# Spherical aberration: circle of least confusion and diffraction limited spot size

I need an estimate of the relative sizes of the circle of least confusion (spherical aberration) and the diffraction limited spot size for typical real lenses. I've seen plenty of discussions, but none address what to expect of a real multi-element photographic lens or simple single element off-the-shelf lens, or an aspheric singlet. Most discussions target wavefront error, but it's not clear to me how that relates to spot size. In short: what is the size of the circle of least confusion in typical real lenses.

To narrow the question, data or information on a double-gauss design would be most helpful.

I might be able to figure this out myself if I had typical aberration coefficients for photographic and simple lenses, but I can't find any published information on the size of those coefficients. If you know where to find these data, please tell me where.

Rules of thumb, estimates based on experience, or educated guesses are welcome. (3x the diffraction spot radius? 10x the diffraction spot radius? ... )

1.- The diffraction limit spot size is often defined as the Airy disk diameter of the aberration-free optical system (even aberrated), and if you know its F-number, it is just 2.44*lambda*F-number, for monocromatic illumination. An interesting question is about polychromatic light, but an estimation could be a weighted sum of the diameters for each wavelentgh. However, I realize that this is a very rough estimation.

2.- It is also common to design real lenses to get the minimum spot size in the image plane when the amount of aberrations is large, and often defined as spot sigma radius. In these situations, and considering that the system is rotational symmetric, the only aberration present for on-axis object is spherical aberration. The spot sigma radius is obtained from the spot diagram data, but also from aberration data. Therefore, if you have typical aberration coefficients of real lenses, you could calculate it. Oh, I should mention that the calculation depends on whether you have wave aberration coefficients or ray aberration coefficients. The expression changes in each case.

3.- In my last paper with Professor Virendra Mahajan, we evaluate the amount of defocus to balance spherical primary aberration (link). Classical balancing for low aberration values of spherical (less than 2 waves) is found in the literature (see Mahajan's books below) which is -1 times the amount of spherical aberration value. When the amount of spherical aberration increases there is also another amount of defocus accepted that balances spherical, which is -4/3. However, in our study we found that for values of the spherical aberration greater that 2 waves, the classical balancing of -4/3 is not useful, but -1/2 and gives a better MTF (modulus of the Optical Transfer function) for moderate spatial frequencies. I am sorry that the study does not include spherical aberration of greater orders that fourth (primary) spherical aberration, which is interesting.

It is worth to take a look at:

Mahajan, V.N., "Optical Imaging and Aberrations" Part I, II and III, SPIE Press. See here

Mahajan, V.N., "Aberration Theory Made Simple", SPIE Press. See here

In these textbooks you could find valuable information about what you are looking for with the theoretical justification of why of those numbers as well as how to calculate them, particularly wavefront error and sigma spot radius.

Hope this helps.

I joined "stack" recently and see that you entered this question a long time ago. I realize that you may have found out all about it in the meantime, and I apologize if what I say is well known to you now or perhaps no longer of interest. But still:

"Handbook of military infrared technology" contains a number of so-called "blur spot charts" displaying the diameter of the "disc of least confusion" for å number of simple lenses depending on bending, index of refraction and f/no. The same is done for curved mirrors and other simple components. These blur spots are obtained by raytracing, making "spot diagrams" and using these to obtain some measure of spot size. It seems that this book can be downloaded free in PDF format. You can find it if you Google the title.

The book "A system of optical design" by Arthur Cox, contains in the last section more than a hundred different designs of 12 different lens types taken from the patent literature. Full design data are given, together with computation of aberration curves of longitudinal spherical aberration, sine condition (coma), s- ant t- surfaces (astigmatism and field curvature), distortion and petzval sum. I think one could derive approximate point image diameters from these curves, but spot diagrams can be obtained if one has access to a computer program for raytracing. Amazon has two copies, but they are very expensive, so it would better to try a library.

Kidger and Wynne has written a paper called : "The design of double Gauss systems using digital computers" in Applied Optics, maybe they would present some spot diameters there.

"Telescope Optics", a book by Rutten and Venrooij, contains spot diagrams of a number of telescope objectives with full design data.

"Star testing Astronomical Telescopes" by H. R. Suiter contains a number of computed pictures of diffraction point spread functions for various aberration types. These are computed from wave-aberrations by algorithms for diffraction calculations.

I don't feel that one could rely on the design data from the patent literature to give the best designs, rather representative of the type.

There is also the question of what quality criterion is most suitable, and this may depend on the application. From a spot diagram one can compute e.g the max diameter, the rms radius, the full width half maximum or the encircled energy, all dependent on focus setting and spectral content of the light.

The Strehl ratio is the max intensity of the spot relative to the intensity of the perfect spot. This is found by a diffraction calculation. For visual instruments used in daylight, this criterion has good correlation with subjective experience of image sharpness.

The optical transfer function is often used for photographic lenses. This must be computed from the wave-aberrations of the lens found by raytracing. This is connected to the "resolving power" sometimes given for the lens.