Subwavelength and Nanometer Diameter Optical Fibers [1st Edition.] 364203361X, 9783642033612

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ADVANCED TOPICS IN SCIENCE AND TECHNOLOGY IN CHINA

ADVANCED TOPICS IN SCIENCE AND TECHNOLOGY IN CHINA Zhejiang University is one of the leading universities in China. In Advanced Topics in Science and Technology in China, Zhejiang University Press and Springer jointly publish monographs by Chinese scholars and professors, as well as invited authors and editors from abroad who are outstanding experts and scholars in their fields. This series will be of interest to researchers, lecturers, and graduate students alike. Advanced Topics in Science and Technology in China aims to present the latest and most cutting-edge theories, techniques, and methodologies in various research areas in China. It covers all disciplines in the fields of natural science and technology, including but not limited to, computer science, materials science, life sciences, engineering, environmental sciences, mathematics, and physics.

Limin Tong Michael Sumetsky

Subwavelength and Nanometer Diameter Optical Fibers With 180 figures

Authors Prof. Limin Tong Department of Optical Engineering Zhejiang University, Hangzhou 310027, China E-mail: [email protected]

Dr. Michael Sumetsky OFS Laboratories Somerset, NJ 08807 USA E-mail: [email protected]

ISSN 1995-6819 e-ISSN 1995-6827 Advanced Topics in Science and Technology in China ISBN 978-7-308-06855-0 Zhejiang University Press, Hangzhou ISBN 978-3-642-03361-2 e-ISBN 978-3-642-03362-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009933160 c Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2010  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is a part of Springer Science+Business Media (www.springer.com)

Preface

A decade ago a book on optical microfibers and nanofibers could be hardly foreseen. In 2003, one of the authors (L.T.), in collaboration with scientists from Harvard and Zhejiang University, published an intriguing paper in Nature on the low-loss waveguiding of silica nanofibers. This paper introduced a new vision of micro/nanofibers as basic elements for miniature photonic devices and initiated numerous scientific publications on the topic of this book. At first glance, microfiber-based photonic technology seems to be a reverse step from the lithographic photonic technology, just like wired circuits in relation to printed-in circuits in electronics. However, there are at least two important advantages of microfibers over lithographically fabricated waveguides: significantly smaller losses for a given index contrast and the potential ability for micro-assembly in three dimensions. These advantages could make possible the creation of micro/nanofiber devices that are considerably more compact and less lossy than devices fabricated lithographically. Furthermore, some microfiber-based devices possess unique functionalities, which are not possible or much harder to achieve by other means. Nowadays research on optical micro/nanofibers is growing rapidly. The authors attempted to write a fairly comprehensive introduction to micro/nanofiber optical properties, fabrication methods and applications. The book will be useful for scientists and engineers who want to learn more about very thin – subwavelength diameter – optical microfibers and, eventually, to be engaged in microfiber photonics research. In particular, the authors hope that the contents of the book will attract students and stimulate their innovative ideas in this fascinating field of optics. L.T. would like to acknowledge a number of his colleagues and students at both Zhejiang University in Hangzhou and Harvard University in Cambridge, MA, USA, for their direct or indirect help in micro/nanofiber research and the writing of this book. Special thanks to Professor Eric Mazur of Harvard University for his indispensable support and advice. Special thanks are also extended to Jingyi Lou, Rafael R. Gattass, Qing Yang, Guillaume Vienne, Jian Fu, Yuhang Li, Xiaoshun Jiang, Zhe Ma, Xin Guo, Shanshan Wang,

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Preface

Fuxing Gu, Zhifang Hu, and Keji Huang for their great help and contribution to the work. M.S. would like to acknowledge the creative “Bell Labs” atmosphere at the OFS Laboratories (formerly the Optical Fiber Research Department of Bell Laboratories), which stimulated his research in micro/nanofibers and the work on this book. Special thanks are extended to his present and former Bell Labs/OFS Labs colleagues David DiGiovanni, Ben Eggleton, Yuri Dulashko, John Fini, Michael Fishteyn, Samir Ghalmi, Siddharth Ramachandran, Paul Westbrook and Andrew Yablon for the fruitful discussions and consultations.

The authors April 2009

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A Brief History of Micro- and Nanofibers . . . . . . . . . . . . . . . . . . . 1.2 Concepts of MNFs and the Scope of this Book . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Optical Waveguiding Properties of MNFs: Theory and Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Guiding Properties of Ideal MNFs . . . . . . . . . . . . . . . . . . . . 2.1.1 Mathematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Single-mode Condition and Fundamental Modes . . . . . . . 2.1.3 Fractional Power Inside the Core and Effective Diameter 2.1.4 Group Velocity and Waveguide Dispersion . . . . . . . . . . . . 2.2 Theory of MNFs with Microscopic Nonuniformities . . . . . . . . . . 2.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Conventional and Adiabatic Perturbation Theory . . . . . . 2.2.3 Transmission Loss Caused by a Weak and Smooth Nonuniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theory of MNF Tapers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Semiclassical Solution of the Wave Equation in the Adiabatic Approximation and Expression of Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Optics of Light Propagation Along the Adiabatic MNF Tapers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Example of a Conical MNF Taper . . . . . . . . . . . . . . . . . . . 2.3.4 Example of a Biconical MNF Taper . . . . . . . . . . . . . . . . . . 2.3.5 Example of an MNF Taper with Distributed Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Thinnest MNF Optical Waveguide . . . . . . . . . . . . . . . . . . . . . 2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Model for FDTD Simulation . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 7 15 15 15 17 22 25 28 28 31 32 33

34 35 36 38 40 42 43 44

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2.5.2 Evanescent Coupling between two Identical Silica MNFs 2.5.3 Evanescent Coupling between two Silica MNFs with Different Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Evanescent Coupling between a Silica MNF and a Tellurite MNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Endface Output Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 MNFs with Flat Endfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 MNFs with Angled Endfaces . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 MNFs with Spherical and Tapered Endfaces . . . . . . . . . . 2.7 MNF Interferometers and Resonators . . . . . . . . . . . . . . . . . . . . . . 2.7.1 MNF Mach-Zehnder and Sagnac Interferometers . . . . . . 2.7.2 MNF Loop Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 MNF Coil Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

51 53 54 57 59 60 60 60 64 69

3

Fabrication of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Taper Drawing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Taper-drawing Fabrication of Glass MNFs . . . . . . . . . . . . . . . . . . 3.2.1 Taper Grawing MNFs Rom Glass Fibers . . . . . . . . . . . . . 3.2.2 Drawing MNFs Directly from Bulk Glasses . . . . . . . . . . . 3.3 Drawing Polymer MNFs from Solutions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 77 78 89 91 94

4

Properties of MNFs: Experimental Investigations . . . . . . . . . . 99 4.1 Micro/Nanomanipulation and Mechanical Properties of MNFs 99 4.1.1 Visibility of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1.2 MNF Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.3 Tensile Strengths of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.1 Optical Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.2 Effect of the Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5

