Subwavelength and Nanometer Diameter Optical Fibers

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Author: Limin Tong | Michael Sumetsky

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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,

VI

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

VIII

Contents

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

50

Contents

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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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(k02 n21 −β 2 )1/2 d(β 2 −k02 n22 )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

0.6961663λ2 0.4079426λ2 0.8974794λ2 + 2 + 2 (2.6) 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 ξ5 ξ6 + ¯ 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

21

Ur a

(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)

F2 −1 F1 −1 F1 −1+2Δ , 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

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

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

1 = 2

ε0 μ0

12

2.1 Basic Guiding Properties of Ideal MNFs

23

and outside the core (a r < ∞):

Sz2

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 K0 K2 cos(2φ) (2.21) 2 a a

1 = 2

ε0 μ0

12

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: a

η = a 0

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 12 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) =

p 12

− 12 γ(z) 1 + (4/π2 ) ln2 2 p

γ(z) Y0 (pρ) × J0 (pρ) − (2/π) ln p

(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
(2.35)

where Rpp (z) is the coupling coeﬃcient between the local fundamental transverse mode Ψp (ρ, z), and the local radiation transverse mode Ψp (ρ, z): ⎛ ⎞⎤ ∞ z dΨp∗ (ρ, z) i 2 2 ⎝ ⎣ ⎠ ⎦ p z + γ (z)dz (2.36) ρΨ (0) (ρ, z)dρ Rpp (z) = πk exp 2k dz ⎡

0

2.2 Theory of MNFs with Microscopic Nonuniformities

31

and Rpp (z) is the coupling coeﬃcient between the local radiation transverse modes Ψp (ρ, z) and Ψp (ρ, z):

i 2 Rpp (z) = πk exp p − p2 z 2k

∞ ρΨp (ρ, z)

dΨp∗ (ρ, z) dρ dz

(2.37)

0

The boundary condition for Eq. (2.35) is −−−− →0 ap (z)− z→−∞

(2.38)

2.2.2 Conventional and Adiabatic Perturbation Theory Let us determine the characteristic variation of γ(z) as Δγ and assume that Δγ is less, or of the order of γ0 . Then the characteristic variation of the integral in the exponent of Eq. (2.36) can be estimated as

1 k

∞ −∞

2 γ1 γ (z) − γ02 dz ∼ ΔγL k

(2.39)

here L is the characteristic length of the MNF nonuniformity discussed in the introduction. The dimensionless parameter ΔγLγ0 /k allows us to deﬁne the condition of validity of the conventional perturbation theory as ΔγL

γ0 1 k

(2.40)

In fact, estimation of the integral terms in Eq. (2.35) shows that they can be neglected under this condition. In this case, Eq. (2.35) can be easily integrated, giving the following expression for the full transmission loss: ∞ Λ=

1 2 |ap (∞)| dp = 2 k

0

!2 ! ! ∞ ! ∞ ! ! ! Rp0 (z)dz !! dp ! ! ! 0

(2.41)

−∞

Eq. (2.41) will be used in subsection 2.2.3 for derivation of the analytical expression for transmission loss caused by weak and smooth nonuniformities. Alternatively, the adiabatic perturbation theory developed in subsection 2.2.4 is applicable if ΔγL

γ0 1 k

(2.42)

32

2 Optical Properties of MNFs: Theory and Numerical Simulations

According to Eqs. (2.40), large L, i.e., slow γ(z), does not necessarily lead to the applicability of the conventional perturbation theory. For example, in the calculation of the microbending losses in a single mode ﬁber it was assumed that k ∼ 5 μm−1 , γ ∼ 0.05k, Δγ ∼ 10−3 μm, and L < 500 μm[28] . These parameters satisfy the condition of Eq. (2.40). However, the parameters of a typical single mode biconical MNF taper, k ∼ 5 μm−1 , γ ∼ Δγ ∼ 1 μm−1 , and L 5 μm, satisfy Eq. (2.42). 2.2.3 Transmission Loss Caused by a Weak and Smooth Nonuniformity The coupling coeﬃcient Rp0 (z) in the Eq. (2.41) for the transmission loss can be calculated using Eqs. (2.32), (2.33), and (2.36): " # 1 p 2 ln 2kp γ(z) dγ(z) Rp0 (z) = × # 12 " dz p π π2 + 4 ln2 γ(z) [γ 2 (z) + p2 ] exp

z i p2 z + λ2 (z)dz 2k

(2.43)

Assuming that the characteristic length of the nonuniformity, L, is large so that γ02 L1 k

(2.44)

we ﬁnd that only p γ0 contribute to the radiation loss deﬁned by the integral over p in Eq. (2.41). Then Eq. (2.43) is simpliﬁed:

Rp0 (z) ≈

1 i 2 dγ(z) 2kp 2 2 p z exp + γ0 dz πγ02 2k

(2.45)

Integrating the internal integral in Eq. (2.41) by parts we have: ∞ −∞

1

2ip 2 Rp0 (z)dz = π

∞ −∞

i 2 2 p + γ0 z dz [γ(z) − γ0 ] exp 2k

(2.46)

The right hand side integral in Eq. (2.46) is the Fourier transform of [γ(z)− γ0 ], which can be calculated asymptotically. In fact, under the condition of Eq. (2.44), the exponent in the right integral of Eq. (2.46) is oscillating much faster than γ(z) changes. Let zpole be the pole of γ(z) in the upper part of

2.3 Theory of MNF Tapers

33

the complex z-plane closest to the real z-axis. The integral in Eq. (2.46) can be calculated by shifting the integration contour as shown in Fig. 2.15: ∞ −∞

1 2 p + γ02 Im (zpole ) (2.47) Rp0 (z)dz ≈ 4p1/2 Res[γ (zpole )] exp − 2k

Fig. 2.15. Contour of integration for the integral in Eq. (2.46)

where Res[γ (zpole )] is the residue of function γ(z) at the pole zpole . Substituting this expression into Eq. (2.41) yields[29] : Λ≈

2 8{Res [γ (zpole )]}2 γ exp − 0 Im (zpole ) kIm (zpole ) k

(2.48)

As a particular case, consider the Lorentzian nonuniformity of the transversal propagation constant γ(z) = γ0 +

Δγ 1 + (z/L)2

(2.49)

with Δγ γ0 . In this case, simple calculation gives zpole = L, Res [γ (zpole )] = ΔγL/2, and 2 2(Δγ)2 L γ Λ= exp − 0 L (2.50) k k Eqs. (2.48) and (2.50) are useful for calculation and estimation of losses caused by microscopic nonuniformities of MNFs. They will be applied to the examination of experimentally measured MNF nonuniformities in Section 4.2.

2.3 Theory of MNF Tapers The conventional perturbation theory considered in the previous subsection fails for relatively large and adiabatic MNF deformations. For this reason, the theory of light propagation along the adiabatically deformed dielectric waveguides and, in particular, MNFs has not been developed until recently. This subsection describes the results of Ref. [29] where the problem of light propagation along the adiabatically tapered MNF has been solved.

34

2 Optical Properties of MNFs: Theory and Numerical Simulations

2.3.1 Semiclassical Solution of the Wave Equation in the Adiabatic Approximation and Expression of Radiation Loss Let us construct the adiabatic solution of the wave equation in the vicinity of an MNF with adiabatically varying parameters. The formal extension of Eq. (2.32) to this case has the form: U (ρ, z)

rρ|γ (0) (z)|

2 πk

1 2

−1

≈

⎡

γ (0) (z)K0 [|γ (0) (z)|ρ] exp ⎣i

Z

γ

k−

(0)

2

(z) 2k

⎤ dz ⎦ (2.51)

where function γ (0) (z) is the transversal propagating constant with index (0) that is added for convenience. Eq. (2.51) is valid in the close neighborhood of an NF. However, it is far away from the MNF at ρ L|γ (0) (z)|/k where L ∼ |γ (0) /(dγ (0) /dz)| is a characteristic length of the MNF nonuniformity. Continuation of Eq. (2.51) to these ρ can be performed using semiclassical approximation. In order to do this, the technique for asymptotic solution of the multidimensional wave equation described in Ref. [31] can be applied. Introduce a parametric system of straight complex classical rays " # ρ(z, γ) = (γ/k) z − z (0) (γ)

(2.52)

with parameter γ. Here z (0) (γ) is a function which is inverse to γ (0) (z). From Eq. (2.51), it is possible to determine the function γ(ρ, z) by expressing γ through ρ and z. This function, as well as the function z (0) (z), is multi-valued and has branches γj (ρ, z). For simplicity, we denote the branch of function (0) z (0) (γ) which corresponds to γj (ρ, z) by the same index j: zj (γ). Then, the semi-classical solutions outside the tapered MNF have the form: Uj (ρ, z)

(0)

≈

|γj (z)|ρ>>1

Aj (ρ, z) exp[iSj (ρ, z)]

(2.53)

where the amplitude Aj (ρ, z) and the action Sj (ρ, z) depend on function γj (ρ, z): Aj (ρ, z) =

iγ 2kρ

12

− 12 !! ! −1 ! (0) ! γ[dzj (γ)/dγ] γ=γ (0)

z − zj (γ)

! ! γ ! γ2 1 (0) Sj (ρ, z) = kz − γ zj (γ)dγ !! z + γρ + 2k k !

(2.54) j (ρ,z)

γ=γj (ρ,z)

(2.55)

2.3 Theory of MNF Tapers

35

(0)

From Eq. (2.51), for ρ L|γ (0) (z)|/k, we have γj (ρ, z) ≈ γj (z). Then the solution (2.53) coincides with Eq. (2.51) in the semi-classical region |γ (0) (z)|ρ 1. In order to determine a global solution for the evanescent ﬁeld near the nonuniform MNF, one has to glue together the solutions corresponding to diﬀerent branches j in Eqs. (2.53)–(2.55). Suppose, for simplicity, that there is only one solution determined by Eq. (2.53) that contributes to the radiation loss. Then we can omit index j and obtain the analytical expression for the radiation loss of the nonuniform MNF in the form[30] :

P =

3 π2

4

! 1 !! (0) ! k 2 ! dz dγ(γ) !

!⎤ ! ! ! 0 ! ! ⎢ 2! !⎥ (0) exp ⎣− !Im γ z (γ)dγ !⎦ ! k! ! ! γ (0) (−∞) ⎡

γ=0

× ! !3 !Im[z (0) (0)]! 2

(2.56)

This equation signiﬁcantly simpliﬁes and corrects the cumbersome expressions for radiation loss involving partial coupling coeﬃcients obtained by other authors and summarized in [2]. In particular, it shows that z (0) (γ) is the only function necessary for determining the radiation loss in a tapered MNF. It is simple to determine this function either numerically or analytically (see examples in the subsection 2.3.3). 2.3.2 Optics of Light Propagation Along the Adiabatic MNF Tapers Before consideration of speciﬁc examples, which are given in the next subsection, let us brieﬂy discuss the structure of the evanescent ﬁeld and the optics of tunneling of light from a tapered MNF, which is determined by the multi-valued function γ(ρ, z). Fig. 2.16 schematically shows a representative surface plot of Re[γ(ρ, z)], which contains branch points indicated by black dots. The branch points belong to a complex caustic of the considered system of rays (i.e., to a focal curve where the amplitude Aj (ρ, z) in Eq. (3.3.4) turns to inﬁnity). The focal points are the projections of the focal circumferences in real 3D space, which are also shown in Fig. 2.16. The optics of light propagation along the tapered MNF can be visualized with Fig. 2.16 as follows. The input mode is launched from the left hand side of Fig. 2.16. In the neighborhood of each focal point, the mode is split into two components. For example, after passing the lower branch point, the ﬁeld splits into the lower component, which is guided by the MNF, and the upper component, which is the radiation component that contributes to the exponentially small radiation loss. Similarly, the lower and upper components of the ﬁeld near the upper focal point are radiating and therefore contribute to the radiation loss. Generally, for adiabatically tapered MNFs, the focal circumferences are responsible for splitting oﬀ the radiating components. The latter interfere with the guiding mode and with each other giving rise to complex behavior

36

2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.16. Characteristic behavior of Re[γ(ρ, z)]. Branch (focal) points indicated by black dots are the projections of focal circumferences in 3D space. They give rise to splitting oﬀ the guiding and radiating components of the propagating electromagnetic ﬁeld.

of the evanescent ﬁeld. These general features of the complex semi-classical dynamics of light in the evanescent region will be illustrated by particular examples of tapered MNFs, which are described in the next subsection. 2.3.3 Example of a Conical MNF Taper Consider an example of a tapered MNF with the following behavior of the transversal propagation constant: γ (0) (z) = γ1 +

γ2 − γ1 1 + exp(−z/L)

(2.57)

Eq. (2.57) describes a tapered MNF, which monotonically changes along the characteristic length L and has γ (0) (z) = γ1 at z → −∞ and γ (0) (z) = γ2 at z → +∞. The propagation loss of this taper can be found from Eq. (2.56) in the form:

1 k 2 |γ1 − γ2 | πL 2 2 P = exp − min |γ1 | , |γ2 | 1 k 4L 2 |γ1 γ2 |

(2.58)

If the taper has the radii a1 and a2 at z → ∓∞ then γ 1 and γ 2 in this equation can be expressed through a1 and a2 , respectively, with Eq. (2.11). Eq. (2.58) is valid only if the exponent is large and therefore P 1. Condition

2.3 Theory of MNF Tapers

37

P ∼ 1 deﬁnes the MNF parameters, for which Eq. (2.58) fails. This condition also deﬁnes the threshold diameter of an MNF corresponding to the transition from waveguiding to the nonwaveguiding regimes that will be considered in subsection 2.4. It follows from Eq. (6.2) that for strong tapering, γ1 γ2 , the loss is determined only by the characteristic length and the transversal propagation constant of the thinnest part of the taper: 1

π2

exp(−S),

S = πL

γ22 k

(2.59) 4S Fig. 2.17 compares the radiation loss calculated by Eq. (2.58) with the numeral simulation by the beam propagation method (BPM) for the wavelength λ = 1.5 μm(k = 4.19 μm−1 ), γ1 = 0.2 μm−1 , and γ2 = 0.4 μm−1 . The agreement with the BPM calculation, as expected, becomes better with decreasing radiation loss, i.e., with an increase in the value of the exponent in Eq. (2.58). Nevertheless, for very small loss, Eq. (2.58) is more accurate than the BMP calculations. P =

1 2

Fig. 2.17. Comparison of radiation loss calculated by Eq. (2.58) (lines) with BPM numerical simulations (dots). Parameters of the MNF taper are shown in the ﬁgure. (Adapted from Ref. [30], with permission from the Optical Society of America)

The optics of light propagation along the considered MNF taper is determined by the multi-branch solution of Eq. (2.52), γ(ρb , z). Function γ (0) (z) in Eq. (2.52) is deﬁned by Eq. (2.57). Though an analytical solution for γ(ρb , z) is not possible, it is possible to ﬁnd the following set of complex caustic surfaces of γ(ρb , z): ± zN (ρ) =

kρ − L ln ± χ (ρ)

γ2 − χ± (ρ) χ± (ρ) − γ1

+ 2πiN

(2.60)

38

2 Optical Properties of MNFs: Theory and Numerical Simulations

where N is an integer, and ±

χ (ρ) =

kρ(γ1 + γ2 ) ±

k 2 (γ1 − γ2 )2 ρ2 − 4kγ1 γ2 (γ1 − γ2 )Lρ 2[L(γ1 − γ2 ) + kρ]

(2.61)

At caustic surfaces determined by Eqs. (2.60) and (2.61), the ampli± tude A(ρ, z) in Eq. (2.54) turns to inﬁnity. Generally, the caustics zN (ρ) are complex-valued and do not correspond to the singularities of the ray dynamics in the real plane (ρ, z). However, there exist a set of real coordinate pairs, (±) (ρ± f N , zf N ). These pairs correspond to the focal points in the plane (ρ, z) and, respectively, to the focal circumferences in the real 3D space surrounding the MNF taper. Consider e.g., the MNT taper with the parameters considered in Fig. 2.17: λ = 1.5 μm, γ1 = 0.2i μm−1 , γ2 = 0.4i μm−1 , and the characteristic length L = 50 μm. Then, with Eqs. (2.60) and (2.61), we ﬁnd that the focal points closest to the MNT are: ⎫ + (ρ+ f 1 , zf 1 ) = (8.943, −47.301) μm ⎬ + (ρ+ f 2 , zf 2 ) = (25.036, −75.016) μm

⎭

(2.62)

Fig. 2.18 shows the surface plot of electromagnetic ﬁeld intensity for the MNT taper with the same parameters, which is found with BPM simulation. The direction of light propagation is from the thinner part to the thicker part of the taper. This ﬁgure exhibits quasiperiodic dips with vanishing ﬁeld intensity, which can be explained as follows. In the neighborhood of the focal point + (ρ+ f 1 , zf 1 ) deﬁned by Eq. (2.62) (indicated by a small circle), the radiating part of the ﬁeld splits oﬀ and localizes along a line starting near the focal point and going to inﬁnity. From this line down, the radiating part exponentially decreases with decreasing ρ, while the guiding part exponentially increases. Therefore, for certain values of ρ these parts become equal in magnitude and their interference causes strong oscillations of the ﬁeld. As illustrated in Fig. 2.16, in real 3D space this focal point corresponds to a focal circumference. Therefore, the points with vanishing ﬁeld correspond to the circumferences with vanishing ﬁeld. 2.3.4 Example of a Biconical MNF Taper Consider now a biconical MNF taper, which is described by the following Lorentzian variation of the transversal propagation constant: γ (0) (z) = γ∞ +

γ0 − γ∞ 1 + (z/L)2

(2.63)

For this taper, the analytical expression for the radiation loss found with Eq. (2.56) has the form[30] :

2.3 Theory of MNF Tapers

39

Fig. 2.18. Distribution of the electromagnetic ﬁeld intensity near the tapered MNF deﬁned by Eq. (2.57) with parameters γ1 = 0.2i μm−1 , γ2 = 0.4i μm−1 , and L = 50 μm, for radiation wavelength λ = 1.5 μm.

⎫

2 ⎪ Lγ∞ ⎪ ⎪ f (δ) B(δ) exp − ⎪ 2 L ⎪ γ∞ 4k ⎪ ⎪ ⎪ * ) ⎬ 1 1 + δ2 1 2 + 2δ (3 − δ) ⎪ f (δ) = (δ + 3)(δ − 1) ln 1 ⎪ ⎪ 1 − δ2 ⎪ ⎪ ⎪ ⎪ γ0 ⎪ − 54 ⎭ B(δ) = (1 − δ)δ , δ = γ∞ 3

π2 P = 8

12

k

(2.64)

The radiation loss predicted by this equation is in good agreement with the BPM numerical simulation[30] . As in the previous section, the agreement improves with growing of the exponent in Eq. (2.64). For a strong taper, |γ0 | |γ∞ |, Eq. (2.64) simpliﬁes: 5

3

P =

π2 1

1

2 × 30 2 S 2

exp (−S) ,

S=

15Lγ02 1

(2.65)

2 8kγ∞

It is seen that, as in the previous subsection, the radiation loss is independent of the value of the transverse propagation constant away from the center of the taper. The detailed investigation of the behavior of the three-branch function γ(ρb , z) is given in Ref. [30]. It has been found that the branch (focal) point of γ(ρb , z) is deﬁned by the equations:

ρb =

! + ! ! L !! 2 − δ3! , Q + Q γ ∞ ! ! k

1 Q = (δ 2 + 18δ − 27), 8

γ0 δ= γ∞

⎫ ⎪ zb = 0 ⎪ ⎬ ⎪ ⎪ ⎭

(2.66)

40

2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.19 shows the ﬁeld intensity distribution near the MNF taper having the parameters k = 4 μm−1 , γ1 = 0.2 μm−1 , γ2 = 0.4 μm−1 , and L = 500 μm. From Eq. (2.66), the focal point, which is shown in Fig. 2.19 by a small circle, is (ρf , zf ) = (8.419, 0). Similar to Fig. 2.18, two semi-classical solutions of the wave equation are split oﬀ near this point. One of them is a radiating component, while the other one is a guiding component. The interference between these components gives rise to the oscillations of the evanescent ﬁeld and to the appearance of quasiperiodic dips as in the case of a conical MNF taper considered in Subsection 2.3.4.

Fig. 2.19. Distribution of the electromagnetic ﬁeld intensity near the MNT taper deﬁned by Eq. (2.63) with parameters γ1 = 0.2μm−1 , γ2 = 0.4 μm−1 , and L = 500 μm for radiation wavelength λ = 1.57 μm(k = 4 μm−1 ). (Adapted from Ref. [30], with permission from the Optical Society of America)

2.3.5 Example of an MNF Taper with Distributed Radiation Loss As it is shown in the previous subsection, for adiabatically tapered MNF, the radiation loss takes place locally in the neighborhood of focal circumferences surrounding the MNF. Away from these locations, a MNF taper is lossless. This qualitatively diﬀerentiates radiation loss in the adiabatic taper from the radiation loss in bent waveguides. In fact, in adiabatically bent waveguides, an eﬀective potential barrier can be introduced[2,32] . The barrier has a ﬁnite width and tunneling through this barrier determines the radiation loss. Usually, the radiation loss is distributed along a bent waveguide and can be described by the attenuation constant, which depends on the coordinate along the waveguide [2] . The MNF taper considered in this subsection represents an interesting exceptional situation when tunneling dynamics of light is similar to that of a bent microﬁber. For this taper, coordinates in the wave equation can be asymptotically separated and the eﬀective potential barrier can be introduced. The existence of an eﬀective potential barrier adjacent to the MNF implies the uniform transmittance of this barrier along the taper, i.e.,

2.3 Theory of MNF Tapers

41

the distributed radiation loss and possibility of the introduction of the local attenuation constant. Mathematically, such a barrier can only be introduced if the wave equation, Eq. (2.26), with the boundary condition determined by the behavior of the transversal propagation constant, γ (0) (z), allows separation of variables. As shown in Ref. [33], the separation of variables is possible for the exceptional shape of MNF, which corresponds to the following variation of the transversal propagation constant: γ0 γ (0) (z) = 1 − (z/L)2

(2.67)

where L is the characteristic nonuniformity of the MNF and γ0 is the value of the transversal propagation constant in the middle of the taper. From Eq. (2.67), the transversal propagation of this taper, which is illustrated in Fig.2.20, has an unphysical singularity at z = ±L.

Fig. 2.20. Illustration of the taper determined by Eq. (2.67) (Adapted from Ref. [33], with permission from the Optical Society of America)

Calculations lead to the following expression for the attenuation constant of this MNF taper[33] :

α(z) =

2πγ 2 (z) πγ 2 L exp − 0 k 2k

(2.68)

It follows from this equation that the attenuation constant grows with the pre-exponential factor, which is proportional to the transversal component of the propagation constant. This factor has a minimum in the center of the taper, where the MNF has the smallest radius, and tends to inﬁnity near the edges of the taper, for z → ±L, where its radius becomes large and Eq. (2.68) fails. The exponent in Eq. (2.68) is independent of z, which indicates the uniform transparency of the eﬀective potential barrier surrounding the MNF taper. Calculations, which can be found in Ref. [33], show that the evanescent part of the fundamental mode of this MNF taper is determined in the form of a singular Gaussian beam.

42

2 Optical Properties of MNFs: Theory and Numerical Simulations

2.4 The Thinnest MNF Optical Waveguide It follows from the consideration of subsection 2.1.2, that the fundamental mode of a uniform lossless dielectric waveguide and, in particular, an MNF exists independently of its thickness. However, in practice, waveguiding ability is constrained by losses due to material absorption and geometric nonuniformities. We apply the term “waveguide” only to a waveguide whose losses are small in comparison to the input power. Subsequently, a basic and practically important question can be asked: how thin can the optical waveguide be[34] ? It is crucial that consideration of waveguide losses is incomplete if input and output losses are ignored. Therefore, in the following[34] we determine the transmission loss of a waveguide by the sum of its own losses and the losses at its input and output connections. Often, there exists a way to minimize input and output losses down to relatively small values and the connection losses can be separated from the losses of the waveguide itself. However, a very thin waveguide is an exception to this rule: its transmission loss is primarily determined by input and output losses, which, in practice, cannot be reduced signiﬁcantly. As a waveguide, Ref. [34] theoretically explored an MNF with adiabatically tapered connections to a regular ﬁber. The radiation loss of an MNF taper can be estimated with Eq. (2.56). In the following[34] we calculate the radiation loss with exponential accuracy for the taper having the Lorentzian radius variation:

a(z) = a∞ −

(a∞ − a0 ) 1 − (z − z0 )2 /L2

(2.69)

where a∞ and a0 are the taper radii at its ends and at the center, respectively. For a silica microtaper (refractive index 1.45) with a small waist diameter 2a λ, the transversal propagation constant is determined by Eq. (2.12). Then the radiation loss P found from Eq. (2.56) depends on the waist radius a0 double-exponentially: ,

0.143λ2 exp − P ∼ exp − 1 1 a20 a02 (a∞ − a0 ) 2 0.51L

(2.70)

According to the deﬁnition of the waveguide given above, the MNF is waveguiding only if P 1, while the condition P ∼ 1 corresponds to the threshold where waveguiding disappears. Fig. 2.21 shows the transmission loss P as a function of the microﬁber diameter 2a0 for radiation wavelength λ = 1.55 μm and a∞ = 0.5 μm. It demonstrates the dramatic threshold behavior of the transmission loss predicted by Eq. (2.71) (curves 1, 2 and 3) and shows no signiﬁcant diﬀerence between the threshold values of the microﬁber diameter for a typical L = 10 km and for a gigantic L = 10 km. They are of the same order of magnitude ∼ 0.2−0.4 μm and only a few times less than the

2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation

43

radiation wavelength. Notice that the loss P is in strongest double-exponent dependence only on the minimum microﬁber radius, a0 . Dependence of P on the parameter a∞ − a0 , which determines the shape of the microﬁber taper, is much weaker. Therefore, it is reasonable to conclude that the speciﬁc shape of the taper, as well as its characteristic length L, does not signiﬁcantly aﬀect the value of the threshold diameter. These predictions have found excellent experimental conﬁrmation both for silica MNF[35] and for subwavelength diameter THz ﬁbers[36] . Comparison of the theory described in this and the previous subsections with the experiment will be considered in Subsection 4.2.1.

2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation Evanescent coupling between MNFs is of special importance for applications such as direct interconnection with external optical systems through ﬁber tapers, energy exchange between two MNFs, and recirculation of optical energy inside ring resonators. Generally, evanescent coupling between adjacent weakly guiding waveguides can be described by perturbation theory[2] , in which the weakly coupled system is assumed. However, for an MNF with air or water cladding, the index contrast is usually high. When the diameter of the MNF goes below the wavelength of the guided light, a strong evanescent ﬁeld presents itself on the surface of the MNF[37] . When two MNFs are brought closely into contact, strong optical coupling may occur. Therefore, optically coupled MNFs do not always indicate a weakly coupled system. Perturbation theory and subsequently the coupled-mode theory for the weakly coupled system may not be applied.

Fig. 2.21. Transmission loss as a function of the microﬁber diameter calculated with Eq. (2.71) for diﬀerent characteristic lengths L of the MNF taper (curves 1, 2, and 3) and a∞ = 0.5 μm. (Adapted from Ref. [34], with permission from the Optical Society of America)

44

2 Optical Properties of MNFs: Theory and Numerical Simulations

The rigorous method for investigating the mode coupling between two MNFs is to solve Maxwell’s equations in the diﬀerent regions and use the boundary conditions to determine the modes of the overall system. However, it is diﬃcult to perform the calculation analytically so that we have to resort to numerical methods. For optically coupled MNFs, with relatively small calculation dimensions, ﬁnite-diﬀerence time-domain (FDTD) simulation is one of the optimal approaches with regard to its accuracy and eﬃciency[38−40] . 2.5.1 Model for FDTD Simulation The mathematic model for numerical analysis is shown in Fig. 2.22, in which two cylindrical MNFs with diameters of D1 and D2 are placed in parallel with a separation H and an overlap L. In the simulation, eigenvalue modes are guided along the input MNF in region I; in region II energy exchanges between the two MNFs; and in region III the output is collected for calculating the coupling eﬃciency. For simplicity, the sidewall of the MNF is assumed to be intrinsically smooth so that the roughness-induced scattering is negligible. Silica and tellurite MNFs, with typical medium and high refractive indices are selected for investigating the coupling behaviors of the MNFs. The index of air is assumed to be 1.0, and the indices of silica and tellurite MNFs at some speciﬁc wavelengths are listed in Table 2.3.

Fig. 2.22. Mathematic model for the coupling of two parallel MNFs. (Adapted from Ref. [43], with permission from the Optical Society of America)

Table 2.3. Refractive indices of silica and tellurite MNFs (Adapted from Ref. [43], with permission from the Optical Society of America) Material Silica Silica Tellurite

Wavelength(nm) 633 1550 633

Refractive index 1.46 1.44 2.08

Since the coupling eﬃciency is polarization-dependent due to the asymmetry of the structure, sources of two polarization states (y- and z-polarization

2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation

45

at the input plane) are used separately. The computational domain is discretized into a uniform orthogonal 3-dimensional mesh with a cell size of one tenth of the MNF diameter, terminated by perfectly matched layer boundaries[41] .There are a number of codes or software packages available for FDTD simulations. For investigation of two coupled MNFs with cylindrical proﬁles, a 3-dimensional FDTD (3D-FDTD) is required. Here the FDTD simulations are performed using a freely available software package Meep [42] . 2.5.2 Evanescent Coupling between two Identical Silica MNFs Using 3D-FDTD simulation based on the model shown in Fig. 2.22, the coupling behavior of parallel MNFs is investigated. As a start, we consider the simplest situation where two identical silica MNFs, operating in single mode, are closely contacted in parallel with a separation of H=0. The results calculated for MNFs with diameters of 350, 400 and 425 nm are shown in Fig. 2.23, where the wavelength of the light is assumed to be 633 nm. The hollow and solid circle points represent the y- and z-polarization, respectively. For better understanding the coupling behavior shown in Fig. 2.23, power maps of evanescent coupling between two parallel 350-nm-diameter silica MNFs with typical L of 0, 2.4 and 4.8 μm are also illustrated in Fig. 2.24. The L-dependent η in Fig. 2.23 shows similar oscillating behavior as that in weakly coupled systems[10] . However, because of the strong coupling between the two MNFs, the minimum transfer length (LT ) for energy exchange is much shorter than that in weakly coupled waveguides. Compared to the z-polarization, the y-polarization has a smaller LT because of stronger overlapping of the optical ﬁelds. Meanwhile, unlike in the weakly coupled system where the minimum coupling eﬃciency (ηmin ) is usually close to zero, ηmin of the strongly coupled MNFs is considerably higher than zero (e.g., about 34% for two 350-nm-diameter silica MNFs shown in Fig. 2.23(a)). This considerably larger ηmin behavior is clearly illustrated in Fig. 2.24(a) and (c), in which L is set to 0 and 4.8 μm to achieve the lowest coupling eﬃciency (as shown in Fig. 2.23(a)). A certain amount of energy is transferred from the upper MNF to the bottom one, leading to the large ηmin observed in Fig. 2.23. The striped intensity variation in Fig. 2.24(a) and (c) is caused by the reﬂection of partial energy at the end of the upper MNF, which results in interference between the forward propagation and backward reﬂection waves. In addition, in Fig. 2.24(a) and (c) obvious radiation loss is observed at the right end of the upper MNF. The maximum coupling eﬃciency ηmax is lower than 100%. Inevitably, a part of the guided mode transfers to radiation modes at the overlapping area due to the breaking of the symmetrical structure. However, this loss is relatively small, as shown in Fig. 2.24(b) where no obvious radiation loss is observed. The maximum coupling eﬃciency approaches 100% for MNFs with diameters around the critical value for single mode operation (e.g. ηmax > 97% for 425-nm diameter MNFs with z-polarization).

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2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.23. Overlapping-length-dependent coupling eﬃciency of two identical MNFs with a 633-nm-wavelength source. The MNF diameters are (a) 350 nm, (b) 400 nm, and (c) 425 nm. The two polarizations of the source are denoted as Ez (z-polarized) and Ey (y-polarized), respectively. (Adapted from Ref. [43], with permission from the Optical Society of America)

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47

Fig. 2.24. Power maps of evanescent coupling between two parallel 350-nmdiameter silica MNFs with overlapping length of (a) 0 μm, (b) 2.4 μm, and (c) 4.8 μm. The source is z-polarized with a wavelength of 633 nm. (Adapted from Ref. [43], with permission from the Optical Society of America) Table 2.4. Minimum transfer length (LT ), maximum coupling eﬃciency (ηmax and minimum coupling eﬃciency (ηmin of two parallel air-clad silica and tellurite MNFs with separation H=0. (Adapted from Ref. [43], with permission from the Optical Society of America) Material Silica Silica Silica Silica Silica Tellurite

Wavelength (nm) 633 633 633 1550 1550 633

D (nm) 350 400 425 900 1000 250

LT (μm) z-p y-p 2.4 2 3.2 2.4 3.4 2.7 6.5 5.1 7.7 6 1.4 1

ηmax z-p y-p 96% 92% 96% 94% 97% 94% 96% 94% 97% 94% 96% 95%

ηmin z-p y-p 34% 41% 31% 33% 30% 11% 38% 43% 36% 38% 30% 34%

Calculated LT , ηmax and ηmin of evanescently coupled silica MNFs with typical diameters are listed in Table 2.4 for reference and typical results of single-mode high-index tellurite MNFs are also provided. It shows that, at the given wavelength of 633 nm or 1550 nm, silica MNFs with a larger diameter show higher ηmax and lower ηmin . For example, using a z-polarized 633-nmwavelength source, when the diameters of silica MNFs decrease from 425 nm to 350 nm, ηmax decreases from 97% to 96%, while ηmin increases from 30% to 34%. Similar behavior is observed with a y-polarized source. In addition, LT of high-index MNFs is much smaller than that of silica MNFs. For example, at the wavelength of 633 nm, LT of the tellurite MNFs (1.4 μm) is about 40% that of the silica MNFs (3.4 μm). The small transfer length (LT ) may suggest the possibility of developing ultra-compact optical devices (e.g., branch couplers) based on the strong coupling of optical MNFs. Meanwhile, at the given wavelength, LT decreases with the decrease in the ﬁber diameter. For example, for silica MNFs operating at 633-nm wavelength, LT decreases from 3.4 μm to 2.4 μm when the ﬁber diameter decreases from 425 nm to 350 nm, indicating the opportunity for achieving smaller LT by using thinner ﬁbers. However, when the ﬁber diameter decreases to a certain value, a similar behavior to antisymmetric supermode cutoﬀ in weakly cou-

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2 Optical Properties of MNFs: Theory and Numerical Simulations

pling systems is observed[44] . As shown in Fig. 2.25(a) oscillating behavior disappears in the two coupled 250-nm-diameter silica MNFs at 633-nm wavelength. This can be explained as follows: when the two MNFs are very thin, the coupled system (that can be viewed as one waveguide in region II in Fig. 2.22) can only support the fundamental mode and works as a single-mode waveguide, and therefore the oscillating behavior disappears. In addition, it is interesting to note that when the two 250-nm-diameter MNFs are separated by a certain distance, e.g., H=70 nm shown in Fig. 2.25(b), oscillation behavior appears, indicating that the overlapping region becomes multimode. The dependence of the coupling eﬃciency (η) on the lateral separation (H) is also investigated. The calculation of η for two 350-nm-diameter silica MNFs with z-polarized 633-nm-wavelength source is shown in Fig. 2.26, in which the square points represent the calculated results with a smoothing B-spline ﬁt. The overlapping length (L) is set to 3.6 μm, so that when the separation H=70 nm, the coupling eﬃciency (η) reaches the maximum value (ηmax ) of 97% with LT = L=3.6 μm. Power maps with lateral separation (H) of 0 (point A), 70 (point B) and 210 nm (point C) are also illustrated in Fig. 2.26(b) for a better understanding of the coupling behavior shown in Fig. 2.26(a). When H=0 (Point A in Fig. 2.26), LT < L, the two MNFs are overcoupled with η ≈70%, and η increases with the increase in H; when H increases to 70 nm, LT = L, η reaches its maximum; further increasing H leads to larger LT (i.e., LT > L), resulting in a monotonous decrease of η. It should be also mentioned that, when the ﬁber diameter approaches or exceeds the single-mode cutoﬀ diameter (DSM ), the coupling of two parallel MNFs shows slightly diﬀerent behavior. Table 2.5 gives the calculated LT , ηmax and η min of evanescently coupled silica MNFs with diameters around DSM . It shows that when the ﬁber diameter is very close to DSM , e.g., 450 nm, ηmax for y-polarization reduces to 93%, while ηmax for z-polarization maintains a high value of 97%. The reduction in y-polarization ηmax can be explained as follows: in the coupling system shown in Fig. 2.22, the interaction of the electric ﬁeld of y-polarization is stronger than that of z-polarization, which makes it possible to excite high-order modes with y-polarization in the overlapping region, resulting in a reduction in coupling eﬃciency. For better understanding, Fig. 2.27 provides power maps of evanescent coupling

Fig. 2.25. Power map of evanescent coupling between two parallel 250 nm silica nanowires with separation of (a) H=0, and (b) H=70 nm. The source is z-polarized with wavelength of 633 nm, the overlapping length (L) is 6 μm. (Adapted from Ref. [43], with permission from the Optical Society of America)

2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation

49

Fig. 2.26. Evanescent coupling behavior of two 350-nm-diameter silica nanowires with a z-polarized 633-nm-wavelength source. (a) H-dependent η, and (b) power maps with lateral separation (H) of 0 (point A), 70 (point B) and 210 nm (point C). The overlapping length (L) is 3.6 μm. (Adapted from Ref. [43], with permission from the Optical Society of America)

between two closely contacted 450-nm-diameter parallel silica MNFs with ypolarization (overlapping length of 3.2 μm) and z-polarization (overlapping length of 4 μm), respectively. While the z-polarization shows a similar power transfer map as that for smaller ﬁber diameters (e.g., 350-nm-diameter silica MNFs in Fig. 2.26), the power map of y-polarization exhibits obvious distortion around the overlapping region, indicating the role of high-order modes. Table 2.5. Minimum transfer length (LT ), maximum coupling eﬃciency (ηmax ) and minimum coupling eﬃciency (ηmin ) of two closely contacted (H=0) parallel air-clad silica MNFs at 633-nm wavelength. Diameter (nm) 450 500

z-p 4 5.2

LT (μm) y-p 3.2 4.4

ηmax z-p 97% 98%

ηmin y-p 93% 94%

z-p 30% 28%

y-p 31% 34%

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2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.27. Power maps of evanescent coupling between two closely contacted 450nm-diameter parallel silica MNFs with (a) y-polarization and overlapping length of 3.2 μm, and (b)z-polarization and overlapping length of 4 μm. The wavelength of the light is 633 nm.

