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· ·
Rosario Martínez-Herrero Pedro M. Mejías
Gemma Piquero
Characterization of Partially
Polarized Light Fields
123
Prof.RosarioMartínez-Herrero Prof.PedroM.Mejías
UniversidadComplutensedeMadrid UniversidadComplutensedeMadrid
FacultyofPhysics FacultyofPhysics
OpticsDepartment,28040Madrid OpticsDepartment,28040Madrid
Spain Spain
r.m-h@fis.ucm.es pmmejias@fis.ucm.es
Dr.GemmaPiquero
UniversidadComplutensedeMadrid
FacultyofPhysics
OpticsDepartment,28040Madrid
Spain
piquero@fis.ucm.es
ISSN0342-4111 e-ISSN1556-1534
ISBN978-3-642-01326-3 e-ISBN978-3-642-01327-0
DOI10.1007/978-3-642-01327-0
SpringerDordrechtHeidelbergLondonNewYork
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Preface
Polarization involves the vectorial nature of light. In theoretical topics and tech-
nological applications of current interest in optical science, the electromagnetic
descriptionofthelightdisturbanceandpolarization-relatedphenomenahaveturned
out to be crucial. In fact, they have gained importance in recent years, attracting
the attention of numerous scientists in the last decade. Photonics devices, biologi-
caloptics,opticalcommunications,atmosphericoptics,andsensortechnologiesare
examplesofresearchareaswherethevectorialfeaturesoflightarerelevant.
In practice, optical radiation exhibits randomness and spatial nonuniformity of
the polarization state. These kinds of realistic and general situations are most fre-
quentlyencounteredintheliterature,anddrasticallydepartfromthesimplestmodel
of harmonic plane waves. Keeping in mind the framework of classical optics, this
bookdealswiththeanalyticalproblemofdescribingandcharacterizingthepolariza-
tionandspatialstructureofnonuniformlypolarizedfields,alongwiththeirevolution
whenlightpropagatesthroughopticalsystems.Thisistheaimofthiswork.
In particular, the scope and the contents of the book are essentially focused on
thecontributionsoftheauthorstofourmainissues,whichcorrespondtoapproach-
ing the problem under study from different angles. Thus, in Chap. 1, after a short
introductiontothemainstandardrepresentationsofthepolarization,weanalyzein
some detail several recently proposed measurable parameters, of practical use for
characterizing, in a global way, the polarization of nonuniformly polarized beam-
like fields. Chapter 2 also proposes an overall description, but now the alternative
formalismallowsacharacterizationofthepolarizationdistributionandtheshapeof
partiallypolarized,partiallycoherentbeams.Afamilyofmeasurableandmeaning-
ful parameters is defined, whose determination involves certain averages over the
regionofthetransverseprofilewherethebeamirradianceissignificant.
On the other hand, in recent years, the coherence theory, well established for
scalar beams, has been investigated with regard to partially polarized fields. It has
been shown that, for stochastic electromagnetic beams, the properties concerning
coherenceandpolarizationfeaturesare,ingeneral,connectedwitheachother.Con-
sequently, fundamental concepts, such as the degree of coherence and the fringe
visibilityinYounginterferometers,shouldberevisited.Thisisdoneinsomedetail
inChap.3.
v
vi Preface
Finally, Chap. 4 is devoted to nonparaxial electromagnetic fields, which arise
whenlightisstronglyfocusedandthebeamreachesawaistsizeevensmallerthan
the wavelength. The theoretical analysis deals with certain kinds of nonparaxial
exactsolutionsoftheMaxwellequations.Theirpolarizationfeaturesarediscussed
and,inthosecasesinwhichtheevanescentwavesaresignificant(highlynonparaxial
regime),theirfieldstructureisalsodescribed.
Although short surveys are provided to review some basic formalisms, the
reader is assumed to be familiar with well-known concepts treated in many optics
textbooks.
Theresearchworkleadingtotheresultsreportedinthisbookhavebeenobtained
in collaboration with a number of researchers. We would like to acknowledge
here their fundamental contribution. We also thank Prof. H. Weber for his inter-
est, helpful suggestions and continuous kindness, which include support during
all the experimental work described in Sect. 2.4.3 and performed at the Optisches
Institut of the Technische Universität in Berlin. We are indebted to Prof. F. Gori,
Dr.M.SantarsieroandDr.R.Borghi,attheUniversityofRome3,forhelpfuldis-
cussionsconcerningpartiallypolarizedGaussianSchell-modelbeams.Inaddition,
we are also grateful to Dr. M. Santarsiero and Profs. S. Bosch and A. Carnicer, at
BarcelonaUniversity,fortheirkindreadingofChaps.3and4.
