How do you incorporate diffraction effects into a merit function?

I have been working with millimeter-wave / radio optics, like the ALMA array, or the ARO, and the most important metric for these types of systems is usually the amount of light that the detector sees that isn’t coming from a cold source. This is called the warm spillover, and is evaluated by using a time-reversed model, where the detectors are treated as the sources. Then, any light that reaches a non-cold source will contribute to the noise in the system. Evaluating this quantity is difficult due to the frequencies of radio systems.

At the long wavelengths of radio optics, we are in a quasi-optical regime where the wave nature of light becomes very relevant, but can still be well represented by geometrical ray tracing. Therefore, to evaluate the spillover metric, we should use a beam propagation method and account for the diffraction that occurs on finite apertures.

However, when designing such systems, it is really challenging to create a merit function that captures this behavior! So, has anyone else dealt with this kind of consideration, and how have you gone about it?

It isn’t possible, in general, to separate diffraction and path length error, since the fourier transform of A e^{(-i \phi)} can’t be broken into a term dealing with A, and a term dealing with phi. However, for small OPD, say less than 1/4 wave PV, you can approximately separate them by treating the PSF as the sum of the diffraction PSF and a scaled PSD of the phase error. In that regime, it may be ‘good enough’ to simply use the size of the airy disk (if your pupil is circular) as a proxy for the effects of diffraction.