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

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1 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 Hulubei National Institute of Physics and Nuclear Engineering, P. O. Box MG-6, Magurele, Bucharest 775, Romania Abstract Simple normalized dispersion relations for transverse magnetic (T M) and transverse electric (T E) propagating modes in parallel-plate waveguides filled with DPS/DPS or DNG/DNG, and DNG/DPS bilayers are presented. The evanescent T E mode of the waveguide filled with a DNG/DPS bilayer is characterized also by a simple normalized dispersion relation. Since an important behavior of the modes in the waveguide filled with a DNG/DPS bilayer is the existence of a turning point (TP) at which the power carried by the respective mode on the propagation direction equals zero and changes the sign, we present also implicit relations for determining the normalized parameters of the T M and T E modes at that TP. We show that the TP begins to exist at certain values of the normalized parameter v characterizing the DPS layer. For both the T M and T E modes, the higher is the mode order, the greater is the v parameter at which the TP begins to exist, but the behavior of the TP is different for the T M and T E modes.. INTRODUCTION The double-negative (DNG) materials having both negative permittivity and negative permeability, enjoy an increased interest, especially because of their physical properties which are different from those of the conventional double-positive (DPS) materials. Veselago [] was the first to study theoretically the DNG materials. Various aspects of this class of metamaterials have been studied for example in [ ]. Interesting characteristics of the guided modes in the parallelplate waveguides filled with bilayers of DPS and DNG materials have Received 6 May, Accepted June, Scheduled 6 July * Corresponding author: E. Cojocaru (ecojocaru@theory.nipne.ro).

2 76 Cojocaru been revealed previously in [9, ] but in terms of different physical parameters. In this paper we present simple normalized dispersion relations for the guided or evanescent modes in the parallel-plate waveguides filled with DPS/DPS, or DNG/DNG, and DNG/DPS bilayers. Numerical examples are given showing dispersion curves of the lower order modes and the total normalized power carried on the propagation direction by the respective modes. Examples of electromagnetic fields inside the waveguide are also given. It has been shown in [9, 5] that an important behavior of the modes in a grounded DNG slab and in the parallel-plate waveguide filled with a DNG/DPS bilayer is the existence of a turning point (TP) at which the power carried by each mode of order m > equals zero and changes the sign. In this paper we present implicit relations for determining the normalized parameters at that TP for the guided modes in the parallel-plate waveguide filled with a DNG/DPS bilayer. Behaviors of the TPs are outlined by numerical examples.. GENERAL RELATIONS We consider a parallel-plate waveguide, made of two infinitely extent perfectly electric conducting plates separated by the distance d = d + d, as shown in Fig.. The waveguide is filled with a pair of parallel layers made of DNG and DPS materials. The two layers are characterized by their thicknesses d and d, and the relative constitutive parameters (ɛ, µ ) and (ɛ, µ ), which are assumed real, nondispersive, and nondimensional, with ɛ µ > ɛ µ. A monochromatic time-harmonic variation exp(iωt) is assumed. The z axis is chosen as direction of propagation of the guided modes whose electromagnetic field varies as exp[i(ωt z)], where is the modal phase constant. For transversal magnetic (T M) modes, the magnetic field is along the y direction, whereas it is the electric field along the x d (ε,µ ) d (ε,µ ) z Figure. Geometry of the parallel-plate waveguide filled with a bilayer of relative material constants (ɛ, µ ) and (ɛ, µ ).

