I spent a very enjoyable hour in a coffee shop with Fergus and applied mathematicians Leslie Greengard, Charlie Epstein, and Mike O'Neil. We discussed the possibility that Fergus and I might do better on our coronograph problems by doing real-live modeling of the scalar or vector wave equations in a physically realizable device. I have an intuition that we will, but I also have a pretty good sense that we can't really model the full electromagnetic field on every surface; that's insane. Some great ideas came up in the discussion. Greengard pointed out that the convolutions (think Green functions) I am doing numerically can be much, much faster with FFT-like techniques (and Greengard should know). Epstein started out by saying
five-meter mirror and micron wavelengths; geometric optics isn't good enough for you?, which is a pretty fair comment, and then followed that by saying that there is a next order to the geometric optics approximation. It is a well-defined limit, after all, so what we think of as being geometric optics (that beautiful theory) is really just the first term in an expansion. That's cool, although the implication is that the next term in the expansion is ugly (
not in the textbooks as Greengard said). Epstein also noted that inference of the phase from an intensity image is probably not a good idea. The conversation was great; but unfortunately it didn't convince me to give up on this crazy idea. Tonight I have to polish up my code and send it to Greengard for re-factoring to non-stupidity.
One paradox still remains in my mind after it all, and it is this: Heuristically (yes, very heuristically) there are trillions of wavelength-squared cells on the entrance aperture (or primary mirror), but heuristically (yes, yes) there are only hundreds or thousands of speckles on the focal plane that get significant illumination. So doesn't this mean that we don't have to model the entrance aperture at full wavelength resolution to precisely model any real speckle pattern? And isn't that somehow odd?