, then the wave number in eq. (8.38) and the phase in eq. (8.35) and can be expressed by: ω ω nc = c vc   x φ0 ðωÞ = ωt − k0 ðωÞx = ω t − vc k0 ðωÞ =

(8:42) (8:43)

Assume t′ = t −

x vc

(8:44)

Since t′ is not a function of ω, the following equation can be obtained by substituting it into the Fourier integral in eq. (8.19): 1 2π

∞ ð

Gc ðωÞe −∞

jωt′

1 dω = 2π =

1 2π

=e

∞ ð

−∞ ∞ ð

Gðω − ωc Þe jωðt − vc Þ dω x

Gðω − ωc Þe jωc ðt − vc Þ e jðω − ωc Þðt − vc Þ dω x

−∞

jωc ðt − vxc Þ

1 2π

∞ ð

−∞ ∞ ð

Gðω − ωc Þe jðω − ωc Þðt − vc Þ dðω − ωc Þ

x 1 GðωÞe jωðt − vc Þ dω 2π −∞       x x x g t− = Eg t − = Ec t − vc vc vc x

= e jωc ðt − vc Þ

x

x

(8:45)

8.3 Effects of dispersion on optical frequency comb

389

Rewriting the last part of eq. (8.45) gives 1 Eg ðt, xÞ = Ec ðt, xÞgðt, xÞ = 2π

∞ ð

Gðω − ωc Þe jφ0 ðωÞ dω

(8:46)

−∞

where   x Eg ðt, xÞ = Eg t − vc   x x = e jωc ðt − vc Þ Ec ðt, xÞ = Ec t − vc   x gðt, xÞ = g t − vc

(8:47) (8:48) (8:49)

It can be seen that the carrier wave and the pulse envelope are now functions of both time t and position x, which can be employed to study the propagation distance of the pulse along the X-axis. As can be seen in eqs. (8.48) and (8.49), in a resonant cavity without dispersion, the carrier wave Ec(t, x) and the pulse envelope g(t, x) travel at the same velocity of vc. Consequently, the pulse wave Eg(t, x) also travels at velocity vc. The propagation distances of the waves, in this case, are shown in Fig. 8.9. In a time period of τrep0, both the carrier wave and the pulse envelope travel a distance of Lcav. τ0 and Lcav have the following relationship: τrep0 =

Lcav Lcav = nc vc c

(8:50)

8.3 Effects of dispersion on optical frequency comb 8.3.1 Effect of phase velocity dispersion In an actual situation, the index of refraction of the lasing medium in the resonant cavity is dependent on frequency and wavelength, which is referred to as the effect of dispersion. In this case, as can be seen in eq. (8.39), the phase velocity vw(ω) of a harmonic light wave traveling in the resonant cavity is also a frequency-dependent function. Due to the phase velocity dispersion, the group velocity of the pulse envelope, which is composed of a group of harmonic waves, will be different from the phase velocity of the pulse carrier wave. In this subsection, this effect of dispersion is discussed. For clarity, the term phase velocity dispersion is used in the subsection title to distinguish the effect of the group velocity dispersion to be discussed in the next subsection. It should be noted that both the phase velocity dispersion and the group velocity dispersion are caused by the dispersion in the index of refraction, which is simply called dispersion for clarity.

390

Chapter 8 Optical frequency comb

Fig. 8.9: The propagation distance of waves in a resonant cavity without dispersion.

Figure 8.10 shows a typical curve of the index of refraction n(ω) as a function of angular frequency ω [7]. At the center of the absorption band with frequency ωa, the index of refraction is 1 and the phase velocity is c, the speed of light in vacuum. In the absorption band, the amplitudes of the light waves increase dramatically, resulting in strong absorption of energy from the light waves. In the areas outside the absorption band, n(ω) increases with the increase of frequency ω or the decrease of wavelength λ(ω). The relationship between ω and k(ω) in the left normal dispersion area in Fig. 8.10 where the cavity modes are located can then be illustrated in Fig. 8.11. The group velocity vg(ω) is defined in the following equation. vg ðωÞ =

dω dkðωÞ

(8:51)

8.3 Effects of dispersion on optical frequency comb

391

Based on eq. (8.38), eq. (8.51) can be rewritten as: vg ðωÞ = =

c c dnðωÞ − kðωÞ 2 nðωÞ n ðωÞ dkðωÞ c kðωÞ dnðωÞ ð1 − Þ nðωÞ nðωÞ dkðωÞ

= vw ðωÞð1 −

(8:52)

kðωÞ dnðωÞ Þ nðωÞ dkðωÞ

In the case of normal dispersion, n(ω) increases with the increase of ω as illustrated in Fig. 8.10 [6]. Therefore dn(ω)/dω is a positive value. From the following equation, it can be seen that dn(ω)/dk(ω) is a positive value. dnðωÞ dnðωÞ dω dnðωÞ = = vg ðωÞ dkðωÞ dω dkðωÞ dω

(8:53)

In eq. (8.52), since k(ω) and n(ω) are also positive values, the group velocity vg(ω) is thus smaller than the phase velocity vw(ω) in the case of normal dispersion. This is demonstrated in Fig. 8.11 [4–8]. Conversely, in the case of anomalous dispersion where dn(ω)/dω is a negative value, vg(ω) is larger than vw(ω).

Fig. 8.10: Dependence of index of refraction on frequency in a medium with dispersion.

392

Chapter 8 Optical frequency comb

Fig. 8.11: S and group velocity with normal dispersion.

Taking the difference between φ(ω) of eq. (8.35) and φ0(ω) of eq. (8.43) gives: ΔφðωÞ = φðωÞ − φ0 ðωÞ = k0 ðωÞx − kðωÞx =

x ω − kðωÞx vc

(8:54)

Then Δφ(ω) can be approximated by Taylor series at ω = ωc as follows:  ΔφðωÞ ≈ Δφðωc Þ +

∂ΔφðωÞ ∂ω

 ωc

ðω − ωc Þ +

  1 ∂2 ΔφðωÞ ðω − ωc Þ2 2 ∂ω2 ωc

(8:55)

where Δφðωc Þ = ðk0 ðωc Þ − kðωc ÞÞx = 0

(8:56)

Denoting the second term and the third term in eq. (8.55) by δφ1(ω) and δφ2(ω), respectively. Based on eq. (8.54), δφ1(ω) and δφ2(ω) can be obtained as       ∂ΔφðωÞ x ∂kðωÞ x x ðω − ωc Þ δφ1 ðωÞ = ðω − ωc Þ = − ðω − ωc Þ = − x ∂ω vc ∂ω vc vg ωc ωc (8:57) δφ2 ðωÞ =

    1 ∂2 ΔφðωÞ 1 ∂2 kðωÞ 2 ðω − ω Þ = ðω − ωc Þ2 x − c 2 ∂ω2 2 ∂ω2 ωc ωc

(8:58)

where vg = vg ðωc Þ

(8:59)

8.3 Effects of dispersion on optical frequency comb

393

At first, we consider the influence of δφ1(ω) on the pulse waves as well as the optical frequency comb. With the existence of δφ1(ω), the phase φ(ω) of a harmonic light wave in the resonant cavity becomes φðωÞ = φ0 ðωÞ + δφ1 ðωÞ

(8:60)

Replacing φ0(ω) with φ(ω) in eq. (8.46) gives ð 1 ∞ Gðω − ωc Þe jδφ1 ðωÞ e jφ0 ðωÞ dω 2π − ∞ −∞

 ∞ ð jðω − ωc Þ vxc − vxg x 1 = Gðω − ωc Þe e jωðt − vc Þ dω 2π −∞

 ∞ ð jðω − ωc Þ vxc − vxg x x 1 = Gðω − ωc Þ e jωc ðt − vc Þ e jðω − ωc Þðt − vc Þ dω 2π −∞

 ∞ ð jðω − ωc Þ t − vxg jωc ðt − vxc Þ 1 =e Gðω − ωc Þe dðω − ωc Þ 2π −∞

 ∞ ð jω t − vxg jωc ðt − vxc Þ 1 =e GðωÞe dω 2π −∞     x x g t− = Ec t − vc vg 1 2π

∞ ð

Gðω − ωc Þe jφðωÞ dω =

(8:61)

Rewriting the last part of eq. (8.61) gives Eg1 ðt, xÞ = Ec ðt, xÞg1 ðt, xÞ

(8:62)

where Ec(t, x) is given by eq. (8.47), and the pulse envelope can be expressed by   x (8:63) g1 ðt, xÞ = g t − vg It can be seen that the moving velocity of the pulse envelope changes from the phase velocity vc to the group velocity vg. Consequently, the pulse envelope and the carrier wave move in different velocities. As shown above, the velocity vg of the pulse envelope is slower than the velocity vc of the carrier wave in the case of normal dispersion. Figure 8.12 illustrates the propagation of the pulse envelope and the carrier wave in this case. The carrier wave Ec(x, t) moves the same distance of Lcav over a time period of τrep0 as that in Fig. 8.9, which is the length of the resonant cavity. Compared with the moving distance Lcav of the pulse envelope Ec(x, t) in Fig. 8.9, the pulse envelope

394

Chapter 8 Optical frequency comb

Ec(x, t) in Fig. 8.11 moves a shorter distance of Lg over τrep0 due to the slower velocity vg. The difference ΔLg between Lcav and Lg can be written as ΔLg = Lcav − Lg = ðvc − vg Þτrep0

(8:64)

Substituting eq. (8.50) into eq. (8.64) gives ΔLg = ðvc − vg Þ

  Lcav vg Lcav = 1− vc vc

(8:65)

Fig. 8.12: The propagation distance of waves in a resonant cavity with normal dispersion.

Now consider the influence of dispersion on the optical frequency comb. As can be seen in Fig. 8.7, the optical frequency comb can be determined by: 1) the center mode frequency, that is, the frequency of the carrier wave, 2) the mode spacing, that is, the repetition frequency of pulse envelope, and 3) the comb envelope. Since the carrier wave shown in eq. (8.61) is exactly the same as that shown eq. (8.45), the center mode frequency in a resonant cavity with dispersion is the same ωc. Here ωc

8.3 Effects of dispersion on optical frequency comb

395

corresponds to the “fixed frequency” in the elastic tape model [9]. By comparing the pulse envelope g1(t, x) in eq. (8.63) with g(t, x) in eq. (8.49), it can be seen that the shape of the pulse envelope does not change with or without dispersion. This indicates the shape of the comb envelope shown in Figs. 8.6 and 8.7 basically will not change. To investigate the change in the mode spacing, the train of pulses observed at a fixed point in a resonant cavity with dispersion is compared with that in a resonant cavity without dispersion. The latter is illustrated in Fig. 8.2. As described before, the pulse makes round-trips within the resonant cavity with a round-trip cavity length Lcav. In the case of a resonant cavity without dispersion where the index of refraction is assumed to be nc of eq. (8.41), both the pulse envelope and the carrier wave move in the same velocity vc (eq. (8.41)). It takes the same time τrep0 (eq. (8.50)) for a point on the pulse envelope and that on the carrier wave to pass through the fixed point once a round-trip. Figure 8.13 shows the train of pulses EP1 (t) observed at a fixed point of x = 0 in a resonant cavity with normal dispersion. According to eq. (8.1), EP1(t) can be expressed by EP1 ðtÞ = gP1 ðtÞEc ðtÞ

(8:66)

where gP1(t) is the train of pulse envelopes. Due to the slower velocity vg (eq. (8.59)) in a resonant cavity with normal dispersion, the time τrep1 for a point on the pulse envelope to make a round-trip will be longer than τrep0, which is the time for a point on the carrier wave to make the same round-trip. τrep1 is expressed by τrep1 =

Lcav vg

(8:67)

The corresponding repetition frequency ωrep1 of the pulse envelope, that is, the mode spacing of the optical frequency comb in a resonant cavity with dispersion becomes: ωrep1 =

2π 2π = vg τrep1 Lcav

(8:68)

In a resonant cavity with normal dispersion, because vg is smaller than vc, the corresponding ωrep1 is smaller than ωrep0 shown in the following equation: ωrep1 < ωrep0

(8:69)

where ωrep0 =

2π 2π = vc τrep0 Lcav

(8:70)

The pulse envelope train gP1(t) can then be obtained by rewriting eq. (8.13) as follows: X gðt − nτrep1 Þ, n = 0, ± 1, ± 2, . . . (8:71) gP1 ðtÞ = n

396

Chapter 8 Optical frequency comb

Fig. 8.13: Trains of waves observed at a fixed point (x = 0) in a resonant cavity with normal dispersion.

Substituting eqs. (8.2) and (8.71) into eq. (8.66) gives X gðt − nτrep1 Þ, n = 0, ± 1, ± 2, . . . EP1 ðtÞ = e jωc t

(8:72)

n

In addition to EP1(t), gP1(t) and Ec(t) observed at the same point of x = 0 are also illustrated in Fig. 8.13. As can be seen in the figure, also in eq. (8.22), there are K periods of the carrier waves within τrep0. Denoting the difference between τrep1 and τrep0 by Δτrep1, the following equation can be obtained from eq. (8.22): ωc = Kωrep0 = K

2π τrep0

2π =K τrep1 − Δτrep1

(8:73)

8.3 Effects of dispersion on optical frequency comb

397

Rewriting eq. (8.73) gives ωc =

Δτ 2πK Δτ ωc + = ωc + Kωrep1 τrep1 τrep1 τrep1

(8:74)

= ωceo + Kωrep1 where ωceo =

Δτrep ωc τrep1

(8:75)

ωceo is referred to as the carrier-envelope offset frequency. From eqs. (8.50) and (8.67), Δτrep can be obtained as   1 1 Lcav Δτrep1 = τrep1 − τrep0 = − (8:76) vg vc Equation (8.75) can then be rewritten as   Lcav 1 1 ωc ωceo = − τrep1 vg vc Substituting eq. (8.67) into eq. (8.77) gives     1 1 vg ωc = 1 − ωc − ωceo = vg vc vg vc

(8:77)

(8:78)

On the other hand, the time period τc of the carrier wave can be expressed by τc =

2π ωc

(8:79)

Substituting eqs. (8.68) and (8.79) into eq. (8.75) gives ωceo =

Δτrep1 ωrep1 τc

(8:80)

As shown in Fig. 8.13, Δτrep1 is smaller than τc. Therefore ωceo < ωrep1 Taking the Fourier transform of the pulse train Ep1(t) in eq. (8.72) gives ð∞ HP1 ðωÞ = gP1 ðtÞe jωc t e − jωt dt = GP1 ðω − ωc Þ

(8:81)

(8:82)

−∞

where HP1(ω) is the optical frequency comb of the pulse envelope train in a resonant cavity with dispersion. Based on eqs. (8.10) to (8.15), GP1(ω) and GP1(ω–ωc) can be expressed by

398

GP1 ðωÞ =

Chapter 8 Optical frequency comb

ð∞ −∞

gP1 ðtÞe − jωt dt = GðωÞωrep1

GP1 ðω − ωc Þ = Gðω − ωc Þωrep1

X

X

δðω − mωrep1 Þ,

m = 0, ± 1, ± 2, . . . (8:83)

m

δðω − ωc − mωrep1 Þ,

m = 0, ± 1, ± 2, . . .

(8:84)

m

Rewriting eq. (8.23) gives the angular frequency ω1m of the mth comb mode in Hp1 (ω) as ω1m = ωc + mωrep1 ,

m = 0, ± 1, ± 2, . . .

(8:85)

ω1m can be further rewritten in the following equation by substituting the ωc of eq. (8.74) into eq. (8.85): ω1m = ωceo + ðK + mÞωrep1 ,

m = 0, ± 1, ± 2, . . .

(8:86)

HP1(ω) of eq. (8.82) can then be expressed as follows by substituting eq. (8.85) into eq. (8.84): X δðω − ω1m Þ HP1 ðωÞ = GP1 ðω − ωc Þ = Gðω − ωc Þωrep1 =

X m

m

ðωrep1 Gðω1m − ωc Þδðω − ω1m ÞÞ =

m = 0, ± 1, ± 2, . . .

X

ðωrep1 Gðmωrep1 Þδðω − ω1m ÞÞ,

m

(8:87)

Figure 8.14 shows the optical frequency comb HP1(ω) of the pulse train in a resonant cavity with normal dispersion. The optical frequency comb HP0(ω) of eq. (8.21) and Fig. 8.7 without dispersion is also shown in the figure for comparison. At first, it can be seen that the dispersion reduces the amplitude of the comb envelope, that is, the amplitude of each comb mode from that without dispersion by a factor of ωrep1/ ωrep0. Since ωrep1/ωrep0 is a constant value, the comb envelope basically does not change its shape. The normal dispersion will also reduce the pulse envelope repetition frequency from ωrep0 to ωrep1, which is the mode spacing of the optical frequency comb. Since the frequency ωc of the center mode does not change, a fraction of ωrep1, that is, the carrier-envelope offset frequency ωceo will be caused. Consequently, the first comb mode will no longer start from ωrep1 but from ωrep1 + ωceo. This is a significant effect of dispersion on the optical frequency comb. In the case without dispersion, the comb mode positions are basically only determined by the comb mode spacing. However, with the presence of the carrier-envelope offset frequency ωceo, it becomes more complicated to identify and stabilize the comb mode positions, which will be addressed in the next section.

8.3 Effects of dispersion on optical frequency comb

399

Fig. 8.14: Optical frequency comb in a resonant cavity with normal dispersion.

8.3.2 Effect of group velocity dispersion Next, we consider the effect of δφ2(ω) in eq. (8.58) on the optical frequency comb. The propagation constants can be expressed by: ∂kðωÞ 1 = ∂ω vg ðωÞ  2      ∂β1 ðωÞ ∂ kðωÞ 1 ∂vg ðωÞ = = − β2 = ∂ω ∂ω2 ωc v2g ∂ω ωc ωc β1 ðωÞ =

(8:88)

(8:89)

It can be seen that β2, which has a unit of s2/m or fs2/mm, is a function of the partial derivative of the group velocity. If the group velocity is dependent on frequency, its partial derivative will have a non-zero value and the lasing medium is regarded to have group velocity dispersion (GVD). β2 is a parameter that is well used to express the degree of group velocity dispersion. Figure 8.15 shows the β2 value of quartz

400

Chapter 8 Optical frequency comb

with respect to wavelength [10, 11]. For example, when an optical fiber made of quartz is used in a fiber ring cavity with a carrier wavelength of 1.5 μm, the β2 value is a negative value of − 71 fs2/mm.

Fig. 8.15: Group velocity dispersion (value) of quartz.

Substituting eq. (8.89) into eq. (8.58) gives δφ2 ðωÞ = −

β2 x ðω − ωc Þ2 2

(8:90)

With the existence of δφ2(ω), the phase φ(ω) of a harmonic light wave in the resonant cavity becomes [12] φðωÞ = φ0 ðωÞ + δφ2 ðωÞ

(8:91)

where δφ1(ω) of eq. (8.57) is not considered for clarity. Replacing φ0(ω) with φ(ω) in eq. (8.46) gives

8.3 Effects of dispersion on optical frequency comb

1 2π = =

ð∞ −∞

1 2π 1 2π

Gðω − ωc Þe

ð∞ −∞

ð∞

−∞

ð∞

Gðω − ωc Þe jδφ2 ðωÞ ejφ0 ðωÞ dω

−∞

Gðω − ωc Þe − j

β2 x 2 jωðt − x Þ vc dω 2 ðω − ωc Þ e

Gðω − ωc Þe − j

β2 x 2 jω ðt − x Þ jðω − ω Þðt − x Þ c vc e vc dω 2 ðω − ωc Þ e c

1 2π ð∞

=e

jωc ðt − vxc Þ

=e

jωc t′

1 2π

1 dω = 2π

jφðωÞ

ð∞

401

(8:92)

β2 x 2 jωðt − vxc Þ 2 ω e dω

GðωÞe − j

−∞ β2 x 2 ′ 2 ω e jωt dω

GðωÞe − j

−∞

where t′ is defined in eq. (8.44). For simplicity, assuming that G(ω) has a Gaussian function shown in eq. (8.6), the term of Fourier integral in the last part of eq. (8.92) can be written as: 1 2π = =

ð∞ −∞

1 2π 1 2π

1 = 2π

G0 e

− 12 ω2 2σω

ð∞ −∞

G0 e

ð∞

−∞

1 + jσ2ω β2 x 2 ω 2σ2ω

′

e jωt dω (8:93)

ð∞

−∞

−

β2 x 2 ′ 2 ω e jωt dω

e−j

G0 e

− 1 2 ω2 2σ′ω

′

e jωt dω

′

G′ðωÞejωt dω

where G′ðωÞ = GðωÞe − j

β2 x 2 2 ω

= G0 e

− 1 2 ω2 2σ′ω

σω σ′ω = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + jσ2ω β2 x

(8:94) (8:95)

σt is given in eq. (8.7). As can be seen in eq. (8.94), G′(ω) is a Gaussian function, its reverse Fourier transform is then also a Gaussian function. Therefore, the Fourier integral in the last part of eq. (8.93) can be obtained as follows: 1 2π

ð∞ −∞

′ G′ðωÞe jωt dω = g′0 e

2 − 1 2 t′ 2σ′t

= g′ðt′Þ

(8:96)

402

Chapter 8 Optical frequency comb

where 2 − 1 2 t′ ′

g′ðtÞ = g′0 e

2σ t

(8:97)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ′ σt= = + jβ x = σ2t + jβ2 x 2 σ2ω σ ′ω

(8:98)

σ ′ω 1 σω 1 g′0 = pffiffiffiffiffi G0 = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi G0 = rffiffiffiffiffiffiffiffiffiffiffiffiffiffi g0 2 β x 2π 1 + jσω β2 x 2π 1 + j σ22

(8:99)

t

g0 is given in eq. (8.8). The last part of eq. (8.92) can then be further rewritten as ð∞ β2 x 2 ′ 1 ′ ′ ejωc t GðωÞe − j 2 ω e jωt dω = e jωc t g′ðtÞ 2π − ∞ =e

jωc t′

=e

jωc t′

=e

g ′0 e

−

1

t 2ðσ2t + jβ2 xÞ

2 β2 x t′ 2ðσ4t + ðβ2 xÞ2 Þ

j

e

β x jðωc + 4 2 t′Þt′ 2ðσt + ðβ2 xÞ2 Þ

′ = e jωc2 t g′0 e

′2

=e

g ′0 e

−

−

g ′0 e

jωc t′

g ′0 e

−

σ2t

σ2t − jβ2 x

2ðσ4t + ðβ2 xÞ2 Þ

t′

2

2

2ðσ4t + ðβ2 xÞ2 Þ 1

t′

t ðβ xÞ2 2σ2t ð1 + 2 4 Þ σt

(8:100) ′2

2 − 12 t′ 2σ

t2

where ωc2 = ωc +

β2 x 2ðσ4t

+ ðβ2 xÞ2 Þ

t′ = ωc + Δωβ2

β2 x

Δωβ2 =

2ðσ4t + ðβ2 xÞ2 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðβ xÞ2 σt2 = σt 1 + 2 4 σt

(8:101)

t′

(8:102)

(8:103)

Equation (8.100) can be obtained by taking the absolute value of the complex number g’0 of eq. (8.99) as follows: e

jωc2 t′

g ′0 e

2 − 12 t′ 2σ

t2

=e

jωc2 t′

1

σt

= Ec2 ðt′Þg2 ðt′Þ = Eg2 ðt′Þ

2

− 2 t′ g0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2σt2 ðβ xÞ2 4 1 + 24

(8:104)

8.3 Effects of dispersion on optical frequency comb

403

where Eg2 ðt′Þ = Ec2 ðt′Þg2 ðt′Þ

(8:105)

′ Ec2 ðt′Þ = ejωc2 t

(8:106)

2

2

− 2 t′ − 2 t′ g0 g2 ðt′Þ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2σt2 = g2 0 e 2σt2 ðβ xÞ2 4 1 + σ2 4 1

1

(8:107)

t

g0 g20 = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðβ xÞ2 4 1 + 24

(8:108)

σt

Here Ec2(t’) and g2(t’) are the pulse carrier wave, the pulse envelope in a resonant cavity with group velocity dispersion, respectively. gc2(t’) is the product Ec2(t’) and g2(t’). Since g2(t’) is a Gaussian function, its Fourier transform G2(ω) is also a Gaussian function and can be expressed by G2 ðωÞ = G20 e

− 12 ω2 2σ

(8:109)

ω2

where σω2 =

1 = σt2

1 σω rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðβ2 xÞ2 ðβ xÞ2 σt 1 + 4 1 + 24 σt

(8:110)

σt

pffiffiffiffiffi pffiffiffiffiffi 2π 2π 1 G0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G20 = g20 = g0 2 σω2 σω2 4 ðβ xÞ ðβ xÞ2 4 1 + 24 1 + 24 σt

(8:111)

σt

It can be seen that the G2(ω), the Fourier transform of the pulse envelope changes with the propagation distance x. The amplitude G20 and the RMS comb width σω2 decrease with the increase of x. Figure 8.16 shows a schematic of the train of pulses observed at a fixed point of x = 0 in the resonant cavity with the existence of phase velocity dispersion. It should be noted that a negative β2 value is taken and the effect of the phase change δϕ1(ω) in eq. (8.57) is not included. The pulse envelopes shown in Fig. 8.16 are numbered as g2(0), g2(1) and g2(2), respectively. The first pulse envelope with n = 0 is at t = 0. Based on eqs. (8.103) and (8.108), the amplitude and RMS width of the nth pulse envelope can be obtained as follow by taking x = nLcav:

404

Chapter 8 Optical frequency comb

Fig. 8.16: Changes of the train of pulses due to the effect of group velocity dispersion.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnβ2 Lcav Þ2 σt2 ðnÞ = σt 1 + , σ4t g0 g20 ðnÞ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðnβ L Þ2 4 1 + 2 4cav

n = 0, 1, 2, . . . n = 0, 1, 2, . . .

(8:112) (8:113)

σt

It can be seen that the RMS width and amplitude of the first pulse envelope are equal to those in the case without dispersion. With the increases of n, the magnitude g20(n) of the pulse envelope will approach zero, which means the pulses will disappear. Similarly, based on eqs. (8.110) and (8.111), the amplitude and RMS width of the Fourier transform G2(ω) in eq. (8.109) of nth pulse envelope can be obtained as follow by taking x = nLcav:

8.3 Effects of dispersion on optical frequency comb

σω σω2 ðnÞ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnβ L Þ2 1 + 2σ4cav

405

(8:114)

t

G0 G20 ðnÞ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnβ L Þ2 4 1 + 2 4cav

(8:115)

σt

It can be seen that the RMS width and amplitude of Fourier transform of the first pulse envelope are equal to those in the case without dispersion. With the increase of n, both the magnitude G20(n) and the RMS width σω2(n) of the Fourier transform will approach zero. The change in the time spacing between two adjacent pulse envelopes, which is referred to as the pulse envelope spacing is investigated. Based on eqs. (8.88) and (8.89), β1(ω) and vg(ω) can be evaluated from β2 as follows: ðω β1 ðωÞ = β2 dω + β1 ðωc Þ = β2 ðω − ωc Þ + β1 ðωc Þ (8:116) ωc

vg ðωÞ =

1 β2 ðω − ωc Þ +

1 vg

= vg

1 1 + vg β2 ðω − ωc Þ

(8:117)

where vg = vg ðωc Þ

(8:118)

Assume β2 is very small, then vg(ω) can be approximated as vg ðωÞ ≈ vg ð1 − vg β2 ðω − ωc ÞÞ

(8:119)

Based on eqs. (101) and (102), the group velocity at ω = ωc2 can then be obtained as vg ðωc2 Þ ≈ vg ð1 − vg β2 Δωβ2 Þ = vg ð1 −

β22 x 2ðσ4t + ðβ2 xÞ2 Þ

vg t′Þ

(8:120)

It can be seen that vg(ωc2) is smaller than vg if β2 is not zero. vg(ωc2) also decreases with the increase of the propagation distance x. Assume the group velocity is kept as the same during one round-trip in the resonant cavity for clarity. Taking x = nLcav and vgt′ = nLcav in eq. (8.113), the time spacing τrep1(n) between the n + 1th and the nth pulse envelopes can be expressed by τrep2 ðnÞ =

Lcav = vg ðωc2 Þ

1 1−

ðβ2 nLcav Þ

2

2ðσ4t + ðβ2 nLcav Þ2 Þ

Lcav = vg 1−

1 1

2 1+

σ4t ðβ2 nLcav Þ2

!

τrep1

(8:121)

406

Chapter 8 Optical frequency comb

where τrep1 is given in eq. (8.67). It can be seen that the time spacing between the first and the second pulse envelopes is equal to τrep1, which is the pulse envelope spacing in the case without dispersion. Then it decreases with the increase of the propagation distance x of the pulse envelope. When n is large enough, τrep1(n) will approach τrep1/2. Since the pulse envelope spacings are not constant and the pulse envelopes are not the same function, the train of pulse envelope as well as that of the pulses, which is the product of the pulse envelope and the carrier wave, are no longer periodic functions. Consequently, the Fourier transform of the train of pulse envelope and that of the train of pulses will not have discrete modes, like those shown in the optical frequency combs in Fig. 8.14. It can be concluded that optical frequency comb cannot be obtained without compensation of the group velocity dispersion.

8.4 Stabilization of optical frequency comb 8.4.1 Stabilization of repetition frequency The repetition frequency, which determines the mode spacing, is an important parameter for an optical frequency comb. As shown in eq. (8.68), repetition frequency is basically determined by the round-trip length of resonant cavity Lcav. Rewrite eq. (8.68) to be ωrep1 = 2πfrep1 = 2π

vg 1 c = 2π Lcav Lcav ng

(8:122)

where ng is referred to as the net index of refraction. When there is a change ΔL in Lcav, the repetition angular frequency becomes 1 c Lcav − ΔL ng

(8:123)

ΔL = ωrep + Δωrep1 Lcav

(8:124)

ω′rep1 = 2πf ′rep1 = 2π If ΔL is small, ω′rep can be approximated by ω′rep1 ≈ ωrep1 + ωrep1 where

Δωrep1 = ωrep1

ΔL Lcav

(8:125)

It can be seen that a change will be caused in repetition angular frequency by ΔL. For a femtosecond fiber laser with a repetition frequency of 100 MHz, letting the net index of refraction of the medium in the resonant cavity is 1.5, Lcav is calculated to

8.4 Stabilization of optical frequency comb

407

be 2 m based on eq. (8.122). Δωrep1/2π can then be calculated to be 0.05 Hz for a ΔL of 1 nm based on eq. (8.125). Equation (8.125) can also be rewritten as Δωrep1 ΔL = ωrep1 Lcav

(8:126)

It can be seen that the ratio of change in the repetition frequency is equal to that of change in the cavity length. A change in cavity length can be caused by vibrations of cavity mirrors. The thermal expansion of the components to construct the resonant cavity is another cause of ΔL. Figure 8.17 shows a schematic for compensation of ΔL by moving one of the cavity mirrors in a Fabry–Pérot cavity. The compensation of ΔL in a fiber ring resonant cavity can be made by gluing a small part of the fiber with a PZT actuator as shown in Fig. 8.18 [13]. The measurement and control of ΔL will be presented in Chapter 11.

Fig. 8.17: Compensation of the change of cavity length in a Fabry–Pérot resonant cavity.

Fig. 8.18: Compensation of the change of cavity length in a fiber ring resonant cavity.

408

Chapter 8 Optical frequency comb

8.4.2 Stabilization of carrier-envelope frequency Meanwhile, ΔL will also make a change in the carrier-envelope frequency. The following equation can be obtained from eq. (8.74): ωc = ω′ceo + K ′ω′rep1

(8:127)

ΔL is assumed to be small enough and will not change the coefficient K. Based on the analysis in Section 8.2, ωc is basically determined the center frequency of the gain curve of the laser medium in the resonant cavity and therefore will also not change. Taking the difference of eq. (8.74) and eq. (8.127) gives Δωceo = ω′ceo − ωceo = Kωrep1 − K ′Δω′rep1

(8:128)

Fig. 8.19: Influence of the cavity length change on the optical frequency comb.

Assume the center wavelength of a femtosecond fiber laser is 1,560 nm. ωc/2π is calculated to be 192,174,652,564,102.56 Hz. When ΔL is zero, the coefficient K and the carrier-envelope frequency are calculated to be 1,921,746 and 52,564,102.56 Hz.

8.4 Stabilization of optical frequency comb

409

When ΔL is 1 nm, the carrier-envelope frequency is calculated to be 92,211,538.48 Hz while the coefficient K′ is calculated to be the same as K. The difference in the carrierenvelope frequencies with and without ΔL is then calculated to be − 96,153.79 Hz, which is approximately 0.1% of the repetition frequency. It can be seen that the change in the carrier-envelope frequency caused by ΔL is significant and cannot be ignored. Figure 8.19 shows a schematic of the influence of ΔL on the carrier-envelope frequency. Reduction of such a change in the carrier-envelope frequency can be made by compensating ΔL with the setups shown in Figs. 8.17 and 8.18. More generally, the change in the carrier-envelope frequency is caused by a change in the group delay time tg of the pulse envelope. The group delay time tg(ω) over one round trip of the pulse envelope is defined as     ∂φðωÞ Lcav Lcav 1 1 (8:129) = − = Lcav − tg ðωc Þ = − vg vc ∂ω ωc vg vc where vc and vg are defined in eq. (8.41) and eq. (8.59), respectively. The carrierenvelope frequency shown in eq. (8.77) can be expressed by tg as follows: ωceo =

ωc tg τrep1

(8:130)

Denoting the change in tg by Δtg, eqs. (8.129) and (8.130) can be rewritten as t′g = tg + Δtg ω′ceo =

(8:131)

ωc ′ ωc ωc tg= tg + Δtg = ωceo + Δωceo τrep1 τrep1 τrep1

(8:132)

ωc Δtg τrep1

(8:133)

where Δωceo =

As can be seen in eq. (8.132), a change Δωceo will be caused in the carrier-envelope frequency by the change Δtg in the group delay time tg. Based on eq. (8.129), it can be seen that Δtg can be caused by either a change in the cavity length or in the group velocity. The former has been discussed at the beginning of this subsection and can be compensated by the setups shown in Figs. (8.17) and (8.18). A more general way is to manipulate the group delay time by using dispersive elements such as diffraction gratings or prisms. Figure 8.20 shows such a compensation setup developed by Kwong et al [13, 14] here a diffraction grating is employed. The setup is composed of a diffraction grating, a lens and a mirror. A pulse laser is projected on the grating at the normal incident. For clarity of description, the light path of the mth mode in the resonant cavity is illustrated in the figure. The light path of the center mode is also illustrated

410

Chapter 8 Optical frequency comb

Fig. 8.20: Compensation of group delay time change by using a grating-based setup.

in the figure. The following grating equation can be obtained for the mth mode with wavelength λm and frequency ωm: P sin θm = λm

(8:134)

where P is the grating pitch and θm is the diffraction angle. It can be seen that the diffraction angle is a function of wavelength and/or frequency. For the center mode with wavelength λc and frequency ωc, eq. (8.134) becomes P sin θc = λc

(8:135)

Denoting the difference between λm and λc by Δλm and that between θm and θc by Δθm gives P sinðθc + Δθm Þ = λc + Δλm

(8:136)

P sin θc + PΔθm cos θc = λc + Δλm

(8:137)

eq. (8.136) can be written as

The following approximations can be made for a small Δθm: cos Δθm ≈ 1 sin Δθm ≈ Δθm

(8:138)

8.4 Stabilization of optical frequency comb

411

Then eq. (8.137) can be further rewritten as Δθm ≈

Δλm P cos θc

(8:139)

As can be seen in the figure, the focal point of the lens is aligned with the incident point A of the pulse laser and diffraction light of the center mode (center diffraction light) is made to align with the optical axis of the lens. The diffraction light of the mth mode (the mth diffraction light) will then be made parallel to the optical axis after passing through the lens based on the principle of autocollimation [15]. Denoting the distance between the incident point B of the mth diffraction light to the center point C of the lens by Δzm, which is also the distance between the optical axis and the incident point D of the mth diffraction light on the mirror. Δzm can be expressed by Δzm ≈ Δθm F =

FΔλm P cos θc

(8:140)

where F is the focal length of the lens. When the mirror is aligned to be orthogonal to the optical axis of the lens, both the center diffraction light and the mth diffraction light will be returned to point A on the grating. When the mirror is tilted at an angle γ about point D, the mth diffraction light will be returned back to point E on the grating. Eventually, the diffraction light rays are merged to form the output pulse laser beam. If γ is small enough, point E will be overlapped with point A and the returning path of the output pulse laser beam will be aligned with that of the incident pulse laser beam. Due to the small γ, an optical-path difference OPDm shown in the following equation will then be generated in the mth diffraction light: OPDm ≈ 2Δzm γ =

2FΔλm γ P cos θc

(8:141)

The corresponding phase change Δφm is then obtained as Δφm =

2πOPDm 4πF Δλm ≈ γ λm P cos θc λm

(8:142)

Denoting the difference between ωm and ωc by Δωm gives Δλm Δωm =− λc ωc

(8:143)

An amount Δtc for group delay time compensation can be obtained as follows by substituting eq. (8.143) into eq. (8.142): Δtc = where

Δφm 4πF ≈− γ = Rd γ Δωm P cos θc ωc

(8:144)

412

Chapter 8 Optical frequency comb

Rd = −

4πF P cos θc ωc

(8:145)

Since Rd is a constant, Δtc, is proportional to the mirror title angle γ. The group time delay then becomes with the additional Δtc as follows: t′g = tg + Δtg + Δtc

(8:146)

The group time delay changes Δtg can then be removed by adjusting γ for a Δtc that is equal to –Δtg. Consequently, the carrier-envelope frequency change Δωceo shown in eq. (8.133) can be removed. Figure 8.21 shows a possible arrangement for integrating the compensation setup into a resonant cavity.

Fig. 8.21: Stabilization of carrier-envelope frequency in a resonant cavity by using the gratingbased setup.

The compensation of group delay time change can also be made by using the combination of a pair of prisms and a mirror as shown in Fig. 8.22 [16]. Similar to that in Fig. 8.20, when a small tilt angle γ′ is applied to the mirror, an optical path difference OPD′m will be generated to the mth mode light. It should be noted that γ′ is small enough so that a light ray reflected by the mirror is regarded to trace the same path of the light ray incident to the mirror. Eventually, the reflected light rays of all the comb modes are merged at the incident point to form the output pulse laser beam. The returning path of the output pulse laser beam will be aligned with that of the incident pulse laser beam. To obtain the relationship between OPD′m and γ′, it is necessary to evaluate the beam distance Δz′m between the mth mode light ray and the center mode light ray. Since the pair of prisms can be optically replaced by a parallelogram, as can be seen in Fig. 8.23, a simple model is then illustrated in Fig. 8.24 for evaluation of Δz′m.

8.4 Stabilization of optical frequency comb

413

Fig. 8.22: Compensation of group delay time change by using a prism-based setup.

Fig. 8.23: Optical correspondence between a parallelogram and a pair of prisms.

The following equations can be obtained for the mth mode and the center mode light rays, respectively, based on the law of refraction: c sin θi = nc sin θ′c = sin θ′c λc sin θi = nm sin θ′m =

c sin θ′m λm

(8:147) (8:148)

where θi is the incident angle, θ′c and θ′m are the refraction angles of the mth mode and the center mode light rays, respectively. Denoting the difference between θ′c and θ′m by Δθ′m and combining eqs. (8.147) and (8.148) give

414

Chapter 8 Optical frequency comb

Fig. 8.24: Simplified model for analyzing the beam spacing Δzm.

λm sin θ′c = λc sin θ′m

(8:149)

ðλc + Δλm Þ sin θ′c = λc sinðθ′c + Δθ′m Þ

(8:150)

eq. (8.150) can be rewritten as follows by taking the approximations in eq. (8.138). Δλm Δθ′m = tan θ′c λc

(8:151)

Taking into consideration that Δθ′m is small, the beam distance Δz′m can then be evaluated as follows: Δz′m =

Δλm W tan θ′c λc

(8:152)

where W is the length of the parallelogram. The optical path difference OPD′m and the corresponding phase change Δφ′m can then be obtained as follows: Δλm ′ OPD′m = 2Δz′m γ = 2W tan θ′c γ λc

(8:153)

2πOPD′m 4πW Δλm ′ Δφ′m = ≈ tan θ′c γ λm λc λm

(8:154)

8.4 Stabilization of optical frequency comb

415

Substituting eq. (8.143) into eq. (8.154) gives 2πOPD′m 4πW Δω′m ′ Δφ′m = ≈− tan θ′c γ λm ωc λm

(8:155)

Fig. 8.25: Stabilization of carrier-envelope frequency in a resonant cavity by using the prism-based setup.

An amount Δt′c for group delay time compensation can then be expressed by Δt′c =

Δφ′m 4πW tan θ′c ′ ≈− γ = Rp γ′ Δωm λm ωc

(8:156)

4πW cos θ′c λm ωc

(8:157)

where Rp = −

Since Rp is a constant, Δt′c, which is proportional to the mirror title angle γ′, can then be adjusted by γ′ for compensating the group time delay change Δtg as well as the carrier-envelope frequency change Δωceo shown in eqs. (8.131) to (8.133). Figure 8.25 shows a possible arrangement for integrating the compensation setup into a resonant cavity. For femtosecond fiber laser with a fiber ring cavity, stabilization of carrierenvelope frequency can be made by adjusting the driving current of the pumping laser diode, which will be explained in Chapter 11.

