Directive radiation from a line source in a metamaterial slab with low permittivity

An investigation is made into the directive radiation patterns produced by an electric line source inside of a low-permittivity metamaterial slab over a ground plane. The metamaterial slab is modeled as a homogeneous slab with a plasma-like dispersive permittivity. For low values of the slab permittivity, a very directive beam pointing at broadside can be obtained. Conditions for the maximization of the power density radiated at broadside are given, and it is shown that the directive radiation effect at broadside is due to a leaky mode supported by the slab with small and nearly equal values of the phase and attenuation constants. The frequency bandwidth and the directivity at broadside are also expressed in a simple closed form in terms of the attenuation constant of this leaky wave. Finally, numerical results are obtained for an actual metamaterial slab composed of metallic rods


I. Introduction and background
It is well known that under suitable conditions, periodic structures can be homogenized and modeled as metamaterials with an effective permittivity and permeability, presenting some interesting features that natural materials do not possess. In particular, a structure composed of a periodic array of metallic cylinders with a spatial period small compared to the wavelength (see Fig. 1(a)) behaves as a homogenous material with an effective permittivity that exhibits a plasma-like behavior [1][2]. Recently, the issue of enhancing the directivity at broadside of a source embedded in an artificial medium and controlling the direction of its radiation has been addressed by several authors, both in the optical and in the microwave ranges [3][4][5]. Some of these effects have been observed previously in a different context [6].
In this paper it is shown how the extremely directive radiation attainable with such low-permittivity metamaterial structures is related to the excitation of a leaky mode with a sufficiently-low value of the attenuation constant, and we derive simple design formulas to optimize radiation at broadside in such periodic structures, based on a modal analysis oftheir equivalent homogenous models. 0-7803-8883-6/05/$20.00 @2005 IEEE I. Analysis and optimization The structure considered here is a grounded plasma slab of height h excited by an electric line source Ji along y, embedded in the plasma medium at a distance h, from the ground plane (see Fig. l Example of EBG material excited by an electric line source: a finite number of periodic rows of metallic cylinders with radius a and spatial period d, small with respect to the wavelength, above a perfectly-conducting ground plane; (b) equivalent homogenized structure, modeled as a grounded plasma slab with relative permittivity r,.
The plasma is modeled as homogeneous and isotropic, with relative permeability A = I and relative permittivity E, = 1 -fp2 / f2, where f, is the plasma frequency. From the spectral Green's function of the problem, known in a simple closed form, the radiated power density at broadside (t= 0) can be obtained: where ko and i7o are the free-space wavenumber and charactenstic impedance, respectively. Since 14 < 1, in order to maximize P(O) we require that the sine term in the denominator of (2) be equal to zero, i.e, k oh.jf = n7r n = 1, 2,... (2) The location of the source should be chosen in order to maximize the numerator in (1), i.e., h. = hl2. From (2), an explicit expression for the optimum frequenciesft can then be obtained in terms off, and h, i.e., fope = (f2 + n2c2 /24h)1 , where c is the speed of light in free space. Moreover, when (2) holds, at the optimum frequency the power density at broadside is inversely proportional to the relative permittivity: and this result clearly shows the importance of metamaterials with low values of pemittivity to obtain strong field values at broadside.
The bandwidth for radiation at broadside has also been studied, defined as the frequency range fB < f < f3dB where the level of power density P(O) radiated at broadside is 3 dB or less lower than its maximum P. Finally, expressions for P., and FBW can be obtained in terms of the normalized attenuation constant & at the optimum frequency fpt. Using (5), the results from (3) and (4)  It is interesting to note that the bandwidth limitation is quite severe when using this type ofmetamaterial leaky-wave antenna.

III. Numerical results
In order to validate the above analysis, numerical results for a specific plasma grounded slab withfp = 20 GHz, h = 60 mm, and h, = h/2 are shown in Fig. 2. In Fig.   2(a), the dispersion curves of the first three TE. leaky modes are reported together with the power density radiated at broadside, P(0), as a function of frequency. As expected, P(O) has its maxima at the frequencies for which a leaky mode has equal values of its phase and attenuation constants. Moreover, the approximate formulas for both the optimum frequencies and the leaky-mode phase and attenuation constants are seen to be in excellent agreement with the exact ones calculated by solving numerically the TE, dispersion equation. In Fig. 2(b), the total radiation pattern is compared with the pattem radiated by the leaky mode excited along the structure; their close agreement confirms the dominant role of the leaky mode in producing the high directivity.
Finally, full-wave numerical results have been obtained through a periodic momentmethod analysis of the structure in Fig. l(a) with d = 20 mm, a = 0.5 mm and N= 6 rows ofcylinders. By considering the equivalent homogeneous grounded-slab model of the structure, with h = Nd (see [2]) andfp = 3.841 GHz (according to the approximate formula reported in [7]), it is found thatfwt = 4.039 GHz (see equation after (2)), which is slightly shifted from the exact value (fwt = 4.01 GHz) obtained from the full-wave numerical analysis of the periodic structure. The pattem and leaky wave phenomenology observed for the homogeneous material are in good agreement with those obtained from the numerical analysis ofthe structure in Fig. l(a).