Application and Optimisation of the Spatial Phase Shifting ...

156 Improvements on SPS
generally necessary in speckle interferometry (nevertheless, it may sometimes be useful to inspect the
speckle phases per se).
In classical interferometry, the variations of the background intensity may reasonably be assumed to be so
low-frequent that the spectrum of the variations of I b is easily separated from the signal in frequency
space. In speckle interferometry however, the high frequencies in O(x,y) cause a significant deficiency of
the FTM: as is clearly seen from Fig. 6.17, O ~ ( ν x , ν y ) is not separated from the sidebands. The speckle
halo overlaps the sidebands at any practicable speckle size, so that a considerable noise background adds
to most of the signal's frequency content. This disturbs the phase reconstruction in a similar way as in the
phase-shifting investigations.
But as familiar as the problems are the ways to cope with them. From (6.19), it is clear that increasing B
will again help to suppress the speckle noise, provided r(x,y) has a narrow spectrum and can be eliminated
in the frequency plane. This can be fulfilled in an excellent way if a fibre is used to illuminate the sensor:
then r(x,y) will be a very broad Gaussian profile, and its spectrum a very narrow Gaussian that will not
overlap with the signal sidebands. It turns out that the performance of the FTM depends on B much in the
same way as for the phase-sampling methods. To quantify this, Fig. 6.1 also contains a plot of σ d as
calculated by (6.19) -(6.21) (black, circle symbols) from the same interferograms as used for the SPS
tests.
Furthermore, we note that the quantity O ~ ( ν x , ν y ) in (6.19) is directly accessible because of
~
O ( νx
, ν y ) = FT O ( x , y ) = FT o ( x , y ) o ( x , y ) , (6.22)
*
( ) ( )
so that we should be able to eliminate the speckle background from the phase calculation if we first record
the speckle pattern alone, calculate its spectrum and subtract it from (6.18). This correction for speckle
intensity is similar to that in 6.2.1, and the remaining phase errors are then mainly from electronic noise
and pixels with insufficient M I . The same would be possible for the reference wave if its spectrum would
overlap the signal spectra. This approach resembles the background subtraction suggested in [Liu97] for
classical interferometry.
Using the linearity of the Fourier transform, we could even subtract the speckle pattern in the space
domain (6.17) before switching to the frequency domain; but the benefit is easier to see in the frequency
representation. Fig. 6.18 provides an example of how the speckle noise is removed in the Fourier plane
when d s =2 d p . The aliased frequencies over ν N remain usable for the FTM by pasting them back to where
they got cut off [Bon86]. The reference wave need not be accounted for, since its spectrum is indeed
easily separated from the sidebands.