Electromagnetic dyadic Green's functions for multilayeredspheroidal structures. I: formulation

Dyadic Green's functions (DGFs) and their scattering coefficients are formulated in this paper for defining the electromagnetic fields in multilayered spheroidal structures. The principle of scattering superposition is applied, in a similar form of the DGF in an unbounded medium under spheroidal coordinates, the scattering DGFs due to multiple spheroidal interfaces are expanded in terms of the spheroidal vector wave functions. For the lack of general orthogonality of the spheroidal radial and angular functions, the Green's dyadics are expressed in a different way where the coordinate unit vectors are also combined in the construction, as compared with the conventional form of vector wave eigenfunction expansion. The matrix equation systems satisfied by the coupled scattering (i.e., reflection and transmission) coefficients of the DGFs are obtained so that these coefficients can be solved uniquely. The DGFs can be employed to investigate effects of spheroidal radomes used to protect the airborne or satellite antenna systems and of handy phone radiation near the spheroid-shaped human head, and so forth. Numerical calculations about the applications of the formulated multilayered DGFs are presented in part II of this paper     

A general expression of dyadic Green's functions in radiallymultilayered chiral media

A general expression of spectral-domain dyadic Green's function (DGF) is presented for defining the electromagnetic radiation fields in spherically arbitrary multilayered and chiral media. Without any loss of the generality, each of the radial multilayers could be the chiral layer with different permittivity, permeability, and chirality admittance, while both distribution and location of current sources are assumed to be arbitrary. The DGF is composed of the unbounded DGF and the scattering DGF, based on the method of scattering superposition. The scattering DGF in each layer is constructed in terms of the modified and normalized spherical vector wave functions. The coefficients of the scattering DGFs are derived and expressed in terms of the equivalent reflection and transmission coefficients, by applying boundary conditions satisfied by the coefficient matrices     

One- and two-dimensional dyadic Green's functions in chiral media

The one-dimensional and two-dimensional dyadic Green's functions are calculated for 1D and 2D electric sources in an unbounded, lossless, reciprocal chiral medium which is electromagnetically described by a set of symmetric constitutive relations. It is shown that in two- and one-dimensional cases, similar to the three-dimensional case, the medium supports two eigenmodes of propagation with two different wavenumbers. One of them corresponds to the right-circularly polarized wave and the other one to the left-circularly polarized wave. The eigenmode amplitudes a and b are similar to those of the three-dimensional case     

Spectral dyadic Green's function formulation for planar integratedstructures

The problem of the complete determination of the dyadic spectral Green's function for an integrated planar structure with a grounded dielectric slab has been considered and solved in a rigorous way by using the spectral theory of the electromagnetic field. The reciprocity theorem and the geometrical symmetry of the structure have demonstrated the different roles played by the independent terms of the spectral Green's function in the evaluation of the electromagnetic characteristics of the grounded slab excited with a general source. Furthermore, an equivalent circuit representation of the structure, allowing a noteworthy simplification in the determination of the total power, has been obtained. These equivalent circuits and the derived spectral Green's function presented here can be used to analyze and design microstrip antennas of arbitrary shape with a general type of loading, such as matched or unmatched loads, parasitic, and shorting pins     

On the electromagnetic dyadic Green's functions for planar multi-layered anistropic uniaxial material media

A complete, plane-wave spectral, vector-wave function expansion of the electromagnetic, electric, and magnetic, dyadic Green's function for electric, as well as magnetic, point currents for a planar, anisotropic uniaxial multi-layered medium is presented. It is given in terms ofz-propagating, source-free vector-wave functions, where ? is normal to the interfaces, and it is developed via a utilization of the Lorentz reciprocity theorem. The electromagnetic dyadic Green's function for periodic electric as well as magnetic point current sources is also presented. Some salient features of the Green's dyadics, along with a physical interpretation are also described.     

