Application and Optimisation of the Spatial Phase Shifting ...

64 Electronic or Digital Speckle Pattern Interferometry
Finally the third step of synchronous detection is the extraction of ϕ(x), using (3.34) and (3.39), by means of
tan ϕ( x)
=
∞
∫
2 Re I ~ ( ν ) iδ ( ν −ν ) dν
0
∞
∫
x x 0x x
2 Re I
~ ( ν ) δ ( ν −ν ) dν
0
x x 0x x
sin ϕ( x)
∝ . (3.40)
cos ϕ( x)
It was shown in [Fre90a] that a correct phase determination requires
~ * ~ ( )
*
S ν = C ( ν )
x
~ * ~ ( )
*
S ν = iC ( ν )
x
x
x
⇔
~ ~
S ( νx
) = C ( νx
)
~ ~
S ( ν ) = − iC ( ν ) ,
x
x
(3.41)
this is, the filter spectra must have equal magnitudes (also called "responses") and be 90° out of phase
(also called "in quadrature"), so that S'(0) represents the sine and C'(0) the cosine of ϕ(x). As a summary
of the involved operations, the whole procedure has been given the name of "quadrature multiplicative
moiré" [Wom84].
In (3.40), (3.41) need only hold for ν 0x , since nothing is detected at other ν x ; but when we confine the
integration to a finite interval (–X,X) instead of (-%,%), the filter responses will broaden around ν 0x . This
need not be a disadvantage, because more signal energy – if present – may be utilised in this way; and as
long as (3.41) remains valid, ϕ(x) can still be correctly determined also for ν x ≠ν 0x . The objective of phase
sampling is now to satisfy (3.41) with only a short sequence of digitised samples of I(x).
3.2.2.2 Digital synchronous detection
Let us now assume that we are working on a discrete pixel grid, where the pixels are assumed to be point
detectors with distance d p . Let M be the number of pixels in x direction and k their individual numbers.
Using the "filter property" of the δ function, the filter outputs are now – with an appropriate choice of the
origin of the co-ordinate system – given by
∞
M
∫ ∑ ∑
S' ( 0) = I ( x) S( x) δ ( x − kd ) dx = I ( kd ) S( kd )
−∞
∞
p
k = 1 k = 1
M
∫ ∑ ∑
C'( 0) = I ( x) C( x) δ ( x − kd ) dx = I ( kd ) C( kd )
−∞
M
M
p
k= 1 k = 1
p
p
p
p
(3.42)
i.e. the signal is being sampled by a sequence of δ functions only. For convenience we retain the
assumption of infinite spatial extent of the signal. To measure ϕ(x) at a given pixel k 0 and thus introduce
the spatial resolution of the phase measurement, the sampling pulse sequence must be "windowed" by
selecting only a few intensity samples at (k 0 +n)d p , with n ∈ {0,..,N-1}, so that, in the simplest case of
using a rectangle function as a window,