Application and Optimisation of the Spatial Phase Shifting ...

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10 Statistical Properties of Speckle Patterns 2.2 First-order speckle statistics To simplify and generalise the treatment of first-order speckle properties, we will first derive a joint probability density function containing all the quantities of interest and then eliminate whatever we want by integrating it. 2.2.1 Basic probability-density function A light-scattering rough object can be regarded as an array of individual, mutually uncorrelated microscopic scatterers, each of which sends an elementary wave into space. The coherent superposition of all these contributions at a certain spatial point determines the speckle intensity and phase at that point. The treatment is also valid for imaging geometries, provided that several point images of scatterers overlap at each point of the image plane. This condition is also referred to as unresolved microstructure. The elementary waves are most conveniently regarded as vectors A in the complex plane, with the squared modulus |A| 2 corresponding to their intensity I and the argument giving the phase ϕ. This representation is well known and very useful; the complex vectors A are usually named phasors, or complex amplitudes. Assumed that (i) the waves´ amplitudes |A| are independent of their phases, (ii) the phases are uniformly distributed over [–π,π), and (iii) the number N of scatterers is large enough (which begins to hold from N 50 on), the summation of the contributions may be visualised as a random walk in the complex plane, putting N phasors together in the manner of vector addition. Then we may use the central limit theorem [Pap65, p. 266] to treat the speckle pattern formation as an asymptotically Gaussian process. If we denote the real and imaginary parts of the phasors by A r and A i , the two are called jointly Gaussian variables [Goo75]. This assumption has been experimentally confirmed in [Mol90b]. As derived in [Pap65, pp. 253 and 475], also the spatial derivatives A r,x , A i,x , A r,y and A i,y ,where A r,x ∂A r /∂x etc., are jointly Gaussian with A r and A i . Assumed the standard deviation of A r and A i is σ, we can establish ( r , i , r, x , i, x , r, y , i, y ) p A A A A A A = 2 2 1 ⎛ A + A ⎞ A A A A exp ⎜ − ⎟ exp ⎝ ⎠ ⋅ 1 ⎛ C − + + + 2 2 2 2 2σ π 2σ 4 ⎜ 0 π ⎝ 2C0 2 2 2 2 r i r, x i, x r, y i, y as detailed in [Och83, Fre95c, Leh98], with denoting the ensemble average; 2σ 2 is indeed the average speckle intensity I as will be derived in (2.7). C 0 depends on the shape and size of the scattering spot or aperture. It is essentially the curvature of the spatial amplitude autocorrelation function at its peak, ∂ 2 (R AA* (∆x,∆y)| ∆x=0, ∆y=0 )/∂(∆x) 2 = ∂ 2 (R AA* (∆x,∆y)| ∆x=0, ∆y=0 )/∂(∆y) 2 , provided the source is symmetrical. For circular scattering spots, we get [Fre95c, Leh98] ⎞ ⎟ ⎠ (2.1)
2.2 First-order speckle statistics 11 C 0 = I π 8 2 2 , (2.2) ⎛ D ⎞ ⎜ ⎟ ⎝ λ z⎠ with D being the diameter of the spot, λ the wavelength and z the distance between scattering plane and point of observation. C 0 is inversely proportional to the square of the speckle size and scales the gradients A r,x etc. For our example image, we find C 0 0.152 grey levels/d 2 p . In order to come from the amplitude description to intensities and phases, we convert the variables to polar co-ordinates: A = I cosϕ A = I sinϕ A A r I x I x = cosϕ − ϕx I sinϕ Ai x = sinϕ + ϕx 2 I 2 I I cosϕ I y I y = cosϕ − ϕy I sinϕ Ai y = sinϕ + ϕy 2 I 2 I I cosϕ r, x , r, y , with the Jacobian GJG=1/8; the procedure is described in more detail in [Och83]. We arrive at ( , ϕ, x , ϕx , y , ϕy ) p I I I = i ( ϕ 2 + ϕ 2) ⎛ I ⎞ ⎛ I 2 + I 2 ⎞ ⎛ 1 1 I exp⎜ − ⎟ ⋅ exp − exp ⎜ I π ⎝ I ⎠ C π ⎜ ⎝ IC ⎟ − 2 16 8 ⎜ ⎠ 2C 0 2 2 0 ⎝ x y x y 0 ⎞ ⎟ ⎟ ⎠ (2.3) (2.4) with 0I
Page 1 and 2: Application and Optimisation of the
Page 3 and 4: Table of Contents 1 1 INTRODUCTION
Page 5 and 6: 3 1 Introduction The present era of
Page 7 and 8: Introduction 5 retrieval can help t
Page 9 and 10: 7 2 Statistical Properties of Speck
Page 11: 2.1 Experimental set-up 9 Gaussian
Page 15 and 16: 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,
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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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3.4 Spatial phase shifting 83 cross
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3.4 Spatial phase shifting 85 Fig.
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3.4 Spatial phase shifting 87 arran
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3.4 Spatial phase shifting 89 infor
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3.4 Spatial phase shifting 91 frequ
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3.4 Spatial phase shifting 93 altho
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3.4 Spatial phase shifting 95 error
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3.4 Spatial phase shifting 97 Since
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100 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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