Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces and its application to the retrieval of self-affine parameters

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces is studied for the purpose of using the intensity scattered from them to obtain the Hurst exponent and topothesy that characterize the self-affine roughness. By the use of the Kirchhoff approximation, a closed-form mathematical expression for the angular dependence of the mean differential reflection coefficient is derived under the assumption that the surface is illuminated by a plane incident wave. It is shown that this quantity can be expressed in terms of the isotropic, bivariate ($\alpha$-stable) Lévy distribution of a stability parameter that is two times the Hurst exponent of the underlying surface. Features of the expression for the mean differential reflection coefficient are discussed, and its predictions compare favorably over large regions of parameter space to results obtained from rigorous computer simulations based on equations of scattering theory. It is demonstrated how the Hurst exponent and the topothesy of the self-affine surface can be inferred from scattering data it produces. Finally, several possible scattering configurations are discussed that allow for an efficient extraction of these self-affine parameters.

Figure 1

Schematics of the scattering geometry that we consider in this work.

Figure 2

The mean DRCs calculated on the basis of Eq. (19) [“Full solution”]; Eq. (23) [“Specular expansion”]; and Eq. (27) [“Diffuse expansion”] for the polar angles of incidence ${\theta }_0=0^{\circ }$ and ${25}^{\circ }$. The self-affine surface parameters were $H=0.70$ and $\ell ={10}^{-5}\lambda$.

Figure 3

The in-plane or out-of-plane angular dependence of the mean DRC of the scattered wave for self-affine surfaces characterized by Hurst exponents $H=0.30$, 0.50, and 0.70 and fixed topothesy $\ell ={10}^{-5}\lambda$, where $\lambda$ denotes the wavelength of the incident plane wave. The results were obtained on the basis of the analytic expression in Eq. (19). The subplots correspond to (a) the polar angle of incidence ${\theta }_0=0^{\circ }$ (in-plane and out-of-plane scattering coincide here due to the assumed isotropy of the surface); (b) ${\theta }_0={50}^{\circ }$, in-plane scattering; and (c) ${\theta }_0={50}^{\circ }$, out-of-plane scattering $[{\varphi }_s={\varphi }_0\pm {90}^{\circ }]$. In all cases, the azimuthal angle of incidence was ${\varphi }_0=0^{\circ }$. The vertical dashed lines in Figs. 3 and 3 indicate the specular direction. The open symbols, added for reasons of clarity, represent the value of the mean DRCs at the specular directions. Note the logarithmic scale used on the second axis.

Figure 4

The in-plane or out-of-plane angular dependence of the mean DRC of the scattered wave for self-affine surfaces characterized by Hurst exponent $H=0.70$ and different values of the topothesy $\ell$. The results were obtained on the basis of the analytic expression in Eq. (19). The subplots correspond to (a) the polar angle of incidence ${\theta }_0=0^{\circ }$ (in-plane and out-of-plane scattering coincide here due to the assumed isotropy of the surface); (b) ${\theta }_0={50}^{\circ }$, in-plane scattering; and (c) ${\theta }_0={50}^{\circ }$, out-of-plane scattering $[{\varphi }_s={\varphi }_0\pm {90}^{\circ }]$. In all cases, the azimuthal angle of incidence was ${\varphi }_0=0^{\circ }$. The vertical dashed lines in the top two subfigures correspond to the specular direction. Note the logarithmic scale used on the second axis.

Figure 5

The scaled in-plane mean DRC, $\langle \partial R\left({\mathbf{q}}_{\parallel }\right|{\mathbf{k}}_{\parallel })/\partial {\mathrm{\Omega }}_s\rangle \times m^{-1}\left({\mathbf{q}}_{\parallel }\right|{\mathbf{k}}_{\parallel })$, defined by Eqs. (19) and (27b), as functions of $|{\mathbf{q}}_{\parallel }-{\mathbf{k}}_{\parallel }|/(\omega /c)$ using a double logarithmic scale. The self-affine parameters were assumed to be $H=0.70,\ell ={10}^{-6}\lambda$ (blue line) and $\ell ={10}^{-5}\lambda$ (orange line) and the polar angle of incidence was ${\theta }_0={50}^{\circ }$. The dashed lines represent inverse power-law function scaling as $|{\mathbf{q}}_{\parallel }-{\mathbf{k}}_{\parallel }{|}^{-2-2H}$ [see Eq. (27a)] for the two values of the Hurst exponent $H=0.70$ and $H=0.50$ as indicated in the figure.

