I think the r^2 term in the polynomial expansion is largely superfluous and can typically be ignored.  Here is what Bentley and Olson say in The Field Guide to Lens Design:

Although as Mark noted, including the r^2 term and using only the polynomial expansion may be useful for some applications.

It’s interesting to note, though, that most smart phone camera lens patents disclose wild aspheres that are described by a base radius, conic, and polynomial terms starting with r^4.  I don’t have enough experience in this space to know whether there is a fundamental reason for doing so, or if it’s just a matter of convention.  In any event, here’s one example that I think may have been used in the iPhone 4S.

Although, in general, there appears to be varied opinions about when to set k=0.  For example, attached are a few pages about aspheres from Lens Design: A Practical Guide by H. Sun.

Thanks to both of you! Great discussion and very helpful!

Markus

Hi Rick,

You could just start with parallel plates and let the optimizer do the work. Or, think of the a^2 term as the curvature of a parabola. Depends on how much data you have at the start, of course. But it’s the same issue with any freeform surface or complex polynomial. 

I wouldn’t make too much of this though. The original poster asked why the r^2 term was there, and this is why. The vast majority of people use R and k and omit a^2, especially when using an asphere for higher order aberration control.

• Mark