Circular Wavegides On-Periphery Filled with Ultralow-Index Artificial Materials

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1 5 Circular Wavegides On-Periphery Filled with Ultralow-Index Artificial Materials A. E. Serebryannikov, A. L. Teplyuk and K. Schuenemann Institut fuer Hochfrequenztechnik Technische Universitaet Hamburg-Harburg, Hamburg, D-2073, Germany Tel: ; Fax: ; Abstract-Characteristics of TE- and TM-polarized modes in circular waveguides with on-periphery placed coaxial inserts made of ultralow-index artificial material are studied. The emphasis is put on the effect of varying permittivity, permeability and thickness of the insert. It is shown that electromagnetic characteristics can substantially be changed in comparison with a hollow waveguide even in case of a very thin insert. At certain conditions, the transverse wavenumbers corresponding to the different polarizations change their locations with respect to each other. Index Terms- circular waveguide, transverse wavenumber, artificial material. I. INTRODUCTION A tremendous progress in theoretical and experimental studies of metamaterials in a wide frequency range, from microwaves to the visible has occurred since several years, e.g., see []-[4]. These materials can be considered as a modern extension of the concept of artificial dielectrics, which were proposed in 50ies for microwave applications [5]. New useful regimes have recently been demonstrated in waveguides and cavities, which are completely or partially filled with artificial materials of different types [6]- []. The most impressive result demonstrated concerns the possibility of miniaturization, and in particular of realizing thin subwavelength cavity resonators using double-negative metamaterials [0,]. An opposite case with cutoff frequency raising with the volume occupied by the singlenegative material insert placed in rectangular waveguide has been considered in [8]. It is associated with the possibility of deminiaturization, which can be important for some applications. Performances and applications of singlenegative, double-negative, and near-zero-index materials have been reviewed in [6]. In particular, to obtain a material with ε< or µ<, wire media or arrays of split-ring resonators (SRR) are used, respectively. Their combination leads to a double-negative behavior [2]. Zero and close-tozero, non-negative values of the refractive index N can be realized in (non-) periodic structures containing metallic constituents [3,4,3], as well as in pure dielectric photonic crystals [4]. The later is also related to negative N [5]. Note that positive and negative values of N 0 can be realized within different frequency ranges of the same material, so that the ultralow-index regime can be obtained in the transition region [2]. An artificial material which is placed into a conventional waveguide can affect the modes of either both or one of polarizations, depending on performance. For example, in case of wire medium, the wires should be present which are parallel to the electric field vector in order to affect electromagnetic characteristics for the corresponding polarization [6]. Different arrangements of the wires, which have been proposed in [7], are in agreement with this rule. In this paper, we study the electromagnetic characteristics of a circular waveguide, which is partially filled with an artificial material with ultralow positive refractive index, i.e., with either 0<ε< and µ= or 0< µ< and ε=. Although a