Figure 6

The full angular distribution of the mean DRC of the scattered wave based on Eq. (19) for self-affine surfaces for a series of Hurst exponents and incident angles. The topothesy is fixed at $\ell /\lambda ={10}^{-5}$ where $\lambda$ denotes the wavelength of the incident plane wave. Row (a) corresponds to $H=0.70$, row (b) to $H=0.50$, and row (c) to $H=0.30$. The polar angle of incidence is constant in each column, and is ${\theta }_0=0^{\circ }, {25}^{\circ }$, and ${50}^{\circ }$ for each column from left to right. Notice the significant difference in values of the color scale between the rows. The $log$ symbol used in the labels denotes the base 10 logarithm.

Figure 7

The in-plane or out-of-plane dependence of the mean DRC of the scattered wave for self-affine surfaces characterized by the same slope $s\left(\lambda \right)=0.0631$. The results were obtained on the basis of the analytic expression in Eq. (19). The subplots correspond to the polar angle of incidence (a) ${\theta }_0=0^{\circ }$ (in-plane and out-of-plane scattering coincide here due to the assumed isotropy of the surface); (b) ${\theta }_0={50}^{\circ }$, in-plane scattering; and (c) ${\theta }_0={50}^{\circ }$, out-of-plane scattering. The vertical dashed lines in panels (a) and (b) correspond to the specular direction. Note the logarithmic scale used on the second axis.

Figure 8

Comparison of the in-plane angular dependence of the mean DRCs obtained on the basis of the Kirchhoff approximation using an incident plane wave, Eq. (19), or a Gaussian beam of width $w/\lambda \in \{4,10,32\}$, for a self-affine surface of parameters $H=0.70$ and $\ell ={10}^{-6}\lambda$. The polar angles of incidence were (a) ${\theta }_0=0^{\circ }$ and (b) ${\theta }_0={50}^{\circ }$. The value of the edges of the surfaces assumed in obtaining these results was $L=3w$.

Figure 9

The in-plane dependence of the mean DRC of the scattered wave for self-affine surfaces characterized by Hurst exponent $H=0.70$ and different values of the topothesy $\ell$ using the Dirichlet boundary condition. The wavelength of the incident wave was $\lambda$. The results shown with solid lines were obtained on the basis of rigorous simulations that assumed an incident Gaussian beam of width $w=10\lambda$, and the dashed lines show results based on the analytic expression in Eq. (19) derived under the assumption of an incident plane wave. The subplots correspond to the topothesy: (a) $\ell ={10}^{-6}\lambda$, (b) $\ell ={10}^{-5}\lambda$, (c) $\ell ={10}^{-4}\lambda$. Each plot contains the in-plane cuts for polar incidence angle ${\theta }_0=0^{\circ }$ and ${\theta }_0={50}^{\circ }$. The vertical dashed lines in panels (a) and (b) correspond to the specular direction. The vertical dashed lines in panels (a) and (b) correspond to the specular direction. Note the logarithmic scale used on the 2nd axis.

Figure 10

Same as Fig. 9 but for a self-affine Neumann surface $[H=0.70]$.

Figure 11

The in-plane or out-of-plane dependence of the mean DRC of the scattered wave for self-affine surfaces characterized by the slope $s\left(\lambda \right)=0.0631$. The results were obtained on the basis of rigorous simulations using the Dirichlet boundary condition. The subplots correspond to the polar angle of incidence (a) ${\theta }_0=0^{\circ }$ (in-plane and out-of-plane scattering coincide here due to the assumed isotropy of the surface); (b) ${\theta }_0={50}^{\circ }$, in-plane scattering; and (c) ${\theta }_0={50}^{\circ }$, out-of-plane scattering. The vertical dashed lines in panels (a) and (b) correspond to the specular direction. Note the logarithmic scale used on the second axis.

Figure 12

The scaled mean DRCs, $〈\partial R\left({\mathbf{q}}_{\parallel }\right|{\mathbf{k}}_{\parallel })/\partial {\mathrm{\Omega }}_s〉/m\left({\mathbf{q}}_{\parallel }\right|{\mathbf{k}}_{\parallel })$, as functions of $|{\mathbf{q}}_{\parallel }-{\mathbf{k}}_{\parallel }|/(\omega /c)$ obtained within the Kirchhoff approximation (solid and dashed lines) or by rigorous computer simulations for Dirichlet and Neumann surfaces (open symbols). The parameters assumed for the self-affine surfaces were $H=0.70,\ell ={10}^{-6}\lambda$ [Figs. 12 and 12] and $\ell ={10}^{-5}\lambda$ [Figs. 12 and 12], the wavelength of the incident wave was $\lambda$ and the polar angles of incidence were ${\theta }_0=0^{\circ }$ [Figs. 12 and 12] and (b) ${\theta }_0={50}^{\circ }$ [Figs. 12 and 12]. The dashed black lines were obtained on the basis of Eq. (19) and therefore assume plane-wave illumination. The result corresponding to the solid lines and rigorous results (open symbols) were obtained under the assumption of incident Gaussian beams of width $w=10\lambda$ and the edges of the surfaces were $L=3w=30\lambda$. In performing the rigorous simulations, the sampling interval assumed for the surfaces was $\mathrm{\Delta }{x}_{\parallel }=\lambda /7$ and the reported results were obtained as averages over $N_{\zeta }=1000$ surface realizations. As a guide to the eye, we have included two sets of gray lines; the solid gray lines of slopes $-2-2H$ represent the tail behavior of the scattered intensity distribution in Eq. (27) and the dashed horizontal gray lines correspond to the specular intensity for a plane incident wave, Eq. (22).

