Consider the simple case of scattering off of a mirror for normal incidence light. Let’s say that the surface reflection coefficient is 0.95, and that the effective RMS surface roughness is 3 nm at a wavelength of 632.8 nm. In addition, let’s say that the surface is characterized by a typical surface wavelength of 0.8 mm and a log-log slope of the BSDF equal to 3. The inputs to the K-correlation model for this surface would be:



For scattering of 300 nm light off of this surface, we can use the formula provided in the previous section to calculate the effective surface roughness:

s²(l₂)/s²(l₁) = (1 - (1 + B²/l₂²)^1-s/2)/ (1 - (1 + B²/l₁²)^1-s/2)
s²(300 nm)/s²(632.8 nm) = (1 - (1 + (5.02655)²/(0.30)²)^1-3/2)/ (1 - (1 + (5.02655)²/(0.6328)²)^1-3/2) = 1.075
s(300 nm) = s(632.8 nm)*√1.075 = (0.003 mm)*(1.037) = 0.00311 mm

This value can be used to calculate the TIS:

TIS ≈ 4p²dn²s²/l² ≈ 4p²*(2)²*(0.00311 mm)²/(0.30 mm)² ≈ 0.017
R*TIS ≈ (0.95)*(0.017) ≈ 0.016

Thus, in this case approximately 1.6% of the incident energy is scattered from the surface in reflection, while the rest of the reflected energy would follow the specular reflected ray path.

A simple ZEMAX file (Simple Example.ZMX) has been set up to test this calculation. This file is provided in the .ZIP folder located at the end of this article. In this file, a small detector (object 3) is placed on axis to measure the specular ray energy. The large detector (object 4) is used to measure the scattered ray energy. We find that the amount of energy that goes into scattering is approximately 1.6%, as expected:



Note: The same calculation could have been done using a single detector and filter strings. More information on the use of filter strings may be found in the article entitled "How to Perform Stray Light Analysis", as well as in the chapter of the ZEMAX manual entitled “Non-Sequential Components”.