416

Chapter 8 Optical frequency comb

8.4.3 Compensation of group velocity dispersion As shown in eq. (8.89), the group-velocity dispersion is caused by a non-zero propagation constant β2. On the other hand, taking the second-order derivative of the phase φ(ω) in eq. (8.35) of a harmonic light wave with angular frequency ω gives ∂2 φðωÞ ∂2 kðωÞ = − x ∂ω2 ∂ω2

(8:158)

As shown in eq. (8.90), the influence of group-velocity dispersion on the laser pulse and optical frequency comb is caused by the product of and the propagation distance x. Compensation of group-velocity dispersion should therefore be made against β2x. β2x can be expressed as follows by combining eqs. (8.158) and (8.89):  2   2  ∂ kðωÞ ∂ φðωÞ β2 x = x= − ∂ω2 ωc ∂ω2 ωc

(8:159)

If an additional phase shift δ(ω) is added to φ(ω), eq. (8.159) then becomes β′ 2 x = −



∂2 ðφðωÞ + δðωÞÞ ∂ω2

 ωc

= β2 x −

 2  ∂ δðωÞ = β2 x − Δβ ∂ω2 ωc

(8:160)

where  Δβ =

∂2 δðωÞ ∂ω2

 (8:161) ωc

Obviously, the influence of group-velocity dispersion caused by β2x can be compensated through manipulating δ(ω) to make Δβ to equal with β2x. This can be realized by using a number of methods [14]. Here, the method using a pair of diffraction gratings, which is shown in Fig. 8.26, is presented as an example [14]. Figure 8.27 shows a diffraction grating with a grating pitch P, on which the mth mode is projected at nonnormal incidence. The grating equation at nonnormal incidence can be rewritten as follows from that at normal incidence in eq. (8.134). P sin θi + P sin θm = λm

(8:162)

Similarly, the corresponding grating equation for the center mode becomes P sin θi + P sin θc = λc

(8:163)

Figure 8.28 shows the optical path of the center mode when it is projected into the pair of gratings with identical grating pitch, which are aligned parallel with each other. Since the incident angle θ′i to the second grating is equal to the diffraction angle θc at the first grating, the diffraction angle θ′c at the second grating will then

8.4 Stabilization of optical frequency comb

417

Fig. 8.26: Compensation of group velocity dispersion by using a pair of gratings.

Fig. 8.27: Grating diffraction at the nonnormal incidence.

be equal to the incident angle θi to the first grating based on the grating equation. Consequently, the incident light to the first grating, which is referred to as the input light, will be parallel to the diffraction light from the second grating, which is referred to as the output light. Now we consider the optical path difference OPDc associated with the propagation of the center mode from point A to point C. Assuming the distance between the two gratings is denoted by Lg, OPDc can be obtained from the following equations: OPDc = AB + BC = ð1 + cosðθi − θc ÞÞAB = ð1 + cosðθi − θc ÞÞ

Lg cos θc

(8:164)

418

Chapter 8 Optical frequency comb

Fig. 8.28: The optical path of the center mode.

where AB =

Lg cos θc

BC = AB cosðθi − θc Þ

(8:165) (8:166)

The corresponding phase shift δc1 can be expressed by δc1 =

2π 2πLg 1 OPDc = ð1 + cosðθi − θc ÞÞ λc cos θc λc

(8:167)

Meanwhile, the phase shift δc2 associated with the lateral position shift of the incident point over the grating surface is expressed as [14] δc2 =

2π 2πLg BD = tan θc P P

(8:168)

The total phase shift for the center mode then becomes δc = δc1 + δc2 =

2πLg 1 2πLg ð1 + cosðθi − θc ÞÞ + tan θc λc cos θc P

(8:169)

As shown in Fig. 8.26, when a pulse laser is projected onto the first grating, the mth mode will be separated from the center mode. The two modes will be parallel with each other after being diffracted by the second grating. The mth mode then has a different optical path from the center mode. Rewriting eq. (8.169) for the phase shift δm of the mth mode gives

419

8.4 Stabilization of optical frequency comb

δm = δm1 + δm2

(8:170)

where δm1 =

2πLg 1 ð1 + cosðθm − θi ÞÞ λm cos θm

(8:171)

2πLg tan θm P

(8:172)

δm2 =

For simplicity of description, the phase shifts of the mth mode and the center mode are combined in the following equation by denoting the diffraction angle of the first grating as θ(ω): δðωÞ = δ1 ðωÞ + δ2 ðωÞ

(8:173)

where δ1 ðωÞ =

2πLg 1 ð1 + cosðθðωÞ − θi ÞÞ λ cos θðωÞ

(8:174)

2πLg tan θðωÞ P

(8:175)

δ2 ðωÞ =

The first-order and the second-order derivatives of δ(ω) are evaluated from those of δ1(ω) and δ2(ω), respectively. Assuming the gratings are in the air with an index of refraction of 1, then the wavelength λ and angular frequency ω have the following relationship: λ=

2πc ω

(8:176)

where c is the speed of light in vacuum. Substituting eq. (8.176) into eq. (8.171) gives δ1 ðωÞ =

ωLg 1 ð1 + cosðθðωÞ − θi ÞÞ c cos θðωÞ

(8:177)

The first-order derivative of δ1(ω) can then be evaluated as follows: ∂δ1 ðωÞ Lg 1 = ð1 + cosðθðωÞ − θi ÞÞ c cos θðωÞ ∂ω + =

ωLg sin θðωÞ ∂θðωÞ ωLg 1 ∂θðωÞ ð1 + cosðθðωÞ − θi ÞÞ − sinðθðωÞ − θi Þ c cos2 θðωÞ c cos θðωÞ ∂ω ∂ω

Lg 1 ωLg 1 ∂θðωÞ ð1 + cosðθðωÞ − θi ÞÞ + ðsin θi + sin θðωÞÞ c cos θðωÞ c cos2 θðωÞ ∂ω (8:178)

420

Chapter 8 Optical frequency comb

Since θi and θ(ω) are the incident and diffraction angles of the first grating, respectively, the grating equation can be expressed as follows by combining with eq. (8.176): P sin θi + P sin θðωÞ =

2πc ω

(8:179)

Combining eq. (8.178) with (8.179) gives ∂δ1 ðωÞ Lg 1 2πLg 1 ∂θðωÞ = ð1 + cosðθðωÞ − θi ÞÞ + c cos θðωÞ P cos2 θðωÞ ∂ω ∂ω

(8:180)

Based on eq. (8.180), the second-order derivative of δ1(ω) can then be evaluated as follows: ∂2 δ1 ðωÞ Lg sinθðωÞ ∂θðωÞ = ð1+ cosðθðωÞ−θi ÞÞ c cos2 θðωÞ ∂ω2 ∂ω

  Lg 1 ∂θðωÞ 4πLg sinθðωÞ ∂θðωÞ 2 2πLg 1 ∂2 θðωÞ − sinðθðωÞ−θi Þ + 3 2 c cosθðωÞ P cos θðωÞ ∂ω P cos θðωÞ ∂ω2 ∂ω   Lg 1 ∂θðωÞ 4πLg sinθðωÞ ∂θðωÞ 2 2πLg 1 ∂2 θðωÞ = + sinθðωÞÞ− + ðsinθ i 2 3 2 c cos θðωÞ ∂ω P cos θðωÞ ∂ω P cos θðωÞ ∂ω2 +

(8:181) Combining eq. (8.181) with eq. (8.179) gives   ∂2 δ1 ðωÞ 2πLg 1 ∂θðωÞ 4πLg sin θðωÞ ∂θðωÞ 2 2πLg 1 ∂2 θðωÞ = + − 2 2 3 2 ωP cos θðωÞ ∂ω P cos θðωÞ P cos θðωÞ ∂ω2 ∂ω ∂ω (8:182) The following equation can be obtained by rewriting eq. (8.179): sin θðωÞ =

2πc − sin θi ωP

(8:183)

Then, ∂θðωÞ 2πc 1 1 =− 2 ∂ω P ω cos θðωÞ  2 ∂θðωÞ 4π2 c2 1 1 = P2 ω4 cos2 θðωÞ ∂ω

(8:184)

(8:185)

∂2 θðωÞ 4πc 1 1 2πc 1 sin θðωÞ ∂θðωÞ 4πc 1 1 4π2 c2 1 sin θðωÞ = − = + P2 ω4 cos3 θðωÞ ∂ω2 P ω3 cos θðωÞ P ω2 cos2 θðωÞ ∂ω P ω3 cos θðωÞ (8:186)

8.4 Stabilization of optical frequency comb

421

Substituting eqs. (8.184) to (8.186) into eq. (8.182) gives ∂2 δ1 ðωÞ 4π2 cLg 1 1 16π3 c2 Lg 1 sin θðωÞ = − − P2 ω3 cos3 θðωÞ P3 ∂ω2 ω4 cos5 θðωÞ +

8π2 cLg 1 1 8π3 c2 Lg 1 sin θðωÞ + 2 3 3 P P3 ω4 cos5 θðωÞ ω cos θðωÞ

=+

4π2 cLg 1 1 8π3 c2 Lg 1 sin θðωÞ − 2 3 3 P P3 ω4 cos5 θðωÞ ω cos θðωÞ

(8:187)

Meanwhile, the first-order derivative of δ2(ω) is evaluated as follows based on eq. (8.175): ∂δ2 ðωÞ 2πLg 1 ∂θðωÞ = 2 P cos θðωÞ ∂ω ∂ω

(8:188)

Then the second-order derivative of δ2(ω) can then be written as   ∂2 δ2 ðωÞ 4πLg sin θðωÞ ∂θðωÞ 2 2πLg 1 ∂2 θðωÞ = + 2 3 2 P cos θðωÞ P cos θðωÞ ∂ω2 ∂ω ∂ω

(8:189)

Substituting eqs. (8.185) and (8.186) into (8.189) gives ∂2 δ2 ðωÞ 8π2 cLg 1 1 24π3 c2 Lg 1 sin θðωÞ = + P2 ω3 cos3 θðωÞ P3 ∂ω2 ω4 cos5 θðωÞ

(8:190)

The second-order derivative of δ(ω) can then be obtained as follows by combining eqs. (8.187) and (8.190). ∂2 δðωÞ ∂2 δ1 ðωÞ ∂2 δ2 ðωÞ 12π2 cLg 1 1 16π3 c2 Lg 1 sin θðωÞ = + = + (8:191) 2 2 2 2 3 3 P P2 ∂ω ∂ω ∂ω ω cos θðωÞ ω4 cos5 θðωÞ It is interesting to observe that the first-order derivative of δ1(ω) in eq. (8.188) has the same magnitude but the opponent signs with the second term in that of δ2(ω) in eq. (8.180). The same result in eq. (8.191) can be obtained more easily by taking the derivative of the sum of the first-order derivatives of δ1(ω) and δ2(ω). As shown in Fig. 8.26, the light wave is returned to the pair of gratings by the mirror. The phase shift is then doubled to 2δ(ω). Based on the fact the second term in eq. (8.191) is sufficiently small, the Δβ in eq. (8.161) for compensation of β2x can then be expressed as follows: 

 ∂2 2δðωÞ 24π2 cLg 1 1 = Δβ = 2 2 3 cos3 θ P ω ∂ω c c ωc

(8:192)

It can be seen that Δβ can be adjusted to compensate for β2x by adjusting the grating distance Lg. Δβ can also be expressed as follows by combining eqs. (8.176) and (8.192)

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Chapter 8 Optical frequency comb

Δβ =

3λ3c Lg 1 πP2 c2 cos3 θc

(8:193)

8.5 Summary A mathematical model based on Fourier transform has been established for analysis of optical frequency comb. It is demonstrated that the optical frequency comb is correspondent to the Fourier series of periodic pulses. An optical pulse generated by a mode-locked laser is composed of a pulse envelope and a carrier wave. Based on the mathematical model, it is made clear the magnitude of the optical frequency comb is proportional to the comb mode spacing, that is, the repetition frequency of the pulse envelope. A change in the repetition frequency, therefore, influences the magnitude of the optical frequency comb. This is not a problem in optical frequency metrology, which only utilizes the comb position along frequency axis. However, this should be taken into consideration when applying optical frequency comb for precision dimensional metrology. The mathematical model has also been utilized to analyze the influence of dispersion on the repetition frequency and the carrierenvelope offset frequency. The mechanism for causing the carrier-envelope offset frequency can be made clear based on that the center comb position is determined by the center frequency of the lasing medium in an ideal resonant cavity. Similarly, the effects of the group-velocity dispersion on the optical frequency comb have been analyzed based on the mathematical model. Finally, stabilization of optical frequency comb through compensation of the effects of dispersions has been mathematically explained in detail.

References [1] [2] [3] [4] [5] [6] [7] [8]

Eckstein JN, Ferguson AI, Hänsch TW. High-resolution two-photon spectroscopy with picosecond light pulses. Phys Rev Lett 1978, 40, 13, 847–850. Udem T, Reichert J, Holzwarth R, Hänsch TW. Absolute optical frequency measurement of the cesium d1 line with a mode-locked laser. Phys Rev Lett 1999, 82, 18, 3568–3571. Udem T, Reichert J, Holzwarth R, Hänsch TW. Accurate measurement of large optical frequency differences with a mode-locked laser. Opt Lett 1999, 24, 13, 881. Jones DJ. Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis. Science (80-) 2000, 288, 5466, 635–639. Jang Y-S, Kim S-W. Distance measurements using mode-locked lasers: A review. Nanomanufacturing Metrol 2018, 1, 3, 131–147. Brigham EO. The Fast Fourier Transform and Its Applications. USA, Prentice-Hall, Inc., 1988. Hecht E. Optics. 5th Pearson, 2017. Minoshima K, Matsumoto H. High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser. Appl Opt 2000, 39, 30, 5512.

References

[9] [10] [11] [12]

[13] [14] [15] [16]

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Droste S, Ycas G, Washburn BR, Coddington I, Newbury NR. Optical frequency comb generation based on erbium fiber lasers. Nanophotonics 2016, 5, 2, 196–213. Kazuo K. Nonlinear Optical Frequency Conversion of Ultrashort Light Pulses. (Accessed February 12, 2021, at http://qopt.iis.u-tokyo.ac.jp/lecture/pdf/NLOtutorial2.pdf) Watanabe S. Ultrafast Optics V. Vol. 132. Watanabe S, Midorikawa K eds., New York, NY, Springer New York, 2007. Inaba H, Daimon Y, Hong F-L, Onae A, Minoshima K, Schibli TR, Matsumoto H, Hirano M, Okuno T, Onishi M, Nakazawa M. Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb. Opt Express 2006, 14, 12, 5223–5231. Kwong KF, Yankelevich D, Chu KC, Heritage JP, Dienes A. 400-Hz mechanical scanning optical delay line. Opt Lett 1993, 18, 7, 558. Kokyo, Inc. | Laser experts. (Accessed February 12, 2021, at https://en.symphotony.com/) Ennos AE, Virdee MS. High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Precis Eng 1982, 4, 1, 5–8. Cundiff ST. Phase stabilization of ultrashort optical pulses. J Phys D Appl Phys 2002, 35, 8, 43–59.

Chapter 9 Angle comb 9.1 Introduction Angle is a fundamental quantity in the geometry of a mechanical component [1, 2] such as the rake/clearance angles of a cutting tool, the taper angle of a machine tool spindle taper, the thread angle of a screw, and the pressure angle of a gear. Angular relationship between the components assembled in a machine such as the perpendicularity of machine tool slides is also often a dominating factor for the performance of the machine. Static angle measurement is typically carried out in these cases by using angular gauges or optical microscopes/projectors [3]. On the other hand, angle of rotation is a fundamental quantity in the motion of a rotating mechanism such as the electric motor for a robot and the workpiece spindle for a machine tool, which typically requires dynamic angle measurement [3]. The tilt error motion of a linear slide or a spindle is another case in which dynamic angle measurement is required [2]. The existing methods for dynamic angle measurement can be basically classified into two types. The first type, such as the autocollimator, is based on the sine principle [4]. In this type, the angle is measured from the ratio of the length of two sides of a right triangle. For example, in an autocollimator, the angle is calculated from the ratio of the displacement of the optical spot on the focal plane of a collimator objective where a position-sensing photodetector is placed, and the focal length of the objective. This type of sensor has a limited measurement range typically less than 1˚ [3]. The second type is called the circle-dividing method or the angle scale method for large angle measurement up to 360˚ [5]. A circular angle scale with line graduations is employed as the measurement standard in this method. The rotary encoder, either optical or inductive, belongs to this type. It is based on the fact that a circle can be divided into any number of parts where the round angle of 360˚ can be used as a natural invariable and error-free angle standard [6]. The advancement in the angle scale manufacturing processes as well as in the optical techniques for reading/dividing the scale graduations has made the optical rotary encoder the most well-used precision angle sensor in production engineering [2, 7]. In manufacturing of an angle scale, a graduation track composed of line structures on a master disk, which are diamond cut by a precision dividing engine, is transferred to a glass or steel substrate disk by optical projection lithography. The angular graduation interval is determined by the ratio of the graduation line spacing typically beyond 2 μm, to the radius of the graduation track typically within 50 mm [8]. A commercially available rotary encoder has an angular graduation interval, often called the signal period, ranging from 3.6 arc-seconds (360,000 pulses/revolution) to 1,296 arc-seconds (900 pulses/revolution) for different applications [8]. The angular https://doi.org/10.1515/9783110542363-009

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Chapter 9 Angle comb

graduation can be further sub-divided electronically for higher resolutions. On the other hand, however, in addition to the high cost of the facility for the angle scale manufacturing, it is difficult to further reduce the graduation line spacing while maintaining the high definition and homogeneity of the edges of the graduation structures, which is the precondition for the high quality of the output signals. The measurement accuracy is also significantly influenced by the mounting errors of the scale disk on the measured shaft when the encoder is employed for angle measurement. In responding to the above problems inherent in rotary encoders, a method is presented in this chapter for the creation of a new type of angle scale, named the angle comb, by using an optical frequency comb [9, 10] and a diffraction grating. This method is based on the fact that a series of optical frequency elements aligned with a frequency interval called the pulse repetition frequency in the optical frequency comb can be separated in space as a comb of first-order diffracted light rays by a diffraction grating, which can be employed as angle graduations [11–14]. The angle comb is applied to the optical lever and laser autocollimator for reliable and wide-range angle measurement.

9.2 Intensity-domain angle comb In an optical lever, a light beam, often a laser beam, is projected onto the measured object. The reflected beam forms a light spot on a light position-sensing photodetector that can detect the linear displacement of the light spot. The angular displacement of the measured object is magnified and converted into a linear displacement of the light spot on the detector, which is a function of the photodetector-object distance and the change of reflection angle of the laser beam. The target angular displacement, which is a half of the reflection angle change, can thus be measured from the photodetector output. By adding a lens in front of the photodetector and locating the photodetector at the focal position of the lens, the angular displacement of the measured object can also be measured without the influence of the change of photodetector-object distance based on the principle of autocollimation [15]. Optical levers have been widely used for measurement of the deflection angle of the cantilever in an atomic force microscope [16], angular motion errors of the spindle and the slide in a machine tool [17]. Recently, optical levers have also been employed for surface form measurement by scanning the local slopes of the surface [18, 19]. The light position-sensing photodetector is the key component to determine the performance of an optical lever in terms of measurement range, resolution and speed. A charge-coupled device (CCD) is often employed in an optical lever to detect the position of the light spot. Although a CCD-type optical lever has the advantage of a large measurement range, the speed is quite low due to the limited readout rate and frame rate for reading out the thousands or even millions of CCD pixels one by one, in a shift-and-read process [20]. For this reason, it can only be used for static or

9.2 Intensity-domain angle comb

427

semi-static measurement such as testing of optical components, measurement of perpendicularity, parallelism and squareness of two surfaces, and calibration of polygon mirrors [20]. Compared with the CCD, a light spot position-sensing photodiode (PD), a single-cell, a bi-cell, or a quadrant photodiode, can detect the linear displacement of a light spot with a much higher speed. For this reason, PD-type optical levers with a continuous wavelength (CW) laser have been applied for dynamic angle measurement [16, 17, 21]. When a position-sensing photodiode is employed for linear displacement of the light spot in the optical lever, the size of the light spot is a critical parameter for determining the measurement range and resolution [19]. The larger the spot size, the wider the range but the lower the resolution. In many cases, it is necessary to maintain a small spot size for achieving the necessary resolution. This reduces the measurement range of the conventional PD-type optical lever with a CW laser to the order of 100 arc-seconds [21]. This is the biggest drawback of the conventional PD-type optical lever with a CW laser compared with the CCD-type optical lever that can have a range larger than 2,000 arc-seconds [20]. To overcome the shortcomings of the conventional PD-type optical levers with a CW laser described above, this section presents an optical lever comb by utilizing the optical frequency comb of a mode-locked ultrashort-pulse laser, which is referred to as the optical lever comb. A schematic of the optical lever comb is shown in Fig. 9.1. A laser beam emitted from a mode-locked laser, which has an optical frequency comb containing a series of discrete modes equally spaced over a wide frequency range [9], is employed as the measurement laser beam for the optical lever. The laser beam is made incident to a reflector having grating patterns with a constant period on its surface. According to the diffraction theory [22], when a laser beam with a light frequency ν is made incident to the diffraction grating with a grating period g, the diffraction angle θ can be calculated by the following equation:   c (9:1) θ = arcsin nair gν where c and nair are the light speed in vacuum and the refractive index in air, respectively. In the optical frequency comb, each mode has a specific frequency. Therefore, each of the first-order diffracted beams has a diffraction angle corresponding to the frequency of each mode. As a result, a group of first-order diffracted beams having discrete angles can be acquired in both the positive and negative directions. In the case of employing a two-axis diffraction grating as the reflector, four groups of positive and negative first-order diffracted beams along both the X- and Yaxes can be acquired. In the figure, only two groups of the first-order diffracted beams are indicated for the sake of clarity. By using the first-order diffracted beams as a ruler for angle measurement, angular motions of the grating reflector about the X- and Y-axes (θX and θY, respectively,) with respect to the incident laser beam can be measured.

428

Chapter 9 Angle comb

Fig. 9.1: A schematic of the angle comb.

Fig. 9.2: Schematic of the conventional PD-type optical lever with a CW laser as the light source.

Compared with the conventional PD-type optical lever with a CW laser, the angle comb can expand the measurement range. As shown in Fig. 9.2, the conventional optical lever with a CW laser can measure the tilt angle of a plane mirror reflector mounted on a measured object by detecting the displacement of the reflected optical beam with a PD. The measurement range is limited by the size of the active cell in the PD and the laser beam diameter. On the other hand, the angle comb can achieve a wider measurement range with the same PD as the light spot positionsensing detector as shown in Fig. 9.3. The frequency of the ith mode νi in the optical frequency comb of the mode-locked laser can be expressed as follows [17]:

9.2 Intensity-domain angle comb

429

Fig. 9.3: Schematic of the optical lever comb with a mode-locked laser as the light source.

νi = vCEO + i · vrep

ði = 1, 2, 3, . . . , nÞ

(9:2)

where νCEO and νrep represent a carrier-envelope offset frequency and a pulse repetition rate, respectively. The mode-locked laser emitted from the fiber cable connected to the laser source is at first collimated by a collimating lens (CL), and the collimated laser beam is made incident to the grating reflector mounted on a measurement target such as a rotary table. According to eqs. (9.1) and (9.2), the diffraction angle of the first-order diffracted beam of the ith mode in the mode-locked laser βi can be expressed as follows:   c ði = 1, 2, 3, . . . , nÞ (9:3) βi = arcsin nair gνi From this equation, the angular period of the first-order diffracted beams Pi, which is defined as the angle difference between the (i + 1)th mode diffracted beam and ith mode diffracted beam, can be calculated as follows:     c c − arcsin ði = 1, 2, 3, . . . , nÞ (9:4) Pi = arcsin nair gνi + 1 nair gνi The frequency of the ith mode νi can be stabilized by phase-locking νCEO and νrep to a frequency standard such as rubidium (Rb) frequency standard [9]. Since the diffraction

430

Chapter 9 Angle comb

angle βi is a function of νi as shown in eq. (9.3), βi can be stabilized by the phase locking. This is one of the remarkable features of the angle comb. More importantly, the angle comb has the potential of assuring the traceability of angle measurement by directly linking the optical frequency comb of the mode-locked laser to the national standard of time and frequency. Figure 9.4 shows a possible diffraction angle error as a function of the uncertainty in fractional frequency calculated from eq. (9.3) in the case of the grating period g of 4 μm. When the light source has the uncertainty in fractional frequency of 10−3, which is the same order as that of a CW laser diode (LD), a huge diffraction angle error of approximately 0.24 arc-second could occur. Meanwhile, on the other hand, in the case of the angle comb by using a stabilized optical frequency comb associated with a frequency standard with an uncertainty of less than 10−9, the diffraction angle error would be smaller than 10−6 arc-second, which is negligibly small.

Fig. 9.4: Estimated diffraction angle error in an angle comb with a stabilized optical frequency comb.

Fig. 9.5: The diffracted beams in the optical lever comb with a low-repetition rate.

9.2 Intensity-domain angle comb

431

Fig. 9.6: The diffracted beams in the optical lever comb with a high repetition rate.

In the optical lever comb, the group of first-order diffracted beams is treated as circular graduations of an angle comb. Each diffracted beam is therefore required to be distinguished from the others. However, in a practical case, the angular distance Pi between the adjacent diffracted beams is quite small. For example, in the case of a mode-locked femtosecond laser, the repetition rate νrep is on the order of 100 MHz, corresponding to Pi of approximately 0.0001 arc-second. When Pi is quite small and the adjacent diffracted beams are overlapped as shown in Fig. 9.5, it is difficult to detect each diffracted beam independently by using the photodetector. To locate the photodetector far away from the diffraction grating will help to distinguish the neighboring diffracted beams in the group of first-order diffracted beams. However, a large space is required. The frequency spectrum of the mode-locked laser is therefore required to be modulated to have a larger pulse repetition rate so that each diffracted beam can be distinguished by the photodetector as shown in Fig. 9.6 when the photodetector is located at the position with a realistic distance from the grating reflector. This operation can be realized by inserting a Fabry–Pérot etalon with an appropriate free spectral range (FSR) in the optical path of the measurement beam, details of which are described in the following section. For separation of the diffracted beams in the optical lever comb, the pulse repetition rate of the mode-locked laser is required to be more than several tens of GHz. In order to achieve such a high pulse repetition rate, a Fabry–Pérot etalon is employed. A Fabry–Pérot etalon consists of a pair of mirror surfaces aligned parallel with each other with a certain amount of gap length. It has an effect of optical band-pass filtering in the frequency domain to modulate the repetition rate of the mode-locked laser [23] by inserting it into the optical path as shown in Fig. 9.7. When the mode-locked laser beam with a repetition rate νrep is made incident to the

432

Chapter 9 Angle comb

Fig. 9.7: Effect of the Fabry–Pérot etalon on the pulse modulation.

Fig. 9.8: Pulse modulation by fully inserted (left) and partially inserted (left) Fabry–Pérot etalons.

9.2 Intensity-domain angle comb

433

Fabry–Pérot etalon where a transparent material having a refractive index nEtalon is located in-between the pair of the mirrors with a gap h as shown in the left part of Fig. 9.8, the repetition rate of the modulated mode-locked laser beam νrep’ passed through the etalon can be expressed by [23] νrep ′ =

c 2hnEtalon

ði = 1, 2, 3, . . . , nÞ

(9:5)

It can be seen that the repetition rate of the pulsed laser is increased to the same value of the FSR of the Fabry–Pérot etalon. Consequently, the angular separation of the diffracted beams in the optical lever comb can be modulated by choosing a proper FSR of the etalon. If half of the light power is incident to the etalon as shown in the right part of Fig. 9.8, the pulse repetition rate can be further increased to about several-fold of the original FSR of the etalon [24]. In this case, the repetition rate of the mode-locked laser νrep'' passed through the etalon can be expressed by [24] νrep ′′ =

c hðnEtalon − nair Þ

ði = 1, 2, 3, . . . , nÞ

(9:6)

To investigate the feasibility of modulating the pulse repetition rate by the abovementioned methods, experiments were carried out. Figure 9.9 shows the experimental setup for verification of the spectral modulation by using Fabry–Pérot etalons. A commercial femtosecond fiber laser (C-Fiber, MenloSystems), which can generate an optical frequency comb with a center wavelength at 1,560 nm, was used as the light source. The pulse repetition rate of the femtosecond laser was 100 MHz, which was stabilized by an Rb frequency standard with an Allan variance of 2 × 10−11 (1 s). At first, the mode-locked laser collimated by using a collimating lens (CL) with a focal length of 4.67 mm was made to pass through an optical isolator composed of a pair of a polarization beam splitter (PBS) and a quarter-wave plate (QWP). After that, the mode-locked laser was made to pass through the Fabry–Pérot etalon. The modulated mode-locked laser was then collected by a fiber coupler, and its spectrum was analyzed by an optical spectrum analyzer (OSA) (AQ6370C, Yokogawa Co.). Figure 9.10 shows the measured spectrum of the mode-locked laser after passing through the Fabry–Pérot etalon. For clarity, only a part of the spectrums is shown in the figure for clarity. The modulated mode-locked laser with the FSRs of 407, 951 and 770 GHz were generated by inserting a part of the etalon as shown in the right part of Fig. 9.8. Since the measurement resolution of the OSA used in this experiment was limited to be 4 GHz, comb modes could not be distinguished without the Fabry–Pérot etalon. However, the femtosecond fiber laser with a high pulse repetition rate equal to the FSR of the Fabry–Pérot etalon was successfully generated from the repetition rate of 25–770 GHz. These results show that the repetition rate of the femtosecond fiber laser can be controlled by choosing Fabry–Pérot etalons having the appropriate FSR.

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Chapter 9 Angle comb

Fig. 9.9: Experimental setup for modulating the pulse repetition rate.

Fig. 9.10: Modulated spectrum of the femtosecond fiber laser.

9.2 Intensity-domain angle comb

435

By using the mode-locked laser, whose pulse repetition rate was modulated by using the Fabry–Pérot etalon, experiments were carried out to demonstrate the feasibility of the proposed PD-type optical lever with a mode-locked laser. A schematic of the experimental setup is shown in Fig. 9.11. The collimated mode-locked laser passed through the Fabry–Pérot etalons was made incident to a one-axis grating reflector with a grating period of 1.67 μm, which was held stationary with respect to the incident mode-laser beam. The mode-locked laser was converted to the group of first-order diffracted beams with the function of the diffraction grating. The group of first-order diffracted beams was then captured by using a beam profiler (BP209IR/M, Thorlabs), which was placed at a distance of approximately 300 mm from the grating surface.

Fig. 9.11: A schematic of the experimental setup with the beam profiler.

The measured intensity distribution of the first-order diffracted beams is shown in the upper part of Fig. 9.12 where the FSR of the Fabry–Pérot etalon was 407 GHz. It can be seen that the light spots of the first-order diffracted beams were arranged regularly in a line on the detector plane of the beam profiler. The cross-section profiles of the intensity distributions of the group of first-order diffracted beams for Fabry–Pérot etalons with different FSRs are shown in the lower part of Fig. 9.12. It

436

Chapter 9 Angle comb

can be seen that the diffracted beams corresponding to the modes of the optical frequency comb of the femtosecond fiber laser were successfully distinguished by the beam profiler when the FSR was larger than 407 GHz. On the other hand, it was difficult for the beam profiler employed in the experiment to distinguish the diffracted beams when the FSR was less than 200 GHz due to the limited lateral resolution of the beam profiler.

Fig. 9.12: Measured intensity distributions of the group of first-order diffracted beams by a beam profiler.

Then grating reflector was mounted on a stepping motor-driven tilt stage having a motion range of ±5° and a titling resolution of 1 arc-second about the Z-axis so that a tilt motion could be applied to it as shown Fig. 9.13. A plane mirror reflector was

9.2 Intensity-domain angle comb

437

also mounted on the other side of the tilt stage so that the stage tilt motion about the Z-axis could be measured by a commercial autocollimator, which was employed as a reference. It should be noted that the measurement range of the commercial autocollimator was much smaller than that of the developed optical lever. The autocollimator was employed only to carry out the precise evaluation of a period of the group of first-order diffracted beams in a limited angle range in the following experiment. To increase the lateral resolution of the beam profiler, a slit with a gap width of approximately 200 μm was prepared in front of the aperture of the beam profiler by using a pair of knife edges. The tilt stage was controlled to make a large angular motion by its full range (5º) to make the optical frequency comb scan across the slit in front of the beam profiler. Figure 9.14 shows the detected intensity output from the beam profiler with respect to the angle of rotation of the tilt stage. As can be seen in the figure, the measurement range was evaluated to be approximately 15,000 arc-seconds, corresponding to 4.2°. Figure 9.15 summarizes the relationship between the FSR of the etalons and the mean period of the group of first-order diffracted beams. In the figure, the mean pe which is riod acquired by the experiment is compared with the theoretical value P, calculated by the following equation: = P

PN

i = 1 Pi

N

=

    N  1X 1 c 1 c − sin − 1 − · sin − 1 − · N i=1 g ði + 1Þ · vrep mod + νCEO g i · vrep mod + νCEO (9:7)

where N is the number of the modes contributed to the PD output, and νrep_mod is the repetition rate of the modulated mode-locked laser. The experimental results showed a good agreement with the theoretical values, which have also verified the feasibility of modulating the repetition rate of the mode-locked laser by the Fabry–Pérot etalon. The etalon is shown in Fig. 9.8. Since the optical lever comb employs a PD to detect the overall intensities of the light spots focused on its sensitive area, it can be referred to as the overall intensitydomain angle comb, simply called the intensity-domain angle comb for clarity. Such an intensity-domain angle comb is then applied to a laser autocollimator, which is referred to be the comb autocollimator. Differing from an optical lever, the angle measurement by a laser autocollimator is not influenced by the distance between the reflector and the detector. As can be seen in Fig. 9.16, a mode-locked laser beam is collimated by a collimating lens (CL) and then projected onto the diffraction grating as a target reflector mounted on a tilt stage. Assume the initial incident angle of the laser beam is zero for simplicity. The optical modes of the incident laser are spatially separated into a group of first-order diffracted beams depending on their optical frequencies, in which the ith mode of the incident mode-locked laser corresponds to the ith first-order diffracted beam. The number of the firstorder diffracted beams is the same as that of the optical modes of the mode-locked

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Chapter 9 Angle comb

Fig. 9.13: Experimental setup for testing the response of optical lever comb to the angle of rotation of tilt stage.

Fig. 9.14: Measure response of the group of first-order diffracted beams to stage angle of rotation.

9.2 Intensity-domain angle comb

439

Fig. 9.15: Relationship between the FSR of the etalon and the typical period of the optical lever comb.

laser, which is N. It can be seen that the group of the first-order diffracted beams emanate from the incident point with corresponding diffraction angles. The diffraction angle βi of the ith first-order diffracted beam is determined by eq. (9.3). Similar to the optical lever comb, a series of focused light spots, which is referred to as the focused light spot array, are employed in the comb autocollimator. The light spots are detected by the PD for extension of the angle measurement range of the laser autocollimator. Assume that the stage is tilted from an initial position of θr1, where the first focused light spot is located at the center of the sensitive area of the PD, to an end position of θri, where the ith focused light spot comes to the center of the sensitive area. The PD will output a periodic signal, which is referred to as the autocollimator output, when the focused light spot array moves across the sensitive area of the PD associated with the tilt angle of the stage. The tilt angle Δθri (=θri–θr1, i = 1, 2,. . ., N) of the stage can thus be obtained from the following equation through counting the period number of the autocollimator output signal: Δθri = βi − β1 ði = 1, 2, ..., NÞ

(9:8)

According to eqs. (9.3) and (9.8), it can be seen that the measurement range of the comb autocollimator is dominated by the spectral range of the mode-locked laser. For this reason, a mode-locked femtosecond laser with a wide spectral range is employed in the following experimental setup. As for a practical application of the comb autocollimator, there would be more issues that should be carefully taken into consideration. The most important issue is the characterization of the focused light spot array. The focused light spot array can be characterized by three parameters that are the diameter dgi of the ith focused

440

Chapter 9 Angle comb

Fig. 9.16: Basic concept of the comb autocollimator.

light spot, the angular intervals Δβi and the separation distance sgi between the ith and i + 1th focused light spots. dgi, Δβi and sgi can be expressed by dgi = 1.22

fc nair Db νi

(9:9)

Δβi = βi + 1 − βi

(9:10)

sgi = 2f · Δβi = 2f · ðβi + 1 − βi Þ

(9:11)

where Db is the diameter of the incident mode-locked laser beam, f is the focal length of the CO.

9.2 Intensity-domain angle comb

441

Fig. 9.17: Modulation of the focused light spot array by using a Fabry–Pérot etalon.

The width of the sensitive area of the PD should be designed to be slightly smaller than the diameter of the focused light spot dgi so that there is always only one light spot to be located on the sensitive area of the PD. In addition, the separation distance sgi between each two neighboring focused light spots is also an important factor. Similar to that shown in Fig. 9.5, a mode-locked laser with a low repetition frequency can lead to overlap between each two neighboring focused light spots. Assume the mode-locked laser has a center frequency of 200 THz and a repetition frequency νrep of 100 MHz, the diameter of the incident beam diameter is collimated to be 1 mm and the focal length of the CO is 20 mm. Under such a condition, a diameter dgi of the focused light spot around the center frequency is approximately

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30 μm. However, a separation distance sgi between each two neighboring focused light spots is only approximately 30 nm, which is much smaller than the diameter of the focused light spot. As a result, each of the focused light spots would be very difficult to be distinguished in the autocollimator output curve. In order to separate the focused light spots from each other, it is necessary to use a mode-locked laser with a high repetition frequency. For example, νrep should be larger than 100 GHz to avoid the overlap of light spots. However, a mode-locked laser with such a high νrep is difficult to be made and not commercially available. To deal with this problem, the modulation method for effective repetition frequency multiplication shown in Fig. 9.7 is employed. A more detailed schematic is shown in Fig. 9.17. As shown in the figure, the density of the spectrum of the modelocked laser is reduced by the filtering effect of the etalon. The effective repetition frequency of the mode-locked laser can be modulated to the free spectral range νFSR of the etalon [23]. Since νFSR of a commercially available etalon is available over a wide range from 2 GHz to 30 THz [25], it provides flexibilities in controlling the focused light spot array for the comb autocollimator. It should be noted that the intensities of the filtered optical modes are not totally reduced to zero depending on the finesse of the etalon as shown in the figure. As a result, the focused light spot array of the first-order diffracted beams will have a continuous intensity distribution on the focal plane of the CO. After the filtering, the total number of the first-order diffracted beams is reduced to M, which is a fraction of N. The corresponding first-order diffracted beams are thus renumbered by j (j = 1, 2,. . ., M). Figures 9.18–9.20 show simulation results of the autocollimator output with etalons of different νFSR. In the simulation, the repetition frequency of the modelocked laser is set to be 100 MHz, the focal length of the CO is set to be 50 mm and the diameter of the incident beam is set to be 1 mm. The simulated autocollimator output in the figure is normalized with respect to its maximum amplitude. For the sake of simplicity, the finesse of the etalon is set to be infinite in the simulation. As can be seen from the figure, it is necessary to use an etalon with a νFSR larger than 200 GHz for distinguishing each of the focused light spots. It should be noted that in a real situation, the resolution and stability of the etalon will be influenced by factors such as the material, finesse, thermal expansion and setting type of the cavity. For achieving an ultra-stable and ultra-sensitive performance, it is necessary to use a stabilization technique such as the Pound-Drever-Hall (PDH) method to stabilize the etalon [1, 26]. Figure 9.21 shows a schematic of the autocollimator output with respect to the angle of the tilt stage. The stage is assumed to have a tilt angle of Δθzj_l about the Zaxis. In responding to Δθzj_l, all of the first-order diffracted beams will have the same angular deviation of Δθzj_l. This causes the focused light spot array to move from the initial position where the first light spot is located at the sensitive area of the PD to the end position where the jth focused light spot comes to the position of

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9.2 Intensity-domain angle comb

Fig. 9.18: Simulation result of the autocollimator output with Fabry–Perot etalons with an FSR of 100 GHz.

Fig. 9.19: Simulation result of the autocollimator output with Fabry–Perot etalons with an FSR of 200 GHz.

Fig. 9.20: Simulation result of the autocollimator output with Fabry–Perot etalons with an FSR of 770 GHz.

the PD. Without loss of generality, assume that the first focused light spot has a distance of Δl1 from the center of the sensitive area of the PD at the initial position, and the jth focused light spot has a distance of Δl2 from the center of the sensitive area of the PD at the end position. The moving distance of the focused light spot array is thus a sum of Δl1-j, Δl1 and Δl2, where Δl1-j is the distance between the first spot and the jth spot. Based on the principle of autocollimation, Δθzj_l can be obtained by Δθzj l =

Δl1 − j − l Δl1 − j + Δl1 + Δl2 = 2f 2f

(9:12)

Substituting eq. (9.11) to eq. (9.12) gives Δθzj l =

2f ðβj − β1 Þ + Δl1 + Δl2 Δl1 Δl2 = ðβj − β1 Þ + + 2f 2f 2f

(9:13)

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where Δl1 and Δl2 can be obtained from the autocollimator output. It can be seen that the measurement range is mainly dominated by the difference between βj and β1, which is determined by the spectral range of the modelocked laser and the grating period. When a mode-locked laser with a spectral range of 10 THz and a center frequency of 200 THz, and a diffraction grating with a pitch period of 2.0 μm are used, the angle measurement range of the comb autocollimator can be as large as 10,000 arc-seconds (2.78°). This makes it possible for the comb autocollimator to have a measurement range significantly expanded from that of a conventional laser autocollimator.

Fig. 9.21: Schematic of the autocollimator output with respect to the tilt angle.

