The second order term is redundant if R and k are used, and you should use either (R, k) or a^2. It’s present so you can use a pure polynomial definition of the surface, which is useful when designing freeforms as distinct to aspheric deviations from a base sphere.

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.

Yep. I think if the designer’s mindset is to produce a small deviation from a base conic asphere, a^2 should be ignored. That’s the typical case when trying to knock down the spherical etc. But if you’re designing a wilder asphere, where the curvature (power) at the vertex is not expected to be an approximate placeholder for the power elsewhere along the sag, then ignore R and k and just use the polynomial terms.

If the radius of curvature isn’t a good approximation to the best-fit sphere, or if there is no meaningful best-fit sphere, then ignore R and k and just use polynomials.

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

Small question on this, how would one just use the polynomials in Zemax? (ie: how do I start a design without putting in some radii)

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