This work was supported by the Ministerio de Educación y Ciencia of Spain,
Project FIS2007-63396, and by the Comunidad de Madrid, Project: CCG07-
UCM/ESP-3070.
Contents
1 RepresentationsofthePolarizationofBeamlikeFields . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 StandardRepresentationsofthePolarization . . . . . . . . . . . 1
1.2.1 TheJonesCalculus . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Coherence-PolarizationMatrices . . . . . . . . . . . . . 4
1.2.3 TheStokes-MüllerCalculus . . . . . . . . . . . . . . . . 6
1.2.4 LocalDegreeofPolarization . . . . . . . . . . . . . . . 8
1.3 WeightedDegreeofPolarization . . . . . . . . . . . . . . . . . 10
1.4 Linear and Circular Polarization Content of Totally
PolarizedBeams . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 KeyDefinitionsandPhysicalMeaning . . . . . . . . . . 12
1.4.2 ApplicationtoanExample. . . . . . . . . . . . . . . . . 13
1.4.3 Measurability . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 RadialandAzimuthalPolarizationContentofTotally
PolarizedBeams . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 KeyDefinitions . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 RelationswiththeStokesParameters . . . . . . . . . . . 19
1.5.3 Relations with the Output of Radial
andAzimuthalPolarizers . . . . . . . . . . . . . . . . . 20
1.5.4 Measurability . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 PartiallyPolarizedGaussianSchell-ModelBeams . . . . . . . . 24
1.6.1 GaussianSchell-ModelBeams:ScalarCase . . . . . . . 24
1.6.2 GaussianSchell-ModelBeams:VectorialCase . . . . . . 26
1.6.3 TheVanCittert-ZernikeTheorem . . . . . . . . . . . . . 28
1.6.4 ExperimentalSynthesisofGSMfields . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Second-OrderOverallCharacterizationofNon-uniformly
PolarizedLightBeams . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Second-OrderOverallCharacterization:ScalarCase . . . . . . . 38
2.2.1 FormalismandKeyDefinitions . . . . . . . . . . . . . . 38
vii
viii Contents
2.2.2 PropagationandMeasurementoftheIrradiance
Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Second-OrderOverallCharacterization:VectorialCase . . . . . 44
2.3.1 TheWignerMatrix. . . . . . . . . . . . . . . . . . . . . 44
2.3.2 TheStokesMatrices . . . . . . . . . . . . . . . . . . . . 45
2.3.3 PropagationLawsandMeasurement . . . . . . . . . . . 48
2.3.4 InvariantParameters . . . . . . . . . . . . . . . . . . . . 51
2.4 GeneralizedDegreeofPolarization . . . . . . . . . . . . . . . . 54
2.4.1 DefinitionandPropertiesofGeneralizedDegree
ofPolarization . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.2 PhysicalMeaningandMeasurement . . . . . . . . . . . 55
2.4.3 GeneralizedDegreeofPolarizationofBeams
EmergingfromOptically-PumpedNd:YAGRods . . . . 58
2.4.4 ClassificationSchemeofPartiallyPolarizedBeams . . . 62
2.5 BeamQualityParameterofPartiallyPolarizedfields . . . . . . . 63
2.6 OverallParametricCharacterizationofPGSMBeams . . . . . . 65
2.7 BeamQualityImprovement:GeneralConsiderations . . . . . . . 69
2.8 BeamQualityImprovementAfterPropagationThrough
OpticalPhaseDevices . . . . . . . . . . . . . . . . . . . . . . . 75
2.8.1 PropagationThroughAnisotropicPure-PhasePlates . . . 76
2.8.2 Propagation of Radially and Azimuthally
PolarizedBeamsThroughQuarticPhasePlates . . . . . . 78
2.8.3 PropagationThroughSpiralPhaseElements . . . . . . . 82
2.9 GlobalBeamShapingwithNon-uniformlyPolarizedBeams . . . 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 PolarizationandCoherenceofRandomElectromagneticFields . . 93
