Electromagnetic Wave Propagation in Waveguide Loaded by Split Ring Resonator of Negative Permeability

This paper aims to study and analyze the electromagnetic propagation in media with negative transverse permeability and how this leads into some physical phenomena such as the appearance of backward waves and the propagation below cutoff. This study is done through the use of metamaterials of split ring resonators. It is shown that the waveguide dimensions needed to transmit a certain frequency band, can be miniaturized to half its dimension. The analytical determination of the propagation inside a waveguide in the presence of two slabs with dielectric permittivity and negative transverse permeability is derived. Finally it is shown by simulation, how to obtain a backward wave with lower loss than reported earlier in the literature.


Introduction
Rectangular waveguides are required for most of applications in microwaves.It can be used as a basic guided structure in radar application.The important application of the waveguide is to radiate element in multi-frequency interlaced antenna arrays.There are several methods to reduce the size of waveguide.One of the most important methods is metamaterial that it is used to reduce the size of the waveguide and to obtain the desired resonant frequency bands.At a particular frequency, metamaterials exhibit both negative permittivity and permeability [1] [2] [3] [4] [5].
In 1968 Veselago [1] analyzed electromagnetic wave propagation through Journal of Electromagnetic Analysis and Applications media with negative electric permittivity ε and magnetic permeability µ .The fields and the wave propagation form a left-handed system in these materials, but the nonexistence of transparent left-handed media in nature made Veselago's results just a theory.Recently, Smith et al. [2] [3] have demonstrated microwave propagation through an artificial left-handed medium (metamaterial).
Several names and terminologies have been suggested for metamaterials with negative permittivity and permeability, such as "left-handed", "backward-wave media" and "double-negative".Nowadays, many researchers are studying various aspects of this class of metamaterials, and several ideas and suggestions for future applications have been proposed [4] [5] [6].
The edge coupled split ring resonators (EC-SRR) are proposed by Pendry et al. [7] and experimentally tested by Smith et al. [2] and Marque's, R., et al. [8] (Figure 1(b)).They are composed of electrically small resonant rings, which show a very high diamagnetic susceptibility above and around its resonance frequency.
The magnetic and electric dipole of the EC-SRR can be expressed by [9]: The Bianistropy terms em yx α & me xy α occurred due to the fact that SRR does not act only as a magnetic dipole [9] [10], but also as an electric dipole.
Avoiding Bianistropy of the EC-SRR by a modification to the Broadside-coupled split ring resonator (Figure 1 backward propagation occurs when the longitudinal permeability is positive and the transversal permeability is negative, but it is noticed that a large insertion loss of almost 25 dB occurs in the S 12 measurement results in the backward wave, and also it has a very narrow bandwidth.
The aim of this paper is to increase the bandwidth and decrease the losses of the backward wave, through maximizing the negative magnetic permeability.

The Proposed Design
From the proposed configuration of Schelkunoff's [17], the magnetic polarizability of a closed metallic loop of radius r loaded by a capacitor is expressed as: where o ω is the resonant frequency of the LC circuit formed by the loop and the capacitor.It is shown from Equation ( 5) that, just above the frequency of resonance, the polarizability becomes negative and very large.Therefore, it is expected that a regular array of capacitive loaded metallic loops will show a negative magnetic permeability just above the frequency of resonance of the loops [11].According to schelkunoff's [17] and Marque's [11], if two or more split rings resonator are formed in a regular array, it will show a large negative magnetic permeability just above the resonance frequency of the rings.By separating the two rings each on a single substrate and with opposite slots as shown in Figure 2, a regular array of capacitive loading ring "the gap capacitance of the slot and the surface capacitance" will show a large negative magnetic permeability in the direction of the magnetic dipole.In addition to the advantage of avoiding the bianistropy, where the electric polarization of the upper half side ( 0 y > ) must equal to the opposite electric polarization of the lower half side ( 0 y < ) of the rings, so the design is not bianisotropic.The magnetic dipole of the proposed design resulting from the regular array of the two rings, can be expressed by: Figure 2. Two separated substrates with opposite single split ring resonator.
where, n is the number of rings in x-direction.

Theoretical Analysis
A rectangular waveguide is loaded by two slabs each of r ε and negative trans- verse permeability tr µ due to the presence of split ring resonator, the two slabs are located in the waveguide as shown in Figure 3.
The electric field y E in the different regions is given as: In air sin e , 0 cos e , 2 sin e , In slab sin sin e , sin sin cos e 2 sin , While, from the basic of Electromagnetic propagation inside a waveguide 0, 0, 0 The magnetic field H is obtained from the Maxwell's equation: And, the magnetic permeability tensor is: Figure 3. Two slabs with negative transverse permeability located in a waveguide.Journal of Electromagnetic Analysis and Applications We can assume a magnetic wall in the middle of the waveguide as shown in Figure 4, while the walls of the waveguide are electric walls.
Then, the wave equation in air region is applied to get: And the boundary condition is applied at x m d = + , we get: (10)   And from Maxwell's equation: Applying wave equation in slab region to get: From Equation ( 11), the wave propagation factor can be expressed as: We define the cutoff frequency of the partially filled waveguide with metamaterial as cp f .It can be shown that: where co f is the cutoff frequency of air-filled waveguide, therefore: In Equation ( 13), if r ε and tr µ are positive, propagation above cutoff occurs.Also if r ε is negative, tr µ must be negative for propagation to occur.The in- teresting case is when the tr µ is negative and cp f f > , where propagation be- low cut off occurs.
A simulation for the propagation constant β versus frequency is shown in Figure 5 at different values of negative transverse permeability tr µ and (m = 2.6 mm, a = 12 mm).
It seems that at a certain frequency, the propagation constant increased as the negative permeability increased.Meanwhile, at same negative permeability, the propagation constant is decreasing with the increasing of the frequency.

Results
We have designed two rings of opposite slots direction at resonance frequency , the S 12 reaches −4 db, while in [14] the S 12 reached −10 db in lossless case and −28 db in lossy case.

GHz
The result of the 3 db bandwidth for the backward wave of Figure 7 is shown in Figure 8.
In Figure 8, it is shown that a bandwidth of 95 MHz has been achieved, which is wider than the bandwidth reported in the literature [12], where it was about 70 MHz.This means that, the bandwidth of the proposed design has increased by 30% relative to that reported in literature [14] [15].By changing the following parameters ( i R , o R , m/a) and applying simulation program CST MW studio, we have got the influences of these parameters on the resonant frequency o f as shown in Table 1 and plotted in Figure 9.

Figure 7 .Figure 8 .
Figure 7.The result of 12 S , the solid line of a waveguide filled with the propose design and the dotted line with adding losses to Cu Clad and substrate with theoretically and simulated by CST MW Studio, and is shown to support a backward propagation below cutoff when the transverse permeability is negative.We have added two slabs symmetrically in waveguide to increase magnetic dipole of the rings.The transverse width of the waveguide can just be miniaturized to smaller than the half of an empty waveguide needed to propagate this frequency.As a result, the losses of the backward wave are decreased to 0 db and the bandwidth becomes wider than that reported in the literature by about 30%.

Table 1 .
Parametric study of different resonant frequency.
i R o R m/a Figure 9.The result of 12S for each case in Table1.