Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 51
filled up by well-modulated data from its π/2-complement. But of course, controlled phase shifts are not
automatically available in an ESPI system.
Fig. 3.2: Left: ESPI correlation fringes from subtraction of two primary speckle interferograms; right: average of
two correlation fringe images with phase offsets of π/2 in the underlying primary interferograms I i , I iπ/2
and I f , I fπ/2 (see text).
Although the optimisation of speckle size and fringe contrast has been the subject of numerous studies
[Tan68, Sle79, Wyk87], the overly – in the sense of (3.9) – speckled appearance of the correlation fringes
still limits the accuracy of ESPI measurements to about 1/10 fringe. Moreover, the fringe profile is an
even function of ∆ϕ, which makes it impossible to determine the sign of the measured displacement
gradient. To get rid of this ambiguity, a-priori information has to be used: either a pre-set bias fringe
pattern with known phase gradient reveals the relative fringe orders when it changes, or the load is applied
in such a way that only one direction of deformation gradient is possible [Wya82, Mat88].
A far more elegant method to retrieve quantitative displacement data is to convert the cosine into a
tangent by means of several phase samples and then to extract the phase mod 2π by a four-quadrant
arctangent. This approach has become very popular under the name of phase sampling – although it relies
on intensity sampling –, or phase shifting. It eliminates completely the difficulties described by (3.9),
which is an important reason for its superior performance.
3.2 Phase-shifting ESPI
The technique of phase sampling or quasi-heterodyning has long been known in information theory and
has first been used in classical interferometry to enhance accuracy [Car66, Bru74, Wya75]. After the
application of phase shifting to holographic interferometry [Har82, Cha85], it was the merit of [Nak85,
Cre85b, Ste85, Rob86] to have realised that also a digital speckle interferogram is an array of independent
"micro-interferometers" that work like classical ones – although some of them suffer from too faint an
object wave.
Hence, the phase information of a speckled wave front, although random per se, nevertheless responds
deterministically to phase changes due to displacement or deformation of the test object, and digital
subtraction of two speckle phase fields yields a difference phase field. The use of phase shifting has