Application and Optimisation of the Spatial Phase Shifting ...

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80 Electronic or Digital Speckle Pattern Interferometry and all have the same I b and M I , which is difficult to achieve [Kuj91a, Het00]. If parts of one and the same sensor are used for the sub-images, resolution is lost; if several full-chip images are taken, they will have to share the light energy available. High expense on the components, great adjustment effort and high sensitivity to misalignment are to be expected when working with set-ups of this type. The phase-ramping or bucket method of SPS works with one detector, on which a dense additional fringe pattern is generated to function as a so-called spatial phase bias or, in the Fourier terminology, carrier frequency. The – low-frequency – signal of interest distorts the carrier pattern and can be retrieved from it by a number of methods [Wom84]. This approach has first been implemented with vertical carrier fringes in [Ich72, Mer83] as analogue real-time processing of TV line signals. (Note here that only SPS lends itself to this technique: TPS requires digital processing since separate TV frames are involved.) The first studies were soon followed by digital implementations [Toy84, Toy86, Sho90, Fre90b,c, Küch90, Küch91, Kuj91b], allowing for arbitrary directions of the carrier fringes. Also, it was demonstrated in [Tak82] that the signal can conveniently be retrieved in the frequency plane by a Fourier-transform method; we defer details to Chapter 6.5. Other methods to retrieve phase from images with a spatial carrier are the phase-locked-loop method [Ser93] and the frequency demodulation technique [Ara96]. Later it was realised that this approach could be applied to speckle interferometry as well [Ste91, Wil91, Gut93]. A standard ESPI set-up is very easily changed to an SPS system; it is sufficient to laterally displace the focus, or source point, of the reference wave to introduce the fringe carrier. Fig. 3.23 shows the modification, with a magnified portion of a speckle interferogram: the fine fringes on the speckles are clearly discernible. z y PC / frame memory reference wave object illumination x ∆z x ∆x y image sensor camera objective aperture stop beam combiner object Fig. 3.23: ESPI set-up slightly modified (cf. standard configuration in Fig. 3.1) for spatial phase shifting. In the following subsections, we will go through some details pertaining especially to SPS to get an overview of the quality criteria for interferograms with a spatial carrier.
3.4 Spatial phase shifting 81 3.4.1 Geometrical description of spatial phase shift The lateral offset ∆x of the reference wave´s origin generates a quasi-linear geometric path and hence phase difference between the object wave O and the reference wave R over the sensor. Fig. 3.24 sketches the principle. sensor y ∆z ∆x source point of R source point of O ∆r y >λ λ 0 -λ ∆z r R ∆x x x r O Fig. 3.24: Left: incidence of two spherical waves with origins displaced by ∆x; right: construction of corresponding pathlength differences ∆r = r O – r R . While a phase shift in the sensor's y-direction may be added by a displacement ∆y, this case is still onedimensional in the appropriate co-ordinate system. Hence it is sufficient to consider the phase difference α(x), given by π ⎛ 2 2 2 x x α( x) = − = ⎛ ∆ ⎞ ⎜ x + y z x y z λ ⎝ ⎠ ⎟ + − ⎛ ⎜ ⎝ − ⎞ ⎞ 2 2 ∆ 2 2 r r ⎜ ⎟ + + O R + ∆ ∆ ⎝ 2 2 ⎠ ⎟ (3.65) ⎠ where x = 0 is defined to be the y axis in the middle between the waves´ source points. This is not generally the central sensor column: since the centre of the aperture should lie on the optical axis over the centre of the sensor, the object wave´s origin cannot be shifted from there. However, y = 0 does lie on the central row of the sensor. The corresponding phase gradient in x-direction, ⎛ ⎜ ∆ x π x αx x y 2 ⎜ + ( , ) = 2 ⎜ λ 2 ⎜ ⎛ ∆ x⎞ ⎜ ⎜ x + ⎟ + y ⎝ ⎝ 2 ⎠ 2 2 + ∆ z − ∆ x x − 2 2 ⎛ ∆ x⎞ ⎜ x − ⎟ + y + ∆ z ⎝ 2 ⎠ 2 2 ⎞ ⎟ ⎟ ⎟ , (3.66) ⎟ ⎟ ⎠ is quasi-constant when ∆z is much larger than everything else, which is quite reasonable to assume when using common imaging optics. Then ∆z will be on the cm scale, whilst the other quantities are on the mm scale. It turns out that the y co-ordinate also has a weak influence on α x ; hence the carrier fringes are not exactly straight. In fact, they have hyperbolic shape, which also follows from the definition of a hyperbola as the set of points for which Fr O F–Fr R F is constant. Fig. 3.25 depicts the situation for an average nominal phase gradient of α x (x,y) = 120° per sensor column and the optical configuration of Fig. 5.1. The spatial dimensions refer to sensor of the camera that was used throughout the work to follow, ADIMEC MX12P with 1024768 pixels of size (7.5 µm) 2 .
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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 31 and 32: 2.2 First-order speckle statistics
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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
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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: 3.4 Spatial phase shifting 79 which
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
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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Page 114 and 115: 112 Comparison of noise in phase ma
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130 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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