# Effective Optical Scattering Range Determination Based on Ray Tracing
## 2 Theoretical Background
### Scattering  
The scattering geometry is defined in Figure 1. The incident beam impinges the surface at the position of (x, y) with the angle of incidence \(\theta_{i}\) and an illuminated area \(dA\). The beam is scattered into a solid angle \(d\omega\) with the polar angle \(\theta_{s}\) and the azimuth \(\varphi_{s}\). The scattered radiance is related to the incident irradiance with:  
\[L(x,y,\theta_{s},\varphi_{s})=\frac{d^{2}\Phi_{s}(x,y,\theta_{s},\varphi_{s})} {d^{2}\Phi_{\text{in}}(x,y,\theta_{\text{in}})\cdot\cos(\theta_{s})\cdot d \omega}\cdot\frac{d^{2}\Phi_{\text{in}}(x,y,\theta_{\text{in}})}{dA} \tag{6}\]  
where \(\Phi_{s}\), \(\Phi_{\text{in}}\) is the scattered flux and the incident flux. The first term on the right-hand side is usually called the Bidirectional Scattering Distribution Function (BSDF) which is usually split into a reflected component (BRDF) and a transmitted component (BTDF). For an isotropic surface, BRDF exhibits rotational symmetry and can be reduced to an in-plane scattering [12]. The radiometric definition of the scattering is straightforward. However, the accurate measurement and modeling can be difficult. The scattering signature is a result of roughness, surface structure, contaminations, scratches, wavelength, and polarization. Furthermore, the digital noise and optical alignment also make the signature even more complicated. Moreover, the scattering can be related to the surface PSD function with the Rayleigh-Rice vector perturbation theory:  
\[\text{BRDF}(\theta_{i},\theta_{s},\varphi_{s},\lambda)=\frac{16\pi^{2}}{ \lambda^{4}}\cdot\cos(\theta_{i})\cdot\cos(\theta_{s})\cdot Q\cdot\text{PSD }(f) \tag{7}\]  
where the Q is called the optical factor which is a function of the incident angle, the scattered angle, the complex refractive index of the surface and the state of polarization. For an isotropic, optically clean and smooth surface, the hemispherical grating equation can be used to evaluate the spatial frequency:  
\[f=\frac{\sin\theta_{s}-\sin\theta_{i}}{\lambda} \tag{8}\]  
Figure 1: Scattering geometry for the definition of BRDF. The total integrated scattering (TIS) is defined as the ratio of light scattered into a hemisphere to the total light reflected by the surface. According to scalar scattering theory, the ratio of specular reflectance \(\mathrm{R_{s}}\) to the total reflectance \(\mathrm{R}\) can be estimated from rms roughness:  
\[\frac{\mathrm{R_{s}}}{\mathrm{R}}=\exp[(-\frac{4\pi}{\lambda}\sigma_{\mathrm{ rms}}\cos\theta_{\mathrm{i}})^{2}] \tag{9}\]  
When it is normal incidence, Equation (9) can be rewritten in combination with TIS:  
\[\mathrm{TIS}=\frac{\mathrm{R-R_{s}}}{\mathrm{R}}\approx\left(\frac{4\pi \sigma_{\mathrm{rms}}}{\lambda}\right)^{2} \tag{10}\]  
However, for a superpolished surface, \(\mathrm{R_{s}}\) / \(\mathrm{R}\) is close to unity and difficult to measure. Therefore, it is reasonable to collect the scattered light instead of the loss of reflectance. After integrating hemispherical scattering, the TIS becomes:  
\[\mathrm{TIS}=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi/2}\mathrm{BRDF}\cdot \sin(\theta_{\mathrm{s}})\cos(\theta_{\mathrm{s}})\mathrm{d}\theta_{\mathrm{s }}\mathrm{d}\varphi_{\mathrm{s}} \tag{11}\]  
Figure 2 shows the relation between the fundamental concepts of roughness and optical scattering. The power spectral density function can be derived either directly from the surface profile through the square of the Fourier transform or indirectly from the optical scattering if the angular distribution of scattered light is measured. However, the surface profile cannot be obtained from the PSD because the phase information about the surface is lost. The rms roughness is the square root of the zeroth term in the autocovariance function. Alternatively, the rms roughness can be obtained by integrating the measured values of the angle-resolved scattering for all angles to yield the TIS and then using Equations (9) and (10) for estimation.