3 Progress In Electromagnetics Research B, Vol., 77 y direction in the case of transversal electric (T E) modes. Thus, the magnetic field H y for the T M modes takes the form { cos(kt x) for < x < d H y = () A cos[k t (x d )]+B sin[k t (x d )] for d <x<d +d where the factor exp( iz) is skipped, and A = cos(k t d ), B = k tɛ sin(k t d ) () k t ɛ Similarly, the electric field E y for the T E modes takes the form { sin(kt x) for < x < d E y = () A sin[k t (x d )]+B cos[k t (x d )] for d <x<d +d where the factor exp( iz) is skipped, and A = k tµ cos(k t d ), B = sin(k t d ) (4) k t µ where k tj, with j =,, is defined by relation { k k tj = ɛ j µ j when < ɛ j µ j i k ɛ jµ j when (5) > ɛ j µ j = /k is the relative phase constant, k = ω ɛ µ is the wavenumber in vacuum. The total normalized power carried by each mode on the z direction takes the form P = (P + P )/( P + P ) (6) where P and P are given up to a common factor by P d H y dx, P d +d ɛ ɛ for the T M modes, and similarly P d E y dx, P d +d µ µ for the T E modes.. DISPERSION RELATIONS d H y dx, (7) d E y dx (8) Dispersion relations are determined either by applying the boundary conditions of the tangential electromagnetic fields at interfaces or from the poles of the complex amplitude reflection coefficients expressed in terms of the transfer matrix elements [], taking into account that the perfectly electric conducting plates of the waveguide are characterized by an infinite permittivity and zero permeability [4].

4 78 Cojocaru.. The Case < ɛ µ < ɛ µ The following dispersion relation results [9, ] for T M modes, and k t ɛ tan(k t d ) + k t ɛ tan(k t d ) = (9) µ k t tan(k t d ) + µ k t tan(k t d ) = () for T E modes. Here we introduce the normalized parameters v j = k d j, u j = ɛ j µ j, j =, () Using relation () in (9) and () gives for T M modes, and ɛ ɛ tan(u v ) + u u tan(u v ) = () µ µ tan(u v ) + u u tan(u v ) = () for T E modes. The normalized dispersion relation of the guided modes is where v = (/u ) { tan [γ tan(u v )] + mπ }, m =,,,... (4) γ = { ɛ u /(ɛ u ) for TM modes µ u /(µ u ) for TE modes (5) Note that, for simplicity, we label the modes by m without any reference to the standard mode labeling of DPS slabs. The cutoff = is determined from (4), where v = (/ ɛ µ ) { tan [γ c tan(v ɛ µ )] + mπ } (6) γ c = { ɛ ɛ µ /(ɛ ɛ µ ) for TM modes µ ɛ µ /(µ ɛ µ ) for TE modes (7) From relations () and () one can see that the cutoff v =, when it exists, is determined by u v = lπ for the T M and T E modes, where

5 Progress In Electromagnetics Research B, Vol., 79 l is an integer number. determined by relation Thus, when the cutoff v = exists, it is = ɛ µ l π /v (8) for the T M and T E modes. From relations () and () one can see also that when u v = u v, a nondispersive mode [9, ] exists in the waveguide filled with a DNG/DPS bilayer, which satisfies the relation for the T M mode, and for the T E mode. ɛ /ɛ + u /u = (9) µ /µ + u /u = ().. The Case ɛ µ < < ɛ µ In this case, k t in (5) is imaginary, and where k t d = ik d ɛ µ = iv ũ () ũ = ɛ µ () The normalized dispersion relation of the guided modes is where v = (/u ) { tan [ γ tanh(ũ v )] + mπ } () γ = { ɛ ũ /(ɛ u ) for TM modes µ u /(µ ũ ) for TE modes (4).. The Case > ɛ µ > ɛ µ In this case, k t and k t in (5) are imaginary, but in the parallel-plate waveguide filled with a DNG/DPS bilayer there is a T E mode with real values of [9, ]. At = ɛ µ, v is determined by relation v = (µ /µ ) tanh(v ɛ µ ɛ µ )/ ɛ µ ɛ µ (5) The mode T E is evanescent and satisfies the normalized dispersion relation v = (/ũ ) tanh [ µ ũ /(µ ũ ) tanh(ũ v )] (6) where ũ is defined by () and ũ is defined similarly. Since µ has negative values, v in (6) has real values, with v when.

