Complex Scaling for the Junction of Semi-infinite Gratings

Abstract

We present and analyze an integral equation method for the scattering of a non-periodic source from a geometry consisting of two semi-infinite, periodic structures glued together in two dimensions. The two structures may involve a periodic wall, several layers of transmission surfaces with a shared period, or periodic sets of obstacles. This integral equation is posed on the infinite interface between the two periodic structures using kernels built out of the Green's function for each structure. To combat the slow decay of the Green's function, we also show that our integral equation can be analytically continued into the complex plane, where it can be truncated with exponential accuracy. A careful analysis of the domain Green's functions far from the periodic structure is then used to prove that the analytically continued equation is Fredholm index zero. Finally, we show that the solution we generate satisfies a radiation condition and demonstrate an efficient and high order solver for this problem.

Figures (12)

• Figure 1: This figure illustrates our problem setup. We have two periodic walls $\gamma_L$ and $\gamma_R$ respectively and the fictitious interface $\Gamma$ that separates the left and right halves of the computational domain.
• Figure 2: The left figure shows the real and imaginary parts of $\tilde{\Gamma}$. The right shows the decay of the densities $\sigma,\tau$ the solve \ref{['eq:comp_IE_2']} when the right hand side is associated to an incoming trapped mode on the left side.
• Figure 3: A few examples of the domain Green's function for a choice of $\gamma$ with $d=1.3$ and $k=1$.
• Figure 4: The error in the \ref{['eq:one_error']} for the point $\boldsymbol{y}=(0,-0.2)$. We see that the solver is accurate to at least 11 digits everywhere away from the corners. The errors in the vicinity of the corners is due to the implementation of the RCIP method used by ChunkIE. The solver also makes errors outside the unit cell because of the $G_{\xi}-G_0$ is nearly singular as $\boldsymbol{x}$ approaches a periodic copy of $\gamma$.
• Figure 5: The trapped modes for our two geometries. The left geometry has a mode at $\tilde{\xi}_1=1.422265877314$. The right geometry has a mode at $\tilde{\xi}_1=1.47762000473$.

Theorems & Definitions (36)

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