MODULATION TRANSFER FUNCTION IN OPTICAL AND ELECTRO-OPTICAL SYSTEMS. [2 ed.] 9781510639379, 1510639373

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Second Edition Glenn D. Boreman

Modulation Transfer Function in Optical and Electro-Optical Systems Second Edition

BOREMAN

This second edition, which has been significantly expanded since the 2001 edition, introduces the theory and applications of the modulation transfer function (MTF) used for specifying the image quality achieved by an imaging system. The book begins by explaining the relationship between impulse response and transfer function, and the implications of a convolutional representation of the imaging process. Optical systems are considered first, including the effects of diffraction and aberrations on the image, with attention to aperture and field dependences. Then electro-optical systems with focal-plane arrays are considered, with an expanded discussion of image-quality aspects unique to these systems, including finite sensor size, shift invariance, sampling MTF, aliasing artifacts, crosstalk, and electronics noise. Various test configurations are then compared in detail, considering the advantages and disadvantages of point-response, line-response, and edge-response measurements. The impact of finite source size on the measurement data and its correction are discussed, and an extended discussion of the practical aspects of the tilted-knife-edge test is presented. New chapters are included on speckle-based and transparency-based noise targets, and square-wave and bar-target measurements. A range of practical measurement issues are then considered, including mitigation of source coherence, combining MTF measurements of separate subsystems, quality requirements of auxiliary optics, and low-frequency normalization. Some generic measurement-instrument designs are compared, and the book closes with a brief consideration of the MTF impacts of motion, vibration, turbulence, and aerosol scattering.

Modulation Transfer Function in Optical and Electro-Optical Systems Second Edition

Modulation Transfer Function in Optical and Electro-Optical Systems

P.O. Box 10 Bellingham, WA 98227-0010

Glenn D. Boreman

ISBN: 9781510639379 SPIE Vol. No.: TT121 TT121

Tutorial Texts Series Related Titles • • • • • • • •

Aberration Theory Made Simple, Second Edition, Virendra N. Mahajan, TT93 Analysis and Evaluation of Sampled Imaging Systems, Richard H. Vollmerhausen, Donald A. Reago, and Ronald Driggers, TT87 Introduction to Optical Testing, Joseph M. Geary, TT15 Laser Beam Quality Metrics, T. Sean Ross, TT96 Modeling the Imaging Chain of Digital Cameras, Robert D. Fiete, TT92 Optical Design: Applying the Fundamentals, Max J. Riedl, TT84 Optical Design for Visual Systems, Bruce H. Walker, TT45 Optical Design of Microscopes, George H. Seward, TT88

(For a complete list of Tutorial Texts, see http://spie.org/publications/books/tutorial-texts.)

Other Related SPIE Press Titles SPIE Field Guides: • • • •

Geometrical Optics, John E. Greivenkamp, FG01 Image Processing, Khan M. Iftekharuddin and Abdul Awwal, FG25 Lens Design, Julie Bentley and Craig Olson, FG27 Microscopy, Tomasz S. Tkaczyk, FG13

SPIE Press Monographs: • • • • • • • • • • • •

Advances in Sampling Theory and Techniques, Leonid P. Yaroslavsky, PM315 Designing Optics Using CODE V®, Donald C. O’Shea and Julie L. Bentley, PM292 Electro-Optical Imaging System Performance, Sixth Edition, Gerald C. Holst, PM278 Electro-Optical System Analysis and Design: A Radiometric Perspective, Cornelius, J. Willers, PM236 Image Acquisition and Preprocessing for Machine Vision Systems, P. K. Sinha, PM197 Introduction to the Optical Transfer Function, Charles S. Williams, and Orville A. Becklund, PM112 Optics Inspections and Tests: A Guide for Optics Inspectors and Designers, Michael Hausner, PM269 Optical Specification, Fabrication, and Testing, Jim Schwiegerling, PM252 Optical Systems Engineering, Keith J. Kasunic, PM235 Photonics Rules of Thumb, Third Edition, John Lester Miller, Edward J. Friedman, John N. Sanders-Reed, Katie Schwertz, and Brian K. McComas, PM314 Signal and Image Restoration: Information Theoretic Approaches, Joseph P. Noonan and Prabahan Basu, PM213 Understanding Optical Systems through Theory and Case Studies, Sijiong Zhang, Changwei Li, and Shun Li, PM276

Library of Congress Cataloging-in-Publication Data Names: Boreman, G. D. (Glenn D.), author. Title: Modulation transfer function in optical and electro-optical systems / Glenn D. Boreman. Description: Second edition. | Bellingham, Washington : SPIE Press, [2021] | Series: Tutorial texts in optical engineering ; Volume TT121 | Includes bibliographical references and index. Identifiers: LCCN 2020043215 (print) | LCCN 2020043216 (ebook) | ISBN 9781510639379 (paperback) | ISBN 9781510639386 (pdf) Subjects: LCSH: Optics. | Electrooptical devices. | Modulation theory. Classification: LCC TA1520 .B67 2021 (print) | LCC TA1520 (ebook) | DDC 621.36‐‐dc23 LC record available at https://lccn.loc.gov/2020043215 LC ebook record available at https://lccn.loc.gov/2020043216

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Introduction to the Series The Tutorial Text series provides readers with an introductory reference text to a particular field or technology. The books in the series are different from other technical monographs and textbooks in the manner in which the material is presented. True to their name, they are tutorial in nature, and graphical and illustrative material is used whenever possible to better explain basic and more-advanced topics. Heavy use of tabular reference data and numerous examples further explain the presented concept. A grasp of the material can be deepened and clarified by taking corresponding SPIE short courses. The initial concept for the series came from Jim Harrington (1942–2018) in 1989. Jim served as Series Editor from its inception to 2018. The Tutorial Texts have grown in popularity and scope of material covered since 1989. They are popular because they provide a ready reference for those wishing to learn about emerging technologies or the latest information within a new field. The topics in the series have grown from geometrical optics, optical detectors, and image processing to include the emerging fields of nanotechnology, biomedical optics, engineered materials, data processing, and laser technologies. Authors contributing to the series are instructed to provide introductory material so that those new to the field may use the book as a starting point to get a basic grasp of the material. The publishing time for Tutorial Texts is kept to a minimum so that the books can be as timely and up-to-date as possible. When a proposal for a text is received, it is evaluated to determine the relevance of the proposed topic. This initial reviewing process helps authors identify additional material or changes in approach early in the writing process, which results in a stronger book. Once a manuscript is completed, it is peer reviewed by multiple experts in the field to ensure that it accurately communicates the key components of the science and technologies in a tutorial style. It is my goal to continue to maintain the style and quality of books in the series and to further expand the topic areas to include new emerging fields as they become of interest to our readers. Jessica DeGroote Nelson Optimax Systems, Inc.

v

Contents Preface to the Second Edition Preface to the First Edition

xi xiii

1 MTF in Optical Systems 1.1 1.2 1.3

1.4 1.5

1.6

1.7

1

Impulse Response Spatial Frequency Transfer Function 1.3.1 Modulation transfer function 1.3.2 Phase transfer function MTF and Resolution Diffraction MTF 1.5.1 Calculation of diffraction MTF 1.5.2 Diffraction MTFs for obscured systems Effect of Aberrations on MTF 1.6.1 MTF and Strehl ratio 1.6.2 Effect of defocus on MTF 1.6.3 Effects of other aberrations on MTF 1.6.4 Minimum modulation curve 1.6.5 Visualizing other MTF dependences Conclusion References Further Reading

2 MTF in Electro-optical Systems 2.1 2.2

2.3 2.4 2.5

1 4 7 9 12 16 18 21 25 27 28 29 29 30 33 35 37 38 39

Detector Footprint MTF Sampling 2.2.1 Aliasing 2.2.2 Sampling MTF Crosstalk Electronic-Network MTF Conclusion References

39 43 44 49 59 61 64 65

vii

viii

Contents

3 Point-, Line-, and Edge-Spread Function Measurement of MTF 3.1 3.2 3.3 3.4 3.5

3.6 3.7

3.8 3.9

Point-Spread Function (PSF) Line-Spread Function (LSF) Edge-Spread Function (ESF) Comparison of PSF, LSF, and ESF Increasing SNR in PSF, LSF, and ESF Tests 3.5.1 Object- and image-plane equivalence 3.5.2 Averaging in pixelated detector arrays Correcting for Finite Source Size Correcting for Image-Receiver MTF 3.7.1 Finite pixel width 3.7.2 Finite sampling interval Oversampled Knife-Edge Test Conclusion References

4 Square-Wave and Bar-Target Measurement of MTF 4.1 4.2 4.3

67 67 68 70 72 73 73 76 78 80 80 81 81 83 84 85

Square-Wave Targets Bar Targets Conclusion References

85 88 94 95

5 Noise-Target Measurement of MTF

97

5.1 5.2 5.3

Laser-Speckle MTF Test Random-Transparency MTF Test Conclusion References

6 Practical Measurement Issues 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

6.9

Measurement of PSF Cascade Properties of MTF Quality of Auxiliary Optics Source Coherence Low-Frequency Normalization MTF Testing Observations Use of Computers in MTF Measurements Representative Instrument Designs 6.8.1 Example system #1: visible edge response 6.8.2 Example system #2: infrared line response 6.8.3 Example system #3: visible square-wave response 6.8.4 Example system #4: bar-target response Conclusion References Further Reading

98 104 110 110 113 113 115 118 120 121 122 122 122 123 123 124 126 126 127 127

Contents

ix

7 Other MTF Contributions 7.1 7.2 7.3 7.4 7.5

Index

Motion MTF Vibration MTF Turbulence MTF Aerosol-Scattering MTF Conclusion References Further Reading

129 129 130 132 134 136 137 137 139

Preface to the Second Edition It had been 19 years since the first edition of this book, when the extended quarantine period of 2020 afforded me the rare opportunity of quiet time away from my usual administrative and research activities. I have significantly expanded the treatment of several topics, including bar-target measurements, noise-target measurements, effects of aberrations, and slant-edge measurements. I have been gratified by the recent industrial and government-lab interest in the speckle techniques, which, after all, comprised a good portion of my dissertation at University of Arizona some 36 years ago. All other topics in the book were reviewed and updated, with recent references added. I have kept my original emphasis on practical issues and measurement techniques. I acknowledge with pleasure discussions about MTF with colleagues and their students here at UNC Charlotte, among whom are Profs. Angela Allen, Chris Evans, and Thomas Suleski. During the writing process, I appreciated receiving daily encouragement by telephone from Dot Graudons, daily encouragement via WhatsApp from Prof. Mike Sundheimer of the Universidade Federal Rural de Pernambuco in Recife Brazil, and weekly encouragement via Zoom from Skye Engel. I am grateful for the permissions granted for reproductions of some of the figures from their original sources, to the two anonymous reviewers for their insightful and helpful comments, and to Dara Burrows of SPIE Press for her expert copyediting. Last but surely not least, I want to thank Maggie Boreman – my wife of 30 years, my main encourager, and technical editor. You have graciously taken time from your equestrian pleasures to struggle, once again, with turning my writing into something approaching standard English. Thanks. Glenn D. Boreman Emerald Rose Farm 23 November 2020

xi

Preface to the First Edition I first became aware that there was such a thing as MTF as an undergraduate at Rochester, scurrying around the Bausch and Lomb building. There was, in one of the stairwells, a large poster of the Air Force bar target set. I saw that poster every day, and I remember thinking. . . gee, that’s pretty neat. Well, more than 25 years later, I still think so. I have had great fun making MTF measurements on focal-plane arrays, SPRITE detectors, scanning cameras, IR scene projectors, telescopes, collimators, and infrared antennas. This book is an outgrowth of a short course that I have presented for SPIE since 1987. In it, I emphasize some practical things I have learned about making MTF measurements. I am grateful for initial discussions on this subject at Arizona with Jack Gaskill and Stace Dereniak. Since then, I have had the good fortune here at Central Florida to work with a number of colleagues and graduate students on MTF issues. I fondly recall discussions of MTF with Arnold Daniels, Jim Harvey, Didi Dogariu, Karen MacDougall, Marty Sensiper, Ken Barnard, Al Ducharme, Ofer Hadar, Ric Schildwachter, Barry Anderson, Al Plogstedt, Christophe Fumeaux, Per Fredin, and Frank Effenberger. I want to thank Dan Jones of the UCF English Department for his support, as well as Rick Hermann, Eric Pepper, and Marshall Weathersby of SPIE for their assistance and enthusiasm for this project. I also appreciate the permissions granted for reproductions of some of the figures from their original sources. Last but surely not least, I want to thank Maggie Boreman – my wife, main encourager, and technical editor. Once again, Meg, you have wrestled with my occasionally tedious expository and transformed it, if not into poetry, then at least into prose. Thanks. GDB Cocoa Beach 15 March 2001

xiii

Chapter 1

MTF in Optical Systems Linear-systems theory provides a powerful set of tools with which we can analyze optical and electro-optical systems. The spatial impulse response of the system is Fourier transformed to yield the spatial-frequency optical transfer function. These two viewpoints are equivalent ways to describe an object—as a collection of points or as a summation of spatial frequencies. Simply expressing the notion of image quality in the frequency domain does not by itself generate any new information. However, the conceptual change in viewpoint—instead of a spot size, we now consider a frequency response— provides additional insight into the behavior of an imaging system, particularly in the common situation where several subsystems are combined. We can multiply the individual transfer function of each subsystem to give the overall transfer function. This procedure is easier than the repeated convolutions that would be required for a spatial-domain analysis, and allows immediate visualization of the performance limitations of the aggregate system in terms of the performance of each of the subsystems. We can see where the limitations of performance arise and which crucial components must be improved to yield better overall image quality. We directly see the effects of diffraction and aberrations at various spatial frequencies. In Chapter 1 we develop the transfer-function concept and apply it to classical optical systems—imaging systems alone without detectors or electronics. We will first define terms and then discuss image-quality issues.

1.1 Impulse Response The impulse response h(x,y) is the smallest image detail that an optical system can form and is called the point-spread function (PSF). It is the blur spot in the image plane when a point source is the object of an imaging system. The finite width of the impulse response is a result of the combination of diffraction and aberration effects. We interpret h(x,y) as an irradiance (W/cm2) distribution that is a function of the image-plane position. Modeling

1

2

Chapter 1

the imaging process as a convolution operation (denoted by *), we express the image irradiance distribution g(x,y) as the ideal image f(x,y) convolved with the impulse response h(x,y): gðx, yÞ ¼ f ðx, yÞ  hðx, yÞ:

(1.1)

The ideal image f(x,y) is the irradiance distribution that would exist in the image plane (taking into account the system magnification) if the system had perfect image quality, in other words, a delta-function impulse response. The ideal image is thus a magnified version of the input-object irradiance, with all detail preserved. For conceptual discussions, we typically assume that the imaging system has unit magnification, so for the ideal image we can directly take f(x,y) as the object irradiance distribution, albeit as a function of imageplane coordinates x and y. We can see from Eq. (1.1) that if h(x,y) ¼ d(x,y), the image would be a perfect replica of the object. A perfect optical system is capable of forming a point image of a point object. However, because of the blurring effects of diffraction and aberrations, a real imaging system has an impulse response that is not a point. For any real system, h(x,y) has a finite spatial extent. It is within this context that h(x,y) is referred to as the pointspread function (PSF)—the image-plane irradiance corresponding to a point source input. The narrower the PSF, the less blurring occurs in the imageforming process. A more compact impulse response indicates better image quality. As Fig. 1.1 illustrates, we represent mathematically a point object as a delta function at location (x0 ,y0 ) in object-plane coordinates: f ðxobj , yobj Þ ¼ dðx0  xobj , y0  yobj Þ:

(1.2)

Assuming that the system has unit magnification, the ideal image is a delta function located at (x0 ,y0 ) in image-plane coordinates: gðx, yÞ ¼ dðx0  x, y0  yÞ:

(1.3)

In a real imaging system, the response to the delta-function object of Eq. (1.2) is the impulse response g(x,y) ¼ h(x0  x, y0  y), centered at x ¼ x0

Figure 1.1 A delta function in the object plane is mapped to a blur function, the impulse response, in the image plane.

MTF in Optical Systems

3

and y ¼ y0 in the image plane. We represent a continuous function f(xobj, yobj) of object coordinates by breaking the continuous object into a set of point sources at specific locations, each with a strength proportional to the object brightness at that particular location. Any given point source has a weighting factor f(x0 ,y0 ) that we determine using the sifting property of the delta function: 0

0

ZZ

f ðx , y Þ ¼

dðx0  xobj , y0  yobj Þf ðxobj , yobj Þdxobj dyobj :

(1.4)

The image of each discrete point source will be the impulse response of Eq. (1.1) at the conjugate image-plane location, weighted by the corresponding object brightness. The image irradiance function g(x,y) becomes the summation of weighted impulse responses. This summation can be written as a convolution of the ideal image function f(x,y) with the impulse response ZZ gðx, yÞ ¼

hðx0  x, y0  yÞf ðx, yÞdx0 dy0 ,

(1.5)

which is equivalent to Eq. (1.1). Figure 1.2 illustrates the imaging process using two methods: the clockwise loop demonstrates the weighted superposition of the impulse responses, and the counterclockwise loop demonstrates a convolution with the impulse response. Both methods are equivalent.

Figure 1.2 Image formation can be modeled as a convolutional process. The clockwise loop is a weighted superposition of impulse responses, and the counterclockwise loop is a convolution with the impulse response.

4

Chapter 1

Representing image formation as a convolutional process assumes linearity and shift invariance (LSI). To model imaging as a convolutional process, we must have a unique impulse response that is valid for any position or brightness of the point-source object. Linearity is necessary for us to be able to superimpose the individual impulse responses in the image plane to form the final image. Linearity requirements are typically accurately satisfied for the irradiance distribution itself (the so-called aerial image). However, certain receivers such as photographic film, vidicon tubes,1 IR detector arrays,2 and xerographic media are particularly nonlinear in their responses. In these cases, the impulse response is a function of the input irradiance level. We can only perform LSI analysis for a restricted range of input irradiances. Another linearity consideration is that coherent optical systems (optical processors) are linear in the electric field (V/cm), while incoherent systems (imaging systems) are linear in irradiance (W/cm2). We will deal exclusively with incoherent imaging systems. Partially coherent systems are not linear in either electric field or irradiance. Thus, their analysis as a convolutional system is more complicated, requiring definition of the mutual coherence function.3 Shift invariance is the other requirement for a convolutional analysis. According to the laws of shift invariance, a single impulse response can be defined that is not a function of image-plane position. Shift invariance assumes that the functional form of h(x,y) does not change over the image plane. This shift invariance allows us to write the impulse response as h(x0 –x, y0 –y), a function of distance from the ideal image point, rather than as a function of image-plane position in general. Aberrations violate the assumption of shift invariance because typically the impulse response is a function of field angle. To preserve a convolutional analysis in this case, we segment the image plane into isoplanatic regions over which the functional form of the impulse response does not change appreciably. Typically, we specify image quality on-axis and at off-axis field locations (typically 0.7 field and full field). It is worth noting that any variation in impulse response with field angle implies a corresponding field-dependent variation in transfer function.

1.2 Spatial Frequency We can also consider the imaging process from a frequency-domain (modulation-transfer-function) viewpoint, as an alternative to the spatialdomain (impulse-response) viewpoint. An object- or image-plane irradiance distribution is composed of “spatial frequencies” in the same way that a timedomain electrical signal is composed of various frequencies: by means of a Fourier analysis. As seen in Fig. 1.3, a given profile across an irradiance distribution (object or image) is composed of constituent spatial frequencies. By taking a one-dimensional profile across a two-dimensional irradiance

MTF in Optical Systems

5

distribution, we obtain an irradiance-vs-position waveform, which can be Fourier decomposed exactly as if the waveform was in the more familiar form of volts vs time. A Fourier decomposition answers the question of what frequencies are contained in the waveform in terms of spatial frequencies with units of cycles per unit distance, analogous to temporal frequencies in cycles/s for a time-domain waveform. Typically for optical systems, the spatial frequency is in cycles/mm. Sometimes we see spatial frequency expressed in lp/mm (line pairs per mm). One cycle ¼ one black/white line pair. An example of one basis function for the one-dimensional waveform of Fig. 1.3 is shown in Fig. 1.4. The spatial period X (crest-to-crest repetition distance) of the waveform can be inverted to find the x-domain spatial frequency denoted by j ≡ 1/X. Fourier analysis of optical systems is more general than for time-domain systems because objects and images are inherently two-dimensional, so the basis set of component sinusoids is also two-dimensional. Figure 1.5 illustrates a two-dimensional sinusoid of irradiance. The sinusoid has a spatial period along both the x and y directions, X and Y, respectively. If we invert these spatial periods, we find the two spatial-frequency components that describe this waveform: j ¼ 1/X and h ¼ 1/Y. Two pieces of information are required for specification of the two-dimensional spatial frequency. An alternative representation is possible using polar (r,u) coordinates, where the minimum crest-to-crest distance of the sinusoid is R, and the orientation of the minimum crest-to-crest distance with respect to the x and y axes is u ¼ tan1(Y/X).

Figure 1.3

Definition of a spatial-domain irradiance waveform.

Figure 1.4

One-dimensional spatial frequency.

6

Chapter 1

Figure 1.5

(a) Two-dimensional spatial period. (b) Two-dimensional spatial frequency.

The corresponding spatial-frequency variables in polar coordinates are r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 þ h2 and f ¼ tan1(h/j). Whatever coordinate system we choose for the description, the basis set over which objects and images are decomposed consists of the whole range of possible orientations and spatial frequencies, where the sinusoid shown in Fig. 1.5 is but a single example. Two-dimensional functions can be separable or nonseparable. A separable function of two variables (such as x and y, or r and u) can be described as a function of one variable multiplied by a function of the other variable. Whether a function is separable depends on the choice of coordinate system. For instance, a function that is rotationally symmetric (just a function of r) is not separable in rectangular coordinates because a slice of the function along the u ¼ 45 deg diagonal is identical to a slice of the function along the horizontal axis, and thus is not a product of x-dependent and y-dependent functions. If a function is separable in the spatial domain, the Fourier transform of that function is separable in the corresponding spatial-frequency domain. Angular spatial frequency is typically encountered in the specification of imaging systems designed to observe a target at a long distance. If the target is far enough away to be in focus for all distances of interest, then it is convenient to specify system performance in angular units, that is, without having to specify a particular object distance. Angular spatial frequency is most often specified in units of cycles/milliradian. These units can initially be a troublesome concept because both cycles (cy) and milliradians (mrad) are dimensionless quantities; however, with reference to Fig. 1.6, we find that the angular spatial frequency jang is simply the range ZR multiplied by the target spatial frequency j. For a periodic target of spatial period X, we define an angular period u ≡ X/ZR, an angle over which the object waveform repeats itself. The angular period is in radians if X and ZR are in the same units. Inverting this angular period gives angular spatial frequency jang ¼ ZR/X. Given the resolution of optical systems, often X is in meters and ZR is in kilometers, for which the ratio ZR/X is then in cy/mrad.

MTF in Optical Systems

7

Figure 1.6

Angular spatial frequency.

1.3 Transfer Function Equation (1.1) describes the loss of detail inherent in the imaging process as the convolution of the ideal image function with the impulse response. The convolution theorem4 states that a convolution in the spatial domain is a multiplication in the frequency domain. Taking the Fourier transform (denoted as F ) of both sides of Eq. (1.1) yields F fgðx, yÞg ¼ F ff ðx, yÞ  hðx, yÞg

(1.6)

Gðj, hÞ ¼ F ðj, hÞ  Hðj, hÞ,

(1.7)

and

where uppercase functions denote the Fourier transforms of the corresponding lowercase functions: F denotes the object spectrum, G denotes the image spectrum, and H denotes the spectrum of the impulse response. H(j,h) is the transfer function; it relates the object and image spectra multiplicatively. The Fourier transform changes the irradiance waveform from a spatial-position function to the spatial-frequency domain but generates no new information. The appeal of the frequency-domain viewpoint is that the multiplication of Eq. (1.7) is easier to perform and visualize than the convolution of Eq. (1.1). This convenience is most apparent in the analysis of imaging systems consisting of several subsystems, each with its own impulse response.

8

Chapter 1

As Eqs. (1.8) and (1.9) demonstrate, each subsystem has its own transfer function—the Fourier transform of its impulse response. The final result of all subsystems operating on the input object distribution is a multiplication of their respective transfer functions. Figure 1.7 illustrates that we can analyze a combination of several subsystems by the multiplication of transfer functions of Eq. (1.9) rather than the convolution of impulse responses of Eq. (1.8): f ðx, yÞ  h1 ðx, yÞ  h2 ðx, yÞ  : : :  hn ðx, yÞ ¼ gðx, yÞ

(1.8)

F ðj, hÞ  H 1 ðj, hÞ  H 2 ðj, hÞ  : : :  H n ðj, hÞ ¼ Gðj, hÞ:

(1.9)

and

For the classical optical systems under discussion in this first chapter, we ignore the effects of noise and typically assume that H(j,h) has been normalized to have a unit value at zero spatial frequency. By the central ordinate theorem of Fourier transforms, this is equivalent to a unit area under the impulse response. This normalization at low spatial frequencies yields a relative transmittance as a function of frequency (ignoring constant attenuation factors such as Fresnel-reflection loss, neutral-density filters, and atmospheric absorption). The lowest spatial frequency (a flat field, that is, a uniform irradiance distribution across the entire field of view) is assumed to come through with unit transmittance. Although this normalization is common, when we use it, we lose information about the absolute signal levels. MTF is thus typically not radiometric in the information it conveys. In some situations, we might want to keep the signal-level information, particularly when electronics noise is a significant factor. When we want to

Figure 1.7 The aggregate transfer function of several subsystems is a multiplication of their transfer functions.