6 8 Cojocaru 4. NUMERICAL EXAMPLES 4.. Parallel-plate Waveguide Filled with a DPS/DPS (or DNG/DNG) Bilayer Consider the parallel-plate waveguide filled with a DPS/DPS bilayer of relative material constants (ɛ, µ ) = (4, ) and (ɛ, µ ) = (,.5). Dispersion curves are shown in Figs. (a) and (b) for the T M and T E modes, respectively. At greater values of v and < ɛ µ, there are composed modes [9]. As for example, in Fig. (a), at v =.5, the second mode is a compound of m = and m = modes. Thus, it is more easy to look after the modes starting from the upper region of the dispersion curves, at > ɛ µ. The cutoffs = are given by v v (a) (b) H y (a.u.) E y (a.u.) - - d d (c) d d Figure. The relative phase constant of (a) T M and (b) T E modes in the parallel-plate waveguide filled with a DPS/DPS bilayer of relative material constants (ɛ, µ ) = (4, ) and (ɛ, µ ) = (,.5), versus the normalized parameter v, when v is constant: v =.5 (dashed line), v =.5 (thin line, marker.), and v = (thick line, marker +). Only the lower order modes are represented: m = (red), m = (blue), m = (magenta), m = (cyan), and m = 4 (green). The fields H y and E y are represented in (c) and (d), at the points shown in (a) and (b) by black markers, when v =.5. For the T M modes in (c), v =, with =.44 (full line), and =. (dashed line). For the T E modes in (d), v =, with =.47 (full line), and =.5 (dashed line). (d)

7 Progress In Electromagnetics Research B, Vol., v =v (a) 4 5 v =v Figure. Dispersion curves for (a) T M and (b) T E modes in the parallel-plate waveguide filled with a DPS/DPS bilayer of relative material constants like in Fig., when v = v. (b) relation (6) and they are the same for the T M and T E modes. For T M and T E modes, does not exceed the value ɛ µ. In Fig. (a), the T M mode has a cutoff v = at = ɛ µ =, which is the same for different values of v. At v =, there is one more cutoff v = at =.796, which is determined from (8) with l =. That cutoff is the same for the T M and T E polarizations. The fields attain maximum values in the first layer. At > ɛ µ, the fields are evanescent in the second layer, as shown by the dashed line in Figs. (c) and (d). Note that, for the parallel-plate waveguide filled with a DNG/DNG bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (,.5), the dispersion curves are the same like in Figs. (a) and (b), the single difference being in the sign of the total normalized power P carried by the modes on the propagation direction: in the DPS/DPS bilayer P = +, the modes representing forward waves, whereas in the DNG/DNG bilayer P =, the modes representing backward waves. Figs. (a) and (b) show dispersion curves for the T M and T E modes in the parallel-plate waveguide filled with a DPS/DPS bilayer of relative material constants like in Fig., when v = v. 4.. Parallel-plate Waveguide Filled with a DNG/DPS Bilayer 4... The Case ɛ µ > ɛ µ Consider the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (,.5). Dispersion curves are shown in Figs. 4(a) and (b) for the T M and T E modes, respectively. Although the relative material constants differ

8 8 Cojocaru 4 5 V (a) 4 5 V (b) P V (c) P V (d) Figure 4. Dispersion curves for (a) T M and (b) T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (,.5), when v is constant: v =.5 (dashed line), v =.5 (thin line, marker.), and v = (thick line, marker +). The colors are kept the same like in Fig.. The TPs are marked by small circles. The total normalized power P carried by each mode on the propagation direction is represented in (c) and (d) for the T M and T E modes, respectively. only by the sign of ɛ and µ, the dispersion curves in Fig. 4 are much different from those in Fig.. Like in the case of a DPS/DPS bilayer, the dispersion curves overlap in the upper region, at > ɛ µ, but in the lower region, at < ɛ µ, there are composed modes. Thus, it is more easy to look after the modes starting from the upper region of the dispersion curves, at > ɛ µ. The cutoffs = are given by relation (6) and they are the same for the T M and T E modes. The cutoffs v = are the same like in the previous case of a DPS/DPS bilayer. In Fig. 4(b), the T E mode allows real solutions at ɛ µ, the starting point being marked by an asterisk. At great values of v, each T M and T E mode with m has at least one TP at which the total power carried on the z direction equals zero and changes the sign. Note that the TPs are more distinctly seen on the P against v curves in Figs. 4(c) and (d) than on the dispersion curves. We denote the values of and v at the TP by TP and v TP, respectively. In Fig. 4(a), the

