Application and Optimisation of the Spatial Phase Shifting ...

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122 Comparison of noise in phase maps from TPS and SPS The conversion factor from phase to displacement is λ/288° or λ/(205 grey levels), which means that one wavelength of in-plane displacement generates 0.8 sawtooth fringes. This has an important consequence: even "good" sawtooth images with low phase σ ∆ϕ error yield a large displacement error σ d after the conversion. Indeed, as Fig. 5.8 shows, the ordinate scale of previously 0.146 λ for the out-of-plane measurements changes to σ d,max 0.36 λ. 0.35 σ d /λ 0.30 5 0 0.25 5 0.20 0 0.15 5 0.10 0 0.05 0.00 N x 0 5 10 15 20 30 40 50 5 60 70 90 100 0 0 2 4 6 8 d s /d p 10 0 2 4 6 8 d s /d p 10 Fig. 5.8: σ d for ESPI displacement measurements with SPS (left) and TPS (right) as a function of speckle size for in-plane displacements. The parameter for each curve is N x , the number of vertical fringes per 1024 pixels, as indicated in the legend box. Since no object tilts are involved here, the image decorrelation is exclusively of type (ii). Evidently, this does not change the qualitative course of the plots: they strongly resemble those of Fig. 5.5. Again, a speckle size between 2.5 and 4 d p is found to be a good choice for SPS and about 1 d p for TPS. The evaluation of σ d with TPS and SPS, respectively, for various N y , led to similar performance as for N x . On comparing the σ d obtained here with those from Chapter 5.4, it turns out that here the σ d are about 2.5 times as large as in 5.4, particularly for the SPS measurements, where the factor is nearly exact. This is a direct consequence of the reduced sensitivity (40% that of the out-of-plane configuration) and tells us that the σ ∆ϕ in the underlying sawtooth images are very similar in both cases. The displacement information is encoded in the interferograms in the same way, but by different displacements, for the outof-plane and in-plane configurations. Hence it is not surprising that also the σ ∆ϕ are on a comparable level. This result confirms that we now have reasonable performance data for smooth-reference ESPI setups with TPS and SPS at our disposal.
5.5 In-plane displacements 123 5.5.2 Purely in-plane sensitive interferometer for TPS In the previous subsection we have seen the disadvantageous effects of a low sensitivity on the σ d of displacement measurements. Besides, it is desirable from a practical point of view to measure the Cartesian components of displacement separately because this simplifies the evaluation greatly. The way to carry out pure in-plane displacement measurements is known since a long time [Lee70] and has become the common choice because of its ease of use and its high sensitivity that is hard to surpass [Sir93, Joe95]. The basic interferometer is modified for symmetrical oblique object illumination as sketched in Fig. 5.9. Component numbers skipped, or not starting from one, indicate that the "original" components are still in place, which helps restoring the former set-up accurately. In particular, the fibre assembly is disabled, but not removed. Object M5 MO4 L4 M8 M4 MO3 L3 M6 M1 BS1 M2/PZT HV amplifier Waveform generator M7 PC frame memory trigger bit A CCD L2 Fig. 5.9: Optical set-up used for pure in-plane TPS measurements. Abbreviations: M, mirrors, BS, beam splitter, L, lenses, MO, microscope objectives, PZT, piezo actuator, A, aperture stop. By BS1, the light is divided into two beams of almost equal power; the "reference" beam is directed into MO4 via M2 and M5. Although there is no distinction of object and reference beam in speckle-reference set-ups, we declare this beam the reference because it is the one to undergo the temporal phase shift by means of the PZT that moves M2. Since M2 reflects the beam at 45°, the phase shift must be recalibrated. Theoretically, the voltage ramp used for normal incidence should be augmented by L2; due to imperfections of the PZT, the true value was 1.33. The "object" beam reaches M4 and then MO3, which is of the same type as MO4. Also the collimating lenses L3 and L4 are of the same type (f =140 mm), and via several other mirrors each beam illuminates the object at an angle of 45°. In this configuration, B is very close to unity to maximise M I . The layout seems somewhat complicated, but is necessary to attain equal paths for both beams, and also facilitates leaving the imaging unit with L2 and A completely unchanged.
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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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Page 75 and 76: 3.2 Phase-shifting ESPI 73 to think
Page 77 and 78: 3.3 Temporal phase shifting 75 Agai
Page 79 and 80: 3.3 Temporal phase shifting 77 addi
Page 81 and 82: 3.4 Spatial phase shifting 79 which
Page 83 and 84: 3.4 Spatial phase shifting 81 3.4.1
Page 85 and 86: 3.4 Spatial phase shifting 83 cross
Page 87 and 88: 3.4 Spatial phase shifting 85 Fig.
Page 89 and 90: 3.4 Spatial phase shifting 87 arran
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Page 93 and 94: 3.4 Spatial phase shifting 91 frequ
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Page 97 and 98: 3.4 Spatial phase shifting 95 error
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Page 102 and 103: 100 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 /
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Page 142 and 143: 140 Improvements on SPS 6.2 Modifie
Page 146 and 147: 144 Improvements on SPS Fig. 6.7 te
Page 148 and 149: 146 Improvements on SPS Finally, bo
Page 150 and 151: 148 Improvements on SPS To use the
Page 152 and 153: 150 Improvements on SPS The improve
Page 154 and 155: 152 Improvements on SPS On the whol
Page 156 and 157: 154 Improvements on SPS In classica
Page 158 and 159: 156 Improvements on SPS generally n
Page 162 and 163: 160 Improvements on SPS uncertain,
Page 164 and 165: 162 Improvements on SPS To obtain p
Page 166 and 167: 164 Improvements on SPS Fig. 6.22:
Page 168 and 169: 166 Improvements on SPS This proced
Page 170 and 171: 168 Improvements on SPS 6.7.2 Long-
Page 172 and 173: 170 Improvements on SPS Fig. 6.28:
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172 Improvements on SPS the heater
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174
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176 Summary method to search for a
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178
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180 References [Bot97] [Bra87] [Bro
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182 References [Ett97] [Fac93] [Far
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184 References [Har94] [Her96] [Het
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186 References [Kuj89] [Kuj91a] [Ku
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188 References [McLa86] [Mer83] J.
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190 References [Rav99] I. Raveh, E.
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192 References [Sur98b] [Sur98c] [S
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194
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196 Appendix A: Counting events and
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198 Appendix B: Real-time phase cal
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200 Appendix C: Derivation of inten
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I 1 a r g ( S ~ ( ) ) 202 Appendix
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204 Appendix D: Alternative error-c
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210 The author List of publications
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