Electric Dipole Antenna in Presence of a Double Wire ... - IEEE Xplore

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014

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Electric Dipole Antenna in Presence of a Double Wire-Mesh Planar Interference Filter Stanislav B. Glybovski, Valeri P. Akimov, Viktor K. Dubrovich, Sergey S. Shchesnyak, and Andrey A. Matskovskiy

Abstract—In this letter, we derive a new full-wave analytical field solution for the problem of two separated planar thin-wire meshes excited by a short horizontal electric dipole (HED). Both meshes are constructed from identical perfectly conducting parallel wires with small period (interwire spacing) compared to wavelength. A fast-converging solution is obtained employing averaged boundary conditions and image theory approach. A set of computed nearand far-field patterns is presented. Index Terms—Boundary value problems, dipole antenna, periodic structures, wire grids.

I. INTRODUCTION Fig. 1. HED over two separated identical meshes of thin parallel wires.

S

TRUCTURES produced by a pair of identical wire meshes are widely used as Fabry–Perot bandpass filters from microwave to far infrared [1] for spectroscopy purposes. Fabry–Perot filters are normally developed for plane wave excitation, however for a number of systems including focal plane arrays, it is attractive to integrate a double-mesh filter with dipole antennas into the same low-profile structure. Therefore, the filter has to be optimized for operation with spherical waves, which requires an advanced analytical field solution. Planar meshes with many configurations of thin metal wires and small interwire spacing have been treated in terms of homogenized Kontorovich averaged boundary conditions, connecting averaged fields with averaged surface current density on a mesh plane, which has given explicit expressions of plane-wave reflection coefficients for single [2] and paired meshes [3]. Spherical wave diffraction has been successfully studied only for single thin-wire meshes [4], [5] employing

Manuscript received May 15, 2014; revised July 07, 2014 and July 10, 2014; accepted July 12, 2014. Date of publication July 18, 2014; date of current version July 30, 2014. This work was supported in part by the Government of the Russian Federation under Grant 074-U01. Numerical simulations of the wire-mesh Fabry–Perot resonator were supported by the Russian Science Foundation under Project No. 14-12-00897. S. B. Glybovski is with the Laboratory “Metamaterials,” University ITMO, St. Petersburg 199034, Russia (e-mail: [email protected]). V. P. Akimov is with the Institute of Physics, Nanotechnology and Telecommucications, St. Petersburg State Polytechnical University, St. Petersburg 195251, Russia. V. K. Dubrovich is with the Scientific Center of Applied Electrodynamics, Special Astrophysical Observatory RAS, St. Petersburg 196140, Russia, and also with the Laboratory of Cryogenic Nanoelectronics, NNSTU, Nizhny Novgorod 603950, Russia. S. S. Shchesnyak and A. A. Matskovskiy are with the Scientific Center of Applied Electrodynamics, St. Petersburg 190103, Russia. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2014.2339635

Exact Image Theory (EIT) approach. Following EIT technique, in this letter we obtain a full-wave solution of the diffraction problem of horizontal electric dipole (HED) over a pair of separated identical meshes of thin parallel wires, suitable for near- and far-field calculation. II. BOUNDARY PROBLEM SOLUTION A. Basic Equations The HED with length and current is located at a height above the upper mesh of the filter, which is at plane as depicted in Fig. 1. Another (lower) mesh is located at plane, so that separation of two meshes is . Both meshes consist of identical thin parallel perfectly conducting wires with spacing . The wires aligned in parallel to -axis have radius . Due to thin-wire approximation, the only -component of Hertzian potential is nonzero. It satisfies Helmholtz equation with delta-like right-hand side, which after Fourier transformation in -plane can be written as (1) where

is a transformed Hertzian potential; is a propagation constant in Fourier space; , are Fourier transform variables; is a free-space wavenumber; and is vacuum permittivity. The considered form of time dependence is . We suppose that , and on the upper mesh plane ( ) the following averaged boundary condition is valid [2]:

