Application and Optimisation of the Spatial Phase Shifting ...

2.3 Second-order speckle statistics 45
The second case we will consider is [Don79]
( ϕ | , , ϕ )
p I I
2 1 2 1
( 1, I2, ϕ1,
ϕ2)
p( I , I , ϕ )
p I
= =
1 2 1
1
4π
2
exp
( z cosϑ)
I ( z)
0
with
z =
2 µ
I
A
I I
1 2
2
( 1 − µ A )
;
(2.60)
as already shown in (2.53) and (2.54), one could eliminate ϕ 1 simply by multiplying p(ϕ 2 FI 1 ,I 2 ,ϕ 1 ) with
2π, yielding p( FI 1 ,I 2 ). Like in 2.3.3.2, the symmetry in gives immediately
ϕ2 | I1, I 2,
ϕ1 = ϕ1
β > 0
= ϕ + π β < 0
. (2.61)
Therefore the variance again depends on |µ A | only, and we get [Don79]
1
σ
| I , I
ϑ 2 1 2
2
( − )
π 4 1
= + ⋅ ∑ I ( z)
3 I ( z)
2 n , (2.62)
n
0
∞
n=
1
n
where I n () are the modified Bessel functions of first kind and n th order. It is now instructive to compare
this function with (2.55) for various speckle intensities. Since the intensities appear together in z, we can
set I=I 1 =1 and vary only I 2 ; Fig. 2.29 covers the range of 0.1<
standard deviation rather than the variance.
I1I2 0, which means that in the very bright
speckles, the phase indeed shows good constancy. At I 1 I 2 =I, the standard deviation is still everywhere
below the "free" (i.e. unconditioned) value of Fig. 2.27. As I 1 I 2 decreases, σ¥|I 1 ,I 2 grows and eventually