# Surface Scattering

## Lambertian Scattering Model

Used for extremely diffusive surfaces. Perfect hemispherical scatter distribution.

#### Validity

* Isotropic Surface Roughness
* Ideal Diffuser.
* Only valid for perfectly diffusing materials.

#### BSDF

$$
\text{BSDF}(\beta-\beta\_0) = \frac{R}{\pi}
$$

Where $$\beta = \sin(\theta\_\text{scatter})$$, $$\beta\_0=\sin(\theta\_\text{scatter})$$ where R is the reflectivity of the sample.

#### Example Values

These values below were provided by the reference below, and re-presented here. These values are built into the "Simplistic Presets", where the idea of this table for this model in Rich Pfisterer's words: "Approximated Scatter Model"

<table><thead><tr><th width="583">Surface</th><th>R</th></tr></thead><tbody><tr><td>Ideal diffuser</td><td>1</td></tr><tr><td>Typical matte paper at normal incidence</td><td>0.85</td></tr><tr><td>Typical diffuse black paint at normal incidence</td><td>0.5</td></tr><tr><td>Perfect absorber</td><td>0</td></tr></tbody></table>

#### References

1. J. E. Harvey, *Understanding surface scatter phenomena: A linear systems formulation*, SPIE Press, Bellingham (2019).
2. [Approximated Scatter Models for Stray Light Analysis, Richard N. Pfisterer](https://photonengr.com/wp-content/uploads/2014/10/ApproximatedScatter1.pdf)

## ABg Scattering Model

Used for approximating smooth surface finishes.

#### Validity

* Isotropic Surface Roughness
* RMS Surface Roughness $$\ll \lambda$$ (Smooth Surfaces)
* Surface Roughness is Bandwidth Limited. I.e. The roughness is not just a single sinusoidal frequency, but a spread of frequencies about some dominant frequency.
* Simplistic Model, Ideal for quick turn-around. Wavelength dependence, and model parameters may differ from reality.

#### BSDF

$$
\text{BSDF}(\beta-\beta\_0) = \frac{A}{B+(\beta-\beta\_0)^g}
$$

Where $$\beta = \sin(\theta\_\text{scatter})$$, $$\beta\_0=\sin(\theta\_\text{scatter})$$ with some fitting parameters: A,B, and g.

#### Example Values

These values below were provided by the reference below, and re-presented here. These values are built into the "Simplistic Presets", where the idea of this table for this model in Rich Pfisterer's words: "Approximated Scatter Model"

{% tabs %}
{% tab title="Mirrors" %}

<table><thead><tr><th width="168">Polish</th><th width="67">g</th><th width="105">B</th><th>A</th></tr></thead><tbody><tr><td>Super</td><td>2.5</td><td>0.00001</td><td><span class="math">1.4\:(\frac{\sigma}{\lambda})^2</span></td></tr><tr><td>Slightly better than Standard</td><td>2.0</td><td>0.0001</td><td><span class="math">5.46\:(\frac{\sigma}{\lambda})^2</span></td></tr><tr><td>Standard</td><td>1.5</td><td>0.001</td><td><span class="math">13.92\: (\frac{\sigma}{\lambda})^2</span></td></tr><tr><td>Slightly worse than Standard</td><td>1.0</td><td>0.01</td><td><span class="math">25.51\: (\frac{\sigma}{\lambda})^2</span></td></tr></tbody></table>
{% endtab %}

{% tab title="Lenses" %}

<table><thead><tr><th width="168">Polish</th><th width="67">g</th><th width="105">B</th><th>A</th></tr></thead><tbody><tr><td>Super</td><td>2.5</td><td>0.00001</td><td><span class="math">0.35\:(\frac{\sigma\Delta n}{\lambda})^2</span></td></tr><tr><td>Slightly better than Standard</td><td>2.0</td><td>0.0001</td><td><span class="math">1.37\:(\frac{\sigma\Delta n}{\lambda})^2</span></td></tr><tr><td>Standard</td><td>1.5</td><td>0.001</td><td><span class="math">3.50\:(\frac{\sigma\Delta n}{\lambda})^2</span></td></tr><tr><td>Slightly worse than Standard</td><td>1.0</td><td>0.01</td><td><span class="math">6.35\: (\frac{\sigma\Delta n}{\lambda})^2</span></td></tr></tbody></table>
{% endtab %}
{% endtabs %}

#### References

&#x20;[Approximated Scatter Models for Stray Light Analysis, Richard N. Pfisterer](https://photonengr.com/wp-content/uploads/2014/10/ApproximatedScatter1.pdf)

## Bidirectional Scattering Distribution Function (BSDF)

Sum of ABg Parameters. See [ABg](#abg-scattering-model) for more information.

## Rayleigh-Rice Scattering Model

#### References

J. E. Harvey, *Understanding surface scatter phenomena: A linear systems formulation*, SPIE Press, Bellingham (2019).

## Beckmann-Kirchoff Scattering Model

#### References

J. E. Harvey, *Understanding surface scatter phenomena: A linear systems formulation*, SPIE Press, Bellingham (2019).

## Harvey-Shack Scattering Model

#### References

J. E. Harvey, *Understanding surface scatter phenomena: A linear systems formulation*, SPIE Press, Bellingham (2019).

## K-Correlation Scattering Model

#### References

J. E. Harvey, *Understanding surface scatter phenomena: A linear systems formulation*, SPIE Press, Bellingham (2019).