# Theory of Phase Shifting Technique

Repeated here, the intensity I(x, y) of an interferogram at a point (x, y) is given as

$$I(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)]$$

There are three unknowns in the equation, namely $I_m$, $I_a$, $\phi$. Three simultaneous equations are needed to evaluate the unknowns. Experimentally, the three equations can be obtained by recording a series of intensity distributions with a uniform change of phase (or fringe order). The three equations can be expressed as

$I_1(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)-\delta]$

$I_2(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)]$

$I_3(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\delta]$

where I1, I2 and I3 are the intensity distributions recorded with a phase change of $-\delta$, 0 and $+\delta$, respectively. From the three equations, the phase $\phi(x, y)$ can be determined as

$$\phi(x,y)=atan\left[\frac{1-cos(\delta)}{sin(\delta)} \frac{I_1(x,y)-I_3(x,y)}{2I_2(x,y)-I_1(x,y)-I_3(x,y)}\right]$$

When $\delta=\frac{2}{3} \pi$, the above expression becomes

$$\phi(x,y)=atan\left[\sqrt{3} \frac{I_1(x,y)-I_3(x,y)}{2I_2(x,y)-I_1(x,y)-I_3(x,y)}\right]$$

For a more accurate phase calculation, other algorithms using more than three phase-shifted images have been developed. The most widely used algorithm uses four phase-shifted images. The set of four images are
$I_1(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)]$
$I_2(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\frac{1}{2}\pi]$
$I_3(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\pi]$
$I_4(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\frac{3}{2}\pi]$
Then, the phase is expressed in a simpler form as
$$\phi(x,y)=atan\left[ \frac{I_4(x,y)-I_2(x,y)}{I_1(x,y)-I_3(x,y)}\right]$$
Another algorithm uses five images [61]. This algorithm was developed to minimize the cases of denominators with zero or near zero values, and thus to reduce uncertainties in the phase calculation. The five different intensities are obtained with a symmetric phase shift as

$I_1(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)-2\delta]$

$I_2(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)-\delta]$

$I_3(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)]$

$I_4(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\delta]$

$I_5(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+2\delta]$

The phase is given by
$$\phi(x,y)=atan\left[\frac{1-cos(\delta)}{sin(\delta)} \frac{I_2(x,y)-I_4(x,y)}{2I_3(x,y)-I_1(x,y)-I_5(x,y)}\right]$$
If $\delta=\frac{1}{2} \pi$, then
$$\phi(x,y)=atan\left[\frac{I_2(x,y)-I_4(x,y)}{2I_3(x,y)-I_1(x,y)-I_5(x,y)}\right]$$

Another notable phase shifting method is the Carré method, where the phase shift amount is also treated as an unknown. The method uses four phase-shifted images as

$I_1(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)-\frac{3}{2} \delta]$

$I_2(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)-\frac{1}{2}\delta]$

$I_3(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\frac{1}{2}\delta]$

$I_4(x,y)=I_m(x,y)+I_a(x,y)cos[\phi(x,y)+\frac{3}{2}\delta]$

Assuming the phase shift is linear and does not change during the measurements, the phase at each point is determined as
$$\phi(x,y)=atan \frac{\sqrt{ \left[ (I_1-I_4)+(I_2-I_3) \right] \left[ 3(I_2-I_3)-(I_1-I_4) \right] } } {(I_2+I_3)-(I_1+I_4)}$$
The advantage of Carré algorithm is clear; it does not require accurate calibration of the phase shifting mechanism as long as it is linear and stable during the measurement.

In spite of the above conventional phase shifting algorithms, there is an advanced algorithm superior to all those algorithms.  It is called avanced iterative alglorithm (AIA).

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