New formulation of dyadic Green's function applied to a microstripline

We have modified the conventional two-layer dyadic Green's function analysis to include finite metal conductivity. The modified Green's function allows us to calculate the current distributions and conductor losses in microstrip circuits. A sample calculation has been applied to a microstrip line     

Time-dependent dyadic Green's functions for rectangular andcircular waveguides

Closed-form expressions for the time-dependent dyadic Green's functions of electric and magnetic types for rectangular and circular waveguides are derived from the dyadic Maxwell equations in the time domain. These functions can be used to obtain the time-dependent electric and magnetic fields propagating in those guides due to any arbitrary time-dependent current distribution inside the guide. Stationary vector wave functions are introduced that separate the space-dependent parts from the time-dependent parts of the Green's functions. Comparison of the results for the rectangular and circular guides reveals that the time-dependent parts are identical. Thus the results can be easily extended to some other cylindrical pipes such as elliptical waveguides and also to coaxial cables     

On the dyadic Green's functions for Beltrami fields

A simple derivation of the Green's functions for Beltrami fields is presented for use with time-harmonic electromagnetism in homogeneous biisotropic media.     

Spectral dyadic Green's function formulation for uniaxial bianisotropic superstrate-substrate structure

In this paper, spectral-domain dyadic Green's functions for the time harmonic electric current source embedded in a two-layer grounded uniaxial bianisotropic media are obtained using Fourier tranform. It is shown that in the uniaxial bianisotropic medium, total spectral electromagnetic field can also be separated into the superposition of transverse electric (TE) and transverse magnetic (TM) wave. Because of the generality of constitutive relations our results include the special cases of achiral, uniaxial, reciprocal and nonreciprocal biisotropic media.

     

Some methods of solving for the coefficients of dyadic Green's functions in isotropic stratified media

Dyadic Green's functions in multi-layered isotropic media are analysed in this paper. Three different kinds of method for obtaining the coefficients of dyadic Green's functions in multi-layered media are given, these are (a) the boundary condition method, which is well-known; (b) the recurrence matrix equation method; and (c) the ray trail method. Using these methods, several examples are considered. Some results are the same as those obtain previously. Some are obtained for the first time.     

Scalarization of dyadic spectral Green's functions and networkformalism for three-dimensional full-wave analysis of planar lines andantennas

A novel and systematic method is presented for the complete determination of dyadic spectral Green's functions directly from Maxwell's equations. With the use of generalized scalarizations developed in this paper, four general and concise expressions for the spectral Green's functions for one-dimensionally inhomogeneous multilayer structures, excited by three-dimensional electric and magnetic current sources, are given in terms of modal amplitudes together with appropriate explicit singular terms at the source region. It is shown that Maxwell's equations in spectral-domain can be reduced, by using dyadic spectral eigenfunctions, to two sets of z-dependent inhomogeneous transmission-line equations for the modal amplitudes. One set of the transmission-line equations are due to the transverse current sources and the other set due to the vertical current sources. Utilizing these equations, network schematizations of the excitation, transmission and reflection processes of three-dimensional electromagnetic waves in one-dimensionally inhomogeneous multilayer structures are achieved in a full-wave manner. The determination of the spectral Green's functions becomes so simple that it is accomplished by the investigation of voltages and currents on the derived equivalent circuits. Examples of singleand multilayer structures are used to validate the general expressions and the equivalent circuits     

The eigenfunction expansion of dyadic Green's functions forchirowaveguides

A general method of formulating eigenfunction expansion of dyadic Green's functions in lossless, reciprocal and homogeneous chirowaveguides is presented. Bohren's decomposition of the electromagnetic field is used to obtain the vector wave functions. The method of G¯¯_m is used to rigorously derive the magnetic and electric dyadic Green's functions. A specific application to the cylindrical chirowaveguide illustrates the method     

On the eigenfunction expansion of dyadic Green's functions

As a result of an error, the singular behavior of the eigenfunction expansion of the dyadic Green's function is not correctly formulated in my book (C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, Scranton, Pa.: International Textbook, 1971). The correct expressions are given here and an improved method for deriving the residue series is presented.     

A note on using dyadic Green's functions on spheres and cones

The objective of this article is to alert antenna engineers to a possible error that may occur when working with dyadic Green's functions associated with conical geometries. For example, such problems can arise when designing microstrip antennas on a conical surface, or when dealing with scattering caused by conical geometries. The dyadic Green's function is a very useful tool for obtaining the interaction of electromagnetic fields with several physical geometries. With this technique, the engineer can formulate various canonical electromagnetic problems in a systematic manner, to avoid having to treat many special cases. However, in dealing with conical geometries, special care must be taken     

Coordinate-independent dyadic formulation of the dispersionrelation for bianisotropic media