MNF-based Photonic Components and Devices . . . . . . . . . . . . 125 5.1 Linear Waveguides and Waveguide Bends . . . . . . . . . . . . . . . . . . . 126 5.1.1 Linear Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.1.2 Waveguide Bends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2 Micro-couplers, Mach-Zehnder and Sagnac Interferometers . . . . 135 5.2.1 Micro-couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.2.2 Mach-Zehnder Interferometers . . . . . . . . . . . . . . . . . . . . . . 138 5.2.3 Sagnac Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.3 MNF Loop and Coil Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.1 MNF Loop Resonator (MLR) Fabricated by Macro-Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.2 Knot MLR Fabricated by Micro-Manipulation . . . . . . . . 146

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IX

5.3.3 5.4 MNF 5.4.1 5.4.2 5.5 MNF 5.5.1 5.5.2

Experimental Demonstration of MCR . . . . . . . . . . . . . . . . 147 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Short-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Add-Drop Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Modeling MNF Ring Lasers . . . . . . . . . . . . . . . . . . . . . . . . . 159 Numerical Simulation of Er3+ and Yb3+ Doped MNF Ring Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.5.3 Er3+ and Yb3+ Codoped MNF Ring Lasers . . . . . . . . . . . 170 5.5.4 Evanescent-Wave-Coupled MNF Dye Lasers . . . . . . . . . . 174 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6

Micro/nanofiber Optical Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.2 Application of a Straight MNF for Sensing . . . . . . . . . . . . . . . . . . 189 6.2.1 Microfluidic Refractive Index MNF Sensor . . . . . . . . . . . . 190 6.2.2 Hydrogen MNF Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.2.3 Molecular Absorption MNF Sensor . . . . . . . . . . . . . . . . . . 192 6.2.4 Humidity and Gas Polymer MNF Sensor . . . . . . . . . . . . . 193 6.2.5 Optical Fiber Surface MNF Sensor . . . . . . . . . . . . . . . . . . 196 6.2.6 Atomic Fluorescence MNF Sensor . . . . . . . . . . . . . . . . . . . 196 6.3 Application of Looped and Coiled MNF for Sensing . . . . . . . . . . 198 6.3.1 Ultra-Fast Direct Contact Gas Temperature Sensor . . . . 200 6.3.2 MCR Microfluidic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.4 Resonant Photonic Sensors Using MNFs for Input and Output Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.4.1 MNF/Microsphere and MNF/Microdisk Sensor . . . . . . . 203 6.4.2 MNF/Microcylinder and MNF/Microcapillary Sensors . 208 6.4.3 Multiple-Cavity Sensors Supported by MNFs . . . . . . . . . 210 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7

More Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.1 Optical Nonlinear Effects in MNFs . . . . . . . . . . . . . . . . . . . . . . . . 215 7.2 MNFs for Atom Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

1 Introduction

In the past 30 years, optical fibers with diameters larger than the wavelength of guided light have found wide applications including optical communication, sensing, power delivery and nonlinear optics[1−6] . For example, by transmission of light through total internal reflection in optical fibers, the power of light has been sent to travel across the sea for telecommunications[1,2] , to creep into buildings for safety monitoring[3,4] , to puncture tissues for laser surgery[5] , as well as many other applications ranging from illumination and imaging to astronomical research[7,8] . Recent advances in nanotechnology and the increasing demand for faster response, smaller footprint, higher sensitivity and lower power consumption have, however, spurred efforts for the miniaturization of optical fibers and fiber-optic devices[8−10] . Therefore, an important motivation for fabricating subwavelength-diameter optical fibers is their potential usefulness as building blocks in future micro- or nanometer-scale photonic components or devices and as tools for mesoscopic optics research. Also, it is always interesting to guide light and watch how it works on those scales that have not been tried yet.

1.1 A Brief History of Micro- and Nanofibers The history of the guided transmission of light can be traced back to the 19th century, when Daniel Colladon and John Tyndall directed beams of light at the path of water[7] , in which light was confined by the internal reflection due to the refractive index change at the water-air interface. In 1880, William Wheeling patented an invention for piping light through pipes relying on mirror reflection[11] . In this idea, light was redirected, branched and delivered using a pipe in the same way that water is poured into and carried along a pipe. On the other hand, shortly after Wheeling’s light pipe, Charles Vernon Boys, a British physicist, reported drawing very thin glass fibers from molten minerals using flying arrows in 1887[12] , which might represent the first written record of taper drawing glass fibers with micro- or nanoscale diameters.

2

1 Introduction

These fibers could be thinner than one micrometer, and were mentioned as “the finest threads” of glasses. Several years later, the approach for drawing these kinds of thin fibers was developed into one of the “laboratory arts”, as documented in the book On Laboratory Arts by Richard Threlfall[13] . However, at that time these “finest threads”, here we call them micro- or nanofibers (MNFs), were not prepared for light transmission, but for mechanical applications such as springs for galvanometers due to their high uniformity and excellent elasticity[12] . Also, due to their small dimensions, it was difficult to precisely determine the thickness of the fiber when its value went below the wavelength of visible light. To the best of the authors’ knowledge, one of the earliest examples of optical guiding in MNFs was reported in 1959 by Narinder S. Kapany, in which a fiber bundle consisting of numerous microand submicrometer-diameter fibers was used for transmission of images[14] . In 1960 Theodore Maiman invented the first laser[15] , and shortly afterwards Charles Kao and George Hockham proposed the possibility of achieving low optical loss in high-purity glasses in 1966[16] , which greatly advanced the establishment of fiber optics for the optical communications industry. From the 1970’s, along with a thriving fiber optics research industry, microfibers tapered from standard glass fibers (usually mentioned as fiber tapers or tapered fibers with waist diameters of several to tens of micrometers) started to play their role as optical waveguides[17−22] . Based on these microfibers, a number of possible applications including optical couplers[23−25] , filters[26,27] , sensors[28,29] , evanescent field amplification[30] and supercontinuum generation[31] were demonstrated. In 1999, a theoretical work on microfibers with subwavelength diameters was reported by J. Bures and R. Ghosh[32] , based on theoretical calculation. They predicted the enhanced power density of the evanescent field in the vicinity of the fiber, which might be used in atomic mirrors. In 2003, L. Tong and co-authors experimentally demonstrated low-loss optical waveguiding in MNFs with diameters far below the wavelength of the guided light[33] , which renewed research interests in optical MNFs as potential building blocks for miniaturized optical components and devices. A few years later, a number of works on the fabrication and/or properties of subwavelength-diameter MNFs were reported[34−62] , and a variety of MNF-based components or devices, ranging from resonators[63−73] , interferometers[36,74] , filters[75−77] and lasers[78−81] to sensors[82−95] , were demonstrated or proposed, together with many other MNF-based applications in nonlinear optics[96−106] and atom optics[107−115] . Besides the above-mentioned glass MNFs, there are a number of other free-standing one-dimensional fiber or wire-like micro- or nanostructures, ranging from crystalline whiskers to semiconductor nanowires and polymer MNFs[116−125] that have been extensively investigated and show potential for optical wave guiding. Among these structures, physically drawn polymer MNFs, although they were not initially targeted for light guidance, exhibit similar properties as glass MNFs regarding extraordinary uniformity and long

1.2 Concepts of MNFs and the Scope of this Book

3

length for low-loss optical waveguiding[126−131] , and are thus within the scope of this book.

