Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 27
As we can see from (2.1), A r (x, y) and A i (x, y) are statistically independent; hence they will independently
fluctuate with a mean value of zero in the speckle field. The zero crossings of either function form closed
contours in the (x, y)-plane; and frequently these lines intersect. In Fig. 2.15, they do so at a right angle,
which is a special case. On moving along the A r (x, y)=0 line in positive y-direction, the phase of the
wavefield remains constant until A i (x, y) vanishes at the singularity and then flips sign, which results in a
phase jump of π. The new phase value also remains constant as we move away from the minimum. Since
A r (x, y) and A i (x, y) can be approximated by planes, the intensity has a quadratic minimum. It has been
shown in [Fre96b] that these "intensity wells" are very narrow: their typical diameter is only 1/7 that of
the speckles. The model singularity shown here is, by definition, positive and of order +1: during a
counterclockwise loop around it, the phase increases by +12π. This non-vanishing rotation of the phase
has led to the term "vortices". If the zero points of A r (x, y) and A i (x, y) are saddle points or extrema, a
dislocation of order N, i.e. with a phase progression of N2π per revolution, can occur [Fre99a,b]; but
these are very unstable [Fre00] and of no practical importance in speckle patterns.
The correspondence of phase dislocations and vanishing field amplitude is indicated in Fig. 2.16. Since
the speckle field is not completely polarised, the dislocations do not always coincide with points of zero
speckle intensity, but they certainly appear at the zeros of interferometric modulation, as the
interferometric phase measurement extracts that state of polarisation from the speckle image which is copolarised
with the reference wave. For this reason, Fig. 2.16 uses the map of modulation rather than the
speckle intensities as the underlying field. As to be seen by comparison with Fig. 2.2, it resembles the
total speckle intensity closely but not exactly. The signs of the dislocations are not indicated here; see Fig.
2.17 for this purpose.
Fig. 2.16: Distribution of phase dislocations (white dots) vs. interferometric modulation of Fig. 2.2, right side.
Phase dislocations of order 1 are topological features in the speckle field [Nye74]; they always appear
and vanish in pairs of opposite sign [Fre93]. In analogy to the intensity map, one can also define a
normalised vector field ∇ϕ /F∇ϕF to find phase dislocations as well as phase minima, maxima and saddle