To demonstrate how measured index data may be fit to the ZEMAX thermal model, we’ll look at an example of data measured with CHARMS measurement system, which has been developed at NASA’s Goddard Space Flight Center. The data that we will look at are for silicon, and can be obtained from an article entitled “Temperature-dependent refractive index of silicon and germanium”.¹  Data at selected values of wavelength and temperature are provided in Table 2 of this article:



As described earlier in the article, the values presented in this table are generated by fitting data measured by the CHARMS system to a model referred to by the authors as the “Temperature-dependent Sellmeier model”. To demonstrate the use of the ZEMAX thermal fitting tools, we will fit the wavelength dependence of this fitted data (i.e. the data presented in the above table) to the Sellmeier 1 formula (as described in the chapter of the manual entitled “Using Glass Catalogs”) and the temperature dependence to the standard ZEMAX thermal model.

It is important to note that each of the dispersion formulas in ZEMAX describes the variation of the relative index of refraction with wavelength. The relative index is simply the index of refraction of the glass normalized to the index of refraction of air. The ZEMAX dispersion formulas describe the relative index at a reference pressure (P₀) of 1 atmosphere (= 101,325 Pa) and at the reference temperature of the glass (T₀). This can be calculated from input data for the relative index that is provided at arbitrary pressure (P) and temperature (T) using the following formulas:

n_rel(P,T,l) = n_input(P,T,l)/n_air(P,T₀,l)
n_rel(P₀,T₀,l) = n_rel(P,T,l)*n_air(P₀,T₀,l) = n_input(P,T,l)*n_air(P₀,T₀,l)/n_air(P,T₀,l)

where l is the wavelength. The index of air may be determined using:



Details regarding the calculation of n_air are provided in the chapter of the ZEMAX manual entitled “Thermal Analysis”. When using these two formulas to convert n_input(P,T,l) values into n_rel(P₀,T₀,l) values, T is always the reference temperature (in degrees C), P is the measured air pressure (in atmospheres), and l is the input wavelength in microns at the reference temperature (T₀) and the measured pressure (P). This is the case because ZEMAX assumes that all inputted index data are relative to air at the measured temperature and pressure, and that all inputted wavelength values are referenced to T=T₀.

The dispersion formulas also require values for the wavelength that are relative to air at atmospheric pressure. Wavelength data inputted at an arbitrary pressure P can be converted to relative wavelengths using a formula which is similar to the one used to convert inputted index data:

l_rel(P₀,T₀) = l_input(P,T₀)*n_air(P,T₀,l)/n_air(P₀,T₀,l)

As an example, let’s take a look at the silicon data provided in the above table. This table lists values for the absolute index of refraction at a series of wavelengths and temperatures. Since the index data are absolute, P=0. In this case n_air(P,T₀,l) = 1.0 for all wavelengths and choices of reference temperature, since the index of refraction of vacuum is always unity.

We now need to choose a reference temperature. In this case, it makes most sense to select 295 K = 21.85 degrees C as the reference temperature, since this temperature is closest to the standard value of T₀ (20 degrees C) used for most other glasses in the ZEMAX glass catalogs. The index of refraction of air at this temperature and at atmospheric pressure may be determined using the formula for n_air provided above.

Over a wavelength range of 1.1-5.5 microns, as provided in the above table, the parameter n_ref varies from 1.00027265 to 1.00027388. At a reference temperature of 21.85 degrees C, the resultant variation in n_air for this same wavelength range is 1.00026630 to 1.00026750. The values of n_air(P₀,T₀,l) can be calculated for each wavelength in the table, allowing the inputted index and wavelength data to be converted into relative index values and wavelengths. In this case the conversion is simplified by the fact that n_air(P,T₀,l) = 1.0 for all wavelengths. However, ZEMAX also handles the general case – i.e. when P is not equal to zero – by calculating the values for n_air(P,T₀,l) at all wavelengths and including these values in the conversion, as described by the above equations. The relative index and wavelength data can then be accurately fit to the standard ZEMAX thermal model and the ZEMAX dispersion model of choice.