Figure 9.22 shows the experimental setup (Setup 1) for testing the feasibility of the comb autocollimator with an expanded angle measurement range. An optical frequency comb was generated from the commercial femtosecond fiber laser source (C-Fiber, MenloSystems Inc.). The repetition frequency νrep of the femtosecond laser was 100 MHz. The spectral range was from approximately 185 THz to approximately 193 THz with a center frequency of approximately 189 THz. The laser light emitted from the fiber output port of the laser source was collimated to a beam with a diameter Db of 0.9 mm by a collimating lens (CL) with a focal length of 4.67 mm. An isolator composed of a polarization beam splitter (PBS) and a quarter-wave plate

9.2 Intensity-domain angle comb

445

(QWP) was located behind the CL to prevent light from being reflected back towards the laser source. The collimated beam was made to pass through a Fabry–Pérot etalon with a νFSR of 770 GHz, a finesse of 6 and an effective diameter of 2 mm. The laser beam after passing through the etalon was then projected onto a diffraction grating, which was mounted on a motorized tilt stage. The period of the diffraction grating was 1.67 μm. An achromatic lens was employed as the collimator objective (CO) of the autocollimation unit. The focal length of the CO was designed to be 50 mm. The light spot diameter and the spot separation distance of the focused diffracted beams near the center frequency were approximately 104 and 475 μm, respectively. The width of the sensitive area of the single-cell PD was selected to be 300 μm, which was smaller than the separation distance. Figure 9.23 shows a measurement result of the output of the comb autocollimator with respect to the tilt angle of the stage. The stage was tilted at a velocity of 0.076 mrad/s and the output was acquired with a sampling rate of 65.2 Hz. The horizontal axis of the figure is the angle of the tilt stage which had been calibrated by using a commercial autocollimator, and the vertical axis is the output of the autocollimator. As can be seen in the figure, the autocollimator output varied periodically when the stage tilted over a range of 4,000 arc-seconds (1.1°). There were approximately 6 periods over the range, each corresponded to an individual focused light spot of the focused light spot array. The average period between two focused light spots was approximately 667 arc-seconds. The continuous variations of the comb autocollimator output could be employed to detect the angles of the tilting stage. It should be noted that the amplitude of every period of the output signal, which was corresponding to the intensity of each focused light spot, was not identical to the other. The visibility of the comb autocollimator output defined in the following equation, which is a typical metric of signal quality, is employed to make a quantitative evaluation of the output of the comb autocollimator: visibility =

Vk max − Vk min Vk max + Vk min

(9:14)

where Vkmax and Vkmin are the maximum and minimum output of the autocollimator in the kth period. It can be seen that the visibility at the 5th period was the maximum, which was 0.21. A possible way to increase the output visibility of the comb autocollimator is to reduce the size of the focused light spot as shown in Fig. 9.24. Since the total intensity of each focused light spot keeps constant which is determined by the intensity of the incident mode-locked laser beam and the diffraction efficiency of the grating, the size reduction of the focused light spot would lead to a steeper slope intensity distribution for each focused light spot, resulting in higher output visibility. According to eq. (9.9), it can be seen that the larger the diameter Db of the incident beam, the smaller the size of the focused light spot. Thus, an experimental setup (Setup 2) with a beam expander as shown in Fig. 9.25 was established for this

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Fig. 9.22: The experimental setup (Setup 1) for feasibility test of the comb autocollimator.

Fig. 9.23: Measured comb autocollimator output in Setup 1.

9.2 Intensity-domain angle comb

447

Fig. 9.24: Influence of the sizes of the focused light spot array on the output visibility.

purpose. The beam expander was composed of a collimator objective (CO) with a focal length of 9.0 mm and a collimating lens (CL) with a focal length of 100 mm. By using the beam expander, the diameter of the incident laser beam was increased from 0.9 mm to 10 mm through controlling the ratio of the focal length of the CO to that of the CL. Figure 9.26 shows a measurement result of the comb autocollimator output based on Setup 2 with the beam expander. As can be seen in the figure, the output visibility of the comb autocollimator did not increase, but decreased instead. One of the main reasons was considered to be the sideband effect of the etalon as shown in Figs. 9.27 and 9.28, in which the side modes could not be suppressed, and the total intensity distribution of the focused light spot array did not change too much with or without the beam expander. The position alignment errors of the CO and the CL were considered as one of the other reasons. However, even with a perfect beam expander, it is still difficult to work well for this case since most of the beam expanders only fit for single-wavelength lasers. When a mode-locked laser with multi-wavelength optical modes is used, it is difficult to focus all the optical modes at the same point by the CO which will lead to light aberration. The experimental setup was then improved by a different method without using the beam expander. In the improved setup, the diffraction grating was kept stationary and a tilting plane mirror was added as the target reflector. In this way, the focused light spots could be further separated as shown in Fig. 9.29, by which both of

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Fig. 9.25: Schematic of the experimental setup (Setup 2) with a beam expander.

Fig. 9.26: Measured comb autocollimator output in Setup 2.

the output visibility and the angle measurement range can be improved. As can be seen in the figure, the group of the first-order diffracted beams from the stationary diffraction grating were projected onto the plane mirror mounted on a motorized tilt stage. The angular intervals between each two neighboring focused light spots could be extended to be twice larger than that in the above experimental setups.

9.2 Intensity-domain angle comb

449

Fig. 9.27: Schematic of the intensity distributions with the beam expander.

Fig. 9.28: Schematic of the intensity distributions without the beam expander.

Fig. 9.29: The improved experimental setup (Setup 3).

Figure 9.30 shows the measurement output of the comb autocollimator with respect to the angles of the tilting stage based on Setup 3. The output signal was processed by using the moving-average method to reduce the electronic noise. The

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angular motion speed and the sampling frequency of the output of the comb autocollimator were set to be the same as those in the experiments described in Setups 1 and 2. Although the absolute value of the output of the comb autocollimator was smaller than that in Fig. 9.23 because some of the power of the diffracted beams was lost at the plane mirror reflector, the maximum output visibility was improved from 0.21 to 0.50, which means the method was effective for increasing the quality of the output signal of the comb autocollimator. In addition, as can be seen in the figure, the output curve had approximately 9 periods covering an angle measurement range up to 11,000 arc-seconds (3.06°), which was a significant extension from that of a conventional laser autocollimator as well as that of the multi-PD array type laser autocollimator. The average period of the periodic output of the comb autocollimator was approximately 1,333 arc-seconds. On the other hand, the amplitudes of the autocollimator output were different over different periods as shown in Fig. 9.30. This was caused by the differences in the intensities of the focused light spots. The comb autocollimator output was then normalized by dividing it by the amplitude of the output at each period. A part of the normalized output is shown in Fig. 9.31.

Fig. 9.30: Measured comb autocollimator output in Setup 3.

Fig. 9.31: Normalized comb autocollimator output.

9.3 Optical frequency-domain angle comb

451

9.3 Optical frequency-domain angle comb In the intensity-domain comb autocollimator presented in the previous section, the angular separation between each two neighboring first-order diffracted beams, which is determined by the mode spacing of the employed optical modes of the mode-locked laser, is an important factor to determine the performance of the autocollimator. Ideally, it is desired to utilize all the optical modes of the mode-locked laser for achieving the utmost sensitivity of angle measurement subject to the conditions that the optical modes can be separated by the diffraction grating reflector to form the group of the first-order diffracted beams and the linear array of the focused light spots can be distinguished by the PD to generate the continuous and periodic output of the autocollimator with respect to a continuous angle variation of the grating reflector. However, even for a mode-locked femtosecond laser, the repetition rate, which is the minimum mode spacing between the optical modes, is typically on the order of 100 MHz [27, 28] and is too small to be separated in space by a diffraction grating. A Fabry–Pérot cavity is thus used in the intensity-domain comb autocollimator to filter the optical modes [22–25]. The mode spacing of the transmitted optical modes, called the effective repetition frequency, can be enlarged to the free spectral range (FSR) of the Fabry–Pérot cavity. Because the FSR can be much larger than the repetition rate of the mode-locked laser, the transmitted optical modes can be employed for the comb autocollimator. The effectiveness of this technique for the comb autocollimator has been demonstrated by using the Fabry–Pérot cavities with FSRs larger than 100 GHz. In this technique, for a certain FSR, the finesse of the Fabry–Pérot cavity must be low enough so that the adjacent light spots focused on the sensitive area of the PD can have a long enough overlapping length with each other for the array of the focused light spots to have a continuous intensity distribution and the autocollimator to have the continuous and periodic output. However, the overlapping of the light spots can significantly increase the minimum value of the output signal, resulting in a reduction in the visibility of the output signal as well as the measurement sensitivity of the autocollimator. Although efforts have been made to improve the output visibility, only maximum output visibility of 0.50 was achieved with a complicated double-reflection configuration. It should be noted that, in the intensity-domain comb autocollimator, the conventional PD-based detection method was employed, which can only detect the overall intensities of the light spots focused on its sensitive area even though the light spots are corresponding to the optical modes with different optical frequencies. This is the fundamental reason for the limited output visibility of the intensity-domain comb autocollimator. An improved angle comb, which is referred to as the optical frequency-domain angle comb (comb autocollimator), is presented in this section to increase the output visibility as well as the sensitivity of the comb autocollimator. Taking into consideration each of the first-order diffracted beams has a determinate and unique optical frequency, the corresponding light spot on the focal plane of the collimator

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objective can be completely distinguished and separated from the other light spots by detecting the light intensity over the optical frequency domain with an OSA, even if the light spots are overlapped with each other in the space domain. As a result, the output visibility of the optical frequency-domain comb autocollimator can be theoretically improved to 100%.

Fig. 9.32: Principle of the optical frequency-domain angle comb (comb autocollimator).

Figure 9.32 shows the principle of the optical frequency-domain angle comb (comb autocollimator). The focused first-order diffracted beams with overlaps are detected by an OSA through a fiber connection instead of detection with a PD. Since the focused first-order diffracted beams have determinate optical frequencies as those of the corresponding transmitted optical modes which are spaced by the repetition rate over the spectrum of the modulated femtosecond laser, the output curve of each first-order diffracted beam with respect to the angle variation of the grating reflector can thus be clearly distinguished with high visibility in the frequency domain by the OSA on the condition that the frequency resolution of the OSA is

9.3 Optical frequency-domain angle comb

453

smaller than the mode spacing of the femtosecond laser. As shown in the figure, by utilizing this method, the output of the femtosecond laser autocollimator provides a special three-dimensional (3D) observation which is different from the twodimensional (2D) observation in the conventional laser autocollimator. In the 3D output observation shown in the figure, the X-axis represents the angle variation of the grating reflector, the Y-axis indicates the optical frequency of the first-order diffracted beam, and the Z-axis represents the light intensity of the focused beams received by the OSA. The YZ-view corresponds to the optical frequency domain which is represented by the optical spectrum of the modulated femtosecond laser. Assume the grating reflector has an angular motion for the OSA to receive the group of the focused first-order diffracted beams one by one from the first beam with an optical frequency of v1 to the last beam with an optical frequency of vn, the light intensity received by the OSA varies periodically with the tilt angle variation of the grating reflector. The output of the femtosecond laser autocollimator consists of a number of curves, each of which is corresponding to each of the first-order diffracted beams. In the optical frequency domain, the output curves are distinguished with the spacing equal to the mode spacing of the modulated femtosecond laser. In the XZ-view, the output curves have gradually shifted one by one with respect to the tilt angle change of the grating reflector, in which the shift amplitude in the angle variation axis (X-axis) is equal to the angular interval of the first-order diffracted beams. In this way, unlike the conventional detection method which suffers from low visibility influenced by the overlapped diffracted beams, the optical frequency domain angle measurement method provides a possibility to obtain good visibility over a large measurement range without compromising the continuous detection capability. A simulation of the optical frequency-domain comb autocollimator is carried out. In the simulation, the femtosecond laser is set to have a central frequency of 193 THz and a repetition rate of 100 MHz. The FSR of the Fabry–Pérot cavity is set to be 100 GHz. For the sake of simplicity, the finesse of the cavity is set to be infinite. The diameter of the incident femtosecond laser beam is set to be 3.0 mm and the focal length of the collimator objective is 15.37 mm. The diffraction grating reflector is set to have a grating period of 1.05 μm. The total light intensity of the incident femtosecond laser is set to be 1. Taking into consideration that the purpose of this simulation is only to demonstrate the feasibility of distinguishing the output curve by the differences of their corresponding optical frequencies, the intensity of each optical mode of the femtosecond laser was set to be constant and the chromatic aberration effect of the collimator objective by which the different wavelengths will be focused on slightly different focal planes was not considered for the sake of simplicity. Figure 9.33 shows a part of the simulation result over a tilt angle variation range of 2,100 arc-seconds around the first-order diffracted beam corresponding to the optical mode at the central frequency. As can be seen in the figure, the visibility of the output of the femtosecond laser autocollimator based on the conventional method, which utilizes a PD to detect the overall intensity of the focused first-order diffracted

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beams on its sensitive area, is only 5.1%. On the contrary, the output of the optical frequency domain comb autocollimator, by which the output curves are distinguished according to the optical frequencies of the first-order diffracted beams, can be up to 100% that is much higher than the intensity-domain comb autocollimator.

Fig. 9.33: Simulated outputs of the comb autocollimators.

The feasibility of the optical frequency-domain comb autocollimator was tested by experiment. Figure 9.34 shows a schematic of the experimental setup. A collimated laser beam from a femtosecond laser generator (C-Fiber 780 HP, MenloSystems GmbH) was employed as the measurement light source. The femtosecond laser had a central wavelength of 1,560 nm and a repetition rate of 100 MHz. The spectral width of the femtosecond laser was 35 nm. The repetition rate of the femtosecond laser was modulated by using a Fabry–Pérot cavity with an FSR of 100 GHz and finesse of 208. After passing through the cavity, the collimated femtosecond laser was projected onto a diffraction grating reflector mounted on an air-bearing spindle with an angle of incidence of approximately 23.3°. The grating period of the diffraction grating reflector was 1.05 μm. The spindle had a rotational resolution of 0.0038 arc-second measured by an embedded rotary encoder. A series of the first-order diffracted beams from the grating reflector was focused by a collimator objective mounted on a fiber alignment stage. The focal length of the collimator objective was 15.37 mm. The focused first-order diffracted beams were received by a single-mode optical fiber and detected by an OSA. The OSA covered a wavelength ranging from 600 nm to 1,700 nm and had a wavelength detection resolution of 0.02 nm. The dynamic range of the OSA was 60 dB. It should be noted that since the focus of this paper was to propose and verify the feasibility of the optical frequency-domain comb autocollimator, for the sake of simplicity, the oscillator of the femtosecond laser was operated at the free-running condition. In the future application of this method, the femtosecond laser can be locked to a frequency standard such as a rubidium oscillator for frequency stabilization. It should be noted that the usage of the OSA and the femtosecond laser generator

9.3 Optical frequency-domain angle comb

455

will not influence the size of the frequency-comb autocollimator, because they can be placed at outward positions and can be connected to the comb autocollimator through the single-mode fibers.

Fig. 9.34: Experimental setup for testing the frequency-comb autocollimator.

Figure 9.35 shows a two-dimensional (2D) profile and a three-dimensional (3D) profile of the first-order diffracted beams detected by using a beam profiler (BP209-IR/M, Thorlabs). It can be seen that each different first-order diffracted beam could not be distinguished from the observation of the beam profiler, which indicated that the first-order diffracted beams had serious overlaps among each other. It was thus difficult to be employed for angle measurement by using the intensity-domain comb autocollimator.

Fig. 9.35: Intensity distribution of the group of the first-order diffracted beams measured by a beam profiler.

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On the other hand, Fig. 9.36 shows a part of the measurement result based on the frequency-domain comb autocollimator. The measurement range shown in the figure was 432 arc-seconds. As can be seen in the figure, although the group of the first-order diffracted laser beams could not be separated in space, in the frequency domain, the output curves corresponding to each of the focused first-order diffracted beams with respect to the change in the tilt angle of the grating reflector were successfully distinguished, which provided full output visibility of 100% for the angle measurement. Within the measurement range shown in the figure, there were five output curves in total detected by the OSA located at the frequencies of 199.875, 199.975, 200.075, 200.175 and 200.275 THz, respectively. The frequency spacing of the output curves in the frequency domain was 100 GHz, which was the same as the repetition rate of the modulated femtosecond laser. The mean angular interval between the apexes of the output curves was evaluated to be approximately 110 arc-seconds. It should be noted that differing from the simulation results, in which a constant intensity was set for all of the optical modes, the light intensity amplitudes of the output curve at different optical frequencies were not constant in the experiment because each of the optical modes of the femtosecond laser had light intensity different with each other. In addition, although the intensity variation could also be influenced by the chromatic aberration of the collimator objective, this can be eliminated by employing a collimator objective well corrected in terms of the chromatic aberration.

Fig. 9.36: Output of the femtosecond laser autocollimator by the optical frequency domain measurement method.

9.3 Optical frequency-domain angle comb

457

Fig. 9.37: Measurement results by the frequency-domain comb autocollimator at spindle angular position of 0°.

Fig. 9.38: Measurement results by the frequency-domain comb autocollimator at spindle angular position of 3°.

Fig. 9.39: Measurement results by the frequency-domain comb autocollimator at spindle angular position of 6°.

Experiments were then extended to a wide measurement range. For reducing the experiment time and the number of the measured data for the sake of simplicity, instead of continuous rotation and sampling, the spindle was controlled to rotate from 0.0° to 6.0° by 6 large steps of 1.0°, and at each large step, it was rotated 0.12°

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by 40 small steps of 0.003°. Figures 9.37–9.39 show the measurement results by the frequency-domain comb autocollimator at the angular positions from 0.0° to 6.0°. It can be seen that over a wide measurement range of 21,600 arc-seconds, all of the output curves could be clearly distinguished by the classification of optical frequencies and the output visibilities were always kept to be as full as 100%. The gray dash line in each figure represents the total intensity output detected by the traditional method. From Figs. 9.37–9.39, the output visibilities by the conventional detection method were evaluated to be 80.9%, 79.1% and 38.5%, respectively. The visibility variations were due to the different angular separations of the first-order diffracted beams at different spectrum locations of the femtosecond laser. A comparison of the output visibilities by the frequency-domain and the intensity-domain comb autocollimators is summarized in Fig. 9.40. As can be seen in the summarized results shown in Fig. 9.40, the output visibilities have been increased from the level of 0.4–0.8 by the intensity-domain comb autocollimator to a constant value at 1.0 over the entire measurement range by the frequencydomain comb autocollimator.

Fig. 9.40: Visibilities of the frequency-domain and the intensity-domain comb autocollimators.

Fig. 9.41: Intensity stability of the optical mode at the central wavelength.

9.3 Optical frequency-domain angle comb

459

Fig. 9.42: FFT of the optical mode at the central wavelength.

The angle measurement stability by the frequency-domain comb autocollimator was then evaluated. Figure 9.41 shows the intensity stability of the first-order diffracted beam corresponding to the optical mode at the central wavelength of 1,560 nm over a time of 15 s. The peak-to-valley (PV) deviation and the standard deviation of the light intensity were evaluated to be 6.40 and 0.88 nW, respectively. The FFT analysis of the intensity stability is shown in Fig. 9.42. The dominating frequencies were mainly below 100 Hz, the root causes of which were considered to be the mechanical vibration of the experimental setup and the influence from the air. Figure 9.42 shows the angle measurement stabilities corresponding to each of the detected optical modes in Figs. 9.37–9.39, which were calculated from the PV deviation and standard deviation of the light intensities, respectively. The minimum measurement stability, which was located at the optical mode with a frequency of 199.97 THz, was evaluated to be 0.03 arc-second based on the calculation by the standard deviation. This shows that the system is possible to have a potential minimum measurement resolution on the order of 0.03 arc-second. However, detailed analysis and experimental verification of the measurement resolution and accuracy should be further carried out in future work. In addition, as can be seen in the figure, the measurement stabilities varied at each optical frequency. This was due to the frequency and intensity instability of the optical modes of the femtosecond laser used in the autocollimator. Although the mode-locked femtosecond laser could be phaselocked to a frequency standard such as an atomic clock for superior frequency and intensity stabilities, the Fabry–Pérot cavity could still induce additional influences on the stability of the angle measurement results. Some available techniques for stability analysis and stabilization of the Fabry–Pérot cavity such as the Pound-DreverHall (PDH) method [26, 1] could be employed for achieving better performances of the autocollimator. The angle measurement sensitivities were evaluated by taking the measurement result shown in Fig. 9.39 corresponding to first-order diffracted beams with optical frequencies of 203.3, 203.4 and 203.5 THz. Figure 9.44 shows the sensitivity evaluation results. The sensitivities were evaluated by calculating the slope at the

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Fig. 9.43: Stabilities in the angle measurement with respect to the optical frequency.

locations of A, B and C marked in the figure. For the traditional method by detecting the overall intensity output as plotted by the gray dash lines in the figure, the measurement sensitivities for the locations A, B and C were evaluated to be 3.16 × 10−7, 3.87 × 10−7 and 3.07 × 10−7 mW/arc-second, respectively. Since the capability of the OSA in detecting small light intensity was higher than that of an ordinary PD, the measurement sensitivities can be much lower in the conventional laser autocollimator than the above-evaluated values. On the other hand, with the frequency-domain comb autocollimator, the measurement sensitivities at locations of A, B and C were improved to be 3.54 × 10−7, 4.06 × 10−7 and 3.60 × 10−7 mW/arc-second with improvements of 11.2%, 10.5% and 11.7%, respectively, compared to those by the intensitydomain autocollimator. Results are also summarized in Tab. 9.1.

Fig. 9.44: Measurement results for calculation of sensitivities at the marked locations.

9.3 Optical frequency-domain angle comb

461

Tab. 9.1: Sensitivities of angle measurement. Location

Sensitivity by the intensitydomain comb autocollimator −mW/arc-second

Sensitivity by the frequencydomain comb autocollimator −mW/arc-second

Improvement, %

A

.

.

.

B

.

.

.

C

.

.

.

It should be noted that non-uniform angle measurement sensitivity/resolution exists over the measurement range of the frequency-domain comb autocollimator because the intensity of the output signal could change with the instability of the incident laser beam and the variation of diffraction efficiency of the grating reflector according to the tilt angle. However, this can be well solved simply by monitoring the light intensity fluctuation of the diffraction beam with a photodiode having a.=relatively large sensitive area through splitting the beam with a beam splitter located in front of the collimator objective [21]. In addition, a Fabry–Pérot cavity with a relatively large FSR and a relatively small finesse was employed in the experiment, which was sufficient for the purpose of verifying the feasibility of the frequency-domain comb autocollimator in visibility improvement. For this reason, the angular pitches of the frequency domain outputs of the mode-locked laser were relatively large and the variation of each of the outputs with respect to the tilt angle variation was relatively smooth. This is why the improvement in the sensitivity and resolution shown in the experiment was not much significant. However, in principle, the sensitivity and resolution can be significantly improved by using a Fabry–Pérot cavity with a small FSR and/or a large finesse. The sensitivity can also be further improved by making use of the relationship between the multiple optical frequency domain outputs with a 100% visibility of the femtosecond mode-locked laser autocollimator. As a consequence of the improvements in visibility and sensitivity, the resolution can be improved as well. More importantly, it is not essential to employ a Fabry–Pérot cavity in the optical frequency-domain comb autocollimator (angle comb) based on a technique shown in Fig. 9.43. The comb autocollimator can be applied to absolute angular position measurement, which is difficult for a conventional laser autocollimator with a single wavelength laser source shown in Fig. 9.45. In the figure, the angle Φ, which is the angular distance between the center axis of the autocollimation unit, is uniquely determined by the geometric relationship between the center of the detector and the collimator objective, and the incident laser beam. Once the laser source and the autocollimation unit are assembled in the laser autocollimator, Φ would be a fixed angle independent from the angular position of the reflector. On the other hand, as a practical matter, it is difficult to accurately identify the orientation of the laser

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source relative to the autocollimation unit during or after the assembly process of the laser autocollimator, resulting in an unknown or inaccurate Φ. It should be noted that Φ is often designed to be zero in commercial autocollimators. However, it is difficult to assure a zero Φ in this case due to the alignment errors of the laser source and the autocollimation unit during the assembly process of the autocollimator. In this paper, Φ is treated to be non-zero aiming not to lose generality.

Fig. 9.45: Conventional laser autocollimator.

In Fig. 9.45, the angle of incidence and that of reflection of the laser beam on the surface of the plane reflector are expressed by θ, which changes with the change in the angular position of the reflector about the Z-axis. Ideally, it is expected for the autocollimator to detect the absolute value of θ, that is, the angle between the normal of the reflector and the incident laser beam or the reflected laser beam, which is referred to as the absolute angular position of the reflector. The position of the

9.3 Optical frequency-domain angle comb

463

focused laser spot Δd with respect to the center point of the photodetector can be obtained by the following equations: Δd = f tan Δψ

(9:15)

Δψ = Φ − 2θ

(9:16)

where f is the focal length of the collimator objective in the autocollimation unit. It can be seen that only the angle Δψ, which is referred to as the relative angular position, can be obtained from eq. (9.16). Since Φ is unknown, θ is also unknown. This means that a conventional autocollimator can only measure the relative angular position Δψ, without being able to make absolute angular position measurement, that is, measurement of the absolute angular position θ. This is not a critical problem for a conventional autocollimator when it is used for measurement of the change of angular position of a reflector, that is, the change of Δψ or the change of θ, which has been the traditional application of an autocollimator, including a laser autocollimator. However, the lack of capability in absolute angular position measurement has excluded the conventional autocollimators from applications such as the measurement of surface directions for alignment of segmented mirrors [29], the measurement of orientations of a moving object [30]. Also because of the small measurement range, a conventional autocollimator cannot be used for absolute angular position measurement of a rotatory or a title stage, which is one of the major drawbacks of conventional autocollimators compared with the absolute type rotary encoders [8]. Taking into consideration that absolute angular position measurement instead of relative angle variation measurement is becoming increasingly important for precision motion systems especially for those in which repetitive positionings are required even when the power is re-started from an interruption of system power supply [2], it is desired to measure the absolute angular position of the reflector. A schematic of the absolute comb-autocollimator for absolute angular position measurement is shown in Fig. 9.46. It should be noted that the zeroth-order diffracted beam, which is not employed for absolute angular position measurement in this study, is not shown in the figure for the sake of clarity. Let the angular position of the incident femtosecond laser beam be the zero angular position. Denote the angle between the normal of the grating reflector and the incident laser beam by θ, which is referred to as the absolute angular position of the grating reflector. The diffraction angle of the ith first-order diffracted beam corresponding to the ith optical mode of the optical frequency comb with respect to the normal of the grating reflector is denoted by βi. According to the diffraction theory [22], there is the following relationship: sin θ + sin βi =

cn g · vi

(9:17)

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Chapter 9 Angle comb

Fig. 9.46: Schematic of an absolute comb autocollimator.

where g is the grating period, and cn is the speed of light in air. The angular position of the ith first-order diffracted beam γi with respect to the zero angle position can be expressed by   cn − sin θ (9:18) γi = θ + βi = θ + arcsin g · vi Now we denote the angle of grating reflector θ as θi when the axis of the ith firstorder diffracted beam coincides with the axis of the autocollimation unit. In this case, the angular distance Φ between the axis of the autocollimation unit and the incident femtosecond laser beam is expressed by   cn (9:19) − sin θi Φ = γi = θi + arcsin g · vi From eq. (9.20), θi can be obtained as follows: Φ cn 1

Φ  · θi = − arcsin 2 2g · cos 2 vi

! (9:20)

As can be seen in eq. (9.21), θi is the absolute angular position of the grating reflector discretized by the comb mode frequency vi of the optical frequency comb under the condition where both Φ and the grating period g are known. Since g can be calibrated with a low uncertainty below 10 pm [31, 32], only the identification of Φ remains to be addressed. Figure 9.47 shows a schematic of a differential method introduced for identification of Φ by utilizing an additional angle sensor which is

9.3 Optical frequency-domain angle comb

465

referred to as the assisted angle sensor. Owing to the advantage of multiple diffracted beams in the femtosecond laser autocollimator instead of a single reflected beam in the conventional laser autocollimator, as shown in the left part of the figure, the grating reflector is first aligned at the position where the ith first-order diffracted beam locates at the center of the photodetector, and then, as shown in the right part of the figure, it is moved to the next angular position where the i + 1th beam locates at the center of the photodetector.

Fig. 9.47: Identification of the angle Φ.

According to eq. (9.21), the angular variation Δθi of the grating reflector between the two angular positions can be expressed as follows: Δθi = θi − θi + 1     cn 1 cn 1 − arcsin = arcsin · · 2g · cosðΦ=2Þ vi 2g · cosðΦ=2Þ vi + 1

(9:21)

where Δθi can be measured by utilizing the assisted angle sensor, while the frequencies of ith- and i + 1th first-order diffracted beams vi and vi +1, respectively, can be detected by the photodetector of the autocollimation unit. Here, for the ideal case in principle, it is assumed that the resolution of the photodetector is enough for the distinction of vi and vi +1. A practical case, in which the resolution of the photodetector is not enough for the distinction of vi and vi +1, will be analyzed and discussed later. Thus, Φ is able to be identified based on eq. (9.22) and the absolute angular position θi of the grating reflector can be consequently determined from the optical frequency vi of the optical frequency comb based on eq. (9.22). It should be noted that the absolute angular position θ, but not the angle Φ, is the final measurand of the proposed method. In another word, the identification of Φ is only for the purpose of measuring θ. Once Φ is identified during the assembly process of the femtosecond

466

Chapter 9 Angle comb

laser autocollimator, the identified value can be used in the subsequent absolute angular position measurement of θ without the assisted angle sensor, since the angular distance Φ is fixed in the femtosecond laser autocollimator. It should be noted that the incident point of the femtosecond laser is not necessarily located at the center of rotation of the grating reflector, which would cause a parallel shift of the group of the first-order diffracted beams when the grating reflector is being rotated, since the position of the focused spot of the first-order diffracted beams on the focal plane will not change based on the principle of autocollimation without influencing the performances of the comb autocollimator. An absolute comb autocollimator shown in Fig. 9.48 was established for the feasibility test. The same femtosecond fiber laser as that in the previous setups was employed as the light source. In the initial verification experiment, the femtosecond laser was operated at a free-running condition without locking to a frequency standard for the sake of simplicity of the setup. A diffraction grating with a period g of 1.052 μm was mounted onto an air-bearing spindle, which served as the target stage whose absolute angle was to be measured. Since this chapter is focused on the proposal and verification of the proposed method for absolute angular position measurement, the value of g provided by the grating manufacturer was employed. In the next step of research, accurate calibration of g is to be made by using the method of Ref. [32]. A rotary encoder with a measurement resolution of 0.0038 arcsecond was embedded with the spindle and employed as the assisted angle sensor for identification of the angular distance Φ. The first-order diffracted beams from the grating reflector were collected by a collimator objective with a focal length of 15.37 mm and were then coupled into a single-mode optical fiber (input fiber), which had a core diameter of 15 μm, for detection by an OSA used as the photodetector to measure the optical frequencies of the first-order diffracted beams. The wavelength range of the OSA was 600–1,700 nm with a wavelength accuracy of ±0.01 nm. It should be noted that chromatic aberration of the collimator objective can be an uncertainty source for the measurement of absolute angular position, since the first-order diffracted beams have different wavelengths, which influence the detected intensity amplitude of each of the first-order diffracted beams by the input fiber. Differing from the ideal case shown in Fig. 9.47, in practice, it is not possible for the diffraction grating to fully separate the first-order diffracted beams in space because of the small repetition rate of the optical frequency comb (90 MHz). Thus, there would be a large overlap between each two neighboring first-order diffracted beams and a series of first-order diffracted beams instead of individual one would be received by the single-mode fiber simultaneously at a specific angular position of the grating. Figure 9.49 shows a schematic of the model used in the simulations for this practical case. Since a period of the grating was 1.052 μm shorter than the wavelengths of the laser, the angle of incidence of the femtosecond laser beam was set to be 37° to generate the first-order diffracted beams. Each of the first-order diffracted

9.3 Optical frequency-domain angle comb

467

Fig. 9.48: An absolute comb autocollimator established for absolute angular position measurement.

beams was focused by the collimator objective as a focused laser spot with a diameter of approximately 3 μm on the focal plane of the collimator objective and was received by the single-mode fiber (input fiber) of the OSA. For the sake of simplicity, consider the case where the first-order diffracted beam with the central wavelength was located in the center of the fiber. According to the geometric relationship, the separation distance between each two neighboring focused laser spots at the central wavelength (1,560 nm) of the optical frequency comb was only 74.3 nm that was much shorter than the diameter of the focused laser spot of the first-order diffracted beam. Due to the small separation, a group of first-order diffracted beams consisting of approximately 200 beams was being received by the single-mode fiber at the same time. Within a diameter of 15 μm of the single-mode fiber, the group of the first-order diffracted beams received by the fiber ranges over 20 GHz, which was only approximately 1/750 of the spectral range (15 THz) of the entire optical frequency comb. Because of the narrow spectral range, the intensity amplitudes of the received group of the first-order diffracted beams were assumed to be uniform in the simulation for the sake of simplicity. Denote the intensity distribution of each of the first-order diffracted beams by Ii, the intensity distribution of the group of the received first-order diffracted beam can be expressed by Ig =

X i

Ii =

X 2J1 ðxi Þ2 i

xi

(9:22)

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Chapter 9 Angle comb

Fig. 9.49: A schematic of the absolute angular position measurement in practical cases.

Fig. 9.50: Peak frequency to be detected by the fiber detector.

Fig. 9.51: Determination of the peak frequency by the detected spectrum.

9.3 Optical frequency-domain angle comb

469

where J1(xi) is the well-known first-order Bessel function, and xi represents the radial distance of the first-order diffracted beam in the optical coordinate that can be expressed by 2π 2π vi ri sin α = v0 ðr0 + ilΔβi Þ sin α cn cn     cn cn Δβi = βi − βi + 1 = arcsin − sin θ − arcsin − sin θ gvi gvi + 1 xi =

(9:23) (9:24)

where ri is the polar coordinate of the radial distance of the ith first-order diffracted beam, r0 is that of the first-order diffracted beam located at the center of the optical fiber, l (50 mm in this research) is the distance from the incident point on the diffraction grating to the center point of the collimator objective of the autocollimation unit, Δβi is the angular separation between the ith and the i + 1th first-order diffracted beams, and sinα (0.25 in this research) is the numerical aperture (NA) of the collimator objective. Figure 9.50 shows the simulated intensity distribution of the group of the firstorder diffracted beams before passing through the single-mode fiber. For the sake of clarity, only 20 intensity distribution curves of the group of the received first-order diffracted beams are shown in the figure. It can be seen that since the intensity distribution curve of each individual diffracted beam was overlapped with each other by a short separation, the whole intensity distribution of the group of the first-order diffracted beams demonstrated a flat distribution shape as shown in the figure. The intensity distribution of the group of the received first-order diffracted beams was then re-shaped by the function of the single-mode fiber whose radial intensity profile typically fits a Gaussian distribution [33, 34] and can be expressed by " # 4w21 w22 2ðΔdÞ2 exp − 2 (9:25) η= 2 w1 + w22 w1 + w22 where w1 and w2 are the radii of the input beam (1.5 μm) and the core of the fiber (7.5 μm), respectively, and Δd denotes the relative radial distance from the center of the fiber. Based on this, the intensity distribution of the received group of the first-order diffracted beams after passing through the fiber, denoted by If, can be expressed by If = Ig × η

(9:26)

Figure 9.51 shows the intensity distribution of the group of the received first-order diffracted beams after passing through the fiber. It can be seen that the group of the received first-order diffracted beam, which demonstrated a flat distribution before the fiber, formed a Gaussian distribution after passing through the fiber. In this case, although the frequency vi of each first-order diffracted beam cannot be distinguished by the OSA, the central frequency vj (j = 1,2,3, . . ., M, where M is the

470

Chapter 9 Angle comb

number of the group and is a fraction of N) of the group of detected first-order diffracted beams can be measured and employed for absolute measurement of the corresponding angular position. The central frequency of the simulation in Fig. 9.51 was evaluated to be equal to the central frequency of the optical frequency comb when the first-order diffracted beam with the central frequency was located in the center of the fiber. For the sake of clarity, the detected groups of the first-order diffracted beams, which could be used for absolute angular position measurement in practical cases, are referred to as the absolute angular codes in the following. In the absolute angular position measurement, the grating reflector was rotated by the spindle for the OSA to detect the absolute angular codes. Figure 9.52 shows a detected absolute angular code by the OSA. One of the horizontal axes in the figure is the optical frequency detected by the OSA, while the other is the output of the rotary encoder recorded synchronously during the rotation of the grating reflector. It should be noted that the output of the rotary encoder was still the relative angle information. As can be seen in the figure, the intensity distribution of the absolute angular code demonstrated a Gaussian distribution as the simulation result described in Fig. 9.51. To decrease the influence of random error caused by the sampling interval frequency in the OSA on the determination of the central frequency of the absolute angular code, the central frequency of the absolute angular code is determined based on the center-of-mass method that can be expressed by the following equation: vcenter =

Pk = kend

Ik × vk 1 Pk = kend Ik 1

(9:27)

where Ik is the light intensity corresponding to the optical frequency vk within the absolute angular code, and kend represents the number of the frequency components within the detected first-order diffracted beams sampled in the OSA. According to this method, the optical frequency of the absolute angular code in Fig. 9.52 was evaluated to be 194.751 THz. In the same manner, the corresponding output of the rotary encoder can also be determined by the same method. The center-of-mass method was applied to the series of absolute angular codes for the determination of their optical frequencies. As described in the principle section, the angular distance Φ between the axis of the autocollimation unit and the incident femtosecond laser beam should be identified as a prerequisite for enabling the absolute angular position measurement. The differential method mentioned in the principle section was applied, in which the rotary encoder of the spindle was used as the assisted angle sensor for identifying Φ. Since a number of absolute angular codes would be generated in the measurement, multiple pairs of angular position intervals of the grating reflector between each two neighboring absolute angular codes can be adopted in the estimation of Φ, which contributes to reducing the estimation uncertainty. Figure 9.53 shows a measurement result of Δθi, that is, the angular variation of the grating reflector between two neighboring absolute

9.3 Optical frequency-domain angle comb

471

Fig. 9.52: Determination of the optical frequency of the absolute angular code based on the center-of-mass method.

angular positions θi and θi +1 as defined in eq. (9.22) over a frequency ranging from 194.65 to 194.85 THz. Because the spindle was controlled to rotate with a constant step of 180 arc-seconds from the output of the rotary encoder, the value of Δθi measured by the output of the rotary encoder (assisted angle sensor) constantly equals 180 arcseconds in this case. Since it is difficult to directly calculate Φ from eq. (9.22), in this chapter, Φ is estimated by fitting the calculated curve with different estimations of Φ, which is obtained with the measured central frequencies of the absolute angular codes based on eq. (9.22), to the assisted angle sensor output. Φ can thus be determined so as to minimize the standard deviation of the calculated values with respect to the values measured by the output of the rotary encoder (180 arc-seconds in this case) reached the minimum. For the sake of clarity, only three curves with three different estimations of Φ are shown in the figure. Based on this method, Φ was characterized to be 20.9345° for this experimental setup. The standard deviation of the differences between the experimental results and the theoretical values was evaluated to be 0.837 arc-second. The obtained value of Φ was adopted in the following experiments. With the identified Φ, the function of an absolute angular position measurement by the femtosecond laser autocollimator was enabled based on eq. (9.21). In order to test the feasibility, the grating reflector was controlled to rotate with a command motion step of 180 arc-seconds by the spindle for angle measurement. Figure 9.54 shows a top view of the light intensity distribution map of a part of the absolute angular position measurement result. Each of the “dots” represented each of the detected intensity distribution of a group of the first-order diffracted beams, which formed an absolute angular code for the determination of the absolute angular positions of the grating reflector. For each of the absolute angular codes detected by the OSA, the corresponding central frequency was determined by the center-of-mass method. The absolute angular position of the grating reflector was thus well decided by the central frequency of the detected absolute code according to eq. (9.21). In this

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Chapter 9 Angle comb

Fig. 9.53: Measurement results of Δθi for determination of Φ.

Fig. 9.54: Absolute angular position measurement result with a motion step of 180 arc-seconds.

9.3 Optical frequency-domain angle comb

473

case, the absolute angular positions of the grating reflector were determined from −37.805° to −37.354°. The angular interval between each two neighboring absolute angular codes was evaluated to be approximately 180 arc-seconds equal to the controlled angular motion step of the grating reflector by the spindle, thus demonstrating the feasibility of the femtosecond laser autocollimator in an absolute angular position measurement. The standard deviation of the measured values determined from the optical frequencies with respect to the controlled value of the spindle (180 arcseconds) was evaluated to be 0.652 arc-second. To investigate the resolution in the absolute angular position measurement by the absolute laser autocollimation, the spindle was controlled to rotate with a small step of 0.003° (10.8 arc-seconds) and a step of 0.0003° (1.08 arc-seconds), respectively. The measurement results are shown in Fig. 9.55 and Fig. 9.58, respectively. For showing the resolution of the absolute angular position measurement more clearly, five consecutive measured curves with step motion intervals of 10.8 arcseconds and 1.08 arc-seconds are shown in Fig. 9.56 and Fig. 9.58, respectively. As can be seen in Fig. 9.56, the detected curves with an absolute angular variation of 0.003° (10.8 arc-seconds), corresponding to the motion step of the spindle, can be clearly detected from the shift of approximately 8.0 GHz of the central peak intensity along the optical frequency axis. In addition, as can be seen from Fig. 9.58, the five curves can also clearly be distinguished along the optical frequency axis. Detection of a much smaller step motion of 0.0003° (1.08 arc-seconds) was achieved from a shift of the central peak intensity of approximately 1.3 GHz along the optical frequency axis. Therefore, the resolution of the femtosecond laser autocollimator in the absolute angular position measurement was evaluated to be about 1.08 arc-seconds. The measurement resolution in the absolute angular position measurement was influenced by the resolution of the OSA, the mechanical stability of the experimental apparatus, etc. It should be noted that this is the resolution in the determination of the discrete absolute angular positions. For relative angle measurement with a higher resolution, the absolute angular positions can be subdivided by detecting the intensity variation as the traditional laser autocollimation method [21]. Since the absolute angular position is determined from the optical frequencies of the first-order diffracted beams, the frequency stability of the femtosecond laser is also an important factor to influence the measurement performance. Figure 9.59 shows the repetition rate stability of the femtosecond laser under the free-running condition, the peak-to-valley (PV) value of the frequency drift over a time of 1,000 s was evaluated to be 15 Hz. Assume a frequency of the first comb mode in the femtosecond laser is 185 THz. Due to the drift of the repetition rate, the mode with a frequency of 192 THz (1,560 nm) in the femtosecond laser will have a corresponding frequency deviation of approximately 1.05 MHz. Figure 9.60 shows the corresponding measurement angle deviation under this condition. The measurement angle deviation was evaluated to be approximately 0.0012 arc-second. Since the measurement angle deviation was small

474

Chapter 9 Angle comb

Fig. 9.55: Top view of the intensity distribution map over a certain measurement range in absolute angular position measurement results for a small angular motion step of 10.8 arc-seconds.

