Application and Optimisation of the Spatial Phase Shifting ...

6.2 Modified phase reconstruction formulae 143
construct our two consecutive sets of samples needed to apply the error compensation of (3.56) from a
sequence of pixels as shown in Fig. 6.7.
I -1 I 0 I 1 I 2
Fig. 6.7: Arrangement of sampling points for a simple phase-shift error compensating formula (3.56) with
α x =90°/column, indicated by the black bars. The intensity readings I -1 to I 2 are taken from consecutive
columns.
If ϕ' O 0 (cf. (3.56)) is constructed from I –1 through I 1 (indicated by the solid-line box) and ϕ' O 1 from I 0
through I 2 (broken-line box) and these two phase measurements are averaged, the error in ϕ' O 0
will be
almost cancelled by that in ϕ' O 1 thanks to their relative offset of 90°. (If the phase offset of ϕ' O0 and ϕ' O1
were exactly 90°, there would be no need for error correction.) It is true that this method of averaging
requires four instead of three pixels and seemingly requires still larger speckles; we will discuss this issue
shortly, in the context of the experimental findings. The improvement of phase calculation by (3.56) is
shown in Fig. 6.8 for B = 30 and d s =3 d p ; both curves have been calculated from the same set of
interferograms.
0.12
σ d /λ
0.10
0.08
0.06
0.04
0.02
3-sample 90°, B=30
0.00
3+3-sample averaging 90°, B=30
0 20 40 60 80 N x 100
Fig. 6.8: σ d for ESPI displacement measurements for B=30 and d s =3 d p by SPS, with and without phase-shift error
compensation, as a function of N x . Triangles, phase calculation by (3.19); squares, phase calculation
according to (3.56).
The modified phase calculation reduces σ d very efficiently; and again the improvement is most relevant at
low fringe densities. The substantial decrease of σ d comes somewhat unexpected in this situation, since