1.2 Concepts of MNFs and the Scope of this Book To introduce the concept of an MNF, it is helpful to compare it with the principles of a standard glass fiber. Shown in Fig. 1.1 is a cross-section view of a typical step-index-profile optical fiber, which consists of two parts (the protective buffer layer is not shown here): a solid cylindrical core, surrounded by a cladding with relatively low refractive index. Depending on various applications, the diameter of the fiber ranges from tens of micrometer (e.g., for fiber-optic sensing) to larger than one millimeters (e.g., for laser power delivery), and correspondingly the core diameter ranges from several micrometers to hundreds of micrometers. In a standard single-mode fiber for optical communications, e.g., Corning SMF28, the fiber and core diameters are 9 and 125 µm, respectively. As illustrated in Fig. 1.2(a), in the view of ray optics, the light conducted along the fiber is confined and guided inside the fiber by means of total internal reflection, as has been well depicted in many textbooks when introducing fiber optics.

Fig. 1.1. Cross-section view of a standard optical fiber.

It is noticeable that in the reflection region where light hits the interface, a certain fraction of light penetrates the boundary of the high-index core, propagates as an evanescent field in the cladding, and finally comes back into the fiber core, forming the reflected ray with a slight shift in the axial direction known as the Goos-Hanchen shift[132,133] . When the diameter of the core decreases, the light penetrates the boundary more frequently, and the probability of propagation outside the core (as evanescent waves) increases, as shown in Fig. 1.2(b). When the core diameter goes below the wavelength of the light, a considerable fraction of the power of the light propagates outside the core, as illustrated in Fig. 1.3. In such a case, the diameter of the fiber core is not thick enough for generating a steady-state electromagnetic field through the interference of reflected light rays, which means that ray optics (as depicted

4

1 Introduction

Fig. 1.2. Optical waveguiding in a standard optical fiber relying on internal total reflection with (a) relatively large and (b) relatively small core diameters.

Fig. 1.3. Optical waveguiding in a micro- or nanofiber with core diameter below the wavelength of the propagating light.

in Fig. 1.2) is no longer applied, and the light ray should be treated as an electromagnetic field. For a fiber with a core diameter below the wavelength of the light, a high index-contrast between the core and the cladding is desired for obtaining a certain degree of optical confinement[34,134] , which is required for light waveguiding in practical applications of these sub-wavelength-diameter optical fibers. Since the refractive index of an optical fiber (mostly made of silica) is not high, low-index media (or environment) such as a vacuum, air, water and polymers are usually used as claddings.

1.2 Concepts of MNFs and the Scope of this Book

5

Similar to the top-down drawing technique for conventional fiber fabrication, the MNFs are usually fabricated by physically drawing viscous melts or solutions, as illustrated in Fig. 1.4. Usually, materials used for drawing MNFs are glass fibers, bulk glasses or polymers[33,35−37,43−48,126−131] . When the starting material is partially melted by heating or dissolved by solvents, it is possible to obtain appropriate viscosity for MNF drawing at a certain area, and high-quality MNFs with diameters down to 30 nm can be obtained when a proper drawing speed is applied. Compared with many other techniques that have been used for MNF or other one-dimensional nanostructure fabrication[116−120] , a physical drawing technique yields MNFs with unparalleled uniformities regarding sidewall smoothness and diameter uniformity. The excellent uniformity of the MNF does not only enable low optical waveguiding loss, but also bestows the MNF with high mechanical strength and flexibility. For example, Fig. 1.5(a) gives an SEM image of a 450-nm-diameter silica MNF (supported on a coated silicon wafer), clearly showing the extraordinary uniformity of the fiber. Fig. 1.5(b) gives an SEM image of a knotted 500-nm-diameter silica MNF. The fiber was first knotted to a size of about 50 µm under an optical microscope, and then transferred onto the sidewall of a human hair. No breakage was observed under these micromanipulations, indicating the high mechanical strength and flexibility of the taper-drawn glass MNF.

Fig. 1.4. Schematic illustration of physical drawing MNFs.

Fig. 1.5. SEM images of typical MNFs. (a) A 450-nm-diameter silica MNF placed on a coated silicon wafer. (b) A knotted 500-nm-diameter silica MNF placed on a human hair.

6

1 Introduction

Due to its tiny endface, the lens-focus and butt-coupling methods for light launching in the conventional fiber are not applicable to the MNF. Instead, taper-squeeze or evanescent coupling is usually employed due to its high efficiency and convenience for managing light in subwavelength-diameter fibers. As shown in Fig. 1.6, for MNFs directly drawn from the starting fiber, taper squeeze is a simple approach for squeezing light from the thick fiber into the thin MNF; while for freestanding MNFs, evanescent coupling between two closely contacting MNFs has proved efficient and convenient for sending light from the launching fiber to the target MNF[33,41,46,56] .

Fig. 1.6. Taper-focus and evanescent coupling approaches for optical launching of MNFs.

Compared with that in a conventional optical fiber, the high index contrast and subwavelength diameter of the MNF make it possible to guide light with a number of interesting properties, such as tight optical confinement[34] , a high fraction of evanescent fields[34] , manageable large waveguide dispersion[34,52] , field enhancement[32] and low optical loss through sharp bends[41] , making the MNF highly potential for a variety of photonics applications. For example, when guiding a 633-nm-wavelength light, a 450-nm-diameter silica MNF confining 80% power inside the fiber core (see Fig. 1.7(a)), makes it possible to guide the light through a 5-µm-radius bend with negligible bending loss[41] , which is desired for the miniaturization of optical circuits and components. When the fiber diameter decreases to 200 nm, more than 90% power moves out of the fiber and is guided as evanescent waves (Fig. 1.7(b)), which may offer MNF-based optical sensing with high sensitivity. In addition, the lowdimension cross section, manageable dispersion and field enhancement have proved helpful in achieving nonlinear optical effects with low threshold on a miniaturized scale, and the abounding evanescent fields have been found useful for atom trapping and guidance with great versatility. This book is intended to provide a general introduction to up-to-date research on subwavelength-diameter optical MNFs. Starting from a brief overview of optical MNFs in this chapter, Chapter 2 is devoted to theo-

References

7

Fig. 1.7. Calculated Poynting vectors of silica MNFs guiding 633-nm-wavelength light with diameters of (a) 450 nm and (b) 200 nm.

retical waveguiding properties of MNFs that may provide a comprehensive understanding of light guiding in subwavelength-diameter MNFs, as well as evanescent coupling between two MNFs and the theory of MNF-based interferometers and resonators. Chapter 3 introduces typical techniques for physical drawing glass and polymer MNFs. Electron microscope investigations of asfabricated MNFs are also presented. Chapter 4 is complementary to Chapter 2, offering experimental properties of MNFs including micromanipulation, mechanical strength, optical losses and effects of the substrate, which are critical to practical usage of MNFs. Chapter 5 introduces various MNF-based photonic components and devices including linear waveguides, waveguide bends, optical couplers, interferometers, resonators, filters and lasers, that have been reported so far. MNF optical sensors, as one of the most widely concerned applications of MNFs, are introduced in Chapter 6. Finally, Chapter 7 provides a brief summary of applications of MNFs in nonlinear optics, atom optics and other possibilities. Although we are trying to provide a comprehensive account of this topic, we do not promise a complete coverage of MNF research. We apologize that we cannot cover all the work in this book. Finally, since optical MNFs or nanowires are frontiers of broad areas including photonics, nanotechnology and materials science, we hope that those who are working in these areas will benefit in some measure from this book, and find it interesting and stimulating.

