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ElEctrical EnginEEring Mathematical Optics Classical, Quantum, and Computational Methods Mathematical Optics Going beyond standard introductory texts, Mathematical Optics: Classical, Quantum, and Computational Methods brings together many new mathematical techniques from optical science and engineering research. Profusely illustrated, the book makes the material accessible to students and newcomers to the field. Classical, Quantum, Divided into six parts, the text presents state-of-the-art mathematical methods and and Computational Methods applications in classical optics, quantum optics, and image processing. • Part I describes the use of phase space concepts to characterize optical beams and the application of dynamic programming in optical waveguides. • Part II explores solutions to paraxial, linear, and nonlinear wave equations. • Part III discusses cutting-edge areas in transformation optics (such as invisibility cloaks) and computational plasmonics. • Part IV uses Lorentz groups, dihedral group symmetry, Lie algebras, and Liouville space to analyze problems in polarization, ray optics, visual optics, and quantum optics. • Part V examines the role of coherence functions in modern laser physics and explains how to apply quantum memory channel models in quantum computers. • Part VI introduces super-resolution imaging and differential geometric methods in image processing. Edited by As numerical/symbolic computation is an important tool for solving numerous real-life problems in optical science, many chapters include Mathematica® code in their appendices. Vasudevan Lakshminarayanan The software codes and notebooks as well as color versions of the book’s figures are available at www.crcpress.com. Maria L. Calvo • Tatiana Alieva K13194 ISBN: 978-1-4398-6960-4 90000 9 781439 869604 K13194_Cover_mech.indd 1 11/7/12 12:11 PM Mathematical Optics Classical, Quantum, and Computational Methods TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Mathematical Optics Classical, Quantum, and Computational Methods Edited by Vasudevan Lakshminarayanan, (cid:45)(cid:65)(cid:82)(cid:73)(cid:65)(cid:0)(cid:44)(cid:14)(cid:0)(cid:35)(cid:65)(cid:76)(cid:86)(cid:79)(cid:0)(cid:115)(cid:0)(cid:52)(cid:65)(cid:84)(cid:73)(cid:65)(cid:78)(cid:65)(cid:0)(cid:33)(cid:76)(cid:73)(cid:69)(cid:86)(cid:65)(cid:0) Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 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 Version Date: 20121031 International Standard Book Number-13: 978-1-4398-6961-1 (eBook - PDF) 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 valid- ity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ..................................................................................................ix Acknowledgments.................................................................................... xiii Editors ................................................................................................. xv Contributors...........................................................................................xvii PART I Special Problems in Ray Optics Chapter1 OrbitalAngularMomentum:ARayOpticalInterpretation ............................3 MilesPadgett Chapter2 WignerDistributionMomentsforBeamCharacterization ........................... 13 TatianaAlieva,AlejandroCámara,andMartinJ.Bastiaans Chapter3 DynamicProgrammingApplicationsinOptics....................................... 53 MariaL.Calvo,JesúsPérez-Ríos,andVasudevanLakshminarayanan PART II Mathematical Formalism in Wave Optics Chapter4 BasisExpansionsforMonochromaticFieldPropagation inFreeSpace............................................................................ 97 MiguelA.AlonsoandNicoleJ.Moore Chapter5 SolutionsofParaxialEquationsandFamiliesofGaussianBeams..................143 EugenyAbramochkin,TatianaAlieva,andJoséA.Rodrigo Chapter6 TheDecompositionMethodtoSolveDifferentialEquations: OpticalApplications...................................................................193 VasudevanLakshminarayanan,SudiptaNandy,andRaghavendraSridhar v vi Contents PART III Plasmonics Chapter7 AnIntroductiontoMathematicsofTransformationalPlasmonics..................235 MuamerKadic,SébastienGuenneau,andStefanEnoch Chapter8 Plasmonics:ComputationalApproach................................................279 MaximSukharev PART IV Applications of Group Theory in Optics Chapter9 LorentzGroupinRayandPolarizationOptics.......................................303 SibelBas¸kalandY.S.Kim Chapter10 ParaxialWaveEquation:Lie-Algebra-BasedApproach............................341 AmaliaTorre Chapter11 DihedralPolynomials ................................................................419 MarlosViana Chapter12 