2 52 dispersionless material with such values of ε and µ is unrealizable, its consideration instead of a dispersive one often allows to better understand the basic features observed. This approach has been exploited in many studies devoted to devices containing components which are made of artificial materials. Following this way, we first consider the case of frequency independent ε and µ. The possibilities of obtaining some of the detected effects in a dispersive material are then discussed for a Drude-type frequency dependence of ε. An approximate characteristic equation has been derived and used for both an a priori analysis of the basic effects and the calculations. The main attention in the numerical study is paid to the search for such regimes that are characterized by a rather strong shift in the transverse wavenumbers, which arises when a small-volume insert is introduced. II. THEORETICAL BACKGROUND We consider circular waveguides which are onperiphery filled with an artificial material. The radii of the inner air-filled region and of the outer metallic shell are denoted by a and b, respectively. The domain with a<ρ<b is assumed to be filled with an ultralow-index material (ULIM). In the TE-case, the transverse wavenumbers have to satisfy the following characteristic equation: with J m'( k a) ξ ( k, a, b) J ( k a) = 0 () m 2 m ξ ( k 2, a, b) = θ Rm / S m (2) m where Rm = J m'( k 2a) Y m'( k 2b) J m'( k 2b) Y m'( k 2a), S m = J m( k 2a) Y m'( k 2b) J m'( k 2b) Y m( k 2a), θ = ε µ / 2 ε 2µ, ε and µ are relative permittivity and permeability of free space, ε = µ =; k 2 = ε 2µ 2k, k =ω/c, ω means angular frequency, c is the velocity of the electromagnetic wave, ε 2 and µ 2 are relative permittivity and permeability of the insert; J m, Y, J ', and ' m m Y m are m-th order Bessel and Neumann functions and their derivatives, respectively. In the TM-case, we obtain with J m( k a) η ( k, a, b) J '( k a) = 0 (3) m 2 m η (, a, b) = / (4) m k 2 θt m U m where T m = J m( k 2a) Y m( k 2b) J m( k 2b) Y m( k 2a) and U m = J m'( k 2a) Y m( k 2b) J m( k 2b) Y m'( k 2a). The equations () and (3) can be simplified in some cases. For example, this is possible for ε 2 =0, what corresponds to an electrostatic electric field at a<ρ<b []. Assume that m>0 and k 2 a<<. Using the small-argument asymptotic expression of the cylindrical functions [8], we obtain instead of () J m( k a) + ( k aω m/ mθ ) J '( k a) = 0, (5) 2 m m where Ω m = [ + ( a / b) ] [ ( a / b) ] 2 2m. Hence the transverse wavenumbers of the TE-modes of a filled waveguide tend to those of the TM-modes of the hollow waveguide of radius ρ=a, provided that k 2 aω m / mθ <<. They vary slightly with a/b 2 if ( a / b ) m <<. Hence the air-ulim interface can play the role of a magnetic wall placed at ρ=a, even at very small thickness of the insert. In turn, (3) is reduced at m>0 and k 2 a<< to J m( k a) + ( k aϑ / mω m) J '( k a) = 0, (6) m where ϑ = µ 2 ε / µ. One can see from (6), that the transverse wavenumbers of the TM-modes are independent of ε 2 and just weakly sensitive to the presence of the insert, at least if k aϑ / m << and/or Ω m <<. Note that the effect of a magnetic wall at ρ=a could be obtained in this case if k aϑ / mω m >>. It is expected that under certain conditions, the transverse wavenumbers of the TE-modes exceed those for the TM-modes,

3 53 what is different from a conventional hollow waveguide. Once a transverse wavenumber χ=k is found as a solution of one of the equations (), (3), (5), or (6), the propagation constant is 2 2 Γ = 0 χ where k 0 is the free- obtained as k space wavenumber. According to perturbation theory, introducing a small lossless body into a hollow cavity leads to an eigenfrequency shift where ω = ω( A + B) / C (7) A = ε 0( ε 2 ε ) V EE 0dV, B = µ ( 0 µ ) 2 µ V HH 0dV, and C = ε EE dv + µ HH dv. 0 V V 0. In the expressions for A, B, and C, E 0 and H 0 are the fields in the unperturbed cavity, E and H are the fields in a perturbed cavity, V 0 and V are the volumes of unperturbed cavity and perturbing body, respectively, ε 0 and µ 0 are absolute permittivity and permeability of free space, ω is here an eigenfrequency of a perturbed cavity, and the asterisk means complex conjugate. The same consideration remains valid for a small variation of the waveguide cross section. In the special case when µ 2 = µ and ε 2 =0, Eq. (7) is reduced to ω = ωε ε EE dv C. (8) 0 0 / V Any such perturbation should result in a shift of χa to larger values, provided that Re EE 0 > 0 and C > 0. At ε 2 = ε and µ 2 =0, we obtain another case but with the same sign of ω if Re HH 0 > 0 and C > 0. Fig.. Effect of ε 2 (a) and b/a (b) on χa at m= and µ 2 =; b/a=.0 in case (a) and ε 2 =0.05 in case (b); solid, dashed, dotted, and dashed-dotted lines are obtained using the rigorous Eqs. () and (3), and approximate Eqs. (5) and (6), respectively. Dashed and dashed-dotted lines almost coincide. III. NUMERICAL EXAMPLES Calculations have been carried out in a wide range of variation of ε 2 and b/a, by using both the rigorous and the approximate characteristic equations. Figure demonstrates the effects of ε 2 and b/a on χa-values for modes TE n and TM n (m=) in the case of a thin ULIM insert. For TEmodes, any variation of ε 2 in the close vicinity of zero results in a substantial variation of χa. In agreement with (6), there is no substantial effect exerted by the variation of ε 2 for TM-modes. One can expect the appearance of hybrid modes in

