On high-frequency radiation scattering sensitivity to surface roughness in particulate media

Abstract

This paper analyzes the sensitivity of high-frequency radiation scattering in particulate media, to particle surface roughness. Ray-tracing theory and computation are employed. Since the magnitude of the Poynting vector ray, the irradiance, is the appropriate quantity to be tracked, the behavior of the reflectance, which controls the ratio of the reflected and incident Poynting vector magnitudes, is of primary concern. The reflectance is a highly nonlinear function of the refractive indices and angle of incidence. The present work first addresses the relationship between a single scatterer’s sensitivity to its surface roughness and then the response of a large number of scatterers to the surface roughness. The analysis indicates that, for a single scatterer, the sensitivity of the response to roughness decreases, up to a point, and then increases again, i.e., it is nonmonotone. However, for a system of multiple scatterers, this effect vanishes, due to multiple internal reflections which dominate the overall response characteristics. While it was relatively straightforward to compute the overall sensitivity of a single scattering body, for example a sphere, when multiple reflecting bodies are considered, numerical simulations are necessary because the reflected rays from one “rough” body will, in turn, be reflected to another “rough” body, etc. Examples are given for a system of randomly distributed scatterers.

Appendix: Generalized Fresnel relations

Following a generalization of the Fresnel relations for unequal magnetic permeabilities in Zohdi [35, 36] and Zohdi and Kuypers [45], we consider a plane harmonic wave incident upon a plane boundary separating two different optical materials, which produces a reflected wave and a transmitted (refracted) wave (Fig. 2). Two cases for the electric field vector are considered: (1) electric field vectors that are parallel (||) to the plane of incidence and (2) electric field vectors that are perpendicular ($\perp$) to the plane of incidence. In either case, the tangential components of the electric and magnetic fields are required to be continuous across the interface. Consider case (1). We have the following general vectorial representations

$$\begin{aligned} \varvec{E}_{||}=E_{||}\cos (\varvec{k}\cdot \varvec{r}-\omega t)\varvec{e}_1 \quad \mathrm{and} \quad \varvec{H}_{||}=H_{||}\cos (\varvec{k}\cdot \varvec{r}-\omega t)\, \varvec{e}_2, \end{aligned}$$

(5.1)

where $\varvec{e}_1$ and $\varvec{e}_2$ are orthogonal to the propagation direction $\varvec{k}.$ By employing the law of refraction ($n_\mathrm{i}\sin \theta _\mathrm{i}=n_\mathrm{t}\sin \theta _\mathrm{t}$), we obtain the following conditions relating the incident, reflected and transmitted components of the electric field quantities

$$\begin{aligned}&E_{||\mathrm{i}}\cos \theta _\mathrm{i}-E_{||\mathrm{r}}\cos \theta _\mathrm{r}= E_{||\mathrm{t}}\cos \theta _\mathrm{t} \quad \mathrm{and} \nonumber \\&H_{\perp \mathrm{i}}+H_{\perp \mathrm{r}}=H_{\perp \mathrm{t}}. \end{aligned}$$

(5.2)

Since, for plane harmonic waves, the magnetic and electric field amplitudes are related by $H=\frac{E}{v\mu },$ we have

$$\begin{aligned} E_{||\mathrm{i}}+E_{||\mathrm{r}}=\frac{\mu _\mathrm{i}}{\mu _\mathrm{t}}\frac{v_\mathrm{i}}{v_\mathrm{t}}E_{||\mathrm{t}}=\frac{\mu _\mathrm{i}}{\mu _\mathrm{t}}\frac{n_\mathrm{t}}{n_\mathrm{i}}E_{||\mathrm{t}}\mathop {=}\limits ^\mathrm{def}\frac{\hat{n}}{\hat{\mu }}E_{||\mathrm{t}}, \end{aligned}$$

(5.3)

where $\hat{\mu }\mathop {=}\limits ^\mathrm{def}\frac{\mu _\mathrm{t}}{\mu _\mathrm{i}},\,\hat{n}\mathop {=}\limits ^\mathrm{def}\frac{n_\mathrm{t}}{n_\mathrm{i}}$ and where $v_\mathrm{i},\,v_\mathrm{r}$ and $v_\mathrm{t}$ are the values of the velocity in the incident, reflected and transmitted directions.⁸ By again employing the law of refraction, we obtain the Fresnel reflection and transmission coefficients, generalized for the case of unequal magnetic permeabilities

$$\begin{aligned}&r_{||}=\frac{E_{||\mathrm{r}}}{E_{||\mathrm{i}}}= \frac{\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{i}-\cos \theta _\mathrm{t}}{\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{i}+\cos \theta _\mathrm{t}} \quad \mathrm{and} \nonumber \\&t_{||}=\frac{E_{||\mathrm{t}}}{E_{||\mathrm{i}}}=\frac{2\cos \theta _\mathrm{i}}{\cos \theta _\mathrm{t}+\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{i}}. \end{aligned}$$

