This article describes the K-correlation surface scattering model and its implementation in OpticStudio. An example comparing and contrasting the model to the Harvey-Shack (ABg) scattering distribution is provided.

**Authored By Sanjay Gangadhara**

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## Introduction

Scattering which results from surface microroughness can often be characterized by the K-correlation model.^{1} This model is similar to the Harvey-Shack (ABg) model except in the region of small scatter angles.

There are a number of inputs required by the K-correlation scatter model which must all be provided by the user if they with so model this surface scattering distribution in OpticStudio.

This article will provide a brief overview of the theory behind the K-correlation scatter model and will run through an example of its use in OpticStudio.

## The K-Correlation Scatter Model

The bi-directional scatter distribution function (BSDF) associated with the K-correlation model is provided by Dittman^{2}:

where s is the effective RMS surface roughness, *s* is the log-log slope of the BSDF at large spatial frequencies, and βis defined as the sine of the scatter angle (*ϑ*_{s}) minus the sine of the specular reflection/transmission angle, i.e. β in the above formula corresponds to the vector x used in OpticStudio:

We find that the K-correlation distribution is very similar to the Harvey-Shack (ABg) model. The main difference is that the K-correlation model provides a roll-off in scattering at small scatter angles:

**Figure 1: A comparison between the K-correlation and Harvey-Shack scattering models. The K-correlation model exhibits a small-angle roll-off, which Dittman indicates is consistent with the observed behavior for many surface finishes. This figure was taken directly from reference [2] (corresponding to Figure 1 in that reference).**

Dittman indicates that this roll-off is consistent with the observed scattering behavior for many surface finishes.

The BSDF for the K-correlation model cannot be analytically integrated. However, a Monte Carlo technique has been used to implement this scattering distribution in OpticStudio. An approximate form for the total integrated scatter (TIS), in which we ignore the cos(*ϑ*_{s}) term in the BSDF equation and set *ϑ _{i}* = 0, is also calculated:

As we can see, there are a number of inputs required by the K-correlation scatter model, and we will examine these in a bit more detail in the next section.

**Note: **When the information available to the user for a particular scattering surface is measured BSDF data rather than parameters that would be obtained from fitting measured surface roughness data to the K-correlation model, we strongly recommend using the measured BSDF data directly in modeling the surface scattering distribution. The process of using measured BSDF data directly in OpticStudio is described in great detail in the article entitled "__How To Use Tabular BSDF Data to Define the Surface Scattering Distribution__".

## Inputs for the K-Correlation Scatter Model

The K-correlation BSDF is described by 6 parameters:

*R* = Surface transmission/reflection coefficient

*dn* = Change in the index of refraction at the surface boundary

*σ* = Total effective RMS surface roughness (µm)

*λ* = "Measurement" wavelength (μm)

*B* = 2πL, where L = typical surface wavelength (mm)

*s* = Log-log slope of the BSDF at large spatial frequencies

Details on each of these parameters are provided in reference [2]. As indicated in that reference, the effective RMS surface roughness is calculated over a range of spatial frequencies from 0 to 1/* λ*, where a non-zero value for *λ* is chosen so as to provide a finite normalization factor for the Total Integrated Scatter (TIS). When determining the K-correlation scatter parameters from experimental measurements of the surface roughness, the choice of *λ* is completely arbitrary; *λ* is simply used to define the inverse cutoff frequency for calculating the power spectral density (PSD) of the measured data, which is in turn used to convert the PSD into a BSDF. Once the value for *λ* is chosen by the experimentalist during analysis of the measured surface roughness data, the effective surface roughness at other wavelengths may be calculated according to:

If the available information for a particular surface is measured BSDF data rather than measured surface roughness data, we strongly recommend using the measured BSDF data directly in modeling the surface scattering distribution in OpticStudio. The procedure for using measured BSDF data to model surface scattering in OpticStudio is described in great detail in the article entitled "__How To Use Tabular BSDF Data to Define the Surface Scattering Distribution__".

