Application and Optimisation of the Spatial Phase Shifting ...

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120 Comparison of noise in phase maps from TPS and SPS Fig. 5.6: Some examples of maps of ∆ϕ(x,y) that have been evaluated for σ d . Upper row, N x =5; lower row, N x =50; left, SPS with d s =3 d p ; right, TPS with d s =1.5 d p . Summarising this subsection, one can state that TPS is significantly more accurate than SPS at low fringe densities. For SPS, the best range of d s is 2.5 to 3.5 d p , with σ d λ/15 for moderate fringe densities; for TPS, we find d s d p to give a typical σ d of λ/20. Imperfections of the test object prevented an extension of the TPS study towards smaller d s . It turns out that in the presence of speckle decorrelation, SPS benefits from larger d s and spatial phase sampling, so that the advantage of TPS fades quickly with increasing object displacement. 5.5 In-plane displacements When carrying out in-plane displacement measurements using SPS and assessing its performance, the reference is the ingenious symmetrical pure-in-plane TPS configuration [Lee70] with its excellent sensitivity. A pure-in-plane SPS configuration using a double aperture has been established [Sir97a], and we will investigate its merits, but it also seems worthwhile to modify the set-up of Fig. 5.1 for more oblique object illumination and to gain in-plane sensitivity in this way, since this arrangement is by far easier to handle. Therefore we start the investigation of in-plane measurement accuracy with a set-up that has a mixed inplane/out-of-plane sensitivity (henceforth abbreviated by "mixed sensitivity") and again offers a
5.5 In-plane displacements 121 possibility to compare TPS and SPS under the same experimental parameters. As soon as pure in-plane sensitivity is demanded, the interferometer assemblies are rather different, also from each other; we will discuss these in the second part of this subsection. 5.5.1 Mixed-sensitivity interferometer As mentioned earlier, the set-up of Fig. 5.1 need only be slightly changed to acquire a non-negligible inplane sensitivity component, which is shown in Fig. 5.7. y x M4 S x z BS1 MO1 L1 M1 M3 PZT M2 HV amplifier BS2 PF Waveform generator MO2 PC frame memory trigger bit A ∆x CCD L2 Fig. 5.7: Mixed-sensitivity set-up for detection of in-plane object displacements by TPS or SPS. To obtain in-plane displacement sensitivity, the object is illuminated obliquely by means of the additional mirror M4, whose centre is placed at co-ordinates (-x M4 , 0, z M4 ). The geometry is chosen to give an angle of incidence of 53° to the surface normal for the object illumination. Thus the sensitivity vector S x is inclined by 26.6° to the normal, and the in-plane sensitivity is half the out-of-plane sensitivity. The latter is not greatly reduced in comparison to the quasi-out-of-plane configuration, but of no concern here. The collimated illumination is particularly important for in-plane geometries, as was shown in [Kun97, Alb99]. M3 has to be rotated to illuminate M4, and since this lengthens the light path in the object arm, M2 is appropriately displaced to bring the temporal coherence back to its maximum. This is rather important because the laser is being operated without an etalon and its coherence length is therefore only 10 cm. This configuration detects in-plane displacements along the x axis; for y-sensitivity, there is another mirror M5 above the object (not shown here) with its centre at (0, x M4 , z M4 ), so that the in-plane components of S x and S y are of equal modulus and orthogonal on the x-y-plane. A rotation of the object about the z axis then yields horizontal (with S x ) or vertical (with S y ) fringe patterns that fulfil the conditions listed in Chapter 4.2. Thanks to the expensive bearing, the reproducibility of the measured σ d was excellent for this type of displacement, also with smaller d s .
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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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Page 77 and 78: 3.3 Temporal phase shifting 75 Agai
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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.
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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
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Page 142 and 143: 140 Improvements on SPS 6.2 Modifie
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Page 156 and 157: 154 Improvements on SPS In classica
Page 158 and 159: 156 Improvements on SPS generally n
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Page 168 and 169: 166 Improvements on SPS This proced
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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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208
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210 The author List of publications
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