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 ScalarFramework . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2.1 SpectralDegreeofCoherence . . . . . . . . . . . . . . . 94
3.2.2 Young’sInterferenceExperiment . . . . . . . . . . . . . 96
3.3 VectorialFramework:KeyDefinitions . . . . . . . . . . . . . . 98
3.3.1 Young’sExperimentRevisited . . . . . . . . . . . . . . . 98
3.3.2 Degrees of Coherence of Random
ElectromagneticFields. . . . . . . . . . . . . . . . . . . 100
3.3.3 Relation Between Degrees of Coherence of
ElectromagneticFields. . . . . . . . . . . . . . . . . . . 103
3.4 MaximumVisibilityUnderUnitaryTransformations . . . . . . . 107
3.5 Position-IndependentStochasticBehaviorofRandom
ElectromagneticFields. . . . . . . . . . . . . . . . . . . . . . . 112
3.6 Mean-SquareCoherentLight:MaximumYoung’sFringe
VisibilityThroughReversibleDevices . . . . . . . . . . . . . . 118
3.7 Comparing Special Types of Random
ElectromagneticFields. . . . . . . . . . . . . . . . . . . . . . . 122
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Contents ix
4 Non-ParaxialElectromagneticBeams . . . . . . . . . . . . . . . . 127
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2 FormalismandKeyDefinitions . . . . . . . . . . . . . . . . . . 128
4.2.1 AngularPlane-WaveSpectrum . . . . . . . . . . . . . . 128
4.2.2 PropagatingandEvanescentWaves . . . . . . . . . . . . 131
4.2.3 TE-andTM-DecompositionofthePropagatingField . . 131
4.2.4 SignificanceoftheLongitudinalFieldComponent . . . . 134
4.3 Propagation of Non-paraxial Beams with Radial or
AzimuthalPolarizationDistributionataTransversePlane . . . . 136
4.3.1 RadialandAzimuthalComponents . . . . . . . . . . . . 136
4.3.2 RadialCase:Free-SpacePropagation . . . . . . . . . . . 138
4.3.3 AzimuthalCase:Free-SpacePropagation . . . . . . . . . 142
4.3.4 ApplicationtoaParticularSetofFields . . . . . . . . . . 145
4.3.5 Radially-PolarizedFieldsatAnyTransversePlane . . . . 148
4.4 ClosestField . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.4.1 DefinitionandMeaning . . . . . . . . . . . . . . . . . . 151
4.4.2 ApplicationtotheUniformly-PolarizedGaussianModel . 152
4.4.3 TransversePolarizationStructureoftheClosest
FieldAssociatedtotheGaussianModel. . . . . . . . . . 157
4.4.4 OtherExamples . . . . . . . . . . . . . . . . . . . . . . 161
4.5 Evanescent Waves Associated to Highly
Non-paraxialBeams . . . . . . . . . . . . . . . . . . . . . . . . 165
4.5.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.5.2 NumericalExamples . . . . . . . . . . . . . . . . . . . . 167
4.6 PartiallyCoherentElectromagneticTE-Fields . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Notation
Symbol Name First
appearancein
thetext
(Subsection)
[αβ]ij,i,j=s,p; 2.3.2
α,β=x,y,u,v
Det determinantofamatrix 1.2.1
E electricfieldvector 1.2.1
˜ ˜
E,H spatialFouriertransformofEandH 4.2.1
e1(φ),e2(ρ,φ) 4.2.3
E˜0(ρ,φ) 4.2.1
Eazim azimuthallypolarizedfield 4.3.3
Eclosest,Hclosest closestelectricandmagneticfields 4.4.2
eev(ρ,φ) 4.5.1
Epr,Eev propagatingandevanescenttermsofE 4.2.2
ER,Eθ radialandazimuthalcomponentsofE 1.5.1
Erad radiallypolarizedfield 4.3.2
Es,Ep transversecomponentsofE 1.2.1
ETE,ETM transverse-electricandtransverse-magnetictermsofE 4.2.3
(cid:7)(r1,r2,τ) mutualcoherencefunction 1.2.2
(cid:7)ˆ(r1,r2,τ) 1.2.2
g12≡g(r1,r2) 3.4
H magneticfieldvector 4.2.1
HTE,HTM transverse-electricandtransverse-magnetictermsofH 4.2.3
h(r,η,z) Wignerdistributionfunction 2.2.1
Hˆ(r˜,η˜,z) Wignermatrix 2.3.1
Jˆ Jonesmatrix 1.2.1
ˆ
J(r1,r2,z) coherence-polarizationmatrix 1.2.2
j(r1,r2) complexdegreeofcoherence 1.6.1
Jsc(r1,r2) mutualintensity 1.6.1
k wavenumber 1.2.1
K Kurtosisparameter 2.9
λ wavelength 1.2.1
M2 beampropagationfactor 2.5
xi