2.5.3 Evanescent Coupling between two Silica MNFs with Diﬀerent Diameters In this section we investigate evanescent coupling of two silica MNFs with different diameters, which may be of practical importance because of the dispersion in diameters of the available MNFs, as well as the desire to couple between MNFs of diﬀerent diameters. With the assumption that the two MNFs are in tight contact (H=0) with a z-polarized 633-nm-wavelength source, the calculated results of L-dependent η are shown in Fig. 2.28, in which MNFs with typical diameters of 350, 400 and 450 nm are used. For reference, L-dependent η of two identical 350-nm-diameter MNFs is also provided. It shows that ηmax of two identical 350-nm-diameter MNFs is about 96%; when the output MNF is replaced by 400- or 450-nm-diameter MNF (that is, 350 → 400 or 350 → 450 in Fig. 2.28), ηmax decreases to 91% (for 350 → 400) or 82% (for 350 → 450). This can be explained as much eﬀective coupling between identical MNFs with matched propagation constants; the mismatch of the propagation constant in-

Fig. 2.28. Overlapping-length-dependent coupling eﬃciency of two silica MNFs with a z-polarized 633-nm-wavelength source. Diameters of the MNFs are denoted as x → y, in which x and y stand for the diameters of the input and output MNFs, respectively. (Adapted from Ref. [43], with permission from the Optical Society of America)

2.5 Evanescent Coupling between Parallel MNFs: 3D-FDTD Simulation

51

Fig. 2.29. Power maps of evanescent coupling between two silica MNFs with zpolarization 633-nm-wavelength sources and overlapping length of 8 μm. (a) Coupling from a 350-nm-diameter MNF to a 450-nm-diameter MNF, and (b) coupling from a 450-nm-diameter MNF to a 350-nm-diameter MNF.

creases with an increasing diﬀerence in diameter, leading to the reduction of ηmax with increasing diameter mismatch. It also shows that the coupling efﬁciency of two diﬀerent MNFs is direction-dependent: coupling light from a thinner MNF to a thicker one (e.g., 350 → 400 or 350 → 450) shows higher efﬁciency (e.g., 91% for 350 → 400 or 82% for 350 → 450) than in the opposite direction (e.g., 88% for 400 → 350 or 71% for 450 → 350). This directiondependent ηmax can be interpreted as meaning that the thicker MNF has a higher eﬀective index, which exhibits a stronger ability to conﬁne and attract the light ﬁeld than the thinner one. For better understanding, Fig. 2.29 gives power maps of evanescent coupling between two silica MNFs with diameters of 350 and 450 nm, respectively. A much stronger attraction of optical intensity in a 450-nm-diameter MNF than that in a 350-nm-diameter MNF is clearly seen. In addition, in spite of the diameter diﬀerence, the minimum transfer lengths (LT ) required for obtaining the maximum coupling eﬃciencies (ηmax ) of the 5 pairs of MNFs are around 2.4 μm in all cases, which is the LT obtained for two 350-nm-diameter silica MNFs at 633-nm wavelength. 2.5.4 Evanescent Coupling between a Silica MNF and a Tellurite MNF Silica MNFs can be directly tapered down from a standard optical ﬁber, providing convenient interconnection with external optical systems. Therefore, silica MNFs can be used as a launching or collecting ﬁber for many other optical MNFs or nanowires such as glass MNFs and semiconductor nanowires. As a typical situation, evanescent coupling between a silica and a tellurite MNF is investigated here. Typical diameters of silica (from 325 to 450 nm) and tellurite (200 nm) MNFs for single-mode operation (at 633-nm wavelength) are used with a zpolarized 633-nm-wavelength source. The calculated results are summarized in Table 2.6. For reference, power maps of evanescent coupling between a 200nm-diameter tellurite MNF and a 450-nm-diameter silica MNF with LT =2.2 μm are also provided in Fig. 2.30. It shows that in spite of the large index

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2 Optical Properties of MNFs: Theory and Numerical Simulations

diﬀerence between silica and tellurite MNFs, when the diameter of the ﬁber is carefully selected, considerable high coupling eﬃciency can be obtained. For example, when a 450-nm-diameter silica MNF is used to couple light out of a 200-nm-diameter tellurite MNF, the coupling eﬃciency can go up to 96%. With the same 200-nm-diameter tellurite nanowire, LT and ηmax increase with the increasing diameter of the silica MNF. For example, when the diameter of the silica MNF increases from 325 nm to 450 nm, LT increases from 1.3 μm to 2.2 μm. This can be explained as that the interaction between the two MNFs becomes weaker with larger diameters. Similar behavior has been presented in Table 2.6. It also shows that ηmax is direction-dependent. For example, ηmax of coupling light from a 200-nm-diameter tellurite MNF to a 450-nm-diameter silica MNF is about 96%, while it decreases to 92% when the light goes in the opposite direction (i.e., coupling light from the silica MNF to the tellurite MNF). Table 2.6. Calculated LT and ηmax of evanescent coupling between silica and tellurite MNFs with H=0 at 633-nm wavelength. Diameter (nm) Tellurite (T) Silica (S) 200 325 200 350 200 400 200 425 200 450

LT (μm) 1.3 1.5 1.8 2.0 2.2

η max T→S 78% 83% 92% 95% 96%

S→T 81% 86% 91% 92% 92%

Fig. 2.30. Power maps of evanescent coupling between a 200-nm-diameter tellurite nanowire and a 450-nm-diameter silica nanowire with a z-polarized 633-nmwavelength source. The overlapping length L = LT =2.2 μm. (a) Coupling light from the tellurite nanowire to the silica nanowire. (b) Coupling light from the silica nanowire to the tellurite nanowire. (Adapted from Ref. [43], with permission from the Optical Society of America)

It is also noticed that matched propagation constants are beneﬁcial for obtaining high coupling eﬃciency between two MNFs of diﬀerent materials, indicating that the MNF with a higher refractive index should have a smaller diameter to obtain a good coupling. For comparison, when a 633-nm-wavelength

2.6 Endface Output Patterns

53

light is coupled from a 450-nm-diameter silica MNF to a 200-nm-diameter tellurite MNF, the coupling eﬃciency goes up to 92% (see Table 2.6); however, if the diameter of the tellurite MNF is increased to 230 nm, the calculated coupling eﬃciency from a 450-nm-diameter silica MNF decreases to about 64%. For reference, Fig. 2.31 gives the power maps of evanescent coupling between a 230-nm-diameter tellurite MNF and a 450-nm-diameter silica MNF with a z-polarized 633-nm-wavelength source. It clearly shows the reﬂection-induced interference patterns inside the launching MNF and radiation loss at the endface of the launching MNF due to the mismatch of the propagation constants of the two MNFs.

Fig. 2.31. Power maps of evanescent coupling between a 230-nm-diameter tellurite nanowire and a 450-nm-diameter silica nanowire with a z-polarized 633-nmwavelength source. The overlapping length L = LT =1.9 μm. (a) Coupling light from the tellurite nanowire to the silica nanowire. (b) Coupling light from the silica nanowire to the tellurite nanowire.

In addition, other factors (e.g., optical conﬁnement) may also aﬀect the coupling loss in these low-dimension structures, and usually a slight diﬀerence in propagation constants of the two MNFs may be required for achieving the maximum coupling eﬃciency (ηmax ).

2.6 Endface Output Patterns Because of the large fraction of diﬀraction and evanescent ﬁelds, light output from a MNF with wavelength or subwavelength diameter behaves diﬀerently with standard optical ﬁbers[45] . Since terminated MNFs, or more generally terminated optical-quality nanowires, are promising structures for manipulating light with micro-/nanometer scale endfaces, such as subwavelength-dimension light beams, optical probes[46,47] and point sources[48,49] , quantitative investigation of output patterns of MNFs may oﬀer valuable references for possible applications of MNFs in a variety of areas such as MNF lasing, sensing, optical trapping, and laser surgery. In this section, the endface output patterns of an MNF with air or water cladding are investigated using the 3D-FDTD method performed by Meep [42] , the same software as used in the previous section.

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2 Optical Properties of MNFs: Theory and Numerical Simulations

The basic model for numerical analysis is shown in Fig. 2.32. The MNF is assumed to have a circular cross-section with a diameter of D and a length of L, an inﬁnite air or water cladding, and a step-index proﬁle. A Cartesian coordinate with its origin located at the center of the output endface of the MNF is used, and the source is assumed to be z -polarized. The index of air is assumed to be 1.0, and the indices of water, silica, and tellurite are obtained from their wavelength-dependent dispersion relations (e.g., 1.33, 1.46 and 2.02 for water, silica and tellurite at 633-nm wavelength, respectively) [3,6] . 2.6.1 MNFs with Flat Endfaces For MNFs with ﬂat endfaces, 400-nm-diameter silica MNFs and 250-nmdiameter tellurite MNFs are used as typical ﬁbers for single-mode operation in air and water at 633-nm wavelength. Figs. 2.33(a)–(d) show the calculated output patterns in x-y plane (z=0) of 4.2-μm-length freestanding silica and tellurite MNFs in air and in wa-

Fig. 2.32. Mathematic model for investigation of the endface output patterns of MNFs

Fig. 2.33. Output patterns in x-y plane (z=0) of (a) a 4.2-μm-length, 400-nmdiameter silica MNF in air; (b) a 4.2-μm-length, 400-nm-diameter silica MNF in water; (c) a 4.2-μm-length, 250-nm-diameter tellurite MNF in air; and (d) a 4.2-μmlength, 250-nm-diameter tellurite MNF in water. The white-line rectangles map the topography proﬁle of the MNFs.

2.6 Endface Output Patterns

55

ter, respectively. Standing wave patterns, similar to those observed in MNFs experimentally[50] , are clearly observed along the length of the MNFs due to the interference between the forward propagating light and backward reﬂection generated at the output endfaces. Compared with MNFs in air, MNFs in water present a higher fraction of evanescent ﬁelds, a smaller divergence in output mode proﬁles and weaker standing wave patterns, which can be explained by tighter conﬁnement and stronger endface reﬂection in air-clad MNFs due to the higher index contrast than that in water-clad MNFs. Normalized intensity distributions along the y-axis in x-y plane (z=0) with distances of 100 nm (x=100 nm, near-ﬁeld) and 3000 nm (x=3000 nm, far-ﬁeld) departed from the end facets of MNFs given in Figs. 2.33(a) – (d), are shown in Fig. 2.34. The intensity peaks at the central axis of the MNF (y=0), and decreases monotonously when departing from the central axis. The spatial concentration of the endface output from an MNF can be estimated by deﬁning a “beam width” as the full widths at half maximum (FWHMs) of the intensity distributions, as shown in Fig. 2.34. For example, the nearﬁeld beam width of a 633-nm light output from a 250-nm-diameter tellurite MNF in water is about 290 nm, indicating that in the near-ﬁeld region (i.e., 100-nm departure from the output endface) the majority of the light energy is conﬁned within a wavelength-scale span. In a relatively distant ﬁeld, e.g., 3000-nm departure from the endface, the beam width spreads to 1.6 μm.

Fig. 2.34. Normalized intensity distributions along the y-axis in x-y plane (z=0) with distances of 100 nm (x=100 nm, ear-ﬁeld, in solid lines) and 3000 nm (x=3000 nm, far-ﬁeld, in dashed lines) departed from the end facets of the silica MNF in air, silica MNF in water, tellurite MNF in air, and tellurite in water. (Adapted from Ref. [45], with permission from the Optical Society of America)

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2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.35 gives the beam width as a function of the normalized ﬁber diameter (D/λ), in which the wavelength of the light is assumed to be 633 nm and the beam width is a measured 100-nm departure from the output endface. It shows that for an MNF with given index and cladding, there exists a certain value of (D/λ) with which the output of the MNF exhibits the lowest divergence in the near ﬁeld. For example, for a silica MNF, when D/λ = 0.82 (i.e., D= 520 nm with 633-nm-wavelength light), the beam width goes to a minimum of 420 nm in air or 480 nm in water. For a high-index tellurite MNF, a smaller beam width (e.g., 320 nm in water) is observed with a smaller ﬁber diameter (367 nm in water).

Fig. 2.35. Beam widths of near-ﬁeld outputs (measured 100-nm departure from the output endfaces) with respect to the normalized ﬁber diameter (D/ λ) at the wavelength of 633 nm for silica MNFs in air or water, and tellurite MNFs in air or water. Open circles denote the critical diameters for single-mode operation. (Adapted from Ref. [45], with permission from the Optical Society of America)

Similar to the diameter of the MNF, the wavelength of the light is another dimension that directly determines the features of the output pattern of an MNF. Fig. 2.36 shows output patterns of a 2.8-μm-length 400-nm-diameter silica MNF in air at 250-nm wavelength (Fig. 2.36(a)) and 800-nm wavelength (Fig. 2.36(b)), respectively. At a short wavelength, the MNF operates in multimode, resulting in multiple lobes in the output pattern (Fig. 2.36(a)); while at a long wavelength, the MNF becomes a single-mode waveguide with a single maximum in output pattern (Fig. 2.36(b)).

2.6 Endface Output Patterns

57

Fig. 2.36. 3D-FDTD simulations of output patterns of a 2.8-μm-length 400-nmdiameter silica MNF in air at (a) 250-nm wavelength, and (b) 800-nm wavelength, respectively. (Adapted from Ref. [45], with permission from the Optical Society of America)

2.6.2 MNFs with Angled Endfaces Optical ﬁbers with angled endfaces are used for decreasing back-reﬂection or redirecting output beams in applications including optical communication, sensing, imaging and laser surgery[51,52] . It is interesting to explore the output properties of the light in an angled MNF with a diameter below the wavelength of the light. In this subsection, as a typical situation, 400-nm-diameter silica MNFs with angled endfaces are employed to investigate the output properties of these kinds of ﬁbers, and the operation wavelength is assumed to be 633 nm to ensure the single-mode operation of the MNFs. Fig. 2.37 shows the calculated output patterns in the x-y plane (z=0) of the 400-nm-diameter silica MNF in air with 15◦ , 45◦ , 75◦ , and 90◦ (ﬂat endface) -angled endfaces, respectively. It shows that the standing wave patterns are more evident in the MNFs with large endface angles (e.g., 75◦ and 90◦ ), while they become dim with decreased angles (e.g., 45◦ ), and ﬁnally disappear with small angles (e.g., 15◦ ). Calculated backward reﬂectances are 1.6×10−4 , 2.9×10−4 , 0.0012, 0.006, 0.017, and 0.022 for 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ -angled endfaces respectively, indicating the feasibility of reducing or eliminating back reﬂection in MNFs by shaping the ﬁber with angled endfaces. It is noted that the reﬂectance in an MNF can be much lower than in a conventional ﬁber (e.g., with a 90◦ -angled endface, the reﬂectance is about 2% in the above-mentioned MNF and 4% in a conventional ﬁber), which can be attributed to the large fraction of diﬀraction and evanescent ﬁelds in an MNF. The output patterns in Fig. 2.37 also show that the angled endfaces in MNFs can no longer eﬃciently redirect the light propagation, but slightly shift the output pattern in the x-y plane.

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2 Optical Properties of MNFs: Theory and Numerical Simulations

Fig. 2.37. Output patterns of 400-nm-diameter air-clad freestanding silica MNFs in x-y plane (z=0) at the wavelength of 633 nm with a (a) 15◦ , (b) 45◦ , (c) 75◦ , and (d) 90◦ (ﬂat endface)-angled endface, respectively. The dashed white lines map the topography proﬁle of the MNFs

Fig. 2.38. Output patterns of MNFs in x-y plane (z=0) at the wavelength of 633 nm. The MNFs have spherical tips with sphere diameters of (a) 400 nm and (b) 800 nm, and tapered tips with tapering angles of (c) 30◦ , (d) 60◦ , and (e) 120◦ , respectively. The white lines map the topography proﬁle of the MNFs

2.6 Endface Output Patterns

59

2.6.3 MNFs with Spherical and Tapered Endfaces Spherical or tapered ﬁber tips are usually used for focusing or dispersing light in conventional ﬁbers [53,54] . Here we investigate the output pattern of an MNF with spherical or tapered tip going down to wavelength or subwavelength level. Fig. 2.38 shows the output patterns of typical MNFs in the x-y plane (z=0), in which the ﬁber diameter and the light wavelength are assumed to be 400 nm and 633 nm, respectively. In Figs. 2.38(a) and (b), the MNFs end with spherical tips 400 nm and 800 nm in diameter, respectively. The light output from the tip spreads out as it propagates along the x-direction, indicating that the spherical tip has no evident focusing eﬀect. In addition, in Figs. 2.38(a), the diameter of the sphere equals that of the MNF, an obvious standing wave pattern inside the MNF is observed, indicating considerable reﬂection at the spherical end. Figs. 2.38(c) – (e) show the output patterns in the x-y plane (z=0) of MNFs with tapered tips, in which the shape of the taper is assumed to be an isoceles triangle with a vertex angle of 30◦ , 60◦ and 120◦ , respectively. It shows that light output from the tapered tip spreads out symmetrically along the x-axis. For tips with a large vertex angle (e.g., 120◦ ), an evident standing wave pattern is formed in the MNF. For reference, Fig. 2.39(b) provides intensity distribution of the MNF output along the y-axis in the x-y plane (z=0) with a distance of 100 nm (x=100 nm, near-ﬁeld) and 3000 nm (x=3000 nm, far-ﬁeld) from the apex of the tip, showing that the tapered tips produce a slightly larger divergence than the spherical tips and the smaller vertex angle yields larger divergence in MNFs with tapered ends.

Fig. 2.39. (a) Coordinate system for MNFs with tapered or spherical tips. (b) Normalized intensity distributions along the y-axis in x-y plane (z=0) with distances of 100 nm (x=100 nm, near-ﬁeld, in solid lines) and 3000 nm (x=3000 nm, far-ﬁeld, in dashed lines) departed from the apex of tapered or spherical MNFs in air. (Adapted from Ref. [45], with permission from the Optical Society of America)

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2 Optical Properties of MNFs: Theory and Numerical Simulations

2.7 MNF Interferometers and Resonators In the previous sections of this chapter we reviewed the transmission properties of uniform MNFs, tapered MNFs and MNF couplers. This section discusses the theory of the simplest devices composed of, or supported by, MNFs. We consider the MNF Mach-Zehnder and Sagnac interferometers, MNF loop and coil resonators, and also MNF-supported microsphere, microdisk, microcylinder and microcapillary resonators. 2.7.1 MNF Mach-Zehnder and Sagnac Interferometers Fig. 2.40 depicts two types of interferometers that can be created with MNFs. The ﬁrst one, shown in Fig. 2.40(a), is a Mach-Zehnder interferometer. It consists of an MNF connected to the input and output and a curved MNF segment, which forms two couplers with the ﬁrst MNF, as shown in Fig. 2.40(a). The input electromagnetic wave splits at the ﬁrst coupler into two waves propagating along the ﬁrst and the second MNFs. After passing the second coupler, these waves interfere and the output wave can be written in the form: Aout = A1 exp[iβ1 (λ)L1 ] + A2 exp[iβ2 (λ)L2 ]

(2.71)

here β1 (λ) and β2 (λ) are the propagation constants of the MNFs and L1 and L2 are their lengths. The amplitudes A1 and A2 in Eq. (2.71) are slow functions of the radiation wavelength λ compared to the exponents. The interference between terms in Eq. (2.71) causes oscillations of the output power |Aout |2 as a function of λ. Another type of interferometer, which can be made of MNF, is a Sagnac interferometer shown in Fig. 2.40(b). This interferometer can be created from a single MNF, which couples itself as illustrated in this ﬁgure. The output wave of this interferometer is Aout = A1 + A2 exp[iβ(λ)L]

(2.72)

here L is the MNF loop length and β(λ) is the propagation constant of the MNF. Similar to Eq. (2.71), the amplitudes A1 and A2 in Eq. (2.72) are slow functions of the radiation wavelength λ compared to the exponent and the output power |Aout |2 is an oscillating function of the radiation wavelength λ. The experimental demonstration of the Mach-Zehnder and Sagnac interferometers fabricated of MNFs will be reviewed in Chapter 5. 2.7.2 MNF Loop Resonators An MNF loop resonator (MLR) is a type of Sagnac interferometer depicted in Fig. 2.41(b) where the coupling between adjacent MNF segments leads to

2.7 MNF Interferometers and Resonators

61

Fig. 2.40. Illustration of the interferometers fabricated of MNFs. (a) Mach-Zehnder interferometer; (b) Sagnac interferometer.

a more or less complete transfer of light from one segment to another. As a result, the electromagnetic wave makes several turns in the MNF loop, forming resonances in the MLR transmission spectrum. Below, in our description of an MLR, we ignore coupling between polarization states. This assumption is usually valid for an MLR shown in Fig. 2.41(a) and may be applicable for a knot MLR shown in Fig. 2.41(b) when the twisting eﬀects can be neglected.

Fig. 2.41. (a) An MNF loop resonator and (b) an MNF knot resonator

The electromagnetic ﬁeld propagating along the MNF is characterized by the propagation constant β(s), which can slowly vary along the MNF length s, e.g., due to MNF diameter variation. In the region of coupling, near point s = 0 (Fig. 2.42), we deﬁne β1 (s) and β2 (s) as the propagation constants of the adjacent MNF segments 1 and 2, respectively. Then the electromagnetic ﬁeld amplitudes in segments 1 and 2, A1 (s) and A2 (s), can be determined by the coupled wave equations[2,33] : ⎛ d ds

A1 A2

⎜ ⎜ = i⎜ ⎜ ⎝

) 0 ) κ(s) exp −i

s

κ(s) exp i * Δβ(s)ds

s s1

0

*⎞ Δβ(s)ds ⎟ ⎟ ⎟ ⎟ ⎠

s1

Δβ(s) = β1 (s) − β2 (s)

(2.73)

62

2 Optical Properties of MNFs: Theory and Numerical Simulations

where k(s) is the coupling coeﬃcient between the adjacent MNF segments. From this equation, the transmission amplitude of the MLR, T , can be found in a form similar to that of the ring resonator[56] : ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

α

T =

e− 2 eiΨ − sin(K) α 1 − sin(K)e− 2 eiΨ

Ψ = Re(Θ), Θ=

1 2

α = 2Im(Θ) Ψ

S [β1 (s) + β2 (s)]ds + ϕ ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕΨ⎪ ⎪ ⎭

(2.74)

0

here, the parameter K and phase ϕ describe the coupling between the adjacent MNF segments, parameter α characterizes the roundtrip attenuation, and S is the circumference of the MLR. The MLR is a non-reﬂecting device and, for lossless propagation, |T | = 1. The resonances in transmission amplitude occur only if the coupling parameter K is close to the values Km =

π (2m + 1), 2

m = 1, 2, · · ·

(2.75)

which correspond to the full transmission of the electromagnetic ﬁeld from one of the adjacent MNF segments to another. Then the resonances in transmission amplitude satisfy the condition: Ψ = Ψmn = π(m + 2n)

(2.76)

Fig. 2.42. Illustration of an MLR. The coupling region is outlined. The numbers 1 and 2 correspond to the coupled microﬁber segments 1 and 2. (Adapted from Ref. c [2006] IEEE) [57],

2.7 MNF Interferometers and Resonators

63

where n is an integer. The group delay of the MLR is determined through the derivative of the phase of T with respect to the radiation wavelength λ:

td =

nf λ2 d ln(T ) 2πc dλ

(2.77)

where c is the speed of light and nf is the eﬀective refractive index of the MNF. If the propagation losses are ignored, then the propagation constant is real and |T | = 1. For a lossless MLR, |T | = 1 and the resonances appear in the group delay, td , only. For relatively low losses, α 1, the resonances are well separated. Then, in the neighborhood of the resonance wavelength λ0 calculated from Eqs. (2.77) and (2.78), the expression for the transmission amplitude is simpliﬁed: (γa − γc ) + i(λ − λ0 ) (γa + γc ) − i(λ − λ0 ) −1 !! −1 !! 1 ∂Ψ 1 ∂Ψ ! ! γa = α, γc = ! ! ! ! 2 ∂λ 2 ∂λ T =

λ=λ0

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ (K − Km )2 ⎪ ⎪ ⎭

(2.78)

λ=λ0

here the parameters γ c and γ a are the coupling and attenuation parameters, respectively. From Eq. (2.78), the transmission power of an MLR is:

P = |T |2 =

(γa − γc )2 + (λ − λ0 )2 (γa + γc )2 + (λ − λ0 )2

(2.79)

The important characteristic of a resonator is its Q-factor, which is determined as the ratio of the radiation wavelength in free space, λ, to the FWHM of the resonance at this wavelength, Δλ: Q = λ/Δλ. From Eq. (2.79) it is simple to ﬁnd

Q=

λ0 2(γa + γc )

(2.80)

Another important characteristic of a resonator is its extinction ratio, which determines the height of the resonance peaks. From Eq. (2.80), the extinction ratio of the resonance is R=

γc + γa γc − γa

2 (2.81)

64

2 Optical Properties of MNFs: Theory and Numerical Simulations

The ﬁnesse of a resonator is the ratio of its free spectral range (FSR) to the resonance FWHM: F = F SR/Δλ. From Eq. (2.74) we ﬁnd:

F =

2π (K − Km )2 + α

(2.82)

The coupling parameter K in Eq. (2.75) is calculated from the coupling wave equations, Eq. (2.73). In Ref. [57] two situations were considered: a case when the propagation constants of the adjacent MNF segments are close, and a case when these propagation constants are equal at a point s = 0 (Fig. 2.42) and their diﬀerence linearly changes near this point. The latter case is studied with the so called Landau-Zener model[58] . 2.7.3 MNF Coil Resonators The MNF coil resonator (MCR)[55] can be created by wrapping an MNF on a dielectric rod with a smaller or same refractive index as illustrated in Fig. 2.43. This device can be used as a basic functional element for the MNF photonics[59] . Similarly to the planar resonant microring structures[56] MCRs can perform complex ﬁltering, time delay, switching, and lasing functions. In addition, MCR-based optical devices have signiﬁcant advantages over planar devices, due to their smaller dimensions, simple low-loss input and output connections, possibility of fabrication with feedback, smaller transmission loss than that of a planar waveguide.

Fig. 2.43. (a) Higher-order MLR and (b) uniform MCR.

A simple case of an MCR is a higher-order MLR, which consists of several independent MLRs as e.g., shown in Fig. 2.43(a). The design of higher-order MLRs is similar to that of all-pass ring resonators[56] . Alternatively, an MCR may be represented by a uniformly wrapped self-coupling coil shown in Fig. 2.43(b)[60] . An MCR wrapped on a nonuniform core was studied in Refs. [61,62]. The experimental demonstration of an MCR containing more than one resonant loop was performed in Ref. [63] (see also Ref. [59]). Independently, an MCR, which exhibited single loop resonances, was demonstrated in Refs. [64,65]. The experimental realization of MCRs will be reviewed in Chapter 5.

2.7 MNF Interferometers and Resonators

65

Consider propagation of light along a coiled MNF. If the characteristic transversal dimension of the propagating mode is much smaller than the characteristic bend radius, then the adiabatic approximation can be applied. For a relatively small pitch of the MNF coil, coupling between co-propagating light in the adjacent turns should be taken into account. Similar to the assumptions made in the consideration of an MLR, we ignore coupling between the TE and TM polarization modes. Then the coupled wave equations, Eq. (2.73), can be generalized in the case of an MCR with M turns[55] : ⎛ d ds ⎛

⎞

A1 A ⎜ ...2 ⎟ ⎜ A ⎟ ⎜ m ⎟ ⎝ ... ⎠ AM −1 AM

=

0 χ12 (s) 0 χ (s) 0 χ (s) ⎜ 210 χ32 (s) 230 ⎜ ... ... i ⎜ ... 0 0 ⎝ 0 0 0 0 0 0 0

⎞⎛

⎞

... 0 0 0 A1 ... 0 0 0 A2 ⎟ ⎜ ... 0 0 0 ... ⎟ ⎟ ⎜ ⎟ ... ... ... ... ⎟ ⎜ Am ⎟ ... 0 χM −1,M −2 (s) 0 ⎠ ⎝ ... ⎠ ... χM −2,M −1 (s) 0 χM −1,M (s) AM −1 0 AM ... 0 χM,M −1 (s)

⎛ χpq (s) = κpq (s) exp ⎝i

s

⎞ [βp (s) − βq (s)]ds⎠

(2.83)

s1

Here, S is the length of a single turn and coeﬃcients Am (s) are deﬁned at the turns s1 < s < s1 + S and satisfy the continuity conditions: s 1 +S

βm (s)ds], m = 1, 2, · · · , M − 1 (2.84)

Am+1 (s1 ) = Am (s1 + S) exp[i s1

With Eqs. (2.83) and (2.84) it is straightforward to numerically calculate the resonance spectrum of MCRs[55,60,61,62] . In some situations these equations allow full analytical solution and complete analysis. In the ﬁrst and simplest case, an MCR is composed of M independent MLR and behaves as a straight waveguide. In the second case the MCR is a higher order MLR, which consists of a sequence of several uncoupled MLR illustrated in Fig. 2.43(a). In the third case, an MCR is composed of a uniformly coiled MNF with M turns[60] . For a higher-order MLR, coupling between adjacent turns of the MF coil is essential in each section containing an MLR, while it is negligible outside these sections. The transmission amplitude of a high-order MLR, T , is simply a product of transmission amplitudes of individual MLRs, U Tm and the total group delay, td is a sum of delay times, tdm , of each MLR:

66

2 Optical Properties of MNFs: Theory and Numerical Simulations

M 0

T =

Tm ,

td =

m=1

M 1

tdm

(2.85)

m=1

The design of the higher-order MLR generating the predetermined group delay and transmission power dependence is similar to the design of the all-pass ring resonators (see Ref. [56] and references therein). For a uniform MCR illustrated in Fig. 2.43(b), Eq. (2.83) simpliﬁes: dA1 = κA2 ds dAm = κ(Am−1 + Am+1 ), ds

m = 1, 2, · · · , M − 1

dAM = κAM −1 ds

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (2.86)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

here κ is the constant coupling coeﬃcient between adjacent turns. Solution of Eqs.(2.86) and (2.84) is[60] :

Am (s) =

M 1

a ¯n1 Amn (s),

−1

amn = amn

π(m − 1)n πmn πn sin − sin M +1 M +1 N +1 πmn πn Amn (s) = exp 2iκs cos sin , m, n = 1, 2, · · · , M M +1 M +1

amn = exp iS β + 2κ cos n=1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (2.87)

Using this equation, the transmission amplitude can be deﬁned as:

T = [A1 (0)]−1

M 1

an,1 AM n (S)

(2.88)

n=1

For the lossless MCR considered in this Subsection, |T | = 1. Eqs. (2.87) and (2.88) show that the optical properties of MCR are fully described by the three dimensionless parameters: the number of turns, M , the dimensionless propagation constant, B = βS, and the dimensionless coupling parameter, K = κS. The transmission spectrum of a uniform MCR with two turns (N = 2) is similar to the transmission spectra of the loop and ring resonators[55,56] . For the uniform MCR with M > 2, more than two turns are coupling simultaneously. This leads to a complex interplay of propagation along the MNF and

2.7 MNF Interferometers and Resonators

67

propagation through interturn coupling. Fig. 2.44 shows the surface relief of the time delay td (B, K) for the MCRs with diﬀerent numbers of turns, M . The top row of plots in this ﬁgure shows that the spectrum behavior is periodic as a function of coupling parameter, K, only for M = 2 and M = 3. For M 4, the dependence on coupling parameter, K, is no longer periodic.

Fig. 2.44. Surface plots of the time delay in the plane (B,K) for the number of turns M equal to 2 through 7, 10, and 20. For M =2, 3, 4, and 5, the points corresponding to MCR eigenmodes are marked by black dots. Upper row of plots shows the time delay for 0 < K < 20. The lower plots show the time delay for 0 < K < 2.5 with higher resolution. In the lower plots, the circles mark similar features (V-shaped and W-shaped). For M → ∞ , all similar features tend to the collapse point, K=0.5. (Adapted from Ref. [60], with permission from the Optical Society of America)

The group delay spectrum experiences interesting evolution with growth of M shown in the lower rows of plots in Fig. 2.44. All the relief features of the spectrum, which appear for a certain number of turns M in the lower row, do not disappear for larger M . These features move along the straight axial lines towards the point of spectral collapse, 1 1 (Bc , Kc ) = (π(2n − ), ) 2 2

(2.89)

where n is an integer and shrink proportionally to the distance from the collapse point. An MCR with a very large number of turns, M → ∞, represents an optical transmission line. In this case, Eq. (2.86) is solved in the form

68

2 Optical Properties of MNFs: Theory and Numerical Simulations

A(±k) m (s) = exp[2iκ cos(ξS)s ± iξSm]

(2.90)

where the integer m is a number of a turn and ξ is an eﬀective propagation constant. The continuity condition, Eq. (2.87), applied to this solution yields the dispersion relation[60] : ω(ξ) =

c [ξ − 2κ cos(ξS)] nf

(2.91)

here ω = cβ/nf is the frequency of the electromagnetic ﬁeld, c is the speed of light, and nf is the eﬀective refractive index of the MNF. It follows from Eq. (2.91) that the MCR transmission line does not have stop bands. The dispersion relation exhibits qualitatively diﬀerent behavior depending on the value of coupling parameter, K. If K < 1/2, the dispersion relation is monotonic as shown in Fig. 2.45(a1) and is qualitatively similar to that of a sequence of

Fig. 2.45. Comparison of characteristic dispersion relations for an MCR, a CROW, and a SCISSOR. (a) Dispersion relations for an MCR with the coupling parameter K =0.25 (a1), K = 0.5 (a2), and K=1.5 (a3); (b) dispersion relation of a CROW; (c) dispersion relation of a SCISSOR. (Adapted from Ref. [60], with permission from the Optical Society of America)

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69

similar ring resonators coupled to a common waveguide (so called SCISSOR) shown in Fig. 2.45(c)[56] . The crossover situation occurs at K = 1/2 (Fig. 2.45(a2)) when the function ω(ξ) has inﬂection points at ξn = π(2n − 12 )/S, where n is an integer. In the vicinity of these points, ω(ξ) ≈ ω(ξn )+α(ξ −ξn )3 , and the group velocity is vanishing simultaneously with the inverse group velocity dispersion. A pulse in the neighborhood of these points experiences dramatic distortion. Finally, in the case of strong coupling illustrated in Fig. 2.45(a3), when K > 1/2, the dispersion relation has the minima and maxima points. At these points, the group velocity is equal to zero. In the neighborhood of these points, the propagation of light is similar to propagation near the band edges of photonics crystals and CROWs[56] , which is illustrated in Fig. 2.45(b). However, in contrast to the CROW, the MCR is an all-pass photonic structure and has no bandgaps.

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56. J. Heebner, T. Ibrahim, R. Grover, Optical Microresonators: Theory, Fabrication, Applications, Springer-Verlag, New York, 2007. 57. M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, D. J. DiGiovanni, The microﬁber loop resonator: theory, experiment, and application, IEEE J. Lightwave Technol. 24, 242–250 (2006). 58. L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Pergamon Press, 1977. 59. M. Sumetsky, Basic elements for microﬁber photonics: micro/nanoﬁbers and microﬁber coil resonators, IEEE J. Lightwave Technol. 26, 21–27 (2008). 60. M. Sumetsky, Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation, Opt. Express 13, 4331–4340 (2005). 61. F. Xu, P. Horak, G. Brambilla, Conical and biconical ultra-high-Q optical-ﬁber nanowire microcoil resonator, Appl. Opt. 46, 570–573 (2007). 62. F. Xu, P. Horak, G. Brambilla, Optimized Design of Microcoil Resonators, J. Lightwave Technol. 25, 1561–1567 (2007). 63. M. Sumetsky, Y. Dulashko, M. Fishteyn, Demonstration of a multi-turn microﬁber coil resonator, In Postdeadline papers, Proceedings of Optical Fiber Communication conference 2007, paper PDP46, Anaheim, 2007. 64. F. Xu., G. Brambilla, Manufacture of 3-D Microﬁber Coil Resonators, IEEE Photon. Technol. Lett. 19, 1481–1483 (2007). 65. F. Xu, G. Brambilla, Embedding optical microﬁber coil resonators in Teﬂon, Opt. Lett. 32, 2164–2166 (2007).

3 Fabrication of MNFs

Similar to standard optical ﬁbers, the materials used for fabrication of optical MNFs are mostly glasses or glass ﬁbers. As one of the fundamental materials in ﬁelds ranging from photonics, electronics and chemistry to biology, glass presents a number of advantages over other materials (e.g., crystal and metal) in homogeneity, transparency, ease of fabrication and excellent solvent properties[1−3] . Recently, micro- or nanostructurized glasses, including lowdimension glasses such as one-dimension nanoﬁber or nanowires, have been attracting growing interest in a variety of ﬁelds such as nanoscale mechanics, photonics, and electronics[4−9] . So far, a number of techniques, including photo- or electron-beam lithography, chemical growth, nanoimprint and taper drawing at high temperature, have been developed for fabrication of low-dimension nanostructures of glasses[4−13] . For example, using vapor-liquid-solid (VLS) growth, amorphous SiOx nanowires have been obtained in a large volume with diameters down to several nanometers[5,9−11] , and a recently reported UV laser pulses assisted nanoimprint technique oﬀers the possibility to fabricate arrays of well-patterned nanoscale silica lines (or gratings) on the surface of a silica substrate[12] . Usually, these techniques are highly productive approaches for nanofabrication. However, for optical waveguiding, excellent geometric uniformity and surface smoothness of the as-fabricated structure is critical for achieving low optical loss, high signal to noise ratio and maintaining coherence of the guided light, which are desired in practical applications including optical communication, sensing, manipulation and computing. Compared with other materials such as crystal or metal, glass of amorphous structure provides a much broader range of viscosity that is strongly dependent on temperature, with a relatively wide range of temperature (corresponding to a relatively wide viscosity range) for ﬁber drawing. At the ﬁber drawing temperature, the existence of the high viscosity—that means a large cohesive force for maintaining the connection between neighboring atoms— makes it possible to balance the breaking tendency caused by the surface tension. For many glasses, this balance is maintained until the ﬁber diameter

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goes to nanometer scale. On the other hand, the high-temperature drawing technique provides a pristine molten-frozen surface of the as-fabricated MNFs, with an intrinsic surface roughness down to sub-nanometer scale that is only limited by the surface capillary waves. Therefore, among the existing techniques, so far a high-temperature taper-drawing approach oﬀers glass MNFs with unparalleled geometric uniformity and surface smoothness[6,14−18] . This chapter begins with an introduction of taper-drawing methods that have been widely used for many years, followed by a review of recently developed taper-drawing techniques that are designed for the fabrication of subwavelength-diameter micro- or nanoﬁbers from glass ﬁbers and bulk glasses, together with electron microscopy investigations of the as-drawn MNFs.

3.1 Taper Drawing Techniques The taper-drawing technique is a top-down approach that reduces largevolume materials to thin ﬁbers by means of taper drawing, as schematically illustrated in Fig. 3.1. The technique can be applied to a variety of materials such as glass and plastic when they oﬀer a certain viscosity suitable for taper drawing. It should also be noticed that, as a natural method, the taperdrawing technique has been widely adopted by living things such as spiders, caterpillars and other creatures.

Fig. 3.1. Schematic diagram of taper-drawingtechnique.

Historically, taper-drawing fabrication of thin ﬁbers was, to our knowledge, ﬁrst reported in the nineteenth century[19] , when C. V. Boys, a demonstrator of physics at the Science School in South Kensington, London, investigated the production, properties and uses of ﬁne threads drawn from glasses. In his pioneering work, Boys used a rapid draw from melts of minerals. In order to obtain long ﬁbers, he devised a pulling system relying on a bow and an arrow, as schematically illustrated in Fig. 3.2. One end of a quartz rod (the starting material) is held in the ﬁngers of the bow, and the other end of the rod is attached to a straw arrow by means of a little sealing wax. When the rod is heated somewhere along its length until there is a minute length that melts, the arrow is let ﬂy and a thread is drawn out following the trajectory of the arrow. Because of the high pulling speed oﬀered by the ﬂying arrow, ultralong (from several to tens of meters) thin ﬁbers were drawn instantaneously

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before the melts cooled down. Thin ﬁbers drawn by this approach were usually micrometers in diameter, with the thinnest ﬁbers as small in diameter as hundreds of nanometers.

Fig. 3.2. Schematic diagram of thin ﬁber drawing with a ﬂying arrow.

The as-drawn quartz ﬁbers showed extraordinary uniformities. Because of their perfect elastic properties, these “ﬁnest threads” were proposed for a number of usages such as springs for a galvanometer, aligned ﬁber arrays for optical gratings, and standard scales for optical microscopes[19,20] . The possible candidates for ﬁber drawing were also investigated. In his work, Boys tried a number of available minerals ranging from glass to crystals, and found that glass-behaving minerals (such as quartz and orthoclase) could be readily drawn into ﬁne threads, while those crystal-behaving minerals (such as sapphire and rutile) were diﬃcult to draw. Besides the ﬂying-arrow drawing technique, other approaches such as electrical spinning and high-pressure ﬂame blowing were also reported for drawing quartz ﬁbers in the early literature[20] , all depending on the large viscosity of the melting quartz for drawing long ﬁbers without them breaking up. Although highly uniform thin ﬁbers could be routinely drawn in the nineteenth century, the use of these ﬁbers for optical wave guiding was not implemented for a long time. It was not until the 1960’s, when optical waveguide theory had been well established, that researchers began to investigate the optical waveguiding properties of taper-drawn ﬁbers. For facilitating fabrication and light launching, as well as the availability of high-purity glasses, standard glass ﬁbers were routinely used as the starting material, with a ﬂameor laser-heated taper-drawing conﬁguration[21−25] . A typical ﬂame-heated taper-drawing system is schematically shown in Fig. 3.3, in which the heating source is a ﬂame-based source. Usually, a hydrogen gas torch is used due to its cleanness, easy control and possibility to provide a high enough temperature for ﬁber stretching. When scanning the ﬂame to-and-fro within a certain region of a standard glass ﬁber that is in axial tension provided by the pulling force applied at the two sides, the ﬁber is elongated with reduced diameter at the hot zone by taper drawing. The process continues with a constant pulling force applied at the two ends, and the ﬁber is stretched gradually until the desired length or diameter of the ﬁber taper is reached. Using this technique, the as-fabricated MNF is usually

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attached to the standard ﬁber through the tapering area at both ends, and is thus mentioned as a “biconical” ﬁber taper.