9 Progress In Electromagnetics Research B, Vol., 8 mode T M at v =.5 has two TPs which are more distinctly seen in Fig. 4(c). Between the two TPs, P has positive values. Beyond the two TPs at m = and at m = and, P has negative values for the T M modes at v =.5. At v =.5, the T M mode with m = has one TP, whereas the modes with m = and in the upper region have two TPs which are more distinctly seen in Fig. 4(c), P having positive values between the two TPs. In Fig. 4(b), at v =, there is one TP for each T E mode with m, the TPs at m = and m = being almost overlapped, but at v <, there is a single TP for the T E mode with m = at v =.5. The fields H y and E y are shown in Figs. 5(a) and (b) for the T M and T E modes, respectively. The behavior of the fields at the DNG-DPS interface is different from that at the interface between the DPS layers in Fig.. Fig. 6 shows the dispersion curves of the T M and T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants like in Fig. 4, when v = v. The dispersion curves in Figs. 6(a) and (b) are much different from those in Fig.. On the same interval of v = v variation, in Fig. 6 there are only two modes for each T M or T E polarization, contrary to Fig. which shows a multitude of modes. Each mode of Figs. 6(a) and (b) has a TP at which P =, as shown in Figs. 6(c) and (d), respectively The Case ɛ µ = ɛ µ with ɛ = ɛ and µ = µ Consider the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (4, ). Since the dispersion curves are the same for the T M and T E modes, H y (a.u.) E y (a.u.) - - d d (a) d d Figure 5. The field H y in (a) and E y in (b) for the T M and T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer at the points shown in Figs. 4(a) and (b) by black markers, when v =. For the T M mode in (a), v =.5, with =.6 (full line), and =.7 (dashed line). For the T E mode in (b), v =.5, with =.4 (full line), and =.4 (dashed line). (b)

10 84 Cojocaru 4 5 v =v 4 5 v =v (a) (b) P.5 P v =v (c) Figure 6. Dispersion curves for (a) T M and (b) T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants like in Fig. 4, when v = v. The total normalized power P carried by each mode on the propagation direction is represented in (c) and (d) for the T M and T E modes, respectively. v =v (d) Fig. 7 refers only to the T E modes. A nondispersive mode satisfying relation () exists at v = v, as shown in Figs. 7(a) and (b), which allows any value of from to. At v > v, the total normalized power P carried on the propagation direction has negative values (the modes are backward waves), whereas at v < v, P has positive values (the modes are forward waves), as shown in Figs. 7(c) and (d) The Case ɛ µ = ɛ µ with ɛ = µ and µ = ɛ Consider the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (, 4). Since the dispersion curves are the same for the T M and T E modes, Fig. 8 refers only to the T E modes. An intricate mode exists at v v, as shown in Figs. 8(a) and (b), which has two TPs in (a) and four TPs in (b). The total normalized power P has positive values between the two TPs of the same order m, as shown in Figs. 8(c) and (d). The modes at v > v have P <, whereas the modes at v < v have P >. In Fig. 8(c), the mode with m = has P at v =, but P does not change the sign, and so, there is not a TP at that value of v.

11 Progress In Electromagnetics Research B, Vol., V (a) V (b) P.5 P V (c) V (d) Figure 7. Dispersion curves for the T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (4, ), when (a) v =.5 and (b) v =.5. The total normalized power P carried by each mode of (a) and (b), is represented in (c) and (d), respectively. 5. IMPLICIT RELATIONS AT THE TP Since the TP is an important characteristic of the modes in the parallelplate waveguide filled with a DNG/DPS bilayer, here we present implicit relations for determining TP when v is constant. 5.. The Case < ɛ µ < ɛ µ At the TP, v is minimum, and dv /d =. Since du /d, we use relation dv /du = with the view to finding an implicit relation at the TP. Thus, we obtain the following implicit relation for determining TP when v is constant, { γ γt = tan + γ t [ σηt + u v ]} ( ) + t u (7)