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72. F. Xu, G. Brambilla, Manufacture of 3-D microfiber coil resonators, IEEE Photon. Technol. Lett. 19, 1481–1483 (2007). 73. M. Sumetsky, Basic elements for microfiber photonics: Micro/nanofibers and microfiber coil resonators, J. Lightwave Technol. 26, 21–27 (2008). 74. Y. H. Li, L. M. Tong, Mach-Zehnder interferometers assembled with optical microfibers or nanofibers, Opt. Lett. 33, 303–305 (2008). 75. X. S. Jiang, Y. Chen, G. Vienne, L. M. Tong, All-fiber add-drop filters based on microfiber knot resonators, Opt. Lett. 32, 1710–1712 (2007). 76. Y. Chen, Z. Ma, Q. Yang, L. M. Tong, Compact optical short-pass filters based on microfibers, Opt. Lett. 33, 2565–2567 (2008). 77. Y. Wu, X. Zeng, C. L. Hou, J. Bai, G. G. Yang, A tunable all-fiber filter based on microfiber loop resonator, Appl. Phys. Lett. 92, 191112 (2008). 78. Y. H. Li, G. Vienne, X. S. Jiang, X. Y. Pan, X. Liu, P. F. Gu, L. M. Tong, Modeling rare-earth doped microfiber ring lasers, Opt. Express 14, 7073–7086 (2006). 79. X. S. Jiang, Q. Yang, G. Vienne, Y. H. Li, L. M. Tong, J. J. Zhang, L. L. Hu, Demonstration of microfiber knot laser, Appl. Phys. Lett. 89, 143513 (2006). 80. X. S. Jiang, Q. H. Song, L. Xu, J. Fu, L. M. Tong, Microfiber knot dye laser based on the evanescent-wave-coupled gain, Appl. Phys. Lett. 90, 233501 (2007). 81. A. AlOrainy, Y. H. Li, L. M. Tong, Evaluation of quenching effects on silicabased erbium doped fiber micro-ring lasers, Opt. Commun. 281, 3000–3003 (2008). 82. J. Y. Lou, L. M. Tong, Z. Z. Ye, Modeling of silica nanowires for optical sensing, Opt. Express 13, 2135–2140 (2005). 83. P. Polynkin, A. Polynkin, N. Peyghambarian, M. Mansuripur, Evanescent fieldbased optical fiber sensing device for measuring the refractive index of liquids in microfluidic channels, Opt. Lett. 30, 1273–1275 (2005). 84. J. Villatoro, D. Monz´ on-Hern´ andez, Fast detection of hydrogen with nano fiber tapers coated with ultra thin palladium layers, Opt. Express 13, 5087–5092 (2005). 85. W. Liang, Y. Huang, Y. Xu, R. K. Lee, A. Yariv, Highly sensitive fiber Bragg grating refractive index sensors, Appl. Phys. Lett. 86, 151122 (2005). 86. S. S. Wang, X. Y. Pan, L. M. Tong, Modeling of nanoparticle induced Rayleigh– Gans scattering for nanofiber optical sensing, Opt. Commun. 276, 293–297 (2007). 87. M. Sumetsky, R. S. Windeler, Y. Dulashko, X. D. Fan, Optical liquid ring resonator sensor, Opt. Express 15, 14376–14381 (2007). 88. L. Shi, Y. H. Xu, W. Tan, X. F. Chen, Simulation of optical microfiber loop resonators for ambient refractive index sensing, Sensors 7, 689–696 (2007). 89. F. Xu, P. Horak, G. Brambilla, Optical microfiber coil resonator refractometric sensor, Opt. Express 15, 7888–7893 (2007). 90. F. Warken, E. Vetsch, D. Meschede, M. Sokolowski, A. Rauschenbeutel, Ultrasensitive surface absorption spectroscopy using sub-wavelength diameter optical fibers, Opt. Express 15, 11952–11958 (2007). 91. X. Guo, L. M. Tong, Supported microfiber loops for optical sensing, Opt. Express 16, 14429–14434 (2008). 92. F. Xu, V. Pruneri, V. Finazzi, G. Brambilla, An embedded optical nanowire loop resonator refractometric sensor, Opt. Express 16, 1062–1067 (2008).

12

1 Introduction

93. L. Zhang, F. X. Gu, J. Y. Lou, X. F. Yin, L. M. Tong, Fast detection of humidity with a subwavelength diameter fiber taper coated with gelatin film, Opt. Express 16, 13349–13353 (2008). 94. G. Vienne, P. Grelu, X. Y. Pan, Y. H. Li, L. M. Tong, Theoretical study of microfiber resonator devices exploiting a phase shift, J. Opt. A: Pure Appl. Opt. 10, 025303 (2008). 95. F. Xu, G. Brambilla, Demonstration of a refractometric sensor based on optical microfiber coil resonator, Appl. Phys. Lett. 92, 101126 (2008). 96. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, M. W. Mason, Supercontinuum generation in submicron fibre Waveguides, Opt. Express 12, 2864–2869 (2004). 97. M. A. Foster, K. D. Moll, A. L. Gaeta, Optimal waveguide dimensions for nonlinear interactions, Opt. Express 12, 2880–2887 (2004). 98. M. A. Foster, A. L. Gaeta, Ultra-low threshold supercontinuum generation in sub-wavelength waveguides, Opt. Express 12, 3137–3143 (2004). 99. M. Kolesik, J. V. Moloney, Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations, Phys. Rev. E 70, 036604 (2004). 100. M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, A. L. Gaeta, Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation, Appl. Phys. B 81, 363–367 (2005). 101. M. A. Foster, A. L. Gaeta, Q. Cao, D. Lee, R. Trebino, Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires, Opt. Express 13, 6848–6855 (2005). 102. A. Zheltikov, Gaussian-mode analysis of waveguide-enhanced Kerr-type nonlinearity of optical fibers and photonic wires, J. Opt. Soc. Am. B 22, 1100–1104 (2005). 103. R. R. Gattass, G. T. Svacha, L. M. Tong, E. Mazur, Supercontinuum generation in submicrometer diameter silica fibers, Opt. Express 14, 9408–9414 (2006). 104. G. Vienne, P. Grelu, Y. H. Li, L. M. Tong, Observation of a nonlinear microfiber resonator, Opt. Lett. 33, 1500–1502 (2008). 105. M. A. Foster, A. C. Turner, M. Lipson, A. L. Gaeta, Nonlinear optics in photonic nanowires, Opt. Express 16, 1300–1320 (2008). 106. S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, M. S. Shahriar, Observation of nonlinear optical interactions of ultralow levels of light in a tapered optical microfiber embedded in a hot rubidium vapor, Phys. Rev. Lett. 100, 233602 (2008). 107. V. I. Balykin, K. Hakuta, F. Le Kien, J. Q. Liang, M. Morinaga, Atom trapping and guiding with a subwavelength-diameter optical fiber, Phys. Rev. A 70, 011401 (2004). 108. F. Le Kien, V. I. Balykin, K. Hakuta, Atom trap and waveguide using a twocolor evanescent light field around a subwavelength-diameter optical fiber, Phys. Rev. A 70, 063403 (2004) . 109. F. Le Kien, S. Dutta Gupta, V. I. Balykin, K. Hakuta, Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light to guided modes, Phys. Rev. A, 72, 032509 (2005). 110. F. Le Kien, V. I. Balykin, K. Hakuta, State-insensitive trapping and guiding of cesium atoms using a two-color evanescent field around a subwavelengthdiameter fiber, J. Phys. Soc. Japan 74, 910–917 (2005).

