Application and Optimisation of the Spatial Phase Shifting ...

50 Electronic or Digital Speckle Pattern Interferometry
( )
2
2 ⎛ ∆ϕ
⎞ 2 ⎛ ∆ϕ
⎞
I f − Ii = 16OR
sin ⎜ϕO
+ ⎟ sin ⎜ ⎟
⎝ 2 ⎠ ⎝ 2 ⎠
= 8OR
sin
2
⎛ ∆ϕ
⎞
⎜ ⎟
⎝ 2 ⎠
( ∆ϕ
)
= 4OR
1−
cos( ) .
(3.8)
This method yields a cosine profile and should be more suitable to generate correlation fringe images as
an input for phase-shifting methods; but in the face of the drawbacks of the correlation fringe method
discussed below, the performance gain will be negligible.
The dark regions of the images are noise-free, while the quality of the bright fringes is degraded by
speckle noise: the visibility of the primary interferometric intensity modulation depends on the individual
speckle brightness and hence fluctuates from point to point. Moreover, there are points where, due to
unfavourable ϕ O , the subsequent phase change does not effect a brightness change:
cos( ϕ ) = cos( ϕ + ∆ϕ)
⇔ ϕ = −
O O O
∆ϕ
2 , (3.9)
which just means that ϕ O and ϕ O +∆ϕ are symmetrical about a – primary, cf. (3.3) – intensity extremum;
and there are many more points coming close to this condition. While (3.9) is true for every
interferometric measurement, it is – besides the fluctuations of O(x,y) – the randomness of the ϕ O that
prevents a spatially uniform detection of ∆ϕ. It is worth noting that in the averages over ϕ O in (3.6) and
(3.8), this loss of signal leads to the factors 2/π and ½, respectively.
If, however, another pair of interferograms were available with phase offsets of, say, π/2 each, we would have
⎛ ⎛
I f Ii OR ϕ
π π
O ϕ π ⎞ π
− = 2 cos⎜
+ + cos ϕ
/ 2 / 2
O
⎝ 2 ⎠
⎟ − ⎛
⎜
⎝
+ ⎞⎞
⎜ ∆
⎟⎟
⎝
2 ⎠⎠
( sin( ϕ ) sin( ϕ ∆ϕ)
)
= 2 OR − +
O
⎛ ⎛ ∆ϕ
⎞ ⎛ ∆ϕ
⎞⎞
= − 4 OR⎜cos⎜ϕO
+ ⎟ sin⎜
⎟⎟
⎝ ⎝ 2 ⎠ ⎝ 2 ⎠⎠
O
(3.10)
and could average the two secondary interferograms to obtain brighter correlation fringes:
I − I + I − I
f i fπ / 2 iπ / 2
ϕ
2
= 4
⎛ ∆ ⎞
OR ⎜
⎝ ⎠
⎟ ⎛ ⎛
⎜
⎝
O + ∆ϕ
⎞
⎠
⎟ + ⎛
⎜
⎝
O + ∆ϕ
⎞⎞
sin ⎜sin ϕ cos ϕ ⎟⎟
2 ⎝ 2 2 ⎠⎠
⎛ ∆ϕ π ⎞ ⎛ ∆ϕ
⎞
= 4 OR 2 sin⎜ϕO
+ + ⎟ sin ⎜ ⎟ .
⎝ 2 4 ⎠ ⎝ 2 ⎠
(3.11)
The improvement of using (3.11) over (3.5) is demonstrated in Fig. 3.2, where on the left-hand side an
image according to (3.5) is shown, and on the right, a superposition according to (3.11); the increase in
brightness should be L2 and is in fact 1.38. In simple words, the disadvantageous points of one image are