Resonance conditions for a substrate-superstrate printed antenna geometry which allow for large antenna gain are presented. Asymptotic formulas for gain, beamwidth, and bandwidth are given, and the bandwidth limitation of the method is discussed. The method is extended to produce narrow patterns about the horizon, and directive patterns at two different angles.

Gain enhancement methods for printed circuit antennas

The classical Sommerfeld half-space problem is revisited, with generalizations to multilayer and plasmonic media and focus on the surface field computation. A new ab initio solution is presented for an arbitrarily oriented Hertzian dipole radiating in the presence of a material half-space with arbitrary horizontal stratification. The solution method combines the vector potential approach and the spectral domain transmission line analog of the medium, which results in the most compact formulation and facilitates the inclusion of any number of layers in the analysis. Following Sommerfeld, the solution is first expressed in terms of the Fourier–Bessel transforms, also known as Sommerfeld integrals. Analytical properties of the integrands in the complex plane are then investigated, including the location of the Sommerfeld pole, which gives rise to the Zenneck wave (ZW) or surface plasmon polariton (SPP), and alternative field representations are developed by a deformation of the integration path and analytic ...

https://www.tandfonline.com/doi/pdf/10.1080/09205071.2015.1093964

The Sommerfeld half-space problem revisited: from radio frequencies and Zenneck waves to visible light and Fano modes

An important extension of the two-level discrete complex image method is proposed to eliminate any concerns on and shortcomings of the approximations of the spatial-domain Green's functions in closed form in planar multilayered media. The proposed approach has been devised to account for the possible wave constituents of a dipole in layered media, such as spherical, cylindrical, and lateral waves, with the aim of obtaining accurate closed-form approximations of Green's functions over all distances from the source. This goal has been achieved by judiciously introducing an additional level into the two-level approach to pick up the contributions of lateral waves in the spatial domain. As a result, three different three-level algorithms have been proposed, investigated, and shown that they work properly over all ranges of distances from the source. In addition to the accuracy of the results at all distances, these approaches also proved to be robust and computationally efficient as compared to the previous algorithms, which can be attributed to the fact that the sampling of the spectral-domain Green's functions in the proposed approaches gives proper emphasis to the associated singularities of the wave types in the spectral domain. However, the judicious choices of the sampling paths may not be enough to get accurate results from the approximations unless the approximating functions in the spectral domain can provide similar wave natures in the spatial domain. To address this issue, the proposed algorithms employ two different approximations; the rational function fitting methods to capture the cylindrical waves (surface waves), and exponential fitting methods to capture both spherical and lateral waves. It is shown and numerically verified that a linear combination of exponential functions in the spectral domain represent the lateral waves at the interface of the involved layers.

Closed-Form Green's Functions in Planar Layered Media for All Ranges and Materials

The closed-form Green's functions (CFGF), derived for the vector and scalar potentials in planar multilayer media, have been revisited to clarify some issues and misunderstandings on the derivation of these Green's functions. In addition, the range of validity of these Green's functions is assessed with and without explicit evaluation of the surface wave contributions. As it is well-known, the derivation of the CFGF begins with the approximation of the spectral-domain Green's functions by complex exponentials, and continues with applying the Sommerfeld identity to cast these approximated spectral-domain Green's functions into the space domain in closed forms. Questions and misunderstandings of this derivation, which have mainly originated from the approximation process of the spectral-domain Green's functions in terms of complex exponentials, can be categorized and discussed under the topics of: 1) branch-point contributions; 2) surface wave pole contributions; and 3) the accuracy of the obtained CFGF. When these issues are clarified, the region of validity of the CFGF so obtained may be defined better. Therefore, in this paper, these issues will be addressed first, and then their origins and possible remedies will be provided with solid analysis and numerical demonstrations.

