Application and Optimisation of the Spatial Phase Shifting ...

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158 Improvements on SPS 0.10 σ d /λ 0.08 0.06 d s=2 d p, B=30 0.04 d s=2 d p, B= 3 d s=2.5 d p, B=30 0.02 d s=2.5 d p, B= 3 d s=3 d p, B=30 d s=3 d p, B= 3 0.00 0 20 40 60 80 N x 100 Fig. 6.19: σ d from ESPI displacement measurements as a function of N x , as calculated by (6.19)-(6.21) (black) and its intensity-correcting extension according to (6.22) (white filled symbols), with various d s and B as listed in the legend box. By the intensity correction, a pronounced improvement is attained for d s =2 d p . This could be expected since the overlap of speckle halo and sidebands is largest at the smallest d s , and hence a subtraction of the speckle noise should have the largest effect. Again, there is not much difference between the curves for d s =2.5 or 3 d p , and the improvement by the intensity correction is similar to that in Fig. 6.16. Generally, the curves shown here are rather similar to those of Fig. 6.16, but a careful comparison reveals a qualitative difference. The FTM yields lower σ d for N x
6.6 Use of depolarisation to eliminate invalid pixels 159 6.6 Use of depolarisation to eliminate invalid pixels A main error source in ESPI and, to a lesser extent, in holographic interferometry, are pixels where M I (x,y) falls below the electronic noise or even vanishes due to low or zero speckle intensity. This occurs quite frequently (cf. Chapter 2.2.5) and leads to a relevant fraction of uncertain or invalid outputs of ∆ϕ(x,y) in displacement measurements. This phenomenon, with the associated discontinuities of the speckle phase, is the origin of the "salt and pepper" noise in ESPI phase maps, and its effect on phase unwrapping has been investigated recently [Hun95]. On the other hand, it is known that the speckle intensity pdf described by (2.6) changes significantly when an incoherent sum of two uncorrelated speckle patterns is considered [Goo75, p.21, Enn75, p.211]. In that case, the maximum of the intensity pdf is shifted away from O(x,y)=0, so that the probability of finding "dark" pixels will decrease. Such a case is encountered in the interferometric investigation of rough objects that give rise to multiple scattering and thus introduce depolarisation, i.e. generate two mutually incoherent speckle fields. In this subsection it will be shown how these can be exploited to improve ESPI measurements [Bro98]. In this context, we call an object depolarising if the state of polarisation (SOP) of the light scattered back from it differs from the SOP of the incident light. In many samples, for instance natural stone, this is a consequence of volume scattering due to the transparency of the material under investigation. Hence, we obtain a scattered wave field with fluctuations of intensity, phase, and polarisation. If the scattered light is split into two orthogonal linearly polarised states (vertical, v, and horizontal, h), two speckle patterns S v (x,y) and S h (x,y) are generated with a normalised correlation coefficient c. As described in [Fre90d], the value of c is chiefly governed by a surface–specific constant, called the depolarisation coefficient ρ. This quantity is defined by the ratio of cross- to co-polarised scattered speckle intensity: 0ρ = S v /S h 1, where h is the SOP of the incident light and v the orthogonal one. Then, we can use c = ( 1− ρ) 1+ ρ 2 2 (6.23) as a very good approximation to determine the correlation of the orthogonally polarised speckle patterns. This theoretical prediction was confirmed by measurements of depolarising natural stones [Ada97]. Such surfaces are for example involved in ESPI-based measurements of deformations and surface changes of historical monuments, which application was developed in the last few years [Gül96]. Even moderate amounts of depolarisation cause a significant decay of c: for ρ >0.5, we find c
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Application and Optimisation of the
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Table of Contents 1 1 INTRODUCTION
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3 1 Introduction The present era of
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Introduction 5 retrieval can help t
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7 2 Statistical Properties of Speck
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2.1 Experimental set-up 9 Gaussian
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2.2 First-order speckle statistics
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2.3 Second-order speckle statistics
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47 3 Electronic or Digital Speckle
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3.1 Subtraction-mode ESPI 49 some 4
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3.2 Phase-shifting ESPI 51 filled u
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3.2 Phase-shifting ESPI 53 known si
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3.2 Phase-shifting ESPI 55 α n =α
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3.2 Phase-shifting ESPI 57 But reme
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3.2 Phase-shifting ESPI 59 ϕ ϕ O,
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3.2 Phase-shifting ESPI 61 These fo
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3.2 Phase-shifting ESPI 63 ∞ ~ ~
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3.2 Phase-shifting ESPI 65 ∞ N
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3.2 Phase-shifting ESPI 67 The spec
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3.2 Phase-shifting ESPI 69 − 0. 2
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3.2 Phase-shifting ESPI 71 ν x 0 i
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3.2 Phase-shifting ESPI 73 to think
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3.3 Temporal phase shifting 75 Agai
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3.3 Temporal phase shifting 77 addi
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3.4 Spatial phase shifting 79 which
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3.4 Spatial phase shifting 81 3.4.1
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3.4 Spatial phase shifting 83 cross
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3.4 Spatial phase shifting 85 Fig.
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3.4 Spatial phase shifting 87 arran
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3.4 Spatial phase shifting 89 infor
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3.4 Spatial phase shifting 91 frequ
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3.4 Spatial phase shifting 93 altho
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3.4 Spatial phase shifting 95 error
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3.4 Spatial phase shifting 97 Since
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100 Quantification of displacement-
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Page 110 and 111: 108 Quantification of displacement-
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Page 114 and 115: 112 Comparison of noise in phase ma
Page 136 and 137: 134 Improvements on SPS intensity o
Page 138 and 139: 136 Improvements on SPS 0.14 σ d /
Page 140 and 141: 138 Improvements on SPS we are conc
Page 142 and 143: 140 Improvements on SPS 6.2 Modifie
Page 144 and 145: 142 Improvements on SPS 0.12 σ d /
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Page 148 and 149: 146 Improvements on SPS Finally, bo
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Page 164 and 165: 162 Improvements on SPS To obtain p
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Page 168 and 169: 166 Improvements on SPS This proced
Page 170 and 171: 168 Improvements on SPS 6.7.2 Long-
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Page 174 and 175: 172 Improvements on SPS the heater
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Page 178 and 179: 176 Summary method to search for a
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Page 182 and 183: 180 References [Bot97] [Bra87] [Bro
Page 184 and 185: 182 References [Ett97] [Fac93] [Far
Page 186 and 187: 184 References [Har94] [Her96] [Het
Page 188 and 189: 186 References [Kuj89] [Kuj91a] [Ku
Page 190 and 191: 188 References [McLa86] [Mer83] J.
Page 192 and 193: 190 References [Rav99] I. Raveh, E.
Page 194 and 195: 192 References [Sur98b] [Sur98c] [S
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Page 198 and 199: 196 Appendix A: Counting events and
Page 200 and 201: 198 Appendix B: Real-time phase cal
Page 202 and 203: 200 Appendix C: Derivation of inten
Page 204 and 205: I 1 a r g ( S ~ ( ) ) 202 Appendix
Page 206 and 207: 204 Appendix D: Alternative error-c
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210 The author List of publications
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Application and Optimisation of the Spatial Phase Shifting ...

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