Missing:
Vortex Laser Beams
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Vortex Laser Beams
V. V. Kotlyar A. A. Kovalev A. P. Porfirev
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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-1385-4211-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
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Contents Preface.......................................................................................................................xi Authors ................................................................................................................... xiii Chapter 1
A Spiral Phase Plate for an Optical Vortices Generation .................... 1 1.1
1.2
1.3
1.4
Chapter 2
Diffraction of a Plane, Finite-Radius Wave by a Spiral Phase Plate .................................................................................1 The Fraunhofer Diffraction of a Finite-Radius Plane Wave ...... 2 Simulation Results .....................................................................3 The Fresnel Diffraction of a Finite-Radius Plane Wave ............ 5 Experiment ................................................................................6 Diffraction of Conic and Gaussian Beams by a Spiral Phase Plate .................................................................................7 Fourier Transform of Helico-Conical-Wave with Infinity ......... 9 Fraunhofer Diffraction of a Gaussian Beam on Spiral Phase Plate ............................................................................... 15 Experimental Studies of Light Diffraction by a Spiral Phase Plate ............................................................................... 17 Diffraction of a Finite-Radius Plane Wave and a Gaussian Beam by a Helical Axicon and a Spiral Phase Plate ............... 19 Description of the Fraunhofer Diffraction of the Limited Plane Wave by a Helical Axicon.............................................. 21 Description of the Fraunhofer Diffraction by a Helical Axicon Using a Bessel Function Series...................................24 Description of the Fraunhofer Diffraction by a Spiral Phase Plate Using a Finite Sum of the Bessel Functions ........25 Description of the Fraunhofer Diffraction of the Gaussian Beam by a Helical Axicon Using the Hypergeometric Functions ...................................................... 31 Numerical Simulation of the Diffraction of the Gaussian Beam by a Helical Axicon and a Spiral Phase Plate ...............34 Fraunhofer Diffraction of the Plane Wave by the Multilevel (Quantized) Spiral Phase Plate......................... 38 Complex Amplitude of an Optical Vortex ............................... 38 Numerical Simulation .............................................................. 41
Elliptic Laguerre-Gauss Beams ......................................................... 45 2.1 2.2
Oblique Paraxial Laguerre-Gaussian Beam............................46 Elliptic Nonparaxial Laguerre-Gaussian Beam ...................... 51 Numerical Simulation of Propagation of the Elliptic Laguerre-Gaussian Beams ....................................................... 59 v
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vi
Chapter 3
Contents
Hypergeometric Vortices ................................................................... 63 3.1 3.2
3.3
3.4
3.5
Chapter 4
Hankel-Bessel Laser Beams ............................................................. 127 4.1
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Hypergeometric Modes ........................................................... 63 Formation of Hypergeometric Modes .....................................66 A Family of Hypergeometric Laser Beams............................. 68 General Form of the Hypergeometric Beams .......................... 69 Hypergeometric Beams in the Near-Field ............................... 71 Hypergeometric Beams in the Far-Field.................................. 72 Particular Cases of the Hypergeometric Beams ...................... 72 Generalized Hypergeometric Modes ....................................... 75 Diffraction of the Gaussian Beam by a Spiral Logarithmic Axicon ................................................................. 76 Laguerre-Gauss Modes of the (0,n)th Order ........................... 79 Numerical Simulation ..............................................................80 Propagation of the Hypergeometric Modes .............................84 Propagation of Hypergeometric Laser Beams in a Medium with the Parabolic Refractive Index ....................... 87 Paraxial Hypergeometric Beams in a Parabolic-Index Medium....................................................................................88 Nonparaxial Modes of a Parabolic-Index Medium ................. 93 A Parabolic-Index Microlens................................................... 95 Binary Parabolic-Index Lens ...................................................97 A Planar Parabolic-Index Lens ................................................ 98 3D Parabolic-Index Lens ....................................................... 100 3D Binary Parabolic-Index Lens ........................................... 100 Nonparaxial Hypergeometric Modes .................................... 103 Angular Spectrum of the Plane-Waves for Nonparaxial Hypergeometric Modes ......................................................... 104 Direct and Inverse Nonparaxial Hypergeometric Modes ............................................. 106 Numerical Simulation ............................................................ 109 Nonparaxial Propagation of a Gaussian Optical Vortex with Initial Radial Polarization ................................. 111 Integral Transforms to Describe the Propagation of Radially and Azimuthally Polarized Laser Beams in Free Space ............................................................................. 112 Propagation of Optical Vortex Beams with Initial Radial Polarization ............................................................................ 114 Relationships for the Longitudinal Component .................... 117 Propagation of Optical Vortex Beams with Initial Elliptical Polarization.................................................. 120 Numerical Simulation ............................................................ 122