(5.4)

Following the same procedure for case (2), where the components of $\varvec{E}$ are perpendicular to the plane of incidence, we have

$$\begin{aligned}&r_{\perp }= \frac{E_{\perp \mathrm{r}}}{E_{\perp \mathrm{i}}}= \frac{\cos \theta _\mathrm{i}-\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{t}}{ \cos \theta _\mathrm{i}+\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{t}} \quad \mathrm{and}\nonumber \\&t_{\perp }=\frac{E_{\perp \mathrm{t}}}{E_{\perp \mathrm{i}}} =\frac{2\cos \theta _\mathrm{i}}{\cos \theta _\mathrm{i}+\frac{\hat{n}}{\hat{\mu }}\cos \theta _\mathrm{t}}. \end{aligned}$$

(5.5)

Our primary interest is in the reflections. We define the reflectances as

$$\begin{aligned} R_{||}\mathop {=}\limits ^\mathrm{def}r^2_{||} \quad \mathrm{and} \quad R_{\perp }\mathop {=}\limits ^\mathrm{def}r^2_{\perp }. \end{aligned}$$

(5.6)

Particularly convenient forms for the reflections are

$$\begin{aligned}&r_{||}=\frac{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}-(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}+(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}} \quad \mathrm{and} \nonumber \\&r_{\perp }=\frac{\cos \theta _\mathrm{i}-\frac{1}{\hat{\mu }}(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}{\cos \theta _\mathrm{i}+\frac{1}{\hat{\mu }}(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}. \end{aligned}$$

(5.7)

Thus, the total energy reflected can be characterized by

$$\begin{aligned} R \mathop {=}\limits ^\mathrm{def}\left( \frac{E_\mathrm{r}}{E_\mathrm{i}}\right) ^2= \frac{E^2_{\perp \mathrm{r}}+E^2_{||\mathrm{r}}}{E^2_\mathrm{r}}= \frac{I_{||\mathrm{r}}+I_{\perp \mathrm{r}}}{I_\mathrm{i}}. \end{aligned}$$

(5.8)

If the resultant plane of oscillation of the (polarized) wave makes an angle of $\gamma _\mathrm{i}$ with the plane of incidence, then

$$\begin{aligned} E_{||\mathrm{i}}=E_\mathrm{i}\cos \gamma _\mathrm{i} \quad \mathrm{and} \quad E_{\perp \mathrm{i}}=E_\mathrm{i}\sin \gamma _\mathrm{i}, \end{aligned}$$

(5.9)

and it follows from the previous definition of I that

$$\begin{aligned} I_{||\mathrm{i}}=I_\mathrm{i}\cos ^2\gamma _\mathrm{i} \quad \mathrm{and} I_{\perp \mathrm{i}}=I_\mathrm{i}\sin ^2\gamma _\mathrm{i}. \end{aligned}$$

(5.10)

Substituting these expression back into the expressions for the reflectances yields

$$\begin{aligned} R=\frac{I_{||\mathrm{r}}}{I_\mathrm{i}}\cos ^2\gamma _\mathrm{i}+\frac{I_{\perp \mathrm{r}}}{I_\mathrm{i}}\sin ^2\gamma _\mathrm{i} =R_{||}\cos ^2\gamma _\mathrm{i}+R_{\perp }\sin ^2\gamma _\mathrm{i}.\nonumber \\ \end{aligned}$$

(5.11)

For natural or unpolarized radiation, the angle $\gamma _\mathrm{i}$ varies rapidly in a random manner, as does the field amplitude. Thus, since

$$\begin{aligned} \left\langle \cos ^2 \gamma _\mathrm{i}(t)\right\rangle _\mathcal{T}=\frac{1}{2} \quad \mathrm{and}\quad \left\langle \sin ^2 \gamma _\mathrm{i}(t)\right\rangle _\mathcal{T}=\frac{1}{2}, \end{aligned}$$

