Prevailing-frequency approximation of the coupling ray theory for electromagnetic waves or elastic S waves

Over the last decade, several methods [1–8] have been developed to obtain the dyadic Green's functions (DGFs) of layered anisotropic media. The most commonly used methods to find the DGFs are the Fourier transform method with differential formulation [6], the method of eigen-function expansion [7], the matrix formulation [8], and the dyadic decomposition techniques [5, 9–11]. In [8], the dyadic Green's function is formulated based on the tangential electric field and current at the interface, which is not suitable for the problem with z-directed current. The dyadic decomposition techniques as discussed in [5, 9–11] are more general and can provide the complete set of the dyadic Green's functions but only the case when the source is in the free space has been considered. If the source is located in a biaxial slab, then reciprocity theorem for the Green's function can be invoked to obtain the DGFs as in [12]. However, the reciprocity holds true only when the slab is filled with reciprocal medium such as uniaxial medium, biaxial medium, etc. For non-reciprocal medium such as the gyroelectric or gyromagnetic medium, the reciprocity relation for the Green's function no longer holds. The DGFs for source located in a non-reciprocal slab cannot be obtained using the reciprocity theorem.

Formulae for the zeroth-order principal term plus the first-order additional term of the qP- and qS-wave Green's functions in the so-called quasi-isotropic (QI) approximation are derived for an unbounded inhomogeneous weakly anisotropic medium. The basic idea of this approximation is to seek the asymptotic solution of the elastodynamic equation as an expansion with respect to two small parameters of the same order: the small parameter used in the standard ray method and a parameter characterizing differences of tensors of elastic parameters of a weakly anisotropic medium and of a nearby 'background' isotropic medium. As a result, the procedure of constructing the Green's functions is split into two steps: (1) calculation of rays, traveltimes, the geometrical spreading and polarization vectors in the background isotropic medium; (2) calculation of corrections of traveltimes, amplitudes and polarization vectors due to the deviation of the weakly anisotropic medium from the isotropic background.

Recent formulations of the dyadic (or generalized) Green's function describe the relationship between sources (both mechanical stresses and electrical charge) and waves (both mechanical displacements and acoustic potential) on the surface of a substrate. The sixteen elements of the function intrinsically describe all propagation modes, whether Rayleigh or leaky, and are therefore extremely useful in the design of surface acoustic wave devices. While requiring little additional computational effort, the dyadic Green's function thus provides much more information than the traditional effective permittivity function. In this paper, we extend the calculation of the dyadic Green's function to multi-layer substrates. We show that its computation involves a simple cascaded matrix multiplication. The resulting function fully contains the substrate characteristics and, once obtained, can be used to describe the surface behavior with no further regard to the substrate's composition. The dyadic Green's function for a sample multi-layer substrate is presented.

The electric field within a source region is derived by use of a vector form of Green's second identity. The expression obtained is identical to that previously derived1 by a less direct method, but the present approach exposes the need to assume continuity of current density and electric field. The analysis gives the extra dyadic which must be added to the customary superposition integral to take account of the source-point singularity of the electric dyadic Green's function.

The dyadic Green's functions for a parallel-plate chirowaveguide are formulated by using the method of eigenfunction expansion. The eigenfunctions of the chirowaveguide are derived first in terms of two transformed fields. The purely solenoidal magnetic type dyadic Green's function is then expanded by these eigenfunctions. The electric type dyadic Green's function, which is solenoidal as well as longitudinal, is subsequently obtained from the magnetic dyadic Green's function through a simple relationship. It is found that the singular term in the electric dyadic Green's function accounting for the electric field in the source point is established through this latter procedure. The dyadic Green's functions obtained can be used to examine the electric and magnetic fields inside the chirowaveguide and to construct other dyadic Green's functions.

The use of a dyadic layered-medium Green's function as the kernel in a method of moments (MoM) modeling problem greatly reduces the complexity of modeling a stratified medium. Compared to the free-space Green's function, there is an additional cost of having to compute a semi-infinite Sommerfeld integral for each call to calculate the dyadic layered-medium Green's function. This letter discusses a method to tabulate and interpolate the Green's function as a method of reducing the impedance matrix filling time. This method can be used in conjunction with existing methods for increasing the computational speed of the Green's functions.

The coupling ray theory provides a continuous transition between the isotropic and anisotropic ray theories, and is particularly important at degrees of anisotropy and frequencies typical in seismic exploration and structural seismology on all scales. In this paper, isotropic-ray-theory, coupling-ray-theory, and anisotropic-ray-theory seismo-grams in a realistic weakly anisotropic model are compared. The most pronounced differences can be observed in polarization. Then, the coupling-ray-theory seismo-grams in four models of differing anisotropy are compared. For weaker anisotropy, just the polarization changes with increasing anisotropy. If anisotropy is increased further, the clear development of S-wave splitting can be observed.

