Application and Optimisation of the Spatial Phase Shifting ...

2.3 Second-order speckle statistics 41
2.3.3.2 Phase statistics
To obtain p(ϕ 1 ,ϕ 2 ), we have to integrate (2.49) over I 1 and I 2 , which is rather complicated, but has
fortunately been taken care of before [Mid60, Goo75]; the result is
1 µ A 2
p( ϕ1 ϕ2) ( 1 )
2
3/
2
β β β β π 2
, = − − ⎛
⎞
− arcsin + + 1 − β
2
⎜
⎟ , (2.53)
4π
⎝
2 ⎠
where β=|µ Α |cos(ϕ 1 –ϕ 2 +ψ) and we deal with ψ as above. To look at the quantity of interest, namely the
phase at (x 2 ,y 2 ) in relation to that at (x 1 ,y 1 ), we can content ourselves with fixing ϕ 1 to some arbitrary
value and varying ϕ 2 from –π to π. For convenience, we introduce the relative phase variable ϕ 1 -ϕ 2 +ψ
and consider p( ), which yields one plot for all ϕ 1 . Repeating the procedure with swapped angles ϕ 1 and
ϕ 2 would teach us nothing new, as β is symmetrical in ϕ 1 and ϕ 2 . The resulting probability distribution of
vs. Fµ Α F is shown in Fig. 2.26.
0.3
0.2
p (¡)
0.1
0
0.5
-3.14
0.25
-1.57
¡
0
¢
0
1.57
¢µ A
3.14
Fig. 2.26: Pseudo-3D plot of p(£) for 0¤¥µ Α ¥