Extended Fresnel
The Fresnel surface described in "Fresnel" models a plane surface which has the refractive (or reflective) power of a curved surface. The Extended Fresnel surface increases the flexibility of this model by supporting a plane, spherical, conic, or polynomial aspheric substrate on which a plane, spherical, conic, or polynomial Fresnel pattern is etched.
The surface sag is identical to the even aspheric surface:
See the section "Even Asphere" for details. The previous expression is used to compute the ray- surface intercept. Once the intercept is found, the refraction (or reflection) of the surface is determined by the local slope of the Fresnel facets, which depends upon both the Fresnel facet shape expression for Zf (below) and the substrate shape expression for Zs (above). The Fresnel facet shape is described by an expression virtually identical to the even asphere sag expression:
The only difference is that the latter expression need not use all 8 terms if n is less than 8, the maximum accepted value. Note that the curvature (symbol c), the conic constant (symbol k), and all of the polynomial coefficients (symbol α) are independent for the substrate sag and Fresnel portions of the surface. The refraction at the surface accounts for both the substrate sag and the Fresnel sag, while the ray-surface intercept depends only upon the substrate sag. The intention was to model a Fresnel lens molded on a plane which is curved or warped into a new substrate shape after it is fabricated.
The substrate sag radius, conic, and polynomial terms are all specified in the Lens Data Editor, just like the even asphere surface. The Fresnel sag terms are specified as extra data values in an extra data file (see the Import section of the Surface Properties). However, the extra data file uses curvature (the reciprocal of radius) rather than radius for the Fresnel sag. The Extra Data Values used are summarized in the following table.
PARAMETER DEFINITIONS FOR EXTENDED FRESNEL SURFACES
| Parameter # | Definition |
| 1 - 8 | α1 - α8 |
| 13 | Maximum aspheric polynomial term number, n. The maximum is 8. |
| 14 | The curvature (NOT RADIUS) of the Fresnel surface. This parameter will effect the refraction of the surface, but not the shape of the substrate. |
| 15 | The conic constant of the Fresnel surface. This parameter will effect the refraction of the surface, but not the shape of the substrate. |
| 16 | Coefficient on r^2 |
| 17 | Coefficient on r^4 |
| ... | ... |
| 23 | Coefficient on r^16 |
The "Maximum aspheric polynomial term number" is used to specify the maximum polynomial term to be used in calculating the surface sag. This number is provided to speed the ray tracing calculation, as terms beyond this number are ignored. As with any complex surface model, extreme care should be taken to evaluate the accuracy and appropriateness of this model, especially where fabrication decisions are concerned.
Because there is no reliable way to compute the phase through a Fresnel surface which is not a plane, any calculation that requires OPD data, such as OPD fans, MTF, and Zernike coefficients, will not be supported if a non-plane substrate Fresnel surface is present in the lens description.
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