Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 55
α n =α(t n ), to obtain a temporal sequence of phase-shifted interferograms I n (x, y, t n ); but once we have
clarified the different methods, the transfer to spatial phase shifting is very simple.
3.2.1.1 Phase-of-differences method
The first approach to think of when processing secondary interferograms is to determine their phases as
familiar from primary interferometric fringes. Given a set of images I n,i of the initial object state, one then
needs only one frame I 0 , f of the final state, so four or five images are sufficient to use the phase-shifting
methods of (3.16) or (3.17), respectively. As only one frame of the final object state is involved, we shall
call the I n,i plus I 0 , f a "reduced" data set. The phase-shifted secondary correlation fringes I n,c are formed
according to
( )
I = I − I
n, c f n,
i
2
( cos( ϕ ∆ϕ) cos( ϕ α ))
= 4OR
+ − +
O O n
2 ⎛ ∆ϕ + αn
⎞ 2 ⎛ ∆ϕ − αn
⎞
= 16ORsin
⎜ϕO
+ ⎟ sin ⎜ ⎟
⎝ 2 ⎠ ⎝ 2 ⎠
= 4OR( 1− cos( 2ϕO + ∆ϕ + αn ))( 1− cos( ∆ ϕ −αn
)) ,
2
(3.20)
where the first cosine describes the speckle noise in the correlation fringes and the second cosine is the
envelope, phase shifted by α n . This approach offers one significant advantage: if the object under test can
initially be observed at rest, the capturing of one interferogram suffices later on to obtain phase-shifted
correlation fringes.
There is another important consequence of (3.20) that has, as far as I know, not been emphasised before:
the first cosine depends on 2ϕ O . This is of course owing to the squaring operation – and would not look
very different if we were dealing with the modulus –, but it means that we cannot distinguish between
positive and negative speckle intensity changes anymore. Thus, half the information delivered by the
intensity changes is discarded, with important consequences for the measured ∆ϕ. The situation is
represented in Fig. 3.3: the black curves show the result of using squared correlation fringes as in (3.20)
for the standard four-sample phase calculation of (3.16).
To the left, a simulation result is shown: for each pre-set ∆ϕ , 64 different ϕ O , uniformly distributed over
[0,2π), were inserted into (3.20) to form the corresponding sets of I n,c , where α=90°. These 64 sets of I n,c
were inserted as the I n in (3.16) to yield 64 values for ∆ϕ , whose average appears as calculated ∆ϕ . The
average over all ϕ O thus gives the expectation value of the calculated vs. the true displacement phase. On
the right, the measured ∆ϕ ∆ϕ(x), i.e. for vertical sawtooth fringes (cf. Fig. 3.4), averaged over 200
rows and represented as grey values, confirms that indeed the extraction of ∆ϕ is almost impossible after
the squaring or rectification process. The white curves refer to the difference-of-phases method and will
be discussed below in 3.2.1.2.