Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication 9819915244, 9789819915248

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Optical Wireless Communication Theory and Technology

Xizheng Ke

Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication

Optical Wireless Communication Theory and Technology Series Editor Xizheng Ke, School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, Shaanxi, China

The book series Optical Wireless Communication Theory and Technology aims to introduce the key technologies and applications adopted in optical wireless communication to researchers of communication engineering, optical engineering and other related majors. The individual book volumes in the series are thematic. The goal of each volume is to give readers a comprehensive overview of how the theory and technology in a certain optical wireless communication area can be known. As a collection, the series provides valuable resources to a wide audience in academia, the communication engineering research community and anyone else who are looking to expand their knowledge of optical communication.

Xizheng Ke

Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication

Xizheng Ke School of Automation and Information Engineering Xi’an University of Technology Xi’an, Shaanxi, China

ISSN 2731-5967 ISSN 2731-5975 (electronic) Optical Wireless Communication Theory and Technology ISBN 978-981-99-1524-8 ISBN 978-981-99-1525-5 (eBook) https://doi.org/10.1007/978-981-99-1525-5 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Optical-wireless communication refers to the use of light waves as information carriers to transmit information in free space between two or more terminals. It has the advantages of a high information-transmission rate and large communication capacity. Space optical–fiber coupling is one of the key technologies in optical-wireless communication. However, it is also a difficulty. This book consists of seven chapters. We review the progress of space light–fiber coupling research and highlight its significance. The basic theory and key issues are analyzed, and methods to improve the space light–fiber coupling efficiency are elaborated. Chapter 1 introduces the structure and development status of an optical-wireless communication system. Chapter 2 describes the optical-fiber structure and the optical-field mode in the fiber; Chap. 3 analyzes lens single-mode-fiber coupling under ideal conditions and calculates the fading of the coupling efficiency caused by assembly errors and non-common optical-path errors. Chapter 4 analyzes spatial plane-wave lens single-mode fiber coupling in weakly turbulent atmospheres and theoretically and experimentally analyzes the spatial optical coupling of lens arrays in atmospheric turbulence. Various methods are presented to improve the spatial light–fiber coupling efficiency, including mode conversion and adaptive-optics wavefront correction. Chapter 5 presents the design of an automatic alignment system, compares and analyzes different control algorithms, and discusses experimental research. Chapter 6 presents theoretical analyses and experimental research on the mode-conversion method. Chapter 7 analyzes the influence of wavefront distortion on coupling efficiency and proposes a method using adaptive optics to improve the coupling efficiency. This book is the result of collective research by the Optoelectronic Engineering Technology Research Center of Xi’an University of Technology. Many graduate students, such as Wu Jiali, Yang Shangjun, Luo Jing, Zhang Xutong, and Li Mengru, participated in the research on this topic and the compilation of this book, and the author appreciates their hard work and dedication. The author consulted many documents and materials in the process of writing this book. He would like to pay high

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tribute to the authors of these documents and materials. Their work inspired and helped the author and thank them for their efforts for the Institute of Science! He is grateful for their contributions to the scientific research community and appreciates their hard work. The publication of this book was funded by the National Natural Science Foundation of China (61377080; 60977054), the Shaanxi Provincial Key Industry Innovation Chain Project (2017ZDCXL-GY-06-01), and the Xi’an Science and Technology Project (2020KJRC0083). This book is a summary of the author’s research related to space light–fiber coupling. Owing to the author’s knowledge level, errors and deficiencies will inevitably occur in the book. Readers are welcome to critique and correct them. This book was completed during the author’s tenure as honorary professor of liberal arts college. I would like to express my gratitude to liberal arts college for its support to the author’s work. Xi’an, China Spring 2022

Xizheng Ke

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Background and Significance . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Optical Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Advantages of Spatial Light–Fiber Coupling . . . . . . . . . . . . . 1.2 Development Status of Optical-Wireless Communication Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Current Foreign-Development Status . . . . . . . . . . . . . . . . . . . . 1.2.2 Domestic Development Status . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Spatial Light–Fiber Coupling Technology . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Research Progress Abroad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Domestic Research Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Spatial Light–Fiber Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Optical Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Hermitian–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Laguerre–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Spatial Light–Fiber Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 6 6 6 8 9 9 13 23 23 26 27 29 31 32

2 Fiber-Optic Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fiber Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Basic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Inverted Parabolic Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Wave-Equation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Light-Wave Propagation Mode in Optical Fiber . . . . . . . . . . . . . . . . . 2.3.1 Vector Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Scalar-Mode Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 39 41 42 43 44 48 48 56

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2.3.3 Normalized Operating Frequency . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Coupling Efficiency of Gaussian Modes [15] . . . . . . . . . . . . . 2.4 Mode Effective Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Effective Refractive Index of a Vector Mode . . . . . . . . . . . . . 2.4.2 Effective Refractive-Index Difference Between Modes . . . . 2.4.3 Dispersion Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Ideal Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Plane-Wave Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Geometrical-Optics Analysis of Coupling Efficiency . . . . . . 3.1.2 Mode-Field Analysis of Coupling Efficiency . . . . . . . . . . . . . 3.1.3 Coupling Efficiency of Lens End Face . . . . . . . . . . . . . . . . . . 3.2 Coupling-Efficiency Decline Caused by Assembly Error . . . . . . . . . 3.2.1 Radial Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Axial Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Axis Tilt Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Adaptive-Optics System Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Calibration Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fitting Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Measurement-Noise Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Bandwidth Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Non-common Path Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Research Status of Non-common Optical-Path-Aberration Calibration . . . . . . . . . . . . . . . . . . . . 3.4.2 Generation of Non-common Optical-Path Aberrations . . . . . 3.4.3 Conversion of Non-common Optical-Path Aberrations . . . . . 3.5 Gaussian-Beam Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Experiment of Manually Eliminating Non-common Optical-Path Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Experiment of Automatically Eliminating Non-common Optical-Path Aberrations . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 63 63 64 64 67 68 69 71 71 72 74 78 79 79 81 83 85 85 86 86 88 89 89 90 91 92 92 93 95 97 98

4 Spatial Plane-Wave Single-Lens Single-Mode Fiber Coupling in Weakly Turbulent Atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 Light-Field Distribution and Refractive-Index Power Spectrum in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.1 Born Solution for the Light-Field Distribution in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Contents

4.1.2 Rytov Solution for the Light-Field Distribution in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Refractive-Index Power-Spectrum Model . . . . . . . . . . . . . . . . 4.2 Lens Coupling in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . 4.2.1 Coupling-Efficiency Model Under the Kolmogorov Turbulence Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Coupling-Efficiency Model Under the Von Kármán Turbulence Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Comparison of the Coupling Efficiency Under the Kolmogorov and Von Kármán Turbulence Spectra . . . . . 4.2.4 Coupling Efficiency of an Oblique-Range Transmission Under the Von Kármán Turbulence Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Relative Fluctuation Variance of the Lens-Coupling Light Power in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Relative Undulation Variance of Single-Lens Single-Mode Fiber-Coupling Power in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Effect of Coupling Efficiency and Coupling-Power Jitter Variance on the BER of a Wireless-Optical Communication System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spatial Optical Coupling of Lens Arrays in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Coupling Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Automatic Fiber-optic-coupling Alignment System . . . . . . . . . . . . . . . . 5.1 Auto-alignment Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Principle of the Auto-alignment System . . . . . . . . . . . . . . . . . 5.1.2 Automatic-alignment System Components . . . . . . . . . . . . . . . 5.1.3 Piezoelectric Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Principles of Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Basic Principles of Simulated-annealing Algorithms . . . . . . 5.2.2 Flow of the Simulated-annealing Algorithm . . . . . . . . . . . . . . 5.2.3 Simulated-annealing Algorithm Features . . . . . . . . . . . . . . . . 5.2.4 Stochastic Parallel Gradient-descent Algorithm . . . . . . . . . . . 5.2.5 Simulation of Different SPGD-algorithm Parameters . . . . . . 5.3 Effect of Alignment Errors on the Efficiency of Spatial Optical-fiber Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Alignment Error and Coupling Efficiency . . . . . . . . . . . . . . . . 5.3.2 Radial, End-face, and Axial Errors . . . . . . . . . . . . . . . . . . . . .

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127 128 130 134 135 136 139 139 139 140 142 142 142 143 145 146 148 151 151 155

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5.4 Two-dimensional Auto-alignment Experiments . . . . . . . . . . . . . . . . . 5.4.1 Piezoelectric-ceramic and Fiber-fixing Method . . . . . . . . . . . 5.4.2 Two-dimensional Alignment Experiments . . . . . . . . . . . . . . . 5.5 Five-dimensional Auto-alignment Experiments . . . . . . . . . . . . . . . . . 5.5.1 Piezoelectric-ceramic Combinations and Methods of Fixing Them to Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Analysis of Experimental Results . . . . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 158 162

6 Mode-conversion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Research Status of Mode Transformations . . . . . . . . . . . . . . . . . . . . . 6.2 Basic Mode-conversion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Spatial Phase-modulation Mode Conversions . . . . . . . . . . . . . . . . . . . 6.3.1 Conversion from High-order Mode to LP01 Mode . . . . . . . . 6.3.2 Conversion-efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Mode-conversion Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Mode Conversion Based on the Simulated-annealing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Comparison of Mode-conversion Effects . . . . . . . . . . . . . . . . 6.5 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Mode-conversion Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Coupling-efficiency Experiment . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 173 176 179 180 181

7 Adaptive-optical Wavefront Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Zernike Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Influence of Wavefront Distortion on Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Power in the Barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Strehl Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Wavefront Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Wavefront Correctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Simulation Analysis and Experimental Research . . . . . . . . . . . . . . . . 7.3.1 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 196 197

162 164 168 169

181 185 187 187 191 193

199 201 201 203 207 213 213 215 225

Chapter 1

Introduction

1.1 Research Background and Significance Optical-wireless communication uses light waves as carriers for high-speed information transmission [1]. Optical signals propagate through free space without waveguides. Optical-wireless communication combines the advantages of microwave and optical-fiber communications and expands the application field of optical communication [2]. The optical-wireless communication system model is shown in Fig. 1.1. It mainly includes optical-wireless communication terminals, optical antennas (telescopes) [1], lasers, signal-processing units, and acquisition, pointing, and tracking (APT) systems. The transmitter light source is a laser diode (LD) or light-emitting diode (LED), and the receiver mainly uses a PIN or avalanche photo diode (APD). Spatial light–fiber coupling is a key technology in optical-wireless communication systems.

1.1.1 Transmitter Some form of information [3] (such as time-varying waveforms, digital symbols, etc.) generated by the source is modulated onto an optical carrier, and the optical carrier (called a light beam or light field) is emitted into free space through an antenna, which is called a transmitter. The transmitter includes source coding, modulation, channel coding, optical-signal amplification, and a transmitting antenna. Channel coding adds redundant symbols [4] to the source data stream to enable error judgment and correction at the receiving end. Improving communication reliability is a basic task of channel coding. The essence of channel coding is to increase the information-transmission reliability; however, it reduces the useful information data-transmission rate, owing to the addition of redundant information [5].

© Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_1

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1 Introduction

Source

Channel coding

Direct modulation (OOK,PPM, etc.) Active modulation Indirect modulation (External Passive modulation modulation)

Laser direct modulation output Semiconductor optical amplifier Optical amplifier Optical fiber amplifier

Single antenna transmission Multi-antenna transmission

Atmospheric channel Outdoor scattering channel UV scattering channel Underwater channel

Direct modulation Accomm odation

Channel decoding

Demodu lation

Fiber coupling detection Coherent detection

Pream plifier

Space optical coupling module

Automatic tracking and aiming system

Single antenna reception Multi-antenna reception

Fig. 1.1 Composition of a wireless-optical communication system [1]

Modulation is the process of transforming a signal. It consistently changes certain characteristic values (such as the amplitude, frequency, or phase) of the opticalcarrier signal according to the coded signal (this is determined by the source signal itself). Modulation places the source onto the optical carrier, which is convenient for transmission in an optical channel; hence, the optical carrier carries the relevant source information [6, 7]. Modulation can be divided into active modulation and passive modulation. If the light source and modulation signal are at the same transmitting end, it is active modulation;If the light source and modulation signal are not at the same end, it is passive modulation; also known as reverse modulation. Controlling the laser-drive power for modulation is direct modulation; modulating the beam or light field emitted by a laser is called indirect modulation; also known as external modulation. If the communication distance is large, the optical power directly output by the laser is insufficient. An optical amplifier is used to amplify the optical signal. Optical amplifiers include semiconductor optical amplifiers and fiber amplifiers. Transmitting antennas include multi-antenna transmission/multi-antenna reception, single-antenna transmission/multi-antenna reception, etc. Multiple (single) antenna transmit/multiple (single) antenna receivers can suppress the influence of atmospheric turbulence [8].

1.1.2 Receiver The receiver includes an optical-signal-collecting antenna, spatial light–fiber coupling unit, preamplifier, detector, and demodulator [8, 9]. The receiving antenna collects the optical signal sent by the transmitter, the spatial light–fiber coupling unit couples the optical signal collected by the receiver into the optical fiber, and the optical-fiber detector converts the photoelectric signal. Energy is lost when an optical signal is coupled into the fiber.

1.1 Research Background and Significance

3

Sometimes, the signal coupled into the fiber is very weak and needs to be amplified before a photoelectric conversion. This amplifier is called a preamplifier. Signal detection includes direct detection by a detector, spatial light–fiber coupling detection, distributed detection, and coherent detection. Direct detection is when the optical detector directly receives the optical signal collected by the antenna. Spatial light–fiber coupling detection is when the space light is coupled into the fiber, and the signal in the fiber is detected by the photodetector. Because the fiber end face is small, the photoelectric converter has a small photosensitive area, and the required optical signal intensity is also small; thus, the spatial light–fiber coupling detection rate is high, and the detection sensitivity is also high. In coherent detection, the signal light and the local oscillator light are mixed in a mixer to amplify the signal light, generally with a gain of 20–23 dB [4]. This book mainly discusses spatial light–fiber coupling technology in wireless-optical communication systems.

1.1.3 Optical Antenna The main optical antenna of a wireless-optical communication system is generally a telescopic system [10]; these include the Newtonian, Kepler, and Cassegrain systems. Among them, the double-reflection Cassegrain system is most commonly used in optical-wireless communication optical systems [11, 12]; it has no chromatic aberration and can easily achieve large-aperture reception. Refractive optical systems are generally suitable for optical-wireless communication in the visible and near-infrared bands, while reflective systems are suitable for all wavelengths. Figure 1.2a–c represent the Newtonian, Gregory, and Cassegrain reflective optical systems, respectively. Reflective optical systems can have multiple focal points, and thus, multiple different relative apertures, field angles, and focal lengths. Commonly used reflective optical systems include the Newtonian, Gregory, and Cassegrain systems. A Newtonian optical system has a relatively large aperture and is often used in optical systems with large apertures; however, the production cost is high, and it produces a coma aberration for off-axis light. The Gregory system can simultaneously eliminate spherical and chromatic aberrations; however, the production process has high requirements, and there are few practical applications. The Cassegrain optical system includes the traditional Cassegrain optical system, the Dall-Kirkham (DK) optical antenna, the Schmidt optical antenna, and the Ritchey-Chrétien optical antenna. Its characteristics are as follows: 1. Achromatic, long focal length, using a wide spectral range, 2. When using an aspherical lens, it has a strong aberration capability; It has a simple optical structure and excellent image quality, which have been widely used in optical-wireless communication systems.

4

1 Introduction Imaging plane

Primary mirror

Secondary reflector

Focus

Primary mirror Second mirror

(a) Newtonian optical system

(b) Gregory optical system

Secondary reflector Primary mirror

Light blocking area Focus Primary mirror focus

(c) Cassegrain optical system Fig. 1.2 Reflective optical systems

As shown in Fig. 1.3, the objective lens at the front of a refractive optical system is a set of cemented lenses. After the light passes through the objective lens, it is imaged before the eyepiece, which enlarges it, and then the image point is received. The refracting telescope has the advantages of a wide field of view, high contrast, and clarity; however, the objective lens causes the refracting telescope to have chromatic aberrations, and the larger the aperture, the more serious the aberration. In addition, the length of the lens barrel determines the focal length, which makes the refracting telescope bulky. Objective lens

Eyepiece Incident light Eyepiece

Correction lens Secondary mirror Primary mirror Imaging

(a) Refractive optical system Fig. 1.3 Refractive and folded optical systems

(b) Folded optical system

1.1 Research Background and Significance

5

The folded optical system combines the objective lens of a refracting optical system with a reflective optical system. The objective lens is a wavy correction lens with a convex center, concave periphery, and complex shape to correct the spherical aberration of the main spherical surface. Catadioptric optical systems mainly include the Schmidt-Cassegrain antenna, Maksutov-Cassegrain antenna structure, and Fresnel optical-lens antenna structure. The Schmidt-Cassegrain antenna structure is named after the Schmidt astronomical observatory and uses a Schmidt correction lens to correct spherical aberrations. In the Cassegrain antenna structure, a convex mirror is used as the secondary reflector of the antenna system. It has many deformable structures, such as double-spherical mirror structures, double-aspherical mirror structures, and a combination of spherical and aspherical mirrors. A typical feature of the Maksutov-Cassegrain (Mak-Cass) antenna is that the secondary reflector is very small, usually an aluminized circular spot on the correction mirror. The catadioptric structure reduces the size of the Mak-Cass antenna while retaining a large aperture and long focal length. When the Mak-Cass antenna is used as a spatial-light coupling system, it can create a large-diameter coupling and receiving technology. It has a strong ability to gather optical power. The focal length is long, and it can easily meet the numerical-aperture requirements of optical fiber. Using the same relative aperture (D/f), the length of the ordinary lens system can be reduced by nearly half. The Fresnel optical antenna is based on the Cassegrain antenna structure, and replaces the refraction-correction lens at the front with a Fresnel lens. Compared with a conventional correction lens, the Fresnel lens has advantages of a strong focusing ability, short focal length, thin lens, and light weight. Its optical efficiency exceeds 90%, which is suitable for most optical-system front-ends with a small field of view and high gain. The advantages and disadvantages of the three structural forms are compared in Table 1.1. Table 1.1 Comparison of refracting, reflective, and reentrant structures Optical system Advantages

Disadvantages

Refractive

Wide field of view; high contrast, high Serious chromatic aberration; pore definition size has a great impact on performance; bulky and bloated

Reflective

Multi-focus, different numerical High production cost, complex apertures, and field of view; eliminate production process; coma aberration spherical and chromatic aberrations; exists in off-axis light wide applicable spectral range; simple optical structure and excellent image quality

Folded

Eliminates refractive chromatic, reflective spherical, and other aberrations; the focal length is longer in the same volume

Correction-lens production is difficult and expensive; the overall assembly and calibration of the antenna are complex and the production cost is high

6

1 Introduction

1.1.4 Advantages of Spatial Light–Fiber Coupling Spatial light–fiber coupling reception technology has become a key research topic in wireless-optical communication systems. It is a receiving method that couples the optical signal into the fiber at the transmitting end and then performs optical amplification, photoelectric detection, and other processes at the receiving end of the fiber. Coupling the signal light into the optical fiber for detection can bring many conveniences to a wireless-optical communication system [13]. 1. The cross-section of the fiber core is small, and a photoelectric conversion can be performed by a detector with a small photosensitive area. The inter-junction capacitance of the detector decreases with a reduction in the photosensitive area, and a higher response rate can theoretically be obtained. 2. The signal-beam characteristics can be controlled in the fiber [14, 15]. For example, erbium-doped fiber amplifiers (EDFAs), semiconductor optical amplifiers, and other devices can be used to amplify the optical power of the signal beam. An optical-fiber polarization controller is used to control the polarization characteristics of the signal beam. 3. Spatial light–fiber coupling can promote the modularization of wireless-optical communication systems and improve the system interchangeability [16, 17]. In general, the photoelectric detector of a wireless-optical communication system converts the signal at the focus of the main antenna, which is subject to strict optical calibration. If the photodetector fails, the entire receiving optical path must be recalibrated after the device is replaced. Wireless-optical communication systems that use spatial light–fiber coupling technology can avoid such problems.

1.2 Development Status of Optical-Wireless Communication Technology Owing to the serious impact of atmospheric turbulence on wireless-optical communication, its development was stagnant in the 1970s [18]. With the continuous progress in laser and detection technology since the 1980s [18], various countries have begun to study wireless-optical communication.

1.2.1 Current Foreign-Development Status In 1880, Bell invented the “optical telephone,” which was the beginning of modern wireless-optical communication. The space-technology-research vehicle 2 (STRV2) [14, 19] experimental-program terminal was funded by the US Ballistic-Missile Defense Organization (BMDO), designed and developed by Astro Terra, and launched at the end of 1998 as a satellite-to-earth wireless-optical communication

1.2 Development Status of Optical-Wireless Communication Technology

7

platform. The design required the stored data to be downloaded from the STRV2 at a bit rate of 155 Mbps, and the uplink data is re-downloaded at a bit rate of 155–1240 Mbps. The optical-communication demonstrator (OCD) [20] laser-communication demonstration system is an experimental platform jointly designed by the National Aeronautics and Space Administration (NASA) and Jet Propulsion Laboratory (JPL). Ground experiments were carried out in May 1998, air–ground experiments began in 1999, and experimental demonstrations were carried out at the International Space Station in 2002. Its goal was to establish a Gbps-level laser-communication downlink from the International Space Station to the ground and to make wireless-optical communication equipment miniature and lightweight, with a high bit rate and low cost. During the development of the OCD platform, NASA and the JPL jointly carried out an experimental study of a 45-km near-Earth optical link [21]. The OCD platform is located at Strawberry Peak in the San Bernardino Mountains, California, USA; it transmits 840-nm laser signals and simultaneously receives 780-nm laser signals from the other end. At the other end is the 0.6-m-aperture telescope at the JPL’s Table Mountain Facility in Wrightwood, California. This experiment was mainly used to determine whether the fading of the average received power measured by each end was within the uncertainty range predicted by the link analysis. The lunar laser-communication demonstration (LLCD) [22, 23] is a moon–Earth wireless-optical communication-demonstration platform developed by NASA. They established a 20-Mbps optical-communication link, which is the longest opticalcommunication link (400,000 km) thus far, and the communication speed is six times that of a traditional radio-frequency link. The semiconductor inter-satellite link experiment (SILEX) [24–28] was carried out under the auspices of the European Space Agency (ESA). It is a comprehensive technology-research and system-experiment project involving the lasercommunication link between the tropospheric emission-monitoring internet service (TEMIS) (geosynchronous equatorial orbit) and the satellite pour l’Observation de la Terre (SPOT 4) (low-Earth orbit) satellite orbits, and is the most representative wireless-optical communication system in Europe. Its purpose is to demonstrate laser communication in a space orbit, and then transmit Earth-observation data from SPOT 4 to the TEMIS satellite, which in turn transmits the data to the ground through its Ka-band channel. The communication distance is 45,000 km. The communication rate from SPOT 4 to TEMIS is 50 Mbps, and the communication rate from TEMIS to SPOT 4 is 2 Mbps. The inter-orbit optical-communication engineering experimental test satellite OICETS [29–31] was developed by the National Space Development Agency of Japan (NASDA) to verify a space laser in an experimental satellite for communication-link technology. It cooperates with the ESA’s SILEX project to conduct intersatellite optical-link experiments. In May 2006, a satellite-to-ground laser-communication experiment was successfully implemented at the National Institute of Information and Communication Technology (NICT) ground station. The laser-utilizing communication equipment (LUCE) laser-communication terminal

8

1 Introduction

carried by OICETS has a communication rate of 49.3 Mbps and a communication distance of 600–1500 km. In 2000, Lucent and Astro Terra [32] successfully realized a wireless-optical communication system with four 1550-nm wavelengths, 10-Gbps wavelengthdivision multiplexing, and a 4.4-km transmission distance. Lucent’s use of opticalfiber amplifiers enabled 20–160 Gbps data communication within a 200-m communication distance. Terabeam [32] used wireless-optical communication equipment for image transmission in the 2000 Sydney, Australia, Olympic Games and used wireless-optical communication equipment to provide a 100-Mbps data link to customers in the Four Seasons Hotel in Seattle, Washington, USA. In 2015, the MB2000 radio-frequency free-space optical (RF-FSO) dual-band wireless-communication product produced by AOptix [33] was also commercialized between Hong Kong’s surrounding islands and Puerto Rico. The communication rate of this product could reach 2 Gbps, with a maximum communication distance of 10 km. In addition, 147-km wireless-optical communication between the Hawaii and Maui islands was also conducted.

1.2.2 Domestic Development Status “Fenghuo” was the beginning of simple optical communication in ancient China. Since the beginning of the twenty-first century, China has gradually conducted research on key technologies for inter-satellite, satellite-to-ground, and terrestrial wireless-optical communication links. Jiang [34] studied satellite-to-ground lasercommunication technology. In October 2011, his satellite-to-ground communication terminal was mounted on the “Ocean-2” satellite, and a satellite-to-ground lasercommunication experiment was carried out. The transfer rate reached 504 Mbps [35, 36]. Most domestic achievements regarding terrestrial optical-communication links and key technologies are still in the research stage. In 2002, the Chengdu Institute of Optoelectronic Technology at the Chinese Academy of Sciences [37] developed a wireless-optical communication terminal with a 10-Mbps transmission rate and a 1– 4-km communication distance. In 2006, the Guilin Institute of Laser Communication [38, 39] completed a 10-km wireless-optical communication system with rates of 155 and 622 Mbps. Yong et al. [40, 41] conducted research on atmospheric laser communication and the capture and tracking of beacon lights. In 2008, a ground-simulation experiment with an automatic-tracking servo system for wireless-optical communication was completed, and in 2011, a 4.6-km wireless-optical communication link was realized on a road [42], using dense wavelength-division multiplexing to complete 5-Gbps data transmissions. In 2013, Jiang Huilin [43, 45] successfully completed a long-distance laser-communication experiment between aircraft and conducted useful exploration research in deep-space optical communication [44].

1.3 Spatial Light–Fiber Coupling Technology

9

Since 2000, Ke Xizheng et al. have conducted research on the transmission characteristics of light in atmospheric turbulence [46, 47], optical design [48], codingmodulation technology [49, 50], coherent-detection technology [4, 51, 52], etc. Since 2018, an 8-kg wireless-optical communication-system terminal was successfully developed. The terminal has a standard Ethernet data-access capability and can reliably transmit duplex, voice, and image data. It has a 1.25-Gbps maximumtransmission rate, 3–10-km transmission distance, and a bit-error rate better than 10–6 under light haze conditions. In 2018, a 100-km coherent optical-communication experiment was conducted.

1.3 Spatial Light–Fiber Coupling Technology Spatial light–fiber coupling technology can process an optical signal in an optical fiber after coupling the light into the optical fiber to improve the interchangeability of the system [1]. Therefore, spatial light–fiber coupling technology is one of the key technologies for wireless-optical communication systems. However, the diameter of a single-mode fiber core is very small, which increases the difficulty of coupling spatial light into the fiber. Moreover, the aberrations and atmospheric turbulence of an optical system will cause angular fluctuations, static angular deviation of the fiber, and positional error of the boresight between the fiber and the spatial light, all of which will affect the fiber coupling efficiency [53]. Improving the efficiency of coupling spatial light into an optical fiber is an important guarantee for improving the reliability and effectiveness of wireless-optical communication and has very important practical value.

1.3.1 Research Progress Abroad (1) Astronomy With the rapid development of free-space optical (FSO) communication, spatial light–fiber coupling technology has gradually achieved certain research results. At the end of the twentieth century, most research and applications of spatial-light singlemode fiber coupling technology were in the field of astronomy. Since then, spatial light–fiber coupling technology has been applied to the field of light detection and ranging (LiDAR). In 1988, Shaklan introduced the problem of coupling starlight into a single-mode fiber and analyzed the average coupling efficiency. Owing to a mismatch between the Airy-disk mode and near-Gaussian mode of the fiber, the maximum efficiency when the LP11 mode was cut off was 78%. When D/r0 = 4, (D: coupling-lens diameter; r0 : atmospheric coherence length) the maximum total coupling power is achieved [54].

10

1 Introduction

In 1998, Winzer et al. derived the coupling-efficiency expression of monochromatic light coupled to a fiber after passing through a lens, and extended it to the fields of coherent and incoherent-fiber laser radar. They optimized the numerical size of the receiving aperture and obtained a maximum coupling efficiency of 42% [55]. (2) Spatial light–fiber coupling auto-alignment Spatial light–fiber coupling automatic-alignment technology is a key issue in coupling-technology research, and scholars have further improved the fiber-coupling efficiency. In 1990, Boroson et al. proposed an active coupling scheme based on fiber nutation applied in the field of laser communication. When the optical signal is received, the single-mode fiber acts as a position-error sensor. According to the coupling optical power, the fiber-position error is estimated by changing the position, and then the error is fed back to the position-control motor to correct the fiber position, such that the coupling loss is reduced to only 2 dB [56]. In 2001, Sayano et al. conducted an experiment on the multichannel multiplexing of optical-code-division multiple access. They coupled the spatial light into a singlemode fiber, using a position sensor to detect the fluctuation of the incident-beam angle, and used a fast-tilt mirror for error compensation. The coupling efficiency was approximately 50% and the coupling loss was approximately 3 dB [57]. (3) Lenses Because microlenses can change the spot size and add control algorithms, lens technology is also widely used to improve the coupling efficiency of fiber coupling. In 1995, Modavis et al. established a Gaussian theoretical model based on laser and fiber fields, analyzed the coupling performance of a deformed fiber microlens between the laser diode and single-mode fiber, and the average coupling efficiency of the lens is measured to be 78% through experiments [58]. In 2001, Ruilier et al. theoretically analyzed the spatial-light single-mode fiber coupling received by a large-aperture telescope and carried out numerical simulations. The results showed that, compared with the traditional theory of the aperturesmoothing effect, the phase fluctuation has a significant effect on the use of large-aperture telescopes. The achieved fiber-coupling efficiency was less affected [59]. In 2002, Sherman M.P et al. designed a wireless-optical communication system utilizing a Ritchey-Chrétien (RC) telescope with aspherical mirrors. The mirror configuration provided a larger focal plane that allowed n × n fiber arrays to be positioned in the focal plane of the RC optical telescope, enabling point-to-multipoint communication with a single optical telescope [60]. In 2002, Oswald W. et al. calculated the coupling efficiency of coherent planewave coupling into a single-mode fiber. Depending on the relationship between the position of the fiber end face and the focal point of the lens, the coupling efficiency of the collimation system can reach 61% [61]. In 2010, Danuel V. H. et al. designed an optical-fiber array as a coupling method for receiving optical signals and used a piezoelectric ceramic driver to drive the

1.3 Spatial Light–Fiber Coupling Technology

11

microlens array to compensate for tilt error. The experimentally measured output optical power increased by 39 dB, and the fiber-coupling efficiency improved [62]. In 2018, Hottinger et al. designed a single-mode fiber-coupled angle sensor with a microlens, which could adjust the position of the fiber by sensing the light energy in the fiber and coupling and aligning the spatial light and the fiber [63]. (4) Multimode fiber The small core diameter of single-mode fiber increases the difficulty of spatial light coupling into the fiber. Therefore, to advance single-mode fiber coupling, scholars have studied multimode fibers, few-mode fibers, and specially designed fiber coupling to improve the coupling efficiency. In 1997, Keming D. et al. studied a technique for coupling the output of a highpower diode laser into a multimode fiber, and the results showed that the coupling efficiency of 20 W of continuous laser power from a laser diode to the fiber was 71% [64]. In 2007, Horton et al. studied the effect of diffraction on the fiber-coupling efficiency of single- and few-mode fibers. The results showed that few-mode fibers offered higher maximum-coupling efficiencies (>90%) than single-mode fibers and that few-mode fibers were less sensitive to obstructions in the telescope pupil than single-mode fibers [65]. In 2013, NASA conducted a lunar-laser communications demonstration (LLCD); the downlink communication rate of the system was 622 Mbit/s, and the uplink communication rate was 20 Mbit/s, using pulse-position modulation (PPM). The ground-receiving end uses a multimode polarization-maintaining fiber to couple the spatial light, and the coupling efficiency of the spatial light to the multimode fiber can reach 92% under weak turbulence conditions [66, 67]. In 2014, Carl M. et al. proposed a device for spatial light-single-mode fiber coupling. It consists of a fiber cone that acts as a mode filter to filter out higherorder modes and then couples the remaining modes into a single-mode fiber with a power loss of only 2.4 dB [68]. In 2019, Vanani et al. established a theoretical coupling model based on the imaging characteristics of free-space beam mode and the inner product between free-space beam mode and few-mode fiber (FMF) mode. Numerical coupling results for mode fibers revealed that to maximize the overall coupling efficiency, the dimensions of the higher-order free-space beam and fiber modes should be approximately matched, and there is an optimal value for the normalized frequency V value of the FMF [69]. (5) Erbium-doped fiber amplifiers An erbium-doped fiber amplifier was added to the front of the spatial-light singlemode fiber-coupling module to improve the coupling efficiency. In 1994, Salisbury M. S., McManamon P. F., Duncan B. D. added an erbium-doped fiber amplifier (EDFA) in front of the spatial-light single-mode fiber-coupling module of a LiDAR system. Compared to traditional PIN detection, the signal-to-noise ratio was improved by 36 dB. The efficiency was increased by 20% [70].

12

1 Introduction

In 2002, Smolyaninov et al. used a 1550-nm laser to conduct a lasercommunication experiment with a 2-km link, and the transmission rate reached 1.2 Gbit/s. The EDFA was used as a receiving preamplifier coupled to a single-mode fiber. The experiment showed that errors occurred at a certain time. The code rate did not exceed 10–4 [71]. (6) Adaptive optics The degradation of the beam quality by atmospheric turbulence also directly affects the efficiency of coupling spatial light to the fiber. Most scholars use adaptive-optics systems that can further improve the fiber-coupling efficiency by compensating for wavefront distortion and correcting the light wave on the coupling end face to a Gaussian-distribution plane wave. In 1998, Ruilier et al. theoretically deduced an analytical formula for the coupling efficiency of a monochromatic-wave–single-mode fiber under atmosphericturbulence conditions, and discussed methods of using adaptive-optics technology to improve the coupling efficiency [72]. In 2002, Weyrauch et al. conducted research on adaptive-fiber coupling using deformable mirrors. The optical power coupled into the fiber was used as an index to measure and correct the experimental system, which mainly used the stochastic parallel gradient-descent (SPGD) algorithm. Finally, the maximum coupling efficiency of single-mode fiber and multimode fiber reached 60% and 70%, respectively [73]. In 2005, Dikmelik et al. conducted an experimental analysis on the influence of atmospheric turbulence on the spatial-optical coupling efficiency, and deduced the relationship expression between the atmospheric-turbulence parameters and the optical-fiber coupling parameters under weak atmospheric turbulence. They calculated that when the coupling efficiency is less than 5%, the communication distance can reach approximately 100 m in strong turbulence; in medium turbulence, the communication distance can reach approximately 800 m; in weak turbulence, the communication distance is more than 1 km [74]. In 2006, Toyoshima et al. analyzed the optimal numerical relationship between the Airy-disk size and the fiber-mode field size in a focusing system in the presence of random angular jitter. The results showed that when the ratio of normalized random angular jitter to the mode-field radius was greater than 0.3, the system average biterror rate (BER) dropped from 10–1 to 10–4 [75]. In 2008, Fidler et al. studied the effects of atmospheric turbulence in different scenarios, including geostationary satellites (GEO), high-altitude platforms (HAP), and optical ground stations (OGS). The effect of phase distortion on the coupling efficiency of a laser beam into a single-mode fiber demonstrated that by correcting for the tilt component, GEO-to-HAP communication could approach diffraction-limited performance [76]. In 2010, Hiderki T et al. established a simulation model of the fiber-coupling efficiency of an Earth-to-satellite laser-communication link in atmospheric turbulence. The fiber-coupling loss exceeded 10 dB under atmospheric turbulence [77].

1.3 Spatial Light–Fiber Coupling Technology

13

In 2012, Takenaka H et al. studied a fast mirror for high-frequency operation under atmospheric-turbulence conditions and verified its spot-position tracking performance in a satellite-to-ground laser-communication experiment. It was experimentally determined that the coupling-efficiency fading in the satellite-to-ground laser-communication link was between 10 and 19 dB, which is consistent with the theoretical value of 17 dB [78]. In 2016, two ground-station optical systems in NASA’s laser-communications relay demonstration (LCRD) project used adaptive optics to compensate for atmospheric turbulence and couple beams into single-mode fibers. The adaptive-optics system adopted a double deformable-mirror design, which corrects the beam coupling caused by low-frequency large-amplitude disturbances and high-frequency small-amplitude disturbances. Experimental results showed that the average coupling efficiency could exceed 50% under general atmospheric conditions [79, 80]. In 2019, Carrizo et al. corrected the wavefront phase by iteratively updating the phase of a single focal-plane speckle to maximize the power coupled to a singlemode fiber. Experimental results showed that a power gain of approximately 4 dB was achieved in less than 60 power-measurement iterations under strong turbulence conditions [81] (The research progress abroad is shown in Table 1.2).

1.3.2 Domestic Research Progress (1) Atmospheric turbulence In terms of spatial light–fiber coupling technology, domestic scholars have conducted in-depth research and made great progress. In the field of space optical communication, owing to the complexity of the transmission channels, atmospheric turbulence also has a significant impact on space-optical coupling. In 2006, Xiang et al. studied the coupling efficiency of spatial-optical coupling to a single-mode fiber. The fluctuation of the optical power was analyzed by considering the influence of turbulent-intensity scintillation, aperture smoothing, turbulent wavefront distortion, and coupling-system tracking errors. The average coupling efficiency and coupling power fluctuation of the uplink and downlink were calculated, and the analysis showed that the influence of the uplink on the optical-coupling efficiency was small. For the downlink, the relative power fluctuation was first due to the aperture average effect. It decreased and then gradually increased, owing to the influence of turbulent wavefront distortion [82]. In 2009, Ma et al. studied the influence of the arrival-angle fluctuation caused by atmospheric turbulence on the fiber-coupling efficiency, and obtained the optimal value of the design parameters of a fiber-coupling optical system in the presence of random jitter; that is, the ratio of the aperture radius to the fiber-mode field radius β = 1.121. The coupling efficiency had a maximum value of 81% [83]. In 2011, Zhao et al. studied the effect of alignment error and random angular jitter on the coupling efficiency of a space laser to a single-mode fiber and obtained the

14

1 Introduction

Table 1.2 Research progress in foreign countries Year

Person/organization

Research progress

1988

S. Shaklan

Coupling starlight into single-mode fiber

1990

Lincoln Laboratory

Active coupling scheme based on fiber nutation for laser communication

1994

Wright and Patterson

An erbium-doped fiber amplifier is added before the fiber-coupling module to improve the coupling efficiency

1995

R. A. Modavis

Built theoretical models of scalar-diffraction and Gaussian-mode shapes for laser and fiber fields

1997

K. Du

A technique for coupling the output of a high-power diode laser rod into a multimode fiber

1998

P. J. Winzer

Optimized the size of the receiving aperture to improve the coupling efficiency

1998

C. Ruilier

Analytical formula for monochromatic-wave single-mode fiber coupling efficiency under atmospheric turbulence

2001

K. Sayano

Building a multi-channel multiplexing optical-code division multiple-access experiment

2001

C. Ruilier

Spatial-light single-mode fiber coupling received by large-aperture telescopes

2002

O. Wallner

Using a collimation system can increase the coupling efficiency

2002

T. Weyrauch

Deformable mirrors are investigated for adaptive fiber coupling

2002

I. Smolyaninov

EDFA is coupled into a single-mode fiber as a receiving preamplifier

2005

Y. Dikmelik

Coherent fiber arrays improve the fiber-coupling efficiency

2006

M. Toyoshima

Spatial optical coupling efficiency is influenced by atmospheric turbulence

2007

A. J. Horton

Effect of diffraction on the fiber-coupling efficiency of single-mode fiber and few-mode fiber

2008

F. Fidler

Influence of wavefront distortion in different scenarios on coupling efficiency

2010

H. Takenaka

Maximum fiber-coupling efficiency in the presence of random angular jitter

2010

D. Hahn

Adding algorithms to microlenses to improve coupling efficiency

2012

H. Takenaka

Compensating for jitter error to improve fiber-coupling efficiency

2013

NASA

Lunar–Earth laser-communication demonstration experiment

2014

C. Weinert

Adding tapered fiber between multimode and single mode improves coupling efficiency

2016

NASA

Compensating for atmospheric turbulence and coupling the beam into a single-mode fiber using adaptive optics

2018

P. Hottinger

Fiber-coupled alignment for spatial light using an angle sensor (continued)

1.3 Spatial Light–Fiber Coupling Technology

15

Table 1.2 (continued) Year

Person/organization

Research progress

2019

A. Fardoost

Calculating the numerical coupling results of six-mode fiber and ten-mode fiber

maximum average coupling efficiency by optimizing the coupling parameters; the maximum coupling efficiency measured experimentally was 65% [84]. In 2016, Zheng et al. studied the coupling efficiency of spatial light and fewmode fiber under the action of a turbulent atmosphere and conducted experiments to simulate atmospheric turbulence indoors using a liquid–crystal light modulator. The results showed that the coupling efficiency of two-mode and four-mode fiber was increased by 4 dB and 7 dB, respectively, compared with single-mode fiber under the same experimental conditions [85]. In 2018, Liu et al. comprehensively discussed the influence of the radial offset on the coupling efficiency from random angular jitter and fiber-alignment error. Through simulations and experiments, they found that the maximum coupling efficiency of the system was 62% and the radial offset capacity was 62%. The difference was 1.52 μm [86]. In 2021, Song et al. established a calculation model of instantaneous coupling efficiency disturbed by atmospheric turbulence and obtained the minimum dimensionless parameters that affect the statistical distribution of the fiber-coupling efficiency. It was found that with an increase in the ratio of the aperture radius to the atmospheric coherence length, the efficiency maximized at a certain point [87]. (2) Adaptive optics Scholars have used adaptive-optics systems composed of deformable mirrors, wavefront sensors, and controllers to suppress the influence of atmospheric turbulence. By compensating for wavefront distortion, the light waves on the coupling end face are corrected to plane waves, which can further improve the efficiency of the fiber coupling. In 2010, Wu et al. combined adaptive-optics technology and coherent fiber-array technology to improve the spatial-optical coupling efficiency. The results showed that this method could increase the efficiency of coupling the signal light into the fiber from 3 to 38%, after passing through atmospheric turbulence [88]. In 2011, Han et al. proposed a model-free blind-optimization wavefront-correction technique to improve the spatial-optical coupling efficiency under the influence of atmospheric turbulence. The results showed that the coupling efficiency of the singlemode fiber increased from 6% to approximately 60% [89]. In 2012, Yang et al. compensated the beam passing through atmospheric turbulence and adopted wavefront phase-mode compensation to make the coupling efficiency of a single-mode fiber close to 81% [90]. In 2013, Xiong et al. used a 37-unit adaptive-optics system to correct atmosphericturbulence aberrations. The results showed that correcting low-order aberrations could increase the coupling efficiency, and that correcting high-order aberrations

