Table Of ContentElEctrical 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
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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
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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.
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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
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