Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 17
We conclude this subsection with another interesting and comparatively easy interpretation of speckle
intensity maps, namely as smooth 2-D surfaces or landscapes. Hence, considerations about the laws of
"twinkling" of a sunlit sea surface [Lon60] are indeed applicable to the spatial intensity structure of a
speckle pattern. This allows one to establish for the relative numbers of speckle (zero and non-zero)
intensity minima, N min , maxima, N max , and saddle points, N sad , respectively [Lon60]:
N min + N max = N sad . (2.16)
More recently, this has been re-derived with the concept of singularities of the normalised vector field
∇I/F∇IF, in which minima, maxima and saddle points appear as topological singularities [Fre95b], and the
evolution rule for speckle fields has been formulated that a new extremum must always appear, or vanish,
together with a saddle point. It has further been found that N min :N max =3:2, this is, we encounter more
minima than maxima in a speckle pattern [Wei82a,b]; the typical spatial arrangement is that of chains of
alternating minima and saddles in the dark valleys between the bright spots (cf. Fig. 2.2). For a circular
aperture, the statistical densities of the intensity features have been determined by computer simulation as
[Fre95b]
ρ( I ) ≅ 0. 46/
A
zero
ρ( I ) ≅ 013 . / A
min
ρ( I ) ≅ 0. 39 / A
max
s
sad s ,
ρ( I ) ≅ 0. 98 / A
s
s
(2.17)
A s being the speckle area defined in (2.36), and ρ(Ι zero ) denoting the density of zero-intensity minima, to
be further investigated in 2.2.5. Thus, the rule (2.16) is confirmed, and in total we have almost two of
these "critical points" of intensity per speckle area. The density of parameters necessary to describe all the
features in (2.17) is almost 6 times the sampling density required to properly resolve the speckle field; this
means that the features cannot really be statistically independent and hence must be more or less
correlated [Fre95b, Fre98a].
It is now interesting to learn at what intensity levels these features occur most frequently; in this respect
the values
I
I
I
max
min
sad
≅ 2.
5 I
≅ 0.
07 I
≅ 05 . I
(2.18)
are given in [Fre96b]; the separate class of zero-intensity minima is here excluded from I min . The most
frequent peak-intensity level (at the centres of the bright speckles) is 1.8I, which supports (2.13):
most of the bright spots stand out strongly and are necessarily associated with large intensity slopes. This
can also be seen in Fig. 2.4.
There are other structural correlations, non-obvious orders and quasi-lattices [Fre95b, Fre95c, Fre97b,
Fre98a] in speckle patterns, again too numerous to describe here; but there should now be no doubt that a