MTF in Optical Systems

9

calculate a spatial-frequency-dependent signal-to-noise ratio, we do not normalize the transfer function. With this normalization, we refer to H(j,h) as the optical transfer function (OTF). Unless the impulse response function h(x,y) satisfies certain symmetry conditions, its Fourier transform H(j,h) is, in general, a complex function, having both a magnitude and a phase portion, referred to as the modulation transfer function (MTF) and the phase transfer function (PTF), respectively: OTF ≡ Hðj, hÞ ¼ jHðj, hÞj expfjUðj, hÞg

(1.10)

and MTF ≡ jHðj, hÞj

PTF ≡ Uðj, hÞ:

(1.11)

1.3.1 Modulation transfer function The modulation transfer function is the magnitude response of the optical system to sinusoids of different spatial frequencies. When we analyze an optical system in the frequency domain, we consider the imaging of sinewave inputs (Fig. 1.8) rather than point objects. Sinewave targets are typically printed as diffuse reflective targets to be viewed under conditions of natural illumination. Alternatively, sinewaves of irradiance can be generated by diffraction as Young’s fringes5 or using two-beam interference techniques.6 A linear shift-invariant optical system images a sinusoid as another sinusoid. The limited spatial resolution of the optical system results in a decrease in the modulation depth M of the image relative to what was in the object distribution (Fig. 1.9). Modulation depth is defined as the amplitude of the irradiance variation divided by the bias level: M¼

Amax  Amin 2  ac component ac ¼ : ¼ Amax þ Amin 2  dc component dc

(1.12)

We can see from Fig. 1.10 that modulation depth is a measure of contrast, with M ! 0 as Amax  Amin ! 0 and M ! 1 as Amin ! 0:

Figure 1.8

Sinewave target containing several spatial frequencies.

(1.13)

10

Chapter 1

Figure 1.9 Modulation depth decreases going from object to image.

Figure 1.10 Definition of modulation depth per Eq. (1.10): (a) high contrast and (b) low contrast.

The M ¼ 0 condition means that no spatial variation of irradiance exists in the image. However, the M ¼ 0 condition does not imply that the imageirradiance level is zero. Another consequence of Eq. (1.13) is that the M ¼ 1 unit-modulation-depth condition is obtained when the irradiance waveform has a minimum value of zero, regardless of the maximum irradiance level. This is a consequence of the usual nonradiometric normalization of MTF. In practice, the variations of irradiance must be sufficient to be seen above any noise level present. The interpretation of modulation depth as a measure of contrast is that a waveform with a small modulation depth is difficult to distinguish against a uniform background irradiance, especially with some level of noise present.

MTF in Optical Systems

11

The finite spatial extent of the impulse response of the optical system causes a filling in of the valleys and a lowering of the peak levels of the sinusoid. This decreases the modulation depth in the image waveform relative to that in the corresponding object waveform. We define the modulation transfer (MT) as the ratio of modulation in the image to that in the object: MT ≡ M image ðjÞ∕M object ðjÞ:

(1.14)

The modulation transfer is, in general, spatial-frequency dependent. The limited resolution of the optics is more important at high spatial frequencies, where the scale of the detail is smaller. When we plot modulation transfer against spatial frequency, we obtain the MTF, generally a decreasing function of spatial frequency, as seen in Fig. 1.11. In the situation shown where the object modulation depth is constant, the MTF is simply the image modulation as a function of spatial frequency: MTFðjÞ ≡ M image ðjÞ∕M object :

(1.15)

The M of the object waveform does not need to be unity. If a lower input modulation is used, then the image modulation will be proportionally lower. Equation (1.14) shows that the modulation of the object waveform does not need to be unity. We can use non-unit-modulation targets (Fig. 1.12) for MTF measurements, but a high-contrast target generally produces the best results.

Figure 1.11 Modulation transfer function is the decrease of modulation depth with increasing frequency.

12

Chapter 1

Figure 1.12 Examples of non-unit-modulation depth waveforms (adapted from Ref. 7, which also includes information on test targets conforming to requirements of ISO-12233).

1.3.2 Phase transfer function Recalling the definition of the optical transfer function in Eq. (1.10), we now proceed to interpret the phase response: OTFðjÞ ≡ MTFðjÞ expfjPTFðjÞg:

(1.16)

For the special case of a reflection-symmetric impulse response centered on the ideal image point, the phase transfer function (PTF) is particularly simple, having a value of either zero or p as a function of spatial frequency. For example, the OTF for a defocused impulse response (Fig. 1.13) exhibits these phase reversals. The OTF, being a complex quantity, can be plotted as a function having both positive and negative values, while the MTF is strictly positive. The PTF shows a phase reversal of p over the spatial frequency range for which the OTF is negative. We can see (Fig. 1.14) the phase reversals associated with defocus using the example of a radial bar target, which has increasing spatial frequency toward the center. Phase reversals (white-to-black line transitions) are seen over certain spatial-frequency (radius) ranges of the defocused image. The MTF is zero at the phase-transition frequency, so we see a uniform gray irradiance at the corresponding radii of the chart where the black/white phase transition occurs—one at a relatively low spatial frequency and one at a higher frequency. The higher-frequency phase transition is of low contrast because the MTF is lower there. We also see this p phase reversal on the image of a radial bar target that was blurred at a 45-deg diagonal direction (Fig. 1.15). Over a certain range of two-dimensional spatial frequencies, what begins as a black bar at the periphery of the target becomes a white bar. This phase shift occurs in a gray transition zone where the MTF goes through a zero. We see more complicated phase transfer functions when the irradiance of the impulse response is not reflection symmetric about the center. For the computed example shown in Fig. 1.16 of a slightly asymmetric impulse

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Figure 1.13 OTF, MTF, and PTF for a defocused impulse response.

Figure 1.14 Radial bar target: comparison of focused and defocused images.

response (Gaussian with a slight linear ramp added), the PTF shows significant variation as a function of spatial frequency. In practice, we most often encounter a plot of the PTF in the output from an optical-design computer program. This is because phase distortion is a sensitive indicator for the presence of aberrations such as coma with an asymmetric PSF for off-axis image points. The PTF is not typically measured directly, but the information is available by Fourier transforming a measured impulse response.

14

Chapter 1

Figure 1.15 Radial bar target that has been blurred in the 45-deg diagonal direction. Phase reversals exist over certain two-dimensional spatial-frequency ranges.

Figure 1.16 Asymmetric impulse responses produce nonlinear PTFs.

Nonlinearity in the PTF causes different spatial frequencies in the image to recombine with different relative phases. This phase distortion can change the shape of the spatial waveform describing the image. We illustrate, using a computed example, that severe phase distortion can produce image irradiance distributions that differ in significant ways from the object irradiance distribution. We thus consider the imaging of a four-bar target with equal lines and spaces (Fig. 1.17). The fundamental spatial frequency jf of the target is the inverse of the center-to-center bar spacing as shown. Although such targets are not periodic in the true sense (being spatially limited), we will speak of nth harmonics of the fundamental spatial frequency as simply njf. In Fig. 1.18 we plot PTF and image-irradiance waveforms for three p-phase step-discontinuity PTFs. To emphasize the effect of the PTF on the image, we let MTF ¼ 1 in this computed example, so that all of the spatial

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Figure 1.17

Four-bar irradiance image.

Figure 1.18 Image-irradiance-vs-position plots for four-bar-target images under specified phase distortions.

frequencies are present in their original amounts and only the relative phases have changed. In the first case, the PTF transition occurs at 4jf. There is no shift for the fundamental or the third harmonic, and a p shift for higher frequencies. We see a phase-reversal artifact as a local minimum at the center of each bar, primarily because the fifth harmonic is out of phase with the third and the fundamental. In the second case, the transition occurs at 2jf, so the only in-phase spatial frequency is the fundamental. The bars are sinusoidal at the center, with secondary-maxima artifacts in the shoulder of each bar and in the space between them, arising primarily from the third and fifth harmonics. The step transition for the third case occurs at 0.9jf, shifting the fundamental and all harmonics with respect to frequencies lower than jf. The most

16

Chapter 1

dramatic artifact is that the image now has five peaks instead of the four seen in the previous cases.

1.4 MTF and Resolution Resolution is a quantity without a standardized definition. Figure 1.19 illustrates the image-irradiance-vs-position plots in the spatial domain, showing a particular separation distance for which images of two points are said to be resolved. A variety of criteria exist for such determination, based on the magnitude of the dip in irradiance between the point images. Resolution can be defined in image-plane distance or in object-space angular measure. In addition to these definitions, resolution can be specified in the spatialfrequency domain as the frequency at which the MTF falls below a particular threshold. A typical value used in practice is 10%, but the required MTF threshold depends on the application. A threshold MTF and hence limiting resolution (Fig. 1.20) can be defined in terms of the noise-equivalent modulation (NEM), which is how much modulation depth is needed to give a unit signal-to-noise ratio, as a function of spatial frequency. We will

Figure 1.19

Resolution can be defined in the spatial domain.

Figure 1.20 Resolution can be defined in the spatial-frequency domain.

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discuss NEM more fully in Chapter 2. The NEM is also referred to in the literature as the demand modulation function or threshold detectability curve. A convenient graphical representation is to plot MTF and the noiseequivalent modulation depth on the same curve. The limiting resolution is the spatial frequency where the curves cross. In general, the best overall image-quality performance is achieved by the imaging system that has the maximum area between the MTF and NEM curves over the spatial frequency range of interest. This quantity, seen in Fig. 1.21, is called MTF area (MTFA), which has been correlated to image quality in human perception tests.8,9 Using either the spatial-domain definition or the spatial-frequencydomain definition, resolution is a single-number performance specification and, as such, it is often seen as being more convenient to use than MTF (which is a function instead of a single number). However, MTF provides more complete performance information than is available from simply specifying resolution, including information about system performance over a range of spatial frequencies. As we can see on the left-hand side of Fig. 1.22, two systems may have an identical limiting resolution but different performances at lower frequencies. The system corresponding to the higher of the two curves would have the better image quality. The right-hand side of Fig. 1.22 shows that resolution

Figure 1.21 MTF area (MTFA) is the area between the MTF and NEM curves. Larger MTFA indicates better image quality.

Figure 1.22

Limiting resolution does not tell the whole story.

18

Chapter 1

Figure 1.23 Comparison of images for detection, recognition, and identification according to the Johnson criteria (top right Fig. adapted from Ref. 11 with permission; bottom right figure adapted from Ref. 12 with permission).

alone can be a misleading performance criterion. The system that has the best resolution (limiting frequency) has poorer performance at the midrange frequencies. A decision about which system has better performance requires us to specify the spatial-frequency range of interest. One way to determine the range of spatial frequencies of interest is to use the Johnson criteria,10 where certain spatial frequencies are necessary for various imaging tasks. Across the smallest dimension of an object, the Johnson criteria state that: detection (an object is present) requires 0.5 to 1 line pair, recognition (the class of object is discerned) requires 4 to 5 line pairs, and identification (the specific type of object) requires 6 to 8 line pairs. These criteria can be visualized in Fig. 1.23. Other imaging tasks, such as lithography, also have critical spatial frequencies. These are needed, for example, to ensure proper edge definition and reproduction of sharp corners.

1.5 Diffraction MTF This introductory section presents results for the circular-pupil case. Because of the wave nature of light, an optical system with a finite-sized aperture can never form a point image. A blur spot in the image plane is formed, even in the absence of aberrations. The smallest spot size that the system can form is determined by diffraction. The diameter of this diffraction blur is d diffraction ¼ 2.44lðF ∕#Þ:

(1.17)

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The irradiance distribution in the diffraction image of a point source is EðrÞ ¼ j2J 1 ðprÞ∕ðprÞj2 ,

(1.18)

where r is the normalized radial distance from the center of the pattern: r ¼ r∕½lðF ∕#Þ:

(1.19)

Figure 1.24 shows a radial plot of Eq. (1.18). In Fig. 1.25 we see a two-dimensional plot that is somewhat overexposed (saturating the center lobe) to better emphasize the ring structure of the diffraction pattern. A twodimensional integration of Eq. (1.18) shows that the blur diameter, defined as

Figure 1.24 Radial plot of diffracted irradiance [Eq. (1.18)].

Figure 1.25 Two-dimensional plot of the diffracted irradiance distribution.

20

Chapter 1

the diameter to the first dark ring of the pattern, 2.4l(F/#), contains 84% of the power in the irradiance distribution. In Eqs. (1.17) and (1.19), the parameter F/# is used as a scale factor that determines the physical size of the diffraction spot. Actually, three different expressions for F/# can be used for diffraction-limited spot size and diffraction-limited MTF calculations. As seen in Fig. 1.26, we can define a working F/# in either object space or in image space in terms of the lens aperture diameter D and either the object distance p or the image distance q: ðF ∕#Þobject-space ¼ p∕D

(1.20)

ðF ∕#Þimage-space ¼ q∕D:

(1.21)

or

For the special case of an object at infinity (Fig. 1.27), the image distance q becomes the lens focal length f, and the image-space F/# becomes ðF ∕#Þimage-space ¼ f ∕D:

(1.22)

Equations 1.20 to 1.22 assume that the lens aperture D is uniformly filled with light. For instances where this is not the case, such as illumination with a Gaussian laser beam, D would be reduced to the effective aperture diameter that the beam illuminates, with a corresponding increase in F/#. Using Eqs. (1.20), (1.21), or (1.22), we can calculate, either in object space or in image space, the diffraction spot size from Eq. (1.17). We can also

Figure 1.26 Definition of working F/# for the finite-conjugate situation.

Figure 1.27 Definition of image-space F/# for the object-at-infinity situation.

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consider the effect of diffraction in terms of the MTF. Conceptually, a diffraction MTF can be calculated as the magnitude of the Fourier transform of the diffraction impulse-response profile given in Eq. (1.18). Diffraction MTF is a wave-optics calculation for which the only variables (for a given aperture shape) are the aperture dimension D, wavelength l, and focal length f. The MTFdiffraction is the upper limit to the system’s performance; the effects of optical aberrations are assumed to be negligible. Aberrations increase the spot size and thus contribute to a reduced MTF. The diffraction MTF is based on the overall limiting aperture of the system. For an optical system with multiple elements, the effect of diffraction is only calculated once per system, at the aperture stop. Diffraction effects are not calculated separately at each optical element, and the diffraction MTF does not accumulate multiplicatively on an element-by-element basis. 1.5.1 Calculation of diffraction MTF We can calculate the diffraction OTF as the normalized autocorrelation of the exit pupil of the optical system, which we will consider for both the circularpupil and square-pupil cases. This is consistent with the definition of Eqs. (1.6), (1.7), and (1.10), which state that the OTF is the Fourier transform of the impulse response. For the incoherent systems we consider, the impulse response h(x,y) is the square of the two-dimensional Fourier transform of the diffracting aperture p(x,y). The magnitude squared of the diffracted electricfield amplitude E in V/cm gives us the irradiance profile of the impulse response in W/cm2: hdiffraction ðx, yÞ ¼ jF F fpðx, yÞgj2 :

(1.23)

From Eq. (1.23), we must implement a change of variables j ¼ x/lf and h ¼ y/ lf to express the impulse response (which is a Fourier transform of the pupil function) in terms of the image-plane spatial position. We then calculate the diffraction OTF in the usual way, as the Fourier transform of the impulse response h(x,y): OTFdiffraction ðj, hÞ ¼ F F fhdiffraction ðx, yÞg ¼ F F fjF F fpðx, yÞgj2 g: (1.24) Because of the absolute-value-squared operation, the two transform operations of Eq. (1.24) do not exactly undo each other: the diffraction OTF is the two-dimensional autocorrelation of the diffracting aperture p(x,y). The diffraction MTF is thus the magnitude of the (complex) diffraction OTF. As an example of this calculation, we take the simple case of a square aperture, seen in Fig. 1.28:

22

Chapter 1

Figure 1.28 Variables for calculating the diffraction MTF of a square-aperture system.

pðx, yÞ ¼ rectðx∕D, y∕DÞ:

(1.25)

The autocorrelation of the square is a (real-valued) triangle-shaped function, MTFðjÞ ¼ 1  j=jcutoff ,

(1.26)

with cutoff frequency defined by jcutoff ¼ 1∕½lðF ∕#Þ:

(1.27)

For the case of a circular aperture of diameter D, the system has the same cutoff frequency, jcutoff ¼ 1/[l(F/#)], but the MTF has a different functional form: MTFðj∕jcutoff Þ ¼

o 2 n 1 cos ðj∕jcutoff Þ  ðj∕jcutoff Þ½1  ðj∕jcutoff Þ2 1∕2 (1.28) p

for j , jcutoff and MTFðj∕jcutoff Þ ¼ 0

(1.29)

for j . jcutoff. These diffraction-limited MTF curves are plotted in Fig. 1.29. The diffraction-limited MTF is an easy-to-calculate upper limit to performance; we need only l and F/# to compute it. An optical system cannot perform better than its diffraction-limited MTF; any aberrations will pull the MTF curve down. We can compare the performance specifications of a given system to the diffraction-limited MTF curve to determine the feasibility of the proposed specifications, to decide how much headroom has been left for

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Figure 1.29 Universal curves representing diffraction-limited MTFs for incoherent systems with either a circular or a rectangular aperture.

manufacturing tolerances, or to see what performance is possible within the context of a given choice of l and F/#. As an example of the calculations, let us consider the rectangular-aperture system of Fig. 1.30. For an object at a finite distance, we use the object-space or image-space F/# as appropriate to calculate the diffraction-limited jcutoff in either the object plane or image plane: jcutoff, obj ¼

1 1 ¼ 800 cy=mm ¼ lðF ∕#Þobj space lðp∕DÞ

(1.30)

jcutoff, img ¼

1 1 ¼ 400 cy=mm: ¼ lðF ∕#Þimg space lðq∕DÞ

(1.31)

or

Figure 1.30 Rectangular-aperture (top) finite-conjugate and (bottom) infinite-conjugate MTF examples.

24

Chapter 1

Because p , q, the image is magnified with respect to the object; hence, a given feature in the object appears at a lower spatial frequency in the image, so the two frequencies in Eqs. (1.30) and (1.31) represent the same feature. The filtering caused by diffraction from the finite aperture is the same, whether considered in object space or image space. For example, knowing the cutoff frequency, we can calculate the image spatial frequency for which the diffraction-limited MTF is 30%. We use Eq. (1.26) to find that 30% MTF is at 70% of the image-plane cutoff frequency, or 280 cy/mm. This calculation is for diffraction-limited performance, and aberrations will narrow the bandwidth of the system. Thus, the frequency at which the MTF is 30% will most likely be lower than 280 cy/mm. Continuing the example for an object-at-infinity condition, we obtain the cutoff frequency in the image plane using Eq. (1.27): jcutoff, img space ¼ 1∕½lðF ∕#Þ ¼ D∕lf ¼ 1200 cy=mm:

(1.32)

A given feature experiences the same amount of filtering, whether expressed in the image plane or in object space. In the object space, we find the cutoff frequency to be jcutoff, obj space ¼ 1∕ðl=DÞ ¼ D∕l ¼ 40 cy=mrad:

(1.33)

Let us verify that this angular spatial frequency corresponds to the same feature as that in Eq. (1.32). Referring to Fig. 1.31, we use the relationship between object-space angle u and image-plane distance X, X ¼ uf :

(1.34)

Inverting Eq. (1.34) to obtain the angular spatial frequency 1/u, we have 1=u ¼ ð1∕X Þf ¼ jf :

(1.35)

Figure 1.31 Relation between object-space angle u and image-plane distance X.

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Figure 1.32 Heuristic representation of the formula for diffraction MTF.

Given that u is in radians, if X and f have the same units, we can verify the correspondence between the frequencies in Eqs. (1.32) and (1.33): ð1=uÞ½cy=mrad ¼ j½cy=mm  f ½mm  ð0.001Þ½rad=mrad:

(1.36)

It is also of interest to verify that the diffraction MTF curves in Fig. 1.29 are consistent with the results of the simple 84% encircled-power diffraction spot-size formula of 2.4l(F/#). As a heuristic verification, in Fig. 1.32 we create a pattern with adjacent bright and dark regions, whose widths are 2.4l(F/#). We set the bright regions at a magnitude of 84% and the dark regions at a magnitude of 16%, consistent with the amount of power inside and outside the central lobe of the diffraction spot, respectively. Overlapping adjacent diffraction spots with this spacing would create an irradiance distribution approximating the situation shown. Considering a horizontal one-dimensional spatial frequency across the pattern, we can calculate both the spatial frequency and the modulation depth. The fundamental spatial frequency of the pattern in Fig. 1.32 is j ¼ 1∕½4.88lðF ∕#Þ ¼ 0.21  ½1∕ðlðF ∕#ÞÞ ¼ 0.21jcutoff

(1.37)

and the modulation depth at this frequency is M ¼ ð84  16Þ∕ð84 þ 16Þ ¼ 68%,

(1.38)

which is in close agreement with Fig. 1.29 for a diffraction-limited circular aperture, at a frequency of j ¼ 0.21 jcutoff, as seen in Fig. 1.33. 1.5.2 Diffraction MTFs for obscured systems Many common optical systems, such as Cassegrain telescopes, have an obscured aperture. We can calculate the diffraction OTF of obscured-aperture systems according to Eq. (1.24), the autocorrelation of the pupil of the system. With an obscuration, an attenuation of image-plane irradiance occurs that is proportional to the fractional blocked area of the pupil. This attenuation affects all spatial frequencies equally. When the autocorrelation is calculated,

26

Chapter 1

Figure 1.33 Verification of the formula for diffraction MTF.

we see a slight emphasis of the MTF at high frequencies, corresponding to an overlap of the clear part of the aperture in the shift-multiply-and-integrate operation of the autocorrelation. This emphasis has come at the expense of the overall flux transfer. In the usual definition of MTF to be unity at j ¼ 0, the attenuation caused by the obscuration is normalized out when MTF is plotted. In Fig. 1.34 the MTF curves for the obscured apertures exceed the diffraction-limited curve for no obscuration (curve A). This behavior is an artifact of the normalization, and there is actually no more image modulation at those frequencies than would exist in the case of an unobscured aperture. If we let the value of curve B at j ¼ 0 be 0.84, the value of curve C at j ¼ 0 be 0.75, and the value of curve D at j ¼ 0 be 0.44, all of these MTF curves would be bounded as an upper limit by the diffraction-limited curve A for no obscuration. The cutoff frequency is the same for all of the obscured apertures, determined by the diameter of the open aperture 2rm and the wavelength l.

Figure 1.34 Diffraction-limited MTF curves for obscured-aperture systems (adapted from Ref. 13).

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1.6 Effect of Aberrations on MTF Diffraction-limited performance is the ideal, representing the best possible imaging achievable from a system with a given F/# and wavelength. The monochromatic (Seidel) aberrations are image-quality defects that arise both from the nonlinearity of Snell’s law and because spherical surfaces are generally used in optical systems. Chromatic aberrations are caused by the variation of refractive index with wavelength. Defects in manufacture or alignment will make aberrations worse. Aberrations will increase the spatial extent of the impulse response, spreading the irradiance into a larger region and lowering the peak irradiance. An impulse response of larger spatial extent reduces the image quality, blurring fine details in the image. Because of the Fourier-transform relationship, an impulse response of larger spatial extent implies a narrower MTF. Aberrations generally get worse with decreasing F/# (wider aperture) and increasing field angle. Defocus and spherical aberration are independent of field angle, so they affect image quality, even for image points near the optical axis. Aberrations reduce the MTF and lower the cutoff frequency, as compared to the diffraction-limited MTF. The MTF at any spatial frequency is bounded by the diffraction-limited MTF curve. Figure 1.35 illustrates that the value of the MTF at any spatial frequency is bounded by the diffractionlimited MTF curve: MTFw=aberr ðjÞ # MTFdiffraction ðjÞ:

(1.39)

We can calculate the OTF of an imaging system in the following way. The wavefront aberration, defined at the exit pupil, is the departure from sphericity of the nominally spherical wavefront proceeding toward a particular image point, bounded by the dimensions of the exit pupil. The OTF is the autocorrelation of this spatially bounded wavefront aberration. We use the change of variables j ¼ x/lf and h ¼ y/lf to convert a spatial shift in the autocorrelation to a spatial frequency. If there are no aberrations, the calculation is simply the autocorrelation of the pupil transmittance function. Any aberrations present in the system will reduce the MTF, given that the positive and negative phase variations on the wavefront will autocorrelate to a

Figure 1.35 The effect of aberrations on MTF is to pull the transfer-function curve down.

