Application and Optimisation of the Spatial Phase Shifting ...

6.3 Modified phase shifting geometry 149
(bsc(ν x ,ν y )=45°), and of course, one point on these lines is ν x =ν y =ν 0 . But also for ν x =ν 0 and ν y =0, and
vice versa, it is easy to see that the phase-extraction formulae will operate correctly, although only onedimensionally
in either case. By the addition of phasors from both directions however, the interesting fact
results that bsc(ν x ,ν y ) has the correct value all along the black lines in Fig. 6.12; this means that
compositions of two "wrong" frequencies can still yield the correct phase. These lines are almost circles
for (6.11); and also (6.13) delivers a similar shape, but only within the range of the signal frequency
bands. We will not go into details as to the theoretical interpretation of these "circles of quadrature"; but
one could argue that the signal bands should be re-positioned to obtain signal frequencies wherever there
are black lines, which would maximise the fraction of signal frequencies yielding correct phases.
Unfortunately, this is not true: one must bear in mind that phase-extraction formulae have weak response
for low spatial frequencies, and none for zero frequency (cf. Chapter 3.2.2), so that signal energy would
be wasted if the sidebands were shifted to touch at ν x =ν y =0. An experimental test confirmed that this
strategy leads to slightly worse measurements than with the nominally correct value of (ν x , ν y ).
The white outlines show those areas for which bsc(ν x ,ν y ) stays within 10° deviation of its nominal
value; as discussed above in Chapter 3.2.2.3, this means that the p-v phase errors δϕ O are confined to 10°
within these regions. They are broadest in the vicinity of ν x =ν y =ν 0 , which, in analogy to Fig. 2.13, shows
that the phase calculation is more stable when the phasors S ~ ( νx , ν y ) and C ~ ( ν x , ν y ) are long, i.e. when
both ν x and ν y contribute to the phase determination.
The measured performance of (6.11) and (6.13) is summarised in Fig. 6.13, where the averaging of
horizontal and vertical phase calculations is abbreviated by "90°/90°". The graphs for the usual 3-sample
90° formula are repeated from Fig. 6.9 to simplify the comparison. The interferograms for the "90°/90°"
error evaluations have d s =3 d p as well.
0.12
σ d /λ
0.10
0.08
0.06
0.04
0.02
0.00
3-sample 90°, B=30
3-sample 90° with intensity correction, B=3
3-sample 90°/90°, B=30
3-sample 90°/90° with intensity correction, B=3
0 20 40 60 80 N x 100
Fig. 6.13: Overview of σ d from ESPI displacement measurements as a function of N x , obtained with merely
horizontal (triangles) and averaged horizontal/vertical phase determination (squares) from four series of
interferograms (two of them already used for Fig. 6.9), all with d s =3 d p but various phase shifts and B values.