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2 Optical Waveguiding Properties of MNFs: Theory and Numerical Simulations

For optical and photonic applications, waveguiding behaviors are the most concerned properties of MNFs, which have been extensively investigated both theoretically and experimentally. This chapter introduces theoretical properties of MNFs based on analytical and numerical approaches.

2.1 Basic Guiding Properties of Ideal MNFs Although the micro/nanofiber (MNF) was not used for optical guiding until the 1960’s, the waveguiding theory for subwavelength-diameter cylindrical fibers has long been well established, based on Maxwell’s equations and boundary conditions[1] . For theoretical investigation, the special advantage of a cylindrical fiber (with a perfect circular cross section) is the possibility to obtain analytical solutions of all the modes supported by the waveguide, greatly facilitating the study and understanding of the guiding properties of MNFs. 2.1.1 Mathematic Model For basic investigation, a straight fiber is assumed to have a circular crosssection, a smooth sidewall, a uniform diameter and an infinite air-cladding with a step-index profile. The length of the fiber is large enough to establish the spatial steady state, and the diameter of the fiber (d) is not very small (e.g., d > 10 nm), so that the permittivity (ε) and permeability (μ) of the bulk material can be used to describe the responses of a dielectric medium to an incident electromagnetic field. With the above-mentioned assumptions, the mathematic model of an airclad fiber is shown in Fig. 2.1, in which the refractive indices of the fiber material and air are n1 and n2 respectively. The index profile of the waveguiding system is then expressed as

16

2 Optical Properties of MNFs: Theory and Numerical Simulations

n(r) =



n1 , 0 < r < a, n2 , a  r < ∞

(2.1)

where a is the radius of the fiber.

Fig. 2.1. Index profile of an air-clad MNF

Usually, both the fiber material and the cladding medium are dielectric, and are used in their transparent spectral range. Therefore, the MNF can be treated as a non-dissipative and source free waveguide, and Maxwell’s equations can be reduced to the following Helmholtz equations: (∇2 n2 k 2 − β 2 )e = 0, (∇2 n2 k 2 − β 2 )h = 0

(2.2)

where k = 2π/λ, and β is the propagation constant. Exact solutions for this model have been well investigated[2] , yielding the eigenvalue equations for the HEvm and EHvm modes 

′

′

Kv (W ) Jv (U ) + U Jv (U ) W Kv (W )



′

′

Jv (U ) n2 K (W ) + 22 v U Jv (U ) n1 W Kv (W )



=



vβ kn1

2 

V UW

4 (2.3)

for the T E0m modes: J1 (U ) K1 (W ) + =0 U J0 (U ) W K0 (W )

(2.4)

and for the T M0m modes: n21 J1 (U ) n22 K1 (W ) + =0 U J0 (U ) W K0 (W )

(2.5)

where Jv is the Bessel function of the first kind, and Kv is the modified d(β 2 −k02 n22 )1/2 d(k02 n21 −β 2 )1/2 , W = , Bessel function of the second kind, U = 2 2 V =

k0 ·d(n21 −n22 )1/2 , 2

and d = 2a is the diameter of the MNF.

2.1 Basic Guiding Properties of Ideal MNFs

17

For an air-clad silica fiber, the index of the air is 1.0, and the index of silica (n1 ) can be obtained by the Sellmeier-type dispersion formula (at room temperature)[3] n2 − 1 =

λ2

0.4079426λ2 0.8974794λ2 0.6961663λ2 (2.6) + 2 + 2 2 2 − (0.0684043) λ − (0.1162414) λ − (9.896161)2

where λ is in µm. By numerically solving the eigenvalue Eqs. (2.3)–(2.5) with refractive indices of air (n2 =1.0) and silica (e.g., n1 =1.46 at λ = 633 nm; and n1 = 1.44 at λ = 1.55 µm), propagation constants (β) of guiding modes supported by these fibers can be obtained. For example, shown in Fig. 2.2 is diameter-dependent β of silica MNFs at the wavelength of 633 nm, where the fiber diameter (d) k ·d(n2 −n2 )1/2

1 2 is directly related to the V -number (V = 0 ). It shows that when 2 the fiber diameter reduced to a certain value (denoted as dSM , corresponding to V =2.405), only the HE11 mode (solid line) exists, corresponding to the single mode operation. When d exceeds dSM , high-order modes appear (denoted as dotted lines). For reference, propagation constants (β) of HE11 modes (the fundamental modes) of several types of glass MNFs operated at 633-nm wavelength are illustrated in Fig. 2.3.

Fig. 2.2. Calculated propagation constant (β) of air-clad silica MNF at 633-nm wavelength. Solid line: fundamental mode. Dotted lines: high-order modes. Dashed line: critical diameter for single-mode operation (dSM ). (Adapted from Ref. [4], with permission from the Optical Society of America)

2.1.2 Single-mode Condition and Fundamental Modes Similar to the weakly guiding optical fibers, the single-mode condition of an airclad “strong guiding” MNF can be obtained from Eqs. (2.4) and (2.5) as[2]

18

2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.3. Calculated propagation constants (β) for HE 11 modes (the fundamental modes) of glass nanowires with refractive indices of 1.46 (silica), 1.48 (fluoride), 1.54 (phosphate), 1.89 (germinate) and 2.02 (tellurite) respectively. A circle marked on each curve locates single-mode cut-off diameter of the nanowire. (Adapted from Ref. [5], with permission from the Optical Society of America)

V =

1 π · dSM  2 · n1 − n22 2 ≈ 2.40 λ0

(2.7)

Single mode conditions (represented by dSM for single-mode operation) of the air- or water-clad silica MNFs with respect to the wavelength and fiber diameters, are illustrated in Fig. 2.3, with dSM for typical wavelength listed in Table 2.1. The wavelength-dependent refractive index of water is obtained from Ref. [6]. n2 − 1 ¯2 ¯ 2 T¯ + ξ4 /λ (1/¯ ρ) = ξ0 + ξ1 ρ¯ + ξ2 T¯ + ξ3 λ n2 + 2 ξ6 ξ5 + ¯ 2 ¯ 2 + ¯ 2 ¯ 2 + ξ7 ρ¯2 λ − λU V λ − λIR

