Application and Optimisation of the Spatial Phase Shifting ...

3.1 Subtraction-mode ESPI 49
some 4 speckle sizes. This value was given in [Tan68] for holography; in ESPI however, the detector's
pixel size plays a role as well.
3.1 Subtraction-mode ESPI
On subtraction of the interferograms obtained from the initial and final object state, we have
I − I = 2 OR(cos( ϕ + ∆ϕ) − cos( ϕ ))
f i O O
⎛ ⎛ ∆ϕ
⎞ ⎛ ∆ ϕ ⎞⎞
, (3.4)
= − 4 OR⎜sin⎜ϕO
+ ⎟ sin⎜
⎟⎟
⎝ ⎝ 2 ⎠ ⎝ 2 ⎠⎠
with the second sine term representing the signal fringe profile and the first sine term the multiplicative
speckle noise on it. Thus, one obtains a – secondary – fringe profile from the subtraction of two
– primary – speckle interferograms. To give these fringes the familiar appearance of interferometric
fringes on, e.g., a monitor, the negative values in the difference image have to be converted into positive
ones. In DSPI, it is easy and customary to use the modulus of the difference,
⎛ ∆ϕ
⎞ ⎛ ∆ϕ
⎞
I f − Ii = 4 OR sin⎜ϕO
+ ⎟ sin⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
; (3.5)
averaging over ϕ O gives a mean fringe intensity of
⎛ ∆ϕ
⎞ ⎛ ∆ϕ
⎞
I
f
− Ii = 4 OR sin⎜ϕO
+ ⎟ sin⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
8 OR
=
π
⎛ ∆ϕ
⎞
sin⎜
⎟
⎝ 2 ⎠
4 2 OR
= 1−
cos( ∆ϕ)
π
(3.6)
in the so-called correlation fringes (note that the fringe envelope is not cosinusoidal and only serves to
visualise the object changes). If an initial speckle interferogram is stored and the difference between it and
the current one is viewed, one gets darkness where the optical phase is the same in both the images (i.e.
the optical path has changed by an integer multiple of the wavelength) and brightness where the difference
is maximum (i.e. the path has changed by an odd multiple of half the wavelength). Thus the digital
secondary interferograms are formed.
Another way to generate the output is to square the fringe signal, in which case the fringe profile is given by
( )
2
2 ⎛ ∆ϕ
⎞ 2 ⎛ ∆ϕ
⎞
I f − Ii = 16ORsin
⎜ϕO
+ ⎟ sin ⎜ ⎟
⎝ 2 ⎠ ⎝ 2 ⎠
(3.7)
and, after averaging over all ϕ O ,