Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 21
positive correlation of I and I x and the anticorrelation of I and ϕ x , an anticorrelation of I x and ϕ x results
that is impressively illustrated by the figure: the worm-like structures of high Fϕ x F circumscribe the bright
speckles (being regions of high FI x F) almost exactly. The white boxes assist in finding examples. The
pinched maxima of Fϕ x F indicate phase singularities (see Fig. 2.17).
Fig. 2.9: Maps of I x (left) and ϕ x (right). White boxes allow comparison of details.
2.2.4 Gradients in two dimensions
The previous treatment, although particularly relevant for our subject of spatial phase measurement, does
not provide a complete insight into the structure of speckle intensity and phase. Therefore we consider
also the two-dimensional gradients,
⎛ I ⎞
∇ I = I
2
+ I
2
y
x y with θI
= arctan⎜
⎟
⎝ Ix
⎠
. (2.27)
⎛ ϕy
⎞
∇ ϕ = ϕ
2
x + ϕ
2
y with θϕ
= arctan⎜
⎟
⎝ ϕ ⎠
The pdf's in terms of F∇IF, F∇ϕF are easily obtained from functions involving I x , I y , ϕ x , ϕ y by integrating
2 2
x y
over θ I and/or θ ϕ on the circles given by I + I and/or ϕ + ϕ , which gives factors of 2πF∇IF
and 2πF∇ϕF, respectively. This changes (2.8) to
from which we can derive
x
2 2
x y
I I
I I
p( I , ∇I
, ∇ ) = exp ⎛ ⎞
⎜−
⎟ ∇ ∇ ϕ ⎛
exp − ∇ 2
1
⎞ ⎛
exp⎜
I ⎝ I ⎠ C
⎜
⎝ IC ⎟ − ∇ ϕ
ϕ
4 ⎠ ⎜
0 2 8 0 ⎝
2C0
2
⎞
⎟ , (2.28)
⎟
⎠