Application and Optimisation of the Spatial Phase Shifting ...

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160 Improvements on SPS uncertain, frequently occur at locations that are different in the v and h fields. This gets clear when we express the probability of finding a "bad" pixel (denoted by subscript b) at (x,y) in either speckle pattern by P vb (x,y) and P hb (x,y) and that of finding a bad pixel in both speckle patterns by P bb (x,y). Then we have Pbb ( x, y) ≅ Pvb ( x, y) Phb ( x, y) , (6.24) which is exact when c=0. Provided both P vb (x,y) and P hb (x,y) are distinctly smaller than unity, this means that it is possible to replace most of the bad pixels from one speckle pattern by valid pixels from the other one. There are of course always several points (even for c = 0) where bad pixels in both speckle fields coincide. But in any case, the number of bad points in the phase map can be minimised by suitable merging of ϕ O,v (x,y) and ϕ O,h (x,y). The merging process is carried out by analysing M I (x,y) in the interferograms between the reference wave (ideally linearly polarised at 45°) and the vertically or horizontally polarised object wave [Cre88], 2 I , vi 1vi 3vi 2vi 1vi 3vi M ( x, y) = 3( I − I ) + ( 2I − I − I ) 2 2 (6.25) M ( x, y) = 3( I − I ) + ( 2I − I − I ) , I , hi 1hi 3hi 2hi 1hi 3hi where α=120°/sample and the index i refers to the initial object state, for reasons to become clear shortly. It should be emphasised that these M I are derived from the "sine" and "cosine" terms of (3.17), which must then be used for the subsequent phase determination; if other α, or phase-calculation formulae, are used, the respective "sine" and "cosine" terms have to be inserted under the square root. Due to the low correlation between the v and h speckle patterns, the two maps of M I,vi (x,y) and M I,hi (x,y) will also be different, and the higher of the two values ought to indicate the prospect of a more accurate phase measurement. Admittedly, (6.25) is not as reliable in SPS as in TPS [Su 94] due to the underlying speckle structure that may yield bogus modulation when the pixel triplet crosses speckle "boundaries"; but as far as a comparison of M I,vi (x,y) and M I,hi (x,y) is concerned, this approach still works rather well, as we shall see. The interferograms are recorded by a CCD camera behind a polariser in the vertical or horizontal position; setting the plane of polarisation of the reference wave to ideally 45° assures P vb (x,y)P hb (x,y). For each point (x,y) in both interferograms we determine M I,vi (x,y) and M I,hi (x,y) and the phase distributions ϕ O,vi (x,y) and ϕ O,hi (x,y). Then, starting with ϕ O,vi (x,y), we replace all phase values in this map by those from ϕ O,hi (x,y) at all the locations where M I,vi (x,y) < M I,hi (x,y). Thus, a pixel is considered "bad" in the sense of (6.24) when a better measurement is available. The locations of replaced pixels are stored in a binary mask B i (x,y). Repeating this modulation analysis for the final object state would lead to a slightly different map B f (x,y) due to speckle decorrelation by the object deformation and statistical temporal fluctuations like camera noise. Therefore, B i (x,y) is used for the final object state too. That is, the phase values in the map ϕ O,vf (x,y) are replaced by those of the map ϕ O,hf (x,y) at the same locations where the replacement is done 2
6.6 Use of depolarisation to eliminate invalid pixels 161 for the initial object state. By this approach, two merged (index m) phase maps ϕ O,mi (x,y) and ϕ O,mf (x,y) are generated, whose correlation is maintained with respect to the pixel replacement. Since the phase offsets N 0v and N 0h (cf. Chapter 4.2) should be the same for both sawtooth images to merge, they should be kept constant during the recording of the two interferogram pairs I vi , I hi and I vf , I hf . In principle, it is possible to correct a phase offset a posteriori and make both fringe systems fit together by subtracting a constant phase from one of the sawtooth images; but the error fringe profiles (cf. Fig. 3.39) are related to the actual physical phase offset, so that such a "makeshift" will produce artefacts and lead to unsatisfactory results. Therefore, a phase stabilisation system to compensate phase fluctuations by, e.g., vibrations or temperature drifts is incorporated in the interferometer as shown in Fig. 6.20. Object M1 L1 POC FC O R POC PZ HV M2 PF L2 A S BS D1 D2 PID CCD Fig. 6.20: ESPI out-of-plane set-up with SPS and active phase stabilisation. Dashed lines: beams for the stabilisation system. Abbreviations: see text. The light of a HeNe laser (25 mW @ 632.8 nm) is coupled with a microscope objective L1 into a standard single mode fibre (Corning Flexcor 633). A fibre coupler FC (Gould) splits the light into an object wave O and a reference wave R with a coupling ratio of 9:1. Both output fibres contain a polarisation controller POC [Lef80] to adjust the SOP at the fibre ends; we use linear 45° polarisation for O and not 45° but 48° for R, which difference will be justified below. Although a standard single mode fibre is used, both SOPs remain almost constant under the given laboratory conditions for a long time. This was verified by longterm measurements with a real-time Stokes polarimeter [Dir97]: for a time period of about four hours, the azimuthal SOP angle changes by less than 2°, and the angle of ellipticity by less than 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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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 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 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 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:
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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