Suppose some (coherent) light propagates onto a one-dimensional barrier with two open slits, each of length $\delta$ and separated by a distance $s$ (we can assume the centroid is known and placed at $x=0$). Diffraction through the barrier can be modeled by a beam splitter unitary $U$ coupling system spatial modes $a(x)$ with orthogonal modes $b(x)$ with a transmittance function $T(x)$ which is equal to one at the openings and zero everywhere else.
What would be the most general way to write an analogous transformation that doesn't just scatter the modes, but also makes sure that the two slits behave as sources of incoherent/partially incoherent light? Since incoherence depends on the relative phase, my guess is we should write something like $$Ua(x)U^\dagger=e^{i\phi(x)}(T(x)a(x)+R(x)b(x)) $$ where for each transmitted photon $\phi(x)$ is a random phase. Several things are not entirely clear to me: first of all, whether we can just take $\phi(x)=\pm\phi/2$ for $x$ belonging to the first and second slit respectively and zero otherwise; secondly, whether there is a dependence on the separation of the sources $s$; thirdly, how to describe sources that are only partially incoherent with the same model.
