Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 29
Fig. 2.17: Left: phase distribution of sample speckle field; [–π,π) represented as grey shades from black to white.
Right: zero crossings of A r (x, y), black lines, and of A i (x, y), white lines. Red dots: positive, green dots:
negative singularities.
The figure shows that the zero-crossings of A r (x, y) and A i (x, y) intersect at all angles between 0 and 90°
[Fre94a], and also explains easily why dislocations always appear and vanish pairwise: it is impossible for
the closed zero contours of A r (x, y) and A i (x, y) to generate only one new intersection. This is also the
reason why they alternate in sign – also called topological charge – on paths along any zero-crossing
contour [Shva94, Fre94b, Fre95d]. When the zero-crossings of A r (x, y) and A i (x, y) touch, they do so
tangentially and generate a zero-amplitude line, or "edge" dislocation [Nye74, Bas95], of infinitesimal
length in the x-y plane, that instantly splits up into the two "screw" dislocations as the zero crossings of
A r (x, y) and A i (x, y) intersect, i.e. as we shift our x-y-plane in z direction and the wavefield evolves in
space. The trajectories of the singularities can be thought of as dark lines that pierce the x-y-plane and are
orientated mostly in z direction [Ber78]. Their shape in space has been referred to as "snake-like" [Bar83];
the process of pair creation or annihilation therefore corresponds to turning points of these trajectories
where the z component of their direction vector changes sign.
The abovementioned z-direction scan of the speckle field gives us the opportunity to track the loci of the
dislocations slice by slice to see whether a pair of dislocations that has appeared together will also vanish
together, and how one should imagine the zero-intensity trajectories in space. Fig. 2.18 presents the zerointensity
lines in the very centre of the sample phase field; the colouring helps to distinguish them. If they
end, it means that they have moved out of the sample volume or that their tracking is discontinued for
clarity of the representation.