4 54 case of weak losses [ Imε 2 0], while Reε 2 tends to zero and χa-values for TE- and TMmodes of the corresponding lossless waveguide tend to coincide [see Fig. (a)]. Varying m, ε 2, and b/a, one can obtain the case that the curves corresponding to different polarizations cross each other and control the number of mode pairs TE mn and TM mn, for which such a crossing takes place. Hence, a decrease of ε 2 can result in that the transverse wavenumbers for TE-modes can be larger than those for TMmodes not only at m=0, as in a conventional hollow circular waveguide, but also at m. As an example, Fig. 2 demonstrates the effect of ε 2 and b/a on χa at rather large m. The dependencies of χa on ε 2 at b/a=.0 and m=6 (not shown) look very similar to those shown in Fig. (a), but are all shifted towards larger χavalues. It is worth noting that the approximate equation (6) always provides good accuracy while (5) does so only if b/a is very small. Figure 3 illustrates the effect exerted by a varying µ 2 and b/a on χa in case of a magnetic insert with 0<µ 2 < and ε 2 =. In both TE- and TM-cases, a variation of either µ 2 or b/a leads to a rather weak effect on χa, whose strength depends on m and n. The realistic ULIM materials are necessarily dispersive. Usually the Drude model is used to appropriately describe the dispersion in these materials. Here we restrict consideration to the case that µ eff =µ 2 = while 2 2 ε eff ( ω) = ε 2 = ω p, eff / ω, (9) Fig.2. Effect of ε 2 (a) and b/a (b) on χa at m=6 and µ 2 =; b/a=. in case (a) and ε 2 =0.05 in case (b); solid, dashed, dotted, and dashed-dotted lines correspond to the same cases as in Fig.. In plot (b), dashed and dashed-dotted lines almost coincide. As follows from the obtained results, an increase of b/a leads to either an increase or decrease of the χa-values for TE-modes and always to a decrease for TM-modes. Note that in most of the cases observed, the change in χa can be interpreted based on (7). In case of m=0, any variation of ε 2 from 0 to does not lead to a substantial variation of χa-values for both TEand TM-modes, at least if b/a is small enough. where ω p,eff is the effective plasma frequency and losses are neglected. Equation (9) is valid for pure metals, where ω p =ω p,eff is typically equal to s, and for an artificial medium composed of metallic rods. According to [4], 9 ω p,eff in the latter case can fall down to 0 s, so that the basic features of plasma-like frequency dependence of ε 2 can be realized in the microwave region. Similar dependence of ε eff on ω has been obtained in [6] in the framework of a quasi-static model. It is worth noting that in the existing materials the condition ω p,eff d/2πc>0., where d is an array period, usually holds whatever the frequency