16

1 Introduction

could be further improved. The experimental results showed that the coupling efficiency increased by 20% when the Strehl ratio was increased from 0.16 to 0.35 [91]. In 2014, Han et al. conducted a theoretical analysis of the influence of satelliteplatform vibration on spatial light–fiber coupling efficiency and established a satellite-platform micro-vibration compensation system. The experimental results showed that the active compensation system of feedback-control technology could effectively suppress low-frequency vibrations. The coupling efficiency of the compensation system could be increased by 54.73% [92]. In 2014, Luo et al. studied the effect of a single aberration on coupling efficiency. When D/r0 (D: coupling-lens diameter, r0: atmospheric coherence length) is small compared with other aberrations, oblique aberration is the main influencing factor, and an adaptive fiber light source collimator (adaptive fiber- optic collimator, AFOC) for correction. The results showed that the average coupling efficiency improved from 30.07 to 61.72% [93]. In 2015, Li et al. developed an adaptive fiber coupler (AFC) that used the stochastic parallel gradient descent (SPGD) algorithm to perform closed-loop control under different atmospheric-turbulence intensities. The results showed that when optimal SPGD control parameters were selected, the fiber coupling efficiency increased from 40 to 76% [94]. In 2017, Zheng et al. studied the effect of using adaptive optics technology to compensate wavefront aberration on the coupling efficiency of space laser-few-mode fiber under moderate atmospheric turbulence. The results show that the coupling efficiency of single-mode fiber and few-mode fiber is improved by 16 dB and 11 dB, respectively, under moderate turbulence conditions [95]. In 2019, Li et al. designed a spatial light-single-mode fiber large-format spatial-light adaptive-coupling system, using a two-dimensional piezoelectric nanopositioning platform, a closed-loop control system composed of a control module, a driving module, a photodetector, and a coupling lens, and a raster-scanning algorithm, to achieve precise positioning and stable tracking of the optimal coupling point. The results showed that the coupling efficiency was improved by 10.6% during dynamic alignment [96]. In 2021, Jiang et al. designed a combination of the SPGD algorithm and a fewmode fiber coupling and demultiplexing system to compensate and correct the wavefront phase distortion caused by dynamic turbulence. The results showed that under different turbulence intensities and wind speeds, the coupling efficiency of the fewmode fiber was 0.5–1.5 dB higher than that of single-mode fiber without the SPGD algorithm correction. The fiber improved the efficiency by 0.4–2.2 dB [97]. (3) Automatic alignment Spatial light-fiber coupling automatic alignment is the process of finding the best coupling position to improve the fiber-coupling efficiency. In 2007, Gao et al. proposed a closed-loop control system composed of piezoelectric ceramics and controllers. After using raster scanning to determine the

1.3 Spatial Light–Fiber Coupling Technology

17

optimal coupling position, a five-point tracking method combined with precise onedimensional translational positioning was used to achieve automatic coupling. The experimental results showed that the method could automatically search for the best position, according to the optical power coupled into the fiber, in a relatively short time, and a maximum coupling efficiency of 59.2% could be obtained [98]. In 2016, Gao et al. proposed an automatic coupling scheme for a spatial lightsingle-mode fiber with laser nutation to reduce the effect of random angular jitter on the coupling efficiency. The experimental study shows that the coupling efficiency of the system is 67% without disturbance. After introducing disturbance and using the control system to compensate the disturbance, the coupling efficiency of the system is increased by 6.5% [99]. In 2017, Wu et al. proposed a coupling scheme based on the grating helical scanning algorithm and SPGD algorithm to improve the spatial light-fiber coupling efficiency. The scanning algorithm could effectively correct the initial alignment errors. After using the SPGD control algorithm, the random lateral offset between the focused spot and the single-mode fiber was corrected, and the coupling efficiency was effectively improved to 81% [100]. In 2019, Zhao et al. designed a nutation coupling algorithm based on a fast mirror combined with a fiber photodetector, built a video-transmission experiment with a 1.65-GHz code rate, and achieved a coupling efficiency of 59.63% under static conditions. The effectiveness and feasibility of the laser nutation algorithm for improving the coupling efficiency were verified [101]. In 2019, Li et al. designed a spatial-light-to-single-mode fiber coupling–optimization scheme based on laser nutation, which could realize an active single-mode fiber under excessive radial error using only a single fast mirror coupling [102]. In 2019, Qi et al. applied the mode-search method to the precise alignment process of an optical fiber and a light-receiving chip. The alignment time of the pattern-search method was within 25 s, and the alignment success rate improved to more than 90% [103]. In 2020, Wu et al. designed a single-mode fiber nutation-tracking coupling system in a fiber laser-communication system to reduce the equivalent effect of atmospheric turbulence on single-mode fiber coupling. The coupling efficiency was 53.5% [104]. In 2020, Zhao et al. studied the effect of the nutation and SPGD algorithm parameters on the stability and convergence speed of the algorithm to compensate for the spatial-light single-mode fiber coupling caused by random vibration of the satellite platform, resulting in a loss of efficiency. The simulation results showed that the average coupling efficiency after compensation by the nutation algorithm was 81.26%, the root-mean-square error was 7.2 × 10–4 ; the average coupling efficiency after compensation by the SPGD algorithm was 80.72%, and the root-mean-square error was 1.9 × 10–3 . Compared to the SPGD algorithm, the nutation algorithm has better stability [105]. (4) Semiconductor lasers When a semiconductor laser adopts the fiber-coupling output method, it not only simplifies the device application and improves the asymmetry of the output beam,

18

1 Introduction

but also easily couples the outputs of multiple semiconductor light sources to obtain a higher power output and fiber-coupling efficiency. In 1996, Wei et al. used a molded aspheric-lens coupling system to study the coupling between a semiconductor laser and single-mode fiber, and the coupling efficiency reached 33% [106]. In 1996, Zhang et al. measured the beam quality of semiconductor lasers, and collected, collimated, shaped, focused, and coupled them into the fiber. The results showed that the coupling efficiency could be as high as 85.7%, using a micro-cylindrical lens and fiber-head processing technology [107]. In 1999, Bo et al. used cylindrical lenses to effectively collect, pre-collimate, and couple the output beams of semiconductor lasers between multimode fibers. The results showed that the coupling efficiency with a 200-μm core-diameter plain-end fiber was as high as 90% or more with a 808-nm-wavelength transmitting unit [108]. In 2002, Lu et al. studied the coupling technology of adding a fiber microlens between a semiconductor laser and optical fiber, which achieved a coupling efficiency of 80% [109]. In 2004, Xu et al. proposed a calculation model using two semi-cylindrical lenses instead of hyperboloid lenses to pair a semiconductor-core laser with a 1 μm × 100 μm light-emitting area. A multimode fiber with a 50-μm diameter was used for coupling, and the total coupling efficiency was 74% [110]. In 2012, Zhang Lin et al. designed a dual-wavelength beam-combining system to combine 980-nm and 880-nm semiconductor lasers at two wavelengths; the combined beam was coupled into a multimode fiber with a 200-μm diameter and a 0.22-μm numerical aperture through a focusing lens that they designed, and the experimental results showed that the coupling efficiency was 78–82% [111]. In 2016, Shi et al. aimed to address the problems of low-coupling efficiency between semiconductor lasers and optical fibers, and a serious loss of light energy combined with the characteristics of the outgoing beams of semiconductor lasers. They used geometric optics to design aspheric lenses and cylindrical lenses and simulated them. The results showed that the coupling efficiency improved from less than 50–61% [111]. In 2017, Liu et al. applied space and polarization coupling technology to develop a high-power semiconductor-laser fiber-coupling module. The experimental results showed that the fiber output power was 234.6 W and the coupling efficiency was 60% [112]. (5) Laser-diode arrays Because the output power of a single-tube semiconductor laser is limited to the order of several watts, to obtain higher output power and higher coupling efficiency, scholars have investigated laser-diode arrays with multiple light-emitting units. In 2000, Shi et al. proposed a beam-shaping technology to realize high-power laser-diode line-array devices with microchip prism stacks, and then produced a fiber-coupling output. This technique coupled the laser output from a high-power laser-diode array into a 600-μm fiber with a total coupling efficiency greater than 50% [113].

1.3 Spatial Light–Fiber Coupling Technology

19

In 2001, Bao et al. used a cylindrical lens to effectively collect and pre-collimate the output beam of a 10-element-array semiconductor laser, as well as a coupling experiment with a multimode fiber. The laser used an emitting unit with an 808-nm wavelength, and the coupling efficiency with a 200-μm core diameter flat-end fiber array was as high as 75% [114]. In 2002, Wang et al. used a section of multimode fiber with a small numerical aperture as a microlens to couple the output beam of a laser-diode line array into a multimode fiber array. The results showed that the coupling efficiency and output optical power reached 75% and 15 W, respectively [115]. In 2004, Zhou et al. proposed a fiber-coupling method for high-power semiconductor-laser array beams that integrated beam collimation, shaping, focusing, and coupling. When the light from a high-power semiconductor laser with an 800-μm core diameter and a numerical aperture of 0.37 was coupled into the fiber, the experimental results showed that the coupling efficiency was greater than 53% [116]. In 2005, Xu et al. used a fiber cylindrical lens and a beam-conversion device to compress the divergence angle of a semiconductor laser-diode array (LDA) and coupled the laser beam into a microsphere lens fiber with a 400-μm core diameter through a focusing lens. The experimental results showed that the highest coupling efficiency was greater than 80% [117]. In 2010, Wang et al. used a stepped mirror group to shape the beam of an 880-nm high-power semiconductor-laser array and developed a high-stability, high-power fiber-coupled module. The results showed a coupling efficiency of 73.8% [118]. In 2015, Xu et al. developed a high-power, high-efficiency multiarray fibercoupled semiconductor-laser module using beam shaping and spatial combining methods. Experimental results showed that the maximum output optical power of the fiber could reach 327 W, and the fiber-coupling efficiency was greater than 93.6% [119]. (6) Errors In an actual spatial light–fiber coupling system, static errors will occur between the incident light wave and the fiber end face, which will also affect the spatial light–fiber coupling efficiency. In 2013, Kang et al. studied the theory of coupling parallel light beams into singlemode fibers, and analyzed the process of parallel-light coupling into single-mode fibers. The results showed that the degree of deviation between the mode matching of the incident light wave of the single-mode fiber and the central axis of the coupling lens reduced the light-coupling efficiency efficiency in the single-mode fiber. Finally, the simulation results showed that the coupling efficiency was significantly affected by the lateral and tilt error, whereas the defocusing effect was relatively small [120]. In 2013, Luo et al. analyzed the coupling efficiency of a Cassegrain system when it was collimated and off-axis. When the off-axis angle of the Cassegrain antenna

20

1 Introduction

was less than 0.055 rad, the coupling efficiency was greater than 80%. When the offaxis of the optical antenna system was 0.08 rad, the coupling efficiency was 69.26% [121]. In 2017, Fan et al. proposed that when the lateral offset of the fiber was 10 μm, 15 μm, and 17 μm, the average coupling efficiency of the multicore fiber was higher than that of the single-core fiber with the same core area by 14.4%, 39.6%, and 36.9%, respectively. When the optical fiber was axially offset by 0.1 mm, the coupling efficiency of the seven-core fiber was approximately 12.9% higher than that of the single-core fiber with the same core area. The results showed that a single-mode multicore fiber could well suppress tilting and defocusing [122]. In 2018, Wang et al. established a theoretical model of the spatial light–fiber coupling efficiency in a non-turbulent environment for two-mode fibers and analyzed the effect of the alignment deviation between the optical axes of the fiber and optical antenna on the coupling efficiency of the system. The results showed that when the lateral offset was 4 μm, the coupling efficiency of the two-mode fiber was 10.23% higher than that of single-mode fiber. When the axial offset was 125 μm, the coupling efficiency of the two-mode fiber is 11.24% higher than that of single-mode fiber. When the standard error of the random jitter amplitude was 5 μm, the coupling efficiency of the two-mode fiber was 12.1% higher than that of the single-mode fiber [123]. (7) Mode-field and phase matching To obtain a high coupling efficiency, it is necessary to simultaneously match the mode field and phase of a fiber lens and single-mode fiber. Therefore, scholars have used fiber-lens methods to achieve high coupling efficiency. In 2003, Zhao et al. established a theoretical model of the coupling between optical fibers and cone-end ball-lens fibers, and conducted coupling experiments with several lens fibers with different cone angles and spherical radii. A maximum coupling efficiency of 78.3% was obtained at 110° [124]. In 2014, Liu et al. studied the coupling of single-mode fibers and semiconductor lasers, and the results showed that making the fiber end face into a wedge-shaped microlens could enable the fiber and semiconductor-laser coupling to match the mode field and phase. The maximum experimentally obtained coupling efficiency was 81.36% [125]. (8) Microlenses Because a microlens can effectively converge a laser output beam, it can change the spot size and add control algorithms. Therefore, lens technology is widely used to improve the fiber-coupling efficiency. In 2003, Wei et al. proposed a method that used a telescope-collimation system to improve the coupling efficiency of laser fibers. Using a 1.6:1 telescope system enabled the coupling efficiency of the single-mode fiber to reach 70%, and the coupling efficiency of the polarization-maintaining fiber reached 67% [126]. In 2015, Wang Yanhong et al. used an inverted front-end optical-amplification system to compress the diameter of a composite beam and used a hexagonal array of

1.3 Spatial Light–Fiber Coupling Technology

21

microlenses as the coupling element to obtain a theoretically lossless high-efficiency fiber-coupling system. A hollow light pipe was used to further homogenize the lightfield distribution, reduce the divergence angle of the edge light, and improve the imaging quality of the edge light. The optimized system coupling efficiency reached 98% [127]. In 2021, Li et al. proposed a method that used a double Gaussian lens as a coupling lens to eliminate aberrations and improve the coupling efficiency. After simulation calculations, the coupling efficiency reached 82.07% [128]. (9) Mode conversions One of the key factors for improving the coupling efficiency is converting to a fundamental-mode form that can be collected and transmitted by the optical fiber. Moreover, the energy distribution is concentrated by the mode-conversion method. In 2015, Qi et al. used a spatial-light modulator (SLM) to convert the free-space optical (FSO) path mode from the LP01 mode to higher-order modes. This scheme uses SLMs, which have good repeatability and low requirements for other devices, and the system is simple and easy to implement [129]. In 2017, Tu et al. converted LP01 mode to LP11 and LP21 modes, based on a simple SLM structure, and coupled them into a few-mode fiber (FMF) for reception and transmission; however, the mode-conversion efficiency achieved by this scheme was low [130]. In 2021, Liu et al. designed single-mode-few-mode fiber (SMF-FMF) mode couplers and single-mode-hollow-core fiber (SMF-HCF) mode couplers for Ybdoped fiber lasers. The results showed that the SMF-FMF mode coupler could convert the LP01 mode to the LP11 /LP21 mode with a coupling efficiency greater than 96%, whereas the SMF-HCF mode coupler could convert the HE11 mode to the HE21 /HE31 mode with a coupling efficiency of over 82% [131]. (10) Structure optimization Scholars have also improved the coupling efficiency by optimizing and improving the optical-fiber structure. On this basis, the design structure is simple, easy to process, and has better anti-interference ability. In 1981, Li et al. adopted a method of coupling a ball-end optical fiber with a light-emitting tube. They set the curvature radius of the ball-end to 45 μm, the diameter of the light-emitting surface of the light-emitting tube to 50 μm, and the fiber-core diameter to 65 μm. When the numerical aperture of the fiber was 0.17 μm, the coupling efficiency was generally 6%, and the maximum value was 13% [132]. In 2011, Ouyang et al. used the gradient-index fiber-lens coupling method to couple semiconductor lasers to single-mode fibers with high efficiency, and on this basis, perfected the ABCD matrix theory of all-fiber coupling of semiconductor lasers. The results showed that the semiconductor laser could be coupled to the singlemode fiber with high efficiency using the focusing characteristics of the gradientindex fiber and selecting an appropriate length; the maximum coupling efficiency was 80.5% [133].

22

1 Introduction

In 2013, Hu et al. established a transmission model of a tapered multimode fiber, simulated the coupling efficiency and transmission mode of a laser in the tapered multimode fiber, and designed and conducted optical experiments. To address the annular light-spot distribution problem, a method of bending the cylindrical optical fiber connected to the output end at a certain angle was proposed. This method improved the light spot to a two-dimensional normal distribution, and the theoretical coupling efficiency was nearly 80% [134]. In 2019, Wang et al. designed a new type of fiber-coupling structure that achieved optimal matching between the focus position and the fiber head and improved the numerical aperture and offset fault tolerance of the fiber end face. The maximum coupling setting of the new fiber-coupling structure was adopted. The efficiency reached 60.3%, which was better than the maximum coupling efficiency of 36.7% of a traditional coupling structure under the same conditions [135]. In 2019, Yan et al. applied tapered fibers to spatial light–fiber coupling to improve its coupling efficiency. The results showed that the transmission efficiency of the tapered fiber was approximately 70%, with low-loss transmission characteristics and good filtering characteristics for matching back-end single-mode optoelectronic devices [136]. (11) Coupling-system optimization The coupling efficiency is improved by further optimizing and improving the coupling system. In 2011, Wang et al. designed a miniaturized optical antenna based on beam-spacediversity transmitting and receiving technology to alleviate the impact of atmospheric turbulence on communication links. When the communication distance of the system was 10 km and the communication rate was 52 Mbit/s, the receiving sensitivity is -35 dBm, and the single-beam transmit power was 14 dBm [138]. In 2014, Zhang et al. coupled free-space light into a single-mode fiber with a larger core size, and then spliced a single-mode fiber with a larger core and a single-mode fiber with a smaller core. Experimental results showed that the coupling efficiency of the improved coupled system was significantly improved, reaching more than 60% [137]. In 2018, to enhance the vibration resistance of spatial light–fiber coupling and improve the coupling efficiency, Hu et al. analyzed the coupling characteristics of space light-fiber in a micro-vibration environment and designed a new conical receiver; the results showed that the coupling efficiency decreased rapidly with an increase in the amplitude, when ordinary fiber was used as the receiver. When the amplitude increased from 0 to 280 μrad, the coupling efficiency decreased from 95 to 10%. When a new tapered fiber was used, the coupling efficiency was reduced from 95% to only 55% [139]. In 2018, Feng et al. proposed a new optical-antenna communication system that simulated the process of coupling a Gaussian beam to a multimode fiber through an optical element and discussed the effect of the coupling-spot offset and the position of the fiber-core diameter on the coupling efficiency. Theoretical calculations showed that when the inverses of the relative aperture of the lens were 0.3100 and 0.3700,

1.4 Spatial Light–Fiber Coupling

23

respectively, the maximum coupling efficiency of a 1310-nm laser was 81.45%, and the maximum coupling efficiency of a 1550-nm laser was 82.54% [140]. In 2021, Ren et al. proposed a front-end flange symmetrical structure with high coupling efficiency and high-temperature adaptability. After a simulation analysis, when the temperature field was 40 °C and the radical change of the focal point of the bottom-mounted structure was −7.32 μm, the coupling efficiency decreased by about 68%. When the radical change of the focal point of the front-end flange symmetrical structure was −1.06 μm, the coupling efficiency decreased by only 5% [141] (The domestic research progress is shown in Table 1.3).

1.4 Spatial Light–Fiber Coupling An optical fiber is a flexible, transparent fiber made of extruded glass or plastic, slightly thicker than a human hair. A single-mode fiber is an optical fiber that transmits optical signals directly in transverse mode. Typically, single-mode fiber is used for long-range signal transmissions. Multimode fiber is primarily used for short-distance fiber-optic communication, such as within buildings or campuses. There are two types of refractive index: graded and step. A few-mode fiber is an optical fiber with a core area sufficiently large to transmit parallel data streams using several independent spatial modes. Ideally, the capacity of a few-mode fiber is proportional to the number of modes. However, few-mode fiber amplifiers are required to extend the transmission distance. The spatial light–fiber coupling efficiency is directly related to the optical mode of the signal light.

1.4.1 Optical Modes There are many optical modes classified at different angles: spatial mode and temporal mode, linearly polarized (LPlm ) and transverse electric and magnetic (TEMmn ) mode, single mode and multimode, transverse mode and longitudinal mode, guided mode and leaky mode, fundamental mode and higher-order mode, packetlayer mode and resonant-cavity mode, Hermitian–Gaussian mode and Laguerre– Gaussian mode, no-cutoff single mode for photonic crystal fiber, wound fiber for mode filtering, mode hopping for single-frequency lasers, and mode locking for ultrashort pulse lasers. An optical mode is a state with a degree of freedom for photons. A mode is the spatial distribution of photons in a cavity or waveguide. A mode is a characteristic function that describes the characteristic problems of a physical system. A mode is a solution to the Maxwell/Helmholtz equations. Modes are the eigenstates of a system; if a system has discrete temporal, spectral, or spatial distributions, they can be called modes.

24

1 Introduction

Table 1.3 Domestic research progress Year

Person/organization Research progress

1981 Li et al.

Coupling with a ball-end fiber and light-emitting tube

1987 Zhou et al.

Single-mode fiber-coupling lens and light-source fiber-coupling device

1996 Zhang et al.

Micro-cylindrical lens and fiber-head processing technology

1999 Bo et al.

Experiment on the output beam of a semiconductor laser with a cylindrical lens

2000 Shi et al.

A microchip prism stack shapes the beam of high-power laser-diode line-array devices

2002 Lu et al.

Adding a fiber microlens between the laser and the fiber improves the coupling efficiency

2002 Wang et al.

The output beam of a laser-diode line array is coupled into a multimode fiber array

2003 Wei et al.

Improving the laser-fiber coupling efficiency with a telescope-collimation system

2003 Zhao et al.

Theoretical model for coupling a flat-end fiber and a taper-end ball-lens fiber

2004 Zhou et al.

High-power semiconductor-laser array beams improve coupling efficiency

2005 Tan et al.

Influence of different optical-fiber oblique-section angles on the coupling efficiency of semiconductor lasers

2006 Xiang et al.

Coupling efficiency of spatial light coupling to single-mode fiber under turbulence

2006 Wang et al.

A practical method for realizing semiconductor- laser and multimode-fiber coupling

2007 Deng et al.

Optimizing the relative aperture of the coupling lens to achieve a high coupling efficiency

2009 Ma et al.

Influence of arrival-angle fluctuation on the fiber-coupling efficiency

2010 Han et al.

Hybrid technology of adaptive optics and coherent fiber-array technology

2011 Han et al.

Model-free blind-optimization wavefront-correction technology improves coupling efficiency

2011 Zhao et al.

Effects of alignment error and random-angle jitter on the coupling efficiency of spatial single-mode fibers

2011 Chen et al.

Effect of the fiber offset on spatial-light single-mode fiber-coupling efficiency

2012 Zhang et al.

A wavelength-beam combining system was designed

2013 Zhang et al.

Coupling of parallel beams into single-mode fiber

2013 Xiong et al.

Adaptive optics to correct atmospheric-turbulence aberrations

2013 Luo et al.

Coupling efficiency of a collimated and off-axis Cassegrain system (continued)

1.4 Spatial Light–Fiber Coupling

25

Table 1.3 (continued) Year

Person/organization Research progress

2014 Hang et al.

Effect of satellite-platform vibration on space optical-fiber coupling efficiency

2014 Luo et al.

Influence of a single wave aberration on the coupling efficiency

2014 Liu et al.

Coupling of a single-mode fiber and semiconductor laser

2015 Liu et al.

Inverted front-end optical-magnification system

2015 Li et al.

Adaptive-fiber couplers improve the coupling efficiency

2015 Xu et al.

Array fiber-coupled semiconductor-laser modules

2016 Gao et al.

Automatic coupling scheme of laser nutation from space light to single-mode fiber

2016 Shi et al.

Design of aspheric and cylindrical lenses to improve the coupling efficiency

2016 Zheng et al.

Coupling efficiency of a spatial laser-few-mode fiber under turbulent atmosphere

2017 Fan et al.

Mathematical model of spatial light coupling into single-mode multicore fibers

2017 Zheng et al.

Influence of adaptive optics on spatial laser–few-mode fiber coupling efficiency in a turbulent atmosphere

2018 Liu et al.

Effects of random angular jitter and fiber alignment errors on coupling efficiency

2018 Liu et al.

High-brightness high-power fiber-coupling module

2018 Wang et al.

A theoretical model of spatial laser-fiber coupling efficiency without turbulence is established

2019 Zhao et al.

A combined fast-mirror and fiber-laser nutation algorithm improves the coupling efficiency

2019 Li et al.

Coupling-optimization scheme of space light to single-mode fiber for coarse-static composite scanning of laser nutation

2019 Li et al.

Spatial-light single-mode fiber large-format spatial-light adaptive-coupling system

2019 Wang et al.

New fiber-coupling structure

2019 Wei et al.

Applying the mode-search method to the precision alignment process of an optical-fiber and light-receiving chip

2020 Wu et al.

Single-mode fiber nutation-tracking coupling system

2021 Liu et al.

Single-mode–few-mode fiber-mode couplers and single-mode–hollow-core fiber-mode couplers

2021 Jiang et al.

Combining the stochastic parallel gradient descent algorithm with a few-mode fiber coupling demultiplexing system

2021 Ren et al.

Front-end-flange symmetrical structure with high coupling efficiency and high temperature adaptability

2021 Li et al.

Using a double Gaussian lens as a coupling lens to eliminate aberrations

26

1 Introduction

Fig. 1.4 Light-propagation modes in fiber

Schema

Primary mode

Secondary mode

The modes of a fiber are the lights that can be transmitted in the fiber, and each mode is a solution that satisfies the Helmholtz equation. Single-mode fiber can only transmit one type of light, that is, light parallel to the axis, whereas a multimode fiber can transmit light with multiple wavelengths. Different wavelengths, numerical apertures, and modes have different transmission paths, as shown in Fig. 1.4.

1.4.2 Hermitian–Gaussian Beams In a square-aperture confocal cavity or a square-aperture stabilized spherical cavity, in addition to the fundamental-mode Gaussian beam, a higher-order Gaussian beam can also be present, and the field distribution in its cross section can be described by the product of a Gaussian function and a Hermitian polynomial. A Hermitian– Gaussian beam propagating in the z-direction can be expressed in the following general form: ) (√ ) ) ] [ ( 2 r2 z 2 2 − ωr 2 −i k z+ 2R −(1+m+n)ar ctg f x Hn y ·e e ω ω (√ ) (√ ) ) ] [ ( 2 1 2 2 −i k z+ r2q −(1+m+n)ar ctg zf x Hn y ·e = Cmn Hm (1.1) ω ω ω

1 ψmn (x, y, z) = Cmn Hm ω

(√

In the above formula, C mn is a√ constant, ω is the beam radius from the beam center to the beam waist z, r = x(2 + y)2 , k is the( wave) vector, R is the beam√ √ wavefront curvature radius, and Hm ω2 x and Hn ω2 y represent m-order and n-order Hermitian polynomials, respectively. The difference between a Hermitian–Gaussian beam and a fundamental-mode Gaussian beam is that the transverse-field distribution of a Hermitian–Gaussian beam is determined by the product of a Gaussian function and a Hermitian polynomial. As shown in Fig. 1.5, a Hermitian–Gaussian beam has m nodal lines in the x direction and n nodal lines in the y direction; an additional phase lead along the transmission axis is relative to the geometric phase shift:

1.4 Spatial Light–Fiber Coupling

27

Fig. 1.5 Hermitian–Gaussian beams in different modes

z Δφmn = (1 + m + n)ar ctg . f

(1.2)

It can be deduced beam is in the x and y directions: { 2that a Hermitian–Gaussian wm = (2m + 1)w02 Waist radius: wn2 = (2n + 1)w02 { 2 wm (z) = (2m + 1)w2 (z) Spot radius at z: wn2 (z) = (2n + 1)w 2 (z) { √ θm = 2m + 1θ0 Far-field divergence angle: . √ θn = 2n + 1θ0 It can be seen from the above formula that the beam-waist size, spot size, and far-field divergence angle all increase with an increase in m and n.

1.4.3 Laguerre–Gaussian Beams In cylindrically symmetric (i.e., cylindrical coordinate) stable cavities (including circular-aperture confocal cavities), the field distribution in the cross section of each high-order Gaussian beam can be described by the product of the Gaussian function and the associated Laguerre polynomial: ψmn (r, ϕ, z) { )m ( 2 ) 2 ( ) ] [ ( r2 cos mϕ r Cmn √ r 2 −(1+m+n)ar ctg zf −i k z+ 2R − ωr 2 m . (1.3) 2 2 Ln 2 2 e ×e = ω ω ω sin mϕ

28

1 Introduction

In the above (r, ϕ, z) represents the cylindrical coordinates of the field ( formula, ) r2 m point and L n 2 ω2 represents the associated Laguerre polynomial. Compared to a fundamental-mode Gaussian beam, the transverse-field distribution of a higher-order Gaussian beam in a cylindrically symmetric system is described by the following function: ( 2) 2{ cos mϕ r r m . (1.4) L n 2 2 e− ω 2 ω sin mϕ As shown in Fig. 1.6, the Laguerre–Gaussian beam has n nodal circles along radius r and m nodal circles along argument ϕ. The additional phase shift of the Laguerre–Gaussian beam is z Δφmn = (1 + m + 2n)ar ctg . f

(1.5)

The Laguerre–Gaussian √ beam can be deduced in the x and y directions: Waist radius: wmn = 1 +√ m + 2nw0 . Spot radius at z: wmn (z) = 1 + m √+ 2nw(z). Far-field divergence angle: θmn = 1 + m + 2nθ0 . It can be seen that both the spot size and the beam-divergence angle increase with an increase in m and n, and the increase in n is faster than that of m. Fig. 1.6 Pull-down Laguerre–Gaussian beams in different modes

1.4 Spatial Light–Fiber Coupling

29

1.4.4 Spatial Light–Fiber Coupling An optical fiber is a cylindrical waveguide. The mode is the spatial distribution of an electromagnetic wave when the waveguide is transmitted. In fact, different spatial distributions of lasers of the same frequency can be propagated simultaneously in the same waveguide. A relatively basic mode is linearly polarized light; the two nearly perpendicular quasi-linearly polarized lights transmitted by a single-mode fiber are two different modes. The reason single-mode fiber avoids multiple modes is related to the inherent cutoff frequency of the cylindrical waveguide. Modes other than the fundamental mode will cause exponential decay in the fiber because they exceed the range of the transmittable mode, rather than following a sinusoidal function transmission. Thus, other modes, even if they exist, will soon disappear. (1) Numerical aperture As shown in Fig. 1.7, the numerical aperture (NA ) is a dimensionless number that measures the angular range of light that a system can collect. The numerical aperture describes the size of the cone angle of the light entering and exiting the fiber. Its size is determined by the following formula: N A = n × sinα, where n is the refractive index of the medium between the observed object and objective lens and α is half of the objective-lens aperture angle (2α). The aperture angle of an objective lens refers to the angle formed by the object point on the optical axis and the effective diameter of the front lens. The numerical aperture reflects the coupling efficiency between the fiber and light source. The numerical aperture describes the size of the cone angle of the light entering and exiting the fiber. (2) Mode-field diameter The width across which the light intensity decreases to 1/e2 of the peak value is the mode-field diameter. The beam maintains a nearly Gaussian intensity profile when f

Fig. 1.7 Numerical-aperture illustration

F

θ

D

30

1 Introduction

propagating along a single-mode fiber, and the width of the profile can be characterized by the mode-field diameter. The typical mode-field diameter is approximately 1.15 times the core diameter. The closer the incident light is to Gaussian light, the higher the coupling efficiency. A high coupling efficiency can be achieved if the incident light is Gaussian and the beam waist is equal to the fiber mode-field diameter. Substituting the mode-field diameter into the beam-waist diameter in the Gaussian-beam formula can accurately calculate the coupling parameters and divergence angles of single-mode fibers. The size of the mode-field diameter is related to the wavelength, and the mode-field diameter increases with the wavelength. (3) Single-mode fiber coupling Improving the coupling efficiency of a single-mode fiber requires that the beam waist of the incident Gaussian beam is located on the fiber end face, and that the beam-waist intensity and mode intensity coincide. The coupling efficiency can be reduced if the beam-waist diameter is not equal to the mode-field diameter, if the beam-intensity profile changes or deviates, or if it is not incident along the fiber axis. (4) Factors affecting the coupling efficiency of single-mode fiber Adjusting the angle, position, and intensity profile of the incident beam can improve the coupling efficiency of single-mode fiber. Assuming that the fiber end face is flat and perpendicular to the axis, a beam that meets the following conditions can achieve the highest coupling efficiency: • • • • •

Gaussian intensity profile Normal incidence from the fiber end face The beam waist is located on the fiber end face The center of the waist is aligned with the center of the fiber core The beam-waist diameter is equal to the fiber mode-field diameter.

If the laser emits only the lowest-order transverse mode, the output is a nearly Gaussian beam that can be efficiently coupled into a single-mode fiber. However, the coupling efficiency of multimode lasers or broadband light sources and singlemode fibers is very low; hence, even if they are focused onto the core region, most of the light will be leaked. This is because only a part of the light of the multimode light source matches the guided-mode characteristics of the single-mode fiber; thus, a multimode light source can provide a higher coupling efficiency with multimode fiber. (5) Fiber-acceptance angle The numerical aperture (N A ) of an optical system is a dimensionless number that measures the angular range of light that the optical system can collect. The relationship between N A and the maximum acceptance angle (θ max ) can be calculated using geometric optics, as shown in Fig. 1.8a. If the incident light is regarded as a ray, θ max represents the ability of the fiber to collect off-axis light. Light with an incident angle less than or equal to θ max will undergo a total internal reflection at

1.5 Summary

31 Fiber Guided Mode ρ

z

NA=no sin θmax

n 2 core

n 2 clad

Single mode fiber End face

Refracted light

TIR rays

I( ρ)

Gaussian beam

I(0)

Mode Strength Section

θmax

I(0)e–2

Core,ncore ρ

External Refractive Index,no

Wrapping,nclad

(a) Schematic diagram of fiber coupling cross-section

Waist Diameter

(b) Optical fiber coupling physical map

Fig. 1.8 Schematic diagrams of a fiber coupling

the interface between the core and the cladding and will be constrained to propagate forward in the core; light rays with an incident angle greater than θ max are eventually lost, owing to refraction. Only light rays incident from the air gap to the fiber end face at an angle of incidence of less than θ max can propagate. θ max is a spatial angle, if light is incident on the fiber end face from a tapered region limited to 2θ max . As shown in Fig. 1.8b, light with an incident angle ≤ θ max is coupled into the guided mode of a multimode fiber. In general, the smaller the angle of incidence, the lower the order of the excited fiber modes. Most of the energy is concentrated in lowerorder modes near the center, with normal incident rays exciting the lowest-order modes. Owing to the limitation of geometrical-optics analyses, N A is not the maximum incident angle of a single-mode fiber and cannot characterize the light-receiving ability of single-mode fibers. Only the lowest-order guided mode excited by the incident light at 0° exists in a single-mode fiber. Using N A to estimate the divergence angle of a single-mode fiber output will produce an inaccurate result. At this time, the beam diverges, owing to diffraction, and geometric optics does not consider this effect; thus, wave optics are needed. This book discusses how to maximize the optical power coupled into a single-mode fiber after a signal beam passes through atmospheric turbulence.

1.5 Summary The development direction of a spatial light–fiber coupling system in wireless-optical communication includes the following aspects.

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1 Introduction

1. A more compact and stable piezoelectric-ceramic composite connection structure was developed to improve the adjustment accuracy and stability of the fiber position [142]. 2. Research was conducted on realizing the automatic alignment of spatial-optical coupling in turbulent environments with different intensities and improving the fiber-coupling efficiency in turbulent environments [134]. 3. A more precise, convenient, and effective adjustment mechanism and method were designed to achieve a better-integrated transceiver antenna [51]. 4. This study considers the simultaneous existence of multiple free-space optical modes and converts multiple modes simultaneously to achieve a higher fibercoupling efficiency [51]. 5. A feedback-optimization algorithm with faster operation was investigated to realize free-space-optical mode conversions more conveniently and improve the fiber-coupling efficiency [143].

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Chapter 2

Fiber-Optic Mode Theory

This chapter describes optical-fiber mode theory, presenting theoretical analyses and deriving formulas for the fluctuation equation, vector modes, normalized cutoff frequency, and coupled mode theory of optical fibers. The solutions to the characteristic equations of the four vector modes and the scalar mode, that is, the line-bias mode, are derived. In addition, the effective refractive index, that is, the relationship between the normalized operating frequency and the number of vector modes, is calculated and compared.

2.1 Fiber Optics 2.1.1 Basic Structure In 1966, Kun G published a paper entitled, “Dielectric fiber-surface waveguide for optical frequencies,” in the Proceedings of the Institution of Electrical Engineers, which theoretically analyzed the possibility of using optical fiber as a transmission medium, and predicted the possibility of manufacturing ultralow-consumption optical fiber for communication. In 1970, Marell, Capron, and Keck of Corning, Inc., which specializes in glass, ceramics, etc., successfully developed an optical fiber with a transmission loss of only 20 dB/km, using a modified chemical-phase deposition method. In 1977, the world’s first fiber-optic communication system was commercially used in Chicago, Illinois, USA, with a rate of 45 Mbit/s. In 1979, Zhao Zisen developed the first practical optical fiber in China and is known as “the father of Chinese optical fiber.” An optical fiber is made of glass or plastic and serves as a light-transmission tool. The transmission principle is ‘total reflection of light’. Most optical fibers must be wrapped in several layers of protective materials before they can be used, and the wrapped cable is known as a fiber-optic cable. The outer protective and insulating © Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_2

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layers of the fiber protect it from the surrounding environment, such as water, fire, and electric shock. Single-mode fiber cores are 8–10 μm in diameter, and the core is surrounded by a glass envelope with a lower refractive index to retain the light inside the core. The outer layer is a thin plastic jacket to protect the envelope. Optical fibers are typically tied into bundles and protected by an outer jacket. The core is usually a double-concentric cylinder with a small cross-sectional area made of quartz glass, which is brittle and prone to breakage; therefore, it requires an additional protective layer. In a multimode fiber, the core diameter is 50–62.5 μm. Figure 2.1 shows the three basic optical-fiber structures. The fiber shown in Fig. 2.1a is a cylindrical dielectric waveguide that connects discrete components; Fig. 2.1b shows a step-index fiber, which is the most basic and widely used fiber in daily applications. The refractive index of the core is higher than that of the cladding from the cross section of the fiber, and the refractive indices of both parts are stepped. Figure 2.1c shows a gradient refractive-index fiber, in which the refractive index from the core to the cladding gradually decreases, and the refractive index of the core is gradually distributed. Optical fiber has become increasingly popular in the field of optical communication, and unlike common plastic optical fiber, the material of optical fiber for optical communication is mainly high-purity quartz (SiO2 ) with a purity of parts-per-million (ppm) magnitude. It has the advantages of low transmission loss, large bandwidth, and light weight. To improve the flexibility and aging resistance of the optical fiber, the outer layer of the fiber is coated with epoxy resin or silicone rubber [1].

x

r b

b a

r

a

ᵩ z

0 a

r n2

b n1

a 0 a

n(r)

b

n2 n1

n(r)

b

y

(a) Coordinate system

(b) Step distribution

(c) Parabolic distribution

Fig. 2.1 Schematic diagram of the basic structure of an optical fiber [1]

2.1 Fiber Optics

41

2.1.2 Inverted Parabolic Fiber In 2014, Wang et al. [2, 3] utilized a core refractive-index gradient-type fiber, which satisfied the high refractive-index and mode-field gradients. In 2015, Huang et al. [4] introduced a high-refractive-index layer into a circular fiber to increase the effective refractive-index difference between the core and cladding, which effectively suppressed intermode crosstalk and optimized the transmission performance of the circular fiber structure. To produce a high-quality vortex beam in a fiber, the structure must have high refractive-index and mode-field gradients to effectively separate the vector modes. The inverted parabolic-distribution type of optical fiber has a sharp refractiveindex difference because of its structure. A low-refractive-index layer is added between the core and cladding, which further increases their refractive-index difference, to accommodate a larger number of modes. The refractive-index distribution of an optical fiber can be expressed as follows:

n(r ) =

⎧ / / 2 2 ⎪ ⎪ ⎨ n 1 1 − 2N (r rcor e ) 0 ≤ r ≤ rcor e (2.1)

n 2 rcor e < r ≤ r2 ⎪ ⎪ ⎩ n 3 r > r2

In Eq. (2.1), n 1 and n 2 are the refractive-index values of the core center (r = 0) and the low-refractive-index layer (r ≤ r2 = 5 μm), respectively. The radius of the inverted-parabolic refractive-index distribution layer is rcor e = 3 μm, and n 3 is the refractive index of the cladding, where r > 5 μm. The solid black line in Fig. 2.2 is the refractive-index distribution of the inverted-parabola graded refractive-index distribution optical fiber. The dotted line shows the original optical-fiber structure [2, 3]. As shown in Fig. 2.2, n 1 = 1.4539, n 2 = 1.440, and n 3 = 1.444. The curvature parameter of the inverted parabola is N = −4, and the maximum refractive-index 1.50

Refractive index distribution

Fig. 2.2 Refractive-index distribution of an optical-fiber structure

na=1.494

1.49

Design Fiber Original Fiber

1.48 1.47 1.46 n1

1.45

n3=1.444

1.44 -6

n2=1.440

-4

-2

0

Radius/μm

2

4

6

42

2 Fiber-Optic Mode Theory

difference appears at the boundary between the inverted parabola core and the lowrefractive-index layer: n a = n 1 − (n 1 − n 2 )N and Δn max = n a − n 2 . N = 0 is an optical-fiber structure with a conventional-step refractive-index distribution.

2.2 Model Theory In an optical-communication system, the optical fiber is an optical waveguide that connect various discrete components. The cross section of the optical fiber can be regarded as a circular dielectric waveguide. The optical fiber can be decomposed into two parts: core and cladding. When analyzing the guided-wave mode field of an optical fiber, its axis is generally set as the z-axis for convenience; the cross-sectional structure is shown in Fig. 2.3. In Fig. 2.3, n1 is the refractive index of the core, n2 is the refractive index of the cladding, a and b respectively represent the radii of the core and cladding, and ϕ is the included angle between a point on the fiber section and the coordinate. The transmission of light in an optical fiber is based on the principle of total light reflection. Ray theory is mainly a geometric ray–analysis method. In a common step fiber, when the beam propagation in the fiber core meets the cladding layer with a low refractive index, it will totally reflect; thus, the beam propagates in a zigzag line within the fiber, as shown in Fig. 1.4. For optical fibers with other structures, such as graded-index fibers, the transmission path of the light beam in the fiber core is generally curved. Waveguide theory is another geometric-analysis method. This method can simply and intuitively obtain the propagation characteristics of light beams in ordinary optical fibers; however, it is only an approximation of wave theory and is not accurate. The core of the optical fiber is on the order of microns, so wave theory needs to be used to analyze the waveguide characteristics of optical fibers. Fig. 2.3 Schematic diagram of the cross section of an optical fiber

y b a n 1 n2

x

2.2 Model Theory

43

2.2.1 Wave Equation In a step fiber, the cladding of the fiber is treated as having an infinite thickness. We solve Maxwell’s equations in the medium, according to the cross section of the optical fiber, in Eq. 2.2 [1, 2]: ∂B ∂t

(2.2)

∂D +J ∂t

(2.3)

∇ ×E=− ∇ ×H =−

∇ ·D=ρ

(2.4)

∇ ·B=0

(2.5)

In Eqs. (2.2)–(2.5), D and B represent the potential shift vector and magneticinduction intensity, respectively; E and H represent the electric-field and magneticfield intensities respectively; ∇ × and ∇· represent the curl and divergence, respectively; and J and ρ represent the current intensity and charge density, respectively. They are combined with constitutive equations [1, 3]: D = ε0 εr E

(2.6)

B = μ0 μ r H

(2.7)

In Eq. (2.6), εr is the relative dielectric constant of the medium, ε0 is the vacuum dielectric constant, and μr and μ0 in Eq. (2.7) are the relative permeability and vacuum permeability of the medium, respectively [3]. The optical waveguide is a medium without charge or current; therefore, ρ = 0, J = 0, and the relative permeability μr = 1. By solving Maxwell’s equation, and combining the relationships between D and E, and B and H in constitutive Eqs. (2.6) and (2.7), the wave equation can be expressed as [4] ∇ 2 E= μ0 ε0 εr

∂ 2E ∂t2

(2.8)

∇ 2 H = μ0 ε0 εr

∂ 2H ∂t2

(2.9)

The special solution in complex form can be obtained from Eqs. (2.8) and (2.9) [4]:

44

2 Fiber-Optic Mode Theory

E(x,y,z,t) = E0 ei(wt−βz)

(2.10)

H(x,y,z,t) = H0 ei(wt−βz)

(2.11)

where ω is the angular frequency of light. The special solution from Eqs. (2.8) and (2.9) is introduced into Eqs. (2.10) and (2.11), respectively, and the Helmholtz equation without time t can be obtained [1]: ∇ 2 E+k 2 E = 0

(2.12)

∇ 2 H + k 2 H =0

(2.13)

Equations (2.12) and (2.13) are vector Helmholtz equations, where k = √ The refractive ω μ0 ε0 = nk0 , and k 0 represents the wavenumber in free √ space.√ √ index of the medium is n, k0 = ω μ0 ε0 = 2π λ, and n = ε/ε0 = εr . When the beam propagates in a limited-size medium or in a material with different refractive indices composed of several media, the components of E and H can be solved using the given boundary conditions.

