Dyadic Green's function in a general rotationally symmetric material with electrical anisotropy using spherical vector wave function expansion

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.

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.

This paper presents a novel eigenfunction expansion of the electric-type dyadic Green's function (DGF) for unbounded gyrotropic bianisotropic media in terms of cylindrical vector wave functions. The DGF is obtained based on the well-known Ohm-Rayleigh method together with dyadic identities formed by the differential, curl and dot product of the constitutive tensors and the cylindrical vector wave functions. Utilization of the dyadic identities greatly simplifies the process of finding the vector expansion coefficients of the DGF for gyrotropic bianisotropic media. The DGF derived is expressed in terms of the contribution from the irrotational vector wave functions and another contribution from the solenoidal vector wave functions, with the /spl lambda/-domain integrals removed using the residue theorem. This result can be used to characterise electromagnetic waves in gyrotropic bianisotropic media and the idea can be extended to the development of DGF for some other media.

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.

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.

In this study, a complete eigenfunction expansion of the dyadic Green's functions (DGF) for an unbounded and a planar, arbitrary multilayered general Faraday chiral media (both the permittivity and permeability are tensors) are formulated in terms of the vector wave functions in cylindrical coordinates, which is a general extension of the gyroelectric cases. 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. These formulations are constructed based on the principle of scattering superposition. Although the dyadic Green's functions for an unbounded chiroplasma medium has been reported in the literature, we here present multi-layered DGFs for the most general Faraday chiral media . The explicit representation of the DGFs after reduction to the chiroplasma or isotropic case agrees well with those existing corresponding results.

This paper presents a novel and rigorous eigenfunction expansion of electric‐type dyadic Green's function for an unbounded uniaxial bianisotropic medium in terms of the cylindrical vector wave functions. The Green's dyadic is obtained on the basis of the well‐known Ohm‐Rayleigh method together with some newly developed vector and tensor identities formed by the differential, curl, and dot product of the constitutive dyadics and the cylindrical vector wave functions. The above identities greatly simplify the process of finding the vector expansion coefficients of the dyadic Green's function of the uniaxial bianisotropic media. The dyadic Green's function derived is expressed in terms of the contribution from the irrotational solenoidal types of vector wave functions, with the λ integrals removed using the residue theorem.

This paper presents an analytic formulation of the near- and far-field distributions due to periodic spherical arrays of dipole antennas on two spherical biisotropic substrates, i.e., on the chiral sphere and on a perfectly conducting sphere coated with one-layer chiral medium, respectively. For an arbitrary orientation of the dipole antenna array, the dyadic Green's functions in the form of an eigenfunction expansion in terms of the normalized spherical vector wave functions are directly employed. Thereby, exact representations of the radiated fields in both near and far zones are obtained in closed form. They are all in a superposition of right- and left-handed circularly polarized waves, and are also applicable for the cases of spherical arrays of dipole antennas on spherical isotropic substrates. Furthermore, the constitutive parameter effects of the chiral medium on the three-dimensional patterns of the far-zone field and its polarized feature are demonstrated numerically.

Propagation and radiation of electromagnetic waves in a lossless, reciprocal, chiral medium is studied in this thesis. Such a medium is described electromagnetically by the constitutive relations D = eE + iγB and H = iγE + (1/µ)B. The constants e, µ, γ are real and have values that are fixed by the size, shape, and the spatial distribution of the elements that collectively compose the medium. The plane wave propagation in an unbounded chiral medium is considered. The propagation constants are obtained and the polarization properties of electromagnetic waves in such a medium are discussed in detail. The problem of reflection from, and transmission through a semi-infinite chiral medium is solved by obtaining the Fresnel equations. The conditions for the total internal reflection of the incident wave from the interface, and the existance of the Brewster angle are obtained. The effects of the chirality on the polarization and intensity of the reflected wave from the chiral half-space are discussed and illustrated by employing the Stokes parameters. The propagation of electromagnetic waves through an infinite slab of chiral medium is formulated for oblique incidence and solved analytically for the case of normal incidence. The radiation emitted by an oscillating dipole in an unbounded, lossless, chiral medium is calculated. From the constitutive relations and from the time-harmonic Maxwell equations ∇ x E = iωB and ∇ x H = J - iωD, it is seen that the wave equation for such a medium is given by ∇ x ∇ x E - ω2µeE - 2ωµγ∇ x E = iωµJ where the source term J is the current density of the oscillating dipole and where E is the electric vector of the radiated field. The desired solution of this wave equation is found by the dyadic Green's function method, that is, by first constructing the dyadic Green's function Γ and then evaluating the expression E = iωµ∫Γ(r,r')•J(r')dV'. The dyadic Green's function Γ and the components of the radiated electric field E are obtained in closed form. The components of the radiated B, D, and H fields can be derived from knowledge of E by using the Maxwell equation B = (1/iω)∇ x E and the constitutive relations. The wave impedance of the medium and the radiation resistance of the dipole are also obtained. The effects of the chiral medium on the polarization and intensity of the dipole radiation are discussed.

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.

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.

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.

An alternative set of Cartesian vector wave functions is presented to simplify the development of eigenfunction expansions of electromagnetic fields and dyadic Green's functions for bianisotropic media. With the eigenfields expanded using these functions, the determination of their expansion coefficients is facilitated via a concise 4×4 matrix method. Based on the complete dyadic discontinuity relations, the source consequents of dyadic Green's functions are also explicitly expressed in terms of these functions. By incorporating the modified reciprocity theorem into the discontinuity relations, more compact source consequents are found to be related directly to the eigenfields in a complementary medium. These complementary eigenfields, in turn, can be deduced readily from their original counterparts by performing certain simple transformations. This manner of incorporating reciprocity provides a clear insight into the significance of each discontinuity relation and reveals the connections between the singularities, discontinuities and the complementary eigenfield components. For illustration, the vector wave functions expansions of the eigenfields for isotropic and uniaxial bianisotropic media are discussed.

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.