Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications

345 Pages·5.626 MB·English

by  Manuel Servin, J. Antonio Quiroga, Moises Padilla

ManuelServin,J.AntonioQuiroga,and
J.Moise´sPadilla
FringePatternAnalysisforOptical
Metrology
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Manuel Servin, J. Antonio Quiroga, and J. Moise´s Padilla
Fringe Pattern Analysis for Optical
Metrology
Theory, Algorithms, and Applications
TheAuthors AllbookspublishedbyWiley-VCHare
carefullyproduced.Nevertheless,authors,
editors,andpublisherdonotwarrantthe
Dr.ManuelServin
informationcontainedinthesebooks,
CentrodeInvestigaciones
includingthisbook,tobefreeoferrors.
enOpticaA.C.
Readersareadvisedtokeepinmindthat
Leo´nGuanajuato
statements,data,illustrations,procedural
Mexico
detailsorotheritemsmayinadvertentlybe
inaccurate.
Dr.J.AntonioQuiroga
UniversidadComplutenseMadrid
LibraryofCongressCardNo.:appliedfor
Dept.deOptica
Madrid
Spain BritishLibraryCataloguing-in-Publication
Data
Acataloguerecordforthisbookisavailable
Dr.J.Moise´sPadilla
fromtheBritishLibrary.
CentrodeInvestigaciones
enOpticaA.C.
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V
Contents
Preface XI
ListofSymbolsandAcronyms XV
1 DigitalLinearSystems 1
1.1 IntroductiontoDigitalPhaseDemodulationinOpticalMetrology 1
1.1.1 FringePatternDemodulationasanIll-PosedInverseProblem 1
1.1.2 AddingaprioriInformationtotheFringePattern:Carriers 3
1.1.3 ClassificationofPhaseDemodulationMethodsinDigital
Interferometry 7
1.2 DigitalSampling 9
1.2.1 SignalClassification 9
1.2.2 CommonlyUsedFunctions 11
1.2.3 ImpulseSampling 13
1.2.4 Nyquist–ShannonSamplingTheorem 14
1.3 LinearTime-Invariant(LTI)Systems 14
1.3.1 DefinitionandProperties 15
1.3.2 ImpulseResponseofLTISystems 15
1.3.3 StabilityCriterion:Bounded-InputBounded-Output 17
1.4 Z-TransformAnalysisofDigitalLinearSystems 18
1.4.1 DefinitionandProperties 18
1.4.2 RegionofConvergence(ROC) 19
1.4.3 PolesandZerosofaZ-Transform 20
1.4.4 InverseZ-Transform 21
1.4.5 TransferFunctionofanLTISystemintheZ-Domain 22
1.4.6 StabilityEvaluationbyMeansoftheZ-Transform 23
1.5 FourierAnalysisofDigitalLTISystems 24
1.5.1 DefinitionandPropertiesoftheFourierTransform 25
1.5.2 Discrete-TimeFourierTransform(DTFT) 25
1.5.3 RelationBetweentheDTFTandtheZ-Transform 26
1.5.4 SpectralInterpretationoftheSamplingTheorem 27
1.5.5 Aliasing:Sub-NyquistSampling 29
1.5.6 FrequencyTransferFunction(FTF)ofanLTISystem 31
1.5.7 StabilityEvaluationintheFourierDomain 33
VI Contents
1.6 Convolution-BasedOne-Dimensional(1D)LinearFilters 34
1.6.1 One-DimensionalFiniteImpulseResponse(FIR)Filters 34
1.6.2 One-DimensionalInfiniteImpulseResponse(IIR)Filters 37
