Superconic
The most common form of a polynomial aspheric surface uses a power series expansion in the radial coordinate r to define the surface sag, where r is defined by
For example, the Even Aspheric surface described in "Even Asphere" uses such an expansion. Since r does not depend upon z, the expansion term is the distance from the vertex to the point on the surface as projected on to the tangent plane. Generally, the departure of the asphere from the tangent plane increases with radial aperture. As the departure increases, the power series expansion parameter r corresponds to a point on the tangent plane which is farther from the point on the surface. This causes the expansion to have poor convergence.
A novel solution proposed by Alan Greynolds of Breault Research Organization is to instead expand in powers of the distance from the vertex to the point on the surface. The expansion is then in terms of
Starting with the conic equation for a surface
where k is the conic constant and R is the radius of curvature, a general power series expansion can be made of the form
The constants are defined as
where U and V are coefficients which define the aspheric shape. Note that if all the U and V terms are zero, a standard conic results. If A is also zero, then the superconic becomes a sphere. The coefficients A, U1, and V1 together form a Cartesian oval. These properties make the superconic stable when optimizing for the coefficients. The superconic can be used to model surfaces which otherwise would require aspheric terms of very high order. OpticStudio models superconics with up to 240 terms, in practice designs rarely use more than 5 terms.
PARAMETER DEFINITIONS FOR SUPERCONIC SURFACES
| Parameter # | Definition |
| 13 | Maximum term number. The maximum is 240, but 4-10 is typical. |
| 14, 15 | U1 and V1 |
| 16, 17 | U2 and V2 |
| ... | ... |
| 252, 253 | U120 and V120 |
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