Electromagnetic scattering from cylindrically and spherically stratified bodies

A method is developed for calculating the electromagnetic field scattered by certain types of bodies. The bodies consist of inhomogeneous media whose constitutive parameters vary only with the distance from some axis or point of symmetry. The method consists in an extension of the invariant imbedding method for treating wave problems. This method, which is familiar in the case of a one-dimensional inhomogeneity, is extended to handle special types of two and three-dimensional inhomogeneities. Comparisons are made with other methods which have been proposed for treating these kinds of problems. Examples of applications of the method are given, some of which are of interest in themselves.

A multiple scattering problem

The present work deals with the problem of the interaction of the electromagnetic radiation with a statistical distribution of nonmagnetic dielectric particles immersed in an infinite homogeneous isotropic, non-magnetic medium. The wavelength of the incident radiation can be less, equal or greater than the linear dimension of a particle. The distance between any two particles is several wavelengths. A single particle in the absence of the others is assumed to scatter like a Rayleigh-Gans particle, i.e. interaction between the volume elements (self-interaction) is neglected. The interaction of the particles is taken into account (multiple scattering) and conditions are set up for the case of a lossless medium which guarantee that the multiple scattering contribution is more important than the self-interaction one. These conditions relate the wavelength λ and the linear dimensions of a particle a and of the region occupied by the particles D. It is found that for constant λ/a, D is proportional to λ and that |Δχ|, where Δχ is the difference in the dielectric susceptibilities between particle and medium, has to lie within a certain range.

The total scattering field is obtained as a series the several terms of which represent the corresponding multiple scattering orders. The first term is a single scattering term. The ensemble average of the total scattering intensity is then obtained as a series which does not involve terms due to products between terms of different orders. Thus the waves corresponding to different orders are independent and their Stokes parameters add.

The second and third order intensity terms are explicitly computed. The method used suggests a general approach for computing any order. It is found that in general the first order scattering intensity pattern (or phase function) peaks in the forward direction Θ = 0. The second order tends to smooth out the pattern giving a maximum in the Θ = π/2 direction and minima in the Θ = 0 , Θ = π directions. This ceases to be true if ka (where k = 2π/λ) becomes large (> 20). For large ka the forward direction is further enhanced. Similar features are expected from the higher orders even though the critical value of ka may increase with the order.

The first order polarization of the scattered wave is determined. The ensemble average of the Stokes parameters of the scattered wave is explicitly computed for the second order. A similar method can be applied for any order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. If the latter is elliptically polarized then the first order scattered wave is elliptically polarized, but in the Θ = π/2 direction is linearly polarized. If the incident wave is circularly polarized the first order scattered wave is elliptically polarized except for the directions Θ = π/2 (linearly polarized) and Θ = 0, π (circularly polarized). The handedness of the Θ = 0 wave is the same as that of the incident whereas the handedness of the Θ = π wave is opposite. If the incident wave is linearly polarized the first order scattered wave is also linearly polarized. The second order makes the total scattered wave to be elliptically polarized for any Θ no matter what the incident wave is. However, the handedness of the total scattered wave is not altered by the second order. Higher orders have similar effects as the second order.

If the medium is lossy the general approach employed for the lossless case is still valid. Only the algebra increases in complexity. It is found that the results of the lossless case are insensitive in the first order of k_imD where k_im = imaginary part of the wave vector k and D a linear characteristic dimension of the region occupied by the particles. Thus moderately extended regions and small losses make (k_imD)² ≪ 1 and the lossy character of the medium does not alter the results of the lossless case. In general the presence of the losses tends to reduce the forward scattering.

Measurements of mantle velocities of P waves with a large array

A large array has been used to investigate the P-wave velocity structure of the lower mantle. Linear array processing methods are reviewed and a method of nonlinear processing is presented. Phase velocities, travel times, and relative amplitudes of P waves have been measured with the large array at the Tonto Forest Seismological Observatory in Arizona for 125 earthquakes in the distance range of 30 to 100 degrees. Various models are assumed for the upper 771 km of the mantle and the Wiechert-Herglotz method applied to the phase velocity data to obtain a velocity depth structure for the lower mantle. The phase velocity data indicates the presence of a second-order discontinuity at a depth of 840 km, another at 1150 km, and less pronounced discontinuities at 1320, 1700 and 1950 km. Phase velocities beyond 85 degrees are interpreted in terms of a triplication of the phase velocity curve, and this results in a zone of almost constant velocity between depths of 2670 and 2800 km. Because of the uncertainty in the upper mantle assumptions, a final model cannot be proposed, but it appears that the lower mantle is more complicated than the standard models and there is good evidence for second-order discontinuities below a depth of 1000 km. A tentative lower bound of 2881 km can be placed on the depth to the core. The importance of checking the calculated velocity structure against independently measured travel times is pointed out. Comparisons are also made with observed PcP times and the agreement is good. The method of using measured values of the rate of change of amplitude with distances shows promising results.

New doubly periodic waves of the (2+1)-dimensional double sine-Gordon equation

New exact solutions of the (2 + 1)-dimensional double sine-Gordon equation are studied by introducing the modified mapping relations between the cubic nonlinear Klein-Gordon system and double sine-Gordon equation. Two arbitrary functions are included into the Jacobi elliptic function solutions. New doubly periodic wave solutions are obtained and displayed graphically by proper selections of the arbitrary functions.

Electromagnetic fields and transmission properties in tapered hollow metallic waveguides

We analyze the electromagnetic spatital distributions and address an important issue of the transmission properties of spherical transverse-electric (TE) and transverse-magnetic (TM) eigenmodes within a tapered hollow metallic waveguide in detail. Explicit analytical expressions for the spatital distributions of electromagnetic field components, attenuation constant, phase constant and wave impedance are derived. Accurate eigenvalues obtained numerically are used to study the dependences of the transmission properties on the taper angle, the mode as well as the length of the waveguide. It is shown that all modes run continuously from a propagating through a transition to an evanescent region and the value of the attenuation increases as the distance from the cone vertex and the cone angle decrease. A strict distinction between pure propagating and pure evanescent modes cannot be achieved. One mode after the other reaches cutoff in the tapered hollow metallic waveguide as the distance from the cone vertex desreases. (C) 2008 Optical Society of America

Beam conditions for radiation generated by an electromagnetic J 0-correlated Schell-model source

Based on the 2 x 2 (electric field) cross-spectral density matrix, a model for an electromagnetic J(0)-correlated Schell-model beam is given that is a generalization of the scalar J(0)-correlated Schell-model beam. The conditions that the matrix for the source to generate an electromagnetic J(0)-correlated Schell-model beam are obtained. The condition for the source to generate a scalar J(0)-correlated Schell-model beam can be considered as a special case. (C) 2008 Optical Society of America

Analysis of electromagnetic propagation in birefringent thin film

In general, the propagating behavior of extraordinary wave in anisotropic materials is different from that in isotropic materials. With the tangential continuity of Maxwell's equations, the electromagnetic propagating behaviors have been investigated at the incident and exit interfaces of the uniaxial anisotropic thin film. The emphasis was placed on two interesting optical phenomena such as homolateral refraction behavior and wide-angle Brewster's phenomenon, which occurred at the interfaces of uniaxial anisotropic thin film.