Talk:Strehl ratio

"Modern definition"?

The article begins "The modern definition of the Strehl ratio is ...". However, nowhere is there any reference to any previous definition. So, can we change this to just "The Strehl ratio is ..."? --Dan Griscom (talk) 01:14, 18 June 2010 (UTC)

Question general validity of formula

Although I think the equation is instructive and should be included in the article, I don't believe the statement "It can be computed from Fresnel’s diffraction theory as:.... " is formally correct. The formula (as you can easily determine) is the average intensity at the central point in the focal plane (the point that would be illuminated according to geometrical optics setting delta=0 along the whole pupil) given by the squared magnitude of the fourier transform at that point (setting the spatial frequency in x and y to zero so that the e^jfx factors become unity). This is what I call the "on-axis strehl" for lack of a better term (is there a term for this?). In high strehl situations it is essentially equal to the (ordinary) strehl, since the peak intensity is always just about at that central point in the focal plane.

In the more general case however the point of maximum instantaneous intensity is not exactly (often not even approximately) at that location, and the Strehl ratio which is based on the point of maximum intensity will be somewhat larger than computed in this manner. In other words the equation is not exactly correct for the strehl ratio and shouldn't be stated as such. But it IS a useful equation that should be in the article, but properly qualified.

On a related note, "Mahajan’s approximate formula" which is next presented is actually (I'm pretty sure) an EXACT formula for the "on-axis Strehl" (defined by the first equation) when the random variable delta has gaussian statistics and is subject to sufficient averaging (such as when the Strehl is due to a zero-mean atmospheric delay averaged over time). But beyond that it differs from the defined strehl ratio as I described above.

Would the person who previously edited these sentences like to try to clarify the text in this regard? Interferometrist (talk) 14:23, 9 April 2013 (UTC)

If you don't mind me saying so, unless User:Edgar.bonet comes forth to volunteer, you're probably the only other qualified user able to rewrite this section. You'll notice that nobody corrected my version of the Majahan formula from March 2011 to December 2012 (changing the "equivalent equals" sign - which was the only one I could find when I learned enough formula wikimarkup to post it - to the correct "approximately equals" sign). This implies that there are very few editors who know both the formulas and the correct markup. I agree with you this is the formula described in the references. But that is the limit of my understanding of it. The other formula added by User:Edgar.bonet last December is unreferenced, which if nothing else implies the dreaded and shunned WP:OR, but who am I to correct somebody who both knows it and knows correct wikimarkup. My level of understanding is merely enough to polish and figure a 1/10 wave telescope mirror, so I'm damned if I know Fresnel's theory from the way to properly mix a martini. So I'm not going to try to corrrect it, though I'm all for you doing so. Cheers. Trilobitealive (talk) 23:26, 9 April 2013 (UTC)

Hi! I have to second User:Trilobitealive here: Interferometrist, you are probably better qualified than I to edit this section. My understanding of the Strehl ratio is still limited, and comes from my researching about the effect of defocus on image quality, and optimizing a pinhole diameter. I did not consider low Strehl situations, nor the case where the optical system loses its axial symmetry. I included the “average over the entrance pupil” formula because it seemed more general than Mahajan’s, without realizing that it is not completely general. And because it proved useful to me in pinhole calculations. I did not source it because it seemed to derive quite straight-forwardly from Fresnel diffraction. I also did not know about Mahajan’s being exact in the gaussian case, but I did know it is not exact for spherical aberration, which is far from gaussian. I will edit the page, to the best of my understanding, if you are reluctant to do so. If so, do you have a reference for Mahajan’s being exact in the gaussian case? It seems to me like a non-trivial yet beautiful result. Regards, Edgar.bonet (talk) 08:22, 10 April 2013 (UTC).

From my viewpoint as a hobbyist mirror grinder, I'm perfectly happy with either one of you making any needed changes. Trilobitealive (talk) 01:45, 12 April 2013 (UTC)

Hi, and thanks for both your comments. Yes, I am willing to edit this page as necessary, but only after I am confident of what I am going to write. I did a search and have on my work computer a few articles referring to "Strehl ratio" in somewhat different contexts, and will look at them again tomorrow. But what I am gathering, is that there might not be a widely accepted definition that answers my original question (whether the intensity is measured on-axis or at the brightest image point). In other words, this is not a term with a strict definition to be found in textbooks for instance, but has been used somewhat ad hoc, so it may be that rather than a single definition one has to refer to its use. It's not even surprising that this issue is unclear since the definitions only diverge appreciably when the Strehl is much less than 1 (which no one wants!). Another issue is that Strehl is used in two rather different situations: 1) To describe the magnitude of FIXED optical aberrations or manufacturing imperfections; and 2) In reference to a varying stochastic optical disturbance, most notably the atmosphere (perhaps with partial correction by AO). In case (2), one can also talk about the short-term Strehl (and its statistics) as would be determined from snapshots of the PSF, and the long-term Strehl as would be found from a long photographic exposure which integrates the average PSF in which case the on-axis definition is implied (since a long exposure in a symmetric system would average out evenly around the on-axis focal point.

And by the way, I made the remark about Mahajan's approximation becoming exact when the wavefront deviations have gaussian statistics because I have been using that result myself for a long time. But looking at it more closely, I see that it only really approaches exactness when the aperture is large compared to the correlation length of the deviations (so that the central limit theorem applies and the averaging is so good that the instantaneous Strehl is about the same as the long-term). Then with the phase x having zero mean and variance sigma^2, the amplitude on-axis is E{e^(jx)} = E{cos(x)} = e^(-sigma^2/2) and the intensity is that squared. But I looked at Mahajan's paper and he wasn't thinking in such broad terms at all, and only advanced this formula as an empirical approximation to the Strehl as a function of Zernike coefficients, so my conjecture won't have any place in the final article! Interferometrist (talk) 17:41, 16 April 2013 (UTC)