Application and Optimisation of the Spatial Phase Shifting ...

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134 Improvements on SPS intensity on adjacent pixels (x,y) and (x1,y), σ O0 ,O 1 , and of that of the imaging system's electronic noise, σ e . * Hence we rewrite (6.1) as σ σ 2 O O + σ 2 0, ± 1 e ϕ O ≅ 2 ⋅ O ⋅ R ⋅ 8 3 . (6.2) In a speckle field, σ O0,O 1 depends on the degree of spatial coherence [Goo75], µ A (x 0 , x 1 ), of the points (k,l) and (k1,l). For a circular aperture and d s = 3d p , we find µ A (x 0 , x 1 ) 0.81. Moreover, σ O0,O 1 is conditioned on O 0 , which relationship is analytically known [Don79]. We can generalise (2.52) to read ( O ) 2 2 0 ± 1 2 2 2 2 σ O , O = ( 1− µ A ) + 2 O µ A ( 1− µ A ) , (6.3) and inserting (6.3) into (6.2), we can calculate σϕ O , which is the same for both object states: σϕ O =σϕ O,i =σϕ O,f . For the phase difference ∆ϕ =ϕ O,f–ϕ O,i we therefore get σ ∆ ϕ = L2σϕ O and from this the corresponding quantity for the displacement, σ d , as a function of the beam ratio B = R/〈O〉. These data can be compared with the experimental results. Fig. 6.1 shows the performance of various evaluation methods for sawtooth images with N x =10, N y =0, d s =3 d p and α x =120°/column from an out-of-plane configuration with SPS; for TPS, d s was set to 1 d p . Curves in Fig. 6.1 that are not addressed here will be discussed later on. The theoretical curve of σ d vs. B for SPS is the bold white line and matches the measured data reasonably if we shift it vertically by adding a constant displacement deviation of σ d 0 = 0.05 λ. This is not an arbitrary adjustment of data: since (6.2) does not account for spatial fluctuations of the phase ϕ O between adjacent pixels, the predicted values of σ d will be too small. Of course, adding a constant σ d 0 relies on the simple assumption that the influence of speckle phase gradients on σ d does not depend on B. From the figure we see that TPS works well from B1 on, and σ d only starts to increase from B100 on, where O is already weaker than the electronic noise. The quasi-constancy of σ d vs. B in TPS has also been reported in [Hun97] for a beam ratio between 0.1B10. For SPS, σ d first decreases as the reference wave gets stronger, and has its minimum around 30. With fading M I , the influence of electronic noise grows and so does σ d . This behaviour agrees reasonably with our theoretical prediction. Hence, in SPS a proper choice of the beam ratio is far more important than it is in TPS. Fortunately the best SPS results turn up in a region of high beam ratio, which alleviates the problem of poor light efficiency somewhat. Based on these results, for most of the investigations in Chapter 5 B was set to 10, at which setting both SPS and TPS operate with near-optimum performance. * With the imaging system used, a realistic value for σ n was ¡ 2.5 grey levels; this corresponds to a resolution of only 6-7 true bits. With optimum intensity resolution, the usable beam ratios would have been even higher.
6.1 Optimisation of experimental parameters 135 0.12 σ d /λ 0.10 0.08 SPS, d s =3 d p SPS with intensity correction SPS with modified intensity correction SPS (theory) FTM FTM with speckle subtraction TPS, d s = d p 0.06 0.04 0.02 0.00 1 10 100 B 1000 Fig. 6.1: σ d for various ESPI measurements of out-of-plane displacements by SPS and TPS as a function of B. All measurements were done with N x =10 and N y =0. Besides the variation of the beam ratio, there is another possibility of reducing σ d : the individual speckle intensities can be accounted for in a modified phase calculation formula. This approach is described in detail in 6.2.1, where also the curves for "SPS with (modified) intensity correction" in Fig. 6.1 will be explained. For a discussion of the Fourier transform method (FTM), see 6.5. 6.1.2 Phase shift In Chapter 3.2.2 we have considered the spectral transfer properties of phase-shifting formulae and discussed some points that are relevant for their application to signals with a broad spectrum. In Chapter 5.2.1, we collected some preliminary evidence that α x =90°/sample should be the better choice. Since it is now our aim to get the best possible performance from SPS, we investigate this issue in more detail by experiment and carry out the same kind of comparison that we did for SPS and TPS. The results are shown in Fig. 6.2, where the left part is the same plot as Fig. 5.5.
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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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Page 102 and 103: 100 Quantification of displacement-
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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
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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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