(2.8)

where T = T /T ∗ , with reference temperature T ∗ = 273.15 K and T the temperature of water; ρ = ρ/ρ∗ , with reference density ρ∗ = 1000 kg·m−3 and ρ the density of water; λ = λ/λ∗ , with reference wavelength λ∗ = 0.589 µm and λ the wavelength of light. The coefficients ξ0 to ξ7 , and the constants λUV and λIR are given as follows: ξ2 = −3.7323499610 × 10−3 , ξ0 = 0.24425773, ξ1 = 9.74634476 × 10−3 , ξ3 = 2.68678472 × 10−4 , ξ4 = 1.58920570 × 10−3 , ξ5 = 2.45934259 × 10−3 , ¯ UV = 0.2292020, λ ¯ IR = 5.432737. ξ6 = 0.900704920, ξ7 = −1.66626219 × 10−2 , λ

2.1 Basic Guiding Properties of Ideal MNFs

19

Table 2.1. dSM for air- and water-clad silica MNFs at typical wavelengths (Index of air=1.0, and index of water is obtained after Wavelength (nm) 325 633 1064 Refractive index of silica 1.482 1.457 1.450 Refractive index of air 1.00 1.00 1.00 Refractive index of water 1.355 1.333 1.325 dSM in air (nm) 228 457 776 dSM in water (nm) 415 824 1383

Ref. [5]) 1550 1.444 1.00 1.316 1139 1996

In Fig. 2.4, the region beneath the lines (solid line for the air-clad fiber and dashed line for the water-clad fiber) corresponds to the single-mode region. For example, at the wavelength of 633 nm (He-Ne laser), for an air-clad fiber dSM is 457 nm; while for a water-clad fiber dSM increases to 824 nm due to the relatively lower index-contrast between the silica and water. For reference, dSM for other types of glass fibers operated at 633-nm wavelength is also provided in Table 2.2.

Fig. 2.4. Single-mode condition for air- or water-clad silica MNFs.

Table 2.2. dSM for air-clad glass MNFs at 633-nm wavelength (Index of air=1.0) Fiber material Fluoride Phosphate Germinate Tellurite Typical refractive index 1.48 1.54 1.89 2.05 444 414 304 275 dSM in air (nm)

When the MNF is thin enough to be single-mode, only the fundamental modes (i.e., HE 11 modes) are supported by the MNF. In this case, Eq. (2.3) becomes

20



2 Optical Properties of MNFs: Theory and Numerical Simulations ′

′

K1 (W ) J1 (U ) + U J1 (U ) W K1 (W )



′

′

J1 (U ) n2 K (W ) + 22 1 U J1 (U ) n1 W K1 (W )



=



β kn1

2 

V UW

4 (2.9)

For a MNF with a very small V -number, V ≪ 1, Eq. (2.9) can be solved in the form[21] : 2πn2 γ 2λ + λ 4πn2

2 1.123 n + n2 λ2 n2 + n2 γ= exp 1 2 2 − 2 1 2 2 2 a 8n2 n2 (n1 − n2 ) (2πa)2

β=

(2.10) (2.11)

This analytical approximation is important for thin MNF when the value of γ is exponentially small and its numerical calculation becomes problematic. In particular, for a glass MNF in air, when n1 = 1.45 and n2 = 1, we have   1.655 λ2 γ= exp −0.0713 2 a a

(2.12)

Fig. 2.5 demonstrates the accuracy of Eq. (2.11) for the wavelength λ = 1530 nm. It is seen that Eq. (2.11) gives a relative error of less than 10% for an MNF radius of less than 300 nm and becomes very accurate when the MNF radius becomes less than 200 nm.

Fig. 2.5. Comparison of transversal propagation constant, γ, calculated by Eq. (2.11) for a silica MNF with exact numerical solution of Eq. (2.9) for the wavelength λ = 1530 nm.

The electric-field components of the fundamental modes are expressed as[2] inside the core (0 < r < a):

2.1 Basic Guiding Properties of Ideal MNFs

  + a2 J2 Uar er = − · f1 (φ) J1 (U )     a1 J0 Uar − a2 J2 Uar eφ = − · g1 (φ) J1 (U )   −iU J1 Uar ez = · f1 (φ) aβ J1 (U ) a1 J 0

 Ur  a

21

(2.13)

(2.14)

(2.15)

and outside the core (a  r < ∞):   − a2 K2 War · f1 (φ) K1 (W )     a1 K0 War + a2 K2 War · g1 (φ) K1 (W )   K1 War · f1 (φ) K1 (W )

U a1 K0 er = − W eφ = − ez =

U W

−iU aβ

Wr  a

(2.16)

(2.17)

(2.18)

F1 −1 F1 −1+2Δ F2 −1 , 2 , a3 = 2 , a5 = 2   F2 +1 F1 +1 F1 +1−2Δ UW 2 , F1 = V [b1 + (1 − 2Δ)b2 ], F2 = a2 = 2 , a4 = 2 , a6 = 2 J0 (U ) J2 (U ) K0 (W ) K2 (W ) 1 V 1 1 U W b1 +b2 , b1 = 2U J1 (U ) − J1 (U ) , b2 = 2U K1 (W ) − K1 (W ) . [2]

where f1 (φ) = sin(φ), g1 (φ) = cos(φ), a1 =

Since the h-components can be obtained from e-components , they are not provided here. When substituting numerical solutions of β obtained from Eq. (2.9) into electromagnetic components in Eqs. (2.13)–(2.18), the total electromagnetic fields of the HE11 modes are obtained as ⎧ ⎪ ⎪ ¯ φ, z) = (er rˆ + eφ φˆ + ez zˆ)eiβz e−iωt , ⎪ E(r, ⎨

⎪ ⎪ ⎪ ¯ φ, z) = (hr rˆ + hφ φˆ + hz zˆ)eiβz e−iωt ⎩ H(r,

(2.19)

Normalized electric components of the HE 11 modes in cylindrical coordination for air-clad silica MNF at 633-nm wavelength are shown in Fig. 2.6. For reference, Gaussian profiles (dashed line) are provided in the radial distributions, and the electric field of an MNF with a diameter of dSM is also provided as a dotted line. Compared to the Gaussian profile, air-clad silica MNF shows much tighter field confinement within a certain diameter range (e.g., around 400 nm), due to the high index contrast between the air and

22

2 Optical Properties of MNFs: Theory and Numerical Simulations

silica. When the fiber diameter reduces to a certain degree (e.g., 200 nm), the field extends to a far distance with considerable amplitude, indicating that the majority of the field is no longer tightly confined inside or around the fiber.

