Generalized Snell's laws for rough interfaces

Abstract

In this paper, we consider the reflection and transmission problem of waves by a rapidly oscillating rough interface that exhibits general mixing properties. Using an asymptotic analysis based on a separation of scales, corresponding to a paraxial (parabolic) scaling regime, we precisely characterize the specular and speckle (diffusive) components of the reflected and transmitted fields. A critically scaled interface is considered, in the sense that the amplitudes of the interface fluctuations and the central wavelength are of the same order. When the correlation length of the interface fluctuations is of the same order as the beam width, random specular components arise in both the reflected and transmitted waves, while no speckle component is observed. Equivalently, the reflected and transmitted fields are essentially confined to the cones formed by the specular components (specular cones) with directions given by the classical Snell's law of reflection and refraction. When the correlation length is smaller than the beam width, a specular homogenization regime emerges. In this case, the rough interface can be approximated by an effective flat interface, yielding deterministic specular reflected and transmitted cones. However, broader cones containing the specular cones appear, within which the wavefields form speckle patterns (speckle cones) whose total energy is of leading order. We provide the two-point correlation functions of these speckle patterns and establish a central-limit-theorem-type result, showing that they can be modeled as Gaussian random fields. These results enable the identification of generalized Snell's laws of reflection and transmission, which depend on an effective scattering operator at the interface.

Figures (15)

• Figure 1: Illustration of the basic physical setup. A source illuminates (gray) a rough surface, producing a reflected wave (red) and a transmitted wave (green). The incident pulse is illustrated by a gray rectangle, and the gray dots represent the associated probing cone. Both the reflected and transmitted waves generally exhibit specular and diffusive components. The directions of the specular components, given by the classical Snell's law of reflection and refraction, correspond to the long arrows, and the associated specular cones are drawn with dotted lines. The diffusive components (speckles), whose directions are represented by small arrows, result from significant scattering of the incident wave by the rough interface. The reflected and transmitted speckle cones are illustrated as light red and green regions, respectively.
• Figure 2: Illustration of the generalized Snell's laws of reflection and refraction. The reflection and transmission angles are parameterized by the scattering slowness vector $\mathbf{p}$.
• Figure 3: The left panel illustrates the reflection and refraction of a pulse by a rough interface. The gray region corresponds to the cone formed by the incident wave, as well as the cones associated with the reflected and transmitted specular components observed at $\mathbf{x}_{obs,ref}$ and $\mathbf{x}_{obs,tr}$, respectively. When the speckle components intersect a horizontal plane, they form ellipses whose shape evolves in time, as illustrated in the right panel using different colors and centered at $\mathbf{x}_{obs}$, which denotes either $\mathbf{x}_{obs,ref}$ or $\mathbf{x}_{obs,tr}$. The width of these ellipses is determined by the initial pulse duration.
• Figure 4: Illustration of the physical setup. The plane $z=0$ contains the source location, while $z=z_{int}$ is the plane around which the rough interface separating $\mathcal{D}^0$ and $\mathcal{D}^1$ is located. The reflected wave is observed at $z=0$, whereas the transmitted wave is observed at $z=z_{tr}$.
• Figure 5: Illustration of the up- and down-going mode amplitudes.

Theorems & Definitions (16)

• Lemma 2.1
• Lemma 4.1
• Proposition 5.1
• Proof 1: of Proposition \ref{['prop:random_travel_time']}
• Proposition 5.2
• Proposition 6.1
• Proof 2: of Proposition \ref{['prop:homog_ref']}
• Proposition 6.2
• Proposition 7.1
• Proposition 7.2