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(2)

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where is a mesh parameter; is free-space wave impedance; is the electric field above the upper mesh ( ), whereas is the field between meshes ( ). The tangential magnetic field’s step is determined by the averaged surface currents . Also, the condition of zero -directed current is required: . A similar boundary condition can be written in the lower mesh plane ( ) for and , where is the field below the lower mesh ( ). The boundary conditions for Hertzian potentials at , corresponding to (2) are (3) where subscripts 1 and 2 have the same meaning as described above. A similar pair of boundary conditions can be written for and at the plane . Inside volumes 1, 2, and 3, separated by meshes, the solutions of (1) one can be found in the form

For

, the function

is explicitly found as

(13) while for

, it gets the rather simpler form (14)

where

means one-sided Dirac delta-function, for which for any smooth function

.

Based on expansion (11) employing (13) and (14), the expression for finally reads (15)

(4) (5) (6)

with

In (4), the second term with is a particular solution representing the field produced by the dipole in free space. The term with describes the scattered field in the upper half-space, whereas the term with in (6) means the transmitted field. and determine the field inside the filter. Substitution of (4)–(6) into (3) after some algebra leads to (7) (8) (9) (10) . with Now instead of applying standard inverse Fourier transform to (which would lead to Sommerfeld-like integrals), let us follow EIT approach. B. Transmitted Field Since for any , the expression for be expanded into a series as

in (6) can

(16) is the Fourier-transformed free-space field of the initial dipole, located at the point . The other terms in (15) correspond to an infinite set of complex images producing the same field in every point below the structure as that produced by all mesh-wire currents. As clearly seen from (15), each complex image of th order has a form of linear distribution of -directed current , distributed along a positive part of the additional -axis. However, in -space, the images are concentrated at the points . The field solution in -space can be directly constructed by comparison of the expression for the initial dipole’s contribution to (15) to that of the th-order image. This finally results into , where the initial dipole’s field is , where is a Green function; is multiple images

. The field of

(11)

(17)

where is a particular transmission coefficient of the wave, which has reflected times within the filter. According to EIT, it should be represented as a Laplace transform of another function of a new variable as

where the complex distance between the points of integration and observation for th image current is

(12)

. The complex square-root should be chosen so that the integrand in (17) branch of decays exponentially. Therefore, the solution shows fast convergence of the integral with respect to . After obtaining , the -component of the electric field is found as .

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GLYBOVSKI et al.: ELECTRIC DIPOLE ANTENNA IN PRESENCE OF DOUBLE WIRE-MESH PLANAR INTERFERENCE FILTER

C. Scattered Field is found similarly as The field above the upper mesh the transmitted one. Expanding (9) into a series gives (18) for and for . The first term in (18) is due to the dipole, whereas the others in the sum are responsible for the scattered field. Performing the same Laplace transform as in (12) for , one can obtain the field in -space above the filter as , where the scattered field is represented by an infinite set of images where

Fig. 2. Convergence of images summing for

;

with respect to . Keeping the terms proportional to some effort one derives

.

, after

(23)

(19) with . The difference in (19) and in (17) is due to different location between of images for scattered and transmitted fields. The th image current distribution reads

(24) where is the structure;

(20) (21) D. Field between Meshes

the

full electric far field above is the transmitted far field; is the far field of the initial dipole in free space; is a spherical angle measured from the positive direction of the -axis in -plane). The transmitted far-field’s magnitude can be exactly equal to one of the dipole in free-space at some angles, determined by the filter’s parameters and frequency. However, there is a small difference in phases. Thus, equating to , one obtains the condition of resonant transmission

and in (7) and (8), Repeating the EIT procedure for one can come to the field between the meshes ( ) with the similar form of

(22) where and are the current distributions for the images located above and under the filter correspondingly