Fig. 2.6. Electric components of HE 11 modes of air-clad silica MNF at 633-nm wavelength with different diameters in cylindrical coordination. Normalizations are applied as: εe r (r=0)=1 and eΦ (r=0)=1. Fiber diameters are arrowed to each curve in units of nm. (Adapted from Ref. [4], with permission from the Optical Society of America)

2.1.3 Fractional Power Inside the Core and Effective Diameter For an ideally straight and uniform MNF with perfect cylindrical symmetry, there is no net flow of energy in the radial (r) or azimuthal (φ) directions. The z-components of Poynting vectors are obtained as[2] inside the core (0 < r < a):

Sz1

 12

    kn21 Ur Ur 2 2 + a2 a4 J2 + × a 1 a 3 J0 βJ12 (U ) a a   

 Ur Ur 1 − F1 F2 J2 cos(2φ) J0 (2.20) 2 a a

1 = 2



ε0 μ0

2.1 Basic Guiding Properties of Ideal MNFs

23

and outside the core (a  r < ∞):

Sz2

 12

    kn21 Wr Wr U2 2 2 + a − × a a K a K 1 5 0 2 6 2 βK12 (W ) W 2 a a    

Wr Wr 1 − 2Δ − F1 F2 (2.21) K0 K2 cos(2φ) 2 a a

1 = 2



ε0 μ0

Fig. 2.7 shows the Poynting vectors of 200- and 400-nm-diameter silica MNFs operated at 633-nm wavelength, where the mesh-profile stands for propagating fields inside the MNF, and the gradient profile stands for evanescent fields guided in air. As one can see, while a 400-nm-diameter MNF confines major power inside its silica core, a 200-nm-diameter MNF leaves a large amount of light guided outside as evanescent waves.

Fig. 2.7. Z -direction Poynting vectors of silica MNFs at 633-nm wavelength with diameters of (a) 400 nm and (b) 200 nm. Mesh, field inside the core. Gradient, field outside the core. (Adapted from Ref. [4], with permission from the Optical Society of America)

With the Poynting vectors obtained in Eqs. (2.20) and (2.21), the fractional power inside the core (η) can be obtained as follows:

η = a 0

a

Sz1 dA ∞ Sz1 dA + a Sz2 dA 0

(2.22)

where dA = r · dr · dφ, and η represents the percentage of the confined light power inside the solid core. η calculated as a function of the fiber diameter (d) for an air-clad silica MNF operated at 633-nm and 1.5-µm wavelengths is shown in Fig. 2.8. It shows that at the critical diameter (dSM , dashed line), η is about 80% in both the 633-nm and the 1.5-µm wavelength cases. The diameter for confining 90%

24

2 Optical Properties of MNFs: Theory and Numerical Simulations

energy inside the core of the fundamental mode is about 566 nm (633-nm wavelength) and 1342 nm (1.5-µm wavelength), while the diameter for confining 10% energy is 216 nm (633-nm wavelength) and 513 nm (1.5-µm wavelength). Tight confinement is helpful for reducing the modal width and increasing the integrated density of the optical circuits with less cross-talk[2,7] , while weaker confinement will be beneficial for energy exchange between MNFs within a short interaction length[8] , as well as for developing high-sensitivity optical sensors[9] .

Fig. 2.8. Fractional power of the fundamental modes inside the core of an air-clad silica MNF operated at (a) 633-nm and (b) 1.5-µm wavelength. Dashed line, critical diameter for single mode operation. (Adapted from Ref. [4], with permission from the Optical Society of America)

For a more intuitive image of the power distribution in the radial direction, Fig. 2.9 provides the calculated effective diameters (deff ) of the 633nm-wavelength light guided by the MNFs; for comparison, the real diameters (dreal ) of the fibers are provided as dotted lines. Here the deff is defined as the diameter within which 1–e2 (about 86.5%) of the total power is confined and is obtained from Eqs(2.23). ⎧  12 deff ⎪ Sz1 dA ⎪ 0 ⎪  ⎪ ⎨ α S dA +  ∞ S dA = 86.5%, (if z1 z2 0 α  21 deff α ⎪ ⎪ Sz1 dA + α Sz2 dA ⎪ ⎪ ∞ = 86.5%, (if ⎩ 0 α Sz1 dA + α Sz2 dA 0

1 2 deff

 a) (2.23)

1 2 deff

> a)

It shows that, when the fiber diameter is very small, deff increases drastically with the decreasing of the fiber diameter, for example, at 633-nm wavelength, deff of a 200-nm-diameter MNF is about 2.3 µm, which is over 10 times larger than the real fiber diameter (d); with increasing d, deff decreases until it reaches a minimum value (deff−min ); after the minimum point, deff increases gradually with the increasing of the fiber diameter. The two lines (solid line

2.1 Basic Guiding Properties of Ideal MNFs

25

for deff and dotted line for dreal ) intersect near the single-mode cutoff diameter (dSM ), and dreal exceeds deff thereafter. Practically, the intersection point may represent the usable fiber diameter for the minimum effective modal diameter. In addition, for an MNF with a real diameter much smaller than the light wavelength (e.g., a 200-nm-diameter working at 633-nm wavelength), maintaining a steady guiding field should be difficult, any small deviation from the ideal condition (such as surface contamination, diameter fluctuation and micro bend) will lead to significant radiation loss of the propagating light. However, the high susceptibility of such an MNF may provide high sensitivity for optical sensing.

Fig. 2.9. Effective diameters (deff ) of the light fields of the fundamental modes of an air-clad silica MNF operated at (a) 633-nm wavelength, and (b) 1.5-µm wavelength. Solid line, deff ; dotted line, real diameter; dashed line,dSM . (Adapted from Ref. [4], with permission from the Optical Society of America)

2.1.4 Group Velocity and Waveguide Dispersion The group velocity of the HE11 mode for the MNF can be obtained as[2]

vg =

cβ n21 k[1 − 2Δ(1 − η)]

(2.24)

Diameter-dependent group velocities of HE11 modes of air-clad silica MNF are calculated at 633-nm and 1500-nm wavelengths. As shown in Fig. 2.10, when the fiber diameter (d) is very small (e.g., 300 nm at 1500-nm wavelength), vg approaches the light speed (c) in air (or vacuum) since most of the guided energy is propagated in air. When d increases, more and more energy enters the fiber core, which increases the effective index of the MNF and reduces vg . As the fiber diameter increases, vg decreases until it reaches a minimum value (e.g., 0.637c at 633-nm wavelength) that is smaller than

26

2 Optical Properties of MNFs: Theory and Numerical Simulations

c/n1 (i.e., 0.688c at 633-nm wavelength, the group velocity of a plane wave in silica), which is reasonable when the majority of the light energy is guided inside the fiber core in a form other than the plane wave. After the minimum, vg increases slowly with d and finally approaches c/n1 at large values of d.

Fig. 2.10. Diameter-dependent group velocities of the fundamental modes of airclad silica MNF at 633-nm and 1.5-µm wavelengths. (Adapted from Ref. [4], with permission from the Optical Society of America)

Wavelength dependence of the group velocity for various fiber diameters are also obtained from Eq. (2.24). As shown in Fig. 2.11, similar to the diameter-dependent behavior, for a given fiber diameter (d), vg approaches c when the wavelength (λ) is very large and c/n1 when λ is very small, with a minimum value smaller than c/n1 .