2.2.2 Wave-Equation Solution An optical fiber is a type of circular dielectric waveguide, which is suitable for solving in cylindrical coordinate systems because of its cylindrical symmetry. In a cylindrical coordinate system, the electric-field intensity E and magnetic-field intensity H can be written as [1] E = er Er + eϕ E ϕ + ez E z

(2.14)

H = er Hr + eϕ Hϕ + ez Hz

(2.15)

The forms of the Ez and Hz solutions in Eqs. (2.14) and (2.15) are [1] E(r, φ,z,t) = E0 ei(wt−βz)

(2.16a)

H(r, φ,z,t) = H0 ei(wt−βz)

(2.16b)

Subsequently, Eq. (2.16a, 2.16b) are introduced into the Helmholtz equation. In a cylindrical coordinate system, Ez and Hz satisfy the waveguide equation, so the longitudinal component Ez of the electric field and Hz of the magnetic field satisfy the wave equation [1]:

2.2 Model Theory

45

∂ 1 ∂ (r r ∂r ∂r

[

]

1 ∂2 )+ 2 2 r ∂ϕ Hz

Ez

[

Ez

]

Hz

[ +

kc2

Ez Hz

] =0

(2.17)

In Eq. (2.17), { kc = w

2

μ0 ε0 n 2c

−β = 2

k02 n 2c

−β = 2

k02 n 21 − β 2 (0 ≤ r ≤ a, c = 1) k02 n 22 − β 2 (r ≥ a, c = 2)

where wavenumber k 0 = 2π/λ. In the cross section, the mode-field distribution of the fiber conforms to the circular-symmetry distribution. We can assume that the periodic function of the solution of the angular distribution field is [1] [

Ez Hz

]

[ ] A = R(r)Φ(ϕ)e−jβz B

(2.18)

where A and B are undetermined coefficients that can be obtained by separating the variables [1]: − jmϕ − jmϕ Φ(ϕ) = Φ+ + Φ− 0e 0e

(2.19a)

2 d2 EZ (r ) 1 dEZ (r ) 2 m +(k + − )E (r ) = 0 c dr 2 r dr r2 Z

(2.19b)

When only forward transmission is considered, { Φ(ϕ) = Φ0 e jmϕ =

sin(mϕ) cos(mϕ)

(2.20)

Equation (2.20) represents two equivalent forms of the solution, which is the same as that in Eq. (2.19a), where Φ0 is a constant; when m = 0, Φ(ϕ) = Φ0 is a constant. It can be seen from Eq. (2.20) that there are sine and cosine terms in the mode, which are called odd and even mode, respectively [1]. In the Bessel Eq. (2.19b), we assume that the form of the core solution is [1] EZ (r ) = A1 Jm (Ur ) + B 1 Nm (Ur )

(2.21a)

The form of the cladding solution is [1] EZ (r ) = C 1 Km (Ur ) + D 1 Im (Ur )

(2.21b)

In Eqs. (2.21a) and (2.21b), A1 , B1 , C1 , and D1 are constants, and Jm and Im are Bessel and imaginary-argument Bessel functions of order m, respectively; Nm and

46

2 Fiber-Optic Mode Theory

Km are Neumann and Hankel functions of order m, respectively. Functions U and W are [1] U2 = k20 n21 −β 2

(2.22a)

W2 = β2 − k20 n22

(2.22b)

Considering that the boundary conditions r < a and EZ (r ) are finite, when r > a, EZ (r ) → 0, according to the form of the Bessel function, and the electric-field form from Eq. (2.21) can only be EZ (r ) = A1 Jm (Ur ) (r ≤ a)

(2.23a)

EZ (r ) = C 1 Km (Wr ) (r > a)

(2.23b)

Similarly, the magnetic-field expression is HZ (r ) = A2 Km (Ur ) (r ≤ a)

(2.24a)

HZ (r ) = C 2 Km (Wr ) (r > a)

(2.24b)

When a mode cutoff occurs, considering the boundary conditions, the propagation constant of the guided-wave mode is equal to that of the cladding. According to the Bessel function and considering the boundary conditions, Eq. (2.19a, 2.19b) is solved. By selecting the first type of Bessel function and the second type of modified Bessel function, the longitudinal components Ez and Hz of the optical-fiber mode field can be obtained as follows [5]: { Ez (r, ϕ, z) = exp[−i (βz ∓ mϕ)] { Hz (r, ϕ, z) = exp[−i (βz ∓ mϕ)]

(

)

A J ur Jm (u) m (a ) r < a A K ur r ≥ a K m (u) m a

(

(2.25a)

)

B J ur Jm (u) m (a ) r < a B K ur r ≥ a K m (u) m a

(2.25b)

In Eq. (2.45), A and B are the electric-field and magnetic-field constants, respectively, a is the fiber-core radius, m is the circumferential mode order, and u is the transverse normalization constant. In an optical fiber, we use E Z and H Z to represent the components in the r and ϕ directions. The equations for Er , E ϕ, Hr , and H ϕ are, respectively, expressed as [6, 7] Er =

) ( ∂Ez −i ωμ0 ∂Hz exp[−i (βz ∓ μϕ)] β + k 2n2 − β 2 ∂r r ∂ϕ

2.2 Model Theory

( ) ∂Ez μ −i β + i ωμ H 0 z exp[−i (βz ∓ μϕ)] k 2n2 − β 2 ∂r r ( ) ∂Hz −i ∂Ez − ωμ0 Eϕ = 2 2 β exp[−i(βz ∓ mϕ)] k n − β2 ∂r ∂ϕ ( ) μ ∂Hz −i iβ ∂Ez − ωμ0 exp[−i (βz ∓ mϕ)] = 2 2 k n − β2 r ∂r ( ) ∂Hz ωε0 n 2 ∂Ez −i β exp[−i(βz ∓ mϕ)] − Hr = 2 2 k n − β2 ∂r r ∂θ ( ) ∂Hz −i 2μ β = 2 2 n − i ωε E 0 z exp[−i (βz ∓ mϕ)] k n − β2 ∂r r ( ) β ∂Hz −i 2 ∂Ez exp[−i (βz ∓ mϕ)] + ωε Hϕ = 2 2 n 0 k n − β 2 r ∂ϕ ∂ϕ ( ) −i μ 2 ∂Ez = 2 2 iβ exp[−i (βz ∓ mϕ)] H + ωε n z 0 k n − β2 r ∂r =

47

(2.26)

(2.27)

(2.28)

(2.29)

The expressions for Ez and Hz (2.25a, 2.25b) are solved using Maxwell’s equations. The expressions of Er , Eϕ , Hr , and Hϕ can be obtained by substituting Eq. (2.25a, 2.25b) into Eqs. (2.26)–(2.29) [1]: ( ( )) ) ⎧ , ( a2 μ Jm ua r u Jm ua r ⎪ ⎪ ⎪ −i 2 Aβ − Bωμ0 exp[−i (βz ∓ mϕ)] r < a ⎪ ⎨ u a Jm (u) r Jm (u) ( Er = ( )) ) , ( ⎪ ⎪ μ K m ωa r ω K m ωa r a2 ⎪ ⎪ − Bωμ0 exp[−i(βz ∓ mϕ)] r ≥ a ⎩ −i 2 Aβ ω a K m (ω) r K m (ω) (2.30) ( ) ) (u ) ( ⎧ , a2 μ Jm ua r μ Jm a r ⎪ ⎪ ⎪ −i − Bωμ Aβ exp[−i (βz ∓ mϕ)] r < a 0 ⎪ ⎨ u2 r Jm (u) a Jm (u) ( Eϕ = ( ) )) , ( ⎪ ⎪ μ K m ωa r ω K m ωa r a2 ⎪ ⎪ − Bωμ0 exp[−i (βz ∓ mϕ)] r ≥ a ⎩ −i 2 Aβ ω r K m (ω) a K m (ω) (2.31) ( ) ) (u ) ( ⎧ , J r a2 μ Jm ua r ⎪ 2μ m a ⎪ ⎪ −i + Bβ n −Aωε exp[−i (βz ∓ mϕ)] r < a 0 1 ⎪ ⎨ u2 r Jm (u) a Jm (u) ( ) Hr = (ω ) , (ω ) 2 ⎪ K K r r ⎪ μ ω a m m ⎪ a a 2 ⎪ + Bβ exp[−i(βz ∓ mϕ)] r ≥ a ⎩ −i 2 −Aωε0 n 1 ω r K m (ω) a K m (ω) (2.32)

48

2 Fiber-Optic Mode Theory

( ( )) ) ⎧ , ( J ur a2 μ Jm ua r ⎪ 2μ m a ⎪ ⎪ −i + Bβ Aωε0 n 1 exp[−i (βz ∓ mϕ)] r < a ⎪ ⎨ u2 a Jm (u) r Jm (u) ( Hϕ = ) ( )) , ( ⎪ ⎪ ω K m ωa r μ K m ωa r a2 ⎪ 2 ⎪ + Bβ exp[−i(βz ∓ mϕ)] r ≥ a ⎩ −i 2 Aωε0 n 1 ω a K m (ω) r K m (ω) (2.33) ,

,

In Eqs. (2.30)–(2.33), Jm and K m are the first derivatives of Jm and K m , respectively, and the dielectric constants of the core and cladding are ε1 and ε2 , respectively. Considering the non-magnetic μ = μ0 of an optical fiber, when the boundarycondition optical fiber r = a is introduced, it is obtained by eliminating A and B from Eqs. (2.30) to (2.33) [1]: ,

m2β 2(

,

,

,

K m (W ) n 21 k02 Jm (U ) n 22 k02 K m (W ) 1 1 2 Jm (U ) + )( + ) + ) = ( U2 W2 U Jm (U ) W K m (W ) U Jm (U ) W W K m (W ) (2.34)

Equation (2.34) is the characteristic equation, also known as the dispersion equation, which describes the relationship between the propagation constant β of the waveguide mode in the optical fiber and the optical frequency. When the opticalfiber parameters are known, the longitudinal-field components Ez and Hz of the optical-fiber core and cladding, respectively, are used, and the transverse component of the electromagnetic field is solved using the longitudinal-field component. The propagation constant of the mode field is obtained according to the boundary and excitation conditions of the optical fiber. Finally, the vector mode that can be transmitted in an optical fiber can be determined using the cut-off conditions [5].

2.3 Light-Wave Propagation Mode in Optical Fiber The light transmitted in an optical fiber must simultaneously satisfy the totalreflection and standing-wave conditions. The total reflection is related to the refractive-index difference between the fiber and cladding, and the standing-wave condition is related to the core size.

2.3.1 Vector Mode Using a rectangular coordinate system to represent the electric and magnetic fields in a cylindrical optical fiber cannot directly reflect the change in the vector mode in the optical fiber. In this study, we used a cylindrical coordinate system. The Laplace operator in a cylindrical coordinate system is [5]

2.3 Light-Wave Propagation Mode in Optical Fiber

49

( ) ∂ 1 ∂ 1 ∂ ∂2 r + 2 2+ 2 ∇ = r ∂r ∂r r ∂θ ∂z 2

(2.35)

We substitute Eq. (2.35) and the E and H components of a cylindrical coordinate system into Eqs. (2.33) and (2.34), respectively. The expressions of Er , Eθ , Ez and Hr , Hθ , Hz in the optical fiber in the cylindrical coordinate system can be obtained as [6] [ ] [ ] [ ] [ ] ( 2 ) Ez 1 ∂ Ez 1 ∂ Ez ∂ 2 Ez 2 + + + k =0 − β Hz ∂r 2 Hz r ∂r Hz r 2 ∂θ Hz ( ) ) ∂ Er 1 ∂ 2 Er 1 ∂ 2 ∂ Eθ Eθ ( r + 2 + 2 2 − 2 − β 2 − k 2 n 2 Er = 0 2 r ∂r ∂r r ∂θ r ∂θ r ( ) ) ∂ Eθ 1 ∂ 2 Eθ Eθ ( 1 ∂ 2 ∂ Er r + 2 − 2 − β 2 − k 2 n2 Eθ = 0 + 2 2 r ∂r ∂r r ∂θ r ∂θ r

(2.36)

(2.37)

(2.38)

Here, only the E component is written; however, it also applies to the H component. β represents the longitudinal-propagation constant. We obtain the vector-mode component in the z direction in the optical fiber from Eq. (2.36) [7]: { E z (r, θ, z) = exp[−i (βz ∓ μθ )] { Hz (r, θ, z) = exp[−i (βz ∓ μθ )]

(

)

A J ur Jμ (u) μ (a ) r < a A K ur r ≥ a K μ (u) μ a

(2.39)

B J ur Jμ (u) μ (a ) r < a B K ur r ≥ a K μ (u) μ a

(2.40)

(

)

where n is the refractive index of the fiber core, u is the transverse-normalization constant, A is the electric-field constant, B is the magnetic-field constant, a is the radius of the fiber core, μ represents the order of the circumferential modulus, J μ (u) is the Bessel equation when the transverse-normalization constant is u, and K μ (u) is the modified Bessel equation when the transverse-normalization constant is u. Using the curl expression in a cylindrical coordinate system, Maxwell’s equations can be written into an expression of the cylindrical coordinate system, where the r, θ, and z components are equal, respectively, and r and θ expressed by Ez and Hz can be obtained. The component form of the direction is as follows [8]: ( ) ∂ Ez ωμ0 ∂ Hz −i β + exp[−i (βz ∓ μθ )] k 2n2 − β 2 ∂r r ∂θ ( ) ∂ Ez −i μ β = 2 2 + i ωμ H 0 z exp[−i(βz ∓ μθ )] k n − β2 ∂r r ( ) ∂ Ez ∂ Hz −i β − ωμ0 exp[−i (βz ∓ μθ )] Eθ = 2 2 k n − β2 ∂r ∂θ Er =

(2.41)

50

2 Fiber-Optic Mode Theory

= Hr = = Hθ = =

( ) μ ∂ Hz −i iβ ∂ E exp[−i (βz ∓ μθ )] − ωμ z 0 k 2n2 − β 2 r ∂r ( ) ∂ Hz ωε0 n 2 ∂ E z −i β − exp[−i(βz ∓ μθ )] k 2n2 − β 2 ∂r r ∂θ ( ) −i ∂ Hz μ β − i ωε0 n 2 E z exp[−i (βz ∓ μθ )] 2 2 2 k n −β ∂r r ( ) β ∂ Hz −i 2 ∂ Ez exp[−i (βz ∓ μθ )] + ωε n 0 k 2 n 2 − β 2 r ∂θ ∂θ ( ) μ −i 2 ∂ Ez iβ exp[−i (βz ∓ μθ )] + ωε n H z 0 k 2n2 − β 2 r ∂r

(2.42)

(2.43)

(2.44)

Substituting Eqs. (2.41) and (2.42) into Eqs. (2.43) and (2.44), we find expressions for Er , Eθ , Hr , and Hθ [7]: ( ( )) ⎧ , (u ) a2 μ Jμ ua r μ Jμ a r ⎪ ⎪ ⎪ −i − Bωμ0 Aβ exp[−i(βz ∓ μθ )] r < a ⎪ ⎨ u2 a Jμ (u) r Jμ (u) ( Er = (ω )) , (ω ) 2 ⎪ ⎪ ⎪ −i a Aβ ω K μ a r − Bωμ μ K μ a r exp[−i (βz ∓ μθ )] r ≥ a ⎪ ⎩ 0 ω2 a K μ (ω) r K μ (ω) (2.45) ( ) (u ) ⎧ , (u ) 2 ⎪ ⎪ −i a Aβ μ Jμ a r − Bωμ0 μ Jμ a r exp[−i (βz ∓ μθ )] r < a ⎪ ⎪ ⎨ u2 r Jμ (u) a Jμ (u) ( ) Eθ = (ω ) , (ω ) ⎪ ⎪ μ Kμ a r ω Kμ a r a2 ⎪ ⎪ −i − Bωμ0 Aβ exp[−i (βz ∓ μθ )] r ≥ a ⎩ ω2 r K μ (ω) a K μ (ω) (2.46) ( ) ( ) ⎧ , (u ) J ur a2 μ Jμ a r ⎪ 2μ μ a ⎪ ⎪ −i 2 −Aωε0 n 1 + Bβ exp[−i (βz ∓ μθ )] r < a ⎪ ⎨ u r Jμ (u) a Jμ (u) ( ) Hr = ( ) , (ω ) ⎪ ⎪ μ K μ ωa r ω Kμ a r a2 ⎪ 2 ⎪ + Bβ exp[−i(βz ∓ μθ )] r ≥ a ⎩ −i 2 −Aωε0 n 1 ω r K μ (ω) a K μ (ω) (2.47) ( (u )) ⎧ , (u ) a2 μ Jμ a r μ Jμ a r ⎪ ⎪ ⎪ −i + Bβ Aωε0 n 21 exp[−i (βz ∓ μθ )] r < a ⎪ ⎨ u2 a Jμ (u) r Jμ (u) ( Hθ = , ( )) , (ω ) ⎪ ⎪ μ Kμ a r ω K μ ωa r a2 ⎪ 2 ⎪ + Bβ exp[−i (βz ∓ μθ )] r ≥ a ⎩ −i 2 Aωε0 n 1 ω a K μ (ω) r K μ (ω) (2.48)

2.3 Light-Wave Propagation Mode in Optical Fiber

51

where ε0 is the conductivity and μ0 is the permeability. At r = a of the optical fiber, the tangential electric and magnetic fields are continuous, and Eqs. (2.46)–(2.48) can be obtained by substituting into the boundary condition [7]: ( ) , , ) ( ωμ0 1 Jμ (u) 1 K μ (ω) iμβ 1 1 + A + 2 −B =0 a u2 ω a u Ju (u) ω K u (ω) ( ) , , ) ( ωε0 n 21 Jμ (u) n 22 K μ (ω) 1 i μβ 1 + A + 2 =0 +B a u Ju (u) ω K u (ω) a u2 ω

(2.49)

(2.50)

For a homogeneous equation composed of Eqs. (2.49) and (2.50), if A and B have nonzero solutions, their coefficient determinant should be zero, and the characteristic equation can be derived [6]: [

,

,

1 K μ (ω) 1 Jμ (u) + u Jμ (u) ω K μ (ω)

][

,

,

n 21 Jμ (u) n 22 K μ (ω) + ω K μ (ω) un 22 Jμ (u)

]

[ =μ

2

n 21 1 1 + 2 2 u2 ω n2

](

1 1 + 2 2 u ω (2.51)

Equation (2.51) is the characteristic equation of the fiber-vector mode, or the dispersion equation. The dispersion equation can be used to determine the u-order mode β value or U value. There are four vector modes in an optical fiber: radial vector beam TE0v , angular vector beam TM0v , hybrid-vector polarized beam HEμv , and mixed-vector polarized beam EHμv . μ represents the order of the circumferential modulus and v represents the order of the radial modulus. In Eq. (2.51), the modes can be determined as follows: • • • •

TE0v mode: μ = 0; Ez = Er = Hθ = 0. TM0v mode: μ = 0; Eθ = Hr = Hz . HEμv mode: μ > 0. EHμv mode: μ < 0.

odd even even EH and HE have odd and even modes, represented by H E μν , H E μν , E Hμν , even and E Hμν , respectively. When the phase factor exp(-j(βmzμθ )) is positive, z increases, θ decreases, and a clockwise rotation indicates right-handed polarization. When the phase factor is negative, z increases, θ increases, and a counter clockwise rotation indicates lefthanded polarization. Figure 2.4 shows the polarization and light-intensity distributions of the basic mode in an optical fiber. It can be observed that the vector modes corresponding to the same light-intensity distribution may be different. Figure 2.4a, b, d, and e belong to the first-order mode in an optical fiber. There is no difference in the light-intensity distribution, but there is a certain difference in polarization.

)

52

2 Fiber-Optic Mode Theory

(a)

(b)

(c)

1

0

(d)

(f)

(e)

1

0

Fig. 2.4 Intensity and polarization distribution of a vector mode in optical fiber. a TM01 ; b TE01 ; c EH11 odd ; d HE21 odd ; e HE21 even ; and f EH11 even

As shown in Fig. 2.4a, the polarization of TE mode has a radial distribution, and as shown in Fig. 2.4b, the polarization of TM mode has an angular distribution. As odd even and H E μν shown in Fig. 2.4d and e, the polarization difference between H E μν is π/2. Comparing Fig. 2.4d and e with Fig. 2.4c and f, the polarization-distribution directions of the odd and even modes of the HE and EH modes are opposite, and the light-intensity distribution is also different. TE is called the transverse-electric mode, which means the electric-field direction is perpendicular to the propagation direction. TM is called the transverse-magnetic mode, which means the magnetic-field direction is perpendicular to the propagation direction. The TE and TM modes can be collectively referred to as linear-polarization (LP) modes. TEM is the transverse-electromagnetic mode, which means that the directions of the electric and magnetic fields are perpendicular to the propagation direction. When an electromagnetic wave is transmitted in an optical fiber, it continuously reflects at the interface between the core and cladding. It is a broken line in a step-index optical fiber. The propagation direction is not the direction of the light in the optical fiber, but the direction of the core. TE and TM modes When solving for the cut-off conditions in different modes, the eigenequation (Eq. 2.34) needs to be used. When m = 0, the left side of Eq. (2.52) is set to 0, and the characteristic equations of TE0n and TM0n are solved respectively [4, 9]: ,

,

1 Jm (U ) 1 K m (W ) + =0 U Jm (U ) W K m (W ) ,

(2.52)

,

n 21 Jm (U ) n 22 K m (W ) + =0 U Jm U W K m (W )

(2.53)

2.3 Light-Wave Propagation Mode in Optical Fiber

53

The fields corresponding to Eq. (2.53) are E ϕ , Hr , and Hz ; the electric field has only angular components, and is called TEmn mode. For TE0n mode, the cut-off condition is J 0 (U) = 0, the eigenvalue of TE0n at the cut-off is U, and the root of J 0 = 0 is the cut-off frequency. In addition, the distance from the cut-off condition ∞ ∞ ) = 0 (U0n /= 0), the eigenvalue of the TE0n mode when it is far from is J1 (U0n ∞ , and the root of J 1 = 0 is the distance from the cut-off the cut-off condition is U0n frequency. When the guided mode is cut off, ω → 0 and J 0 (u) = 0 can be obtained from the recurrence formula and the asymptote of the modified Bessel function. The Bessel curves are shown in Fig. 2.5. Fig. 2.5 Relationship between transverse-normalization constant u and Bessel function J(u)

1

0

(a)

m = 2,n = 1

(b)

m = 3,n = 1

(c)

m = 4,n = 1

(d)

m = 2,n = 2

(e)

m = 3,n = 2

(f)

m = 4,n = 2

Fig. 2.6 Light-intensity distribution of HE mode

54

2 Fiber-Optic Mode Theory

Table 2.1 Roots of Bessel function (orders 0–7) [10] n

m 0

1

2

3

4

5

7

1

2.4048

3.8317

5.1356

6.3802

7.5883

8.7715

11.0864

2

5.5201

7.0156

8.4172

9.7610

11.0647

12.3386

14.8213

3

8.6537

10.1735

11.6198

13.0152

14.3725

15.7002

18.2876

4

11.7915

13.3237

14.7960

16.2235

17.6160

18.9801

21.6415

5

14.9309

16.4706

17.9598

19.4094

20.8269

22.2178

24.9349

6

18.0711

19.6159

21.1170

22.5827

24.0190

25.4303

28.1912

7

21.2116

22.7601

24.2701

25.7482

27.1991

28.6266

31.4228

8

24.3525

25.9037

27.4206

28.9084

30.3710

31.8117

34.6371

As can be seen from Fig. 2.5, the Bessel curve is in the form of oscillations, and there are multiple roots. After the Bessel order is determined, a certain transversenormalization constant u has only one root. The roots of the Bessel function are listed in Table 2.1. The fields corresponding to Eq. (2.53) are Hϕ , Er , and E z , and there are only transverse components, which is called TMmn mode. Similarly, for TM0n mode, the 0 0 ) = 0, the eigenvalue of TM0n at the cutoff is U0n , and the cutoff condition is J0 (U0n ∞ )=0 cutoff frequency is the root of J0 = 0. Away from the cut-off condition, J1 (U0n ∞ ∞ (U0n /= 0), the eigenvalue of TM0n mode away from the cut-off condition is U0n , and the root of J1 = 0 is away from the cut-off frequency. When the cut-off conditions of TE mode and TM mode are obtained by solving Eqs. (2.52) and (2.53), it is found that TE mode and TM mode have the same eigenvalue, and the two modes are in a degenerate state [5]. HE mode The characteristic equation of HEmn mode can be written out from Eq. (2.52) [4]: ,

,

1 Jm (U ) 1 K m (W ) 1 1 + = −m( 2 + 2 ) U Jm (U ) W K m (W ) U W

(2.54)

The above formula is simplified using the recursive formula of the Bessel function. When m = 1, Eq. (2.54) can be expressed as [10] 1 J0 (U ) 1 K 0 (W ) − =0 U J1 (U ) W K 1 (W )

(2.55)

0 For HEmn mode, when m = 1, the cut-off condition is J1 (U1n ) = 0, the eigenvalue 0 at the cut-off of HE1n mode is U1n , and the root of the same J1 (U ) = 0 is the cut-off frequency of HE1n mode. The eigenvalue away from the cut-off condition is

2.3 Light-Wave Propagation Mode in Optical Fiber

55

∞ ∞ J0 (U1n ) = 0, the eigenvalue away from the cut-off condition is U1n , and the root of J0 (U ) = 0 is the HE1n mode away from the cut-off frequency [5]. It can be seen from Table 2.1 that the cut-off frequencies of HE11 , HE12 , and HE13 are far from the cut-off frequency; the first root 0 of the cut-off frequency of J1 (U ) is HE11 , and the first root is 2.4084 away from the cut-off frequency of J0 (U ); the second root with the cut-off frequency of HE12 is 3.831, and 5.52021 away from the second root with a cut-off frequency of J0 (U ). The cut-off frequency of HE13 is 3.831 for the third root with a cut-off frequency of J1 (U ) and 8.6537 away from the third root with a cut-off frequency of J0 (U ). The cut-off and far cut-off frequencies of HE can be analyzed in turn. By solving the cut-off frequency and away from the extent of the cut-off frequency, it is found that the cut-off frequency of HE11 is 0 at any wavelength and optical fiber; thus, it is the fundamental mode in an optical fiber. When m > 1, Eq. (2.55) is expressed as [4]

Jm−1 (U ) U = Jm (U ) 2(U − 1)

(2.56)

The cutoff frequencies of higher-order modes can be obtained from Eq. (2.56). 0 0 ) = 0(Umn /= 0), the eigenvalue of the For m > 1, the cut-off condition is Jm−2 (Umn 0 , and the cut-off frequency is Jm−2 (U ) = 0. The eigenvalue cut-off condition is Umn ∞ ∞ ) = 0(Umn /= 0), the eigenvalue away away from the cut-off condition is Jm−1 (Umn ∞ from the cut-off condition is Umn , and the root away from the cut-off frequency is Jm−1 = 0. We substitute m and n into Eq. (2.25a) to obtain the light-intensity distributions of HE mode in Fig. 2.3. Figure 2.6a, b, and c show the light-intensity distribution of HE mode when n = 1, and Fig. 2.6d, e, and f show the light-intensity distribution of HE mode when n = 2. By comparing Fig. 2.6a, b, and c, it is found that, with an increase in m, the larger the area of the central dark spot, and the more concentrated the energy on the ring. By comparing the upper and lower light intensities, it is found that n determines the radial distribution of the vector mode; that is, the number of turns of the vector-mode light intensity. EH mode The characteristic equation of vector mode EHmn is simplified using Eq. (2.34) [4]: ,

,

1 Jm (U ) 1 K m (W ) 1 1 + = m( 2 + 2 ) U Jm (U ) W K m (W ) U W

(2.57)

The solution of Eq. (2.57) is the same as that of HE mode. For EHmn mode, when 0 0 ) = 0(Umn /= 0), and the far cut-off condition m > 1, the cut-off condition is Jm (Umn ∞ ∞ is Jm+1 (Umn ) = 0(Umn /= 0). Similarly, the polarization of the vector mode in an optical fiber is the same as that of HE mode [6]. The cutoff frequency and guided-mode cutoff can be obtained using the recurrence formula and asymptote of the modified Bessel function. Because the Bessel curve is in the form of oscillations, there are multiple roots, as shown in Table 2.1. At the

56

2 Fiber-Optic Mode Theory

root, the cut-off frequencies of TE01 and TM01, and TE02 and TM02 correspond to 2.4084, 5.52021, 8.6537, …, respectively. When n = 1, the first roots of J0 (u) are 2.4084, 3.8317, 5.1356, …, corresponding to the U values when HE11 , HE21 and EH11 , HE31 and EH21 are cut off, respectively.

2.3.2 Scalar-Mode Solution The mode of the actual solution in an optical fiber can be obtained from Maxwell’s equations. The solution of the scalar mode in an optical fiber is the degeneracy of the vector mode of the same order; it is called the linear-biased (or polarized) mode, that is, LPmn mode. The field distribution in the optical fiber adopts a scalar approximation, and the light field of the LP mode along the z-axis is [1] Ψ (r, φ, z) = Ψ 0 (r, φ)ej(wt−βz)

(2.58)

In Eq. (2.58), the angular frequency is w, and Ψ 0 (r, φ) is the transverse field. The electric field is expressed in the following form [1]: Ψ (r, φ, z) = Cm Jm (Ur)ejmφ (r < a) Ψ (r, φ, z) = Cm

Jm (Wr) Km (Wr)ejmφ (r > a) Km (Wa)

(2.59a) (2.59b)

Considering the continuity of the electric field at the boundary between the fiber core and cladding, the dispersion Eqs. (2.59a and 2.59b) can be obtained by setting the equal sign on the left and right. We can decompose Ψ 0 (r, φ) into Eqs. (2.59a and 2.59b) using the Euler formula. From the decomposition formula, we can observe that the distribution of the electromagnetic field along the circumference is composed of two linear polarizations. The solution form can also appear as [1] {

sin(mφ) (r < a) cos(mφ) { sin(mφ) Jm (Wr) Km (Wr) Ψ (r, φ, z) = Cm (r > a) Km (Wa) cos(mφ) Ψ (r, φ, z) = Cm Jm (Ur)

(2.60a)

(2.60b)

The waveforms expressed in Eqs. (2.59a, 2.59b) and (2.60a, 2.60b) are LPmn modes, that is, linearly polarized. The polarization direction remains unchanged over time. This approximate theory is often referred to as linear-polarization mode theory. Figure 2.7 shows the mode-field distribution of an optical fiber after considering the polarization state.

2.3 Light-Wave Propagation Mode in Optical Fiber

57

x y

LP01x

LP11ex

LP11ox

LP11ex

LP11ey

LP21ex

LP21ox

LP21ey

LP21oy

LP01y

Fig. 2.7 Mode field in an optical fiber, considering polarization (weak-conductance approximation) 1

0 (b) LP21

(c) LP31 1

(a) LP11

0 (d) LP12

Fig. 2.8 Light-intensity distributions of LP modes

(e) LP13

58 Table 2.2 Relationships between LP mode and vector mode in an optical fiber

2 Fiber-Optic Mode Theory Degenerate mode (LP)

Vector mode

LP-mode merger degree

LP01

HE11

2

LP11

TE01 /HE21/ TM01

4

LP21

HE31/ EH11

4

LP31

HE41/ EH21

4

In the case of a weak derivative approximation, a partial linear mode is formed by the merger of several vector modes with similar propagation constants. In Fig. 2.7, superscripts o and e represent the odd and even modes of LP mode in the optical fiber, respectively. Superscripts x and y of LP mode respectively represent the coordinates of the polarization direction, and the arrow in the figure represents the polarization direction of the LP mode field. Traditionally, subscripts O and E are also used to represent odd and even modes in an optical fiber, respectively. Linear-biased odd and even modes can also be expressed as LPmn,O and LPmn,E . Table 2.2 shows the corresponding relationship between the LP and vector modes in an optical fiber. The light-intensity distributions of linearpolarization modes LPl,m in the optical fiber are shown in Fig. 3.4. Modes LP11 , LP21 , and LP31 in Fig. 2.8a, b, and c can reflect the influence of the circumferential-mode order l on the intensity distribution of the scalar mode. There are two bright spots in LP11 mode along the circumferential direction, four bright spots in LP21 mode, and six bright spots in LP31 mode. Modes LP11 , LP12 , and LP13 in Fig. 2.8a, d, and e reflect the influence of radialmode order m on the mode light-intensity distribution. The electromagnetic field of LP11 mode has only one circle of bright spots along the radius, that of LP21 mode has two circles of bright spots, and that of LP31 mode has three circles of bright spots. Therefore, we can draw the following conclusions: the larger l is, the more bright spots are on the circumference of the light-intensity distribution of the LP mode; the number of bright spots is related to l, which is 2l. Similarly, the larger m is, the more turns of the bright spot in the radial direction, because the number of turns is equal to m. The composition of LP mode in an optical fiber is as follows [10]: L Pl,m = H El+1,m ± E Hl−1,m

(2.61)

L Pl,m = H El+1,m ± T Ml−1,m

(2.62)

L Pl,m = H El+1,m ± T El−1,m

(2.63)

According to Eqs. (2.61)–(2.63), the linearly polarized LP mode in the optical fiber can be composed of the superposition of different vector modes HEl+1,m , EHl-1,m , TEl-1,m , and TMl-1,m in the optical fiber. Because different vector modes have different

2.3 Light-Wave Propagation Mode in Optical Fiber

59

effective refractive indices, that is, different longitudinal-propagation constants, the superposition of different modes will cause the “mode walk-off” phenomenon.

2.3.3 Normalized Operating Frequency There are four vector modes in an optical fiber, and each mode corresponds to a fixed cutoff frequency with a given range. The normalized cutoff frequency corresponding to each order vector mode is also called the normalized working frequency. The normalized operating frequency can be expressed as V =

/ 2π a n 21 − n 22 < 2.4048 λ

(2.64)

Equation (2.64) shows the transmission condition of a single-mode optical fiber. When the value of the optical-fiber working frequency v is less than 2.4048, only HE11 , that is, the fundamental mode, can be transmitted in the optical fiber. When the fundamental mode does not meet the transmission requirements, an optical fiber that can accommodate and transmit high-order vector modes is needed; therefore, a less-mode optical fiber or multimode optical fiber should be used. When solving for the vector mode in a multimode fiber, the effective refractive index of each order vector mode / is solved using the characteristic equation. n e f f = β k0 is the effective refractive-index formula for an optical fiber. The relationship between U, W, and V can be solved by using the effective refractive index neff and the characteristic equation of the optical fiber [14]: ) ( U 2 = k02 n 21 − k02 n 2e f f a 2

(2.65)

) ( W 2 = k02 n 2e f f − k02 n 22 a 2

(2.66)

V 2 = U2 + W2

(2.67)

In Eqs. (2.65)–(2.67), β is the longitudinal propagation constant, k 0 is the wavenumber, and U and W are the normalized constant and normalized attenuation coefficient, respectively. We set U to the cut-off frequency of the vector mode in Table 2.1, and then use Eqs. (2.62) and (2.64) to obtain the effective refractive index of each order of the vector mode. Figure 2.9 shows the effective refractive-index curve between the normalized working frequency and each vector-mode order in the optical fiber. It can be seen intuitively that the number of vector modes in the optical fiber increases with an increase in the normalized working frequency. HE11 is the basic mode and has no cut-off frequency. When v > 2.4048, the first-order mode begins to appear, including HE21 , TM01 , and TE01 , collectively referred to as LP1n mode. When v > 3.8317, the

60

2 Fiber-Optic Mode Theory

Fig. 2.9 Relationship between the effective refractive index and normalized operating frequency (0–3 mode orders)

second-order mode begins to appear. By analogy, the corresponding vector modes at the corresponding normalized cutoff frequencies can be obtained.

2.3.4 Coupling Efficiency of Gaussian Modes [15] When the normalized frequency V of a single-mode fiber is in the range of 1.9– 2.4, the beam transmitted in the fiber can be approximated by a Gaussian beam. For step-type single-mode fiber, the size of the mode spot in the Gaussian-beam approximation ω is [15] ) ( ω = a 0.65 + 1.619V −3/2 + 2.8798V −6

(2.68)

where (a is the )fiber-core radius. V is the normalized frequency and is defined as V 2 = n 21 − n 22 k02 a 2 . n1 is the refractive index of the core, n2 is the refractive index of the cladding, and k 0 is the wavenumber in free space. In a Gaussian-beam approximation, all types of coupling can be reduced to the coupling between two Gaussian modes. The smaller the value of V, the weaker the ability of the optical fiber to limit optical leakage and the fewer modes can be transmitted. When only one mode of V < 2.045 exists, it can be called a single-mode fiber. As shown in Fig. 2.10, Gaussian beam 1 propagates from left to right and Gaussian beam 2 propagates from right to left. In the plane perpendicular to the z-axis at crossing point 0, beam 2 has a transverse deviation x 0 and angular deviation θ relative to beam 1. If the coordinate system is established with the symmetry axis of beam 1

2.3 Light-Wave Propagation Mode in Optical Fiber

61

Fig. 2.10 Gaussian-mode coupling [15] X

2

2

R1 01

2

1

2

02

2

x0 R2 O

Z

as the z axis, the complex amplitude of the light field describing Gaussian mode 1 is normalized [15]. ) ( ( )1/4 ( )1/2 1 x2 2 x2 1 ϕ1 = exp − 2 − ik0 π ω1 2 R1 ω1

(2.69)

where ω1 is the spot radius of beam 1 at plane 0, R1 is the curvature radius of the corresponding wave surface, and k 0 is the propagation constant of free space. For beam 2, when the deviation angle θ is very small, its influence on the complex amplitude is only an additional phase factor exp(ik 0 θ x). Therefore, under a smallangle approximation, the normalized complex amplitude of Gaussian mode 2 is [15] ( )1/4 ( )1/2 1 2 1 (x − x0 )2 (x − x0 )2 ik0 ϕ1 = exp[− − ] · exp{ik0 θ x} (2.70) π ω2 2 R2 ω22 The coupling efficiency of two Gaussian modes is [15] η=q

1/2

[ ( x2 x0 k02 ω22 x02 1 2 2 2 3k02 ω12 x0 k ω θ − exp q − 02 − − + 8 0 0 2ω2 2ω12 8R22 8R22 1 k 2 ω2 x 0 θ k 2 ω2 x 2 k 2 ω2 x 0 θ k 2 ω2 x 2 − k0 ω12 θ 2 + 0 1 + 0 1 0 − 0 1 − 0 12 0 8 2R2 2R1 R2 4R1 8R1

)] (2.71)

where { q=4

[( ω12 ω22

1 1 + 2 2 ω1 ω2

)

( ) ]}−1 1 2 1 1 2 + k0 − 4 R2 R1

(2.72)

Equation (2.72) is the Gaussian-mode coupling-efficiency formula that considers both lateral and angular deviations.

62

2 Fiber-Optic Mode Theory

(1) Fully aligned modes When x0 = 0 and θ = 0, the two Gaussian modes are fully aligned, and the coupling efficiency is [15] [(( ηa = q 1/2 = 2/ω1 ω2

1 1 + 2 ω12 ω2

)

( ) )]1/ 2 1 2 1 1 2 + k0 − 4 R2 R1

(2.73)

(2) Angular deviation When x0 = 0, there is only angular deviation, and the coupling efficiency is [15] ) ( η = ηa exp −(θ/θe )2

(2.74)

( )1/2 θe = 23/2 /k0 ηa ω12 + ω22

(2.75)

where

According to Eq. (2.75), θe represents the corresponding angular deviation when η drops to ηa /e. (3) Lateral deviation When θ = 0, there is only lateral deviation, and the coupling efficiency is [15] η=q

1/2

)] [ ( 2 x0 x02 k02 ω22 x02 3k02 ω12 x02 k02 ω12 x0 k02 ω12 x02 exp q − − − + − 2R1 R2 2ω22 2ω12 8R22 8R22 8R12 (2.76)

In fact, the waists of the two Gaussian modes are very close, that is, R1 and R2 are very large, 1/R12 → 0 and 1/R22 → 0. Therefore, η≈q

1/2

]] [ [ ] [ x02 x02 == ηa exp −(x0 /xe )2 exp q − 2 − 2 2ω2 2ω0 ( xe = 2

1/2

/ηa

1 1 + 2 2 ω1 ω2

)1/

2

where x e represents the corresponding lateral deviation when η drops to ηa /e.

(2.77)

(2.78)

2.4 Mode Effective Refractive Index

63

2.4 Mode Effective Refractive Index 2.4.1 Effective Refractive Index of a Vector Mode The dispersion, limiting loss, and nonlinear coefficient of an optical fiber are closely related to the effective refractive index. By using the effective refractive index, we can understand the various characteristics of optical fibers when transmitting signals. There are different effective refractive indices and different propagation constants for modes. The effective refractive index neff can be expressed as [16] ne f f =

β k0

(2.79)

where k 0 = 2π/λ is the wavenumber in a vacuum, λ is the wavelength of the light wave, and β is the propagation constant. Figure 2.11 shows the effective refractive index of each mode of a photonic-crystal fiber, and the changes in the effective refractive index of HE and EH modes. In the actual propagation process, the effective refractive index changes with a change in the light wavelength. As shown in Fig. 2.11, in the wavelength range of 1.15–2.0 μm, when the light wavelength increases, the field-intensity distribution of the light beam gradually diffuses to the cladding part. Therefore, the effective refractive index neff of each vector mode in the photonic-crystal fiber gradually decreases with the increasing wavelength. At the same wavelength, the larger the mode order is, the easier it is for the light field to diffuse to the cladding part. Therefore, with the increase in wavelength and mode order, the effective refractive index decreases faster, and the curve becomes more inclined.

Fig. 2.11 Relationship between the effective refractive index and wavelength of vector modes. a HE mode; b EH mode

64 -3

×10

5.5

Effective refractive index difference

Fig. 2.12 Relationship between the effective refractive-index difference between modes and the wavelength in a photonic-crystal fiber

2 Fiber-Optic Mode Theory

5.0 4.5 4.0 3.5 3.0 2.5 2.0

HE3,1-EH1,1 HE4,1-EH2,1 HE5,1-EH3,1 HE6,1-EH4,1 HE7,1-EH5,1 HE8,1-EH6,1 HE9,1-EH7,1 HE10,1-EH8,1 HE11,1-EH9,1 HE12,1-EH10,1 HE13,1-EH11,1 HE14,1-EH12,1 1E-4

1.5 1.0 0.5 0.0 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Wavelengthl/mm

2.4.2 Effective Refractive-Index Difference Between Modes Figure 2.12 shows the relationship between the effective refractive-index difference neff between HE mode and EH mode and the wavelength. Different modes with the same characteristic (transmission) parameters are called degenerate modes. There are two types of degeneracy in a circular waveguide: polarization and mode. Mode degeneracy seriously affects the quality of transmitted signals in optical fibers. Therefore, the effective refractive-index difference neff between modes should meet the basic condition that it is greater than 1 × 10–4 . As can be seen from Fig. 2.12, the effective refractive-index difference neff will gradually increase with the increase of wavelength. At the 1.55-μm wavelength, the effective refractive-index difference neff between the HE3,1 and EH1,1 modes can reach 2.6 × 10–3 ; by comparison, the basic condition of 1 × 10–4 is increased by one order of magnitude. The increase in the effective refractive-index difference will reduce the possibility of mode coupling. Because each mode is transmitted separately, the intermode crosstalk is reduced.