12 86 Cojocaru V (a) V (b) P.5 P V (c) V (d) Figure 8. Dispersion curves for the T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (, 4), when (a) v =.5 and (b) v =.5. The total normalized power P carried by each mode of (a) and (b), is represented in (c) and (d), respectively. where γ is defined by relation (5), t = tan(u v ), η = u /u, and { for TM modes σ = (8) for TE modes Once TP is determined from (7), the respective value v TP is obtained from (4). 5.. The Case ɛ µ < < ɛ µ Using dv /du = gives the following implicit relation for determining TP when v is constant, { [ σ γ γτ = tan + γ τ ητ σ u v ]} ( ) τ ũ (9) where γ is defined by relation (4), τ = tanh(ũ v ), η = + u /ũ. Once TP is determined from (9), the respective value v TP is obtained from ().

13 Progress In Electromagnetics Research B, Vol., The Case ɛ µ = ɛ µ with ɛ = µ and µ = ɛ Replacing u = u and η = in (7) gives the following implicit relation for determining TP of the T M and T E modes, when v is constant, γt = tan [ ( ) ( γu v + t / + γ t )] () where γ = ɛ /µ Numerical Examples Using relations (7) and (9) gives at v =.5 in Fig. 4(a) [ TP, v TP ] = [.646,.977] at m =, TP = [.8779,.77] with v TP = [.74,.97] at m =, and TP = [.799,.4] with v TP = [4.94, 4.459] at m =, the values of m referring to the upper region of the dispersion curves. Using relation () gives for the TPs in Fig. 8(a) TP = [.88,.46] with v TP =..9 TP TP v (a) v (b).4 TP.. TP v (c) v (d) Figure 9. The relative phase constant TP versus v for (a) T M and (b) T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer of relative material constants like in Fig. 4, that is, (ɛ, µ ) = ( 4, ) and (ɛ, µ ) = (,.5). The same in (c) and (d), but for the relative material constants (ɛ, µ ) = (,.5) and (ɛ, µ ) = (.5, ).

14 88 Cojocaru [.668,.94], whereas in Fig. 8(b), for the intricate mode at v v, TP = [.658,.4,.749,.678] with v TP = [.566,.78,.495,.65], and for the next mode on the side v > v in that figure, TP = [.576,.9988] with v TP = [.755,.7899]. The dependence of TP on v is illustrated in Fig. 9 for two sets of relative material constants, with ɛ µ > ɛ µ. When two TPs there are for a given mode, the principal TP corresponding to the maximum value of TP is considered. The TP appears at certain values of v, the higher is the mode order, the greater is the v parameter at which the TP begins to exist. One can see that the behavior of the TPs is different for the T M and T E modes. In Fig. 9(d), the TPs at m = and m = are overlapped. 6. CONCLUSION In this paper, we analyzed the T M and T E modes of the parallelplate waveguide filled with DPS/DPS (or DNG/DNG), and DNG/DPS bilayers. Simple normalized dispersion relations for the guided and evanescent modes were presented. Numerical examples were given showing dispersion curves, the total normalized power carried by the modes on the propagation direction, and the fields inside the parallelplate waveguide. Interesting results were obtained in the specific case when ɛ µ = ɛ µ, as for example, the nondispersive mode in Fig. 7, at v = v, and the intricate mode in Fig. 8, at v v. Although the TP has been evidenced by other authors [9, 5], there is no relation (upon our knowledge) for the normalized parameters at the TP. In this paper we presented implicit relations for determining the normalized parameters at the TP for the T M and T E modes in the parallel-plate waveguide filled with a DNG/DPS bilayer. We showed that the TP appears at certain values of the DPS layer parameter v, the higher the mode order, the greater the v parameter at which the TP begins to exist. The behavior of the TP is different for the T M and T E modes, as shown in Fig. 9. We showed also that a given mode could have several TPs, as for example, the intricate mode in Fig. 8(b) at v v has four TPs, the plots of the total normalized power carried by each mode on the propagation direction being very useful in the designation of the TPs. For simplicity, we considered ɛ µ ɛ µ and several examples were given, but relations presented in terms of normalized parameters in this paper could be applied also to other combinations for the relative material constants of the DPS and DNG materials [].

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