5 55 planes [9, Ch. -4] is considered as an alternative. Besides, the possible performances leading to ultralow values of µ eff are now under consideration. One should keep in mind that the continuous dependencies shown correspond to hypothetic materials, since the frequency dependence of ε 2 and/or µ 2 has not been taken into account. An example of the transverse wavenumber spectrum for a waveguide filled on-periphery with a dispersive material with ε 2 = ε eff given by (9) and µ 2 = is presented in Fig. 4. Here ε 2 =0. corresponds to the mode, which is referred to as the strongly shifted waveguide mode. This is a TE-mode, whose χa-value is strongly shifted towards larger values owing to the effect of the insert. This shift is associated with the wavenumber raising observed at small ε 2, as in Fig. (a). Strictly speaking, χa-values are shifted due to the insert for all modes. However, only one of them is strongly shifted, which corresponds to ε 2 being close to 0. Examples of the shifted waveguide modes are presented in Table, which correspond to various values of ε 2 <. Fig.3. Effect of µ 2 (a) and b/a (b) on χa for modes TE n (solid line), TM n (dashed line), TE 6n (dotted line), and TM 6n (dashed-dotted line); ε 2 =, µ 2 =0.05 in case (b) and b/a=. in case (a). The first four and two modes are shown at m= and m=6. range used, e.g. see [4,5,0,6]. If the condition According to the above presented reasoning, the insert parameters, which have been used to obtain Fig. 4 and Table, would rather correspond to larger χa-values, at least for the existing volumetric ULIM s. However, all features observed are expected to remain if τ is increased up to more realistic values. ω p, eff ( b a) / 2π c > 0.M (0) should be satisfied for M=4...6 in order to obtain appropriate effective behavior, the case when χa, ε 2 0 and (b-a)/b<< are simultaneously satisfied cannot be realized. The possibilities of satisfying these conditions are now under study. In particular, the case of M= for the circular rods and various noncircular shapes of a rod cross section are studied. The use of recently suggested artificial ground Fig.4. Normalized transverse wavenumbers σ =χa (circles) for modes TE n of a circular waveguide filled with a Drude-type material at ε 2 >0; ε 2 is given by (9) where ω p,eff =2c/a, µ 2 = and b/a=..

6 56 Table : Effect of the insert on shift of transverse wavenumbers. h is radial index of the shifted waveguide mode, κ is transverse wavenumber of a hollow waveguide, τ =ω p,eff a/c. τ mode b/a ε 2 χa χb κb 2 TEh TEh TMh TE2h TE6h TM6h TEh TEh TEh TEh TE2h TE2h IV. CONCLUSIONS We studied the effect exerted by geometrical and material parameters of an ULIM coaxial ring inserted into a circular waveguide on its electromagnetic characteristics. It has been shown that even introducing a very thin ULIM insert with close-to-zero permittivity can lead to a substantial increase of the transverse wavenumbers in the TE-case. This mechanism of varying the wavenumbers can be assigned to surface effects, or strictly speaking to quasisurface effects, since it allows to keep the biggest part of the waveguide cross section unfilled. This is a difference in comparison with most of earlier studies of waveguides filled with artificial materials. This effect can be used in those applications, where an increase of a wavenumber or/and a larger waveguide cross section is needed. Such a possibility appears due to that the air-ulim interface plays the role of a magnetic wall. For instance, it should allow applying cavities formed by finite-length sections of the considered waveguides in fast-wave electron devices. Inserting an ULIM ring made of a magnetic material does not lead to any strongly pronounced effect exerted on the transverse wavenumbers. Sample numerical results confirm that the shift of the wavenumbers of TE-modes, which has been predicted in the framework of the dispersionless model, can occur when using a material with Drude-type frequency dependence of the permittivity. The obtained results have been discussed in the context of possible realization of the detected effects. REFERENCES [] R. W. Ziolkowski, Propagation in and scattering from a matched metamaterial having a zero index of refraction, Phys. Rev. E, Vol. 70, , [2] R. W. Ziolkowski and C.-Y. Cheng, Lumped element models of double negative metamaterialbased transmission lines, Radio Science, Vol. 39, RS207, [3] N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, Zero permittivity materials: band gaps at the visible, Appl. Phys. Lett., Vol. 80, pp , [4] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Extremely low frequency plasmons in metallic mesostructures, Phys. Rev. Lett., Vol. 76, pp , 996. [5] J. Brown, Artificial dielectrics having refractive indices less than unity, Proc. IEE, Vol. 00C, pp. 5-62, 953. [6] N. Engheta and R. Ziolkowski, A positive feature for double-negative metamaterials, IEEE Trans. Microwave Theory Tech., Vol. 53, pp , April [7] H. Cory and A. Shtrom, Wave propagation along a rectangular metallic waveguide longitudinally loaded with a metamaterial slab, Microwave Opt. Technol. Lett., Vol. 4, pp , [8] Y. Xu, Wave propagation in rectangular waveguide filled with single negative metamaterial slab, Electron. Lett., Vol. 39, pp , [9] H. Dong and T. X. Wu, Analysis of discontinuities in double-negative (DNG) slab waveguides, Microwave Opt. Technol. Lett., Vol. 39, pp , [0] N. Engheta, An idea for thin sub-wavelength cavity resonators using metamaterials with negative permittivity and permeability, IEEE Antennas Wireless Propag. Lett., Vol., pp. 0-3, [] L. Shen, S. He, and S. Xiao, Stability and quality factor of a one-dimensional subwavelength cavity resonator containing a left-handed metamaterial, Phys. Rev. B, Vol. 69, 5, [2] R. A. Shelby, D. R. Smith, and S. Schultz, Experimental verification of a negative index of refraction, Science, Vol. 292, 554, pp , 200. [3] B. T. Schwartz and R. Piestun, Total external reflection from metamaterials with ultralow refractive index, J. Opt. Soc. Am. B, Vol. 20, pp , 2003.