28

Chapter 1

lower value than would unaberrated wavefronts without such variations. This calculation is the typical means of computing MTF in optical-design software programs, accounting for aberrations (by means of the wavefront error) and diffraction (by means of the pupil dimension and the wavelength). Alternately, we can conveniently use a back-of-the-envelope method to estimate MTF. If we have a raytraced spot diagram for the system or a spot size based on geometrical-optics aberration formulae, we take this as the PSF h(x,y) and calculate a geometrical MTF in the manner of Eqs. (1.6) and (1.7) by Fourier transformation. We can approximate the overall MTF by multiplying the geometrical and diffraction MTFs. This multiplication will reduce the MTF below the diffraction-limited curve and will lower the cutoff spatial frequency. By the convolution theorem of Eqs. (1.6) and (1.7), this is equivalent to convolving the diffraction irradiance distribution with the PSF of the aberration.14,15 This approach is suitable for encircled-energy calculations but does not capture the fine spatial irradiance fluctuations of the aberrated diffraction image. 1.6.1 MTF and Strehl ratio A useful single-number performance specification is the Strehl ratio s, defined as the on-axis irradiance produced by the actual system divided by the on-axis irradiance that would be formed by a diffraction-limited system of the same F/#: s≡

hactual ðx ¼ 0, y ¼ 0Þ : hdiffraction ðx ¼ 0, y ¼ 0Þ

(1.40)

A Strehl ratio in excess of 0.8 indicates excellent image quality (l/4 of wavefront aberration). We can obtain an alternative interpretation of the Strehl ratio using the central-ordinate theorem for Fourier transforms, which says that the volume under a two-dimensional function in the transform domain (or area under a one-dimensional function) equals the on-axis value of the function in the spatial domain: ZZ f ðx ¼ 0, y ¼ 0Þ ¼ F ðj, hÞdjdh: (1.41) Using this relation, we can express the Strehl ratio as the volume (area) under the actual OTF curve divided by the volume (area) under the diffractionlimited OTF curve: RR OTFactual ðj, hÞ s ¼ RR : OTFdiffraction ðj, hÞ

(1.42)

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A large Strehl ratio implies a large area under the MTF curve and high irradiance at the image location. Aberration effects such as those seen in Fig. 1.35 can be interpreted directly as the decrease in volume (area) under the MTF curve. 1.6.2 Effect of defocus on MTF In Fig. 1.36 we compare a diffraction-limited OTF to that for systems with increasing amounts of defocus. The amount of defocus is expressed in terms of optical path difference (OPD), which is the wavefront error at the edge of the aperture in units of l. In all cases, the diffraction-limited curve is the upper limit to the OTF. For small amounts of defocus, the OTF curve is pulled down only slightly. For additional defocus, a significant narrowing of the transfer function occurs. For a severely defocused system, we observe the phase-reversal phenomenon that we saw in the defocused image of the radial bar target of Fig. 1.14. We can visualize the relationship between Strehl ratio and MTF by comparing the PSFs and MTFs for the diffraction-limited and l/4-of-defocus conditions. For l/4 of defocus, the transfer function is purely real, so OTF ¼ MTF. From Fig. 1.37, we see that the defocus moves about 20% of the power from the center of the impulse response into the ring structure. The onaxis value of the impulse response is reduced, with a corresponding reduction in the area under the MTF curve. 1.6.3 Effects of other aberrations on MTF To further visualize the effects of aberrations on MTF, let us compare the PSF and MTF for different amounts of wavefront error (WFE). The graphs on pages 31 to 33 show the PSFs (both as irradiance as a function of position and as a gray-level irradiance distribution), along with corresponding MTF plots.

Figure 1.36 Effect of defocus on the OTF of a diffraction-limited circular-aperture system (adapted from Ref. 13).

30

Chapter 1

Figure 1.37 Comparison of diffraction-limited (DL) performance and quarter wavelength of defocus for (left) PSFs (adapted from Ref. 16 with permission) and (right) MTFs (adapted from Ref. 13).

We begin with defocus and spherical aberration in Fig 1.38. Both aberrations produce radially symmetric PSFs and MTFs, which can be conveniently represented as one-dimensional functions. We can see the reduction in on-axis irradiance and corresponding reduction in area under the MTF curve, both being more pronounced with increasing wavefront error. Next, in Fig. 1.39 we compare PSF and MTF for various amounts of coma. The effects of coma are asymmetric, so the PSFs and MTFs are both two-dimensional functions. The narrowest and widest profiles of the PSFs are shown, along with the best and worst MTFs, all as one-dimensional plots for the noted values of WFE. Because of the asymmetry of the PSF, the reduction in MTF depends on the orientation of the aberrated PSF with the twodimensional spatial frequencies of interest. As seen in Fig. 1.40, the MTF is highest if the wide direction of the PSF is along the constant direction of the spatial-frequency sinusoid, and is lowest for the orthogonal orientation. Finally, in Fig. 1.41 we compare PSF and MTF for different amounts of astigmatism. Again, the image quality depends on the orientation of the asymmetric PSF with the two-dimensional spatial frequencies of interest, as seen in Fig. 1.42. 1.6.4 Minimum modulation curve From a specification viewpoint, we often want to ensure that a given minimum MTF is met at certain spatial frequencies. Usual practice is to

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Figure 1.38 Comparison of PSF and MTF for defocus and spherical aberration, for different amounts of wavefront error (adapted from Ref. 17 with permission).

specify horizontal and vertical slices of the two-dimensional MTF. However, this approach can miss important information if there is significant asymmetry in the MTF, as seen in the previous section. To address this situation, Aryan, Boreman, and Suleski recently developed a summary descriptor of the two-dimensional MTF, the minimum modulation curve (MMC).18 The two-dimensional MTF(j, h) can be expressed in polar coordinates using r ¼ (j2 þ h2)1/2 and azimuth angle f ¼ tan1(h/j), yielding MTF(r,f). The radial spatial frequency r is the distance from the center of the two-dimensional polar plot, and f is measured from the horizontal (j) axis. To generate the MMC, the MTF is evaluated for all values of f, for a given value of r. The minimum MTF value at that r for any value of f becomes the MMC value for that r:

32

Chapter 1

Figure 1.39 Comparison of PSF and MTF for coma, for different amounts of wavefront error. Best and worst one-dimensional profiles are shown of the two-dimensional PSF and MTF functions for different amounts of wavefront error (adapted from Ref. 17 with permission).

Figure 1.40 Image quality depends on the orientation of the asymmetric PSF with respect to the two-dimensional spatial-frequency components.

MMCðrÞ ¼ min fMTFðr, fÞg: f∈½0, 2p

(1.43)

So, the MMC presents the minimum MTF found for any azimuth angle, as a function of r. This displays information in a familiar one-dimensional form. If used as a performance specification, the MMC would guarantee that a given MTF specification is met for any possible orientation of spatial frequencies in the image. We illustrate this concept using the example of a Cooke triplet at a field angle of 20 deg. The two-dimensional MTF for this situation has significant

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Figure 1.41 Comparison of PSF and MTF for astigmatism, for different amounts of wavefront error (adapted from Ref. 17 with permission). Best and worst one-dimensional profiles are shown of the two-dimensional PSF and MTF functions for different amounts of wavefront error.

Figure 1.42 Image quality depends on the orientation of the asymmetric PSF with respect to the two-dimensional spatial-frequency components.

asymmetry, as seen in the color plot and wireframe plots of Fig. 1.43. We can see qualitatively from those plots that the horizontal and vertical slices of the MTF are an inadequate representation of the overall performance. In Fig. 1.44 we plot the horizontal and vertical MTFs for this lens, along with the MMC. This shows the potential utility of MMC as a practical performance specification. 1.6.5 Visualizing other MTF dependences Although a typical MTF plot is a function of spatial frequency, sometimes the dependence of MTF on field position or axial location of the image plane are of interest. Aberrations typically get worse with increasing aperture and field,

34

Chapter 1

Figure 1.43

Two-dimensional MTF for the example lens.

Figure 1.44 Comparison of horizontal and vertical MTFs and MMC for the example lens (reprinted from Ref. 18).

so we usually find that the quality of the image is best near the optic axis and not as good toward the edge of the image. In Fig. 1.45 we see some MTF plots for an example lens at 10, 20, and 40 lp/mm spatial frequencies, as a function of location of the image with respect to the optic axis. MTF falls for higher spatial frequencies and for larger field heights, as expected. The solid lines are for the optimal orientation of the two-dimensional spatial frequency, and the dashed lines are for the worst orientation. The PSF is increasingly asymmetric as the field height increases. Of course, the two plots converge for small field heights. MTF also depends on the axial position of the image plane with respect to the optical system, known as the through-focus MTF. In Fig. 1.46 we see

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Figure 1.45 Best and worst MTFs for an example lens at three specific spatial frequencies, as a function of image-plane field height (adapted from Ref. 19 with permission).

MTF plots for an example lens at three specific spatial frequencies as a function of axial focus position. When the lens operates at the wide aperture setting of F/1.4, spherical aberration is the primary limitation to image quality. The image plane location is such that 20 lp/mm is at best focus. From the MTF plots, we see that the 10 lp/mm and 40 lp/mm spatial frequencies have their best focus positions at different axial locations. In this situation, the image quality is not the same on either side of best focus. If we stop down this lens to a smaller aperture (F/4), the best focus shifts away from the lens, consistent with the behavior of spherical aberration seen in Fig. 1.47. Also, for the smaller aperture setting, the peak MTFs at each spatial frequency increase because of the reduction in spherical aberration. For the F/4 case, the three spatial frequencies are in best focus at about the same axial position, indicating more symmetry in the through-focus image quality.

1.7 Conclusion Expression of image quality in terms of a transfer function provides additional insight into the performance of an optical system, compared to describing the irradiance of a blur spot or a specification of resolution. We can conveniently account for the various contributions to image quality by multiplying transfer functions for the different subsystems. The transfer function approach allows us to directly see the effects of diffraction and aberrations at various spatial frequencies.

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Figure 1.46 MTFs for an example lens at three specific spatial frequencies as a function of axial image-plane location. The zero defocusing location is set as best focus for F/1.4 at 20 lp/mm. Top plots are for F/1.4 aperture, and bottom plots are for F/4 aperture, with defocusing in millimeters (adapted from Ref. 19 with permission).

MTF in Optical Systems

37

Figure 1.47 Ray trace showing the presence of spherical aberration. Best focus is the image plane location resulting in the smallest spot size. It is located between the marginalray focus and the paraxial-ray focus. A reduced aperture diameter will tend to shift the position of best focus away from the lens.

References 1. G. D. Boreman, A. B. James, and C. R. Costanzo, “Spatial harmonic distortion: a test for focal plane nonlinearity,” Opt. Eng. 30, pp. 609–614 (1991) [doi: 10.1117/12.55832]. 2. G. D. Boreman and C. R. Costanzo, “Compensation for gain nonuniformity and nonlinearity in HgCdTe infrared charge-coupleddevice focal planes,” Opt. Eng. 26, pp. 981–984 (1987) [doi: 10.1117/12. 7974184]. 3. M. Beran and G. Parrent, Theory of Partial Coherence, Prentice-Hall, Englewood Cliffs, New Jersey (1964). 4. J. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons, New York (1978). 5. K. J. Barnard, G. D. Boreman, A. E. Plogstedt, and B. K. Anderson, “Modulation-transfer function measurement of SPRITE detectors: sine-wave response,” Appl. Opt. 31(3), 144–147 (1992). 6. M. Marchywka and D. G. Socker, “Modulation transfer function measurement technique for small-pixel detectors,” Appl. Opt. 31(34), 7198–7213 (1992). 7. N. Koren: www.normankoren.com/Tutorials/MTF.html; www.imatest. com. 8. L. M. Biberman, “Image Quality,” Chapter 2 in Perception of Displayed Information, L. M. Biberman, Ed., Plenum Press, New York, pp. 52–53 (1973). 9. H. L. Snyder, “Image Quality and Observer Performance,” Chapter 3 in Perception of Displayed Information, L. M. Biberman, Ed., Plenum Press, New York, pp. 87–117 (1973).

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10. J. Johnson, “Analysis of image forming systems,” in Image Intensifier Symposium, AD 220160, Warfare Electrical Engineering Department, U.S. Army Research and Development Laboratories, Fort Belvoir, Virginia, pp. 244–273 (1958). 11. A. Richards, FLIR Systems Inc., private communication. 12. www.flir.com/discover/security/thermal/5-benefits-of-thermal-imagingcameras/ 13. W. Wolfe and G. Zissis, Eds., The Infrared Handbook, Environmental Research Institute of Michigan/Infrared Information and Analysis Center, Ann Arbor, U.S. Office of Naval Research, Washington, D.C. and SPIE Press, Bellingham, Washington (1985). 14. E. H. Linfoot, “Convoluted spot diagrams and the quality evaluation of photographic images,” Optica Acta 9(1), p. 81–100 (1962). 15. K. Miyamoto, “Wave Optics and Geometrical Optics in Optical Design,” in Progress in Optics, Vol. 1, E. Wolf, Ed., North-Holland, Amsterdam, pp. 31–40, 40a, 41–66 (1961). 16. J. Wyant, class notes, Wyant College of Optical Sciences, University of Arizona. 17. www.telescope-optics.net/mtf.htm, with graphics credit to Cor Berrevoets (http://aberrator.astronomy.net/). 18. H. Aryan, G. D. Boreman, and T. J. Suleski, “The minimum modulation curve as a tool for specifying optical performance: application to surfaces with mid-spatial frequency errors,” Opt. Exp. 27(18), 25551–25559 (2019). 19. H. Nasse, “How to read MTF curves,” https://lenspire.zeiss.com/photo/ en/article/overview-of-zeiss-camera-lenses-technical-articles

Further Reading Baker, L. Selected Papers on Optical Transfer Function: Foundation and Theory, SPIE Milestone Series, Vol. MS59, SPIE Press, Bellingham, Washington (1992). Williams, C. S., and Becklund O. A., Introduction to the Optical Transfer Function, Wiley, New York (1989); reprinted by SPIE Press, Bellingham, Washington (2002). Williams, T., The Optical Transfer Function of Imaging Systems, Institute of Physics Press, Bristol (1999). https://www.imatest.com/docs/ https://lenspire.zeiss.com/photo/en/article/overview-of-zeiss-camera-lensestechnical-articles

Chapter 2

MTF in Electro-optical Systems In Chapter 1 we applied a transfer-function-based analysis to describe image quality in classical optical systems, that is, systems with optical components only. In this chapter we will examine the MTF of electro-optical systems, that is, systems that use a combination of optics, scanners, detectors, electronics, signal processors, and displays. To apply MTF concepts in the analysis of electro-optical systems, we must generalize our assumptions of linearity and shift invariance. Noise is inherent in any system with electronics. Linearity is not strictly valid for systems that have an additive noise level because image waveforms must be of sufficient irradiance to overcome the noise before they can be considered to add linearly. The classical MTF theory presented in Chapter 1 does not account for the effects of noise. We will demonstrate how to broaden the MTF concept to include this issue. Electro-optical systems typically include detectors or detector arrays for which the size of the detectors and the spatial sampling interval are both finite. Because of the shift-variant nature of the impulse response for sampled-data systems, we will develop the concept of an average impulse response obtained over a statistical ensemble of source positions to preserve the convenience of a transfer-function analysis. We will also develop an expression for the MTF impact of irradiance averaging over the finite sensor size. With these modifications, we can apply a transfer-function approach to a wider range of situations.

2.1 Detector Footprint MTF We often think about the object as being imaged onto the detectors, but it is also useful to consider where the detectors are imaged. The footprint of a particular detector, called the instantaneous field of view (IFOV), is the geometrical projection of that detector into object space. We consider a scanned imaging system in Fig. 2.1 and a staring focal-plane-array (FPA) imaging system in Fig. 2.2. In each case, the flux falling onto an individual detector produces a single output. Inherent in the finite size of the detector elements is some spatial averaging of the image irradiance. For the

39

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Chapter 2

Figure 2.1

Figure 2.2

Scanned linear detector array.

Staring focal-plane-array imaging system.

configurations shown, we have two closely spaced point sources in the object plane that fall within one detector footprint. The signal output from the sensor will not indicate the fact that there are two sources. Our first task is to quantify the spatial-frequency filtering inherent in an imaging system with finite-sized detectors. A square detector of size w
w performs spatial averaging of the scene irradiance that falls on it. When we analyze the situation in one dimension, we find that the integration of the scene irradiance f(x) over the detector surface is equivalent to a convolution of f(x) and the rectangle function1 that describes the detector responsivity: w∕2 Z

f ðxÞ dx ¼ f ðxÞ  rectðx∕wÞ:

gðxÞ ¼ w∕2

(2.1)

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41

By the convolution theorem,1 Eq. (2.1) is equivalent to filtering in the frequency domain by a transfer function:    sin ðpjwÞ :  MTFfootprint ðjÞ ¼ jsincðjwÞj ¼ (2.2) pjw  Equation (2.2) shows us that the smaller the sensor photosite dimension, the broader the transfer function. This equation is a fundamental MTF component for any imaging system with detectors. In any given situation, the detector footprint may or may not be the main limitation to image quality, but its contribution to a product such as Eq. (1.9) is always present. Equation (2.2) is plotted in Fig. (2.3), where we see that the sinc-function MTF has its first zero at j ¼ 1/w. Let us consider the following plausibility argument to justify the fact that the footprint MTF ¼ 0 at j = 1/w. Figure 2.4 represents spatial averaging of an input irradiance waveform by sensors of a given dimension w. The individual sensors may represent either different positions for a scanning sensor or discrete locations in a focal-plane array. We will consider the effect of spatial sampling in a later section. Here we consider exclusively the effect of the finite size of the photosensitive regions of the sensors. We see that at low spatial frequencies there is almost no reduction in modulation of the image irradiance waveform arising from spatial averaging over the surfaces of the photosites. As the spatial frequency increases, the finite size of the detectors becomes more significant. The averaging leads to a decrease in the maximum values and an increase in the minimum values of the image waveform, decreasing the modulation depth. For the spatial frequency j ¼ 1/w, one period of the irradiance waveform just fits onto each detector. Regardless of the position of the input irradiance waveform with respect to the photosite boundaries, each sensor will collect

Figure 2.3

Sinc-function MTF for detector of width w.

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Figure 2.4

At a frequency of j ¼ 1/w the modulation depth goes to zero.

exactly the same power (spatially integrated irradiance). The MTF is zero at j ¼ 1/w because each sensor reads the same level and there is no modulation depth in the resulting output waveform. Extending our analysis to two dimensions, we consider the simple case of a rectangular detector with different widths along the x and y directions: hfootprint ðxÞ ¼ rectðx∕wx , y∕wy Þ ¼ rectðx∕wx Þrectðy∕wy Þ:

(2.3)

By Fourier transformation, we obtain the OTF, which is a two-dimensional sinc function: OTFfootprint ðj, hÞ ¼ sincðjwx , hwy Þ and

(2.4)

MTF in Electro-optical Systems

  sinðpjwx Þ MTFfootprint ðj, hÞ ¼ pjwx

43

   sinðphwy Þ    phw y

  : 

(2.5)

The impulse response in Eq. (2.3) is separable, that is, hfootprint(x,y) is simply a function of x multiplied by a function of y. The simplicity of the separable case is that both h(x,y) and H(j,h) are products of two one-dimensional functions, with the x and y dependences completely separated. Occasionally, a situation arises in which the detector responsivity function is not separable.2,3 In that case, we can no longer write the MTF as the product of two onedimensional MTFs, as seen in Fig. 2.5. The MTF along the j and h spatial frequency directions is affected by both x and y profiles of the detector footprint. For example, the MTF along the j direction is not simply the Fourier transform of the x profile of the footprint but is MTFfootprint ðj, 0Þ ¼ jH footprint ðj, h ¼ 0Þj ≠ jF fhfootprint ðx, y ¼ 0Þgj:

(2.6)

Finding MTF in these situations requires a two-dimensional Fourier transform of the detector footprint. The transfer function can then be evaluated along the j or h axis, or along any other desired direction.

2.2 Sampling Sampling is a necessary part of the data-acquisition process in any electrooptical system. We will sample at spatial intervals Dx ≡ xsamp. The spatial sampling rate is determined by the location of the detectors in a focal-plane

Figure 2.5

Example of a nonseparable detector footprint (adapted from Ref. 3).

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array. The process of spatial sampling has two main effects on image quality: aliasing and the sampling MTF. 2.2.1 Aliasing Aliasing is an image artifact that occurs when we insufficiently sample a waveform. We assume that the image irradiance waveform of interest has already been decomposed into its constituent sinusoids. Therefore, we can consider a sinusoidal irradiance waveform of spatial frequency j. If we choose a sampling interval sufficient to locate the peaks and valleys of the sinewave, then we can reconstruct that particular frequency component unambiguously from its sampled values, assuming that the samples are not all taken at the same level (the 50%-amplitude point of the sinusoid). Thus, the two-samples-per-cycle minimum sampling rate seen in Fig. 2.6 corresponds to the Nyquist condition: Dx ≡ xsamp ¼ 1∕ð2jÞ:

(2.7)

If the sampling is less frequent [xsamp . 1/(2j)] than required by the Nyquist condition, then we see the samples as representing a lower-frequency sinewave (Fig. 2.7). Even though both sinewaves shown are consistent with the samples, we will perceive the low-frequency waveform when looking at the sampled values. This image artifact, where samples of a high-frequency waveform appear to represent a low-frequency waveform, is an example of aliasing. Aliasing is symmetric about the Nyquist frequency of jNyquist ¼ 1/(2Dxsamp), which means that the amount by which a waveform’s spatial frequency exceeds 1/(2Dxsamp) is the amount by which we perceive it to be below the Nyquist frequency. So, a frequency transformation of ðjNyquist þ DjÞ ! ðjNyquist  DjÞ takes place between the input waveform and the aliased image data.

Figure 2.6

Nyquist sampling condition of two samples per cycle.

(2.8)

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45

Figure 2.7 Aliasing of a high-frequency waveform (solid line) to a lower-spatial-frequency waveform (dashed line) by insufficient sampling.

Figure 2.8 Example of aliasing in the image of a radial bar target.

Figure 2.8 shows an example of aliasing for the case of a radial bar target, for which the spatial frequency increases toward the center. The right-hand image has been sampled using a larger sampling interval. With an insufficient spatial-sampling rate, we see that the high frequencies near the center are aliased into the appearance of lower spatial frequencies. Figure 2.9 is a three-bar target that shows aliasing artifacts. The left image was acquired with a small spatial-sampling interval, and we see that the bars have equal lines and spaces, and are of equal density. The right image was acquired with a larger spatial-sampling interval. Although bar targets are not periodic in the true sense, we can consider the nth harmonics of the fundamental spatial frequency jf as njf. Some of these frequencies are above jNyquist and are not adequately sampled. The fact that not all of the bars in a

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Figure 2.9

Bar-target pattern showing aliasing artifacts.

given three-bar pattern are of the same width or density in the undersampled image on the right is evidence of aliasing. Figure 2.10 shows another example of aliasing using three versions of a scene. Part (a) is a 512
512 image, which appears spatially continuous without significant aliasing artifacts evident. Part (b) has been downsampled to 128
128 pixels, and aliasing artifacts in sharp edges begin to be visible because of lower jNyquist. In part (c), the image has been downsampled to 64
64 pixels, and we see extensive aliasing artifacts as low-frequency banding in the folds of the shirt and the sharp edges.

Figure 2.10 Pictorial example of aliasing.

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47

After the irradiance waveform has been sampled, aliasing artifacts cannot be removed by filtering because, by Eq. (2.8), the aliased components have been lowered in frequency to fall within the main spatial-frequency passband of the system. Thus, to remove aliasing artifacts at this point requires the attenuation of broad spatial-frequency ranges of the image data. We can avoid aliasing in the first place by prefiltering the image, that is, bandlimiting it before the sampling occurs. The ideal anti-aliasing filter, seen in Fig. 2.11, would pass at unit amplitude all frequency components for which j , jNyquist and attenuate completely all components for which j . jNyquist. The problem is that neither the detector MTF (a sinc function) nor the optics MTF (bounded by an autocorrelation function) follows the form of the desired antialiasing filter. An abrupt filter shape such as the one in Fig. 2.11 can be implemented in the electronics subsystem. However, at that stage the image irradiance has already been sampled by the sensors, so the electrical filter cannot effectively serve an anti-aliasing function. The optics MTF offers some flexibility as an anti-aliasing filter but, because it is bounded by the autocorrelation function of the aperture, it does not allow for the abrupt-cutoff behavior desired. By choosing l and F/# we can control the cutoff frequency of the optics MTF. However, this forces a tradeoff of reduced MTF at frequencies less than jNyquist against the amount of residual aliasing. Using the diffraction-limited MTF as in Eq. (1.26) or (1.28) and Fig. 1.29 as an anti-aliasing filter requires setting the cutoff so that MTF (j $ jNyquist) ¼ 0. This results in a loss of considerable area under the MTF curve at frequencies below Nyquist. If we set the cutoff frequency higher, we preserve additional modulation depth for j , jNyquist at the expense of nonzero MTF above Nyquist (leading to some aliasing artifacts). The choice of a higher cutoff frequency for the linear MTF function preserves more modulation below Nyquist but results in higher visibility of aliasing artifacts. A small amount of defocus is occasionally used

Figure 2.11 Comparison of an ideal anti-aliasing filter to filters corresponding to diffractionlimited optics MTF, showing a tradeoff of MTF below Nyquist with an amount of residual aliasing.

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in a bandlimiting context, but the MTF does not have the desired functional form either, and hence a similar tradeoff applies. Birefringent filters that are sensitive to the polarization state of the incident radiation can be configured to perform an anti-aliasing function,4 although still without the ideal abrupt-cutoff MTF shown in Fig. 2.11. A filter of the type shown in Fig. 2.12 is particularly useful in color focal-plane arrays, where different spectral filters (red, blue, green) are placed on adjacent photosites. Because most visual information is received in the green portion of the spectrum, it is radiometrically advantageous to set the sampling interval for the red- and blue-filtered detectors wider than for the green-filtered detectors. If we consider each color separately, we find a situation equivalent to the sparse-array configuration seen in Fig. 2.12, where the active photosites for a given color are shown shaded. The function of the birefringent filter is to split an incident ray into two components. A single point in object space maps to two points in image space, with a spacing equal to one-half of the detectorto-detector distance. The impulse response of the filter is two delta functions: hfilter ðxÞ ¼

1 fdðxÞ þ dðx þ xsamp ∕2Þg: 2

(2.9)

The corresponding filter transfer function can be found by Fourier transformation as MTFfilter ðjÞ ¼ j cos½2pðxsamp ∕4Þj j,

(2.10)

which has its first zero at 1/(xsamp) ¼ 2jNyquist. The birefringent filter thus provides a degree of prefiltering, in that the bandlimiting function is applied before the image is sampled by the detector array. The blur obtained using a

Figure 2.12

Mechanism of a birefringent anti-aliasing filter.

