Gradient 5
The shape of the Gradient 5 surface is the same as the Standard surface plus a "tilt" term in both x and y directions:
where c is the curvature (the reciprocal of the radius), r is the radial coordinate in lens units, k is the conic constant, and tan α and tan β are the tangents of the tilt angles in x and y. Note this is not the same surface shape as a tilted Standard surface, but it is a close approximation as long as the curvature is small, or if the tilt angles are small. The Gradient 5 surface has the following gradient profile:
where r2 = x2+y2. Ten parameters are required: the maximum step size, Δt, the base index, n0, and the remaining six coefficients of the preceding equation, and the two tilt terms. Note that some of the coefficients have units.
The important feature of the Gradient 5 model is that the model allows specification of dispersion properties of the media. The dispersion data is user defined and is stored in an ASCII file called SGRIN.DAT. The format for SGRIN.DAT will be described shortly.
The material name is entered in the glass column of the Gradient 5 surface type. If the glass column is blank, then dispersion effects are ignored.
To perform ray tracing, OpticStudio first computes the index at a "reference" wavelength using the preceding formula for nref. The index at any other wavelength is then computed using the following method based upon a general expansion of the Sellmeier formula:
Where
The coefficients Kij and Lij define the dispersion of the material, whereas the gradient coefficients specified by parameters 2-8 (see the following table) define the index gradient profile at the reference wavelength. This very general dispersion model permits modeling of nearly arbitrary gradient index dispersion over a wide wavelength band. The parameters K_MAX and L_MAX may be as little as 1 term, or up to 8 terms for more accurate modeling.
The dispersion data is stored in the ASCII file SGRIN.DAT which must reside in the <glass> folder (see "Folders").
The SGRIN.DAT file contains blocks of 10 lines each. The file format has the following structure:
MATERIALNAME MIN_WAVELENGTH MAX_WAVELENGTH REF_WAVELENGTH K_MAX L_MAX K11 K12 K13 ... K1K_MAX K21 K22 K23 ... K2K_MAX K31 K32 K33 ... K3K_MAX L11 L12 L13 ... L1L_MAX L21 L22 L23 ... L2L_MAX L31 L32 L33 ... L3L_MAX
Multiple materials may be defined in the same file by placing additional blocks of 10 lines one after the other, with no blank lines between. The supplied file SGRIN.DAT contains coefficients to describe some gradient materials offered by LightPath Technologies.
The maximum step size Δt determines the trade-off between ray tracing speed and accuracy. For details see "Discussion on maximum step size for GRIN surfaces". Also see "Restrictions on surfaces following GRIN surfaces".
PARAMETER DEFINITIONS FOR GRADIENT 5 SURFACES
| Parameter # | Definition |
| 1 | Δt |
| 2 | n0 |
| 3 | nr2 |
| 4 | nr4 |
| 5 | nz1 |
| 6 | nz2 |
| 7 | nz3 |
| 8 | nz4 |
| 9 | tan α |
| 10 | tan β |
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