Rectangular Boundary Conditions Research Articles

Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges

We present surprisingly simple closed-form solutions for electric fields and electric potentials at arbitrary position ( x , y ) within a plane crossed by infinitely long line charges at regularly repeating positions using angular or elliptic functions with complex arguments. The lattice sums for the electric-field components and the electric potentials could be exactly solved, and the duality symmetry of trigonometric and lemniscate functions occurred in some solutions. The results may have relevance in calculating field configurations with rectangular boundary conditions. Several series related to Gauss’s constant are presented, established either as corollary results or via parallel investigations conducted in the spirit of experimental mathematics.

Time domain transport equations of light in multilayered rectangular biological tissue with semi-infinite medium

In the study of tissue optics, physical models with the boundary of finite size are rarely established. Based on the diffusion equation, under the rectangular boundary conditions, this paper adopts extrapolation boundary conditions to establish a frequency domain model of multilayer medium diffusion equation, according to Fourier transform of frequency domain, turns the frequency domain solution into time domain model. On the basis of the deduced equation, the corresponding calculation process is written and the time resolved diffuse reflectance is calculated. The established model is compared with Monte Carlo simulation of time domain and the traditional medium with semi-infinite thickness, the result shows that our theory is correct. Concurrently, the effects of different boundary conditions on the diffusion equation are compared, it shows that our equations can not only solve the finite boundary problems, when the boundary is infinite, it can replace multilayer medium diffusion model.

Optical properties of mirrors for focusing of non-normal incidence atom beams

Calculations of focal properties and third-order aberration coefficients are presented for atom mirrors based on elastically deformed single-crystal surfaces. Biaxial loading of the mirrors is achieved by a combination of a uniform applied pressure and rectangular boundary conditions. The results are calculated for readily available single-crystal wafers of Si(111) with a thickness of 50 μm and a usable free standing diameter of 18 mm. Focusing of beams at non-normal incidence requires the principal radii of curvature, in planes normal to the mirror plane, to be different. The ratio of these principal radii of curvature is shown to be insensitive to the applied pressure and can be varied by a factor greater than 10 by changing the boundary conditions from square (1:1) to rectangular (1:2).

Determination of Green's Function Matrix for Multiconductor and Anisotropic Multidielectric Planar Transmission Lines: A Variational Approach

In this paper, a set of simple recurrence formulas to evaluate the Green's function matrix for a generic multiconductor and multidielectric planar transmission system with arbitrary rectangular boundary conditions is obtained. Combining these formulas with the variational technique in the spectral domain, two useful algorithms to calculate the capacitance matrix of a very wide range of practical configurations are proposed. Upper and lower bounds on mode capacitances are obtained by using both algorithms. A number of practical structures have been analyzed and their most interesting features discussed, The method is very versatile and can handle a large class of MIC configurations, no matter how complex the planar structure.