SCATMECH > Classes and Functions > Surface Scattering Models > Polydisperse_Sphere_BRDF_Model

class Polydisperse_Sphere_BRDF_Model

The class Polydisperse_Sphere_BRDF_Model is a BRDF_Model that uses the Double_Interaction_BRDF_Model with MieScatterer and with a diameter distribution given by SurfaceParticleSizeDistribution. This model is useful for estimating scatter from contaminated surfaces.

Rendered image showing many spheres on a surface with different diameters.
This model evaluates the Mueller matrix BRDF ${\bf f}_{\rm r}$ by integrating over the particle size distribution $N(D)$, \begin{equation} {\bf f}_{\rm r} = \int_{D_{\rm start}}^{D_{\rm end}} {\bf f}_{\rm r}^{\rm sphere}[D,N(D)]\; {\rm d}D. \label{EqA} \end{equation} where ${\bf f}_{\rm r}^{\rm sphere}[D,N(D)]$ is the Mueller matrix BRDF for spheres of diameter $D$ and surface particle density $N(D)$. Double_Interaction_BRDF_Model with MieScatterer is used to evaluate ${\bf f}_{\rm r}^{\rm sphere}[D,N(D)]$. A change of variables can be made in Eq. $(\ref{EqA})$, so that it can be written as \begin{equation} {\bf f}_{\rm r} = \int_{\log D_{\rm start}}^{\log D_{\rm end}} {\bf f}_{\rm r}^{\rm sphere}[D,N(D)]\; D \; {\rm d}(\log D). \label{EqB} \end{equation} Finally, Eq. $(\ref{EqB})$ is approximated by the sum \begin{equation} {\bf f}_{\rm r} = \sum_{i=0}^n {\bf f}_{\rm r}^{\rm sphere}[D_i,N(D_i)]\;D_i \; s. \label{EqC} \end{equation} where $D_i=(1+s)^i D_{\rm start}$ and $n={\rm floor}[\log(D_{\rm end}/D_{\rm start})/\log(1+s)]$. The user chooses $D_{\rm start}$, $D_{\rm end}$, and $s\ll 1$ and should check that the result has converged.

Parameters:

Parameter Data Type Description Default
lambda double Wavelength of the light in vacuum [µm].
(Inherited from BRDF_Model.)		 0.532
type int Indicates whether the light is incident from above the substrate or from within the substrate and whether the scattering is evaluated in reflection or transmission. The choices are:
0 : Light is incident from the above the substrate, and scattering is evaluated in reflection.
1 : Light is incident from the above the substrate, and scattering is evaluated in transmission.
2 : Light is incident from the within the substrate, and scattering is evaluated in reflection.
3 : Light is incident from the within the substrate, and scattering is evaluated in transmission.
For 1, 2, and 3, the substrate must be non-absorbing.
(Inherited from BRDF_Model).		 0
substrate dielectric_function The optical constants of the substrate, expressed as a complex number (n,k) or, optionally, as a function of wavelength.
(Inherited from BRDF_Model.)		 (4.05,0.05)
distribution SurfaceParticleSizeDistribution The distribution of sphere diameters and the total number of spheres per unit area . SurfaceParticleSizeDistribution
stack StackModel_Ptr Description of any stack of films deposited on the substrate. No_StackModel
particle dielectric_function The optical constants of the spheres, expressed as a complex number (n,k) or, optionally, as a function of wavelength. (1.5,0.0)
Dstart double The starting diameter [$D_{\rm start}$ in Eq. $(\ref{EqA})$] for the integration [µm]. This parameter should be at least as small as the smallest diameter in the distribution that will contribute significantly to the BRDF. 0.1
Dend double The ending diameter [$D_{\rm end}$ in Eq. $(\ref{EqA})$] for the integration [µm]. This parameter should be at least as large as the largest diameter in the distribution that will contribute significantly to the BRDF. 100
Dstep double The fractional step size [$s$ in Eq. $(\ref{EqC})$] for the integration. This parameter should be much less than 1. 0.01
fractional_coverage double If fractional_coverage is non-zero, then the model will scale the distribution so that the fractional area coverage of the spheres on the surface is fractional_coverage. The fractional area coverage $f_A$ of the distribution is determined by integrating \begin{equation} f_A = \int_{D_{\rm start}}^{D_{\rm end}} N(D)\; \pi (D/2)^2\; {\rm d}D \approx \sum_{i=0}^n N(D_i) \;\pi (D_i/2)^2 \; D_i \;s. \end{equation} If fractional_coverage is zero, then the model will assume that the distribution is appropriately normalized to give the fractional area coverage. 0
antirainbow double Non-absorbing spheres usually exhibit rainbows. This parameter adds a small amount of absorption to the spheres. The minimum extinction coefficient is determined to be $k=a\lambda/(4D\pi)$, where $a$ is the antirainbow parameter, $\lambda$ is the wavelength, and $D$ is the diameter. That is, if $a=1$, the particles will have an absorption coefficient given by $1/D$. To eliminate rainbows, a value of $a=10$ is usually sufficient. 0

See also:

SCATMECH Home,   Conventions,   BRDF_Model,   dielectric_stack,   Double_Interaction_BRDF_Model,   MieScatterer,   SurfaceParticleSizeDistribution.

K. B. Nahm and W. L. Wolfe, "Light-scattering models for spheres on a conducting plane: comparison with experiment," Appl. Opt. 26, 2995-2999 (1987).

R.V. Peterson, P.G. Magallanes, and D.F. Rock, "Tailored particle distributions derived from MIL-STD-1246," Proc. SPIE 4774, 79-98 (2002).

M. G. Dittman, "Contamination scatter functions for stray-light analysis," Proc. SPIE 4774, 99-110 (2002).

J. Fleming, B. Matheson, M.G. Dittman, F. Grochocki, and B. Firth, "Modeling particle distributions for stray light analysis," Proc. SPIE 6291, 62910T (2006).

K. Balasubramanian, S. Shaklan, and A. Give'on, "Stellar coronagraph performance impact due to particulate contamination and scatter," Proc. SPIE 7440, 74400T (2009).

Include file:

#include "polydisperse.h"

Source code:

polydisperse.cpp

For More Information

SCATMECH Technical Information and Questions
Sensor Science Division Home Page
Sensor Science Division Inquiries
Website Comments

Current SCATMECH version: 7.22 (April 2021)