1.7 Convolution-Basedtwo-dimensional(2D)LinearFilters 39
1.7.1 Two-Dimensional(2D)FourierandZ-Transforms 39
1.7.2 StabilityAnalysisof2DLinearFilters 40
1.8 RegularizedSpatialLinearFilteringTechniques 42
1.8.1 ClassicalRegularizationforLow-PassFiltering 42
1.8.2 SpectralResponseof2DRegularizedLow-PassFilters 46
1.9 StochasticProcesses 48
1.9.1 DefinitionsandBasicConcepts 48
1.9.2 ErgodicStochasticProcesses 51
1.9.3 LTISystemResponsetoStochasticSignals 52
1.9.4 PowerSpectralDensity(PSD)ofaStochasticSignal 52
1.10 SummaryandConclusions 54
2 SynchronousTemporalInterferometry 57
2.1 Introduction 57
2.1.1 HistoricalReviewoftheTheoryofPhase-ShiftingAlgorithms
(PSAs) 57
2.2 TemporalCarrierInterferometricSignal 60
2.3 QuadratureLinearFiltersforTemporalPhaseEstimation 62
2.3.1 LinearPSAsUsingReal-ValuedLow-PassFiltering 64
2.4 TheMinimumThree-StepPSA 68
2.4.1 AlgebraicDerivationoftheMinimumThree-StepPSA 68
2.4.2 SpectralFTFAnalysisoftheMinimumThree-StepPSA 69
2.5 Least-SquaresPSAs 71
2.5.1 Temporal-to-SpatialCarrierConversion:Squeezing
Interferometry 73
2.6 DetuningAnalysisinPhase-ShiftingInterferometry(PSI) 74
2.7 NoiseinTemporalPSI 80
2.7.1 PhaseEstimationwithAdditiveRandomNoise 82
2.7.2 NoiseRejectioninN-StepLeast-Squares(LS)PSAs 85
2.7.3 NoiseRejectionofLinearTunablePSAs 86
2.8 HarmonicsinTemporalInterferometry 87
2.8.1 InterferometricDatawithHarmonicDistortionandAliasing 88
2.8.2 PSAResponsetoIntensity-DistortedInterferograms 91
2.9 PSADesignUsingFirst-OrderBuildingBlocks 95
2.9.1 MinimumThree-StepPSADesignbyFirst-OrderFTFBuilding
Blocks 97
2.9.2 TunableFour-StepPSAswithDetuningRobustnessat𝜔=−𝜔 100
0
2.9.3 TunableFour-StepPSAswithRobustBackgroundIllumination
Rejection 101
2.9.4 TunableFour-StepPSAwithFixedSpectralZeroat𝜔=π 102
2.10 SummaryandConclusions 104
Contents VII
3 AsynchronousTemporalInterferometry 107
3.1 Introduction 107
3.2 ClassificationofTemporalPSAs 108
3.2.1 Fixed-Coefficients(Linear)PSAs 108
3.2.2 Tunable(Linear)PSAs 108
3.2.3 Self-Tunable(Nonlinear)PSAs 109
3.3 SpectralAnalysisoftheCarre´PSA 110
3.3.1 FrequencyTransferFunctionoftheCarre´PSA 112
3.3.2 Meta-FrequencyResponseoftheCarre´PSA 113
3.3.3 Harmonic-RejectionCapabilitiesoftheCarre´PSA 114
3.3.4 Phase-StepEstimationintheCarre´PSA 116
3.3.5 ImprovementofthePhase-StepEstimationinSelf-Tunable
PSAs 118
3.3.6 ComputerSimulationswiththeCarre´PSAwithNoisy
Interferograms 120
3.4 SpectralAnalysisofOtherSelf-TunablePSAs 122
3.4.1 Self-TunableFour-StepPSAwithDetuning-ErrorRobustness 123
3.4.2 Self-TunableFive-StepPSAbyStoilovandDragostinov 126
3.4.3 Self-TunableFive-StepPSAwithDetuning-Error
Robustness 128