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49

birefringent filter is approximately independent of the lens F/#, which is not the case when defocus is used as a prefilter. Typically, F/# is needed as a variable to control the image-plane irradiance. 2.2.2 Sampling MTF We can easily see that a sampled-imaging system is not shift invariant. Consider the FPA imager seen in Fig. 2.13. The position of the imageirradiance function with respect to the sampling sites will affect the final image data. If the image is aligned so that most of the image irradiance falls completely on one single column of the imager, then a high-level signal is produced that is spatially compact. If the image irradiance function is moved slightly so that it falls on two adjacent columns, the flux from the source is split in two, and a lower-level signal of broader spatial extent is produced. If we compute the MTF for such a sampled system, the Fourier transform of the spatial-domain image will depend on the alignment of the target and the sampling sites, with the best alignment giving the broadest MTF. This shift variance violates one of the main assumptions required for a convolutional analysis of the image-forming process. To preserve the convenience of a transfer-function approach, the concept of impulse response can be generalized to define a shift-invariant quantity. Following Park, Schowengerdt, and Kaczynski,5 we define a spatially averaged impulse response and a corresponding MTF component that is inherent in the sampling process itself by assuming that the scene being imaged is randomly positioned with respect to the sampling sites. This random alignment corresponds to the situation where a natural scene is imaged with an ensemble of individual alignments. Park’s original work was in the context of star-field images. For a two-dimensional rectangular sampling grid, the

Figure 2.13 A sampled image-forming system is not shift invariant.

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sampling impulse response is a rectangle function whose widths are equal to the sampling intervals in each direction: hsampling ðx, yÞ ¼ rect ðx∕xsamp , y∕ysamp Þ:

(2.11)

We see from Eq. (2.11) that wider-spaced sampling produces an image with poorer image quality. An average sampling MTF can be defined as the magnitude of the Fourier transform of hsampling(x, y): MTFsampling ¼ j F frect ðx∕xsamp , y∕ysamp Þgj,

(2.12)

which yields a sinc-function sampling MTF: MTFsampling

   sinðpjxsamp Þ sinðphysamp Þ : ¼ jsincðjxsamp , hysamp Þj ¼ physamp  pjxsamp

(2.13)

The sampling MTF is equivalent to the average of the MTFs that would be realized for an ensemble of image locations, uniformly distributed with respect to the sampling sites. As Fig. 2.13 demonstrates, when the alignment is optimum, the MTF is broad, but for other source positions, the MTF is narrower. The sampling MTF is the average over all possible MTFs. Thus defined, the sampling MTF is a shift-invariant quantity, and we can proceed with a usual transfer-function-based analysis. The sampling MTF of Eq. (2.13) is a component that multiplies the other MTF components for the system. However, this sampling MTF does not contribute in an MTF-measurement setup where the test target is aligned with the sampling sites because the central assumption in its derivation is the random position of any image feature with respect to the sampling sites. In typical MTF test procedures, we adjust the position of the test target to yield the best output signal (most compact output, best appearance of bar-target images). In the typical test-setup case, the sampling MTF equals unity except where random-noise test targets6 that explicitly include the sampling MTF in the measurement are used. Because typical test procedures preclude the sampling MTF from contributing to the measurements, the sampling MTF is often forgotten in a system analysis. However, when the scene being imaged has no net alignment with respect to the sampling sites, the sampling MTF will contribute in practice and should therefore be included in the system-performance modeling. The combined MTF of the optics, detector footprint, and sampling can be much less than initially expected, especially considering two common misconceptions that neglect the detector and sampling MTFs. The first error is to assume that if the optics blur-spot size is matched to the detector size then there is no additional image-quality degradation from the finite detector size. We can see from Eq. (1.9) that the optics and detector MTFs multiply, and hence both terms contribute. Also, it is quite common to forget the sampling

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MTF, which misses another multiplicative sinc contribution. As an illustration of the case where the optics blur-spot size is matched to the detector size in a contiguous FPA, we assume that each impulse response can be modeled in a one-dimensional analysis as a rect function of width wx. Each of the three MTF terms is a sinc function, and the product is proportional to sinc3(jwx), as seen in Fig. 2.14. The difference between the expected sinc (jwx) and the actual sinc3(jwx) at midrange spatial frequencies can be 30% or more in terms of absolute MTF and, at high frequencies the MTF can go from an expected value of 20% to essentially zero, with the inclusion of the two additional sinc-function contributions. The recent trend in visible and IR FPAs is that pixels are very small (on the order of a wavelength), with a near-contiguous FPA pitch. This has several image-quality advantages. Finer spatial sampling pushes the Nyquist frequency higher, increasing the un-aliased bandwidth. For a given system MTF, increasing the Nyquist frequency lowers the MTF at the onset of aliasing, decreasing the visibility of aliasing artifacts. Additionally, small contiguous pixels increase the overall system MTF by increasing the footprint MTF and the sampling MTF at a given spatial frequency. However, there is a decreased amount of performance to be gained in terms of overall system MTF because the upper limit to the overall MTF is the (diffraction-limited) optics MTF. Nonetheless, given the reduced cost of large-pixel-count FPAs, it is worth pushing into the range of “diminishing returns” to get slightly better overall MTF, which directly translates into increased detection- and recognition-range performance for IR systems. We see a direct demonstration of the sampling MTF contribution in the following discussion. This method itself is of decreasing importance nowadays because FPA pixel densities are so high, but it is included here to illustrate the contribution of the sampling MTF. The pixel spacing on a FPA is of course fixed, but we can decrease the spatial sampling interval by microscanning (also called microdither).7–10 The image falling on the FPA is moved by a piezoelectric actuator that moves optical elements or by a liquid crystal beam

Figure 2.14 MTF contributions multiply for detector footprint, optics blur spot, and sampling.

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steerer, in either case slightly displacing the line of sight. Successive frames of displaced image samples are obtained and interlaced with appropriate spatial offsets. Usually, we collect four frames of data at half-pixel relative shifts, as shown in Fig. 2.15. In part (a) we see the original image of the object superimposed on the detector array. In part (b) the image location with respect to the detector array is shifted by a half-pixel spacing in the horizontal direction. In part (c) the image location is shifted by a half-pixel spacing in the vertical direction. In part (d) the image location is shifted by a half-pixel spacing in both the horizontal and vertical directions. The four frames are interlaced to produce an output frame with twice the effective sampling rate in each direction. Finer sampling yields better sampling MTF along with higher Nyquist frequencies. The fact that microscanning produces better pictures (Fig. 2.16) is intuitive proof of the existence of a sampling MTF contribution because the detector size and the optics MTF are both unchanged. The single MTF component that is improved by the microscan technique is the sampling MTF. The drawback to microscanning, from a systems viewpoint, is that the frame takes longer to acquire for a given integration time. Alternatively, if we keep the frame rate constant, the integration time decreases, which can have a negative impact on

Figure 2.15 Illustration of the microscan process (adapted from Ref. 9).

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53

Figure 2.16 Microscan imagery: (left) the original image, (center) the four shifted images, and (right) the interlaced image (adapted from Ref. 10).

the signal-to-noise ratio. There is also an additional data-processing complexity involved in interlacing the frames together. We now compare several sampled-image situations with FPAs, where we consider the overall MTF as the product of sampling MTF and footprint MTF. We consider only the MTF inherent in the averaging-and-sampling process arising from finite-sized detectors that have a finite center-to-center spacing. In actual practice, other MTF contributions such as those arising in the fore-optics or electronics subsystems would multiply these results. For simplicity, we analyze only the x and y sampling directions. Recall that the Nyquist frequency is the inverse of twice the sampling interval in each direction. In each of the cases considered, we keep the sensor dimension constant at w and investigate the aggregate MTF as we vary the sampling situation. In each of the following FPA examples, the image quality in the x and y directions is identical. We do not account for the finite dimension of the array as a whole in any of the examples. First, we consider the sparse FPA shown in Fig. 2.17. In this case, the sampling interval is twice the detector width. Because the Nyquist frequency is

Figure 2.17 MTF for a sparse focal-plane array.

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rather low at j ¼ 0.25/w, the MTF is high at the aliasing frequency, which means that aliasing artifacts such as those seen in Figs. 2.8 through 2.10 are visible with high contrast. In addition, the large sampling interval places the first zero of the sampling MTF at j ¼ 0.5/w, narrowing the MTF product considerably compared to the detector MTF, which has a first zero at j ¼ 1/w. In Fig. 2.18 the sensor size remains the same as in Fig. 2.17, but now the detectors are contiguous, with a sampling interval equal to the detector width. The Nyquist frequency is thus raised to j ¼ 0.5/w, which has two related effects. First, because the aliasing frequency is higher, the MTF is lower at the aliasing frequency, so aliasing artifacts are not as visible. Also, the usable bandwidth of the system, from dc to the onset of aliasing, has been increased by the finer sampling. The sinc-function MTF for the detectors and for the sampling is identical, with a first zero at j ¼ 1/w for each. Their product is a sinc-squared function, which has considerably higher MTF than did the MTF of the sparse array seen in Fig. 2.17. In Fig. 2.19 we consider a situation that would arise in the context of microdither. The sensors are physically contiguous, and their size remains the same; but now samples are taken at half-pixel intervals in each direction. The Nyquist frequency rises to j ¼ 1/w, which increases the available un-aliased bandwidth. Also, in this case the MTF is zero at Nyquist. Thus, as spatial frequency increases, any aliasing artifacts will come in slowly and initially with very low visibility. The sinc-function MTF for the detectors has its first zero at j ¼ 1/w, while the sinc-function MTF for the sampling is twice as wide, having its first zero at j ¼ 2/w. Thus, the product of these two MTFs is wider than it is for the overall MTF seen in Fig. 2.18.

Figure 2.18 MTF for a contiguous focal-plane array.

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55

Figure 2.19 MTF for a contiguous focal-plane array with samples at half-pixel intervals in x and y directions.

Now we explore analogous sampling situations with a linear array of scanned detectors. The effective scan velocity relates temporal and spatial variables, and similarly relates temporal and spatial frequencies. Given a single detector scanned in the horizontal direction, the instantaneous field of view (IFOV) is that portion of the object scene being looked at by the detector at one instant in time. The IFOV is the moving footprint of the detector, and it has (spatial) dimensions of distance. As seen in Fig. 2.20, when the IFOV scans across a localized (delta function) feature in the scene, there will be a signal on the detector for a time period called the dwell time td ¼ IFOV∕vscan while the IFOV continuously moves across this feature. We are free to decide the sampling interval for the time-domain analog waveform arising from a continuously scanned detector. Given td ¼ IFOV∕vscan , the waveform sample spacings in time are directly related to spatial samplings in the scene’s spatial variable x by Dt ¼ Dx∕vscan . Thus, temporal frequencies and spatial frequencies are related by

Figure 2.20 Two positions of a detector: at the start and at the end of the overlapping of the IFOV with a delta function scene feature.

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f ½Hz ¼ vscan ½mm∕sec
j ½cy∕mm:

(2.14)

Typical practice in scanned sensor systems is to sample the analog signal from the detector at time intervals equivalent to two samples per detector width, known as “twice per dwell.” Finer sampling is certainly possible, but we obtain the maximum increase in image quality by going from one to two samples per dwell. Beyond that spatial sampling rate, there is a diminishing return in terms of image quality. Let us see why two samples per dwell has been such a popular operating point, with reference to Fig. 2.21. With a sampling interval of w/2, the x-direction Nyquist frequency has been increased to j ¼ 1/w. This higher aliasing frequency is beneficial because the usable bandwidth has been increased, but the other factor is that now the MTF of the detector footprint goes through its first zero at the Nyquist frequency. Because the transfer function is zero at Nyquist, the image artifacts arising from aliasing are naturally suppressed. A final advantage is that the x-direction MTF has been increased because of the broader sampling-MTF sinc function, which has its first zero at j ¼ 2/w. Since the detectors are contiguous in the y direction, the aliasing frequency is h ¼ 0.5/w and the overall MTF as a function of h is just the sinc-squared function seen in the analysis of the contiguous FPA in Fig. 2.18. In Fig. 2.22 we extend this analysis to a pair of staggered linear arrays offset by half of the detector-to-detector spacing, which is a commonly used configuration. Once again, we must perform additional data processing to interlace the information gathered from both sensor arrays into a highresolution image. The advantage we gain is that an effective twice-per-dwell sampling in both the x and y directions is achieved, with wider h-direction MTF, higher h-direction aliasing frequency, and additional suppression of

Figure 2.21 MTF for a scanned linear array of sensors.

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57

Figure 2.22 MTF for a staggered pair of scanned linear arrays.

aliasing artifacts compared to the situation in Fig. 2.21. Although it is possible to interlace more than two arrays to increase the signal-to-noise ratio in the context of time delay-and-integration (TDI), the greatest image-quality benefit in terms of MTF is gained in going from a single array to two. As a final example of this type of analysis, we consider a fiber array (Fig. 2.23), which can transmit an image-irradiance distribution over the length of the fibers in the bundle.11,12 If we assume that the arrangement of the fibers is preserved between the input and output faces (a so-called coherent array), we can describe the MTF as the product of the MTF of the fiber footprint (because there is no spatial resolution within an individual fiber, only a spatial averaging of irradiance) and the sampling MTF (which depends on the details of the arrangement of the fibers). In this example, we consider a hexagonal packed array with a sampling lattice having different sampling intervals in the x and y directions: an x-sampling interval of D/2 and a pffiffiffi y-sampling interval of 3D∕2. This yields different Nyquist frequencies and different sampling MTFs in the x and y directions. Other spatial arrangements of the fibers are possible. Once the center-to-center spacing of the fibers in each direction is fixed, the sampling MTF along j and h can be found from Eq. (2.13), under the assumption that any scene irradiance distribution is randomly positioned with respect to the fibers. Calculation of the footprint MTF requires a two-dimensional Fourier transform of the fiber footprint because it is not a separable function in x and y. This gives us the Besselfunction footprint MTF shown in Fig. 2.23. So far, we have considered only the nearest-neighbor sampling distances along the x and y directions. If we extend this sampling to any direction in the x-y plane, we can extend our analysis to any sampled-image system where

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Chapter 2

Figure 2.23

MTF for a coherent fiber-bundle array.

directions other than x and y are important for image formation, such as hexagonal focal-plane arrays, fiber bundles, and laser printers. Once the twodimensional sampling MTF is in hand, we multiply it by the two-dimensional Fourier transform of the pixel footprint to yield the overall sampling-andaveraging array MTF.13,14 A one-dimensional sinc-function sampling MTF along the lines of Eq. (2.13) applies to the spacing between the nearest neighbors in any direction because the distance between samples in any direction can be modeled as a rect-function impulse response (assuming a random position of the scene with respect to the sampling sites). The width of the rect function depends on the particular direction u in which the nextnearest neighbor is encountered: hsampling ðuÞ ¼ rect ½x∕xsamp ðuÞ

(2.15)

and MTFsampling ðju Þ ¼ j F frect ½x∕xsamp ðuÞg ¼ j sinc½ju xsamp ðuÞ j,

(2.16)

where ju is understood to be a one-dimensional spatial frequency parameterized on the u direction. Directions with more widely spaced next-nearest neighbors will have poorer image quality. For a finite-dimension sampling array, a nearest neighbor does not exist at all in some directions, so the MTF is necessarily zero in that direction (because the image data of that particular spatial frequency cannot be reconstructed from the samples). We find that the MTF of Eq. (2.16) is thus a discontinuous function of angle (Fig. 2.24).

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59

Figure 2.24 The nearest-neighbor distance, and therefore the sampling MTF, is a discontinuous function of angle u.

2.3 Crosstalk Crosstalk arises when the signal of a particular detector on a FPA contributes to or induces a spurious signal on its neighbor. Origins of crosstalk include charge-transfer inefficiency, photogenerated carrier diffusion, inter-pixel capacitance caused by coupling of close capacitors inherent in FPA pixel structures, and channel-to-channel crosstalk caused by the wiring harness and readout electronics. One way to measure inter-pixel crosstalk is by illuminating a single pixel of the FPA with an image of an x-ray source. The x-ray photons will generate charge carriers that give a signal from the illuminated pixel (and adjacent pixels). If we use a short-wavelength source, we can generate a spot that is smaller than the pixel of the FPA to be measured, which would usually not be possible because of diffraction if we chose a source that was within the response band of the FPA. As Fig. 2.25 shows, crosstalk can be approximately modeled with an impulse response of a Gaussian or negative-exponential form. Typically, there is not a large number of sample points because only the few closest channels will have an appreciable crosstalk signal. Thus, we have some flexibility in picking the fitting function, as long as the samples that are present are appropriately represented. If we Fourier transform the impulse response, we obtain a crosstalk MTF component. We then cascade this crosstalk MTF component with other system MTF contributions such as footprint and sampling MTFs.

Figure 2.25 Modeling the inter-pixel crosstalk response. Only the central sensor is illuminated using a short-wavelength source.

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Figure 2.26 (left) Image of the charge injection pattern on the FPA. (right) Image of a magnified view of the signal read out from nearby pixels (adapted from Ref 15).

In Ref. 15 the channel-to-channel crosstalk was measured for an IR FPA by directly injecting charge into a single channel and measuring the signal at all other channels. Figure 2.26 shows examples of the resulting impulse response function, which are of significant extent in the horizontal direction. Charge-transfer inefficiency,16 seen in charge-coupled devices (CCDs), is caused by incomplete transfer of charge packets along the CCD delay line. A smearing of the image occurs that is spatial-frequency dependent. The image artifact seen is analogous to a motion blur in the along-transfer direction. For n charge transfers, ε fractional charge left behind at each transfer, and Dx pixel spacing, the crosstalk MTF from charge-transfer inefficiency (Fig. 2.27) is given by MTFðjÞ ¼ enε½1cosð2pjDxÞ

for 0 # j #

1 : 2Dx

Figure 2.27 Crosstalk MTF caused by charge-transfer inefficiency.

(2.17)

MTF in Electro-optical Systems

61

Charge-carrier diffusion also leads to crosstalk effects.17 The absorption of photons in a semiconductor material is wavelength dependent and is high for short-wavelength photons and decreases for longer-wavelength photons as the band-gap energy of the material is approached. With less absorption, long-wave photons penetrate deeper into the material and their photogenerated charges must travel farther to be collected. The longer propagation path leads to more charge-carrier diffusion and hence more charge-packet spreading, and ultimately poorer MTF for long-wavelength illumination. Figure 2.28 shows a family of MTF curves parameterized on wavelength for a Si-based FPA; the decrease in MTF we see for longer wavelengths is caused mainly by charge-carrier diffusion. We see another example of charge-carrier diffusion MTF18 in signalprocessing-in-the-element (SPRITE) detectors, where an optical image is scanned along a long horizontal semiconductor detector to increase the dwell time and hence the signal-to-noise ratio of the detected signal. The increased dwell time allows additional charge-carrier diffusion, which reduces the MTF, as seen in Fig. 2.29.

2.4 Electronic-Network MTF Electronic networks are essential to electro-optical imagers. They are present in data-acquisition (i.e., frame grabbers),19 signal-processing, and display subsystems, and establish a baseline noise level. To cast the electronics transfer function as an MTF and to cascade it with the MTFs for other subsystems, we must convert temporal frequency [Hz] into spatial frequency. As seen in Fig. 2.20, these frequencies are related by a quantity having units of

Figure 2.28 Variation of carrier-diffusion MTF with illumination wavelength for a Si FPA.

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Chapter 2

Fig. 2.29 MTF vs integration length for a SPRITE detector (adapted from Ref. 18).

an effective scan velocity vscan in either image-plane spatial frequency or object-space angular spatial frequency: f ½Hz ¼ vscan, image-plane ½mm∕s
j ½cy∕mm

(2.18)

f ½Hz ¼ vscan, angular ½mrad∕s
j ½cy∕mrad:

(2.19)

or

We can easily visualize the meaning of scan velocity for a scanned-sensor system such as that seen in Fig. 2.1 because the IFOV is actually moving across the object plane. It is not as easy to visualize scan velocity for a staring system like that seen in Fig. 2.2 because there is no motion of the IFOV. However, we can calculate a quantity having units of scan velocity if we know the field of view and the frame rate. In practice, it is not necessary to explicitly calculate the scan velocity to convert from temporal to spatial frequencies. We can determine the multiplicative factor experimentally using an electronic spectrum analyzer as seen in Fig. 2.30. First, we set up a bar target of known fundamental frequency that will create an image-plane spatial frequency which we can calculate (knowing the optical magnification) or which we can measure directly from the output signal (knowing the pixel-to-pixel spacing of the detector array). Taking the output video signal from the detector array into the spectrum analyzer will give us a readout of the electrical frequency corresponding to the fundamental image-plane spatial frequency of the bar target. The transfer function of an electronic network can be tailored in ways that the transfer function of an optical system cannot. We can implement a boost filter that preferentially amplifies a particular band of frequencies, which can help to compensate for losses in modulation depth incurred in the optical

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63

Figure 2.30 Experimental determination of temporal-to-spatial frequency correspondence.

portion of the system. The usefulness of a boost filter is limited by the effects of electronics noise. At any given frequency, an ideal boost filter would amplify signal and noise equally, but in practice, a boost filter increases the electrical noise-equivalent bandwidth and hence decreases the image signal-tonoise ratio (SNR).20 Both a high MTF and a high SNR are desirable, so in the design of a boost filter we need to decide how much gain to use and what frequencies we want to emphasize. An image-quality criterion that we can use to quantify this tradeoff is the MTF area (MTFA), which has been validated by field trials to correlate well with image detectability.21 MTFA is the area between the MTF curve and the noise-equivalent modulation (NEM) curve. The NEM characterizes the electronics noise in terms of modulation depth, being defined as the amount of modulation depth needed to yield an SNR of unity. The ratio of MTF to NEM at any spatial frequency can be interpreted as an SNR. Because the electronics noise is frequency dependent, the NEM is usually a function of spatial frequency. A convenient representation is to plot the MTF and the NEM on the same graph, as seen in Fig. 2.31. The limiting resolution is the spatial frequency where the curves cross.

Figure 2.31 Relationship of MTF, NEM, and MTFA.

64

Chapter 2

The power spectral density (PSD) is a common way to describe the frequency content of the electronics noise. The PSD is expressed in units of W/Hz. Because the PSD is in terms of power and the NEM, being in modulation-depth units, is proportional to the voltage, we can relate NEM and PSD by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NEMðjÞ ¼ CðjÞ PSDðjÞ, (2.20) where the frequency-dependent proportionality factor C(j) accounts for the display and observer. Taking MTFA as the quantity to be maximized, we can write the MTFA before application of a boost filter: Zj2 MTFAbefore-boost ¼

fMTFðjÞ  NEMðjÞg dj:

(2.21)

j1

An ideal boost filter with gain function B(j) amplifies signal and noise equally at any frequency, so after the boost, the MTFA becomes Zj2 MTFAafter-boost ¼

BðjÞfMTFðjÞ  NEMðjÞg dj:

(2.22)

j1

From Eq. (2.22) we see that MTFA is increased, enhancing image detectability, if the range of application of the boost is restricted to those frequencies for which the system MTF is greater than the NEM. This confirms that we cannot amplify noisy imagery and obtain a useful result. The MTFA actually achieved will be somewhat lower than Eq. (2.22) indicates. To keep the increase in noise-equivalent bandwidth (and the consequent degradation of SNR) to a minimum, the boost should be implemented at only those frequencies most important to the imaging task, and the magnitude of the boost should be limited to avoid oscillation artifacts at abrupt light-todark transitions in the image.22

2.5 Conclusion We can apply a transfer-function analysis that was originally developed for classical optical systems to electro-optical systems by generalizing the assumptions of linearity and shift invariance. Linearity is not strictly valid for systems that have an additive noise level because image waveforms must be of sufficient irradiance to overcome the noise before they can be considered to add linearly. The definition of NEM allows us to consider a spatialfrequency-dependent signal-to-noise ratio rather than simply a transfer function. Shift invariance is not valid for sampled-data systems; however, to

MTF in Electro-optical Systems

preserve the convenience of a transfer-function analysis, average response of the system to an ensemble of waveforms, each with a random position with respect sampling sites. With the above-mentioned modifications, transfer-function approach to a wider range of situations.