Figure 13

Inversion of in-plane scattering data obtained in computer experiments for Gaussian incident beams scattered from self-affine surfaces of Hurst exponent $H=0.70$ and several values of the topothesy $\ell$. (a) Monte Carlo simulated input data (blue open circles) for polar angle of incidence ${\theta }_0=0^{\circ }$ generated within the Kirchhoff approximation for a self-affine surface of topothesy $\ell ={10}^{-6}\lambda$ where $\lambda$ is the wavelength of the incident field. These data were obtained by averaging the results over $N_{\zeta }=10000$ surface realizations and the incident Gaussian beam had width $w=32\lambda$. The orange sold mean DRC curve was produced by inverting the input data to obtain the reconstructed self-affine parameters of values $H^{★}=0.70$ and ${\ell }^{★}=1.25\times {10}^{-6}\lambda$. (b) In-plane scattering data (blue open symbols) generated by rigorous simulations for self-affine Neumann surfaces of topothesy $\ell ={10}^{-6}\lambda$ and obtained by averaging the results over $N_{\zeta }=5000$ surface realizations. Incident Gaussian beams of width $w=10\lambda$ and polar angles of incidence ${\theta }_0=0^{\circ }$ and ${\theta }_0={50}^{\circ }$ were assumed in generating these results. Inversion of the computer generated data set for polar angle of incidence ${\theta }_0=0^{\circ }$ produced the orange solid line and the reconstructed self-affine parameters $H^{★}=0.71$ and ${\ell }^{★}=1.41\times {10}^{-6}\lambda$. The dashed orange line was produced from Eq. (19) using the reconstructed values for $H$ and $\ell$. (c) Same as Fig. 13 but computer-simulated data sets (open blue symbols) corresponding to a Dirichlet surface of topothesy $\ell ={10}^{-5}\lambda$. The inversion of the ${\theta }_0=0^{\circ }$ date set was performed for $|q_1|<0.5\omega /c$; this produced the solid orange line and the reconstructed parameter values $H^{★}=0.71$ and ${\ell }^{★}=7.0\times {10}^{-6}\lambda$. Using these values and ${\theta }_0={50}^{\circ }$ in Eq. (19) produced the orange dashed line.

Figure 14

Rocking scan curves for self-affine surfaces characterized by (a) several values of the Hurst exponent (see legend) and topothesy $\ell ={10}^{-5}\lambda$ and (b) Hurst exponent $H=0.70$ and several values of the topothesy $\ell$ (see legend) for ${\theta }_0={\theta }_s={45}^{\circ }$. The solid lines were obtained on the basis of Eq. (19) while the red open symbols were produced by rigorous computer simulations assuming a self-affine Neumann surface of parameters $H=0.70$ and $\ell ={10}^{-5}\lambda$ and an incident Gaussian beam of width $w=10\lambda$. The rocking angle $\vartheta$ is defined so that the effective polar angle of incidence is ${\theta }_0+\vartheta$, and the effective polar angle of scattering is ${\theta }_s-\vartheta$.

Figure 15

Rescaled in-plane mean DRCs from Fig. 9 that were generated by rigorous computer simulations of $w=10\lambda$ wide normally incident Gaussian beams scattered from self-affine Dirichlet surfaces of Hurst exponent $H=0.70$ and topothesies $\ell$ as indicated in the legend (open symbols). The rescaling was done according to Eq. (30) so what is presented is $〈\partial R\left({\mathbf{q}}_{\parallel }\right|{\mathbf{k}}_{\parallel })/\partial {\mathrm{\Omega }}_s〉M^{-1}({\mathbf{q}}_{\parallel },{\mathbf{k}}_{\parallel };H,\ell )$ as function of $\chi ({\mathbf{q}}_{\parallel },{\mathbf{k}}_{\parallel };H,\ell )$. The blue solid line is the “master curve” ${\mathcal{L}}_{2H}(\chi ,1/2)$ that is predicted by Eq. (30). The data collapse brings out the Lévy shape in accordance with Eq. (19).