Electromagnetic-wave propagation along curved surfaces

We show that Maxwell’s equations for a nonmagnetic, isotropic, but electrically inhomogeneous medium in the absence of charges or current sources lead to a wave equation governing surface electromagnetic wave propagation along a general curved, smooth surface which, when recasted using an appropriate choice of curvilinear coordinates $u^1,u^2,u^3$, can be fully separated in the spatial dimensions. It is shown that surface electromagnetic wave solutions decay exponentially away from the surface (along the $u^3$ coordinate) with the same decay rate independent of the shape of the surface. Transmission and reflection coefficients governing scattering of electromagnetic waves on a varying surface shape are derived. Two test cases of a Gaussian-shaped and a sinusoidal-shaped surface are solved in details and discussed numerically in terms of transmission and reflection coefficients including dependencies on surface-shape parameters in the wavelength range 250–750 nm. The present method for determining surface electromagnetic wave propagation along complex-shaped metal-dielectric surfaces allows better insight into the importance of surface geometry as well as considerably faster computational speeds than those provided by standard numerical methods.

Figure 1

Schematic of the surface as a function of $u^1$.

Figure 2

(Color online) Surface electromagnetic wave propagation along a Gaussian-shaped surface as given by Eq. (51). The upper plot shows the SPP solution subject to the boundary condition $f\left(u^1\right)=\text{exp}\left(i\sqrt{-\alpha }{u}^1\right)$ as $u^1\to \infty$. Parameters used in the calculation are: $A_1=100\text{nm}$, $L=1\mu \text{m}$, and $\delta =33\text{nm}$. Permittivity values $-6.7$ and 1 at a wavelength of 400 nm are considered (gold and air, respectively). The lower plot shows the metal-dielectric Gaussian surface function $z\left(u^1\right)$ plotted in the $x\text{−}z$ plane.

Figure 3

(Color online) Transmission coefficient plotted vs. electromagnetic surface wavelength (Gaussian-shaped surface). The upper-left, upper-right, lower-left, and lower-right subplots correspond to Gaussian amplitude values $A_1$ equal to 10, 23, 37, and 50 nm, respectively. The Gaussian broadening parameter is $\delta =13\text{nm}$. The wavelength of the incoming surface wave is varied in the range between 250 and 750 nm ($x$ axis).

Figure 4

(Color online) Reflection coefficient plotted vs electromagnetic surface wavelength (Gaussian-shaped surface). Data correspond to those in Fig. 3.

Figure 5

(Color online) Transmission coefficient plotted vs electromagnetic surface wavelength (Gaussian-shaped surface). The upper-left, upper-right, lower-left, and lower-right subplots correspond to Gaussian width values $\delta$ equal to 13, 49, 70, and 100 nm, respectively. The Gaussian amplitude parameter is $A_1=300\text{nm}$. The wavelength of the incoming surface wave is varied in the range between 250 and 750 nm ($x$ axis).

Figure 6

(Color online) Reflection coefficient plotted vs electromagnetic surface wavelength (Gaussian-shaped surface). Data correspond to those in Fig. 5.

Figure 7

(Color online) Transmission coefficient plotted vs electromagnetic surface wavelength (sinusoidal-shaped surface). The upper-left, upper-right, lower-left, and lower-right subplots correspond to a varying number of sinusoidal periods ($N=2$, 3, 4, and 5, respectively). The sinusoidal corrugation length is fixed at $L_2-L_1=417\text{nm}$. The wavelength of the incoming surface wave is varied in the range between 250 and 750 nm ($x$ axis).

Figure 8

(Color online) Reflection coefficient plotted vs electromagnetic surface wavelength (sinusoidal-shaped surface). Data correspond to those in Fig. 7.

Figure 9

(Color online) Transmission coefficient plotted vs electromagnetic surface wavelength (sinusoidal-shaped surface). The upper-left, upper-right, lower-left, and lower-right subplots correspond to a length of the sinusoidal corrugation equal to 167, 222, 278, and 333 nm, respectively. The number of periods over a length $L$ is fixed at $N=5$. The wavelength of the incoming surface wave is varied in the range between 250 and 750 nm ($x$ axis).

Figure 10

(Color online) Reflection coefficient plotted vs electromagnetic surface wavelength (sinusoidal-shaped surface). Data correspond to those in Fig. 9.