Application and Optimisation of the Spatial Phase Shifting ...

88 Electronic or Digital Speckle Pattern Interferometry
3.4.5 Spectral side-effects of spatial phase calculation
As mentioned above, the findings of 3.2.2 require some experimental inspection because they were
derived, so to speak, in the absence of speckle. In particular, the spatial phase calculation in SPS is
influenced by the spatial correlation of the pixels selected for processing, as demonstrated in Fig. 3.30 for
intensity sampling by (3.18) and (3.19), respectively.
S'(x k ) S'(x k )
I 0 I 1 I 2
I 0 I 1 I 2
C'(x k ) C'(x k )
Fig. 3.30: Processing of intensity samples by (3.18) (left) and (3.19) (right), where the smaller outlines in white
and black indicate the smaller coherence areas required for S'(x k ) and C'(x k ) alone.
In (3.18), the first and last intensity sample are used for S'(x k ) and all three samples for C'(x k ); the terms
are balanced with respect to the central pixel, being the target pixel of both calculations. In (3.19), both
S'(x k ) and C'(x k ) are constructed from only two consecutive samples; hence they make lower demands on
the spatial coherence of the pixels. Of course, the complete sampling window is still three pixels wide,
and S'(x k ) and C'(x k ) are associated with slightly different portions of the speckle field, so that their spatial
correlation may suffer. The same line of argument applies to all other phase-shifting formulae, where
different a n and/or b n are large, small, or vanish, in different representations of the formulae.
To find out the significance of this consideration, we study S ~ ( νx , ν y ) and C ~ ( ν x , ν y ) experimentally.
First, we generate two separate arrays I(x,y)S x (n) and I(x,y)C x (n), this is, we use (3.68) to process 2-D
images with a 1-D phase shift. The results, when visualised as images, should yield two fringe patterns
that look very much like the speckle interferogram, but have a phase lag of 90° and hence deserve the
names of "sine" and "cosine" image. This processing method has been used in [Sin94] in the context of
phase demodulation.
The power spectra of the "sine" and "cosine" images, ~ ( , ) ~ 2 ~ ~
2
I ν ν ⋅ S ( ν ) and I ( ν , ν ) ⋅ C(
ν ) , can be
x y x
x y x
compared with that of the original interferogram, I ~ ( νx , νy
)
2 , to reveal the changes * . This yields
* Realising that the phase-shift is one-dimensional, it would suffice to investigate the ν x only; but since we will be concerned
with full 2-D information in Chapter 6.3, we include the ν y here already, bearing in mind that they contribute little information
now.