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.

Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of `generalized boundary blow-up' in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial data at the boundary faces in the form of a refinement of the `basic monoidal complex' of the manifold. This data specifies which notion of homogeneity is realized at each of the boundary hypersurfaces in the blown-up space. As an application of this theory, the existence of fiber products is examined for the natural smooth maps in this context, the b-maps. Transversality of the b-differentials is shown to ensure that the set-theoretic fiber product of two maps is a `binomial variety'. Properties of these (extrinsically defined) spaces, which generalize manifolds but have mild singularities at the boundary, are investigated and a condition on the basic monoidal complex is found under which the variety has a smooth structure. Applied to b-maps this additional condition with transversality leads to a universal fiber product in the context of manifolds with corners. Under the transversality condition alone the fiber product is resolvable to a smooth manifold by generalized blow-up and then has a weaker form of the universal mapping property requiring blow-up of the domain.Comment: 53 pages, to appear in Transactions of the AMS. Includes revisions suggested by the refere

Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution. , and replaces our previous heuristic proposal for anisotropic-material interfaces [5]. Ordinarily, the presence of discontinuous material interfaces degrades the accuracy of FDTD to first-order ͓O͑⌬x͔͒ from the usual second-order ͓O͑⌬x 2 ͔͒ accuracy [6], but our work demonstrates how an appropriate choice of subpixel smoothing can both restore second-order asymptotic accuracy and give the lowest errors compared with competing schemes even at modest resolutions. Subpixel smoothing has an additional benefit: it allows the simulation to respond continuously to changes in the geometry, such as during optimization or parameter studies, rather than changing in discontinuous jumps as interfaces cross pixel boundaries. This technique additionally yields much smoother convergence of the error with resolution, which makes it easier to evaluate the accuracy and enables the possibility of extrapolation to gain another order of accuracy [4]. The ability to handle anisotropic materials is becoming increasingly important via the use of anisotropic materials to represent arbitrary coordinate transformations in Maxwell's equations [7], most prominently to design cloaking metamaterials [8]. Our smoothing scheme requires preprocessing of the materials and does not otherwise modify the FDTD algorithm. It is therefore particularly simple to implement (free software is available [9]).Our basic approach, as described previously [2,3], is to smooth the structure to eliminate the discontinuity before discretizing, but because the smoothing itself changes the geometry we use first-order perturbation theory to select a smoothing with zero firstorder effect. For isotropic materials, this approach made rigorous a smoothing scheme that had previously been proposed heuristically [10][11][12] and explained its second-order accuracy [3]. Advances in perturbation theory have enabled us to extend this scheme to interfaces between anisotropic materials, initially for a plane-wave method [2]. Here, we adapt the technique to FDTD, combined with a recent FDTD scheme with improved stability for anisotropic media [4]. Although this Letter focuses on the case of anisotropic electric permittivity , exactly the same smoothing and discretization schemes apply to magnetic permeabilities µ owing to the equivalence in Maxwell's equations under interchange of /µ and E / H.We define an interface-relative coordinate frame as in Fig. 1, so that the first component "1" is the direction normal to the interface. ...