Fig. 3.3. Schematic diagram of taper drawing of optical ﬁbers using a scanning ﬂame heated technique.

Generally, the hydrogen ﬂame is applied in open air at room temperature, with a ﬂow rate of the order of 100 sccm. Due to the scanning mode, the ﬂame works as a brush and is thus mentioned as a “ﬂame brush” in some of the literature. Occasionally, in order to control the temperature and size of the ﬂame, sometimes oxygen is supplied to the torch. In addition, to reduce the E-band OH peak attenuation around 1380-nm wavelength, deuterium gas has been suggested to replace hydrogen gas as the fuel for the torch[26] . Using a carbon dioxide laser beam as a heating source is another elegant approach to reduce or eliminate the 1380-nm-wavelength OH absorption in tapered ﬁbers, as well as to avoid problems such as contamination and random turbulence-induced non-uniformity of a ﬂame-heated system[22,24,26,27] . Since there is no combustion involved, and no requirement for the oxygen supply, laser heating is a very clean scheme and can be completely isolated from the environment, making it possible to avoid most of the contamination related to the combustion or surrounding atmosphere. Furthermore, compared with the “ﬂame brush” that tends to be instable due to the convection or other kinds of ﬂuctuation, the laser beam can be made with high stability in power and high repeatability in operation. Fig. 3.4 shows a schematic diagram of a typical laser-heated taper-drawing system for MNF fabrication. A continuous wave carbon dioxide laser, about tens of Watts in power, is used as the heating source. The wavelength of the laser is around 10.6 μm, that falls within the strong absorptive band of most glasses. The laser beam is ﬁrst directed through a variable attenuator for controlling the power used for ﬁber heating, focused with a ZnSe lens, and is then redirected to focus on the ﬁber by a scanning mirror. The scanning mirror redirects the focused laser beam to the “hot zone” shown in Fig. 3.4 and scans the beam to-and-fro rapidly along this zone. When the temperature of the “hot zone” becomes high enough for ﬁber tapering, the ﬁber is elongated with a reduced diameter at the hot zone on account of the pulling forces applied at both ends. However, unlike in the ﬂame-heated method, with constant pulling forces, a laser-heated taper drawing procedure presents a “self-regulating” eﬀect[28] that automatically stops the stretching process when the ﬁber diameter de-

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creases to a certain value. This can be explained or predicted by vector diﬀraction theory (Mie theory)[29] : with the decrease in the ﬁber diameter, the ﬁber shows a decreasing ability to absorb laser energy. When the ﬁber diameter decreases to a certain point at which the ﬁber cannot absorb suﬃcient energy for maintaining the softening temperature, the tapering drawing process stops. The “self-regulating” eﬀect of the laser-heated approach oﬀers the possibility of quantizing the ﬁnal diameter of the taper-drawn microﬁber: one can predict the diameter of the attained microﬁber by presetting the heating power and the pulling force, making it possible to repeatedly fabricate MNFs with a particular diameter. However, on the other hand, the self-regulation presents a limitation on the attainable ﬁber diameter. Usually, the diameters of MNFs fabricated by a laser-heated taper drawing technique cannot go below one micrometer.

Fig. 3.4. Schematic diagram of taper drawing of optical ﬁbers using a scanning carbon dioxide laser beam.

3.2 Taper-drawing Fabrication of Glass MNFs Based on the taper-drawing approach described above, a number of taperdrawing techniques have been reported for the fabrication of glass MNFs with diameters around or below one micrometer. Depending on the desired applications and geometric parameters of the ﬁnal MNF, various “preforms” (e.g., optical ﬁbers, ions doped glasses) and heating sources (e.g., ﬂames, lasers and electric heating elements) have been used for taper-drawing fabrication of glass MNFs. This section reviews typical techniques that have been reported for MNFs fabrication.

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3.2.1 Taper Grawing MNFs Rom Glass Fibers Taper drawing of glass ﬁbers represents a general and simple route to the fabrication of MNFs that fulﬁlls the requirements of many applications. When a standard ﬁber is used as the preform, the as-fabricated MNF is usually connected to the ﬁber at one or both ends, which greatly facilitates both optical launching and mechanical handling of the MNF through the tapering region. Depending on the desired parameters (e.g., diameter, length or tapering shape) of the ﬁnal MNF and other considerations, several techniques have been demonstrated for taper-drawing MNFs from glass ﬁbers, as introduced below. (1) Flame-heated taper drawing of optical ﬁbers As mentioned before, the ﬂame-heated taper drawing approach (e.g., the system shown in Fig. 3.3) is capable of directly drawing MNFs from optical ﬁbers. Usually, the eﬀective length of the MNF required for most micro- or nanophotonic applications is small (e.g., less than one millimeter), and the diameter of the as-fabricated MNF is much smaller than the preform, which means only a very short segment of preform is consumed for drawing a usable MNF..Therefore, for the fabrication of MNFs with diameters below one micrometer, an immobile ﬂame is usually used instead of a scanning ﬂame. A typical ﬂame-heated taper drawing system is schematically illustrated in Fig. 3.5. A glass ﬁber mounted on two translation stages is used as a preform. Depending on the applications of the MNFs, the ﬁber preform can be either a standard single-mode ﬁber or a large-core multimode ﬁber, and the ﬁber material can be silica or other glasses (e.g., phosphate glass). An immobile ﬂame from a torch is used as the heating source. The two motorized translation stages, with pulling force and speed controlled by a controller, are used to stretch the ﬁber when it is heated to softening temperature. By precisely controlling the pulling forces, speeds and traveling distances of the translation stages, the geometric parameters of the as-fabricated MNFs (e.g., diameters, length and shape of the tapering region) can be roughly predicted. The ﬂame used here is relatively moderate and small in size (e.g., less than one centimeter in both height and width), burnt in a torch with a millimeter-scale nozzle. The fuel of the torch is supplied by well-controlled (e.g., controlled by a mass ﬂow controller) hydrogen gas or low-carbon-content liquid such as methanol or ethanol. For in-situ monitoring, the pulling process and measuring of the transmission behavior of the MNFs, a visible or infrared laser is launched into and guided through the ﬁber and the MNFs, with a detector receiving output at the other end of the ﬁber. The ﬂame-heated taper-drawing system works well for drawing MNFs with diameters ranging from hundreds of nanometers to several micrometers, and sometimes MNFs with diameters down to 100 nm can be obtained with elaborate conditions for ﬁber pulling and special carefulness in operation. It should be noted that MNFs fabricated using the above-mentioned technique keep both ends connected to the standard ﬁber through the tapering

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Fig. 3.5. Schematic diagram of ﬂame-heated taper-drawing system for drawing MNFs from optical ﬁbers.

region, which is very convenient for launching light into and/or picking light up from the MNF in practical applications. The one-step approach is simple and convenient. However, when drawing ﬁbres directly from a ﬂame-heated melt, the instability of the ﬂame usually makes it diﬃcult to control the temperature gradient in the drawing region, and consequently very thin MNFs are diﬃcult to obtain. In order to maintain a steady working temperature for drawing thin MNFs with high repeatability, a two-step taper drawing technique can be applied[6] . A schematic view of a two-step taper drawing method is illustrated in Fig. 3.6. First, a microﬁber with a diameter of several micrometers is taper drawn using a laser- or ﬂameheated pulling. Second, to obtain a steady condition for the further reduction of the ﬁber diameter, a tapered sapphire ﬁber with a tip diameter of around 100 μm is used to absorb the thermal energy from the ﬂame. The sapphire ﬁber taper conﬁnes the heating to a small volume and helps maintain a steady temperature distribution via its thermal inertia during the drawing. When one end of the micrometer-diameter ﬁber is wound around the sapphire tip at the

Fig. 3.6. Schematic diagram of drawing MNF from a micrometer-diameter (μmdiameter) ﬁber assisted with a sapphire taper. The sapphire taper is heated with a ﬂame, and the micrometer-diameter ﬁber is wound around the tip of the sapphire taper. The MNF is drawn in a direction perpendicular to the sapphire taper.

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softening temperature of the glass (e.g., about 2000 K for silica), the sapphire tip is moved about 0.5 mm out of the ﬂame to prevent the melting of the microﬁber coil. By applying a certain drawing force perpendicular to the axis of the sapphire tip, the microﬁber is drawn in the horizontal plane at a speed of 1–10 mm/s to form the MNF. With this technique, the diameter of an as-fabricated MNF can go down to 50 nm. To fabricate even thinner MNFs, a self-modulated drawing force can be used instead of the constant drawing force in the second-step taper drawing process[16] . As shown in Fig. 3.7, to introduce the self modulation, an elastic bend around the taper area of the microﬁber can be produced by holding the microﬁber parallel to the sapphire taper and by tautening the connected microﬁber between the glass and the sapphire taper. The tensile force generated by the elastic bend can be used for self modulation: during the initial stage, when drawing a thick MNF that requires a relatively large force, the bending center (around which the sharpest bend centered) occurs at the thicker part of the taper. As the ﬁber is elongated and the MNF diameter goes down, the bend loosens and the bending center moves towards the thin end of the taper, resulting in smaller forces for drawing thinner MNFs, which is helpful for drawing uniform MNFs with very small diameters. The self-modulation can also instantly smoothen unpredictable undulations such as temperature ﬂuctuation (that may cause a large variation in the viscosity of glass) by shifting the bending center to-and-fro to avoid sudden changes in MNF diameter, whereas a constant-force drawing may cause an abrupt taper or even breakage of the MNF in these cases. With the self-modulated pulling scheme, MNFs with diameters down to 20 nm have been obtained.

Fig. 3.7. Taper-drawing of thin MNFs with self-modulated pulling force. (a) Closeup photograph of the MNF drawing assisted with a bent taper for self modulation. The light shining around the MNF and the taper is a guided He-Ne laser for monitoring. (b) Schematic diagram of the self-modulation realized by shifting the bending center. The bending center is marked around the sharpest bend with a dotted circle. (Adapted with permission from Ref. [16], IOP Publishing Ltd.)

When the drawing is completed, the MNF is connected to the starting ﬁber at one end and is free-standing on the other end. When the working

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temperature is kept below the melting temperature of sapphire (about 2320 K), the sapphire tip can be used repeatedly. In addition, for real-time monitoring of the drawing process, a visible light (e.g., He-Ne laser of 633-nm wavelength) can be launched into and guided along the starting ﬁber, ﬁber taper and MNF to illuminate the taper and MNF. Generally, the MNF obtained with a ﬂame-heated taper drawing technique contains three parts: an obviously tapered start connected to the starting microﬁber with a length in millimeters, a uniform MNF with a length up to some centimeters, and an abruptly tapered end that is usually several to tens of micrometers in length. For reference, Fig. 3.8 shows a uniform part and distal end of a typical MNF, with the diameter of the uniform MNF centered around 390 nm. The excellent uniformity of the MNF is clearly seen. Depending on the fabrication conditions (e.g., softening temperature, pulling force and speed), the diameter of the as-drawn MNF can be roughly predicted.

Fig. 3.8. SEM images of a typical MNF with its (a) uniform part and (b) distal end. The diameter of the uniform MNF is centered around 390 nm.

Because of the amorphous structure of the glass and the surface tension during the melting drawing fabrication, the cross section of the taper-drawn MNF is circular. Fig. 3.9 shows an SEM image of the circular cross-section of a 480-nm diameter MNF. The cylindrical geometry of the MNF makes it possible to obtain exact expressions of the guided modes by solving Maxwell’s equations analytically[30] . Available lengths of the MNFs depend on their diameters. Generally, for MNFs thinner than 100 nm, the length of the uniform part can go up to 1 millimeter; while for MNFs thicker than 200 nm, they can be as long as several to hundreds of millimeters. For example, Fig. 3.10 shows an SEM image of a 4-mm-long MNF with a diameter of 260 nm; the MNF is roughly coiled up on the surface of a silicon wafer to show its length. Besides the large length, taper-drawn MNFs also provide excellent uniformities in terms of diameter variation and surface roughness, as has been shown in Fig. 3.8. Diameter uniformities of the MNFs can be quantitatively investigated by measuring the diameter variation (ΔD) along a length (L) with an SEM or a TEM. Shown in Fig. 3.11 are measured diameter D and

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Fig. 3.9. SEM image of cross-section of a 480-nm diameter silica MNF.

Fig. 3.10. SEM image of a 4-mm-long MNF with a diameter of 260 nm. The MNF is roughly coiled up on the surface of a silicon wafer. (Reprinted by permission from Macmillan Publishers Ltd. Ref [11], copyright 2003)

diameter uniformity UD of a typical MNF with respect to its length (starting from the thin end), where we deﬁne UD = ΔD/L, with L the length over which the MNF gives a maximum diameter deviation ΔD from the central diameter D. Although it exhibits an overall monotonic tapering tendency, neglecting the obvious initial and end tapered regions, the MNF shows a high uniformity. For example, with D >30 nm, U D < 10−4 , which means that if we cut a 100-μm-length MNF, the maximum diameter diﬀerence between the two ends is less than 10 nm, and such a tiny tapering eﬀect is acceptable in most applications. Thicker MNFs show even better uniformities. For example, for the 260-nm-diameter MNF shown in Fig. 3.10, the maximum diameter variation ΔD is about 8 nm over the 4-mm-length L of the ﬁber, giving U D =2×10−6 . Recently, with a ﬂame-heated taper drawing technique using high-purity oxygen and isobutene as fuel, G. Brambilla et al. reported U D lower than 5×10−7 in a 320-nm-diameter 20-mm-length silica MNF[14] . The small diameter of the MNF makes it possible to investigate the surface roughness with a transmission electron microscope (TEM). Shown in Fig. 3.12 is a typical TEM image taken at the edge of a 330-nm-diameter silica MNF.

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Fig. 3.11. Measured diameter D and diameter uniformity UD of a self-modulated taper-drawn silica MNF with respect to its length. The length starts from the distal end. (Adapted with permission from Ref. [16], IOP Publishing Ltd.)

Fig. 3.12. TEM image of the edge of a 330-nm-diameter silica MNF.

No obvious irregularity and defect can be found along the sidewall of the ﬁber. The typical sidewall root-mean-square (RMS) roughness of the MNF can go down to 0.2 nm [16] , approaching the intrinsic roughness of melt-formed glass surfaces[32,33] . Considering that the length of Si-O bond is about 0.16 nm[34] , such a roughness represents an atomic-level smoothness of the MNF surface, and is much lower than those of silica nanowires, tubes or strips obtained using other fabrication methods[4,5,7,9−13] . (2) Laser-heated taper drawing of optical ﬁbers There are several advantages of taper drawing with laser heating. Laser heating fully eliminates the ﬂow of the burning gas, reduces the air convection, solves the problem of cleanness, and makes it easier to achieve reproducibility by controlling the laser power. However, drawing of very thin MNFs using direct heating by a laser beam demands impractically large laser power. Actually, for an MNF heated with a CO2 laser, the power acquired from the laser beam is proportional to the volume of the ﬁber, i.e., it drops proportionally to

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the ﬁber radius squared. At the same time, the power dissipated by the ﬁber due to air convection is proportional to the surface area of the ﬁber, i.e., it drops linearly with the ﬁber radius. At a certain radius the dissipated power equalizes the acquired power and further heating of the ﬁber is no longer possible. For this reason, the minimum diameter of the microﬁber is limited by the power of the laser beam[28,22] . Fig. 3.13 shows the ﬁber temperature as a function of the incident power calculated in Ref. [28] for a CO2 laser with a Gaussian beam having 0.6 mm FWHM. These calculations are in qualitative agreement with the experimental data shown in Fig. 3.14, which was obtained in Ref. [22] for a CO2 laser beam having 0.82 mm FWHM. This ﬁgure shows a calibration curve used for producing single ﬁber tapers of a given diameter. Figs. 3.13 and 3.14 conﬁrm the impracticality of fabricating silica ﬁbers with a submicron diameter using a CO2 laser. Nevertheless, there exists a way of fabricating the MNFs by indirect melting of a silica ﬁber using a laser. Fig. 3.15 illustrates a setup for the MNF fabrication developed in Ref. [35]. An optical ﬁber was placed into the sapphire capillary tube (microfurnace), which was heated by a CO2 laser beam. The beam size along the tube was controlled by focusing with a lens. The setup is operated by displacement of four translation stages, which allow fabrication of the biconical tapers with predetermined diameter variation. The ﬁber ends were ﬁxed at stages 1 and 2. Stage 3 was used to translate the ﬁber as a whole with respect to the laser beam. The sapphire tube was translated with respect to the ﬁber and the laser beam by stage 4. This stage allows the removal of the tube from the tapered ﬁber segment after drawing is com-

Fig. 3.13. Variation of the ﬁbre temperature as a function of the applied CO2 laser power for the perpendicular polarisation. (Adapted from Ref. [28],with permission from Elsevier )

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Fig. 3.14. Calibration curve showing the desired laser power as a function of taper diameter. (Adapted from Ref. [22],with permission from the Optical Society of America )

Fig. 3.15. Illustration of the setup for drawing MNF using a sapphire tube heated with a CO2 laser. (Adapted from Ref. [35],with permission from the Optical Society of America )

pleted. The outer and inner diameters of the tube can be chosen depending on the taper parameters and are usually of the order of a millimeter. For these tube dimensions, the mass of the melting ﬁber is much smaller than the mass of the corresponding heated section of the sapphire capillary. Therefore, the temperature inside the microfurnace is not aﬀected by the ﬁber radius variation in the process of microﬁber drawing. As a simple illustration of the result of the drawing process, Fig. 3.16 shows an SEM image of an MNF, which was drawn from a regular single mode ﬁber by simple pulling with stage 1 only. In this ﬁgure, the MNF diameter decreases from 700 nm to less than 100 nm. Notice that heating the sapphire tube from only one side introduces a temperature gradient along the cross-section of the tube. The latter eﬀect is small for a thin microﬁber and can be further reduced by various axially symmetric heating techniques illustrated in Fig. 3.17. Fig. 3.17(a) is an illus-

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Fig. 3.16. SEM image of an MNF drawn in sapphire microfurnace by translation of stage 1 only.(Adapted from Ref. [35], with permission from the Optical Society of America)

tration of a setup for the axially symmetric heating of a silica preform, which was developed for drawing of telecommunication optical ﬁbers several decades ago (see Ref. [36] and references therein). This setup employs two orthogonal mounted mirrors, driven by galvanometers to distribute the laser energy around the preform. The setup shown in Fig. 3.17(b) was used in Ref. [37] for drawing axially symmetric long period gratings. In order to generate the axially symmetric heating, the authors of Ref. [37] ﬁrst created a collimated beam by defocusing the CO2 laser beam and then symmetrically focused this beam onto the ﬁber using a parabolic mirror. Fig. 3.17(c) illustrates another method for axially symmetric heating of the ﬁber suggested in Ref. [37]. As opposed to the approaches illustrated in Fig. 3.17(a) and (b), in this technique the laser beam is not translated or modiﬁed. Instead, the axially symmetric heating is achieved by the rotating of the ﬁber preform. Direct application of this method to the drawing of MNFs could be problematic because of the rotary stage eccentricity errors. The indirect laser heating setup illustrated in Fig. 3.15 can be combined with one of the axially symmetric heating setups shown in Fig. 3.17 to arrive at a very accurate and eﬀective system for MNF fabrication. (3) Electric heater taper drawing of optical ﬁber Electrical heaters are beneﬁcial compared to ﬂame and laser heaters due to their simple design, convenience in applications, and controllability. These heaters do not generate a disturbing gas ﬂow like the ﬂame burners and they do not generate long-range damaging radiation like the laser heaters. The disadvantages of electric heaters are their limitation in achieving high temperature and volatilization of the heater material. The latter takes place at high temperatures and contaminates the MNF in the process of drawing. For these reasons, electrically heated microfurnaces are primarily used for draw-

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Fig. 3.17. Illustrations of setups for the axially symmetric laser heating of an optical ﬁber preform. (a) A setup employs two orthogonal mirrors driven by galvanometers to distribute the laser energy around the preform; (b) A setup for symmetrical focusing of the collimated laser beam onto the ﬁber preform using a parabolic mirror; (c) A setup having a rotating ﬁber preform.(Adapted from Refs. [36] and [37],with permission from the Optical Society of America )

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ing the MNFs from materials with a relatively low softening temperature like, for example, polymers and compound glasses[15] . In addition, using electric heaters for drawing microﬁbers from regular silica optical ﬁbers has been demonstrated as well[38] . The application of electric micro-heaters for drawing micro/nanowires from compound-glass ﬁbers was demonstrated in Ref. [15]. The standard silica ﬁber coupler fabrication setup, shown in Fig. 3.5, was modiﬁed as follows. The ﬂame burner was replaced with a current controlled graphite microheater, which operated in the temperature range of 200 to 1700 ◦ C and provided much better temperature control than a ﬂame burner. In addition, the whole fabrication rig was housed in a Plexiglass box to avoid air turbulence. The ﬁber was pulled during the heating to form a taper in two translation stages each providing submicron precision. MNFs from a lead-silicate (LS) ﬁber and from a bismuth-silicate (BS) ﬁber were made. The fabrication of MNFs having a length of up to 100 mm with diameters as low as 90 nm (see inset in Fig. 3.18), was demonstrated. The diameter uniformity sampled along a few millimeters was within the SEM resolution (about 5 nm). The results of the transmission loss measurements are presented in Fig. 3.18. The loss at 1550 nm was a fraction of 1 dB/mm, i.e., slightly higher than that measured for silica telecom ﬁbers.

Fig. 3.18. Loss measurement at wavelength 1550 nm in lead-silicate (LS) and bismuth-silicate (BS) MNFs. Inset: SEM picture of LS MNF with radius 90 nm. c (Adapted from Ref. [15],[2005] IEEE )

In Ref. [38], an electric strip heater was used in the fabrication of silica MNFs. The designed MNF drawing rig is illustrated in Fig. 3.19. One side of the drawing ﬁber was ﬁxed at the edge of a stationary stage. The melting part of the ﬁber was placed into an electric strip heater, and the other side of the ﬁber was ﬁxed at the edge of the rotary stage. The rotary stage served for simultaneous pulling and wrapping of an MNF. Using this setup, the authors of Ref. [38] fabricated long MNFs of several tens of cm, which have a diameter

3.2 Taper-drawing Fabrication of Glass MNFs

89

Fig. 3.19. Experimental setup for fabrication of long MNF using electric strip heater and rotary stage. Inset: SEM image of an MNF with a diameter of 900 nm. (Adapted from Ref. [38],with permission from the Optical Society of America )

of 900 nm (see inset in Fig. 3.19) and an optical loss of about 0.1 dB/cm at 532 nm wavelength. As in the previous example[15] , the diameter uniformity of fabricated MNFs was within the SEM resolution, ∼ 5 nm. 3.2.2 Drawing MNFs Directly from Bulk Glasses All the above-mentioned MNF fabrication starts from glass ﬁber, which has been proved to be a very eﬃcient technique for obtaining MNFs with extraordinary uniformities and uninterrupted connection with standard ﬁbers. However, it requires an optical ﬁber as the starting point. Therefore, the available materials that can be drawn into MNFs are limited to those that have been drawn into ﬁbers. As mentioned in Section 3.1, drawing glasses into very thin ﬁbers has long been explored before the invention of the optical ﬁber[19,20] . However, the tools and conditions used at that time (e.g., arrows and bows) are not suitable for precise control and repeatable fabrication. This subsection introduces a simple technique for directly drawing MNFs from bulk glasses reported recently[18] , in which heated sapphire ﬁbers are used to melt the glass and draw MNFs with high uniformity and repeatability. The fabrication process for direct drawing of glass MNFs is schematically illustrated in Fig. 3.20. First, a CO2 laser or ﬂame is used to heat a sapphire ﬁber (hundreds of micrometers in diameter) to a temperature high enough for melting the glass, and the glass is moved towards the ﬁber. When the ﬁber is immersed in the glass through local melting, the glass is withdrawn with a portion of melt left on the ﬁber. Secondly, another sapphire ﬁber is brought into contact with the glass-coated sapphire ﬁber end, and the heating power is reduced or removed to cool down the melt to a proper temperature (e.g., 800 –1000 K for phosphate glass) for drawing, and then the second sapphire ﬁber is withdrawn at a speed of 0.1–1 m/s to draw tapers from the melt until breakage of the taper. When the process is ﬁnished, glass MNF of considerable length is formed at the freestanding side of the taper. An inert gas (e.g., argon or nitrogen) atmosphere with laser heating could be applied

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for drawing MNFs from chemically unstable materials. It should be noted that this technique can also be applied to pulverized glasses, allowing for tailoring of the optical properties of the MNF by the composition of the initial powder. In addition, using this technique, MNF can be drawn from extremely low starting quantities of glass, greatly reducing the quantity requirement of the starting material.

Fig. 3.20. Schematic diagram illustrating the direct draw of MNFs from bulk glasses. (a) A glass is moved towards a sapphire ﬁber heated by a CO2 laser or ﬂame. (b) The ﬁber end is immersed into the glass through local melting. (c) A portion of molten glass is left on the end of the ﬁber when the glass is withdrawn. (d) A second sapphire ﬁber is brought into contact with the molten-glass-coated end of the ﬁrst sapphire ﬁber. (e) The heating power is reduced and the second sapphire ﬁber is withdrawn. (f) A MNF is formed on the freestanding side of the glass taper. (Adapted from Ref. [18],with permission from the Optical Society of America )

Using this technique, a variety of glasses, including phosphate, ﬂuoride, silicate and tellurite glasses, have been drawn into highly uniform MNFs with diameters down to 50 nm and lengths up to tens of millimetres. Shown in Fig. 3.21 are typical MNFs drawn from phosphate, silicate and tellurite glasses. Fig. 3.21 (a) shows a scanning electron microscope (SEM) image of a 100-nmdiameter tellurite glass MNF. The uniform diameter and defect-free surface of the MNF is clearly visible. The high uniformity and integrity give the MNF considerable strength and pliability for manipulation. For example, Fig. 3.21(b) shows an elastically bent 320-nm-diameter silicate glass MNF with a minimum bending radius of 5 μm. The MNF shows a tensile strength high enough for withstanding such a sharp bend. Fig. 3.21(c) shows an SEM image of the fracture face of a 400-nm-diameter tellurite glass MNF, showing the high-symmetric circular cross section after the MNF is cut using the bendto-fracture process[39] . As-drawn MNFs can also be plastically bent with an annealing-after-bending procedure[39] . For reference, Figs. 3.21(d) and 3.21(e) show a spiral plastic bend of an 80-nm-diameter Er and Yb co-doped phosphate glass MNF and sharp turns made on a 170-nm-diameter tellurite glass MNF. The surface roughness of the MNF is examined using a transmission electron microscope (TEM). Figure 3.21(f) shows a TEM image of the sidewall of a 210-nm-diameter Er and Yb co-doped phosphate glass MNF, showing no visible defects or irregularities on the surface. Typical sidewall root-mean-

3.3 Drawing Polymer MNFs from Solutions

91

square roughness of these MNFs is around 0.3 nm, which is of the same order as silica MNFs drawn from optical ﬁbers and approaching the intrinsic roughness of melt formed glass surfaces[32,33] . Excellent diameter uniformity, a defect-free surface, high strength and hospitality to ion dopants, as well as the low dimension for single-mode operation, make these MNFs ideal and versatile for photonic applications.

Fig. 3.21. Electron microscopic characterizations of as-drawn glass MNFs. (a) SEM image of a 100-nm-diameter tellurite glass MNF. (b) SEM image of an elastically bent 320-nm-diameter silicate glass MNF. (c) SEM image of the cross section of a 400-nm-diameter tellurite glass MNF. (d) SEM image of a spiral plastic bend of an 80-nm-diameter phosphate glass MNF. (e) SEM image of a 170-nm-diameter tellurite glass MNF with sharp plastic bends. (f) TEM examination of the sidewall of a 210-nm-diameter phosphate glass MNF. (Adapted from Ref. [18],with permission from the Optical Society of America )

3.3 Drawing Polymer MNFs from Solutions Besides glass, polymer is another excellent material for photonics that can be easily molded with high ﬂexibility. Actually, in some respects polymer presents special advantages over other materials. For example, inherited from the perm-selective nature of polymer materials[40] , gas molecules can be easily diﬀused into the polymer matrix which may be diﬃcult for other materials such as crystal and glass. Also, polymers are hospitable to a variety of functional dopants ranging from metal oxides and ﬂuorescent dyes to enzymes, and are ready to oﬀer plentiful choices for photonic applications based on their favorable optical and electrical properties[40−42] . In addition, when dissolved in solutions, polymer shows high pliability for molding and drawing at room temperature, exempting the requirement of high temperature that is

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indispensable for glasses. Beneﬁting from these favorable properties, in past years polymers have been widely used as building blocks for optical components and photonic devices[43−47] . Among the various shapes and dimensions of polymer photonic structures, one-dimensional optical waveguiding polymer MNFs are of great interest for micro/nanophotonics applications, as has been extensively investigated very recently[48−52] . So far a number of techniques, including chemical synthesis[53] , electrochemical synthesis[54] , nanolithography[55] , electrospinning[41,49,56−59] and physical drawing[48,60] have been reported for the fabrication of polymer MNFs. Among these techniques, physical drawing is one of the simplest methods for achieving polymer MNFs with excellent sidewall smoothness and diameter uniformity that are critical to low-loss optical wave guiding. This subsection brieﬂy introduces the fabrication of polymer MNFs by physically drawing solvated polymers at room temperature. The basic procedure for the physical drawing of polymer MNFs is schematically illustrated in Fig. 3.22. First, the polymer to be drawn is dissolved in a certain kind of solvent that can be easily volatilized later for ﬁber drawing. To functionalize the MNF, functional materials can also be added into the solvent at this stage. Secondly, after the polymer is completely dissolved and well mixed or blended with functional materials, a droplet of the solution is cast

Fig. 3.22. Basic procedure for physical drawing of polymer MNFs. First, dissolve the polymer to be drawn in a solvent. Secondly, after the polymer is completely dissolved and well mixed or blended with the functional materials, cast a droplet of the solution on a plate. Thirdly, dip a probe into the droplet of the solution; when the solvent is volatilized to a certain degree so that the viscosity of the solvated polymer is suitable for drawing, withdraw the probe to draw a polymer MNF.

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93

on a plate (e.g., glass slide or silicon wafer) for MNF drawing. Then a probe (e.g., an AFM probe[48] or a tapered ﬁber probe[51] ) is dipped into a droplet of the solution. When the solvent is volatilized to a certain degree so that the viscosity of the solvated polymer is suitable for drawing, the probe is withdrawn to draw a polymer MNF. After the solvent is completely volatilized, a polymer MNF is obtained. Using this technique, various polymer MNFs such as polymethyl methacrylate (PMMA), polystyrene (PS) and polyacrylamide (PAM) MNFs can be routinely drawn with high uniformity and small diameter[52] . Usually, the diameter of the as-drawn MNFs can be roughly controlled by the drawing speed and concentration of the solution at the time of drawing. Fig. 3.23 shows an SEM image of typical as-drawn PMMA MNFs, which are drawn from a PMMA solution prepared by dissolving PMMA (weight-average molecular weight, MW =350000; Alfa Aesar) in chloroform[51] . As shown in the ﬁgure, the diameter of the as-drawn MNF can be as small as 60 nm, and the sidewall of the MNF is very smooth.

Fig. 3.23. SEM image of PMMA MNFs drawn from a PMMA (Mw = 350000) dissolved chloroform solution.(Adapted from Ref. [51])

Due to the hospitability to exotic dopants of polymer matrix, polymer MNFs can be ﬂexibly doped with various functional materials without sacriﬁcing good integrity and uniformity. Shown in Fig. 3.24 is a typical SEM image of a 290-nm-diameter bromothymol blue (BTB)-doped PMMA MNF. The MNF is drawn from a chloroform solution containing 0.5 wt% BTB (Alfa Aesar) and 5 wt% PMMA (Mw = 350000; Alfa Aesar). Excellent diameter uniformity and surface smoothness are clearly seen. In addition, by incorporating electrical, optical and magnetic nanoparticles, multifunctionalized composite polymer MNFs have also been synthesized and investigated recently[49,58,59] .

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Fig. 3.24. SEM image of a 290-nm-diameter bromothymol blue (BTB)-doped PMMA MNF drawn from a chloroform solution containing 0.5 wt% BTB and 5 wt% PMMA (Mw = 350000).

References 1. T. Izawa, S. Sudo, Optical Fiber: Materials and Fabrication, Kluwer Academic Publishers, Dordrecht, 1987. 2. M. Yamane, Y. Asahara, Glasses for Photonics, Cambridge University Press, Cambridge, 2000. 3. K. Hirao, T. Mitsuyu, J. Si, J. Qiu, Active Glasses for Photonic Devices: Photoinduced Structures and Their Applications, Springer-Verlag, New York, 2001. 4. D. P. Yu, Q. L. Hang, Y. Ding, H. Z. Zhang, Z. G. Bai, J. J. Wang, Y. H. Zou, W. Qian, G. C. Xiong, S. Q. Feng, Amorphous silica nanowires: Intensive blue light emitters, Appl. Phys. Lett. 73, 3076–3078 (1998). 5. Z. L. Wang, R. P. P. Gao, J. L. Gole, J. D. Stout, Silica nanotubes and nanoﬁber arrays, Adv. Mater. 12, 1938–1940 (2000). 6. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, E. Mazur, Subwavelength-diameter silica wires for low-loss optical wave guiding, Nature 426, 816–819 (2003). 7. Z. L. Wang, Functional oxide nanobelts: Materials, properties and potential applications in nanosystems and biotechnology, Annu. Rev. Phys. Chem. 55, 159–196 (2004). 8. E. C. C. M Silva, L. M. Tong, S. Yip, K. J. van Vliet, Size eﬀects on the stiﬀness of silica nanowires, Small 2, 239–243 (2006). 9. G. Bilalbegovic, Electronic properties of silica nanowires, J. Phys.: Condens. Matter 18, 3829–3836 (2006). 10. Y. N. Xia, J. A. Rogers, K. E. Paul, G. M. Whitesides, Unconventional methods for fabricating and patterning nanostructures, Chem. Rev. 99, 1823–1848 (1999). 11. Z. W. Pan, Z. R. Dai, C. Ma, Z. L. Wang, Molten gallium as a catalyst for the large-scale growth of highly aligned silica nanowires, J. Am. Chem. Soc. 124, 1817–1822 (2002). 12. S. Y. Chou, P. R. Krauss, P. J. Renstrom, Imprint lithography with 25nanometer resolution, Science 272, 85–87 (1996).

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35. M. Sumetsky, Y. Dulashko, A. Hale, Fabrication and study of bent and coiled free silica nanowires: Self-coupling microloop optical interferometer, Opt. Express 12, 3521–3531 (2004) 36. R. C. Oehrle, Galvanometer beam-scanning system for laser ﬁber drawing, Appl. Opt. 18, 496–500 (1979). 37. S. T. Oh, W. T. Han, U. C. Paek, Y. Chung, Azimuthally symmetric longperiod ﬁber gratings fabricated with CO2 laser, Microwave Opt. Technol. Lett. 41, 188–190 (2004). 38. L. Shi, X. F. Chen, H. J. Liu, Y. P. Chen, Z. Q. Ye, W. J. Liao, Y. X. Xia, Fabrication of submicron-diameter silica ﬁbers using electric strip heater, Opt. Express 12, 5055–5060 (2006). 39. L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. F. Chen, L. Liu, E. Mazur, Assembly of silica nanowires on silica aerogels for microphotonic devices, Nano Lett. 5, 259–262 (2005). 40. B. Adhikari, S. Majumdar, Polymers in sensor applications, Prog. Polym. Sci. 29, 699–766 (2004). 41. D. Li, Y. N. Xia, Electrospinning of nanoﬁbers: Reinventing the wheel? Adv. Mater. 16, 1151–1170 (2004). 42. H. Ma, A. K.-Y. Jen, L. R. Dalton, Polymer-based optical waveguides: Materials, processing, and devices, Adv. Mater.14, 1339–1365 (2002). 43. D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, Demonstration of 110 GHz electro-optic polymer modulators, Appl. Phys. Lett. 70, 3335–3337 (1997). 44. J. P. D. Cook, G. O. Este, F. R. Shepherd, W. D. Westwood, J. Arrington, W. Moyer, J. Nurse, S. Powell, Stable, low-loss optical waveguides and micromirrors fabricated in acrylate polymers, Appl. Opt. 37, 1220–1226 (1998). 45. D. C. An, Z. Z. Yue, R. T. Chen, Dual-functional polymeric waveguide with optical ampliﬁcation and electro-optic modulation, Appl. Phys. Lett. 72, 2806– 2807 (1998). 46. C. L. Callender, J. F. Viens, J. P. Noad, C. Eldada, Compact low-cost tunable acrylate polymer arrayed-waveguide grating multiplexer, Electron. Lett. 35, 1839–1840 (1999). 47. S. W. Ahn, S. Y. Shin, S. S. Lee, Polymeric digital optical modulator based on asymmetric branch, Electron. Lett. 37, 172–174 (2001). 48. S. A. Harfenist, S. D. Cambron, E. W. Nelson, S. M. Berry, A. W. Isham, M. M. Crain, K. M. Walsh, R. S. Keynton, R. W. Cohn, Direct drawing of suspended ﬁlamentary micro- and nanostructures from liquid polymers, Nano Lett. 4, 1931–1937 (2004). 49. H. Q. Liu, J. B. Edel, L. M. Bellan, H. G. Craighead, Electrospun polymer nanoﬁbers as subwavelength optical waveguides incorporating quantum dots, Small 2, 495–499 (2006). 50. D. O’Carroll, I. Lieberwirth, G. Redmond, Microcavity eﬀects and optically pumped lasing in single conjugated polymer nanowires, Nat. Nanotechnol. 2, 180–184 (2007). 51. Q. Yang, X. S. Jiang, F. X. Gu, Z. Ma, J. Y. Zhang, L. M. Tong Polymer micro or nanoﬁbers for optical device applications, J. Appl. Polymer Sci. 110, 1080–1084 (2008). 52. F. X. Gu, L. Zhang, X. F. Yin, L. M. Tong, Polymer single-nanowire optical sensors, Nano Lett. 8, 2757–2761 (2008).

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4 Properties of MNFs: Experimental Investigations

Although the theoretical calculations presented in Chapter 2 can be used to predict the optical waveguiding properties of an MNF in ideal situations, in practice the high fractional evanescent ﬁelds guided outside the ﬁber core usually lead to optical coupling with exotic objects such as microparticles and supporting substrates, resulting in distortion in modal ﬁelds and optical loss in the guided power to some extent. Also, for an MNF with a much smaller diameter, the slight ﬂuctuation in the ﬁber diameter and microbending of the ﬁber, that are unavoidable in some cases, may contribute signiﬁcantly to the modiﬁcations of the guiding properties. Therefore, experimental properties of MNFs are of critical importance for estimating and utilizing MNFs for practical applications. Due to the small size, with a diameter below the diﬀraction limit of an optical microscope, MNF is diﬃcult to handle for experimental measurement, especially for some very thin MNFs. Fortunately, taper drawing fabrication usually yields long MNFs with very large aspect ratios, making them possible to be visible under an optical microscope or even with the naked eye with appropriate illumination. Usually, to handle the MNF for characterization, tailoring, or assembly, high-precision micro/nanomanipulation under an optical microscopy is indispensable. Apart from optical properties, mechanical behavior is another property much involved in experimental testing, tailoring and applications of the MNF. This chapter experimentally investigates the manipulation, mechanical behavior and optical properties of MNFs.

