From the formulas provided in the previous section, it is clear that when the system temperature and pressure equal the reference temperature and pressure that n_air(P_s,T_s) = n_air(P₀,T₀). If the glass temperature is also equal to the system temperature, then Dn_abs = 0, yielding:

n_rel(P,T) = n_rel(P₀,T₀)

as we would expect. Now let’s take a look at a slightly more complicated example. Let’s consider the case of BK7 glass at a temperature of 30 degrees C and a pressure of 2 atmospheres. In this example, we shall set the system temperature and pressure equal to the glass temperature and pressure, and the input wavelength will be 0.55 mm.

The reference temperature for BK7 glass is 20 degrees C, as provided in the Schott glass catalog:



For T₀ = 20 degrees C and P₀ = 1 atmosphere, n_air(P₀,T₀) = 1.00027308. For the given system temperature (30 degrees C) and pressure (2 atmospheres), n_air(P_s,T_s) = 1.00052810 (this is the absolute index of the air, of course). Using these two values we calculate that the relative wavelength is 0.55014022 mm.

At this wavelength, the relative index of BK7 at the reference temperature and pressure is 1.51851533 (calculated from the Sellmeier 1 formula, using the wavelength dispersion coefficients K₁, L₁, … as provided in the glass catalog). The absolute index of BK7 at T₀ and P₀ is calculated by dividing this relative index value by n_air(P₀,T₀), yielding a value of 1.51810078.

The change in absolute index due to difference between the glass temperature and the reference temperature is calculated from the ZEMAX thermal model (as described in the previous section), using the thermal dispersion coefficients (D₀, D₁, …) provided in the glass catalog. For this case, we find that Dn_abs = 0.00001547. Thus, the absolute index at the glass temperature and pressure is 1.51810078 + 0.00001547 = 1.51811625. The relative index at T and P is then calculated from the product of this absolute index value with n_air(P_s,T_s), yielding a value of 1.51891796. This final value agrees exactly with the value used in ZEMAX, as shown in the Prescription Report:

INDEX OF REFRACTION DATA: 

System Temperature: 30.0000 Celsius 
System Pressure   :  2.0000 Atmospheres 
Absolute air index:  1.000528 at wavelength 0.550000 µm 
Index data is relative to air at the system temperature and pressure. 
Wavelengths are measured in air at the system temperature and pressure. 


 

Surf     Glass Temp  Pres 0.550000 
0              30.00 2.00 1.00000000 
1          BK7 30.00 2.00 1.51891796 
2              30.00 2.00 1.00000000