Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods

We derive a correct first-order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted. Most previous perturbative methods give incorrect results for this case, even to lowest order, because of the complicated discontinuous boundary conditions on the electric field at such an interface. Our final expression is simply a surface integral, over the material interface, of the continuous field components from the unperturbed structure. The derivation is based on a “localized” coordinate-transformation technique, which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary. Not only is our result potentially useful in evaluating boundary perturbations, e.g., from fabrication imperfections, in highly anisotropic media such as many metamaterials, but it also has a direct application in numerical electromagnetism. In particular, we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media, in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes.

Figure 1

(Color online) Schematic of an interface perturbation: the interface between two materials ${ϵ}^a$ and ${ϵ}^b$ (possibly anisotropic) is shifted by some small position-dependent displacement $h$.

Figure 2

Schematic of an interface perturbation as in Fig. 1, magnifying a small portion of the interface where the surface is locally flat. A local coordinate frame $\left(x_1,x_2,x_3\right)$ is chosen so that $x_1$ is perpendicular to the surface, and so that $x_1=0$ denotes the location of the perturbed surface $S$ (shifted perpendicularly by $\mathbf{h}$ from the original surface $S_0$).

Figure 3

(Color online) Numerical validation of perturbation-theory formula, applied to compute the derivative $d\omega /dh$ for a Gaussian “bump” of height $h$ on a square lattice (period $a$) of anisotropic-$ϵ$ rectangles (inset) with an eigenfrequency $\omega$ (corresponding to $\lambda \sim 3a$). Positive (negative) $h$ indicate bumps (indentations) (see lower right and left insets for $h=\pm 0.15a$), respectively. Solid lines are numerical differentiation of the eigenfrequency, and dots are from perturbation theory. The different lines correspond to different random dielectric tensors ${ϵ}^a$ and ${ϵ}^b$.

Figure 4

(Color online) Relative error $\Delta \omega /\omega$ for an eigenmode calculation with a square lattice (period $a$) of 2D anisotropic ellipses (green inset), versus spatial resolution, for a variety of subpixel smoothing techniques. Straight lines for perfect linear (black dashed) and perfect quadratic (black solid) convergence are shown for reference.

Figure 5

(Color online) Relative error $\Delta \omega /\omega$ for an eigenmode calculation with cubic lattice (period $a$) of 3D anisotropic ellipsoids (green inset), versus spatial resolution, for a variety of subpixel smoothing techniques. Straight lines for perfect linear (black dashed) and perfect quadratic (black solid) convergence are shown for reference.