65

we consider the image-irradiance to the array of we can apply a

References 1. J. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons, New York (1978). 2. G. D. Boreman and A. Plogstedt, “Spatial filtering by a nonrectangular detector,” Appl. Opt. 28(6), 1165–1168 (1989). 3. K. J. Barnard and G. D. Boreman, “Modulation transfer function of hexagonal staring focal plane arrays,” Opt. Eng. 30(12), 1915–1919 (1991) [doi: 10.1117/12.56012]. 4. J. E. Greivenkamp, “Color-dependent optical prefilter for suppression of aliasing artifacts,” Appl. Opt. 29(5), 676–684 (1990). 5. S. K. Park, R. Schowengerdt, and M.-A. Kaczynski, “Modulationtransfer-function analysis for sampled image systems,” Appl. Opt. 23(15), 2572–2582 (1984). 6. A. Daniels, G. D. Boreman, A. Ducharme, and E. Sapir, “Random transparency targets for modulation transfer function measurement in the visible and IR,” Opt. Eng. 34(3), 860–868 (1995) [doi: 10.1117/12.190433]. 7. K. J. Barnard, E. A. Watson, and P. F. McManamon, “Nonmechanical microscanning using optical space-fed phased arrays,” Opt. Eng. 33(9), 3063–3071 (1994) [doi: 10.1117/12.178261]. 8. K. J. Barnard and E. A. Watson, “Effects of image noise on submicroscan interpolation,” Opt. Eng. 34(11), pp. 3165–3173 (1995) [doi: 10.1117/12/213572]. 9. E. A. Watson, R. A. Muse, and F. P. Blommel, “Aliasing and blurring in microscanned imagery,” Proc. SPIE 1689, pp. 242–250 (1992) [doi: 10. 1117/12.137955]. 10. J. D. Fanning and J. P. Reynolds, “Target identification performance of superresolution versus dither,” Proc. SPIE 6941, 69410N (2008) [doi: 10. 1117/12.782274]. 11. L. Huang and U. L. Osterberg, “Measurement of cross talk in orderpacked image-fiber bundles,” Proc. SPIE 2536, pp. 480–488 (1995) [doi: 10.1117/12.218456]. 12. A. Komiyama and M. Hashimoto, “Crosstalk and mode coupling between cores of image fibers,” Electron. Lett. 25(16), 1101–1103 (1989). 13. O. Hadar, D. Dogariu, and G. D. Boreman, “Angular dependence of sampling modulation transfer function,” Appl. Opt. 36(28), 7210–7216 (1997).

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14. O. Hadar and G. D. Boreman, “Oversampling requirements for pixelated-imager systems,” Opt. Eng. 38(5), 782–785 (1999) [doi: 10. 1117/1.602044]. 15. A. Waczynski, R. Barbier, and S. Cagiano, et al., “Performance overview of the Euclid infrared focal plane detector subsystems,” Proc. SPIE 9915, 991511 (2016) [doi: 10.1117/12.2231641]. 16. D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE 63(1), 38–67 (1975). 17. E. G. Stevens, “A unified model of carrier diffusion and sampling aperture effects on MTF in solid-state image sensors,” IEEE Trans. Electron Devices 39(11), 2621–2623 (1992). 18. G. D. Boreman and A. E. Plogstedt, “Modulation transfer function and number of equivalent elements for SPRITE detectors,” Appl. Opt. 27(20), 4331–4335 (1988). 19. H. A. Beyer, “Determination of radiometric and geometric characteristics of frame grabbers,” Proc. SPIE 2067, pp. 93–103 (1993) [doi: 10.1117/12. 162117]. 20. P. Fredin and G. D. Boreman, “Resolution-equivalent D* for SPRITE detectors,” Appl. Opt. 34(31), 7179–7182 (1995). 21. J. Leachtenauer and R. Driggers, Surveillance and Reconnaissance Imaging Systems, Artech House, Boston, pp. 191–193 (2001). 22. P. Fredin, “Optimum choice of anamorphic ratio and boost filter parameters for a SPRITE based infrared sensor,” Proc. SPIE 1488, pp. 432–442 (1991) [doi: 10.1117/12.45824].

Chapter 3

Point-, Line-, and Edge-Spread Function Measurement of MTF There are several ways we can measure MTF using targets that have an impulsive nature, each with positive aspects as well as drawbacks. In this chapter, we first develop the mathematical relationships between the data and the MTF for the point-spread function (PSF), line-spread function (LSF), and edge-spread function (ESF). One item of notation in this section is that we use * to denote a one-dimensional convolution, and ** to denote a twodimensional convolution. We then compare the measurement techniques and consider options for increasing the signal-to-noise ratio and extending the spatial-frequency range of the measurements.

3.1 Point-Spread Function (PSF) In the idealized arrangement of Fig. 3.1, we use a point source, represented mathematically by a two-dimensional delta function, as the object: f ðx, yÞ ¼ dðx, yÞ:

(3.1)

We initially assume that the image receiver is continuously sampled; that is, we do not need to consider the finite size of pixels nor the finite distance between samples. We will address these aspects of the measurementinstrument response later in this chapter. Here we assume that we can measure the image-irradiance distribution g(x,y) to the necessary spatial precision. If the object is truly a point source, the two-dimensional imageirradiance distribution g(x,y) equals the impulse response h(x,y). This is also called the point-spread function (PSF): gðx, yÞ ¼ hðx, yÞ ≡ PSFðx, yÞ:

(3.2)

The PSF can be Fourier transformed in two dimensions to yield the two-dimensional OTF. Taking the magnitude yields the MTF: 67

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Chapter 3

Figure 3.1

Point-spread-function measurement configuration.

jFF fPSFðx;yÞgj ¼ MTFðj, hÞ:

(3.3)

We can evaluate this two-dimensional transfer function along any desired profile, for example, MTF(j,0) or MTF(0,h).

3.2 Line-Spread Function (LSF) Figure 3.2 is a schematic of the measurement setup for the line-spread function (LSF), also called the line response. Instead of the point source used in Fig. 3.1, the LSF test uses a line-source object. A line source acts as a delta function in the x direction and is constant in the y direction (sufficient to overfill the measurement field of view of the lens under test): f ðx, yÞ ¼ dðxÞ 1ðyÞ:

(3.4)

The two-dimensional image irradiance distribution g(x,y) is the LSF, which is a function of one spatial variable (the same variable as that of the impulsive behavior of the line source—in this case, the x direction): gðx, yÞ ≡ LSFðxÞ:

Figure 3.2 Line-spread-function measurement configuration.

(3.5)

Point-, Line-, and Edge-Spread Function Measurement of MTF

69

Figure 3.3 The LSF is the two-dimensional convolution of the line-source object with the PSF (adapted from Ref. 1 with permission; © 1978 John Wiley & Sons).

Each point in the line source produces a PSF in the image plane. These displaced PSFs overlap in the vertical direction, and their sum forms the LSF. As seen schematically in Fig. 3.3, the LSF is the two-dimensional convolution (denoted by **) of the line-source object with the impulse response of the image-forming system: gðx, yÞ ≡ LSFðxÞ ¼ f ðx, yÞ   hðx, yÞ ¼ ½dðxÞ1ðyÞ   PSFðx, yÞ: (3.6) The y-direction convolution with a constant in Eq. (3.6) is equivalent to an integration over the y direction: Z` gðx, yÞ ¼ LSFðxÞ ¼

hðx, y0 Þdy0 ,

(3.7)

`

which verifies that the LSF is a function of x alone. It must be independent of y because the object used for the measurement is independent of y. The object, being impulsive in one direction, provides information about only one spatialfrequency component of the transfer function. We can find one profile of the MTF from the magnitude of the one-dimensional Fourier transform of the line response: jF fLSFðxÞgj ¼ MTFðj, 0Þ:

(3.8)

We can obtain other profiles of the transfer function by reorienting the line source. For instance, if we turn the line source by an in-plane angle of 90 deg, we get f ðx, yÞ ¼ 1ðxÞ dðyÞ, which yields a y-direction LSF that transforms to MTF(0,h).

(3.9)

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Figure 3.4 Comparison of the x-direction functional forms of the PSF and LSF for a diffraction-limited system. The Airy-disc radius is 1.22 l(F/#).

It is important to note that because of the summation along the constant direction in the line-source image, the LSF and PSF have different functional forms. The LSF(x) is not simply the x profile of the PSF(x,y): LSFðxÞ ≠ PSFðx, 0Þ:

(3.10)

In Fig. 3.4 we compare the PSF and LSF for a diffraction-limited system. We see that, while the PSF(x,0) has zeros in the pattern, the LSF(x) does not.

3.3 Edge-Spread Function (ESF) In Fig. 3.5 we see the configuration for the measurement of the edge-spread function. We use an illuminated knife-edge source (a step function) as the object: f ðx, yÞ ¼ stepðxÞ1ðyÞ:

(3.11)

The ESF is the convolution of the PSF with the unit-step function: gðx, yÞ ≡ ESFðxÞ ¼ PSFðx, yÞ   stepðxÞ1ðyÞ:

(3.12)

The y convolution of the PSF with a constant produces an LSF, and the x convolution with the step function produces a cumulative integration, as seen schematically in Fig. 3.6:

Point-, Line-, and Edge-Spread Function Measurement of MTF

Figure 3.5

71

Edge-spread-function measurement configuration.

Figure 3.6 The ESF is the two-dimensional convolution of the edge-source object with the PSF (adapted from Ref. 1 with permission; © 1978 John Wiley & Sons).

Zx ESFðxÞ ¼ PSFðx, yÞ   stepðxÞ1ðyÞ ¼

LSFðx0 Þdx0 :

(3.13)

`

The ESF is a cumulative, monotonically increasing function. Figure 3.7 illustrates the ESF for a diffraction-limited system.

Figure 3.7

Plot of the ESF for a diffraction-limited system. The Airy-disc radius is 1.22l(F/#).

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We can understand the ESF in terms of a superposition of LSFs.2,3 Each vertical strip in the open part of the aperture produces an LSF at its corresponding location in the image plane. These displaced LSFs overlap in the horizontal direction and sum to form the ESF. We can write this superposition as ESFðxÞ ≈

` X

LSFðx  xi Þ:

(3.14)

i¼1

In the limit of small displacements, the summation becomes an integral, consistent with Eq. (3.13). To convert ESF data to the MTF, we first take the spatial derivative of the ESF data to invert the integral: d d fESFðxÞg ¼ dx dx

Zx

LSFðx0 Þdx0 ¼ LSFðxÞ,

(3.15)

`

as seen schematically in Fig. 3.8. With the LSF in hand, the magnitude of the one-dimensional Fourier transform yields one profile of the MTF by means of      F d ESFðxÞ  ¼ MTFðj, 0Þ:   dx

(3.16)

We can obtain any one-dimensional profile of the MTF by appropriately reorienting the knife edge.

3.4 Comparison of PSF, LSF, and ESF When we compare the advantages and disadvantages of the PSF, LSF, and ESF tests, we find that the PSF test provides the entire two-dimensional OTF in one measurement. The major drawback to the PSF test is that point-source objects often provide too little flux to be conveniently detected. This is particularly true in the infrared portion of the spectrum, where we typically

Figure 3.8

Spatial derivative of ESF data produces an LSF.

Point-, Line-, and Edge-Spread Function Measurement of MTF

73

use blackbodies as the flux sources. Having sufficient flux is usually not an issue in the visible because hotter sources are typically used. The LSF method provides more image-plane flux than does the PSF test. The ESF setup provides even more flux and has the added advantage that a knife edge avoids slit-width issues. However, the ESF method requires a spatial-derivative operation, which accentuates noise in the data. If we reduce noise by convolution with a spatial kernel, the data-smoothing operation itself has an MTF contribution. In any wavelength region, we can use a laser source to illuminate the pinhole. The spatial coherence properties of the laser do not complicate the interpretation of PSF data if the pinhole is small enough to act as a point source (by definition, spatially coherent regardless of the coherence of the illumination source). An illuminated pinhole acts as a point source if it is smaller than both the central lobe of the PSF of the system that illuminates the pinhole and the central lobe of the PSF of the system under test, geometrically projected (with appropriate magnification) to the source plane. Even with a laser-illuminated pinhole, the PSF measurement yields an incoherent MTF because the irradiance of the PSF is measured rather than the electric field. For sources of extended spatial dimension, such as those for LSF and ESF tests, we must ensure that the coherence properties of the illumination do not introduce interference-fringe artifacts into the data.

3.5 Increasing SNR in PSF, LSF, and ESF Tests We can use a variety of image-averaging techniques to increase the signal-tonoise ratio (SNR) in PSF, LSF, and ESF tests. These averaging techniques can be implemented in either the object plane or the image plane, provided we take steps to ensure that the data are the same in one direction and we average over that direction. We usually configure the illumination level in the measurement apparatus such that the shot noise of the signal dominates the noise in the measurement. This means that the root-mean-square (rms) noise grows in proportion to the square root of the signal. In this case, the SNR increases in proportion to the square root of the signal level. In an averaging procedure, the signal is proportional to the number of independent samples. Therefore, we can achieve a considerable SNR advantage if we average a large number of samples (such as the rows in a typical CCD array). In certain instances (such as operation in the amplifier-noise limit or another signalindependent noise floor), the SNR can grow as fast as linearly with respect to the number of samples. 3.5.1 Object- and image-plane equivalence This technique requires that we assume that we match the object and receiver symmetry for optimum flux collection. We equate higher flux collection to

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better SNR. Note, however, that larger detectors generally exhibit more noise than smaller detectors; the rms noise is proportional to the square root of the sensor area. This dependence on area reduces the SNR gain somewhat if a larger detector is not fully illuminated. But, if the collected power (and hence the signal) is proportional to the detector area and the rms noise is proportional to the square root of the detector area, the more flux we can collect, the better our measurement SNR, even if that means we must use a larger detector. In the configurations illustrated in Figs. 3.1, 3.2, and 3.5, we assumed a continuously sampled image receiver, which is analogous to a point receiver that scans continuously. If we want to obtain PSF data, our test setup must include a point source and this type of point receiver. The only option to increase SNR if we use a PSF-test setup is to increase the source brightness or to average over many data sets. However, for an LSF measurement, we can accomplish the measurement in a number of ways, some of which yield a better SNR than others. We can use a linear source and a point receiver, such as the configuration seen in Fig. 3.2. This system will give us a better SNR than PSF measurement because we are using a larger-area source. Equivalently, as far as the LSF data set is concerned, we can use the configuration seen in Fig. 3.9: a point source and a slit detector (or a slit in front of a large-area detector). The data acquired are equivalent to the data for an LSF test because of the averaging in the vertical direction. Similar to the situation using a line source and a point receiver, this collects more flux than a PSF test and has a better SNR. However, since we are using a linear detector, we might also want to use a linear source (Fig. 3.10). This configuration still provides data for an LSF test, but now the source and the receiver have the same geometry. This arrangement collects the most flux and will provide the best SNR of any LSF test setup. A number of different configurations will work for ESF tests, and some are better than others from an SNR viewpoint. We begin with a knife-edge source and a scanned point receiver (Fig. 3.5). We can collect more flux with the configuration of Fig. 3.11, where the ESF measurement is performed with

Figure 3.9 A PSF test performed with a scanning linear detector produces data equivalent to an LSF test.

Point-, Line-, and Edge-Spread Function Measurement of MTF

75

Figure 3.10 An LSF test performed with a linear detector produces a better SNR than when performed using a point receiver.

Figure 3.11 ESF test setup using a point source and a scanning knife edge with a largearea detector.

a point source and where a knife edge in front of a large detector serves as the image receiver. We will obtain a better SNR (and the same ESF data) using the setup illustrated in Fig. 3.12, which involves a line source and a scanning knife edge in front of a large-area detector. We can also use a knife-edge source and a scanning linear receiver (Fig 3.13) or a knife-edge source and a scanning knife edge in front of a large-area detector (Fig. 3.14) because the data set is constant in the vertical direction. The measurement configuration

Figure 3.12 ESF test configuration using a slit source and a scanning knife edge with a large-area detector.

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Figure 3.13 ESF test configuration using an edge source and a scanning linear detector.

Figure 3.14 ESF test configuration using an edge source and a scanning knife edge with a large-area detector.

of Fig. 3.14 should produce the highest SNR, assuming that the detector is of an appropriate size to accommodate the image-irradiance distribution (that is, it is not significantly oversized). 3.5.2 Averaging in pixelated detector arrays The acquisition of PSF, LSF, and ESF data with a detector array has MTF implications in terms of pixel size and spatial sampling, as we will see later in this chapter. But the use of a detector array facilitates data processing that can increase the SNR by averaging the signal over the row or column directions. Beginning with the PSF-test configuration seen in Fig. 3.15, we can sum PSF data along the y direction, which yields an LSF measurement in the x direction: LSFðxi Þ ¼

M X

PSFðxi , yj Þ:

(3.17)

j¼1

Summing the PSF data along the y direction and accumulating them along the x direction yields an ESF measurement in the x direction:

Point-, Line-, and Edge-Spread Function Measurement of MTF

77

Figure 3.15 PSF test configuration using a two-dimensional detector array can be used to produce PSF, LSF, and ESF data.

ESFðxi0 Þ ¼

M X i0 X

PSFðxi , yj Þ :

(3.18)

j¼1 i¼1

Because of signal averaging, the LSF and ESF test data will have a better SNR than the original PSF test. Similarly, using a line source oriented along the y direction, summing (or averaging) the LSF data along y yields an LSF measurement with better SNR. Accumulating the LSF data along x yields an ESF measurement. If we sum (or average) the ESF data along y, we obtain an ESF measurement with better SNR. However, when using the signal-averaging techniques just described, we must be sure to subtract any background-signal level in the data. Expressions such as Eqs. (3.17) and (3.18) assume that the detector data are just the imageplane flux. Residual dark-background signal at each pixel, even if low-level, can become significant if many pixels are added. Another item we must consider is whether or not to window the data before a summation is performed. Often, we only use data from the central region of the image, where the flux level is highest and the imaging optic has the best performance. If the lens under test has field-dependent aberrations, it is particularly important that we use data from the region of the sensor array that contains the data from the FOV of interest. Also, if the signal-processing procedure involves a summation of data over columns, we must ensure that each column has the same data, i.e., there is no unintended in-plane angular tilt of a slit or edge source with respect to columns. In taking a summation, spatial broadening of the measured response will occur if the slit or edge is not precisely parallel to the columnar structure. If there is a tilt, two (or more) adjacent columns can receive significant portions of the signal irradiance. If the tilt is accounted for, and the data from successive rows are interlaced with the appropriate spatial offset, then the data do not suffer from unintended broadening. We will take up that issue later in this chapter.

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3.6 Correcting for Finite Source Size Source size is an issue for PSF and LSF measurements. We must have a finite source size to have a measurable amount of flux. For a one-dimensional analysis, recall Eq. (1.1), where f, g, and h are the object irradiance distribution, image irradiance distribution, and impulse-response irradiance distribution, respectively: gðxÞ ¼ f ðxÞ  hðxÞ,

(3.19)

and Eq. (1.7), where F, G, and H are the object spectrum, the image spectrum, and the transfer function, respectively: GðjÞ ¼ F ðjÞ HðjÞ:

(3.20)

If the input object f(x) ¼ d(x), then the image g(x) is directly the PSF h(x): gðxÞ ¼ dðxÞ  hðxÞ ¼ hðxÞ:

(3.21)

For a delta-function object, the object spectrum F(j) ¼ 1, a constant in the frequency domain, and the image spectrum G(j) is directly the transfer function H(j): GðjÞ ¼ F ðjÞHðjÞ ¼ HðjÞ:

(3.22)

Use of a non-delta-function source f(x) effectively bandlimits the input object spectrum. A narrow source f(x) implies a wide object spectrum F(j), and a wider source implies a narrower spectrum. The object spectrum F(j) will fall off at high frequencies rather than remain constant. Usually, a onedimensional rect function is convenient to describe the source width, and the corresponding sinc function is convenient to describe the object spectrum: f ðxÞ ¼ rectðx∕wÞ

(3.23)

F ðjÞ ¼ sinðpjwÞ∕ðpjwÞ:

(3.24)

and

In the case of a non-delta-function source, we need to divide the measured image spectrum by the object spectrum to solve Eq. (3.20) for H(j):

Point-, Line-, and Edge-Spread Function Measurement of MTF

HðjÞ ¼ ½GðjÞ ∕ ½F ðjÞ,

79

(3.25)

which is equivalent to a deconvolution of the source width from the imageirradiance data. Obtaining the transfer function by the division of Eq. (3.25) would be straightforward if not for the effects of noise. We cannot measure the image spectrum G(j) directly; the measured spectrum is the image spectrum added to the noise spectrum: Gmeas ðjÞ ¼ GðjÞ þ NðjÞ,

(3.26)

where the noise spectrum N(j) is defined as the square root of the power spectral density (PSD) of the electronics noise: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PSDðjÞ:

(3.27)

Gmeas ðjÞ GðjÞ þ NðjÞ NðjÞ ¼ ¼ HðjÞ þ , F ðjÞ F ðjÞ F ðjÞ

(3.28)

NðjÞ ¼ The division of Eq. (3.25) becomes HðjÞ ¼

which yields valid results at spatial frequencies for which F ðjÞ .. NðjÞ,

(3.29)

such that the last term in Eq. (3.28) is negligible. For frequencies where the input spectrum is near zero, the deconvolution operation divides the finite noise spectrum by a very small number. In this frequency range, the calculated MTF will exhibit significant noise artifacts and will increase with frequency as seen in Fig. 3.16. The MTF data are obviously not valid for these frequencies. To extend the frequency range of the test as far as possible, we want a wide source spectrum, which requires us to use a narrow input source. There is a practical tradeoff here because a smaller source gives less flux and a poorer signal-to-noise ratio. We want to use a source that is sufficiently narrow that the source spectrum has appreciable magnitude at the upper end of the spatial-frequency band of interest. If we obtain a poor signal-to-noise ratio with a small source, then we must either use a brighter source or employ signal-averaging techniques to overcome the electronics noise.

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Figure 3.16 Once the source spectrum has fallen to the level of the noise spectrum, MTFtest results are invalid because of division-by-zero artifacts.

3.7 Correcting for Image-Receiver MTF In any MTF test, we need an image receiver to acquire the image-irradiance function g(x,y). We would like to perform this image acquisition on a continuous basis, with infinitely small pixels and infinitely small spatialsampling intervals, which were the original conditions for the development of the PSF, LSF, and ESF (seen in Figs. 3.1, 3.2, and 3.5). However, from a practical viewpoint, there is always a finite-sized averaging area for an imageplane sensor and a finite distance between data samples. The finite dimensions of the image receiver also contribute to the MTF of the test instrument according to Eq. (1.9), and those contributions must be divided out from the measured data in a similar manner to Eq. (3.25) for us to obtain the MTF of the unit under test. 3.7.1 Finite pixel width The detector used to acquire the image can be staring or scanning, but the finite dimension of its photosensitive area results in a convolution of the image irradiance g(x) with the pixel-footprint impulse response hfootprint(x). The MTF component of finite-footprint pixels is the same as was discussed previously in Section 2.1. For most measurement situations, we use a one-dimensional rect function of width w to describe the pixel footprint, leading to the MTF seen in Eq. (2.2):        sinðpjwÞ  : MTFfootprint ðjÞ ¼sincðjwÞ ¼ pjw 

(3.30)

Point-, Line-, and Edge-Spread Function Measurement of MTF

81

This term is one component of the instrumental MTF that should be divided out from the measured MTF to obtain the MTF of the unit under test. 3.7.2 Finite sampling interval For a staring-array image receiver, the sampling interval is the center-tocenter pixel spacing of the array. For a scanning sensor, the sampling interval is the spatial distance at which successive samples are taken. We can determine this sampling distance if we know the sampling-time interval used to sample the analog video waveform and convert from time to distance units, as discussed in Chapter 2. For manually scanned sensors such as a photomultiplier tube on a micropositioner, the sampling interval is simply the distance we move the sensor between samples. We can apply Eq. (2.13) to determine the sampling MTF, which again represents an instrumental-MTF component that should be divided out from the final MTF results. However, note that sampling-MTF effects are not always present in the data, so we do not always need to divide them out. Measurement procedures for PSF, LSF, and ESF tests typically involve alignment of the image with respect to sampling sites. We tweak the alignment to produce the best-looking image, typically observing the data on a TV monitor or an oscilloscope as we perform the alignment. In this case, a sampling MTF does not filter the image data because the image irradiance is not randomly aligned with the sampling sites. Thus, any correction for a sampling MTF using the approach of Eq. (3.25) will unrealistically bias the MTF measurement, giving artificially high values. The targets used for PSF, LSF, and ESF contain very high frequencies; therefore, even with fine alignment of the target with respect to the sampling sites, MTF results are typically only accurate to j  1/(4Dxsamp)—half the Nyquist frequency—because of aliasing artifacts. If we want to measure MTF at higher frequencies, a good option is to use an oversampled knife-edge test.

