Visibilities

In radio astronomy, the quantity analogous to the modulation pattern is the fringe pattern, which is the response of an interferometer to point sources in the sky. This is the fourier transform, and it is usually represented by a sum of complex exponentials in the u,v plane:

$$V_j = C_j\ e^{2\pi i(u_j\ x + v_j\ y)} \eqno{(3)}$$

The above function is similar to the waveform in equation (1), except that it has an imaginary part, and its real and imaginary parts can be negative. Translating from radio into x-ray language, the u_j and v_j are simply coordinates in the modulation pattern, with units 1/pitch. So for a single subcollimator oriented at an angle $\phi$with respect to some reference plane (such as the solar equator), $u_j=cos(\phi)/p$ and $v_j=sin(\phi)/p$. Then it is easy to see that the sum of terms like (3) might be made to add up to the function M in equation (1) with judicious choices of coefficients so that the imaginary parts are forced to cancel out, and with the addition of a constant term, so the sum is never negative. The result, however, would not be appropriate to a single point source at (x₀,y₀), since the only way to make the imaginary parts cancel completely is to add a ``mirror'' source on the other side of the spin axis at (-x₀,-y₀).

The function analogous to the modulation profile is the sequence of terms V_j on a circle in the (u,v) plane. Letting $u_j=cos(\phi)/p$ and $v_j=sin(\phi)/p$, we get the visibility of a point source at (x₀,y₀):

$$V(\phi) = C(\phi) e^{2\pi i (x_0\ cos(\phi)+y_0\ sin(\phi))/p} \eqno{(4)}$$

The complex coefficients $C(\phi)$ have both amplitude and phase: $C(\phi) = A(\phi) e^{i\alpha}$, so the real part of V in (4) can be made to look very much like the modulation profile in (2).