4.1 Micro/Nanomanipulation and Mechanical Properties of MNFs To experimentally investigate properties of MNFs, the abilities to individually identify and manipulate MNFs on micro or nanometer scale are required. Meanwhile, mechanical strength and pliability are critical to the manipulation

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and assembly of MNFs to desired shapes. This section introduces the visibility, manipulation and mechanical strength of single MNFs. 4.1.1 Visibility of MNFs For low-dimension nanostructures such as nanoparticles and nanowires, precise manipulation or assembly remains a great challenge. Various techniques, incorporated with sophisticated facilities such as electric ﬁelds, magnetic ﬁelds, atomic force microscopes and optical traps or tweezers have been reported[1−5] . Fortunately, because of their large available lengths, MNFs are easily identiﬁed under an optical microscope, and for an MNF with a diameter larger than 200 nm, it is possible to see it by the naked eye if it is properly illuminated. Fig. 4.1 shows typical images of a 60-nm-diameter MNF taken by an optical microscope. The MNF is placed on a silicon wafer to obtain a large index contrast between the silica (about 1.46 at visible spectral range) and silicon (about 3.5), which greatly enhances the visibility of the MNF. The 60-nm-diameter MNF is visible in both images, and the dark-ﬁeld image (Fig. 4.1(b)) provides a much clearer proﬁle than the bright-ﬁeld image (Fig. 4.1(a)). Usually, with a high-quality 50× (or higher) objective, an optical microscope can be used for identiﬁcation of long MNFs with diameters down to 20 nm, although it is unable to measure the exact diameter of the ﬁber due to the diﬀraction limit. In addition, the use of silicon as a substrate also facilitates the geometrical investigation of the MNF under an SEM. Silicon provides good electric conductance, therefore no metallic evaporation is required for the glass MNF, especially when a ﬁeld-emission SEM is operated under a relatively low voltage (e.g., <5 kV).

Fig. 4.1. Optical microscope images of a 60-nm-diameter silica MNF taken using (a) reﬂection mode with bright-ﬁeld illumination and (b) reﬂection mode with darkﬁeld illumination. Scale bars, 10 μm. The microscope used is a Nikon ME600 with a SLWD 20× objective.

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101

4.1.2 MNF Manipulation The optical visibility of the MNF greatly facilitates the identiﬁcation and manipulation of MNFs under an optical microscope in the open-air, which also makes it possible to introduce other techniques during the manipulation, for example sending light into the MNF through the untapered ﬁber (which should otherwise be removed to adapt the sample requirement in an SEM) for in-situ investigation of optical near-ﬁeld interaction between the MNF and other objects in the vicinity. A typical experimental setup for manipulation of MNFs is schematically shown in Fig. 4.2. An optical microscope, equipped with a super-long-workingdistance (SLWD) objective that is critical for oﬀering highly ﬂexible space (especially for high-magniﬁcation objectives) for probe manipulation, is used for imaging and monitoring the manipulation process. A CCD camera and a monitor are used to facilitate remote control and observation when a special vibration-free condition is required. The MNF samples, usually placed on a substrate (e.g., silicon or sapphire wafer), are mounted beneath the objective. Tungsten or ﬁber taper probes, mounted on precision 3-dimensional stages, are used to hold and manipulate the MNFs. Usually, tungsten STM probes fabricated by means of electrochemical etching are used because of their small tip sizes (they can go below ten nanometers) and high mechanical strength[6] . During the manipulation, visible light from a laser source can be launched into and guided through the MNF (see inset of Fig. 4.2), which greatly enhances the visibility of the MNF, as well as facilitating investigation of the dynamic evolution of the optical behavior of the MNF.

Fig. 4.2. Schematic diagram of a typical experimental setup for MNF manipulation. Shown on the right is an optical microscope image captured from the monitor, in which a 500-nm-diameter silica MNF is held and bent by two STM probes.

Using the micromanipulation system shown in Fig. 4.2, MNFs with diameters ranging from tens of nanometers to a few micrometers can be bent, cut, twisted, tied, aligned or transferred, which is very helpful (sometimes indispensable) for testing, tailoring or the assembly of MNFs for various purposes.

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Fig. 4.3(a) shows a typical SEM image of a bent MNF, in which a 190-nmdiameter silica MNF is bent to a 2.5-μm-radius curve. To perform the bending process, the MNF is ﬁrst placed on a silicon wafer, and then being pushed in steps at one end with a ﬁnely controlled STM probe, as schematically shown in Fig. 4.3(b). Because of the friction between the MNF and the substrate, the as-bent MNF keeps its shape when the probe is withdrawn, while recoiling to its straight form when it is lifted up from the substrate, due to the elastic nature of the bend.

Fig. 4.3. Elastical bending of MNFs. (a) SEM image of an elastically bent 190-nmdiameter silica MNF with a bending radius of 2.5 μm. (b) Schematic illustration of the bending process using a ﬁnely controlled STM probe from step (1) to step (4). The MNF to be bent is assumed to be supported on a substrate.

To keep the bend in a free-standing MNF, as well as to avoid long-term fatigue and fracture of an elastically bent MNF due to bending stress[7,8] , the elastic bend can be annealed to a permanent plastic deformation without a change in surface smoothness or diameter uniformity[9] . Fig. 4.4 shows plastic bends in 620- and 330-nm-diameter silica MNFs (a) and multiple plastic bends in a 940-nm-diameter silica MNF (b), in which both MNFs are ﬁrst elastically

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103

bent on a sapphire wafer and then annealed for 2 h at 1400 K in a vacuum (2×10−3 Pa). Usually, before annealing, the elastically bent MNF is held tightly on the substrate, and the geometry of the assembly is hardly aﬀected by the annealing process, making it possible to lay out a ﬁnal design before annealing.

Fig. 4.4. Plastic bending of MNFs. (a) SEM image of plastic bends in 620- and 330-nm-diameter silica MNFs. (b) SEM image of multiple plastic bends in a 940nm-diameter silica MNF. (Adapted with permission from Ref. [27], copyright 2005, American Chemical Society)

The possibility of achieving elastic or plastic bending of a single MNF with high precision makes it possible to form a waveguide bend (refer to chapter 5 for details) or to investigate the mechanical properties (will be introduced in the next section of this chapter) of these thin ﬁbers. Using the bending technique, it is also possible to cut the MNF with highly symmetrical ﬂat endfaces by a bend-to-fracture approach. Fig. 4.5 schematically illustrates the bend-to-fracture technique with the help of three tungsten STM probes. While two STM probes are used to hold the MNF on a silicon or sapphire substrate, a third probe is used to bend the ﬁber to fracture at the desired point. Since the taper-drawn MNF is highly uniform in both geometry

Fig. 4.5. Schematic diagram of the bend-to-fracture process performed using three STM probes. Two probes are used to hold the MNF, and the third is used to push and bend one end of the MNF till the fracture of the ﬁber.

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4 Properties of MNFs: Experimental Investigations

and integrity, a fracture with a symmetrical ﬂat endface requires minimum energy, the bend-to-fracture process usually yields ﬂat end faces at the fracture point with good symmetry. Fig. 4.6, for example, gives SEM images of typical endfaces of silica MNFs cut with the bend-to-fracture technique. In Fig. 4.6(a), three silica MNFs, of 140, 420 and 680 nm in diameter, are cut with ﬂat endfaces. And in Fig. 4.6(b), a 160-nm-diameter 1.5-μm-length segment of MNF is cut oﬀ from a long MNF, yielding ﬂat endfaces at both ends, indicating the possibility of high-precision tailoring for very thin single MNF of short length.

Fig. 4.6. SEM images of typical endfaces of silica MNFs cut with the bend-tofracture technique. (a) Flat endfaces of three silica MNFs with diameters of 140, 420 and 680 nm, respectively. (b) A 160-nm-diameter MNF segment cut with a length of 1.5 μm. (Adapted with permission from Ref. [27], copyright 2005, American Chemical Society)

Besides the tungsten probes, ﬁber tapers with relatively lower stiﬀness and higher pliability can be used to manipulate MNFs with high ﬂexibility. For instance, a tilted ﬁber taper can be used to twist MNFs by rubbing the MNF on the surface of the substrate[10] , as schematically shown in Fig. 4.7, in which a 180◦ -bent MNF is rubbed by a ﬁber taper (usually several to tens of micrometers in diameter) perpendicularly to its length. Fig. 4.8 shows SEM images of a rope-like twist of a 480-nm-diameter MNF formed by the above-

Fig. 4.7. Schematic diagram of the twisting of an MNF by rubbing it on a substrate with a ﬁber taper.

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105

mentioned method, indicating that the MNF can withstand shear deformation to a certain degree.

Fig. 4.8. SEM image of a rope-like twist of a 480-nm-diameter silica MNF formed by rubbing the MNF on a silicon substrate with a ﬁber taper. Inset, close-view of the twisted area.

Due to its large length and high pliability, the MNF can also be tied into a knot using micromanipulation. With the help of two probes, the knot is ﬁrst assembled with a large diameter (e.g., several millimeters), freestanding in air, and is then tightened by a moving stage, which drags the knot to the desired size. Fig. 4.9 shows an SEM image of a 15-μm-diameter knot tied from a 520-nm-diameter silica MNF. Due to the high tensile strength (higher than 5 GPa, with details investigated in the next section), the MNF does not break when being tied to such a small knot.

Fig. 4.9. SEM image of a 15-μm-diameter knot tied from a 520-nm-diameter silica MNF. (Reprinted by permission from Macmillan Publishers Ltd. Ref [11], copyright 2003)

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4 Properties of MNFs: Experimental Investigations

Apart from the above-mentioned assembly or patterning, many other manipulations can be applied. For example, Fig. 4.10 shows an SEM image of a bundle of MNFs aligned using MNFs with diameters of 30, 140 and 510 nm,respectively. The bundle is placed on a silicon substrate and is selfsupporting by van der Waals attractions between MNFs. Shown in Fig. 4.11 is an SEM image of a 40-μm-diameter knot placed on a 60-μm-diameter human hair. The knot is ﬁrst assembled from a 500-nm-diameter silica MNF, and then transferred onto the surface of the hair using micromanipulation, demonstrating the possibility of transferring patterned MNFs for device assembly.

Fig. 4.10. SEM image of a self-supporting bundle of MNFs assembled with MNFs with diameters of 30, 140 and 510 nm, respectively. (Adapted with permission from Ref. [16], IOP Publishing Ltd.)

Fig. 4.11. SEM image of a 40-μm-diameter knot placed on a 60-μm-diameter human hair. The knot is ﬁrst assembled from a 500-nm-diameter silica MNF, and then transferred onto the surface of the hair using micromanipulation.

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107

4.1.3 Tensile Strengths of MNFs Using the above-mentioned micromanipulation, it is also possible to quantitatively investigate the mechanical behavior of single MNF. This section introduces a simple approach to estimate the tensile strength of a silica MNF based on a bend-to-fracture process. The bend-to-fracture process has been frequently reported for a strength test of ﬁber-like structures in forms of a 3-point bend or 4-point bend[12] . The basic model of a bending ﬁber is schematically shown in Fig. 4.12.

Fig. 4.12. Basic model of a bending ﬁber for mechanical evaluation.

When a ﬁber with diameter D is bent to a radius RB that represents the minimum allowed bending radius without fracture, the tensile strength is estimated as[13]

σ=

ED 2RB

(4.1)

where E is Young’s modulus of the ﬁber material. Since the taper-drawn MNF has excellent uniformity, integrity and an extremely low ﬂaw population, generally it is possible to withstand a large strain (e.g., > 5%) without breakage. Therefore, for better accuracy, the Young’s modulus should take a nonlinear form[14] E(ε) = E0 (1 + αε + βε2 )

(4.2)

where ε is strain, E0 = 72.2 GPa, α = 3.2, and β = 8.48. To perform the bending process, two STM probes are used for holding and pressing a 180◦ -bend MNF under an optical microscope. First, a 180◦ -bend MNF with relatively large bending radius is assembled using micromanipulation on a silicon substrate, then two STM probes are applied to press the two wings of the MNF from the opposite direction, until the fracture of the ﬁber. During the process, a CCD camera is used for in-situ monitoring of the bending radius of the MNF, and to capture the minimum allowed bending radius before the breakage of the MNF. Also, the STM probes are ﬁnely controlled

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4 Properties of MNFs: Experimental Investigations

by high-precision moving stages to avoid any abrupt wobble. Fig. 4.13, for example, provides microscope images of a bend in a 160-nm-diameter silica MNF before and after fracture.

Fig. 4.13. Microscope images of a bend in a 160-nm-diameter silica MNF. (a) Optical microscope image of the bend before fracture. (b) SEM image of the bend after fracture. The white arrows indicate the bend being investigated.

Fig. 4.14. Tensile strengths of typical MNFs estimated by the bend-to-fracture process.

Tensile strengths obtained in silica MNFs with typical diameters by the bend-to-fracture process are given in Fig. 4.14. The test is performed in air at room temperature with a relative humidity around 50%. It shows that the average tensile strength of the MNF is around 5 or 6 GPa, which is higher than that reported in silica ﬁbers with larger diameters (usually around 3 GPa under similar conditions[15] ). The large tensile strength obtained in MNFs can be explained by a low ﬂaw population in an MNF of small diameter, as well as by the short length used in the test.

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109

4.2 Optical Properties For an MNF with high fractional evanescent ﬁelds and open-air clad, optical transmission of the ﬁber is highly sensitive to a variety of parameters (e.g., surface roughness, diameter ﬂuctuation and surface contamination of the ﬁber) that may be neglected in theoretical analysis. Experimentally, the imperfect transmission can be readily evaluated by measuring the optical loss of the MNF. In addition, for practical use, the MNF is usually supported by a certain kind of substrates, which may aﬀect waveguiding behaviors of the MNF. This section introduces experimental investigation of optical losses and substrate-induced eﬀects in MNFs. 4.2.1 Optical Losses The minimum possible optical loss of the conventional telecommunication ﬁber is determined by the fundamental scattering and absorption in highpurity glasses. The loss depends on the radiation wavelength is illustrated in Fig. 4.15 and can be as small as about 0.15 dB/km in the low-loss window, which is located near the radiation wavelength of 1.5 μm. Commercially fabricated single mode ﬁbers have a transmission loss which is closely approaching this limit. Alternatively, the transmission loss of an MNF is determined by its surface nonuniformities and contamination. In addition, the MNF transmission loss cannot be smaller than the fundamental value originating from its fabrication method. For silica MNF, as explained in Subsection (1), this value is about 0.01 dB/m, which is two orders of magnitude larger than the loss of a state of the art telecommunication ﬁber. Nevertheless, this MNF transmission loss is signiﬁcantly smaller than that of the lithographically fabricated planar photonic circuits. The experimentally demonstrated transmission loss of an MNF is about 1 dB/m[16,17] . This value is still two orders of magnitude greater than the smallest possible loss predicted theoretically. Fig. 4.16 shows typical experimental data for the transmission power of a biconical taper with MNF waist considered as a function of the MNF radius in the process of drawing (the MNF length in this example is 4 mm and the radiation wavelength is 1.53 μm). Small oscillations correspond to weak coupling between the fundamental mode and higher order modes. It is seen that the transmission power only slightly reduces when the MNF radius decreases down to about 0.5 μm, i.e., ∼ quarter of radiation wavelength. The transmission loss of MNFs with the radius exceeding this value, considered in Subsections (1) and (2), is primarily determined by the MNF surface roughness and local nonuniformities[18−25] . A simple and comprehensive technique for investigation of MNF nonuniformities[25] is described in Subsection (3). For MNFs with a smaller radius, considered in Subsection (4), the transmission loss drops dramatically, which is primarily caused by input and output losses[26−28] .

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4 Properties of MNFs: Experimental Investigations

Fig. 4.15. Transmission spectrum of a telecommunication optical ﬁber.

Fig. 4.16. Transmission power of an MNF in the process of drawing.

(1) Frozen capillary waves The fundamental limitation of the MNF transmission loss originates from its fabrication method, which consists in heating an MNF preform to a melting temperature followed by drawing and cooling of the created MNF. The drawn MNF possesses a very small residual surface roughness. Fig. 4.17 shows an example of a surface proﬁle of a 1 μm × 1 μm square region of an optical ﬁber, which was measured immediately after its drawing[19] . The measurement was performed with an AFM having a tip of about 10 nm diameter. The peak-to-valley value for this plot is 1.5 nm and RMS roughness

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111

is about 0.2 nm. The subnanometer-scale nonuniformities of the silica surface are caused by the frozen capillary waves, which solidify in the melting temperature range[18,20−23] .

Fig. 4.17. Proﬁle of a silica ﬁber surface measured with an AFM. (Adapted from Ref. [19],with permission from Elsevier )

A surface with frozen capillary waves is determined by the height distribution function h(x ). The expressions of the surface roughness, σ, of the spatial correlation function, G(x), and of the height-height correlation function, H(x) can be obtained in the form[18,23] : 2 3! σ = h2 (x) !x = 2

kT 2πγ

ln

ζM ζm

G(x) = h(x + y)h(y) |y =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

kT 2πγ

ln

3! 2 H(x) = |h(x + y) − h(y)|2 !y =

ζM x

kT πγ

ln

x ζm

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.3)

where x and y are the two-dimensional vectors with absolute values x = |x| and y = |y|, k is the Boltzman constant, and γ is the surface tension. The temperature T in these expressions is taken in the transition region. For example, for silica the transition region is around T ∼ 1500 ◦ C. The expressions for G(x) and H(x) are logarithmically divergent. Therefore, an upper (ζM ) and a lower (ζm ) spatial cut-oﬀ must be introduced. Physically, the upper and lower cut-oﬀ parameters are determined from the restrictions of the theory yielding Eq. (4.3). For example, the neglected gravity eﬀects deﬁne the upper cut-oﬀ given by the capillary length (γ/(ρg))1/2 , where ρ is the glass density and g

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4 Properties of MNFs: Experimental Investigations 2

3

is the gravity constant. For silica, γ = 0.3 J/m and ρ = 2.2 kg/m , so that this length is about 4 mm. The lower cut-oﬀ is estimated by a characteristic molecular size, which is, for silica, about 0.5 nm. It follows from Eq. (4.3) that the typical roughness of a silica surface in the glass transition region due to capillary waves is about σ ∼ 0.4nm. Molecular and capillary lengths deﬁne the ultimate physical bounds of the expected logarithmic scaling. However, in roughness measurement, the cut-oﬀ lengths are determined by the bandwidth of the experimental set-up. For example, in the AFM measurements, the upper cut-oﬀ length, μM , is usually in the micrometer range, while the lower cut-oﬀ length μm is of the order of the nano-sized AFM tip diameter. Fig. 4.18 depicts the results of AFM measurement of the height-height correlation function H(x) for the silica surface after annealing at diﬀerent temperatures in the transition region. It is seen that the experimental curves ﬁt well the logarithmic dependence predicted by Eq. (4.3). The measured surface roughness σ ∼ 0.2nm is in good agreement with the theoretically predicted value.

Fig. 4.18. Height variogram H(x) obtained from AFM measurements on silica surfaces after annealing treatments at diﬀerent temperatures in the glass transition regime (symbols). The lines correspond to the logarithmic behavior predicted by Eq. (4.3) induced by the presence of frozen capillary waves. (Adapted from Ref. [23],with permission from Springer Science + Business Media)

The measured height-height correlation function allows us to estimate the transmission loss of an MNF using Eq. (2.50). Consider a silica MNF transmitting radiation of 1.5 μm wavelength. For the radius of an MNF between 0.3 and 0.6 μm, the linear ﬁt of the numerical dependence in Fig. 2.5 gives simple expression for the transversal component of the propagation constant γ as a function of the MNF radius r in the form: γ(r) = 4.9r − 1.2. Here r is expressed in μm and γ is expressed in μm−1 . Substituting this expression into Eq. (2.50) and assuming the characteristic variation length L ∼ 1 μm we get the following estimate for the propagation constant of a silica MNF through the characteristic MNF radius variation Δγ (expressed in μm):

4.2 Optical Properties

α ∼ 4(Δr)2 dB/μm

113

(4.4)

Setting Δγ equal to the measured value of surface roughness, Δr = σ = 2 × 10−4 μm, we arrive at the estimate for the smallest possible propagation constant of a silica MNF α ∼ 0.01 dB/m. (2) MNF nonuniformities Ideally, the transmission loss of an MNF can be as small as predicted by the capillary wave theory considered in the previous subsection. In addition, nonuniformities are introduced in the process of controlled MNF fabrication. For example, in the ﬂame brushing method, the nonuniformities are generated by the noise and acceleration of the translation stages as well as by the ﬂame ﬂuctuations. These nonuniformities make a signiﬁcant contribution to the MNF transmission loss. As explained in the previous subsection, to evaluate ultra-low transmission losses, MNF nonuniformities should be characterized at the nanometer level. However, the SEM and AFM measurements of only the local surface nonuniformity are insuﬃcient to characterize the total MNF loss. These methods are unable to perform the comprehensive characterization of MNFs, which have millimeter-range length and a micron and sub-micron size cross-section. On the other hand, application of transmission electron microscopy, scanning near-ﬁeld optical microscopy and other side-scattering methods are problematic due to alignment and focusing diﬃculties caused by small transversal geometry and vibration of the MNF. In Ref. [25], a simple measurement tool for the comprehensive characterization of the MNF nonuniformities was developed. The authors experimentally determined the eﬀective MNF radius variation (combining the eﬀects of MNF surface and index variations) and achieved subnanometer accuracy with remarkable reproducibility. The measurement setup, illustrated in Fig. 4.19, consisted of a partly stripped optical ﬁber segment used as a probe.

Fig. 4.19. The setup for measurement of the MNF nonuniformities. A partly stripped ﬁber probe slides along the MNF. Diﬀerent sides of the MNF can be tested by rotation of the MNF with respect to the probe. (Adapted from Ref. [25], with permission from the Optical Society of America)

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4 Properties of MNFs: Experimental Investigations

Fig. 4.20. Transmission power of an MNF as a function of the probe position compared to the MNF radius variation measured by an SEM. (Adapted from Ref. [25],with permission from the Optical Society of America )

The investigated MNF was a part of a biconical ﬁber taper with a uniform waist of about 4 mm length and about 0.53 μm radius. At the measurement wavelength (1530 nm), the taper’s waist was a single mode MNF. The MNF is positioned in direct contact with the ﬁber probe and normal to its axis. The probe was translated along the MNF and, simultaneously, the transmission power was detected by an OSA. The MNF was tested from diﬀerent sides by rotation of the MNF with respect to the probe as illustrated in Fig. 4.20. For small MNF radius variation, it is natural to assume that the experimentally measured change in the logarithm of transmission power of the ﬁber taper is proportional to the radius variation at the point of the probeMNF contact. The constant of proportionality was determined in Ref. [25] experimentally by comparison of the MNF radius variation measured using SEM with the transmission power variation along the MNF regions. Fig. 4.20 demonstrates good correspondence between the transmission power and SEM measurements. Curves 1 and 2 in Fig. 4.21(a) show the reproducibility of the eﬀective radius variations measured along the 500 μm segment of an MNF. Fig. 4.21(b) shows a magniﬁed comparison of these curves along a 100 μm length. The RMS diﬀerence between these curves 1 and 2 in Fig. 4.21(a) is 0.37 nm, while for their parts shown in Fig. 4.21(b) it is only 0.19 nm. Curve 3 in Fig. 4.21(a) is the scan of the same MNF segment rotated by 90 ◦ . The similarity of curves 1 and 2 to curve 3 provides evidence for the axial symmetry of observed microdeformations. These curves also show two pronounced asymmetrically localized defects (sharp peaks) magniﬁed in Fig. 4.21(c). Using Eq. (4.4), and the experimental curves in Fig. 4.21, the MNF attenuation constant was estimated as α ∼ 0.01 dB/cm, which was in reasonable agreement with experimental data α ∼ 0.05 dB/cm. The described experimental technique can be further developed by miniaturization of the probe, e.g., by replacing the ﬁber probe with a sharp glass blade. This will allow us to improve the longitudinal resolution of measure-

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115

Fig. 4.21. MNF radius variation along a 500 μm segment. (a) curves 1 and 2 – two measurements of this segment, which are shifted along the vertical axis for visibility; 3 – measurement of the same segment after the MNF was rotated by 90 ◦ ; (b) magniﬁed plot of curves 1 and 2 along the 100 .μm segment; (c) magniﬁed plot of peaks in Fig. 4.21(a). (Adapted from Ref. [25],with permission from the Optical Society of America )

ments. The described technique opens broad opportunities for investigation of MNF nano-scale optical and physical properties, and can be applied to investigation of MNFs with diﬀerent radii and complex internal structure (e.g., photonic crystal microﬁbers) by varying the radiation wavelength and considering higher order modes. (3) MNF surface contamination The fabricated MNF is subject to surface contamination, which progresses in time and causes a further increase in the MNF transmission loss. The contamination includes electrostatic, chemical, adsorptive and diﬀusive adhesion of ambient microparticles and molecules. Illustration of a signiﬁcant degradation in transmission of an MNF, situated outside a clean room, is given in Fig. 4.22[24] . This ﬁgure shows the transmission spectra of an MNF with 375 nm radius at diﬀerent times after fabrication. The spectral dependence of induced loss has been studied by launching light from a broadband light

116

4 Properties of MNFs: Experimental Investigations

source and collecting the transmitted light with an OSA. The authors of Ref. [24] found that the use of conventional chemical cleaning compounds (acetone, isopropanol, methanol, water) allowed only a partial recovery of the induced loss. Therefore, it is unlikely that the observed loss is caused only by dust or microparticles deposited on the surface. After 24 hours the ﬂame burner that has been scanned below the taper allowed us to recover the initial transmission properties of the MNF, as depicted in Fig. 4.22. Thus, post-processing of MNFs by heating is an eﬃcient method for restoration of their transmission characteristics.

Fig. 4.22. Transmission spectra of an MNF with radius 375 nm at diﬀerent times after fabrication: (0 h) – recorded at end of fabrication; (2 h) – 2 hours later; (10 h) – 10 hours later; (24 h) – after ﬂame brush cleaning performed 24 hours later. c (Adapted from Ref. [24],[2006] IEEE)

In the considered example, surface contamination dramatically aﬀected the transmission loss of an optical MNF. This happened primarily due to the fact that a considerable fraction of the fundamental mode was propagated in the vicinity of the MNF surface. For MNFs with a larger radius, this fraction becomes signiﬁcantly smaller and, consequently, the surface contamination is much less signiﬁcant. (4) Transmission loss of very thin MNFs Consideration of waveguide losses should include the input and output losses, which are determined by the way the input and output of light is performed. For a conventional optical communication ﬁber, the input and output losses can be minimized down to relatively small values. For this reason, these losses are considered separately from the losses of the ﬁber itself. However, for an MNF whose diameter is signiﬁcantly smaller than the radiation wavelength, the transmission loss is primarily determined by input and output

4.2 Optical Properties

117

losses. These losses, in practice, cannot be reduced signiﬁcantly as explained in Subsection 2.4[26,27] . Experimental results[16,17,28,29] indicate the existence of a steep transition between the highly lossless and highly lossy regimes in the MNF transmission. In this subsection, following the results of Ref. [29], we show that the experimentally determined radius of the thinnest MNF waveguide is in good agreement with predictions of the theory of adiabatic MNF tapers described in Subsection 2.3[27] . In addition, a simple method for the determination of the MNF radius from its transmission spectrum is demonstrated. In experiment[29] , the sample MNFs were fabricated by indirect laser heating described in Subsection 3.2.1[30] . The MNFs were drawn using the laser beam brushing method, which consisted of multiple cycles of pulling the MNF from a conventional telecommunication ﬁber with an initial radius of 62.5 μm. After a cycle of drawing, the ﬁber radius is set to be reduced by a factor f = 0.75. Thus, after the N th cycle of drawing, the radius of the taper MNF waist was r (N ) = Rf N = 62.5 × 0.75N (μm)

(4.5)

The drawn MNF radius variation was measured using the technique described in the previous Subsection[25] . The transmission loss of MNFs was calculated with the theory[27] described in Subsection 2.3.4. The tapered sections of fabricated MNFs were approximated by Eq. (2.57) with characteristic length parameter L. In Fig. 4.23(a), the transmission loss deﬁned by Eq. (2.58) is plotted as a function of the MNF radius for diﬀerent wavelengths. This plot determines the radii, in the neighborhood of which the transmission loss starts a noticeable deviation from zero. The solid and dashed lines correspond to the MNF nonuniformity L = 250 μm and L = 500 μm, respectively. The Table below Fig. 4.23(a) determines the corresponding cycle numbers and the MNF radii found from Eq. (4.5). The predictions of Fig. 4.23(a) were tested experimentally by monitoring the transmission power of the MNF taper in the process of drawing. Fig. 4.23(b) shows the results of measurements[29] . The x-axis in this ﬁgure is the eﬀective MNF radius, which was calculated by the continuation of Eq. (4.5) to the non-integer values of N . The characteristic step-shaped behavior of the curves near the threshold radii is explained by temporal nonuniformity of MNF drawing, which consisted of multiple cycles. Comparison of the theoretical prediction for the threshold radii of Fig. 4.23(a) with the experimental data of Fig. 4.23(b) demonstrates a reasonable agreement of the threshold radius value. It is important to note that in the regions of the threshold behavior of the transmission power, the loss originating from the tapered sections (serving as the input and output connections) of MNFs is signiﬁcantly greater than the local transmission losses due to MNF micrononuniformities. Similarly, there exists a threshold radiation wavelength for an MNF with a ﬁxed radius. For the wavelengths that are smaller than the threshold wave-

118

4 Properties of MNFs: Experimental Investigations

Fig. 4.23. (a) Theoretically calculated transmission loss as a function of MNF radius determined for diﬀerent wavelengths (1230, 1320, 1430, and 1530 nm); (b) Transmission loss of MNFs measured in the process of drawing for the same transmission wavelengths (1230, 1320, 1430, and 1530 nm) as a function of eﬀective MNF radius. In order to illustrate the reproducibility of measurements, two measurements for each wavelength are shown. The curves are shifted along the vertical axis for visibility. The table between ﬁgures shows the number of cycles and MNF diameters corresponding to the regions separated by the vertical dashed lines. (Adapted from Ref. [29],with permission from the Optical Society of America )

length, the MNF has small losses while, for larger wavelengths, the radiation losses grow rapidly. This threshold wavelength can be determined from Eqs. (2.12) and (2.58). In order to show this experimentally, the authors of Ref. [14] fabricated an MNF with a radius of r(21) = 149 nm, corresponding to

4.2 Optical Properties

119

Fig. 4.24. Experimentally measured transmission spectrum of an MNF fabricated with 21 cycles of drawing. Inset: magniﬁed experimental spectrum and theoretical spectra. The parameters of theoretical spectra are shown in the ﬁgure. (Adapted from Ref. [29],with permission from the Optical Society of America )

N = 21 cycles in Eq. (4.5). The transmission spectrum of this MNF is shown in Fig. 4.24. The inset in this ﬁgure magniﬁes the region of the spectrum where the propagation loss starts, noticeably departing from zero. In the inset, the experimental curve is compared with the theoretical dependences found in Eq. (2.58) with L = 100, 250, 500, and 1000 μm. It is seen that although the chosen characteristic lengths L diﬀer by up to a factor of 10, the diﬀerence between the corresponding threshold radii is less than 9%. The calculated radius values are in reasonable agreement with the value r(21) = 149 nm predicted by Eq. (4.5). Therefore, the transmission spectrum of an MNF allows a fairly simple and accurate estimate of its radius. 4.2.2 Eﬀect of the Substrate Due to the small diameter, a free-standing MNF is highly pliable and is easily inﬂuenced by environmental airﬂow. For practical characterization or application of these kinds of thin ﬁbers, a certain kind of physical support is required in most cases for maintaining the desired geometry or assembly, and the easiest way to do this is to integrate the MNF with a solid substrate. Whatever the material is, a solid substrate usually has a refractive index higher than air (i.e., n=1.0) for dielectrics, or has a large imaginary part (that may lead to high optical loss) for metals. An MNF with high fractional evanescent ﬁelds is very likely to suﬀer from the mode transition (from guide mode to leakage mode) that leads to power leakage of the guided light or

120

4 Properties of MNFs: Experimental Investigations

radiation loss due to imperfection of the substrate that has been reported in both MNFs and oxide nanowires[9,11,31,32] . Fig. 4.25 shows the intensity distribution of a MgF2 -supported 500-nm-diameter silica MNF guiding 633nm-wavelength light. Refractive indices of silica and MgF2 are 1.46 and 1.39, respectively. It shows that when the light is guided across the edge of the MgF2 , a serious power leakage occurs.

Fig. 4.25. 3D-FDTD simulation of intensity distribution of an MgF2 -supported 500-nm-diameter silica MNF guiding 633-nm-wavelength light. Refractive indices of silica and MgF2 are 1.46 and 1.39, respectively.

Fig. 4.26. Optical microscope image of an MgF2 -supported 440-nm-diameter PS MNF guiding a broadband light from a supercontinuum source. The color change in input (orange) versus output (green) indicates wavelength-dependent transmission of a supported MNF. (Adapted with permission from Ref. [29], copyright 2008, American Chemical Society)

With a given diameter, the fractional power of evanescent ﬁelds outside an MNF is in proportion to the wavelength of the guided light[33] . Therefore, the power leakage in a supported MNF (similar to the one shown in Fig. 4.25) is wavelength-dependent: light of a longer wavelength suﬀers higher leakage loss

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than that with a shorter wavelength. The above-mentioned short-pass ﬁltering eﬀect has been observed in various types of nanowires or nanoﬁbers, such as oxide nanowires, polymer MNFs and glass MNFs[31,34,35] . For example, Fig. 4.26 shows an optical microscope image of an MgF2 -supported 440-nmdiameter PS MNF (Refractive index about 1.59) guiding a broadband light from a supercontinuum source. The color change in input (orange) versus output (green) indicates wavelength-dependent transmission of a supported MNF. To quantify this eﬀect, Fig. 4.27 gives the transmission spectrum of an MgF2 -supported 730-nm-diameter silica MNF, with the segment supported on the MgF2 of about 0.69 mm in length. The spectrum is measured using a supercontinuum that spans a broad spectral range (dotted line). The transmittivity shows a clear drop around the 650-nm wavelength, with an attenuation as high as –40 dB at the long wavelength side.

Fig. 4.27. Transmission spectrum of an MgF2 -supported 730-nm-diameter silica MNF, with the segment supported on the MgF2 of about 0.69 mm in length.

The substrate also changes the single-mode condition of the MNF[36] . For a freestanding MNF, the single-mode condition is well deﬁned by Eq. (2.7). When it is supported by a substrate, the MNF and the substrate may be evanescently coupled and form a composite waveguide that exhibits a red-shift of cutoﬀ wavelength, or equivalently a larger cutoﬀ diameter at a given wavelength. For example, for a freestanding air-clad silica MNF, the single-mode cutoﬀ diameter is about 380 nm at 532-nm wavelength; when it is supported by a MgF2 substrate, the single-mode cutoﬀ diameter moves to a much larger diameter, which can be attributed to the increase in the eﬀective index of the cladding due to the presence of the substrate. Fig. 4.28(a) shows a near-ﬁeld scanning optical microscope (NSOM) image of the output pattern of an MgF2 supported 800-nm-diameter silica MNF guiding a 532-nm-wavelength light. A line cut of the near-ﬁeld mode proﬁle parallel to the facet (Fig. 4.28(a)) is

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Fig. 4.28. Output pattern of an MgF2 -supported 800-nm-diameter silica MNF guiding a 532-nm-wavelength light. (a) Near-ﬁeld scanning optical microscope (NSOM) image of the output pattern. (b) A line cut of the near-ﬁeld mode proﬁle parallel to the facet (100 nm away from the facet).

given in Fig. 4.28(b). The mode proﬁle shows that the MNF is single-mode, although the 800-nm diameter is much larger than the single-mode cutoﬀ diameter (380 nm) of a freestanding MNF. Therefore, in addition to the stable support, the presence of the substrate releases the diameter requirement for single-mode operation, which may also facilitate the handling and practical applications of single-mode MNFs due to their larger usable diameters.

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10. L. M. Tong, J. Y. Lou, Z. Z. Ye, G. T. Svacha, E. Mazur, Self-modulated taper drawing of silica nanowires, Nanotechnology 16, 1445–1448 (2005). 11. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, E. Mazur, Subwavelength-diameter silica wires for low-loss optical wave guiding, Nature 426, 816–819 (2003). 12. R. C. Hibbeler, Mechanics of Materials, Prentice Hall, 2004. 13. G. N. Morscher, H. Sayir, Bend properties of sapphire ﬁbers at elevatedtemperatures. I. Bend survivability, Mater. Sci. Eng. A 190, 267–274 (1995). 14. J. T. Krause, L. R. Testardi, R. N. Thurston, Deviations from linearity in the dependence of elongation upon force for ﬁbers of simple glass formers and of glass optical lightguides, Phys. Chem. Glasses 20, 135–139 (1979). 15. M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, Strength measurement of optical ﬁbers by bending, J. Am. Ceram. Soc. 69, 815–821 (1986). 16. G. Brambilla, V. Finazzi, D. J. Richardson, Ultra-low-loss optical ﬁber nanotapers, Opt. Express 12, 2258–2263 (2004). 17. 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, 2864–2896 (2004). 18. J. Jackle, K. Kawasaki, Intrinsic roughness of glass surfaces, J. Phys.: Condens. Matter 7, 4351–4358 (1995). 19. P. K. Gupta, D. Innis, C. R. Kurkjian, Q. Zhong, Nanoscale roughness of oxide glass surfaces, J. Non-Cryst. Solids 262, 200–206 (2000). 20. T. Seydel, M. Tolan, B. M. Ocko, O. H. Seeck, R. Weber, E. DiMasi, W. Press, Freezing of capillary waves at the glass transition, Phys. Rev. B 65, 184207 (2002). 21. M. Sprung, T. Seydel, C. Gutt, R. Weber, E. DiMasi, A. Madsen, M. Tolan, Surface roughness of supercooled polymer melts, Phys. Rev. E 70, 051809 (2004). 22. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, T. A. Birks, J. C. Knight, P. St. J. Russell, Ultimate low loss of hollow-core photonic crystal ﬁbres, Opt. Express 13, 236–244 (2005). 23. T. Sarlat, A. Lelarge, E. Søndergard, D. Vandembroucq, Frozen capillary waves on glass surfaces: an AFM study, Eur. Phys. J. B 54, 121–126 (2006). 24. G. Brambilla, F. Xu, X. Feng, Fabrication of optical ﬁber nanowires and their optical and mechanical characterization, Electron. Lett. 42, 517–518 (2006). 25. M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, J. W. Nicholson, Probing optical microﬁber nonuniformities at nanoscale, Opt. Lett. 31, 2393–2396 (2006). 26. M. Sumetsky, How thin can a microﬁber be and still guide light, Opt. Lett. 31, 870–872 (2006). 27. M. Sumetsky, Optics of tunneling from adiabatic nanotapers, Opt. Lett. 31, 3420–3423 (2006). 28. A. M. Clohessy, N. Healy, D. F. Murphy, C. D. Hussey, Short low-loss nanowire tapers on singlemode ﬁbres, Electron. Lett., 41, 954–955 (2005). 29. M. Sumetsky, Y. Dulashko, P. Domachuk, B. J. Eggleton, Thinnest optical waveguide: experimental test, Opt. Lett. 32, 754–756 (2007). 30. M. Sumetsky, Y. Dulashko, A. Hale, Fabrication and study of bent and coiled free silica nanowires: Self-coupling microloop optical interferometer, Opt. Express 12, 3521–3531 (2004).

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5 MNF-based Photonic Components and Devices

Optical ﬁber based components and devices have been very successful in the past 30 years and will surely continue to thrive in a variety of applications including optical communications, optical sensing, power delivery and nonlinear optics[1−3] . With increasing requirements for higher performance, wider applicability and lower energy consumption, there is a strong demand for the miniaturization of ﬁber-optic components or devices. When operated on a smaller spatial scale, a photonic circuit can circulate, process and respond to optical signals on a smaller time scale. Only at wavelength or subwavelength size does the photonic structure manifest evident near-ﬁeld features that can be utilized for interlinking and processing optical signals highly eﬃciently. For example, it was estimated that to reach an optical data transmission rate as high as 10 Tb/s, the size of photonic matrix switching devices should be reduced to 100-nm scale[4] . At the same time, to perform a given function that relies on a certain kind of light-matter interaction, usually less energy is required when smaller quantities of matter are involved. Optical MNFs featured at subwavelength scale, provide a number of interesting properties such as tight conﬁnement, high fractional evanescent ﬁelds, large and manageable waveguide dispersion that are highly desirable for functionalizing ﬁber-optic circuits with great versatility on a micro/nanoscale[5−7] . At the same time, taper-drawn MNFs provide excellent compatibility with standard optical ﬁber systems. This chapter gives an up-to-date review of MNF-based photonic components/devices that have been investigated very recently. More particularly, starting from the basic components such as linear waveguides and waveguide bends, we introduce optical couplers, interferometers, resonators, ﬁlters and lasers that have been assembled using MNFs. Fig. 5.1 illustrates basic methods of MNF device fabrication. Often, these methods can be divided into micromanipulation employing positioning of MNFs at the low-index substrate with miniature tips of an STM (Fig. 5.1(a)) and macro-manipulation employing translation and rotation of the ends of a ﬁber taper with an MNF waist (Fig. 5.1(b)). MNF-based optical sensors will be introduced in the next chapter.