This paper presents a coordinate-independent dyadic formulation of the dispersion relation for general bianisotropic media. The dispersion equation is expanded with the aid of dyadic operators including double-dot, double-cross and dot-cross or cross-dot products. From the dispersion relation, the Booker quartic equation is derived in a form well-suited for studying multilayered structures. Several deductions are made in conjunction with the bianisotropic media satisfying reciprocity and losslessness conditions. In particular, for reciprocal bianisotropic media, the dispersion equation is biquadratic in wave vector while for lossless bianisotropic media, all dispersion coefficients are of real values. As an application example, the dispersion equation for gyrotropic bianisotropic media is considered in detail     

On the eigenfunction expansion of electromagnetic dyadic Green's functions

A relatively simple approach is described for developing the complete eigenfunction expansion of time-harmonic electric (bar{E}) and magnetic (bar{H}) fields within exterior or interior regions containing an arbitrarily oriented electric current point source. In particular, these results yield directly the complete eigenfunction expansion of the electric and magnetic dyadic Green's functionsbarbar{G}_{e}andbarbar{G}_{m}that are associated withbar{E}andbar{H}, respectively. This expansion ofbarbar{G}_{e}andbarbar{G}_{m}contains only the solenoidal type eigenfunctions. In addition, the expansion ofbarbar{G}_{e}also contains an explicit dyadic delta function term which is required for making that expansion complete at the source point. The explicit dyadic delta function term inbarbar{G}_{e}is found readily from a simple condition governing the behavior of the eigenfunction expansion at the source point, provided one views that condition in the light of distribution theory. These general expressions for the eigenfunction expansion ofbarbar{G}_{e}andbarbar{G}_{m}reduce properly to those obtained previously for special geometries by Tai.     

Analysis of microstrip wraparound antennas using dyadic Green's functions

The radiation pattern of microstrip wraparound antennas was obtained here using a theory based on dyadic Green's functions for concentric-cylindrical layered media. The dielectric layer that is usually neglected as a first-order approximation was considered here. An asymptotic expression for the dyadic Green's function that takes into account only the space wave is first obtained. Radiation patterns for various radii, permittivities, and thicknesses of the dielectric layer of a microstrip wraparound antenna were obtained using as a source a uniform annular magnetic current obtained by means of a cavity model with conducting magnetic walls. The calculated values of the percent pattern coverage decreases as the thickness and the permittivity of the dielectric layer increase. The influence of the dielectric layer is more pronounced for radiation direction near that of the axis of the cylindrical surface. It is also shown that the radiation patterns at a frequency of 2.0 GHz are not much dependent on the diameter of the antenna for values from 3 to 120 in.     

Transient and time-harmonic dyadic Green's functions for a perfectly conducting cone

New representations for the time-dependent dyadic Green's functions for a perfectly conducting semi-infinite cone are presented. For the special case of small cone angles and an on-axis source, simplified expressions are given for both the time-dependent and time-harmonic regimes.     

Closed-form Green's functions for cylindrically stratified media

A numerically efficient technique is developed to obtain the spatial-domain closed-form Green's functions of the electric and magnetic fields due to z- and φ-oriented electric and magnetic sources embedded in an arbitrary layer of a cylindrical stratified medium. First, the electric- and magnetic-field components representing the coupled TM and TE modes are derived in the spectral domain for an arbitrary observation layer. The spectral-domain Green's functions are then obtained and approximated in terms of complex exponentials in two consecutive steps by using the generalized pencil of function method. For the Green's functions approximated in the first step, the large argument behavior of the zeroth-order Hankel functions is used for the transformation into the spatial domain with the use of the Sommerfeld identity. In the second step, the remaining part of the Green's functions are approximated on two complementary parts of a proposed deformed path and transformed into the spatial domain, analytically. The results obtained in the two consecutive steps are combined to yield the spatial-domain Green's functions in closed forms     

On the dyadic Green's function for a planar multilayereddielectric/magnetic media

A complete plane wave spectral eigenfunction expansion of the electric dyadic Green's function for a planar multilayered dielectric/magnetic media is given in terms of a pair of the (zˆ)-propagating solenoidal eigenfunctions, where (z ˆ) is normal to the interface, and it is developed via a utilization of the Lorentz reciprocity theorem. This expansion also contains an explicit dyadic delta function term which is required for completeness at the source point. Some useful concepts such as the effective plane wave reflection and transmission coefficients are employed in the present spectral domain eigenfunction expansion. The salient features of this Green's function are also described along with a physical interpretation