Application and Optimisation of the Spatial Phase Shifting ...

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94 Electronic or Digital Speckle Pattern Interferometry go back to a simple representation of the formula, which, as we have seen, will not affect its performance. For the case of d s =3 d p , it was indeed found that the best ν c,geom for, say, (3.19) was α75°/sample, and for, e.g., (3.17), the minimal error occurred around α100°/sample. But the difference in performance was only 2-3%; and since also these values may change with the sensor and electronics used, we do not further pursue this detail. 3.4.6 Distorted phase distributions due to miscalibrated phase shift None of the formulae investigated, including the error-compensating ones, will perfectly suppress the oscillating phase errors sketched in Fig. 3.14. This has important consequences for the statistics of the measured phases. In the presence of a systematic phase-shift deviation, the uniform distribution of the speckle phases (cf. (2.6)) will be modified by the faulty measurements, as pointed out in [Kad91]. The arrows in Fig. 3.14 indicate the direction to which the true ϕ O values are biased by measuring ϕ O +δϕ O under excessive phase shift: phase readings of 0
3.4 Spatial phase shifting 95 error"; in classical interferometry, it causes a fine ripple in the measured phase map [Cre96]. In speckle interferometry, where sawtooth images ∆ϕ(x,y) are generated by subtracting speckle phases ϕ O,i (x,y) and ϕ O,f (x,y), δϕ O alternately cancels and doubles, depending on ∆ϕ(x,y). Independent of the miscalibration's sign, δ(ϕ O ) =δ(ϕ O +π) –δ(ϕ O +π/2) (cf. Fig. 3.14); therefore δϕ O cancels when ∆ϕ(x,y) is 0 or π, and doubles in between. This is in contrast to (3.55), where δϕ O is averaged out by addition of two sampling sequences offset by π/2; the ∆ϕ(x,y) maps however are formed by subtraction. The effect of this can be seen in Fig. 3.39 for d s =3 d p : the phase-measurement error depends on the phase to be found. The error images on the right are the absolute differences between the actual phase maps and their least-squaresfitted noise-free counterparts (see Chapter 4.2). The inserted white error curves allow a quantitative comparison of δϕ O vs. ∆ϕ(x,y). δϕ Ο /° 70 0 π ∆ϕ(x) 2π 0 70 0 π ∆ϕ(x) 2π δϕ Ο /° 0 0 π ∆ϕ(x) 2π 0 π ∆ϕ(x) 2π Fig. 3.39: Fringe profiles (left) and corresponding δϕ O (right) from a displacement measurement with α=90°/column and various phase-extraction formulae: top, (3.19); bottom, (3.57). All images come from the same region of interest and have 256 64 pixels; the error curves on the right are vertically averaged. When the phase is calculated by (3.19) (upper row), a significant ripple is produced around ∆ϕ(x)=π/2 and ∆ϕ(x)=3π/2 that also leads to a higher average error. In the lower row, the error compensation of (3.57) does not remove the ripple completely, but is good enough to suppress it to approximately the level of the speckle noise: almost no high-frequency oscillations can be seen in the sawtooth image. The remaining profile of δϕ O is still periodical but the frequency is halved to 1/(2π). The importance of phase-shift deviations and their correction is clearly emphasised by Fig. 3.39. Moreover, we observe that in both of the error graphs, δϕ O is higher for ∆ϕ(x)=π than for ∆ϕ(x)=0. This reminds of correlation fringes, which also have low noise for ∆ϕ=0, and also the reason is similar: a phase difference of zero is always measured reliably because both interferograms simply look the same in this case. It is then unimportant what formula is used: the error minima are almost identical in both δϕ O images. At ∆ϕ =π, the carrier fringe pattern is "inverted" from the first to the second interferogram, so that phase-calculation errors with a period of 2π will change their sign and introduce the largest deviations upon subtraction of the speckle phase maps. It is now interesting to see what fringe profiles we can obtain with our 120° formulae. The error suppression by (3.58) is theoretically somewhat inferior to (3.57); the ratio of remaining errors of the
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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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Page 45 and 46: 2.3 Second-order speckle statistics
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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 and 60: 3.2 Phase-shifting ESPI 57 But reme
Page 61 and 62: 3.2 Phase-shifting ESPI 59 ϕ ϕ O,
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
Page 93 and 94: 3.4 Spatial phase shifting 91 frequ
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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
Page 144 and 145: 142 Improvements on SPS 0.12 σ d /
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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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158 Improvements on SPS 0.10 σ d /
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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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