Application and Optimisation of the Spatial Phase Shifting ...

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3 Electronic or Digital Speckle Pattern Interferometry
The speckled wave scattered off an object bears a random intensity and phase structure that, in itself, will
not reveal information about the object’s macroscopic shape or deformation. By superposition with a
reference wave, it becomes phase sensitive and the intensity modulation of each speckle is deterministic.
It obeys the relationship
I( x, y) = O( x, y) + R( x, y) + 2 O( x, y) R( x, y) cos( ϕO
( x, y) −ϕ R ( x, y))
(3.1)
where I denotes the interferogram intensity, O that of the object wave and R that of the reference wave,
and ϕ O and ϕ R the respective phases; x and y are the co-ordinates of the image plane. While both O and ϕ O
fluctuate strongly with x and y, the spatial variations of R and ϕ R are generally negligible. For the sake of
readability, we will henceforth omit the spatial dependence of all variables. If two speckle interferograms
are recorded, we have
I
i
= O
i+ R
i+ 2 O
i
R
i
cos( ϕO, i
−ϕ
R,
i
)
, (3.2)
I = O + R + 2 O R cos( ϕ −ϕ
)
f f f f f O, f R,
f
with subscript i for the initial and f for the final object state. On assuming that the intensities do not
change during the experiment – which is easy to assure for R but requires the absence of speckle
decorrelation for O –, we can reasonably rewrite this as
Ii
= O + R + 2 OR cos( ϕO
)
, (3.3)
I = O + R + 2 OR cos( ϕ + ∆ϕ)
f
where we have set ϕ R,i =0 without loss of generality, and omitted the "initial" subscript for the speckle
phase. The deterministic phase change ∆ϕ is caused by object displacements, but includes possible global
fluctuations of ϕ R,f as well. From (3.3), the difference to classical interferometry is clear: for diffusely
reflecting objects, no reference surface is available, and we need to compare it with itself. This can be
done either by holography [Har94], where we have true interference of two object wavefronts, or by
acquisition and subtraction of digitised interferogram intensity data [Løk87, Dov00]. The latter is
commonly called digital or electronic speckle pattern interferometry (DSPI or ESPI), and the digital
difference images are sometimes referred to as secondary interferograms, which is to distinguish them
from direct, or primary, interferometric images.
Compared to holographic interferometry, the resolution of the pixel array-based digital method is poor,
but with appropriate magnification of the object surface, still sufficient for many purposes. Among the
advantages of an electronic system are versatility and very quick carrying out of experiments. Fig. 3.1
sketches the basic parts of an ESPI set-up.
O