Application and Optimisation of the Spatial Phase Shifting ...

2.2 First-order speckle statistics 25
Apart from statistical considerations, a very simple explanation of the phenomenon is the phasor
interpretation suggested in [Bur98a, Leh98] and shown in Fig. 2.13.
A i
A i
A p (∆x, ∆y)
A p (∆x, ∆y)
A 2 (x 2 ,y 2 )
A 2 (x 2 ,y 2 )
A 1 (x 1 ,y 1 )
A 1 (x 1 ,y 1 )
ϕ 2 (x 2 ,y 2 )
ϕ 1 (x 1 ,y 1 )
A r
ϕ 2 (x 2 ,y 2 )
ϕ 1 (x 1 ,y 1 )
A r
Fig. 2.13: Variation of a speckle phasor A 1 due to a perturbation A p for different amplitudes A 1 (x 1, y 1 ) . ϕ 1 (x 1, y 1 )
and A p (∆x, ∆y) are the same in both cases.
If a phasor A 1 (x 1, y 1 ) undergoes a change A p (∆x, ∆y) while we move from (x 1, y 1 ) to (x 2, y 2 ) in the speckle
field, then the phase change will greatly depend on the length of A 1 (x 1, y 1 ). In the sketch to the left, the
phase ϕ changes considerably on the way from (x 1, y 1 ) to (x 2, y 2 ), since FA 1 (x 1, y 1 )F is relatively small. The
drawing to the right demonstrates the higher stability of brighter regions against changes: when FA 1 (x 2, y 2 )F
is large, the same A p (∆x, ∆y) leads to a distinctly smaller phase change. This is valid for all arguments of
A p except ϕ 1 . Unfortunately, this model is not suitable to understand the correlation of intensity and
intensity gradients. To conclude with, Fig. 2.14 gives an impression of the relation between intensities and
phases in the sample speckle field.
Fig. 2.14: Intensity (black/white) and phase (coloured isolines with 45° spacing) of a speckle field.