Application and Optimisation of the Spatial Phase Shifting ...

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124 Comparison of noise in phase maps from TPS and SPS The sensitivity vector lies in the object's plane in horizontal direction; in this case, only x-displacements can be measured. The object rotation generates 1.4 horizontal sawtooth fringes per wavelength of in-plane displacement. Consequently, the conversion factor from phase to displacement is λ/509° or λ/(362 grey levels), which is approximately halfway between the out-of-plane and the in-plane sensitivity that we have previously been dealing with. (It would however be easy to increase this value: if both incidence angles were 53°, as in 5.5.1, we would get 1.6 sawtooth fringes per wavelength of displacement.) The measured σ d are shown in Fig. 5.10. σ d /λ 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 30 40 50 60 70 90 100 0 2 4 6 8 d s /d p 10 N y Fig. 5.10: σ d for ESPI displacement measurements with pure in-plane TPS as a function of speckle size for inplane displacements. The parameter for each curve is N y , the number of horizontal fringes per 1024 pixels, as indicated in the legend box. Again, the ordinate reflects the change in sensitivity: σ d,max 0.20 λ for this geometry. But since only 57% of the displacement of 5.5.1 are necessary to generate the same number of fringes, there is less speckle decorrelation present than in Fig. 5.8. This improves the performance significantly for higher fringe densities, although the optimum of the speckle size still wanders towards two or more pixels for larger rotations. Over the whole range of fringe densities, the accuracy is comparable to that obtained in the outof-plane TPS study. Apart from the comparison with SPS that we are to continue, this shows that specklereference ESPI is not very much inferior to the smooth-reference configurations and that a 3-D TPS system with Cartesian sensitivities would have well-balanced systematic σ d in each of the directions. 5.5.3 Purely in-plane sensitive interferometer for SPS A set-up that facilitates exclusive in-plane displacement detection also with SPS has been described in [Sir97a]. Because the sensitivity of this configuration is also adjustable, we are able to compare the merits of SPS and TPS also with pure in-plane interferometers of equal sensitivity. Fig. 5.11 shows a schematic of the interferometer.
5.5 In-plane displacements 125 Object M3 M4 M1 MO3 L4 M2 MP ∆x Aperture shape M5 DA L3 M6 ∆x D DA PC frame memory CCD Fig. 5.11: Optical set-up used for pure in-plane SPS measurements. Abbreviations: M, mirrors, MP, mirror prism, L, lenses, MO, microscope objective, DA, double aperture; lower left: detailed view of DA as seen from the direction of the camera. The laser beam is expanded, collimated and directed normally onto the object by M4, which is located so as to be out of the viewing paths. By M5 and M6, some of the scattered light is directed towards the aluminium coated prism MP. It is attached directly in front of the double aperture DA so that each "object beam" finds its own aperture to reach the sensor. In this case, the imaging lens (f=140 mm) is located immediately behind the apertures, but still we can use the (equal) diameters of the apertures D for the determination of speckle sizes by means of (2.43). Like the set-up of 5.5.2, this in-plane configuration generates horizontal fringes only. By means of the distance ∆x between the centres of the apertures, each of diameter D, the two speckle fields interfere at an angle on the sensor, which introduces the spatial phase shift. Due to the spatial extent D of both the sources of "reference" and "object" light, the power spectrum of the interference sideband that carries the signal is twice as broad for a given speckle size as it is for the interferogram of one speckle field and a point source. In other words, there will be twice the phase shift miscalibrations and nearly twice the number of phase singularities disturbing the interferogram. Moreover, B is fixed to unity, which makes all the improvements for smooth-reference SPS (see Chapter 5) inapplicable. It is quite instructive to compare the power spectra of interferograms from the set-up in Fig. 5.11 with those from a smoothreference configuration (see Chapter 3.4.4). Fig. 5.12 shows the spatial frequency content of specklereference SPS interferograms for two different speckle sizes. The double aperture generates signal sidebands that are of the same extent as the speckle halo itself, and at least 50% of the spectral power is inevitably contained in the speckle halo, in contrast to smooth-reference interferograms. Hence, if the signal frequencies are to be well separated from the speckle noise and to remain below the Nyqvist limit, the speckle size must be twice that which was derived for a point-source reference in Chapter 3.4.4.
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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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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.
Page 89 and 90: 3.4 Spatial phase shifting 87 arran
Page 91 and 92: 3.4 Spatial phase shifting 89 infor
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 /
Page 140 and 141: 138 Improvements on SPS we are conc
Page 142 and 143: 140 Improvements on SPS 6.2 Modifie
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Page 148 and 149: 146 Improvements on SPS Finally, bo
Page 150 and 151: 148 Improvements on SPS To use the
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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-
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Page 174 and 175: 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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