Fig. 9.56: Five consecutive measured curves with motion step of 10.8 arc-seconds.

enough compared with the absolute angular position measurement resolution, the frequency instability was not a critical influencing factor for the absolute comb autocollimator. It should be noted that the influence by the drift of the carrier-envelope offset frequency, which would be about several Hz under the free-running condition over this measurement time, was not considered in this calculation due to its ignorable influence. Figure 9.61 shows the repetition rate stability of the femtosecond laser when the optical frequency comb was phase-locked to a rubidium frequency standard to stabilize both the repetition rate and the carrier-envelope offset frequency. Figure 9.62 shows the angle measurement drift corresponding to the frequency deviation. The repetition rate stability under this condition was evaluated to be 2.264 mHz over a measurement time of 1,000 s, which was much smaller than that under the free-running condition and its influences in the absolute angular position measurement can be totally ignored as shown in Fig. 9.62.

9.3 Optical frequency-domain angle comb

475

Fig. 9.57: Top view of the intensity distribution map over a certain measurement range in absolute angular position measurement results for a small angular motion step of 1.08 arc-seconds.

Fig. 9.58: Five consecutive measured curves with motion step of 1.08 arc-seconds.

Fig. 9.59: The repetition rate deviation under the free-running condition.

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Chapter 9 Angle comb

Fig. 9.60: Influences of the repetition rate deviation on the absolute angular position measurement with the first-order diffracted beam at the central frequency of 192 THz.

Fig. 9.61: The repetition rate deviation of the stabilized femtosecond laser by phase locking to a rubidium frequency standard.

Fig. 9.62: The influences of repetition rate deviation of the stabilized femtosecond laser by phase locking to a rubidium frequency standard on the absolute angular position measurement with the first-order diffracted beam at the central frequency of 192 THz.

9.4 Summary

477

In addition to the frequency stability of the optical frequency comb, according to eq. (9.23), the measurement of the absolute angular position of the grating reflector was also influenced by the grating period g, the angular distance Φ and the light speed in air cn. The calibration uncertainty of g, which can be below 10 pm by the method introduced in [32], corresponded to an angle measurement uncertainty of approximately 2.2 arc-seconds. The identification of Φ was also influenced by the wavelength accuracy of the OSA (± 0.01 nm), the resolution (0.0038 arc-second) of the rotary encoder of the spindle serving as the assisted angle sensor for the identification, the uncertainty caused by the center-of-mass method to determine the central frequency, the misalignment of the optical components, and the thermal drift during the measurement, etc. The deviation of light speed cn due to the atmospheric conditions can also be an uncertainty source.

9.4 Summary A concept of angle comb based on the optical frequency comb from a femtosecond laser has been presented to realize reliable angle measurement over an expanded angle measurement range. The overall intensity-domain angle comb has been applied to an optical lever. In the optical lever comb, a femtosecond laser beam is projected onto a grating reflector mounted on a target stage. A group of the reflected first-order diffracted beams in responding to the optical modes equally spaced over wide spectra are utilized as the scale graduations of the angle comb. A photodiode (PD) is employed to detect the overall intensities of the light spots focused on the sensitive area of the PD. The mode-locked laser with an extended mode spacing up to 770 GHz is generated in combination with a Fabry–Pérot etalon. Experimental results have demonstrated that the optical lever comb can realize a measurement range up to 15,000 arc-seconds (4.2°). The achieved measurement range is much wider than that of the conventional PD-type optical levers. The intensity-domain angle comb has also been applied to laser autocollimation for realizing a comb autocollimator with an extended angle measurement range. An optical frequency-domain angle comb has then been presented to provide a higher signal visibility and improved measurement sensitivity. In such a frequencydomain comb autocollimator, the group of the first-order diffracted beams reflected by the grating reflector, which have one-to-one correspondence relationship with the optical modes of the femtosecond laser, are focused by a collimator objective and detected by an optical spectrum analyzer (OSA) through a single-mode optical fiber. Differing from the overall intensity-domain angle comb which suffers from low output visibility and limited measurement sensitivity influenced by the overlapped first-order diffracted beams, the optical frequency domain angle measurement by the OSA can clearly distinguish the output curves, which are corresponding to each first-order diffracted beam, with respect to the angle change of the grating reflector in the frequency

478

Chapter 9 Angle comb

domain. In this way, both the signal visibility and measurement sensitivity are improved over the whole measurement range. The feasibility of the frequency-domain comb autocollimator has been verified by experimental results showing full signal visibility of 100% and an improved measurement sensitivity over a large measurement range of 21,600 arc-seconds (6°). The frequency-domain angle comb has been applied for absolute angular position measurement. Differing from the conventional laser autocollimation which can only make relative angle measurement since the angle Ø between the axis of the autocollimation unit and the incident laser beam is unknown, the absolute angular position measurement can be achieved by the angle comb based on a differential method to accurately identify the angle Φ. Because each of the first-order diffracted beams has a deterministic unique angular position depending on its optical frequency, which well corresponds to that of the optical frequency comb mode of the femtosecond laser, the absolute angular position information of the measured target with respect to the optical axis of the incident femtosecond laser can thus be well established by the detected optical frequencies of the first-order diffracted beams received by the OSA. The performances of an established absolute comb autocollimator have been presented. The absolute angular position measurement resolution and the influences by the optical frequency stability on the performances during the angle measurement have also been investigated and discussed. It has been verified that the absolute comb autocollimator can achieve an absolute angular position measurement resolution of 0.0003° (1.08 arc-seconds) and measurement stability of 0.0012 arc-second under the free-running condition of the optical frequency comb.

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Renishaw plc. RESOLUTE absolute optical encoder with Biss serial communications. (Accessed March 17, 2021, at https://resources.renishaw.com/en/details/data-sheetresolute-absolute-optical-encoder-with-biss-serial-communications--111510) Udem T, Holzwarth R, Hänsch TW. Optical frequency metrology. Nature 2002, 416, 6877, 233–237. Kanda Y, Matsukuma H, Yang S, Shimizu Y, Inaba H, Gao W Development of a fiber-laserbased frequency comb for precision dimensional metrology. 2018 IEEE International Conference on Advanced Manufacturing (ICAM), IEEE, 2018, 18–19. Kudo Y Angle scale comb using optical frequency comb. Tohoku University, Master thesis, 2015. Tamada J Angle Sensor Using Optical Frequency Comb. Tohoku University, Master Thesis, 2017. Gao W. angle measurement by using optical frequency comb. J Japan Soc Precis Eng 2018, 84, 8, 696–700. Nakamura K, Matsukuma H, Shimizu Y, Gao W Angle measurement using a diffraction of optical frequency comb. 2018 IEEE International Conference on Advanced Manufacturing (ICAM), IEEE, 2018, 26–27. Bryan JB. The Abbé principle revisited: An updated interpretation. Precis Eng 1979, 1, 3, 129–132. Moore WR, Special M, Co T Foundations of Mechanical Accuracy., 1970. Gao W, Saito Y, Muto H, Arai Y, Shimizu Y. A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann – Manuf Technol 2011, 60, 1, 515–518. Siewert F, Buchheim J, Boutet S, Williams GJ, Montanez PA, Krzywinski J, Signorato R. Ultraprecise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry. Opt Express 2012, 20, 4, 4525. Gao W, Huang PS, Yamada T, Kiyono S. A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers. Precis Eng 2002, 26, 4, 396–404. MÖLLER-WEDEL OPTICAL. Electroninc autocollimators. (Published 2007Accessed March 14, 2021, at www.moeller-wedel-optical.com) Shimizu Y, Tan SL, Murata D, Maruyama T, Ito S, Chen Y-L, Gao W. Ultra-sensitive angle sensor based on laser autocollimation for measurement of stage tilt motions. Opt Express 2016, 24, 3, 2788. Hecht E. Optics, 5th Pearson, 2017. Steinmetz T, Wilken T, Araujo-Hauck C, Holzwarth R, Hänsch TW, Udem T. Fabry-Pérot filter cavities for wide-spaced frequency combs with large spectral bandwidth. Appl Phys B Lasers Opt 2009, 96, 2–3, 251–256. Wada K, Sakai M, Watanabe H, Matsuyama T, Horinaka H Two-wavelength oscillation from laser resonator partially-inserted an optical etalon. 2009 Conference on Lasers & Electro Optics & The Pacific Rim Conference on Lasers and Electro-Optics, IEEE, 2009, 1–2. Properties of SSI’s Liquid Crystal Tunable Filter. (Accessed December 23, 2020, at https:// www.cpi.com/assets/img/LC-Properties1.pdf) Akbulut M, Davila-Rodriguez J, Ozdur I, Quinlan F, Ozharar S, Hoghooghi N, Delfyett PJ. Measurement of carrier envelope offset frequency for a 10 GHz etalon-stabilized semiconductor optical frequency comb. Opt Express 2011, 19, 18, 16851. Kim Y-J, Kim Y, Chun BJ, Hyun S, Kim S-W. All-fiber-based optical frequency generation from an Er-doped fiber femtosecond laser. Opt Express 2009, 17, 13, 10939. Wei D, Takahashi S, Takamasu K, Matsumoto H. Analysis of the temporal coherence function of a femtosecond optical frequency comb. Opt Express 2009, 17, 9, 7011.

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[29] Choi H, Trumper I, Dubin M, Zhao W, Kim DW. Simultaneous multi-segmented mirror orientation test system using a digital aperture based on sheared Fourier analysis. Opt Express 2017, 25, 15, 18152. [30] Creuze V Distance and orientation measurement of a flat surface by a single underwater acoustic transducer. Proceedings of the 19th European Signal Processing Conference (EUSIPCO 2011), IEEE, 2011, 1790–1794. [31] Gao W, Kimura A. A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating. CIRP Ann – Manuf Technol 2010, 59, 1, 505–508. [32] Korpelainen V, Iho A, Seppä J, Lassila A. High accuracy laser diffractometer: Angle-scale traceability by the error separation method with a grating. Meas Sci Technol 2009, 20, 8, 084020. [33] Ankiewicz A, Peng GD. Generalized gaussian approximation for single mode fibers. J Light Technol 1992, 10, 1, 22–27. [34] Marcuse D. Gaussian approximation of the fundamental modes of graded-index fibers. J Opt Soc Am 1978, 68, 1, 103.

Chapter 10 Chromatic confocal comb 10.1 Introduction Confocal microscopy [1, 2] is a technique that can be used for measurement of the three-dimensional (3D) profiles of microstructures. A unique function of the confocal microscopy referred to as the depth-sectioning effect enables it to achieve better performances compared with conventional optical microscopy. A laser scanning confocal microscopy (LSCM) is one of the typical types of confocal microscopy, in which a monochromatic laser source is employed [3]. In the confocal microscope, the optical head detects the height position of a measured point in focus. Therefore, it is necessary to apply a relative motion between the optical head and the target object for measurement of the 3D profile of a target surface; this is one of the drawbacks of the LSCM since time-consuming mechanical scanning is required. In addition, a high-precision scanning system is mandatory for accurate measurement. Chromatic confocal microscopy [4–13] is a candidate for addressing the above issue. With the employment of an objective lens having chromatic dispersion or a diffractive Fresnel lens, as well as a broadband light source in the chromatic confocal probe, scan-less 3D profile measurement can be achieved. In the chromatic confocal probe, the height information of a measured point is provided by the peak wavelength observed in the spectral analysis of the reflected light, since the focal length of the chromatic objective lens depends on the light wavelength. Meanwhile, the axial range of dimensions is constrained by both the dispersive properties of the chromatic objective lens and the spectral distance of the light source. As a consequence, a white light source with a wide spectrum, such as a Xenon lamp, is also used for a wide range of measurements. However, the weak spatial coherence of white light sources often leads to low illumination quality and restricts the sensitivity of the chromatic confocal probes. Another drawback that degrades the output of chromatic confocal probes is the variability of the origin of the white light spectrum. A mode-locked femtosecond laser source [14–16], consisting of a series of specific optical frequency modes equally spaced by a certain phase spacing, has an exceptionally stable optical range of high spatial coherence and is designed to overcome the above drawbacks of conventional white light sources and is intuitively suited for chromatic confocal imaging. In addition, from the point of view of finding the peak wavelength, the high intensity of the mode-locked laser source is useful for chromatic confocal imaging. There is also an exciting prospect for using femtosecond laser frequency comb technology [14] to measure the absolute wavelengths needed for traceable chromatic confocal microscopy measurement. However, the mode-locked laser source also has an exceptionally non-smooth spectrum with many peaks such as the Kelly sideband peaks induced by non-linear pulse-forming https://doi.org/10.1515/9783110542363-010

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effects [17]. The spectral width is therefore relatively narrow. This limits the use of a mode-locked laser source for a wide range of depth measurements in chromatic confocal probes. The supercontinuum laser source, which is often generated by coupling the femtosecond laser pulse to the nonlinear photonic crystal fiber, can expand the spectral range of a laser as an improved version of the mode-locked laser source [18]. It has been successfully adopted for chromatic confocal imaging [8, 9]. Nevertheless, spectral non-smoothness also limits the complete discovery of a broader axial measurement range in chromatic confocal systems of the entire spectrum of the supercontinuum laser source [9]. Nevertheless, spectral non-smoothness also limits the expansion of a broader axial measurement range in chromatic confocal systems of the entire spectrum of the supercontinuum laser source [9]. A mode-locked femtosecond laser confocal probe is developed in this study. A fiber-based optical configuration based on the dual-detector confocal configuration is designed to remove the spectral non-smoothness of the mode-locked laser source. The reflected beam is separated into two sub-beams in the optical setup, which are then made to travel through two optical paths of separate confocal configurations where two similar fiber detectors are positioned at the focal position and a defocus position for the acquisition of two confocal signals, respectively. For axial displacement estimation, an axial response known as the strength ratio of the two confocal signals is then added. Finally, the depth data is obtained by calculating the peak position of the axial response. The introduced axial response in this study is found to be more robust in eliminating the spectral response compared to the signal processing method in a chromatic confocal setup using a supercontinuum laser source, where the axial response is defined as the ratio of the intensity collected by a detector and the intensity of a reference beam obtained by removing the chromatic objective form of the setup. In addition, the axial reaction added is efficient in eliminating the reflection of the specimen, which makes our established chromatic confocal probe feasible for inhomogeneous reflectivity imaging samples. For the chromatic setup in Ref. [9], it will be difficult to work with samples with our understanding of inhomogeneous reflectivity. We are successfully expanding the working spectral range of the employed mode-locked laser source to the entire spectrum with the planned confocal configuration and the introduced axial response. In this chapter, the introduced signal processing method is actually inspired by the method of signal processing in the so-called differential confocal systems [19– 26]. We should remember, however, the disparity between the axial response adopted and its equivalents in Refs. [19–24]. The conventional differential confocal setups can be divided into three groups, according to the particular axial responses. In the first form, the axial response is defined as the ratio of the difference between the intensities gathered by the two pinhole detectors and the total [19–21] or maximum [22] of the two intensities gathered, in which the two pinhole detectors are located at two symmetrical defocus positions relative to the focal plane. In the second form, the axial response is defined as the ratio of intensities gathered by two pinhole

10.2 Optical comb chromatic confocal probe

483

detectors, both positioned at focal positions but with different sizes of pinhole [23]. To prevent conventional axial scanning, the differential confocal setups of the first and second types were developed for LSCM. Usually, though, the depth measuring range is very limited (about several micrometers) for these two types of differential confocal setups. The third method is a mixture of the approaches to signal processing in the first two types and the chromatic confocal microscopy to increase the range of depth measurement, where the axial response is defined as the ratio of the difference between the intensities obtained by two pinhole detectors placing both the focal points but with different pinhole sizes and the amount of the two intensities obtained [24]. It should also be remembered that in the latter three types of differential confocal configurations, the depth information is no longer collected by calculating the peak position of the axial response, but by a monotonous curve intercepted away from the peak position of the axial response. The range of the monotone curve determines the final depth measurement range, and the linearity of the monotone curve determines the final measurement accuracy. The long-range monotone curve typically has poor linearity and hence produces low measurement precision. In addition, differential confocal configurations also have inconsistent signal-to-noise ratios and measurement accuracy for the points close and far from the peak location of the axial response on the monotone curve, especially for the second and third types of differential confocal configurations. In contrast, the axial response introduced in this study is defined as the ratio of confocal signals acquired by two equal focal point fiber detectors and, respectively, the defocus point, which is a new differential confocal configuration [27–29]. In addition, the depth information is derived by calculating the peak position of the axial response, which in the above differential confocal settings will be more stable than the depth measurement method centered on an intercepted monotone curve from the axial response. In Section 10.2, an optical comb chromatic confocal probe with the signal processing method referred to as the tracking local minimum method is introduced. In the signal processing method, not only the information of the main lobe but also those of the side lobes can be employed for measurement. In Section 10.3, a technique of expanding the measurement range of an optical comb chromatic confocal probe with the employment of the side lobes is described. Stabilization of the optical comb chromatic confocal probe is also described in Section 10.4, followed by the improvement of the signal processing method described in Section 10.5.

10.2 Optical comb chromatic confocal probe A schematic of the optical setup for the chromatic confocal probe is presented in Fig. 10.1. A mode-locked laser source is used in the setup, while single-mode fibers are used to transfer the observed electric fields to the spectrometers. Every mode has a deterministic frequency in the mode-locked laser. By using a pulse

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repetition rate νrep and a carrier-envelope offset frequency νCEO, the frequency νi and wavelength λi of the ith optical mode can be expressed as follows [14–16]: νi = νCEO + i · νrep λi =

c c = νi νCEO + i · νrep

(10:1) (10:2)

In the above equation, c is the speed of light. As can be seen in the figure, an optical fiber detector with a spectrometer is employed for signal detection instead of a pinhole, which is often employed in confocal microscopy. In this optical configuration, the imaging can be treated as coherent due to the fiber dimensions [30], and the intensity distribution can be described as follows [30]:  2 Iðu, vÞ = h1 ðu, vÞh2eff ðu, vÞ#τðvÞ

(10:3)

In the above equation, the symbol # corresponds to the convolution operation, while τ(ν) represents the amplitude reflection or transmittance of the object under inspection. The terms h2eff and hm (m = 1,2) in the above equation can be represented as follows: h2eff ðu, vÞ = h2 ðu, vÞ#eðvÞ Z 1 Pm ðu, ρÞJ0 ðρvÞρdρ hm ðu, vÞ =

(10:4) (10:5)

0

It should be noted that h1(u,v) and h2(u,v) in the above equations are the point spread functions (PSFs) for the objective lens and the collector lens, respectively, Pm(u,p) is the pupil function of the lens, e(v) is the eigenfunction of the fundamental mode of the optical fiber [31]. J0() is the zeroth-order Bessel function of the first kind. The parameters u and v are the normalized optical coordinates of the real radial and axial coordinates r and z, respectively, and can be expressed by the following equations [32]: v=

2π r sin α λ

(10:6)

u=

2π zsin2 α λ

(10:7)

where sinα stands for the numerical aperture (NA) of the chromatic objective. A new dual-detector configuration in the modified chromatic confocal probe is used to resolve the non-smoothness of the spectrum of the mode-locked femtosecond laser source, as shown in Fig. 10.1. Ir(u,v) is the ratio of the intensities I1(u,v) and I2(u,v) collected by the two fiber detectors having an identical mode-field diameter. The intensity ratio Ir(u,v) becomes 1 in the ideal case where each of the fiber detectors is placed on the focal plane of the corresponding objective lens. In a practical case, either of the

10.2 Optical comb chromatic confocal probe

485

Fig. 10.1: A femtosecond laser chromatic confocal probe with a dual-detector configuration.

detectors is defocused with a distance ud along the optical axis to avoid Ir(u,v) being 1. In this case, Ir(u,v) can be modified as follows: Ir ðu, vÞ =

I1 ðu, vÞ I2 ðu, v; ud Þ

(10:8)

where I2 can be expressed as follows:  2 I2 ðu, v; ud Þ = h1 ðu, vÞh2eff ðu + ud , vÞ#τðvÞ

(10:9)

It should be noted that ud could have a positive or negative sign in accordance with the displacement of Detector 2 with respect to the focal plane of the lens. The depthsectioning effect, which is one of the most significant characteristics of the confocal system, can be explained by the measurement of a plane surface that is scanned in the axial direction [33]. The intensity ratio in eq. (10.8) is the axial response of the chromatic confocal probe that can be expressed as follows: ð 1 2   2  I1 ðuÞ =  expðjuρ ÞPeff ðρÞρdρ

(10:10)

ð 1 2  I2 ðu; ud Þ =  exp½jðu + ud 2Þρ2 Peff ðρÞρdρ

(10:11)

0

0

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Chapter 10 Chromatic confocal comb

The symbol j corresponds to the imaginary unit. Peff(ρ), the effective pupil function can be given as follows by using the first-order Bessel function of the first kind J1(): Peff ðρÞ =

ρvc J1 ðρvc Þ − GðUÞJ0 ðρvc Þ ½ðρvc Þ2 − U 2 ½ðρvc Þ2 + W 2 

(10:12)

Where W is that of the cladding. In the above equation, vc is the optical fiber core radius given by using the numerical aperture sinβ of the focusing lens (L1 and L2) as follows: vc = ð2π=λÞrc sin β

(10:13)

It should be noted that G(U) is given as follows: GðUÞ =

UJ1 ðUÞ J0 ðUÞ

(10:14)

A schematic of the prototype of the chromatic confocal probe developed in this study is shown in Fig. 10.2. A light source unit with a mode-locked femtosecond laser (Menlo Systems GmbH, C-Fiber) with vrep of 100 MHz having a spectrum ranging from 1.48 to 1.64 μm was employed for illumination. The laser beam generated from the mode-locked femtosecond laser source unit was made to pass through a pinhole, and was then collimated by a collimating lens (CL) to create a collimated beam having a diameter of 3.6 mm. Since the optical fiber of the laser source could have a pinhole effect, the pinhole is employed to make the confocal system simpler. The collimated beam’s polarization orientation was changed so that it could pass through a polarized beam splitter (PBS). The collimated beam was made to focus on a plane mirror (Edmund Optics, #67–484), which was used as the target under test and was driven by a one-axis piezoelectric (PZT) positioning stage along the optical axis, by using a chromatic objective. To calculate the axial displacement along the Z-direction, a capacitive displacement sensor, whose measurement axis was oriented to coincide with the optical axis, was placed behind the plane mirror. For the separation of the light source from the reflected beam, a quarter-wave plate was also mounted in the optical path from the CL to the plane mirror. The reflected beam was then bent by the PBS and separated by a non-polarized beam splitter (BS) into two sub-beams: Beam 1 and Beam 2. Beam 1 was directly coupled into one end of a single-mode optical fiber (Corning Inc., SMF-28e) with a core diameter of 8.2 μm and a NA of 0.14, respectively, by using a focusing (achromatic) lens (L1, Edmund Optics, # 30–046) having a focal length f of 16.6 mm. It should be noted that a fiber launch system (Thorlabs Inc., MBT612D) was employed as shown in Fig. 10.2 to carry out the fine adjustment of the fiber detector with respect to the lens. An optical spectrum analyzer (Yokogawa Co., Ltd., AQ6370D) was attached to the other end of the fiber. In the same manner, Beam 2 was coupled into the other fiber detector having identical optical components and positioning system, in the same manner as Beam 1. One edge of the fiber for Beam 1 was mounted on the back focal plane of

10.2 Optical comb chromatic confocal probe

487

the corresponding focusing lens (L1), while that for Beam 2 was relocated along the optical axis with a defocus of d to the back focal plane of the corresponding lens (L2). In the optical coordinates, according to eq. (10.7), the defocus d in the real coordinate can be represented as follows: (10:15) ud = ð2π λÞd sin2 α In the adjustment process, the edge of the fiber for Beam 2 was at first placed at the back focal plane of the lens by monitoring the spectrum. After that, the defocus d was given to the fiber edge by using the fiber launch system with a sub-micrometric resolution. It should be noted that the positive direction of d was defined as the one from the back focal plane of L2 to L2. In the same manner, the positive direction of the axial displacement z of the plane mirror was defined as the one from an initial position z0 to the chromatic lens.

Fig. 10.2: A setup for the femtosecond laser chromatic confocal probe.

The light source spectrum observed by linking the light source directly to the spectrum analyzer is shown in Fig. 10.3. As can be seen in the figure, the spectrum of the femtosecond laser beam from the light source is not flat, and contains multiple peaks. Based on the conventional signal processing in the confocal setup where a smooth spectrum is required for accurate measurement, the possible spectral range is restricted to be approximately 50 nm, as indicated in Fig. 10.3. Figure 10.4 shows the spectra of Beam 1 and Beam 2 under the condition of z = 40 μm and d = 150 μm.

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As can be seen in the figure, it is difficult to accurately detect the peak wavelengths from the captured spectra; these results indicate the necessity of addressing the non-smooth spectrum of the light source. To address the issue, the technique of employing the normalized intensity ratio of the obtained two spectra has been introduced. Figure 10.5 shows the normalized intensity ratio of Beam 1 to Beam 2 that has been calculated from the spectra shown in Figs. 10.3 and 10.4. For a fair comparison, the normalized intensity ratio of Beam 1 to the spectrum of the light source, which is calculated based on the conventional signal processing method explained in [9], is also plotted in Fig. 10.6. As can be seen in the figures, it becomes much easier to determine the peak wavelength in the newly proposed signal processing method based on eq. (10.8).

Fig. 10.3: Spectrum of the Beam 1.

Fig. 10.4: Spectrum of the Beam 2.

A plano-convex lens made of N-SF11 was used as the chromatic objective in the femtosecond laser chromatic confocal probe developed in this study. The focal length of the chromatic objective fλ can, according to the thin-lens equation, be expressed by the following equation: fλ =

1 ðnλ − 1Þð1=R1 − 1=R2 Þ

(10:16)

10.2 Optical comb chromatic confocal probe

489

Fig. 10.5: Normalized intensity ratio calculated from the spectra of Beams 1 and 2.

Fig. 10.6: Normalized intensity ratio of Beam 1 to the spectrum of the light source.

In the above equation, R1 and R2 are the curvature radii of the chromatic objective, while the parameter nλ is the refractive index of the chromatic objective that can be expressed as follows based on the Sellmeier formula [29]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 λ2 B2 λ2 B3 λ2 + 2 + 2 nλ = 1 + 2 (10:17) λ − C1 λ − C2 λ − C3 The parameters Bi and Ci (i = 1,2,3) are coefficients associated with the lens material. Parameters of the chromatic objective are summarized in Tab. 10.1. The variation of nλ with respect to λ calculated based on the above equation is shown in Fig. 10.7. As can be seen in the figure, there exists a linear relationship between nλ and λ. The parameter nλ can be expressed by an approximate expression nλ = kλ + b, where k and b are the slope and the intercept, respectively. Through the linear fitting operation, k and b were found to be –0.01793 μm−1 and 1.7712, respectively (R2 = 1). The parameter z, the axial displacement of a target, can be expressed as z = fλ0–fλ, where fλ0 and fλ are the focal lengths of the chromatic objective at the wavelengths of λ0 and λ, respectively. It should be noted that λ0 is the peak wavelength at the initial position of the plane mirror (z0). Regarding eq. (10.15), the following equation can be obtained:

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Tab. 10.1: Physical parameters for the objective lens. Item

Symbol

Value

Symbol

Value

B

.

C

.

B

.

C

.

B

.

R

. mm

Sellmeier coefficients

Surface radii of the lens

R

dz = − dfλ =

dnλ fλ nλ − 1

∞

(10:18)

By using eqs. (10.16) and (10.18), the sensitivity of the chromatic confocal probe dλ/ dz can be estimated as follows: dλ 1=R1 − 1=R2 b−1 ðkλ + b − 1Þ2 ≈ = k dz kfλ0

(10:19)

As can be seen in the above equation, the parameters k, b and fλ0, are included in the equation; this means that the sensitivity is intimately connected with NA and the material of the chromatic objective. Now the parameters k and b can be estimated from the graph shown in Fig. 10.7, and the sensitivity can be estimated at any central wavelength λ0. Meanwhile, the measurement sensitivity could slightly change with the change in the central wavelength. In the case of λ0 = 1.56 μm a sensitivity will be approximately –4.07 nm/μm. Regarding the spectral range of the femtosecond laser ranging from 1.48 μm to 1.64 μm, the chromatic confocal probe is expected to have a measurement range of approximately 39.3 μm.

Fig. 10.7: Relationship between nλ and λ.

10.2 Optical comb chromatic confocal probe

491

The sensitivity of the chromatic confocal probe is associated with its depth resolution and the noise components in the obtained signal. The wavelength resolution of the spectrometer employed in the fiber detector unit could also affect the depth resolution. On the assumption that the spectrometer has a wavelength resolution of 0.02 nm, which is almost the same as that of the optical spectrum analyzer employed in this study, a depth resolution could reach 5 nm in the ideal case with the signal having a high signal-to-noise ratio. Meanwhile, in a practical case, the resolution becomes worse than 5 nm due to noise components in the captured signal. The full-width half-maximum (FWHM) of the axial response of a confocal imaging system is an important indicator for the evaluation of the depth resolution of the system. In the conventional confocal system, the axial response can be expressed by eq. (10.10). On the other hand, in the method proposed in this study, the axial response can be expressed by eq. (10.9); this fact means that the FWHM of the axial response defined by eq. (10.9) is different from that in the conventional confocal imaging system. Meanwhile, a narrower axial response still contributes to carrying out accurate detection of the peak wavelength, and the FWHM of the axial response could still be employed to evaluate the performance of the confocal imaging system, as well as to estimate the depth resolution of the system. According to the numerical calculations based on eq. (10.9), defocus d in Fig. 10.2 (, which corresponds to ud in Fig. 10.1) could give a difference in the FWHM of the axial response. Therefore, an attempt was made to find out optimal d for obtaining smaller FWHM. The variation of FWHM with respect to d is shown in Fig. 10.8. As can be seen in the figure, FWHM was found to become minimum at d = 150 μm. The corresponding variation of the FWHM observed in experiments is shown in Fig. 10.9. The trend of the FWHM observed in experiments was found to well agree with that predicted in the numerical calculations. It should be noted that the central wavelength λ0 was set to be 1.56 μm in the experiments and numerical calculations. Results of the repetitive experiments also indicated that the minimum FWHM was always found when d was set to be in a range of 140–170 μm. Regarding the results, in the following experiments, d was set to 150 μm.

Fig. 10.8: Variation of FWHM with respect to d (in simulation).

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Fig. 10.9: Variation of FWHM with respect to d (in experiments).

To investigate the measurement sensitivity and measurement range of the produced chromatic confocal probe, experiments were carried out. The plane mirror was kept traveling along the optical axis in a step of 2 μm by a PZT stage in experiments. For comparison, the plane mirror’s axial displacements were evaluated simultaneously by a capacitive displacement sensor. From the center-of-mass position of the corresponding normalized axial response, the peak wavelength was determined at each axial position of the mirror. The peak wavelengths derived from the normalized axial responses at various axial displacements are shown in Fig. 10.10. In the figure, linear fitting of the results was carried out to determine the sensitivity of the instrument. The sensitivity was evaluated to be −3.73 nm/μm with a coefficient of determination of R2 = 0.9991. In the experiments, a measurable range was evaluated to be 40 μm. The sensitivity and the range observed in the experiments showed a fair agreement with those forecasted in theoretical calculations.

Fig. 10.10: Peak wavelengths obtained from the normalized axial responses.

In order to evaluate the actual depth resolution of the femtosecond laser chromatic confocal probe, additional tests were then carried out. The plane mirror was driven in much smaller steps by the PZT stage along the optical axis in the experiments. Variations of the peak wavelengths in the normalized axial responses at different axial displacements in steps of 40, 30 and 20 nm are plotted in Figs. 10.11, 10.12 and 10.13, respectively. As can be seen in the figures, the steps of the mirror were successfully observed in each of the cases, and a resolution of 30 nm was successfully

10.2 Optical comb chromatic confocal probe

493

achieved. Meanwhile, although the mirror displacement in steps of 20 nm was successfully resolved in some Z-regions in Fig. 10.13, a coefficient of determination of R2 was reduced to 0.969. These results have demonstrated the feasibility of the developed femtosecond laser chromatic confocal probe.

Fig. 10.11: Variation of the peak wavelengths in the normalized axial responses due to the small axial steps of a target in a step of 40 nm.

Fig. 10.12: Variation of the peak wavelengths in the normalized axial responses due to the small axial steps of a target in a step of 30 nm.

Fig. 10.13: Variation of the peak wavelengths in the normalized axial responses due to the small axial steps of a target in a step of 20 nm.

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10.3 Range-expanded chromatic confocal comb A new signal processing approach for the axial response from a fiber-based dualdetector chromatic confocal probe with a mode-locked femtosecond laser source is investigated as the second stage of the study. The approach is planned to extend the measuring spectrum further without changing the dual-detector chromatic confocal probe optical configuration. There is often a main-lobe and side-lobes in the axial response of a confocal probe due to the diffraction effect of a point detector during confocal microscopy. The main-lobe is the peak having the highest intensity in the axial response, while the side-lobes are the peaks with local maximum intensities besides the main-lobe in the axial response. Many efforts have been made so far to reduce [34–42] or remove [43, 44] the influences of the side-lobes in the axial response, since they reduce the light intensity of the main-lobe and also degrades the depth resolution of confocal microscopy. On the other hand, the irritating sidelobes can be employed to extend the measurement range of the femtosecond laser chromatic confocal probe with a dual-detector configuration. A schematic of the femtosecond laser chromatic confocal probe is shown in Fig. 10.14, in which the optical configuration with fiber-based dual-detectors is employed [27]. As the light source, a mode-locked femtosecond laser is used in the setup. For the transferring of the optical signals, single-mode fibers are employed. In the mode-locked femtosecond laser, the kth optical mode has a particular frequency vk corresponding to the deterministic working wavelength λk, that can be expressed as follows: νk = νceo + k · νrep λk =

(10:20)

c νk

(10:21)

where vrep and vceo stand for the pulse repetition rate and the carrier-envelope offset frequency, respectively [14–16]. In eq. (10.21), the speed of light in vacuum is indicated as c. PSF and the pupil function of the imaging lens are important for the image creation of confocal microscopy. These two items are employed in the equation of the final axial response of confocal microscopy. PSF can be expressed by the following equation [2, 27, 30, 33]: ð1 hi ðu, vÞ = ( Pi = hi

Pi ðu, ρÞJ0 ðρvÞρdρ,

i = 1, 2

(10:22)

0

1, inpupil 0, else

,

i = 1, 2

eff ðu, vÞ = hi ðu, vÞ#e1 ðu, vÞ,

(10:23) i = 1, 2

(10:24)

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where hi(u,v) and hi_eff(u,v) are the PSF and the effective PSF, respectively, while Pi (u,ρ) is the pupil function of the imaging lens. From the above equation, the complex amplitude on the confocal plane, Uf, can be expressed by the following equation: Uf ðu, vÞ = h1

eff ðu, vÞh2 eff ðu, vÞ#t ðu, vÞ

(10:25)

The definitions of the optical coordinates u and v are indicated in eqs. (10.6) and (10.7). The light intensity I to be captured by the detector, which is placed on the focal plane of the objective lens, can be represented as follows by using the amplitude reflection/transmittance of the object t(u,v), since the confocal configuration can be treated to be coherent: I = jh1

eff ðu, vÞh2 eff ðu, vÞ#tðu, vÞj

2

(10:26)

The light intensity distribution Im(u) on the focal plane of the objective lens can be given as follows, where the fiber detector is located, based on the above analyses: ð 1 

 Im ðuÞ =  exp juρ2 P1 0

2   eff P2 eff ρdρ

(10:27)

Fig. 10.14: A femtosecond laser chromatic confocal probe with an improved signal processing algorithm.

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A dual-detector differential confocal configuration shown in Fig. 10.14 was proposed to resolve the non-smoothness of the laser spectrum in the mode-locked femtosecond laser source [27]. On a plane with a distance d from the focal plane of the objective lens, the light intensity distribution Ir(u, ud) can be given as follows: ð 1 h  ud  2 i Ir ðu, ud Þ =  exp j u + ρ P1 2 0

2   eff P2 eff ρdρ

(10:28)

In the above equation, the defocus d can be represented by the optical distance ud that can be expressed by eq. (10.14). Typical spectra Im(u) and Ir(u, ud) to be captured by the two identical fiber detectors are shown in Fig. 10.15. Since a modelocked femtosecond laser source has a non-smooth spectrum, as seen in Fig. 10.16, it is hard to find out a peak in the spectrum. It should be noted that the spectrum seen in Fig. 10.16 is that of a particular femtosecond laser used in the following experiments, and the spectral non-smoothness is not the same as the other generic laser sources. The normalized intensity ratio Ia(u, ud) and the inverted normalized intensity ratio In(u, ud) defined by the following equations are used in the proposed method for solving the aforementioned problem: Ia ðu, ud Þ =

Im ðuÞ Ir ðu, ud Þ

(10:29)

In ðu, ud Þ =

Ir ðu, ud Þ Im ðuÞ

(10:30)

The normalized intensity ratio and the reversed normalized intensity ratio derived from the light intensity range displayed in Fig. 10.15 are shown in Fig. 10.17. A defocus d was set to 150 μm. As can be seen in the figure, main-lobes can clearly be observed in Ia and In, while side-lobes are unnoticed; this gives a high resolution of 30 nm and strong linearity for displacement calculation along the depth direction for the developed fiber-based dual-detector chromatic confocal probe. A defocus d was thus set to 110 μm in the following simulations and experiments so that the side-lobes can be seen and can be employed to expand the measurement range.

Fig. 10.15: Spectra obtained by the detectors.

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497

Fig. 10.16: A spectrum of the light source.

Fig. 10.17: Normalized intensity ratios obtained from the spectra.

The spectral bandwidth of the mode-locked laser source and the chromatic objective used in the optical setup defines the measurement range of the proposed fiber-based dual-detector chromatic confocal probe. A new approach using both the main-lobes and the side-lobes of axial responses obtained from the normalized intensity ratio Ia(u, ud) and the inversed normalized intensity ratio In(u, ud) is proposed in this study to further extend the measurement range of the fiber-based dual-detector chromatic confocal probe without any changes to the optical setup. A variance of the normalized intensity Ia is seen in Fig. 10.18 as a function of the Zdirectional location of a measurement target at a particular wavelength λ, defined as the axial response [27]. The axial response obtained from the invert normalized intensity ratio at the same wavelength λ is also indicated in the figure. As can be seen in the figure, the axial responses can easily distinguish not only the main-lobe but also the side-lobes of the axial response suggested in [27], which cannot be found in the single axial responses of Im and Ir without the normalization operation. This is another advantage of the newly proposed axial response; contrary to the single axial response predicted from a Lorentzian form in which the side-lobe is blurry and cannot be readily recognized, the side-lobes are smoother. The normalized intensity ratio Ia(u,ud) and the inverted normalized intensity ratio In(u, ud) with distinct defocus d are evaluated, and the spacings between side-lobes and that between the main-lobe and the side-lobe at different d is summarized in Fig. 10.19.

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We set a defocus d to be 110 μm in order to make effective use of the side lobes to expand the measurement range, in which the side lobes are more noticeable and recognizable than the cases in which d is set to be within a range of 50 μm to 80 μm. As a result, the use of side-lobes allows the measuring spectrum of the fiber-based dual-detector chromatic confocal probe to be extended, and theoretical measurements are then carried out as follows.

Fig. 10.18: A variance of the normalized intensity ratio.

Fig. 10.19: Main-lobe and the side-lobes in the normalized intensity ratios.

With regard to the non-smoothness of the laser spectrum seen in Fig. 10.16, based on eqs. (10.28) and (10.29), the axial responses of the fiber-based dual-detector chromatic confocal probe are calculated. The proposed axial response in the calculation is defined as the intensity ratio of signal 1 and signal 2 and no longer as a single axial response. Therefore, the distance between the main-lobe and the side-lobe is measured by the relative magnitude of the signal 1 and signal 2 ratios at the same target location. Furthermore, the relative height of the main-lobe and the side-lobes is also controlled by the relative magnitude of the ratios of signal 1 and signal 2 at the same target position that is further driven by d shown in eq. (10.27). As can be seen in eq. (10.27), technically, when a defocus d is set to 0, the values of eqs. (10.29) and (10.30) still become 1.

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499

To express the influence of the chromatic objective, the parameters summarized in Tab. 10.1 are used for the calculations. The axial responses of the normalized intensity ratio and the inverted normalized intensity ratio measured at wavelength region ranging from 1,550 to 1,580 nm are seen in Figs. 10.20 and 10.21. As indicated in the figures, with respect to those from the normalized intensity ratio, the main lobe and side lobes of the axial response from the inverted normalized intensity ratio are seen to be shifted along the axial (Z-) direction. Only the main-lobe information was used to detect a displacement in the axial direction in the normalized intensity ratio in the previous word described in [27], which has limited the measurement range to be 40 μm. In the meanwhile, the measuring spectrum can be increased by using the data from not only the main-lobe but also the side-lobes in the axial responses from both the intensity ratios. The difference of the adjacent lobes (including the main lobe and side lobes) in the axial response of Ia(u, ud) can be filled by the corresponding lobes in the axial response of In(u, ud), as can be seen in Figs. 10.22 and 10.23.

Fig. 10.20: The measured axial responses of the normalized intensity ratio.

