Application and Optimisation of the Spatial Phase Shifting ...

82 Electronic or Digital Speckle Pattern Interferometry
α x / deg/column
120.2
120.0
119.8
119.6
-2.31 0.09 x /mm 2.49 4.89
-2.85
-0.05
y /mm
2.75
Fig. 3.25: Spatial distribution of α x (x,y) on the CCD sensor area for z =10.2 cm and ∆x =2.9 mm.
Clearly, the continuous phase progression over the sensor leads to an integration over the pixels, so that
this is an integrating-bucket method. When the phase runs along columns or rows only, the recorded I n are
described by (3.59), since also the camera pixels are rectangular integration windows, only in space
instead of time. The factors given in 3.3 for the decrease of M I remain valid in this case.
If, however, the carrier fringes are slanted with respect to the Cartesian sensor axes, the situation is
different: for instance, if ∆x=∆y, the slant is 45° and the function over which the phase progression is
"windowed" becomes a triangle; for values below 45°, it acquires trapezoidal shape. Fortunately, the
windows remain symmetrical in any case, from which it follows that the detected phase angles will
remain correct [Wom84]. To determine the loss of M I due to a "composite" phase ramp (i.e. for phase
shift in x and y direction), it is easiest to integrate over its components separately, which gives
α x
In = Ib + M I ⋅ 2 sin( )
⋅ 2 sin( )
2 2
⋅ cos( ϕO
+ αn
) ; (3.67)
α α
x
not surprisingly, this reflects the theoretical 2D-MTF for square pixels. For α x =α y , and hence a triangular
envelope of the phase integration, the factor becomes 4 sin 2 (α x /2)/α 2 x and is indeed the transfer function
of a triangle. We will be concerned with such a case in Chapter 6.3.
The choice of the carrier frequency is influenced by contradictory requirements: on the one hand, it should
be as high as possible to allow a broad range of signal frequencies to be measured. On the other hand,
aliasing of too high frequencies must be avoided. In general, α x must have the same sign in the whole
measuring field to keep the phase extraction unambiguous: a reversed, or aliased, phase shift leads to the
wrong sign of the calculated phase, cf. 3.2.2. In classical interferometry, this means that closed
interferometric fringes are not allowed, and the complete fringe pattern must be properly sampled; in
speckle interferometry, the requirements are different and we will discuss them in 3.4.4.
Since the I n are arranged as adjacent pixels on the sensor, it is clear that the speckles must be enlarged to
obtain sufficient spatial correlation of speckle intensity and phase within the sampling pixel cluster, so
that the modulation detected by a phase-extraction formula comes more from the phase shift than from
α y
y