Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 23
calculation formula. However, it has been shown that the integration over the pixels' finite apertures can
alleviate the problem somewhat [Bar91].
Considering the two-dimensional phase gradients, we derive from (2.28)
which is plotted in Fig. 2.11.
1 I I I
p( I , ∇ ) = exp ⎛ ⎞ ∇ϕ
⎜−
⎟ exp
⎛− ⎜ ∇ ϕ
ϕ
I ⎝ I ⎠ C0
⎜
⎝
2C0
2
⎞
⎟ , (2.31)
⎟
⎠
2.5
¡I¢C 0 £ p(I , ¤∇ϕ ¤)
0.3
0.2
1.25
0.1
∇ϕ /2C 0
0
I / ¢ ¡I
3
2
1
0
0
Fig. 2.11: Pseudo-3D plot of p(I, ¥∇ϕ ¥).
In this figure, the stationary points of the phase (extrema and saddle points, for which F∇ϕF= 0) lie on the
I axis and the zero-intensity minima on the F∇ϕF axis. They are both existent but of measure zero, again
in qualitative difference to the one-dimensional case. The phase gradient alone obeys
p
( ϕ )
∇ = 4
I C
0
∇ϕ
( 2C0
+ I ∇ϕ
)
π C
with ∇ ϕ = ,
2 I
2 2 0
(2.32)
which results in F∇ϕF 172° per speckle size. But like Fig. 2.8, Fig. 2.11 clearly reveals
anticorrelation between intensity and phase gradient, so that we can expect F∇ϕF to fall below F∇ϕF in
the brighter regions of the field. This is demonstrated in [Shva95]: bright speckles tend to lie close to, but
not exactly over, the stationary points of phase; the phase is found to vary by typically 45-90° over the
half width of a speckle, with F∇ϕF I max 49°/d s at the intensity maxima. Most of the stationary points of
phase are saddles; phase extrema contribute only 1/15. This distinct qualitative difference between
phase and intensity field will be briefly interpreted in 2.2.5.