Solution to the Phase Problem

In our case, the crucial assumption is that the phase and amplitude change on slower time scales than the modulation cycle. At two (or more) closely-spaced orientation angles, i.e., within one cycle of modulation, we assume that the amplitude and phase does not change. Then we can solve the following equations for A and $\alpha$, given M₁ and M₂:

$$Re[ V(\phi_1) ] = A\ cos(\psi_1 + \alpha) = M_1 \eqno{(5a)}$$

$$Re[ V(\phi_2) ] = A\ cos(\psi_2 + \alpha) = M_2 \eqno{(5b)},$$

where we define $\psi_1$ and $\psi_2$ by

$$\psi_1=2\pi (x_0\ cos(\phi_1)+y_0\ sin(\phi_1) )/p$$

and

$$\psi_2=2\pi (x_0\ cos(\phi_2)+y_0\ sin(\phi_2) )/p.$$

The solution of the equations (5a,b) for $\alpha$ is:

$$tan(\alpha) ={ {M_1 cos(\psi_2) - M_2 cos(\psi_1) } \over {M_1 sin(\psi_2) - M_2 sin(\psi_1) } }$$

From the solution $\alpha$, the amplitude A may be determined from either (5a) or (5b).