Theoretical calculations were applied to a mode-locked laser with a working range of 1.48–1.64 μm and a target location of over 250 μm. The axial responses of the normalized intensity ratio of Ia and the inverted normalized intensity ratio of In are shown in Figs. 10.24 and 10.25. In both axial responses, side-lobes are distinctly observed over a target location range of 250 μm, as can be seen in the figures. Meanwhile, in both axial responses, the presence of a distance between each adjacent lobe is also confirmed. Furthermore, the findings of the theoretical equation demonstrate that the distance gets larger as the target position increases. The main-lobes and side lobes of

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Fig. 10.21: The measured axial responses of the inverted normalized intensity ratio.

Fig. 10.22: Normalized intensity ratio and the inverted normalized intensity ratio.

the normalized intensity ratio Ia are seen in Fig. 10.26, plotted along with those of the reversed normalized intensity ratio In. The two axial responses have almost 180° phase difference, as shown in the figure. The differences between each adjacent lobes in the normalized intensity ratio can be filled by the reversed normalized intensity ratio over a target location range of 150 μm.

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Fig. 10.23: The measured axial responses of the normalized intensity ratio and the inverted normalized intensity ratio.

Fig. 10.24: Normalized intensity ratio predicted in the numerical calculations.

It should be noted that calculation of partial derivatives is necessary to utilize the side-lobes of the axial responses, as illustrated in Fig. 10.23, for the expansion of the axial measurement range by the following equation:

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Fig. 10.25: Inverted normalized intensity ratio predicted in the numerical calculations.

 pffiffiffiffiffi 

pffiffiffiffi ∂Ia ∂Ia ∂u 4πsin2 β Ir Im expðjuρ2 Þ − Im Ir exp j u + = = ∂z ∂u ∂z Ir 2 λ

ud  2  2 ρ

(10:31)

To obtain the relative maximum intensity values for the wavelength-to-displacement encoding of the side-lobes, let eq. (10.31) be equal to zero as follows and get the relationship between the target position z and wavelength λ: ∂Ia ∂Ia ∂u = =0 ∂z ∂u ∂z

(10:32)

The main-lobe and side-lobes are used in the wavelength-to-displacement encoding, as seen in Fig. 10.23, but differences between the main-lobe and side-lobes make the final measurement spectrum discontinuous. Side-lobes of the axial response of the reversed normalized pressure ratio are used in order to remove the differences. The following partnership can, thus, also be satisfied:

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Fig. 10.26: Two ratios predicted in the numerical calculations.

∂In ∂In ∂u = =0 ∂z ∂u ∂z

(10:33)

In order to investigate the feasibility of the proposed system of measurement range extension, experiments were carried out. To obtain the axial responses, the plane mirror was made to travel along the Z-direction in a step of 2 μm. It should be remembered that the displacement of the plane mirror was calculated for comparison simultaneously by a capacitive displacement sensor. The peak wavelengths at each Z-position of the mirror are determined from the centroid positions of the respective normalized intensity ratio, as well as from the reversed normalized intensity ratio. The normalized intensity ratio obtained at each Z-position over a scanning range of 250 μm is seen in Fig. 10.27. It was clearly indicated not only the main-lobe but also the side-lobes in Fig. 10.27. While the sub-lobes’ peak widths were observed to become broader as the plane mirror’s Zdisplacement increased, the sharpness of the sub-lobes was adequate to be employed for the encoding of wavelength to displacement. Figure 10.28 indicates the peak wavelength at each Z-directional location of the plane mirror obtained from the normalized strength ratio in Fig. 10.27. For the main lobe and each of the side-lobes, a linear relationship between the peak wavelength and the axial Z-directional displacement of the

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plane mirror was observed, as can be seen in the figure. Meanwhile, a distance between both of the adjacent lobes was observed, as expected in the results of the above theoretical calculations. The inverted normalized intensity ratio obtained at each Z-position, determined from the same intensities used to measure the corresponding normalized intensity ratio as shown in Fig. 10.27, is shown in Fig. 10.29. In the inverted normalized intensity ratio obtained, the main lobe and the side lobes were clearly observed, as can be seen in Fig. 10.29. The relationship between the peak wavelength and the axial Zdirectional displacement of the flat mirror was explored in the same way as in the case of the normalized intensity ratio Ia, by removing the peak wavelengths in the inverted normalized intensity ratio shown in Fig. 10.29. Figure 10.30 summarizes the results. The peak frequency in the invert normalized intensity ratio at a particular Z-directional location of the plane mirror was observed to change along the axial direction compared with that of the normalized intensity ratio shown in Fig. 10.28, as expected in the results of the above theoretical calculations.

Fig. 10.27: Axial responses of the normalized intensity ratio observed in experiments.

The wavelength-to-displacement encoding curves obtained from the axial responses of Ia normalized intensity ratio and the invert normalized intensity ratio are summarized in Fig. 10.31. The axial response from the invert normalized intensity ratio successfully fills the distance between each opposing lobe in the axial response of the normalized intensity ratios, as can be seen in the figure. These experimental findings revealed that the measurement range of the fiber-based dual-detector chromatic confocal probe was successfully extended by the proposed method from 40 to 250 μm without any alteration of the optical structure of the confocal probe being created.

10.4 Stabilized chromatic confocal comb

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Fig. 10.28: λ-Z curve from the normalized intensity ratio observed in experiments.

Fig. 10.29: Axial responses of the normalized intensity ratio observed in experiments.

10.4 Stabilized chromatic confocal comb As mentioned above, through theoretical calculations and experiments, the feasibility of the “normalized” spectrum, which can be used as a highly sensitive axial response, has been confirmed. Experimental studies have shown that a vertical resolution of 30 nm and a measuring length of 40 μm can be obtained by the proposed method [27]. In addition, it was also shown that by using side-lobes in the normalized spectrum, the measurement range can be increased to 250 μm [45]. Meanwhile, because an optical spectrum analyzer is used in the built setup to read

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Fig. 10.30: λ-Z curve from the inverted normalized intensity ratio observed in experiments.

Fig. 10.31: λ-Z curves from the intensity ratios.

the spectrum over a wavelength range of 160 nm, it takes a relatively long time up to several seconds to evaluate a target sample at a single height location. Thus, when the established probe is applied for long-term displacement measurement or surface profile measurement where long-term scanning is needed, the thermal stability of the probe is a critical problem. Thus, experimental testing of the thermal stability of the first femtosecond laser chromatic confocal probe prototype [27] is carried out where a major thermal instability of the probe is tested for displacement measurement. The potential explanations for the probe’s thermal instability, such as the thermal instability of the confocal lens refractive index and the thermal expansion of the probe’s mechanical jigs, are then quantitatively evaluated. A second prototype femtosecond laser chromatic confocal probe is newly developed and installed, based on the research results. The design was carried out in such a way as to increase the thermal stability of the displacement measurement by minimizing

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507

the laser beam’s optical path length and by using a low thermal expansion material for mechanical jigs in the optical configuration. Also presented are experimental findings to show the increased thermal stability of the second prototype probe. It should be remembered that not only the thermal stability of the chromatic confocal probe itself, but also the thermal stability of the instrument, the sample jigs and the mounting pad contribute to the thermal stability of the overall chromatic confocal measurement setup. As the thermal stability of the sample, the sample jigs and the mounting plate shift when the sample is used for various applications, the thermal stability of the chromatic confocal probe is thus taken into account. Figure 10.32 displays a diagram of the overall chromatic confocal measurement configuration, which is referred to as the first prototype measurement system. The configuration consists of a chromatic confocal femtosecond laser mode-locked probe, referred to as the first prototype probe, a sample device consisting of the sample and the jigs of the sample, and the mounting plate on which the probe and the sample unit are mounted. A photograph of the overall first prototype measuring setup is shown in Fig. 10.33. The probe and the sample unit are mounted on a stainless-steel mounting plate.

Fig. 10.32: A diagram of the overall chromatic confocal measurement configuration.

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Fig. 10.33: A photograph of the overall first prototype measuring setup.

A fiber-based dual-detector is used in the proposed chromatic confocal probe to remove the influence of the non-smooth spectrum of the mode-locked laser source [27]. A normalized axial response can be obtained on the basis of the Imea and Iref spectra obtained by Detector1 and Detector2, respectively, represented by the following equation: In =

Imea Iref

(10:34)

An example of the spectrum obtained by a pair of single-mode fibers is seen in Fig. 10.34. The result of the convolution of the axial reaction of the confocal setup and the spectrum of the light source is the obtained spectra (Imea and Iref). The influences of the reflection of the target surface may affect the spectrum received. The transmittances of optical components in the setup could also influence the obtained spectra. The Imea and Iref spectra obtained are also non-uniform as the femtosecond laser source has a non-uniform spectrum. This indicates that the peaks in the light spectrum that are needed to acquire axial location information of the measurement target surface under inspection are difficult to collect directly. Meanwhile, the effects of non-uniformity in the spectrum of the light source as well as the surface reflectance can be cancelled by the normalization process by adding the normalized axial response expressed in eq. (10.34). In addition, the transmittance effects of these optical elements can also be reduced by the normalization operation by using equivalent fiber coupling lenses and single-mode fibers for both Imea and Iref. The normalized axial response obtained by Imea and Iref, as shown in Fig. 10.34, is shown in Fig. 10.35. The valley in Iref, as can be seen in this figure, is detected as the peak in the normalized axial response. As can be seen in the figure, the peak is

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10.4 Stabilized chromatic confocal comb

sharper in the normalized axial response than in Imea and Iref. The sharper peak in the axial response in the confocal method leads to the realization of higher precision in the detection of the focused wavelength λfocused, and thus the higher resolution in the axial displacement calculation. The method for obtaining the axial location information from the normalized axial response of the measurement target surface under inspection is shown in Fig. 10.36. The normalized axial response In is seen in the figure. In the normalized axial response, which is referred to as the centroid wavelength, the wavelength λfocused at the peak can be obtained by the following equation: P ðI ðλ Þ × λi Þ Pn i λfocused = (10:35) In ðλi Þ By the following equation, the Z-displacement ΔZ of the target sample surface can be calculated as follows: ΔZ =

dZ · Δλfocused dλ

(10:36)

In the above equation, dZ/dλ is the detection sensitivity of the Z-displacement with respect to the focused wavelength variation Δλfocused. dZ/dλ can be determined theoretically, since it is possible to measure the relationship between the fλi, the focal length of the chromatic objective lens and the corresponding λi, light wavelength, on the basis of the eqs. (10.16) and (10.17). The sensitivity df/dλ can be calculated as 255 nm/nm, on the basis of the specification of the chromatic objective lens employed in the first prototype (the details of which are represented in Ref. [27]). Since the positive direction of the Z-displacement of the target sample surface is set as the direction the sample approaches the chromatic objective lens, sensitivity dZ/dλ = – (df/dλ) = − 255 nm/nm.

Fig. 10.34: Spectra to be obtained by a pair of single-mode fibers and the normalized intensity.

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Fig. 10.35: Normalized axial response curve.

Fig. 10.36: Obtaining the axial location information from the normalized axial response.

Experimental results have demonstrated that the first prototype confocal probe has an axial resolution of 30 nm over a measurement range of 40 μm by fully utilizing a spectral range of 160 nm of the femtosecond laser [27]. The reduction of the influences of the surface reflectance of a target sample, as well as the transmittance of optical components in the probe, was also realized. On the other hand, it takes quite a long time (about eight seconds) to take the Imea and Iref spectra at a position in a sample plane because of the poor measurement throughput of the optical spectrum analyzer. In order to apply the probe for long-term displacement measurement or for surface profile measurement with multi-point scanning, the good thermal stability of the probe is thus necessary. The thermal stability of the formed probe in axial displacement measurement is important to define for this reason. It should be noted that the thermal stability of the sample, the sample jig and the mounting plate could also affect the chromatic confocal measurement, as well as the thermal stability of the chromatic confocal probe itself. It is thus important to distinguish the thermal stability of the probe from that of the sample, the sample jig and the mounting plate from the point of view of the formed femtosecond laser chromatic confocal probe. Meanwhile, because the thermal stability of the probe is difficult to calculate explicitly, the thermal stability of the overall measurement configuration

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511

shown in Fig. 10.32 is determined first. Theoretical work is then performed to describe the thermal stability of the first prototype probe. The thermal stability of the overall measurement configuration for axial displacement measurement (dZ/dT) is difficult to evaluate directly. Meanwhile, as mentioned above, the Z-λ susceptibility (dZ/dλ) can be theoretically determined. DZ/dT can be determined from the following equation by obtaining the thermal stability of the detection of centered wavelength (dλ/dT) in experiments: ΔZ =

dZ · Δλfocused dλ

(10:37)

Experiments were performed by using the setup shown in Fig. 10.33 to obtain the thermal stability of centered wavelength (dλ/dT) detection. A flat mirror, which was used as the target sample, was kept stationary in the setup while a change of the focused wavelength λfocused was observed over a duration of 5,000 s at each 100 s. A temperature sensor was installed in the optical setup to monitor the temperature variance, as can be seen in Fig. 10.33. The result of the thermal stability of the detection of focused wavelength λfocused is shown in Fig. 10.37. Not only the influences of thermal stability of the chromatic confocal probe itself but also those of the thermal stability of the sample device and the mounting layer are included in the results. The tests were carried out in a laboratory room with a regulated temperature of 20°C ± 0.5°C, while the temperature deviated regularly within a few hours. As can be seen in the figure, a direct link between the λfocused variance and the temperature shift can be observed. A mean value of dλ/dT of − 20.4 nm/°C was obtained through three repeated tests. The thermal stability of the overall measurement setup for axial displacement measurement (dZ/dT) was then assessed from eq. (10.37) as dZ/dT = (−20.4 nm/°C) ·(−255 nm/nm) = 5.20 μm/°C.

Fig. 10.37: Thermal stability of the detection of focused wavelength λfocused.

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This finding implies that in the total chromatic confocal measurement configuration, a temperature difference in the laboratory room varying from 19.5 to 20.5°C will result in a variance of the observed axial displacement of ±2.60 μm. The thermal stability of the developed chromatic confocal probe needs to be improved with regard to long-term displacement measurement or surface profile measurement where longtime scanning is needed. In order to isolate and describe the thermal stability of the femtosecond laser chromatic confocal probe based on the measurement outcome of the overall measurement setup in Fig. 10.33, a thorough analysis is therefore conducted in the following. Table 10.2 summarizes the potential factors leading to the overall measurement setup’s thermal stability. To characterize the thermal stability of the probe itself, the contribution of each element is evaluated. Tab. 10.2: Contributors in the thermal stability of the developed setup. Item

Factor

Symbol

Probe

Refractive index instability of the chromatic objective lens

P1 = ðdZ=dT ÞP1

Refractive index instability of the surrounding air

P2 = ðdZ=dT ÞP2

Thermal expansion of mechanical jigs

P3 = ðdZ=dT ÞP3

Thermal expansion of optical path length

P4 = ðdZ=dT ÞP4

Thermal stability of the probe

P = ðdZ=dT ÞProbe = P1 + P2 + P3 + P4

Mounting Thermal expansion of the mounting plate (corresponding plate to the thermal stability of the mounting plate)

M = ðdZ=dT ÞMountingplate

Sample unit

Thermal expansion of the sample

S

Thermal expansion of the sample jig

S

Thermal stability of the sample unit

S = ðdZ=dT ÞSampleunit = S1 + S2

Thermal stability of the overall setup

O = ðdZ=dT ÞOverall = P + S + M

(a) Contribution of the chromatic objective lens‘ refractive index instability (P1) The following equation indicates the influence of the temperature and the lens refractive index in a vacuum [46]: ! dnabs ðλi , T Þ n2 ðλi , T0 Þ − 1 E0 + 2E1 ΔT 2 (10:38) D0 + 2D1 ΔT + 3D2 ΔT + = dT 2nðλi , T0 Þ λi 2 − λi 2TK The above equation contains the temperature deviation ΔT from a reference temperature T0 (ΔT = T–T0). The parameter associated with the lens material: Dp (p = 0,1,2), Eq (q = 0,1) and λTK are also included. Table 10.3 summarizes the physical parameters of

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513

the employed chromatic objective lens [47]. Refractive index instability is estimated to be −1.0629 × 10−6°C−1 at λi = 1,560 nm, based on eq. (10.38). It should be noted that, according to eq. (10.17), this value corresponds to dλ/dT of 0.0594 nm/°C. The contribution of the chromatic objective lens’ refractive index instability to the thermal stability of the probe is then determined as P1 = (0.0594 nm/°C) ·(−255 nm/nm) = −15.1 nm/°C from eq. (10.37). Tab. 10.3: The physical parameters used in the theoretical calculations for N-SF11 [47]. Symbol

Value

D

−. × −

D

. × −

D

−. × −

E

. × −

E

. × −

λTK

.

(b) Contribution of the instability in the refractive index of the surrounding air (P2) Equation (10.16) can be rewritten as follows when describing the air refractive index nair and the lens refractive index nlens: fλ i =

1 ðnlens =nair − 1Þð1=R1 − 1=R2 Þ

(10:39)

The following equation can be obtained by taking the derivative of fλ1 with respect to nair: df 1 nlens nlens = = ·f dnair ðnlens =nair − 1Þ2 ð1=R1 − 1=R2 Þ nair 2 nair ðnlens − nair Þ The following equation is then given by modifying the above equation:   df nlens dnair dZ ·f · =− = dT dT nair ðnlens − nair Þ dT

(10:40)

(10:41)

Here, at the reference temperature, the air refractive indices of the lens (nlens) and the air (nair) are 1.7432 (at λ = 1,560 nm) and 1.00027, respectively. The refractive index volatility of the air (dnair/dT) is approximately −1.0 × 10−6/°C according to Ciddor’s theoretical formula [33]. The contribution of the refractive index instability of the surrounding air to the thermal stability of the probe is therefore determined in eq. (10.41) as P2 = −24.7 nm/°C at λ = 1,560 nm.

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(c) Thermal expansion contributions of the mechanical jigs (P3) and mounting plate (M) A detailed diagram of the chromatic confocal lens and the target specimen in the measuring configuration seen in Fig. 10.33 is shown in Fig. 10.38. The difference in the interval between the target sample and the chromatic lens in the setup is now taken into consideration. The distance between the chromatic objective lens and the instrument can be modified by the thermal expansion of components such as the probe, the sample, the sample jig and the mounting plate in the measuring setup. In the meanwhile, their contribution to probe instability from the thermal expansion of the sample and that of the sample jig is predicted to be comparatively smaller than that of Jig-B, which is much longer than the other jigs in the probe. Therefore, these contributions are not considered in the first stage of the study for the sake of clarity. Generally, the following equation will explain the linear thermal expansion of an object: d, = α · , · dT

(10:42)

In the above equation, d, and α are the change in the length of the object with the original length , and the linear coefficient of thermal expansion of the object, respectively [48]. As the object is supposed to extend by the same amount from its core in both directions, |dZ| can be treated as equal to |d,/2|. The contribution of a component’s thermal expansion in the calculation setup can be represented from this equation by the following equation: dZ d,A ,A = ± αB · =± dT 2 dT

(10:43)

where the subscripts A and B indicate the component name and the component material, respectively. In the calculation configuration, the sign “±” in the equation is determined by the object’s position. The sign is determined as positive as the chromatic objective lens reaches the target sample due to the object’s thermal expansion. The contributions of thermal expansion of mechanical jigs in the probe (P3) and the mounting plate (M) are studied based on the above equation. The contribution of thermal expansion of mechanical jigs in the probe (P3) is first discussed. Jig-B, which ties the lens holder and Jig-A, is the key contributor to the mechanical jigs, since its length is the longest of the other jigs in the probe. It should be noted that at this point, the influences of other contributors such as JigA or lens holders are not taken into account, as the effect of Jig-B is dominant. Regarding a diameter of the focused beam (20 μm) and out-of-flatness of the flat mirror employed in the following experiment (λ/10 at a wavelength of 633 nm), the displacement of the flat mirror in the X-direction due to the thermal expansion was expected to have a small influence compared with the main factor. As the temperature increases, owing to the thermal expansion of Jig-B, the chromatic objective lens

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Fig. 10.38: A detailed diagram of the chromatic confocal lens and the target specimen.

enters the target sample. Jig-B, whose length is set to be 430 mm, is made of aluminum having a linear thermal expansion coefficient of 23.1 × 10−6°C−1 [48]. The contribution of Jig-B to the thermal stability of the probe is assessed as P3 = 4.96 μm/°C from eq. (10.43). In the same manner, an investigation was carried out for the contribution of the mounting plate M. Unlike in the case of Jig-B, because of the thermal expansion of the mounting disk, the chromatic objective lens travels away from the sample. The mounting plate, whose length is designed to be 100 mm, is made of stainless steel having a linear thermal expansion coefficient of 14.7 × 10−6°C−1 [48]. Therefore, the contribution of the mounting plate is determined from eq. (10.43) as M = (dZ/dT) = − 0.735 μm/°C. (d) Contribution of optical path length thermal expansion (P4) A schematic of the lens configuration of the chromatic confocal probe is shown in Fig. 10.39. The axial distance between the focal position of the optical mode with the wavelength λi +1 and that with the wavelength λi is defined as dZλi. The defocus of the ith optical mode at the fiber detector dZλi’ corresponding to dZλi can be represented by the following equation, according to the geometric relationship in the optical configuration: ′

dZλi =

2F2 dZλi 2dZλi ðL − fλi − FÞ + fλi 2

(10:44)

In the above equation, the distance between the chromatic objective lens and the fiber coupling lens is indicated as L. The parameter F is the distance between the fiber coupling lens and the fiber detector, and is equal to the focal length of the fiber coupling lens. The sum of L and F is regarded as the optical path length in this study.

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L and F were designed to be 500 mm and 16.6 mm, respectively, in the first prototype chromatic confocal measurement setup. As can be seen in the equation, L and F influence dZλi’, leading to a transition in the axial reaction In; in other words, the focused λfocused wavelength is influenced by the optical path length’s thermal expansion.

Fig. 10.39: The lens configuration of the chromatic confocal probe.

The theoretical investigation of the contribution of the thermal expansion of the optical path length (P4) is, however, not a simple task, since the normalized axial solution equation In [27] is quite complex. Therefore, experiments were performed in this chapter to quantitatively evaluate P4. The contribution of the gap F between the fiber coupling lens in P4 can be obtained by the following equation: dZ dF dλ dZ = · · dT dT dF dλ

(10:45)

In the above equation, from theoretical estimates, dF/dT and dZ/dλ can be calculated. The fiber coupling lens and the detector were mounted on a jig made of aluminum with a linear thermal expansion coefficient of 23.1 × 10−6°C−1 [48] in the first prototype measurement system. In the first prototype measurement setup, the line length of the jig which corresponded to F was planned to be 16.6 mm. The thermal expansion sensitivity of the jig dF/dT was assessed from eq. (10.42) to be 0.383 μm/°C. In addition, from the theoretical investigation mentioned previously, dZ/dλ is already known as dZ/ dλ = −255 nm/nm. Meanwhile, for the evaluation of dZ/dT, the term dλ/dF in eq. (10.45) still remains to be discussed. Experiments were conducted to obtain dλ/dF by purposely adjusting F in the calculation setup. As the target sample, a flat mirror was used, which was left stationary in the setup. The experiments were carried out in such a way that the parameter d was modified from 0 to 300 μm in a 20 μm step, while the centered wavelength was observed at each step. It should be noted that parameter d corresponded to the defocus of Detector2 from the focal plane of the fiber coupling lens. The variance of the parameter d (Δd) corresponded to -dF in eq. (10.45) in the experiments. It should be

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remembered that the positive orientation of d was set to be that of the fiber coupling lens from the fiber detector. The relationship between d and the focused wavelength λfocused derived from the normalized axial responses in Fig. 10.40, while the relationship between d and the corresponding normalized axial responses obtained in the experiments is shown in Fig. 10.41. dλ/dF was determined to be 0.5686 nm/μm from the experimental data. Therefore, the contribution of F in P4 from eq. (10.45) was assessed as dZ/dT = (0.383 μm/°C) ·(0.5686 nm/μm) ·(−255 nm/nm) = −55.5 nm/°C.

Fig. 10.40: The relationship between d and the corresponding normalized axial responses obtained in the experiments.

Fig. 10.41: The relationship between d and the corresponding focused wavelength in the normalized axial responses obtained in the experiments.

It also examines the contribution of the distance L between the chromatic objective lens and the fiber coupling lens in P4. The sum of the lengths of the mounting plate and Jig-A corresponds to L (=500 mm). A finite element model (FEM) shown in Fig. 10.42 is employed in this chapter to approximate the thermal expansion of L

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(ΔL). ΔL is calculated to be approximately 10 μm from the product of FEM analysis (by the Autodesk Nastran In-CAD); this value is small relative to L (=500 mm), and the effect of ΔL is negligibly small. It can be said from these findings that the contribution of F to the thermal expansion of optical path length P4 is much greater than that of L. P4 was measured to be −55.5 nm/°C as a consequence. It should be remembered that the measurement uncertainty of the femtosecond chromatic laser confocal probe could be increased by the uncertainty of L. For the stability of the femtosecond chromatic confocal probe, thus, the optical path length L is supposed to be decreased.

Fig. 10.42: Finite element model for the estimation of the thermal deformation of each component in the setup.

The contribution of each element to the overall thermal stability of the set-up is summarized in Tab. 10.4. In summary, from the experimental investigation, the thermal stability of the overall measurement setup (dZ/dT) is assessed to be 4.13 μm/°C overall. The discrepancy between the thermal equilibrium predicted in principle (4.13 μm/°C) and that obtained in the experiments (5.20 μm/°C) mentioned in the previous section of this chapter can be found at approximately 1 μm/°C. The discrepancy is taken to be the amount of the minor contributions from the other variables not taken into account in the theoretical equations. From this result, it can be concluded that the femtosecond laser chromatic confocal probe is required to stabilize the reduction of the line lengths of Jigs-A and-B, and the mounting plate, as well as the use of low thermal expansion materials for the Jig-B and jigs in the overall setup. The second prototype setup was newly designed and developed with the intention of improving the thermal stability of the chromatic confocal probe. The thermal stability of the chromatic confocal probe itself leads to the thermal stability of the

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Tab. 10.4: Contribution of each factor in the setup. Item

Factor

value

Probe

Refractive index instability of the chromatic objective lens

P = −. μm/°C

Refractive index instability of the surrounding air

P = −. μm/°C

Thermal expansion of mechanical jigs

P = . μm/°C

Thermal expansion of optical path length

P = −. μm/°C

Thermal stability of the probe

P = P  + P + P + P =. μm/°C

(Thermal expansion of a sample)

(S)

(Thermal expansion of sample jig)

(S)

(Thermal stability of the sample unit)

(S = S + S) Not considered

(Sample unit)

Mounting Thermal expansion of the mounting plate,corresponding to plate the thermal stability of the mounting plate

M = −. μm/°C

Thermal stability of the overall setup

O=P+S+M =. μm/°C

overall chromatic confocal measurement setup. Contributions could also be made to the thermal stability of the instrument by the sample jig and mounting plate. This study focuses on improving the thermal stability of the chromatic confocal probe itself, as the probe is used for various applications, the thermal stability of the sample, the sample jigs and the mounting plate shift. In order to increase the thermal stability of the femtosecond laser chromatic confocal probe, attention was paid to choosing a material for the main mechanical part (Jig-B) in the chromatic confocal probe with a low thermal expansion coefficient. Attention was also paid to the optical path length so that it would be as short as possible. The newly developed second prototype measurement setup is shown in Fig. 10.43. The size of the second prototype measurement setup, as can be seen in the figure, is reduced to 1/6 of that of the first prototype measurement setup shown in Fig. 10.33. Optical devices equivalent to those used in the first prototype measurement setup were used in the second prototype measurement setup for fair correlation. Jig-B material was also changed from aluminum to Super Invar, with a much smaller coefficient of thermal expansion relative to aluminum [49, 50]. On an aluminum optical breadboard, all the optical modules and jigs were mounted. In theoretical calculations, the increase in the thermal stability of the second prototype measurement configuration is tested first. The optical path length in the second prototype measurement setup is shortened by the changes mentioned above to be approximately 1/6 of that of the previous first prototype measurement setup. The thermal

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Fig. 10.43: The previous first prototype measurement setup and the newly developed second prototype measurement setup.

stability is calculated in the same way as the previous first prototype measurement setup mentioned in the previous section of this chapter for the newly developed second prototype measurement setup. The contribution of each element in the thermal stability of the probe to the second prototype measurement setup is summarized in Tab. 10.4. In the second prototype measurement setup (0.0197 μm/°C), the thermal stability of the chromatic confocal probe is much greater than that of the first prototype measurement setup (4.86 μm/°C). This outcome means that the second prototype measuring configuration will maintain the sub-micrometric stability of the femtosecond laser chromatic confocal probe. In order to check the improvement of the thermal stability of the second chromatic confocal probe prototype in the same way as the first prototype mentioned in the previous section, experiments were then carried out. Figure 10.44 shows one of the results in three repetitive experiments. As a result, a mean value of 7.9 nm/°C was obtained for dλ/dT. The thermal stability of the overall second prototype measurement setup for axial displacement measurement (dZ/dT) was then assessed from eq. (10.37) as (dZ/dT) = (7.9 nm/°C) ·(−255 nm/nm) = −2.01 μm/°C. The contribution of thermal expansion of the mounting plate is included in this result. The contribution of the mounting plate, whose linear thermal expansion coefficient is 23.1 × 10−6°C−1 [48], is measured in theory to determine the thermal instability of the probe. The configuration of the chromatic confocal lens and the target sample in the second prototype measurement setup is shown in Fig. 10.45. The contribution of the mounting plate from eq. (10.42) is determined as (dZ/dT)Mounting plate = (23.1 × 10−6°C−1) ·(175/2 mm) = −2.02 μm/°C with respect to the length of the mounting plate (175 mm). Taking into account the results of

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the experimental equation above, the thermal stability of the second prototype chromatic confocal probe is evaluated as follows: ðdZ =dT ÞProbe ≈ ðdZ =dT ÞOverall − ðdZ =dT ÞMounting plate = ð − 2.01 μm= CÞZ − ð − 2.02 μm= CZÞ = 0.01 μm= C

(10:46)

As can be seen in the equation, relative to that of the first prototype chromatic confocal probe (4.86 μm/°C), the thermal stability of the second prototype chromatic confocal probe (0.01 μm/°C) was effectively enhanced. In the theoretical equation (0.0197 μm/°C), the outcome obtained through the experiments was also similar to that expected. The effect of the thermal stability improvement in the newly developed second prototype chromatic confocal probe is demonstrated by the result.

Fig. 10.44: Thermal stability of the second chromatic confocal probe prototype.

Fig. 10.45: The configuration of the chromatic confocal lens and the target sample in the second prototype measurement setup.

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10.5 Improved chromatic confocal comb The effectiveness of the signal processing method referred to as the “tracking local minimum method” (TLM), in which the normalized output is obtained by employing the spectra captured by the two identical fiber detectors but having different confocal setup, has been verified by theoretical analysis. Experiments have also shown that with this approach, the evolved femtosecond laser chromatic confocal probe could calculate the displacement over a range of 40 μm with a resolution of 30 nm [27], as described in Sections 10.3 and 10.4. In the meanwhile, the amplitude of the confocal axial response recorded by the reference detector at the first local minimum is usually quite small and is therefore readily compromised by background variations caused by electromagnetic noises and mechanical vibrations. Furthermore, due to the asymmetry around the first local minimum of the confocal axial response, the normalized output curve around the peak position is asymmetric. This could cause a tracking error influencing the calculation of the Z-directional displacement. In this section, a new differential algorithm proposed for obtaining the normalized output is introduced. The newly proposed algorithm, which is referred to as the “tracking intersection method” (TIM), is expected to solve the above-mentioned problem in order to boost the efficiency of the femtosecond laser chromatic confocal probe. In the process, by detecting the corresponding wavelength at a peak, the wavelength at the intersection of the confocal axial reaction curves collected by the two detectors is monitored for the Z-directional displacement measurement of the object. One of the advantages of the method is a symmetric peak shape in the normalized output curve. Because at the intersection wavelength, the intensities of the two detector outputs are considerably greater than the background fluctuations, a higher signal-to-noise (S/N) ratio can be obtained relative to the previous tracking local minimum method. A schematic diagram of the chromatic axial responses IM and IR obtained by the measurement detector and the reference detector, respectively, is shown in Fig. 10.46. Both the chromatic axial responses have main-lobe and side-lobes, as can be seen in the figure [45]. The optical mode focusing on the surface of the measuring object is focused on the measurement detector in the setup. The maximum amplitude of the chromatic axial response IM, which can be obtained by the measurement detector, can be detected at the corresponding mode wavelength. On the other hand, the maximum amplitude of the chromatic axial response IR to be captured by the reference detector can be measured at a wavelength shorter than that of the optical mode focused on the surface of a measurement target, owing to the defocus d of the reference detector. The first local minimum between the main-lobe and the primary side-lobe in IR is used in the traditional tracking local minimum method to achieve the uniform output for calculating the Z-directional displacement

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of the measurement target. On the contrary, in the newly proposed tracking intersection method, the intersection point between IM and IR is used. In the following section, details of these two strategies are described.

Fig. 10.46: A schematic diagram of the chromatic axial responses obtained by the measurement detector and the reference detector.

A schematic of the traditional tracking local minimum method [27] is shown in Fig. 10.47. By using the chromatic axial responses IM and IR of the measurement detector and the reference detector, respectively, the normalized output ITLM can be obtained via the calculation algorithm defined as follows: ITLM =

IM IR

(10:47)

In Fig. 10.47, a schematic of ITLM is shown to be obtained by the above calculation algorithm. In the chromatic axial response IR, ITLM adopts the maximum value at the wavelength of the first local minimum. it should be noted that, because of the asymmetric feature of IR between the main lobe and the primary side lobe, the peak in ITLM has the asymmetric profile, as can be seen in Fig. 10.48. In a functional case where the normalized output has noise components, this asymmetric peak profile makes the estimation of the peak wavelength in ITLM challenging. The centroid method [27] is then applied, in which the centroid wavelength λTLM_C of the peak is obtained by the following equation: P i ½ITLM ðλi Þ · λi  (10:48) λTLM C = P i ITLM ðλi Þ

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In the above equation, by setting the cut-off intensity to be half of the maximum intensity, ITLM(λi) can be expressed as follows: ( 0 ITLM ðλi Þ < 0.5 · maxfITLM g ITLM ðλi Þ = (10:49) ITLM ðλi Þ ITLM ðλi Þ ≥ 0.5 · maxfITLM g The Z-directional displacement of a measurement object can be detected in the same way as the traditional chromatic confocal probes by tracking the centroid wavelength λTLM_C. It should be remembered that, due to the asymmetric profile of the ITLM_max, the centroid wavelength λTLM_C changes from the peak wavelength in ITLM.

Fig. 10.47: A schematic of the tracking local minimum method.

Fig. 10.48: A peak to be observed in the tracking local minimum method.

To approximate the peak profile that is to be found in a normalized output, numerical calculations are performed. The parameters used in the equations are the same as those used in [27]. The defocus d is set to be a particular value (80 μm) used in the sequence of experiments in this section. A measurement object under inspection

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525

is considered to have a flat profile in the following numerical calculations, and is moved in a step of 1 μm along the Z-direction. The confocal axial responses of the measurement detector and the reference detector are calculated at each Z-position of the measurement target, and then the normalized output ITLM is obtained on the basis of eq. (10.47). Results are shown in Fig. 10.49. Figure 10.50 shows a cross-sectional profile of ITLM at a certain Z-position in Fig. 10.49. The peak has an asymmetric profile, as can be observed in the figure. As a result, the centroid wavelength λTLM_C becomes different from the actual peak wavelength λTLM. The variance of the centroid wavelength λTLM_C is summarized in Fig. 10.51 as the change in the measurement object’s position in the Z-direction. For comparison, the real peak wavelength λTLM is also plotted in the diagram. As indicated in the figure, the asymmetric feature of the peak profile in the normalized output could influence the linearity of the variance of λTLM_C with respect to the Z-directional displacement.

Fig. 10.49: The observed normalized intensity ratio.

Fig. 10.50: Normalized intensity ratio at a certain Z-position.

A new algorithm for the estimation of the normalized output referred to as the tracking intersection method is proposed to solve the aforementioned issue in the traditional tracking local minimum method. A schematic of the signal processing method

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Fig. 10.51: The variance of the centroid wavelength λTLM_C.

proposed is shown in Fig. 10.52. One of the key features of the tracking intersection method is that, based on the following equation, the algorithm uses the intersection point between the measurement detector’s chromatic axial response IM and that of the reference detector IR, respectively, to obtain the normalized output ITIM: ITIM =

IM + IR jIM − IR j

(10:50)

In Figs. 10.52 and 10.53, a diagram of ITIM is shown to be obtained by the above calculation algorithm. At the wavelength of the intersection point between IM and IR, the normalized output ITIM takes the highest value. The peak in the normalized ITIM becomes much narrower than that in ITLM to be achieved by the traditional tracking local minimum method, owing to the steep slopes of IM and IR around the intersection point. The discrepancy between the traditional tracking local minimum method and the tracking intersection method is outlined in Figs. 10.54 and 10.55. In the normalized output ITIM, the narrower peak is effective in improving the accuracy in the determination of the peak centroid wavelength, which leads to improving the precision and resolution of the calculation of the Z-directional displacement. Another benefit of the newly proposed tracking intersection method is that the normalized ITIM output can be achieved by using much higher intensity confocal axial responses relative to those used with the traditional tracking local minimum method. Against the background disturbances introduced by electromagnetic noises and mechanical vibrations, even greater intensities of the confocal axial responses are required to make the normalized output stable. Numerical calculations are carried out, in the same manner, to approximate the peak profile to be observed in the tracking intersection method. The conditions are set to be the same in numerical equations as those for the conventional tracking local minimum method. The normalized output ITIM obtained at each Z-position of the measurement object is shown in Fig. 10.56, and the cross-sectional profile of ITIM at a given Z-position in Fig. 10.56 is shown in Fig. 10.57. The peak in the normalized output becomes much narrower than that in the traditional tracking local

10.5 Improved chromatic confocal comb

Fig. 10.52: A schematic of the proposed tracking intersection method.

Fig. 10.53: A schematic of the proposed signal processing method.

Fig. 10.54: A schematic of the traditional tracking local minimum method.

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Fig. 10.55: A schematic of the tracking intersection method.

minimum method, as can be seen in the figure. The difference of the centroid wavelength λTIM_C and the peak wavelength λTIM is summarized in Fig. 10.58 as the change of the measurement object’s Z-position. The discrepancy between λTIM_C and λTIM is found to be much smaller than in the traditional local minimum method, due to the much narrower peak in the normalized output.

Fig. 10.56: The normalized output in simulation.

Fig. 10.57: The normalized output at a certain Z-position in simulation.

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Fig. 10.58: The difference of the centroid wavelength λTIM_C and the peak wavelength λTLM.

Experiments were carried out to check the viability of the tracking intersection method. A schematic diagram of the setup built in this study is shown in Fig. 10.59. Setup parameters, such as the spectral distance of the femtosecond laser and the optical elements, were the same as those used in their previous setup [45]. The measurement detector output and the reference detector output were obtained in sequence, since the optical spectrum analyzer (OSA) employed in this study had a single input fiber port. In order to obtain the detector outputs sequentially, a fiber switch (OSW-1X2-1310/1550-5V-N-900-1-1-FC, Optei Communications Co.) was thus employed, and was placed between the dual-detector unit and the OSA. The output of the detector can be acquired as the product of the light source optical spectrum and the confocal axial response collected by the detector. The measurement light spectrum IM and the reference light spectrum IR at each measuring point required eight seconds to be received by the OSA. The employment of a high-speed spectrometer is expected to improve the performance. However, in general, a trade-off relationship can be found between the wavelength resolution and the measurement throughput in a spectrometer. The problem is supposed to be solved by enhancing the resolution of the Z-position calculation by the configuration adjustment of the chromatic objective [24]. A diagram and a photograph of the setup of the dual-detector differential chromatic confocal probe established in this study are shown in Figs. 10.60 and 10.61. The key sections used in the setup were the same as those used in the traditional setup [27]. In this study, meanwhile, attempts have been made to reduce the optical path length in the chromatic confocal probe in order to increase the thermal stability of the system [25, 27]. The total size of the modified design was decreased to 200 mm × 250 mm (in the XZ-plane), which was roughly one-third of that of the standard one (350 mm × 480 mm), thus enabling the synchronization of each of the system’s optical components.

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Fig. 10.59: A schematic diagram of the setup.

Fig. 10.60: A diagram of the setup of the dual-detector differential chromatic confocal probe.

In order to explore the viability of the proposed algorithm seen in eq. (10.50), tests have been carried out. A defocus d of the reference detector was set to be 80 μm in the subsequent experiments with respect to the results of a preliminary experiment where a reasonably good sensitivity was obtained. The optical spectrum analyzer bandwidth

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Fig. 10.61: A photograph of the setup of the dual-detector differential chromatic confocal probe.

was set to be 0.5 nm with a sampling interval of 0.1 nm; the capturing of a single detector output takes around four seconds with this configuration. To capture the measurement detector output and the reference detector output sequentially, the target mirror was initially kept stationary at a certain Z-position in the middle of the Z-directional measurement spectrum of the probe. The outcomes are shown in Fig. 10.62. Both detector outputs were observed to be highly influenced by the non-uniform strength of the light source, as can be seen in the figure. As a result, the first local minimum of the reference detector output in the figure is difficult to find out. There is a small difference between the two outputs of the detectors; this was mostly due to the defocus d provided to the reference detector. Normalized outputs based on the traditional and newly proposed differential algorithms were determined by using the obtained detector outputs. The normalized output determined based on the traditional tracking local minimum method is shown in Fig. 10.63. The wavelength of the first local minimum of the confocal axial response can be clearly observed as the peak wavelength in the figure under the extreme influence of the non-uniform intensity of the femtosecond laser beam because of the effect of the differential algorithms. In the normalized output, a full width at half limit (FWHM) of the peak was evaluated to be 11.4 nm. A normalized output was also determined based on the newly proposed tracking intersection method by using the detector outputs. The outcome is shown in Fig. 10.64. A sharp peak with an FWHM of less than 0.1 nm was reached, as can be seen in the figure; this is far narrower than that obtained by the conventional tracking local minimum method. It should also be emphasized that the light intensities at the point of intersection in the detector outputs were observed to be much greater than those at the first local minimum in the reference detector output. The signal-to-noise (S/N) ratio of the chromatic confocal probe produced is supposed to increase the greater light intensities at the intersection point.