LieAlgebraandLiouville-SpaceMethodsinQuantumOptics....................439 MasashiBan PART V Quantum Optics Methods Chapter13 FromClassicaltoQuantumLightandViceVersa:Quantum Phase-SpaceMethods................................................................483 AlfredoLuis Chapter14 CoherenceFunctionsinClassicalandQuantumOptics............................507 ImranaAshrafZahidandVasudevanLakshminarayanan Chapter15 QuantumMemoryChannelsinQuantumOptics...................................533 TomásˇRybár,MárioZiman,andVladimírBuzˇek Contents vii PART VI Computational Optics/Image Processing Chapter16 AnIntroductiontoSuper-ResolutionImaging......................................555 JonathanD.SimpkinsandRobertL.Stevenson Chapter17 TheDifferentialStructureofImages................................................581 BartM.terHaarRomeny Index..................................................................................................599 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Preface TheNobel Prize–winning physicist Eugene Wignerinafamouspaper entitled“The unreasonable effectivenessofmathematicsinthephysicalsciences”wrote“themiracleoftheappropriatenessof the language of mathematics for the formulation of the laws is a wonderful gift (that) we neither understandnordeserve”[1].Twentyyearslater,thecomputerscientistRichardHammingposedand triedtoanswerthequestion“howcanitbethatsimplemathematicssufficestopredictsomuch?” [2].Thisunreasonableeffectivenessisalsotrueforopticalscienceandengineering. OnecangobackintimetoremembertheancientGreekphilosopherswhowereinterestedinthe descriptionofnaturalphenomena,suchasthevisualprocessandastronomicalandmechanicallaws. Ptolemy(ClaudiusPtolemaeus:c.AD90–c.AD168)isanexcellentexample.Heinfactwrotetexts onmathematicaloptics,whichreliedmostlyongeometricaltheorems[3].Theaimofthisbookis topresentvarioussophisticatedmathematicaltechniquesandconceptsthatare“stateoftheart”and areusedtodescribeavarietyofopticalphenomena. Theoriginofthisbookgoesbackabouteightyearswhenoneofus(V.L.)proposedaworkshopon mathematicalmethodsinopticstobeheldattheAbdusSalamInternationalCenterforTheoretical Physics (ICTP) at Trieste, Italy. The proposal stated “There have been many advances in various sophisticated mathematical methods to analyze and solve a wide variety of problems encountered inopticsandphotonics.Many,ifnotmostofthesetechniquesarefoundprimarilyintheresearch literature. We feel the time has come for wider dissemination of the knowledge to students and researchers workingindisparateareas ofoptics...(thestudent)willbeintroduced toseveral math- ematicaltechniquesusedinadiverserangeofopticalproblems.Thetechniqueswereselectedfor theirconnectionwithphysicalintuitionandfortheirusefulnessinthepracticalsolutionofproblems. Theyfindapplicationsthroughoutray,waveandquantumoptics”(theproposedweeklongworkshop wasrenamedasthePreparatorySchoolandisheldeveryyearbeforetheannualWinterCollegein OpticsheldatICTP.Thefirstworkshopwasheldin2006.Alinktothatworkshopisavailableat: http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a05378). The justification stated in the aforementioned paragraph still holds. We believe there is a need for a book such as this, and this need is likely to increase. The simple reason is the large and growing number of academic programs in optical science and engineering around the world (see e.g.,http://www.opticseducation.org).Manyopticsprogramsrequireagraduatelevelmathematical methods course (e.g., Complex analysis and math methods for optics at the Institute of Optics, University of Rochester; Mathematical methods for photonics and applications at the University of Arizona, Optical Sciences Center; Theoretical foundations of optics at CREOL, University of CentralFlorida,andMathematicalmethodsforscienceandengineeringattheDepartmentofPhysics and Optical Sciences, University of North Carolina at Charlotte) for students majoring in optics. Other similar courses are given around the world in various optics/optical engineering programs. With the exception of a recently published book [4], there is no book that specifically addresses optical applications (one can make a strong case for the book by Barrett and Myers [5], which is usedasatextbookinvariouscourses).ThesearealsothebooksbyBarnettandRadmore[6]andPuri [7],whichdealwithquantumoptics.Mostcoursesusestandardmathematicalphysicstextssuchas Arfken[8],MathewsandWalker[9],Boas[10],orHassani[11].Thisbookgoesbeyondthestandard introductory courses (and associated textbooks) and will introduce the reader to some techniques currently being used in research. This is a broad overview and as such can be used for self-study, as a textbook in advanced courses, or as supplementary reading. A book similar in scope/aims is ix