2.4.3 Dispersion Characteristics According to guided-wave optics theory, an optical signal transmitted in an optical fiber propagates at different speeds. After reaching a certain distance, the transmission signal distorts. This effect is called fiber dispersion. Owing to the limitations of its structure and material, the dispersion of ordinary optical fiber is uncontrollable and cannot meet the optical-fiber dispersion requirements for optical-fiber communication.

2.4 Mode Effective Refractive Index

65

The constituent materials of photonic-crystal fiber are relatively simple and have a flexible structure. By reasonably adjusting the diameter and spacing of the hollow pores of the photonic-crystal fiber, the dispersion performance can be controlled, and photonic-crystal fiber structures that meet the actual dispersion requirements can be produced. The dispersion in an optical fiber mainly includes waveguide dispersion, Dw , and material dispersion, Dm . Material dispersion is mainly caused by the base material and has little effect on the total dispersion. This is mainly because the base material of most photonic-crystal fibers is composed of a single material that is independent of the fiber structure. Therefore, the material dispersion is constant and can be ignored [17]. The waveguide dispersion is mainly affected by the structure of the photoniccrystal fiber, which is determined by its geometric size and structure. By reasonably changing the structural parameters of the fiber, the fiber dispersion can be effectively controlled, and a photonic-crystal fiber structure that meets various practical needs can be designed. The dispersion coefficient D in the photonic-crystal fiber is expressed as follows [18]: | ( )| λ ∂ 2 |Re n e f f | D= − c ∂λ2

(2.80)

where c is the speed of light in a vacuum, Re(neff ) is the real part of the effective refractive-index of the mode, and λ is the wavelength. Figure 2.13 shows the variation curve of the dispersion coefficient of each vector mode with the wavelength in the wavelength range of 1.15–2.0 μm (850 nm) of the photonic-crystal fiber. It can be seen from Fig. 2.13 that when the wavelength of incident light is in the range of 1.15–2.0 μm (850 nm), the dispersion coefficient of low-order modes (l < 6) changes little with the increase of wavelength, and the dispersion coefficient curve tends to have a flat distribution. However, the dispersion coefficient of high-order modes is larger than that of low-order modes, and gradually increases with the increase of wavelength. This also verifies the unstable-transmission characteristics of high-order modes in optical

Fig. 2.13 Dispersion-coefficient variation of vector modes. a HE mode; b EH mode

66

2 Fiber-Optic Mode Theory

fiber. Although the dispersion coefficient of high-order modes is larger, dispersioncompensation technology can be used to compensate in practical applications. At a 1.55-μm wavelength, the dispersion coefficient of HE3,1 mode is 46.9649 ps (nm km)−1 , and that of HE4,1 mode is 57.4461 ps (nm km)−1 . Photonic-crystal fibers have a special waveguide structure. Air holes of different shapes and sizes are present in the fiber. Part of the energy diffuses into the cladding area of the light field transmitted in the fiber. Because the cladding area is finite, part of the energy will disappear during the transmission, causing energy loss; this is called limiting loss [19] or leakage loss. The limiting loss L can be expressed as [20] L=

( ) 2π 20 Im n e f f λ ln10

(2.81)

where Im(neff ) is the imaginary part of the effective refractive index of the mode, λ is the wavelength, and the unit of L is dB m−1 . Table 2.3 shows the limiting loss of the vector mode in a photonic-crystal fiber at a 1.55-μm wavelength. It can be seen from Table 2.3 that at a 1.55-μm wavelength, each vector mode in the photonic-crystal fiber has a relatively low limiting loss, which is in the range of 10–11 dB m−1 –10–9 dB m−1 . This is mainly owing to the existence of a highrefractive-index circular region, which makes the limiting loss of each vector mode ideal. The limiting loss of HE4,1 mode is only 1.30 × 10−10 dB m−1 . The small limiting loss provides favorable conditions for the practical use of this optical-fiber structure for transmission. Photonic-crystal fibers can improve the structural parameters of the fiber, making the limiting loss more ideal. Table 2.3 Limiting loss of vector modes in a photonic-crystal fiber at a 1.55-μm wavelength Vector mode

HE1,1

HE2,1

HE3,1

HE4,1

HE5,1

HE6,1

Confinement 4.8 × 10–10 1.4 × 10–10 1.2 × 10–9 1.3 × 10–10 2.5 × 10–10 7.5 × 10–11 loss (dB m−1 ) Vector mode

HE7,1

HE8,1

HE9,1

HE10,1

HE11,1

HE12,1

Confinement 2.8 × 10–11 3.3 × 10–10 5.9 × 10–10 7.5 × 10–10 3.1 × 10–10 3.1 × 10–10 loss (dB m−1 ) Vector mode

HE13,1

Confinement 9.0 × loss (dB m−1 )

10–10

Vector mode EH5,1 Confinement 7.5 × loss (dB m−1 )

HE14,1

EH1,1

2.0 ×

3.7 ×

10–9

EH6,1 10–10

Vector mode Confinement loss (dB

3.1 ×

EH2,1 10–10

EH7,1 10–10

3.1 ×

EH8,1 10–10

EH10,1 m−1 )

5.90 ×

2.8 ×

EH3,1 10–11

2.8 ×

3.3 × EH7,1

10–11

3.1 ×

EH11,1 10–10

EH4,1 10–11

7.50 ×

5.9 × 10–10 EH9,1

10–10

3.31 × 10–10

EH12,1 10–10 0

3.19 × 10–10

2.4 Mode Effective Refractive Index

67

2.4.4 Nonlinear Effects The nonlinear effect is also an important optical property of photonic-crystal fiber. The nonlinear effect in photonic-crystal fiber can be expressed using a nonlinear coefficient. When the nonlinear coefficient is small, it can effectively suppress the nonlinear effect to improve the transmission quality of the communication system. Before discussing the characteristics of the nonlinear coefficient, the effective modefield area of the photonic-crystal fiber is analyzed. The effective mode-field area can be expressed as [21] (˜ Ae f f =

˜

|E(x, y)|2 d xd y

)2

|E(x, y)|4 d xd y

(2.82)

where E(x,y) represents the transverse electric-field intensity distribution of the optical fiber. The nonlinear coefficient can be expressed as [22] γ =

2π n 2 λAe f f

(2.83)

where n2 is the nonlinear refractive index of the base material of the photonic-crystal fiber; different materials correspond to different values. According to Eqs. (2.66) and (2.67), the variation trend of the nonlinear-coefficient characteristics of photonic-crystal fiber is opposite that of the effective mode-field area. In general, the effective mode-field area of a photonic-crystal fiber can first be calculated, and then its nonlinear coefficient can be obtained, according to the inverse relationship. The relationship between the effective mode-field area of each vector mode in a photonic-crystal fiber and the wavelength is shown in Fig. 2.14. In the 1.152.0 μmwavelength range, when the light wavelength gradually increases, the effective modefield area Aeff of the vector mode in the photonic-crystal fiber also gradually increases. Similarly, when the mode order increases, the effective mode-field area Aeff also increases because the energy of the higher-order mode can easily leak into the air cladding, resulting in an increase in the effective mode-field area with an increase in the mode order. The variation trend of the nonlinear coefficient of photonic-crystal fiber is opposite that of the effective mode-field area, as shown in Fig. 2.15. In the 1.15–2.0-μm wavelength range, the nonlinear coefficient of a vector mode in photonic-crystal fiber gradually decreases with the increase of wavelength and mode order. The nonlinear coefficient of a low-order mode is relatively large, whereas that of a high-order mode is relatively small. The nonlinear coefficients of vector modes in photonic-crystal fiber are relatively small, all within 0.6 W−1 :km−1 –1.6 W−1 :km−1 . At a 1.55-μm wavelength, the nonlinear coefficient of HE8,1 mode is only 0.80175 W−1 :km−1 , compared with the results of Ref. [21] (2.08 W−1 :km−1 ) and Ref. [23] (0.833 W−1 :km−1 ). The mode

68

2 Fiber-Optic Mode Theory

Fig. 2.14 Schematic diagram of the effective mode-field area changes of vector modes in photoniccrystal fiber. a HE mode; b EH mode

Fig. 2.15 Variation diagram of the nonlinear mode coefficients in photonic-crystal fiber. a HE mode; b EH mode

nonlinear coefficient of photonic-crystal fiber in this study is relatively small, which is conducive to reducing the nonlinear effect in the fiber. This photonic-crystal fiber with a small nonlinear effect has many applications and broad prospects.

2.4.5 Ideal Mode The electromagnetic field in an actual optical fiber can be expressed, not only by the linear combination of a group of ideal modes, but also by the linear combination of a group of LP modes. An LP mode is not a degenerate mode of an ideal fiber but a coupled wave with residual coupling. The ideal mode of an ideal fiber can be obtained by solving the coupled-wave equation containing only the residual coupling. If there is no degeneracy between the coupled LP modes, the LP mode itself is an approximation of the corresponding ideal mode. If the LP-guided modes degenerate, only the composite mode is a good approximation of the ideal mode. If the guided LP mode and radiation mode degenerate, the corresponding ideal mode will have leakage loss (i.e., attenuation). Solving the coupled-wave equation, including

References

69

residual coupling and inherent coupling, can solve more complex problems, such as anisotropic optical fibers with various irregularities (e.g., micro-bending).

References 1. Wei J (2017) Theoretical research and design of vortex fiber. Beijing Jiaotong University, BeiJing, pp 30–35 2. Ung B, Vaity P, Wang L et al (2014) Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes. Opt Express 22(15):18044–18055 3. Wang L, Vaity P, Ung B et al (2014) Characterization of OAM fibers using fiber Bragg gratings. Opt Express 22(13):53–61 4. Zhang X (2016) Study on generation and regulation of vortex beam in optical fiber. University of Science and Technology of China, Anhui, pp 2–36 5. Zhang W, Huang L, Wei K et al (2016) High-order optical-vortex generation in a few-mode fiber via cascaded acoustically driven vector mode conversion. Opt Lett 41(21):2–5 6. Okamoto K (2001) Fundamentals of optical waveguides. Academic Press, New York, pp 51–70 7. Chen J (2002) Discussion on Maxwell equations. Phys Eng 12(4):18–20 8. Qiao H, Wang Y, Chen Z et al (2013) Analysis of waveguide modes with arbitrary cross section by full vector finite difference method. J Phys 62(7):24–31 9. Sun P (2016) Generation of vector vortex beam in optical fiber. Harbin University of Technology, Harbin, pp 22–27 10. Sun Y (2006) Optical fiber technology: theoretical basis and application. Beijing University of Technology Press, Beijing, pp 51–52 14. Orlov S, Stabinis A (2003) Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams. Opt Commun 226(1–6):97–105 15. Tian W (2017) Design of optical fiber carrying photon orbital angular momentum light wave mode. Beijing University of Posts and Telecommunications, Beijing, pp 19–20, 37–40 16. Xie X, Xu S (1989) General formula of Gaussian mode coupling efficiency. J Shaanxi Normal Univ (Natural-Sci Ed) 17(04): 1–16 17. Ding R (2018) Design and transmission performance of terahertz photonic crystal fiber. Lanzhou University of Technology, Lanzhou, pp 15–16 18. Dashti PZ, Alhassen F, Lee HP (2006) Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber. Phys Rev Lett 96(4):1–4 19. Kaneshima K (2006) Numerical investigation of octagonal photonic crystal fibers with strong confinement field. IEICE Trans Electron 89(6):830–837 20. Maji PS, Chaudhuri PR (2013) Circular photonic crystal fibers: numerical analysis of chromatic dispersion and losses. ISRN Opt 4(13):1–9 21. Xu H, Wu J, Xu K (2011) Ultra-flattened chromatic dispersion control for circular photonic crystal fibers. J Opt 13(5):994–1001 22. Tian W, Zhang H, Zhang X (2016) A circular photonic crystal fiber supporting 26 OAM modes. Opt Fiber Technol 30(6):184–189 23. Bai X, Chen H, Lingfei (2019) Annular photonic crystal fiber with orbital angular momentum mode transmission. Infrared Laser Eng 48(02):224–231

Chapter 3

Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

Under ideal conditions, the coupling performance of a spatial plane wave and Gaussian beam coupled into a single-mode fiber through a single lens is analyzed. The calculation formula for the coupling efficiency is deduced, and the influence of the relative aperture of the lens on the coupling efficiency is analyzed. The specific mathematical model of coupling-efficiency fading caused by assembly error is given and numerically analyzed. This lays a theoretical foundation for the installation of spatial-light single-lens single-mode fiber-coupling systems. The influence of noncommon optical-path errors on spatial light–fiber coupling efficiency is analyzed, and provides an experimental basis for spatial light–fiber coupling technology. Finally, the coupling efficiency of a Gaussian beam coupled into a single-mode fiber through a single lens is analyzed.

3.1 Plane-Wave Coupling Optical fiber is a dielectric waveguide and its cross section is usually circular, as shown in Fig. 1.7. The optical fiber uses the total-reflection principle to restrict the light-wave energy at the interface and guide the light wave to travel along the axis of the optical fiber. Its transmission characteristics are determined by the structure (core diameter Df ) and material (refractive indices n1 and n2 of the core and cladding, respectively). Owing to limitations at the core boundary, the electromagnetic-field solution is not continuous. This discontinuous field solution is called the mode [1]. In Fourier optics, the lens is the phase converter of a light wave [2]. Under ideal conditions, after passing through an ideal thin lens (without considering the influence of lens aberration), the plane wave becomes a converged spherical wave and converges at the lens focus. As shown in Fig. 3.1, if the optical-fiber axis coincides with the optical axis of the lens, and the end face of the optical fiber is placed at the focal plane of the lens, the light energy can be constrained in the optical-fiber © Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_3

71

72

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

Lens Lens focal plane

Incident light

Single mode fibe

f

Fig. 3.1 Schematic diagram of spatial plane-wave single-lens single-mode fiber coupling

interface using the total-reflection principle, and the light wave can travel along the axis of the optical fiber. If the fiber placed at the focus is a single-mode fiber, it belongs to the “spatial plane-wave single-lens single-mode fiber coupling” process. As shown in Eq. (3.1), the coupling efficiency η is defined as the ratio of the optical power P coupled into the optical fiber to the optical power P0 incident on the receiving plane of the optical system. The coupling efficiency is an important index for measuring the performance of coupling systems. / η = P P0

(3.1)

3.1.1 Geometrical-Optics Analysis of Coupling Efficiency As shown in Fig. 3.2, the plane wave is incident on the receiving end face, shaped and converged by a single lens, and then incident on the end face of the optical fiber. The main factors affecting the coupling efficiency are matching the signallight divergence angle and optical-fiber numerical-aperture angle and matching the spot diameter and optical-fiber core diameter. Without considering the reflection, absorption loss, and lens-phase difference, all the light incident within the fiber core and an incident angle that is less than the fiber numerical-aperture angle 2θ c can be coupled into the fiber. / 2θc = arcsin

n 21 − n 22 n0

(3.2)

where n0 , n1 , and n2 are the refractive indices of the environment where the optical fiber is located, the optical-fiber core, and the cladding, respectively.

3.1 Plane-Wave Coupling

73

Core:

Lens

refractive Index n1

Incident light

χ0

DA/2

θc

θd

I(x,y,z) f

Single mode Cladding: fibe refractive index n2

Fig. 3.2 Geometrical-optics diagram of spatial plane-wave single-lens single-mode fiber coupling

After setting the transmission distance L of the optical-signal beam, the lightintensity distribution at the end face of the lens is I(x, y, L) and the end face of the optical fiber is located at the focus of the single lens. The total optical power incident on the lens is { P0 =

DA/ 2

{

−D A / 2

DA/ 2 −D A / 2

I (x, y, L)d xd y

(3.3)

where DA is the diameter of the receiving lens. The optical power coupled into the fiber is given by { P=

f tan θc

− f tan θc

{

f tan θc − f tan θc

I (x, y, L)d xd y

(3.4)

where f is the focal length of the lens. Then, the coupling efficiency is { f tan θc { f tan θc P − f tan θ − f tan θ I (x, y, L)d xd y = { D 2c { D 2 c η= A/ A/ P0 −D A / 2 −D A / 2 I (x, y, L)d xd y

(3.5)

Analyzing the coupling efficiency using geometrical optics is intuitive; however, it is not suitable for the structural design and optimization of a coupling system. Geometrical-optics analysis is based on the premise that light is transmitted in a straight line, without considering the wave nature of light, especially the diffraction effect. The mode-field analysis method can obtain a more complete solution and can intuitively observe the influence of various physical parameters of the lens and optical fiber on the coupling system, which is convenient for the optimal design of various parameters.

74

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

3.1.2 Mode-Field Analysis of Coupling Efficiency As shown in Fig. 3.3, when coupling spatial light into an optical fiber, the diffraction effect of the optical-system aperture should be considered. In particular, when the incident-light wavelength is equal to the fiber-core diameter (when coupled with a single-mode fiber with a small core diameter) or the aperture of the optical system, the diffraction effect is large, and the analysis result using the mode-field matching method is more in line with the actual situation. In the mode-field analysis method, the coupling efficiency η is defined as [3] |2 |{ { ∗ | Ui (r )U f (r )r dr dθ | {{ η= {{ Ui (r )Ui∗ (r )r dr dθ × U f (r )U ∗f (r )r dr dθ

(3.6)

where U i (r) is the amplitude distribution of the signal light on the end face of optical fiber U i,B (r) or lens U i,A (r); and U f (r) is the distribution of the electromagnetic field of the single-mode optical fiber on the surface of optical fiber U f,B (r) or lens U f,A (r). (1) Electromagnetic-field distribution U f,B (r) of optical fiber As shown in Fig. 3.4, a single-mode fiber can only transmit the fundamental mode LP01 [1]. The electromagnetic-field distribution of the fundamental mode LP01 in fiber cross-section B is a zero-order Bessel function, which can be approximately expressed by a Gaussian distribution [4]: / U f,B (r ) =

[ ( ) ] 2 1 r 2 exp − π Wm Wm

(3.7)

Fig. 3.3 Schematic diagram of the coupled mode field of a spatial plane-wave single-lens singlemode fiber

3.1 Plane-Wave Coupling

(a) λ = 1550 nm

75

(b) λ = 1310 nm

Fig. 3.4 Electromagnetic-field distribution at the end face of a single-mode fiber

√ where r = x 2 + y 2 is the radial distance from any point on the cross section of the optical fiber to the center, and W m is the mode-field radius of the single-mode optical fiber. Usually, when λ = 1550 nm, the typical value of W m is 5.25 μm ± 0.5 μm; when λ = 1310 nm, the typical value of W m is 4.60 μm ± 0.25 μm; and when λ = 632.8 nm, the typical value of W m is 2.20 μm ± 0.25 μm [4]. (2) Electromagnetic-field distribution Ui,B (r ) of a plane-light incident-light field at the focus of a single lens (end face of optical fiber) In wireless-optical communication systems, a laser light source is often used as the signal light source. Because of the good directivity and monochromaticity of a laser, its wavefront can be regarded as a monochromatic plane wave after a long-distance transmission. Considering the limitation of the aperture DA of the receiving lens and the influence of lens aberration, the light-field distribution of the incident plane wave on the focal plane of the receiving lens can be expressed [5] using the Fresnel diffraction formula: ( )⎞ ⎛ ( ) 2J1 π D A r 2 λf DA ⎝ ikr 1 ⎠ exp(ik f ) exp( )π Ui,B (r ) = (3.8) π DAr iλf 2f 2 λf where f is the focal length of the coupling lens, DA is the diameter of the coupling lens, J 1 (·) is the first-order Bessel function of the first kind, and k = 2π/λ is the spatial-angular frequency (also known as the wave vector) of the wave. It can be seen that the electromagnetic-field distribution of the plane wave at the focus is related to the relative aperture DA /f of the lens and the wavelength λ of the incident light. As shown in Fig. 3.5, when DA = 10 mm and λ = 1550 nm, the light-field distribution of a plane wave passing through a single lens with different relative apertures at the focal plane is calculated using Eq. (3.8). It can be seen that if the lens diameter remains unchanged and the lens focal length increases, the light spot is increasingly seriously diffracted, and the light-field distribution is increasingly dispersed.

76

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

(a) λ = 1550 nm, DA = 10 mm), f = 20 mm

(b) λ = 1550 nm, DA = 10 mm, f = 50 mm

(c) λ = 1550 nm, DA = 10 mm, f = 100 mm Fig. 3.5 Electromagnetic-field distribution of a plane-wave incident-light field at the focal plane of a single lens [6]

(3) Coupling efficiency of a monochromatic plane-wave single-lens single-mode fiber The coupling efficiency η is calculated at the cross section of the optical fiber, ignoring the lens phase difference and the reflection and absorption losses. Assuming the ˜ monochromatic plane wave is incident and there is U f,B (r )U f,B ∗ (r )r dr dθ = 1, replace the electromagnetic field distribution at the end face of the optical fiber in Eq. (3.7) and the plane wave light field distribution at the focal plane in Eq. (3.8) into Eq. (3.6) to calculate the coupling efficiency: | |˜ ∗ | U (r )U f,B (r )r dr dθ |2 i,B η= ˜ ∗ Ui,B (r )Ui,B (r )r dr dθ | |{ | exp(−ik f ) exp[−ikr 2 /(2 f )]J1 [π D A r/(λ f )] exp[−(r/Wm )2 ]dr |2 {∞ (3.9) =4 Wm2 0 {J1 [π D A r/(λ f )]}2 /r dr As shown in Fig. 3.6, the influence of different lens relative apertures and different wavelength beams on coupling efficiency is simulated using Eq. (3.9). During the

3.1 Plane-Wave Coupling

77

simulation, DA=10mm is taken. If the relative aperture increases from small to large, the lens focal length decreases from large to small. From Fig. 3.6, it can be seen that: (1) ) As the relative aperture of the lens increases from small to large, there will be a peak in coupling efficiency. This is because the relative aperture is small, the focal length of the lens is long, and the energy distribution of the light spot is affected by diffraction effects and dispersed. From the perspective of geometrical optics, although the divergence angle of the beam is within the numerical aperture of the fiber, the area of the incident light spot is mismatched with the area of the fiber end face, resulting in a decline in the coupling efficiency. As the relative aperture gradually increases, the focal length becomes shorter, and the influence of diffraction decreases, the coupling efficiency gradually increases. However, if the relative aperture continues to increase and the focal length continues to decrease beyond a certain value, it will cause a mismatch between the beam incidence angle and the fiber numerical aperture, and also lead to a decrease in coupling efficiency. (2) Different wavelengths of incident beams require different coupling structures to cooperate with them. When the wavelength of the incident beam changes, in order to achieve a maximum coupling efficiency of 81.4%, a lens with a relative aperture of 0.211 needs to be selected for 1550nm light waves; For 1310nm light waves, a lens with a relative aperture of 0.203 needs to be selected.

Fig. 3.6 Electromagnetic-field distribution of a plane-wave incident-light field at the focal plane of a single lens [6]

78

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

3.1.3 Coupling Efficiency of Lens End Face As shown in Fig. 3.3, the light-field distribution at the end face of the lens and the light-field distribution at the focal plane satisfy a Fourier-transform relationship [7]. According to Parseval’s theorem [8], the following two coupling-efficiency calculation processes are equivalent: • Converting the incident-light field distribution to the end face of the optical fiber through a Fourier transform, and • Converting the electromagnetic-field distribution at the end face of the optical fiber to the end face of the lens through a Fourier transform. Therefore, the integration shown in Eq. (3.6) can be performed at the receivingaperture plane: |{ { ∗ |2 | Ui, A (r )U f, A (r )r dr dθ | {{ η= {{ Ui, A (r )Ui,∗ A (r )r dr dθ × U f, A (r )U ∗f, A (r )r dr dθ

(3.10)

where Ui, A (r ) is the light-field distribution on the receiving-lens plane and U f, A (r ) is the electromagnetic-field distribution of the single-mode fiber on the receiving-lens plane. It is assumed that the cross section of the optical fiber is perpendicular to the incident-light field and is located at the center of the focal plane of the lens. Because the Fourier transform of a Gaussian function is still a Gaussian function, the normalized optical-fiber mode-field distribution is converted to mode-field distribution U f,A (r ) on the receiving-lens surface as [9, 10] [ ( ) ] kWm 2 2 kWm U f, A (r ) = √ r exp − 2f 2π f

(3.11)

where k = 2π/λ is the wave vector, Wm is the radius of the optical-fiber mode field, f is the focal length of the lens, and r is the distance from any radial point on the lens surface to the center of the lens. Assuming that the fully coherent unit monochromatic-plane light wave is incident, that is, Ui, A (r ) = 1, the coupling efficiency of Eq. (3.10) can be expressed as

ηA =

| |{ 2π { D A / 2 | | 0 0

√kWm 2π f

|2 [ ( )2 ] | 2 m | r dr dθ exp − kW r | 2f

{ 2π { D A / 2 0

0

(3.12)

r dr dθ

Let a = π Wm D A /(2λ f ) be the coupling parameter [11], where D A is the diameter of the receiving aperture. Then, Eq. (3.12) can be simplified as )]2 [ ( 2 1 − exp −a 2 ηA = a2

(3.13)

3.2 Coupling-Efficiency Decline Caused by Assembly Error

79

Fig. 3.7 Coupling-efficiency curve on the end face of the optical fiber (ηB ) and the end face of the receiving lens (ηA )

The coupling-efficiency expression of Eq. (3.13) is simpler and more intuitive than that of Eq. (3.9). It can be seen from Eq. (3.13) that under ideal conditions, the coupling efficiency ηA is a function of the coupling parameter a. As shown in Fig. 3.7, if λ = 1550 nm and W m = 5.25 μm, the couplingefficiency curve is calculated using Eqs. (3.9) and (3.13), respectively. The coupling efficiency calculated by the two methods is the same; however, the expression for the coupling efficiency on the lens plane ηA is simpler. When DA /f = 0.211 and a = π Wm D A /(2λ f ) ≈ 1.12, a maximum coupling efficiency of 81.4% can be obtained.

3.2 Coupling-Efficiency Decline Caused by Assembly Error Assembly errors are inevitable during the actual installation of a coupling system. Assembly errors can be roughly divided into three categories [12]: radial error Δr , axial error Δz (defocus), and axis tilt error Δϕ.

3.2.1 Radial Error As shown in Fig. 3.8, the radial error indicates that when the coupling system is assembled, the optical axis of the optical fiber does not coincide with the optical axis of the coupling system—there is a radial error Δr —but the cross section of the optical fiber coincides with the focal plane of the lens. The electromagnetic-field distribution at the end face of the optical fiber is shown in Eq. (3.14):

80

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

/ U f,B (r ) =

[ ( ) ] 2 1 r + Δr 2 exp − π Wm Wm

(3.14)

Substituting Eqs. (3.14) and (3.8) into Eq. (3.6), the coupling-efficiency fading caused by radial error Δr can be expressed as

η B (Δr ) =

| |{ ∞ | | 0

[ ( ) ( ) )2 ] ||2 π DAr r +Δr exp(−ik f ) exp J1 λ f exp − Wm dr || [ ( / (3.15) )2 ] { ∞ ( π D A r )2 {∞ r +Δr r dr × 0 exp −2 Wm r dr 0 J1 λf (

1 −i

−ikr 2 2f

Figure 3. shows the declining curve of the coupling efficiency caused by a radialinstallation error Δr simulated using Eq. (3.15). It can be observed that the coupling efficiency decreases with an increase in the radial error. For a short-wavelength beam, the reduction in the coupling efficiency caused by the radial error is more significant. Fig. 3.8 Schematic diagram of radial error

Lens end face

Radial error

Optical fiber end face

Coupling efficiency η

Fig. 3.9 Coupling-efficiency fading caused by radial deviation [6]

Radial error

3.2 Coupling-Efficiency Decline Caused by Assembly Error

81

3.2.2 Axial Error As shown in Fig. 3.10, the axial error refers to the axial error (also known as defocus [13]) when the cross section of the optical fiber does not coincide with the focal plane of the lens, during the assembly of the coupling system. Without considering the influence of lens aberrations, the electromagnetic-field distribution at the end face of the fiber during defocus remains unchanged [as shown in Eq. (3.7)], and the light-field distribution of the incident light at the end face of the fiber changes. When considering the aperture limitation, the light-field distribution at the axial error Δz of the focus of a single lens is expressed as ] [ ikr 2 1 exp[ik( f + Δz)] exp Ui,B (r, Δz) = i λ( f + Δz) 2( f + Δz) ) [ ] [ ]} { ( 2 ikr0 ikr02 r0 × exp × exp − (3.16) × B circ D A /2 2( f + Δz) 2f where B{•} is the Fourier Bessel transform [14, 15]. It can be seen that when Δz → 0, the light-field distribution is the same as that in Eq. (3.8); that is, the light-field distribution at the focal plane of the plane light wave considering the aperture limit. Order: ) ( )} { ( ikr 2 Δz r0 × exp − 0 G(ρ) = B circ D A /2 2 f ( f + Δz) { D A /2 = 2π r0 0 ) ( Δz ikr 2 r × J0 (2πr0 ρ)dr0 |ρ= λ( f +Δz) × exp − 0 (3.17) 2 f ( f + Δz) ( ikr 2 We expand exp − 2 0 Fig. 3.10 Schematic diagram of axial error

Δz f ( f +Δz)

) at zero using Taylor’s formula:

Lens end face

Axial error

Optical fiber end face

82

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

) ( Δz ikΔz ikr02 =1+0− r2 + 0 + · · · exp − 2 f ( f + Δz) f ( f + Δz) 0

(3.18)

and use the first three items as approximations: ) ( Δz ikΔz ikr 2 ≈1− r2 exp − 0 2 f ( f + Δz) f ( f + Δz) 0

(3.19)

Then, G(ρ) {

DA/ 2

≈ 2π 0

ikΔz r0 × J0 (2πr0 ρ)dr0 − f ( f + Δz)

{

DA/ 2 0

r03

| | | × J0 (2πr0 ρ)dr0 | |

r ρ= λ( f +Δz)

(3.20) and by the properties of the Bessel function [16]: { x J0 (x) = x J1 (x) + C

(3.21)

x 3 J0 (x) = x 3 J1 (x) − 2x 2 J2 (x) + C

(3.22)

{

G(ρ) can be further simplified as ) ( J1 (πρ D A ) J1 (πρ D A ) D A 4 ikΔz G(ρ) ≈ − πρ D A 2 f ( f + Δz) πρ D A | ) ( J2 (πρ D A ) || ikΔz DA 3 + | 2 πρ f ( f + Δz) πρ D A | r π D 2A

(3.23)

ρ= λ( f +Δz)

Substituting Eq. (3.23) into Eq. (3.16), and Eq. (3.7) into Eq. (3.6), the couplingefficiency fading caused by the axial error Δz can be expressed as

η B (Δz) ≈

|{ | ∞ 4| 0

r i λ( f +Δz)

|2 ] [ [ ( )] 2 | G(ρ)dr exp − Wrm exp[ik( f + Δz)] exp 2( fikr | +Δz) {∞ 2 r Wm2 0 λ2 ( f +Δz) 2 G(ρ) dr (3.24)

Figure 3.11 shows the fading curve of the coupling efficiency caused by axial error Δz, calculated using Eq. (3.24). It can be observed that the coupling efficiency decreases with an increase in the axial deviation. Compared with Figs. 3.9 and 3.11,

3.2 Coupling-Efficiency Decline Caused by Assembly Error

83

Coupling efficiency η

Fig. 3.11 Coupling-efficiency fading caused by axial error [6]

Axial error

it can be seen that the coupling efficiency caused by radial error decreases more significantly, and the radial-alignment requirements are higher when assembling the coupling system.

3.2.3 Axis Tilt Error As shown in Fig. 3.12, when the coupling system is assembled, the cross section of the optical fiber is at the focal plane of the lens; however, an angle Δϕ is included between the optical axis of the optical fiber and the optical axis of the lens; this is the axis tilt error. At this time, the light-field distribution of the incident light on the end face of the optical fiber remains unchanged, and the electromagnetic-field distribution of the end face of the optical fiber deviates. As shown in Fig. 3.12, the end face of the optical fiber is circularly symmetrical. To simplify the calculation, only the rotation around the Y-axis is considered. The electromagnetic-field distribution of the rotating optical-fiber end face cannot be Fig. 3.12 Schematic diagram of the axis tilt error

Lens end face

On axis tilt error

84

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

Coupling efficiency η

Fig. 3.13 Coupling-efficiency fading caused by axis tilt error Δϕ [6]

Axis tilt error

expressed by a unified function in the original coordinate system; however, there is a relationship between the coordinates of the original electromagnetic-field distribution U f,B (x, ( y) of the) optical-fiber end face and the electromagnetic-field distribution U f,B x0 , y0 , Δϕ y after rotation, as shown in Eq. (3.25): ) ) ( ( ⎧ ⎨ x0 = x cos Δϕ y − U f,B (x, y) sin Δϕ y y =y ) ) ( ( ⎩ 0 U f,B (x0 , y0 , Δϕ y ) = x sin Δϕ y + U f,B (x, y) cos Δϕ y

(3.25)

Using Eq. (3.25), we recalculate the incident-light field distribution Ui,B (x0 , y0 ) in the coordinate system after rotation Δϕ y . The numerical simulation was performed using MATLAB, and the results are shown in Fig. 3.13. It can be seen that the short-wavelength beam has high requirements for on-axis yaw alignment. By studying the coupling-efficiency fading caused by radial, axial, and axis tilt errors, it was found that the coupling efficiency was most sensitive to radial error Δr , followed by axis tilt error Δϕ, and finally axial error Δz. When the wavelength of the communication beam is greater than 632.8 nm, to reduce the coupling-efficiency fading caused by installation to less than −5 dB, the three errors should be controlled ◦ within the ranges of Δr ≤ 2.2 μm, Δϕ ≤ 0.3 , and Δz ≤ 70 μm. When the communication wavelength increases, the installation tolerance requirements can be relaxed appropriately. Automatic alignment can also be adopted.

3.3 Adaptive-Optics System Errors

85

3.3 Adaptive-Optics System Errors Common errors in adaptive-optics (AO) systems include calibration, fitting, measurement, and bandwidth errors.

3.3.1 Calibration Errors The calibration error of an adaptive-optics system refers to the residual-beam error in the system in the absence of external aberrations [17]. Calibration errors σC2 AL I B generally include non-conjugate errors caused by the non-conjugate position of the wavefront detector and wavefront corrector, and non-common optical-path aberrations caused by the non-common optical path of the wavefront detector and imaging camera. When using adaptive-optics technology to correct a distorted beam, not only should the wavefront aberration measured by the wavefront sensor be corrected, but also the calibration error in the system should be compensated. When there is no calibration error or aberration in the adaptive-optics system, the point light source should be able to produce a perfect diffraction-limit image in the camera; however, this cannot occur because of optical aberrations in the common and imaging paths. In the calibration process, deformations can be added to the deformable mirror to compensate for these aberrations. The process of estimating and applying the required deformations is called image sharpening. Because some errors in the camera can be eliminated by the image-sharpening process, the camera is associated with calibration errors. Deformations introduced on the deformable mirror and defects in the lens array cause the wavefront sensor point to shift its standard position. The centroid generated by the standard position is defined as the reference centroid and subtracted from the measured actual centroid when the adaptive-optics loop is closed. If the reference centroid is inaccurate, for example, if the optical system in the adaptiveoptics system is misaligned, or the measurement of the reference centroid is noisy, additional calibration errors will be generated. The calibration error can be measured by comparing the closed loop and the Strehl ratio of point-source imaging without external aberrations. However, the image of a point light source contains significantly more information than the wavefront error. The phase-recovery algorithm estimates the amplitude and phase of the pupil using the intensity and pupil-size information on the image plane. The disadvantage of this type of algorithm is that if the pupil is symmetrical, the sign and direction of the phase are fuzzy; therefore, it is difficult to accurately measure the image difference using the phase-recovery algorithm. The phase-difference algorithm can solve this problem. The concept is to take two images, one on the focal plane and the other slightly out of focus. The additional information obtained can solve this blur problem. If the target is not a point light source, the target area can also be estimated.

86

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

Non-common path aberrations (NCPA) arise from the fact that the wavefront sensor and imaging camera are located in different optical paths. However, this static aberration cannot be measured using a wavefront sensor.

3.3.2 Fitting Errors A fitting error refers to a wavefront-aberration component that cannot be fitted by the wavefront corrector [17]. This error depends on the spatial characteristics of the aberration to be corrected and on the spatial characteristics of the wavefront corrector. A wavefront corrector can be regarded as a high-pass spatial filter in the frequency domain. The cutoff spatial frequency is given by the Nyquist criterion of the driver position; that is, the cutoff frequency is equal to the reciprocal of twice the spacing between adjacent drivers. Then, the wavefront corrector corrects the part whose spatial frequency is lower than the Nyquist criterion, and any spatial frequency higher than the Nyquist criterion will directly cause a fitting error. The calculation formula of the cut-off frequency f c can be expressed as [17] fc =

1 2δa

(3.26)

where δa is the distance between adjacent drives. When the influence function and wavefront aberration of the actuator are known, the fitting error can be obtained by making the actuator-influence function process the wavefront using the least-squares method.

3.3.3 Measurement-Noise Errors Measurement noise may occur when the wavefront sensor measures the wavefront slope. The error caused by this noise is the measurement error σ N2 O I S E in an adaptiveoptics system [18]. The dynamic characteristics of the AO system were modeled using the closed-loop control-block diagram shown in Fig. 3.14. As shown in Fig. 3.14, the control system has two inputs, wavefront aberration X ( f ) and noise N ( f ), which are assumed to be white noise. Similarly, there are two outputs: wavefront M( f ) after deformation-mirror correction and wavefront D( f ) after reconstruction. The reconstructed wavefront D( f ) in the control loop occurs immediately after the noise N ( f ) is added. For simplicity, we assume that the noise is input before the camera acquisition, which has little effect on the transfer function of the control loop. First, the camera of the wavefront sensor captures the wavefront to be corrected in a sampling period and then calculates the delay time τc , which corresponds to the delay between the time when the camera stops integrating and the time when the

3.3 Adaptive-Optics System Errors

Wavefront aberrationX(f)

87

+

+ _

Wavefront corrected by deformable mirrorM(f)

Zero order hold

+ Compensator

+

Delay

Camera acquisition

+ Reconstructed wavefrontD(f)

NoiseN(f)

Fig. 3.14 Closed-loop control-block diagram of an adaptive-optics system

voltage in the deformable mirror is updated. This includes the time required to read the charged-coupled device (CCD), calculate the centroid, multiply the centroid by the reconstruction matrix, and calculate the new voltage. The controller calculates the voltage to be applied, based on the previous voltage and reconstructed wavefront. Generally, the difference equation of the integral controller is [17] y[n] = y[n − 1] + K u[n]

(3.27)

where K is the variable-loop gain, y[n] is the output of the controller, and u[n] is the input of the controller at time n. The transfer function of the integral controller can be written as [17] HC O M P (z) =

K 1 − z −1

(3.28)

where z is the complex z-transform variable. Finally, the voltage of the input deformable mirror is kept constant during the sampling period, which is called the zero-order hold. The transfer functions of each module are as follows. 1. Camera acquisition and zero-order hold; sampling period T = 1/ f s , where f s is the sampling frequency: HST A R E (s) = H Z O H (s) =

1 − e−sT sT

(3.29)

2. Delay time τc : H D E L AY (s) = e−sτc

(3.30)

3. Integral compensator with gain: HC O M P (s) =

K 1 − e−sT

(3.31)

88

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

In the above equation, s = i2π f is the complex frequency variable, f represents the frequency, and i refers to the imaginary unit. In the following, all modules will take f as the parameter, because f has a more intuitive meaning than s and can be calculated directly by a discrete Fourier transform. Thus, the transfer function H ( f ) of the entire loop can be written as the product of all modules [17]: H ( f ) = HST A R E ( f )H Z O H ( f )H D E L AY ( f )HC O M P ( f )

(3.32)

Therefore, wavefront M( f ) after the deformation-mirror correction, and wavefront D( f ) after reconstruction can be expressed as [17] M( f ) =

H( f ) (X ( f ) + N ( f )) 1 + H( f )

(3.33)

(X ( f ) + N ( f )) 1 + H( f )

(3.34)

D( f ) =

In summary, we can use the following formula to express σ N2 O I S E : σ N2 O I S E =

∑ || H ( f ) ||2 2 | | | 1 + H ( f ) | |N ( f )|

(3.35)

3.3.4 Bandwidth Errors The reason for bandwidth errors is that the adaptive-optical system fails to correct the input dynamic-aberration component in time, which depends on the dynamic response of the wavefront controller and the dynamic change in aberration. The 2 can be expressed using Eq. (3.11 ) [17]: bandwidth error σ BW 2 σ BW

|2 ∑ || | H( f ) | | = | X ( f ) − 1 + H ( f ) X ( f )| | | ∑ | X ( f ) |2 | | = |1 + H( f )|

(3.36)

Combining Eq. (3.34) with Eq. (3.36), we obtain the bandwidth error: 2 σ BW

=

∑

(

) |2 | | | 1 2 | | |N ( f )| |D( f )| − | 1 + H( f )| 2

(3.37)

3.4 Non-common Path Aberrations

89

In general, the bandwidth error can be calculated by substituting the measured values of |N ( f )|2 and |D( f )|2 into Eq. (3.37).

3.4 Non-common Path Aberrations Unlike assembly errors, non-common path aberrations (NCPA) are caused by different optical paths of wavefront sensors and imaging cameras [19].

3.4.1 Research Status of Non-common Optical-Path-Aberration Calibration Different methods exist for detecting and calibrating non-common path aberrations: phase retrieval (PR) [20], phase diversity (PD) [21], and phase-diversity phaseretrieval (PDPR) [22]. In 1979, Gonsalves proposed the PR method, which is a theoretical and computer-simulation method for wavefront estimation using focalplane images. It estimates the wavefront passing through the imaging aperture by measuring the point-spread function of the system. This is called phase recovery because it determines the phase of the complex function by observing the modulus of the complex function [23]. In 1998, Kendrick calibrated the non-common optical-path aberration of a telescope adaptive-optical system using a phase-difference phase-recovery method [24]. In 2003, Blanc and Fusco analyzed the phase-difference technology in telescope static-aberration calibration, provided an experimental basis for the influence of static aberration on wavefront estimation [25, 26], and obtained a method to calibrate the static aberration of an adaptive-optics system. In 2013, Wang Zongyang et al. proposed a phase-difference technology for calibrating adaptive-optics telescopes. First, the system aberration was measured, and then the measured aberration was converted into the initial surface shape of the deformable mirror. The experiment showed that phase-difference technology is very important for the accurate detection and calibration of static aberrations [19]. In 2015, Wang Liang et al. used the PD method to convert non-common optical-path aberrations into the reference-centroid offset of a wavefront sensor and experimentally verified the procedure [27]. The calibration methods for non-common optical-path aberrations mainly include focal-plane sharpening and the phase-difference method. Focal-plane sharpening is a very useful method in non-common-path aberration calibration, which can increase the Strehl ratio by 40%. The phase-difference method is a promising calibration method. It does not require any continuous image-generation process and only needs to obtain a set of images in a few seconds.

