Application and Optimisation of the Spatial Phase Shifting ...

3.4 Spatial phase shifting 87
arrange the signal bands advantageously in the spatial frequency plane. An example of how to obtain very
large, "clean" (i.e. non-overlapping) sidebands will be given in Chapter 6.5.
The speckle size is determined from power spectra with a spatial phase shift by
122 . 0.
61
d s = =
ν+ − ν- νc − ν , (3.71)
-
where Fν + F is the largest and Fν - F the smallest spatial frequency of a sideband and Fν c F=(Fν + F+Fν - F)/2. This
of course needs to be modified when Fν + F>Fν N F: due to aliasing, (ν N +ν a ), where subscript a denotes the
aliased contributions above ν N , will appear in the Fourier plane at (ν N –ν a ). To find the minimum
permissible speckle size when Fν c Fis given and no aliasing is to occur, we find
0.
61
νN ≤ νc
+ ; (3.72)
considering the examples of Fig. 3.29, we have ν c =1/(3 d p ) for α x =120°/column; therefore,
0.61/d s 1/(6 d p ), which gives the condition that d s 3.66 d p . Similarly, for ν c =1/(2 d p ), d s 2.44 d p .
For real sensors, the merely geometrical notion of ν c is a more or less accurate approximation: the higher
spatial frequencies will usually be attenuated by the falling pixel MTF and the read-out electronics. This is
not visible in Fig. 3.29 due to the logarithmic display; examples may be found in Fig. 3.31 and Fig. 3.34.
This "low-pass" behaviour shifts the actual ν c , or the "centre of gravity" of the sidebands' detected
power, below their geometrical centre, ν c,geom . This raises the question whether an advantage can be
gained by calibrating the phase shift on ν c , which minimises the actual phase-shift deviations. However
we retain the geometrical definition for three reasons: (i) With respect to the high spatial frequencies, it is
indispensable to operate the camera with its pixel clock activated. Unfortunately, this damps ν x much
more strongly than ν y (for the camera used, the pixels are read out in x direction at a rate of 20 MHz as
independent video lines, whose frequency is only 15.625 kHz), which would greatly complicate the
treatment of composite x-y-phase shifts if we used ν c,x and ν c,y for calibration. (ii) Shifting a sideband
outward, until the measured ν c,x reaches its nominal value, is a waste of signal energy, because more
and more of the sideband then comes to lie in the low-MTF regions of the frequency plane. (iii) The
problem affects the methods for ν c,x =1/(3 d p ) more than those with ν c,x =1/(4 d p ); but we have seen from
Fig. 3.13 and Fig. 3.18 that phase-shifting errors are less severe for ν x