Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media

The scattering of polarized light incident from one dielectric medium on its two-dimensional randomly rough interface with a second dielectric medium is studied. A reduced Rayleigh equation for the scattering amplitudes is derived for the case where p- or s-polarized light is incident on this interface, with no assumptions being made regarding the dielectric functions of the media. Rigorous, purely numerical, nonperturbative solutions of this equation are obtained. They are used to calculate the reflectivity and reflectance of the interface, the mean differential reflection coefficient, and the full angular distribution of the intensity of the scattered light. These results are obtained for both the case where the medium of incidence is the optically less dense medium and in the case where it is the optically more dense medium. Optical analogs of the Yoneda peaks observed in the scattering of x rays from metal surfaces are present in the results obtained in the latter case. Brewster scattering angles for diffuse scattering are investigated, reminiscent of the Brewster angle for flat-interface reflection, but strongly dependent on the angle of incidence. When the contribution from the transmitted field is added to that from the scattered field it is found that the results of these calculations satisfy unitarity with an error smaller than ${10}^{-4}$.

Figure 1

A sketch of the scattering geometry assumed in this work. The figure also shows the coordinate system used, angles of incidence (${\theta }_0,{\varphi }_0$) and scattering (${\theta }_s,{\varphi }_s$), and the corresponding lateral wave vectors ${\mathbf{k}}_{\parallel }$ and ${\mathbf{q}}_{\parallel }$, respectively.

Figure 2

The contribution to the incoherent component of the mean differential reflection coefficient from the in-plane, co-polarized scattering of p- and s-polarized light incident normally $[{\theta }_0=0^{\circ }]$ on (a) a random vacuum-dielectric interface $[{\varepsilon }_1=1.0,{\varepsilon }_2=2.64]$ and (b) a dielectric-vacuum interface $[{\varepsilon }_1=2.64,{\varepsilon }_2=1.0]$ as a function of the polar angle of scattering ${\theta }_s$. The solid curves were obtained on the basis of numerically solving the reduced Rayleigh equations [Eq. (27)] for an ensemble of 4500 surface realizations. The dashed curves are results from small-amplitude perturbation theory [Eq. (53)], included for comparison. The specular direction of reflection is indicated by the vertical dash-dotted line at ${\theta }_s=0^{\circ }$, and in (b) the dotted lines at $|{\theta }_s|={\theta }_c={sin}^{-1}\sqrt{{\varepsilon }_2/{\varepsilon }_1}\approx {38.0}^{\circ }$ indicate the positions of the critical angle for total internal reflection (as expected for a flat surface system). Results for cross-polarized scattering have not been indicated since they are generally suppressed in the plane of incidence. The wavelength of the incident light in vacuum was $\lambda$. The rough interface was assumed to have a root-mean-square roughness of $\delta =\lambda /40$, and it was characterized by an isotropic Gaussian power spectrum [Eq. (4)] of transverse correlation length $a=\lambda /4$. In the numerical calculations it was assumed that the surface covered an area $L\times L$, with $L=25\lambda$, and the surface was discretized on a grid of $321\times 321$ points.

Figure 3

Same as Fig. 2, but for the root-mean-square roughness $\delta =\lambda /20$.

Figure 4

The incoherent component of the mean differential reflection coefficient, showing the full angular intensity distribution as a function of the lateral wave vector of the light scattered by a rough interface between vacuum and a dielectric. The light was incident on the surface from the vacuum, $[{\varepsilon }_1=1.0,{\varepsilon }_2=2.64]$. The angles of incidence were $({\theta }_0,{\varphi }_0)=(0^{\circ },0^{\circ })$. The position of the specular direction in reflection is indicated by white dots. The parameters assumed for the scattering geometry and used in performing the numerical simulations had values that are identical to those assumed in obtaining the results of Fig. 2. The in-plane intensity variations in panels (b) and (f) are the curves depicted in Fig. 2. The star notation, e.g., $p\to ☆$, indicates that the polarization of the scattered light was not recorded; hence, panel (a) is the sum of panels (b) and (c), and panel (d) is the sum of panels (e) and (f). Furthermore, for the subfigures in the third column the open circle in, e.g. $\circ \to ☆$ symbolizes that the incident light was unpolarized; these simulation results were obtained by taking the arithmetic average of the other two subfigures in the same row. The roughness parameters assumed in obtaining these results were $\delta =\lambda /40$ and $a=\lambda /4$.

Figure 5

The same as Fig. 4, but for light incident from the dielectric side onto the interface with vacuum, $[{\varepsilon }_1=2.64$; ${\varepsilon }_2=1.0]$. The in-plane intensity variations in panels (b) and (f) are the curves depicted in Fig. 2. Notice the rapid changes in intensity around the polar angle ${\theta }_s={\theta }_c={sin}^{-1}\sqrt{{\varepsilon }_1/{\varepsilon }_2}$ [or $q_{\parallel }=\sqrt{{\varepsilon }_2}\omega /c]$.