90

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

3.4.2 Generation of Non-common Optical-Path Aberrations Figure 3.15 shows the optical path of an adaptive-optics system based on a coherent optical-communication system. It can be seen from the figure that the light output from the deformable mirror is first divided into two parts by a light-splitting prism. A part of the reflected beam enters the wavefront-sensor equipment to measure the slope-distribution signal, and another part enters the optical-communication terminal through the coupling lens. Owing to the difference in optical elements contained in the detection optical path and communication optical path, the aberrations caused by passing through the detection optical path and communication optical path are different, which leads to a difference between the wavefront detected by the wavefront sensor and the wavefront entering the optical-communication terminal. This wavefront difference is the non-common optical-path aberration. Although adaptive-optics technology can correct a distorted wavefront measured by the wavefront sensor, because of the non-common optical-path aberration, the wavefront entering the communication optical path still has an aberration, which results in the uncertainty of the communication optical-path wavefront and affects the communication quality of the communication system; hence, it must be calibrated. If the influence of non-common optical-path aberrations is not considered, the optical power coupled into a single-mode fiber will decrease after the system is a closed loop. Figure 3.16 shows the optical-power change curve of a 1550-nm signal light coupled into a single-mode fiber, directly after the closed-loop control-room experiment of a traditional adaptive-optics system when the non-common opticalpath aberration was not calibrated. It can be seen from the figure that when the system is a closed loop, the optical power decreases from −28 dBm before correction to − 52 dBm with an increase in iteration time. Fig. 3.15 Optical-path diagram of the adaptive-optics system of a coherent optical-communication system

Distorted wavefront Spectroscopic prism

Deformable mirror Control loop

Lens

Single mode fiber

Photodetector Communication optical path Detection light path

Wavefront controller

Wavefront sensor

3.4 Non-common Path Aberrations

91

Fig. 3.16 Optical-power variation curve during an adaptive-optics closed loop when a non-common optical-path aberration is uncalibrated

Optical power (dBm)

-25 -30 -35 -40 -45 -50 0

160

320 480 640 Time (Frames)

800

3.4.3 Conversion of Non-common Optical-Path Aberrations The non-common optical path aberration ϕncpa can be expressed as the difference between the wavefront ϕw f s at the wavefront sensor and the wavefront ϕsm f at the single-mode fiber [7]: ϕncpa = ϕw f s − ϕsm f

(3.38)

As shown in Fig. 3.14, if the wavefront at the wavefront corrector is ϕdm , the wavefront at the coupling lens is ϕw f s − ϕdm , and the wavefront at the lens of the AO control-correction loop is ϕw f s − ϕdm . Under ideal conditions, we should maximize the optical power coupled into the single-mode fiber; that is, the optical wave coupled into the single-mode fiber is a plane wave and the wavefront phase at the single-mode fiber is zero. At this time, the wavefront ϕsm f at the single-mode fiber requires a phase-shape variable of −ϕsm f , and the phase at the wavefront corrector is ϕdm − ϕsm f . After passing through the splitting prism and a 4f system composed of two lenses in the closed-loop control loop, the wavefront that finally reaches the wavefront detector is ,

ϕw f s = ϕdm − ϕsm f + ϕw f s − ϕdm = ϕw f s − ϕsm f

(3.39)

Combining Eqs. (3.38) and (3.39), we convert the non-common optical-path aberration into reference-point information measured by the wavefront detector. First, the initial control voltage of the deformable mirror is set to zero, a random disturbance voltage is applied to the actuator of the deformable mirror, and the reference-point information of the system is determined by an intelligent algorithm. When the optical power or voltage value coupled into the single-mode fiber reaches a stable state or the threshold set by the system, the iterative process is terminated. At the end of the iteration, the wavefront sensor is used to measure the wavefront slope, and the obtained wavefront slope information is the reference slope for the subsequent wavefront-correction system.

92

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

3.5 Gaussian-Beam Coupling The transmission distance of an optical signal in space is relatively short. If the beam emitted by a semiconductor laser is directly coupled into an optical fiber, the laser beam remains Gaussian [28, 29].

3.5.1 Coupling Efficiency As shown in Fig. 3.17, assuming that the collimated Gaussian beam is perpendicular to the lens, the electromagnetic-field distribution of the Gaussian light field on the end face of the lens is [30, 31] ) ( r2 UG,i (r ) = exp − 2 Ws

(3.40)

where Ws denotes the waist-spot radius of the Gaussian light field on the end face of the lens. The electromagnetic-field distribution of the optical fiber is converted to the surface of the receiving lens to calculate the coupling efficiency. Substituting Eqs. (3.40) and (3.11) into Eq. (3.10), the coupling efficiency of a Gaussian beam coupled into a single-mode fiber through a single lens can be expressed as follows: |2 | [ ( )2 ] ( 2) | |{ 2π { D A /2 kWm kWm r 2 | √ exp ω2 2π f exp − 2 f r r dr dθ || | 0 0 s ( 2) ηG = { 2π { D A /2 2r r dr dθ exp 0 0 ω2 s

The integral in Eq. (3.41) can be expressed as Fig. 3.17 Schematic diagram of a collimated Gaussian beam coupled into a single-mode fiber through a single lens

Lens

Optical fiber

(3.41)

3.5 Gaussian-Beam Coupling

93

Coupling efficiency

Fig. 3.18 Change of Gaussian-beam coupling efficiency with the relative aperture of the lens [6]

[ ( 2 ) ]2 D exp 4WA2 − a 2 − 1 s ηG = ( )2 [ ( 2 ) ] DA f kWm Ws exp − 1 − 2 kWm Ws 4f 2W

(3.42)

s

where a = π Wm D A /(2λ f ) is the coupling parameter, DA is the lens diameter, f is the focal length of the lens, and W m is the mode-field radius of the optical fiber. As shown in Fig. 3.17, the coupling efficiency of a spatially collimated Gaussian beam coupled into a single-mode fiber through a single lens was simulated using Eq. (3.42), with λ = 1550nm and W m = 5.25 μm. The lens diameter DA = 10 mm changes the lens focal length and obtains the coupling-efficiency curve under different relative apertures. The black-square curve in Fig. 3.18 is the coupling-efficiency curve of the incident plane beam, and the rest are the coupling efficiency curves of Gaussian beams with different mode-spot radii. It can be observed that the larger the waist-spot radius of the incident Gaussian beam, the higher the coupling efficiency, and the closer it is to the coupling efficiency when the plane light is incident. With a decrease in the waist-spot radius, the coupling efficiency gradually decreases.

3.5.2 Experiment of Manually Eliminating Non-common Optical-Path Aberrations The coupling efficiency of a Gaussian beam was studied experimentally using a He–Ne laser, a single lens with different relative apertures, and a single-mode fiber. The diameter of the single-mode fiber core/cladding used in the experiment was 0.072 μm. The radius of the mode field was 3.6–5.3 μm (at 633 nm), the numerical

94

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

aperture was approximately 0.10–0.14, and the single-mode working wavelength was 633–780 nm. The laser beam emitted by the He–Ne laser was expanded and collimated by a Newtonian telescope system composed of two lenses to form a spot with a ~20mm diameter, which was incident on the coupled single lens with different relative apertures. The optical fiber was placed at the focal point of the coupling lens (K9 glass Plano convex lens; see Table 3.2 for the parameters). The maximum coupling power was obtained by adjusting the optical-fiber adjusting frame. The optical power at the focal plane of the lens and the tail end of the optical fiber was measured with an optical power meter to calculate the coupling efficiency. See Table 3.1 for the specific parameters of the experimental instrument. See Table 3.2 for the experimental results. As shown in Fig. 3.19, the solid circle is the coupling-efficiency data measured in the experiment, the hollow circle is W s = 10 mm, DA = 10 mm, W m = 2.2 μm, and λ = 632.8 nm. The theoretical coupling-efficiency curve is calculated using Eq. (3.28), and the red-square curve is the coupling-efficiency curve fitted by the experimental data. It can be seen that the fitted coupling-efficiency curve is almost consistent with the theoretically calculated coupling-efficiency curve. From an experimental point of view, it is verified that there is an extreme value with an increase in the relative aperture of the lens; that is, there is an optimal relative aperture. Table 3.1 Experimental equipment for Gaussian-beam coupling efficiency [6] Component

Model

Main parameter

Laser

GY-10

Wavelength: 632.8 nm; exit pupil power: 3 mW

Optical power meter

PD-300UV (OPHIR)

Wavelength: 200–1100 nm; power detection: 20 pW–3 mW

Optical-fiber adjusting frame

GCX-M0101FC

Provides XY-direction deflection

Table 3.2 Experimental results of the Gaussian-beam coupling efficiency measurement [6] Sequence

Relative aperture of lens (DA /f )

Measured fiber-coupling optical power (mW)

Focal-plane power (mW)

Measured coupling efficiency

(a)

10/20 = 0.50

0.341

1.686

0.20

(b)

10/30 = 0.33

1.058

1.633

0.65

(c)

10/40 = 0.25

1.637

2.009

0.81

(d)

10/75 = 0.13

1.736

2.128

0.82

(e)

10/100 = 0.1

0.783

2.007n

0.39

Fig. 3.19 Variation curve of Gaussian-beam coupling efficiency measured experimentally with the relative aperture of the lens [6]

95

Coupling efficiency η

3.5 Gaussian-Beam Coupling

Measured single-mode fiber coupling efficiency point Theoretical curve of coupling efficiency of single-mode fiber with different numerical apertures Different numerical aperture, single-mode fiber coupling efficiency fitting curve

Relative aperture

3.5.3 Experiment of Automatically Eliminating Non-common Optical-Path Aberrations A coherent optical-communication system is shown in Fig. 3.20. The closed-loop correction effect of the adaptive optics directly affects the optical power coupled into a single-mode fiber, and the size and fluctuation of the optical power directly affect the coherence efficiency of the coherent receiver. This section focuses on the influence of the AO closed-loop correction after non-common optical-path aberration calibration on the optical power coupled into a single-mode fiber in a field experiment of a coherent optical-communication system under different weather conditions. Figure 3.21a–c show the variation curves of optical power coupled into a singlemode fiber with iteration times in open and closed loops on cloudy, sunny, and rainy days, respectively. As shown in Fig. 3.21a, in the closed-loop process, the optical

Transmitting antenna Single mode fiber

Atmospheric turbulence

Receiving antenna

Spectroscopic prism Lens L1

Control box Deformable mirror Mixer Focusing lens

Double balance detector

Lens L2

PC Laser

Wavefront sensor

Fig. 3.20 Schematic diagram of a coherent optical-communication system [32]

3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

Optical power (dBm)

(a) -25

Open loop Closed loop

-30 -35 -40 -45 -50

0

120

240 360 Time (Frames)

480

600

(b) -25 Optical power (dBm)

96

Open loop Closed loop

-30 -35 -40 -45 -50

0

200

400 600 Time (Frames)

800

1000

Optical power (dBm)

(c) -25 Open loop Closed loop

-30 -35 -40 -45 -50

0

200

400 600 Time (Frames)

800

1000

Fig. 3.21 Optical-power curve coupled into a single-mode fiber under different weather conditions [32]. a Clouds; b sun; c rain

power coupled into the single-mode fiber continues to increase and finally reaches a stable state after frame 70 of the closed-loop control. Upon a statistical analysis, after the adaptive optics stabilized the closed loop, the optical power increased from −41.54 dBm in the open loop to −30.03 dBm in the closed loop, an increase of 11.51 dBm, and the variance decreased from 0.270 to 0.052. It can be seen from Fig. 3.21b that during the closed-loop process, the system basically reaches a stable state after the initial 76-frame closed-loop adjustment. Upon a statistical analysis, after the adaptive optics stabilized the closed loop, the optical power increased from −44.20 dBm in the open loop to −33.41 dBm in the closed loop, increasing by 10.79 dBm, and the variance decreased from 1.81 to 0.97, a decrease of 0.84. As can be seen from Fig. 3.21c, in the closed-loop process, the optical power coupled into the single-mode fiber reaches a stable state after the initial 127-frame closed-loop correction. Upon a statistical analysis, after adaptive optics stabilized the closed loop, the optical power increased from −43.72 dBm in the open loop to − 34.60 dBm in the closed loop, an increase of 9.12 dBm, and the variance decreased from 2.82 to 1.35. Comparing the improvement effect of adaptive-optics closed-loop correction on cloudy, sunny, and rainy days, it was found that the improvement effect on cloudy days

3.6 Summary

97

was significantly better than that on sunny and rainy days, and that on sunny days was better than that on rainy days. This improvement is mainly reflected in the improvement of the average optical power and reduction of the wavefront jitter. Through experiments, it was found that if the optical power coupled into the single-mode fiber decreased sharply, the intermediate-frequency signal output by the coherent receiver also decreased sharply, and the adaptive-optics closed-loop correction after non-common optical-path aberration calibration significantly improved the optical power value coupled into the single-mode fiber.

3.6 Summary This chapter focused on an analysis of the coupling efficiency of spatial plane waves and Gaussian beams coupled into a single mode through a single lens under ideal conditions. The coupling-efficiency expression of a spatial plane wave coupled into a single-mode fiber through a single lens when only considering the aperture limitation was derived, and the method of calculating the Gaussian-beam coupling efficiency under the same condition was given. The following conclusions were drawn: 1. Under ideal conditions, only the aperture-diffraction effect was considered. When the spatial plane wave was coupled into the single-mode fiber through a single lens, the coupling efficiency changed with the coupling parameter, a = π Wm D A /(2λ f ). When a = 1.12, the coupling efficiency reached an extreme value of 81.4%. 2. The coupling efficiency of the spatial plane-wave single-mode fiber-coupling structure was most sensitive to radial error Δr , followed by axis tilt error Δϕ, and finally, axial error Δz. When the wavelength of the communication beam was greater than 632.8 nm, to reduce the coupling-efficiency fading caused by installation to less than −5 dB, the three errors should be controlled within the ◦ range of Δr ≤ 2.2 μm, ·Δϕ ≤ 0.3 , and Δz ≤ 70 μm. 3. The coupling efficiency of a spatial Gaussian beam coupled into a single-mode fiber through a single lens was related, not only to the relative aperture of the lens and the mode-field radius of the fiber, but also to the mode-spot radius of the incident Gaussian beam on the surface of the lens. When the other parameters were constant, a larger mode-spot radius resulted in a higher coupling efficiency. 4. The non-common optical-path error in an adaptive-optics system can be automatically eliminated using a specific program to improve the spatial light–fiber coupling efficiency.

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3 Single-Lens Single-Mode Fiber Coupling Under Ideal Conditions

References 1. Yang Y (2007) Fiber optics. Tsinghua University Press, Beijing, pp 13–15 2. Lv N (2006) Fourier optics. China Machine Press, Beijing, pp 95–95 3. Leeb WR, Winzer PJ, Kudielka KH (1998) Aperture dependence of the mixing efficiency, the signal-to-noise ratio, and the speckle number in coherent LiDAR receiver. Appl Opt 37:3143– 3148 4. Shao X, Zhang X, Wu J et al (1986) Study on mode field radius of single-mode fiber. J Commun (03):49–54 5. Ke Deng, Bingzhong Wang, Xv Wang, et al. Factor analysis of coupling efficiency between space light and single-mode fiber. J Univ Electron Sci Technol 36(5):889–891 6. Lei S (2016) Research on optical coupling and beam control technology in free space optical communication. Xi’an University of Technology, Xi’an 7. Steward EG (1983) Fourier optics. Halsted Press, New South Wales 8. Li Z (1986) Application of Parseval theorem in Fourier optics. J Sichuan Normal Univ (NaturalSci Edn) 12(01):75–80 9. Li S (2014) Research on key technologies of all fiber spectroscopic rotational Raman temperature measurement LiDAR system. Xi’an University of Technology, Xi’an 10. Gisin N, Passy R, Perny B (1993) Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field. J Lightwave Technol 11(11):1875–1883 11. Dikmelik Y, Davidson FM (2005) Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence. Appl Opt 44(23):4946–4952 12. Chen H, Yang H, Li T et al (2011) Effect of fiber offset on coupling efficiency of spatial light single-mode fiber. Laser Infrared 41(1):75–78 13. Du Y, Wang X (2016) Evolution of defocus aberration in unstable cavity and intracavity compensation. J Opt 36(02):129–134 14. Candel SM (1981) An algorithm for the Fourier-Bessel transform. Comput Phys Commun 23(04):343–353 15. Ghobber S, Jaming P (2011) Strong annihilating pairs for the Fourier-Bessel transform. J Math Anal Appl 377(02):501–515 16. Liu Z, Wang S (2003) Complex function and mathematical equation. Hubei Science and Technology Press, Wuhan 17. Porter J (2005) Adaptive optics for vision science: principles, practices, design and applications. Wiley-Interscience 17(09):189–202 18. Wang L (2016) Research on adaptive optics testing and system optimization. Graduate School of Chinese Academy of Sciences, Changchun Institute of Optics, Precision Machinery, and Physics 19. Wang Z, Wang B, Wu Y et al (2012) Calibration of static aberration of non common optical path by phase difference technique. J Opt 32(07):41–45 20. Fienup JR (1993) Phase-retrieval algorithms for a complicated optical system. Appl Opt 32(10):1737–1746 21. Paxman RG, Schulz TJ, Fienup JR (1992) Joint estimation of object and aberrations by using phase diversity. J Opt Soc Am A 9(7):1072–1085 22. Dean BH, Bowers CW (2003) Diversity selection for phase-diverse phase retrieval. J Opt Soc Am A 20(8):1490–1504 23. Wild WJ (1998) Linear phase retrieval for wave-front sensing. Opt Lett 23(8):573–575 24. Yue D, Xu S, Nie H (2015) Co-phasing of the segmented mirror and image retrieval based on phase diversity using a modified algorithm. Appl Opt 54(26):7917–7924 25. Blanc A, Fusco T, Hartung M et al (2003) Calibration of NAOS and CONICA static aberrations application of the phase diversity technique. Astron Astrophys 399(1):373–383 26. Blanc A, Mugnier LM, Idier J (2003) Marginal estimation of aberrations and image restoration by use of phase diversity. J Opt Soc Am 20(6):1035–1045

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Chapter 4

Spatial Plane-Wave Single-Lens Single-Mode Fiber Coupling in Weakly Turbulent Atmospheres

The basic concept of the turbulence spectrum and the method of calculating the optical-field distribution of light transmission in a turbulent flow are introduced, and the basic definitions of the plane-wave correlation and cross-correlation functions are provided. The inter-correlation and cross-correlation functions are used to derive the coupling efficiency and the relative variance of the coupling optical power of the “spatial plane-wave single-lens single-mode fiber” coupling structure under the von Kármán turbulence spectrum, and to analyze their effects on the bit-error rate of an on–off keying (OOK)-modulated wireless-optical communication system.

4.1 Light-Field Distribution and Refractive-Index Power Spectrum in Atmospheric Turbulence The diffraction, attenuation, atmospheric-turbulence, and thermal-corona effects of an atmospheric channel all affect the signal beam; however, the influence mechanisms are different. This section focuses on the random characteristics of the optical-field distribution of a laser beam in atmospheric turbulence and its statistical description.

4.1.1 Born Solution for the Light-Field Distribution in Atmospheric Turbulence As shown in Fig. 4.1, atmospheric turbulence is an irregular random motion in the atmosphere that commonly exists within the boundary layer at the bottom of the atmosphere, inside cloud masses in the troposphere, and within westerly rapids in the upper troposphere [1]. © Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_4

101

102

4 Spatial Plane-Wave Single-Lens Single-Mode Fiber Coupling … Turbulence Optical fiber Laser

Collimation system

Transmitting plane Z=0

Z=L Receiving plane

Fig. 4.1 Laser beams encountering atmospheric turbulence

Atmospheric turbulence randomly perturbs the propagation of light waves through the atmosphere. The random rise and fall of physical properties, such as pressure, velocity, and temperature, at each point of turbulence cause random fluctuations in the atmospheric refractive index. The random fluctuation of the refractive index directly creates different light distances through which different rays of the beam will pass. { The optical range is the integral ( L n(s)ds) of the curve of the refractive index n(s) over the beam-transmission path L at each point s on the transmission path and can characterize the amount of delay in the wavefront phase; it is also called the optical thickness. The difference in the optical range of different rays in a beam causes aberrations in the complex amplitude distribution of the incident beam in the receiving plane (wavefront aberration). Wavefront distortion directly causes a mismatch between the incident-light field distribution and the electromagnetic field (EMF) distribution at the fiber end face, causing a decrease in the coupling efficiency. Random fluctuations of the atmospheric refractive index caused by turbulence also result in beam drift, reduced spatial coherence, and random fluctuations of the light intensity and phase [1]. It has been found that large-scale turbulence (greater √ than the beam diameter) causes the beam to wander, and small-scale turbulence ( L/k) in the first Fresnel region causes light-intensity scintillations [2]. The fundamental properties of light waves propagating in an unbounded continuous medium with a slowly varying refractive index can be described by Maxwell’s equations. In polar coordinates and without considering cross-polarization effects, the electromagnetic-field distribution U(R) in its cross section perpendicular to the transmission direction (Z-axis) satisfies the stochastic Helmholtz equation [2] ∇ 2 U (R) + k 2 n 2 (R)U (R) = 0

(4.1)

where R = (x, y, z) is the space vector and ∇ 2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 + ∂ 2 /∂ z 2 is the Laplace operator. n(R) = 1 + 77.6 × 10−6 (1 + 7.52 × 10−3 λ−2 )P(R)/T (R) is the refractive index at any point in space (the change in refractive index with time can be neglected for light waves), where P(R) is the pressure at point R (millibars), and T (R) is the temperature at point R (Kelvin). Owing to the presence of turbulence, the temperature and pressure vary randomly at each point in the atmospheric channel; therefore, n(R) is generally described statistically in turbulence.

4.1 Light-Field Distribution and Refractive-Index Power Spectrum …

103

Green’s function has been used to solve Eq. (4.1) [2], and later the Born and Rytov perturbation methods were proposed to approximate the solution [3]. In turbulent atmospheres, n(R) is a random field that is usually described statistically and can be expressed as [2] n(R) = n 0 + n 1 (R)

(4.2)

where n 0 = ⟨n(R)⟩ ∼ = 1 (⟨•⟩ represents the average of the system), n 1 (R) represents the randomly varying part of the refractive index, and ⟨n 1 (R)⟩ = 0. Then n 2 (R) in Eq. (4.1) can be expressed as [2] n 2 (R) = [n 0 + n 1 (R)]2 ∼ = 1 + 2n 1 (R) |n 1 (R)| 1, the optical power of the array coupled into the fiber exceeds the coupled optical power of the lens, and a conclusion consistent with Fig. 4.18 can be drawn. As shown in Fig. 4.20, Eqs. (4.73) and (4.60) were substituted into Eq. (4.74) for a communication distance of L = 5 km, while varying the turbulence intensity, and numerical simulations were performed to derive the curve of the power-boost coefficient versus the communication distance. It can be seen that the coupling efficiency of the array will be higher than that of a single lens of equal area, when the communication wavelength is 650 nm and the turbulence intensity exceeds 1.14 × 10–15 m−2/3 , when the communication wavelength is 1310 nm and the turbulence intensity exceeds 3.75 × 10–15 m−2/3 , and when the communication wavelength is 1550 nm and the turbulence intensity exceeds 5.3 × 10–15 m−2/3 . The above three turbulence intensities are in the range of moderately

133

Power improving factor K

4.4 Spatial Optical Coupling of Lens Arrays in Atmospheric Turbulence

Distance L(m) Fig. 4.19 Power-improving factor K (Parray /Plens ) as a function of the communication distance under moderately strong turbulence [15]

Power improving factor K

Fig. 4.20 Power-improving factor K (Parray /Plens ) as a function of turbulence intensity (communication distance L = 5 km) [15]

134

4 Spatial Plane-Wave Single-Lens Single-Mode Fiber Coupling …

strong turbulence; therefore, lens arrays can be applied to wireless-optical systems at long distances and with high turbulence intensities.

4.4.2 Coupling Experiment The experimental scheme shown in Fig. 4.21a was designed. The beam was collimated and expanded using a Mak-Cass antenna so that it could completely cover the array and lens-coupled system placed side-by-side at the receiving end. A multimode fiber with a 1-mm core diameter was used for the outfield experiments to facilitate the alignment and reduce the addition of arrays engineering precision (single-mode fiber radial-alignment error < 2.2 μm). As shown in Fig. 4.21b, experimental site 1 was from the east gate of Xi’an University of Technology to the south side of the soccer field, with a straight-line distance of 407 m, using a semiconductor laser with 10 mW of power and a 650nm wavelength. Experimental site 2 was from Lab 820 on the sixth floor of the teaching building of Xi’an University of Technology to the 16th floor of the Kaisen Fujingayuan building on the second ring road of Xi’an city, with a straight-line distance of 1.3 km, using a semiconductor laser with 30 mW of power and a 650-nm wavelength. The experimental times were December 13 and 14, 2014. The average values of the optical power of the lens array and tail fiber of the lenscoupling system within 1 min were measured every 50 m in the range of 0–400 m

Test site L=437m

XAUT Football Field

East Gate of XAUT

Laboratory 820, Sixth floor, XAUT

Array coupling

Beam expanding system

Test site

Lens coupling Laser

Kaisen community

Power Detection

Power Detection

(a) Fig. 4.21 Experiment system with lens array and lens

(b)

L=1.3 km

4.5 Summary

135

Table 4.4 Power-improvement factor K, theoretically and experimentally * Laser power

Distance

10 mW λ= 6 nm

5m

30 mW

Theoretical calculation of K

Experimentally measured K

C n 2 = 10–14 (m−2/3 )

C n 2 = 10–16 (m−2/3 )

⟨PZ ⟩ Array (mW)

⟨PA ⟩ Lens (mW)

K

0.80

0.7778

1.21

1.57

0.77

100 m

0.82

0.7778

1.03

1.39

0.74

150 m

0.85

0.7778

0.85

1.13

0.75

200 m

0.88

0.7787

0.68

0.86

0.79

250 m

0.91

0.7787

0.47

0.64

0.73

300 m

0.95

0.7787

0.29

0.35

0.83

400 m

1.02

0.7797

0.17

0.19

0.89

1300 m

1.68

0.7807

2.77

2.67

1.03

at experimental site 1, and an overhead long-distance experiment was conducted at experimental site 2. In the experiment, the effective unit of the lens array was seven single lenses with 10-mm diameters and 50-mm focal lengths, and the total diameter of the array structure was 30 mm. The lens-coupling system was a single lens with a 30-mm diameter and 150-mm focal length. Owing to the experimental turbulence-intensity diagnostic and the same commissioning method, it is obvious from the data in Table 4.4 that the power decreases significantly faster when using a single-lens coupling structure coupled into the fiber. The experimental data from experimental site 2, that is the 1.3-km outfield experiment, directly demonstrate that the mean value of the optical power coupled into the lens array is significantly higher than that of a single lens of equal area. The coupling effect of the lens array under strong turbulence at long distances is significantly better than that of a single lens with equal area.

4.5 Summary In this chapter, the coupling effect of space plane waves coupled into singlemode fibers via lenses and lens arrays under atmospheric-turbulence conditions was analyzed. The following conclusions were drawn. The coupling efficiency of a space plane-wave single-lens single-mode fibercoupling structure in a turbulent atmosphere was inversely proportional to the ratio of the lens diameter to the atmospheric coherence length. The relative variance of the coupled optical power of the spatial plane-wave singlelens single-mode fiber-coupling structure in a turbulent atmosphere increased with an increase in the turbulence intensity. When the transmission distance and turbulence intensity were constant, the relative variance of the coupled optical power undulation,

136

4 Spatial Plane-Wave Single-Lens Single-Mode Fiber Coupling …

with the increase in lens diameter, showed a trend of first decreasing, then increasing and then decreasing. The experiments verified that the coupling-power probability-density distribution function obeyed the lognormal distribution under weak turbulence. When choosing a suitable lens structure, the coupling structure only affected the mean value of the coupling power and did not affect the size of the relative variance of the coupling optical-power undulation. Moreover, the size of the variance was consistent with the size of the optical-power scintillation in the focal plane of the lens. The coupling efficiency of the lens array for a transmission distance over a certain range or turbulence intensity exceeded a certain value, and the coupling efficiency of the array coupling structure was greater than that of a single-lens coupling structure of equal area.

References 1. Rao RC (2005) Light propagation in turbulent atmospheres. Anhui Science and Technology Press, Hefei, pp 117–149 2. Andrews LC, Phillips RL (2005) Laser beam propagation through random media. Oxford University Press, Washington, pp 145–181 3. Shi W (1986) Wave propagation and scattering in random media. Translated by Huang Ruiheng. Science Press, Beijing, pp 383–387 4. Parry G, Pusey PN (1994) K distributions in atmospheric propagation of light laser. J Opt Soc Am 69(5):796–798 5. Wang LG (2014) Characteristics of reflected wave from targets illuminated by laser beams in turbulent atmosphere. Xidian University, Xi’an, pp 14–17 6. Nor NAM, Islam MR, Al-Khateeb W et al (2013) Atmospheric effects on free space earth-tosatellite optical link in tropical climate. Int J Comput Sci Eng Appl 3(1):17 7. Zhang YX (2002) Propagation and imaging of light in random media. National Defense Industry Press, Beijing, pp 114–161 8. Tatarskii VI (1961) Wave propagation in a turbulent medium. Dover Publication, New York, pp 89–101 9. Tan L, Zhai C, Yu S, Cao Y et al (2014) Fiber-coupling efficiency for optical wave propagating through non-Kolmogorov turbulence. Opt Commun 23(6):291–296 10. Dikmelik Y, Davidson FM (2005) Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence. Appl Opt 44(23):4946–4952 11. Andrews LC (1992) Special functions of mathematics for engineers, 2nd edn. Oxford University Press, Washington, pp 10–87 12. Andrews LC, Vester S (1993) Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations. J Mod Opt 40(5):931–938 13. Mei HP, Wu XQ, Rao RZ et al (2006) Measurement of inner and outer scale of atmospheric optical turbulence in different areas. High Power Laser Particle Beams 18(3):362–366 14. Tatarskii VI (1961) Wave propagation in a turbulent medium. McGraw–Hill, New York 15. Lei SC (2016) The coupling and beam control technology in free-space optical communication. Xi’an University of Technology, Xi’an, pp 40–45 16. Wang ML, Heng Y (2001) The sample mean Monte Carlo method for approximate integral using measure theory. J South China Normal Univ (01):50–52 17. Li F, Wu Y, Hou ZH (2012) Analysis and Experimental research on bit error rate for free-space optical communication systems through turbulent atmosphere. Acta Optical Sinica 39(06):28– 33

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18. Durbin PA, Pettersson Reif BA (2011) Statistical theory and modeling for turbulent flows. John Wiley & Sons, Hoboken 19. Ji Y, Yue P, Yan RQ et al (2016) BER performance analysis of the atmospheric laser communication system on the slant path in weak turbulence. J Xidian Univ (Natural-Science Edition) 43(1):66–70 20. Du WH, Chen FZ, Yao ZM et al (2013) Influence of non-Kolmogorov turbulence on bit-error rates in laser satellite communications. J Russ Laser Res 34(4):351–355 21. Lei SC, Zheng XZ (2015) Coupling efficiency of lens array spatial light in turbulence. Chin J Lasers 42(06):179–186

Chapter 5

Automatic Fiber-optic-coupling Alignment System

Spatial optical coupling is a key technology in wireless-optical communication systems. Highly efficient coupling can directly improve communication quality, and using automatic alignment can significantly reduce the coupling-alignment difficulty. This chapter presents the design of a closed-loop device for optical fiber coupling based on piezoelectric ceramics. Piezoelectric ceramics, driver circuits, controllers, photodetectors, and other peripheral devices form an optoelectronic circuit, and a simulated-annealing algorithm is used to automatically align spatial light with an optical fiber.

5.1 Auto-alignment Systems This chapter presents the design of an automatic-alignment system for spatial optical coupling, including the design of a piezoelectric-ceramic power supply, driver circuit, and operational amplifier circuit.

5.1.1 Principle of the Auto-alignment System The technical focus of spatial optical single-mode fiber coupling is to determine the exact position of the light spot on the end face of the fiber. A two-dimensional piezoelectric-ceramic piece is used to control the optical fiber; the voltage applied to the piezoelectric ceramic causes motion on the order of microns, which adjusts the position of the fiber. A simulated-annealing algorithm is used to automatically align the spatial optical-fiber coupling and uses precise positioning to find the best coupling point.

© Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_5

139

140

5 Automatic Fiber-optic-coupling Alignment System

The simulated-annealing algorithm accepts the Metropolis criterion and, with a certain probability, is able to select some edge states that make the result worse; i.e., this enables the algorithm to jump out of a local minimum and thus, find the global optimum. The high accuracy of the piezoelectric-ceramic drive circuit and the reliability of the piezoelectric-ceramic fixation method designed in this spatial optical coupling auto-alignment system are of great importance for the study of optical-fiber coupling. As shown in Fig. 5.1, the designed automatic spatial optical-fiber couplingalignment system mainly consists of a light source, optical system, and optical fiber, and an alignment control system that consists of a two-dimensional piece of piezoelectric ceramic, a feedback system, and a control algorithm. A photodetector is used as an evaluation indicator to provide a real-time feedback voltage, which in turn drives the two-dimensional piezoelectric-ceramic piece, forming a complete photoelectric closed-loop control system. Piezoelectric ceramics convert electrical energy into mechanical energy. Depending on the voltage applied to the piezoelectric ceramic, the piezoelectric crystal stretches and changes; thus, driving its displacement. The specific working principle is as follows: The two-dimensional piezoelectric-ceramic piece and the optical fiber are fixed to the coupling end of the optical fiber, and the piezoelectric ceramic produces micro-displacements in the optical fiber. The optical signal coupled into the optical fiber acts as an output to the surface of the photodetector, thereby completing the photoelectric conversion. The control unit detects the strength of the electrical signal, according to the photodetector, and uses an optimization algorithm to output a voltage-signal command to act on the piezoelectric ceramic piece. The resulting voltage drives the displacement of the fiber-optic end surfaces affixed to the two-dimensional piezoelectric ceramic, thus enabling it to find the optimum coupling position. The control process is as follows: The specific value of the voltage signal is acquired from the photodetector as a feedback quantity, and combined with a simulated-annealing algorithm to control the movement of the two-dimensional piezoelectric ceramic piece. This enables the real-time adjustment of the fiber position to match the optical and mode fields in real time to achieve the maximum coupling efficiency.

5.1.2 Automatic-alignment System Components As shown in Fig. 5.2, the hardware module of the system described in this chapter consists mainly of an STM32F103 microcontroller, piezoelectric ceramic piece, photodetector, and piezoelectric-ceramic driver circuit, protection circuit, and voltage-amplifier circuit. In terms of the overall system function, the system is divided into three modules: data acquisition, data processing, and driving the piezoelectric-ceramic piece.

5.1 Auto-alignment Systems

141

Optical systems Control systems Incidence

Feedback system

Optical fiber

Light source

Optical Power Meters

Intelligent algorithms

Two-dimensional stepper motors/ Piezoelectric ceramics

(a) Optical systems Control systems

Feedback system

Incidence Light source

Photoelectric detectors

Intelligent algorithms

Two-dimensional stepper motors/ Piezoelectric ceramics

(b) Fig. 5.1 Spatial optical-fiber coupling auto-alignment system DC regulated power supplies

Computers

Protection circuits

Piezoelectric ceramic drive power supplies

STM32 systems

A/D

Fig. 5.2 System hardware-module design diagram

Photoelectric detectors

PZT

142

5 Automatic Fiber-optic-coupling Alignment System

The functions of the three modules are as follows: The acquisition module acquires the voltage from the photodetector into the microcontroller in real time and converts the digital quantity into the corresponding analog voltage (digital-to-analog conversion). The output high-impedance, small-voltage signal is amplified using the amplification circuit to produce a control voltage in the driving range to drive the piezoelectric ceramic piece (PZT); the voltage-amplification circuit provides a proportional amplification link. In the main processor, the piezoelectric-ceramic drive circuit applies voltage to both ends of the piezoelectric ceramic piece to cause micro-displacements. The control system uses a photodetector to capture this micro-displacement signal, and then converts the light signal into an electrical signal and sends it to the microcontroller to form a closed-loop system.

5.1.3 Piezoelectric Ceramics Piezoelectric ceramics are special ceramic materials that convert electrical energy into mechanical energy based on their internal piezoelectric effect. Piezoelectric ceramics are piezoelectric, dielectric, and cause spontaneous polarization. Depending on the magnitude and direction of the electric field applied to the piezoelectric ceramic, the piezoelectric ceramic can be deformed to different degrees of displacement. Because of the dielectric nature of piezoelectric ceramics, when the applied electric field changes, if the applied electric field is in the same direction as the ceramic polarization, the polarization strength increases, causing the piezoelectric ceramic to move in the polarization direction; conversely, if the opposite voltage is applied, it will move in the opposite direction. The magnitude of the displacement is proportional to the voltage applied to the piezoelectric ceramic. By precisely adjusting the displacement of the piezoelectric ceramic, the charge distribution inside the piezoelectric ceramic can be steered to regulate the optical-fiber coupling.

5.2 Basic Principles of Control Algorithms 5.2.1 Basic Principles of Simulated-annealing Algorithms The simulated–annealing (SA) algorithm, first proposed by Metropolis and inspired by the annealing process [1], is a stochastic optimization search algorithm based on the Monte Carlo iterative-solution strategy [2]. This can enable the system to overcome the optimization process falling into a local minimum by jumping out of it, or it can enable the system to overcome the initial value dependence.

5.2 Basic Principles of Control Algorithms

143

The simulated-annealing method uses the Metropolis probability criterion to perform a random search in the feasible-solution space, repeatedly sampling and comparing the difference between two evaluation functions to decide whether to keep the new solution; thus, it obtains the global optimum. The key to the simulatedannealing algorithm is the Metropolis criterion, which has some marginal states that make the result worse; thus, enabling the process to jump out of a local optimum. Compared to other optimization algorithms, the simulated-annealing algorithm is not as demanding on the objective function, and is able to find the optimal solution with a higher probability, while having a lower probability of falling into a local optimum. It is suitable for both continuous and discrete variables and has no requirements for the objective function or constraints.

5.2.2 Flow of the Simulated-annealing Algorithm A flowchart of the simulated-annealing algorithm is shown in Fig. 5.3. When the system is in state x old , its state changes to x new after being perturbed by external influences. The objective function and acceptance probability of the system change from f (x old ) to f (x new ) and state x old to state x new , respectively [3]: { P=

1 ( ) f (xnew ) ≥ f (xold ) f (xnew )− f (xold ) exp − f (xnew ) < f (xold ) T

(5.1)

If the difference between the two objective functions is Δf ≥ 0, the state of x new is received as the current state. If the difference between the two objective functions is Δf < 0, some edge states that worsen the result are received with a probability determined by the Metropolis criterion, which determines whether the state of x new is received as the current state. A uniformly distributed random number is generated in (0–1), the magnitudes of random numbers p and q, calculated according to the Metropolis criterion, are compared, that is, Eq. (5.1), and the next-state process is run according to both values. When p > q, state x new is the current state; otherwise, state x old is kept as the current state. After p and random number q have been compared and the states have migrated several times, the equilibrium state at the current temperature is reached. Then, the temperature continues to be lowered to find the equilibrium state at each current temperature. The process is repeated until the final “annealed” temperature is close to 0°C, which is the process of jumping out of local extrema to obtain the maximum value. The simulated-annealing algorithm and the actual system are combined as follows. Usually, for piezoelectric ceramics, the full voltage range is 30 V; hence, it can be considered that the x- and y-directions each have an applied voltage of 15 V. At this time, the spatial states of left and right, up and down have equal symmetry with the adjustment range. Therefore, the efficiency of the optical-fiber coupling can be

144

5 Automatic Fiber-optic-coupling Alignment System Start

Set initial temperature T0 Set the temperature drop factor Qr Set the number of outer loops M Set the number of inner loops N Generate random matrix q ( M , N ) ~U (0 1 )

Generate random numbers in the feasible range x0 Calculate the objective function value f old = f x0

Initialize the number of loops j = 0

N

j>M

End

Y Initialize the number of loops i = 0 i = i +1

Metropolis criterion, accept with probability some edge states that make the outcome worse

Y Generate random numbers in the feasible range xi Calculate the objective function f new = f ( xi ) value Δ f = fnew − fold N

Δ f >0

f max = f new

q ( i, j ) < exp( −|Δf | /Ti)

Y

fold = fnew

N

Y f max = f old

i>N N j=j+1 Update temperature

Fig. 5.3 Flowchart of the SA algorithm to find the maximum value [4]

5.2 Basic Principles of Control Algorithms

145

directly reflected in the optical-power output. After the signal passes through the photoelectric linear conversion and embedded AD acquisition system, the value can be used as the evaluation index of the target function. When the target function reaches its maximum value, the optical-fiber coupling efficiency can be considered to be at its maximum. The steps of the actual simulated-annealing algorithm, applied to the optical-fiber coupling system, are as follows. 1. Given that both DA outputs are 1.5 V, the values are amplified to 15 V. 2. The electrical signal is collected from the output of the photoconverter and an AD acquisition is performed, denoted as AD1. 3. Given a constant-step trial in the x-direction, let the trial direction be the xpositive direction. The photoelectric-conversion signal is collected at this point, and an AD acquisition is performed, recorded as AD2. 4. The magnitudes of AD1 and AD2 are compared. If AD2 > AD1, the voltage is still applied in the x-positive direction. If AD2 < AD1, the voltage is applied in the x-positive direction with probability p and in the x-negative direction with probability (1 − p), according to the Metropolis criterion. 5. The process is repeated for the y-direction, and the iteration of the annealing simulation is completed. 6. The iterations continue until |AD2 − AD1| is less than a given threshold, at which point, the position where the fiber is located is the best coupling position.

5.2.3 Simulated-annealing Algorithm Features Compared to other optimization algorithms, simulated-annealing algorithms have the following properties [3, 5]. 1. The simulated-annealing algorithm allows for a limited number of solutions that “worsen” the objective function, according to the Metropolis criterion, allowing the algorithm to jump out of local extrema and find the optimal solution. The Metropolis criterion assumes that in state x old , owing to external factors, the state changes to x new ; the energy of the system also changes from E(x old ) to E(x new ), and the system changes from state x old to state x new with acceptance probability P [3]: { P=

1 ( ) E(xnew ) ≥ E(xold ) . E(xnew )−E(xold ) E(xnew ) < E(xold ) exp − T

(5.2)

The implication of the above equation is that the current state is received with probability 1 (i.e., completely) when the energy at the previous moment is less than the energy at the current moment. It is received with the probability in Eq. (5.2) when the energy at the previous moment is greater than the energy at

146

5 Automatic Fiber-optic-coupling Alignment System

the current moment; that is, it is beneficial to jump out of the local extremum and thus, find the global optimum. 2. The simulated-annealing algorithm introduces the annealing-temperature parameter T(t), which allows the algorithm to accept a limited number of solutions that make the objective function “worse”, when the temperature is high at the beginning, so that it is beneficial to jump out of the local extremum. The algorithm gradually converges to the optimal solution along with the cooling process; thus, the entire process has a higher probability to find the global optimum. Assuming that T(t) represents the temperature at a given moment, the cooling process can be expressed as the following function [3]: T (t) =

T0 . ln(1 + t)

(5.3)

In addition, to simplify the complexity of the algorithm, Ingber [6] proposed a functional expression for the cooling process of the approximate simulation solution: T (t) =

T0 . 1+t

(5.4)

Both classical and fast algorithms allow the algorithm to reach a global maximum. When the temperature starts high, the algorithm accepts a limited number of solutions that “worsen” the objective function, which helps to jump out of local extrema. When the temperature is lowered, the algorithm slowly reaches the global optimum, which in turn leads to a steady state. 3. Compared with other optimization algorithms, such as hill climbing and stochastic parallel-gradient descent, simulated-annealing algorithms can find the optimal solution in a shorter time with lower algorithm complexity. They have lower hardware-configuration requirements. They can fall into a local optimal solution with a lower probability and find the global optimum with a higher probability. In addition, they can be applied to a variety of optimization combination problems with relatively relaxed requirements for the objective function and constraints.

