The Gradient 5 surface type has parameters to model variations of the refractive index in r and z. Other gradient surface types can model these variations as well, but what makes Gradient 5 different is that it includes dispersion parameters which mimic the often-used Sellmeier 1 formula. The dispersion formula for the Gradient 5 surface type is:



while the dispersion given by the Sellmeier 1 formula is:



If one is modeling a Sellmeier 1 glass that also has r and/or z variations (say, because of temperature gradients), then only the first K term and the first L term are necessary, since the Sellmeier K and L parameters do not vary with the index at the reference wavelength. In other words, the only terms needed for the Gradient 5 surface will be K₁₁, K₂₁, K₃₁, L₁₁, L₂₁, and L₃₁. Note that the convention that I’m using here is that K’s and L’s with one index are Sellmeier constants, and those with two indices are Gradient 5 constants.

The L_i1’s are defined in the same way as the L_i terms in the Sellmeier formula, which makes sense because the index formulae should have singularities at the same wavelengths (assuming small changes in index). However, the K_i1’s (which generally indicate the strength of the absorption line) are not defined the same way as the K_i terms in the Sellmeier formula. If one evaluates the Sellmeier formula for a wavelength λ and a wavelength λ_ref, takes the difference between the expressions, and compares to the Gradient 5 formula, one finds that:



with the other K_ij and L_ij terms being zero.

One item to note is that it does not matter what wavelength is used as λ_ref as long as the K_i1’s are calculated using the same wavelength, and that same wavelength is used in finding the base index n₀ in the Gradient 5 surface type.

If one has a glass that is not described by a Sellmeier 1 formula, then one can use the index fitting feature (Tools > Catalogs > Glass Fitting) to generate a Sellmeier 1 model. More information on the Glass Fitting Tool may be found in the article entitled “Fitting Index Data in ZEMAX”.

This example uses fused silica as the glass to be modeled by the Gradient 5 surface type. The Glass Catalog entry for fused silica is shown below, followed by the relevant 10 lines from the SGRIN.DAT file. Note that the K’s are different, but that the L’s are the same. Also, note that the K’s and L’s are in different orders.
 


Gfusedsilica   ← name of Gradient 5 glass
0.21 3.71      ← wavelength range
0.58756180     ← reference wavelength
1 1            ← # of K_ij and L_ij terms
-0.00956531    ← K₁₁
-0.01661702    ← K₂₁
0.900654310    ← K₃₁
0.004679148    ← L₁₁ (same as L₁ in Sellmeier formula)
0.013512060    ← L₂₁ (same as L₂ in Sellmeier formula)
97.93400000    ← L₃₁ (same as L₃ in Sellmeier formula)

To check that the model is correct, put both the original glass and the Gradient 5 glass in your lens file and plot the index versus wavelength for the two surfaces using the Universal Plot 1D feature (the usual dispersion plot feature found under the Analysis > Glass and Gradient Index > Dispersion Diagram menu will not work because it will not accommodate the Gradient 5 feature). Be sure that the Gradient 5 surface has zeros for the r and z parameters and that the n₀ term is set according to the reference wavelength in the SGRIN.DAT file. If the model is correct and the horizontal/vertical scales are the same, then overlaying the two index plots will show no difference between the indices of the two surfaces:



The lens file which illustrates this comparison is provided as an attachment at the end of this article.