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Fig. 10.62: Spectra obtained by the setup.

Fig. 10.63: Normalized output by the tracking local minimum method.

Fig. 10.64: Normalized output by the tracking intersection method.

To test the basic characteristics of the newly proposed algorithm, experiments were expanded. The target mirror was made to pass over a range of 70 μm in a step of 2 μm along the Z-direction, and the outputs of the detector were obtained at each Z-position. After that, depending on the algorithms, the normalized outputs at each Z-position are determined. The results obtained by the traditional tracking local minimum method are shown in Figs. 10.65 and 10.66. Figure 10.65 indicates a variation of the centroid wavelength of the peak of the normalized output as the shift of the Z-position of the target mirror. In a Z-directional range of 44 μm, strong linearity can be observed. The FWHM of the peak in the normalized output at every Z-position of the target mirror is seen in

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Fig. 10.66. In the Z-directional measurement range, the FWHM was observed to deviate, as can be seen in the figure. The FWHM of the peak in the normalized output has been found to be wide at the edges of the Z-directional measurement spectrum. It should be noted that the light intensities in the spectrum of the femtosecond laser employed in the experiments were low at the wavelengths. In the determination process of the centroid wavelength of the peak, the broader FWHM of the peak, as well as the lower S/N ratio, could cause an increase in uncertainty and is thus beneficial to be decreased.

Fig. 10.65: The results obtained by the traditional tracking local minimum method.

Fig. 10.66: FWHM of the peak obtained by the traditional tracking local minimum method.

In the same manner, the newly proposed tracking intersection method processed the normalized outputs produced from the same setup. The variance of the centroid wavelength of the peak of the normalized output is seen in Fig. 10.67 as the shift in the Z-position of the target mirror. In a Z-directional range of 50 μm, which is broader than that achieved by the traditional approach, strong linearity can be found. The FWHM of the peak in the normalized output at each Z-position is shown in Fig. 10.68. The FWHM of the peak over the Z-directional measurement spectrum was found to be narrower than 0.5 nm, which is far narrower than that observed by the traditional approach. In the newly proposed tracking intersection method, these features of the normalized output led to increasing the Z-directional measurement range.

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It should be noted that in the normalized output in Fig. 10.64, a slight peak can be found at 1,625 nm. This peak corresponds to the point at which the main-lobe and the primary side-lobe intersect [45]. The measurement sensitivities of the probe obtained by the traditional method and the newly proposed method were evaluated from the results shown in Figs. 10.67 and 10.65 to be −3.02 and −2.99 nm/μm, respectively. The sensitivities obtained are identical to one another. It should be remembered that the sensitivities found in the experiments were lower than those expected in the theoretical calculations (−3.92 nm/μm for the newly proposed method); the discrepancy is mostly due to the deviation in the theoretical equations of the dispersions of the optical components used in the practical configuration.

Fig. 10.67: The results obtained by the traditional tracking intersection method.

Fig. 10.68: FWHM of the peak obtained by the tracking intersection method.

Finally, tests were conducted based on the recently proposed tracking intersection method to explore the resolution of the fiber-based dual-detector differential chromatic confocal probe. The target mirror was made to move in a step of several-ten nm along the Z-direction, and at each Z-position the measurement detector output and the reference detector output were obtained. The variances of the centroid wavelength of the peak, obtained in the experiment with the Z-directional mirror displacement in steps of 40, 30 and 20 nm, respectively, are shown in Figs. 10.69–10.71, respectively. These experimental findings revealed that, based on the newly proposed tracking intersection method, the fiber-based dual-detector differential chromatic

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confocal probe could carry out more stable peak detection in the normalized output compared to the traditional tracking local minimum method. This leads to the achievement of a broader measuring spectrum and higher resolution under the same physical configuration of the dual-detector chromatic confocal probe.

Fig. 10.69: Z-directional mirror displacement in a step of 40 nm measured by the tracking intersection method.

Fig. 10.70: Z-directional mirror displacement in a step of 30 nm measured by the tracking intersection method.

Fig. 10.71: Z-directional mirror displacement in a step of 20 nm measured by the tracking intersection method.

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The uncertainty analysis was also carried out based on the proposed tracking intersection method on the determination of the peak wavelength λTIM_C. λTIM_C can be estimated, as stated in the previous section of this article, by using ITIM calculated on the basis of eq. (10.50) with the detectors’ IM and IR chromatic axial responses obtained. The centroid approach mentioned in eq. (10.48) is, meanwhile, used in the estimation process; this makes the study of uncertainty difficult. A Monte-Carlo approach [51] was then used in this study to estimate the uncertainty of the proposed algorithm. Using the experimental setup shown in Fig. 10.72 where the target mirror was kept stationary, the contributions of IM and IR were first investigated by repeated experiments, since ITIM can be derived using IM and IR. For the calculation of the standard uncertainty of the light intensity data at each light wavelength in IM and IR, 200 repetitive trials were made. IM and IR have been observed to have light intensity fluctuation of approximately 0.0204 percent at each light wavelength from the experimental results. This was largely attributed to the fluctuation of the light source, though it involves a contribution from the variability of the reading in the OSA (1.0 × 10−4%). Based on the estimated fluctuation of IM and IR, a Monte Carlo method was used for the propagation of the distributions of IM and IR based on eq. (10.50). It used a trial number of 106. The outcome of the Monte Carlo methods is indicated in Fig. 10.72. In the outcome, an asymmetric distribution was found; this was attributable to the peak asymmetry in ITIM. An uncertainty uλTIM-C of the estimation of the peak wavelength λTIM-C was assessed from the distribution to be approximately 7.10 × 10−4 nm. As far as the sensitivity of the chromatic confocal probe developed (−2.99 nm/μm) is concerned, this ambiguity leads to a Z-displacement of approximately 0.24 nm, which is less than the resolution of the developed chromatic confocal probe (20 nm).

Fig. 10.72: Result of the Monte-Carlo method.

References

537

10.6 Summary In this chapter, a fiber-based dual-detector chromatic confocal probe developed with a mode-locked femtosecond laser source has been described. In addition, for the femtosecond laser chromatic confocal probe, an improved data processing method referred to as the tracking intersection method has also been presented. The method has been designed in such a way that the intersection point of the two fiber detector outputs is utilized for obtaining the Z-directional displacement of a measurement object. To simulate the peak profile in a normalized output to be obtained by the two fiber detector outputs, computational calculations were performed. The findings have shown that the proposed tracking intersection method can obtain a much narrower peak in the normalized output compared to that of the traditional local minimum tracking method. Using a redesigned fiber-based dualdetector chromatic confocal probe, tests were also carried out. In the optical configuration, for more reliable estimation, the optical path length has been shortened to confirm the viability of the newly proposed method of data processing. Experimental findings have confirmed that an improvement in the normalized output with a full-width half-maximum (FWHM) of less than 0.5 nm can be reached by the proposed tracking intersection method, which is far smaller than that of the traditional method. Experimental studies have shown that a measurement length of 50 μm and a resolution of 20 nm, which is better than that of the traditional method, can be obtained by the proposed method. The feasibility of the newly proposed tracking intersection method has successfully been verified from the results of numerical calculations and experiments.

References [1] [2]

[3] [4] [5] [6] [7]

[8]

Marvin M. U.S. patent 3, 0130467:Microscopy apparatus. Wilson T, Sheppard C. Orlando San Diego San Francisco New York Toronto Montreal Sydney Tokyo Sao Paulo L. Theory and Practice of Scanning Optical Microscopy with Contributions from ACADEMIC PRESS, 1984. Davidovits P, Egger MD. Scanning laser microscope. Nature 1969, 223, 5208, 831–831. Browne MA, Akinyemi O, Boyde A. Confocal surface profiling utilizing chromatic aberration. Scanning 1992, 14, 3, 145–153. Tiziani HJ, Uhde H-M. Three-dimensional image sensing by chromatic confocal microscopy. Appl Opt 1994, 33, 10, 1838. Luo D, Kuang C, Liu X. Fiber-based chromatic confocal microscope with Gaussian fitting method. Opt Laser Technol 2012, 44, 4, 788–793. Brand U, Beckert E, Beutler A, Dai G, Stelzer C, Hertwig A, Klapetek P, Koglin J, Thelen R, Tutsch R. Comparison of optical and tactile layer thickness measurements of polymers and metals on silicon or SiO2. Meas Sci Technol 2011, 22, 9. Shi K, Li P, Yin S, Liu Z. Chromatic confocal microscopy using supercontinuum light. Opt Express 2004, 12, 10, 2096.

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Chapter 10 Chromatic confocal comb

Minoni U, Manili G, Bettoni S, Varrenti E, Modotto D, De Angelis C. Chromatic confocal setup for displacement measurement using a supercontinuum light source. Opt Laser Technol 2013, 49, 91–94. Dobson SL, Sun P, Fainman Y. Diffractive lenses for chromatic confocal imaging. Appl Opt 1997, 36, 20, 4744. Lin PC, Sun P-C, Zhu L, Fainman Y. Single-shot depth-section imaging through chromatic slit-scan confocal microscopy. Appl Opt 1998, 37, 28, 6764. Cha S, Lin PC, Zhu L, Sun P-C, Fainman Y. Nontranslational three-dimensional profilometry by chromatic confocal microscopy with dynamically configurable micromirror scanning. Appl Opt 2000, 39, 16, 2605. Yang C, Shi K, Zhou M, Zheng S, Yin S, Liu Z. Z-microscopy for parallel axial imaging with micro mirror array. Appl Phys Lett 2012, 101, 23, 231111. Jones DJ, Cundiff ST, Fortier TM, Hall JL, Ye J. Carrier–Envelope Phase Stabilization of Single and Multiple Femtosecond Lasers. Few-Cycle Laser Pulse Generation and Its Applications, Berlin, Heidelberg, Springer, 2004, 317–343. Chen Y-L, Shimizu Y, Kudo Y, Ito S, Gao W. Mode-locked laser autocollimator with an expanded measurement range. Opt Express 2016, 24, 14, 15554. Shimizu Y, Kudo Y, Chen Y-L-L, Ito S, Gao W. An optical lever by using a mode-locked laser for angle measurement. Precis Eng 2017, 47, 72–80. Kelly SMJ. Characteristic sideband instability of periodically amplified average soliton. Electron Lett 1992, 28, 8, 806–807. Tauser F, Leitenstorfer A, Zinth W. Amplified femtosecond pulses from an Er: fibersystem: Nonlinear pulse shortening and selfreferencing detection of the carrier-envelope phase evolution. Opt Express 2003, 11, 6, 594. Kobayashi K, Akiyama K, Suzuki T, Yoshizawa I, Asakura T. Laser-scanning imaging system for real-time measurements of 3-D object profiles. Opt Commun 1989, 74, 3–4, 165–170. Butler DJ, Horsfall A, Hrynevych M, Kearney PD, Nugent KA. Confocal profilometer with nanometric vertical resolution. Opt Commun 1993, 100, 1–4, 87–92. Liu J, Tan J, Bin H, Wang Y. Improved differential confocal microscopy with ultrahigh signal-tonoise ratio and reflectance disturbance resistibility. Appl Opt 2009, 48, 32, 6195–6201. Jang S, Shimizu Y, Ito S, Gao W. Development of an optical probe for evaluation of tool edge geometry. J Adv Mech Des Syst Manuf 2014, 8, 4. Qiu L, Liu D, Zhao W, Cui H, Sheng Z. Real-time laser differential confocal microscopy without sample reflectivity effects. Opt Express 2014, 22, 18, 21626. Chen L-C, Nguyen DT, Chang Y-W. Precise optical surface profilometry using innovative chromatic differential confocal microscopy. Opt Lett 2016, 41, 24, 5660. Sato R, Shimizu Y, Chen C, Matsukuma H, Gao W. Investigation and Improvement of Thermal Stability of a Chromatic Confocal Probe with a Mode-Locked Femtosecond Laser Source. Appl Sci 2019, 9, 19, 4084. Shimizu Y, Nakamura T, Chen Y-L, Gao W. A confocal microscope with a mode-locked laser source. Proceedings of International Conference on Leading Edge Manufacturing in twenty-first Century: LEM21, Japan Society of Mechanical Engineers, 2017, 1–4. Chen X, Nakamura T, Shimizu Y, Chen C, Chen Y-LY-L, Matsukuma H, Gao W. A chromatic confocal probe with a mode-locked femtosecond laser source. Opt Laser Technol 2018, 103, 359–366. Nakamura T. Confocal microscopy using optical frequency comb. Tohoku University, Master thesis, 2018. Sato R. A Confocal Probe with a Femtosecond Laser. Tohoku University, Master thesis, 2021.

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[30] Kimura S, Wilson T. Confocal scanning optical microscope using single-mode fiber for signal detection. Appl Opt 1991, 30, 16, 2143. [31] Snyder AW, Love JD. Optical Waveguide Theory, Boston, MA, Springer US, 1984. [32] Born M, Wolf E, Bhatia AB, Clemmow PC, Gabor D, Stokes AR, Taylor AM, Wayman PA, Wilcock WL. Principles of Optics, Cambridge University Press, 1999. [33] Wilson1 T, Carlini AR. Size of the detector in confocal imaging systems. Opt Lett 1987, 12, 4, 227. [34] Sheppard CJR, Gu M. Aberration compensation in confocal microscopy. Appl Opt 1991, 30, 25, 3563. [35] Martínez-Corral M, Caballero MT, Pons A, Andrés P. Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings. Micron, Vol. 34, Elsevier Ltd, 2003, 319–325. [36] Suhre DR, Gupta N. Acousto-optic tunable filter sidelobe analysis and reduction with telecentric confocal optics. Applied Optics, Vol. 44, OSA – The Optical Society, 2005, 5797–5801. [37] Segawa S, Kozawa Y, Sato S. Resolution enhancement of confocal microscopy by subtraction method with vector beams. Opt Lett 2014, 39, 11, 3118. [38] Brakenhoff GJ, Blom P, Barends P. Confocal scanning light microscopy with high aperture immersion lenses. J Microsc 1979, 117, 2, 219–232. [39] Neil MAA, Juškaitis R, Wilson T, Laczik ZJ, Sarafis V. Optimized pupil-plane filters for confocal microscope point-spread function engineering. Opt Lett 2000, 25, 4, 245. [40] Wang TD, Contag CH, Mandella MJ, Chan N, Kino GS. Dual-axes confocal microscopy with post-objective scanning and low-coherence heterodyne detection. Opt Lett 2003, 28, 20, 1915. [41] Gweon D, Kang D. US Patent US20070070496A1: Confocal self-interference microscopy from which side lobe has been removed. [42] Stelzer EHK, Lindek S. Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: Confocal theta microscopy. Opt Commun 1994, 111, 5–6, 536–547. [43] Nagorni M, Hell SW. 4Pi-confocal microscopy provides three-dimensional images of the microtubule network with 100- to 150-nm resolution. J Struct Biol 1998, 123, 3, 236–247. [44] Cox IJ, Sheppard CJR, Wilson T. Improvement in resolution by nearly confocal microscopy. Appl Opt 1982, 21, 5, 778. [45] Chen C, Sato R, Shimizu Y, Nakamura T, Matsukuma H, Gao W. A Method for Expansion of Z-Directional Measurement Range in a Mode-Locked Femtosecond Laser Chromatic Confocal Probe. Appl Sci 2019, 9, 3, 454. [46] Ciddor PE. Refractive index of air: New equations for the visible and near infrared. Appl Opt 1996, 35, 9, 1566. [47] Schott AG. Technical Information [TIE-19] Temperature Coeffcient of the Refractive Index., 2016. (, at https://www.schott.com/d/advanced_optics/3794eded-edd2-461d-aec5 -0a1d2dc9c523/1.1/schott_tie-19_temperature_coefficient_of_refractive_index_eng.pdf) [48] Hayashi M. Rika Nenpyo., 2018. [49] Berthold JW, Jacobs SF. Ultraprecise thermal expansion measurements of seven low expansion materials. Appl Opt 1976, 15, 10, 2344. [50] Jacobs SF. Dimensional stability of materials useful in optical engineering. Opt Acta (Lond) 1986, 33, 11, 1377–1388. [51] Joint Committee For Guides In Metrology. Evaluation of measurement data – Guide to the expression of uncertainty in measurement. Int Organ Stand Geneva ISBN 2008, 50, September, 134.

Chapter 11 Position comb 11.1 Introduction Precision positioning is one of the important operations in manufacturing equipment of semiconductors and flat panel displays, state-of-the-art measuring instruments and precision machine tools [1, 2]. Accurate measurement of position is essential for precision positioning. Position measurement can be made by using interferometers. The position of an object in a plane (two-axis) or a space (three-axis) can also be measured by using multiple interferometers. Laser interferometers have the advantages of high resolution, large measurement range, fast measurement speed, flexible arrangement of optical paths, direct linkage to the length definition, etc. However, operation in a vacuum is required for the interferometers to avoid influences of air pressure, air temperature and relative humidity on the refractive index. Taking into account that in most cases the position measurement is required to be made in air, it is difficult to maintain the measurement accuracy of the interferometer. Compared with a laser interferometer, an optical linear encoder is more robust to measurement environment and good in cost performance, which is more suitable for one-axis position measurement in industry. As presented in Chapters 3 and 4, surface encoders have also been developed for two-axis and/or three-axis position measurement.

Fig. 11.1: Schematic of the conventional absolute-type linear encoder.

Position measurement is often required to be made with respect to a reference point, which is typically the origin of a given coordinate system. Such a measurement is referred to as the absolute position measurement. One-axis absolute position measurement can be made by using an absolute-type linear encoder. Figure 11.1 shows an https://doi.org/10.1515/9783110542363-011

542

Chapter 11 Position comb

Fig. 11.2: Schematic of the absolute position comb.

absolute-type linear encoder with a single absolute track of M-codes [1, 3]. Multiple tracks of binary codes or grey codes can also be employed for the encoding of the absolute position [4, 5]. In Fig. 11.1, an additional incremental track is added for the subdivision of the position codes so that the absolute position measurement can be made in a high resolution. On the other hand, however, the two-axis and/or threeaxis position measurement by a surface encoder, in which a scale grating with a constant grating pitch is read by an optical head with a CW laser source, is limited to an incremental measurement mode, which can only provide the information on changes of position. In this chapter, an absolute-type surface encoder, which is referred to as the position comb, is presented. The position comb, a schematic of which is shown in Fig. 11.2, is presented by combining the optical frequency comb of a femtosecond laser and a planar scale grating having variable periods along the X- and Y-directions. Meanwhile, the stability of the optical frequency comb is an important factor for the position comb. In Section 11.2, the stability of a free-run optical frequency comb from a commercial femtosecond laser is confirmed by using a real-time measurement system for the comb spacing of the optical frequency comb, that is, the repetition frequency of the pulse envelope from the femtosecond laser. Section 11.3 then presents the construction of a stabilized optical frequency comb based on a femtosecond fiber laser with integrated measurement and feedback control capability of both the repetition frequency and the carrier-envelope offset frequency, followed by the position comb with the stabilized optical frequency comb in Section 11.4.

11.2 Real-time monitoring of repetition frequency

543

11.2 Real-time monitoring of repetition frequency When an optical frequency comb is applied to measure the optical frequency of a continuous wave (CW) laser, the CW laser is “locked” to a comb mode of the optical frequency comb, which is employed as the frequency ruler. The frequency of the CW laser can be obtained from the comb frequency and the beat frequency produced from the interference of the comb mode and the CW laser. In such an optical frequency measurement, stabilization of the optical frequency comb is basically required through feedback control of the repetition frequency frep and the carrier-envelope offset frequency fceo, which is often associated with a hardware-based control unit with a high-cost. Careful adjustment and tuning of the control parameters are also necessary and the stability of the optical frequency comb is directly influenced by the control unit. On the other hand, when an optical frequency comb is applied to precision dimensional metrology, in addition to the employment of a stabilized optical frequency comb realized by hardware-based feedback control, there is also a possibility to utilize a free-run optical frequency comb without stabilization, depending on the principle and configuration of the dimensional measurement system. In this case, the repetition frequency frep and/or the carrier-envelope offset frequency fceo can be monitored in real-time for software compensation of the results of dimensional measurement. A free-run optical frequency comb is much less expensive and easier to use compared with a stabilized optical frequency comb. It is also expected to achieve a higher measurement resolution of frep and/or fceo without the existence of the electronic noises and instabilities of the hardwarebased frequency feedback control unit. The software compensation approach based on real-time comb frequency monitoring of frep and/or fceo can therefore be an effective alternative for dimensional metrology. In this section, the feasibility of this approach is demonstrated by real-time frep measurement of a commercial femtosecond fiber laser [6].

Fig. 11.3: Direct measurement of repetition frequency by using a frequency counter.

The repetition frequency of a mode-locked femtosecond laser can be directly measured by using a frequency counter as shown in Fig. 11.3 [7]. Denoting the electric field of the pulse train from a mode-locked laser by EP(t), EP(t) can be expressed as

544

Chapter 11 Position comb

EP ðtÞ = e

− j2πfc t

gP ðtÞ = e

− j2πfc t

 X  1 , g t−n frep n

n = 0, 1, 2, . . .

(11:1)

where fc is the frequency of the carrier wave. g(t) and gP(t) are the pulse envelope and the train of pulse envelopes, respectively. gP(t) is a periodic function of fundamental frequency frep, which can be described by the following Fourier series with Fourier coefficients of am: X

 am cos 2πmfrep t , m = 0, 1, 2, . . . (11:2) g P ðt Þ = m

EP(t) is then converted into an intensity signal IP(t) by the photodiode. IP(t), which is also a periodic function of fundamental frequency frep, can be expressed by Fourier series with Fourier coefficients of bm as follows: X

 bm cos 2πmfrep t , m = 0, 1, 2, . . . (11:3) IP ðtÞ = EP ðtÞEP* ðtÞ = gP2 ðtÞ = m

where E*P(t) is the complex conjugate of EP(t). As presented in Chapter 8, the pulse envelope g(t) has a short pulse width covering a wide frequency range. The intensity signal IP(t) is then composed of a massive number of high-order harmonics of the fundamental frep. In a frequency counter, the number of periodic signals occurring within a certain period of time is counted to measure the fundamental frequency of the input signal [8]. After a preset period known as the gate time, the counter is reset to zero after the measurement result is output. If the input signal repeats itself with sufficient stability and the frequencies included in the input signal are considerably lower than that of the clock oscillator, that is, the time base of the counter [9, 10], the fundamental frequency of the input signal can be obtained with an adequate resolution from the gate time. However, if the input signal contains high-order harmonics of the fundamental frequency, such as the case of the intensity signal IP(t), the measurement resolution and accuracy of the fundamental frequency will be reduced. Due to this reason, the resolution and accuracy of the method for measurement of frep shown in Fig. 11.3 are limited although the method is simple. The measurement of frep can be improved by using the system shown in Fig. 11.4, which is referred to as the real-time repetition frequency (frep) monitoring (RTRFM) system based on the heterodyne method [11]. A frequency mixer and a low-pass filter are employed to generate a beat frequency, which is the difference between frep and a reference frequency from a high-accuracy radio frequency (RF) synthesizer [12]. The beat frequency can be measured with a higher resolution by the frequency counter since it is much lower than frep by setting the reference frequency to be close to the nominal value of frep. The influence of the higher harmonics of frep included in the PD output is also removed by the low-pass filter. Both the RF synthesizer and the frequency counter are phase-locked to a rubidium (Rb) frequency standard [13], which is employed to assure the measurement accuracy of frep.

11.2 Real-time monitoring of repetition frequency

545

Fig. 11.4: A real-time repetition frequency monitoring (RTRFM) system for a commercial femtosecond fiber laser.

Fig. 11.5: Frequency manipulation in the RTRFM system for measurement of frep.

As shown in Fig. 11.5, the pulse intensity signal IP(t), or simply called the pulse signal, which carries the information of the repetition frequency frep, is input to a frequency mixer. The harmonics of frep included in the pulse signal are not illustrated in the figure for clarity based on the fact the harmonics of frep will be removed

546

Chapter 11 Position comb

by the low-pass filter. For the commercial femtosecond fiber laser used in the experiment, the nominal value of frep is 100 MHz. A signal IRF(t) output from the RF synthesizer with a frequency fRF, which is set to be consistent or close to the nominal value of frep, is input into the frequency mixer as the local oscillator (LO) signal. The output signal of the frequency mixer, which is referred to as the intermediate frequency (IF) signal IIF(t), is a product of the pulse signal and the LO signal. The IF signal IIF(t) contains two frequencies: a sum frequency frep + fRF and a differential frequency frep–fRF. The former is approximately two times of frep and the latter is a small fraction of frep. The latter with a very low frequency can then be separated from the former by using the low-pass filter for high-resolution and high-accuracy measurement of frep. An avalanche photodiode (APD, Hamamatsu C5658) module is employed as the PD in the RTRFM system, which is a highly sensitive photodiode consisting of a Si-APD, a bias power supply and a low-noise amplifier. It has a high response speed up to 1 GHz and a low noise level down to −48 dB in combination with the low-noise amplifier. Figure 11.6 shows the output of the APD, that is, the pulse signal, which was observed by an oscilloscope (Yokogawa DLM2024). It can be seen that the pulse signal has a period of approximately 10 ns, corresponding to the repetition frequency of 100 MHz.

Fig. 11.6: Waveform of the pulse intensity signal from APD.

An RF synthesizer is an electric circuit for generating harmonics of a standard frequency. Most RF synthesizers employ a phase-locked loop (PLL) [14] for feedback control of the output frequency. Figure 11.7 and Tab. 11.1 show a schematic and the specifications of a PLL-based commercial frequency synthesizer used in the RTRFM system. The output frequency after passing through a frequency divider is fed back to compare with the standard frequency. The error signal is then lowpass filtered to drive a voltage-controlled oscillator (VCO) for producing a compensated output frequency. Consequently, the output frequency fRF will be locked to the

11.2 Real-time monitoring of repetition frequency

547

Fig. 11.7: Schematic of the PLL synthesizer generating the local oscillator (LO) signal for the frequency mixer.

standard frequency and the output signal IRF(t) can be expressed by the following equation: IIF ðtÞ = cosð2πfRF tÞ

(11:4)

where the amplitude of IRF(t) is assumed to be 1 for clarity. A commercial frequency mixer of double-balanced passive type (Tab. 11.2) is employed in the RTRFM system. The dividing number K of the frequency divider is programmable and can be selected based on the specific output frequency. In the following experiment, K is selected to be 10 so that the output frequency fRF is 100 MHz, which is consistent with the nominal value of frep. The accuracy of an RF synthesizer is basically determined by the standard reference. In the RTRFM system, a commercial rubidium frequency standard, the specifications of which are shown in Tab. 11.3, is employed to provide an external standard reference signal to the RF synthesizer. In the rubidium (Rb) frequency standard, a specified hyperfine transition of electrons in rubidium-87 atoms is used to provide an accurate and stable frequency [15]. It is one of the most widely used atomic clocks with the advantages of low cost and compactness. The intermediate frequency (IF) signal IIF(t) output from the frequency mixer, which is a product of the pulse signal IP(t) from the photodiode and the local oscillator (LO) signal IRF(t) from the RF synthesizer can be expressed as follows based on eqs. (11.3) and (11.4):

548

Chapter 11 Position comb

IRF ðtÞ = IP ðtÞIIF ðtÞ =

X

 bm cos 2πmfrep t cosð2πfRF tÞ

m

=

X bm m

=

2

ðcos 2πðmfrep + fRF Þt + cos 2πðmfrep − fRF ÞtÞ

b0 b1 b1 + cos 2πðfrep − fRF Þt + cos 2πðfrep + fRF Þt 2 2 2 +

b2 ðcos 2πð2frep + fRF Þ + cos 2πð2frep − fRF ÞÞ + · · · 2

Tab. 11.1: Specifications of RF synthesizer [6]. Maker

Tektronix Co, Ltd

Model name

TSGA

Frequency resolution

 μHz at any frequency

Switching speed

> + > < (11:21) Eg after ðtÞ = gðtÞECS ðtÞ = gðtÞΔESPM ðtÞ, − tS ≤ t ≤ tS > > > > + > > : gðtÞEc0 ðtÞ, tS ≤ t ≤ tP where ΔESPM(t) is the electric field of the part of the carrier wave with chirped frequency by SPM, and ΔESPM ðtÞ = ej2πðfc0 + γtÞt − tS ≤ t ≤ tS

(11:22)

Based on the assumption that tS is sufficiently small, the magnitude of the pulse envelope over the time period of t = −tS to tS can be approximated to be the maximum magnitude g0 of g(t) at t = 0. Equation (11.21) can then be rewritten as:

11.3 Fiber frequency comb

8 gðtÞEC0 ðtÞ, − tP ≤ t < − tS > > > > > + > < Eg after ðtÞ = gðtÞECS ðtÞ = g0 ΔESPM ðtÞ, − tS ≤ t ≤ tS > > > > + > > : gðtÞEC0 ðtÞ, tS ≤ t ≤ tP Equation (11.23) can be further expanded to 8 gðtÞEC0 ðtÞ, − tP ≤ t < − tS > > > > > > + > > > > > > > gðtÞEC0 ðtÞ, − tS ≤ t ≤ tS > > < Eafter ðtÞ = gðtÞECS ðtÞ = + > > > > > g0 ΔESPM ðtÞ − gðtÞEC0 ðtÞ , − tS ≤ t ≤ tS > > > > > > + > > > > : gðtÞEC0 ðtÞ, tS ≤ t ≤ tP 8 8 gðtÞEC0 ðtÞ, − tP ≤ t < − tP < < Eg before ðtÞ, − tP ≤ t < − tP + = + = : : ΔEg ðtÞ, − tS ≤ t ≤ tS g0 ΔESPM ðtÞ − g0 EC0 ðtÞ , − tS ≤ t ≤ tS

569

(11:23)

(11:24)

where ΔEg ðtÞ = g0 ΔESPM ðtÞ − g0 EC0 ðtÞ = g0 ðej2πðfc0 + γtÞt − ej2πfc0 t Þ 2

(11:25)

= ej2πfc0 t g0 ðej2πγt − 1Þ, − tS ≤ t ≤ tS Taking into consideration that the electric wave is a real function of time, then eq. (11.25) can be rewritten as follows by taking the real part of the complex function in the last brackets of eq. (11.25): ΔEg ðtÞ = g0 ΔESPM ðtÞ − g0 EC0 ðtÞ = g0 ðej2πðfc0 + γtÞt − ej2πfc0 t Þ = ej2πfc0 t g0 ðcos 2πγt2 − 1Þ, − tS ≤ t ≤ tS

(11:26)

Letting wðtÞ = g0 ðcos 2πγt2 − 1Þ, − tS ≤ t ≤ tS

(11:27)

ΔEg ðtÞ = ej2πfc0 t wðtÞ, − tS ≤ t ≤ tS

(11:28)

gives

570

Chapter 11 Position comb

For simplicity, w(t) is referred to as the SPM modulation time function. The Fourier transform W(f) of w(t), which is similarly referred to as the SPM modulation spectrum, can be obtained by taking the following Fourier integral. ð tS ð∞ wðtÞe − j2πft dt = g0 ðcos 2πγt2 − 1Þe − j2πft dt Wðf Þ = −∞

= g0

ð tS − tS

" = g0

− tS

! 2 2 ej2πγt + e − j2πγt − 1 e − j2πft dt 2 2

ej2πðγt − ftÞ j4πð2γt − f Þ

#t S

" + g0

− tS

2

e − j2πðγt + ftÞ − j4πð2γt + f Þ

#tS − tS

 t e − j2πft S − g0 − j2πf − t

S

(11:29)

2   ej2πγtS j2 sinð2πftS Þ j2 sinð2πftS Þ j2 sinð2πftS Þ − g0 + j4π f − 2γtS f + 2γtS j2πf   2 2 f − 1 sincð2πftS Þ = 2tS g0 ej2πγtS 2 f − 4γ2 tS2

= g0

= Aðf Þsincð2πftS Þ where  2 Aðf Þ = 2tS g0 ej2πγtS

 f2 − 1 f 2 − 4γ2 tS2

(11:30)

It can be seen that the SPM modulation spectrum W(f) is a sinc function with an amplitude of A(f). The sinc function W(f) reaches its maximum at f = 0 with an amplitude of W0 (=A0 = 2tSg0). The sinc function W(f) reaches its first zero points at frequencies of ±fS = ± 1/2tS. fS is an indicator showing the spectral bandwidth of W(f), which is referred to as the SPM modulation bandwidth for clarity. Obviously, a shorter tS will generate a wider fS. This is the reason why the output pulse from the fiber ring oscillator is amplified by an EDFA before it is input to the HNLF for expansion of spectral bandwidth. In the pulse amplification by EDFA, the pulse magnitude g0 is enlarged and tS is shortened. Consequently, the bandwidth fS (=1/2tS) of the SPM modulation spectrum W(f) can be widened while its magnitude W0 (=2tSg0) is maintained. Meanwhile, A(f) is a function of f 2, and the term of sinc(2πfts) is an even function of f centered at f = 0. W(f) is therefore also an even function of f centered at f = 0. Denoting the Fourier transform of ΔEg(t) in eq. (11.28) by ΔHg(f), ΔHg(f) can be expressed to be W(f − fc0) with a constant frequency shift fc0 from W(f) based on eq. (11.28). Let the Fourier transform of the electric field Eg_after(t) of the single pulse after SPM be Hg_after(f). Based on eq. (11.24), Hg_after(f) can be expressed as Hg

after ð f Þ = Hg before ð f Þ + ΔHg ðf Þ = Gðf

− fc0 Þ + Wðf − fc0 Þ

(11:31)

11.3 Fiber frequency comb

571

As can be seen in eq. (11.31), in Hg_after(f), that is, the spectrum of the pulse after SPM, the SPM modulation spectrum ΔHg(f) or W(f − fc0) is added to the spectrum of the pulse before SPM, that is, Hg_before(f) or G(f − fc0). If the SPM modulation time width ts is sufficiently small and the SPM modulation spectral bandwidth fS is wide enough, Hg_after(f) can then be modulated to cover a wider spectral bandwidth with large enough magnitudes. This is the mathematics behind the effect of SPM for the expansion of pulse spectral bandwidth. Since the motivation of the above discussion is to make a simple mathematical explanation for spectral bandwidth expansion of pulse associated with SPM, only the effect of the linear frequency chirp occurring in the center portion of pulse is considered. It should be noted that the nonlinear frequency chirps occurring in the other portions of pulse also contribute to the expansion of pulse spectral bandwidth. Consequently, the SPM occurring over the entire pulse range will have a more significant effect on the expansion of pulse spectral bandwidth, compared with the effect by the SPM occurring only in the center position of pulse shown in the above discussion. Meanwhile, if the pulse envelope g(t) is a Gaussian function with a peak magnitude of g0 and an infinite pffiffiffiffiffipulse full-time width tP, its Fourier transform G(f) has a (root-mean-square) width of g(t) peak magnitude G0 of 2πσt g0 where σt is the RMS pffiffiffiffiffi (see Chapter 8). Since tS is smaller than σt, G0 (= 2πσt g0 ) is, therefore, larger than W0 (=2tS g0 ). Now we consider how the optical frequency comb of a pulse train with a constant repetition time (pulse interval) is changed by SPM. Based on the discussions made in Chapter 8, the spectrum, that is, the Fourier series of a pulse train, which is the optical frequency comb, can be mathematically obtained by digitizing the spectrum, that is, the Fourier integral of a single pulse with a constant comb spacing frep where the center comb is located at the center frequency of the single pulse spectrum. First, assume the pulse train before SPM has a repetition time τrep and it takes a time period of ΔT for a single pulse to pass through the SPM medium. The single pulse then has a time delay of ΔT before and after it passes through the SPM medium. Since all the single pulses have the same time delay of ΔT, the entire pulse train after SPM then has a time delay of ΔT with respect to the pulse train before SPM. However, for the same reason, the repetition time, that is, the time interval between two subsequent/neighboring pulses will not change from τrep. Therefore, the comb spacing frep, which is equal to 1/τrep, will keep being the same value for the pulse trains before and after SPM. Second, since the SPM modulation spectrum W(f − fc0) is an even function centered at f = fc0, W(f − fc0) will have a center frequency of fc0, which is overlapped with that of Hg_before(f) or G(f − fc0), that is, the spectrum of single pulse before SPM. Hg_after(f), that is, the spectrum of single pulse after SPM will then have the same center frequency of fc0.

572

Chapter 11 Position comb

As described above, since the comb spacing frep and the center comb position fc0 are not changed before and after SPM, the optical frequency combs before and after SPM can thus be obtained in the following equations based on Chapter 8: X X HPbefore ð f Þ=2πfrep Hgbefore ðmfrep Þδðf −fc0 −mfrep Þ=2πfrep Gðmfrep Þδðf −fc0 −mfrep Þ m

m

(11:32) HPafter ð f Þ = 2πfrep

X

ðGðmfrep Þ + Wðmfrep ÞÞδðf − fc0 − mfrep Þ

(11:33)

m

Consequently, the spectral bandwidth of the optical frequency comb after passing through the SPM medium is expanded without changing the center comb position and comb spacing. Meanwhile, a wider spectral bandwidth of pulse implies a pulse envelope with a shorter temporal/time width. As shown in Fig. 11.30, the pulse after SPM can thus be expressed by an effective pulse with a compressed pulse width, the carrier wave frequency of which is fc0. It should be noted that the loss in the energy of pulse due to SPM is not considered in the above discussion. On the other hand, as shown in Chapter 8, a group velocity dispersion will also cause a time-variant chirp in the frequency of the pulse carrier wave, which is similar to SPM. However, the group velocity dispersion will cause a significant change in the shape of the pulse envelope. It will also cause a change in the interval between pulse envelopes in a train of pulses, that is, pulse repetition time, as demonstrated in Chapter 8. Therefore, the effect of frequency chirp associated with group velocity dispersion cannot be utilized for the expansion of comb spectral bandwidth. However, practically, the SPM medium is also associated with group velocity dispersion and must be compensated when it is employed for spectral bandwidth expansion of a pulse. For example, the HNLF employed in the fiber frequency comb presented in this chapter had a positive group velocity dispersion. A single-mode fiber with a negative group velocity dispersion was then employed to compensate for that of the highly nonlinear fiber. The length of the single-mode fiber, which was referred to as dispersion compensator, was adjusted in a trial-and-error manner. Figure 11.31 shows the measurement results of an expanded comb spectral that had a bandwidth over an octave of wavelengths from 1,000 nm to 2,000 nm. Compared with the spectral bandwidth before expansion in Fig. 11.28, the bandwidth in Fig. 11.31 was significantly expanded. Since the optical spectrum employed in the measurement only had a wavelength measurement range of 1,700 nm a monochromator with a wider measurement range was also employed. The results by the two measuring instruments are shown in the figure. 4–2) Second harmonic generation (SHG) of 2fm signal by using PPLN The second harmonic generation shown in Fig. 11.32 is an effective way of producing the 2fm signal [29], where a fundamental wave with a frequency fm enters a nonlinear optical medium along the X-direction, a second harmonic wave with a frequency 2fm

11.3 Fiber frequency comb

573

Fig. 11.31: Measurement results of expanded spectral bandwidth by HLNF. α: by an optical spectrum analyzer, β: by a monochromator.

will be generated in the medium if the condition of phase matching is satisfied. Figure 11.33 shows a schematic of phase matching between the SHG waves generated at different positions in the nonlinear optical medium. A fundamental wave enters the medium at point P0, and leaves the medium after passing through points P1 to Pi along the X-direction with a phase velocity v1. At each of the positions, a 2fm wave is generated. The 2fm waves then propagate along the X-direction in the medium with a phase velocity v2. The output 2fm wave from the medium is a superposition of all the 2fm waves generated in the medium. For clarity, only a part of each wave is illustrated in the figure and the change in the amplitude of the electric field of each wave during the propagation is not considered. Denoting the electric field of the input fundamental fm wave in the medium at time t and position x by E1(t, x), E1(t, x) can be expressed by E1 ðt, xÞ = Aejφ1 ðt, xÞ

(11:34)

where A is the amplitude and φ1(t, x) is the phase of the fundamental wave. φ1(t, x) can be written as φ1 ðt, xÞ = 2πfm t − k1 x + ε1

(11:35)

where ε1 is the initial phase and k1 is the wavenumber. k1 has the following relationship with the wavelength λ′1 in the medium and λ1 in vacuum: k1 =

2π 2π = n1 λ′1 λ1

(11:36)

Here n1 is the index of refraction of the medium at fm. fm and λ1 have the following relationship:

574

Chapter 11 Position comb

Fig. 11.32: A schematic of second harmonic generation (SHG) for generating 2fm.

Fig. 11.33: Phase mismatching between 2fm waves generated at different positions in the medium.

