Application and Optimisation of the Spatial Phase Shifting ...

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58 Electronic or Digital Speckle Pattern Interferometry The left image in Fig. 3.5 was calculated according to [Kuj89] from a data set with α=120°, which reduced σ ∆ϕ = 62.6° as obtained from raw correlation fringes (image not shown) to σ ∆ϕ = 24.0°. To the right, the method of [Moo94] was applied to the previous data set with α=90° that led to the results in Fig. 3.4, and σ ∆ϕ dropped to 27.2°. In both cases, the accuracy is more than doubled and most of the initial spatial resolution is maintained. The price for this is increased computational effort: a reference phase map must be generated first, whose lower resolution may influence the choices for ϕ O somewhat, and one out of four phase values must be selected for every pixel. Since generally no ideal reference image will be available, the errors in it will also influence the choice of ϕ O and propagate into ∆ϕ. Finally, the histogram distortion can in neither case be removed. Another method that uses I {0,1},i and I {0,1},f with α=90° has been proposed in [Own88]; while it is obviously not suitable for highly dynamic phenomena, it does find ∆ϕ unambiguously. The result of this calculation can be seen in Fig. 3.6. Both the phase map (σ ∆ϕ = 53.2°) and the histogram of the phase distribution show that this method is rather susceptible to noise; therefore it has been used in [Own88, Own91b] with smoothing the sine and cosine terms before calculating ∆ϕ. The argument of calculation speed that led to the development of this method is not important anymore; but interestingly, the very same scheme has meanwhile been applied in temporal phase unwrapping, again for reasons of, inter alia, speed [vBru98, vBru99]. Fig. 3.6: Result of calculating ∆ϕ with the method of [Own88]. As these considerations have shown, the use of ESPI correlation fringes for phase-shifting purposes is problematic when we are considering raw, i.e. unfiltered, phase data. This is because one uses only one set of phase-shifted data to determine ∆ϕ . Nevertheless, this approach may sometimes be a good way to perform phase measurements when dynamic objects are studied. 3.2.1.2 Difference-of-phases method Provided it is possible to record two sets of phase-shifted interferograms I n,i and I n,f for both object states, one can calculate two speckle phase maps by, e.g., (3.16):
3.2 Phase-shifting ESPI 59 ϕ ϕ O, i O, f I3i ( x, y) − I1i ( x, y) ( x, y) mod 2π = arctan I ( x, y) − I ( x, y) 0i 2i I3 f ( x, y) − I1f ( x, y) ( x, y) mod 2π = arctan I ( x, y) − I ( x, y) 0 f 2 f , (3.22) and then determine the phase change ( O, f O, i ) ∆ϕ ( x, y) mod 2 π = ϕ ( x, y) mod 2π − ϕ ( x, y) mod 2π mod 2π . (3.23) Admittedly, this requires more information than the phase-of-difference approach – 8 images with (3.16), and 6 with (3.17) –, but eliminates all the problems brought about by the ambiguity of intensity differences. Also, the pixels are truly regarded as independent entities, which accounts appropriately for the speckle nature of the wavefront to determine. In this case, the phase calculation reproduces the expected fringe profile rather well, as the white curves in Fig. 3.3 demonstrate. The displayed sawtooth edges are somewhat blurred by the averaging over the residual speckle noise; but the corresponding measured phase map, shown in Fig. 3.7, is of excellent quality when we compare it with the other unfiltered results obtained so far. In that case, σ ∆ϕ = 18.2° without any low-pass filtering. Also, the pdf of measured phases is now uniform, which shows that computational biases are negligible for this method. Fig. 3.7: Result of calculating ∆ϕ with the difference-of-phases method. This confrontation clearly indicates that it is necessary to genuinely measure the speckle phases twice to get the best sawtooth image. While an approximate recovery of information from reduced data sets is possible, the performance of this approach remains restricted. Therefore, the performance data given in chapters 5 and 6 are based on sawtooth images from the difference-of-phases method without exception. A practical merit of keeping ready the speckle phase distributions for every recorded object state is that one need not compare all data to the initial state anymore. In other words, it becomes possible to track phase differences incrementally even if the first and last state show decorrelated speckle patterns [Flo93]. We will come back to this issue in Chapter 6.7.
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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
Page 9 and 10: 7 2 Statistical Properties of Speck
Page 11 and 12: 2.1 Experimental set-up 9 Gaussian
Page 13 and 14: 2.2 First-order speckle statistics
Page 35 and 36: 2.3 Second-order speckle statistics
Page 49 and 50: 47 3 Electronic or Digital Speckle
Page 51 and 52: 3.1 Subtraction-mode ESPI 49 some 4
Page 53 and 54: 3.2 Phase-shifting ESPI 51 filled u
Page 55 and 56: 3.2 Phase-shifting ESPI 53 known si
Page 57 and 58: 3.2 Phase-shifting ESPI 55 α n =α
Page 59: 3.2 Phase-shifting ESPI 57 But reme
Page 63 and 64: 3.2 Phase-shifting ESPI 61 These fo
Page 65 and 66: 3.2 Phase-shifting ESPI 63 ∞ ~ ~
Page 67 and 68: 3.2 Phase-shifting ESPI 65 ∞ N
Page 69 and 70: 3.2 Phase-shifting ESPI 67 The spec
Page 71 and 72: 3.2 Phase-shifting ESPI 69 − 0. 2
Page 73 and 74: 3.2 Phase-shifting ESPI 71 ν x 0 i
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
Page 91 and 92: 3.4 Spatial phase shifting 89 infor
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Page 95 and 96: 3.4 Spatial phase shifting 93 altho
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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108 Quantification of displacement-
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110
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112 Comparison of noise in phase ma
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134 Improvements on SPS intensity o
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136 Improvements on SPS 0.14 σ d /
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138 Improvements on SPS we are conc
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140 Improvements on SPS 6.2 Modifie
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144 Improvements on SPS Fig. 6.7 te
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146 Improvements on SPS Finally, bo
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148 Improvements on SPS To use the
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150 Improvements on SPS The improve
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152 Improvements on SPS On the whol
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154 Improvements on SPS In classica
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156 Improvements on SPS generally n
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160 Improvements on SPS uncertain,
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162 Improvements on SPS To obtain p
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164 Improvements on SPS Fig. 6.22:
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166 Improvements on SPS This proced
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168 Improvements on SPS 6.7.2 Long-
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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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