Application and Optimisation of the Spatial Phase Shifting ...

52 Electronic or Digital Speckle Pattern Interferometry
greatly extended the possibilities of ESPI and enhanced the attainable accuracy of phase measurements by
a factor of about 10. Whereas quantitative evaluation of correlation fringes requires sophisticated
automation algorithms (see Chapter 4.1) or laborious interactive procedures, the phase shifting method
automatically yields complete phase maps, so that today the correlation fringe methods have mostly been
superseded by phase-shifting ESPI.
To introduce temporal phase sampling, or stepping, we establish the expression
In ( x, y, tn ) = Ib ( x, y) + M
I
( x, y) ⋅ cos( ϕO ( x, y, tn ) + α
n
( x, y, tn
))
(3.12)
with
n: number of phase sample
I n : measured intensity in the n th frame
I b : bias intensity; corresponds to O+R
M I : intensity modulation; corresponds to 2LOR
ϕ O : speckle phase
α n : additional (known) shift of ϕ R ; generally, α n =nα and n ∈ {0,..,N-1}.
For now, we restrict ourselves to static phase shifts, since a distinction between temporal and spatial
phase ramping must be made that will be described in 3.3 and 3.4.4, respectively. Also, ϕ R has been set to
zero as above. All quantities depend on x and y due to the underlying speckle field. The phase shift
α n (x,y,t n ) may be, but in practice seldom is, spatially uniform; various numbers N of phase samples can be
used. Assuming O(x,y) and R(x,y) to remain temporally quasi-stable, we still have to account for possible
temporal fluctuations of ϕ O and ϕ R . For convenience we put them all into ϕ O .
The set of equations given by (3.12) can easily be linearised; the principle is outlined in Appendix C. It
contains three unknowns, namely I b , M I , and ϕ O , and hence we need at least three linearly independent
measurements of the I n (N3), with pairwise different α n , to solve unambiguously for ϕ O . This can be
done by generating an expression that gives tan(ϕ O )=sin(ϕ O )/cos(ϕ O ); i.e. one needs a numerator
proportional to the sine and a denominator proportional to the cosine of ϕ O . To achieve this, the I n are put
together as
ϕ
O
N −1
∑a I
modπ
= arctan
n=
0
N −1
∑b I
n=
0
n n
n n
with
∑
n
a
n
∑
= b = 0, (3.13)
n
n
which is valid for any phase-sampling scheme. In all of such formulae, the coefficients in numerator and
denominator add up to zero, which cancels the contribution from I b . The simplest expression to evaluate
the recorded data relies on equally spaced α n that are uniformly distributed in the interval [0,2π); it is well