11.3 Fiber frequency comb

fm =

c λ1

where c is the speed of light in vacuum. Substituting eqs. (11.36) and (11.37) into (11.35) gives   k1 c n1  x + ε1 = 2π t − x + ε1 φ1 ðt, xÞ = 2πfm t − 2πfm c λ1

575

(11:37)

(11:38)

Denoting the electric field of the second harmonic 2fm wave generated at point P0 by E20(t, z), E20(t, z) can be expressed by E20 ðt, xÞ = Bejφ20 ðt, xÞ

(11:39)

where B is the amplitude and φ20(t, x) is the phase of the 2fm wave generated at point P0. Assuming that the initial phase of the 2fm wave is the same as the phase of the fundamental fm wave at the position where the 2fm wave is generated, φ20(t, x) can then be written as φ20 ðt, xÞ = 4πfm t − k2 x + ε1

(11:40)

where k2 is the wave number. k2 has the following relationship with the wavelength λ′2 in the medium and λ2 in vacuum: k2 =

2π 2π = n2 λ′2 λ2

(11:41)

Here n2 is the index of refraction of the medium at the frequency of 2fm. Taking into consideration that λ2 is half of λ1, Equation (11.41) can be written as k2 =

4π n2 λ1

Substituting eqs. (11.37) and (11.42) into eq. (11.40) gives   k2 c n2  x + ε1 = 4π t − x + ε1 φ20 ðt, xÞ = 4πfm t − 4πfm c λ1

(11:42)

(11:43)

Now consider the phase φ21(t, x) of the second harmonic 2fm wave generated when the fundamental fm wave reaches point P1 with a distance λ′1 from P0. Denoting the time for the fm wave to propagate from P0 to P1 by T1, T1 can be obtained as T1 = The phase velocity v1 is defined by

λ′1 v1

(11:44)

576

Chapter 11 Position comb

v1 =

2πfm = fm λ′1 k1

(11:45)

Substituting eq. (11.45) into eq. (11.44) gives: T1 =

λ′1 λ1 = ′ fm λ 1 c

(11:46)

φ21(t, x) can then be written as φ21 ðt, xÞ = 4π

c n2 c ððt − T1 Þ − ðx − λ′1 ÞÞ + ε1 = 4π c λ1 λ1

    λ1 n2 λ1 − + ε1 t− x− c c n1 (11:47)

Similarly, the phase φ2i(t, x) of the 2fm wave generated when the fm wave reaches point Pi with a distance iλ′1 from P0:      c n2 c λ1 n2 λ1 ′ − + ε1 φ2i ðt, xÞ = 4π ðt − iT1 Þ − ðx − iλ 1 Þ + ε1 = 4π t−i x−i c c c n1 λ1 λ1 (11:48) Since the output 2fm wave is a superposition of all the generated 2fm waves, it is important to make the generated second harmonic waves to be in-phase or nearly inphase so that a strong output 2fm wave can be obtained. Now we consider the phase difference Δϕ20-2i between the 2fm waves generated at point P0 and Pi. When the 2fm wave generated at point P0 reaches Pi, it will have a phase as follows:     c n2 ′  c λ1 n2 λ1 n2 ′ + ε1 = 4πi 1 − + ε1 φ20 ðiT1 , iλ 1 Þ = 4π iT1 − iλ 1 + ε1 = 4π i − i c c c n1 n1 λ1 λ1 (11:49) The 2fm wave generated at point Pi has the following phase: c n2 φ2i ðiT1 , iλ′1 Þ = 4π ððiT1 − iT1 Þ − ðiλ′1 − iλ′1 ÞÞ + ε1 = ε1 c λ1 The phase difference Δϕ20–2i can then be obtained as   n2 Δφ20 − 2i = 4πi 1 − n1

(11:50)

(11:51)

Δϕ20–2i can be made to be zero, that is, phase-matched, by setting n2 to be equal to n1. Under the phase-matching condition, all the generated 2fm waves, as well as the fundamental fm wave, will be in-phase. Phase-matching can be made by using a birefringent crystal such as a BaB2O4 (BBO) crystal as shown in Fig. 11.34 [30]. Such a crystal, however, has some drawbacks such as hygroscopy. A periodically poled lithium niobate (PPLN) can be

11.3 Fiber frequency comb

577

Fig. 11.34: Phase-matching of 2fm waves by using BBO crystal.

employed for quasi-phase-matching of the second harmonic waves. As can be seen in eq. (11.51), the two 2fm waves will be out-phase for Δϕ20-2i of π where i can be obtained as follows: i=

n1 4ðn1 − n2 Þ

(11:52)

The distance Li in the out-phase condition is called the coherence length, which is denoted by Lcoherence. Lcoherence can be expressed by Lcoherence = iλ′1 =

n1 λ1 λ1 = 4ðn1 − n2 Þ n1 4ðn1 − n2 Þ

(11:53)

In a PPLN, the orientation of the Lithium Niobate crystal is periodically inverted with a period of Lcoherence to provide an additional phase of π so that the superimposed waves, that is, the output second harmonic wave, will have the maximum electric field and intensity. Figure 11.35 shows a schematic of PPLN [16]. A PPLN was selected to produce a target second harmonic 2fm signal (λ2 = 1,030 nm) of a fundamental fm signal (λ1 = 2,060 nm) in the setup shown in Fig. 11.36. Figure 11.37 shows the measured spectral of the PPLN output by an optical spectrum analyzer with a measurement range of 1,700 nm. A long-pass filter with a cutoff wavelength was used to cut off the fundamental wave components with wavelengths shorter than 1,450 nm. It can be seen that the target 2fm signal (λ2 = 1,030 nm) was successfully produced.

578

Chapter 11 Position comb

Fig. 11.35: Quasi-phase-matching of 2fm waves by using PPLN.

Fig. 11.36: Configuration of PPLN setup for 2fm wave generation.

4–3) Detection of fceo by 2fm–f2m interferometer The output of the PPLN setup shown in Fig. 11.36 is composed of the f2m and 2fm waves. The two waves are overlapped with each other, and therefore naturally can interfere with each other. As a result, the 2fm–f2m interferometer can be established by simply adding a photodiode to detect the intensity of the interference signal between the f2m and 2fm waves. Figure 11.38 shows the measured radio frequency (RF) spectra of the photodiode output in the 2fm–f2m interferometer. Three peaks are

11.3 Fiber frequency comb

579

Fig. 11.37: Measured spectral of PPLN output by an optical spectrum analyzer.

observed in the RF spectra, which are corresponding to the beat frequencies shown in Fig. 11.39. The strongest peak presents the RF spectrum of the frep component numbered by 1 in Fig. 11.39. The other two peaks present the RF spectra of the frep–fceo component numbered by 2 and the fceo component numbered by 3. The results demonstrated that the detection of fceo was successfully carried out. It should be noted that there are actually more beat frequencies generated in the higher RF range, which are not indicated in the figures.

Fig. 11.38: Output of the photodiode in the 2fm–f2m interferometer.

4–4) Feedback to ring oscillator for stabilization of fceo The output of the photodiode in the 2fm–f2m interferometer is then sent to the ring oscillator for feedback control of fceo. Figure 11.40 shows the block diagram for the feedback control where PLL was employed. The electric current applied to the pump laser diode was changed to stabilize f ceo [16, 26, 28]. Figure 11.41 shows the

580

Chapter 11 Position comb

Fig. 11.39: Beat frequencies generated in the 2fm–f2m interferometer.

measurement result of fceo without stabilization by the feedback control. It can be seen that the variation of fceo over a period of 30 min was larger than 2 MHz. Figure 11.42 shows the result with stabilization by the feedback control. The dividing number of the frequency divider was set to be 20 and the frequency from the function generator (FG) was set to be 525,000 Hz. fceo was then locked to a reference frequency of 10,500,000 Hz (20 × 525,000 Hz). It can be seen from Fig. 11.42 that fceo was successfully locked to the reference frequency. The standard deviation of the variation of the locked fceo was approximately 11 mHz. The stabilized fceo and frep were stable enough for use in the position comb to be addressed in the following section.

Fig. 11.40: Diagram of feedback control for stabilization of fceo.

11.4 Position comb

581

Fig. 11.41: Instability of fceo without stabilization.

Fig. 11.42: The result of fceo stabilization.

11.4 Position comb The position comb is an absolute surface encoder for in-plane position measurement. A fundamental principle of measuring the absolute in-plane position in the position comb is shown in Fig. 11.43. Instead of a monochromatic laser employed in the conventional surface encoder [31–35], a mode-locked femtosecond laser is newly employed as the light source. In addition, a planar scale grating having variable periods in the orthogonal two axes is newly employed in the absolute surface encoder. The X- and Y-directional grating periods dX(X) and dY(Y) are designed to be expressed as monotonically increasing or decreasing functions of the absolute Xand Y-positions (X and Y), respectively. By projecting a collimated mode-locked femtosecond laser beam onto the planar scale grating at a right angle, groups of the positive and negative first-order diffracted beams can be obtained.

582

Chapter 11 Position comb

Fig. 11.43: Geometric relationship between the detector units and the groups of the X-directional positive and negative first-order diffracted beams in the position comb.

According to the diffraction grating equation [36], the angle of diffraction of the mth mode in the X-directional first-order diffracted beams βX_m and that of the nth mode in the Y-directional first-order diffracted beams βY_n can be expressed as follows: sin βX

m

=

c , νm dX ðXÞ

sin βY

n

=

c νn dY ðYÞ

(11:54)

where νm and νn are the optical frequencies of the mth and nth modes in the firstorder diffracted beams, respectively, and c is the speed of light in vacuum. In the optical head, the groups of the X- and Y-directional positive and negative first-order diffracted beams are detected by each of the detector units composed of an objective lens and a polarization-maintaining single-mode optical fiber (PM fiber) from an optical spectrum analyzer (OSA). It should be noted that the detector units for capturing the Y-directional first-order diffracted beams are not indicated in Fig. 1 for the sake of clarity. Each of the detector units is aligned in such a way that its optical axis and the mode-locked femtosecond laser beam intersects on the scale grating at the angle of θX (θY). Now we consider the case where the mth modes in the X-directional first-order diffracted beams and the nth modes in the Y-directional first-order diffracted beams emanated from the absolute in-plane position (Xm, Yn) on the planar scale grating are captured by the detector units as shown in Fig. 11.43. In this case, the following equations can be obtained based on eq. (11.54): sin θX =

c , νm dX ðXm Þ

sin θY =

c νn dY ðYn Þ

(11:55)

11.4 Position comb

583

By modifying eq. (11.55), the following equations can be obtained:         c λm c λn = dX− 1 , Yn = dY− 1 = dY− 1 (11:56) Xm = dX− 1 sin θX sin θY νm sin θX νn sin θY Since dX(X) and dY(Y) are monotonically increasing or decreasing functions of X and Y, respectively, the absolute in-plane position (Xm,Yn) can be uniquely identified by the optical frequencies νm and νn (or the corresponding light wavelengths λm and λn) of the modes captured by the X- and Y-directional detector units, respectively. In other words, the equally spaced modes of the mode-locked femtosecond laser in the frequency domain are converted into the graduations of the X- and Y-directional absolute scales in the spatial domain by using the planar scale grating having variable periods. In a practical case of measurement, due to the diffraction limit and the modefield diameter of fiber detectors, several modes are captured together by the detector units. Meanwhile, the received group of the first-order diffracted beams after passing through the fiber forms a Gaussian light intensity distribution due to the light propagation principle of the single-mode fiber [37]. As a result, the centroid frequency obtained from the Gaussian distribution becomes almost equal to the optical frequency of the mode located in the center of the fiber detector [37]. Therefore, by employing the centroid frequencies of the spectra as the mode frequencies νm and νn in eq. (11.55), the absolute in-plane position (Xm,Yn) can be determined as shown in Fig. 11.44(a). Numerical calculations are carried out based on eq. (11.50) to estimate the variation of the optical spectrum of the coupled first-order diffracted beams to be received by the detector units. The parameters employed in the calculations are summarized in Tab. 11.7. For the sake of simplicity, the mode-locked femtosecond laser beam from the light source is assumed to have a flat optical spectrum over its spectral range, and each of the focused optical modes on the fiber detector has a Gaussian light intensity distribution. In addition, the grating period is assumed to change linearly with respect to the X-position. Figure 11.44(b) shows the variation of the centroid frequency obtained from the spectra calculated at each absolute X-position. As can be seen in the figure, the absolute X-position can be uniquely identified from the obtained centroid frequency (wavelength). It should be noted that the principle of measuring linear absolute position based on the variable line spacing (VLS) grating (sometimes referred to as the chirped grating [38]) with an incoherent chromatic light source has already been reported [39, 40]. Meanwhile, two major differences can be found between the conventional method and the method newly proposed in this chapter. One of the major differences is that the optical head is designed to have dual detector units for capturing both the positive and negative first-order diffracted beams simultaneously. This modification enables the newly proposed method to carry out robust absolute position measurements. Figure 11.45 shows a schematic of the case where the planar scale grating has an angular error motion α(X) about the Y-axis. In this case, the

584

Chapter 11 Position comb

Fig. 11.44: Theoretical calculation of the relationship between the absolute position and the peak frequency/wavelength in the spectrum of the captured optical modes. (a) Variation of the optical spectrum and (b) variation of the peak frequency of the spectrum in (a).

Tab. 11.7: Parameters for theoretical calculations. Item

Value

Spectral range of the mode-locked femtosecond laser

,–, nm

Focal length of the lens in the detector units

 mm

Diameter of the collimated laser beam

 mm

Coefficients of the variable grating period: g(X) = a + aX

a

. μm

a

 nm/mm

Angular position of the fiber detector (θ)

.°

Mode-field diameter of the fiber detector

 μm

11.4 Position comb

585

centroid frequencies in the spectra of the captured positive and negative firstorder diffracted beams νPos and νNeg, respectively, are affected by α(X) as follows: νPos = vm − γðαðXm ÞÞ, νNeg = vm + γðαðXm ÞÞ

(11:57)

where γ(α) denotes the influence of the angular error motion on the centroid frequency. Since the absolute position is obtained from νPos or νNeg in the previous method, the angular error motion of the scale directly affects the measurement result. On the other hand, in the newly proposed method, the centroid frequency νC_m in the spectrum of the coupled first-order diffracted beams is employed to obtain the absolute position. As a result, the influence of the angular error motion of the scale grating can be reduced.

Fig. 11.45: Influences of the misalignments of detector units and the angular error motion of the scale grating on the absolute position measurement.

Furthermore, with the employment of a mode-locked femtosecond laser as the coherent light source for the optical head, which is another major difference in the newly proposed method, interference signals capable of improving the resolution of absolute in-plane position measurement can be generated by coupling the groups

586

Chapter 11 Position comb

of the positive and negative first-order diffracted beams. In the next, the principle of the generation of interference signals is introduced. In the newly proposed absolute surface encoder, a mode-locked femtosecond laser, where each of the optical modes can be treated as a coherent monochromatic laser, is employed as the light source in the optical head. With the enhancement of the coherence of the mode-locked femtosecond laser beam, interference signals capable of improving the resolution of absolute in-plane position measurement can be obtained by coupling the groups of positive and negative first-order diffracted beams. Since the Y-directional absolute position can be measured in the same manner as the X-directional one, the X-directional absolute position measurement is treated in the following for the sake of simplicity. Now the electric fields of the received jth positive and negative first-order diffracted beams UPos_j and UNeg_j, respectively, can be expressed as follows [36]:  UPos j = APos j · exp ið2πνj t + kj LPos + δj + ΩðxÞÞ , (11:58)  UNeg j = ANeg j · exp ið2πνj t + kj LNeg + δj − ΩðxÞÞ where APos_j and ANeg_j are the real electric field amplitudes, k is the wavenumber (=2πνj/c), LPos and LNeg are the optical path lengths of the positive and negative first-order diffracted beams, respectively, δj is the initial phase of the jth first-order diffracted beams, and Ω(x) is the phase shift generated by the X-directional scale displacement with respect to the optical head. Since several modes are included in the received positive and negative first-order diffracted beams, the light intensity I of the coupled groups of the positive and negative first-order diffracted beams can be obtained as follows: I=

m Max X

m Max X

p = mmin q = mmin

=

m Max X

m Max X

p = mmin q = mmin

ðUPos p + UNeg q Þ · ðUPos p + UNeg q Þ h i A2Pos p + A2Neg q + 2APos p ANeg q cos Φ







 Φ = 2π νp − νq t + kp LPos − kq LNeg + δp − δq + 2ΩðxÞ where mMax and mmin are the maximum and minimum mode numbers of the optical modes in the received groups of the first-order diffracted beams, respectively. On the assumption that APos_p = ANeg_p = Ap and LPos = LNeg, the above equation can be rewritten as follows:

11.4 Position comb

I=

m Max X

m Max X

p = mmin q = mmin

=

m Max X

n

h

i A2Pos p + A2Neg q + 2APos p ANeg q cos Φ

o 2A2j ½1 + cosð2ΩðxÞÞ +

j = mmin m Max X

+

587

p = mmin , q = mmin

m Max X p = mmin , q = mmin , p≠q

A2Pos p + A2Neg q



    

δp − δq + 2ΩðxÞ L + 2APos p ANeg q cos 2π ðp − qÞνrep t + 2π c , p≠q (11:60)

As can be seen in eq. (11.60), the interference signal contains components having frequencies of the multiple of the γrep. Meanwhile, these high-frequency components can be eliminated when being observed in the optical spectrum analyzer. As a result, the interference signal I to be observed by the optical spectrum analyzer becomes as follows: I=

m Max X j = mmin

=

m Max X

n n

o 2A2j ½1 + cosð2ΩðxÞÞ +

m Max X p = mmin , q = mmin , p≠q

o 2A2j ½1 + cosð2ΩðxÞÞ + I0

A2Pos p + A2Neg q



(11:61)

j = mmin

=

m Max X

Ij + I0

j = mmin

where Ij denotes the contribution from the jth mode. Figure 11.46 shows a schematic of the spectrum of the coupled groups of positive and negative first-order diffracted beams at each X-position. Since the period of the interference signal only depends on the grating period and is independent of the mode frequency in the same manner as the conventional linear encoder [41], the interference signals generated from different optical modes have the same phase (and hence, the same period).

Fig. 11.46: Interference signals to be generated by the groups of the positive and negative firstorder diffracted beams. (a) Variation of the spectrum over a long X-range and (b) variation of the spectrum in a short X-range.

588

Chapter 11 Position comb

It should be noted that attention should be paid to the phase shift Ω(x) generated by the X-directional scale displacement with respect to the optical head, since the planar scale grating has grating pattern structures with variable periods. Figure 11.47 shows a schematic of how to derive Ω(x). Denoting the ith period of the interference signal as pi and the corresponding graduations of the incremental scale as xi, the following equation should be satisfied: xi = xi − 1 + pi ði > 0Þ

(11:62)

From the theory of the interferential scanning-type optical encoder [41], the period of the interference signal becomes a half of the grating period. Therefore, the following equation should also be satisfied: 1 pi = gðxi Þ 2

(11:63)

where g(xi) is the grating period at X = xi. As can be seen in eqs. (11.62) and (11.63), pi can be derived once g(X) and the initial X-position x0 are determined. Ω(x) can thus be expressed as follows: 2π ðΔxÞ gðxÞ=2   i−1 X 4π pr ðx − x0 Þ − = 2π · ði − 1Þ + gðXÞ r=1

ΩðxÞ = 2π · ði − 1Þ +

where Δx = x-xi-1. Equation (11.61) can thus be rewritten as follows:    n  i−1 X X 8π 2 + I0 2Aj 1 + cos pr I= ðx − x0 Þ − gðXÞ r=1 j=1

(11:64)

(11:65)

Since the contribution from the jth mode in the interference signal is synchronized with the ones from the other modes, the continuous interference signal can be obtained over the whole measurement range of the absolute surface encoder. The generated interference signal is expected to improve the resolution of the absolute position measurement with the signal interpolation technique [41] in the case where the graduations of the absolute scale (Xm) are synchronized with the graduations of the incremental scale (xi). Especially, in the case where the following equation is satisfied, moving direction of the planar scale grating can be distinguished with the combination of the absolute position information and the interference signal: xi = XN · m ðN:Even numberÞ

(11:66)

Figure 11.48 shows an example of the case of N = 2. Since the amplitude of the interference signal monotonically increases or decreases in between the neighboring graduations of the absolute scale, the moving direction of the scale can be easily distinguished. Meanwhile, since Xm and xi are determined independently based on

11.4 Position comb

589

Fig. 11.47: Relationship between the period of the interference signal and the graduations of the incremental scale.

the theoretical equations described above, attention should be paid to the design of the grating periods. Due to the characteristics of the incremental scale xi, it is difficult to derive g(X) directly from eqs. (11.56), (11.62), (11.63) and (11.66). However, a polynomial expression of g(X) that well satisfies eq. (11.60) can easily be found out through numerical calculations. Figure 11.49(a) shows an example of the interference signal and the absolute scale obtained under the conditions summarized in Tab. 11.8 with N = 4. g(X) is expressed as a polynomial of the fourth degree so that the absolute and the incremental scales can be well synchronized. Figure 11.49(b) shows the difference between the absolute and relative scales over a range of 10 mm. As can be seen in the figure, the absolute and incremental scales are well synchronized with each other with a deviation of 5 nm.

Fig. 11.48: Relationship between the period of the interference signal and the graduations of the absolute scale.

590

Chapter 11 Position comb

Fig. 11.49: Interference signal, incremental and absolute scales calculated by using the parameters in Table 11.2. (a) X = 5 mm and (b) difference between the graduations of the absolute and incremental scales.

In a practical system, it is difficult to distinguish neighboring graduations of the absolute scale due to the limited optical frequency (wavelength) resolution of the detector units. As a result, several cycles of the interference signal will exist between the neighboring graduations of the absolute scale. Figure 11.50 shows the calculated absolute scale and the interference signal in the case where a wavelength resolution of the detector units is 0.002 nm. Parameters shown in Tab. 11.8 are employed for the calculations. As can be seen in the figure, many cycles of interference signal exist in between the neighboring graduations of the absolute scale. It should be noted that even in this case the absolute scale is synchronized with the incremental scale, and the interference signal can be employed to improve the resolution of absolute position measurement in the same manner as the conventional absolute linear encoder having an incremental track [42].

11.4 Position comb

591

Fig. 11.50: Interference signal, incremental and absolute scales calculated under the condition in a practical case where the frequency resolution of the detector units is limited to be 4 GHz.

Tab. 11.8: Parameters for the theoretical calculation of the interference signal. Item

Value

Pulse repetition rate (νrep)

 MHz

Carrier-envelope offset frequency (νCEO)

 MHz

Minimum mode number (mmin)

,,

Coefficients of the variable grating period: g(X) = a + aX + aX + aX + aX

Angular position of the fiber detector (θ)

a

. μm

a

−. μm/m

a

. μm/m

a

−. μm/m

a

. μm/m .°

To verify the feasibility of the proposed method for measurement of the absolute in-plane position, an optical head was designed and constructed for carrying out experiments. In the proposed method, absolute X- and Y- positions are measured by employing the received X- and Y-directional diffracted beams, respectively, from the same point on the scale grating. By designing the optical head to have two sets of dual detector units for capturing both the X- and Y-directional diffracted beams simultaneously, measurement of the absolute X- and Y-positions can be carried out with a single measurement laser beam and a single optical head. In this chapter, the optical head with a set of dual detector units was designed for measurement of the absolute linear X-position for the sake of simplicity.

592

Chapter 11 Position comb

Figure 11.51(a) and 11.51(b) shows a schematic and a photograph of the developed optical head designed for reading the fabricated VLS grating. The optical head was mainly composed of a collimating lens and a pair of detector units having objective lenses and PM fiber detectors. The detector units were symmetrically arranged with respect to the measurement laser beam projected onto the VLS grating at a right angle so that the same optical mode in the groups of the positive and negative first-order diffracted beams could be received. The optical axis of each detector unit was aligned to have an angle of 45˚ with respect to the measurement laser beam. The design parameters of the developed optical head are summarized in Tab. 11.9. The mode-locked femtosecond laser beam generated in the laser source was delivered to the optical head through a single-mode fiber, and was then collimated to have a beam diameter of 3 mm. After that, the collimated beam was projected onto the VLS grating. The groups of the positive and negative first-order diffracted beams from the VLS grating were coupled into each of the PM fibers in the detector units, and were then combined at the polarization-maintaining fiber coupler (PN1550R5F2, Thorlabs). The optical spectrum of the combined first-order diffracted beams was then analyzed by the optical spectrum analyzer (AQ6370D, Yokogawa Co.). In the setup, the VLS grating was mounted on the table of a precision air-bearing linear slide. A corner cube was also mounted on the table so that the linear displacement of the VLS grating along the X-direction could be observed by using a reference laser interferometer (Agilent 5517C). Experiments were carried out by using the developed setup where a linear scale having a variable period was employed, which is referred to as the variable line space (VLS) grating [43]. At first, the X-directional displacement was given to the VLS grating from the X-position of x = 0 mm to x = 12 mm in a step of 1 mm through observing the optical spectrum of the coupled first-order diffracted beams. In the following experiments, the optical spectrum analyzer was operated with the setting parameters summarized in Tab. 11.10. Figure 11.52(a) shows the optical spectrum of the coupled first-order diffracted beams observed at each X-position. As can be seen in the figure, the peak wavelength in the optical spectrum was found to change as the change in the absolute X-position. Figure 11.52(b) shows the centroid wavelength calculated from the spectrum observed at each absolute X-position. The centroid wavelength λc was calculated based on the following equation: Ð λc =

PN λ × IðλÞdλ j = 1 λj · Iðλj Þ · Δλ Ð = PN IðλÞdλ j = 1 Iðλj Þ · Δλ

(11:67)

where I(λj) is the light intensity data of the coupled first-order diffracted beams at the wavelength λj observed in the optical spectrum analyzer with the wavelength resolution Δλ. As can be seen in the figure, a linear relationship between the detected centroid wavelength and the absolute X-position of the scale grating was observed over a range of ±6 mm. These results have demonstrated the feasibility of

11.4 Position comb

593

Fig. 11.51: Experimental setup: (a) a schematic of the setup and (b) a picture of the setup. Tab. 11.9: Design parameters of the optical head. Item

Value

Diameter of the laser beam projected onto the grating

 mm

Focal length of the objective lenses in the detector units

 mm

Mode field diameter of the PM fibers

. μm

Spectral range of the mode-locked laser

,–, nm

Angular position of the detector units (θX)

°

594

Chapter 11 Position comb

detecting the scale absolute position through obtaining the spectrum of the coupled groups of the positive and negative first-order diffracted beams.

Fig. 11.52: Variation of the peak frequency due to the X-directional scale displacement: (a) variation of the obtained spectrum and (b) variation of the centroid wavelength calculated from the spectrum shown in (a).

Tab. 11.10: Setup parameters for the optical spectrum analyzer. Item

Value

Number of the captured data

 points

Wavelength resolution

. nm

Resolution bandwidth

. nm

Sweep range

.–. μm

Sensitivity mode

Normal

11.4 Position comb

595

Experiments were extended to verify the possibility of obtaining interference signals by coupling the groups of positive and negative first-order diffracted beams. The X-directional linear displacement was given to the VLS grating in a small step of 25 nm through obtaining the optical spectrum of the coupled first-order diffracted beams at each X-position. Figure 11.53 shows the variations of the light intensity data I(λj) as the change in the absolute X-position observed at λj = 1,531.8, 1,532.0 and 1,532.2 nm, respectively.

Fig. 11.53: Obtained interference signal: (a) λ = 1,531.8 nm, (b) λ = 1,532.0 nm and (c) λ = 1,532.2 nm.

As can be seen in the figures, the intensity data were found to be modulated with a period of approximately 1.1 μm, corresponding to a half of the period of scale grating (approximately 2,200 nm), while being synchronized with each other. Figure 11.54 shows the result. As can be seen in the figure, the X-directional scale displacement was successfully detected by using the modulated intensity data with a nonlinear error component of approximately ±100 nm; the nonlinear error was mainly caused by the fluctuation in the variable period of the fabricated VLS grating. These results demonstrated the feasibility of the proposed method for measurement of the absolute position with a high resolution.

596

Chapter 11 Position comb

Fig. 11.54: X-displacement obtained from the interference signal and the interpolation error.

11.5 Summary A real-time repetition frequency monitoring (RTRFM) system for high-resolution measurement of the repetition frequency of mode-locked laser has been constructed. The system is composed of a photodiode, a frequency mixer, an RF synthesizer and a frequency counter. Instead of employing the frequency counter to measure the repetition frequency directly, the RTRFM system uses the frequency mixer to convert higher harmonic components of the repetition frequency to a low-frequency component based on the heterodyne method, through which the resolution for measurement of the repetition frequency can be improved. A sub-mHz resolution has been demonstrated by experimental results on a commercial femtosecond laser. A fiber ring laser for a fiber optical frequency comb has been constructed by using an erbium-doped fiber as a laser medium. By adopting the NPR (Nonlinear Polarization Rotation) method for mode-locking, the laser oscillator was composed of only fibers. It can achieve high stability and is maintenance-free. The constructed laser oscillated in the wavelength range of 20–30 nm at 1,550 nm and pulsed at a repetition frequency of about 100 MHz. An 2fm–f2m interferometer was also constructed to detect the carrier-envelope offset frequency of the optical frequency comb. Stabilization control of the optical frequency comb based on phase-locked loop (PLL) was carried out through stabilization of the repetition frequency and the carrier-envelope offset frequency to achieve a highly stable fiber frequency comb, which is suitable for use as the light source in the position comb for absolute position measurement. The kHz-order frequency instability of the constructed ring fiber oscillator was reduced to a sub-mHz level by performing PLL. Meanwhile, the mechanism of self-phase modulation (SPM) using a highly nonlinear fiber (HNLF) for expansion of the spectral bandwidth of optical frequency comb, as well as the mechanism of phase-matching for second harmonic generation (SHG) of the 2fm signal in the 2fm–f2m interferometer, is demonstrated mathematically.

References

597

The constructed fiber ring in combination with a scale grating with variable periods has then been applied to a position comb, which is an absolute surface encoder. In the position comb, the optical modes in the mode-locked femtosecond laser can be converted into the graduations of absolute scale for absolute in-plane position measurement due to the dispersive characteristics of the planar scale grating with variable periods. The graduations of the absolute scale can be detected as the centroid frequency (wavelength) in the spectrum of the coupled first-order diffracted beams received by the dual detector units. The dual detector configuration contributes to carrying out robust absolute in-plane position measurement against the angular error motion of the planar scale grating. In addition, with the enhancement of the spatially coherent characteristic of the mode-locked femtosecond laser, interference signals capable of being employed to improve the resolution of absolute in-plane position measurement through signal interpolation can be generated. The feasibility of the position comb has been verified by experimental results.

References [1]

Gao W, Kim SW, Bosse H, Haitjema H, Chen YL, Lu XD, Knapp W, Weckenmann A, Estler WT, Kunzmann H. Measurement technologies for precision positioning. CIRP Ann – Manuf Technol 2015, 64, 2, 773–796. [2] Gao W, Haitjema H, Fang FZ, Leach RK, Cheung CF, Savio E, Linares JM. On-machine and inprocess surface metrology for precision manufacturing. CIRP Ann 2019, 68, 2, 843–866. [3] Matsuzoe Y. High-performance absolute rotary encoder using multitrack and M-code. Opt Eng 2003, 42, 1, 124–131. [4] Renishaw plc. RESOLUTE absolute optical encoder with Biss serial communications. (Accessed March 17, 2021, at https://resources.renishaw.com/en/details/data-sheet-reso lute-absolute-optical-encoder-with-biss-serial-communications–111510) [5] Tamiya H, Taniguchi K, Yamazaki K, Aoyama H. Detection principle and verification of noncontact displacement meter with pico-meter resolution. J Adv Mech Des Syst Manuf 2018, 12, 5, 1–7. [6] Yang S Measurement of Optical Frequency Stability of Mode-Locked Laser. Tohoku University, Master thesis, 2017. [7] Lee J, Kim S-W, Kim Y-J. Repetition rate multiplication of femtosecond light pulses using a phase-locked all-pass fiber resonator. Opt Express 2015, 23, 8, 10117. [8] Rubiola E, Lenczner M, Bourgeois PY, Vernotte F. The Ω Counter, a frequency counter based on the linear regression. IEEE Trans Ultrason Ferroelectr Freq Control 2016, 63, 7, 961–969. [9] Johansson S New frequency counting principle improves resolution. Proceedings of the IEEE International Frequency Control Symposium and Exposition, Vol 2005. Institute of Electrical and Electronics Engineers Inc., 2005, 628–635. [10] Schaad TP Nano-Resolution Oceanic, Atmospheric, and Seismic Sensors With Parts-PerBillion Resolution., 2016. [11] Li X, Liu Y, Wang D, Bian Y Heterodyne frequency measurement method based on virtual instrument. Proceedings of the IEEE International Frequency Control Symposium and Exposition, IEEE, 2007, 220–222.

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[12] ARTECH HOUSE USA: Frequency Synthesizer Design Handbook. (Accessed December 22, 2020, at https://us.artechhouse.com/Frequency-Synthesizer-Design-Handbook-P651.aspx) [13] Holzwarth R, Udem T, Hänsch TW, Knight JC, Wadsworth WJ, Russell PSJ. Optical frequency synthesizer for precision spectroscopy. Phys Rev Lett 2000, 85, 11, 2264–2267. [14] Gupta SC. Phase-Locked Loops. Proc IEEE 1975, 63, 2, 291–306. [15] Hashi T, Chiba K, Takeuchi C, Miniature A. High-Performance Rubidium Frequency Standard. Institute of Electrical and Electronics Engineers (IEEE) 2008, 646–650. [16] Kanda Y Prism angle measurement by using optical frequency comb angle sensor. Tohoku University, Master thesis, 2020. [17] Inaba H, Ikegami T, Hong F-L, Onae A, Koga Y, Schibli TR, Minoshima K, Matsumoto H, Yamadori S, Tohyama O, Yamaguchi S-I. Phase locking of a continuous-wave optical parametric oscillator to an optical frequency comb for optical frequency synthesis. IEEE J Quantum Electron 2004, 40, 7, 929–936. [18] Zhu W, Qian L, Helmy AS. Implementation of three functional devices using erbium-doped fibers: An advanced photonics lab. Nantel M ed, Tenth International Topical Meeting on Education and Training in Optics and Photonics, Vol. 9665, SPIE, 2015, 966511. [19] Nelson LEE, Jones DJJ, Tamura K, Haus HAA, Ippen EPP. Ultrashort-pulse fiber ring lasers. Appl Phys B Lasers Opt 1997, 65, 2, 277–294. [20] Ippen EP, Haus HA, Liu LY. Additive pulse mode locking. J Opt Soc Am B 1989, 6, 9, 1736. [21] Maker PD, Terhune RW, Savage CM. Intensity-dependent changes in the refractive index of liquids. Phys Rev Lett 1964, 12, 18, 507–509. [22] Stolen RH, Botineau J, Ashkin A. Intensity discrimination of optical pulses with birefringent fibers. Opt Lett 1982, 7, 10, 512. [23] Stolen RH, Ashkin A. Optical Kerr effect in glass waveguide. Appl Phys Lett 1973, 22, 6, 294–296. [24] Li J, Wang Y, Luo H, Liu Y, Yan Z, Sun Z, Zhang L. Kelly sideband suppression and wavelength tuning of a conventional soliton in a Tm-doped hybrid mode-locked fiber laser with an allfiber Lyot filter. Photonics Res 2019, 7, 2, 103. [25] Mears RJ, Reekie L, Jauncey IM, Payne DN. Low-noise erbium-doped fibre amplifier operating at 1.54μm. Electron Lett 1987, 23, 19, 1026. [26] Ishizuka R Two-axis Absolute Encoder Using Optical Frequency Comb. Tohoku University, Master thesis, 2020. [27] Jones DJ. Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis. Science (80-) 2000, 288, 5466, 635–639. [28] Inaba H, Daimon Y, Hong F-L, Onae A, Minoshima K, Schibli TR, Matsumoto H, Hirano M, Okuno T, Onishi M, Nakazawa M. Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb. Opt Express 2006, 14, 12, 5223–5231. [29] Xu L, Hänsch TW, Spielmann C, Poppe A, Brabec T, Krausz F. Route to phase control of ultrashort light pulses. Opt Lett 1996, 21, 24, 2008. [30] Kazuo K. Nonlinear Optical Frequency Conversion of Ultrashort Light Pulses. (Accessed February 12, 2021, at http://qopt.iis.u-tokyo.ac.jp/lecture/pdf/NLOtutorial2.pdf) [31] Gao W, Kimura A. A Three-axis Displacement Sensor with Nanometric Resolution. CIRP Ann – Manuf Technol 2007, 56, 1, 529–532. [32] Kimura A, Gao W, Kim W, Hosono K, Shimizu Y, Shi L, Zeng L. A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement. Precis Eng 2012, 36, 4, 576–585. [33] Li X, Gao W, Muto H, Shimizu Y, Ito S, Dian S. A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage. Precis Eng 2013, 37, 3, 771–781.

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[34] Shimizu MG. Optical Sensors for Multi-Axis Angle and Displacement Measurement Using Grating Reflectors. Sensors 2019, 19, 23, 5289. [35] Matsukuma H, Ishizuka R, Furuta M, Li X, Shimizu Y, Gao W. Reduction in cross-talk errors in a six-degree-of-freedom surface encoder. Nanomanufacturing Metrol 2019, 2, 2, 111–123. [36] Hecht E. Optics, 5th Pearson, 2017. [37] Chen Y-L, Shimizu Y, Tamada J, Kudo Y, Madokoro S, Nakamura K, Gao W. Optical frequency domain angle measurement in a femtosecond laser autocollimator. Opt Express 2017, 25, 14, 16725. [38] Sanchez-Brea LM, Torcal-Milla FJ, Morlanes T. Near-field diffraction of chirped gratings. Opt Lett 2016, 41, 17, 4091. [39] Spillman WB Jr., Patriquin DR, Dhc. Fiber optic linear displacement sensor based on a variable period diffraction grating. Appl Opt 1989, 28, 17, 3550–3553. [40] Zhou J, Ouyang M, Shen Y, Liu DH. Study of a displacement sensor based on transmission varied-line-space phase grating. Optoelectron Lett 2008, 4, 3, 217–222. [41] Teimel A. Technology and applications of grating interferometers in high-precision measurement. Precis Eng 1992, 14, 3, 147–154. [42] Li X, Wang H, Ni K, Zhou Q, Mao X, Zeng L, Wang X, Xiao X. Two-probe optical encoder for absolute positioning of precision stages by using an improved scale grating. Opt Express 2016, 24, 19, 21378. [43] Shimizu Y, Ishizuka R, Mano K, Kanda Y, Matsukuma H, Gao W. An absolute surface encoder with a planar scale grating of variable periods. Precis Eng 2021, 67, 36–47.

Chapter 12 Optical comb applied metrology 12.1 Introduction The function/performance of an optical component can be affected by its surface form [1]. For the assurance of the quality of optical components, the evaluation of their surface form is an important task. For this purpose, many measuring instruments have been developed so far. Such measuring instruments can be classified into those based on non-optical methods and those based on optical methods. As an example of non-optical methods, a stylus profiler can be mentioned [2]. Reliable profile measurement of the optical components of complex pattern structures can be carried out with the use of a stylus having a micrometric tip radius. Submicrometric surface structures can also be examined by atomic force microscopes. One of the drawbacks of the non-optical techniques, meanwhile, is that the stylus could cause damage to the examined surface. In industries where surface damage by surface inspection should be minimized as far as possible, non-contact surface profile approaches are preferred, though some measuring instruments are designed to prevent significant damage to the surface by reducing the load to be added to the surface under inspection [3]. For the measurement of optical materials, measuring instruments based on phase-shifting interferometry are also used for many types of optical methods [1, 4]. Measurement of the surface profile is performed in phaseshifting interferometry by using fringe patterns produced by superimposing the laser beams reflected from the inspected surface and the reference mirror, respectively. Surface profile measurement over a wide area with a nanometric resolution can be achieved in a short time with the use of the large-aperture reference mirror. The measured region of the tools, meanwhile, is constrained by the scale of the reference mirror employed. While the observable region could be expanded by the stitching technique [5], the stitching procedure may cause additional measurement uncertainty. Therefore, it is desirable to create a system for testing large-sized optical components. For this reason, a measurement system based on an optical angle sensor has gained interest [6–8]. In this chapter, optical comb applied metrology for form measurement of precision optical components such as aspheric mirrors and/or prisms [9–12]. In Section 12.2, form measurement of an aspheric surface based on the mode-locked femtosecond laser angle sensor described in Chapter 9 is introduced. Also, In Section 12.3, the application of the concept of the mode-locked femtosecond laser autocollimator to the measurement of the apex angle of a prism in a small dimension, which is difficult to be evaluated by the conventional optical methods, is also presented Furthermore, an optical comb has also been applied to the evaluation of a grating pitch [10, 13], which is described in Section 12.4. https://doi.org/10.1515/9783110542363-012

602

Chapter 12 Optical comb applied metrology

12.2 Multi-beam form measurement A method for surface profile measurement of a precision optical component such as a large optical flat or an aspheric mirror based on an optical angle sensor has gained interest in recent years [6–8]. The approach is to measure surface slopes in order to determine the profile of the surface under inspection. The surface profile can be reconstructed by scanning the measurement laser beam of the angle sensor and obtaining the local slope of the surface being examined at each in-plane location. One of the benefits of the angle sensor-based approach is that there is no requirement for a reference mirror that is required for measuring instruments based on phase-shifting interferometry. This benefit helps the procedure to test a largescale optical component. On the other hand, one of the disadvantages of the optical angle sensor system is that the laser beam calculation must be scanned over an inspected surface. The surface profile inspection of a large-scale optical component thus takes a long time for measurement. A new method with a femtosecond laser angle sensor to calculate the surface profile of an optical component is proposed in this study in response to the context mentioned above. Based on the concept of laser autocollimation [14], the femtosecond laser angle sensor is designed to use a single-mode fiber as its optical detector. In addition, in order to achieve the local slopes of a surface under observation at several in-plane locations simultaneously, the femtosecond laser beam is separated into a set of diffracted beams. A prototype optical setup for the femtosecond laser angle sensor with several measurement beams is created as the early stage of the study, and the basic characteristics of the optical setup are tested in experiments. It is possible to obtain the local slope of a surface under inspection at each location by scanning the measurement beam of an optical angle sensor, as can be seen in Fig. 12.1. The effect of the angular motion error of the scanning stage about the Y-axis can be minimized by utilizing the function of a pentaprism. It should be noted that, with the function of the optical angle sensor based on laser autocollimation, the straightness error of the scanning stage in the Z-direction can be ignored [15]. The measuring beam of the angle sensor is expected to be scanned over the surface under inspection in this process. The mechanical scan of the measurement laser beam may cause measurement uncertainty to increase, as well as the measurement throughput to degrade. A new approach using an optical angle sensor with a mode-locked femtosecond laser source is proposed in this study to achieve the measurement of a surface profile based on an angle sensor without the scanning of a measurement laser beam. A modification has been made to the mode-locked femtosecond laser autocollimator [16, 17] described in Chapter 9. A single-mode fiber (SMF) detector collects the measurement beam reflected from a grating reflector in the developed optical angle sensor, and observes the angular displacement of the reflector in the light-frequency field. In the approach suggested in this study, this principle is also employed.

