Application and Optimisation of the Spatial Phase Shifting ...

54 Electronic or Digital Speckle Pattern Interferometry
well. This has been successfully applied in practice (cf. Chapter 6.7) and simplifies the account of
[Nak95], where a 3-D phase LUT was used. But in general, the LUT approach works only if all the
coefficients a n , b n can be integrated in the LUT; hence the requirement is that the coefficients, or at least
their ratios, be expressible by integers; an example is given in Appendix B.
For all these practical reasons, we will restrict ourselves to standard three- or four-sample formulae in this
work. From (3.14), we get the widespread four-step formula for α=90°,
ϕ
O
I − I
mod 2 π = arctan 3 1
I − I
0 2
(3.16)
and the three-step formula for α=120°,
ϕ
O
I I
mod 2 π = arctan 3 2 − 1
2 I − I − I
0 1 2
,
(3.17)
where a factor of ½ has been cancelled from the fraction. Note that this formula follows likewise from
(3.15) because, for α=120°, I -1I 2 . If however α=90°, (3.15) delivers the three-step (non-DFT) formula
ϕ
O
I−1 − I1
mod 2 π = arctan
2I − I − I
0 −1 1
.
(3.18)
To simplify (3.18), it is usual to accept a phase offset – which is hardly relevant in classical, and less so in
speckle interferometry – and choose a representation in which the coefficients are equal for all intensity
samples:
I
− ° mod 2π
= arctan
I
( ϕ 45 )
O
− I
− I
2 1
0 1
(3.19)
As mentioned above, the phases obtained from such calculations can be mapped onto a grey scale of, say,
256 steps. When ϕ O crosses a 2π boundary, it jumps back to zero, and so do the associated grey levels;
this is why the images thus generated are known as sawtooth images. Since speckle interferometry is
about comparing phases, we will dedicate the following subsection to finding out the best way to do so.
3.2.1 Calculation of phase changes in ESPI
There are several ways to come from interferograms to ∆ϕ(x,y), the displacement phase map which is
represented in a sawtooth image; and since the accuracy in measuring ∆ϕ(x,y) is the pivotal issue in this
work, it is certainly worthwhile to investigate the different strategies in detail.
In what follows, we will refer to the first two approaches by the handy terms "phase of difference" and
"difference of phase"; this nomenclature follows [Moo94], one of the relatively few papers on ESPI
concerned with quantitative performance issues. For the third method, I propose the term "complex
division". All of the methods have been introduced together with phase-shifting ESPI [Nak85, Cre85b,
Ste85]. First of all, the treatment concerns temporal phase shifting, i.e. we shift the phase in time,