Solution of the Helmholtz Equation in the Parabolic Coordinates............................................................................ 127
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Contents
4.2
Chapter 5
Accelerating Beams.......................................................................... 141 5.1 5.2
Chapter 6
Accelerating Airy Beams ...................................................... 141 Hyperbolic Airy Beams ......................................................... 142 Numerical Results ................................................................. 143 Transformation of Decelerating Laser Beams into Accelerating Ones ................................................................. 145 Accelerating Beams ............................................................... 146 Airy Beams with a Hyperbolic Path ...................................... 147 Hermite-Gauss Beams ........................................................... 148 Decelerating Beams ............................................................... 150 Diffraction of a Plane Wave by an Opaque Semi-Infinite Screen .............................................................. 150 Two-Dimensional Hypergeometric Beams and Bessel Beams ......................................................................... 151 Transformation of Decelerating Beams into Accelerating Ones ................................................................. 153 Diffraction of a Gaussian Beam by a Semi-Infinite Opaque Screen ....................................................................... 156
Hermite-Gaussian Vortices .............................................................. 159 6.1
6.2
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Peculiarities of the Hankel-Bessel Beams............................. 133 Particular Cases of the Hankel-Bessel Beams ....................... 133 Simulation Results ................................................................. 135 Focusing of the Hankel-Bessel Beams .................................. 137
Hermite-Gaussian Modal Laser Beams with Orbital Angular Momentum .............................................................. 159 Generalized Hermite-Gaussian Laser Beams ........................ 160 Orbital Angular Momentum of a Linear Combination of Two Hermite-Gaussian Modes .......................................... 162 Orbital Angular Momentum of a Linear Combination of Two Elegant Hermite-Gaussian Beams ............................. 164 Orbital Angular Momentum of a Linear Combination of Two Hybrid Hermite-Gaussian Beams..............................165 Numerical Simulation ............................................................ 166 Discussion.............................................................................. 168 Vortex Hermite-Gaussian Laser Beams ................................ 174 Hermite-Gaussian Beam with Complex Argument .......... 174 Orbital Angular Moment of Vortex Hermite-Gaussian Beam....................................................176 Computer Simulation............................................................. 177 Isolated Optical Nulls of Vortex Hermite-Gaussian Beam ................................................ 177 Orthogonality of Vortex Hermite-Gaussian Beams ............179 Experiment ............................................................................ 180
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Chapter 7
Contents
Asymmetric Vortices Bessel Beams ................................................ 183 7.1
7.2
7.3
7.4
7.5
Chapter 8
Pearcey Laser Beams ....................................................................... 239 8.1
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Asymmetric Bessel Modes .................................................... 183 Superposition of Bessel Modes ............................................. 184 Numerical Simulation ............................................................ 185 Asymmetric Modes with a Left-Side Optical Crescent .........187 Orbital Angular Momentum of Asymmetric Bessel Modes .................................................................. 189 Orthogonality of Asymmetric Bessel Modes ........................ 190 Asymmetric Bessel-Gauss Beams......................................... 190 Linear Combination of Bessel-Gaussian Beams ................... 191 Fourier Spectrum of the Asymmetric Bessel-Gaussian Beam .......................................................... 198 Orbital Angular Momentum of Asymmetric Bessel-Gaussian Beam .......................................................... 199 Orthogonality of Asymmetric Bessel-Gaussian Beams ........200 Experiment ............................................................................ 201 Lommel Modes...................................................................... 203 Complex Amplitude of Lommel Modes ................................204 Orbital Angular Momentum of the Lommel Modes...........208 Orthogonality of Complex Amplitudes of Lommel Modes ...... 210 Lommel Modes with Complex Shift in the Cartesian Plane..... 211 Superpositions of Asymmetrical Bessel Beams .................... 212 Asymmetrical Bessel Modes of the Second Type ...................213 Linear Combination of the Asymmetrical Bessel Modes ...... 215 Orbital Angular Momentum of Superposition of Asymmetrical Bessel Modes of the First and the Second Type...........................................................................220 Shifted Nondiffractive Bessel Beams ................................... 223 Fourier Spectrum of a Shifted Bessel Beam..........................224 Relation between the Amplitudes of Spectra of Shifted and Nonshifted Bessel Beams ............................................... 225 Orbital Angular Momentum of a Shifted Bessel Beam ......... 226 Orbital Angular Momentum of the Superposition of Shifted Bessel Beams ............................................................ 227 Superposition of Three Shifted Bessel Beams ...................... 229 Superposition of Identical Bessel Beams Found at the Vertices of a Regular Polygon ............................................... 231 Superposition of a Large Number of Bessel Beams Centred on a Circle ................................................................ 235
