Application and Optimisation of the Spatial Phase Shifting ...

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156 Improvements on SPS generally necessary in speckle interferometry (nevertheless, it may sometimes be useful to inspect the speckle phases per se). In classical interferometry, the variations of the background intensity may reasonably be assumed to be so low-frequent that the spectrum of the variations of I b is easily separated from the signal in frequency space. In speckle interferometry however, the high frequencies in O(x,y) cause a significant deficiency of the FTM: as is clearly seen from Fig. 6.17, O ~ ( ν x , ν y ) is not separated from the sidebands. The speckle halo overlaps the sidebands at any practicable speckle size, so that a considerable noise background adds to most of the signal's frequency content. This disturbs the phase reconstruction in a similar way as in the phase-shifting investigations. But as familiar as the problems are the ways to cope with them. From (6.19), it is clear that increasing B will again help to suppress the speckle noise, provided r(x,y) has a narrow spectrum and can be eliminated in the frequency plane. This can be fulfilled in an excellent way if a fibre is used to illuminate the sensor: then r(x,y) will be a very broad Gaussian profile, and its spectrum a very narrow Gaussian that will not overlap with the signal sidebands. It turns out that the performance of the FTM depends on B much in the same way as for the phase-sampling methods. To quantify this, Fig. 6.1 also contains a plot of σ d as calculated by (6.19) -(6.21) (black, circle symbols) from the same interferograms as used for the SPS tests. Furthermore, we note that the quantity O ~ ( ν x , ν y ) in (6.19) is directly accessible because of ~ O ( νx , ν y ) = FT O ( x , y ) = FT o ( x , y ) o ( x , y ) , (6.22) * ( ) ( ) so that we should be able to eliminate the speckle background from the phase calculation if we first record the speckle pattern alone, calculate its spectrum and subtract it from (6.18). This correction for speckle intensity is similar to that in 6.2.1, and the remaining phase errors are then mainly from electronic noise and pixels with insufficient M I . The same would be possible for the reference wave if its spectrum would overlap the signal spectra. This approach resembles the background subtraction suggested in [Liu97] for classical interferometry. Using the linearity of the Fourier transform, we could even subtract the speckle pattern in the space domain (6.17) before switching to the frequency domain; but the benefit is easier to see in the frequency representation. Fig. 6.18 provides an example of how the speckle noise is removed in the Fourier plane when d s =2 d p . The aliased frequencies over ν N remain usable for the FTM by pasting them back to where they got cut off [Bon86]. The reference wave need not be accounted for, since its spectrum is indeed easily separated from the sidebands.
6.5 Fourier transform method of phase determination 157 Fig. 6.18: Interferogram power spectra for d s =2 d p and B=3 without (left) and with (right) speckle subtraction. The speckle halo is larger than the frequency plane; the attenuation of high horizontal frequencies is mostly due to the pixel clock (cf. 3.4.5). Spatial frequency scales are as in, e.g., Fig. 6.12 on the left. This approach eliminates the problem of growing overlap of speckle halo and signal band with decreasing speckle size, so that a very large part of the frequency plane can now conveniently be utilised. Also, the "crosstalk" of the sidebands addressed in 6.1.2 (cf. Fig. 6.3) is avoided. The setting of ν x =ν y = ½ ν N , chosen for convenience of phase sampling (cf. 6.3), also appears to be the optimum choice in frequency space: it has been used in [Küch91] for a high-performance interferometer, and a computer simulation in [Che91] showed it to yield the error minimum. The vacant regions of the frequency spectrum can even be used to record further information [McLa86, Hor90, Sim93, Pir95, Ped97a, Ped97b, Tak97a, Tak97b, Sched99], e.g. about a second deformation direction; this approach has become popular under the name of spatial frequency multiplexing. Including time as a parameter enables spatio-temporal frequency multiplexing with one [Tak90a] or two [Tak92, Mor94b] spatial dimensions. The improvement of speckle subtraction over the non-correcting FTM for varying B is also shown in Fig. 6.1 for d s =3 d p (black, white circle symbols). The behaviour of the correction is the same as for the phaseshifting method: the effect vanishes for B30. When the same interferograms as in Chapter 6.4 are processed by the FTM, again at B=30 without and B=3 with the intensity correction, one comes to the results plotted in Fig. 6.19. To use a 10241024 pixel FFT, the standard input images consisting of 1024768 pixels were padded with zeros in the last 256 rows. In a comparison of genuine 1024 2 pixel images processed entirely and partly, the difference in the σ d values remained within 1%.
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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 108 and 109: 106 Quantification of displacement-
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 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 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:
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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