Figure 6

(a) Same as Fig. 2, but for angles of incidence $({\theta }_0,{\varphi }_0)=({66.9}^{\circ },0^{\circ })$. (b) Same as panel 6, but for out-of-plane scattering $[{\varphi }_s=\pm {90}^{\circ }]$. Results for combinations of the polarizations of the incident and scattered light for which the scattered intensity was everywhere negligible have been omitted. Parameters: ${\varepsilon }_1=1.0,{\varepsilon }_2=2.64$; $\delta =\lambda /40,a=\lambda /4$.

Figure 7

Same as Fig. 6, but for root-mean-square roughness $\delta =\lambda /20$.

Figure 8

Same as Fig. 4, but for the angles of incidence $({\theta }_0,{\varphi }_0)=({66.9}^{\circ },0^{\circ })$. Parameters: ${\varepsilon }_1=1.0,{\varepsilon }_2=2.64$; $\delta =\lambda /40,a=\lambda /4$.

Figure 9

(a) Same as Fig. 2, but for angles of incidence $({\theta }_0,{\varphi }_0)=({34.5}^{\circ },0^{\circ })$. (b) Same as panel (a), but for out-of-plane scattering $[{\varphi }_s=\pm {90}^{\circ }]$. Results for combinations of the polarizations of the incident and scattered light for which the scattered intensity was everywhere negligible have been omitted. Parameters: ${\varepsilon }_1=2.64,{\varepsilon }_2=1.0$; $\delta =\lambda /40,a=\lambda /4$.

Figure 10

Same as Fig. 9, but for root-mean-square roughness $\delta =\lambda /20$.

Figure 11

Same as Fig. 5, but for the angles of incidence $({\theta }_0,{\varphi }_0)=({34.5}^{\circ },0^{\circ })$. As can be seen from the position of the white dot, this figure captures the scattering distribution when the polar angle of incidence ${\theta }_0$ is close to the critical angle ${\theta }_c={sin}^{-1}\sqrt{{\varepsilon }_2/{\varepsilon }_1}$ for a corresponding flat-interface system. Parameters: ${\varepsilon }_1=2.64$; ${\varepsilon }_2=1.0$; $\delta =\lambda /40,a=\lambda /4$.

Figure 12

Same as Fig. 11, but for the angles of incidence $({\theta }_0,{\varphi }_0)=({45.5}^{\circ },0^{\circ })$. Parameters: ${\varepsilon }_1=2.64,{\varepsilon }_2=1.0$; $\delta =\lambda /40,a=\lambda /4$.

Figure 13

(a) The reflectivities ${\mathcal{R}}_{\alpha }\left({\theta }_0\right)$ of a two-dimensional randomly rough vacuum-dielectric interface $[{\varepsilon }_1=1.0,{\varepsilon }_2=2.64]$ for p- and s-polarized light as functions of the polar angle of incidence. (b) The same as in panel (a), but for a dielectric-vacuum interface $[{\varepsilon }_1=2.64,{\varepsilon }_2=1.0]$. The quantity ${\mathcal{R}}_{\alpha }^F\left({\theta }_0\right)$ indicates the Fresnel reflection coefficient (flat surface reflectivity). The critical angle ${\theta }_0={\theta }_c={sin}^{-1}\sqrt{{\varepsilon }_2/{\varepsilon }_1}$ for total internal reflection for an equivalent flat-interface system is indicated by a vertical dashed line in panel (b). Several simulations were run with small perturbations in the surface length $L$ in order to obtain reflectivity data with higher angular resolution. The roughness parameters assumed in obtaining these results were $\delta =\lambda /40$ and $a=\lambda /4$.

Figure 14

The ${\theta }_0$ dependence of the contribution to the reflectance from p- and s-polarized incident light that has been scattered incoherently from a two-dimensional randomly rough surface. This quantity is for $\beta$-polarized incident light defined as ${\mathcal{R}}_{\beta }{\left({\theta }_0\right)}_{\mathrm{incoh}}={\mathcal{R}}_{\beta }\left({\theta }_0\right)-{\mathcal{R}}_{\beta }\left({\theta }_0\right)$. (a) The reflectances for a vacuum-dielectric interface $[{\varepsilon }_1=1.0,{\varepsilon }_2=2.64]$ for p- and s-polarized light as functions of the polar angle of incidence. (b) Same as panel (a), but for a dielectric-vacuum interface $[{\varepsilon }_1=2.64,{\varepsilon }_2=1.0]$. As in Fig. 13, the critical angle for total internal reflection in a corresponding flat-interface system, ${\theta }_c$, is indicated by a vertical dashed line in Fig. 14. Several simulations were run with small perturbations in the surface length $L$ in order to obtain reflectance data with higher angular resolution. The roughness parameters assumed in obtaining these results were $\delta =\lambda /40$ and $a=\lambda /4$.