Half Pearcey Laser Beams .................................................... 239 Three-Dimensional Half-Pearcey Beams ..............................240 Two-Dimensional Half-Pearcey Beams ................................ 245
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Contents
8.2
Chapter 9
Asymmetric Gaussian Vortices ........................................................ 259 9.1
9.2
9.3
9.4
9.5
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Vortex Pearcey Beams........................................................... 247 Autofocusing Hypergeometric Beams...................................248 Autofocusing Hypergeometric-Gaussian Beams................... 253 Experiment ............................................................................ 254
An Asymmetric Gaussian Optical Vortex............................. 259 Gaussian Beam with a Displaced Optical Vortex ..................260 Orbital Angular Moment of an Asymmetric Gaussian Optical Vortex ........................................................................ 262 Experiment ............................................................................ 263 Elliptic Gaussian Optical Vortices ........................................266 Computation of the Orbital Angular Momentum ....................267 Computing the Field Complex Amplitude............................. 270 Experiments on Generating an Elliptic Gaussian Beam........ 272 Controlling Orbital Angular Momentum of an Optical Vortex by Varying Its Ellipticity ........................................... 279 Orbital Angular Momentum of an Elliptic Gaussian Beam with an Embedded Intensity Null............................................... 280 Partial Cases .......................................................................... 281 Numerical Simulation ............................................................ 286 Asymmetric Laguerre-Gaussian Beams ............................... 287 Complex Amplitude of an Asymmetric Laguerre-Gaussian Beam ...................................................... 289 Power of a Shifted Laguerre-Gaussian Beam........................ 291 Orbital Angular Momentum of a Shifted Laguerre-Gaussian Beam ...................................................... 293 Paraxial Laguerre-Gaussian Beams in the Form of a Rotating Crescent .................................................................. 296 Experimental Generation of an Asymmetric Laguerre-Gaussian Beam Using a Spatial Light Modulator..........................................................................301 Rotating Superpositions of Asymmetric Laguerre-Gaussian Beams .....................................................302 Astigmatic Laser Beams with a Large Orbital Angular Momentum ..............................................................305 Vortex-Free Beam with the Orbital Angular Momentum .......307 The Elliptic Gaussian Beam is Rotating after Passing the Cylindrical Lens ..............................................................308 Vortex Beam after Two Crossed Cylindrical Lenses .............309 Elliptical Gaussian Beam after ABCD System .......................309 Cylindrical Lens is Placed in Arbitrary Plane ....................... 311 Generation of an Elliptic Gaussian Beam ............................. 312 Orbital Angular Momentum of an Astigmatic Beam ............ 313 Numerical Simulation ............................................................ 317
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Contents
Chapter 10 Perfect Vortices ................................................................................ 321 10.1
10.2
An Optimal Phase Element for Generating a Perfect Optical Vortex...................................................................... 321 Generating the Perfect Optical Vortex Using an Amplitude-Phase Optical Element....................................... 322 Generating the Perfect Optical Vortex Using an Optimal Phase Optical Element ......................................................... 326 Generating a Perfect Optical Vortex Using a Conical Axicon .................................................................................. 328 Simulation Results ............................................................... 331 Experiment ........................................................................... 335 Elliptic Perfect Optical Vortices .......................................... 339 Phase Optical Element .........................................................340 Numerical Simulation .......................................................... 342 Experiment ........................................................................... 343
Chapter 11 Hankel Optical Vortices ................................................................... 349 11.1
11.2
