Half-plane diffraction problems on a triangular lattice

Abstract

We investigate thin-slit diffraction problems for two-dimensional lattice waves. Namely, Dirichlet problems for the two-dimensional discrete Helmholtz equation on the triangular lattice in a half-plane are studied. Using the notion of the radiating solution, we prove the existence and uniqueness of a solution for the real wave number $k\in (0,3)\backslash \{2\sqrt{2}\}$ without passing to the complex wave number. Besides, an exact representation formula for the solution is derived. Here, we develop a numerical calculation method and demonstrate by example the effectiveness of our approach related to the propagation of wavefronts in metamaterials through two small openings.

Appendix A: Sparse matrices

The sparse matrices $\alpha _n(k)$, $\beta _n(k)$, and $\gamma _n(k)$ are defined as follows: if $n=2p$ then $\alpha _{2p}(k)$ is a $(p+1)\times p$ matrix such that $\alpha _{2p}(k)\mid _{i,i}=1$, $i=\overline{1,p}$, $\alpha _{2p}(k)\mid _{i,i-1}=1$, $i=\overline{2,p}$, while $\alpha _{2p}(k)\mid _{p+1,p}=2$, and all other matrix elements are zero. The $\beta _{2p}(k)$ is a $(p+1)\times (p+1)$ matrix such that $\beta _{2p}(k)\mid _{i,i}=1$, $i=\overline{1,p}$, $\beta _{2p}(k)\mid _{i,i+1}=1$, $i=\overline{2,p}$, while $\beta _{2p}(k)\mid _{p+1,p+1}=\beta _{2p}(k)\mid _{1,2}=2$, and all other matrix elements are zero. The $\gamma _{2p}(k)$ is a $(p+1)\times (p+1)$ matrix such that $\gamma _{2p}(k)\mid _{i,i}=6-k^2$, $i=\overline{1,p+1}$, $\gamma _{2p}(k)\mid _{i,i+1}=\gamma _{2p}(k)\mid _{i,i-1}=-1$, $i=\overline{2,p}$, and $\gamma _{2p}(k)\mid _{1,2}=\gamma _{2p}(k)\mid _{p+1,p}=-2$.

If $n=2p+1$ then $\alpha _{2p+1}(k)$ is a $(p+1)\times (p+1)$ matrix such that $\alpha _{2p+1}(k)\mid _{i,i}=1$, $i=\overline{1,p+1}$, $\alpha _{2p+1}(k)\mid _{i,i-1}=1$, $i=\overline{2,p+1}$, and all other matrix elements are zero. The $\beta _{2p+1}(k)$ is a $(p+1)\times (p+2)$ matrix such that $\beta _{2p+1}(k)\mid _{i,i}=1$, $i=\overline{1,p+1}$, $\beta _{2p+1}(k)\mid _{i,i+1}=1$, $i=\overline{2,p+1}$, while $\beta _{2p+1}(k)\mid _{1,2}=2$, and all other matrix elements are zero. The $\gamma _{2p+1}(k)$ is a $(p+1)\times (p+1)$ matrix such that $\gamma _{2p+1}(k)\mid _{i,i}=6-k^2$, $i=\overline{1,p}$, $\gamma _{2p+1}(k)\mid _{p+1,p+1}=5-k^2$, while $\gamma _{2p+1}(k)\mid _{i,i+1}=-1$, $i=\overline{2,p}$, $\gamma _{2p+1}(k)\mid _{i,i-1}=-1$, $i=\overline{2,p+1}$, and $(\gamma _{2p+1}(k)\mid _{1,2}=-2)$. Finally, $\gamma _1(k)$ is a $1\times 1$ matrix with an element $4-k^2$.