Sinusoidal phase modulation

In this technique, the phase is modulated by a sine wave in the frequency range of 10Hz to several hundreds of Hz [10]. The phase modulation therefore reads:

where is the frequency of the phase modulation, φ its synchronisation phase, and its amplitude. We still consider a sinusoidal phase profile, of constant amplitude, contrast, and phase offset. The spatial dependency being implied, we have:

The signal is integrated successively during 4 quarters of the modulation period , therefore:

We develop I(t) in series of Bessel functions of the first kind:

We can write the result of the temporal integration as:

with:

and for each interferogram:

which gives the following system:

from which we deduce:

and the phase is obtained by a arctangent function:

can be calculated using the equations:

A study of the algorithm taking into account an additive noise on the integrated intensity shows that the noise influence is minimum for and [10]. For example, for a real time acquisition with a camera giving 200 images/s over a matrix of 256 x 256 pixels, the modulation frequency must be of 50 Hz to register a quarter of period in 2 ms.