(25) In order to pass the dipole’s radiation to the direction with the angle at frequency , the filter must have mesh separation given by (25), in which is the order of resonance. Based on (22), it can be also shown that in far-field region, the field between the meshes is equal to one of the initial dipole having a form of a spherical wave because, due to radiation, excited guided modes within the structure decay exponentially. IV. RESULTS

The derived formulas completely describe the full-wave field solution of the system. III. FAR FIELDS In order to obtain far-field asymptotic formulas, let us set , , and in (17) and (19), which is possible due to good convergence of the series and the integral

In this section, we show examples of calculated near- and farfield plots and compare them to results of full-wave numerical simulations. Let us consider the frequency at which the mesh separation is half a wavelength: , where is the light velocity. First, we test convergence of summing image contributions in (17) by computing a near-field value at the point . The relative error depending on the number of preserved images is shown in Fig. 2 for ; , where the theory provides accuracy of 0.07% with only . Generally, due to smaller , the convergence slows down as interwire spacing decreases and wires get thicker. For

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Fig. 3.


distributions at and (right)

(at first resonance) for (left) .

Fig. 6. Far-field pattern at


(below resonance).

(above resonance).

lobe is suppressed when the frequency is lower than . At , the broadside transmission level of the far field ( ) is equal to one of the free-space dipole. However, even at the resonance, the pattern shape is affected by the meshes because different spatial harmonics of the dipole’s field meet the requirement (25) at different frequencies. For , (25) is satisfied at angles , where the lobes are observed in Fig. 6 producing a conical-beam pattern in transmission. Numerical simulations provide good validation of the predicted frequency-dependent patterns. However, there is a 5% difference between the theoretical resonant frequency and the simulation result, which is explained by the presence of open radiation boundaries in CST. It is seen in Fig. 5, where the CST pattern for is not equal to the free-space dipole pattern in contrast to the theoretical one. Moreover, several additional sidelobes appear in simulations due to the restricted structure’s dimensions of in -plane. V. CONCLUSION


(at resonance).

plots in this section, we keep . In numerical integration with respect to , we restrict the upper limit to , which provides sufficient accuracy. Fig. 3 shows the theoretical distributions of for , , mA, and at (resonant broadside transmission). Two dipole positions are considered: and . The -plane far-field patterns of the dipole at the height over the structure with and , calculated with (23) and (24) are depicted in Figs. 4–6 for , , and correspondingly. Each pattern is compared to simulation results by commercial finite-difference time-domain (FDTD) solver CST Microwave Studio as well as with free-space dipole patterns. It is seen that the transmission

In this letter, a problem of an HED antenna in presence of a filter with two identical meshes of parallel wires was solved analytically employing averaged boundary conditions and EIT. The derived expressions are validated through numerical simulations and allow high-speed numerical calculation. The considered filter excited by a spherical wave has bandpass properties with respect to the far-field observation point regardless of the dipole-to-filter distance. The aspects of frequency-selective field transmission through the filter have been studied in both near- and far-field regions. REFERENCES [1] R. Ulrich, K. Renk, and L. Genzel, “Tunable submillimeter interferometers of the Fabry-Perot type,” IEEE Trans. Microw. Theory Tech., vol. MTT-11, no. 5, pp. 363–371, Sep. 1963. [2] M. I. Kontorovich, M. I. Astrakhan, V. P. Akimov, and G. A. Fersman, Electrodynamics of Grid Structures (in Russian). Moscow, Russia: Radio Svyaz’, 1987. [3] J. L. Adams and L. C. Botten, “Double gratings and their applications as fabry-perot interferometers,” J. Opt., vol. 10, no. 3, p. 109, 1979. [4] I. Lindell, V. P. Akimov, and E. Alanen, “Image theory for dipole excitation of fields above and below a wire grid with square cells,” IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 2, pp. 107–110, May 1986. [5] V. Akimov and L. Babenko, “Field and input admittance of a vertical magnetic dipole placed above a plane grid screen,” Tech. Phys. Lett., vol. 26, no. 8, pp. 682–685, 2000.