3.8 Oversampled Knife-Edge Test This test requires the introduction of a slight tilt to the knife edge. The data sets from each line are interlaced with an appropriate subpixel shift, corresponding to the position of the knife edge with respect to the column. This creates a measurement data set with a very fine sampling interval, essentially setting the sampling MTF equal to unity, and pushing jNyquist to very high spatial frequencies. There are some subtle points involved in the implementation of this measurement.4 First, with the data set in hand, we estimate the position of the knife edge for each horizontal row of data. Next, we use a least-squares fitting procedure to fit a line through the individual edge-location estimates. We then use this fitted line as a more accurate estimate of the edge location for each

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Figure 3.17 (left) Registering scans with a tilted knife edge. (a) Sampling grid with the knife edge skewed from perpendicular. (b) Knife-edge shift in successive scans. (c) Combined scan with reregistered edges. (right) Tilted-knife-edge scan. (Left diagram reprinted from Ref. 4.)

row, allowing the proper position registration of the individual ESF data sets. The superposition of the data sets will thus have a very fine sampling (Fig. 3.17). The method assumes that the ESF of the system under test is constant over the measurement region, so the measurement FOV in the alongedge direction should be small enough that this condition is satisfied. Because computation of MTF will require a spatial derivative to be performed on the ESF data (Fig. 3.8), we may want to smooth and resample the high-resolution ESF data set with a moving-window average to reduce noise. A nearest-neighbor smoothing is usually sufficient and will have a minimal MTF impact. Another technique that we can use to reduce the noise before taking the derivative is to fit the high-resolution ESF data set to a suitable functional form such as a cumulative Gaussian, a sigmoid function, or a low-order polynomial. We can then take the derivative on the functional form rather than on the data itself. The other fine point in the design of the measurement involves deciding what tilt to use. To ensure a uniform distribution of positions of the edge with respect to the columnar structure, we should use at least a one-pixel difference

Point-, Line-, and Edge-Spread Function Measurement of MTF

83

Figure 3.18 A knife-edge tilt that is too large can produce a small number of redundant data-registration positions.

between the top and bottom rows of the scan data. Such fine control of the angular position of the knife edge is not required, and we can use a somewhat more pronounced tilt. Reference 4 suggests one pixel of horizontal position difference of the knife edge over 64 scan lines. This criterion should produce a good oversampling, and the overlap of data-point positions over an entire data set of more than 64 rows should be helpful in reducing noise. We should avoid significantly more pronounced tilts because we want a uniform distribution of positions of the edge with respect to the columnar structure. Consider the extreme example of a one-pixel horizontal offset over four scan lines (14-deg tilt), as shown in Fig. 3.18. If the edge were centered on one pixel and on its adjacent neighbor four rows down, this situation would result in only four edge positions in the data set, rather than the desired uniform distribution. Thus, we want to avoid this possibility entirely by using a sufficiently small tilt that any such repetition of data would be spaced by many rows. We should avoid pronounced tilts for another reason—because we want to measure MTF in a particular direction (which we assume to be horizontal in this discussion), and using a slight tilt provides the closest approximation to that measurement, in the context of the tilted knife-edge test.

3.9 Conclusion There are several ways we can measure MTF using targets that have an impulsive nature, each with positive aspects as well as drawbacks. For point sources and slit sources, we need to consider the dimensions of the source object. Edge sources are a versatile method that can measure MTF past the Nyquist frequency of a sensor array. We can employ a variety of averaging methods to increase signal-to-noise ratio, with appropriate caveats to ensure high-fidelity data sets.

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References 1. J. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons, New York (1978). 2. B. Tatian, “Method for obtaining the transfer function from the edge response function,” JOSA 55(8), 1014–1019 (1965). 3. R. Barakat, “Determination of the optical transfer function directly from the edge spread function,” JOSA 55(10), 1217–1221 (1965). 4. S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices;,” Opt. Eng. 30(2), pp. 170–177 (1991) [doi: 10.1117/12/55783].

Chapter 4

Square-Wave and Bar-Target Measurement of MTF Although MTF is defined in terms of the response of a system to sinusoids of irradiance, we commonly use binary targets in practice because sinusoidal targets require analog gray-level transmittance or reflectance. Fabricating targets with analog transmittance or reflectance usually involves photographic or lithographic processes with spatial resolutions much smaller than the period of the sinusoid so that we can achieve an area-averaged reflectance or transmittance. Sinusoidal targets used in MTF testing should have minimal harmonic distortion so that they present a single spatial frequency to the system under test. This is difficult to achieve in the fabrication processes. Conversely, binary targets of either 1 or 0 transmittance or reflectance are relatively easy to fabricate. We can fabricate binary targets for low spatial frequencies by machining processes. For targets of higher spatial frequency, we can use optical-lithography processes of modest resolution because the metallic films required to produce the binary patterns are continuous on a micro scale. In this chapter, we will first consider square-wave targets and then three-bar and four-bar targets. Square-wave targets not only consist of the fundamental spatial frequency, but also contain higher harmonic terms. Bar targets contain both higher and lower harmonics of the fundamental. Because of these harmonics, we must correct modulation-depth measurements made with binary targets to produce MTF data, either with a series approach or with digital filtering.

4.1 Square-Wave Targets Targets that have an infinite number of square-wave cycles are simple to analyze mathematically. Figure 4.1 shows a portion of an infinite square wave (equal lines and spaces) and its spectrum. The spectrum consists of a series of delta functions at dc, the fundamental frequency (the inverse of bar spacing), and third, fifth, and higher odd harmonics. The one-sided amplitude of the

85

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Figure 4.1

Square-wave target and its spectrum.

harmonics is 2/(pn), which decreases with order number n. For this case, the amplitude at dc is 0.5, which is the average value of the transmittance waveform. Of course, we never use an infinite square wave in practice. However, it is quite feasible to fill the field of view of the system under test with square-wave targets, using, for instance, a set of Ronchi rulings of appropriate dimension located at the object plane. In this case, the delta-function spectrum of the infinite square wave will be convolved with a function that represents the Fourier transform of the field of view. For targets of high spatial frequency, the resultant spectral broadening is negligible, and a series representation is still accurate. For targets of lower spatial frequency, the bar spacing may be an appreciable fraction of the field of view. In those cases, we may find that the frequency-domain width of the broadened spectral line is significant compared to the harmonic spacing, and a series representation is consequently less accurate. For infinite square-wave targets, we can define a contrast transfer function (CTF) as a function of the square-wave fundamental spatial frequency: CTFðjf Þ ¼

M output ðjf Þ M input square wave ðjf Þ

:

(4.1)

The CTF is not a transfer function in the true sense because it is not defined in terms of sine waves. CTF cannot be cascaded with MTF curves without first converting CTF to MTF. The modulation depth of the input square wave is usually 1 for all targets in the set, and for an infinite-square-wave target, the maxima of irradiance Amax are all equal and the minima of irradiance Amin are all equal, allowing an unambiguous calculation of the output modulation depth Moutput as

Square-Wave and Bar-Target Measurement of MTF

M output ¼

Amax  Amin : Amax þ Amin

87

(4.2)

We can express the modulation depth and, hence, the CTF at any frequency as a two-sided summation of Fourier-series harmonic components. These components are weighted by two multiplicative factors in the summation: their relative strength in the input waveform and the MTF of the system under test at each harmonic frequency. This process yields an expression1 for CTF in terms of MTF:   4 MTFðj ¼ 3jf Þ MTFðj ¼ 5jf Þ CTFðjf Þ ¼ þ  ::: MTFðj ¼ jf Þ  p 3 5 (4.3) and inversely, for MTF in terms of CTF:   p CTFðjf ¼ 3jÞ CTFðjf ¼ 5jÞ MTFðjÞ ¼  þ : : : : (4.4) CTFðjf ¼ jÞ þ 4 3 5 As defined in Eqs. (4.3) and (4.4), both CTF and MTF are normalized to unity at zero spatial frequency. We can see from Eq. (4.4) that, if we want to use the series conversion, calculating MTF at any particular spatial frequency requires us to have CTF data at a series of frequencies that are harmonically related to the frequency of interest. Typically, the procedure to accomplish this is to measure the CTF for a sufficient number of fundamental frequencies (over a range from low frequencies up to where the CTF is negligibly small) so that we can interpolate a continuous curve between the measured values. This allows us to find the CTFs at the frequencies needed for computing an MTF curve from the CTF data. Owing to the higher harmonics present in a square wave, it is not accurate to directly take the square-wave CTF measurements as MTF measurements. A built-in bias makes the CTF higher at all frequencies than the corresponding MTF. We can see this bias from the first term in the series summation of Eq. (4.3). This first term, Eq. (4.5), is a high-frequency approximation to the series because, for high fundamental spatial frequencies, the MTF at the harmonic frequencies is low: CTFðjf Þ  ð4∕pÞ MTFðjf Þ:

(4.5)

For lower fundamental frequencies, we must include more harmonic terms for an accurate representation because the MTF at the harmonic frequencies is higher, but the CTF always exceeds the MTF. We compare plots of CTF and MTF in Fig. 4.2 for the case of a diffraction-limited circular aperture. We calculated the CTF values directly

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Figure 4.2 Comparison of MTF, CTF, three-bar IMD, and four-bar IMD for a diffractionlimited circular-aperture system (reprinted from Ref. 2). [IMD is image modulation depth, as defined in Eq. (4.6) in the next section.]

from the series in Eq. (4.3). The pronounced hump in the CTF near j/jcutoff  0.3 arises from the 4/p multiplier and from the fact that that the third harmonic, which adds into the series with a negative sign, is nearing the cutoff frequency and therefore is not contributing negatively to the CTF. The small oscillations in the CTF at lower frequencies arise from the fifth harmonic (which adds with a positive sign) nearing cutoff and the seventh harmonic (which adds with a negative sign) nearing cutoff.

4.2 Bar Targets A very common target type for measuring MTF is the three-bar or four-bar binary transmission pattern, as seen in Fig. 4.3. Any particular bar target is specified in terms of its fundamental frequency jf, which is the inverse of the center-to-center spacing of the bars. Typically, the widths of the lines and the spaces are equal. Using a target set consisting of a number of different sizes of this type of target, we measure the modulation depth M of the non-sinusoidal image waveform as a function of jf, using maximum and minimum values of imageplane irradiance for each target to yield the image modulation depth (IMD): IMDðjf Þ ¼

M output ðjf Þ : M input bar target ðjf Þ

(4.6)

This IMD does not equal the MTF at jf because of the extra frequency components at both higher and lower frequencies3 than jf. These components contribute to the IMD, biasing the measurement toward higher modulation values than would be measured with a sinusoidal input. In practice, we

Square-Wave and Bar-Target Measurement of MTF

Figure 4.3

89

Binary (left) three-bar and (right) four-bar transmission targets.

remove these non-fundamental-frequency components by computation, or we filter them out electronically or digitally to yield MTF data. We determine the modulation depth of the output image from the maximum and minimum values of the irradiance waveform of the three- or four-bar target. Often the effects of aberrations, shading, or aliasing produce maxima and minima that are not equal for each bar in the output image, as seen in Fig. 4.4. We calculate the modulation depth using the highest peak (maximum) of one or more of the bars and the lowest inter-bar minimum to occur in any particular image. As a practical note, if we are measuring the modulation depth of a bar target and cannot adjust the fine positional alignment of the target so that all four bars have the same height at the output, then we are probably trying to measure MTF at too high a frequency. The measurement accuracy of the

Figure 4.4

Measurement of three-bar IMD from unequal bar data (adapted from Ref. 4).

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modulation depth will be reduced. Bar targets contain a range of frequencies, and even if the fundamental is not aliased, some of the higher frequencies will be. We saw this bar-target aliasing effect in Fig. 2.9. Because their Fourier transforms are continuous functions of frequency rather than discrete-harmonic Fourier series, the series representations of Eqs. (4.3) and (4.4) are not strictly valid for three-bar and four-bar targets. To explore the validity of the series conversion, we compare2 four different curves (the MTF, CTF, three-bar IMD, and four-bar IMD) for four optical systems of interest: a diffraction-limited circular-aperture system, a system with Gaussian MTF, a system with exponential MTF, and a diffraction-limited annular-aperture system with a 50% diameter obscuration. If the IMD curves are close to the CTF curve, then we can use the series to convert bar-target data to MTF for systems with that type of MTF dependence. To produce the three-bar and four-bar IMD curves shown in Figs. 4.2, 4.5, 4.6, and 4.7, we calculated spectra for 120 bar targets of various fundamental frequencies, which were then filtered by each of the MTF curves. We inverse transformed the resulting filtered spectra and calculated the IMDs from the resulting image data according to Eq. (4.6). We plotted these IMDs as a function of the fundamental frequency of the bar target. For the diffraction-limited circular-aperture system, we already considered the behavior of the CTF in Fig. 4.2. The three-bar and four-bar IMD curves we show there are very close to the CTF curve, and the series conversions are sufficiently accurate. The small difference would not be measurable in practice. The small amount of modulation depth past cutoff seen for the three-bar case is consistent with Fig. 6 of Ref. 3 because spatial frequencies just below cutoff will contribute to a residual modulation near the

Figure 4.5 Comparison of MTF, CTF, three-bar IMD, and four-bar IMD for a system with a Gaussian MTF ¼ exp{–2(j/j0)2}. The CTF and the three-bar and four-bar IMD curves are identical for this case (reprinted from Ref. 2).

Square-Wave and Bar-Target Measurement of MTF

91

Figure 4.6 Comparison of MTF, CTF, three-bar IMD, and four-bar IMD for a system with an exponential MTF ¼ exp{–2(j/j0)} (reprinted from Ref. 2).

Figure 4.7 Comparison of MTF, CTF, three-bar IMD, and four-bar IMD for a diffractionlimited, annular-aperture system with a 50% diameter obscuration (reprinted from Ref. 2).

original fundamental spatial frequency of the three-bar target, even though the MTF is zero at the fundamental frequency. In Figs. 4.5 and 4.6 we present a comparison of MTF, CTF, three-bar, and four-bar IMD curves for the cases of a Gaussian and an exponential MTF, respectively. For the Gaussian MTF case, the CTF and the three-bar and four-bar curves are identical. For the exponential case, the three-bar and four-bar IMDs are slightly larger than the CTF, with the three-bar IMD as the highest, similar to the behavior of the diffraction-limited system seen in Fig. 4.2. Despite the similar shape of the MTFs in the diffraction-limited and

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exponential cases, we do not observe the mid-frequency hump of Fig. 4.2 with the exponential MTF because there is no cutoff frequency at which a given harmonic ceases to contribute to the series representing the CTF. The situation is a bit different in Fig. 4.7, where we consider the case of an obscured-aperture system. The MTF curve is not as smooth as in the case of no obscuration, and the discontinuities in its derivative make the weighting of the different terms of the series a more complicated function of frequency. The CTF curve still exceeds the MTF curve. At some frequencies, the three-bar and four-bar IMD curves are nearly identical to the CTF, and at other frequencies, there is as much as a 10% difference in absolute modulation depth or a 40% relative difference between the curves. So, in some cases, a bartarget-to-MTF conversion using a series is not accurate. The examples shown indicate that, if the MTF curve is a smooth, monotonic decreasing function, the series conversion will be reasonably accurate. But in a measurement context, we do not know the MTF curve beforehand, and we want to have a procedure that is valid, in general, for conversion of bar-target data to MTF. When digitized image data are available, we can perform a direct bartarget-to-MTF conversion that makes no prior assumptions about the MTF curve being measured. For either a three-bar or four-bar target with equal bars and spaces, we know the magnitude spectrum of the input as a mathematical function, for any fundamental frequency jf:      1 j j 1 (4.7) S input, 3-bar ðjÞ ¼ sinc cos 2p þ jf 2jf jf 2       1 j j j : S input, 4-bar ðjÞ ¼ sinc cos 3p þ cos p jf 2jf jf jf

(4.8)

These spectra are plotted in Fig. 4.8. We can find the MTF at the fundamental frequency of any particular target by taking the ratio of output to input magnitude spectra:   S output ðj ¼ jf Þ MTFðj ¼ jf Þ ¼ : (4.9) S input-bar-target ðj ¼ jf Þ We take the absolute value of the Fourier transform of the digitized bar-target image data to produce the output magnitude spectrum Soutput(j). Figure 4.9 shows a measured three-bar-target magnitude spectrum and the corresponding input spectrum Sinput-bar-target(j), both normalized to unity at j ¼ 0. We perform the bar-target MTF calculation at the fundamental frequency of the particular target being used. Note that the image-plane fundamental frequency is not simply the frequency of the (j . 0) maximum of the

Square-Wave and Bar-Target Measurement of MTF

Figure 4.8

93

Normalized magnitude spectra: three-bar (top), four-bar (bottom).

measured spectrum. The measured output spectrum has been filtered by the system MTF. Because this MTF decreases with frequency, we see that the peak of the output spectrum occurs at a slightly lower frequency than the fundamental jf of the input target. The ratio of Eq. (4.9) is to be calculated at the fundamental frequency. Without knowing the MTF curve, we cannot say how much the peak was shifted. So, to determine the fundamental spatial frequency of the input target, we use the first zero of the spectrum, the location of which is not shifted by the MTF. In Eq. (4.7) describing the threebar target, the term in square brackets first goes to zero at jfirst-zero ¼ jf ∕3,

(4.10)

and in Eq. (4.8) describing the four-bar target, the term in square brackets first goes to zero at jfirst-zero ¼ jf ∕4:

(4.11)

Once we determine the fundamental frequency of the particular target being used, we can make the calculation in Eq. (4.9), with the output spectrum evaluated at jf. We repeat this process for each target used in the measurement

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Figure 4.9 Measured three-bar-target magnitude spectrum Soutput(j) (dotted curve) and the corresponding calculated input spectrum Sinput-bar-target(j) (solid curve) (reprinted from Ref. 2).

to generate the MTF curve from digitized bar-target data without the need for a series conversion. Thus, we still make the bar-target MTF measurement one frequency at a time, but now without concern about the accuracy of a series conversion because we use a digital-filtering technique to isolate jf from the continuous spectrum of the bar target. From Fig. 4.9, we can see that there is more information present than just at the fundamental frequency, and it is tempting to try to extend the range of the measurement beyond the fundamental. We found experimentally that accuracy suffered when we tried to extend the frequency range of the measurement on either side of the fundamental. The spectral information is strongly peaked, and dividing the output curve by the input curve tends to emphasize any noise present in the measured data because the input decreases so rapidly on either side of the peak. However, it may be possible to use the lower-frequency subsidiary maxima seen in Fig. 4.7 to at least get another measurement frequency for any given target, assuming a good signal-to-noise ratio. Higher-frequency data will have generally been attenuated by the MTF to a degree such that taking the ratio of Eq. (4.9) will not produce results of good quality.

4.3 Conclusion We often use bar targets in MTF measurements. It is important to realize that we must correct modulation depth measurements made with bar targets to produce MTF data, either with a series approach or with digital filtering. If the MTF curve of the system under test is relatively smooth, the agreement between CTF and bar-target data is often quite close. If the MTF curve of the

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system under test has significant slope discontinuities, we find more differences between CTF and bar-target data. In this situation, we prefer the digital-filtering approach.

References 1. J. W. Coltman, “The specification of imaging properties by response to a sine wave input,” JOSA 44(6), 468–471 (1954). 2. G. D. Boreman and S. Yang, “Modulation transfer function measurement using three- and four-bar targets,” Appl. Opt. 34(34), 8050–8052 (1995). 3. D. H. Kelly, “Spatial frequency, bandwidth, and resolution,” Appl. Opt. 4(4), 435–437 (1965). 4. I. de Kernier, A. Ali-Cherif, N. Rongeat, O. Cioni, S. Morales, J. Savatier, S. Monneret, and P. Blandin, “Large field-of-view phase and fluorescence mesoscope with microscopic resolution,” J. Biomedical Optics 24(3), 036501 (2019) [doi: 10.1117/1.JBO.24.036501].

Chapter 5

Noise-Target Measurement of MTF Measurement of a system’s transfer function by means of its response to random-noise inputs has long been a standard procedure in time-domain systems.1 If the input is white noise, which contains equal amounts of all frequencies, the action of the transfer function is to impart a nonuniformity of frequency content that can be assessed at the output of the system by means of a Fourier analysis. This concept has not historically been employed in the measurement of optical systems, with the exception of an initial demonstration2 using (non-white) film-grain noise. Noise-like targets of known spatial-frequency content are useful for MTF testing, particularly for spatially sampled systems such as detector-array image receivers. Noise targets have a random position of the image data with respect to sampling sites in the detector array and measure a shift-invariant MTF that inherently includes the sampling MTF. Noise targets measure the MTF according to PSDoutput ðj, hÞ ¼ PSDinput ðj, hÞ  ½MTFðj, hÞ2 ,

(5.1)

where PSD denotes power spectral density, defined as the ensemble average of the square of the Fourier transform of object or image data. The PSD is a measure of spatial-frequency content for random targets or random images. We calculate the output PSD from the image data. Generally, we calculate the finite-length Fourier transform of a row of image data and square the result. This is an estimate of the PSD, but because the calculation is performed on a data record of finite length, there is noise in the estimate. When we perform this operation on other rows of image data, we generate other PSD estimates. Averaging over these additional estimates gives a more accurate estimation3 of the PSD of the underlying random process. Noise targets usually measure the MTF averaged over a system’s whole field of view. However, we can calculate PSDs from various subregions of the image. If we use smaller data 97

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sets in this way, we will likely need to average over additional independent data sets to obtain PSD estimates of sufficient signal-to-noise ratio. Accuracy of an MTF measurement using noise targets depends critically on how well we know the input PSD. This is a central consideration in the design of the specific measurement apparatus to be used. The two main methods we use for generating noise targets are laser speckle and random transparencies. We generally use the laser-speckle method to measure the MTF of a detector array alone because the method relies on diffraction to deliver the random irradiance pattern to the receiver, without the need for an intervening optical system. Because a transparency must be imaged onto the detector array using optical elements, we find the random-transparency method to be more convenient for MTF measurement of a complete imager system, including both the detector array and the fore-optics. In both cases, we can generate a variety of input PSDs, depending on the specific instrumentation details.

5.1 Laser-Speckle MTF Test Laser speckle is the spatial variation of irradiance that is produced when coherent light is reflected from a rough surface. We can also generate speckle by transmitting laser light through a phase-randomizing component. We see the general geometry in Fig. 5.1, where the coordinates xs, ys denote the illuminated rough surface (the source), and xobs, yobs denote the plane in which we observe the speckle. If the distance z between the source and observation planes satisfies the Fresnel conditions,4 the PSD of the speckle irradiance consists of a delta function at zero frequency, along with an extended-frequency component proportional to the normalized autocorrelation of the irradiance distribution at the plane of the source.5 This relation allows us to tailor the frequency content of the speckle pattern by the design of the illuminated scattering aperture.

Figure 5.1

General geometry for speckle formation.

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There is a Fourier transform relationship between a power spectrum and an autocorrelation. Inverse transforming the speckle PSD produces the autocorrelation of the speckle pattern irradiance. This autocorrelation function is, except for a constant bias term, proportional to the irradiance that would be present in the diffraction pattern of the clear aperture (i.e., the square of the Fourier transform). The speckle pattern thus has a frequency content similar to the diffraction pattern of the clear aperture, since the autocorrelation width of the speckle equals the first-zero width of the diffraction pattern. We visualize this relationship in Fig. 5.2 for a single rectangular aperture of width W. In generating laser speckle for MTF measurements, we randomize the phase of the laser light either by passing it through a transmissive groundglass diffuser, or by passing it through an integrating sphere. A binarytransmittance aperture then follows, whose geometry determines the functional form of the speckle PSD. In either case, it is important to have the aperture uniformly illuminated so that the input PSD is known. A linear polarizer follows the aperture to increase the contrast of the speckle pattern by eliminating the incoherent sum of two speckle patterns of orthogonal polarization. We see the basic layout in Fig. 5.3. Although not radiometrically efficient, the integrating-sphere method produces nearly uniform irradiance at its output port,6 as long as we use

Figure 5.2 Relationship between the frequency content of a speckle pattern and a diffraction pattern, given a single aperture of width W.

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Figure 5.3 Laser-speckle setup for MTF tests, using an integrating sphere to illuminate the aperture with phase-randomized laser light.

appropriate baffling to preclude direct transmission of the laser beam. Smaller integrating spheres are more flux efficient, at the expense of irradiance uniformity at the output port. In our experimental work, we found that a 25-mm-diameter sphere was a suitable compromise. Using a transmissive ground-glass diffuser requires that the laser illumination be uniform across the full extent of the source aperture. We can accomplish this by expanding the laser beam so that the uniform region near the peak of the Gaussian is used, although at a considerable loss of power. We may increase the overall flux efficiency of the specklegeneration process using microlens diffusers7 or holographic techniques.8 For any of the laser-speckle MTF tests, we should confirm the input PSD by imaging the illuminated aperture to verify the uniformity of illumination across each aperture, and for multi-aperture configurations, to verify the uniformity between the apertures. There will be some speckle artifacts in any image of the laser-illuminated aperture. We can reduce these artifacts by averaging the image irradiance data over a small moving window, or by averaging over several aperture images with independent speckle patterns. We can conveniently generate new speckle patterns by very small position changes of the ground-glass diffuser or the integrating sphere with respect to the input laser beam. If the large-scale illumination of the aperture is uniform, the geometry of the aperture, along with the wavelength and distance, will determine the functional form of the speckle PSD. In the design of the measurement instrumentation, we choose the aperture configuration. The rectangular aperture of width W and its PSD were shown in Fig. 5.2. This aperture allows assessment of MTF over a range of spatial frequencies in one measurement.9 The upper frequency limit is W/lz. To avoid aliasing, this upper frequency limit must be less than the Nyquist frequency of the array. This aperture has the disadvantage that the input PSD is decreasing linearly with frequency. Thus, when we use the division of Eq. (5.1) to compute MTF, any noise in the measurement will limit the frequency range of validity because of division-by-zero artifacts, as discussed in Section 3.6. We show a dual-slit aperture of separation L and slit width l1 in Fig. 5.4. This aperture will project random fringes of narrowband spatial frequency

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Figure 5.4

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Dual-slit aperture and the corresponding speckle PSD.

onto the detector array.10 In a manner analogous to the formation of a twoslit diffraction pattern, the distance z from the dual-slit aperture to the detector array controls the spatial frequency content of the speckle pattern according to jpeak ¼ L∕lz,

(5.2)

where jpeak is the peak spatial frequency of the narrow high-frequency component of the speckle PSD seen in Fig. 5.4. The baseband component at the low spatial frequencies is not useful in the measurement, and its finite width limits the lowest frequency at which a measurement can be made. The spectral width of all features in the PSD scales inversely with z. Smaller slit widths will result in a narrowing of both the baseband and the high-frequency components, at the expense of less flux reaching the detector array. We can use the dual-slit-aperture arrangement to measure MTF past the Nyquist frequency of the detector array because the input PSD is largely narrowband. If we know the distance z, we know its peak frequency from Eq. (5.2). Thus, we can interpret the speckle pattern even if jpeak is aliased because of the narrowband nature of the speckle produced by the dual-slit aperture. If jpeak ¼ jNyquist + Dj, there is no spatial-frequency content at jNyquist  Dj that would complicate the computation of the strength of the peak. The solid curve in Fig. 5.5 is the speckle PSD for a distance z such that the spatial frequency jpeak was below the aliasing frequency of the sensor array. The dotted line is the speckle PSD for a shorter z, for which the spatial frequency of the fringes was above the aliasing frequency. We see the falloff in MTF between these two frequencies as a decrease in the peak height of the PSD between those two values of jpeak. We plot MTF as the peak strength of the narrowband component as a function of spatial frequency, with representative results shown in Fig. 5.6. We can reduce the noise seen in those MTF results by implementing additional PSD averaging over independent speckle patterns.