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Fig. 5.1. Illustration of MNF manipulation: (a) micro-manipulation, (b) macromanipulation.

5.1 Linear Waveguides and Waveguide Bends As has been mentioned before, MNFs fabricated by taper drawing technique can serve as free-standing or supported waveguides. This subsection introduces the possibility of using MNFs as linear waveguides and waveguide bends, which are usually categorized as basic optical components for optical circuits and devices. 5.1.1 Linear Waveguides The linear waveguide is one of the simplest optical components for optical circuits and devices. So far, there are a number of linear waveguides with micro/nano (or subwavelength) feature sizes that have been welldeveloped including silicon-on-insulator (SOI) waveguides or silicon photonic wires[8−10] , photonic bandgap structures[11−13] , plasmonic waveguides[14−16] , semiconductor/oxide crystalline nanowires[17−19] , and physically drawn optical MNFs[20−24] . Among these structures, MNF is featured with ultralow optical loss, simple structure, easy fabrication and high ﬂexibility and strength for manipulation and assembly[25−27] . Also, the MNF materials, usually glasses or polymers, bestow the MNF with great versatility for tailoring optical properties with exotic dopants and coatings [22,24,28−30] . As a linear waveguide, MNF can serve as both free-standing and supported waveguides. Table 5.1 lists the possible schemes for using an MNF as a linear waveguide classiﬁed by its supporting form. (1) Freestanding MNFs An as-drawn MNF freestanding in air or vacuum can serve as an optical waveguide in its simplest form. As it has an ideal cylindrical symmetry, the waveguiding properties of a straight freestanding MNF can be well predicted (see Chapter 2). Meanwhile, since there is no substrate-induced eﬀect (e.g., short-pass ﬁlter eﬀect, see Section 4.3), light can be guided along a freestanding MNF with a diameter much smaller than the wavelength of the light, providing the possibility for optical wave guiding with a higher fraction

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Table 5.1. Possible schemes for using an MNF as a linear waveguide Supporting form Typical supporting Medium Gas Freestanding Liquid Nanostructures Surface support Solids Embedded Solidiﬁcation-after-embedded

Examples of supporting materials Air Water Silica aerogel MgF2 , UV-cured polymer UV-cured polymer, CDMS

of evanescent waves. Fig. 5.2 shows a broadband transmission spectrum of a freestanding 300-nm-diameter poly-(methyl methacrylate) (PMMA) MNF. With a diameter of 300 nm, the MNF can guide light at a wavelength larger than 1000 nm with low optical loss. Calculated optical conﬁnement of the MNF (refractive index of about 1.49) at 1000 nm is about 15% (refractive index of the PMMA is assumed to be 1.49), indicating that 85% of the optical power is guided outside the ﬁber as evanescent waves. For reference, the inset of Fig. 5.1 gives optical microscope images of the 300-nm-diameter PMMA MNF guiding a broadband supercontinuum and monochromatic lasers (wavelengths of 488, 532, 660, and 980 nm, respectively), clearly showing the light guiding through the whole length of the MNF.

Fig. 5.2. Broadband transmission spectrum of a freestanding 300-nm-diameter PMMA MNF. Inset, optical microscope images of the MNF guiding a broadband supercontinuum (SC) and monochromatic lasers with wavelengths of 488, 532, 660 and 980 nm, respectively. Scale bar, 50μm. (Adapted with permission from Ref. [29], copyright 2008, American Chemical Society)

MNFs can also serve as freestanding waveguides in liquid. Shown in Fig. 5.3 is an optical microscope image of an 800-nm-diameter silica MNF guiding

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a 633-nm-wavelength light from air into water. Compared with the strong scattering of the guided light at the interface between air and water, the weak scattering along the length of the MNF in water indicates low-loss waveguiding of the water-clad MNF. Assuming the refractive index of water to be 1.33 at 633-nm wavelength, the single-mode cut-oﬀ diameters of air- and water-clad silica MNFs are 456 nm and 805 nm respectively[6] , therefore the multimode guidance of the MNF in air transits to single-mode guiding in water when the MNF enters into water, leading to strong radiation loss at the interface.

Fig. 5.3. Optical microscope image of an 800-nm-diameter silica MNF guiding a 633-nm-wavelength light from air into water.

A unique advantage of a guiding light using a freestanding MNF is its high ﬂexibility: the high mechanical strength and pliability of a taper-drawn MNF (see Section 4.1) make it possible to be ﬂexibly patterned with elastic bends, and the high index contrast between the ﬁber core and cladding provides tight optical conﬁnement for a guiding light along these curvy structures with low optical loss. Shown in Fig. 5.4 is a 570-nm-diameter silica MNF guiding a 633-nm-wavelength light through a 3-dimension curved structure with bending

Fig. 5.4. Optical microscope image of a 570-nm-diameter silica MNF guiding a 633-nm-wavelength light through a 3-dimension curved structure with bending radii around 100 μm. The white arrows indicate the direction of light propagation.

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radii around 100 μm. The shape of the pattern can be changed freely without incurring mechanical breakdown and optical loss, indicating the possibility for highly ﬂexible optical connecting using this kind of guiding structure on a micro- or nanoscale. Besides the above-mentioned passive waveguides, MNFs can also be functionalized into active waveguides by exotic dopants. Compared with crystals, glasses and polymers are excellent hosts for a variety of dopants ranging from rare earth ions to chemical indicators. Fig. 5.5(a) shows an optical microscope image of a 320-nm-diameter 0.1 mol% Er-doped ZBLAN glass MNF excited by a 975-nm-wavelength light. The MNF is fabricated by direct draw of Er-doped ZBLAN glass (See Section 4.3). The pumping light is evanescently coupled into the MNF by a ﬁber nanotaper. The strong up-conversion green luminescence is clearly seen and is picked up by another nanotaper about 100 μm along the MNF. The emission spectrum given in Fig. 5.5(b) shows strong emission peaks centered around 550, 658 and 848 nm, which can be assigned to the transitions between 2 H11/2 , 4 F9/2 , 4 I9/2 and ground state 2 I15/2 of Er3+ ions, respectively. Shown in Fig. 5.6 is a photoluminescence (PL) spectrum of a 190-nm-diameter polyethylene oxide (PEO) polymer MNF doped with Tris(2,2’-bipyridine) ruthenium(II) chloride (Ru(bpy)3 Cl2 ). The MNF is excited by a 488-nm-wavelength light. Strong ﬂuorescent light around 600-nm wavelength generated and guided along the MNF is clearly seen in the optical microscope image, given in the inset.

Fig. 5.5. Er-doped ZBLAN glass MNF. (a) Optical microscope image of a 320-nmdiameter 0.1 mol% Er-doped ZBLAN glass MNF excited by a 975-nm-wavelength light coming from the nanotaper on the left-hand side. The up-conversion luminescence is clearly visible. (b) Photoluminescent spectrum of the Er-doped ZBLAN glass MNF shown in (a). (Adapted from Ref. [22],with permission from the Optical Society of America)

(2) Supported MNFs The freestanding MNF is easily aﬀected by the surrounding environment such as gravity and airﬂow, especially when the aspect ratio of the MNF is large. Therefore, for many applications the MNF needs a certain kind of support.

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Fig. 5.6. Photoluminescence (PL) spectrum of a 190-nm-diameter PEO polymer MNF doped with Tris(2,2’-bipyridine) ruthenium(II) chloride (Ru(bpy)3 Cl2 ) excited by a 488-nm-wavelength light. Inset, optical microscope image of strong ﬂuorescent light around 600-nm wavelength generated and guided along the MNF.

As has been mentioned in Section 4.2.2, due to the small diameter, light guided along a MNF leaves a high fraction of evanescent waves, which is very likely to suﬀer from the mode transition (from guide mode to leakage mode) that leads to wavelength-dependent power leakage of the guided light when the eﬀective index of the MNF is close to or lower than that of the supporting substrate. Therefore, in order to reduce or eliminate the substrate-induced loss for eﬀective support, the eﬀective index of the supporting structure should be lower (usually much lower) than that of the MNF. There are several types of low-index supporting schemes, the simplest approach is surface supporting: directly put a freestanding MNF on the surface of a low-index substrate. Because of the van der Waals and electrostatic attraction between the MNF and the substrate, the MNF can be tightly held on the surface of the substrate. As listed in Table 5.1, two types of typical low-index substrates have been investigated: porous nanostructures such as aerogels, and low-index solids such as MgF2 and UV-cured low-index polymer. For MNFs drawn from materials with moderate refractive indices (e.g., 1.46 of silica), it is desirable to have a substrate with a very low index. For this purpose one of the possible choices is silica aerogel[27] , which is a tenuous porous silica network of silica nanoparticles of about tens of nanometers in size, much smaller than the wavelength of the guided light, and has a transparent optical spectral range similar to that of silica[31,32] . Because the aerogel is mostly air, its refractive index is very close to that of air (1.0). Fig. 5.7(a) shows a close-up view of a 450-nm diameter silica MNF supported on a substrate of silica aerogel. Because the index diﬀerence between the silica aerogel and air (about 0.03 in this work) is much lower than the index dif-

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ference between the silica MNF and air (about 0.45), the optical waveguiding properties of the aerogel-supported MNF are similar to those of air-clad ones. Fig. 5.7(b) shows an aerogel- supported 380-nm-diameter silica MNF guiding 633-nm-wavelength light. The uniform and virtually unattenuated scattering along the nearly 0.5-mm length of the MNF and the strong output at the end face show that the scattering loss is small relative to the guided intensity, indicating low optical loss of the supported MNF. Fig. 5.7(c) gives the measured loss of silica MNFs supported by a silica aerogel substrate, which shows that for MNF with a diameter near the single-mode cutoﬀ diameter, typical loss is lower than 0.06 dB/mm, and light can be conﬁned and guided along the silica MNF with a diameter as small as 250 nm.

Fig. 5.7. Silica aerogel-supported silica MNFs. (a) SEM image of a 450-nm-diameter silica MNF supported by silica aerogel. (b) Optical microscopy image of a 380-nm diameter silica MNF guiding 633-nm-wavelength light on the surface of silica aerogel. The left-side arrow indicates the direction of light propagation; at the right end of the MNF, the light spreads out and scatters on the aerogel surface. (c) Measured optical loss of aerogel-supported silica MNFs at a wavelength of 633 nm. (Adapted with permission from Ref. [27], copyright 2005, American Chemical Society)

For MNF drawn from materials with high refractive indices (e.g., 2.0 of tellurite glass), it is possible to directly use low-index solids such as MgF2 and silica as eﬀective substrates[22,33−35] . Shown in Fig. 5.8 is an optical microscope image of a MgF2 -crystal-supported 260-nm-diameter tellurite glass MNF (refractive index higher than 2.0); the MNF guides 633-nm-wavelength light on the surface of the MgF2 from bottom left to the top right end. No scattering is observed along the whole length of the ﬁber in spite of the strong guided intensity output at the top right end, indicating the low leakage and scattering losses, which can be explained by lower scattering loss in the highindex MNF of tighter optical conﬁnement. Besides the surface support, the MNF can also be fully embedded inside a certain kind of low-index medium, as has been reported by several groups recently[36−41] . The basic idea for this method is as follows: ﬁrst, a freestanding MNF, either in straight form or assembled into a certain kind of pattern (e.g., a bend or a loop), is embedded in a liquid medium (e.g., dissolved polymer). When the MNF is placed in position, the liquid is solidiﬁed by a certain kind of physical or chemical process (e.g., UV curing or heating), and the MNF assembly is frozen inside the substrate. Shown in Fig. 5.9

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Fig. 5.8. Optical micrograph of a 260-nm-diameter tellurite glass MNF guiding 633nm-wavelength light on the surface of an MgF2 crystal. The white arrow indicates the propagation direction of light along the ﬁber. (Adapted from Ref. [22],with permission from the Optical Society of America)

Fig. 5.9. Optical microscope image of a 1.5-μm-diameter silica MNF (assembled into a knot structure) enclosed in a UV-cured ﬂuoropolymer. (Adapted with permission c 2007 IEEE) from Ref. [38],

is an optical microscope image of a 1.5-μm-diameter silica MNF (assembled into a knot structure) enclosed in a UV-cured ﬂuoropolymer. The MNF is ﬁrstly assembled into a freestanding knot with a diameter of about 180 μm by micromanipulation in air, and then immersed into a drop of ﬂuoropolymer (EFIRON PC-373) that has a refractive index of about 1.38. Finally the polymer was UV cured for solidiﬁcation of the whole assembly. Since the MNF is entirely enclosed inside the polymer matrix, the surface contamination of the MNF, that usually happened in a freestanding MNF[20,27] , is eliminated

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in this scheme and thus may provide high mechanical and/or optical stability for practical applications. 5.1.2 Waveguide Bends Changing the direction of light propagation is indispensable in most optical circuits, for which a guiding light with a waveguide bend is a basic and general approach. Therefore, waveguide bends are essential structures in optical components and devices. Moreover, the possibility of guiding light through sharp bends is critical for the miniaturization of waveguide-based optical components and devices[42−44] . Air-clad MNFs, usually with refractive index contrast (between the ﬁber core and air cladding) higher than 0.4, provide tight optical conﬁnement that is critical for guiding light through sharp bends with low radiation loss; also, the small diameter and large aspect ratio of the MNF make it highly pliable. Therefore, MNFs are suitable for being assembled into low-loss waveguide bends with small bending radii. Fig. 5.10 shows a three-dimensional ﬁnite-diﬀerence time-domain (3DFDTD) simulation of a 633-nm-wavelength light guided along a 450-nmdiameter silica MNF with a 5-μm-radius bend. The intensity distribution of the electric ﬁeld on the cross-section shows that there is virtually no leakage of light through such a tight bend.

Fig. 5.10. 3D-FDTD simulations of light intensity distribution in a 5-μm radius bend of a 450-nm diameter silica MNF. The wavelength of the guided light is 633 nm. The electric ﬁelds are polarized perpendicular to the paper. (Adapted with permission from Ref. [27], copyright 2005, American Chemical Society)

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Experimentally, MNF bends have been demonstrated when supported by low-index substrates. For example, Fig. 5.11 shows waveguide bends assembled by integration of silica MNFs with silica aerogel substrates. A 530-nmdiameter MNF is ﬁrst bent to a radius of about 8 μm on a sapphire wafer, annealed in a high-temperature oven to obtain a plastic deformation, and then transferred to the surface of a silica aerogel by micromanipulation. An SEM image of the as-assembled waveguide bend is shown in Fig. 5.11(a), in which the 8-μm bending radius is well maintained after the transfer. Fig. 5.11(b) gives an optical microscope image of the above-mentioned MNF bend guiding a 633-nm-wavelength light, showing the successful light guiding along such a sharp bend. The measured bending losses of a 633-nm-wavelength light guided through a 90 ◦ bend in a 530-nm-diameter silica MNF are given in Fig. 5.11(c). It shows that, when the bending radius is larger than 5 μm, the bending losses are relatively low. For example, the bending loss of a 12-μm-radius MNF bend is less than 0.1 dB, which is low enough for optical circuits in most cases.

Fig. 5.11. Silica aerogel-supported silica MNF for waveguide bends. (a) SEM image of an aerogel-supported 530-nm-diameter silica MNF with a bending radius of 8 μm. (b) Optical microscopy image of the aerogel-supported 530-nm-diameter MNF guiding light around the 8-μm-radius bend. (c) Measured bending loss in a 90 ◦ bend in aerogel-supported 530-nm-diameter MNFs at a wavelength of 633 nm. (Adapted with permission from Ref. [27], copyright 2005, American Chemical Society)

Compared to glass, which is a brittle material that needs high-temperature annealing after elastic bending to achieve permanent plastic deformation to avoid fatigue in sharply bending structures, polymer is a ductile material that presents much better pliability and ﬂexibility for bending and twisting at room temperature. Therefore, polymer MNFs are ideal candidates for waveguide bends. Shown in Fig. 5.12 is an optical microscope image of a 500-nm-diameter polystyrene MNF guiding a 532-nm-wavelength light through a 1.5-μm-radius bend. The MNF bends are supported on a MgF2 substrate. Because of the large index contrast between the polystyrene (1.59) and the MgF2 substrate (1.39), the tight optical conﬁnement of the MNF conveys the waveguiding along such a sharp bend assembled with the subwavelength-diameter MNF. Compared with waveguide bends of many other types such as photonic bandgap structures[45−48] , the MNF bends oﬀer the advantages of compact overall size, low coupling loss, simple structure and easy fabrication. Also, when the transparent spectral range of the MNF and substrate is properly

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Fig. 5.12. Optical microscope image of a 500-nm-diameter polystyrene MNF guiding a 532-nm-wavelength light through a 1.5-μm-radius bend. The MNF bends are assembled on a MgF2 substrate by micromanipulation.

matched, MNF bends can be used for guiding light through sharp bends over a broad range of wavelengths.

5.2 Micro-couplers, Mach-Zehnder and Sagnac Interferometers Due to their strong evanescent ﬁelds, two MNFs can be eﬃciently coupled within a short interaction length when they come close enough, as has been introduced in Section 2.5. This kind of evanescent coupling can be used to derive a series of MNF-based components and devices with high compactness. This subsection introduces two types of basic evanescent-coupling-based components: optical couplers and interferometers. 5.2.1 Micro-couplers In its simplest forms, MNF-based couplers are formed by freestanding MNFs in air. Shown in Fig. 5.13 is an optical microscope image of a freestanding optical coupler assembled by two 460-nm-diameter silica MNFs, which tightly attract each other to form the coupler in air. The detailed structure of the coupler is schematically illustrated in the inset, in which the two MNFs are partially contacted in parallel to form a highly eﬃcient coupling region, with an interaction length of tens of micrometers. When a 633-nm-wavelength light is sent into one MNF from the left side, it bifurcates into two at the overlapping region and propagates along the two branches of the coupler, serving as an optical splitter.

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Fig. 5.13. Optical microscope image of a freestanding optical coupler assembled by two 460-nm-diameter silica MNFs in air. Inset, schematic illustration of the detailed structure.

For a coupler assembled with given MNFs, the coupling eﬃciency can be tuned by changing the interaction length, as has been investigated in Section 2.5. Experimentally, a tunable X-coupler can be assembled using two freestanding MNFs, with the coupling eﬃciency tuned by the cross angle of the two ﬁbers, which in turn equivalently changes the eﬀective interaction length of the coupler. Fig. 5.14 shows the cross-angle-dependent coupling eﬃciency of a freestanding X-coupler assembled by two silica MNFs with diameters of 470 and 560 nm, respectively (see Inset). When the cross angle increases from 0 to 80◦ , the coupling eﬃciency decreases from 0 to about –40 dB, demonstrating the possibility to acquire high eﬃcient coupling with small cross angles and excellent isolation at large angles.

Fig. 5.14. Cross-angle-dependent coupling eﬃciency of a freestanding X-coupler assembled by two silica MNFs with diameters of 470 and 560 nm, respectively. Inset, SEM image of the X-coupler.

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When integrated with a low-index substrate, supported optical couplers can be assembled with silica MNFs. Fig. 5.15 shows an X-coupler assembled from two 420-nm-diameter silica MNFs. The two MNFs are ﬁrst plastically bent using a annealing-after-bending process (see Section 4.12), and then transferred to the surface of a silica aerogel and assembled into the coupler using micromanipulation. When a 633-nm-wavelength light is launched into the bottom left arm, the coupler splits the ﬂow of light in two. With an overlap of about 5 μm, the coupler works as a 3-dB splitter with an excess loss of less than 0.5 dB.

Fig. 5.15. Optical microscope image of a micrometer-scale X-coupler assembled from two 420-nm-diameter silica MNFs with an overlap of about 5 μm at the center. Inset, SEM image of the coupling region of the coupler. The arrows indicate the direction of light propagation.

Using MNFs drawn from high-index materials, a more robust coupler can be formed on the surface of a solid substrate. Fig. 5.16 shows an optical coupler

Fig. 5.16. Optical microscope image of an optical coupler assembled using two tellurite glass MNFs on the surface of a silicate glass. The diameters of the MNFs are 350 and 450 nm, respectively. The white arrow indicates the direction of light launching and propagation. Inset, SEM image of the coupling region. (Adapted from Ref. [22],with permission from the Optical Society of America)

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assembled using two tellurite glass MNFs supported by an MgF2 substrate. The diameters of the two MNFs are 350 and 450 nm, respectively. When a 633-nm-wavelength light is launched into the bottom left arm (indicated by a white arrow), the coupler splits the ﬂow of light in two with an interaction length of less than 4 μm. Because of the large index contrast between the tellurite (2.02) and MgF2 (1.39), the light is well conﬁned on the surface of the substrate and guided along the MNFs. Also, there is no scattering observed around the coupling area, indicating very low excess loss of the coupler. Compared with other types of microscopic couplers, such as fused couplers made from ﬁbre tapers using conventional methods that usually require an interaction length of the order of 100 μm[49] , MNF-based couplers reduce the device size by more than an order of magnitude. 5.2.2 Mach-Zehnder Interferometers The Mach-Zehnder interferometer (MZI) is one of the most widely used structures in optical components and devices ranging from telecom to medical diagnostics to spectroscopy. For example, optical sensors and modulators based on MZIs have long been demonstrated due to their phase sensitivity to the refractive index change of the waveguiding arms themselves or their surrounding media[51−55] , and add/drop ﬁlters have also been realized using cascaded MZIs for optical communications[56] . So far, a variety of MZIs relying on various waveguiding structures, including silicon-on-insulator (SOI) planar waveguides[57−59] , photonic crystal[60] , photonic crystal ﬁber[61] , metal heterowaveguides[62] and microﬂuidic channels[63] , have been realized, and miniaturization of the MZI structure is desirable for achieving faster response and high sensitivity in the relative devices. The small dimension, low optical loss, high fractional evanescent ﬁelds and mechanical ﬂexibility of the MNF make it possible to achieve MZI with a small footprint and high ﬂexibility, and MZI assembled with MNFs has also been proposed for optical sensing with high sensitivity[64] . When two micro-couplers are connected in cascade, an MZI is formed. To realize this with MNFs, one simple way is to form two identical couplers with two identical MNFs taper-drawn form the same ﬁber. First, taper drawing an optical ﬁber into a biconical taper with the MNF located at the center of the waist, cut the uniform waist at the center to obtain two identical MNFs each connected to the original ﬁber on one side and freestanding on the other side. Place the two MNFs on a low-index substrate in parallel with a certain length of overlap, and assemble two couplers by means of micromanipulation under an optical microscope. Fig. 5.17 shows a schematic diagram of an asassembled MZI. The theory of this device is outlined in Subsection 2.7.1. The MZI structure can be attracted to the surface of the substrate by van der Waals force and electrostatic attraction, and the two standard ﬁbers connected with the MNFs, used for optical launching and signal collection, are ﬁxed to the substrate for robust operation of the device.

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Fig. 5.17. Schematic diagram of an MZI assembled with two MNFs on the surface of an MgF2 substrate. (Adapted from Ref. [35],with permission from the Optical Society of America)

Fig. 5.18 shows an optical microscope image of a typical MZI assembled using two silica MNFs with diameters of 1 μm, with a whole dimension of around 300 μm. The transmission spectrum of the MZI is shown in Fig. 5.19, in which a clear interference with an extinction ratio of about 10 dB is observed.

Fig. 5.18. Optical microscope image of an MZI assembled using two silica MNFs with diameters of 1 μm. (Adapted from Ref. [35],with permission from the Optical Society of America)

Since the couplers in the above-mentioned MZI are formed by side-by-side coupling and sustained by van der Waals force and electrostatic attraction, the path-length diﬀerence of the two arms can be changed by shifting the contact region using micromanipulation. Shown in Fig. 5.20 is the transmission spectra of a MZI assembled using two silica MNFs with diameters of 1.1 and 1.2 μm, respectively. The path-length diﬀerence is changed from 8.3 to 58 μm. Typical interference fringes of the MZI with path-length diﬀerence of about 8.3, 13, 19 and 58 μm are provided, demonstrating the ﬂexibility for tuning the freespectral range of this type of MZI. The noisy signal around 680 nm observed in all the four spectra comes from the supercontinuum source that serves as the probing light in this work.

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Fig. 5.19. Transmission spectrum of the MZI shown in Fig. 5.18. (Adapted from Ref. [35],with permission from the Optical Society of America)

Fig. 5.20. Transmission spectra of an MNF-assembled MZI with path-diﬀerences of (a) 8.3, (b) 13, (c) 19 and (d) 58 μm. The spectral intensities of (a)–(c) are oﬀset for clarity. The MZI is assembled from two silica MNFs with diameters of 1.1 and 1.2 μm, respectively. (Adapted from Ref. [35],with permission from the Optical Society of America)

At a given wavelength, when the refractive index of a MNF increases, the diameter of the MNF for obtaining a given conﬁnement and the minimum interaction length of a micro-coupler for obtaining a certain coupling eﬃciency reduces[6,50] . Therefore, the size of the MZI can be reduced when high-index MNFs are used. Fig. 5.20(a) shows an optical microscope image of an MZI assembled with two 480-nm-diameter tellurite MNFs that are drawn from a

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tellurite glass with refractive index of 1.76. The total size of the MZI is around 50 μm×50 μm with a coupling length of about 5 μm. Steep ﬁber tapers are used to eﬃciently launch light into and couple light out of the MZI, as shown in Fig. 5.21(a). The transmission spectrum in Fig. 5.21(b) shows interference fringes with an extinction ratio of about 8 dB and the path-length diﬀerence of about 29 μm.

Fig. 5.21. MZI assembled with tellurite glass MNFs. (a) Optical microscope image of an MZI assembled with two 480-nm-diameter tellurite MNFs. White light from a supercontinuum source is launched into and picked up from the MZI by two silica ﬁber tapers. The white arrows indicate the direction of light propagation. (b) Transmission spectrum of the MZI shown in (a). (Adapted from Ref. [35],with permission from the Optical Society of America)

5.2.3 Sagnac Interferometers The interference phenomenon can be also observed when a coiled MNF touches itself and causes self-coupling of propagating light. Transmission of light

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through a loop having relatively large losses and/or small self-coupling amplitude can be described with the theory of subsection 2.7.1, Eq. (2.74). As illustrated in Fig. 2.40(b), the power spectrum of light transmitted through the loop is a result of the interference of two coherent beams. Even if the coupling amplitude A1 is small it still can be compared to the amplitude A2 for a loop with large propagation loss. If the amplitudes of beams 1 and 2 are close to each other and their phases are opposite the beams will cancel each other. This case is known as the condition of critical coupling in the theory of microresonators (see e.g., Ref. [64]). With the macro-manipulation of MNFs illustrated in Fig. 5.1(b), one can create an MNF Sagnac interferometer shown in Fig. 2.40(b). Fig. 5.22 shows the results of bending an MNF of about 700 nm diameter from Ref. [66]. First, a bent MNF was created by moving the taper ends towards each other (Fig. 5.22(a)). It is seen that the bent MNF has strong bend losses growing from 1.5 to 7.5 dB in the bandwidth considered. With further stage translation, a less bent loop having smaller bend losses was created (Fig. 5.22(b)). After the MNF touched itself, the oscillations in the transmission spectrum were observed (Fig. 5.22(c)–(e)). The experimental transmission spectrum shown in Fig. 5.22(e) was ﬁtted assuming that the amplitude of propagation along the loop, A2 , decreases exponentially with the wavelength while the coupling amplitude, A1 , is constant.

5.3 MNF Loop and Coil Resonators Fabrication of an MNF loop resonator (MLR) consists of drawing an MNF and bending it into a self-coupling loop. Two types of MLR have been demonstrated experimentally: a regular MLR shown in Fig. 5.23 [66−69] and a knot MLR shown in Fig. 5.9[20,36,38,70−72] . The theory of MLR was considered in Subsection 2.7.2. Methods of fabrication and transmission properties of MLRs are reviewed in Subsection 5.3.1. An MNF coil resonator (MCR) can be fabricated by wrapping an MNF around an optical rod. The theory of MCR was considered in Subsection 2.7.3. Methods of fabrication and transmission properties of MCR are reviewed in Subsection 5.3.2. 5.3.1 MNF Loop Resonator (MLR) Fabricated by Macro-Manipulation The setup for the MRL fabrication by macro-manipulation is illustrated in Fig. 5.1(b). It consists of three translation stages, which can translate the ends of a biconical taper with an MNF waist with respect to each other, and a rotation stage, which can twist the MNF. In Refs. [68,69] a biconical taper was fabricated with the CO2 laser indirect drawing technique described in Section 3.2.1[66] . In the experiment[68] an MLR with a Q-factor around 15000 was fabricated with an MNF of diameter 0.66 μm. In Ref. [69] an MLR

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Fig. 5.22. Transmission spectrum and optical microscope images for diﬀerent conﬁgurations of MNF. (a), (b) bent and coiled MNF without self coupling; (c),(d), and (e) self-coupling MNF with successively decreasing bend radius. (e) shows comparison of the experimental data with the theory. (Adapted from Ref. [66],with permission from the Optical Society of America)

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c [2006] Fig. 5.23. Optical microscope image of an MLR. (Adapted from Ref. [69], IEEE)

was fabricated with a 0.9 micron diameter MNF. The circumference length of the MLR was around 2 mm and the total length of the biconical taper was 25 mm. Controlling the shape of the MNF with an optical microscope, the authors coiled the taper waist into a self-touching loop. Fig. 5.23 shows an optical microscope image of this loop with the input and output ends aligned parallel to each other. The parallel alignment of the adjacent MNF segments was achieved with the help of the surface attraction forces (Van der Waals and electrostatic), which kept the ends together, apparently overcoming the elastic forces that would tend to straighten out the MNF. For this MLR, the MNF diameter variation in the coupling region was slower than for the MLR of Ref. [68], which resulted in smaller losses, higher coupling eﬃciency and, eventually, a higher Q-factor of the MLR. Fig. 5.24 shows an example of very uniform and smooth transmission spectra in the C-band (from 1520 to 1570 nm), which were obtained by translation of the MLR ends[69] . In order to calculate the Qfactor of an MLR, the experimental data has been analyzed using the theory presented in Section 2.7.2. It was found that the loaded/intrinsic Q-factor and ﬁnesse of the resonances in Fig. 5.24(a) achieve the values of 22000/60000 and 9. In Fig. 5.24(b) these values are 95000/630000 and 42, and in Fig. 5.24(c) these values are 120000/155000 and 35, respectively. The loaded Q-factor demonstrated in Ref. [69] was close to the largest Q-factors achieved for the planar ring resonators[73−75] . The unpolarized light used in Ref. [69] showed up in small asymmetry in each of the resonances in Fig. 5.24(a) due to the superposition of TE and TM modes, which are shifted in phase and have diﬀerent Q-factors. In Fig. 5.24(b) and (c), the resonances of diﬀerent Polarization states are narrower and are well separated. The amplitudes of resonances are slowly changing with the wavelength. The change is caused by the weak dependence of coupling and attenuation parameters in Eq. (2.80) on the wavelength. Narrow spectral resonances of an MLR are very sensitive

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Fig. 5.24. Transmission spectra of MCR tuned to loaded Q-factor of (a) 22000, (b) c [2006] IEEE) 98000, and (c) 120000. (Adapted from Ref. [69],

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to variations in the eﬀective refractive index and dimensions of the resonator and the refractive index of the ambient medium changing with temperature, pressure, and applied radiation. Therefore, an MLR can be used as a fast and eﬀective sensor of the ambient medium. Applications of an MLR as an optical sensor will be described in Chapter 7. 5.3.2 Knot MLR Fabricated by Micro-Manipulation A knot MLR was fabricated in Refs. [20,36,38,70-72] with the micro-manipulation technique illustrated in Fig. 5.1. As opposed to the regular MLR, a knot MLR cannot be fabricated from a biconical ﬁber taper because an MNF open end should be available. This end is bent into a relatively large loop about a few millimeters in diameter, which is then tightened into a smaller knot by pulling the free end of the MNF. Fig. 5.25(a) shows an SEM image of a 290-μm-diameter MNF knot from Ref. [70]. An advantage of a knot MLR compared to a regular MLR is that a knot makes the loop much more stable. This allows us to fabricate robust MLR devices and, in particular, to immerse them in a polymer matrix[36] . Recently, an MNF knot resonator was used to demonstrate an MNF knot laser[71,72] (see Subsection 5.5.3). For optical characterization of the knot MLR, its open end is coupled to a second MNF as illustrated in Fig. 5.25(b)[70] . Fig. 5.26 shows the experimentally measured transmission spectra of MNF knots with diameters of 396 μm and 850 μm[70] . The 396-μm-diameter MLR (Fig. 5.26(a)), created from a 2.66-μm-diameter MNF, had a Q-factor and a ﬁnesse of about 10000 and 9.2, respectively. The 850 μm diameter knot (Fig. 5.26(b)), created from a 1.73-μm-diameter MNF, had a Q-factor and a ﬁnesse of about 57000 and 22, respectively. The authors of Ref. [70] demonstrate a knot MLR in water with a Q-factor of 31000. The robustness of a knot MLR makes it convenient to use as a microﬂuidic chemical and biological sensor. In addition, a knot MLR can be safely imbedded into

Fig. 5.25. (a) SEM image of the inter-twisted overlap region of a knot MLR with diameter of 290 μm. The MLR is fabricated from a 2.66-μm-diameter MF. (b) Schematic diagram of an MNF knot. (Reprinted with permission from copyright 2006, American Institute of Physics)

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a UV curable polymer. After curing, the device is secured in a solid polymer matrix and exhibits the resonant spectrum[36] . An optical microscope image of a knot MLR embedded in ﬂuoroacrylate from Ref. [36] is shown in Fig. 5.27.

Fig. 5.26. Transmission spectra of (a) a 396-μm diameter knot MLR assembled using a 2.66-μm diameter MNF and (b) a 850-μm diameter knot MLR assembled using a 1.73-μm diameter MNF. The inset in (b) shows a single resonance peak. (Reprinted with permission from copyright 2006, American Institute of Physics)

Fig. 5.27. Optical microscope image of a knot MLR embedded in ﬂuoroacrylate. The MF diameter is 5 μm. (Adapted from Ref. [36],with permission from Oxford University Press)

5.3.3 Experimental Demonstration of MCR Manipulation of an MNF in free space allows us to create high Q-factor loop MLRs reviewed in the previous subsection. However, it is not possible to create a multi-turn MCR by moving the ends of an MNF in free space without

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touching and holding the intermediate part of the MNF. In order to fabricate an optical MCR, an MNF should be wrapped on a central rod. This subsection describes the ﬁrst experimental demonstrations of an MCR[23−26] . (1) Fabrication of MCR in air

Fig. 5.28. Illustration of the setup for fabrication of an MCR in air. (Adapted from c Ref. [76],[2008] IEEE)

Fig. 5.29. Transmission spectrum of an MCR fabricated in air. Curve 1 – a free MNF; Curve 2 – a straight MNF touching the optical rod; Curve 3 – an MCR tuned to the highest Q-factor; Curve 4 – an MCR tuned to the highest extinction ratio. c (Adapted from Ref. [76],[2008] IEEE)

A setup used in Ref. [76] for wrapping a silica MNF on a silica rod in air is illustrated in Fig. 5.28. The MNF was a part of a 1.7-μm-diameter waist of

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a biconical taper fabricated from a conventional single mode ﬁber. A 345-μmdiameter silica ﬁber was used as an optical rod. The taper was connected to a broadband light source and to an optical spectrum analyzer, which monitored the MNF transmission spectrum. The transmission spectrum of a free taper is shown in Fig. 5.29, curve 1. Initially, the silica rod was oriented perpendicular to the MNF and put in contact with it. At this step the transmission spectrum of the MNF (Fig. 5.29, curve 2) exhibited whispering gallery mode resonances generated in the rod. At the next step the MNF was wrapped on a silica rod with the rotation and 3D translation stages. In the process of wrapping, the angle between the MNF and the rod was gradually increased so that it achieved about 90◦ when the uniform MNF part touched the rod. After an MNF closed loop was formed, the transmission spectrum was tuned to arrive at resonances with the greatest Q-factor and extinction ratio shown in Fig. 5.29, curves 3 and 4. In Fig. 5.29, curve 3 is the transmission spectrum of the MNF after it was wrapped on the rod and the resonant loop was aligned to maximize the resonance Q-factor of the device. The Q-factor of about 20000 has been achieved. The insertion loss of the fabricated MCR was about 9 dB, which was primarily generated outside the resonant loop. The loss was caused by discontinuities in the contact between the rod and the MNF, contamination of the silica surface and non-adiabaticity of wrapping. Curve 4 in Fig. 5.29 is the transmission spectrum of the same MNF after it was aligned in order to arrive at the condition of critical coupling (γa = γc in Eq. (2.80)), which maximized the extinction ratio of transmission power oscillations. The achieved extinction ratio was greater than 20 dB. Wrapping of an MCR on a central rod with the refractive index lower than that of an MNF may be more preferable because, presumably, then the coupling between the MNF mode and the rod modes can be better suppressed. In Ref. [77] an MCR was created by wrapping a silica MNF on a Teﬂon AF rod with refractive index 1.29. The maximum achieved Q-factor of MCR was about 10000, i.e., still less than that achieved for a silica rod. Potentially, the achieved Q-factor can be increased by improving the smoothness and uniformity of the surface of the low-index rod and adiabaticity of the MNF wrapping. (2) Fabrication of MCR in low-index polymer The problems of MCR fabrication in air, which are mentioned at the end of the previous subsection, can be solved if the MCR is completely immersed in a liquid or solid matrix, e.g., a polymer. In particular, the MCR can be imbedded in an environment having the refractive index equal to the refractive index of the rod as demonstrated in Ref. [78] and Refs. [76,79]. In this case, the MCR behaves as though in uniform space because the MNF does not “see” the rod optically. Therefore, the MCR losses can be signiﬁcantly decreased. The experimental setup for fabrication of MCRs in liquid used in Refs. [76,78] is shown in Fig. 5.30. One of the ends of the biconical taper used in this experiment was glued to the Γ-shaped leg, which was ﬁxed at the rotating shaft. The other end was ﬁxed at the ﬁber spring, which maintained the taper

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Fig. 5.30. A sketch of a setup for the fabrication of an MF coil resonator in indexc matching liquid. (Adapted from Ref. [76],[2008] IEEE)

in a strained condition. The rod was fabricated of a silica ﬁber coated with a ﬁlm of a cured, low-index polymer with a low refractive index 1.384. The MNF and the rod were immersed in a pool of the same uncured liquid polymer. The convex meniscus at the pool edges, shown in Fig. 5.30, allowed us to immerse the MNF into the liquid and perform wrapping. Due to the small diﬀerence between the refractive indices of the cured and uncured polymer, their interface was almost invisible for the light propagating along the MNF. After the MNF was immersed into the liquid polymer, the transmitted power decreased by about 0.5 dB (curves 1 and 2 in Fig. 5.31). Presumably the

Fig. 5.31. Transmission spectrum of the MNF: 1 – in air; 2 – in liquid polymer; 3 – at 180◦ of rotation; 4 – at 360◦ of rotation; 5 – at 450◦ of rotation; 6 – at 720◦ c of rotation. (Adapted from Refs. [76,78],[2008] IEEE)

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loss was caused by scattering of light at the meniscus surface. In the process of wrapping of the ﬁrst turn of an MCR, no noticeable transmission losses were observed. Curves 3 and 4 in Fig. 5.31 show the transmission spectrum of the taper at rotation angles 180◦ and 360◦ , respectively. After the rotation angle of 360◦ was reached, the taper ends were translated in order to create a closed loop and to initiate self-coupling. The self-coupling showed itself at ∼450◦ when oscillations in the transmission spectrum appeared (curve 5 in Fig. 5.31). At larger angles the resonant character of the spectrum was enhanced. Curve 6 in Fig. 5.31 shows the resonant spectrum exhibiting the Q-factor of 61000, observed at 720 ◦ , i.e., for the two-turn MCR. The total loss of the resonant loop was about 0.06 dB[76,78] . The resonance structure of this spectrum corresponded to a double loop MCR. In fact, the group of four peaks, which appears periodically in curve 6, corresponds to two resonances of diﬀerent polarization of the ﬁrst and second loops of the MCR. An application of an imbedded MCR as an optical sensor will be considered in Chapter 7.

