This discussion will have little to no practical value to most, but will hopefully contribute to the forum information which falls under the category of Chris S' "esoteric knowledge". (Rik, feel perfectly free to delete this post if it falls outside of the forum's focus.)

I kept having that gnawing feeling that the photos in the thread should not look as good as they do with 2 waves of error on the wavefront, that much aberration should be fatal by our standards.

Then a phrase of Rik's caught my attention,

Recalculating the total optical path lengths then shows a difference of 2 wavelengths *between center and edge of the aperture.*"

Emphasis mine. This would imply that the approach which Rik used solved the focus of the

*paraxial ray*, and that a tiny refocus solved so as to achieve the smallest RSM error across the wavefront should give a lower total P-V error. In other words, best focus is typically found to be a situation in which the

*edges of the aperture are level with the center* (measured in this case at the lens' focus or at any point after the acrylic plate) when referenced against a perfect (spherical) wavefront.

I need to make it clear that the methodologies used by Rik and myself were quite different.

More gory details ...

I, as Rik, had no info as to the design specs of the lens, and so I made some simple assumptions which I knew would be incorrect but which should still exhibit a

*relative* discrepancy between the two focus solves. I used as the lens' focal length merely its working distance, an assumption which is most certainly wrong (the focal length is actually measured from a theoretical point somewhere

*within* the lens), but which would none-the-less reveal a fairly accurate

*proportional difference* in the wavefront errors arrived at by the two approaches.

As the model for the theoretical perfect lens I just used a parabolic reflector, inherently fully corrected for both SA of the axial light cone as well as for all color errors . Just as eyepieces are raytraced in reverse, wavefront errors will have the same values when raytraced in either direction. So I worked this in reverse to Rik's approach, beginning with a perfect (spherical and converging) wavefront and solved so as to indicate the RMS wavefront error at any point after the light exits the acrylic plate.

These are the system specs which I used.

Light from infinity reflects from the objective's pupil as a perfectly spherical convergent wavefront, just as would be the emergent beam exiting the microscope objectives

*entrance pupil* (light raytraced in reverse), considered for our purposes to be perfect. This light cone then encounters the acrylic plate (which had been ignored by light approaching the mirror). After exiting the plate, the rays will have an aberrated wavefront which will, depending on the focus point for which the system is solved, display either the wavefront of the paraxial focus (a simple curve profile to the wavefront) or the wavefront of a system solved instead for the lowest RMS value (a compound curve profile).

The system layout ...

The relative wavefronts. (In the following illustrations Rik's solution in terms of my own will always be the first graph shown.) Since Rik's solution resulted in a P-V (peak to valley) OPD (optical path difference) of 2 waves, my own solve for the paraxial ray (1.367 waves P-V) is almost surely the consequence of the degree of error in my initial assumptions for the lens specs, and so the OPD value of the system solved for least RMS (0.4224 P-V OPD) needs also to be adjusted proportionally resulting in a value of 0.618 P-V OPD. The relatively high values of both are the result of SA introduced by the acrylic plate into the light path.

The next two graphs indicate the profiles of the axial airy disks of our two solves. The central peak represents the signal, the rings surrounding this peak are noise. While even a perfect system also exhibits these surrounding rings, in the presence of aberrations light is stolen from the central disk and thrown into the rings, quickly reducing the system's S/N ratio. While a P-V OPD of 0.25 waves is generally considered to be

*diffraction-limited*, or observationally perfect according to the Rayleigh criterion, a reduction of the OPD to 0.125 is still quite easily detected visually as an improvement in IQ.

Finally we have the MTF graphs for the two solves, with which many of us are familiar.

These graphs clearly illustrate Rik's statement,

Spherical aberration (SA) manifests as reduced contrast, particularly for moderately fine detail but not so much for the very finest detail.

Assuming the reasoning in my own approach to be at least loosely valid, this to my mind solves the question as to why the images look somewhat better than what the focus solve arrived at by Rik's approach would indicate.

One more assumption, by both of us I believe, is that the surfaces of the acrylic are optically plane, which is almost certainly not the case, but which the images seem to indicate is not an error of significance in relation to the SA introduced by the plate. But I expected this surface error to be worse than that which the images appear to exhibit, hence my surprise at the image quality which resulted.

My apologies for any mistakes in synax or terminology in the discussion, I'm only an amateur.

Cheers,