5.2.4 Stochastic Parallel Gradient-descent Algorithm 1. Description The stochastic parallel gradient-descent (SPGD) algorithm was proposed by Vorontsov of the US Army Research Laboratory (ARL) in the late twentieth century. It was developed on the premise of perturbing both the stochastic approximation algorithm and the stochastic error-descent algorithm. The algorithm determines a search direction during each iteration, in which, the direction of

5.2 Basic Principles of Control Algorithms

147

the negative gradient of the objective function is set as its search direction. It advances by one step during each iteration, eventually approaching the extreme value of the objective function. To increase the accuracy of the gradient estimate, positive and negative perturbations are added to the control covariates in a manner known as bilateral perturbation. After the positive and negative perturbations have been added, the system’s performance metrics are measured, the difference in the performance metrics is recorded, and the difference is the final gradient estimate of the system’s performance metrics. In practice, when the difference is negative, the objective function is optimized in a greater direction, and when the difference is positive, the objective function is optimized in a smaller direction. In this study, the evaluation metric is the optical power coupled into a single-mode fiber; therefore, the difference in the objective function is considered to be negative. The stochastic parallel gradient-descent algorithm is widely used in fields such as neural networks and automatic control; therefore, this algorithm was also used in this study. 2. Flowchart The voltage signal is the device-control quantity for fiber-optic coupling autoalignment, and the flowchart of the SPGD algorithm is shown in Fig. 5.4. The specific implementation steps of the SPGD algorithm are as follows. 1. Initialization: A random set of initial control-voltage signals U0 = [u 1 , u 2 , u 3 , . . . , u N ] is generated, which corresponds to the output voltage of the Nth channel of the DA module. Perturbation: A set of randomly perturbed voltage signals Δu i = [σ u 1 , σ u 2 , . . . , σ u N ] is generated, where σ u N corresponds to the randomly perturbed voltage of the Nth channel. The voltage signals σ u i are independent of each other and obey a Bernoulli distribution. 2. The voltage sequence U0 + Δu i is loaded onto the actuator piezoelectric ceramic, and the coupled optical power of the feedback is recorded as a function of the system evaluation as J+(i) . 3. The voltage sequence U0 − Δu i is loaded onto the actuator piezoelectric ceramic, and the coupled optical power of the feedback is recorded as a function of the system evaluation as J−(i) . 4. The amount of change is calculated in the evaluation function σ J (i+1) = J+(i) − J−(i ) . 5. The control voltage is updated according to the voltage-iteration formula Ui+1 = Ui − μσ J (i) Δu i , where μ is the parameter for the iteration step. 6. The value of the evaluated function is calculated using the system performance corresponding to Ui+1 . If J (i+1) does not satisfy the system, steps 2, 3, 4, 5, and 6 are repeated until the system requirements for the evaluation function are satisfied.

148

5 Automatic Fiber-optic-coupling Alignment System

Fig. 5.4 Flowchart of the SPGD algorithm

Start

Initialize, and calculate the system objective function value J

Generate initial voltage value

Generate disturbance

U 0 = [u1 , u2 ,..., u N ]

Δui = [σ u1 , σ u2 ,

,σ uN ]

U i+ = U i + Δui

Calculate the voltage after perturbation

U i− = U i − Δui

Calculate the value of J after perturbation, respectively J +i , J −i count as Calculate the difference between two J values

δ J i = δ J +i − δ J −i

Calculate iterative voltage U i +1 = U i − μδ J (i ) Δui

Update initial voltage U 0 = U i +1 and calculate the system J value at the new voltage

The objective function meets the system convergence requirements

No

Yes End

5.2.5 Simulation of Different SPGD-algorithm Parameters (1) Fixed-gain SPGD algorithm It can be observed that the gain coefficient is an important parameter in the SPGD algorithm iteration. In addition, the coupling efficiency is used as the algorithm evaluation function to numerically simulate the automatic spatial-optical couplingalignment process. The effects of different gain coefficients on the algorithm were analyzed when the SPGD algorithm was used for coupling alignment. Because the radial deviation has the strongest effect on the coupling efficiency, its adjustment is the most important, when performing spatial-optical single-mode fiber coupling. In addition, because there is always an alignment error in the specific operation of spatial-optical single-mode fiber coupling and the radial-alignment error of a single-mode fiber cannot exceed 10 μm, the error range of the initial alignment

5.2 Basic Principles of Control Algorithms Fig. 5.5 Fiber-coupling efficiency variation curve with the number of iterations for different gain factors

149

1.0 0.9

Coupling efficiency

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

20

40

60

80

100 120 140 160 180 200

Number of iterations

) ) ( ( error is set to Δx ∈ −10−5 , 10−5 m and Δy ∈ −10−5 , 10−5 m in the simulation. The wavelength was set to 1550 nm, the lens diameter was 2 cm, the numerical aperture was 0.211, and the algorithm perturbation was set to 0.002. The effects of different gain coefficients μ on the performance of the algorithm were compared at the end of the algorithm with 200 iterations. Figure 5.5 shows the variation curve of the number of iterations versus coupling efficiency. As can be seen from Fig. 5.5, the algorithm converges more slowly when the gain factor μ is low and the extremes cannot converge before the iteration condition terminates. As the gain coefficient increases, the algorithm-convergence speed is accelerated; however, the fluctuation of the coupling efficiency during the convergence will also increase; therefore, it is necessary to choose a suitable gain coefficient in the actual application system to achieve a faster convergence speed and better system stability. (2) Variable-gain SPGD algorithm The traditional SPGD algorithm uses a fixed iteration step, and the performanceevaluation function has two processes in the general system: fast convergence, which occurs at the beginning of the algorithm, and slow, high-precision convergence, which occurs at the end of the algorithm. When the gain coefficient is a definite value, the larger the value, the faster the algorithm will converge; however, the evaluation function may oscillate, and the system will become less stable in convergence. When the gain coefficient is small, the algorithm converges more slowly. To achieve faster convergence and weaken the oscillation, it is necessary to adaptively adjust the gain coefficient during the algorithm iteration. Thus, a variable-gain SPGD algorithm was designed, which differs from the fixed-gain SPGD algorithm in that the gain coefficient is updated in each step of the iteration process. The gain-coefficient update formula is as follows: μ(i+1) = C1

/(

J (i) − ε

)C2

(5.5)

150

5 Automatic Fiber-optic-coupling Alignment System

where μ is the iteration step after the ith algorithm is updated using Eq. (5.5) and J (i) is the system-evaluation function after the ith iteration. C 1 and C 2 are constants. When the objective function increases, set C 1 > 0 and C 2 > 0. In the variable-gain SPGD algorithm, set C 2 = 1. In the fixed-gain SPGD algorithm, it is necessary to select the appropriate C 1 , according to the actual system. The gain of each iteration depends on the previous system-evaluation function, J (i) . The gain coefficient tends to stabilize as the evaluation function increases, which is conducive to system convergence and stability. As shown in Fig. 5.6, using the same simulation conditions as in the fixedgain section, the coupling-efficiency curve, as a function of the number of iterations for/different parameters, is obtained by adding a gain-coefficient update step ( ) μ = C1 J (i) − 0.003 to the algorithm. The algorithm converges slowly when 0.1 is used, and convergence is not achieved before the iterative condition terminates. The algorithm convergence gradually accelerates as μ is increased to 0.15, 0.2, 0.25, and 0.3. It can also be seen from Fig. 5.6 that the overall convergence trend of the variable-gain SPGD algorithm with different parameters is faster at the beginning of the algorithm iterations and slower in the later iterations of the algorithm. This is related to the change in the gain coefficient μ. The variation curve of the variable-gain SPGD algorithm with the number of iterations for different parameters is shown in Fig. 5.7. Under different μ, the gain C 1 gradually decreases and finally stabilizes with the algorithm iterations. C 1 is larger at the beginning of the algorithm and gradually decreases at the end of the iterations. This makes the algorithm converge faster at the beginning and have better stability at the end of the iterations. 1.0 0.9

Coupling efficiency

Fig. 5.6 Variable-gain SPGD algorithm coupling efficiency as a function of the number of iterations for different parameters

0.8 0.7 0.6 C1=

0.5

C1= C1=

0.4

C1=

0.3 0.2

C1=

0

20

40

60

80

100 120 140 160 180 200

Number of iterations

5.3 Effect of Alignment Errors on the Efficiency of Spatial Optical-fiber … 1.6 1.4 C1=

1.2

Coupling efficiency

Fig. 5.7 Variable-gain SPGD coupling efficiency as a function of the number of iterations for different parameters

151

C1= C1=

1.0

C1= C1=

0.8 0.6 0.4 0.2 0.0

0

20

40

60

80

100 120 140 160 180 200

Number of iterations

5.3 Effect of Alignment Errors on the Efficiency of Spatial Optical-fiber Coupling 5.3.1 Alignment Error and Coupling Efficiency A single-mode fiber-core diameter is usually 8–10 μm. The actual spatial optical single-mode fiber alignment varies, owing to manufacturing errors, mounting errors, external-perturbation errors, etc. The incident light through the focal length f , DA coupling-lens focus, single-mode fiber end face, and lens focal plane will produce alignment errors, as shown in Figs. 5.8 and 5.12. Alignment errors are usually divided into radial deviation Δr, end-face tilt deviation Δϕ, and axial deviation Δz [7]. In plane B, the single-mode-fiber tilt direction 0.65

coupling efficiency η

Fig. 5.8 Effect of the directional angle Ω between Δϕ and Δr on the coupling efficiency [11]

η ηΔr+ηΔϕ+ηΔz

0.60

0.55

0.50

0.45

0

90

180

Ω/ o

270

360

152

5 Automatic Fiber-optic-coupling Alignment System

is defined as the X-axis and the direction perpendicular to it as the Y-axis; the radial deviation Δr is projected along the X- and Y-axes as Δx and Δy, respectively. Ω is the angle between Δr and the Δ-axis; r a is the distance from the point (x a , ya ) on the receiving aperture to the optical axis; and w0 is the single-mode-fiber mode-field radius. A single-mode-fiber mode-field distribution with alignment errors can be described by an eccentric Gaussian distribution. The single-mode-fiber mode-field distribution FB (x b , yb ) at plane B is [8] FB (xb , yb ) } { ] 1 ik [ 2 2 = exp − − x + i x + − y + i y (xb (yb d 0) d 0) q0 2q0

(5.6)

where q0 is the q parameter of the incident Gaussian beam (q0 = z + i z 0 , z is the / 2 λ z = Δϕπ w is the Rayleigh length), k is the wavenumber transmission distance, 0 0 / (k = 2π λ, λ is the beam wavelength), xd = Δx and yd = Δy are the Gaussianbeam waist center positions, x0 = Δϕ × z 0 and y0 = 0. The single-mode-fiber mode field is transmitted from plane B. The ABCD transmission matrix, when transmitted in the reverse direction to the receiving plane A, is [

A B C D

]

[ =

1 / 0 −1 f 1

][

1 f + Δz 01

]

[ =

] 1 / f + Δz / . −1 f −Δz f

(5.7)

Based on the Collins diffraction-integral formula, the single-mode-fiber modefield distribution FA (x a , ya ) at receiving plane A can be obtained as [9] ¨ ik exp(−ikz) FB (xb , yb ) FA (xa , ya ) = 2π B ) ⎧ ⎫ ⎡ ( 2 ⎤ 2 ⎪ ⎪ ⎨ ik A xb + yb ⎬ ⎢ ⎥ × exp − ⎣ −2(xa xb + ya yb )⎦ dxa dya ⎪ ⎪ ) ( ⎩ 2B ⎭ +D xa2 + ya2

(5.8)

where A, B, C, and D are ABCD transmission-matrix elements. The simplification shows the single-mode-fiber mode-field distribution FA(x a , ya ) at lens plane A with a complex amplitude:

5.3 Effect of Alignment Errors on the Efficiency of Spatial Optical-fiber …

) B −1 FA (xa , ya ) = E 0 A + q0 exp[−ik( f + Δz)] ⎧ ⎫ [ ] −ik ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎨ 2q( f + Δz) (xa − xd ( f + Δz)) + (ya − yd ( f + Δz)) ⎪ ⎬ ] [ exp −ik εx ( f + Δz)xa + ε y ( f + Δz)ya ⎪ ⎪ ⎪ ⎪ ⎪ [ ⎪ ] ⎩ ⎭ −i φx ( f + Δz) + φ y ( f + Δz)

153

(

(5.9)

/ where q( f + Δz) = (Aq0 + B) (Cq0 + D) is the q parameter of the Gaussian beam at lens plane A; εx and εy are the tangent values of the angle between the peak-intensity axis of the eccentric beam and the transmission axis in the XOZ and / / YOZ planes, respectively; εx = x0 z 0 and ε y = y0 z 0 , φ x and φ y are the phase factors of the eccentric Gaussian beam, and E 0 is the mode-field distribution of the single-mode fiber at the origin at transmission distance z = 0. [

] ] [ ][ xd (z + Δz) yd (z + Δz) A B xd yd = εx (z + Δz) ε y (z + Δz) C D εx ε y [ ] ][ A B xd yd = C D Δϕ 0 / ] [ ] [ x ε + Δz) + Bε + Δz)] φx (z + Δz) k[C x (z (z d d x x ]/ 2 = [ φ y (z + Δz) k C yd yd (z + Δz) + Bε y ε y (z + Δz) 2 i E0 = − z {0 } ] k [ 2 2 x + y0 + 2i(xd x0 + yd y0 ) exp 2z 0 0

(5.10)

(5.11)

(5.12)

After simplification and normalization, the mode-field distribution of a singlemode fiber with a Gaussian mode field plus reverse transmission to the plane of the receiving aperture can be expressed as /

] [ 2 1 (xa − Δϕ f )2 + ya2 FA (xa , ya ) = exp − π wa wa2 { ┌ ( ]} ) Δz xa2 + ya2 xa Δx + ya Δy + × exp ik 2f2 f

(5.13)

/ where wa = λ f (π w0 ) is the radius of the Gaussian beam waist at the plane of the lens. The polar form is applied to Eq. (5.13), such that {

xa = ra cos θ ya = ra sin θ

{

Δx = Δr cos Ω Δy = Δr sin Ω

(5.14)

154

5 Automatic Fiber-optic-coupling Alignment System

where θ is the angle between r a and the x-axis, which gives /

] [ 2 2 1 r + (Δϕ f )2 − 2ra Δϕ f cos θ exp − a π wa wa2 { [ ]} Δzra2 ra Δr cos(θ − Ω) × exp ik + 2f2 f

FA (ra , θ ) =

(5.15)

When the effect of atmospheric turbulence on the projected light is not considered, the optical field E A (r, θ) of the incident beam at receiving plane A is { E A (r, θ ) =

/ 1 r ≤ DA/ 2 0 r > DA 2

(5.16)

Substituting Eqs. (5.15) and (5.16) into the coupling-efficiency expression [10] yields a reduced coupling efficiency of |{ { |2 | 1 || R 2π η= FA (ra , θ )ra dra dθ || | 2 πR 0 0

(5.17)

/ / / / where R = D A 2; thus, p = ra R, and β = R wa = Rπ w0 λ f . Equation (5.17) can be simplified as ηΔr

|{ | = 8β | 2|

1

0

|2 ( ) | ( 2 )2 2πρβwa Δr ρdρ || exp −β ρ J0 λf

(5.18)

When the three alignment errors are present individually, the coupling efficiency can be reduced to ηΔr

|{ | = 8β | 2|

0

1

|2 ) ( | ( 2 2) 2πρβwa Δr ρdρ || exp −β ρ J0 λf

(5.19)

ηΔϕ

|2 )|{ ( ) ( | ( 2 2) 2ρβΔϕ f (Δϕ f )2 || 1 ρdρ || = 8β exp −2 exp −β ρ I0 | wa2 w a 0 |2 |{ 1 ) ( | | ( 2 2) i π Δzρ 2 β 2 wa2 2| | ρdρ ηΔz = 8β | exp −β ρ exp | λf 2 0 2

(5.20)

(5.21)

where J 0 is the first-class zero-order Bessel function and I 0 is the first-class zero-order modified Bessel function.

5.3 Effect of Alignment Errors on the Efficiency of Spatial Optical-fiber …

155

5.3.2 Radial, End-face, and Axial Errors For illustrative purposes, the parameters were set as follows: incident-beam wavelength λ/= 1550 nm, coupling-lens diameter DA = 10 mm, coupling-lens focal length f = D 0.211, single-mode-fiber mode-field radius w0 = 5.25 m, direction of endface tilt Δ∅ and lateral offset Δz with an angle Ω = 180° between the direction of the end-face tilt Δϕ and the direction of the lateral offset Δr. From Eq. (5.18), it can be concluded that the effects of the radial error Δr, endface inclination error Δϕ, and axial error Δz on the coupling efficiency are related to each other. When Δr = 2 μm, Δϕ = 2°, and Δz = 50 μm, the relationship between the coupling efficiency and the clamping angle Ω is shown in Fig. 5.8. It can be seen that when the clamping angle Ω is 90° and 270°, the coupling efficiency is the sum of the three alignment errors alone, and the three alignment errors can be considered to have an approximately independent effect on the coupling efficiency. The coupling efficiency was greatest at an angle Ω of 180°. Figures 5.9, 5.10, 5.11 show the curves of the effects of three different alignment errors on the coupling efficiency. It can be seen from the graphs that the coupling efficiency decreases as the fiber-alignment error increases. The radial error Δr has the strongest effect on the coupling efficiency, followed by the end-face tilt error Δϕ, and finally, the axial error Δz. The very small core diameter of single-mode fibers makes manual alignment difficult, and the effect of different alignment errors on the coupling efficiency can be compensated for by designing an automatic coupling-alignment system.

0.8

0.8

coupling efficiency

(b) 1.0

coupling efficiency

(a) 1.0

0.6

0.4

0.4

0.2

0.2

0.0

0.6

0

2

4

6

8

10

0.0

0

2

4

6

8

10

Fig. 5.9 Effect of end-face errors on the coupling efficiency [11] a different radial errors; b different axial inclination errors

156

5 Automatic Fiber-optic-coupling Alignment System

0.8

0.8

coupling efficiency

(b) 1.0

coupling efficiency

(a) 1.0

0.6

0.4

0.4

0.2

0.2

0.0

0.6

0

50

100

150

0.0

200

0

50

100

150

200

Fig. 5.10 Effect of axial errors on the coupling efficiency [11] a different radial errors; b different end-face inclination errors

0.8

0.8

coupling efficiency

(b) 1.0

coupling efficiency

(a) 1.0

0.6

0.4

0.4

0.2

0.2

0.0

0.6

0

2

4

6

8

10

0.0

0

2

4

6

8

10

Fig. 5.11 Effect of radial errors on the coupling efficiency [11] a different end-face inclination errors; b different axial errors

Fig. 5.12 Piezoelectric-ceramic fixation diagram

5.4 Two-dimensional Auto-alignment Experiments

157

5.4 Two-dimensional Auto-alignment Experiments 5.4.1 Piezoelectric-ceramic and Fiber-fixing Method The specific parameters of the two-dimensional (2D) piezoelectric ceramics are presented in Table 5.1. This experiment used NAC2710 piezoelectric ceramic, which has two axes, X and Y, to control its motion. The supply voltage is a constant positive 30 V and a constant negative 30 V. Applying different voltages between 0 and + 30 V, and − 30 V and 0 to the X and Y axes, respectively, will produce different displacements, which, in turn, will control the movement of the piezoelectric ceramic. Here, a positive voltage refers to a positive movement along the axis in which it is located, while a negative voltage refers to a similar negative movement. The different voltages cause the piezoelectric ceramic to drive the optical fiber with different micro-displacements. Epoxy resin can be used as a clamping device for piezoelectric ceramics. It demonstrates advantages in terms of insulation, adhesion, dielectric properties, and high voltage resistance. In addition, the cured epoxy resin has low shrinkage and generates little internal force, enabling the front-end piezoelectric ceramics to produce more accurate displacements while ensuring the stability of the tail end. Two epoxyresin plates were used to clamp the rear end of the piezoelectric ceramic, as shown in Fig. 5.12. Two-thirds of the length of the piezoelectric ceramic is left at the clamped end; hence, the displacement generated by the upper power is not affected by the clamping, allowing the piezoelectric ceramics’ micro-displacement to drive the optical fiber. The control unit is based on the strength of the electrical signal detected by the photodetector. The output voltage signal applies commands to the reference voltage, which acts on the piezoelectric-ceramic drive circuit, which controls the two-dimensional piezoelectric ceramic piece to produce micro-displacements and adjusts the position of the optical fiber to find the optimum value. The optical power values at the focal plane of the lens and at the end of the fiber were measured by an optical power meter to obtain the ratio of the two; that is, the fiber-coupling efficiency, to align the optical axes. Table 5.1 Basic parameters of piezoelectric ceramics

Parameter

Requirement

Length

36.5 mm

Width

1.75 mm

Height

1.75 mm

Maximum stroke

± 90 μm

Supply voltage

± 30 V

158

5 Automatic Fiber-optic-coupling Alignment System

5.4.2 Two-dimensional Alignment Experiments As shown in Fig. 5.13, when an optimal control algorithm is not present, the electrical signal from the photodetector fluctuates to extreme values, which is the process of finding the maximum value. However, the entire process cannot exit the extreme value; that is, it cannot find the best coupling position. After adding an intelligent optimization algorithm, that is, the simulated-annealing algorithm, as shown in Fig. 5.14, the electrical signal from the photodetector varies from 0 to 3 V, with fluctuations and extreme values at the beginning. This process shows that the system has been looking for the maximum value; and after approximately 1500 ms, it jumps out of the local extreme value and finds the maximum value, which tends to a more stable state; that is, the best coupling position is found. The value drifts up and down by 1.67 V with a magnitude of no more than 0.05 V, at which point the average voltage is 1.658 V and the variance is 0.0132 V2 . As shown in Fig. 5.15, the simulated-annealing algorithm is applied when a stable trend is reached. The voltage value fluctuates up and down by 2.05 V, with a magnitude of no more than 0.1 V, at which point the mean voltage value is 2.025 V and the variance value is 0.01672 V2 . As shown in Fig. 5.16, the voltage range is within 0–3 V, with fluctuations at first, always looking for the best coupling position, until, and it tends to stabilize after about 1350 ms. This means that the best coupling position was found. The voltage value fluctuated up and down by 2.02 V, with changes of no more than 0.1 V, the average voltage value was 1.99 V and the variance value was 0.01552 V2 . As shown in Fig. 5.17, the output voltage of power supply 1 was measured after adding the simulated-annealing algorithm. This voltage value represents the voltage added by the adjustable piezoelectric ceramic all the way to the X-axis. The range is 0–30 V, which corresponds to the DA1IN input by a factor of approximately 10 because the adjustable part of the piezoelectric ceramic has an amplification module. 2.0

Measured data

Photodetector voltage output /V

Fig. 5.13 Output voltage of the photodetector when adding the algorithm [11]

1.5

1.0

0.5

0.0 0

10

20

30

40

Number of samples

50

60

5.4 Two-dimensional Auto-alignment Experiments

159

Fig. 5.14 Output voltage of the photodetector when no algorithm is used [11]

2.5

DA1IN voltage output /V

Fig. 5.15 DA1IN voltage during the addition algorithm [11]

Measured data

2.0 1.5 1.0 0.5 0.0 0

10

20

30

40

50

60

Number of samples

As can be seen from the graph, the curve fluctuates at the beginning; that is, the entire system is looking for the maximum value. The fluctuations are the process of finding the best coupling position, after which it is in a stable state; that is, the best coupling position is found. At this time, the voltage value drifts up and down by 20.68 V with an amplitude of no more than 0.1 V. The average voltage value is 20.65 V and the variance value is 0.21052 V2 . As shown in Fig. 5.18, this voltage value, together with the output voltage of power supply 2, measured after the simulatedannealing algorithm, represents the voltage added to the piezoelectric-ceramic piece, which is adjustable all the way to the Y-axis; the range is − 30 – 0 V. Here the negative voltage only represents the movement direction of the piezoelectric ceramic and the opposite direction of the applied voltage; the size of the

160 2.5

Measured data

DA2IN voltage output /V

Fig. 5.16 DA2IN voltage during the addition algorithm [11]

5 Automatic Fiber-optic-coupling Alignment System

2.0 1.5 1.0 0.5 0.0 0

10

20

30

40

50

60

Number of samples 30

Power1 voltage output /V

Fig. 5.17 Power supply 1 voltage during the addition algorithm [11]

Measured data

25 20 15 10 5 0 0

10

20

30

40

50

60

Number of samples -30

Power2 voltage output /V

Fig. 5.18 Power supply 2 voltage during the addition algorithm [11]

Measured data

-25 -20 -15 -10 -5 0 0

10

20

30

40

Number of samples

50

60

5.4 Two-dimensional Auto-alignment Experiments

161

value reflects the size of the piezoelectric-ceramic displacement. The correspondence with DA2IN is about 10 times the relationship, because the adjustable part of the piezoelectric ceramic has an amplification module. As can be seen from the graphs, the curve fluctuates at the beginning, that is, the entire system searches for a maximum value. The fluctuation is the process of finding the best coupling position, after which, it is in a stable state, that is, the best coupling position was found. The average voltage value is − 20.323 V, and the variance value is 0.1792 V2 . These voltage plots reflect the process of determining the optimum coupling position for the entire system in real time. As shown in Fig. 5.19, the trajectories of the voltage outputs tested with multiple runs of the program are plotted. After executing the simulated-annealing algorithm, a digital multimeter was used to measure the output voltage values, which control the two adjustable voltages of the piezoelectric ceramics. This can reflect the movement of the piezoelectric ceramics in real time. The movement trajectory, which is unstable at the beginning of the final movement trajectory, tends to be stable in the middle of the diagram; that is, it reflects the entire process of seeking and finding the system. Through analyzing and calculating the experimental data, the results show that automatic alignment and positioning can be achieved in a relatively short time using the optimized algorithm, with a coupling efficiency of 51.4%. Fig. 5.19 Multiple tested motion trajectories [11]

2.5

DA2 voltage output /V

Measured data

2.0 1.5 1.0 0.5 0.0 0.5

1.0

1.5

2.0

DA1 voltage output /V

2.5

3.0

162

5 Automatic Fiber-optic-coupling Alignment System

5.5 Five-dimensional Auto-alignment Experiments 5.5.1 Piezoelectric-ceramic Combinations and Methods of Fixing Them to Optical Fibers In practice, we used three different piezoelectric ceramics, whose specific parameters are listed in Table 5.2. The three piezoelectric ceramics had different displacement modes, and the combination of the three piezoelectric ceramics allowed for five degrees of freedom for adjusting the fiber end face. Figures 5.20, 5.21, and 5.22 show the displacement modes of the NAC2710, NAC2910, and MPT150 piezoelectric ceramics, respectively. As shown in Fig. 5.20, the NAC2710 displacement adjustment is controlled by the drive voltage. This piezoelectric ceramic acts as a deflection adjustment for the end face of the fiber. The displacement end of the NAC2910 piezoelectric ceramic can be driven by a corresponding control voltage to achieve a linear displacement with two degrees of freedom, as shown in the schematic diagram in Fig. 5.21. This piezoelectric ceramic applies radial-deviation adjustments to the end face of the optical fiber. Table 5.2 Piezoelectric-ceramics parameters

Fig. 5.20 NAC2710 piezoelectric-ceramic displacement schematic

Parameter

NAC2710

NAC2910

MPT150

Itinerary

± 90 μm

± 35 μm

40 μm

Supply voltage

± 30 V

± 100 V

None

Max. working voltage

60 V

200 V

150 V

Static capacity

x = 700 nF, y = 1300 nF

400 nF

4 nF

Length/width/ height (mm)

36.5/1.75/1.75

28/2.5/2.5

40/5/5

5.5 Five-dimensional Auto-alignment Experiments

163

Fig. 5.21 NAC2910 piezoelectric-ceramic displacement schematic

Fig. 5.22 MPT150 piezoelectric-ceramic displacement schematic

Figure 5.22 shows a physical view of the MPT150 stacked piezoelectric ceramic. The displacement can be generated by applying the corresponding control voltage at the displacement control end, and its displacement is a one-dimensional linear motion. This piezoelectric ceramic performs single-mode-fiber axial-deviation adjustments. The three types of piezoelectric ceramics exhibit different movement modes. By combining the three piezoelectric ceramics in a certain connection design, the displacement characteristics of each ceramic can be used to adjust the position deviation of the fiber end face and focus spot. Considering the need to place the end face of the optical fiber at the front end and the fact that the three piezoelectric ceramics have different maximum forces, the connection structure shown in Fig. 5.23 was used, according to the maximum forces of the piezoelectric ceramics. The operating principle is that the NAC2710 drives the optical fiber fixed at its displacement end to adjust the angular deviation, the NAC2910 drives the NAC2710 and the optical fiber through the connector to adjust the radial deviation, and the final MPT150 drives the NAC2910 and NAC2710 through the connector to adjust the axial deviation of the optical-fiber position. The

164

5 Automatic Fiber-optic-coupling Alignment System

Fig. 5.23 Schematic diagram of the three piezoelectric-ceramic connection structures

Fig. 5.24 Piezoelectric ceramic and optical-fiber connection

three types of piezoelectric ceramics are connected to form a single unit, which adjusts the alignment deviation of the fibers. Figure 5.24 shows the piezoelectric ceramic and optical-fiber connection.

5.5.2 Analysis of Experimental Results In the experiment, a random parallel gradient-descent (RPGD) algorithm was used to align the coupling between the spatial light and the single-mode optical fibers. The algorithm parameters have a significant impact on the performance of the algorithm. Different algorithm parameters were set in the experiment, and the light power coupled into the single-mode optical fibers during the algorithm execution was recorded. The light power at the focal plane was 1.46 mW, as measured in the experiment. In the experiment, the gain factor was set to a fixed value of 0.3, the perturbation voltage was set to Δu = 0.001 and Δu = 0.005, and the number of iterations of the algorithm was set to 200. The change in the coupling optical power during the iteration of the algorithm was recorded, as shown in Fig. 5.27. From Fig. 5.25, it can be observed that the RPGD algorithm can automatically find the optimal coupling position, and the optical power value of the single-mode

0

-2

-4

-6

-8

-10

0

50

100

150

Number of iterations

(a) Δ u = 0.001

200

Optical power coupled into a single -mode fiber (dBm)

Optical power coupled into a single -mode fiber (dBm)

5.5 Five-dimensional Auto-alignment Experiments

165

0 -2 -4 -6 -8 -10

0

50

100

150

200

Number of iterations

(b) Δ u = 0.005

Fig. 5.25 Variation curve of coupling optical power with the number of iterations for different perturbation voltages [12]

optical fiber is improved significantly with the iterations of the algorithm. It can also be seen from the diagram that when the gain factor μ is fixed, the algorithm converges after about 60 iterations when Δu = 0.001, and the converged light power fluctuates up and down near − 1.24 dBm, where the coupling power variance is 0.0021 dBm2 . When Δu = 0.005 is used, the algorithm converges to the maximum value after 25 iterations. After convergence, the coupling power fluctuates up and down near − 1.23 dBm2 , and the coupling-power variance is 0.0023 dBm2 . It can be seen that when the gain factor μ is fixed, the number of iterations required for the algorithm to converge decreases with an increase in the disturbance voltage Δu; however, the power fluctuation increases when the algorithm converges and stabilizes. A smaller disturbance voltage Δu is beneficial for power stability after convergence. In the next part of the experiment, the fixed disturbance voltage Δu was 0.001, gain coefficients were set to μ = 0.1 and μ = 0.5, and the number of iterations was set to 200. The relationship between the coupling-light power collected in the experiment and the number of iterations is shown in Fig. 5.26. It can be seen from Fig. 5.26a that when gain factor μ = 0.1 is used, convergence is achieved after 160 iterations. The converged coupling optical-power value fluctuates up and down near − 1.22 dBm, and the coupling optical-power variance is 0.0013 dBm2 . From Fig. 5.26b, it can be seen that when the gain factor μ = 0.5, the algorithm converges to the vicinity of the maximum coupled optical power after about 30 iterations, and the converged coupled optical-power value fluctuates up and down near − 1.23 dBm, where the variance is 0.0051 dBm2 . To make the algorithm converge faster and more stably, a smaller perturbation voltage can be used to ensure the stability of the algorithm after convergence. The convergence speed of the algorithm is improved/by the gain factor. The perturbation voltage Δu = 0.001 and gain factor μ = 0.2 (J − 0.003) are set. The curve of

5 Automatic Fiber-optic-coupling Alignment System

0 -2 -4 -6 -8 -10

0

50

100

150

200

Number of iterations

(a) μ = 0.1

Optical power coupled into a single-mode fiber(dBm)

Optical power coupled into a single -mode fiber(dBm)

166

0 -2 -4 -6 -8 -10

0

50

100

150

200

Number of iterations

(b) μ = 0.5

Fig. 5.27 Curve of coupled light power with number of iterations, / gain factor μ = 0.2 (J − 0.003) [12]

Optical power coupled into a single -mode fiber(dBm)

Fig. 5.26 Curve of coupling optical power with number of iterations under different gain coefficients [12]

0

-2

-4

-6

-8

-10

0

50

100

150

200

Number of iterations

the coupling-light power collected experimentally with the number of iterations is shown in Fig. 5.27. From Fig. 5.27, it can be seen that when using the variable-gain RPGD algorithm for spatial optical coupling auto-alignment, the algorithm converges after approximately 50 iterations. The convergence speed is faster at the beginning of the convergence and slows at the later stage. The coupling optical power fluctuates around − 1.17 dBm after convergence; the coupling optical-power variance is 0.0014 dBm2 . With a variable gain factor, the algorithm can consider both the convergence speed and stability, and the maximum coupling efficiency after convergence is 53.2%. To verify the practicability of the coupling device, its ability to adjust the position of the optical fiber when different alignment errors were generated was tested. In the experiment, different alignment errors were created by fine-tuning the displacement

5.5 Five-dimensional Auto-alignment Experiments

167

Fig. 5.29 Radial-deviation control [12]

Optical power coupled into a single-mode fiber (dBm)

Fig. 5.28 Axial-deviation control [12]

Optical power coupled into a single-mode fiber (dBm)

/ table and the angle of incident light. μ = 0.2 (J − 0.003) and Δu = 0.001 were selected as the algorithm parameters for the experiment. Radial, axial, and azimuthal deviations of the optical-fiber end face from the focal spot were caused by adjusting the incident direction of the micro-displacement table and incident light, respectively, as shown in Figs. 5.9, 5.10, and 5.11. The curves of the coupling optical power with the number of iterations are shown in Figs. 5.28, 5.29, and 5.30 when the axial, radial, and azimuthal deviations are introduced, respectively. From Fig. 5.28, it can be seen that the initial coupling optical power was − 12.37 dBm when introducing the axial bias. After approximately 75 iterations of the algorithm, the optical power converged to − 1.15 dBm. The small fluctuation in the optical power coupled into the optical fiber indicates that the axial bias was adjusted.

0 -2 -4 -6 -8 -10 -12 -14

0

50

100

150

200

150

200

Number of iterations

0 -2 -4 -6 -8 -10 -12 -14 -16

0

50

100

Number of iterations

Fig. 5.30 End-face-inclination deviation control [12]

5 Automatic Fiber-optic-coupling Alignment System Optical power coupled into a single -mode fiber (dBm)

168

0 -2 -4 -6 -8 -10 -12 -14 -16 -18

0

50

100

150

200

Number of iterations

From Fig. 5.29, it can be observed that the initial coupling optical power when a radial deviation was introduced was − 15.51 dBm, and convergence was achieved by the algorithm after approximately 90 iterations of coupling optical power. After convergence, the coupling optical power fluctuated slightly around − 1.21 dBm, indicating that the radial deviation was compensated. From Fig. 5.30, it can be observed that the initial coupling power was − 17.4 dBm, when the angle deviation was introduced by adjusting the direction of the incoming light. After 100 iterations, the coupling power gradually converged. Finally, the angle deviation was compensated by fluctuating around − 1.19 dBm. The experimental results show that the coupling device can compensate for different alignment deviations.

5.6 Summary Owing to the existence of various optical and adjustment errors, alignment errors, such as radial, end-face, and axial deviation, occur on the end faces of incoming beams and single-mode optical fibers. By analyzing the effects of different alignment errors on the coupling efficiency of spatial-light single-mode optical fibers, it was found that when the angle between the end-face deviation and radial deviation was 90° or 270°, the effects of the three alignment errors on the coupling efficiency were independent. On this basis, an optical coupler with a five degrees-of-freedom orientation adjustment was designed, based on piezoelectric ceramics, and a spatial optical single-mode coupling-alignment experiment was completed. The results showed that the optical coupler with a five degrees-of-freedom orientation adjustment could compensate

References

169

for the effects of various alignment errors on the single-mode optical-coupling efficiency. The single-mode optical-coupling power could reach 53.2% after the system was closed loop.

References 1. Metropolis N, Rosenbluth AW, Rosenbluth MN et al (1953) Equation of state calculations by fast computing machines [J]. J Chem Phys 21(6):1087–1092 2. Shuyou L, Zhihui D, Mengyue W et al (2015) Parallel implementation of simulated annealing algorithm and its application [J]. J Phys 31(6):96–103 3. Shuyou L, Zhihui D, Yue WM et al (2001) Parallel implementation and application of simulated annealing algorithm [J]. J Phys 50(7):1260–1263 4. Sichen L (2016) Optical coupling and beam control in free space optical communication [D]. Xi’an: Xi’an University of Technology 5. Yan W, Kaigui X (1999) Simulated annealing algorithm [J]. J Mengzi Teach Coll 1(4):7–11 6. Ingber L (1989) Very-fast simulated re-annealing [J]. Math Comput Model 12(8):967–973 7. Xueyan D, Pingxue L, Xi Z, Weixin Y (2020) Alignment error analysis of beam expanding optical fiber connector based on ZEMAX [J]. Progress in Laser and Optoelectronics 57(15):187–201 8. Yaxiong L, Yapei Y, Shufen C (1999) Laser beam transmission and transformation technology [M]. University of Electronic Science and Technology Press, Chengdu 9. Collins SA (1970) Lens-system diffraction integral written in terms of matrix optics [J]. J Opt Soc Am 60(9):1168–1177 10. Dikmelik Y, Davidson FM (2005) Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence [J]. Appl Opt 44(23):4946–4952 11. Jing L (2018) Research on spatial optical coupling automatic alignment technology [D]. Xi’an: Xi’an University of Technology 12. Benkang Y (2018) Experimental research on spatial optical fiber coupling automatic alignment and control algorithm [D]. Xi’an: Xi’an University of Technology

Chapter 6

Mode-conversion Methods

In wireless laser communication systems, owing to the influence of atmospheric turbulence on the spatial-light mode, the beam mode degrades, which leads to incomplete matching with the mode field of the single-mode fiber used for coupling at the receiving end. This reduces the coupling efficiency of the spatial-light singlemode fiber. Therefore, it is particularly important to match the beam mode with the fiber mode. This chapter discusses a spatial-light mode-conversion method based on a liquid–crystal spatial-light modulator (LC-SLM), and optimizes the conversion-transfer function.

6.1 Research Status of Mode Transformations By converting a degraded high-order mode into the required fundamental mode to improve the content of the fundamental mode, the coupling efficiency of a singlemode fiber can be improved directly and efficiently; hence, the system can achieve more stable coupling conditions. In 2007, Tsekrekos of Eindhoven University of Science and Technology in the Netherlands proposed a spatial filter with a mode-selection function that can control the mode [1, 2]. A prism was placed between the end of the fiber core of a multimode fiber and the photodetector for filtering processing, and the required mode was selected for the output. In 2010, Leon-Saval et al. of the University of Sydney, Australia, applied “photonic lanterns” as a multimode single-mode converter [3] to a mode-conversion system to complete a multimode single-mode conversion. In 2011, Carpenter et al. of the photoelectric center of Cambridge University in the UK realized a mode-conversion scheme [4] that improved the mode selection by using spatial-light modulation to modulate a phase plate. In 2011, Professor Amphawan of Oxford University in the UK proposed a method for multimode-fiber-input modefield matching. This method uses an SLM and prism to generate and modulate the © Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_6

171

172

6 Mode-conversion Methods

amplitude of the light field [5]. When combined with the optimization algorithm, the multimode-fiber mode field was matched to convert the mode. In 2012, Fontaine et al. of Bell Laboratories in the United States proposed a method based on photonic integrated circuits (PIC) to convert from one mode to another in one direction [6]. In 2012, Birks et al. at the University of Bath, UK, integrated a fiber Bragg grating into the single-mode core diameter of a multi-core fiber [7], connected the fiber to multimode fiber through a tapered fiber and applied it to a “photon lantern” filter. In 2012, Leon et al. of the University of Sydney, Australia, designed and produced “photon lanterns” to convert from single-mode to multimode, fused the single-mode to multimode, and compared and analyzed the spectra obtained at the output end of the photon lanterns with those obtained through step-index single-mode and multimode optical fibers. It was found that the modal noise generated using photon lanterns for conversion was less than that of a step-index multimode fiber with the same core diameter. The experiment established the feasibility of using photon lanterns for mode conversion between single-mode fiber and multimode fiber [8]. However, the manufacturing-process requirements of photon lanterns are relatively high, and their application has certain limitations. In 2013, Ding et al. of the Technical University of Denmark improved the PIC scheme and converted between six lower-order modes by using the selective excitation of three spatial modes [9]. In 2013, Hanzawa of the Nippon Telegraph and Telephone (NTT) Corporation in Japan deduced the transfer function between the mode conversion of an asymmetric-planar optical waveguide, combined it with a planar lightwave circuit (PLC), mutually converted between LP01 mode and LP11 mode [10], and then converted from LP01 mode to LP11 mode and LP21 mode [11]. In 2014, Uematsu et al. converted from LP11a mode to LP11b mode through a PLC LP11 analog converter [12]. Mode conversion based on a PLC is characterized by high stability, low insertion loss, and good integration. However, the design and manufacture of PLCs consider only a single conversion. In addition, PLC has high requirements for precision and cost, and lacks flexibility and repeatability. In 2014, Gao Li et al. took the lead in China to convert between low-order modes, based on pure-phase SLMs, by using the reusability and infinite programming of SLMs in a mode-division multiplexing system [13]. In 2015, Qi Xiaoli et al. converted a free-space optical-path mode from LP01 mode to high-order modes using SLM [14]. Owing to the use of SLM, the scheme has good repeatability and low requirements for other devices, and the system is simple and easy to implement. In 2016, Aymen et al. connected two optical fibers by embedding a section of an air-silicon microstructure between a single-mode fiber (SMF) and few-mode fiber (FMF) and selectively excited the fundamental mode. The results showed that it achieved a better selective conversion of the fundamental mode in FMF without loss [15]. In 2017, Tu Jiajing et al. converted from LP01 mode to LP11a mode and LP21a mode based on a simple SLM structure coupled to FMF for receiving and transmission [16]; however, the mode-conversion efficiency of this scheme was low.

6.2 Basic Mode-conversion Theory

173

In 2018, Shen et al. proposed a novel optical-waveguide LP01–LP02 mode converter with a two-photon structure, based on the coupled mode theory [17]. It was composed of cladding, a conical core, and a two-dimensional conical structure. The results showed that the working bandwidth of the mode converter was 1350– 1700 nm, the conversion efficiency was 90% (0.5 dB), and crosstalk with other modes was low. Optical-mode conversion technology is mainly divided into two categories: optical waveguides and free-space optical paths. Optical-waveguide mode conversion mainly realizes the mode-conversion scheme by designing the device structure, mainly including PLC, PIC, “photon lanterns”, etc. Therefore, the optical-waveguide system has high complexity and poor flexibility, and its application scenarios are limited to converting from one mode to another.