5.4 MNF Filters Fiber-based optical ﬁlters, which ﬁlter out noise or unwanted signals, fatten or suppress the gain proﬁle after ampliﬁcation, are one of the most important optical components in ﬁber-optic devices and optical ﬁber networks[2,80−82] . Optical MNFs, with small dimension, low optical loss and strong wavelengthdependent evanescent interaction, are promising building blocks for highly miniaturized optical ﬁlters that are compatible with outer ﬁber systems. This subsection introduces two types of recently developed micro ﬁlters based on silica MNFs: short-pass ﬁlters relying on strong wavelength-dependent evanescent leakage[83] , and add-drop ﬁlters based on MNF loop resonators[84] . 5.4.1 Short-Pass Filters Short-pass ﬁlters are one of the essential optical components for photonic systems and networks[85−87] , and a variety of ﬁber-optic short-pass ﬁlters have been demonstrated based on dispersive ﬁbers[85,86,88] , side polished ﬁbers with polymer overlay[89] and tapered ﬁbers[90] in the past years. Recently, a shortpass ﬁlter eﬀect was reported in semiconductor nanoribbons due to substrateinduced leakage[17,19] , indicating a promising approach to the realization of compact short-pass ﬁlters based on MNFs for high compatibility with standard ﬁber systems. As has been mentioned in Section 4.2.2, in an air-clad MNF with a diameter close to or smaller than the wavelength of the guided light, the fraction of power guided outside the ﬁber (as evanescent waves) is strongly dependent on the wavelength: the longer the wavelength, the larger the fraction of evanescent power. If we deﬁne an eﬀective index of the MNF as neﬀ = β/k0 (where β is the propagation constant, and k0 = 2π/λ), when the eﬀective

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index of the MNF approaches the index of the substrate, light leaks into the substrate. In a given MNF, the longer the wavelength, the lower the eﬀective index of the guided light. Therefore, light with a longer wavelength will be more easily ﬁltered out. Shown in Fig. 5.32 is an optical microscope image of an MgF2 -supported 1-μm-diameter silica MNF guiding a supercontinuum light. When the supercontinuum light (orange) is launched from the left side, the color of the scattering light (which represents the spectral components of the local guided mode) changes from orange to yellow, green and ﬁnally blue on the right side, clearly showing the short-pass ﬁlter eﬀect in the supported MNF.

Fig. 5.32. Optical microscope image of an MgF2 -supported 1-μm-diameter silica MNF guiding a supercontinuum light.

Based on the above-mentioned ﬁlter eﬀect, a short-pass ﬁlter can be constructed, as schematically illustrated in Fig. 5.33. A taper-drawn biconical silica MNF (drawn from SMF-28e, corning), with a uniform-waist of about 1 μm in diameter, is placed in close contact with a piece of MgF2 wafer of about 1 mm in thickness. The surface of the MgF2 crystal is ﬁnely polished and cleaned for tight contact with the MNF. The MNF is attracted to the substrate through van der Waals and electrostatic attractive force. Both sides of the as-drawn MNF are continuously connected to the standard ﬁbers for light launching and collection. To characterize the ﬁlter, light from a supercontinuum source is launched into and guided along the MNF through the tapered ﬁber (see Fig. 5.33), and the output from the other end of the MNF is sent to an optical spectrum analyzer. Typical transmission spectra of the short-pass ﬁlter are shown in Fig. 5.34, in which the diameter of the MNF is 1.04 μm and the interaction length (overlap of the MNF and the MgF2 substrate) is about 1.1 mm. The shortpass eﬀect is clearly shown in the transmission spectrum (bold black line). For reference, Fig. 5.34 also provides original outputs from the freestanding MNF in air and the MgF2 -supported MNF without normalization. The ﬁlter cuts oﬀ around 1110 nm if the cutoﬀ level is assumed to be −30 dB. There is no notable optical loss in the pass band, indicating that the insertion loss is very low. The rejection loss is higher than 50 dB, which is better than that obtained in many other ﬁlter structures[86,88,90] . To investigate the possibility of shifting the cutoﬀ wavelengths of the ﬁlter, the interaction length between the MNF and the substrate is changed by means of micro-manipulation under an optical microscope. Fig. 5.35(a) shows

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Fig. 5.33. Schematic diagram of a short-pass ﬁlter assembled with an MgF2 substrate-supported MNF. The biconically taper-drawn MNF is connected to the standard ﬁbers for light launching and collection. A supercontinuum source (SCS) is launched into the left-hand side, and the output from the right-hand side is sent to an optical spectrum analyzer (OSA). (Adapted from Ref. [83],with permission from the Optical Society of America)

Fig. 5.34. Typical transmission spectra (the line labeled “Transmission”) of a silica MNF short-pass ﬁlter. The lines labeled “Air” and “MgF2 ” represent the output spectra before and after the integration with the MgF2 substrate, respectively. The diameter of the MNF is 1.04 μm and the interaction length is 1.1 mm. (Adapted from Ref. [83],with permission from the Optical Society of America)

typical transmission spectra of a ﬁlter based on a 0.99-μm-diameter silica MNF. With the decreasing interaction length from 3 to 0.65 mm, the cutoﬀ wavelength increases from 640 to 1064 nm. Fig. 5.35(b) plots the interactionlength-dependent cutoﬀ wavelength of the ﬁlter, the monotonous and near linear dependence over a 400-nm span indicates the possibility of wideband shifting of the cutoﬀ wavelength. With a given interaction length, the cutoﬀ wavelength is also strongly dependent on the diameter of the MNF. Fig. 5.36 characterizes the diameter

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5 MNF-based Photonic Components and Devices

dependence of the ﬁlter eﬀect. While the interaction length is kept at 1.1 mm, MNFs with diameters ranging from 0.75 to 1.96 μm are used for this test. Fig. 5.36(a) gives normalized transmission spectra of the ﬁlter assembled using MNFs with diameters of 0.75, 0.88, 1.17, 1.29, 1.42, 1.72, 1.82 and 1.96 μm, respectively. Short-pass ﬁlter eﬀects are observed with a cutoﬀ wavelength ranging from 570 to 1480 nm. Fig. 5.36(b) plots the cutoﬀ wavelength with respect to the diameter of the MNF, showing a monotonous and near linear dependence, which may be helpful for predicting the cutoﬀ wavelength from the MNF diameter over a wide spectral range. In addition, since the overlap of transparent bands of the ﬁber material (silica) and the substrate (MgF2 ) are much broader than those provided in Fig. 5.35 and Fig. 5.36, the ﬁlter can be applied to a much broader spectral range. Compared with other types of ﬁber-based ﬁlter structures[88−90] , an MNFbased ﬁlter is very compact in size and simple in structure, with favorable properties including wideband applicability, high rejection loss, and compatibility with miniaturized ﬁber devices, and may ﬁnd applications in wideband photonic circuits or devices with high compactness. 5.4.2 Add-Drop Filters Add-drop ﬁlters are important components for optical signal processing, especially in modern wavelength division multiplexing (WDM) networks[2] . One typical scheme of add-drop ﬁltering is handling signals with optical resonance, as has been implemented in various forms of resonators including microrings[91−95] , microdisks[92,96] , microspheres[97,98] , microtoroids[99] , and photonic crystal resonators[100] . Previously, ﬁber-based add-drop ﬁlters were constructed either by combining ﬁber Bragg grating with a polarization splitter[101] or with a Mach-Zehnder interferometer[102] , or by recording a Bragg grating in the merged region of two adiabatically tapered and fused ﬁbers[103] . Recently developed MNF loop or knot resonators[68−70] , owing to their high Q-factors, simple structures and compatibility with optical ﬁber systems, are promising candidates for the realization of all-ﬁber add-drop ﬁlters with miniaturized sizes and large free-spectral ranges, as have been introduced in Section 5.3. This subsection introduces MNF-based add-drop ﬁlters that have been reported recently[84] . A typical add-drop ﬁlter including an MNF knot (serves as a resonator) and two evanescently coupled MNFs (which serve as add and drop ﬁbers) is schematically illustrated in Fig. 5.37(a). The MNF knot is fabricated by assembling the freestanding end of a taper-drawn silica MNF under an optical microscope. When the knot is tightened to the desired size, a second MNF (collection ﬁber 1) is used to collect the transmitted power by means of evanescent coupling, while the third MNF (collection ﬁber 2) placed tangentially to the MNF knot is used to drop the resonant signal. The dropping MNF can be placed either in parallel (Fig. 5.37(b)) or perpendicularly (Fig. 5.37(c)) to the input port and is joined together with the MNF knot through

5.4 MNF Filters

155

Fig. 5.35. Interaction-length dependence of the cutoﬀ wavelength of the ﬁlter. (a) Normalized transmission spectra of an MNF-assembled short-pass ﬁlter with 1 2.99, 2 2.57, 3 2.04, 4 1.82, 5 1.42, 6 1.02, and 7 interaction lengths of 0.65 mm. The diameter of the silica MNF is 0.99 μm. (b) Cutoﬀ wavelength versus interaction length. (Adapted from Ref. [83],with permission from the Optical Society of America)

van der Waals and electrostatic attractive force. The resonant power can be transferred from the input port (port 1) to the drop port (port 3), while the nonresonant power is transferred from the through port (port 2) without being signiﬁcantly aﬀected. Fig. 5.38 shows the transmission spectra of a 308-μmdiameter add-drop ﬁlter, assembled with a 2.7-μm-diameter MNF, with the dropping taper placed parallel to the input port of the ﬁlter at the through port (Fig. 5.38(a)) and the drop port (Fig. 5.38(b)).

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5 MNF-based Photonic Components and Devices

Fig. 5.36. MNF-diameter dependence of the cutoﬀ wavelength of the ﬁlter. (a) Normalized transmission spectra of short-pass ﬁlters assembled using MNFs with 1 0.75, 2 0.88, 3 1.17, 4 1.29, 5 1.42, 6 1.72, 7 1.82, and 8 diameters of 1.96 μm. The interaction length is kept to 1.1 mm. (b) Cutoﬀ wavelength versus MNF diameter. (Adapted from Ref. [83],with permission from the Optical Society of America)

Similar to the vertically coupled microring resonator add-drop ﬁlter composed of a ring resonator stacked above a cross-grid node[94] , the MNF knot resonator based add-drop ﬁlter also works when the dropping taper is placed perpendicularly to the input port. As shown in Fig. 5.37(c), the distal end of the dropping taper crosses the through port of the ﬁlter. Because of excellent isolation of two vertically crossed MNFs[27,94] , the resonant signal in this structure is well retrieved without crosstalk. Fig. 5.39 shows a typical signal dropped from a vertically coupled 65-μm-diameter MNF knot, which is assembled with a 1.8-μm-diameter silica MNF. The measured FSR is about 8.1 nm with Q-factor and ﬁnesse of about 3300 and 17.3, respectively.

5.5 MNF Lasers

157

Fig. 5.37. (a) Schematic diagram of the MNF knot resonators based add-drop ﬁlter. (b), (c) optical microscope image of a ∼200 μm-diameter microknot add-drop ﬁlter with the dropping taper placed (b) parallel or (c) perpendicular to the input port of the ﬁlter respectively. (Adapted from Ref. [84],with permission from the Optical Society of America)

To achieve larger FSR, a smaller knot structure with thinner MNFs can be used. Fig. 5.40 shows the transmission spectra of a 35-μm-diameter knot resonator assembled with a 1.0-μm-diameter MNF. Measured FSR is about 14.9 nm, a Q-factor of about 2700 and a ﬁnesse of about 28. Considering that Q-factors as large as 4×108 have been demonstrated in silica toroidal microcavities with similar diameters[104] and that the bending loss of the MNF is very low[105] , the Q-factor can be greatly increased by further improvements such as ﬁnely tuning the coupling strength at the twisted region and decreasing the contamination-induced scattering loss along the ﬁber. Since the MNF-based add-drop ﬁlter is constructed solely with ﬁbers (standard ﬁbers, tapered ﬁbers and MNFs), it oﬀers easy connection to both standard ﬁber systems and miniaturized ﬁber devices. For practical applications, the device can be embedded in a low-index matrix for high stability and easy handling[36,38] . In addition, high-index low-loss MNFs (e.g., tellurite MNFs[22] ) can also be used for constructing this type of ﬁlter with small size and larger FSR.

5.5 MNF Lasers As introduced in Section 3.2.2, a special advantage of drawing glasses or polymers is the possibility of doping MNFs for active devices such as lasers. To construct a laser, the typical approach is to incorporate an active ﬁber into

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5 MNF-based Photonic Components and Devices

Fig. 5.38. Transmission spectra of a 308-μm-diameter add-drop ﬁlter with the dropping taper placed parallel to the input port of the ﬁlter at the through port (a) and the drop port (b). The knot resonator is assembled using a 2.7-μm-diameter MNF. (Adapted from Ref. [84],with permission from the Optical Society of America)

a cavity structure. The MNF ring/knot resonator, with small size and high Q-factor, provides an ideal cavity structure for lasing. This subsection introduces MNF lasers based on MNF ring resonators in the following arrangement: ﬁrstly, numerical simulations of MNF ring/knot lasers in a steady state using rate equations with erbium and ytterbium dopants[106] ; secondly, experimental demonstration of knot lasers with erbium and ytterbium doped phosphate MNFs[71] ; ﬁnally, MNF knot dye lasers based on the evanescent-wave-coupled

5.5 MNF Lasers

159

Fig. 5.39. Typical signal dropped from a vertically coupled 65-μm-diameter MNF knot, which is assembled with a 1.8-μm-diameter silica MNF. (Adapted from Ref. [84],with permission from the Optical Society of America)

gain[72] , in which the active medium for light emission is not directly doped inside the MNF, but is distributed in the evanescent ﬁelds outside the MNF. 5.5.1 Modeling MNF Ring Lasers The laser modeling is based on two sets of equations: coupling equations to analyze resonances in a ring resonator, and rate-equations to describe the transitions in active ions. The pump light is assumed to be consumed by the absorption of active ions and the scattering loss in the coupling region. Similarly, the signal light is either ampliﬁed as a result of population inversion or suﬀers loss by scattering. For simplicity, a three-level system is used in analysis. Energy transfer eﬀects which are detrimental to the quantum eﬃciency

Fig. 5.40. Transmission spectrum of a 35-μm-diameter knot resonator assembled using a 1.0-μm-diameter silica MNF. (Adapted from Ref. [84],with permission from the Optical Society of America)

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5 MNF-based Photonic Components and Devices

of lasers, such as concentration quenching, are not taken into account and the numerical examples are chosen for concentrations where these eﬀects are not prevalent. Also, the strong signal case where gain saturation may occur is not considered. (1) Ring resonator equations The MNF ring resonator is schematically illustrated in Fig. 5.41. When in stationary operation, the relation between complex ﬁeld amplitudes for the pump light in the MNF is[107]

1

1

1

E3,p = (1 − γp ) 2 [(1 − Kp ) 2 E1,p + jKp2 E2,p ] 1

1

1

E4,p = (1 − γp ) 2 [jKp2 E1,p + (1 − Kp ) 2 E2,p ]

(5.1)

D

and E2,p = E3,p e−αp π 2 ejβπD , where Kp , γp are the intensity coupling coeﬃcient and coupling loss for the pump light, respectively. Ei,p (i=1,2,3,4) are the complex ﬁeld amplitudes corresponding to Ii,p (i =1,2,3,4) shown in Fig. 5.41, D is the diameter of the ring, and β is the longitudinal propagation constant of the pump light in the MNF. When the resonance condition for β is satisﬁed, the intensity relations are[107] I3,p I1,p I4,p I1,p

= =

(1−γp )(1−Kp ) 1

1

[1−(1−γp ) 2 Kp2 e−αp π 1 (1−γp )[Kp2

D 2 ]2

D 1 −(1−γp ) 2 e−αp π 2 ]2 1 1 D [1−(1−γp ) 2 Kp2 e−αp π 2 ]2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(5.2)

Since the pump light experiences attenuation along the ring, the average intensity is

Ip =

1 πD

πD 0

I3,p e−zαp dz =

I3,p (1 − e−αp πD ) αp πD

(5.3)

The intensity enhancement factor E for the resonant pump is given

E=

(1 − γp )(1 − Kp ) Ip (1 − e−αp πD ) = 1 1 D I1,p αp πD [1 − (1 − γp ) 2 Kp2 e−αp π 2 ]2

(5.4)

The enhancement factor E for MNF rings with diﬀerent αp πD is plotted in Fig. 5.42(a). It can be seen that no enhancement occurs when Kp =0. When Kp reaches its optimum value, which corresponds to critical coupling, E increases to its maximum. In Fig. 5.42(b), the values of E at critical coupling are plotted against αp πD. It can be seen that Emax increases dramatically as αp πD becomes smaller.

5.5 MNF Lasers

161

Fig. 5.41. Schematic of an MNF ring resonator. (a) The pump light I1,p is partially coupled into the ring by a ﬁber taper at the coupling region of the ring, of intensity coupling coeﬃcient Kp , and of coupling loss γp . The pump experiences loss of coeﬃcient αp , as well as dephasing while propagating along the ring, and is partially coupled outside at the coupling region, resulting in a transmitted intensity, I4,p . Loss for the pump light is due to absorption and scattering, thus αp = αabs,p + αsc,p . For a glass MNF suspended in air, the refractive index contrast is at least 30% and the bending loss is neglected here. (b) The signal light resonates in the ring in two directions, I+,s and I−,s , and is coupled out at the coupling region, resulting in transmitted intensities I1,s and I4,s . The loss coeﬃcient, the gain coeﬃcient, the coupling loss at the coupling region and the signal coupling factor, are denoted as αs , G, γs , and Ks , respectively. Loss is assumed to be solely due to scattering, so αs = αsc,s .

For the signal, light is coupled out in two directions, as is shown in Fig. 5.41(b), so that I4,s I1,s = = (1 − γs )(1 − Ks ) I−,s I+,s

(5.5)

where Ks and γs are the intensity coupling coeﬃcient and coupling loss for the signal light, respectively.

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5 MNF-based Photonic Components and Devices

Practically, when the FSR of a MNF ring resonator (usually less than 10 nm) is narrower than the emission bandwidth of rare-earth ions in glasses, it is possible to make the ring resonant for both the pump and the signal light. (2) Rate equations Atomic transitions in lasers can generally be treated as three-level transitions (TLT) or four-level transitions (FLT). For simplicity a three-level system is used, as illustrated in Fig. 5.43. The similar analysis is not diﬃcult to extend to a four-level system, as provided in the appendix of Ref. [106]. Assuming that the pump ﬁeld and the dopant distribution are uniform across the ﬁber, that the degenerations of the upper and lower transition levels are equal, that τ32 is several orders of magnitude smaller than τ , and that excited state absorptions and energy transfers between ions are not present, the population rate equations can be expressed as[108] dN1 dt

= −RN1 +

N2 τ

+ N2 W21 − N1 W12

(5.6)

N1 + N2 = N Γ I

Γ (I

+I

)

Γ (I

+I

)

p p −,s −,s where R = hν σabs,p , W21 = s +,s σem , W12 = s +,s σabs . hνs hνs p In these equations, N is the dopant concentration, N1 , N2 are the populations per unit volume for |E1 > and |E2 >, respectively. σ abs,p is the absorption cross sectional area for the pump light, and σabs , σem are the cross sectional area for re-absorption and stimulated emission of the signal light, respectively. Γp and Γs stand for the fractional intensity inside the MNF for the pump and signal light[6] . τ denotes the spontaneous emission lifetime for

Fig. 5.42. (a) Intensity enhancement factor E versus pump coupling coeﬃcient Kp . Each curve corresponds to diﬀerent αp πD for the pump light, see labels. Kp =0 is the nonresonant case, whereas the peak values correspond to critical coupling. (b) Maximum of E versus αp πD. The coupling loss is assumed to be 0.003 for both (a) and (b). (Adapted from Ref. [106],with permission from the Optical Society of America)

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163

Fig. 5.43. Schematic diagram for three-level transitions (TLT) with corresponding rates. σabs,p , σabs , σem stand for the cross sectional areas for the pump absorption, the signal re-absorption, and the stimulated emission, respectively. Since only a fraction of light propagates in the subwavelength ﬁber[19] , the average pump and signal intensities in the ﬁber core are reduced by the overlap factors Γp and Γs . τ32 and τ are the spontaneous radiation lifetimes from |E3 > to |E2 > and |E2 > to |E1 >, respectively. (Adapted from Ref. [106],with permission from the Optical Society of America)

|E2 > I+,s , I−,s are the signal light intensities circulating in the ring in two directions and νp , νs are the frequencies of the pump and signal light. To be precise, the populations are dependent on the position along the MNF; here it is assumed that the pump absorption is weak along a single path of the ring, so that an average pump value is considered within the ring and the spatial variation is neglected. In the steady state (dNi /dt=0), the gain compensates for the loss so that Ks (1 − γs )e−αs πD eGπD = e(GπD−αtot,s πD) = 1

(5.7)

where G=Γ s (N2 σ em − N1 σ abs ) is the gain factor, and αtot,s is the total loss of the signal light per round trip distributed over length,

αtot,s = αsc,s +

1 1 1 1 ln ln + πD Ks πD 1 − γs

(5.8)

From Eqs. (5. 3) and (5. 6), the signal light intensity inside the ring is

I+,s + I−,s

1 hνs (1 − = Γs σem

αtot,s Γp I3,p 1−e−αp πD αtot,s σabs 1 Γs N σem ) hνp αp πD σabs,p −( σem + Γs N σem ) τ α (1 + σσabs ) tot,s em Γs N σem

(5.9) (3) Lasing conditions and quantum eﬃciency The lasing threshold is reached when the signal loss equals the gain, which should not exceed Γ s σ em N of the full inversion, resulting in the lasing condition

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5 MNF-based Photonic Components and Devices

αtot,s < Γs σem N

(5.10)

2πn λΓs σem N

(5.11)

or equivalently,

Qs >

where the quality factor of the signal light Qs of a resonant cavity is given by Q = 2πn/(λα), in which n is the refractive index, λ is the free space wavelength and α is the loss factor[109] . Setting I+,s + I−,s = 0 in Eq. (5.9), the threshold for the pump power is α

Pth =

σ + Γs σtot,s 1 1 hνp 1 σabs em em N A αtot,s Γp σabs,p τ 1 − Γs σem N E

(5.12)

where A is the cross sectional area of the MNF. Using Eqs. (5.6) and (5.7), it gives

αabs,p = Γp N1 σabs,p =

αtot,s Γs N σem + σσabs em

1− 1

Γp N σabs,p

(5.13)

The quantum eﬃciency for steady output can be obtained from Eqs. (5.1), (5.3) and (5.9) (assuming I+,s = I−,s ):

ηq =

d(

I3,s +I4,s ) hνs Ip d( hνp )

αtot,s

=

Γp [(1 − γs )(1 − Ks )][1 − Γs N σem ] N σabs,p E Γs αtot,s (1 + σσabs )/Γs em (5.14)

In order to get more physical insight into ηq , Eq. (5.14) can be rewritten in the following form

ηq =

αabs,p (1 − Ks )(1 − γs ) 1 αtot,s E

(5.15)

thus ηq

αabs,p πD(1 − γp ) (1 − Ks )(1 − γs ) αabs,p πD (1 − Ks )(1 − γs ) = . 1 α πD 1 − (1 − γp )e−πDαp αtot,s πD tot,s Emax

When πDαp 1, 1 − Ks 1, γs 1

5.5 MNF Lasers

ηq ≈

γp πD

1−Ks πD (1 − γs ) 1 ln K1s ln 1−γ αp α s sc,s + πD + πD 1−Ks αabs,p πD (1 − γs ) γs s αabs,p + αsc,p αsc,s + 1−K πD + πD

165

αabs,p γp 1 1−γp πD + +

(5.16)

It can be seen that the ﬁrst term in Eq. (5.16) is related to “useful” absorption, i.e., the fraction of pump attenuation used to excite the rareearth ions to a metastable state; the second term is the fraction of signal light extracted out of the cavity. 5.5.2 Numerical Simulation of Er3+ and Yb3+ Doped MNF Ring Lasers To apply the theoretical model to real systems, erbium and ytterbium ions are chosen as examples of dopants because of their technological importance. They also illustrate the diﬀerence between the weak and the high absorption cases. Erbium ions have the property of eﬃciently storing energy for light emission around 1.55 μm, the wavelength of choice for telecommunications, but present a relatively low absorption cross-section at practical pump wavelengths. Furthermore, erbium ions in glasses are readily aﬀected by detrimental concentration eﬀects[110] . Ytterbium ions oﬀer high quantum eﬃciency with low heat dissipation as well as a high peak absorption cross-section and low concentration quenching. This makes them the ideal dopants for high power or compact ﬁber lasers[111] . The host glass for erbium ions is assumed to be aluminosilicate, the most common host for erbium doped ﬁbers[110] , and the host for ytterbium ions is assumed to be a phosphate based composition (40P2 O5 -19SiO2 -40B2 O3 -1Yb2 O3 ) due to its capability for high concentration doping. (1) Eﬀect of pump coupling factor (Kp ) on threshold power (Pth ) and quantum eﬃciency (ηq ) As indicated before, resonance of pump light that is determined by the pump coupling factor (Kp ), plays an important role in the threshold power (Pth ) and the quantum eﬃciency (η q ) of the MNF ring laser. To demonstrate this, numerical results of Kp -dependent Pth and η q of Er3+ and Yb3+ doped MNF ring lasers are provided in Figs. 5.44(a) and (b), respectively. The spectroscopic parameters used here are listed in Table 5.2, and the ring parameters are listed on the graphs. Various Ks have been tried and the optimum case, where the quantum eﬃciency is highest (corresponding to critical coupling), is given here. (2) Eﬀect of coupler losses γs and γp on optimum threshold power and quantum eﬃciency As shown in Fig. 5.44, when Kp is set to achieve the critical coupling condition, optimal values for the threshold power and the quantum eﬃciency are reached simultaneously. However, the magnitudes of these optima are strongly

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5 MNF-based Photonic Components and Devices

Fig. 5.44. Threshold power Pth (dotted line) and quantum eﬃciency η q (solid line) for glass MNF rings made of (a) Er3+ doped Al2 O3 -SiO2 and (b) Yb3+ doped phosphate glass of spectroscopic parameters listed in Table 5.2. The diameters of the rings are assumed to be 1 mm for both (a) and (b). The diameters of the ﬁbers are chosen to be 1.0 and 0.63 μm for (a) and (b) respectively, in order to maintain singlemode operation for the signal light and the same Γs in both cases. The coupling loss γp (γs ) at the coupling region is assumed to be 0.3%. The intensity coupling coeﬃcient for the signal light (Ks ) is set to the optimum values, 0.979 for (a) and 0.937 for (b). The scattering loss is assumed to be the same for the pump and the signal light, and is set to 0.001 dB/mm (0.002 cm−1 ) according to Ref. [26]. The fractions of pump and signal intensities (Γp and Γs ) inside the ﬁber are calculated according to Ref. [6], and are listed on the graphs. (Adapted from Ref. [106],with permission from the Optical Society of America)

5.5 MNF Lasers

167

Table 5.2. Spectroscopic parameters for Er/Al/Si glass and Yb/P glass Glass N (1020 cm−3 ) σabs,p (pm2 ) σabs , σem (pm2 ) T (ms) Ref. Er/Al/Si glass 0.5 0.19@980 nm 0.48@1535 nm, 0.58@1535 nm 10.2 [112] Yb/P glass 1.0 1.5@980 nm 0.1@1022 nm, 0.6@1022 nm 0.69 [113]

aﬀected by the coupling losses, as shown in Fig. 5.45, in which it is assumed that γ p = γ s . In Ref. [114], Cai et al. have reported that γp (γs ) can be as low as 0.3%, and it is unknown at this stage whether γp (γs ) can be further decreased. The optimum value of Ks depends both on D and γp (γs ) In Fig. 5.45, the curve Op corresponds to the optimum cases for γ s =0.003 and D =1 mm. Because there exists uncertainty regarding the achievable value of γp (γs ) , curves for values of Ks below and above this optimum are also shown. Since γ s contributes to αtot,s there is an upper limit of γ s in order to reach the threshold, according to condition Eq. (5.10). This limit is indicated as γ max in Fig. 5.45. For a given γ s , a larger Ks results in a larger αtot,s , and thus a larger Pth according to Eq. (5.12). But the situation is more complex for the quantum eﬃciency: with regard to η q , there is an optimum of Ks , and the optimum value varies with γ s . This explains the crossings observed in Fig. 5.45 between the quantum eﬃciency curves for diﬀerent Ks . In Fig. 5.45 it is observed that the optimal values for Pth and η q depend more sharply on γp (γs ) in the weak absorption case (Er3+ case) than in the strong absorption case (Yb3+ case). This is expected considering Eqs. (5.12) and (5.16): the coupling loss γp (γs ) contributes much more to the total loss of both pump and signal light in the weak absorption case than in the strong absorption case. In the latter case, the rare-earth absorption is by deﬁnition larger, and larger γp (γs ) can be tolerated. In order to obtain an eﬃcient MNF ring laser, γp (γs ) should be as low as possible, and its value is particularly critical in the weak absorption case. (3) Eﬀect of ring diameter on threshold power and quantum eﬃciency For the purpose of miniaturization, which is one motivation for developing MNF-based devices, it is important to evaluate the minimum possible ring size before the laser performance is signiﬁcantly reduced. From Eqs. (5.8) and (5.10), the minimum possible diameter for lasing can be obtained as

Dmin =

1 ln K1s + ln 1−γ s

(Γs N σem − αsc,s )π

(5.17)

Optimal threshold power Pth and quantum eﬃciency η q against ring diameter D for rings made of Er3+ doped and Yb3+ doped glasses are shown in Figs. 5.46(a) and (b), respectively. It is clear in Eq. (5.17) that a high doping concentration N is desirable for miniaturization, although a high erbium concentration may result in concentration quenching. On the other hand,

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5 MNF-based Photonic Components and Devices

Fig. 5.45. Optimal threshold power Pth (dotted line) and quantum eﬃciency ηq (solid line) against the coupling loss γp (γs ) for (a) Er3+ doped Al2 O3 -SiO2 glass, and (b) Yb3+ doped phosphate glass MNF rings. Diﬀerent Ks are labeled on the curves; the maximum values of γ (labeled as γmax ) are also shown. The parameters used for this simulation are listed on the graphs. (Adapted from Ref. [106],with permission from the Optical Society of America)

5.5 MNF Lasers

169

Fig. 5.46. Threshold power Pth (dotted line), and quantum eﬃciency ηq (solid line) v.s. ring diameter D for three diﬀerent values of Ks , in (a) Er3+ doped Al2 O3 -SiO2 glass and (b)Yb3+ doped phosphate glass MNF rings. In (b), the concentration of Yb3+ ions is 5.0×10−20 cm−3 and Ks =0.939 is the optimum for a 0.2 mm diameter ring. Other parameters used for this simulation are listed in Fig. 5.1 and Table 5.2. The lowest limit of D (Dmin ) for the three diﬀerent values of Ks is also indicated. The region below 50 μm is shaded to indicate that below this diameter radiation losses may need to be taken into account. (Adapted from Ref. [106],with permission from the Optical Society of America)

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5 MNF-based Photonic Components and Devices

ytterbium ions are much more resilient to the quenching eﬀect and their concentration can be increased to 5.0×1020 cm−3 , the value used to plot Fig. 5.46(b). As in Fig. 5.45, a set of curves for three diﬀerent values of Ks has been plotted. It should be noted that the optimum value of Ks for ytterbium has changed compared to Fig. 5.45 because the ytterbium concentration has been increased. It should also be noted that for the ytterbium doped MNF ring laser the optimum Ks corresponds to a diameter (D) of 0.2 mm, instead of 1 mm in Fig. 5.45. The curve for this optimum Ks is marked as Op. Since the diameter of 0.2 mm was chosen somewhat arbitrarily to deﬁne this optimum, plots for values of Ks below and above this optimum are also shown. Fig. 5.46 shows that ηq reaches a peak and then decreases slightly although the ﬁrst term in Eq. (5.16) increases with D. This decrease is due to the fact that the second term in Eq. (5.16) slowly decreases as Dincreases, as more signal light is lost by scattering. This decrease in ηq is moderate, whereas Pth increases more rapidly after reaching its lowest value. Indeed, Pth depends on αtot,s , and the term regarding αtot,s (see Eq. (5.12)), which changes much more dramatically when D increases. It can be inferred from Eq. (5.17) that Dmin can be much smaller when Ks approaches 1, and that for a givenKs , Dmin is determined by γs . Moreover, it is anticipated that even higher ytterbium concentrations are possible before the onset of concentration quenching, so that even smaller ring diameters are possible. It has been estimated that in a silica microsphere of a diameter exceeding 20 μm, the contribution of purely radiative loss can be neglected[115] . The refractive indices of the glasses used here are close to the refractive index of pure silica. Therefore, it is reasonable to assume the upper limit of the ring diameter, where radiation losses may come into play, is below 50 μm, which is marked as a shaded area in Fig. 5.46. In Fig. 5.46(b) the quantum eﬃciency curve for Ks = 0.990 indicates that an ytterbium doped MNF ring laser with a ring diameter of only 50 μm can achieve a ηq as high as 70%. Comparing Fig. 5.46(a) to Fig. 5.46(b) it is clear that erbium doped MNF ring lasers cannot be made as small as their ytterbium doped counterparts. However, it is possible to obtain the compactness of ytterbium doped MNF ring lasers combined with an emission around 1.5 μm by using an erbium- ytterbium co-doped glass[116] . 5.5.3 Er3+ and Yb3+ Codoped MNF Ring Lasers Experimentally, MNF ring lasers have been realized in MNF knots assembled with Er3+ and Yb3+ codoped phosphate glass MNF[71] . The knot resonator is employed due to its mechanically robust structure and easy fabrication, and the active MNF is fabricated by direct drawing of Er3+ and Yb3+ codoped phosphate glass[22] , as have been introduced in Subsection 3.2.2.

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The starting material is a piece of bulk phosphate glass doped with erbium (1.25 mol%) and ytterbium (2.35 mol%) ions. The refractive index of the glass is 1.52 around 1.5-μm wavelength. The as-drawn MNFs, with diameters of one to several micrometers, show high diameter uniformity and excellent sidewall smoothness. Typical absorption of the as-drawn MNF, measured at the pump wavelength of 975 nm, is about 3.4 dB/mm. To assemble the lasing system, the MNF is ﬁrst tied to a knot and tightened to the desired size with a knot diameter of several millimeters. Since the MNF is directly drawn from the doped glass, it is not naturally connected to a standard ﬁber as that in the conventional ﬁber tapers. The knot is then hung between two holders for mechanical support, as shown in Fig. 5.47. For MNF directly drawn from doped glass, the whole length of the ﬁber (both the knot and the two arms) is homogeneously doped with ions. In such a case, to pump the knot with high eﬃciency, the pump light is launched into the knot using an evanescent coupling method: the pump light is ﬁrst coupled into and guided along a standard silica ﬁber, then squeezed into the silica ﬁber taper (about 1μm in diameter) and ﬁnally evanescently coupled into the knot by joining the silica launching taper and the knot at “port 1” through van der Waals and electrostatic attractive forces. Another silica ﬁber taper, positioned outside but in close proximity to the knot, is used to couple light out of the doped MNF at “port 3”.

Fig. 5.47. Schematic diagram of the structure of an MNF knot laser with pump and signal light paths. The pump light is launched into the knot from port 1, and the signal light is collected from port 2. Inset, an optical microscope image of an Er:Yb-doped phosphate glass MNF knot pumped at a wavelength of 975 nm. The green up-converted photoluminescence is clearly seen. (Reprinted with permission from Ref.[71], copyright 2006,American Institute of Physics)

The optical path followed by the pump light is indicated by the arrows shown in Fig. 5.47. When the pump light is coupled into the knot at port 1,

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it propagates in the anticlockwise direction. After passing through the intertwisted region, part of the pump light is recirculated into the knot, while the rest exits the knot at port 2. The recirculation of the light (both the pump and luminescent light) inside the circle forms the resonant cavity for the lasing operation, and the doped ﬁber knot serves as both active medium and laser cavity. The signal is generated and propagated in both a clockwise and counter-clockwise direction. To pump the knot, light from a semiconductor diode laser operating at a wavelength of 975 nm with a linewidth of about 1 nm is launched into the knot. A typical optical microscope image of a pumped knot is shown in the inset of Fig. 5.47, in which the diameters of the knot and the MNF are about 1.1 mm and 3.3 μm, respectively. The strong green emission, due to spontaneous emission from the up-converted F9/2 level to the ground state of Er3+ , is observed along the whole loop. In the experiment, the power of the pump light is increased in small steps (0.05 mW per step), and the output signal from the collection taper is sent to an optical spectrum analyzer. Lasing action was observed for knots with diameters ranging from 1 to 3 mm. Typical output spectra measured around 1540nm wavelength for a 2-mm-diameter knot assembled with a 3.8-μm-diameter MNF are shown in Fig. 5.48. For a pump power below threshold, resonant broadband luminescence is observed (Fig. 5.48(a)). The measured free-spectral range (FSR) of the resonance is about 0.24 nm, corresponding to a knot diameter of 2.0 mm. When the pump power is increased to a certain value (the threshold), a single and narrow peak dominates the luminescence spectrum, indicating lasing of a single longitudinal mode. A typical single-mode lasing spectrum is shown in Fig. 5.48(b). The peak is centered around 1541.1 nm with a side-mode suppression of about 47 dB. The measured linewidth of the lasing mode is about 0.05 nm (6.3 GHz), which may be limited by the spectral resolution available from the optical spectrum analyzer (0.05 nm). When the pump power is increased, a slight red-shift of the peak wavelength (a 1-μW increment in output leads to about a 0.1-nm shift in peak wavelength) is also observed. This shift can be attributed to the temperature rise in the knot resulting from the increased pump power, as has been observed in rare-earthdoped microsphere lasers[117,118] . In addition, by intentionally changing the coupling of the knot structure, the single-mode lasing action can be switched to multimode with depressed intensity of each mode, as shown in Fig. 5.48(c). Collected single-mode laser output power versus pump power in the knot (same knot as used for Fig. 5.48) is plotted in Fig. 5.49. The pump power is calibrated using direct coupling between two MNFs with the same diameter as used for the knot. It shows that the lasing threshold for the knot laser is about 5 mW. The maximum single-mode output is about 8 μW at the maximum available pump power of about 12.8 mW. Considering that a part of the output signal exits from port 4 (see Fig. 5.47), the output from the knot laser should be higher than 8 μW.