Fig. 2.11. Wavelength-dependent group velocities of the fundamental modes of airclad silica MNFs. Fiber diameters are labeled on each curve in units of nm. (Adapted from Ref. [4], with permission from the Optical Society of America)

From the group velocity obtained above, dispersion (D) of the MNF is obtained as[10]

2.1 Basic Guiding Properties of Ideal MNFs

D=

d(vg−1 ) dλ

27

(2.25)

Diameter- and wavelength-dependent waveguide dispersions of air-clad silica MNFs are illustrated in Fig. 2.12 and Fig. 2.13. For reference, in Fig. 2.13,

Fig. 2.12. Diameter-dependent waveguide dispersion of fundamental modes of airclad silica MNFs at 633-nm and 1.5-µm wavelengths. (Adapted from Ref. [4], with permission from the Optical Society of America)

Fig. 2.13. Wavelength-dependent waveguide dispersion of fundamental modes of air-clad silica MNFs for various fiber diameters (the fiber diameter is labeled on each curve in units of nm). Material dispersion is plotted on dotted line. (Adapted from Ref. [4], with permission from the Optical Society of America)

28

2 Optical Properties of MNFs: Theory and Numerical Simulations

material dispersions of fused silica calculated from Eq. (2.6) are also provided. It shows that the waveguide dispersion D of the MNF can be very large compared with those of weakly guiding fibers and bulk material. For example, when operated at 633-nm wavelength, a 270-nm-diameter silica MNF shows a negative dispersion as large as –4.9 ns·nm−1 ·km−1 (see Fig. 2.12). Furthermore, by choosing the appropriate fiber diameter, the waveguide dispersion (here is the combined material and waveguide dispersions) of an MNF can be made positive, zero, or negative within a given spectral range. For example, at 490-nm wavelength, dispersion of bulk silica is about –800 ps·nm−1 ·km−1 ; when it is made into a 400-nm-diameter MNF, the waveguide dispersion shifts to zero (see Fig. 2.13). Controlling light propagation and nonlinear effects by tailoring the dispersion of the waveguide is widely used in optical communications and nonlinear optics[11−14] , therefore the small diameter of the MNFs offers new possibilities for these purposes, as has been reported elsewhere [15−20] .

2.2 Theory of MNFs with Microscopic Nonuniformities The electromagnetic wave propagating along the optical waveguide experiences changes at nonuniformities. The short-range nonuniformity, which has a relatively small characteristic length L and a characteristic value ε, can be considered as perturbation and results in change, which scales with ε and L linearly or by the power law[2] . However, the perturbation theory fails for long-range perturbations when changes in the propagating mode are accumulated at a long spatial interval L and can scale with the perturbation in a quite complex way. The fiber tapers drawing process and MNFs [22,23,24,15,19,25] suggest that the fiber diameter changes smoothly and can be adiabatically slow as a function of the fiber length. Consequently, the amplitudes of transitions between transversal modes propagating along the adiabatic MNF, and in particular the amplitude of their radiation decay, should be exponentially small. In this subsection, calculation of the radiation loss caused by relatively small and smooth MNF nonuniformities is performed. 2.2.1 Basic Equations Consider the propagation of light along a smoothly deformed weakly guiding MNF of which diameter, d, is significantly less than the radiation wavelength λ. The fundamental mode in this situation is propagating mostly outside the weakly guiding MNF and is very sensitive to the MNF nonuniformity. Assume that the nonuniformity is axially symmetric with respect to the fiber axis z. In the cylindrical coordinates (ρ, ϕ, z), propagation of light outside the MNF can be described by the scalar wave equation:

2.2 Theory of MNFs with Microscopic Nonuniformities

Uzz + Uρρ + (1/ρ)Uρ + k 2 U = 0

29

(2.26)

where k = 2πn2 /λ, n2 is the refractive of the surrounding medium, and λ is the radiation wavelength in a vacuum. Here, for convenience, we have included the refractive index in the definition of k (compare with Eqs. (2.2)). Usually, the surrounding medium is air and k = 2π/λ as in the previous section. The weakly guiding condition, d ≪ λ, implies smallness of the local transversal component of the propagation constant in the evanescent region, γ(z), compared to the longitudinal component, β(z), i.e.,

β(z) =



k 2 + γ(z)2 ≈ k +

γ 2 (z) 2k

(2.27)

Functions β(z) and γ(z) can be determined from the MNF local parameters by the solution of transcendental Eq. (2.9). Adiabatically slow nonuniformity causes transmission from the fundamental mode to the radiation modes. The transversal propagation constant p of these radiation modes is much smaller than their longitudinal propagation constant β:

β=



k 2 − p2 ≈ k −

p2 2k

(2.28)

The solutions of the wave Eq. (2.26) with small transversal propagation odinger equation: constants satisfy the Schr¨ ikΨz = Hρ Ψ, 1 Hρ Ψ = −Ψρρ − Ψρ , Ψ (ρ, z) = U (ρ, z) exp(−ikz) ρ

(2.29)

Substitution of Eq. (2.29) for Eq. (2.26) is also known as paraxial approximation[26] and parabolic equation method[27] . In the region of the nonuniformity, γ(z) slowly depends on z, as illustrated in Fig. 2.14. It is assumed that the fiber is uniform at z → ±∞ so that −−−− →γ γ(z)− z→±∞ 0

Fig. 2.14. Illustration of a nonuniform MNF

(2.30)

30

2 Optical Properties of MNFs: Theory and Numerical Simulations

Let us introduce the local transverse fundamental mode, Ψ (0) (ρ, z), and radiation modes, Ψp (ρ, z), which satisfy equations: Hρ Ψ (0) = −γ(z)2 Ψ (0) Hρ Ψp = p2 Ψp

(2.31)

Normalized solutions of these equations are 1

Ψ (0) (ρ, z) = 2 2 γ(z)K0 [λ(z)ρ]

(2.32)

and

Ψp (ρ, z) =



− 12 γ(z) 1 + (4/π2 ) ln2 2 p

  γ(z) Y0 (pρ) × J0 (pρ) − (2/π) ln p  p  21 

(2.33)

Here K0 (x), J0 (x), and Y0 (x) are the Bessel functions and the radiation mode Ψp (ρ, z) is chosen to be orthogonal to the fundamental mode Ψ (0) (ρ, z). Using the method of the coupling wave equations [2] , the solution of Eq. (2.29) can be found in the form (0)

Ψ (ρ, z) = a

(z)Ψ

(0)

(ρ, z) +



∞

ap (z)Ψp (ρ, z)dp

(2.34)

0

where a(0) (z) is the amplitude of that remaining in the fundamental mode Ψ (0) (ρ, z) and ap (z) is the amplitude of transition from the fundamental mode to the radiation mode Ψp (ρ, z). Substitution of Eq. (2.34) into Eq. (2.29) yields the following coupled wave integro-differential equations for the amplitudes ap (z)[2] : ⎛ ⎞ z ∞ ∞ 1 1 1 dap (z) ∗ ⎝ ⎠ = Rp0 (z) 1 + dz Rp0 (z)ap (z)dp + Rpp′ ap′ (z)dp′ dz k k k −∞

0

0

0