class Growth_Roughness_Stack_BRDF_Model
The class Growth_Roughness_Stack_BRDF_Model uses Roughness_Stack_BRDF_Model to calculate the scatter from a stack of films, whose roughness evolves from that of the substrate. The model employs a linear growth model to determine the roughness of each interface and the cross-correlation statistics. The model takes two power spectral density (PSD) functions as input: $S_0(f)$ for the substrate and $S_{\rm int}(f)$ for the intrinsic roughness of the coatings (in the large thickness limit), where $f$ is spatial frequency. A single intrinsic roughness is used and applies to all of the coating materials. A replication function with two adjustable parameters, a relaxation parameter $l_r$ and a spatial frequency exponent $n$, are used to describe propagation of roughness from one interface to the next and for development of the finite thickness intrinsic roughness. This model is based upon that of Stearns. The replication and growth factor is given by \begin{equation} a(f)=\exp{[-l_r^{n-1}t_i(2\pi f)^n] } \end{equation}
where $t_i$ is the thickness of layer $i$. The PSD of inteface $i$ is given by \begin{equation} S_i(f) = a(f)^2 S_{i-1}(f) + [1-a(f)^2] S_{\rm int}(f) \end{equation} The function $a(f)$ also expresses the correlation between respective interfaces.
When $l_r =0$ and $S_{\rm int}(f)=0$, the results of the model will match that of Correlated_Roughness_Stack_BRDF_Model. When $l_r\rightarrow\infty$ and $S_{\rm int}(f)= S_0(f)$, the results of the model will match that of Uncorrelated_Roughness_Stack_BRDF_Model.
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) |
| psd | PSD_Function_Ptr | The
two-dimensional power spectrum of the surface height
function for the substrate, $S_0(f)$ [µm4]. (Inherited from Roughness_BRDF_Model). |
ABC_PSD_Function |
| stack | StackModel_Ptr | Description of the stack of films deposited on the substrate. | No_StackModel |
| intrinsic | PSD_Function_Ptr | The two-dimensional power spectrum of the surface height function for the coatings in the thick coating limit, $S_{\rm int}(f)$. Note that, for simplicity, $S_{\rm int}(f)$ is assumed to be the same for all coating materials. [µm4]. | ABC_PSD_Function |
| relaxation | double | The relaxation parameter $l_r$ for the growth process, corresponding to the characteristic length for which roughness from preceeding layers is damped out. Note that, for simplicity, this value is assumed to be the same for all coating materials. [µm] | 0.05 |
| exponent | double | The relaxation exponent $n$ for the growth process. Different values of $n$ correspond to different growth mechanisms: $n=1$ indicates viscous flow, $n=2$ indicates the condensation and re-evaporation, $n=3$ indicates bulk diffusion, and $n=4$ indicates surface diffusion. Note that, for simplicity, $n$ is assumed to be the same for all coating materials. | 2 |
See also:
SCATMECH Home, Conventions, BRDF_Model, Roughness_BRDF_Model, Roughness_Stack_BRDF_Model,
dielectric_stack
D.G. Stearns, "Stochastic model for thin film growth and erosion," Appl. Phys. Lett. 62(15), 1745-1747 (1993).
E. Spiller, D. Stearns, and M. Krumrey, "Multilayer x-ray mirrors: Interfacial roughness, scattering, and image quality," J. Appl. Phys. 74(1), 107-118 (1993).
J. M. Elson, "Multilayer-coated optics: guided-wave
coupling and scattering by means of interface random
roughness," J. Opt. Soc. Am. A 12(4), 729
(1995).
J. M. Elson, "Theory and Software for Light Scattering From Multilayer Optical
Components with Interfacial Roughness," Naval Air Warfare Center Weapons Division (NAWCWPNS)
Technical Publication 8084 (1992).
T.A.Germer, "Measuring Interfacial Roughness by Polarized Optical Scattering," in Light Scattering and Nanoscale Surface Roughness, Ed. A.A. Maradudin, (Springer,New York, 2007).
E. Spiller, S. Baker, E. Parra, and C. Tarrio, "Smoothing of mirror substrates by thin-film deposition," in EUV, X-Ray, and Neutron Optics and Surfaces, C.A. MacDonald, et al., Eds., Proc. SPIE 3767, 143-153 (1999).
Include file:
#include "allrough.h"
Source code:
allrough.cpp
Definition of public elements:
class Growth_Roughness_Stack_BRDF_Model : public Roughness_BRDF_Model { };
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)