To characterize electromagnetic waves in complex media has been an important topic because of the useful applications and the scientific significance of the physical mechanism. Dyadic Green's functions, as a mathematical kernel or a dielectric medium response, relate directly the radiated electromagnetic fields and the source distribution. In terms of the vector wave functions in cylindrical coordinates, dyadic Green's functions in an unbounded and a planar, multilayered gyroelectric chiral media are formulated. After a general representation of the Green's dyadics is obtained, the scattering coefficients of the Green's dyadics are determined from the boundary conditions at each interface and are expressed in a greatly compact form of recurrence matrices. In the formulation of the Green's dyadics and their scattering coefficients, three cases are considered, i.e. the current source is impressed in (1) the first, (2) the intermediate, and (3) the last regions, respectively. Although the dyadic Green's functions for an unbounded gyroelectric chiral medium have been reported in the literature, the results are either incorrect or suspicious. As compared to the existing results, the current work basically contributes (1) a correct form of the dyadic Green's function for a unbounded gyroelectric chiral medium, (2) the general representation of the dyadic Green's functions for a multi-layered gyroelectric chiral medium, and (3) a convincing and direct derivation of the irrotational Green's dyadic.

The dyadic Green's functions of the cylindrical chirowaveguide are derived by modal expansion. Bohren's decomposition of the electromagnetic field is used to obtain the vector wave functions. The magnetic dyadic Green's function that is purely solenoidal is first derived and the electric dyadic Green's function is then obtained by manipulating the magnetic dyadic Green's function. The singular term in the expression of the electric dyadic Green's function is reinstated through the manipulation procedure.

We propose a fast and efficient method to build anisotropic seismic velocity models by combining forward modeling and traveltime inversion of 2-D walkaway VSP data. Applying this approach, we derived anisotropic seismic velocity models for a 1200 m thick reservoir overburden consisting of horizontally layered shales and shaley sands in the Western Canadian basin. The models were determined by iteratively minimizing the misfit between the measured and modeled VSP first break times. The modeled times were calculated by 2-D ray tracing through isotropic and anisotropic velocity models. Once we achieved a fairly good match between the data and the model we then used inversion for fine-tuning. The proposed new method can help to provide good anisotropic velocity models for both surface and borehole seismic anisotropic migration.

This paper presents an approach to analyze the microstrip lines using Vector Dyadic Green's function. The vector Dyadic spectral Green's function is obtained directly from the Spectral domain immitance approach thus leading to a discretized system of equations which can be utilized in the semi analytical analysis of the planar microwave structures and resonators. Finally, the equations so derived have been used to analyze the characteristics of microstrip line in simple and suspended configurations. The current distribution on the strip is also calculated. The results thus obtained were found to be in good agreement with results using Spectral Domain Approach (SDA). Ill. 5, bibl. 9 (in English; summaries in Lithuanian, English and Russian).

This paper presents a systematic and unified formulation of the dyadic Green's functions for multilayered biisotropic media. The media are stratified in a general direction which includes the canonical cases of planar, cylindrical and spherical geometries. Both electric and magnetic dyadic Green's functions are derived simultaneously based on the principle of scattering superposition. For the scattered dyadic Green's functions, their scattering coefficients are determined using the effective reflection and transmission concepts. This approach has avoided cumbersome operations and has also provided good physical insights to the scattering mechanism. The resulting general expressions are written in very compact and convenient forms. These expressions have been applied explicitly to the three canonical cases. For some common sets of biisotropic constitutive relations, their dyadic Green's functions are discussed and compared. The special case of multilayered isotropic media are considered as well.

Dyadic Green's Function Microstrip Circulator Theory for Inhomogeneous Ferrite with and without Penetrable Walls

A spectral-domain dyadic Green's function for electromagnetic fields in cylindrically multilayered media with circular cross section is derived in terms of matrices of the cylindrical vector wave functions. Some useful concepts, such as the effective plane wave reflection and transmission coefficients, are extended in the present spectral domain eigenfunction expansion. The coupling coefficient matrices of the scattering dyadic Green's functions are given by applying the principle of scattering superposition. The general solution has been applied to the case of axial symmetry (n=0, n is eigenvalue parameter in /spl phi/ direction) where the scattering coefficients are decoupled between TM and TE waves. Two specific geometries, i.e., two- and three-layered media that are frequently employed to model the practical problems are considered in detail, and the coupling coefficient matrices of their dyadic Green's functions are given, respectively.

This paper presents an eigenfunction expansion of the dyadic Green's function in a general rotationally symmetric medium with electrical anisotropy using spherical vector wave functions. As usual, the dyadic Green's function in the spectral domain consists of the solenoidal and nonsolenoidal contributions from the rotational and irrotational wave eigenfunctions. It is visible from the final expansion that the parameters of the medium have close and intricate relationships with the properties of the Green's function, hence, influencing the nature of the propagating waves. The formulation in the paper reveals that the singularity term due to the irrotational vector wave function present in the full eigenfunction expansion is somewhat similar to that of an isotropic case except that the permittivity in the expansion is now a tensor. Like the chiral medium, this particular anisotropic medium exhibits the ability to support the propagation of oppositely polarized waves. This result is due to the fact that the permittivity tensor has some specific off-diagonal non-zero components. Although not straightforwardly implied from its constitutive relations, the mode splitting can be observed in the final expression of the Green's functions.