Once the inputted data have been converted to relative values at the reference temperature and pressure, this data is ready to be fit. The wavelength variation of the data at the reference temperature is first fit to one of the following ZEMAX dispersion models: Schott, Herzberger, Conrady, or Sellmeier 1. For the case of the silicon data provided in the previous section, we choose the Sellmeier 1 formula:



The standard algorithms used in ZEMAX for optimization are also used to determine the best-fit dispersion coefficients (K₁, K₂, …) for the reference temperature data. Data at other temperatures are then fit to the ZEMAX thermal model, which describes the variation of the absolute index of a glass with temperature:



In this formula n is the relative index at the reference temperature and pressure, and DT is the change in temperature relative to the reference temperature. Values for n are taken directly from the fits to the Sellmeier 1 formula (or in general, the dispersion model of choice) obtained for data at the reference temperature. The standard optimization algorithms used by ZEMAX are then used to determine the best-fit thermal coefficients (D₀, D₁, …) for the data.

Fits to the dispersion and thermal models are performed using the Glass Fitting Tool, located under Tools > Catalogs > Glass Fitting. Input data are provided to this tool via an ASCII text file, which must have the following format:

PRESSURE P
T₀  l₁₀  n₁₀
T₀  l₂₀  n₂₀
.
.
.
T₀  l_f0  n_f0
T₁  l₁₁  n₁₁
T₁  l₂₁  n₂₁
.
.
.
T₁  l_f1  n_f1
T₂  l₁₂  n₁₂
.
.
.
T_f  l_ff  n_ff

where “f” stands for “final” (i.e. the last data set). For example, for the silicon data provided in the previous section, the table would look like:

PRESSURE 0.0
21.85  1.1  3.54394
21.85  1.2  3.52253
21.85  1.3  3.50616
21.85  1.4  3.49335
21.85  1.5  3.48314
21.85  2.0  3.45352
21.85  2.5  3.44011
21.85  3.0  3.43293
21.85  3.5  3.42864
21.85  4.0  3.42589
21.85  4.5  3.42404
21.85  5.0  3.42274
21.85  5.5  3.42180
-243.15  1.1  3.51113
-243.15  1.2  3.49071
-243.15  1.3  3.47508
-243.15  1.4  3.46285
-243.15  1.5  3.45309
-243.15  2.0  3.42478
-243.15  2.5  3.41196
-243.15  3.0  3.40509
-243.15  3.5  3.40100
-243.15  4.0  3.39837
-243.15  4.5  3.39661
-243.15  5.0  3.39537
-243.15  5.5  3.39449
-233.15  1.1  3.51123
.
.
.
-73.15  4.5  3.40982
-73.15  5.0  3.40855
-73.15  5.5  3.40765
-23.15  1.1  3.53555
-23.15  1.2  3.51437
-23.15  1.3  3.49818
-23.15  1.4  3.48551
-23.15  1.5  3.47540
-23.15  2.0  3.44609
-23.15  2.5  3.43283
-23.15  3.0  3.42572
-23.15  3.5  3.42148
-23.15  4.0  3.41877
-23.15  4.5  3.41694
-23.15  5.0  3.41566
-23.15  5.5  3.41474

where all of the temperatures have been converted to degrees C, and we have explicitly indicated that the data are for P = 0. The file containing this raw data is named SI.TID (all input data files used for thermal fitting must have the .TID extension, as described in the chapter of the ZEMAX manual entitled “Tools Menu”), and can be found in the .ZIP file located at the end of this article.

The Glass Fitting Tool is then used to fit this data to the Sellmeier 1 dispersion formula and the ZEMAX thermal model. A new glass corresponding to the fitted data may also be added to one of the existing user-defined glass catalogs (we do not allow new glasses to be added to any of the glass catalogs provided with the ZEMAX installation; those catalogs get overwritten every time a new version of ZEMAX is installed, and thus any user-defined glasses added to these catalogs would be deleted during each upgrade of ZEMAX!). For example, in this case a new glass named SI_THERMAL will be added to the SG_MISC catalog:



A summary of the fit results is provided in a text window. These results may be saved to a file. The text file which is generated when fitting the data provided in SI.TID to the Sellmeier 1 dispersion formula and the standard ZEMAX thermal model is also included in the .ZIP file located at the end of this article (ZEMAX_fit_Si_data.txt). The text output includes best fit values for the dispersion and thermal coefficients, the maximum and RMS errors associated with the fitting in each case, and a comparison between the original relative index data and the fitted value for each input point:



As indicated in the file, the original index and wavelength values are converted to relative index and wavelength values for a reference temperature of 21.85 degrees and a reference pressure of 1.0 atmosphere. This conversion uses the formulas provided in the previous section. For example, for a wavelength of 1.1 microns, the refractive index of air at 21.85 degrees C and 1 atmosphere is 1.00026630. The relative index values at this wavelength are then calculated by multiplying the original index data by 1.00026630 (in general you would also need to divide by the refractive index of air at 21.85 degrees and the measured pressure, but since the pressure is zero the corresponding refractive index is always 1.0). The relative wavelength is calculated by dividing the original value by 1.00026630, i.e. l_rel = 1.1/1.00026630 = 1.09970715. The value used by ZEMAX in this case is actually 1.09970583, as shown in the text output; the difference is approximately 1.0E-06, and simply arises because we have rounded-off the value for the index of refraction in our simple calculation here, while ZEMAX uses a more exact value for the index of refraction (to 16 decimal places).

The results indicate a very good fit of the reference temperature data to the Sellmeier 1 dispersion formula, with an RMS error of 2.5E-06 and a maximum error of 4.3E-06. The results also show that a good fit of the data to the ZEMAX thermal model has been obtained; the RMS error in this case is 2.9E-05, and the maximum error is 9.0E-05. The small values for the fit errors indicate that the ZEMAX thermal model plus the Sellmeier 1 dispersion model provide an accurate description of the input data in this case.