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Figure 5.5 (left) Typical narrowband laser-speckle pattern and (right) the corresponding image PSD plot in which both aliased and non-aliased PSDs are shown (reprinted from Ref. 10).

Figure 5.6 MTF results from a dual-slit speckle measurement of a detector array, showing measured data beyond the Nyquist frequency (reprinted from Ref. 10).

Implementation of the dual-slit aperture measurement requires either dual-slit apertures of different spacings L or changing the source-to-detectorarray distance z to generate the range of spatial frequencies needed. We can implement a mirror-based optical delay line to vary z without motion of the source or detector. The mirror apertures of the delay line must be sufficiently large that they do not limit the spatial extent of the source as viewed from the detector over the range of required distances. Figure 5.7 shows a two-dimensional slanted-dual-slit aperture10 that generates a constant input PSD over an extended frequency range along the j direction. An autocorrelation of the aperture in the x direction produces the PSD form as shown. Figure 5.8 shows a plot of the two-dimensional PSD and a speckle pattern resulting from this aperture.

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Figure 5.7 Ref. 10).

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(left) Slanted-dual-slit aperture and (right) its j-direction PSD (reprinted from

Figure 5.8 (left) Two-dimensional PSD plot and (right) a speckle pattern from the slanteddual-slit aperture of Fig. 5.7 (reprinted from Ref. 10).

Figure 5.9 shows another two-dimensional aperture11 that generates a constant input PSD over an extended frequency range along the j direction. An autocorrelation of the aperture in the x direction produces the PSD form as shown. Figure 5.10 shows a plot of the two-dimensional PSD and a speckle pattern resulting from this aperture. This configuration has been used for measurement of detector array MTF beyond the Nyquist frequency in the following manner. A measurement of MTF out to Nyquist was first performed using the flat region of the PSD. The aperture-to-detector distance was then decreased so that the flat region of the PSD ranged from Nyquist to twice Nyquist. A new output PSD was acquired, which aliased the highfrequency information onto the original un-aliased PSD. A PSD subtraction was performed, which yielded only the high-frequency PSD. The un-aliased PSD and the high-frequency PSD were then concatenated, and Eq. (5.1) was

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Figure 5.9 (left) 45-deg cross aperture and (right) its j-direction PSD (reprinted from Ref. 11).

Figure 5.10 (left) Two-dimensional PSD and (right) speckle pattern from the 45-deg cross aperture of Fig. 5.9 (reprinted from Ref. 11).

used to yield the MTF. Figure 5.11 shows the results of this measurement. The Nyquist frequency of the tested array was 107 cy/mm. The discontinuity in the MTF plot near Nyquist resulted from the absence of data in that vicinity because of the finite width of the baseband feature in the PSD.

5.2 Random-Transparency MTF Test We can create transparency targets with random patterns of known PSD. Unlike the laser-speckle patterns discussed in the last section, these targets must be imaged onto the detector array using optical elements and, therefore, are most naturally used for characterizing a complete camera system—optics and focal-plane array together. We can use these with broadband flux such as from a blackbody or lamp source. Uniformity of the backlighting source is important for the accuracy of the measurement, with integrating spheres or

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Figure 5.11 Speckle MTF measurement to twice the Nyquist frequency of a detector array, using the aperture of Fig. 5.9. jNyquist ~107 cy/mm (adapted from Ref. 11).

large-area blackbodies having suitable characteristics. We can fabricate the random patterns on different types of substrates.12 For the visible and the 3- to 5-mm MWIR band, we use either a laser-printing or photographic process on plastic film. For the MWIR and the 8- to 12-mm LWIR band, we use optical or electron-beam lithography involving a metal film on an IR-transparent substrate. Because it is difficult to obtain gray-level transmittance in the IR, our approach is to use an array of nominally square apertures of various sizes in a metallic film to achieve the desired transmittances. To avoid diffraction-induced nonlinearity in transmittance as a function of aperture area, the minimum aperture dimension should be approximately five times larger than the wavelength to be used.13 Figure 5.12 presents the usual setup for projecting a random-transparency object scene into a system under test, which consists of both the imaging optics and a sensor array. We position a transparency at the focus of a collimator and backlight it using a uniform source. The MTF of the collimator should be such that it does not limit the frequency range of the measurement; alternatively, this MTF should be known and corrected for in the data processing. We begin our discussion by considering the generation of a random pattern with a white-noise PSD up to a predetermined cutoff frequency. This cutoff frequency on the transparency should correspond to the Nyquist frequency of the system under test, considering the geometrical magnification (m ¼ fsensor/fcollimator) of the two-lens relay in Fig. 5.12. Given that the pixel spacings in the object and image planes are related by Dximg ¼ Dxobj  m, we find that jcutoff ¼ jNyquist

f sensor f collimator

:

(5.3)

We can generate data for the transparency as follows. We create a set of N random numbers taken from a range of 256 gray-level values, where N is the

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Figure 5.12 Setup for the random-transparency MTF test (adapted from Ref. 12).

number of pixels in a row of the detector array. We repeat this for the M rows of the detector array so that we have an N  M matrix of uncorrelated random numbers. If we render these data values as square contiguous pixels of transmittance values on the substrate at the desired spacing Dxobj, we will have spatial white noise bandlimited to the desired spatial frequency. We see an example of this type of pattern in Fig. 5.13, along with a PSD computed from the N random numbers for one line of data. Because the image data is random, there are fluctuations in the PSD. This noise would average out if we used more lines of data to compute the PSD, but the single-line PSD estimate shown demonstrates the white-noise characteristic of the transparency. In situations where the required spatial resolution of the pattern is well within the capabilities of the system that generates the transparency, it may be acceptable to confirm the input PSD in this manner directly from the pattern

Figure 5.13 (left) Bandlimited white-noise pattern and (right) a sample of the PSD computed from the pattern (reprinted from Ref. 12).

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data. For other situations, we recommend experimental confirmation of the input PSD by acquiring a high-resolution image of a portion of the pattern and computing the PSD from that image. We illustrate this procedure using the example of an infrared transparency fabricated by electron-beam lithography, where the transmittance of each pixel was determined by the fractional open area not covered by a metal film. Figure 5.14 shows a small region of the original design data, a small region of the as-fabricated transparency on ZnSe, and the input PSD calculated from the image of the asfabricated transparency. We see a falloff in the PSD arising from imperfections in the fabrication process. These include misplaced metal spots and some apertures that should be clear, being covered with metal. We used the fourth-order polynomial fit to the PSD as the input PSD in Eq. (5.1). With that correction, we obtained MTF results that agreed well with LSF-derived measurements.12

Figure 5.14 (Upper left) A small region of design data for the IR transparency; (upper right) microscopic image of a small region of the as-fabricated transparency (pixel size 46 mm); (lower) the input PSD calculated from a microscopic image of the as-fabricated transparency, where the smooth curve is a fourth-order polynomial fit (reprinted from Ref. 12).

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Figure 5.15 MTF measurement results for bandlimited white-noise pattern (reprinted from Ref. 12).

Figure 5.15 shows MTF measurement results for the pattern in Fig. 5.13. In the left figure, the dots are data points from the random-transparency MTF technique, and the solid line is a fourth-order polynomial fit to the data. The dashed lines are measured data using a line-response method. The upper dashed curve corresponds to the situation where the image of the line source is centered on a column of the detector array; the lower MTF curve corresponds to the situation where the line image is centered between the columns. Individual data points are the result of a particular spatial-frequency component in the image, which have a random position with respect to the photosite locations. The data points thus fall between the maximum and minimum LSF-derived MTF curves, and the fitted curve falls midway between the two line-response MTF curves. In the right figure of Fig. 5.15, we compare the fourth-order data fit seen in the left figure to an average lineresponse MTF, which was measured as follows. The line source was moved randomly, and 15 LSF-derived MTFs were calculated and averaged. The comparison between the random-transparency MTF (solid curve) and the average LSF-derived MTF (dotted curve) shows excellent agreement, consistent with the analysis of Park et al.,14 where a shift-invariant MTF is calculated as the average over all possible test-target positions. We confirmed the repeatability of the random-transparency method by making in-plane translations of the transparency and comparing the resulting MTFs. We found a variation in MTF of less than 2%. Thus, we demonstrated the random-transparency method to be shift invariant. This shift invariance, which applies to all noise-target methods, relaxes alignment requirements in the test procedure, as compared to methods not employing noise targets, where the test procedure generally requires fine positional adjustment of components to achieve the best response.

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Other PSD dependences are possible. For example, using the frequencydomain filtering process seen in Fig. 5.16, we can modify white-noise data to yield a pattern with several discrete spatial frequencies of equal PSD magnitude. Figure 5.17 shows a resulting random pattern, along with the frequency filtering function we used. The PSD is no longer white, resulting in a discernable inter-pixel correlation. We show the MTF measurements in Fig. 5.18 by comparing the MTF results from the random discrete-frequency pattern and the white-noise pattern. Also we show the amplitude spectrum of

Figure 5.16 Generation process of a discrete-frequency pattern (adapted from Ref. 12).

Figure 5.17 (left) Random discrete-frequency pattern and (right) its corresponding filter function (reprinted from Ref. 12).

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Figure 5.18 Comparison of MTF measurements with the random discrete-frequency pattern (solid line) and the white-noise pattern (dotted line). Also shown is the amplitude spectrum of the system noise (dot-dashed line) (reprinted from Ref. 12).

the system noise, which we measured by taking the magnitude of the FFT of the array data, averaged over rows, for a uniform irradiance input equal to the average value of the discrete-frequency target image. The discrete-frequency method allows a single-frame measurement of both MTF and spatial noise at several discrete spatial frequencies.

5.3 Conclusion Random-noise targets are useful for measuring a shift-invariant MTF. The two primary methods for generating these targets are laser speckle and transparencies. We generally use laser speckle to test detector arrays because no intervening optical elements are required. If the input noise PSD is narrowband, we can use laser speckle to measure MTF past the Nyquist frequency of the detector array. We generally use transparencies to test camera systems consisting of fore-optics and a detector array. In both cases, we can generate a variety of input PSDs, depending on the specifics of the instrument design.

References 1. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, pp. 346–350 (1965). 2. H. Kubota and H. Ohzu, “Method of measurement of response function by means of random chart,” JOSA 47(7), 666–667 (1957).

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3. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley-Interscience, New York, pp. 189–193 (1971). 4. M. Born and E. Wolf, Principles of Optics, Fifth edition, Pergamon Press, Oxford, p. 383 (1975). 5. J. W. Goodman, “Statistical Properties of Laser Speckle,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., Springer-Verlag, Berlin, Heidelberg, pp. 38–40 (1984). 6. G. D. Boreman, Y. Sun, and A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990) [doi: 10. 1117/12.55601]. 7. A. D. Ducharme, “Microlens diffusers for efficient laser speckle generation,” Opt. Exp. 15(22), 14573–14579 (2007). 8. A. D. Ducharme and G. D. Boreman, “Holographic elements for modulation transfer function testing of detector arrays,” Opt. Eng. 34(8), 2455–2458 (1995) [doi: 10.1117/12.207144]. 9. G. D. Boreman and E. L. Dereniak, “Method for measuring modulation transfer function of charge-coupled devices using laser speckle,” Opt. Eng. 25(1), 148–150 (1986) [doi: 10.1117/12/7973792]. 10. M. Sensiper, G. D. Boreman, A. D. Ducharme, and D. R. Snyder, “Modulation transfer function testing of detector arrays using narrowband laser speckle,” Opt. Eng. 32(2), 395–400 (1993) [doi: 10.1117/12. 60851]. 11. A. D. Ducharme and S. P. Temple, “Improved aperture for modulation transfer function measurement of detector arrays beyond the Nyquist frequency,” Opt. Eng. 47(9), 093601 (2008) [doi: 10.1117/1.2976798]. 12. A. Daniels, G. D. Boreman, A. D. Ducharme, and E. Sapir, “Random transparency targets for modulation transfer function measurement in the visible and IR,” Opt. Eng. 34(3), 860–868 (1995) [doi: 10.1117/12.190433]. 13. A. Daniels, G. D. Boreman, and E. Sapir, “Diffraction effects in IR halftone transparencies,” Infrared Phys. Technol. 36(2), 623–637 (1995). 14. S. K. Park, R. Schowengerdt, and M.-A. Kaczynski, “Modulationtransfer-function analysis for sampled image systems,” Appl. Opt. 23(15), 2572–2582 (1984).

Chapter 6

Practical Measurement Issues In this chapter, we will consider a variety of practical issues related to MTF measurement, including the cascade property of MTF multiplication, the quality of the auxiliary optics such as collimators and relay lenses, source coherence, and normalization at low frequency. We will conclude with some comments about the repeatability and accuracy of MTF measurements and the use of computers for data acquisition and processing. At the end of the chapter, we will consider four different instrument approaches that are representative of commercial MTF equipment and identify the design tradeoffs.

6.1 Measurement of PSF We begin with a useful approximate formula. For a uniformly distributed blur spot of full width w, the resulting MTF, shown in Fig. 6.1, is sinðpjwÞ : MTFðjÞ ¼ (6.1) ðpjwÞ We can use this simple approach as a handy reality check, comparing a measured spot size to computer-calculated MTF values. Often when computers are part of the data-acquisition and data-processing procedures, we cannot check and verify each step of the MTF calculation. In these situations, it is a good idea to manually verify the results of the computation by using a measuring microscope to assess the size of the impulse response formed by the system. Figure 6.2 shows a convenient configuration1 for a visible-band PSF measurement that is easy to assemble. A microscope objective is paired with a Si charge-coupled-device (CCD) camera head by means of threaded adapter rings. Even if the distance between the objective and the CCD image plane is not the standard 160 mm for a microscope, an in-focus image will be formed when the impulse response being viewed is at some small distance in front of the objective. It is convenient to use a ground-glass screen on which to project 113

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Figure 6.1 MTF for a uniform blur spot of width w.

Figure 6.2 Measuring-microscope apparatus for measuring PSF.

the impulse response of the system under test. This allows us to measure the PSF irradiance distribution at the plane of the ground glass without regard for whether incident rays at all angles would be captured by the finite numerical aperture of the microscope objective. The objective and CCD combination should be first focused on the ground glass, and the whole assembly should allow precise axial motion to find the best focus of the impulse response being measured. The objective and FPA combination should have a three-axis micropositioner, allowing motion perpendicular to the optic axis, to allow for centering the PSF in the field of the camera and measurement of the impulse response width. We can obtain a quick estimate of the width of the blur spot by positioning the center of the image at one side of the blur spot and noting the amount of cross-axial micropositioner motion required to place the center of the image on the other side of the blur spot. Surely the blur spot is not a uniform irradiance distribution, and there is some arbitrariness in the assessment of the spot width in this manner. Nevertheless, we can obtain a back-of-the-envelope estimate for the MTF using that estimate of w. When we compare Fig. 6.1 [using the manual measurement of w in Eq. (6.1)] to the computer-calculated MTF, the results should be reasonably close in magnitude (if not in acutal functional form). If we do not find a suitable correspondence, we should re-examine the assumptions made in the computer calculation before certifying the final measurement results. Common errors

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include missed scale factors of 2p in the spatial-frequency scale, incorrect assumptions about the wavelength or F/# in the computer calculation of the diffraction limit, incorrect assumed magnification in a re-imaging lens, or an inaccurate value of detector-to-detector spacing in the image receiver array. Of course, a more complete MTF measurement can be made if the irradiance image of the impulse response falling on the ground glass is captured. A frame grabber digitizes the video signal coming from the detector array (some cameras will have the option for direct digital output). We can then Fourier transform the PSF data, being careful to first account for the magnification of the PSF irradiance distribution by the microscope-objective/ FPA system. The microscope objective is being used nearly on-axis, so its own MTF should usually not affect the PSF measurement.

6.2 Cascade Properties of MTF We often account for the combination of several subsystems by simply multiplying MTFs. That is, we calculate the overall system MTF as a pointby-point product of the individual subsystem MTFs, as in Eq. (1.9) and Fig. 1.7. This is a very convenient calculation but is sometimes not correct. We want to investigate the conditions under which we can multiply MTFs. In consideration of incoherent imaging systems, the MTF multiplication rule can be simply stated as follows: We can multiply MTFs if each subsystem operates independently on an incoherent irradiance image. We will illustrate this rule using examples. First, we consider the example seen in Fig. 6.3, the combination of a lens system (curve a) and a detector (curve b) in a camera application. The MTF of the combination (curve c) is a point-by-point multiplication of individual MTFs because, surely, the detector responds to the spatial distribution of irradiance (W/cm2) in the image plane without any partial-coherence effects.

Figure 6.3 Cascade of MTFs of a lens system (curve a) and a detector (curve b) produces the product MTF (curve c).

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A relay lens pair with a diffusing screen at the intermediate image plane is another case where we can cascade MTFs by means of a point-by-point multiplication (Fig. 6.4). The exitance [W/cm2] on the output side of the diffuser is proportional to the irradiance [W/cm2] at the input face. Any pointto-point phase correlation in the intermediate image is lost in this process. The diffuser forces the two systems to interact independently, regardless of their individual state of correction. The two relay stages cannot compensate for the aberrations of the other because of the phase-randomizing properties of the diffuser. The MTFs of each stage multiply, and the product MTF is correspondingly lower. This relay-lens example is a bit contrived because we typically do not have a diffuser at an intermediate image plane (from a radiometric point of view, as well as for image-quality reasons), but it illustrates the MTF multiplication rule by presenting the second stage with an incoherent irradiance image formed by the first stage. Figure 6.5 illustrates a case in which cascading MTFs will not work. This is a two-lens combination where the second lens balances the spherical aberration of the first lens. Neither lens is well-corrected by itself, as seen by the poor individual MTFs. The system MTF is higher than either of the

Figure 6.4 Relay-lens pair with a diffuser at the intermediate image plane. The MTF of each stage is simply multiplied.

Figure 6.5

Pair of relay lenses for which the MTFs do not cascade.

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individual-component MTFs, which cannot happen if the MTFs simply multiply. The intermediate image is partially coherent, so these lenses do not interact on an independent irradiance basis.2 Lens 1 does not simply present an irradiance image to Lens 2. Specification of an incoherent MTF for each element separately is not an accurate way to analyze the system because that separate specification does not represent the way that the lenses interact with each other. However, if two systems have been independently designed and independently corrected for aberrations, then the cascade of geometrical MTFs is a good approximation. We consider the example seen in Fig. 6.6 of an afocal telescope combined with a final imaging lens. This is a typical optics configuration for an infrared imager. Each subsystem can perform in a well-corrected manner by itself. As noted in Chapter 1, the limiting aperture (aperture stop) of the entire optics train determines the diffraction MTF. This component does not cascade on an element-by-element basis. The diffraction MTF is included only once in the system MTF calculation. This distinction is important when measured MTFs are used. For example, suppose that separate measured MTF data are available for both subsystems. Simply multiplying the measured MTFs would count diffraction twice because diffraction effects are always present in any measurement. Because the front end of the afocal telescope in Fig. 6.6 is the limiting aperture of the system, we can use the measured MTF data for subsystem #1: MTF1, meas ðjÞ ¼ MTF1, geom ðjÞ  MTF1, diff ðjÞ:

(6.2)

To find the geometrical MTF of subsystem #2, we must separate the geometrical and diffraction contributions3 in the measured MTF of the subsystem:

Figure 6.6 Combination of an afocal telescope and a detector lens.

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MTF2, meas ðjÞ ¼ MTF2, geom ðjÞ  MTF2, diff ðjÞ

(6.3)

and MTF2, geom ðjÞ ¼

MTF2, meas ðjÞ : MTF2, diff ðjÞ

(6.4)

The only diffraction MTF that contributes to the calculation of the total MTF is that of subsystem #1, so for the total system MTF, we have MTFtotal ðjÞ ¼ MTF1, diff ðjÞ  MTF1, geom ðjÞ  MTF2, geom ðjÞ:

(6.5)

6.3 Quality of Auxiliary Optics Figure 6.7 is a schematic of the typical MTF setup, where the unit under test (UUT) images a point-source object. However, in practice, we often use auxiliary optical elements in the test setup, as seen in Fig. 6.8. For instance, we use additional optics to simulate a target-at-infinity condition in order to magnify a small image-irradiance distribution before acquiring it with a detector array, or to test an afocal element. To prevent the auxiliary optics from impacting the MTF results, the following conditions must be met. First, the aperture of the collimator must overfill that of the UUT so that all aperture-dependent aberrations of the UUT are included in the measurement. Second, in the case of the re-imager, the relay-lens aperture must be sufficiently large that it does not limit the ray bundle focused to the final image plane. In both instances, the aperture stop of the end-to-end measurement system must be at the UUT. We want the auxiliary optics to be diffraction-limited; the geometricalaberration blur of the auxiliary optics should be small compared to its diffraction blur. Because the previous condition requires that the auxiliary optics overfill the aperture of the UUT—putting the aperture stop at the UUT—the diffraction limit will also occur in the UUT. If the auxiliary-optics angular aberrations are small compared to the diffraction blur angle of the auxiliary optics, then the aberrations of the auxiliary optics will be even smaller compared to the larger diffraction blur angle of the UUT. Under these conditions, we can directly measure the image blur of the UUT because both the diffraction and aberration blur angles of the auxiliary optics are small

Figure 6.7

MTF test where the source is directly imaged by the unit under test.

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Figure 6.8 Auxiliary optics used in MTF-test setups.

compared to the UUT blur. If the auxiliary optics is not diffraction-limited, then its aberration blur must be characterized and accounted for in the MTF product that describes the measured data. Then, we divide out this instrumental aberration contribution to MTF in the frequency domain, providing that the aberrations of the auxiliary optics are small enough to not seriously limit the spatial-frequency range of the measurement. In a typical test scenario, we have a number of impulse responses that convolve together to give the measured PSF: the diffraction of the UUT, the aberration of the unit under test, the aberration of the collimator, the detector footprint, the source size, and the sampling. We want to isolate the PSF of the UUT, which is PSFUUT ¼ PSFUUT, diff  PSFUUT, aberr :

(6.6)

The other PSF terms should be known and should be much narrower than the PSF of the UUT so that we can divide them out in the frequency domain using Eq. (1.9) without limiting the frequency range of the MTF test.

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6.4 Source Coherence In MTF tests that involve the imaging of a back-illuminated extended target (for instance, LSF, ESF, or bar-target tests), the illumination must be spatially incoherent to avoid interference artifacts between separate locations in the image that can corrupt MTF data. If the extended source is itself incandescent (as in a hot-wire LSF test for the infrared), the source is already incoherent. It is when an extended aperture is back-illuminated that coherence effects may become important. When incandescent-bulb sources are used with a low-F/# condenser-lens setup to back-illuminate the source aperture, partialcoherence interference effects can be present in the image. The usual way we reduce the source coherence for an incandescent source is to place a groundglass diffuser on the source side, adjacent to the aperture being illuminated, as seen in Fig. 6.9. The phase differences between various correlation cells of the diffuser are generally sufficient for reduction of the source coherence. If we illuminate the slit aperture of Fig. 6.9 with a collimated laser beam, the spatial coherence across the slit is nearly unity, and we will need a more elaborate setup to reduce the coherence. We again use a ground-glass diffuser at the illuminated aperture, but we must mechanically move the diffuser on a time scale that is short compared to the integration time of the sensor to average out the speckles in the image data. Piezoelectric transducers are a common choice to implement such motion. Concerns about the spatial coherence of the input object generally do not apply to pinholes because, if a source is small enough to behave as a point source with respect to the system under test, it is essentially only a singlepoint emitter, which is self-coherent by definition. We consider spatial coherence only when adjacent point sources might interfere with each other. A pinhole aperture can be illuminated by a laser source without appreciable

Figure 6.9 A ground-glass diffuser is used to reduce the partial coherence of an incandescent source.

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spatial-coherence artifacts. The incoherent MTF will be measured because the irradiance of the PSF is used for analysis.

6.5 Low-Frequency Normalization A practical issue is how to normalize a set of measurements so that MTF ¼ 1 at j = 0. If we are calculating MTF from PSF data, the normalization issue is straightforward because the Fourier transform of the PSF is set to 1 at j = 0. However, if we are making measurements with sinewaves, bar targets, or noise targets, we have a set of modulation data at various frequencies. We do not have access to a target of arbitrarily low frequency, as seen in the data of Fig. 6.10. We should try to acquire data at as low a frequency as possible in any test scenario. Sometimes it is feasible to measure the modulation depth of “flat-field” data, where the lowest spatial frequency for which the modulation depth can be measured is a flat field of maximum and minimum irradiance filling the FOV of the system under test. Thus, the lowest frequency that can possibly be measured is j = 1/(FOV). The resulting modulation depth would provide the normalization value for which the MTF can be set to unity, and negligible error would result from this approximation to zero frequency. If flat-field modulation data are not available (for instance, in some speckle measurements), the issue of low-frequency normalization becomes more subtle. The normalization is important because it sets the MTF value for all data points taken. Obviously, we cannot simply set the lowest frequency MTF value in Fig. 6.10 to 1 because that will unduly inflate all of the MTF values in the data set. An a priori analytical function to which the MTF should be similar can provide guidance regarding the slope of the MTF curve as j = 0 is approached. Otherwise, a straight line joining the lowest data point and j = 0 is a reasonable assumption.