/pdf/clarification-of-issues-on-the-closed-form-green-s-functions-8m4xndf5ij.pdf

Clarification of issues on the closed-form Green's functions in stratified media

A review is presented of the most effective methods for the computation of Sommerfeld integral tails. Such integrals, which are often oscillatory, singular and divergent, commonly arise in layered media Green functions. The mathematical foundations of various pertinent methods are discussed in detail and their performance is illustrated by relevant numerical examples. The associated algorithms are given in pseudocode for their easy computer implementation.

https://tandfonline.com/doi/pdf/10.1080/09205071.2015.1129915

Efficient computation of Sommerfeld integral tails – methods and algorithms

Microstrip elements of arbitrary shape are modeled in multilayered media. The Green's function for the multilayered structure is developed in a form useful for efficient computation for interacting microstrip elements, which may be located at any substrate layer and separated by an arbitrarily large distance. This result is of significant value to a variety of applications in wave propagation besides those discussed in this paper. The mixed-potential integral-equation (MPIE) method is developed in the spatial domain. Examples for regularly/arbitrarily shaped geometries in single and multilayered media are presented. These involve the optimization of an open-end microstrip, a radial-stub microstrip, a five-section overlay-gap-coupled filter, and a circular-patch proximity-coupled microstrip antenna. Very good agreement with measurement and other published data is observed.

Modeling planar arbitrarily shaped microstrip elements in multilayered media

This paper presents a technique for electromagnetic synthesis of microstrip overlap-gap-coupled filters. The algorithm uses a rigorous analysis, which solves an electric field integral equation in the spectral domain using the method of moments, inside an outer loop that optimizes the filter dimensions. A piecewise synthesis procedure generates initial dimensions that provide a good starting point for direct electromagnetic optimization of the complete filter. Measured results for a five-section filter are shown that verify the accuracy of this design method. >

Electromagnetic synthesis of overlap-gap-coupled microstrip filters

An efficient numerical integration algorithm is developed to evaluate the spatial-domain Green's function of vector and scalar potentials. A rigorous formulation for the Green's function in multi-layered media is derived in the spectral domain involving Sommerfeld-type integrals. This approach requires little analytic derivation for singular terms, and reduces the computation effort by using the weighted-average method to evaluate the oscillatory tail integrals. It is found that this algorithm is very efficient and suitable to model interconnects and microstrip elements for printed circuits/antennas with arbitrary size and any number of layers.

Sommerfeld Integrals in Modeling Interconnects and Microstrip Elements in Multi-Layered Media

The printed aperture of arbitrary shape is introduced into multi-port microstrip circuits as a vertical transition within a multi-layered structure. Its use proves to be practical in the design as well as manufacturing process of multi-layered circuits. With the help of the mixed-potential integral equation based moment method, it becomes possible to analyze and optimize the performance of this arbitrary shape aperture transition for multi-port circuit applications. Bandwidth enhancement is obtained by changing the shape of the slot for a two-port back-to-back microstrip transition. Minimized size and mutual coupling have been thoroughly studied for optimized circuit performance. Using this transition, a 3-port power divider with -3 dB amplitude and 180 degree phase difference and a 4-port 3 dB directional coupler have been designed.

Optimization of aperture transitions for multi-port microstrip circuits

In the past, integral-equation modeling techniques such as the spectral-domain electric-field integral equation (EFIE) and the mixed-potential integral equation (MPIE) have been employed to accurately characterize microstrip patch antennas. However, due to an insufficient segmentation scheme for current densities inside a finitely thick microstrip, these methods are limited to an idealized microstrip patch with infinitesimal thickness. In the present paper, the spectral-domain EFIE formulation in conjunction with a suitable current expansion mechanism (first introduced in Becks and Wolff, 1992) for analysis of a microstrip via-holes and air-bridges, is used to evaluate the finite metallization thickness effects. For the first time, very rigorous theoretical results are demonstrated to illustrate the influence of metallization thickness on resonant frequency, reflection coefficient and current distribution of a microstripline-fed patch antenna. >

http://ieeexplore.ieee.org/iel2/3222/9160/00407923.pdf

Ming-Ju Tsai

Papers

Modeling planar arbitrarily shaped microstrip elements in multilayered media

Electromagnetic synthesis of overlap-gap-coupled microstrip filters

Sommerfeld Integrals in Modeling Interconnects and Microstrip Elements in Multi-Layered Media

Optimization of aperture transitions for multi-port microstrip circuits

The influence of metallization thickness on a microstripline-fed patch antenna