6.2 Basic Mode-conversion Theory As shown in Fig. 2.3, an optical fiber is composed of three parts: cladding, core, and coating [18]. In Fig. 2.3, a represents the core radius and b represents the cladding radius. As the light beam propagates in the core, it is bound, and the light beam propagates forward along the core, owing to the total reflection on the boundary between the core and cladding [19]. Depending on the refractive-index distribution of the fiber core, optical fibers can be generally divided into two categories: graded index (GI) [20] and step index [21]. The refractive-index distribution of the core with a graded refractive index has quadratic-function characteristics. The part with a high refractive index is farther away from the cladding, and the part with a low refractive index is farther away from the core. The refractive index of the core is a constant value, expressed by n1 , and the refractive index of the cladding is also a constant value, expressed by n2 . In addition, the refractive index n1 of the core is higher than that of the cladding, that is, n2 < n1 . At the junction of graded-index fibers, the refractive index n1 of the core is reduced to the refractive index n2 of the cladding. For a step-index fiber, the refractive index at the junction of the cladding and core changes step-by-step, and the refractive index n1 at the core remains constant. Owing to the different refractive-index distributions, the light-propagation modes in optical fibers are also different. The expression of a mode solution in a graded index fiber is a Laguerre–Gaussian function, whereas for the step-index fiber, the expression of the mode solution is a Bessel function [22]. The refractive-index distribution in the graded refractive index is expressed as [23] { n 2 (r ) =

( )( )α n 21 − n 21 − n 22 ar 0 ≤ r ≤ a . n 22 r >a

(6.1)

174

6 Mode-conversion Methods

where n1 is the core refractive index in the graded refractive index, n2 is the cladding refractive index in the graded refractive index, a is the core radius, and α is the refractive index of the core. The relative refractive-index difference between the cladding and the core is Δ. The specific expression of Δ is Δ=

n 21 − n 22 . 2n 21

(6.2)

If Eq. (6.2) is substituted into Eq. (6.1), then { n (r ) = 2

[ ( )α ] 0≤r ≤a n 21 1 − 2Δ ar 2 r >a n 1 (1 − 2Δ)

(6.3)

For the field distribution in optical fiber ψ(r, θ, z), when the cylindrical coordinate condition is satisfied, the wave equation is expressed as ∇ 2ψ + k 2n2ψ = 0

(6.4)

where k is the free-space wavenumber, and n is the refractive-index distribution of the GI fiber. When the beam propagates in the z-axis direction, ψ(r, θ, z) decomposes into ψ(r, θ, z) = ϕ(r, θ )e− jβz

(6.5)

When a linear-polarization (LP) mode exists, it is transversely distributed, ϕ(r, θ ). It can be further decomposed to separate r and θ. These two variables are ϕ(r, θ ) = ϕ(r )e jmθ

(6.6)

where m is the number of angular modes whose value is a non-negative integer. It corresponds to cos(m); in the case of a real field, cos(mθ ) and sin(mθ ) are two degenerate solutions, as shown in Eq. (6.7): { cos mθ ϕ(r, θ ) = ϕ(r ) sin mθ

(6.7)

Substituting Eq. (6.7) into the wave equation of Eq. (6.4), we obtain ) ( ∇ 2ϕ + k 2n2 − β 2 ϕ = 0 In a cylindrical coordinate system, ∇2ϕ =

1 ∂ 2ϕ ∂ 2 ϕ 1 ∂ϕ + + ∂r 2 r ∂r r 2 ∂θ 2

(6.8)

6.2 Basic Mode-conversion Theory

175

( =

) m 2 jθ d 2ϕ 1 dϕ − 2 e + dr 2 r dr r

(6.9)

Then, Eq. (6.9) can be expressed as {

] } [ d 2ϕ 1 dϕ m2 2 2 2 ϕ e jθ = 0 + k + n − − β (r ) dr 2 r dr r2

(6.10)

namely, ] [ d 2ϕ 1 dϕ m2 2 2 2 ϕ=0 + k + n − − β (r ) dr 2 r dr r2

(6.11)

where m is the number of angular modes with a non-negative integer value, and n is the number of angular modes with a positive integer value [24]. The refractive-index distribution of the core in a graded refractive index follows a quadratic-term distribution [25]. Therefore, the core-field distribution defined by Eq. (6.11) has the characteristics of a Laguerre–Gaussian function [26]. For the graded-refractive index, the mode-field distribution decreases linearly to zero with an increase in the cladding radius. Therefore, the field distribution in the cladding exhibits the characteristics of a Bessel function. The mode-transverse field distribution in optical fiber φ(r ) is shown in Eq. (6.12) after simplification: ϕm,n (r ) ⎧ ( ) ( 2) m − r 22 ⎨ r r m 2ω0 e L C 1 n−1 ω02 0 ≤ r ≤ a ω0 = ⎩ C K (wr ) r >a 2 m

(6.12)

Coefficients C 1 and C 2 are determined by the boundary conditions, ω0 represents the waist radius of the fundamental-mode Gaussian beam, and W is the waist radius of the target beam. The specific expressions of ω0 and parameter W are shown in Eqs. (6.13) and (6.14), respectively: / ω0 = w=

/

a √ kn 1 2Δ

(6.13)

2 − k 2n2 βm,n 2

(6.14)

The waist radius ω0 of the fundamental mode in the optical fiber is determined by Eq. (6.13), where ω0 depends on the core refractive index n1 , core radius a, and the relative refractive-index difference Δ between the cladding and core. Equation (6.13) is substituted into Eq. (6.14) to obtain [24]

176

6 Mode-conversion Methods

ϕm,n (r, θ ) ( )m r 2 ( ){ − 2 m r2 r cos mθ 2ω0 = C1 e Ln sin mθ ωo ω02

(6.15)

Equation (6.15) is the field-distribution expression of the LP mode, which is the transverse mode in a weak-guide fiber core. In Eq. (6.15), when m = 0, LP0n has only one non-degenerate mode; when m /= 0, there are two degenerate modes in LPmn: cos(mθ ) and sin(mθ ).

6.3 Spatial Phase-modulation Mode Conversions The mode-conversion input is a laser to obtain the required beam of each high-order conversion mode, the middle filter selects the liquid–crystal spatial light modulator (LC-SLM), and the receiving end uses a beam analyzer to collect the converted mode. A schematic of the mode conversion based on an object-transform image (OTI) is shown in Fig. 6.1. The object–image-conversion mode-conversion system in Fig. 6.1 takes a liquid– crystal (LC)-SLM as the core [27] and converts between modes through spatialspectrum filtering. The complex amplitude of the input light field in( plane)P1 is Ui (r0 , θo ) and the complex amplitude of the light (field in)plane P2 is U f r f , θ f . The transfer function on LC-SLM is as H f r f , θ f , the complex amplitude of ( expressed ) the light field in plane P3 is U ,f r f , θ f , plane P4 is the converted light-field complex amplitude U0 (r, θ ), and the complex amplitude of the target-mode light field under ideal conditions is U D (r, θ ). According to the two-dimensional fast Fourier transform (FFT2) property of the lens, the calculated transfer-function phase hologram was loaded onto the LC-SLM for spatial-spectrum filtering, and the converted mode was collected by a chargecoupled device (CCD) at the receiver. The complex amplitude of the mode light field emitted from plane P1 is Ui (r0 , θ0 ), after the FFT2 transformation of Lens 1 [22], and the complex amplitude on plane

Optical signal Electrical signal He-Ne laser

Diaphragm Polarizer 1

Polarizer 2

Fig. 6.1 Schematic diagram of an object–image-conversion mode-conversion system

6.3 Spatial Phase-modulation Mode Conversions

177

P2 is ( ) U f r f , θ f = FFT2[Ui (r0 , θ0 )]

(6.16)

( ) The phase-modulation( function is H f r f , θ f , the complex amplitude after the ) LC-SLM conversion U ,f r f , θ f is ( ) ( ) ( ) U ,f r f , θ f = U f r f , θ f · H r f , θ f

(6.17)

After passing through the FFT2 of Lens 2, the complex amplitude distribution U0 (r, θ ) of the target spot on plane P4 is )] [ ( Uo (r, θ ) = FFT2 U ,f r f , θ f

(6.18)

The field distribution Ui (r0 , θ0 ) of the conversion mode LPmn is expressed [20] by the associated Laguerre–Gaussian function of Eq. (6.15): Ui (r0 , θ0 ) (

√ r = 2 w f_in

) ( r2 ){ − 2 r2 cos(mθ ) wf _in 2 2 e sin(mθ ) w f_in

(

)m Lm n

(6.19)

Similarly, the field distribution U D (r, θ ) is also expressed [20] by the associated Laguerre–Gaussian function in Eq. (6.15): U D (r, θ ) ( ) ( )p ( ) { r2 2 − √ 2 r r wf cos( pθ ) − out 2 L qp 2 2 = e sin( pθ ) w f− out w f out

(6.20)

−

In Eqs. (6.19) and (6.20), L pq (x) and L mn (x) correspond to the associated Laguerre polynomials in the conversion mode and the target mode, respectively; f _in and f _out are the focal lengths of the front and rear focal planes of the LC-SLM, respectively; and w f _in and w f _out are the waist radii of the input conversion mode and target mode, respectively. m and n are the radial and angular indices of the conversion mode, respectively. p and q correspond to the radial and angular indices of the target mode, respectively. cos(mθ ), sin(mθ ), cos( pθ ), and sin( pθ ) represent two degenerate solutions, m and p, respectively. When the waist radii w f _in and w f _out equal the spatial-spectrum radii of ω f _in,m,n and ω f _out, p,q , the states match. ω f _in,m,n and the spatial-spectral radius ω f _out, p,q are finite. When ω f _out, p,q > ω f _in,m,n , useful high-frequency information will be lost after conversion. If ω f _out, p,q < ω f _in,m,n , high-frequency noise will be introduced after conversion. Therefore, only when ω f _out, p,q = ω f _in,m,n will the spatial spectrum of the target mode and the conversion mode reach a matching state. The

178

6 Mode-conversion Methods

spatial-spectrum radii of the conversion mode ω f _in,m,n and target mode ω f _out, p,q are determined by Eq. (6.21):

ω f_out, p,q

√

λ f in 2π w f _in √ λ f out = p + 2q − 1 2π w f_out

ω f_in,m,n =

m + 2n − 1

(6.21)

When the spatial-spectrum radius matches, i.e., ω f _in, p,q = ω f _out,m,n , the spatialspectrum radius can be obtained from Eq. (6.21), using ω f _in, p,q and ω f _out,m,n , and the mode order and the light wavelength λ. The waist radius w f _in of the conversion mode and the waist radius w f _out of the output mode are related to the focal length of the lens. For a system with known input and output modes, the light wavelength λ, the waist radii for the conversion and target modes, and the focal length of the lens are known. Therefore, when the spatial spectrum of the conversion mode ω f _in, p,q and the spatial spectral radius of the target mode ω f _out, p,q match the waist radii w f _in and w f _out , he relationship between them is as follows: w f− in w f− out

√ m + 2n − 1 = √ p + 2q − 1

(6.22)

The mode-conversion requirement in wireless optical communication is mainly to convert from a high-order mode to a fundamental mode. Therefore, the output light is a fundamental-mode Gaussian beam, that is, the waist radius of the output beam is 25 μm. The waist radius of the input beam is √

m + 2n − 1 · w f− out w f− in = √ p + 2q − 1

(6.23)

The conversion-transfer function of the LC-SLM is obtained by the ratio of the inverse fast Fourier transform (IFFT) of the target-mode field distribution of the output plane to the FFT of the converted-mode field distribution, expressed as ) IFFT2[U D (r, θ )] ( H rf,θf = FFT2[Ui (r0 , θ0 )]

(6.24)

U D (r, θ ) and Ui (r0 , θ0 ) are expressed in amplitude and phase forms, that is, ) IFFT2[U D (r, θ )] ( H rf,θf = FFT2[Ui (r0 , θ0 )] | | |u f , p,q | o | ei [ϕ fo (r f ,θ f )−ϕ fi (r f ,θ f )] =| |u f ,m,n | i

(6.25)

6.3 Spatial Phase-modulation Mode Conversions

179

Equation (6.25) is the intermode-conversion transfer function H in the ideal case. The transfer function is composed of two parts: amplitude and phase. If the amplitude and phase information in H in Eq. (6.25) are modulated simultaneously, the insertion loss cannot be ignored [21]. A hologram for mode-conversion phase modulation can be obtained from Eq. (6.25). For a mode in optical fiber, it can be seen from the expressions of the spatial-domain distribution and the spatial frequency-domain distribution in Eq. (6.15), both are real numbers, and the phase has only the values of 0 and π. Therefore, the phase value of the conversion-transfer function can start from the most basic case containing only 0 and π. The RL-P2-SLM used in the experiment is a reflective-phase spatial light modulator, which ignores the amplitude information and retains only the phase information. Eq. (6.25) can be used to obtain the conversion transfer function ) ( A simplified H rf,θf : } { ) ( IFFT2[U D (r, θ )] H r f , θ f = arg FFT2[Ui (r0 , θ0 )] ) ( )]} { [ ( = exp j ϕ f0 r f , θ f − ϕ fi r f , θ f

(6.26)

where FFT represents a fast Fourier transform and IFFT represents a fast inverse Fourier transform.

6.3.1 Conversion from High-order Mode to LP01 Mode Based on the object-image-conversion mode-conversion system, a simulation analysis of the 0- and π-phase mode conversions from the high-order mode to LP01 mode was carried out. The simulation results for the mode conversion of the pure binary phase of the high-order LP01 mode are shown in Fig. 6.2. In Fig. 6.2, after the mode of the first row is converted by the corresponding conversion phase of the second row, the target light-field LP01 mode of the third row is obtained. It can be seen from each column in the figure that the mode after the binary phase modulation has an obvious conversion, and the spot energy is concentrated after conversion, which is similar to the Gaussian fundamental-mode LP01 mode. After the LP11 mode of the first column was converted, the radius of the central spot remains at 25 μm. The spot size remains unchanged; however, there is a highorder diffraction component on the periphery. The line formation of higher-order diffraction components is caused by the sudden change of the liquid–crystal molecular arrangement and the diffraction of the liquid–crystal structure itself during the phase loaded by the LC-SLM. After the radial light intensity of the second column LP21 is converted, part of it is concentrated in the central position, leaving incomplete conversion parts and high-order diffraction components in the radial position of the mode itself. After the

180

6 Mode-conversion Methods

Conversion mode

Conversion phase

Target mode

Fig. 6.2 Simulation results of the high-order-mode LP01-mode pure-binary-phase mode conversion

binary phase conversion, most of the light in the radial and angular positions of the third column LP31 mode is also concentrated in the center of the spot. In addition, the radial angular position of the mode itself has incomplete conversion and high-order diffraction components. The conversion effect of LP41 is more obvious; however, there are more highorder diffraction components around the center of the spot, which is still different from the ideal target LP01 mode. According to the results of the high-order to loworder conversion, the corresponding high-order diffraction components increase with increasing mode order.

6.3.2 Conversion-efficiency Analysis The conversion effect after mode conversion is quantitatively analyzed by obtaining the autocorrelation function between the converted mode and target mode under ideal conditions [20]: | |˜ | Uo (x, y) · U D (x, y)ds |2 ˜ CE = ˜ |Uo (x, y)|2 ds · |U D (x, y)|2 ds

(6.27)

6.4 Mode-conversion Improvements

181

Table 6.1 Conversion efficiency corresponding to the LP01-mode–high-order mode binary phase conversion Conversion mode

LP01

Target mode LP02 LP03 LP04 LP11 LP12 LP13 LP21 LP22 LP23 LP31 LP32 LP41 Conversion 22.7 efficiency/%

9.43

9.42

38.3

18.6

12.9

7.95

5.89

4.79

1.05

0.99

0.1

where Uo (r, θ ) is the converted mode, U D (r, θ ) is the target mode in an ideal situation, and the conversion efficiency is expressed by the correlation degree after mode conversion. CE is the conversion efficiency of the calculated mode-conversion system. The larger the CE value, the higher the accuracy of the corresponding mode-conversion system and the better the conversion effect. According to Eq. (6.27), the conversion efficiencies of LP01 mode, high-order mode, and high-order LP01 mode are shown in Tables 6.1 and 6.2. According to Tables 6.1 and 6.2, the conversion efficiency decreases with increasing order. Under the LP01-mode–high-order mode conversion, the highest conversion efficiency of the LP02 mode was 22.73%, the highest mode order was LP41, and the corresponding conversion efficiency was only 0.1%. For the conversion of the high-order LP01 mode, the highest conversion efficiency of the LP02 mode was 82.75% and the lowest conversion efficiency of LP51 mode was 59.88%. Considering Tables 6.1 and 6.2, with an increase in mode order, the corresponding high-order diffraction component increases after conversion. With a decrease in conversion efficiency, the obtained conversion transfer-function phase hologram has a distortion in the conversion accuracy, and the conversion efficiency needs to be improved. Therefore, it is necessary to choose an algorithm that can jump out of the local optimum and select a global optimum to transform the transfer-function hologram to improve the optimized initial-phase hologram to improve the conversion efficiency and accuracy.

6.4 Mode-conversion Improvements 6.4.1 Mode Conversion Based on the Simulated-annealing Algorithm In the mode-conversion system, the information loaded onto the LC-SLM is zero or π phase information, which ignores the amplitude information. To improve the conversion efficiency, the ignored amplitude information is compensated. Considering that converting the amplitude information to the frequency domain is the change between adjacent pixels in a hologram, this rate of change can be used to indirectly

LP02

LP01

82.75

Conversion mode

Target mode

Conversion efficiency/%

81.82

LP03 84.21

LP04 79.18

LP11 69.94

LP12 71.98

LP13 69.96

LP21 67.48

LP22

71.66

LP23

Table 6.2 Conversion efficiency corresponding to the high-order–mode fundamental-mode binary phase conversion

64.90

LP31

68.40

LP32

61.30

LP41

59.88

LP51

182 6 Mode-conversion Methods

Input face to be converted

Before annealing

6.4 Mode-conversion Improvements

183 Output plane target mode Fourier transform via lens

Ideal target mode

After annealing

Converted transfer function hologram optimized by simulated annealing algorithm

Fourier transform via lens

Output plane target mode

Fig. 6.3 Optimized transfer function using the simulated-annealing algorithm

describe the amplitude information [26]. Therefore, the amplitude information is compensated for by changing the speed of the adjacent pixels. The value of any pixel on the phase matrix of the conversion-transfer function can be perturbed by 0 or π; that is, the spectrum information is processed. The change in spectrum information affects the conversion target mode, making the converted mode closer to the ideal target mode. Therefore, combined with the global-optimization characteristics of the simulated annealing (SA) algorithm, the phase at a random pixel is perturbed by 0 or π. Finally, the optimized conversion transfer-function phase hologram is obtained to realize an efficient conversion between modes. The simulated-annealing algorithm shown in Fig. 6.3 optimizes the transfer-function scheme flow. The simulated-annealing algorithm optimizes the mode-conversion transfer function. The annealing result of the phase hologram is related to the selection of the initial temperature, annealing coefficient, number of iterations, and annealing interval of the pixels. The specific analysis is as follows. As shown in Fig. 6.4, the conversion-efficiency value corresponding to the initial solution of LP02 mode after the binary-phase mode conversion is 82.75%. The SA algorithm optimizes the conversion-transfer function in this state as the initial value. According to the conversion efficiency under different initial temperatures in Fig. 6.4a, when the initial temperature T = 5, the conversion efficiency reaches 96.66% before the conversion is complete. The annealing process is too fast and does not converge to a stable conversion efficiency. When the initial temperature T = 10, the conversion efficiency increases for a long time; however, the conversion efficiency finally reaches a stable state of 98.64%. When T = 20, the conversion effect improves to 96.81%, compared with T = 5; however, it does not reach a stable state of 98.64% when T = 10. Therefore, the initial temperature setting is too high, when T = 20. When T = 50, as with T = 20, the initial temperature is too high and the conversion efficiency is only 95.42%. Figure 6.4b shows conversion efficiency under different annealing coefficients. When the annealing coefficient α = 0.9, the conversion is complete and reaches a

6 Mode-conversion Methods

Conversion efficiency (%)

Conversion efficiency (%)

184

Number of iterations

Number of iterations (b) Effect of annealing coefficient on conversion efficiency

Conversion efficiency (%)

Conversion efficiency (%)

(a) Effect of initial temperature on conversion efficiency

Number of iterations (c) Influence of disturbance times on conversion efficiency

Number of iterations (d) Effect of annealing region on conversion efficiency

Fig. 6.4 Conversion efficiency under different annealing conditions

stable final value of 98.60%. When α = 0.95, the annealing process is too slow. Although it reaches a stable conversion-efficiency value of 98.31%, its convergence speed is too slow, and the calculation time required is too long. Therefore, the annealing coefficient should be 0.9. Figure 6.4c shows the conversion efficiency under different numbers of disturbances. When the number of disturbances k = 200, the conversion efficiency increases the fastest, up to 97.64%; however, it does not reach the convergence state. When the number of disturbances k = 500 and k = 800, the conversion efficiencies are higher than that at k = 200 : 97.78 and 98.24%, respectively; however, they also do not converge. When k = 1000, the conversion efficiency finally reaches a stable convergence state of 98.89%. Therefore, the number of disturbances k = 1000 was selected. Figure 6.4d shows the conversion efficiency of different annealing intervals. When the annealing intervals were 450 × 450 and 550 × 550, the conversion efficiency increased at the same speed, and the time to reach a stable conversion-efficiency value of 98.86% was the same. The influence of the two annealing intervals on the coupling efficiency was the same, and it was concluded that the effective interval during mode conversion was 450 × 450.

6.4 Mode-conversion Improvements

185

The results in Fig. 6.4 determined the annealing parameters when SA optimizes the transfer function: initial temperature T = 10, annealing coefficient α = 0.9, number of disturbances k = 1000, and the optimization area is 450 × 450. In addition, because the phase of the SA algorithm can be randomly reversed when optimizing the conversion-transfer function, the final value of each conversion is not completely consistent; however, the conversion efficiency after each conversion is stable at more than 98.85%.

6.4.2 Comparison of Mode-conversion Effects Using LP02, LP12, LP22, and LP32 as examples, a simulation experiment converting from a high-order mode to the fundamental mode was carried out. SA was added to locally optimize the phase hologram of the conversion-transfer function, and the conversion results were compared with those of the binary phase. In Fig. 6.5, in the conversion of LP02, the SA randomly perturbed the phase of the central pixel point. The phase value of each point determines whether to accept the change, according to the Metropolis criterion, to obtain the optimal phase distribution and maximize the conversion accuracy. There is a circle of high-order diffraction components around the transfer function of LP02 mode in the second column after the binary phase conversion. Observing the third column, after the SA phase optimization, the high-order diffraction component at the center of the phase-conversion transfer function was significantly reduced over that without SA, the energy of the central spot of the converted mode was more concentrated and symmetrically distributed, and the accuracy of the converted mode was improved. After the SA optimization, the mode accuracies of LP12, LP22, and LP32 were also improved. The improvement in the conversion efficiency after SA optimization compared with the pure binary conversion is shown in Fig. 6.6. In Fig. 6.6, using LP31 mode as an example, the conversion efficiency after the binary phase conversion is CEold = 64.90%. After applying the SA algorithm, the value of the objective function tends to be stable, and the conversion efficiency from LP31 mode to the LP01 mode reaches 86.84%. It was verified that the SA algorithm is insensitive to the initial value, and the conversion efficiency of each order mode was improved after optimization by the SA algorithm.

186

6 Mode-conversion Methods Conversion mode

Binary conversion

SA optimized conversion Transform transfer function holographic phase

Target mode

Transform transfer function holographic phase

Target mode

Transform transfer function holographic phase

Target mode

Transform transfer function holographic phase

Target mode

Fig. 6.5 Phase holograms of the transfer function after SA conversion and conversion results

6.5 Experimental Study

Binary conversion SA optimization

Conversion efficiency

Fig. 6.6 Comparison of the conversion efficiency when converting a high-order mode to LP01, before and after SA

187

Conversion type

6.5 Experimental Study 6.5.1 Mode-conversion Experiment (1) Experimental platform The experimental system includes a 632-nm He–Ne laser, two lenses (Lens 1 and Lens 2), an aperture, a Spiricon beam analyzer, two reflective LC-SLM-R2, and a polarization controller. Because a high-order mode cannot be obtained directly through the laser, pretreatment is required in the experiment. The purpose of the pretreatment is to use a spatial light modulator (SLM) to generate a high-order mode degraded after turbulence. Then, the basic-mode Gaussian beam is obtained through the modulation of the mode-conversion system. The optical-path diagram of the spatial-light mode conversion based on the LC-SLM is shown in Fig. 6.7. The specific operation process of the mode-conversion experiment is as follows. The fundamental-mode Gaussian beam emitted by the He–Ne laser passes through the diaphragm to filter out peripheral stray spots, after which the polarization state is adjusted to a state parallel to the long axis of the liquid–crystal plate of the SLM through Polarizer 1. LC-SLM 1 is located on the rear focal plane of Lens 1 and the front focal plane of Lens 2. Therefore, Lens 1, Lens 2, and LC-SLM 1 form an OTI spatial-filtering system. First, the preprocessing process excites the fundamental mode to a high-order mode. Then, the obtained high-order mode is passed through LC-SLM 2, and the loaded converted holographic-phase diagram is changed using the driving software of the LC-SLM 2 controller to realize different modulations of different high-order incident-light modes. (2) Experimental results The two-dimensional and three-dimensional distributions of the degraded mode spot measured by the beam analyzer are shown in Fig. 6.8.

188

6 Mode-conversion Methods Optical signal Electrical signal Diaphragm Polarizer 1

He-Ne laser

LENS1

LC-SLM1

Mode conversion system

LENS3

Beam analyzer

Fig. 6.7 Optical-path diagram of a spatial-light mode conversion based on LC-SLM Three dimensional distribution

Two dimensional distribution

LP11

a(1)

a(2)

b(1)

b(2)

Simulation

Experiment

Fig. 6.8 LP11 mode-preprocessing results [27]

6.5 Experimental Study

189

In Fig. 6.8, the two-dimensional experimental results after the LP11 mode pretreatment are shown in Fig. 6.8b(1), and the two-dimensional pattern of LP11 mode under ideal conditions is shown in Fig. 6.8a(1). Comparing a(1) and b(1) in Fig. 6.8, the diffraction around the mode obtained by pre-treatment is consistent with the randomness of the mode distribution after turbulence. The three-dimensional experimental results after the LP11 mode pretreatment are shown in Fig. 6.8b(2), and the three-dimensional pattern of the LP11 mode under ideal conditions is shown in Fig. 6.8a(2). Considering Fig. 6.8a(2) and b(2), there are two symmetrical peaks in the LP11 mode. The mode-energy distribution obtained by pretreatment was consistent with the ideal energy distribution of the LP11 mode, and the randomness of the peripheral diffraction was consistent with the mode distribution after turbulence. In Fig. 6.9, the first column shows the spot distribution of the ideal high-order mode and the second column shows the two-dimensional spot distribution of the excitation mode obtained by experimental preprocessing collected by the beam analyzer. The third column shows the three-dimensional spot distribution of the excitation mode obtained by the collected preprocessing. As shown in Fig. 6.9, the energy distribution of the high-order mode obtained by preprocessing is consistent with that of the ideal high-order mode, the distribution

Fig. 6.9 High-order modes generated by pretreatment of the mode-conversion experiment [27]

190

6 Mode-conversion Methods

symmetry of the three-dimensional image is consistent with the position of the peak in each mode and the target high-order mode under ideal conditions, and the effect of the high-order mode, after turbulence degradation generated by preprocessing, is remarkable. Owing to the uncontrollable filling factor of liquid–crystal devices during conversion, peripheral high-order diffraction components are inevitable. With an increase in the mode order, the higher-order diffraction component in the periphery is consistent with the random dispersion state formed after turbulence. After the excited turbulence degenerates, higher-order diffraction components around the mode increase. The diffraction increases with an increase in the mode order. In Fig. 6.10, the second behavior is the high-order mode after turbulence degradation obtained by pretreatment, and the third behavior is the target LP01 mode generated after passing through the mode-conversion system. As shown in Fig. 6.10, the LP02 mode distribution in the first column is compared with the ideal light-intensity distribution. The distribution and symmetry of the light-field energy are consistent with the ideal situation; however, the inevitable high-order diffraction components are present. The degraded mode then passes through LC-SLM 2 to generate the final fundamental-mode LP01 mode. The converted result is shown in the third line, and a high-order diffraction component also exists after secondary modulation. However, the energy of the LP01-mode light-field distribution is concentrated, which is consistent with the ideal-mode field distribution. Moreover, with a decrease in the mode order, the light-field distribution after conversion becomes more concentrated, which is consistent with the simulation results in Table 6.2, in which the conversion efficiency decreases with an increase in the mode order. It can be observed that after the spatial-mode conversion based on the LC-SLM combined with the SA algorithm, the obtained LP01 mode distribution is consistent

Ideal target mode

Incentive mode

Conversion mode

Fig. 6.10 Mode-conversion results [27]

6.5 Experimental Study

191

with that under ideal conditions, and the energy is concentrated at the center, which has an obvious conversion effect. The other high-order modes also have an obvious conversion effect after passing through the mode-conversion system.

6.5.2 Coupling-efficiency Experiment (1) Mode-conversion coupling experiment As shown in Fig. 6.11, the specific process of the coupling experiment is as follows. The fundamental-mode Gaussian beam of the laser is used as the emitted light. Using the polarization controller, the polarization state is parallel to the long axis of the liquid–crystal plate of the spatial-light modulator (SLM). The phase hologram obtained by preprocessing on LC-SLM 1 modulates the fundamental mode into a higher-order mode after turbulence through LC-SLM 1. It converges at the center through Lens 1. LC-SLM 1 is loaded to generate high-order mode information after turbulence. When LC-SLM 2 is powered off, it can be regarded as a plane mirror. The beam analyzer can detect the light-field distribution of the higher-order mode after turbulence. The field distribution of the detected degraded high-order mode is shown in Figs. 6.9 and.6.10. LC-SLM 2 is placed in the rear focal plane of Lens 2. Therefore, Optical signal Electrical signal Diaphragm Polarization 1 LENS 1 He-Ne laser

LC-SLM 1

Mode conversion system

Spectroscopic prism

Coupling device

Optical power meter

Beam analyzer

Fig. 6.11 Schematic diagram of the mode-conversion system for a coupling experiment

192

6 Mode-conversion Methods

Lens 2, the spectroprism, and LC-SLM 2 form a mode-conversion system to convert the high-order degraded mode into the basic mode. By changing the loaded hologram through the driving software of the LC-SLM 2 controller, different high-order modes can be phase modulated, and finally converted into the basic mode. The light converted by LC-SLM 2 is divided into two beams through the splitting prism, one of which is reflected on the beam analyzer for observation, and the other is transmitted to the coupling device for single-mode opticalfiber coupling. The optical-power meter is used to detect the optical power at the end face of the optical fiber and coupled into the single-mode optical fiber. (2) Analysis of mode-conversion results The output power P f at the end of the optical fiber, collected by the optical power meter, the power P0 at the end of the optical fiber, and the coupling efficiency calcuP lated according to the formula η = P0f are shown in Table 6.3. According to the coupling experiment under mode conversion shown in Fig. 6.11, the influence of mode conversion on the coupling efficiency of the spatial-light single-mode fiber was analyzed and compared. The fundamental-mode power was measured using an optical-power meter when the beam was coupled from the coupling device to the SMF. In Table 6.3, when the degraded mode obtained by pretreatment is LP22, the coupling power during a pure binary conversion is − 18.53 dBm, and the coupling efficiency optimized by the SA algorithm increases from 55.92 to 66.21%. It was concluded that mode conversion plays an important role in improving the basic-mode content and coupling efficiency. An LC-SLM is used to realize the mode conversion, increase the content of the fundamental mode, and indirectly improve the coupling efficiency of the spatial-light single-mode fiber from mode conversion. The coupling efficiency of LP02 mode with the smallest order improved to 69.18%. The Table 6.3 Mode-conversion coupling efficiency [27] Conversion mode

LP02

Target mode

Lp01

Fiber end power/dBm

LP22

LP23

LP71

Binary conversion

− 18.13

− 18.18

− 18.53

− 19.56

− 19.54

SA optimization

− 15.62

− 15.65

− 15.85

− 15.86

− 16.36

− 20.49

− 20.69

− 21.06

− 21.10

− 22.62

− 17.22

− 17.30

− 17.64

− 17.67

− 18.56

Optical power Binary meter/dBm conversion SA optimization Coupling efficiency

LP12

Binary conversion

58.09%

56.18%

55.92%

55.71%

49.19%

SA optimization

69.18%

68.38%

66.21%

65.91%

60.29%

References

193

corresponding order of the LP71 mode was the largest, and the coupling efficiency was improved from the 49.12% of a binary conversion to 60.29%.

References 1. Tsekrekos CP, Koonen AMJ (2007) Mode-selective spatial filtering for increased robustness in a mode group diversity multiplexing link [J]. Opt Lett 32(9):1041–1043 2. Tsekrekos CP, Koonen AMJ (2008) Mitigation of impairments in MGDM transmission with mode-selective spatial filtering [J]. IEEE Photonics Technol Lett 20(13):1112–1114 3. Leon-Saval SG, Argyros A, Bland-Hawthorn J (2010) Photonic lanterns: a study of light propagation in multimode to single-mode converters [J]. Opt Express 18(8):8430–8439 4. Carpenter J, Wilkinson TD (2011) Precise modal excitation in multimode fibre for control of modal dispersion and mode-group division multiplexing [C]. In: 37th European conference and exposition on optical communications, OSA technical digest (CD). Optica Publishing Group 5. Amphawan A (2011) Holographic mode-selective launch for bandwidth enhancement in multimode fiber [J]. Opt Express 19(10):9056–9065 6. Fontaine NK, Doerr CR, Mestre MA et al (2012) Space-division multiplexing and all-optical MIMO demultiplexing using a photonic integrated circuit [C]. In: OFC/NFOEC, Los Angeles, CA, USA. IEEE, pp 1–3 7. Birks TA, Mangan BJ, Díez A et al (2012) “Photonic lantern” spectral filters in multi-core fiber [J]. Opt Express 20(13):13996–14008 8. Ding Y, Ou H, Xu J et al (2013) Silicon photonic integrated circuit mode multiplexer [J]. IEEE Photonics Technol Lett 25(7):648–651 9. Hanzawa N, Saitoh K, Sakamoto T, Matsui T et al (2014) PLC-based mode multi/demultiplexer for MDM transmission [C]. In: Proceedings of the next-generation optical communication: components, sub-systems, and systems III. SPIE OPTO, San Francisco, California, United States, vol 9009, pp 8–16 10. Hanzawa N, Saitoh K, Sakamoto T et al (2013) Two-mode PLC-based mode multi/ demultiplexer for mode and wavelength division multiplexed transmission [J]. Opt Express 21(22):25752–25760 11. Uematsu T, Hanzawa N, Saitoh K et al (2014) PLC-type LP11 mode rotator with single-trench waveguide for mode-division multiplexing transmission [C]. In: Optical Fiber Communications Conference (OFC), San Francisco, California, USA. IEEE, pp 1–3 12. Gao L, Shang XH, Lan MY et al (2015) An accurate arbitrary mode conversion method based on multi-phase simulated annealing algorithm.Chinese patent: CN105007545A 13. Qi XL (2015).Research on precise control and selection of optical mode using spatial light modulator in space division multiplexing [D]. Beijing: Beijing University of Posts and Telecommunications 14. Taher AB, di Bin P, Bahloul F et al (2016) Adiabatically tapered microstructured mode converter for selective excitation of the fundamental mode in a few mode fiber [J]. Opt Express 24(2):1376–1385 15. Tu JJ, Zhang H, Li H et al (2017) Design of the third mock examination multiplexer/ demultiplexer based on multi-core fiber [J]. J Opt 37(3):162–169 16. Shen D, Wang C, Ma C et al (2018) A novel optical waveguide LP01 /LP02 mode converter [J]. Opt Commun 418:98–105 17. Sun XK, Zhang JJ (2012) Optical fiber communication [M]. People’s Posts and Telecommunications Press, Beijing, pp 41–43 18. Gu WY (2013) Optical fiber communication system [M], 3rd edn. Beijing Posts and Telecommunications Press, Beijing, pp 169–172 19. Stepniak G, Maksymiuk L, Siuzdak J (2011) Binary-phase spatial light filters for mode-selected excitation of multimode fibers [J]. J Lightwave Technol 29(13):1980–1987

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20. Li G, Bai N, Zhao N et al (2014) Space-division multiplexing: the next frontier for highly efficient spatial mode conversion [J]. Opt Express 22(10):11610–11619 21. Ke XZ (2018) Principle and application of wireless optical orthogonal frequency division multiplexing [M]. Science Press, Beijing, pp 12–16 22. Wang ZH, Wu ZY (1997) Calculation of mode field and propagation constants for graded-index optical fibers by using coupled mode theory [J]. J Photonics 26(2):115–120 23. Stepniak G, Maksymiuk L, Siuzdak J (2010) Increasing multimode fiber transmission capacity by mode selective spatial light phase modulation [C]. In: 36th European conference and exhibition on optical communication, Torino, Italy. IEEE, pp 1–3 24. Kitayama K, Tateda M, Seikai S et al (1979) Determination of mode power distribution in a parabolic-index optical fiber: theory and application [J]. IEEE J Quantum Electron 15(10):1161–1165 25. Frey RW, Burkhalter JH, Zepkin N et al (2002) Apparatus and method for objective measurements of optical systems using wavefront analysis [P]. US Patent: US 6460997 B1 26. Lan M, Gao L, Yu S et al (2015) An arbitrary mode converter with high precision for mode division multiplexing in optical fibers [J]. J Mod Opt 62(5):348–352 27. Zhang XT (2020) Study on improving coupling efficiency of single-mode fiber by mode conversion method [D]. Xi’an: Xi’an University of Technology

Chapter 7

Adaptive-optical Wavefront Correction

Wireless-optical communication includes two signal-detection methods: intensity modulation/direct detection (IM/DD) and coherent detection. Compared with direct detection, coherent detection has a detection sensitivity of approximately 20 dB gain and is more suitable for long-distance spatial laser communication. Coherent detection uses a mature fiber mixer in optical-fiber communication. A fiber mixer is used in a spatial-optical communication system to efficiently couple spatial light into the fiber. This is an effective method to improve the sensitivity of coherent detection by using adaptive-optics technology to improve the coupling efficiency of spatial light into optical fiber.

7.1 Introduction According to the pattern-matching principle, theoretical calculations show that the maximum coupling efficiency of a spatial-light–single-mode fiber in the 1550-nm band can reach 82.69% [1]. Radial deviation has the greatest influence on the coupling efficiency [2]. Array-fiber–coupling beam-synthesis technology and adaptive-optics technology can improve the coupling efficiency by changing the beam quality [3]. The coupling efficiency of a hexagonal array fiber is higher than that of a single fiber [4]. Adaptive-optics technology can increase the coherence length of spatial light and thus improve the coupling efficiency [5]. A fast reflector is used to correct the wavefront tilt component, which can directly improve the coupling efficiency of spatial light to an optical fiber [6]. The combination of adaptive-optics technology and array-fiber–coupling technology can effectively improve the coupling efficiency by correcting only the first 3–20 orders of the wavefront Zernike coefficient; the coupling efficiency can be increased from 15 to 40% under medium-turbulence conditions [7]. Wavefrontcorrection technology has also been used in wireless-optical communications. It © Science Press 2023 X. Ke, Spatial Optical-Fiber Coupling Technology in Optical-Wireless Communication, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-1525-5_7

195

196

7 Adaptive-optical Wavefront Correction

improves the communication quality, based on adaptive-optics technology, to correct the wavefront of a signal light by modifying the wavefront-measurement results of the beacon light [8]. For an actual system, experiments showed that after the adaptive-optics system correction, the Strehl ratio of the beam improved from 8 to 33%, and the coupling efficiency improved from 12.5 to 29.5% for moderate turbulence, and from 6.6 to 21.8% for strong turbulence [9]. However, owing to the existence of non-common opticalpath aberrations, the wavefront phase at the end face of the fiber is not corrected ideally after the traditional adaptive-optics closed loop [10–12]. Compared with the phase-difference method, the wavefront-based adaptiveoptical algorithm is more mature in the practical-engineering application of eliminating non-common optical-path aberrations [13–15]. The Strehl ratio is typically used to estimate the coupling efficiency [16], an optimization algorithm is used to improve the coupling efficiency of the array fiber to detect a distorted wavefront phase with static aberrations, and the relationship between the array coupling efficiency and the wavefront phase can be obtained [17, 18]. In this chapter, the influence of wavefront distortion on the coupling efficiency is analyzed in detail.

7.2 System Composition Figure 7.1 shows the spatial optical–fiber coupling structure. The transmitting end is composed of a laser light source, electro-optic modulator, fiber amplifier, transmitting antenna, coarse-aiming platform, and fine-aiming platform. The laser-seed source is a narrow line-width laser in the 1550-nm band. The signal is modulated by an electrooptic phase modulator and amplified by an optical-fiber amplifier. The transmitting antenna is a Kepler optical-transmission antenna. The emission structure is a double-emission structure—that is, a seed light source is used—and two identical optical antennas are used to collimate the output laser

Laser

Transmitting antenna

Atmospheric turbulence

Receiving antenna

Rapid reflector

Computer

Source

Accommodation

Deformation mirror Wavefront sensor

Local oscillator laser

Balance detector

Electrical signals Optical signals

Optical fiber coupling

Mixer Balance detector

Fig. 7.1 Schematic diagram of spatial optical–fiber coupling for a fast reflector with a deformation mirror

7.2 System Composition

197

beams after amplification via beam splitting. The coarse-aiming platform, microaiming platform, and optical-antenna focusing system are composed in a composite axis-linked focal-length adjustment structure to track the alignment and capture the beam. The receiving end is mainly composed of a receiving antenna, adaptive-optical system, and coherent-detection system. The receiving antenna is a Cassegrain antenna. The adaptive-optical system is composed of a fast reflector, deformation mirror, and wavefront sensor to correct the optical signal. After the wavefront correction, the beam is coupled to a mixer and balance detector using an optical fiber, and the signal recovery is completed after mixing with the local oscillator.

7.2.1 Zernike Polynomial A Zernike polynomial describes optical-system aberrations in the form of a powerseries expansion. Wavefront w(r, θ ) can be decomposed into a Zernike polynomial function-expansion form: w(r, θ ) =

N ∑

ai Z i (r, θ ) + Δ w

i=1

= Δ Z (r, θ ) = A00 +

N ∑

An0 Rn0 (r )

n=2

+

N ∑ n ∑

Rnm (Anm cos(mθ ) + Bnm sin(mθ ))

(7.1)

n=1 m=1

where Z i is the i th-term Zernike polynomial, ai is its coefficient, N is the number of Zernike polynomials used, and Δ w is the residual introduced using finite-term Zernike polynomials. To facilitate the use of Zernike polynomials, single-index patterns Z i and Z nm (r, θ ) are double-index polynomials used in the wavefront reconstruction of the pattern method. The corresponding relationship between the double-index numbers n and m and the single-index number i is not strictly required because the order of the Zernike polynomials has no effect on their coefficients. Among them, Z nm (r, θ ) ⎧ √ ⎨ Z odd• j (r, θ ) = √2(n + 1)Rnm (r ) sin mθ, m < 0 = Z even• j (r, θ ) = 2(n + 1)Rnm (r ) cos mθ, m > 0 ⎩√ n + 1Rnm m=0

(7.2)

198

7 Adaptive-optical Wavefront Correction

In Zernike polynomials, n is the order of the Zernike polynomial, m is the angular frequency, and n and m satisfy m ≤ n, n − |m| = even.

(7.3)

Rnm (r ) can be expressed as Rnm (r ) ⎧ (n−m) ⎪ 2 ⎨ ∑ (−1)s (n−s)! r n−2s , n − m is even n−m = s=0 s![ n+m 2 −s ]![ 2 −s ]! ⎪ ⎩ 0, n − m is odd

(7.4)

The mapping between single index i and double index n and m is as follows: ⎧ n(n+1) n−m ⎪ ⎨ j = [ 2 √+ 2 + ] 1 −3+ 9+8(i−1) n= 2 ⎪ ⎩ m = n 2 + 2(n − i + 1)

(7.5)

The Zernike polynomial’s root-mean-square value in the unit circle is ┌ | {1 {2π ( )2 | | Z nm (r, θ ) r dr dθ |0 0 | =1 | {1 {2π | r dr dθ

(7.6)

0 0

Zernike polynomials are closely related to Seidel aberrations in optical design (such as spherical, astigmatism, comet, etc.), where 0 is a constant term, namely, the translation term, and this coefficient represents the average optical-path difference. Terms 1 and 2 represent oblique aberrations in the x- and y-directions, respectively. The third term is the defocus aberration. Terms 1–3 represent the Gaussian or paraxial characteristics of the wavefront. Terms 4 and 5 represent astigmatism and defocus, respectively. Terms 6 and 7 represent coma and tilt, respectively. Term 8 represents a spherical aberration. Terms 4–8 are third-order aberrations. The other higher-order aberrations are ordered in sequence. Table 7.1 shows the expressions and physical meanings of the low-order Zernike polynomials.