7 57 [4] B. Gralak, S. Enoch, and G. Tayeb, Anomalous refractive properties of photonic crystals, J. Opt. Soc. Am. A, Vol. 7, pp , [5] P. V. Parimi, W. T. Lu, P. Vodo, J. B. Sokoloff, and S. Sridhar, Negative refraction and lefthanded electromagnetism in microwave photonic crystals, Phys. Rev. Lett., Vol. 92, 2740, [6] S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, Wire media with negative effective permittivity: A quasi-static model, Microwave Opt. Technol. Lett., Vol. 35, pp. 47-5, [7] W. Rotman, Plasma simulation by artificial dielectrics and parallel-plate media, IRE Trans. Antennas Propag., Vol. 0, pp , Jan [8] M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, 972. [9] Metamaterials: Physics and Engineering Explorations. N. Engheta and R. W. Ziolkowski, Eds., IEEE Press, Piscataway, NJ, A. E. Serebryannikov was born in Kharkov, Ukraine, in 967. He graduated from the Kharkov Polytechnic University in 990 and received the PhD degree from the Kharkov National University in 996. In and since 2004 he has been with the Technische Universitaet Hamburg-Harburg, Hamburg, Germany. His current research interests include electromagnetic theory of periodic structures, applications of artificial materials, and millimetre-wave tubes. Dr. Serebryannikov is a Member of IEEE. A. L. Teplyuk was born in Kiev region, Ukraine, in 977. He graduated from the Kharkov Aerospace University in 2000 and since that time has been with the Institute of Radio-Physics and Electronics of National Academy of Sciences of Ukraine. Now he is working towards PhD degree. His current research interests include antenna design, waveguides, and periodic structures. K. Schuenemann was born in Braunschweig, Germany, in 939. He received the Dipl.-Ing. degree in Electrical Engineering and the Doktor-Ing. degree from the Technische Universitaet Braunschweig in 965 and 969, respectively. Since 983, he has been a Full Professor of Electrical Engineering at the Technische Universitaet Hamburg-Harburg, Germany. His current research interests are concerned with the application of millimeter waves in geoscience, transport phenomena in submicron devices, CAD of planar millimeter-wave circuits, optoelectronics, and high-power millimetre-wave tubes. Prof. Schuenemann is a Fellow of IEEE.

WAVEGUIDES FILLED WITH BILAYERS OF DOUBLE- NEGATIVE (DNG) AND DOUBLE-POSITIVE (DPS) METAMATERIALS

Progress In Electromagnetics Research B, Vol., 75 9, WAVEGUIDES FILLED WITH BILAYERS OF DOUBLE- NEGATIVE (DNG) AND DOUBLE-POSITIVE (DPS) METAMATERIALS E. Cojocaru * Department of Theoretical Physics, Horia