Application and Optimisation of the Spatial Phase Shifting ...

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195 Appendix A: Counting events Intensity level crossings per unit length In Chapter 2 we have encountered two occasions where probabilistic events had to be counted. The derivation is similar for both of them. The level-crossing problem of (2.14) starts from the integral [Bar80] Nd ( It ) = ∫ δ ( I − It ) ∂ I / ∂ x dx ; (A.1) d here N d (I t ) is the number of times that the intensity crosses the value I t on a path d (the probability for the point I=I t being an extremum has measure zero on a straight line). The δ function assures that the integral responds only when I=I t ; to make each such contribution equal to one, i.e. to establish a counting function, the integration over x must be undone by the derivative ∂I/∂x; the modulus signs ensure that +1 is being counted for each event. However, since now I x appears, which is not independent of I, we need to know its expectation value at a given I, which requires the joint pdf p(I, I x ) and changes the integral to N ( I ) = δ ( I − I ) p( I, I ) I dI dx d t t x x x d Ix = ∫ ∫ ∫ ∫ d Ix p( I , I ) I dI dx t x x x on integrating over unit length, one obtains the density of the level-crossings, ( ) ρ( I ) = ∫ p I , I I dI , t t x x x I x ; (A.2) (A.3) which is (2.14). Intensity zero points per unit area By the same line of argument as above, we can start from [Ber78, Bar81] N ( A ) ( A ) ( A , A ) / ( x , y ) dxdy , (A.4) = ∫∫ δ δ ∂ ∂ disl r i r i S where the dislocation is expressed by the vanishing of A r and A i , the integral is over an area S and the quantity between the modulus signs is the Jacobian GJG=FA r,x A i,y – A r,y A i,x F. Obviously, we need p(A r , A i , A r,x , A i,x , A r,y, A i,y ) to evaluate this integral, or, more specifically, p(0 , 0 , A r,x , A i,x , A r,y , A i,y ) after the δ functions are accounted for. In analogy to above, we write N disl ( ) = ∫∫ ∫ ∫ ∫ ∫ p 0, 0, A r, x , A i, x , A r, y , A i, y ∂( Ar , Ai ) / ∂ ( x, y) dA r, xdA i, xdA r, ydA i, ydxdy , (A.5) S Ar , x Ai , x Ar , y Ai , x
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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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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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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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142 Improvements on SPS 0.12 σ d /
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Page 148 and 149: 146 Improvements on SPS Finally, bo
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Page 160 and 161: 158 Improvements on SPS 0.10 σ d /
Page 162 and 163: 160 Improvements on SPS uncertain,
Page 164 and 165: 162 Improvements on SPS To obtain p
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Page 168 and 169: 166 Improvements on SPS This proced
Page 170 and 171: 168 Improvements on SPS 6.7.2 Long-
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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
Page 184 and 185: 182 References [Ett97] [Fac93] [Far
Page 186 and 187: 184 References [Har94] [Her96] [Het
Page 188 and 189: 186 References [Kuj89] [Kuj91a] [Ku
Page 190 and 191: 188 References [McLa86] [Mer83] J.
Page 192 and 193: 190 References [Rav99] I. Raveh, E.
Page 194 and 195: 192 References [Sur98b] [Sur98c] [S
Page 198 and 199: 196 Appendix A: Counting events and
Page 200 and 201: 198 Appendix B: Real-time phase cal
Page 202 and 203: 200 Appendix C: Derivation of inten
Page 204 and 205: I 1 a r g ( S ~ ( ) ) 202 Appendix
Page 206 and 207: 204 Appendix D: Alternative error-c
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Page 212 and 213: 210 The author List of publications
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