Vortex Hankel Laser Beams ................................................ 349 Scalar Hankel Laser Beams ................................................. 349 Angular Spectrum of the Hankel Beam ............................... 352 Linearly Polarized Hankel Beams ....................................... 358 Linearly Polarized Hankel Beam in Far Field ..................... 361 Circularly Polarized Hankel Vortices .................................. 363 Projections of the Electromagnetic Field Vectors for Clockwise and Anticlockwise Circular Polarization............364 Vector Vortex Hankel Beams with Circular Polarization in the Far-Field ..................................................................... 371
Chapter 12 Conclusion ........................................................................................ 379 References ............................................................................................................. 385 Index ......................................................................................................................403
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Preface Derivation of analytical solutions of Maxwell’s equations, as well as other mathematical tools widely employed in optics—such as a Helmholtz equation or paraxial Schrodinger-type equations—has always been in the focus of interest of optical researchers. The said solutions describe in detail properties of coherent electromagnetic light fields and laser beams, which have found numerous practical applications. In particular, widely known light fields that can be described analytically include plane or spherical waves and Gaussian and Bessel beams [1], to name just a few. Recently, new promising light beams that can be described by exact analytical relations have been proposed. These include Hermite-Gaussian and LaguerreGaussian modal beams [2], Hermite-Laguerre-Gauss beams [3], elliptic Mathieu and Ince beams [4,5], hypergeometric beams [6], accelerating Airy beams [7], and self-focusing Pearcey beams [8]. Further research of elegant Laguerre-Gaussian and Hermite-Gaussian beams is currently underway [9,10], with their behavior being described using polynomials with complex argument. The elliptic Laguerre-Gaussian beams have been studied using a number of approaches [11,12]. Recent years have seen an increase of interest in deriving exact solutions of paraxial Schrodingertype equations in cylindrical coordinates. More recently, hypergeometric Gaussian beams [13] and circular beams [14] have been proposed. A number of well-studied light beams, such as conventional and elegant Laguerre-Gaussian modes, quadratic Bessel-Gaussian beams [15], and Gaussian optical vortices [16], have been shown to be a particular case of the circular beams [14]. Light fields can be grouped into two classes: those that carry orbital angular momentum (OAM) [17] and those devoid of OAM. Beams that carry OAM are termed as vortex or singular beams. The vortex laser beams are characterized by a helical or spiral phase, wavefront dislocations, and isolated intensity nulls. Currently, vortex laser beams have been put to many practical uses, including turbulent atmosphere probing [18], wireless optical communications [19], fiber optic communication channel multiplexing [20], astronomy [21], quantum informatics [22], and micromanipulation [23]. It has not been our goal to offer a comprehensive review of all exact solutions of Maxwell’s, Helmholtz, and Schrodinger equations that are currently utilized in optics. Instead, our book covers in detail only the vortex laser beams that were personally proposed by the current authors over years of research, which are quite many. For instance, these include hypergeometric beams, Hankel-Bessel beams, half Pearcey beams, asymmetric Bessel modes, Lommel modes, asymmetric LaguerreGaussian beams, vortex Hermite-Gaussian beams, vector Hankel beams, and others. It is worth noting that a vortex laser beam is most easy to generate via illuminating a spiral phase plate (SPP) by a conventional Gaussian beam. Hence, diffraction of light by a SPP is given particular attention in this book. The book contains research results financially supported by the Russian Science Foundation grant No. 17-19-011. xi
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Authors V. V. Kotlyar is a Head of Laboratory at the Image Processing Systems Institute AQ 1 (Samara) of the Russian Academy of Sciences and a professor in the Computer Science department at Samara National Research University. He received his MS, PhD, and DrSc degrees in physics and mathematics from Samara State University (1979), Saratov State University (1988), and Moscow Central Design Institute of Unique Instrumentation, the Russian Academy of Sciences (1992). He is a co-author of 300 scientific papers, 5 books, and 7 inventions. His current interests are diffractive optics, gradient optics, nanophotonics, and optical vortices. A. A. Kovalev (b. 1979) graduated (2002) from Samara National Research University, majoring in Applied Mathematics. He received his Doctor in Physics & Maths degree in 2012. He is a senior researcher in the Laser Measurements laboratory at IPSI RAS, a branch of the FSRC “Crystallography and Photonics” RAS. He is a co-author of more than 150 scientific papers. His current research interests are mathematical diffraction theory and photonic crystal devices. A. P. Porfirev (b. 1987) graduated (2010) from Samara National Research University, majoring in Applied Physics and Mathematics. He is a candidate in Physics and Mathematics (2013). Currently he is an associate professor in the Technical Cybernetics department of Samara National Research University and a researcher in the Micro- and Nanotechnologies laboratory of the IPSI RAS, a branch of the FSRC “Crystallography and Photonics” RAS. His current research interests include diffractive optics, optical manipulation, and structured laser beams.
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Author Query Sheet Chapter No.: Front Matter Query No.
Query
AQ 1
Please confirm the surname and forename used in the author bios are fine.
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Response
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