In OpticStudio, the surface transmission/reflection coefficient (*R*) is determined by the coating (or lack thereof) on the surface, while the change in index at the surface boundary (*dn*) is calculated directly. Each of the 4 remaining parameters (*σ*, * **λ*,* B, s*) of the K-correlation BSDF must then be specified as inputs to the K-correlation scatter model in OpticStudio:

An additional parameter (SFV1) is required by the DLL to provide the value of *dn* to the Scatter Function Viewer (SFV). Although *dn* is calculated directly in the DLL for purposes of scattering off of a known object, this information cannot be made available to the SFV feature, as this feature is not tied to any particular object in the design. Thus, to ensure accurate calculation of the BSDF in the SFV feature, *dn* must be explicitly provided (again, this has no effect on the actual scattered ray distribution in the OpticStudio model). The associated DLL (K-CORRELATION.DLL) is included with OpticStudio, and is located in the appropriate installation folder ({Zemax}\DLL\SurfaceScatter\).

The inputted value for s corresponds to the value at the reference wavelength (= "measurement" wavelength), as defined by the “Ref. Wave.” parameter, while the value of λ used in the K-correlation BSDF equation corresponds to the wavelength of the ray. The effective RMS surface roughness at the ray wavelength is calculated from the inputted values for s and l using the above formula (relating σ(*λ*_{2}*) *to σ(*λ*_{1}*)*. Once the scaled value of σ is determined, the total integrated scatter is calculated using:

When Importance Sampling is turned off, the TIS value calculated in the DLL is used to determine the amount of energy that goes into scattering; the rest of the incident energy follows the specular ray path. To ensure that the TIS calculation works correctly for this case, make sure to set the “Scatter Fraction” parameter equal to 1, as shown above.

When Importance Sampling is turned on, the amount of energy that goes into scattering is determined explicitly by the "Scatter Fraction" parameter. Therefore, for this case the user should manually set the "Scatter Fraction" parameter equal to the TIS, as determined using the above equation.

In either case, if the input parameters are defined in such a way that the TIS > 1, the DLL will perform no scattering, and all of the incident energy will follow the specular ray path.

## A Simple Example

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:

This value can be used to calculate the TIS:

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 OpticStudio 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 OpticStudio manual entitled “Non-Sequential Components”.

## Comparing the K-Correlation and ABg Models

We can see from the BSDF equation that the K-correlation scatter model is very similar to the ABg model. In fact, we find nearly exact agreement between the models when the following inputs are used for each case:

*ϑ*_{i} (= incident angle) = 0 degrees

K-correlation model: B = λ; S = 2

ABg model: A = [4π • R • d*n*^{2} • *σ*^{2}] / [lin(2) • *λ*^{2}]; B = 1; g = 2

In this case, the BSDF equation for the ABg model is:

The BSDF equation for the K-correlation model is almost identical, as it simply contains an additional cos(*ϑ*_{s}) term. It is the presence of this additional term that leads to small angle roll-off in the K-correlation BSDF, which Dittman indicates is consistent with the observed behavior for many surface finishes.

An OpticStudio file (K-correlation vs. ABg.ZMX) has been set up to investigate the structure of the radiant intensity profile for emission resulting from scattering off of a mirror at normal incidence. This file is provided in the .ZIP folder which is located at the end of this article. As with the previous example, the file contains two detector objects: one to record the specular energy and the other to record the scattered intensity profile. The inputs for the K-correlation model in this case are:

These inputs correspond to a TIS value of 0.117 (ray wavelength = reference wavelength = 0.55 mm in this case). An ABg scatter profile was created to mimic these K-correlation inputs and yields the same TIS value:

Radiant intensity data at Y = 0 degrees were obtained for both scattering profiles, and the results were plotted on a single graph in Excel:

As expected, results from the K-correlation model are more peaked, due to the presence of the cos(*ϑ*_{s}) term in the BSDF. The peak intensity is also higher for the K-correlation model, as expected, since this is needed to provide agreement for the TIS values (= integral over the distribution) between the two models. If we were to scale the ABg model by the ratio of peak amplitudes (≈ √2) and by the cosine of the scatter angle, we would find exact agreement in the results:

## References

1. Peter G. Nelson, "An Analysis of Scattered Light in Reflecting and Refracting Primary Objectives for Coronagraphs", Coronal Solar Magnetism Observatory Technical Note 4 (https://opensky.ucar.edu/islandora/object/reports%3A12/datastream/PDF/view).

2. Michael G. Dittman, "K-correlation power spectral density & surface scatter model", Proceedings of SPIE, Vol. 6291, p. 62910R (2006).

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