What is the mathematical equation for an aspheric lens surface?

[Drakkith]

I'm messing around with an optical design program, and it's got an option to enter a custom lens surface using something similar to this equation.

z = ax^2 + bx^4 + cx^6 + dx^8

The problem is, I have no idea what this means. Looking around online I found this equation and a partial description of what Z and X mean, but that's it. Any help available?
Thanks.

 

[Andy Resnick]

Aspheres are generally defined in terms of their departure from a spherical surface. In your case, it seems to be an expression for the actual surface sag- if the height off the optical axis is 'x', the departure from a plane z= 0 is given by your expansion (which is really a truncated portion of the infinite series).

As an example, consider a Schmidt corrector plate, commonly used in large aperture telescopes, trackers, and projectors. For distant objects, the reflecting surface free of spherical aberration is a paraboloid, which has a sag of ax^2. So, to correct a spherical surface one must cancel the term bx^4 (and higher terms), so the sag of a Schmidt corrector has the form z = ax^2+bx^4, where all the a's and b's are specific to the particular surface.

Use of aspheres in optical systems is generally frowned upon (although with new manufacturing techniques they are becoming more accepted) due to the expense of manufacture and specialized alignment tooling. There are good discussions in:

Malacara "Optical Shop Testing", Wiley (ch. 18)
Schulz, "Aspheric Surfaces", Progress in Optics vol. XXV, Wolf (ed.), 1988
Shannon, "Aspheric Surfaces", Applied Optics and Optical Engineering vol. VIII, 1980
Menchaca and Malacara, "Toroidal and Sphero-Cylindrical Surfaces" Appl. Opt. 25, 3008-3009 (1986).
Malacara et. al., "Axially Astigmatic Surfaces: Different Types and Their Properties" Opt. Eng 35 3422-3426 (1996).

 

[Drakkith]

Thanks Andy. It looks like my Telescope Optics book isn't going to cut it and I'll have to invest in a much more detailed book, because I barely understood any of that lol. Thanks for the references too.