The Off-Axis Conic Freeform surface is the sum of an off-axis conic surface with freeform polynomial terms. The coordinate system for the surface is decentered and tilted to the center of the off-axis conic, as shown by the (y, z) coordinate system sketched below. Polynomial freeform terms are added to the surface at the center of the off-axis part. The edges of a mirror defined by this surface shape are parallel to the z axis.

The sag of the surface can be written as:

where:

and *R* is the radius of curvature, *k* is the conic constant, *Yo *is the off-axis distance measured along the *y* axis of the on-axis conic, and *Rn* is the normalization radius for the freeform polynomial coefficients. The position coordinates *x* and *y* are in the off-axis coordinate system.

The freeform polynomial allows 230 terms, up to *y20*. The *A00 *term is omitted. The coefficients are arranged by increasing total power *(j+k)* and then by decreasing powers of *x^j*, as shown in the equation below. Note that the polynomial coefficients *Aj,k* are in lens units because the *(x,y) *coordinates are dimensionless.

The freeform coefficients are specified in the Lens Data Editor beginning with parameter 13.

More details on the Off-Axis Conic Freeform surface derivation and design examples can be found in *D. Reshidko, J. Sasian, "A method for the design of unsymmetrical optical systems using freeform surfaces," Proc. SPIE 10590, International Optical Design Conference 2017, 105900V (27 November 2017). *