Diffraction on the Two-Dimensional Square Lattice

We solve the thin-slit diffraction problem for two-dimensional lattice waves. More precisely, for the discrete Helmholtz equation on the semi-infinite square lattice with data prescribed on the left boundary (the aperture), we use lattice Green's functions and a discrete Sommerfeld outgoing radiation condition to derive the exact solution everywhere in the lattice. The solution is a discrete convolution that can be evaluated in closed form for the wave number $k=2$. For other wave numbers, we give a recursive algorithm for computing the convolution kernel.

[1]  J. Marsden,et al.  Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines , 2006 .

[2]  Montr ´ eal.  Novel broadband conventional- and dual-composite right/left-handed (C/D-CRLH) metamaterials: properties, implementation and double-band coupler application , 2007 .

[3]  T. Itoh,et al.  Characteristics of the composite right/left-handed transmission lines , 2004, IEEE Microwave and Wireless Components Letters.

[4]  T. Morita.  Useful Procedure for Computing the Lattice Green's Function‐Square, Tetragonal, and bcc Lattices , 1971 .

[5]  B. Vainberg,et al.  Radiation conditions for the difference schrödinger operators , 2001 .

[6]  Eleftherios N. Economou,et al.  Green's functions in quantum physics , 1979 .

[7]  Fritz Albregtsen,et al.  Fast and exact computation of moments using discrete Green's theorem , 1994 .

[8]  Sakari Inawashiro,et al.  Lattice Green's Functions for the Rectangular and the Square Lattices at Arbitrary Points , 1971 .

[9]  C. Caloz,et al.  Dual Composite Right/Left-Handed (D-CRLH) Transmission Line Metamaterial , 2006, IEEE Microwave and Wireless Components Letters.

[10]  E. Afshari,et al.  Harmonic Generation Using Nonlinear LC Lattices , 2010, IEEE Transactions on Microwave Theory and Techniques.

[11]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[12]  Ehsan Afshari,et al.  Ultrafast analog Fourier transform using 2-D LC lattice , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[13]  E. H. Linfoot.  Principles of Optics , 1961 .

[14]  E Afshari,et al.  Nonlinear constructive interference in electrical lattices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  B. Osting,et al.  Thin slit diffraction in conventional and dual composite right/left-handed transmission line metamaterials , 2008, 2008 Asia-Pacific Microwave Conference.

[16]  Vladimir M. Shalaev,et al.  Electrodynamics of Metamaterials , 2007 .

[17]  O. Buneman.  Analytic inversion of the five-point poisson operator , 1971 .

[18]  O. Momeni,et al.  Electrical Prism: a high quality factor filter for mm wave and terahertz frequencies , 2008, 2008 Asia-Pacific Microwave Conference.

[19]  Christophe Caloz,et al.  Novel broadband conventional- and dual-composite right/left-handed (C/D-CRLH) metamaterials: properties, implementation and double-band coupler application , 2007 .

[20]  Patrick Joly,et al.  Numerical Diffraction by a Uniform Grid , 1988 .

[21]  Gilbert Labelle,et al.  The discrete Green Theorem and some applications in discrete geometry , 2005, Theor. Comput. Sci..

[22]  B. Vainberg,et al.  Large Time Behavior of Solutions to Difference Wave Operators , 2005, math/0503128.

[23]  N. Engheta,et al.  Metamaterials: Physics and Engineering Explorations , 2006 .

[24]  Paul A. Martin,et al.  Discrete scattering theory : Green's function for a square lattice , 2006 .

[25]  Xiaofeng Li,et al.  Electrical funnel: A broadband signal combining method , 2006, 2006 IEEE International Solid State Circuits Conference - Digest of Technical Papers.

[26]  Tatsuo Itoh,et al.  Electromagnetic metamaterials : transmission line theory and microwave applications : the engineering approach , 2005 .