Paraxial Direction Research Articles

Target scattering calculations with the parabolic equation method

The parabolic equation technique is used to solve the Helmholtz equation in the presence of scatterers of arbitrary shape, in two and three dimensions. The scattered field is computed directly, using non-homogeneous boundary conditions on the scattering object to represent the incident field. Effectively this decouples the PE paraxial direction from the direction of incidence. For convex objects the whole range of scattering angles can be covered with a small number of narrow-angle calculations. Finite-difference implementations involve tridiagonal matrices in two dimensions and more general sparse matrices in three dimensions. The resulting codes can be used to solve scattering problems for objects ranging in size from a few wavelengths to hundreds of wavelengths. The method has been tested against analytical solutions for soft and rigid circular cylinders in 2D and soft and rigid spheres in 3D, showing good agreement at all scattering angles.

Modeling acoustic waves with paraxial extrapolators

Modeling by paraxial extrapolators is applicable to wave‐propagation problems in which most of the energy is traveling within a restricted angular cone about a principal axis of the problem. Using this technique, frequency‐domain finite‐difference solutions accurate for propagation angles out to 60° are readily generated for both two‐dimensional (2-D) and three‐dimensional (3-D) models. Solutions for 3-D problems are computed by applying the 2-D paraxial operators twice, once along the x‐axis and once along the y‐axis, at each extrapolation step. The azimuthal anisotropy inherent to this splitting technique is essentially eliminated by adding a phase‐correction operator to the extrapolation system. For heterogeneous models, scattering effects are incorporated by determining transmission and reflection coefficients at structural boundaries within the media. The direct forward‐scattered waves are modeled with a single pass of the extrapolation operator in the paraxial direction for each frequency. The first‐order backscattered energy is then modeled by extrapolation (in the opposite direction) of the reflected field determined on the first pass. Higher order scattering can be included by sweeping through the model with more passes. The chief advantages of the paraxial approach are (1) active storage is reduced by one dimension compared to solutions which must track both forward‐scattered and backscattered waves simultaneously; thus, realistic 3-D problems can fit on today’s computers, (2) the decomposition in frequency allows the technique to be implemented on highly parallel machines, (3) attenuation can be modeled as an arbitrary function of frequency, and (4) only a small number of frequencies are needed to produce movie‐like time slices.

High-frequency radiation from an electromagnetic source located on a finite cylinder--A spectral domain approach

A technique for analyzing the problem of radiation from a point electromagnetic source located on a perfectly conducting cylinder of finite length is described. The method of attack is based on the spectral domain approach, generalized for the case of a curved surface with a large radius of curvature and used in conjunction with appropriate asymptotic theories. Special situations, such as radiation in the paraxial direction, or along caustics, are treated via modifications introduced in the conventional geometrical theory of diffraction (GTD) formulas. Some typical numerical results are presented.