Application and Optimisation of the Spatial Phase Shifting ...

6.6 Use of depolarisation to eliminate invalid pixels 159
6.6 Use of depolarisation to eliminate invalid pixels
A main error source in ESPI and, to a lesser extent, in holographic interferometry, are pixels where
M I (x,y) falls below the electronic noise or even vanishes due to low or zero speckle intensity. This occurs
quite frequently (cf. Chapter 2.2.5) and leads to a relevant fraction of uncertain or invalid outputs of
∆ϕ(x,y) in displacement measurements. This phenomenon, with the associated discontinuities of the
speckle phase, is the origin of the "salt and pepper" noise in ESPI phase maps, and its effect on phase
unwrapping has been investigated recently [Hun95].
On the other hand, it is known that the speckle intensity pdf described by (2.6) changes significantly when
an incoherent sum of two uncorrelated speckle patterns is considered [Goo75, p.21, Enn75, p.211]. In that
case, the maximum of the intensity pdf is shifted away from O(x,y)=0, so that the probability of finding
"dark" pixels will decrease. Such a case is encountered in the interferometric investigation of rough
objects that give rise to multiple scattering and thus introduce depolarisation, i.e. generate two mutually
incoherent speckle fields. In this subsection it will be shown how these can be exploited to improve ESPI
measurements [Bro98].
In this context, we call an object depolarising if the state of polarisation (SOP) of the light scattered back
from it differs from the SOP of the incident light. In many samples, for instance natural stone, this is a
consequence of volume scattering due to the transparency of the material under investigation. Hence, we
obtain a scattered wave field with fluctuations of intensity, phase, and polarisation.
If the scattered light is split into two orthogonal linearly polarised states (vertical, v, and horizontal, h),
two speckle patterns S v (x,y) and S h (x,y) are generated with a normalised correlation coefficient c. As
described in [Fre90d], the value of c is chiefly governed by a surface–specific constant, called the
depolarisation coefficient ρ. This quantity is defined by the ratio of cross- to co-polarised scattered
speckle intensity: 0ρ = S v /S h 1, where h is the SOP of the incident light and v the orthogonal one.
Then, we can use
c =
( 1−
ρ)
1+
ρ
2
2
(6.23)
as a very good approximation to determine the correlation of the orthogonally polarised speckle patterns.
This theoretical prediction was confirmed by measurements of depolarising natural stones [Ada97]. Such
surfaces are for example involved in ESPI-based measurements of deformations and surface changes of
historical monuments, which application was developed in the last few years [Gül96].
Even moderate amounts of depolarisation cause a significant decay of c: for ρ >0.5, we find c