Application and Optimisation of the Spatial Phase Shifting ...

136 Improvements on SPS
0.14
σ d /λ
0.12
14
12
0.10
10
0.08
08
0.06
06
0.04
0.02
0.00
x
0 5 10 15
20 30 40 50
02
60 70 90 100
N
00
0 2 4 6 8 d s /d p 10
04
N x
0 5 10 15
20 30 40 50
60 70 90 100
0 2 4 6 8 d s /d p 10
Fig. 6.2: σ d for ESPI displacement measurements by SPS with α x =120°/column (left) and α x =90°/column (right)
as a function of speckle size for out-of-plane displacements. The parameter for each curve is N x , the
number of vertical fringes per 1024 pixels, as indicated in the legend boxes.
The figure shows clearly that α x =90°/column yields indeed better performance over the whole range of
fringe densities. The difference is small at low fringe densities, whereas it gets significant over N x 30; it
is most pronounced at the optimum speckle size of 3 d p .
In the face of these findings, it seems more appropriate to set α x =90°/sample. As already hinted in 3.2.2.3,
it was found out that the phase calculation with the 90° formula (e.g. (3.19)) tolerates large
miscalibrations of α x ; there is practically no loss in performance for deviations of α x of up to
15°/sample. Moreover, the error-compensating 90° formulae are more suitable than those with α=120°
for the averaging procedures described in 3.2.2.4.
The phase determination with α x =120°/sample quickly loses accuracy when α x >120°/sample and functions
even slightly better when α x 100°/sample. This can be attributed to the facts that (i) the sidebands of the
interferogram's power spectrum already contain aliased super-Nyqvist frequencies Fν x F>Fν N F at ν x0 =1/(3 d p )
and d s =3 d p (cf. 3.4.4), and (ii) also the horizontal MTF of the camera that I used drops considerably for
higher spatial frequencies. Hence, the signal power is utilised more efficiently by the 90° method, where the
sidebands are neatly centred in the (f x ,f y ) half-planes, as depicted in Fig. 6.3.