(5.12)

and therefore for natural radiation

$$\begin{aligned} I_{||\mathrm{i}}=\frac{I_\mathrm{i}}{2} \quad \mathrm{and} \quad I_{\perp \mathrm{i}}=\frac{I_\mathrm{i}}{2}. \end{aligned}$$

(5.13)

and therefore

$$\begin{aligned} r^2_{||}=\left( \frac{E^2_{|| \mathrm{r}}}{E^2_{|| \mathrm{i}}}\right) ^2 =\frac{I_{||\mathrm{r}}}{I_{||\mathrm{i}}} \quad \mathrm{and} \quad r^2_{\perp }=\left( \frac{E^2_{\perp \mathrm{r}}}{E^2_{\perp \mathrm{i}}}\right) ^2 =\frac{I_{\perp \mathrm{r}}}{I_{\perp \mathrm{i}}}.\nonumber \\ \end{aligned}$$

(5.14)

Thus, the total reflectance becomes

$$\begin{aligned} R=\frac{1}{2}\left( R_{||}+R_{\perp }\right) =\frac{1}{2}\left( r_{||}^2+r_{\perp }^2\right) , \end{aligned}$$

(5.15)

where $0\le R\le 1.$ For the cases where $\sin \theta _\mathrm{t}=\frac{\sin \theta _\mathrm{i}}{\hat{n}}>1,$ one may rewrite reflection relations as

$$\begin{aligned}&r_{||}=\frac{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}-j(\sin ^2\theta _\mathrm{i}-\hat{n}^2)^{\frac{1}{2}}}{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}+j(\sin ^2\theta _\mathrm{i}-\hat{n}^2)^{\frac{1}{2}}} \quad \mathrm{and}\nonumber \\&r_{\perp }=\frac{\cos \theta _\mathrm{i}-\frac{1}{\hat{\mu }}j(\sin ^2\theta _\mathrm{i}-\hat{n}^2)^{\frac{1}{2}}}{\cos \theta _\mathrm{i}+\frac{1}{\hat{\mu }}j(\sin ^2\theta _\mathrm{i}-\hat{n}^2)^{\frac{1}{2}}}, \end{aligned}$$

(5.16)

where, $j=\sqrt{-1},$ and in this complex case⁹

$$\begin{aligned} R_{||}\mathop {=}\limits ^\mathrm{def}r_{||} \bar{r}_{||}=1, \quad \mathrm{and} \quad R_{\perp }\mathop {=}\limits ^\mathrm{def}r_{\perp } \bar{r}_{\perp }=1, \end{aligned}$$

(5.17)

where $\bar{r}_{||}$ and $\bar{r}_{\perp }$ are complex conjugates. Thus, for angles above the critical angle $\theta ^*_\mathrm{i},$ all of the energy is reflected. Notice that as $\hat{n}\rightarrow 1$ we have complete absorption, while as $\hat{n}\rightarrow \infty$ we have complete reflection. The total amount of absorbed power by the particles is $(1-R)I_\mathrm{i}.$ Thermal (infrared) coupling effects, which are outside of the scope of this paper, have been accounted for in Zohdi [35, 36] and Zohdi and Kuypers [45].

In order to understand the dependency of the results on $\hat{n},$ recall the fundamental relation for reflectance

$$\begin{aligned} R= & {} \frac{1}{2}\left( \left( \frac{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}-(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}{\frac{\hat{n}^2}{\hat{\mu }}\cos \theta _\mathrm{i}+(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}\right) ^2\right. \nonumber \\&\quad +\left. \left( \frac{\cos \theta _\mathrm{i}-\frac{1}{\hat{\mu }}(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}{\cos \theta _\mathrm{i}+\frac{1}{\hat{\mu }}(\hat{n}^2-\sin ^2\theta _\mathrm{i})^{\frac{1}{2}}}\right) ^2\right) , \end{aligned}$$

(5.18)

whose variation as a function of the angle $\theta _\mathrm{i}$ is depicted in Fig. 3. For all but $\hat{n}=2,$ is there discernible nonmonotone behavior. The nonmonotone behavior is slight for $\hat{n}=4,$ but nonetheless present. Clearly, as $\hat{n}\rightarrow \infty ,\,R\rightarrow 1,$ no matter what the angle of incidence’s value. Also, as $\hat{n}\rightarrow 1,$ provided that $\hat{\mu }=1,\,R\rightarrow 0,$ i.e., all incident energy is absorbed. With increasing $\hat{n},$ the angle for minimum reflectance grows larger.