3.4.4 Self-TunableFive-StepPSAwithDoubleZeroesattheOriginandthe
TuningFrequency 130
3.4.5 Self-TunableFive-StepPSAwithThreeTunableSingleZeros 131
3.4.6 Self-TunableFive-StepPSAwithSecond-HarmonicRejection 133
3.5 Self-CalibratingPSAs 136
3.5.1 IterativeLeast-Squares,theAdvancedIterativeAlgorithm 137
3.5.2 PrincipalComponentAnalysis 140
3.6 SummaryandConclusions 145
4 SpatialMethodswithCarrier 149
4.1 Introduction 149
4.2 LinearSpatialCarrier 149
4.2.1 TheLinearCarrierInterferogram 149
4.2.2 InstantaneousSpatialFrequency 152
4.2.3 SynchronousDetectionwithaLinearCarrier 155
4.2.4 LinearandNonlinearSpatialPSAs 159
4.2.5 FourierTransformAnalysis 164
4.2.6 Space–FrequencyAnalysis 170
4.3 CircularSpatialCarrier 173
4.3.1 TheCircularCarrierInterferogram 173
4.3.2 SynchronousDetectionwithaCircularCarrier 174
4.4 2DPixelatedSpatialCarrier 177
4.4.1 ThePixelatedCarrierInterferogram 177
4.4.2 SynchronousDetectionwithaPixelatedCarrier 180
4.5 RegularizedQuadratureFilters 186
VIII Contents
4.6 RelationBetweenTemporalandSpatialAnalysis 198
4.7 SummaryandConclusions 198
5 SpatialMethodsWithoutCarrier 201
5.1 Introduction 201
5.2 PhaseDemodulationofClosed-FringeInterferograms 201
5.3 TheRegularizedPhaseTracker(RPT) 204
5.4 LocalRobustQuadratureFilters 215
5.5 2DFringeDirection 216
5.5.1 FringeOrientationinInterferogramProcessing 216
5.5.2 FringeOrientationandFringeDirection 219
5.5.3 OrientationEstimation 222
5.5.4 FringeDirectionComputation 225
5.6 2DVortexFilter 229
5.6.1 TheHilbertTransforminPhaseDemodulation 229
5.6.2 TheVortexTransform 230
5.6.3 TwoApplicationsoftheVortexTransform 233
5.7 TheGeneralQuadratureTransform 235
5.8 SummaryandConclusions 239
6 PhaseUnwrapping 241
6.1 Introduction 241
6.1.1 ThePhaseUnwrappingProblem 241
6.2 PhaseUnwrappingby1DLineIntegration 244
6.2.1 LineIntegrationUnwrappingFormula 244
6.2.2 NoiseToleranceoftheLineIntegrationUnwrappingFormula 246
6.3 PhaseUnwrappingwith1DRecursiveDynamicSystem 250
6.4 1DPhaseUnwrappingwithLinearPrediction 251
6.5 2DPhaseUnwrappingwithLinearPrediction 255
6.6 Least-SquaresMethodforPhaseUnwrapping 257
6.7 PhaseUnwrappingThroughDemodulationUsingaPhase
Tracker 258
6.8 SmoothUnwrappingbyMaskingout2DPhaseInconsistencies 262
6.9 SummaryandConclusions 266
AppendixAListofLinearPhase-ShiftingAlgorithms(PSAs) 271
A.1 BriefReviewofthePSAsTheory 271
A.2 Two-StepLinearPSAs 274
A.2.1 Two-StepPSAwithaFirst-OrderZeroat−𝜔 (𝜔 =π∕2) 274
0 0
A.3 Three-StepLinearPSAs 275
A.3.1 Three-StepLeast-SquaresPSA(𝜔 =2π∕3) 275
0
A.3.2 Three-StepPSAwithFirst-OrderZerosat𝜔={0,−𝜔 }
0
(𝜔 =π∕2) 276
0
A.4 Four-StepLinearPSAs 277
A.4.1 Four-StepLeast-SquaresPSA(𝜔 =2π∕4) 277
0