7.2 System Composition

199

Table 7.1 Zernike polynomial coefficients and their significance Z nm

n

Z 00

0

Z 11 Z 1−1 Z 20 Z 2−2 Z 2−2 Z 3−1 Z 31 Z 3−3

m

Noll index j

Zi

Name

0

0

1

Translation

1

1

1

2r cos θ

x tilt

1

−1

2

y tilt

2

0

3

2

−2

4

2

2

5

3

−1

6

3

1

7

3

−3

8

2r sin θ √ ( 2 ) 3 2r − 1 √ 2 6r sin 2θ √ 2 6r cos 2θ √ ( 3 ) 8 3r − 2r sin θ √ ( 3 ) 8 3r − 2r cos θ √ 3 8r sin 3θ

Defoucs Astigmatism Astigmatism Coma in y direction Coma in x direction Spherical aberration

7.2.2 Influence of Wavefront Distortion on Coupling Efficiency A distorted wavefront phase can typically be expanded according to the Zernike coefficient. Considering the accuracy of fitting the Zernike polynomial to the wavefront, 30 terms of the Zernike polynomial of the wavefront were considered. Therefore, the Zernike-expansion expression of the discrete wavefront phase ϕ(x, y) is ϕ(x, y) = 1 +

30 ∑

ai · z i (x, y)

i=1

x = 0, 1, · · · , M − 1

(7.7)

y = 0, 1, · · · , N − 1 In Eq. (7.7), M and N are the discretization of the radial and angular coordinates, respectively, ai is the Zernike coefficient, and z i (x, y) is the discretization of the Zernike polynomial. Because the distorted wavefront phases ϕ(x, y) usually do not have circular-domain symmetry, the light-field distribution in the focal plane after the lens cannot be directly solved using a Fourier–Bessel transform. A two-dimensional discrete Fourier transform in polar coordinates is required to solve it [19, 20]. The two-dimensional discrete Fourier transform of the polar coordinates of a distorted wavefront phase ϕ(x, y) is expressed as follows: M−1 N −1 ∑∑

x 2 · MN M x=0 y=0 [ (y ) (u D ) (x )] v a · cos · exp − j2π Wm · 2π − 2π M M 2λ f N N ( ) u = 0, 1, · · · , M − 1 v = 0, 1, · · · , N − 1

φ(u, v) =

ϕ(x, y) ·

(7.8)

200

7 Adaptive-optical Wavefront Correction

In Eq. (7.8), Wm is the mode-field radius of the single-mode fiber, λ is the beam wavelength, f is the focal length of the coupling lens, Da is the diameter of the coupling lens, u and v are the radial and angular coordinate points in the frequency domain, respectively, and the points of the discrete two-dimensional Fourier transform are the same as the wavefront phase points, for easy calculation. When ai = 0, the equation is the two-dimensional discrete Fourier-transform expression of the plane-wavefront phase polar coordinates with a unit piston. According to the Fourier-optics principle, the light field on the focal plane of a lens is an image formed by the Fraunhofer diffraction of a beam with a distorted wavefront. Therefore, the light-field distribution Ui (u, v) located in the focal plane of the lens can be expressed as follows: Ui (u, v) exp(ik f ) = ik f [ ( )2 ] ( )2 ik Mu Wm Da · exp · φ(u, v) · 2f 2

(7.9)

/ In Eq. (7.9), k = 2π λ is the wave vector, and the mode-field distribution U f (u, v) of the discrete single-mode fiber is expressed as / U f (u, v) =

[ ( ) ] 2 1 u 2 · exp − π Wm M

(7.10)

Considering the diffraction effect, the mode-field-matching analysis method is typically used to calculate the coupling efficiency of the distorted wavefront phase. When ignoring various alignment errors of the optical-fiber coupling and aberrations of the lens itself, according to the mode-field matching principle, the spatial optical– fiber coupling efficiency of the discrete distorted wavefront η can be solved as follows: |2 | M−1 N −1 | | ∑ ∑ W 2π u ∗ m | · N · U f (u, v) · Ui (u, v) · M Wm || | M u=0 v=0 η = ( M−1 N −1 ) ∑ ∑ Wm 2π u ∗ · · U f (u, v) · U f (u, v) · Wm M N M u=0 v=0 ) ( M−1 N −1 ∑ ∑ Wm 2π u · · Ui (u, v) · Ui∗ (u, v) · Wm M N M u=0 v=0

(7.11)

7.2 System Composition

201

7.2.3 Power in the Barrel The power in the barrel (PIB) can be used to represent the degree of energy aggregation of the focal-plane spot affected by the distorted wavefront. The definition of the wavefront distortion for the normalized power in the barrel is as follows M1 ∑ N ∑

PIB =

u=0 v=0 M2 ∑ N ∑ u=0 v=0

Ui (u, v) · Ui∗ (u, v) (7.12) Ui (u, v)

· Ui∗ (u, v)

/ Let d = M1 M2 · Da be the barrel diameter. ) Note that the covariance of ai and ( ai, between the Zernike coefficients is E ai , ai, ; then, ) ( E ai , ai,

/ √ , 2.2698(−1)(n+n −2m ) 2 (n + 1)(n , + 1) [( )/ ] ( / )5 3 5 · δZ · ┌ n + n, − 2 · D r0 / 3 )/ ] = [( 17 2 ┌ n − n, + 3 )/ ] [( )/ ] [( 17 23 , , 2 ·┌ n+n + 2 ·┌ n −n+ 3 3

(7.13)

In Eq. (7.13), n, n , , m, and m , are the Zernike polynomial order and angular frequency, respectively, of coefficients ai and ai, , δz is the Kronecker function, δz is the aperture diameter of the optical system, and r0 is the atmospheric coherence length (Fried constant). By constructing a/statistic independent Karhunen-Loève function, under the conditions of different D r0 turbulence intensities, the wavefront Zernike coefficient under the corresponding conditions can be generated. It can be substituted into Eq. (7.11) to obtain the relationship between the turbulence intensity and the power in the normalized barrel.

7.2.4 Strehl Ratio The Strehl ratio (SR) was introduced by a German scholar in the study of optical imaging and is often used as a general performance-evaluation standard in the adaptive-optics field. The optical-transfer function (OTF) parameter is generally used in optical systems to measure system indexes; however, the OTF generally considers the average value, while the adaptive-optics error comes from randomly disturbed turbulence, so this index is not appropriate. Therefore, the Strehl ratio is

202

7 Adaptive-optical Wavefront Correction

the parameter used in adaptive-optics systems. The Strehl ratio is defined as the ratio of the actual spot peak intensity to the far-field spot peak intensity of the ideal wave surface: SSR

| | | A(ρ)e jϕ(ρ) |2 I (x0 , y0 ) | | = = I0 (x0 , y0 ) | A(ρ) |

(7.14)

In Eq. (7.14), I (x0 , y0 ) is the peak intensity of the far-field light spot of the distorted wavefront, I0 (x0 , y0 ) is the peak intensity of the far-field light spot of the ideal wavefront, that is, the peak intensity of an ideal aberration-free light spot, and I (x0 , y0 ) < I0 (x0 , y0 ). A(ρ) is the complex amplitude corresponding to the ideal wavefront and ϕ(ρ) is the distorted wavefront. The Strehl ratio is a dimensionless number between 0 and 1. The closer the Strehl ratio is to 1, the better the correction result. The closer the Strehl ratio is to zero, the smaller the peak light intensity and the worse the beam quality. Therefore, the Strehl specific energy can directly reflect the peak light intensity in the far field during a laser-beam transmission; however, it cannot provide the light-intensity distribution of energy laser applications. The Maclaurin series expansion of Eq. (7.14) can be obtained as SR ≅ 1 − σϕ2 ,

(7.15)

where σϕ2 is the root-mean-square error of the wavefront, denoted as RMS, which is defined as RMS

┌ | 2π 1 | { { ( )2 |1 ϕ(ρ, θ ) − ϕ(ρ, θ ) ρdρdθ =| π 0 0 / 2 = ϕ 2 (ρ, θ ) − ϕ(ρ, θ ) .

(7.16)

If the distorted-wavefront phase follows a Gaussian distribution and because σϕ2 is generally small, the Strehl ratio can be approximated as ) ( SR ≅ exp −σϕ2 .

(7.17)

) ( SR ≅ exp −RMS2 .

(7.18)

Namely,

In this case, the Strehl ratio is only a function of the wavefront difference, which is very sensitive to the difference and suitable for evaluating the correction quality of adaptive-optical systems. The Strehl ratio can also be approximated using the

7.2 System Composition

203

following formula: [ ( / )5 3 ]−6/ 5 S R ≅ 1 + D r0 / .

(7.19)

There is a good / fitting relationship / between the coupling efficiency and the Strehl ratio when D r0 > 1. When D r0 = 1, , the Strehl ratio is proportional to the intensity of the light field entering the single-mode fiber, which can be approximated to the fiber coupling efficiency in practice. Therefore, the Strehl ratio or RMS can be used as an indicator to measure the received signal quality and the correction effect of the free-space optical (FSO) system. In short, the quality of an FSO system spot can be examined using the above indicators.

7.2.5 Wavefront Sensors Wavefront sensors are modern photoelectric testing instruments that use wavefrontdetection technology to measure wavefronts [21]. As the “eye” of an adaptive-optical system, a sensor can detect an aberration in the input wavefront in real time, and then transmit a voltage-control signal to the control system to correct the interference to the beam phase caused by atmospheric turbulence. To ensure the accuracy of the distorted-wavefront correction by the adaptiveoptical system, the spatial and temporal resolution of the wavefront sensor must match the time and spatial scale of the disturbance signal; that is, the subaperture size of the wavefront sensor must be smaller than the coherence length of the atmosphere, and the sampling frequency of the CCD must match the coherence time of the atmosphere. There are four types of common wavefront sensors: (1) (2) (3) (4)

Shack–Hartmann wavefront sensors, curvature wavefront sensors, point-diffraction interferometers, and transverse-shear interferometers.

(1) Shack–Hartmann wavefront sensor The Shack–Hartmann wavefront sensor (SHWFS) is the most widely used wavefront sensor for adaptive optics. It originated from the Hartmann screen invented by the German scientist Johannes Franz Hartmann. It is a highly porous mask used in astronomical exploration. Later, Roland V. Shack et al., from the University of Arizona, transformed it [22] to form the structure commonly observed today. Current Shack–Hartmann wavefront sensors consist of a microlens array and a photodetector (camera). The microlens array divides the incident light beam into subapertures and focuses the incident light beam within each subaperture onto the camera at the focal plane. As shown in Fig. 7.2, the camera records the light-energy distribution of the spot, focusing on the focal plane in each subaperture. The focal

204

7 Adaptive-optical Wavefront Correction f

Δx Δy

Turbulent Micro wavefront lens

Reference position Probe location

Imaging

Detector

Fig. 7.2 Schematic diagram of a Shack–Hartmann wavefront sensor

length of the microlens can be determined by calibration. When the incident light is distorted, the centroid of the subaperture light spot deviates with respect to the focus of the microlens. The spot coordinates are obtained using the centroid algorithm of the light spot. Figure 7.3 shows a plane diagram of the optical path in a single subaperture of the wavefront sensor, which can intuitively show the relationship between the light-spot offset and the wavefront tilt. d is the diameter of the microlens and f is the focal length of the microlens. The wavefront of the beam incident on the lens is expressed as ϕ(x, y), and the relationship between the beam and wavefront is expressed by the local average slope, as follows: { ∂ϕ(x,y) = gx ∂x (7.20) ∂ϕ(x,y) = gy ∂y

d

Fig. 7.3 Schematic diagram of a single-subaperture optical path

δ

θ

f

Micro lens

x Δx

7.2 System Composition

205

Wavefront phase

Focal length f

Focal plane

Fig. 7.4 Schematic diagram of a curvature-wavefront sensor [24]

Then, Δ x and Δ y are used to calculate the slopes gx and g y of the local wavefront: {

gx = tan θx = g y = tan θ y =

Δ x f Δ y f

(7.21)

The wavefront error on the subaperture is expressed by δ; then, g = tan θ ≈ sin θ =

δ d

(7.22)

(2) Curvature-wavefront sensing The principle diagram of the curvature wavefront sensor is shown in Fig. 7.4. The wavefront-phase distribution is calculated by measuring the intensity distribution on both sides of the two focal planes. When a distorted wavefront exists, the focal spot moves along the optical axis, and the wavefront-phase distribution can be obtained by comparing the light-intensity distribution of the two defocused planes [23]. The curvature sensor measures the wavefront-curvature information and has high resolution and high measurement accuracy. Its output signal can directly control the distortion mirror to compensate for the distortion aberration, which improves the system feedback speed and significantly reduces the calculation time of the adaptive optics system. However, curvature sensors are suitable for the detection and correction of loworder aberrations, and their processing ability of high-order aberrations has certain limitations. Meanwhile, the large number of wavefront calculations is not conducive to real-time processing [24]. (3) Wavefront sensors based on the interference principle Wavefront sensors based on the principle of interference have various structures. According to the different reference light-wave generation modes, they are mainly divided into point-diffraction interferometers and transverse-shear interferometers.

206

7 Adaptive-optical Wavefront Correction

Mask(Pinhole) Camera

Wavefront under test

Fig. 7.5 Schematic diagram of a point-diffraction interferometer [25]

Figure 7.5 shows a schematic diagram of a point interferometer. The pointdiffraction interferometer has a translucent mask plate with a pinhole, and the measured light beam is focused on the translucent mask. The measured light beam carries phase information through the mask and forms a reference spherical light beam with the pinhole diffraction. The two beams produce interference through the point-diffraction plate, and the distortion information of the measured wavefront can be obtained by analyzing the interference fringes of the two wavefronts. The optical-path difference between the two interferential beams is fixed, so the coherence of the point-diffraction interferometer is not high. A point-diffraction interferometer has the advantages of a simple structure and high sensitivity. The reference light comes from the measured beam and exhibits good anti-interference performance. However, point-diffraction interferometers are only suitable for wavefrontdetection systems with good light intensity, and their performance is low in weak light environments. Figure 7.6 shows a schematic diagram of the transverse-shear interferometer. The reference light in the transverse-shear interferometer is no longer an ideal sphere or plane wave, but is generated by changing the wave to be measured. A beam of incident light emitted from the wave surface is measured and it is split using a specific prism to obtain a second beam. The original beam has a certain lateral dislocation of the wave surface. The wave surfaces of the two beams overlap, with interference fringes in the overlapping area. The phase difference between the wavefront to be measured and the misaligned wavefront is analyzed through a study of interference fringes [27]. The advantages of a transverse-shear interferometer are that the standard reference optical surface can be omitted, and its structure is simple and stable. Table 7.2 compares the characteristics of the four wavefront sensors.

7.2 System Composition

207 Original waveW

Parallel-light Original waveW transverse shear interferometer

Wave surface after dislocationW1 Original waveW

Will focus the Original waveW transverse shear interferometer Wave surface after dislocationW2

Fig. 7.6 Schematic diagram of a transverse-shear interferometer [26]

Table 7.2 Comparison of common wavefront sensors Name of wavefront sensor

Output data type

Advantages

Disadvantages

Shack–Hartmann wavefront sensor

Slope

High utilization rate of light energy; large detection range

Owing to the limited spatial resolution due to the size of subaperture, the optimal wavefront-reconstruction order should be selected when using the mode method

Curvature sensor

Curvature

The deformation of the Low measurement accuracy deformation mirror can be directly controlled, with good real-time performance and low price

Point-diffraction interferometer

Phase

The requirements for coherence are not high; The utility model relates to a common optical-path interferometer with good anti-interference performance

Low light-energy utilization

Transverse-shear interferometer

Light intensity

High signal-to-noise ratio

Low light-energy utilization

7.2.6 Wavefront Correctors Wavefront-corrector types include deformation mirrors and a liquid–crystal spatiallight modulator (LC-SLM). The wavefront corrector generates a surface shape to compensate for the measured wavefront aberration, according to the control of the computational control processor. It is the core component of an adaptive-optical system and is generally conjugated to a wavefront sensor in the optical path. The

208

7 Adaptive-optical Wavefront Correction

wavefront corrector is controlled by the voltage signal of the computing controller to produce the corresponding deformation, and the phase structure of the incident-light wavefront is changed by changing the optical sign-in path or the refractive index of the transmission medium to correct the wavefront phase. To ensure the accuracy of a distorted-wavefront correction of the adaptive-optical system, the wavefront corrector is required to have sufficient spatial freedom to allow the aberration to be corrected well. In addition, the response speed should far exceed the time-varying frequency of the disturbed wavefront. The linear responsiveness of the wavefront corrector, calibration range, cost, and other factors are also key points to consider. Wavefront correctors include the following types: 1. 2. 3. 4. 5. 6.

Separation-actuator continuous-surface deformation mirror, Splicing sub-mirror deformation mirror, Thin-film deformation mirror, Double-piezoelectric deformation mirror, Microelectromechanical system (MEMS) deformation mirror, and Liquid–crystal spatial light modulator.

The separation-actuated continuous-surface deformation mirror is composed of a base, actuator, and continuous mirror, as shown in Fig. 7.7. The facesheet is a continuous mirror, and the actuators are actuators and baseplate bases. The actuator produces a push–pull displacement under the action of the control-signal voltage. Because the base is much stiffer than the mirror, the displacement generated by the actuator causes the mirror to deform. The change in the actuator length causes a local Gaussian-like deformation of the mirror, and the surface shape of the overall mirror can be expressed by the following formula:

Surface

Actuator

Base

Fig. 7.7 Schematic diagram of the structure of a separated actuated continuous-surface deformation mirror

7.2 System Composition

209

Fig. 7.8 (top) Single degree–of-freedom and (bottom) multiple degree-of-freedom splicing sub-mirror variant-mirror diagrams

r (x, y) =

∑

Vi ri (x, y)

(7.23)

where Vi represents the voltage applied to the i th actuator, and ri (x, y) is the response function of the mirror at the i th actuator. The principle diagram of the splicing sub-mirror deformation mirror is shown in Fig. 7.8. The mirror is composed of multiple sub-mirrors, each of which has one or more actuators to adjust the mirror-surface shape. The single-degree-of-freedom variant mirror (one actuator) can only correct the optical-path difference in the direction of beam propagation, whereas a multidimensional variant mirror (multiple actuators) can produce more complex surface shapes and correct more complex wavefront aberrations in different directions. Owing to their advantages of light weight, relaxed tolerance, and foldability, splicing deformation mirrors have been used in lightweight, large-aperture, and highresolution space-optical telescopes and other adaptive-optical systems. The gap between the continuous sub-mirrors reduces the light-energy utilization rate and increases the difficulty of adjusting the surface shape. In addition, the continuous wavefront structure must be ensured only when the edges of the adjacent submirrors are in phase. Therefore, common phase errors are the main limiting factor for splicing sub-mirrors, and the diffraction elements introduce additional wavefront aberrations into the system. In 1976, Yellin et al. of PerkinElmer Corporation invented an electrostatic-driven thin-film deformation mirror using thin films as mirrors, as shown in Fig. 7.9. Compared with the traditional piezoelectric deformation mirror, the cost of the thin-film deformation mirror is significantly reduced; however, the film-preparation process needs to be improved. Owing to the thin-film mirror’s own stiffness, the electrostrictive actuator only needs to produce a small force under the action of the control circuit to deform the mirror. The static deformation of thin films can be described by the following

210

7 Adaptive-optical Wavefront Correction

Incident light

v0

Protecting window Transparent electrode Membrane mirror

Optical aperture

Electrode separation

v0 ± vi Connection control circuit Fig. 7.9 Schematic diagram of a thin-film deformation mirror structure

formula: → = − p(r ) ∇z Tm

(7.24)

where p(r ) is the stress on the film related to the control voltage (unit: Nm−2 ) and Tm is the linear tension of the film (unit: Nm−1 ). The double-piezoelectric deformation mirror consists of two pieces of piezoelectric ceramic glued together, with a control electrode sandwiched between the two pieces, and a common external electrode. Under the action of the voltage signal applied by the electrode, the two piezoelectric-ceramic pieces produce transverse displacements in the opposite directions, which deforms the mirror bonded to the piezoelectric-ceramic plate. The structure is shown in Fig. 7.10. Its deformation can be expressed as ∇r (x, y) = −A∇V (x, y)

(7.25)

where V (x, y) is the electrode voltage distribution on the plane of the piezoelectricceramic plate, and A is a constant related to the material characteristics of the piezoelectric-ceramic plate. Owing to the limitations of the deformation characteristics of piezoelectric ceramics, the spatial resolution of double–piezoelectric-ceramic deformation mirrors is always low; however, its correction amount is of a higher order than that of other types of correctors, so it is used to correct low-order aberrations. The MEMS deformation mirror has many micro-correction units made using lithography technology, similar to electronic chips. The MEMS deformation mirror

7.2 System Composition

211

Fig. 7.10 Double– piezoelectric-ceramic deformation mirror

Z

Silicon mirror h h

V

Piezoelectric ceramic

Common electrode

has two corrector types: One is similar to the membrane-deformation mirror, and the other is similar to the separation-actuator deformation mirror for surface micromachining, as shown in Fig. 7.11. This type of deformation mirror can produce deformations in two directions. It can also produce a maximum deformation that is much higher than that of other types of correctors, using a small driving voltage. Therefore, the application of a MEMS deformation mirror to an adaptive optical system produces good results. The incident light is divided into O light and E light inside the liquid–crystal material; the two light beams correspond to different refractive indices, as shown in Fig. 7.12. The liquid–crystal spatial-light modulator uses the electronic birefringence effect of the liquid–crystal material to control the refractive index and adjust the wavefront phase. The structure is shown in Fig. 7.13. Incident light is loaded onto the liquid–crystal material in the form of a voltage signal, and each pixel on the liquid–crystal layer responds separately to the light; the emitted light is the light after modulation. The liquid–crystal molecule produces a deflection angle under the action of the driving voltage: θ=

Fig. 7.11 Structure diagram of a thin-film deformation mirror based on MEMS technology

( ) π − 2 arctan e−V 2

(7.26)

Reflection mirror Silicon ~400um height

Silicon nitride epitaxial layer

Drive electrode UV photosensitive adhesive

Silicon or glass 30um depth

212

7 Adaptive-optical Wavefront Correction

Input light

Input light

Parallel arrangement structure

Output light

Distorted structure

Output light

Fig. 7.12 Schematic diagram of a birefringence structure

Incident light

Reflected light Coating Conductive films

Glass plate

Liquid crystal layer Silicon substrate

Aluminum electrode layer

Fig. 7.13 Schematic diagram of a liquid–crystal spatial-light modulator

The refractive index of E light changes with the voltage: n e(v) = /

ne no

(7.27)

n 2e sin2 θ + n 20 cos2 θ

When the polarization direction of the incident light is parallel to the liquid–crystal optical axis, the liquid–crystal spatial-light modulator can modulate the wavefront phase under the action of the driving voltage: )/ ( δ = 2π d n α(v) − n o λ

(7.28)

where d is the thickness of the liquid–crystal layer and λ is the wavelength of the incident light. As a new wavefront-correction device, a liquid–crystal spatial-light modulator has many correction units, low cost, a short fabrication period, and high correction accuracy. However, it has some defects, such as polarized incident light, low correction

7.3 Simulation Analysis and Experimental Research

213

Table 7.3 Common wavefront correctors Name

Drive mechanism

Advantages

Disadvantages

Separation-actuated continuous-surface deformation mirror

Electro-deformation telescopic device

High light-energy utilization rate; small nonlinear hysteresis; long life

Thin mirror; high-density actuator processing; production requirements are high

Splicing sub-mirror deformation mirror

Voltage-driven extender

Simple structure and light weight; high resolution

Low light-energy utilization; co-phases of adjacent gaps need to be adjusted

Double-piezoelectric deformation mirror

Piezoelectric ceramic

Large correction amount

Small number of correction units

Membrane deformable mirror

Electrostatic force

Large correction amount

Low spatial resolution

MEMS deformable mirror

Microelectronic devices produce stress

Deformation occurs in two directions; high operating bandwidth

Limited correction travel; residual stress affects surface shape

Liquid–crystal spatial-light modulator

Liquid crystal birefringence

Many correction units; low price; short production cycle; high correction accuracy

Polarized-light incidence; low calibration frequency; dispersion in correction

frequency, and dispersion. The performance of a liquid–crystal spatial-light modulator is evaluated by the phase-modulation depth, corresponding time, and degree of polarization correlation (Table 7.3).

7.3 Simulation Analysis and Experimental Research 7.3.1 Simulation Analysis We substitute the following values into Eq. (7.11): Wavelength λ = 1550 nm, optical-fiber mode-field radius Wm = 5.25 μm, lens focal length f = 125 mm, lens diameter D A = 25.4 mm, and discretization points M = N = 100. Figure 7.14 shows the influence of the distortion of the first-order Zernike coefficient of the wave predecessor on the coupling efficiency of the fiber. As can be seen from Fig. 7.14, taking a spherical aberration as an example, the coupling efficiency of the distorted wavefront decreases with an increase in the wavefront Zernike coefficient. Moreover, there is a minimum extreme value of the influence of the wavefront-defocusing term distortion on the coupling efficiency. Noll has shown [28] that in the Kolmogorov turbulence spectrum, the tilt component of the wavefront accounts for approximately 82% of the total distortion. Therefore, it is

214

7 Adaptive-optical Wavefront Correction

Fig. 7.14 Influence of wavefront Zernike coefficient and coupling efficiency

1.0

tilt x defocus oblique astigmatism vertical coma oblique trefoil

Coupling efficiency η

0.8

0.6

0.4

0.2

0.0 0.0π

0.5π

1.0π

1.5π

2.0π

Zernike coefficient / rad

necessary to use a large-stroke tilt mirror to independently correct the tilt component of the wavefront to improve the coupling efficiency. With barrel diameters d = 1, 2, and 3 mm, the influences of different turbulence intensities on the power in the normalized barrel were calculated using Eq. (7.12). It can be seen from Fig. 7.15 that the power in the normalized barrel decreases with an increase in turbulence intensity, and at a certain turbulence intensity, the power in the normalized barrel increases with an increase in the barrel diameter. However, the core diameter of a single-mode fiber is in microns; therefore, it is necessary to modify the wavefront to improve the coupling efficiency of a spatial light-to-single-mode fiber with a distorted wavefront. 1.0

PIB(normalized)

0.8

0.6

0.4

d = 1 mm d = 2 mm d = 3 mm

0.2 0

20

40

D/r0

60

80

Fig. 7.15 Impact of different turbulence intensities on the power in a bucket

100

7.3 Simulation Analysis and Experimental Research

215

7.3.2 Experimental Study Figure 7.16 shows an adaptive-optical wavefront-correction fiber-coupling control system. The collimating beam output by the 1550-nm fiber laser and transmitted through the atmospheric channel was first fully reflected by the fast reflector and then fully reflected by the deformation mirror. The fast reflector was used to correct the oblique component of the wavefront, and the deformation mirror was used to correct the higher-order component of the wavefront. The beam corrected by the wavefront is divided into two collimating beams with a power ratio of 1:1. The beam transmitted through it acts on the wavefront sensor to monitor the current distorted wavefront. The reflected beam is collected by an infrared camera or directly coupled into a single-mode fiber by a coupling lens after convergence. The real-time power value coupled into the single-mode fiber was measured using an optical-power meter for optical mixing and coherent-detection communication at the back end. The adaptive-optical system is composed of a fast reflector, deformation mirror, wavefront sensor, and computer, and the fiber-coupling system is composed of a splitting prism, coupling lens, single-mode fiber, optical-power meter, and infrared camera. The fiber-coupling power is controlled through the wavefront closed-loop control of the adaptive-optical system. Table 7.4 lists the main device parameters. Atmospheric turbulence

Laser

Rapid reflector

Incident wavefront deformable mirror

Wavefront sensor

Optical field distribution

Deformation mirror control box

Optical field distribution Wavefront phase Optical power meter

Rapid reflector control box

PC

Fig. 7.16 Adaptive-optical wavefront-correction fiber-coupled control system

216 Table 7.4 Main device parameters

7 Adaptive-optical Wavefront Correction

Device name

Parameters

ALPAO 69-unit deformation mirror

Number of drives: 69 Resonant frequency: 800 Hz Effective surface diameter: 10.5 mm

Shack–Hartmann wavefront sensor HASO4 NIR

Clear aperture: 3.6 mm × 4.5 mm Number of microlenses: 32 × 40 Sampling frequency: 100 Hz Working wavelength: 1500–1600 nm Absolute accuracy (RMS): λ/35

Bobcat-640-GIGE infrared camera

Basal type: InGaAs Response range: 0.9–1.7 m Pixel resolution: 320 × 256 Pixel size: 20 m

PT2M60 rapid reflector

Resonant frequency: 1000 Hz Effective surface diameter: 25.4 mm

Corning single-mode fiber SMF-28e

Mode-field diameter: 10.4 m

(1) Coupling power Because fast and deformation mirrors can correct the distorted wavefront phase, we can use them to generate different degrees of distorted wavefront phases to analyze the effect of wavefront distortion on fiber coupling. / Different D r0 values were selected to represent different turbulence intensities. / According to Eq. (7.13), D r0 was converted into Zernike coefficients corresponding to order 30. The 1–2 order coefficients (tilt-x and tilt-y) are converted into a voltage that drives the fast mirror through the linear relationship between the tilting coefficient of the mirror face and the voltage. The coefficients of order 3–30 are converted into a voltage that drives the deformation mirror using the Zernike2cmd matrix (a transformation matrix from the wavefront Zernike coefficients to the voltage) of the deformation mirror. The coupling power was measured by optical-fiber coupling on the focal plane, and the distorted light spot was collected by an infrared camera. To ensure the randomness of the collected data, the generation rate of the distorted wavefront should be greater than the sampling frequency of the infrared camera. / Under different D r0 conditions, the focal-plane spots of 100 frames under different conditions were collected by observing the focal-plane spots to plot their centroid distribution. Table 7.5 shows the variance of the distribution of the centroids of the distorted spots in the x- and y-directions under different D/r / 0 conditions. As can be seen from Fig. 7.17 and Table 7.5, with an increase in D r0 , the distortion

7.3 Simulation Analysis and Experimental Research

217

degree of the light spot and the distribution of the light-spot centroid increase, which is not conducive to improving the spatial light–fiber coupling efficiency. Figure 7.18 shows the/spot patterns collected by the infrared camera in the focal / plane under different D r0 conditions. With an increase in D r0 , namely, the enhancement of turbulence, the spot dispersion in the focal plane becomes more serious, which is not conducive to coupling between the spatial light and optical fiber. Therefore, it is necessary to modify the wavefront to improve the coupling efficiency. (2) Closed-loop control The adaptive-optical system consists of a deformation mirror, plano-convex lens 3, plano-convex lens 4, wavefront sensor, and computer. The interaction matrix (IM) and command matrix (CM) can be calculated using the traditional extrapolation method. Table 7.5 Focal-plane-spot centroid distribution in the x- and y-directions / / / / / D r0 = D r0 = D r0 = D r0 = D r0 = 0 0.2 0.5 1 2

/ D r0 = 5

x-direction variance (pixel2 )

0.3246

0.4817

0.7814

1.6985

6.3083

20.6361

y-direction variance (pixel2 )

0.3347

0.6684

0.7747

1.6091

8.7942

13.4836

250

(a)

Centroid D/r0 = 0

y pixel

150

200

150

150

100

100

50

50

0

0 0

50

100

150 x pixel

200

100

0 50

100

150 x pixel

200

250

0

250

(d) 150

150

y pixel

200

100

100

50

50

50

100

150 x pixel

200

250

100

150 x pixel

(f)

Centroid D/r0 = 2

(e)

200

250

Centroid D/r0 = 5

200 150 100 50

0 0

50

250

Centroid D/r0 = 1

200

0

Centroid D/r0 = 0.5

50

0

250

250

y pixel

(c)

Centroid D/r0 = 0.2

y pixel

y pixel

200

250

(b)

200

y pixel

250

0 0

50

100

150 x pixel

200

250

0

50

100

150 x pixel

200

250

Fig. 7.17 Focal-plane focal-spot centroid distribution with different turbulence intensities (1 pixel = 20 μm)

218

7 Adaptive-optical Wavefront Correction -25

0.8

0.6

0.4

D/r0=5

μPIB=0.816

D/r0=2

μPIB=0.899

D/r0=1

μPIB=0.970

Optical Power/dBm

PIB(normalized)

1.0

D/r0=0.5 μPIB=0.952

-30 -35 D/r0 = 10 μ = -38.88dBm

-40 -45

0

200

400

μ = -35.31dBm μ = -32.05dBm

D/r0 = 1

μ = -30.79dBm

D/r0 = 0.5 μ = -31.42dBm

D/r0=0.2 μPIB=0.979

0.2

D/r0 = 5 D/r0 = 2

D/r0 = 0.2 μ = -31.97dBm

600

0

200

400

Sampling times/s

Sampling numbers

/ Fig. 7.18 Different D r0 turbulence conditions. a Power in the barrel; b coupled power

IMcmd2slope = V −1 · Slope )−1 ( CMslope2cmd = IMcmd2slope ,

(7.29)

where slope and V are the slope value collected by the wavefront sensor and the voltage value applied by the deformation mirror, respectively, and -1 represents the inverse of the matrix. Figure 7.19 shows a diagram of the adaptive-optical system, where Star = 0 represents the controlled wavefront slope target, Smea and Serr represent the measured and error wavefront slopes, respectively, and Sout represents the output slope. The error slope Serr is converted into an error voltage using the command matrix CMslope2cmd and sent to the deformation mirror through an integral operation. The wavefront sensor collects the slope again and completes the closed-loop iteration after the system delay. Wavefront reconstruction is an optional part of an adaptive-optical closed-loop system. Therefore, the nth voltage closed-loop iteration of the adaptive-optical system can be expressed as

Star + S err

Smea−

CM slope 2 cmd

DM

WFS

Wavefront Reconstruction Unnecessary Part

Delay Fig. 7.19 Adaptive-optics closed-loop control block diagram

Sout

7.3 Simulation Analysis and Experimental Research

219

V(nT + T ) = V(nT ) + ki · (Star (nT ) − Smea (nT )) · CMslope2cmd .

(7.30)

In Eq. (7.30), ki represents the integral gain and T represents the system delay. Considering the algorithm complexity and calculation speed of the real-time control system, the wavefront can be controlled by eliminating the steady-state error using a separate integral controller. The adjustment time and state stability of the closed-loop control of the adaptiveoptical system are usually determined by the closed-loop gain ki and the system delay T . The closed-loop gain is mainly reflected in the peak–valley adjustment of each reconstructed wavefront during the wavefront-control process. The system delay must ensure that the velocity of each wavefront adjustment is greater than the change in velocity of the incident wavefront. Figure 7.20a shows the peak–valley and root-mean-square values of the wavefront in the open-loop and closed-loop states of the 1550-nm wavefront. Figure 7.20b shows the change in the optical-power values coupled into the single-mode fiber in the closed-loop process. Figure 7.20c and d show the reconstructed-wavefront phase diagrams at 1550 nm in the open-loop and closed-loop states. open-PV μ=5.00μm σ2=0.35μm2 close-PV μ=2.05μm σ2=0.46μm2 open-RMS μ=1.01μm σ2=0.0066μm2 Close-RMS μ=0.36μm σ2=0.01μm2

(a) 6

PV & RMS(μm)

AO-close PV AO-close RMS

PV & RMS (μm)

10 8

4

2

0 0

50

100

150

Sampling numbers

6 4

AO-Close A plane optical power

(b) -33

Optical Power/dBm

12

-36

-39

2

-42

0 0

2000

4000

sampling numbers

6000

8000

0

200

400

600

800

Sampling Numbers

Fig. 7.20 a Wavefront-change curve of adaptive-optics open and closed loops; b power-change curve of adaptive-optics closed-loop coupling; c adaptive-optics open-loop wavefront diagram; d adaptive-optics closed-loop wavefront diagram

220

7 Adaptive-optical Wavefront Correction

It can be observed from Fig. 7.20 that, after approximately 100 frames of closedloop control iteration, the system’s wavefront peak and valley values and root-meansquare values reach a stable state. The peak and valley values of the wavefront drop from open loop, 5 m, to closed loop, 2.05 m, and the root-mean-square values drop from open loop, 1.01 m, to closed loop, 0.36 m. The smoothness of the wavefront after the closed loop was significantly better than that after the open-loop state. This indicates that the wavefront at position A in Fig. 7.20 was corrected. Figure 7.20c shows that the coupling power of the singlemode fiber decreases as the closed-loop tends to stabilize. Although the wavefront at position A in Fig. 7.20 is modified to a plane wave through the adaptive-optics system, the wavefront correction at position B is only a static wavefront after the traditional adaptive-optics closes the loop, owing to the influence of the aberration of the non-common optical path. Meanwhile, the assembly error and aberration of the non-common optical path at the fiber-coupling point are considered. As a result, when the wavefront at position A completes the plane-wave correction, the coupled optical power cannot be significantly improved, and the reduced coupling efficiency directly affects the subsequent communication quality (Fig. 7.21). (3) Coupling efficiency

Optical Power/dBm

The Zernike coefficients of the wavefront were measured, and the coupling efficiency was calculated after the light wave was transmitted at different distances, such as -25 -30

Stable Adjustment

-35 AO-Close Optical Power

-40 0

200

400

600

Sampling Numbers

PV(μm)

20

AO-Close PV

15 10 5 0

Reference PV 0

2000

4000

6000

8000

Sampling Numbers Fig. 7.21 Power curve and wavefront-change curve of the adaptive-optics closed loop before and after eliminating the static error and relative wave

7.3 Simulation Analysis and Experimental Research

221

indoors, 600 m, 1 km, 5 km, 10 km, and 100 km. Figure 7.22 shows the 10-km and 100-km experimental links. The communication ends of the 10-km experimental link are located in Bailu Plain and the sixth building of Xi’an University of Technology. The communication ends of the 100-km experimental link are located in Qinghai Lake’s Erlangjian scenic spot and Quanji Township, Gangcha County. Table 7.6 lists the parameters of the experimental environment. Figure 7.23 shows the variances of the Zernike coefficients of the wavefront collected at different communication distances. The variances of the tilt-x and tilt-y orders are higher than those of other orders for the short and medium ranges (0.5 m, Rx Quanji Township

(b)

(a) Rx Xi'an University of Technology

Tx Bailu Plain

Tx Erlangjian

Fig. 7.22 Communication test links: a 10 km; b 100 km

Table 7.6 Experimental environmental parameters [29] Distance

Indoors (0.5 m)

600 m

1 km

5 km

10 km

100 km

Time

2020.4.20 2020.8.13 2018.4.13 2018.10.1 2018.9.25 2019.8.19 17:00–18:00 21:00–22:00 14:00–15:00 23:00–24:00 1:00–2:00 4:00–5:00

Weather

Overcast

Overcast

Rain

Sunny

Cloudy

Overcast

Average 14 ambient temperature (°C)

24

15

12

28

9

Altitude (m) 400

400

400

400

700

3200

Wind direction and speed (m/s)

Southwest 1.7

Southeast 0.4

Southeast 1.5

Northwest Northeast 3.2 3.7

r0 (average) 60 (cm)

22.7

28.3

14.4

13.6

1.3

Greenwood frequency (Hz)

3.29

0.61

4.58

10.51

121.53

Southwest 0.3

0.21

222

7 Adaptive-optical Wavefront Correction

600 m, 1.3 km, and 5 km), because the tilt-x and tilt-y orders account for most of the wavefront distortion. For long-distance communication (10 and 100 km), the variance difference of each wavefront order is small. After the beam is transmitted over a long distance, the proportion of high-order aberration increases, and the general trend of the variance jitter of the Zernike coefficient for each order of aberration increases, indicating that the amplitude and change rate of the wavefront also accelerate with the increase in turbulence. Figure 7.24 shows the coupling-efficiency calculation results based on the measured wavefront Zernike coefficients at different distances. It can be seen from Fig. 7.24 that, after the wavefront is corrected, when the light wave is transmitted at distances of indoors, 600 m, 1 km, 5 km, 10 km, and 100 km, the influence of wavefront distortion on the coupling efficiency increases from 10.1, 1.3, 11.4, 2.8, 7.2, and 8.2% to 95.2, 83.4, 93.5, 56.7, 42.4, and 13.1%, respectively. In addition, the lifting effect of the short-distance correction was better than that of the long-distance correction. The numerical fluctuation in the coupling efficiency corrected at short distances is smaller than that corrected over long distances. For laboratory calibration, the coupling curve was relatively smooth. For wavefront corrections of 600 m and 1 km, the correction curve fluctuated; the fluctuation at 1 km was less than that at 600 m because the communication distance is only one of many factors affecting the turbulence intensity. At 5 km, the intensity scintillation under moderate turbulence discontinued the wavefront-phase acquisition information, and the correction fluctuation was large. For the coupling efficiency before and after wavefront correction at a distance of 10 km, the coupling efficiency after correction numerically improved the coupling before correction; however, the coupling efficiency fluctuated excessively, which is not conducive to the stability of the communication system. However, there was no significant improvement in the coupling efficiency of the 100-km wavefront after modification, because of the light-intensity flicker caused by the strong turbulence; the light spot was completely broken, which directly led to the discontinuous data collection of the Zernike coefficient of the wavefront. The distortion of the 100-km wavefront was discontinuous and too large to ensure real-time correction of the wavefront. Therefore, improving the coupling efficiency of spatial light transmitted over a long distance is a complicated process that is affected by the beam alignment, beam mode, optical-fiber alignment error, wavefront-measurement accuracy, detection speed, closed-loop correction bandwidth, and atmospheric-measurement environment.

1

1

2

(d)

2

3

3

5

6

7

5

6

7

Zernike orders

4

Zernike orders

4

9

10

8

9

10

coeff variance 5 km

8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.2

0.4

0.6

1

1

2

(e)

2

(b)

3

3

5

6

7

5

6

7

Zernike orders

4

Zernike orders

4

9

10

8

9

10

coeff variance 10km

8

coeff variance 600m

0

10

20

30

40

50

0.0

0.2

0.4

0.6

1

1

(f)

2

2

(c)

3

3

6

5

6

7

7

Zernike orders

4

5

Zernike orders

4

9

10

8

9

10

coeff variance 100 km

8

coeff variance 1km

Fig. 7.23 Variances of Zernike coefficients of the wavefront collected at different communication distances [29]: a 0.5 m; b 600 m; c 1 km; d 5 km; e 10 km; f 100 km

0.0

0.5

1.0

1.5

2.0

0.0

0.1

Coeffience variance (μm)

Coeffience variance (μm)

0.2

coeff variance indoor (0.5 m) Coeffience variance (μm) Coeffience variance (μm)

Coeffience variance (μm)

Coeffience variance (μm)

(a)

7.3 Simulation Analysis and Experimental Research 223

0

0

(d)

500 Iteration times

5km

500 Iteration times

1000

1000

Indoors

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0

0

(e)

(b)

500 Iteration times

Iteration times

500

10km 1000

1000

600m

0.0

0.2

0.4

0.6

0.8

(f)

(c)

0

0

1.0

0.0

0.2

0.4

0.6

0.8

1.0

500 Iteration times

500 Iteration times

100km

1000

1000

1km

Fig. 7.24 Influence curve of wavefront correction on the coupling efficiency at different distances [29]: a 0.5 m; b 600 m; c 1 km; d 5 km; e 10 km; f 100 km

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8 Cupling efficiency η

Coupling efficiency η

Coupling efficiency η

Coupling efficiency η

(a) Coupling efficiency η

Coupling efficiency η

1.0

224 7 Adaptive-optical Wavefront Correction

References

225

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