Abstract
We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green's function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green's function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 105 unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.
Highlights
Periodic geometries and multilayered media are both essential for the manipulation of waves in modern optical and electromagnetic devices
Related wave scattering problems appear in photonic crystals [30], process control in semiconductor lithography [44], in the electromagnetic characterization of increasingly multilayered integrated circuits, and in models for underwater acoustic wave propagation
In this paper we introduce a simpler class II boundary integral equations (BIE) method which combines the free-space Green’s function for the unit cell and neighbors, with a ring of proxy point sources to represent the far-field contributions which “periodize” the field
Summary
Periodic geometries (such as diffraction gratings and antennae) and multilayered media (such as dielectric mirrors) are both essential for the manipulation of waves in modern optical and electromagnetic devices. The solution is unique at all but a discrete set of frequencies ω when θinc is fixed [13, Thm. 9.4]; these frequencies correspond to guided modes of the dielectric structure, where resonance makes the physical problem ill-posed They are distinct from (but in the literature sometimes confused with) Wood anomalies [57], which are scattering parameters (θinc,ω) for which kUn = 0 or kDn = 0 for some n, making the upper or lower nth Rayleigh–Bloch mode a horizontally traveling plane wave. In this paper we introduce a simpler class II BIE method which combines the free-space Green’s function for the unit cell and neighbors, with a ring of proxy point sources (i.e. the method of fundamental solutions, or MFS [12, 7]) to represent the far-field contributions which “periodize” the field This combines ideas in [9, Sec. 3.2] and the fast direct solver community [42, Sec. 5], and has been independently proposed recently for Laplace problems by Gumerov–Duraiswami [24].
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