12.2 Multi-beam form measurement

603

Fig. 12.1: Surface profile measurement by an optical angle sensor with the enhancement of a pentaprism.

A diagram of the proposed approach with the angle sensor providing a femtosecond laser source is shown in Fig. 12.2. The femtosecond laser beam collimated by a collimating lens (L1) is made incident to a diffraction grating having equally spaced micropattern structures. A group of first-order diffracted beams can be obtained since the femtosecond laser beam comprises a set of optical modes evenly spaced in the optical-frequency domain. Using another lens (L2), several laser beams arranged to be parallel with each other can be obtained from these diffracted beams. Multiple laser beams are used as measuring laser beams for the optical angle sensor in the proposed method and are projected onto a surface under examination. The reflected measurement beam group concentrates on the single-mode fiber detector. It is detected concurrently in the optical-frequency domain. Each measurement laser beam is projected at a different position on a specimen surface, while having an identical optical frequency. Therefore, local slopes of the measured surface can be obtained simultaneously, without the mechanical scan of the measuring laser beams, by observing the optical spectrum of the group of reflected measurement beams. The single-mode fiber is expected to be useful from the theoretical calculation as the detector for the optical angle sensor. It should be remembered that in practical situations, the center of the focused laser beam is positioned to be at the edge of the fiber core. The design parameters for the optical setup are summarized in Tab. 12.1. Figure 12.3 shows an optical setup for the multi-beam femtosecond laser angle sensor. The femtosecond laser used in this chapter was a fiber-based one [18] with a pulse repetition rate of around 100 MHz and a spectral width of approximately 160 nm. To produce the group of measurement rays, a diffraction grating was used with a constant pitch of 1.67 μm. A diameter of approximately 0.9 mm was defined for the collimated femtosecond laser beam projected onto the grating reflector. The multi-beam generated by the setup was observed by a commercial beam profiler, as shown in Fig. 12.4. A spatial pitch of approximately 64 nm was designed for each neighboring mode. Due to the limited lateral resolution of the beam profiler, each of the measurement beams cannot be distinguished. However, the generation of the group of the multiple measurement beams was observed. The measurement beam

604

Chapter 12 Optical comb applied metrology

Fig. 12.2: Surface profile measurement by a multi-beam mode-locked femtosecond laser autocollimator.

Tab. 12.1: Parameters for the optical design. Incident beam diameter

D

. [mm]

Focal length of L

f

 [mm]

Focal length of L

f

. [mm]

Mode field diameter of SMF

w

. [μm]

group is projected onto the target surface, and a single-mode fiber detector is used to detect the reflected measurement beams. A flat mirror was employed as the objective in the following experiment. The mode-field diameter of the single-mode fiber detector was measured initially. In the radial direction, the single-mode fiber detector was shifted while the target mirror was left stationary after the careful alignment of the target mirror. The light intensity of the laser beam with a wavelength of 1,560 nm detected by the commercial spectrometer (Yokogawa AQ6370D) is shown in Fig. 12.5. The variation of the intensity was well fit by the Gaussian function, as predicted in theory [19]. Meanwhile, a mode-field diameter was estimated to be marginally smaller than the designed value (10.4 μm) according to the outcome. In order to determine the feasibility of the built setup as an optical angle sensor, tests were extended. Around the Y-axis, the target mirror was rotated in Fig. 12.3 in a step of roughly 10 arc-seconds, when the light spectrum collected by the singlemode fiber of the directed laser beams was tracked. By using a commercial laser autocollimator, which was used as a measuring reference, the angular motion of the flat mirror was also measured. The variation of the light intensity of the mode in a

12.2 Multi-beam form measurement

605

Fig. 12.3: Developed multi-beam mode-locked femtosecond laser angle sensor.

Fig. 12.4: Multi-beam observed by a commercial beam profiler.

wavelength of 1,560 nm observed in the experiment is shown in Fig. 12.6. The angular motion of the flat mirror was successfully observed. In the figure, the outcome of the theoretical equation is also plotted. Between the findings of the experiment and the statistical estimation, a strong consensus can be identified. In the same way, the angular displacement of the target mirror was also observed by other optical modes. The experimental outcome showed that the prototype optical configuration with multiple laser beams produced from the femtosecond laser as an optical angle sensor for profile measurement was feasible. Figure 12.7 shows the profile of a mirror having a concave profile with a curvature radius of 1,000 mm measured by the proposed multi-beam femtosecond laser angle sensor. The profile was reconstructed by the angular data obtained by the multi-beam femtosecond laser autocollimator. For the verification of the reconstructed mirror profile, the mirror was also measured by a commercial optical profiler (Zygo NewView

606

Chapter 12 Optical comb applied metrology

Fig. 12.5: Light intensity detected by the fiber detector.

Fig. 12.6: Light intensity variation due to the angular displacement of the target mirror.

Fig. 12.7: A comparison of the measured profile of a concave mirror by the multi-beam femtosecond laser angle sensor and a commercial optical profiler.

12.3 Micro-prism angle measurement

607

7300). As can be seen in the figure, a good agreement can be found between the results. These results demonstrated the feasibility of the multi-beam femtosecond laser autocollimator for profile measurement of an optical component.

12.3 Micro-prism angle measurement In many industrial areas, optical angle sensors play an important role in measuring the motion of the angular error of a moving object [20], evaluating the surface shape of an object with a reflective surface [1, 7] and so on. To measure the dihedral angle of two intersecting plates, optical angle sensors may also be used with the help of a precision rotary table in which a precision rotary encoder is integrated [21]. An autocollimator, among many types of angle sensors, is used to measure an object’s angular motion without having a rotational axis kept stationary in space [15, 22–24], as well as a multilateration-based technique [25]. It is also possible to measure the surface form of an optical part by employing a precision scanning device such as a precision air-bearing linear slide. With the employment of a retroreflector, the influence of the angular error motion in the measurement light beam scanning can be reduced [7, 26]. With a white light source and an image sensor such as a charge-coupled device (CCD), traditional autocollimators can achieve a high resolution over a broad measurement range [22]. Meanwhile, a certain amount of measuring beam diameter is needed for high-precision measurement. A collimator objective having a long focal length is also required for highly sensitive angle measurement. These drawbacks prohibit the traditional autocollimator from assessing tiny optical objects such as a small aspheric lens or a prism in a millimetric dimension. In the meanwhile, the high-precision measurement can be carried out by a laser autocollimator having a single-mode laser source and a photodiode as the detector based on laser autocollimation [27], even when a measurement laser beam having a small beam diameter is employed. A highly sensitive angle measurement can be achieved with the use of single-cell photodiodes [28]. Through detecting the displacement of the focused laser spot on the photodiode, the angular displacement of the surface under measurement can be measured in the conventional laser autocollimator [27]. The photodiode can measure the spot displacement by the change in the captured laser beam intensity. Because the reflected beam’s low light intensity could result in the quality deterioration of the reading signal in the optical configuration, in measuring a surface with a poor reflectivity, the sensitivity of the traditional laser autocollimator could be degraded. On most uncoated optical surfaces with a reflectivity of a few percent, it is thus impossible for traditional laser autocollimators to carry out highly sensitive angle measurements. In addition, it is known that the measurement beam diameter and the sensitivity are in a trade-off relationship in the laser autocollimation [28]. Therefore, highly sensitive angle measurement becomes more difficult with the decrease

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of the irradiating area of the measurement laser beam. Another problem to be considered is the smaller measurement range of the conventional laser autocollimator [28] compared with conventional autocollimators. Meanwhile, in the field of dimensional metrology, the optical frequency comb [29] has been expanding its usages in the last two decades [20, 30–32], since nowadays a fiber-based mode-locked femtosecond laser source can be constructed even in a laboratory having limited budgets and facilities for experiments. In recent years, optical angle sensors using a mode-locked femtosecond laser source have also been developed [16, 17, 33–36]. With the combination of a mode-locked femtosecond laser and a reflective-type diffraction grating, the “angle scale comb,” a series of scale graduations for angle measurement [37] has been generated. In the angle scale comb, a series of modes evenly spaced in the optical frequency domain can be transformed into an angle scale comb on the basis of the dispersive function of a grating reflector. In addition, a mode-locked femtosecond laser autocollimator with a high resolution over a wide measurement range can be realized by integrating the angle scale comb with the laser autocollimation [34]. A stable angle measurement with a high visibility reading output can also be accomplished with the use of a detector device consisting of an optical fiber detector and a spectrometer [16, 17]. Experiments have confirmed the viability of the femtosecond laser autocollimator. On the other hand, the femtosecond laser autocollimator has not been employed for form measurement. In the conventional mode-locked femtosecond laser autocollimator, due to its principle, the measurement beam is required to be projected onto a grating reflector; this restriction on the optical configuration has prevented it from being employed for form measurement. In this study, for measuring the apex angle of a small right-angle prism, the optical layout of the conventional mode-locked femtosecond laser autocollimator has been substantially changed. The followings are major modifications made to the optical setup: the optical setup is at first modified so that the measurement laser beam is projected onto a prism, which is a target of measurement. Then, the laser beam reflected at the prism surface was made incident to a grating reflector. This modification makes it possible to measure the inclination of the object surface, since all first-order diffracted beams emanated from the grating surface undergo the change in the direction of propagation due to the object surface inclination. In the meanwhile, the mode-locked femtosecond laser autocollimator with a right-angle incidence optical design suffers from the effect of the internally reflected beam in the examined prism. Furthermore, for accurate measurement of a right-angle prism, it is important to consider the misalignment of the target prism. The oblique-incidence femtosecond laser autocollimator is thus designed and developed to resolve the problem of the influence of the internally reflected beam. Basic characteristics of the developed oblique-incidence femtosecond laser autocollimator are at first verified in experiments. After that, the oblique-incidence femtosecond laser autocollimator is used to measure the apex angles of small, uncoated prisms. It should be noted that, since a small prism allows only limited beam size and provides a reflected laser beam in a weak intensity,

12.3 Micro-prism angle measurement

609

it has been a difficult task to evaluate such a small prism. A method using Brewster’s angle is also introduced to the setup to further decrease the influence of the internally reflected beam. In this study, a target measurement uncertainty is set to be sub-arcsecond, regarding the fabrication tolerance of the commercially available high-end prisms (1 arc-second) employed in the state-of-the-art scientific and industrial fields. Figure 12.8 shows a diagram of the optical setup for measuring the apex angle of the prism by a conventional laser autocollimator based on the laser autocollimation [14] with a single-mode laser source. By measuring the resulting movement of a reflected laser beam focused on a light detector, the inclination angle of a prism surface can be detected. The relationship between the inclination angle of a prism surface Δθ and the corresponding displacement Δd of the focused beam on the detector can be represented by the following equation: Δd = f tan 2Δθ

(12:1)

In the above equation, the focal length of the collimator objective is denoted by f. This equation tells us that Δd relies only on f and Δθ; this also means that the working distance will not affect the angle measurement. The traditional laser autocollimator has been innovated with the use of a mode-locked femtosecond laser and reflective diffraction grating [34, 37]. A schematic of the optical configuration for the measurement of the apex angle of a prism by the traditional femtosecond laser autocollimator is shown in Fig. 12.9. Instead of a single-mode laser source, a mode-locked femtosecond laser is used as the light source. A grating reflector is also employed. The grating reflector’s dispersive function helps the optical setup to transform the mode-locked femtosecond laser’s evenly spaced modes in the frequency domain into the group of the first-order diffracted beam referred to as the angle scale comb [37]. A high-resolution angle displacement measurement over a wide measuring range can be accomplished with the combination of the angle scale comb and the laser autocollimation [34]. Furthermore, through observing it in the frequency domain, the visibility of the reading signal from the optical head can be increased by using a detector device consisting of a collimator target, a single-mode optical fiber and a spectrometer [38]. In the meanwhile, the femtosecond autocollimator with a right-angle incidence optical arrangement suffers from the effect of the internal reflection beam from an examined prism.

Fig. 12.8: A conventional single-mode laser autocollimator.

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Chapter 12 Optical comb applied metrology

Fig. 12.9: A conventional mode-locked femtosecond laser autocollimator with the optical configuration of a right-angle incidence.

A significant modification to the traditional femtosecond laser autocollimator for the calculation of the apex angle of a right-angle prism is thus made in this study. A diagram of the oblique-incidence femtosecond laser autocollimator is shown in Fig. 12.10. With oblique incidence, the collimated laser beam from the mode-locked femtosecond laser source is projected onto a prism. To generate the first-order diffracted beams for measurement of a prism surface angle, the laser beam reflected at the prism surface is then projected on the grating reflector. This setup enables it to measure the angle of the prism apex, thus reducing the effect of the internally reflected beam.

Fig. 12.10: An oblique-incidence mode-locked femtosecond laser autocollimator for measuring a small right-angle prism.

12.3 Micro-prism angle measurement

611

Figures 12.11 and 12.12 illustrate the geometric relationship in the obliqueincidence femtosecond laser autocollimator between the measurement laser beam, a target item, the reflective-type diffraction grating and the detector device. The initial setup is shown in Fig. 12.11. With regard to the grating reflector, the reflected beam from the target surface has an angle of incidence of θG. The angle of the optical axis of the detector unit with respect to the normal of the grating reflector is set to θD. In this setup, the following relationship should be satisfied regarding the grating equation [39]: g½sinðθG Þ + sinðθi

init Þ = λi

(12:2)

In the above equation, g and λi are the grating pitch and the light wavelength of the ith mode, respectively. The angle of diffraction of the ith mode is denoted by θi init . From the geometric relationship, the reflected beam will have the angle of incidence θG + 2Δθ with respect to the grating reflector as shown in Fig. 12.12 under the condition where the target surface has an angular displacement Δθ. Meanwhile, the change in the diffraction angle of each mode in the first-order diffracted beams could also be included by Δθ, and the following equation can be obtained from the grating equation [39]: g½sinðθG + 2ΔθÞ + sinðθi Þ = λi

(12:3)

In the above equation, the angle of diffraction of the ith mode is denoted by θi. Because the angle of diffraction of the ith mode becomes equal to (θD–θG) due to the geometrical relationship, the following equation can be obtained: g½sinðθG + 2ΔθÞ + sinðθD − θG Þ = λdet

(12:4)

In this equation, λdet is the peak wavelength to be captured by the detector unit. The above equation tells us that, once the geometric relationship is established between the optical head and the surface under measurement, it is possible to create a oneon-one relationship between λdet and Δθ; namely, by detecting the change in λdet, the angular displacement Δθ can be measured. The relationship between λ and Δθ estimated in the numerical calculation on the basis of eq. (12.4) with the parameter summarized in Tab. 12.2 is shown in Fig. 12.13. It is possible to observe an almost linear interaction between λ and Δθ with a sensitivity of 0.0301 nm/arc-second over a range of ±1,800 arc-seconds. From the obtained sensitivity value, a resolution of better than 0.67 arc-second can be expected in terms of a wavelength resolution of the spectrometer in the detector unit (0.02 nm). In addition, from the result of numerical calculations, a measuring range of ±1,800 arc-seconds can also be expected with the employment of a mode-locked femtosecond laser whose spectral ranging is spreading from 1,510 to 1,620 nm. Two major approaches can be found for the measurement of a prism’s apex angle. One is the laser interferometer method [40, 41]: In order to obtain the relative angle between them the two prism surfaces forming the apex angle are determined.

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Chapter 12 Optical comb applied metrology

Fig. 12.11: Oblique-incidence mode-locked femtosecond laser autocollimator.

Fig. 12.12: Change in the optical modes to be detected by the fiber detector due to the angular displacement of a target surface.

However, since the laser beam propagated through a target prism is employed in the method, this method experiences the measurement error associated with the deviation of the prism refractive index. In addition, the prism having a coated surface cannot be measured by the method. Another major one is employing an autocollimator [21]. A precision rotary table equipped with a high-precision rotary encoder is used to evaluate the apex angle of a prism. In the method, the apex angle measurement will

12.3 Micro-prism angle measurement

613

Fig. 12.13: Relationship between the angular displacement and the peak wavelength in the oblique-incidence mode-locked femtosecond laser autocollimator.

Tab. 12.2: Setup parameters for the simulation. θD

.°

θG

°

g

. mm

λi

, nm

θi_init

.°

be carried out as follows: at first, the autocollimator is used to measure the inclination of a prism surface. The prism is then rotated for the inclination angle measurement of another prism surface. This technique can be extended to the arbitrary angles of the apex of the prism by changing the angular orientation of the prism using the rotary precision table. Since only the reflected beams from the prism surface are employed, a prism having coated surfaces can also be evaluated by the method. However, it becomes difficult to measure a prism if it does not have a reflective coating on its surfaces. This drawback can be overcome by the oblique-incidence femtosecond laser autocollimator, since the reflected beam is observed in the spectral domain, and the information of the peak wavelength is employed for angle measurement. As shown in Fig. 12.14, the two intersecting prism surfaces are now defined as Surfaces-A and -B, respectively. On a precision rotary table having a high-precision rotary encoder, a prism was mounted. Align the angular orientation of the prism first such that the Surface-A is in the measurement range of the oblique-incidence femtosecond laser autocollimator. At first, align the angular position of the prism so that the Surface-A is in the measurement range of the oblique-incidence femtosecond laser autocollimator. The readings of the oblique-incidence femtosecond laser autocollimator and the rotary encoder are now denoted as θA and ϕA, respectively, at this state. The rotary table then rotates the prism roughly 90° such that the

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Chapter 12 Optical comb applied metrology

Surface-B can be determined by the oblique incidence femtosecond laser autocollimator as shown in Fig. 12.15. Denoting the readings of the oblique-incidence femtosecond laser autocollimator and the rotary encoder at this state as θB and ϕB, respectively. By the following equation, the prism apex angle can be expressed as follows: θ = ðθA − θB Þ + ðϕA − ϕB Þ + 90

(12:5)

Using the measurements of the oblique-incidence femtosecond laser autocollimator and the rotary encoder obtained in measurement of Surfaces-A and -B, the prism apex angle can be evaluated by the above equation.

Fig. 12.14: Measurement of Surface-A of a prism angle by the oblique-incidence mode-locked femtosecond laser autocollimator.

Fig. 12.15: Measurement of Surface-B of a prism angle by the oblique-incidence mode-locked femtosecond laser autocollimator.

12.3 Micro-prism angle measurement

615

Light rays emitted from the top of the prism are used to determine the apex angle of a right-angle prism by the oblique-incidence mode-locked femtosecond laser autocollimator. A schematic of the externally reflecting beam and the internally reflected beam is shown in Fig. 12.16 in the case of measuring a right-angle prism. A prism’s typical surface reflectivity is approximately 4.0%. Regarding this value, 3.7% of the incident beam could come out from the measuring surface as the internally reflected beam. Due to the geometric relationship, the directly reflected beam and the internal reflection beam come from different locations on the surface of the prism. However, when sensed by the autocollimation unit, the internally reflected beam could impact the apex angle measurement. As can be seen in Fig. 12.16, now we denoted the deviations of the apex angles of a prism from the original design as α and β, respectively. In this case, from the geometric relationship, the angle of the internally reflected beam θInt can be expressed by the following equation:     sin θS (12:6) θInt = sin − 1 n sin 2α + 4β + sin − 1 n In the above equation, the refractive index of the prism is denoted by n. eq. (12.6) can be rewritten as follows, on the basis that α and β are small: θInt ≈ nð2α + 4βÞ + θS

(12:7)

The discrepancy between the propagating paths of the externally reflected beam and the internally reflected beam depends on the apex-angle deviations in the prism under examination, as shown in the above equation. Dividing these two reflected beams before being captured by the laser autocollimation unit is not a realistic way. As a result, interference fringes at the detector [42] are created by the two superimposed beams. It should be noted that where the two pulse trains in the two reflected beams intersect with each other in the spatial domain will interference fringes be detected. Meanwhile, a pulse width could be stretched since grating reflectors are employed in both the laser autocollimator and the spectrometer [43]. As a result, interference fringes on the detector in the spectrometer could be produced by the two pulse trains. By treating the two beams as plane waves, the light intensity of the superimposed beams I can be expressed as follows in the case where the two beams with light intensities I1 and I2 propagate in the same direction: pffiffiffiffiffiffiffi    2 I1 I2 2πnLPrism (12:8) cos I = ð I1 + I2 Þ 1 + ðI1 + I2 Þ λ In the above equation, the prism size is denoted by LPrism. The interference between the two beams thus could affect the spectrum of the laser beam to be captured by the laser autocollimation unit, as shown in the above equation, when measuring a prism in a small dimension. The characteristic of Brewster’s angle is used to resolve the above problem in this study.

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Chapter 12 Optical comb applied metrology

Fig. 12.16: Internally reflected laser beam in a right-angle prism.

An optical setup of the femtosecond laser autocollimator with an obliqueincidence configuration was planned and developed. In this study, regarding the performances of conventional optical autocollimators in a market [22–24], a resolution of 1 arc-second is set as the target over a measurement range of 3,600 arcseconds (1°). A schematic of the experimental setup established is seen in Fig. 12.17. As the light source in the system, a handmade Er-doped fiber-based mode-locked femtosecond laser with a pulse repetition rate of 96 MHz was used. A collimating lens with a numerical aperture (NA) of 0.53 was used to collimate the femtosecond laser beam directed through the setup. At first, the collimated laser beam was made to travel through a polarizer. A diameter was 0.9 mm at a wavelength of 1,550 nm. After that, a target surface was irradiated by the laser beam. In the setup, θS, the angle of incidence of the laser beam with respect to the prism surface was adjusted to be 21.9°. The reflected laser beam was projected onto a grating reflector. The angle of incidence θG of the laser beam was adjusted to be 20°. A blazed grating (blaze angle: 8°36′) with a grating pitch of 3.3 μm was used as the grating reflector. A detector unit, which consists of a lens having a focal length of 18.75 mm and a single-mode fiber (SMF), captured a group of diffracted first-order beams emanating from the grating surface. On the focal plane of the lens, the SMF was placed so that the lens can be employed as the collimator objective in the principle of laser autocollimation [27]. The detector unit captured the group of first-order diffracted beams, and the captured laser beam was analyzed in the optical frequency domain by a spectrometer. In the setup, a commercial optical spectrum analyzer (AQ6370C, Yokogawa, Co., Ltd) was employed as the spectrometer. In the optical spectrum analyzer, a Gaussian fitting was applied to the obtained spectrum for the peak wavelength detection. By using the Ethernet connection, the peak wavelength data was then transferred to a personal computer. It should be noted that a three-axis positioning system was employed for the fine alignment of the optical head in the setup. A target prism was placed on a two-axis tilt stage so that its angular

12.3 Micro-prism angle measurement

617

position about the X- and Y-axes can be adjusted. A rotary table with a high-precision rotary encoder (resolution: 0.11 arc-second/pulse) was employed in the system. The two-axis tilt stage was mounted on the rotary table so that the target prism on the tilt stage can be rotated 360°.

Fig. 12.17: An oblique-incidence mode-locked femtosecond laser autocollimator in the setup.

Experiments were carried out to verify the basic characteristics of the developed optical setup. By using a prism having a low reflectivity as the measuring objective, the following experiments were performed. The stability of the reading performance of the established setup was first assessed. The variance of the observed peak wavelength in the spectrum of the captured group of diffracted first-order beams for 300 s is seen in Fig. 12.18. A measurement period of 300 s was set with regard to the necessary time for angle measurement in the experiments mentioned in the following. Also plotted in the figure is the change in the room temperature. Primarily due to the thermal deformation of the set-up mechanical components, a decrease of 0.0035 nm in the light wavelength was observed. The thermal sensitivity of the reading output was measured to be −0.011 nm/°C. In experiments, the sensitivity of the established angle sensor was then assessed. As the measurement target, a right-angle prism in a size of 3 mm × 3 mm with a λ/4 flatness error was used. The right-angle prism was rotated by the rotary table over an angular range of ±2,200 arc-seconds around the Z-direction in a step of 110 arcseconds. During the rotation of the prism, at each angular position, the peak wavelength in the obtained spectrum was detected. The angular position where a peak wavelength of 1,565 nm was detected by the setup was treated to be the origin of the prism angular position. Table 12.3 summarizes the setup parameters employed in the

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Chapter 12 Optical comb applied metrology

Fig. 12.18: Variation of the sensor reading in a period of 300 s.

optical spectrum analyzer. The detected peak wavelengths during the experiments were shown in Fig. 12.19. The peak wavelength has been observed to change over a range of 3,300 arc-seconds in proportion to the angular displacement of the rightangle prism, as can be seen in the figure. From the results, the sensitivity of the shift in the peak wavelength relative to the angular displacement of the measurement target was thus measured as 0.0294 nm/arc-second. In the numerical approximation, a sensitivity of 0.0301 nm/arc-second was predicted on the basis of eq. (12.4). A good agreement can be found between the experimental findings and the ones predicted in the numerical calculation. In order to evaluate the resolution of the oblique-incidence femtosecond laser autocollimator, tests were also carried out. In a small step of 0.11 arc-seconds, the right-angle prism was rotated around the Z-axis. It should be noted that this value is comparable to the minimum resolution of the rotary table in the setup. At each angular position of the prism, the peak wavelength of the obtained spectrum is analyzed. At each angular position, the right-angle prism was held stationary in a period of 60 s for measurement. The other experimental parameters were the same as those for the sensor sensitivity test in the previous experiments. It took about ten seconds for each optical spectrum to be collected. The experimental results are summarized in Fig. 12.20. The developed femtosecond laser autocollimator has successfully resolved an angular displacement in a 0.11 arc-second step, as shown in the figure. A transition in the peak wavelength was measured as approximately 0.03 nm in response to the single step of the angular displacement. This result was well in line with the value calculated from the sensitivity of the angle sensor obtained in the previous experiments. These experimental findings revealed a resolution greater than 0.11 arc-second for the developed angle sensor.

12.3 Micro-prism angle measurement

619

Fig. 12.19: Relationship between the peak wavelength and the angular displacement of a surface under inspection.

Tab. 12.3: Setup parameters for the optical spectrum analyzer (AQ6370C). Sensitivity

Normal

Resolution bandwidth

 nm

Sampling interval

 pm

Observed spectral range

,–, nm

In Fig. 12.20, the movement of the diffracted beam in the radial direction may be caused by the difference in the distance between the measuring target and the optical head. The angle of the beam axis with respect to the collimator objective does not depend on Δh. As a result, the focused laser beam position on the focal plane of the collimator objective is independent of Δh, as well. In the conventional autocollimator and laser autocollimators employing an image sensor, angle measurement will not be affected by Δh. In the meanwhile, in the case of the femtosecond laser autocollimator where a single-mode fiber combined with the collimator objective is employed as the detector, the change in the angle of incidence of the focused laser beam to the fiber associated with Δh could affect the measurement. To address the issue, a theoretical investigation is carried out. Now we consider the case, a schematic of which is shown in Fig. 12.21, where the diffracted beam from the grating reflector is projected onto the collimator objective with an angle of θ regarding the optical axis of the laser autocollimation unit. In addition, for the collimator objective and the single-mode fiber, we describe the local coordinate systems (u-v and x-y), respectively. For the sake of convenience, the rotation of the calculation target along

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Chapter 12 Optical comb applied metrology

Fig. 12.20: Evaluation of the sensor resolution.

the Y-axis in Fig. 12.21 is not included in the following theoretical investigation. At the position of the collimator objective the light intensity IBeam(u,v) can be described as follows [39]: i9 8 h < − 2 u2 + ðv − L sin θ − w0 Þ2 = (12:9) IBeam ðu; vÞ ¼ I0 exp : ; ðD=2Þ2 In the above equation, the maximum intensity at the beam center is denoted by I0. The distance between the collimator objective and the grating reflector is denoted by L. The diameter of the diffracted beam is denoted by D, while the position of the diffracted beam from the axis of the laser autocollimator is denoted by w0. By using IBeam(u,v), the power of the captured diffracted beam PNA can be described as follows [44, 45]: ðð (12:10) PNA ðw0 , θÞ = IBeam ðu, vÞ · aNA ðu, v, θÞdudv The parameter aNA in the above equation, corresponding to the attenuation due to the fiber numerical aperture, can be expressed by the following equation [44]: ( ) − 2½u2 + v2  aNA ðu, v, θÞ = exp (12:11) ðf tan θÞ2 In the above equation, the focal length of the collimator objective is denoted by f. In eq. (12.10), the influences of the attenuations due to the spot position at the fiber detector, as well as that of the mode field diameter DMFD of the fiber, are not taken

12.3 Micro-prism angle measurement

621

into consideration. The normalized light intensity I(x,y) of the focused laser beam can be represented by the following equation at the location (x,y) on the fiber detector [39]: i9 8 h < − 2 x2 + ðy − w1 Þ2 = (12:12) I ðx; yÞ = exp

2 : ; Dspot 2 In the above equation, the distance between the focused beam and the optical axis of the laser autocollimation unit axis is denoted by w1(=f · tanθ). Dspot denotes the diameter of the focused beam, and can be obtained by using the light wavelength λ as Dspot = 4λf/(πD). Also, the attenuation by the mode-field diameter of the singlemode fiber aCore can be described by the following equation [39]: ( ) 2½x2 + y2  aCore ðx; yÞ ¼ exp (12:13) ðDMFD =2Þ2 On the basis of these equations, the total laser power Pdet to be captured by the single-mode fiber can be calculated by the following equation [44, 45]: ðð (12:14) Pdet ðw0 , θÞ = PNA ðw0 , θÞ · I ðx, yÞ · aCore ðx, yÞdxdy

Fig. 12.21: Influence of the beam offset in the setup.

On the basis of eqs. (12.10) and (12.14), the influence of the lateral shift of the diffracted beam on the change in the peak wavelength is estimated. Results are summarized in Fig. 12.22. For the calculations, the conditions shown in Tab. 12.4 are applied. Two plots for the cases with f = 4.67 mm and f = 18.75 mm can be found in the figure. With a smaller f, the effect of the beam shift is predicted to be considerable, as shown in the figure. In order to validate the validity of the theoretical estimation, tests were conducted using the setup seen in Fig. 12.17. Rather than providing a change in the radial direction to the diffracted beam, the laser autocollimation unit was moved

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Chapter 12 Optical comb applied metrology

along the Y-direction to simulate the radial beam shift. For the verification of the influence of the focal length on the angle measurement, two collimator objectives with different focal lengths of 18.75 mm and 4.67 mm were used. Results are summarized in Fig. 12.23. A shift of approximately 6 nm with a displacement of the laser autocollimation unit of 2,000 μm in the radial direction was observed at the peak wavelength in the case of f = 4.67 mm. On the contrary, in the case of f = 18.75 mm, there was no noticeable shift in the peak wavelength. Sensitivities with respect to the radial beam shift are evaluated to be 6.8 arc-second/mm for the case with f = 18.75 mm. From this result, it can be concluded that, as long as the target misalignment is suppressed to be less than 100 μm, its contribution to the measurement uncertainty could be less than 0.68 arc-second.

Fig. 12.22: Theoretical calculation of the influence of beam-shift on the peak wavelength.

Using the developed setup, an attempt was made to measure the apex angle of a prism after the evaluation of basic features of the oblique-incidence mode-locked femtosecond laser autocollimator. In the setup, only a mean inclination angle of the measured area can be obtained by the oblique-incidence femtosecond laser autocollimator. In the meanwhile, this information can be employed to carry out the inspection of the prism products having a flatness better than λ/4. The established setup was employed to measure the apex angles of right-angle prisms. On a vibration-isolating table, both the optical head of the oblique-incidence femtosecond laser autocollimator and the rotary table are mounted. The right-angle prism in a size of 44 mm × 44 mm was at first measured. The spectra obtained in the measurements of Surfaces-A and -B are shown in Fig. 12.24. Without the demounting operation of the right-angle prism from the rotary table, five repeated trials were carried out. The apex angle of the prism was assessed to be 6.46 arc-seconds. A standard

12.3 Micro-prism angle measurement

623

Fig. 12.23: Influence of beam-shift on the peak wavelength observed in experiments. Tab. 12.4: Parameters employed in the setup for evaluating the sensor resolution. Item

Value

Beam diameter (D)

. mm

Length prism – grating (L)

 mm

Focal length (f)

. mm 

Fiber NA (at /e )

.

Spot diameter (Dspot)

. μm

Center wavelength (λcenter)

, nm

Mode field diameter (MFD)

. μm

deviation of the five repetitive measurements was 0.11 arc-second. Experiments were extended to conduct the measurement with demounting and remounting operations of the right-angle prism at each measurement to determine the reproducibility of the measurement. A peak-to-peak deviation of 4.40 arc-seconds is obtained for the prism apex angle. The right-angle prism of 3 mm × 3 mm was also measured in the same way. Through the five repetitive trials, measurement repeatability was assessed to be 0.99 arc-second. A deviation of the apex angle measurement was 11.80 arc-seconds in five trials including the demounting and remounting operations of the right-angle prism. As can be seen in the results, the alignment of the right-angle prism has a significant impact on the measurement of the apex angle. The size of the right-angle prism may also affect the repeatability of the apex angle measurement and its reproducibility.

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Chapter 12 Optical comb applied metrology

Fig. 12.24: Spectra obtained in measurement of Surfaces-A and -B (prism size: 44 mm).

In the autocollimation unit, the internally reflected beam could influence the measurement of the apex angle; in particular in the case where a small prism is measured as explained in the measurement principle of the proposed method. In order to minimize the effect of the interaction between the two lights, the function of Brewster’s angle is used. It is well understood that with respect to the angle of incidence of the beam, the p-polarized and s-polarized waves have different reflectance. At Brewster’s angle θB satisfying the following equation, the reflectance of the p-polarized beam becomes zero: tan θB =

nglass nair

(12:15)

In the above equation, the refractive indices of the prism and the air are dented by nglass and nair, respectively. The influence of the light interference is expected to be reduced by the effect of Brewster’s angle. The relationship between θS, the angle of incidence of the laser beam, and θR, the angle of the internally reflected beam with respect to the hypotenuse of the right-angle prism, is summarized in Figs. 12.25 and 12.26. The angle θR can be represented by the following equation from the geometric relationship:   nair ′ (12:16) sin θS θR = 45 − θS = 45 − sin − 1 nglass From eqs. (12.15) and (12.16), the following equation should be satisfied when θR is set to be Brewster’s angle:

12.3 Micro-prism angle measurement

θS = sin

−1



   nglass π nair −1 sin − tan nair nglass 4

625

(12:17)

On the basis of the above equation, the internally reflected beam is expected to vanish away when θR is adjusted to be 21.9°, under the condition of a refractive index nglass of 1.62 at a wavelength of 1,530 nm.

Fig. 12.25: Optical configuration designed for the reduction of the influence of internal reflection based on Brewster’s angle in the case of the measurement of Surface-A.

Fig. 12.26: Optical configuration designed for the reduction of the influence of internal reflection based on Brewster’s angle in the case of the measurement of Surface-B.

626

Chapter 12 Optical comb applied metrology

Experiments were carried out to verify the decrease of the influence of the internally reflected beam by the proposed method. The spectrum obtained by inserting a measurement laser beam through the surface-A of a right-angle prism at a right angle at a size of 2 mm × 2 mm is seen in Fig. 12.27. A significant light interference was found between the externally reflected beam and the internally reflected beam. The period of spectrum amplitude modulation was found to be 0.37 nm. A good agreement could be found between the value observed in the experiments and that (0.375 nm) predicted based on eq. (12.14). Similarly, tests were carried out with the same right-angle prism with a 21.9° measurement laser beam incidence angle. For two cases where the polarization orientation of the measurement laser beam was changed to be p-polarization and s-polarization, respectively, using the polarizer in the optical head, tests were carried out to validate the viability of the proposed process. The obtained results are shown in Fig. 12.28. The effect of the interference, especially in the case where the polarization orientation of the measurement laser beam was changed to be p-polarization, was decreased, as can be seen in the figure. Owing to the weak polarization extinction ratio of the polarizer used in the configuration the influence of the interference still remained in the obtained spectrum. The apex angle of the prism in a size of 2 mm × 2 mm was measured under the optimized condition where the polarization orientation of the measurement laser beam was changed to be p-polarization while the angle of incidence of the measurement laser beam was set to 21.9°. Results of the tenrepetitive measurements are summarized in Fig. 12.29. A standard deviation of the apex angle measurement observed in the experiments (0.22 arc-second) was found to be better than that obtained with the prism having a larger size of 3 mm × 3 mm. The effect of the suggested approach based on the characteristics of Brewster’s angle was demonstrated by these results. On the contrary, in the case of measuring a large prism, as can be seen in Fig. 12.24, the influence of the interference was not observed. This is mainly attributable to the long optical path difference (OPD) between the two reflected beams.

Fig. 12.27: Spectra of the first-order diffracted beams in the case with a prism in a size of 2 mm at a right-angle incidence.

627

12.3 Micro-prism angle measurement

Fig. 12.28: Spectra of the first-order diffracted beams in the case with a prism in a size of 2 mm at oblique incidence with the Brewster’s angle.

Fig. 12.29: Measurement repeatability under the condition of Brewster’s angle (prism size: 2 mm).

Based on GUM [46], measurement uncertainty of the prism apex angle by the built oblique-incidence femtosecond laser autocollimator is also studied. The following equation can define a model of the prism angle measurement used in this chapter: Φ = ðθA + φA Þ − ðθB + φB Þ + 90

(12:18)

In the above equations, symbols Φ, θi and φi (i = A, B) denote the apex angle of a prism, the reading of the oblique-incidence femtosecond laser autocollimator, and the positioning error of the rotary table, respectively. The contributions of the sources of uncertainty are summarized in Tab. 12.5. An expanded uncertainty U was estimated to be 0.82 arc-second (k = 2, 95% confidence) for the prism apex angle measurement. The target uncertainty (1 arc-second) of this study has successfully been achieved by the proposed method.

628

Chapter 12 Optical comb applied metrology

Tab. 12.5: Measurement uncertainty. Source of uncertainly

Symbol

Type

Standard uncertainly u

Sensitivity coefficient C

|u × C|

Sensor output

u(θi)

–

. arc-second



. arc-second

Wave length detection

u(λ)

A

. nm

. arc-second/nm

(. × −)

Temperature

u(ΔT)

B

. °C

−. arc-second/°C

(.)

Rotary stage

u(φi)

B

. arc-second



. arc-second

Combined uncertainty

uc

Expanded uncertainly

U

. arc-second k=

. arc-second

12.4 Diffraction grating calibration In many industrial fields such as the precision machining industry and the semiconductor industry, precision positioning is one of the fundamental technologies [20]. It is necessary for precision positioning to apply appropriate measuring instruments. Optical encoders such as linear encoders and/or planar encoders are major ones employed for the purpose. In such optical encoders, reflective-type or transparent-type diffraction gratings are employed as the scale for measurement, while the periodic pattern structures are employed as the scale graduation for measurement. Scale gratings are thus important components determining the measurement accuracy, as well as the measurement range. Therefore, the evaluation of the period of pattern structures is an important task to assure measurement accuracy in optical encoders. As described in Chapter 6, many efforts have been made so far to realize the calibration of a scale grating. Among them, a method based on laser diffraction is capable of measuring the absolute grating period with the enhancement of a precision rotary table equipped with a high-precision rotary encoder. In the method, the calibration accuracy of the period of pattern structure is mainly governed by the detection accuracy of the angles of diffraction of the diffracted laser beams in the Littrow configurations. The stability of the optical frequency of the laser beam also affects the calibration accuracy. For the improvement of the measurement accuracy, the angles of multiple diffracted beams in different Littrow configurations are often observed to obtain the mean value of the grating periods to be acquired in the different Littrow

12.4 Diffraction grating calibration

629

configurations. However, in the case of the evaluation of a diffraction grating with a small period, the number of diffracted beams to be obtained will be quite limited. In addition, the frequency fluctuation of a monochromatic laser source in the method could degrade the measurement accuracy. To solve the above problems, a method employing a mode-locked femtosecond laser beam is proposed [10]. With the employment of a mode-locked femtosecond laser as the measurement laser beam, a group of first-order diffracted beams corresponding to each of the optical modes can be obtained. By observing multiple diffracted beams in the group of the first-order diffracted beams, the measurement accuracy of the grating period is expected to be improved due to the averaging effect. The stable optical frequencies of the optical modes in the mode-locked femtosecond laser are also expected to improve the measurement accuracy. Figure 12.30 shows a typical setup for the conventional laser diffraction method with a monochromatic laser source [47]. The angular displacement of a diffraction grating under measurement is detected by a precision rotary encoder in the rotary table. A laser autocollimator is often employed to observe diffracted beams from the diffraction grating.

Fig. 12.30: Optical setup for the conventional laser diffraction method.

The diffraction grating is at first positioned in the zeroth-order Littrow configuration, where the zeroth-order diffracted beam is retro-reflected. After that, the diffraction grating is rotated by the rotary table so that the grating can be positioned in the first-order Littrow configuration. Denoting the angle of diffraction of the first-order diffracted beam by θ1, the grating period can be calculated as follows:

630

Chapter 12 Optical comb applied metrology

P=

1 c 2 sin θ1 f

(12:19)

In the above equation, c is the speed of light in air, and f is the optical frequency of the monochromatic laser source employed in the setup. It is worth noting that P represents the average grating period for an area equal to the beam diameter. In the same manner, by using the observed angle of diffraction of the mth-order Littrow configuration θm, P can be calculated by the following equation: P=

m c P 2 sin θm f

(12:20)

By taking the average of P obtained at each of the Littrow configurations based on the following equation, the measurement accuracy of the grating period can be improved: P=

X m

m c 2 sin θm f

(12:21)

Meanwhile, the maximum diffraction order mMax capable of being observed by the setup is limited in theory, and should satisfy the following equation: mMax