Temporal phase shifting
This method requires a linear and uniform phase modulation over the field of view during the exposure time of the detector. This imposes:
A sequence of fringe patterns with a phase increment is registered and the phase difference covered during the exposure time must be identical from one acquisition to the next [1,4]. If the optical phase is varying very rapidly with time, the acquisition must be done with a very high temporal bandwidth and in this case we are confronted with a technology problem, as the camera speed is rarely above 1000 images/s with a spatial resolution of 1024 x 1024 pixels. In the case where the phase is static (motionless) or quasi-static, the applied linear modulation can be “slow”, and therefore the bandwidth can be lower, meaning that we register one image per camera cycle, typically 25 images/s. The detector temporally averages the fringes signal .
Let us consider the following sinusoidal phase profile:
The mean level (a) and contrast (b) are time independent. The spatial dependencies have been implied.
is the phase we want to determine (and is also time independent). Each image is obtained by integrating over a time interval ΔT :
This technique is known as "integrating bucket". Figure 6 illustrates the acquisition process by showing the interferometric signal and the area under the curve which corresponds to the temporal integration over each time interval of duration ΔT.
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The calculation of the temporal integration yields:
We note
,
is the phase increment between each image. We then obtain:
The values registered during each integration time are represented on figure 7. The samples follow a sine wave (dashed line) of smaller modulation than the initial signal.
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On a mathematical point of view, the expression of the temporally integrated interference signal has three unknowns a, b, Δφ, among which Δφ is the main unknown to determine. To solve this problem, we therefore need at least three equations, thus generating a system of at least three equations with three unknowns. Solving this system will give the three parameters. If we register at least three interferograms E n, , then we will be able to determinate Δφ.
This phase stepping method therefore consists in a tridimensional sampling of the fringe pattern: two planar dimensions with the coordinates on the image and one longitudinal dimension with the optical phase variation
. The choice of
must be done according to Shannon theorem, which tells us that we need to register at least two values of the signal per fringe period, thus imposing
.
Let us consider for example the case where
is equal to
. Using a 25 Hz acquisition system, we have Δt = 40 ms and we will need to impose ƒ0
= 6.25 Hz ; registering the 4 interferograms will give:
Fringe modulation is reduced by approximately 10% since
. We obtain Δφ using a arctangent function:
The amplitude of fringe oscillation is:
and the constant component of the signal is given by:
We can thus determine the modulation ratio of the fringe pattern:
The determination of this modulation ratio can be used to extract the useful region for processing interferograms. Indeed, only pixels for which the modulation is above a certain threshold (for example m > 10%) should be processed and therefore they will be affected a value of 1 in the image mask, whereas the other pixels (m < 10%) will be affected to a value of 0 in the mask. The analysing software will process only pixels for which the mask value is 1.