Application and Optimisation of the Spatial Phase Shifting ...

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82 Electronic or Digital Speckle Pattern Interferometry α x / deg/column 120.2 120.0 119.8 119.6 -2.31 0.09 x /mm 2.49 4.89 -2.85 -0.05 y /mm 2.75 Fig. 3.25: Spatial distribution of α x (x,y) on the CCD sensor area for z =10.2 cm and ∆x =2.9 mm. Clearly, the continuous phase progression over the sensor leads to an integration over the pixels, so that this is an integrating-bucket method. When the phase runs along columns or rows only, the recorded I n are described by (3.59), since also the camera pixels are rectangular integration windows, only in space instead of time. The factors given in 3.3 for the decrease of M I remain valid in this case. If, however, the carrier fringes are slanted with respect to the Cartesian sensor axes, the situation is different: for instance, if ∆x=∆y, the slant is 45° and the function over which the phase progression is "windowed" becomes a triangle; for values below 45°, it acquires trapezoidal shape. Fortunately, the windows remain symmetrical in any case, from which it follows that the detected phase angles will remain correct [Wom84]. To determine the loss of M I due to a "composite" phase ramp (i.e. for phase shift in x and y direction), it is easiest to integrate over its components separately, which gives α x In = Ib + M I ⋅ 2 sin( ) ⋅ 2 sin( ) 2 2 ⋅ cos( ϕO + αn ) ; (3.67) α α x not surprisingly, this reflects the theoretical 2D-MTF for square pixels. For α x =α y , and hence a triangular envelope of the phase integration, the factor becomes 4 sin 2 (α x /2)/α 2 x and is indeed the transfer function of a triangle. We will be concerned with such a case in Chapter 6.3. The choice of the carrier frequency is influenced by contradictory requirements: on the one hand, it should be as high as possible to allow a broad range of signal frequencies to be measured. On the other hand, aliasing of too high frequencies must be avoided. In general, α x must have the same sign in the whole measuring field to keep the phase extraction unambiguous: a reversed, or aliased, phase shift leads to the wrong sign of the calculated phase, cf. 3.2.2. In classical interferometry, this means that closed interferometric fringes are not allowed, and the complete fringe pattern must be properly sampled; in speckle interferometry, the requirements are different and we will discuss them in 3.4.4. Since the I n are arranged as adjacent pixels on the sensor, it is clear that the speckles must be enlarged to obtain sufficient spatial correlation of speckle intensity and phase within the sampling pixel cluster, so that the modulation detected by a phase-extraction formula comes more from the phase shift than from α y y
3.4 Spatial phase shifting 83 crossing speckle "boundaries". The ideal case is sketched in Fig. 3.26 for d s =3 d p . Depending on the orientation and density of the carrier fringe pattern, various phase-extraction formulae can be applied. I 0 I 1 I 2 I 0 I 1 I 2 Fig. 3.26: Acquisition of three intensity samples I 0 , I 1 , I 2 for SPS. Small squares: sensor pixels, irregular outlines: speckles. Direction and spacing of the carrier fringes are indicated by the vertical black bars; left: α=90°/d p , right: α=120°/d p . The speckles, and also the sampling pixel cluster, should be as small as possible for the sake of spatial resolution; on the other hand, a somewhat larger pixel cluster can lead to more reliable phase measurements even when the speckle size is not increased. We will consider this point in detail in Chapter 6.2.2. In any case, the aperture must be smaller than in TPS; a first guess for the minimal speckle size would be d s =3 d p , because in (3.12), we need n3. So small an aperture entails some drawbacks: first, significantly less object light is available; and second, ∆x can usually not be chosen freely, since for R to reach the sensor, ∆x must not exceed D/2 (cf. Fig. 3.21). The latter problem can be solved with customised imaging optics: a narrow slit beside the diaphragm hole, allowing the focus of R to pass, will broaden the possible range of ∆x (cf. Fig. Fig. 5.1). Finally, when the test surface undergoes a tilt, the decorrelation of the speckle field proceeds faster with narrow than with wide apertures: that portion of the speckle field which is collected by the aperture is being panned "out of view" sooner when D is small. Once these problems are overcome, it becomes possible to study dynamic phenomena; using (double-)pulsed illumination, even very rapid transients can be frozen [Ped93, Ped94, Sched97, Ped97c, Pet98, Pet99]. Moreover, the decrease in spatial resolution is in practice more than offset by the low data storage requirements, since mostly the sawtooth images are smoothed anyway during data processing. 3.4.2 Evaluation of SPS interferograms The intensity samples for the phase calculation are picked from an interferogram in analogy to (3.12) which we rewrite as a spatial version (restricting ourselves to α=α(x), i.e. the phase changes from column to column of the image): In ( xk + n , y, t) = Ib ( xk + n , y) + M I ( xk + n , y) ⋅ cos( ϕO ( xk + n , y, t) + α n ( xk + n , y)) , (3.68) this is, to find the phase at a pixel in column k of the image, some neighbouring pixels are needed to provide the phase-shifted interference data. The equation system expressed by (3.68) then imposes 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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Page 33 and 34: 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 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: 3.4 Spatial phase shifting 81 3.4.1
Page 87 and 88: 3.4 Spatial phase shifting 85 Fig.
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Page 91 and 92: 3.4 Spatial phase shifting 89 infor
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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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Page 114 and 115: 112 Comparison of noise in phase ma
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132 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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