Application and Optimisation of the Spatial Phase Shifting ...

2.3 Second-order speckle statistics 39
p (I 1 ,I 2 )
0.4
0.3
0.2
0
0.1
0.25
0.5
µ A 0.75
0.95
1
I 2 /I 1
2
0
3
0
Fig. 2.23: Pseudo-3D plot of p(I 1 ,I 2 ).
To find out the influence of a fixed I 1 , we write down the pdf of I 2 conditioned on I 1 , which is
( | I )
p I
2 1
( , I )
p I1 2
= =
p I
1
⎛
⎜
µ
2
A I1 + I2
exp
I
( ) 2
⎜−
2
⎟ 0⎜
2
1 I ( 1−
µ A ) ⎜
⎝
I ( 1−
µ A ) ⎟ ⎜
⎠ ⎝
I ( 1−
µ A )
⎞
⎟
⎛
⎜
2
I I
1 2
µ
A
⎞
⎟
⎟ . (2.51)
⎟
⎠
As to be seen, the coupling between I 2 and I 1 depends on Fµ Α F and I 1 . To understand the role of I 1 , we
visualise three cases with Fµ Α F=0.1, 0.6, and 0.95, respectively.
0
0.5
0
1
2
0
I 3 1
0
1
3
2 I 2
1
p (I 2 |I 1 )
p (I 2 |I 1 )
1.5
1
0.5
0
1
2
0
I 3 1
0
1
3
2 I 2
1
2
10
p (I 2 |I 1 )
0
3 1
0
3
2
I 1 I 2
5
Fig. 2.24: Pseudo-3D plots of p(I 2 |I 1 ) for ¡I¢=1 and µ Α =0.1 (left), 0.6 (centre), and 0.95 (right).
While it is not surprising that I 2 almost remains a negative exponential when Fµ Α F is small, we find an
interesting behaviour for intermediate values of Fµ Α F. When I 1 is small, the distribution of I 2 is only
slightly altered, which means that the dark portions of the speckle field are narrow structures: their
influence does not reach very far. Then, at large I 1 , the maximal probability of I 2 reluctantly moves away
from zero, but remains quite low. This means that bright spots do cause their surroundings to get brighter,
but that the latter will nonetheless be considerably darker than the bright spots themselves, in agreement
with the positive correlation of the intensity and its gradient that we found in 2.2.3.1. The last example