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Fig. 5.48. Output spectra of an Er:Yb-doped phosphate glass MNF knot pumped at a wavelength of 975 nm. The 2-mm-diameter knot is assembled with a 3.8-μmdiameter MNF. (a) Resonant luminescence obtained with pump power slightly below the threshold. (b) Single-longitudinal-mode laser emission with pump power above the threshold. (c) Multi-longitudinal mode laser emission observed when intentionally changing the coupling eﬃciency between MNFs. (Reprinted with permission from Ref.[71], copyright 2006,American Institute of Physics)

Although it is predicted theoretically from the numerical simulation (see Section 5.5.2) that for lasing action the knot diameter can be well below 1 mm, experimentally the attempts to reduce the knot diameter (by tightening the knot) to below 1 mm failed to produce a laser. This is believed to be the consequence of insuﬃcient pump absorption or signal gain within the microscale knot. Signiﬁcant improvement to this kind of laser is expected by

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Fig. 5.49. Collected laser output power versus pump power of the MNF knot laser (same knot as used for Fig.5.48) operated in a single-longitudinal mode. (Reprinted with permission from Ref.[71], copyright 2006, American Institute of Physics)

using a pump laser with narrow linewidth , and/or by further optimization of the knot conﬁguration in order to make the pump light resonate under the critical coupling, as has been predicted elsewhere[106] . Compared to conventional ﬁber lasers or compact lasers based on other kinds of microscale cavities such as microdisks[119] , microspheres[114,117,120] and microtoroid[121,122] , the MNF knot laser may present special advantages including simple fabrication (e.g., no additional coupler needed), stable operation (e.g., robust coupling), and compact size (e.g., compared with conventional ﬁber lasers), and suggests a simple approach to highly compact ﬁber lasers. 5.5.4 Evanescent-Wave-Coupled MNF Dye Lasers In the previous section, ring lasers have been realized using active MNFs with dopants contained inside the ﬁber[71] , in which erbium and ytterbium ions were codoped into the MNFs to form active cavities, and were pumped by the light conﬁned inside the MNFs. While doping active media inside MNFs is an eﬀective approach to MNF lasers, the large fractional evanescent waves of an MNF may oﬀer more ﬂexibility for achieving lasing devices based on MNFs. For example, this section introduces an evanescent-wave-coupled MNF dye laser[72] , in which active “dopants” are not directly doped into the MNF, but are “doped” in the evanescent ﬁeld surrounding the MNF. That is, the evanescent wave of the MNF extends the physical boundary of the MNF to an eﬀective boundary (near-ﬁeld boundary) deﬁned by the evanescent ﬁeld, and the dopants doped within the near-ﬁeld boundary are functionalized in a similar way as those actually doped inside the MNF. The structure of the knot dye laser is schematically shown in Fig. 5.50(a). A silica MNF, taper drawn from one end of a standard single-mode ﬁber, is

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tied into a micro knot and immersed into a rhodamine 6G (R6G) dye solution. The untapered end of the standard ﬁber is used as an input port, and the freestanding end of the MNF after the knot structure is used as the output port of the dye laser, which is connected by a ﬁber taper through evanescent coupling for signal collection.

Fig. 5.50. (a) Schematic diagram of the structure of an MNF knot dye laser. (b) Optical microscope image of a typical 450-μm-diameter knot immersed in a 5mM/L rhodamine 6G dye solution. The dye is evanescently pumped by 532-nm-wavelength light guided along the knot. Strong yellow photoluminescence is clearly seen along the MNF knot. (Reprinted with permission from Ref.[72], copyright 2007, American Institute of Physics)

To operate the dye laser, the pump light is ﬁrst sent into the input port (see Fig. 5.50(a)) by lens-focused launching, and then squeezed into the MNF and the knot structure through the taper. Because of the small diameter of the MNF and relatively low index contrast between the MNF and the dye solution (1.46 v.s. 1.43), a certain fraction of the pump light is guided outside the ﬁber core as evanescent waves. The dye molecules distributed in the near ﬁeld of the microknot are thus excited by evanescent waves, leading to strong visible photoluminescence (see Fig. 5.50(b)) that may lead to lasing action when recirculating inside the knot cavity. Since the MNF used here is directly drawn from standard silica ﬁber and is naturally connected to the standard ﬁber, it is very convenient for eﬃciently launching the pump light. Meanwhile, the

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laser light generated around the knot structure can be evanescently coupled back into the MNF, making it very convenient to collect the laser output. In a practical experiment[72] , a 350-μm-diameter MNF knot assembled with a 3.9-μm-diameter silica MNF, was immersed into an ethylene glycol doped with 5 mM/L Rh6G dye. Laser pulses from a frequency-doubled Nd:YAG laser (532-nm wavelength, 6-ns pulses duration, and 10-Hz repetition rate) were used as the pump source, and the signal from the output port of the knot structure was sent into an optical spectral analyzer with a resolution of about 0.03 nm. When the pump power exceeded a certain level (the threshold for lasing oscillation), a laser signal was clearly observed. Fig. 5.51(a) shows three lasing groups centered around 567-, 570- and 580-nm wavelength with a pumping power of 23.5 μJ/pulse. Close-up views of groups B and C in Fig. 5.51(a) are provided in Figs. 5.51(b) and 5.51(c), respectively. The freespectral range measured from the lasing group B (see Fig. 5.51(b)) is about 0.21 nm, corresponding to a knot diameter of about 350 μm. The linewidth of a typical laser peak (e.g., the one centered around 570.6 nm wavelength, marked by an arrow in Fig. 5.51(b)) measured in group B is about 0.06 nm, indicating a Q-factor of about 10 000. The dependence of the output versus pumping power is shown in Fig. 5.52, in which the output sums up signals spectrally ranging from 564 to 584 nm, and the pumping energy is gradually increased from 4 to 60 μJ/pulse. The lasing threshold estimated from Fig. 5.52 is about 9.2 μJ/pulse, and a good linear dependence of the output versus pump power is obtained when the pump energy exceeds the threshold. Considering that the energy loss by absorption before the knot (see Fig. 5.50(b) where strong photoluminescence is produced before the knot by this unintended absorption) has not been deducted here, the real threshold for the knot dye laser should be lower than 9.2 μJ/pulse. A lower threshold may be obtained by optimizing the fraction of the evanescent ﬁeld outside the MNF (usually this can be achieved by optimizing the diameter of the MNF or the refractive index of the dye solution), improving the Q-factor of the knot structure, and eﬃciently resonating both the pumping and the signal light in the same knot. Previously, evanescent-wave-pumped lasers have been realized using various structures such as planar waveguides[123] , ﬁber tapers with external cavities[124,125] , cylindrical microcavities and microspheres[126−131] . The use of MNF microknot for dye laser generation has a number of advantages including compact size, easy fabrication and is compatible with out ﬁber systems, and may provide new opportunities for achieving compact ﬁber lasers with high ﬂexibility, as well as for investigating lasing properties of active media in a simple manner. In addition, beneﬁting from its small size and robust structure, this kind of knot dye laser maybe incorporated with optoﬂuidic technology[132] for lab-on-chip applications.

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Fig. 5.51. (a) Laser emission from a 350-μm-diameter MNF knot pumped by a pulsed frequency-double Nd:YAG laser (532-nm wavelength, 6-ns pulses, 10-Hz repetition rate). The diameter of the MNF is 3.9 μm, and the pumping power is 23.5 μJ/pulse. Typical lasing groups centered around 567 (Group A), 570 (Group B) and 580-nm (Group C) wavelength are clearly observed. Close-up views of groups B and C are provided in (b) and (c), respectively. The linewidth of a typical laser peak marked by an arrow in (b) is about 0.06 nm. (Reprinted with permission from Ref.[72], copyright 2007, American Institute of Physics)

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Fig. 5.52. Integrated emission intensity vs. pump energy of the MNF knot dye laser (same laser as used for Fig. 5.51). The lasing threshold is estimated to be about 9.2 μJ/pulse. (Reprinted with permission from Ref.[72], copyright 2007, American Institute of Physics)

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6 Micro/nanoﬁber Optical Sensors

6.1 Introduction This chapter considers applications of MNFs and photonic devices that can be fabricated from or using MNFs as sensors that detect changes in the ambient medium by monitoring changes in the transmission power of light propagation through MNFs. The changes in the medium may be caused by variation in temperature, radiation, concentration of chemical or biological species, microparticles, etc. As has been described in Chapter 2, an electromagnetic ﬁeld mode supported by an MNF has an evanescent part, which is distributed outside the MNF. The evanescent part may include a signiﬁcant part of the propagating mode and, therefore, can be very sensitive to changes which take place at or near the surface of the MNF. Fig. 6.1 illustrates the principle of optical MNF sensor performance. An MNF either directly performs sensing (Figs. 6.1(a) – (f)) or serves as a waveguide, which connects a photonic sensor to the source and the detector of light (Figs. 6.1(c) and (g) – (k)). Fig. 6.1(a) depicts the simplest uncoated MNF sensor, whose thickness is usually less than the radiation wavelength. For this sensor it is assumed that a signiﬁcant fraction of the fundamental mode of radiation propagates in the ambient medium. Transmission power of this sensor changes with its optical properties, which depend on the properties of the ambient medium. The sensor illustrated in Fig. 6.1(b) is an MNF, which is coated with chemical or biological reagents. The reagents are supposed to be sensitive to selective chemical or biological species. Fig. 6.1(c) illustrates a generic photonic sensor having MNF input and output. Figs. 6.1(d) – (j) depict the straight MNF sensor[1−8] , the MNF loop sensor[9−12] , the MNF coil sensor[12−18] , the MNF/microsphere sensor[19−30] , the MNF/microdisk sensor[31−34] , the MNF/microcylinder sensor[4] , and the MNF/microcapillary sensor[35−39] , respectively. Fig. 6.1(k) illustrates the possible assembly of several MNF-based sensors. The performance of sensors shown in Fig. 6.1 can either be dependent or independent of the interaction of the ambient medium with the evanescent ﬁeld. The sensors of the ﬁrst group are devices whose transmission characteristics vary due to changes in

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optical and/or geometrical properties of the MNF itself and/or of the mode propagating along the MNF. Changes of this kind can be caused by incident radiation, variation of temperature, mechanical vibration, etc. For example, an MNF loop resonator illustrated in Fig. 6.1(e) has been demonstrated as a fast temperature sensor and infrared radiation sensor[9] . The simplest sensor of the second group is a single MNF shown in Fig. 6.1(b), which can detect changes caused by proximity of chemical/biological species and microparticles at the MNF surface. This sensor was demonstrated for silica MNFs[1−7] and polymer MNFs[8] .

Fig. 6.1. Illustration of diﬀerent types of MNF-based photonic sensors. (a) Straight MNF sensor with surrounding evanescent ﬁeld; (b) Straight MNF sensor coated with bio- or chemical layer and surrounding evanescent ﬁeld; (c) Generic structure of MNF-based optical sensor with surrounding evanescent ﬁeld; (d) Straight MNF sensor; (e) MNF loop resonator (MLR) sensor; (f) MNF coil resonator (MCR) sensor; (g) MNF/microsphere sensor; (h) MNF/microdisk sensor; (i) MNF/microcylinder sensor; (j) MNF/microcapillary sensor; (k) A sensor composed of an MNF coupled to a series of optical microcylinders.

Fig. 6.2 compares the MNF evanescent sensor with other types of optical waveguide evanescent sensors. They are a planar waveguide sensor (Fig. 6.2(a)) and a polished optical ﬁber sensor (Fig. 6.2(b)). The advantage of the MNF evanescent sensor, compared to other sensors shown in Fig. 6.2 [40−43] , is that the MNF is more open and more sensitive to ambient changes. In addition, the sensors based on MNF illustrated in Fig. 6.1, can often be more compact than similar optical sensors fabricated with planar lithographic technology.

6.2 Application of a Straight MNF for Sensing

189

Fig. 6.2. Comparison of diﬀerent types of evanescent sensors. (a) A planar waveguide; (b) A polished optical ﬁber; (c) An MNF taper.

Section 6.2 of this chapter considers the simplest MNF sensors fabricated with straight MNFs. These sensors probe the environment by monitoring changes in the evanescent ﬁeld and also of the refractive index and geometric parameters of a straight MNF. They include microﬂuidic sensors, hydrogen sensors, molecular absorption sensors, humidity and gas sensors, optical ﬁber surface sensors and atomic ﬂuorescence sensors. Section 6.3 describes optical sensors fabricated of looped and coiled MNFs. This section considers a direct-contact gas temperature sensor, an infrared radiation sensor and a microﬂuidic sensor. Section 6.4 describes more complex sensors, which consist of MNFs coupled to microspheres, microdisks, microcylinders, and microcapillaries. These photonic devices were demonstrated as refractive index sensors, nanolayer sensors, surface sensors and also as individual atom and molecule sensors. Section 6.5 summarizes this chapter.

6.2 Application of a Straight MNF for Sensing In order to perform physical, chemical and biological sensing of the ambient medium, an MNF should change its transmission properties in response to variation of the sample properties. To do this, an MNF must be coated with reagents that change the refractive index/light absorption in the process of interaction with the chemical/biological objects under test. An evanescent MNF hydrogen sensor was demonstrated in Ref. [1]. The MNF was coated with a thin palladium ﬁlm. The exposure of the coated MNF to hydrogen gave rise to the creation of the palladium hydride layer and changed the MNF transmission loss. Ref. [2] demonstrated another type of MNF sensor which has been immersed into a lower-index cured polymer having a channel in close proximity to the MNF. This MNF sensor measured the refractive index of liquid in the channel. In Ref. [5], an MNF was demonstrated as a molecular

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absorption sensor. In Refs. [3,4], an MNF was used as a sensor of the optical ﬁber radius variation and surface contamination. In Ref. [8], diﬀerent types of polymer MNFs were used for humidity and gas sensing. The authors of Refs. [6,7] suggested and demonstrated an MNF for probing and trapping of atoms. This section reviews the above mentioned applications of straight MNFs. 6.2.1 Microﬂuidic Refractive Index MNF Sensor An MNF-based optical sensor, which measures the refractive index of liquids propagating in a microﬂuidic channel, was demonstrated in Ref. [2]. The MNF was fabricated from a conventional single-mode optical ﬁber and immersed in a transparent curable soft polymer. The sensing device, which is illustrated in Fig. 6.3(a), was fabricated as follows. A rectangular bath was ﬁlled with the transparent curable soft polymer (PDMS). The channel for the liquid analyte, which had a semicircular cross section, was formed by a glass rod positioned in the bottom of the bath. After the bath was ﬁlled with liquid PDMS, the ﬁber taper oriented orthogonally to the glass rod was slowly lowered into the uncured polymer from the top while the optical transmission through the taper was monitored. After the PDMS was cured, the device was ﬂipped over and the glass rod was removed, exposing the channel for the sample liquid. The principle of operation of this sensor was based on the fact that the shape of the fundamental optical mode, which travels through the MNF, was modiﬁed depending on the index contrast between the solution in the channel and the polymer. Consequently, the change of the fundamental mode resulted in variation of the MNF transmission. The sensitivity of the device was measured by monitoring the optical transmission through the MNF with the channel ﬁlled with solutions of glycerol in water. The refractive index of the solution was changed by using diﬀerent concentrations of glycerol. The resulting calibration curves are shown in Fig. 6.3(b) for two diﬀerent sensors, fabricated with (1.6±0.2) μm diameter and with (0.7±0.2) μm diameter MNF sections. In both cases, the transmission achieves maximum when the refractive index of the liquid matches the index of the surrounding polymer, and the thinner MNF is more sensitive. The results of measurements, which are plotted in Fig. 6.3(b), show that the potential measurement accuracy of the refractive index is about 10−4 . 6.2.2 Hydrogen MNF Sensor An optical MNF sensor of hydrogen concentration was demonstrated in Ref. [1]. This sensor monitored the variation of transmission power of a biconical taper with an MNF waist shown in Fig. 6.4(a). An MNF was coated with an ultra thin palladium ﬁlm. The operational principle of this sensor was based on the fact that if a palladium ﬁlm is exposed to hydrogen, its refractive index and, in particular, absorbance changes. The change in the refractive index causes a change in transmission power of an MNF. The MNF fabricated in Ref.

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191

Fig. 6.3. (a) Illustration of the MNF sensor with liquid channel. (b) Measured optical transmission as a function of refractive index of the liquid in the channel for two sensors with diﬀerent MNF diameter at the waist: curve 1 – 1.6 μm; curve 2 – 700 nm. Circles indicate the measurement data points. The arrow shows the refractive index of the surrounding polymer. (Adapted from Ref. [2], with permission from the Optical Society of America)

[1] had a palladium ﬁlm of 4 nm thickness and 2 mm length. In Fig. 6.4(b), the transmission power of the MNF is shown as a function of time when the sensor was exposed successively to a 3.9% concentration of hydrogen. The response time calculated from the plot was about 10 s. This response time is 3 to 5 times faster than that of other optical hydrogen sensors and about 15 times faster than that of some electrical nano hydrogen sensors. The fast response of the sensor was due to the ultra small thickness of the palladium ﬁlm that is rapidly ﬁlled with hydrogen. Fig. 6.4(c) shows the transmission of this sensor as a function of time for diﬀerent concentrations of hydrogen. A calibration curve of the sensor is shown in Fig. 6.4(d). It was suggested[1] that with thinner tapers the detected changes in transmission power could be larger, meaning that the length of the palladium ﬁlm can be made even shorter, probably of a few hundred microns. In addition, the authors of Ref. [1] found

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Fig. 6.4. (a) Illustration of a tapered optical ﬁber (inset – schematic cross section of the MNF); ρ0 is the initial diameter, ρ is the waist diameter, L0 is the length of the waist, t is the maximum thickness of the palladium ﬁlm (shadowed area) and λ is radiation wavelength. (b) Time response of the sensor to periodic cycles from a pure nitrogen atmosphere to a mixture of 3.9% hydrogen in nitrogen. (c) Time response of a sensor when it was exposed to diﬀerent hydrogen concentrations. (d) Transmission versus hydrogen concentration; sensor parameters: ρ = 1300 nm, L = 2 mm, and t= 4 nm. (Adapted from Ref. [1], with permission from the Optical Society of America)

that the demonstrated sensor has preserved its characteristics, sensitivity and fast response time, after some months of its fabrication. 6.2.3 Molecular Absorption MNF Sensor The authors of Ref. [5] developed an optical MNF sensor of sub-monolayers of perylene tetracarboxylic dianhydride molecules (PTCDA, see inset in Fig. 6.5(a)). A 500-nm diameter MNF waist of a tapered single mode ﬁber was used for measurements. Fig. 6.5(a) shows the experimental setup using an

6.2 Application of a Straight MNF for Sensing

193

absorption spectrometer conﬁguration with a tungsten light source and a CCD spectrograph. The molecules are deposited on the MNF by placing a crucible with PTCDA crystals below the ﬁber and by heating it up to 250 ◦ C. The sublimated molecules are carried by air to the MNF where they are adsorbed. The transmission spectrum of the MNF was recorded during deposition with an integration time of 1 s. Fig. 6.5(b) shows evolution of the spectrum of sub-monolayer deposition at 1, 2, 4, and 8 s after the beginning of molecule exposure. Qualitatively, they agree well with spectra of sub-monolayers of PTCDA on mica. The absorbance spectrum can be measured very rapidly and with an excellent signal to noise ratio. Fig. 6.5(c) displays a series of spectra that monitor the post-deposition evolution, i.e., the ripening of the ﬁlm after stopping the deposition of the molecules. It demonstrates that a PTCDA monolayer on a glass surface is meta-stable in ambient conditions and transforms into islands with a thickness of at least two monolayers within minutes. The authors of Ref. [5] found that the sensitivity of measurements with the MNF sensor exceeded the sensitivity of previous studies by two orders of magnitude. 6.2.4 Humidity and Gas Polymer MNF Sensor The authors of Ref. [8] demonstrated polymer MNFs that were used for humidity sensing with a response time of 30 ms, and for NO2 and NH3 detection down to sub-parts-per-million level. The polymer MNFs were fabricated by drawing of solvated polymers. The polymer MNFs were connected to the input and output with two ﬁber tapers drawn from a single mode ﬁber as illustrated in Fig. 6.6(a). The ends of these tapers, which had a diameter of about 500 nm, were placed in parallel and close contact with the ends of a polymer MNF supported by a low-index substrate. Due to the strong evanescent coupling between the polymer MNF and the ﬁber taper, light can be eﬃciently launched into and picked up from the MNF within a few micrometers overlap. The authors employed a PAM MNF for relative humidity sensing. The response of the MNF was tested by alternately cycling 75% and 88% relatively humid air inside the chamber, with an excellent reversibility shown in Fig. 6.6(b). The estimated response time (baseline to 90% signal saturation) of this sensor was about 25–30 ms. This is one or two orders of magnitude faster than existing relative humidity sensors. The remarkably fast response of this sensor is caused by the small diameter and large surface-to-volume ratio of the MNF that enables rapid diﬀusion or evaporation of the water molecules. When blended or doped with other functional materials, polymer MNFs can be used for highly fast and eﬃcient optical sensing. As an example, in Ref. [8] a PANI/PS MNF was applied for gas sensing. When exposed to NO2 , the increase in the oxidation degree of PANI results in a change of the absorption of the propagating light. The absorbance of the MNF at room temperature in response to periodic in time NO2 /nitrogen exposure with NO2 concentration

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Fig. 6.5. (a) Illustration of the molecular absorbance MNF sensor (inset – PTCDA molecule); (b) Evolution of spectrum of sub-monolayer deposition at 1, 2, 4, and 8 s after beginning of molecule exposure; (c) Subsequent evolution of the spectral absorption at constant molecule number at 13, 53, 413, and 2393 s after beginning of molecule exposure. (Adapted from Ref. [5], with permission from the Optical Society of America)

changing between 0.1 and 4 ppm is given in Fig. 6.6(c). This ﬁgure demonstrates good reversibility of the MNF response. Since polymers can be doped and coated with a wide range of functionalized materials, the polymer MNF optical sensors can be realized for detecting various other chemical and biological species.

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195

Fig. 6.6. (a) Illustration of a supported polymer MNF with two ends coupled to ﬁber tapers. (b)Time dependence of the MNF transmittance to alternately cycling air with 75% and 88% relative humidity. (c)Time-dependence of the MNF absorbance to cyclic NO2 /nitrogen exposure with NO2 concentration from 0.1 to 4 ppm. Inset, dependence of the absorbance over the NO2 concentration ranging from 0.1 to 4 ppm. (Adapted with permission from Ref. [8], copyright 2005, American Chemical Society)

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6.2.5 Optical Fiber Surface MNF Sensor Refs. [3,4] developed a very accurate method for sensing the optical ﬁber surface and, in particular, for measurement of the ﬁber radius variation, which is important for understanding the range of diameter variation of the optical ﬁber, the surface contamination and adhesion. In addition, it was mentioned[4] that the optical ﬁber can be used as a substrate for sensing the absorbates disposed on its surface, e.g., thin ﬁlms, microparticles and biological species. The developed method is based on detection of the whispering gallery mode (WGM) resonances in the optical ﬁber using an MNF moving along the ﬁber surface. The experimental setup used in Refs. [3,4] is illustrated in Fig. 6.7(a). It consists of a ﬁber under test and an MNF touching this ﬁber and aligned normally to the ﬁber axis. The MNF was connected to a broadband light source at one end and to the OSA at the other. An example of the measured WGM transmission spectrum is shown in Fig. 6.7(b). In order to determine the ﬁber radius variation, a well-pronounced resonance (circled in the inset of Fig. 6.7(b)) was chosen. The position of this resonance was monitored during the translation of the MNF along the ﬁber and used to calculate the variation in the eﬀective diameter of the optical ﬁber. In addition, the change in the broadband spectrum was used for advanced sensoring of the optical ﬁber surface. The optical ﬁber eﬀective diameter variation, Δd, was calculated from the displacement of the chosen resonance, Δλ, with a simple equation, Δd = dΔλ/λ, where d = 125 μm is the diameter of the ﬁber and λ = 1.55 μm is the wavelength of the resonance. Figs. 6.8(a), (b), and (c) show the results of the measurement of the ﬁber diameter variation with 250 μm, 50 μm, and 5 μm steps, respectively[4] . In order to estimate the accuracy of the method, each measurement was performed two times. Small local spatial shifts between the two similar measurements were caused by the stick-slip problem. Fig. 6.8 demonstrates the achieved angstrom accuracy of measurement of the optical ﬁber diameter variations. More detailed information about the properties of the optical ﬁber surface can be obtained by treating the broadband spectrum of light transmitted through this MNF/optical ﬁber system. As an example, Figs. 6.9(a) and (b) show two measurements of the same segment of a contaminated optical ﬁber with 5 μm steps in the wavelength interval from 1552.3 nm to 1552.8 nm. For comparison, Fig. 6.9(c) shows the spectrum variation at the 250 μm segment of a clean ﬁber in a wavelength interval from 1548.5 nm to 1549 nm. Figs. 6.9(a) and (b) demonstrate that the developed method has a very good spatial resolution and clearly distinguishes features on the 5 μm scale. The developed method can be useful for the highly accurate sensing of the optical ﬁber surface and, in particular, the ﬁber diameter variation. 6.2.6 Atomic Fluorescence MNF Sensor The authors of Refs. [6,7] suggested and demonstrated an optical MNF as a sensor of atomic ﬂuorescence. The experimental setup used in Ref. [7] is illustrated in Fig. 6.10(a). A magneto-optical trap (MOT) for cold Cs-atoms was

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Fig. 6.7. (a) Illustration of the experimental setup. (b) Transmission spectrum of an MNF touching the optical ﬁber; inset shows the spectral band and transmission dip (circled) used for measurement of the ﬁber diameter variation. (Adapted from Ref. [4], with permission from the Optical Society of America)

used as an atom source. The ﬂuorescence of MOT atoms around the MNF was detected by the measurement of ﬂuorescence photons with an avalanche photodiode connected to one end of the biconical ﬁber taper with an MNF waist. Signals were accumulated and recorded on a PC using a photon-counting PCboard. It was shown that a very small number of atoms can be detected by monitoring the transmission of an MNF under strong resonant laser irradiation. Fig. 6.10(b) shows the coupling eﬃciency of spontaneous emission into the MNF propagation mode. These results show that the optical MNF may serve as an eﬃcient tool for quantum optics. The approach developed in Ref. [7] can naturally be extended to systems other than atoms, e.g., molecules or quantum dots.

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Fig. 6.8. Demonstration of angstrom accuracy and reproducibility of the measurement of the eﬀective ﬁber diameter variation. (a) Measurement along the 12 mm segment with 250 μm steps. (b) Measurement along the 2.5 mm segment with 50 μm steps. (c) Measurement along the 250 μm segment with 5 μm steps. Solid curve – ﬁrst measurement, dashed curve – second measurement. (Adapted from Ref. [4], with permission from the Optical Society of America)

6.3 Application of Looped and Coiled MNF for Sensing The sensitivity of an MNF sensor can be signiﬁcantly increased by employing interferometric and resonant MNF structures. Fig. 6.1(e) shows an MNF loop resonator (MLR) sensor representing a Sagnac interferometer described in Sections 2.7 and 5.3. Fabrication of the MLR consists of drawing an MNF and bending it into a self-coupling loop. Two types of MLR have been demonstrated experimentally: a regular MLR illustrated in Fig. 6.1(d)[9] and a knot

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199

Fig. 6.9. Variation of the transmission power spectrum along the 250 μm segments of dirty (a, b) and clean (c) optical ﬁbers. (a) and (b) are two measurements of the same segment, which demonstrate the reproducibility of our measurements. (Adapted from Ref. [4], with permission from the Optical Society of America)

MLR, which has a twisted self-coupling region[10,11] . The MNF coil resonator (MCR) can be fabricated by wrapping an MNF on a central optical rod [12,15−18] . This section reviews the application of an MLR for gas temperature sensing and an MCR for microﬂuidic sensing.

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Fig. 6.10. (a) Illustration of the experiment on detection of atomic clouds with an MNF. APD is the avalanche photodiode. MOT is the magneto-optical trap. (b) Coupling eﬃciency of spontaneous emission into each direction of the MNF propagation mode, ηg , versus atom position r/a, where r and a are distance from nanoﬁber axis and radius of nanoﬁber, respectively. (Adapted from Ref. [7], with permission from the Optical Society of America)

6.3.1 Ultra-Fast Direct Contact Gas Temperature Sensor Narrow resonances of the MLR transmission spectrum are very sensitive to variations of the optical properties of the ambient medium, which change with temperature, pressure and applied radiation. In this subsection the performance of an MLR as an ambient gas local temperature sensor[9] is brieﬂy reviewed. Fig. 6.11(a) shows the transmission spectrum of an MNF loop resonator, which was demonstrated in Ref. [9] as a fast direct gas temperature sensor and infrared radiation sensor. The resonator was periodically heated by a CO2 laser beam transmitted through a beam chopper, as illustrated in Fig. 6.12. Figs. 6.11(b) and (c) show the transmission power variation of this resonator at wavelengths corresponding to a ﬂat region of the spectrum (no eﬀect) and to a resonance region of the spectrum (strong oscillations). From the transmission power oscillations shown in Fig. 6.11(c), the magnitude of temperature variation was estimated as 0.4 K, and the thermal equilibration time constant was estimated as 3 ms, in agreement with the theoretical prediction. For a regular optical ﬁber, which has a diameter 100 times greater, this constant is, proportionally, 0.3 s. The measured temperature range in this experiment was about 1 K. The temperature range can be increased by decreasing the quality

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factor of the MLR or by choosing a region of the transmission spectrum which has less steepness.

Fig. 6.11. (a) Transmission spectrum of MLR used as a temperature sensor. Transient behavior of MRL periodically heated by CO2 laser at wavelength of (b) 1550.4 nm (ﬂat spectrum) and (c) 1550.12 nm (steep spectrum). (Adapted from Ref. [9], c [2006] IEEE)

Fig. 6.12. Periodic on/oﬀ heating of a MLR using a CO2 laser beam and a beam c chopper. (Adapted from Ref. [9], [2006] IEEE)

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6.3.2 MCR Microﬂuidic Sensor As was already mentioned in Section 5.3, the design of an optical sensor by embedding an MNF into the curable liquid can increase the robustness of the device as well as broaden its functionality. Fabrication of a knot MLR embedded into a polymer matrix was considered in Ref. [11]. The MCR wrapped on a supporting optical rod imbedded in low-index polymer was demonstrated in Refs. [12,16,18]. The authors of Refs. [17,18] suggested and demonstrated the MCR microﬂuidic sensor that was created from an embedded MCR by removing the center rod. The MCR was fabricated as follows. An 2.5-μmdiameter MNF was wrapped on a 1-mm-diameter poly (methylmethacrylate) (PMMA) rod. The PMMA refractive index was in the range of 1.49–1.51. Then the MCR was coated with the low-index Teﬂon solution 601S1-100-6 (DuPont, USA). Then the solution was dried in air and solidiﬁed. Next, the MCL was left soaking in acetone to dissolve the PMMA rod. Finally, the microﬂuidic MCR sensor with a ∼ 1-mm-diameter microchannel was fabricated. The sensor consisted of a MCR with ﬁve turns, which contained a channel inside, as illustrated in Fig. 6.13. The thickness of the Teﬂon ﬁlm separating the MCR from the channel was very small. The sensitivity of this device was measured by inserting the sensor in a beaker containing mixtures of isoproply and methanol with ratios 60%, 61.5%, 63%, 64.3%, 65.5%, 66.7% and 67.7%. Fig. 6.14 shows the transmission spectra of the MCR. It is seen that the resonance peak is shifted towards longer wavelengths with growth in the refractive index. The calculated sensitivity of this MCR sensor was about 40 nm/RIU.

Fig. 6.13. The cross section of an MCR microﬂuidic sensor. (Reprinted with permission from Ref. [18], copyright 2008, American Institute of Physics)

6.4 Resonant Photonic Sensors Using MNFs for Input and Output Connections Assembling MNFs with other photonic elements broadens their functionality and can make them more robust. The simplest elements that can be joined

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Fig. 6.14. Output spectrum of the MCR microﬂuidic sensor in mixtures of isopropyl 1 60%, 2 61.5%, 3 63%, 4 64.3%, 5 of methanol. The isopropyl fraction is 6 66.7%, and 7 67.7%, respectively. (Reprinted with permission from Ref. 65.5%, [18], copyright 2008, American Institute of Physics)

with MNFs are optical microresonators such as microspheres, microdisks, microcylinder and microcapillary resonators. This section brieﬂy considers several designs of optical sensors where an MNF performs coupling of light into a microresonator, which probes the environment with excited whispering gallery modes (WGMs). The principle of operation of these devices is based on controlling the shift and width (generally the shape) of transmission resonances, which respond to changes in the ambient medium. 6.4.1 MNF/Microsphere and MNF/Microdisk Sensor An adiabatic optical ﬁber taper having an MNF waist is convenient for coupling light into a microsphere or a microdisk and exciting the WGMs[19−34] . The MNF/microsphere and MNF/microdisk photonic devices, which are illustrated in Figs. 6.1(g) and (h), have been suggested and demonstrated for sensing of physical, chemical, and biological properties of the environment situated near or at the microsphere (microdisk) surface[19−34] . In particular, these devices were used for sensing of the complex-valued refractive index of the ambient medium, of deposited nanolayer ﬁlms and of absorbed microparticles, molecules and atoms. (1) Refractive index sensor The performance of the microsphere and microdisk photonic sensors is similar to that of the MNF resonant sensors described in Section 5.3. The evanescent part of the WGM excited by an MNF is partly propagating in the ambient medium. For this reason, the resonant peaks in the transmission spectrum change with variation in the refractive index of the environment. In the

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simplest situation, the sample distribution can be assumed to be uniform in space surrounding the resonator. Then, the only parameter which aﬀects the resonant spectrum is the refractive index of the sample. The narrow transmission resonance of a microsphere (microdisk), which is described by Eq. (2.82), has three parameters: λ0 , γa , and γc . These parameters are simply determined from Eqs. (2.83) and (2.84) using the experimentally measured values of the resonance peak position, Q-factor and extinction ratio. Variation of these parameters with the refractive index of the ambient medium can be calibrated using a known sample and then used for determination of the real and imaginary parts of the refractive index of unknown samples. For a small change in the refractive index, Δn, variations of λ0 , γa and γc are proportional to Δn. As an example, Ref. [32] demonstrated very accurate determination of the concentration of heavy water D2 O in H2 O with an MNF coupled to a microdisk resonator. The authors used the fact that the refractive indices of H2 O and D2 O are the same. For this reason, the resonant wavelength λ0 does not change with the D2 O concentration (Δλ0 = 0). However, the absorption of H2 O is larger than that of D2 O and, therefore, the Q-factor of the resonator immersed in H2 O is smaller than when it is immersed in D2 O. The corresponding variation in parameters γa and γc can be calibrated and used for determination of the D2 O concentration. The results of the D2 O concentration measurement are shown in Fig. 6.15. By monitoring the quality-factor of the microdisk, concentrations of 0.0001% of D2 O in H2 O have been detected. Reversibility of detection was demonstrated by periodic introduction and ﬂushing of D2 O.

Fig. 6.15. Variation of the Q-factor in response to a periodic change in the concentration of D2 O in H2 O. (Adapted from Ref. [32],with permission the Optical Society of America)

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(2) Nanolayer sensor The refractive index analysis considered in the previous subsection was based on monitoring of a single resonance in the transmission spectrum of a MNF/microresonator structure. The information collected from a larger region of the transmission spectrum allows us to obtain more information about the medium adjacent to the surface of the microresonator. Generally, the transmission spectrum depends on wavelength λ and on the parameters of the ambient medium. The dependence of the transmission spectrum on these parameters can be deﬁned theoretically and by calibration measurements. As an example, Ref. [21] suggested a method that allows simultaneous detection of the refractive index and thickness of a nanolayer adsorbed to the surface of a silica microsphere. As opposed to the case considered in the previous subsection, only the shifts in the resonances in the transmission spectrum (j) were taken into account. It was found that the shift Δλ0 of the resonance (j) Δλ0 due to the change Δn in the refractive index of the ambient medium, which has thickness t, can be estimated by the equation: (j)

Δλ0

(j)

λ0

, ) *(j) λ0 nm 2 2 1/2 t 1 − exp −4π(ns − nm ) Δn (6.1) = (j) 2πR(n2s − n2m )3/2 λ0

here ns and bm are the refractive indices of the microsphere and ambient medium, respectively, and R is the microsphere radius. In order to deter(j) mine Δn and t from this equation, it is suﬃcient to measure Δλ0 for two (1) (2) (1) wavelengths, Δλ0 and Δλ0 . In Ref. [21] this was done for λ0 = 760 (2) nm and λ0 = 1310 nm. As a result, the authors characterized a hydrogel nanolayer with 110 nm thickness and an extremely small excess refractive index of 0.0012, which was formed in situ in an aqueous environment. (3) Surface sensor In Ref. [27], the MNF/microsphere resonator was applied for sensing of ﬂat surfaces. The corresponding setup is illustrated in Fig. 6.16(a). It consists of an MNF positioned in direct contact with a microsphere contacting a sample surface. In order to enhance the response of this device, the transmission power was recorded at the wavelength corresponding to the high gradient slope of a resonant peak. A quartz diﬀraction grating (phase mask) was chosen as a sample surface. The sample was scanned by the MNF/microsphere along the direction normal to the grooves. Variation of the transmission power, which is shown in Fig. 6.16(b), exhibited oscillations which characterize the grooves. It is seen that the segment with defective grooves is followed by the segment with good periodic grooves. The micron resolution, which was achieved in Ref. [27], can be further improved by attaching a metal nanoparticle to the bottom of the microsphere. An MNF/microsphere resonator with the attached nanoparticle is a promising type of near-ﬁeld probe with subwavelength resolution, which is much more robust and eﬃcient than the conventional ultra-small probes used in near-ﬁeld microscopy.

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Fig. 6.16. (a) Illustration of MNF/microsphere tool for surface sensing. (b) Transmission power dependence on the position of the microsphere at the diﬀraction grating surface. Comparison of two measurements of the same grating region shows the reproducibility of measurements. (Adapted from Ref. [27],with permission the Optical Society of America)

(4) Individual atom and molecule sensor The sensitivity of the MNF/microresonator devices can be dramatically increased with their quality factor. In Ref. [33], it was shown that coupling between individual caesium atoms and the ﬁelds of a high-quality toroidal microresonator, illustrated in Fig. 6.1(h) and in the inset in Fig. 6.17(a), can be detected by monitoring the transmission spectrum of this resonator. As the result, the authors detected transit events for single atoms falling through the resonator’s evanescent ﬁeld. In the experiment[33] , the MNF/microtoroid resonator was tuned to the condition of critical coupling which the transmission spectrum shows in Fig. 6.17(a). Cold atoms were delivered to the vicinity of the resonator from a small cloud of caesium atoms cooled to T < 10 mK.

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Fig. 6.17. (a) Transmission power of microtoroid resonator near the condition of critical coupling (inset shows the MNF/microtoroid sensor). (b) Single-photon counting events C(t) as a function of time t after the release of the cold atom cloud at t =0. (Reprinted by permission from Macmillan Publishers Ltd: Ref. [33], copyright 2006)

The interaction of an individual atom with the evanescent ﬁeld destroyed the condition of critical coupling, leading to an increase in transmission power. Fig. 6.17(b) shows records C(t) for the number of single photon detection events within time bins of 2 ms as functions of time t. This ﬁgure shows sharp peaks of duration <2 ms that were attributed to the transit of a single atom through the resonant mode of the microtoroid. In Ref. [34], a similar MNF/microtoroid sensor was used to detect the single molecule adsorption at the surface of the toroidal microresonator. The eﬀect was demonstrated with the interleukin-2 (IL-2) molecules that are instrumental in the body’s natural response to microbial infection. The shifts caused by individual molecules binding to the surface of the microtoroid were resolved by optimizing the concentration and injection rate of the solution of these molecules as well as acquisition rates. The following new mechanism of the eﬀect of absorbed molecules on the transmission spectrum of the micro-

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toroid was suggested. The high circulating ﬁeld intensity within the resonator locally heats molecules. The corresponding temperature increase results in a red-shift of the resonant wavelength through the thermo-optic eﬀect, when the microresonator material is heated by the molecule. 6.4.2 MNF/Microcylinder and MNF/Microcapillary Sensors A particular case of a MNF/microcylinder sensor has already been considered in Subsection 6.2.5 where an MNF was used for probing nonuniformities of an optical microcylinder. The same MNF/microcylinder conﬁguration can be used for the sensing of changes which take place in the medium adjacent to the circumference of this cylinder in contact with the MNF. An interesting approach for evanescent ﬁeld sensing employing an optical microcapillary was suggested in Refs. [35,36]. The authors suggested a liquid core WGM optical sensor, which consists of an optical capillary coupled to an MNF or another optical waveguide. This device, which is illustrated in Fig. 6.18(a), was called a liquid-core optical ring-resonator sensor (LCORRS). If the walls of the capillary are thin enough, the WGM transmission spectrum of the MNF can detect changes, which happen along the internal surface of the capillary. The LCORRS was demonstrated as an eﬃcient tool for chemical and biological sensing[35−39] .

Fig. 6.18. (a) Illustration of an LCORRS. (b) Illustration of an LCORRS imbedded in a low-index polymer matrix.

In order to increase the sensitivity, the wall of the capillary has to be decreased, which increases the fragility of this device. In addition, the waist of the optical ﬁber is prone to contamination and corrosion, which reduces the transmission power of the detected light. These problems were solved in Ref.