Figure 6.10 MTF data points without low-frequency data.

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6.6 MTF Testing Observations When we measure the MTF by two different methods, for instance, a bar-target test and an impulse-response test, the results should be identical. When the measured MTFs do not accurately agree, some portion of the system (typically, the detector array) is not responding in a linear fashion. The accuracy of commercial MTF measurement systems ranges from 5% to 10% in absolute MTF with 1% accuracy possible. Radiometric measurements are difficult to perform to the 1% level, and MTF testing combines both radiometric- and position-accuracy requirements. The position-accuracy and repeatability requirements are often quite demanding—often less than 1 mm in many cases. Good micropositioners and solid mounts are required if we are to achieve high-accuracy MTF measurements. Even with the best components, lead-screw backlash errors can occur, and high-accuracy motion must be made in only one direction.

6.7 Use of Computers in MTF Measurements Most commercial systems come with computer systems for data acquisition, processing, and experimental control. Precise movement control of scanning apertures and other components is very convenient, especially for throughfocus measurements, which are tedious if done manually. The graphicsdisplay interface allows us to immediately visualize the data and lets us adjust parameters in real time. Digitized data sets facilitate further signal processing, storage, and comparison of measurement results. There is, however, a drawback to the use of computers: software operations typically divide out instrumental MTF terms, using assumed values for the slit width, collimator MTF, wavelength range, F/#, and other parameters. In practice, we must ensure that these settings are correct. We can check the final results for realism as discussed in Section 6.1. Also, we must remember that the electronics of the data-acquisition system can trigger from a variety of spurious signals. It is possible to take erroneous MTF measurements on a clock-feed-through waveform from the detector array electronics, rather than from the PSF of the system. As a test, we can cover the detector array and see if an MTF result is displayed on the monitor. Occasionally this does happen, and it is worth checking for if the measurement results are in question.

6.8 Representative Instrument Designs We now consider four different instrument approaches taken in commercial MTF-measurement systems. We examine the design tradeoffs and compare the systems for their viability in visible and infrared applications.

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Figure 6.11 Example system #1: visible ESF.

6.8.1 Example system #1: visible edge response In the visible ESF design (Fig. 6.11), intended for MTF measurement of visible-wavelength lenses, a pinhole is illuminated by either a laser or filtered arc lamp. A relatively high flux level and good SNR allow for a simple system. A tilt control for the lens under test allows for off-axis performance measurement. A diffraction-limited refractive collimator limits the aperture size of lenses that can be tested. Larger apertures can be obtained using reflective collimators. A scanning “fishtail” blocking device is used as a knife edge in the image plane. The obtained data are therefore ESF, even with a pinhole source. The configuration of the knife edge allows for a measurement in either the x or y direction without changing the apparatus, requiring a twodimensional (2D) scan motion. A photomultiplier tube detector is used, consistent with the visible-wavelength response desired. This sensor typically gives a high SNR. Because of the high SNR, the derivative operation required to go from ESF to LSF is not unduly affected by electronics noise. A fastFourier transform (FFT) of the LSF data produces one profile of the MTF. Scanning in the orthogonal direction produces the other profile. 6.8.2 Example system #2: infrared line response The infrared line response system (Fig. 6.12) is designed for LSF measurement of lenses in the LWIR portion of the spectrum. A number of techniques are employed to enhance the SNR, which is always a concern at IR wavelengths. A heated ceramic glowbar oriented perpendicular to the plane of the figure illuminates a slit, which acts as an IR line source. This configuration gives more image-plane flux than a pinhole source, allowing for the use of the relatively low-temperture glowbar source consistent with the LWIR band of interest. A narrow linear detector oriented parallel to the source direction is scanned to acquire the LSF image. This HgCdTe sensor is cyrogenically cooled with liquid nitrogen (LN2). The detector serves the function of a scanning slit as the detector moves. This avoids the difficulty of implementing a movable cooled slit inside the liquid-nitrogen dewar. A diffraction-limited reflective collimator allows for larger UUT apertures and broadband IR

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Figure 6.12 Example system #2: infrared line response.

operation without chromatic aberrations. A mechanical chopper is used for modulating the source, allowing for narrowband synchronous amplification. This reduces the noise bandwidth of the measurement and increases the SNR. The LSF is measured directly. This approach avoids the derivative operation needed for an ESF scan (which is problematic at low SNR) and also requires less dynamic range from the sensor. 6.8.3 Example system #3: visible square-wave response The visible square-response system4 is rather special in that it approximates a square-wave test rather than a bar-target test. The system is configured for testing the MTF of lenses in the visible portion of the spectrum. The object generator seen in Fig. 6.13 serves as the source of variable-frequency square waves. The object generator consists of two parts, a radial grating that rotates at a constant angular velocity and a slit aperture. The center of rotation of the grating is set at an angle (0 , u , 90 deg) with respect to the slit aperture. This allows control of the spatial frequency of the (moving) square waves. The response of the system is measured one frequency at a time.

Figure 6.13 The object generator for the visible square-wave test.

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Figure 6.14 Example system #3: visible square-wave response.

The object generator is backlit and, as seen in Fig. 6.14, is placed at the focus of a collimator and re-imaged by the lens under test. A long, narrow detector is placed perpendicular to the orientation of the slit image, and the moving square waves pass across the detector in the direction of its narrow dimension. The analog voltage waveform output from the sensor is essentially the time-domain square-wave response of the system to whatever spatial frequency the object generator produced. To avoid the necessity of using the series conversion from CTF to MTF, a tunable-narrowband analog filter is used, eliminating all harmonics and allowing only the fundamental frequency of the square wave to pass through. The center frequency of the filter is changed when, with different u settings, the object frequency is changed. The waveform after filtering is sinusoidal at the test frequency, and the modulation depth is measured directly from this waveform. Filtering the waveform electronically requires the electronic frequency of the fundamental to stay well above the 1/f-noise region (1 kHz and below), which implies a fast rotation of the object-plane radial grating. This necessitates a relatively high-power optical source to back-illuminate the object generator because the detector has to operate with a short integration time to acquire the quickly moving square waves. This design works well in the visible, but was not suitable in the infrared, where the SNR would be lower because of the lower source temperatures used in that band. Compensating for the low SNR would

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Figure 6.15

Example system #4: infrared bar-target response.

require a longer integration time. Lowering the speed of the rotating grating would put the electronic signal into the 1/f-noise region. 6.8.4 Example system #4: bar-target response Figure 6.15 shows an IR bar-target system that is intended to measure the bar-target response of lenses at LWIR wavelengths. The four-bar patterns are made of metal and are backlit with an extended blackbody source. A separate bar target is used for each fundamental spatial frequency of interest. The resulting image is scanned mechanically with a cryogenically cooled single-element HgCdTe detector. Because the image stays stationary in time, a slow sensor-scan speed allows for signal integration, improving the SNR. The bar-target data is processed using the series correction of Eq. (4.4). Analog filtering as seen in example system #3 is not used because the slow scan speed of the sensor would put the resulting analog waveforms into the 1/f-noise region at low frequencies.

6.9 Conclusion Measurement of MTF requires attention to various issues that can affect the integrity of the data set. Most importantly, the quality of any auxiliary optics must be known and accounted for. The optics other than the unit under test should not significantly limit the spatial frequency range of the test. The positional repeatability of the micropositioners used is critical to obtaining high-quality data. We should take care to ensure that the coherence of the source does not introduce interference artifacts into the image data that could bias the MTF computation. It is also important that we carefully consider the issue of low-frequency normalization because this affects the MTF value at all frequencies. Computers are ubiquitous in today’s measurement apparatus. We should make sure that all default settings in the associated software are consistent with the actual measurement conditions.

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References 1. Prof. Michael Nofziger, University of Arizona, personal communication. 2. J. B. DeVelis and G. B. Parrent, “Transfer function for cascaded optical systems,” JOSA 57(12), 1486–1490 (1967). 3. T. L. Alexander, G. D. Boreman, A. D. Ducharme, and R. J. Rapp, “Point-spread function and MTF characterization of the kinetic-kill-vehicle hardware-in-the-loop simulation (KHILS) infrared-laser scene projector,” Proc. SPIE 1969, pp. 270–284 (1993) [doi: 10.1117/12.154720]. 4. L. Baker, “Automatic recording instrument for measuring optical transfer function,” Japanese J. Appl. Science 4(suppl. 1), 146–152 (1965).

Further Reading Baker, L., Selected Papers on Optical Transfer Function: Measurement, SPIE Milestone Series, Vol. MS 59, SPIE Press, Bellingham, Washington (1992).

Chapter 7

Other MTF Contributions We now consider the MTF contributions arising from image motion, image vibration, atmospheric turbulence, and aerosol scattering. We present a first-order analysis of these additional contributions to the system MTF. Our heuristic approach provides a back-of-the-envelope estimate for the imagequality impact of these effects, and a starting point for more advanced analyses.

7.1 Motion MTF Image-quality degradation arises from movement of the object, image receiver, or optical-system line of sight during an exposure time te. We consider uniform linear motion of the object at velocity vobj and a corresponding linear motion of the image at velocity vimg, as shown in Fig. 7.1. Over an exposure time te, the image has moved a distance vimg
te. This one-dimensional motion blur can be modeled as a rect function:

Figure 7.1

Linear motion blur is the product of image velocity and exposure time.

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hðxÞ ¼ rect½x∕ðvimg
te Þ, leading to an MTF along the direction of motion, sinðpjvimg te Þ : MTFalong-motion ðjÞ ¼ ðpjvimg te Þ

(7.1)

(7.2)

7.2 Vibration MTF Sometimes the platform on which the optical system is mounted vibrates. Typically, we analyze the effect of the vibration one frequency at a time, thus assuming sinusoidal motion. The most important distinction is between high-frequency and low-frequency motion. We compare the temporal period of the motion waveform to the exposure time of the sensors te. The case of high-frequency sinusoidal motion, where many oscillations occur during exposure time te, is the easiest for us to analyze. As seen in Fig. 7.2, we model the vibration by assuming that a nominally axial object point undergoes sinusoidal motion of amplitude D perpendicular to the optic axis. The corresponding image-point motion will build up a histogram impulse response in the image plane. This impulse response will have a minimum at the center of the motion and maxima at the image-plane position corresponding to the edges of the object motion because the object is statistically most likely to be found near the peaks of the sinusoid where the object motion essentially stops and turns around. The process of stopping and turning leads to more

Figure 7.2 High-frequency sinusoidal motion builds up a histogram impulse response in the image plane.

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residence time of the object near those locations and thus a higher probability of finding the object near the peaks of the sinusoidal motion. If the sinusoidal object motion has amplitude D, the total width of h(x) is 2D, assuming unit magnification of the optics. There is zero probability of the geometrical image point being found outside of this range, which leads to the impulse response depicted in Fig. 7.3. If we take the Fourier transform of this h(x), we obtain the corresponding vibration MTF seen in Fig. 7.4.1 For low-frequency sinusoidal vibrations, the image quality depends on whether the image-data acquisition occurs near the origin or near the extreme points of the object movement. As stated earlier, the velocity slows near the extreme points and is at its maximum near the center of the motion. In the case of low-frequency sinusoidal vibrations, we must perform a more detailed

Figure 7.3

Impulse response for sinusoidal motion of amplitude D.

Figure 7.4 MTF for sinusoidal motion of amplitude D.

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analysis to predict the number of exposures required to get a single lucky shot where there is no more than a prescribed degree of motion blur.2

7.3 Turbulence MTF Atmospheric turbulence results in image degradation. We consider a random phase screen (Fig. 7.5) with an autocorrelation width w that is the size of the refractive-index eddy and a phase variance s2. The simplest model is one of cloud-like motion, where the phase screen moves with time but does not change form—the frozen-turbulence assumption. We model an average image-quality degradation over exposure times that are long compared with the motion of the phase screen. Image quality estimates for short exposure times require more complicated models. We consider diffraction from the eddys to be the cause of MTF degradation (blurring). We assume that w ≫ l, which is consistent with the typical eddy sizes (1 cm , w , 1 m) encountered in practice. Refractive ray-deviation errors can be determined from a separate angle-of-arrival analysis, resulting in image-plane motion (distortion). As seen in Fig. 7.6, the impulse response h(x) consists of a narrow central core from the unscattered radiation and a wide diffuse region from the scattered radiation. Larger phase variation in the screen leads to more of the impulse-response power being contained in the scattered component. The narrow central core of the impulse response contributes to a broad flat MTF at high frequencies. The broad diffuse scattered component of the impulse response will contribute to an MTF rolloff at low frequencies. We can write3 the turbulence MTF as     2  lj 2 MTFðjÞ ¼ exp s 1  exp  , (7.3) w where j is the angular spatial frequency in cycles/radian. This MTF is plotted in Fig. 7.7, with the phase variance as a parameter. For a phase variance near zero, turbulence contributes minimal image-quality degradation because most

Figure 7.5

Frozen-turbulence model for a phase screen.

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Figure 7.6

Impulse-response form for atmospheric turbulence.

Figure 7.7

Turbulence MTF parameterized on phase variance.

of the light incident on the phase screen passes through unscattered. As the phase variance increases, more light will be spread into the diffuse halo of the impulse response seen in Fig. 7.6, and the MTF will roll off at low frequencies. For all of the curves in Fig. 7.7, the MTF is flat at high frequencies. The atmospheric turbulence MTF is only one component of the MTF product, and the high-frequency rolloff typically seen for overall system MTF will be caused by some other MTF component. For a large phase variance, the turbulence MTF of Eq. (7.3) reduces to a Gaussian form:   2  2 lj MTFðjÞ ¼ exp s , (7.4) w with a 1/e rolloff frequency of j1/e ¼ w/(ls). In this limit it is straightforward to identify the issues affecting image quality. The transfer function has a higher rolloff frequency (better image quality) for larger eddy size w

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Figure 7.8

Chapter 7

Comparison of field imagery and turbulence simulations (adapted from Ref. 4).

(less diffraction), shorter l (less diffraction), and smaller s (less phase variation). The effect of turbulence is less for shorter propagation paths. Recent advances in modeling image degradation caused by atmospheric turbulence have been made at the U.S. Army Night Vision Lab4 by researchers who compared model predictions and field-measurement videos. They found good agreement for low and medium turbulence strength (values of the refractive-index structure constant Cn2 between 5 and 10
1014) and for short and medium ranges (300 and 650 m), as seen in Fig. 7.8. The main points of the model are the use of a short-exposure atmospheric MTF expression to account for blurring and an angle-of-arrival expression to account for the motion of image features.

7.4 Aerosol-Scattering MTF Forward scattering from airborne particles (aerosols) also causes image degradation. We consider a volume medium with particles of radius a, as seen in Fig. 7.9, and assume that the particle concentration is sufficiently low that multiple-scattering processes are negligible. We consider diffraction from the particles to be the primary beam-spreading mechanism.5,6 According to Eq. (7.5), attenuation of the transmitted beam power f is caused by both absorption (exponential decay coefficient ¼ A) and scattering (exponential decay coefficient ¼ S). Thus, the 1/e distance for absorption is 1/A, and for scattering is 1/S:

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Figure 7.9 Forward-scattering model for aerosols.

fðzÞ ¼ f0 eðAþSÞz :

(7.5)

Absorption is not a diffraction process and does not depend on spatial frequency. Because when we set MTF(j ¼ 0) ¼ 1 the absorption effects will be normalized out, only the scattering process is important for development of an aerosol MTF. We typically assume that the particle size a . l, consistent with the usual range of particle sizes of interest for aerosols 100 mm , a , 1 mm. Thus, the size of the diffracting object a for scattering is much smaller than was w for the case of atmospheric turbulence, with consequently larger forward-scatter diffraction angles. If a  l, which is important only for very small particles, or for imaging in the infrared portion of the spectrum, the scattering is nearly isotropic in angle. In this case, the image quality is significantly degraded. Within the limitation of the forward-scatter assumptions, the impulse response for aerosol scatter has a narrow central core from the unscattered radiation, along with a diffuse scattered component. This is similar in functional form to the plot seen in Fig. 7.6, but different in scale. The spatial extent of the halo is wider for aerosol scattering than for turbulence because of the smaller size of the scattering particles. The aerosol-scattering MTF has two functional forms: one corresponding to the low-frequency rolloff region resulting from the Fourier transform of the broad halo and one corresponding to the flat high-frequency region resulting from the transform of the central narrow portion of the impulse response. The transition spatial frequency jt marks the boundary between these two functional forms. For the aerosol MTF we have MTFðjÞ ¼ expfSzðj∕jt Þ2 g, for j , jt

(7.6)

MTFðjÞ ¼ expfSzg, for j . jt ,

(7.7)

and

where z is the propagation distance, and the transition frequency is

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Figure 7.10 Increasing the propagation path decreases the aerosol-scattering MTF at all frequencies.

Figure 7.11 Increasing the scattering coefficient decreases the aerosol-scattering MTF at all frequencies.

jt ¼ a∕l

(7.8)

in angular spatial frequency [cycles/radian] units. Shorter wavelengths and larger particles yield less diffraction and result in better image quality. For longer paths (larger z), the MTF decreases at all frequencies, as shown in Fig. 7.10. For more scattering (larger S), MTF decreases at all frequencies, as shown in Fig. 7.11.

7.5 Conclusion This chapter provided a brief consideration of MTF contributions from motion, vibration, turbulence, and aerosols. The treatment is intended as a first-order approach to a complex topic. Additional information can be found in the references and further-reading list.

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References 1. O. Hadar, I. Dror, and N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations. Part IV: real-time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33(2), 566–578 (1994) [doi: 10.111/12.153186]. 2. D. Wulich and N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations,” Opt. Eng. 26(6), 529–533 (1987) [doi: 10.1117/12. 7974110]. 3. J. W. Goodman, Statistical Optics, John Wiley and Sons, New York (1985). 4. K. J. Miller, B. Preece, T. W. Du Bosq, and K. R. Leonard, “A dataconstrained algorithm for the emulation of long-range turbulence-degraded video,” Proc. SPIE 11001, 110010J (2019) [doi: 10.1117/12.2519069]. 5. Y. Kuga and A. Ishimaru, “Modulation transfer function and image transmission through randomly distributed spherical particles,” JOSA A 2(12), 2330–2336 (1985). 6. D. Sadot and N. S. Kopeika, “Imaging through the atmosphere: practical instrumentation-based theory and verification of aerosol MTF,” JOSA A 10(1), 172–179 (1993).

Further Reading Andrews, L., and R. Phillips, Laser Beam Propagation through Random Media, Second edition, SPIE Press, Bellingham, Washington (2005) [doi: 10.1117/3.626196]. Bohren, C. F., and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (1998). Ishimaru, A., Wave Propagation and Scattering in Random Media, WileyIEEE Press (1997). Kopeika, N., A System Engineering Approach to Imaging, SPIE Press, Bellingham, Washington, Chapters 14, 16, and 17 (1998) [doi: 10.1117/3. 2265069]. Zege, E. P., A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium, Springer-Verlag, Berlin (1991).

Index diffraction-limited MTF, 22 diffraction MTF, 25, 117 division by zero, 79

A aberrations, 24, 27, 29, 33 aerosol scattering, 135 aliasing, 44, 90 angular spatial frequency, 6, 24 astigmatism, 30 atmospheric turbulence, 132 autocorrelation, 21, 25, 99 auxiliary optics, 118

E edge-spread function (ESF), 70, 74, 123 electro-optical systems, 39 electronic networks, 61 electronics noise, 64, 79

B bar target, 126 bar-target-to-MTF conversion, 92 birefringent filters, 48 boost filter, 62

F fiber bundles, 58 finite source size, 78 flat field, 8, 121 focal-plane array (FPA), 39, 51 four-bar target, 89 frame grabbers, 61

C charge-carrier diffusion, 61 charge-transfer inefficiency, 60 coherence effects, 120 coma, 30 contrast transfer function (CTF), 86 convolution theorem, 7, 41 critical spatial frequencies, 18 crosstalk MTF, 59 cutoff frequency, 24, 47, 105

G geometrical MTF, 28, 117 ground glass, 113–114 ground-glass diffuser, 100, 116, 120 I image modulation depth (IMD), 88 impulse response, 1 instantaneous field of view (IFOV), 55 integrating spheres, 99–100, 104 interlacing, 53

D defocus, 12, 29 detection, recognition, and identification, 18 detector arrays, 39 detector footprint, 41 diffraction, 18

J Johnson criteria, 18 139

140

L laser speckle, 98 line-spread function (LSF), 68, 74, 123 linearity, 4, 39 M microdither, 51 microscanning, 51 minimum modulation curve (MMC), 31 modulation depth, 9, 42, 89 modulation transfer (MT), 11 modulation transfer function, 11 motion blur, 129 MTF area (MTFA), 17, 63 multiplication of transfer function, 8, 115 N noise-equivalent modulation (NEM), 16, 63 noise targets, 97 normalization, 8, 10, 87, 121 Nyquist frequency, 44, 51, 100–101, 105 O obscured-aperture systems, 25 optical transfer function (OTF), 9 oversampled knife-edge test, 81 P phase reversal, 12, 29 phase transfer function (PTF), 9, 12

Index

point-spread function (PSF), 1, 67, 113 power spectral density (PSD), 64, 79, 97 R random-transparency target, 104 resolution, 16 S sampling, 43 sampling MTF, 50 scan velocity, 55, 62 separability, 43 separable function, 6 shift invariance, 4, 39, 49, 97, 108 signal-averaging techniques, 77 signal-processing-in-the-element (SPRITE) detectors, 61 spatial frequency, 5 spherical aberration, 30 square wave, 124 square-wave targets, 86 Strehl ratio, 28 T through-focus MTF, 34 time-delay-and-integration (TDI), 57 V vibration, 129 W wavefront error, 30 white noise, 106

Glenn D. Boreman is Professor and Chairman of the Department of Physics & Optical Science and Director of the Center for Optoelectronics & Optical Communications at the University of North Carolina at Charlotte. He is cofounder and Board Chairman of Plasmonics, Inc. (Orlando). From 1984 to 2011 he was on the faculty of the University of Central Florida, where he is now Professor Emeritus. He has supervised to completion 35 MS and 27 PhD students. He has held visiting research positions at IT&T (Roanoke), Texas Instruments (Dallas), US Army Night Vision Lab (Ft. Belvoir), McDonnell Douglas Astronautics (Titusville), US Army Redstone Arsenal (Huntsville), Imperial College (London), Universidad Complutense (Madrid), Swiss Federal Institute of Technology (Zürich), Swedish Defense Research Agency (Linköping), and University of New Mexico (Albuquerque). He received the BS in Optics from the University of Rochester, and the PhD in Optics from the University of Arizona. Prof. Boreman served as Editor-in-Chief of Applied Optics from 2000 to 2005, and Deputy Editor of Optics Express from 2014 to 2019. He is coauthor of the graduate textbooks Infrared Detectors and Systems and Infrared Antennas and Resonant Structures (SPIE Press), and author of Modulation Transfer Function in Optical & Electro-Optical Systems (SPIE Press) and Basic Electro-Optics for Electrical Engineers (SPIE Press). He has published more than 200 refereed journal articles in the areas of infrared sensors and materials, optics of random media, and image-quality assessment. He is a fellow of SPIE, IEEE, OSA, and the Military Sensing Symposium. He is a Professional Engineer registered in Florida. Prof. Boreman served as the 2017 President of SPIE.

Second Edition Glenn D. Boreman

Modulation Transfer Function in Optical and Electro-Optical Systems Second Edition

BOREMAN

This second edition, which has been significantly expanded since the 2001 edition, introduces the theory and applications of the modulation transfer function (MTF) used for specifying the image quality achieved by an imaging system. The book begins by explaining the relationship between impulse response and transfer function, and the implications of a convolutional representation of the imaging process. Optical systems are considered first, including the effects of diffraction and aberrations on the image, with attention to aperture and field dependences. Then electro-optical systems with focal-plane arrays are considered, with an expanded discussion of image-quality aspects unique to these systems, including finite sensor size, shift invariance, sampling MTF, aliasing artifacts, crosstalk, and electronics noise. Various test configurations are then compared in detail, considering the advantages and disadvantages of point-response, line-response, and edge-response measurements. The impact of finite source size on the measurement data and its correction are discussed, and an extended discussion of the practical aspects of the tilted-knife-edge test is presented. New chapters are included on speckle-based and transparency-based noise targets, and square-wave and bar-target measurements. A range of practical measurement issues are then considered, including mitigation of source coherence, combining MTF measurements of separate subsystems, quality requirements of auxiliary optics, and low-frequency normalization. Some generic measurement-instrument designs are compared, and the book closes with a brief consideration of the MTF impacts of motion, vibration, turbulence, and aerosol scattering.

Modulation Transfer Function in Optical and Electro-Optical Systems Second Edition

Modulation Transfer Function in Optical and Electro-Optical Systems

P.O. Box 10 Bellingham, WA 98227-0010

Glenn D. Boreman

ISBN: 9781510639379 SPIE Vol. No.: TT121 TT121