Application and Optimisation of the Spatial Phase Shifting ...

2.3 Second-order speckle statistics 33
2.3 Second-order speckle statistics
In SPS, and in TPS with unresolved speckles, it occurs that the distances over which the spatial structure
of the speckle field changes are not much larger – or even very much smaller – than the pixel size. Then
one needs to know the spatial relation of speckle intensity and phase between two points P 1 =(x 1 , y 1 ) and
P 2 =(x 2 , y 2 ) in the speckle field, p(I 1 , I 2 , ϕ 1 , ϕ 2 ), or simplifications thereof. We will proceed from the most
general concept, the spatial autocorrelation of intensity and phase, to the somewhat more complicated
topic of the relation between intensity and phase.
2.3.1 Intensity autocorrelation
Probably the most popular and indeed very useful second-order quantity is the concept of the mean
speckle size in terms of intensity. We start with the autocorrelation of the complex amplitude,
R * ( x , y , x , y ) = A( x , y ) A ( x , y ) , (2.39)
AA
*
1 1 2 2 1 1 2 2
which is also referred to as mutual intensity of the speckle field [Goo75, p. 36]. For our purposes, it may
suffice to remember that this function is essentially the Fourier transform of the intensity distribution
within the scattering spot or the aperture shape, depending on whether objective or subjective speckles are
concerned. For the latter case however, the treatment is correct only if the imaging aperture contains a
large number of speckles. Then the aperture may be thought of as another rough surface, whose shape
plays the same role for the formation of subjective speckles as does the scattering spot in the case of
objective speckles.
The mutual intensity is usually normalised to yield the complex coherence factor
µ A ( x , y , x , y )
1 1 2 2
R * ( x , y , x , y )
AA
1 1 2 2
R * ( x , y , x , y ) R * ( x , y , x , y )
AA
1 1 1 1 AA 2 2 2 2
, (2.40)
which is unity for x 1 =x 2 and y 1 =y 2 and decays as the points move away from each other; when it becomes
zero, the points are said to be one spatial correlation length or speckle size apart.
It can be shown [Goo75, pp. 36-38] that the intensity autocorrelation R I (x 1 , y 1 , x 2 , y 2 ) is given by
( )
RI
( ∆x, ∆y) = I
⎛⎜
⎝
1 + µ A ∆x,
∆y
2 2
⎞
⎠
⎟
(2.41)
with ∆x = x 2 –x 1 and ∆y = y 2 –y 1 . That is, the shape of the µ A curve determines that of a typical speckle area,
or correlation cell, in the speckle field. If the scatterer or aperture is a uniformly bright circle, we get
2J
( α)
D
µ A( ∆x,
∆y)
= 1
with α = π ∆x
2
+ ∆y
2
,
α
λ z
(2.42)
J 1 denoting the first-kind Bessel function of first order. This can very easily be generalised to the elliptical
apertures that are also used in the experimental work. For circular apertures, µ A is the well-known Airy