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[39] where an LCORRS, which exhibited the highest sensitivity and robustness, was demonstrated. To do this, the LCORRS was immersed in a curable low index polymer. After curing of the polymer, the taper and the capillary surfaces of the LCORRS, which is illustrated in Fig. 6.18(b), were fully protected from corrosion and contamination. Furthermore, because the capillary was imbedded into the solid polymer matrix, the authors of Ref. [39] were able to etch the capillary wall down to submicron values. Eventually, the silica capillary was fully eliminated. In this sensor, a signiﬁcant part of resonant WGMs is located inside the tested liquid (rather than propagating primarily along the capillary wall, as in the free-standing LCORRS), which considerably increases its sensitivity. The sensitivity of the device was measured with optical refractive index matching liquids. Fig. 6.19(a) shows an example of measurement of the sensitivity. The spectrum of the LCORRS was recorded for close refractive indices n1 = 1.444 and n2 = 1.446 and the sensitivity found from the shift of the dips in these spectra, Δγ, as S = Δλ/Δn. For

Fig. 6.19. (a) Illustration of measurement of the LCORRS sensitivity. (b) The sensitivity of the LCORRS measured when the silica microcapillary was not removed completely (curve 1) and for the removed microcapillary (curve 2). Curve 3 is the sensitivity predicted for the Teﬂon AF matrix. (Adapted from Ref. [39],with permission from the Optical Society of America)

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the spectra depicted in Fig. 6.19(a), Δλ ≈ 1nm and Δn = n2 − n1 = 0.002 resulting in S ≈ 500nm/RIU. The plot of sensitivity of the LCORRS is shown in Fig. 6.19(b), curve 1. For the test liquids with larger refractive indices, nc > 1.5, the sensitivity was as high as about 700 nm/RIU. For smaller indices, the sensitivity decreased and was about 25 nm/RIU at nc = 1.296. In order to further increase the sensitivity of the LRROS, the microcapillary was completely removed by etching. The corresponding sensitivity of the LRROS is given by the curve 2 in Fig. 6.19(b). It is seen that, as a result, the sensitivity at larger refractive indices increased by up to about 800 nm/RIU and also signiﬁcantly increased at lower indices. The authors of Ref. [39] suggested that if the polymer matrix used in their experiment (having a refractive index of about 1.38) is replaced with a lower index Teﬂon AF, which has a refractive index of about 1.29, the sensitivity of the LCORRS can be further increased to the sensitivity given by curve 3 in Fig. 6.19(b). 6.4.3 Multiple-Cavity Sensors Supported by MNFs The previous subsections considered the simplest MNF-based sensing devices, which are assembled from MNFs and individual optical cylinders, disks and microspheres. In addition, these elements can be assembled in more complex multifunctional structures. One such structure is illustrated in Fig. 6.20(a). This is a matrix of parallel MNFs positioned in touch with parallel optical microcylinders. Each of the MNFs excites WGMs in each of the cylinders. The transmission spectrum of each MNF contains resonances from WGMs excited at the intersection of this MNF with each of the cylinders. If the cylinders are exactly uniform in diameter, it is not possible to identify the changes in particular cylinders from the MNF transmission spectrum. However, if the diameters of the cylinders, di , are diﬀerent from each other (which always happens in practice) then the contributions of individual cylinders to the transmission spectrum are linear independent. A small change in the refractive index, localized near an intersection of a certain cylinder with a certain MNF, will result in a complex linear shift of the transmission spectrum of this cylinder. Due to linear independency, the positions of local changes and their values can be separated and determined. In other situations, the cylinders in Fig. 6.20(a) can be replaced by microcapillaries, microspheres or microdisks. If the resonances in the transmission spectrum, which are introduced by separate microcylinders (microspheres, microdisks, etc.) coupled to the same MNF are not overlapping then these resonances can be easily separated and identiﬁed. In this case, variations in the transmission spectrum recorded from an individual MNF gives information about changes that happen locally at each of the microcylinders coupled to this MNF. A sensor based on this principle was demonstrated in Ref. [23] where the DNA detection was performed by using two microspheres connected to a single MNF illustrated in Fig. 6.20(b). The multiplexed signal from two microspheres allowed the authors to deter-

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mine a single nucleotide mismatch in an 11-mer oligonucleotide with a high signal-to-noise ratio of 54.

Fig. 6.20. (a) Illustration of a set of parallel MNFs and parallel microcylinders coupled to each other. (b) Two microspheres coupled to an MNF (from Ref. [23]). (c) Two microcapillaries coupled to an MNF imbedded in a low index polymer (from Ref. [44],with permission from the Optical Society of America ).

Generally, the resonances of individual microresonators connected to a single input/output MNF overlap and their separation in the transmission spectrum may not be simple. However, in the case of relatively small changes, identiﬁcation of the transmission spectrum, which belongs to the individual resonator, can be performed by the method suggested in Ref. [44]. The transmission power of an optical microresonator sensor, P (λ, n1 , n2 , · · · , nK , T, p) depends on the wavelength, λ, refractive indices of components, nk , temperature, T , and other parameters, e.g., the pressure inside the channels, p. If refractive index variations, Δnk = nk −n0k , temperature variation, ΔT = T −T0 , and pressure variations, Δp = p − p0 are relatively small then the corresponding variation of P (λ, n1 , n2 , · · · , nK , T, p) is a linear function of Δnk , ΔT , and Δp:

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P (λ, n1 (T ), n2 (T ), · · · , nK (T ), T, p) − P (λ, n01 , n02 , · · · , n0K , T0 , p0 ) K 4 (6.2) = Qnk (λ)Δnk + QT (λ)ΔT + Qp (λ)Δp k=1

Here functions Qn (λ), QT (λ), and Qp (λ) can be determined experimentally with calibration samples. If these functions are linear independent then the parameters Δnk , ΔT and Δp can be uniquely determined from the variation of P (λ, n1 , n2 , · · · , nK , T, p), which are considered as a function of λ. Using this approach, the side eﬀects, i.e. the temperature and pressure dependences, can be eliminated from the transmission spectrum. The sensing method based on this simple idea was applied in Ref. [44] to the measurement of microﬂuidic refractive index changes in two microcapillaries coupled to a single MNF illustrated in Fig. 6.20(c). The developed approach allowed us to compensate the side temperature and pressure variation eﬀects.

6.5 Summary This chapter reviewed applications of MNFs for optical sensing. It is shown that the MNF can serve as a basic element as well as a light input/output waveguide of miniature photonic sensors. One of the advantages of MNFs as optical sensors is their free-standing position, enabling much larger contact with the environment compared to other types of evanescent waveguide sensors. In addition, a large evanescent ﬁeld, extremely small cross-sections and weight, very small losses and the availability of miniature MNF-based optical resonators, including MNF supported optical microresonators, allow the MNFs to perform the detection of extremely small variations in physical, chemical and biological properties of the ambient medium.

References 1. 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). 2. 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). 3. T. A. Birks, J. C. Knight, T. E. Dimmick, High-resolution measurement of the ﬁber diameter variations using whispering gallery modes and no optical alignment, IEEE Photon. Technol. Lett. 12, 182–184 (2000). 4. M. Sumetsky, Y. Dulashko, Sensing an optical ﬁber surface by a microﬁber with angstrom accuracy, in Optical Fiber Communication conference, Anaheim, 2006. 5. 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).

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6. V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, M. Morinaga1, Atom trapping and guiding with a subwavelength-diameter optical ﬁber, Phys. Rev. A 70, 011401 (2004). 7. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, K. Hakuta, Optical nanoﬁber as an eﬃcient tool for manipulating and probing atomic Fluorescence, Opt. Express 15, 5431–5438 (2007). 8. F. X. Gu, L. Zhang, X. F. Yin, L. M. Tong, Polymer single-nanowire optical sensors, Nano Lett. 8, 2757–2761 (2008). 9. M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, D. J. DiGiovanni, The microﬁber loop resonator: theory, experiment, and application, J. Lightwave Technol. 24, 242–250 (2006). 10. X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, Demonstration of optical microﬁber knot resonators, Appl. Phys. Lett. 88, 223501 (2006). 11. G. Vienne, Y. H. Li, L. M. Tong, Microﬁber resonator in polymer matrix, IEICE Trans. Electron. E90, 415–421 (2007). 12. M. Sumetsky, Basic elements for microﬁber photonics: micro/nanoﬁbers and microﬁber coil resonators, J. Lightwave Technol. 26, 21–27 (2008). 13. F. Xu, P. Horak, G. Brambilla, Conical and biconical ultra-high-Q optical-ﬁber nanowire microcoil resonator, Appl. Opt. 46, 570–573 (2007). 14. F. Xu, P. Horak, G. Brambilla, Optimized design of microcoil resonators, J. Lightwave Technol. 25, 1561–1567 (2007). 15. F. Xu, G. Brambilla, Manufacture of 3-D Microﬁber Coil Resonators, IEEE Photon. Technol. Lett. 19, 1481–1483 (2007). 16. F. Xu, G. Brambilla, Embedding optical microﬁber coil resonators in Teﬂon, Opt. Lett. 32, 2164–2166 (2007). 17. F. Xu, P. Horak, G. Brambilla, Optical microﬁber coil resonator refractometric sensor, Opt. Express 15, 7888–7893 (2007) 18. F. Xu, G. Brambilla, Demonstration of a refractometric sensor based on optical microﬁber coil resonator, Appl. Phys. Lett. 92, Art. 101126 (2008). 19. K. J. Vahala, Optical microcavities, Nature 424, 839–846 (2003). 20. F. Vollmer, D. Braun, A. Libchaber, Protein detection by optical shift of a resonant microcavity, Appl. Phys. Lett. 80, 4057–4059 (2002). 21. M. Noto, F. Vollmer, D. Keng, I. Teraoka, S. Arnold, Nanolayer characterization through wavelength multiplexing of a microsphere resonator, Opt. Lett. 30, 510– 512 (2005). 22. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, F. Vollmer, Shift of whisperinggallery modes in microspheres by protein adsorption, Opt. Lett. 28, 272–274 (2003). 23. F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, Multiplexed DNA quantiﬁcation by spectroscopic shift of two microsphere cavities, Biophys. Journ. 85, 1974–1979 (2003). 24. I. Teraoka, S. Arnold, F. Vollmer, Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium, J. Opt. Soc. Am. B20, 1937–1946, (2003). 25. N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. M. White, X. Fan, Refractometric sensors based on microsphere resonators, Appl. Phys. Lett. 87, 201107 (2005).

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26. I. M. White, N. M. Hanumegowda, X. Fan, Subfemtomole detection of small molecules with microsphere sensors, Opt. Lett. 30, 3189–3191 (2005). 27. M. Sumetsky, Y. Dulashko, D. J. DiGiovanni, Optical surface microscopy with a moving microsphere, in Nanophotonics Topical Meeting, OSA, Uncasville, 2006. 28. H. Ren, F. Vollmer, S. Arnold, A. Libchaber, High-Q microsphere biosensor – analysis for adsorption of rodlike bacteria, Opt. Express 15, 17410–17423 (2007). 29. A. B. Matsko, V. S. Ilchenko, Optical resonators with whispering-gallery modespart I: basics, IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006). 30. V. S. Ilchenko, A. B. Matsko, Optical resonators with whispering-gallery modespart II: applications, IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). 31. D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Ultra-high-Q toroid microcavity on a chip, Nature 421, 925–928 (2003). 32. A. M. Armani, K. J. Vahala, Heavy water detection using ultra-high-Q microcavities, Opt. Lett. 31, 1896–1898 (2006). 33. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, H. J. Kimble, Observation of strong coupling between one atom and a monolithic microresonator, Nature 443, 671–674 (2006). 34. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, K. J. Vahala, Labelfree, single-molecule detection with optical microcavities, Science 317, 783–787 (2007). 35. I. M. White, H. Oveys, X. Fan, Liquid core optical ring resonator sensors, Opt. Lett. 31, 1319–1321 (2006). 36. I. M. White, H. Oveys, X. D. Fan, T. L. Smith, J. Y. Zhang, Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reﬂecting optical waveguides, Appl. Phys. Lett. 89, Art. 191106 (2006). 37. X. D. Fan, I. M. White, H. Y. Zhu, J. D. Suter, H. Oveys, Overview of novel integrated optical ring resonator bio/chemical sensors, Proc. SPIE 6452, 1–20 (2007). 38. H. Y. Zhu, I. M. White, J. D. Suter, P. S. Dale, X. D. Fan, Analysis of biomolecule detection with optoﬂuidic ring resonator sensors, Opt. Express 15, 9139–9146 (2007). 39. M. Sumetsky, R. S. Windeler, Y. Dulashko, X. Fan, Optical liquid ring resonator sensor, Opt. Express, 15, 14376–14381 (2007). 40. P. H. Paul, G. Kychakoﬀ, Fiber-optic evanescent ﬁeld absorption sensor, Appl. Phys. Lett. 51, 12–14 (1987). 41. C. Piraud, E. K. Mwarania, J. Yao, K. O’Dwyer, D. J. Schiﬀrin, J. S. Wilkinson, Optoelectrochemical transduction on planar optical waveguides, J. Lightwave Technol. 10, 693–699 (1992). 42. V. Ruddy, B. D. MacCraith, J. A. Murphy, Evanescent wave absorption spectroscopy using multimode ﬁbers, J. Appl. Phys. 67, 6070–6074 (1990). 43. S. W. James, R. P. Tatam, Fibre optic sensors with nano-structured coatings, J. Opt. A: Pure Appl. Opt. 8, S430–S444 (2006). 44. M. Sumetsky, Y. Dulashko, R. S. Windeler, Temperature and pressure compensated microﬂuidic optical sensor, in Conference on Lasers and ElectroOptics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) Optical Society of America, paper CMJ6, 2008.

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Besides the microphotonic components and devices introduced in Chapters 5 and 6, more applications of MNFs, including optical nonlinear eﬀects and atom manipulation, have manifested themselves and been explored in recent years. This chapter gives a brief review of the recent progress in these interesting and exciting areas of MNF research.

7.1 Optical Nonlinear Eﬀects in MNFs Nonlinear eﬀects in conventional optical ﬁbers have been extensively explored in past years[1−3] , which stimulated numerous opportunities and applications in broad areas. Generally, the nonlinearity in an optical ﬁber can be characterized using a nonlinear parameter[1]

γ=

n 2 ω0 2πn2 = cAeﬀ λAeﬀ

(7.1)

where c is the velocity of the light, n2 is the nonlinear index in units of m2 /W, ω 0 is the frequency of the light, and Aeﬀ is the eﬀective mode area of the guided light. In a micro- or nanoscale MNF, when the ratio of the wavelength and ﬁber diameter is properly chosen, the small diameter and the high-index-contrast[4,5] induced tight optical conﬁnement yield very small Aeﬀ . For example, when guiding a 1550-nm-wavelength light,Aeﬀ of a 1.1-μm-diameter silica MNF is as small as 1 μm2 , about two orders of magnitude smaller that of a standard single-mode ﬁber, which may signiﬁcantly enhance nonlinear eﬀects in such a thin ﬁber. Manageable dispersion is another merit of the MNF. In a conventional ﬁber with a core diameter far beyond the wavelength of the guided light, the total dispersion of the ﬁber is dominated by material properties, with a certain degree of tunability within a narrow spectral range[6] . In contrast, in a

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subwavelength-diameter MNF, waveguide dispersion is signiﬁcantly enhanced due to the low dimension and high index-contrast[4] . The enhanced waveguide dispersion, which is strongly diameter-dependent, can be used to compensate the material dispersion by choosing a proper MNF diameter. For example, when a standard silica ﬁber is drawn into a 500-nm-diameter silica MNF, the zero dispersion point can be shifted from about 1.3-μm to around the 550-nm wavelength. Therefore, by reducing the ﬁber diameter to the subwavelength scale, it is possible to extend the nonlinear ﬁber optics to previously inaccessible ranges of wavelength.

Fig. 7.1. (a) Power localization and (b) eﬀective nonlinearity and mode ﬁeld diameter (MFD) of an air-cladding silica MNF. (Adapted from Ref. [7], with permission from the Optical Society of America)

The optimal waveguide dimensions for nonlinear interactions in MNFs were theoretically reported by Foster et al.[7] , in which they investigated strong light conﬁnement in these high core-cladding index contrast waveguides with dimensions smaller than the wavelength of incident light, and demonstrated that an optimal core size exists that maximizes the eﬀective nonlinearity. Fig. 7.1 shows the calculated power localization, the eﬀective nonlinearity and mode ﬁeld diameter (MFD) of an air-cladding silica MNF. The unit of the

7.1 Optical Nonlinear Eﬀects in MNFs

217

abscissa is normalized as D/λ, in which D stands for the physical diameter of the MNF and λ is the wavelength of the light. It is clearly seen that there exists a maximum nonlinearity around D/λ=0.7 (see Fig. 7.1(b)), according to about 80% power conﬁned inside the core of the MNF (see Fig. 7.1(a)). The maximum nonlinearity occurs at the point that is very close to the minimum modal ﬁeld diameter (MFD), manifesting the important role of tight conﬁnement of the MNF in the enhancement of the nonlinearity. More signiﬁcantly, Zheltikov investigated pulse propagation in a nonlinear MNF using a Gaussian approximation of the guiding modes[8] , and concluded that the maximum enhancement of nonlinearity requires an eﬀective mode area slightly larger than its maximum. This is reasonable since in an MNF, only the core is nonlinear. More reports on nonlinear pulse propagation in subwavelength-diameter MNFs can be found elsewhere[9−11] . The above-mentioned theoretical prediction is helpful for choosing optimal parameters for generating nonlinear eﬀects in MNFs. For example, using typical ultrafast pulses from a Ti:sapphire laser centered around the 800-nm wavelength, the nonlinearity in a silica MNF can be maximized when its diameter is set to around 550 nm[7] . For ultrafast pulses, the dispersion of the MNF should be considered. Fig. 7.2 shows the dispersion of the diameter-dependent total dispersion of an air-cladding silica MNF at the wavelength of 800 nm. For reference, nonlinearity of the MNF is also provided. The dispersion goes to zero when the diameter of the MNF approaches 700 nm, where the nonlinearity remains considerably high (though not the maximum), demonstrating the possibility of simultaneously achieving high eﬀective nonlinearity and low dispersion in a subwavelength-diameter MNF. Furthermore, dispersion of a given MNF can be engineered by thin coatings on its surface[12] . For example, when guiding a 500-nm-wavelength light, the overall dispersion of the fundamental modes in a 400-nm-diameter silica MNF is about −480 ps·nm−1 ·km−1 , well below the zero-dispersion level; when a 5-nm-thickness ﬁlm with refractive index of 2.7 is coated on the surface of the same MNF, a positive dispersion of 485 ps·nm−1 ·km−1 arises, shifting the dispersion of the coated MNF slightly beyond the zero-dispersion point. Besides the miniaturized modal area and engineerable dispersion, large available length, low waveguiding loss, easy connection to a standard ﬁber optic system with adiabatic taper are also highly desired properties of the MNF for nonlinear optical applications. Experimentally, the ﬁrst nonlinear eﬀect reported in a subwavelength- diameter MNF is supercontinuum generation. Using nanosecond pulses from a 532-nm Nd:YAG microchip laser as pumping source, Leon-Saval et al. observed single-mode supercontinuum that is broad enough to ﬁll the visible spectrum[13] . Compared with that in a 920-nm-diameter silica MNF, the pumping power required for supercontinuum generation in a “subwavelength” 510-nm-diameter MNF reduced from 3 mW to 1.5 mW, and the required length reduced from 90 mm (for the 920-nm-diameter MNF) to 20 mm (for

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Fig. 7.2. Diameter dependence of the dispersion and nonlinearity of an air-cladding silica MNF at 800-nm wavelength. (Adapted from Refs. [4,7], with permission from the Optical Society of America)

the 510-nm-diameter MNF), demonstrating the considerable nonlinear enhancement in subwavelength- diameter MNFs. Supercontinuum generation using femtosecond pulses has also been reported by Gattass and Mazur et al.[14] . Compared with the nanosecond-pulse-pumped eﬀect, in which stimulated Raman and parametric processes gave rise to the broadening, supercontinuum generation with femtosecond pulses is mainly attributed to self-phase modulation and four-wave mixing. Pulses from a Ti:sapphire laser, with the wavelength centered around 800 nm and a pulse duration of about 100 fs, were sent into silica MNFs with diﬀerent diameters. Strong broadening was observed with MNF diameter between 400 and 700 nm, which agrees well with the spectral region for strongly enhanced nonlinearity shown in Fig. 7.2. For reference, Fig. 7.3 shows the supercontinuum spectrum generated in a 700-nm-diameter MNF with a length of about 10 mm, which is pumped with pulse energy of 4 nJ. In addition to supercontinuum generation, more nonlinear eﬀects have been investigated in MNFs. Relying on the broad region of anomalous group velocity dispersion and the large eﬀective nonlinearity of MNFs, Foster and Gaeta demonstrated soliton-eﬀect self-compression of 70-fs pulses from an 80MHz Ti:Sapphire oscillator down to 6.8 fs in a 980-nm-diameter glass MNF[15] . When pumped by pico-second pulses from a mode-lock laser at 532-nm wavelength, Shi and Chen et al. reported enhanced stimulated Raman scattering in a 12-cm-length 900-nm-diameter silica MNF[16] . Very recently, You et al. theoretically investigated a subwavelength-diameter MNF in a vacuum chamber surrounded by rubidium vapor[17] , and predicted an obvious enhancement in the rate of two-photon absorption due to the small mode volume and relatively large overlap of the evanescent ﬁeld with the atomic vapor. As silica glass is relatively low in optical nonlinearity, besides the silica MNF mentioned above, MNFs fabricated from many other materials with

7.2 MNFs for Atom Optics

219

Fig. 7.3. Supercontinuum spectrum of a 10-mm-length 700-nm-diameter silica MNF pumped by 100-fs pulses centered around 800-nm wavelength with pulse energy of 4 nJ. (Adapted from Ref. [14], with permission from the Optical Society of America)

high nonlinearities have been proposed or employed for nonlinear optical applications. Usually, high nonlinearity is accompanied by a high refractive index, which may further enhance the nonlinearity due to the tighter optical conﬁnement in an MNF with higher index contrast. By taper-drawing of lead-silicate and bismuth-silicate optical ﬁbers, Brambilla et al. reported low-loss compound glass MNFs that may provide nonlinear indices 50 times higher than in silica MNFs[18] . Recently, Magi and Eggleton et al. reported considerably enhanced Kerr nonlinearity in subwavelength-diameter highly nonlinear As2 Se3 chalcogenide MNFs[19] . By elaborately tapering low-meltingtemperature As2 Se3 chalcogenide ﬁbers with initial diameters of 165-μm using electrical heating, MNFs with diameters of 1.2 μm were obtained. Based on self phase modulation measurement, the authors reported an enhanced nonlinearity of 68 W−1 · m−1 at the 1545-nm wavelength, which is about 62000 times higher than in standard silica single-mode ﬁbers. More recently, Pelusi et al. reported 160 Gb/s demultiplexing via four wave mixing in a 1.9-μmdiameter MNF tapered from As2 S3 chalcogenide glass single-mode ﬁber[20] , which may represent the ﬁrst all-optical signal processing device based on highly-nonlinear MNFs.

7.2 MNFs for Atom Optics In past years, signiﬁcant progress has been achieved in manipulating single atoms using optical waveguiding geometrics. Relying on light-induced dipole force exerted by a gradient optical ﬁeld, neutral atoms have been successfully trapped or guided along optical waveguides such as hollow waveguides[21−25] . Compared with those waveguides used before, subwavelength-diameter MNFs oﬀer a strong transverse conﬁnement of the guided ﬁber mode while exhibiting a pronounced evanescent ﬁeld surrounding the ﬁber[4,26,27] . This tightly

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conﬁned evanescent ﬁeld with high spatial gradients can be used to eﬃciently trap atoms near the surface of the MNF and couple radiation from atoms to the guided modes of the MNF, making the MNF a powerful tool for atom optics. Balykin et al. proposed atom trapping and guiding with a subwavelengthdiameter MNF[28] . As shown in Fig. 7.4, when a red-detuned evanescentwave ﬁeld in the fundamental mode is launched in an MNF with a diameter smaller than half of the wavelength, the attractive gradient force exerted by the evanescent-wave ﬁeld can be used to balance the centrifugal force of a moving atom when the component of the angular momentum of the atoms along the ﬁber axis is in an appropriate range. For reference, Balykin et al. predicted that cesium atoms colder than 0.29 mK can be trapped and guided along a 400-nm-diameter silica MNF with a 27-mW-power 1.3-μm-wavelength light[28] .

Fig. 7.4. Schematic diagram of an atom trapping around an optical MNF.

To produce a deeper potential for atoms trapping with large coherence time and trap lifetime, Le Kien et al. reported a two-color scheme using faroﬀ-resonance lights with substantially diﬀering evanescent decay lengths[29] , in which an MNF carries a red-detuned light and a blue-detuned light, both far from the resonance wavelength of the atom. The possibilities of conﬁning atoms to a cylindrical shell around the ﬁber with circularly polarized light ﬁelds and conﬁning atoms to two straight lines parallel to the ﬁber axis with linearly polarized light ﬁelds were demonstrated theoretically. In addition to the large depth of the net potential for atom trapping, the scheme also oﬀers large coherence time and large trap lifetime. For example, with a 30-mW-power 1.06-μm-wavelength red-detuned light and a circularly polarized 29-mW-power 700-nm-wavelength blue-detuned light guided in a 400nm-diameter silica MNF, Le Kien et al. predicted a trap depth of 2.9 mK, a coherence time of 32 ms, and a recoil-heating-limited trap lifetime of 541 s for cesium atoms[29] .

7.2 MNFs for Atom Optics

221

While light guided along the MNF can be used to trap and guide the atoms, light emitted from the atoms can also be coupled back into the guiding mode of the MNF. Based on theoretical calculation of the spontaneous emission of a cesium atom in the vicinity of a subwavelength-diameter silica MNF, Le Kien et al. demonstrated that a fraction power up to 28% of the spontaneous emission (around the 852-nm wavelength) by the atom can be channeled into guided modes of a 400-nm-diameter silica MNF[30,31] , showing the potential for using MNFs as nanoprobes for atom optics. Furthermore, they showed that the strong conﬁnement of the guided modes in the MNFs signiﬁcantly contributes to the enhancement of the spontaneous emission, which has not been reported before without a cavity. Experimentally, Hakuta et al. reported eﬃcient coupling of atomic ﬂuorescence to the guided mode of a subwavelength-diameter MNF[32] . The cold Cs-atom trapped by a magneto-optical trap (MOT) had a number density of 2×1010 atoms/cm3 with a temperature of the order of 102 μK. When the MOT was overlapped with the MNF, ﬂuorescence of a very small number of atoms that was coupled into the guiding mode of the MNF was observed at the output of the ﬁber, making the MNF an promising tool for manipulating and probing atomic ﬂuorescence. With the capability of manipulating atoms, collecting and guiding radiation from the atoms using a single MNF, new possibilities for atom optics or quantum optics have been proposed. For example, based on the full quantization of both the radiation and guided modes of the MNF, Le Kien et al. investigated spontaneous emission from a pair of two-level atoms near an MNF, and theoretically predicted a substantial radiative exchange between distant atoms mediated by the guided modes of the MNF, demonstrating the possibility of entangling two distant atoms through the guided mode of the MNF[33] . Using high resolution spectroscopy, Sague et al. experimentally investigated the interaction of a small number of cold cesium atoms with the guided mode and ﬁber surface of a subwavelength-diameter silica MNF[34] , and observed light-induced dipole forces, van der Waals interaction and a signiﬁcant enhancement of the spontaneous emission rate of the atoms due to the modiﬁcation of the vacuum modes by the MNF. More recently, by overlapping cold Cs-atoms in an MOT with a 400-nm-diameter silica MNF and observing ﬂuorescence photons through the MNF, Nayak and Hakuta experimentally demonstrated that single atoms could be detected using a subwavelengthdiameter MNF, and that single photons spontaneously emitted from the atoms could be guided into a single-mode optical ﬁber that connected the MNF through a tapering region[35] . These possibilities that bridge atom physics and ﬁber-optic technologies, may open new approaches for manipulating a single atom or a single photon, and provide a new perspective for quantuminformation technologies.

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7.3 Other Applications Apart from the above-mentioned ﬁelds, a number of new possibilities have been reported recently. For example, using a piece of waveguiding silica MNF as a tiny and high-sensitive probe, She et al. reported a direct observation of the inward push force on the free end face of the MNF exerted by the outgoing light[36] . The simple and elegant experiments oﬀered direct evidence for supporting the Abraham’s momentum, resolving a century-old mystery in optics. Using low-dimensional and high-uniformity silica MNFs as templates, Zhang et al. fabricated nanochannels with widths ranging from 100 to 900 nm and lengths up to several millimeters[37] , and established a simple, fast and cost-eﬀective method for nanochannel fabrication in microﬂuidic chips. More applications such as using two dissimilar MNFs for tunable slow and fast light in coupled structures[38] , and using a light-driven bend eﬀect in MNFs for a low-power all-optical switch[39] , have also been reported very recently. As an excellent research platform merging ﬁber-optic technology and nanotechnology, there is no doubt that MNF will continue to open up new opportunities in broad areas including micro- and nanoscale photonics, nonlinear optics and quantum optics in the near future. This promises to go beyond the scope of this book, incorporating new technology and creativity.

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31. 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 (vol 72, pg 032509, 2005), Phys. Rev. A72, 049904 (2005). 32. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. Le Kien, V. I. Balykin, K. Hakuta, Optical nanoﬁber as an eﬃcient tool for manipulating and probing atomic ﬂuorescence, Opt. Express 15, 5431–5438 (2007). 33. F. Le Kien, S. D. Gupta, K. P. Nayak, K. Hakuta, Nanoﬁber-mediated radiative transfer between two distant atoms, Phys. Rev. A72, 063815 (2005). 34. G. Sague, E. Vetsch, W. Alt, D. Meschede, A. Rauschenbeute, Cold-Atom Physics Using Ultrathin Optical Fibers: Light-Induced Dipole Forces and Surface Interactions, Phys. Rev. Lett. 99, 163602 (2007). 35. K. P. Nayak, K. Hakuta, Single atoms on an optical nanoﬁbre, New J. Phys. 10, 053003 (2008). 36. W. L. She, J. H. Yu, R. H. Feng, Observation of a Push Force on the End Face of a Nanometer Silica Filament Exerted by Outgoing Light, Phys. Rev. Lett. 101, 243601 (2008). 37. L. Zhang, F. X. Gu, L. M. Tong, X. F. Yin, Simple and cost-eﬀective fabrication of two-dimensional plastic nanochannels from silica nanowire templates, Microﬂuid. Nanoﬂuid. 5, 727–732 (2008). 38. L. Shi, X. F. Chen, L. A. Xing, W. Tan, Compact and tunable slow and fast light device based on two coupled dissimilar optical nanowires, J. Lightwave Technol. 26, 3714–3720 (2008). 39. J. H. Yu, R. H. Feng, W. L. She, Low-power all-optical switch based on the bend eﬀect of a nm ﬁber taper driven by outgoing light, Opt. Express 17, 4640–4645 (2009).

Index

Symbols V -number, 17, 20

A absorption absorption cross sectional area, 162 absorption cross-section, 165 excited state absorptions, 162 molecular absorption, 189, 190, 192 pump absorption, 163, 173 two-photon absorption, 218, 223 adiabatic adiabatic approximation, 34 adiabatic MNF deformations, 33 adiabatic perturbation theory, 31 adiabatic solution, 34 adiabatic taper, 40, 217 air-cladding, 15 all-optical signal processing, 219 all-optical switch, 222, 224 analytical solution, 15, 37, 65 angled endface, 57 atom optics, 7 atomic ﬂuorescence, 196, 213 atoms trapping, 220 axial direction, 3 azimuthal direction, 22

B backward reﬂectance, 57 bandwidth, 112, 142, 162 beam propagation method (BPM), 37 bend-to-fracture process, 103, 104, 107 Bessel function

Bessel function of the ﬁrst kind, 16 blue-detuned light, 220 Boltzman constant, 111 boundary condition, 15, 31, 41, 44 Bragg grating, 11 broadening, 218

C capillary wave, 110–113, 123 carbon dioxide laser, 76, 77 cavity knot cavity, 175 laser cavity, 172 resonant cavity, 164, 172 cesium atom, 12, 13 chalcogenide, 219, 223 coherence time, 220 coil resonator, 64, 188, 213 core diameter, 3, 4 coupled MNFs, 43, 45 Coupled-mode theory, 43 coupler micro-coupler, 138, 140 X-coupler, 136, 137 coupling coeﬃcient, 30–32, 62 coupling eﬃciency, 44–53 coupling equation, 159 coupling factor, 161, 165 coupling length, 141 coupling loss, 53 coupling region, 62 coupling strength, 157 coupling wave equations, 30

226

Index

critical coupling, 206, 207 critical diameter, 17, 24 cross angle, 136 cross section, 81, 90, 91, 190, 192, 202, 212 cross sectional area, 162–164 cross talk, 24 crosstalk, 156 cylindrical coordinates, 28 cylindrical core, 3 cylindrical ﬁber, 15 cylindrical symmetry, 126

D demultiplexing, 219 detector, 78, 187 diameter ﬂuctuation, 25 diameter variation, 81, 196–198, 212 dielectric waveguide, 42 diﬀraction, 53, 57, 205, 206 diﬀraction limit, 99, 100, 179 dispersion dispersion relation, 68, 69 dispersion shift, 10, 179, 223 engineerable dispersion, 217 group velocity dispersion, 218 manageable dispersion, 6 material dispersion, 27, 28, 216 negative dispersion, 28 positive dispersion, 217 Sellmeier-type dispersion, 17 total dispersion, 215, 217 waveguide dispersion, 27, 28

E eﬀective coupling, 50 eﬀective diameter, 196 eﬀective index, 25 eﬀective length, 78 eﬀective MNF radius, 113, 117, 118 eﬀective mode area, 215, 217 eﬀective nonlinearity, 216–218 eﬀective propagation constant, 68 eigenvalue equation, 16 electrostatic attraction, 130, 138, 139 electrostatic attractive force, 152, 155, 171 endface output pattern, 10, 53, 54, 71 endface reﬂection, 55

enhanced nonlinearity, 218, 219 erbium doped, 11 evanescent evanescent coupling, 6, 7, 47, 48, 50–53, 193 evanescent ﬁeld, 2, 3, 9, 35, 71, 95, 187–189, 206–208, 212, 214 evanescent wave, 214

F ﬁber doped ﬁber, 165, 172 ﬁber array, 75 ﬁber taper, 75, 81, 95, 190, 193, 195, 197, 203, 212 glass ﬁber, 75, 78 silica ﬁber, 84, 88 tapered ﬁber, 59, 84 weakly guiding ﬁber, 28 ﬁber coupler, 88 ﬁber laser, 165, 174, 176, 184 ﬁber optics, 2 ﬁeld intensity, 39, 40, 208 ﬁlter add-drop ﬁlter, 11, 151, 154, 156–158, 182, 183 ﬁlter eﬀect, 152, 154 MNF ﬁlter, 151 short-pass ﬁlter, 11, 126, 151–156, 182, 183 ﬁnesse, 64, 144, 146, 156, 157, 183 ﬁnite-diﬀerence time-domain (FDTD), 44 ﬂuorescence, 189, 197, 221, 224 ﬂuorescent dye, 91 focal circumference, 35, 36 focal circumferences, 35, 38, 40 focal point, 35, 38 Fourier transform, 32 free-spectral range, 139, 154, 172, 176, 183 fundamental mode, 24, 28, 30, 41, 48, 109, 116, 182, 187, 190, 217, 220 fused silica, 28

G gain coeﬃcient, 161 gain saturation, 160 Goos-Hanchen shift, 3

Index

227

group velocity, 25, 26, 69

numerical solution, 9, 71

H

O

height distribution function, 111 height-height correlation function, 111, 112 Helmholtz equation, 16

OH absorption, 76 optical conﬁnement, 127, 128, 131, 133, 134, 215, 219 optical coupler, 125, 135–137 optical ﬁlter, 151 optical interferometer, 123, 181 optical resonator, 10, 72 optical sensor, 7, 24, 96, 124, 125, 138, 146, 151, 180, 187–190, 194, 202, 203, 208, 212–214 optical splitter, 135 overlapping length, 47, 49, 50

I index proﬁle, 15 intensity coupling coeﬃcient, 160, 161, 166 intensity distribution, 40, 59, 120, 133, 223 interferometer, 60, 125, 135, 180, 181, 198

P K Kerr nonlinearity, 219, 223

L Laser-heated taper drawing, 83 linear waveguide, 125–127 longitudinal component, 29 Lorentzian nonuniformity, 33

M Mach-Zehnder interferometer, 138, 154, 181 material absorption, 42 Maxwell’s equations, 15, 16, 44, 71, 81 melting temperature, 81, 110, 111 microbending, 99 micromanipulation, 101, 105–107, 132, 134, 135, 137–139 modal diameter, 25 mode ﬁeld diameter, 216 Modiﬁed Bessel function Bessel function of the second kind, 16 multi-valued function, 35

N nanotechnology, 1, 7, 122, 123, 222 nonlinear eﬀect, 215, 217, 218 nonlinear ﬁber optics, 216, 222 nonlinear index, 215 nonlinear parameter, 215 numerical, 20, 21, 37, 39, 44, 54, 112, 160, 165 numerical simulation, 158, 165, 173

parabolic equation method, 29 paraxial approximation, 29 perturbation theory, 33 PMMA, 93, 94, 127, 202 Polarization, 144 polarization, 9, 44, 61, 65, 151, 154, 183, 223 potential barrier, 41 Poynting vector, 7 propagation constant, 17, 29, 60, 61, 63, 112, 113, 151, 160

Q quality factor, 201, 206 quality factor (Q-factor), 147, 164, 185 quantum eﬃciency, 159, 163–170

R radial direction, 24 radiation loss, 25, 28, 35, 37, 39, 40, 42, 45, 53, 120, 128, 133 radiation mode, 30 rate equation, 158, 162 red-detuned light, 220 refractive index, 4, 11, 18, 19, 29, 63, 64, 68, 119, 127, 128, 130–133, 138, 140, 141, 146, 149, 150, 161, 164, 170, 171, 176, 209–212, 217, 219 resonance spectrum, 65 ring laser, 159, 165, 167, 170, 174, 182, 183 ring resonator, 11, 144, 156, 158–162, 178, 182, 183, 214

228

Index

S Sagnac interferometer, 135, 141, 142 scalar wave equation, 28 scanning electron microscope, 5, 81, 85, 88–91, 93, 100–106, 108, 113, 114, 131, 134, 136, 137, 146 scanning near-ﬁeld optical microscopy, 113 scattering loss, 131, 157, 159, 166 Schr¨ odinger equation, 29 self modulation, 80 self phase modulation, 218, 219 self-compression, 218 self-coupling, 64, 141–143, 151, 198, 199 self-modulated, 80 self-regulating, 76, 77 semiclassical approximation, 34 single-mode condition, 17, 19, 121 single-mode cutoﬀ diameter, 48 single-mode ﬁber, 8, 174, 215, 219 softening temperature, 80, 88 spatial gradient, 220 spatial steady state, 15 spontaneous emission, 162, 172, 197, 200, 221 step-index-proﬁle, 3 stimulated Raman scattering, 218 strain, 107 supercontinuum, 2, 120, 121, 127, 139, 141, 152, 153, 217, 218, 222, 223 supercontinuum generation, 123, 179, 218, 223 surface contamination, 109, 115, 116, 132, 190, 196 surface roughness, 109–113, 123 surface tension, 111

tellurite MNF, 51–55, 140, 141, 157 tensile strength, 90, 105, 107, 108 three-level system, 159, 162 total internal reﬂection, 1, 3 transfer length, 45, 47, 49 transmission amplitude, 62, 63, 65, 66 transmission electron microscope, 81–83, 90, 91, 179 transmission loss, 31, 42, 64, 88, 109, 110, 112, 113, 115–118, 189 transmission spectrum, 66, 110, 117, 119, 121, 127, 139–143, 148–152, 159, 193, 196, 197, 200, 201, 203, 205–208, 210–212 transversal propagation constant, 20, 29, 33, 36–38, 41, 42

V van der Waals attraction, 106 van der Waals force, 138, 139

W waveguide waveguide bend, 103, 125, 126, 133, 134, 180 waveguide dispersion, 125, 216 waveguide loss, 42, 116 weakly coupled system, 43, 45 weakly coupled waveguide, 45 weakly guiding condition, 29 whispering gallery mode, 149, 183, 184, 196, 203, 212

Y ytterbium doped, 158, 170

T

Z

taper-drawing, 74, 75, 77–79, 219

zero dispersion, 216