## Abstract

A rigorous full wave analysis of bianisotropic split ring resonator (SRR) metamaterials is presented for different electromagnetic field polarization and propagation directions. An alternative physical explanation is gained by revealing the fact that imaginary wave number leads to the SRR resonance. Metamaterial based parallel plate waveguide and rectangular waveguide are then examined to explore the resonance response to transverse magnetic and transverse electric waves. It is shown that different dispersion properties, such as non-cutoff frequency mode propagation and enhanced bandwidth of single mode operation, become into existence under certain circumstances. In addition, salient dispersion properties are imparted to non-radiative dielectric waveguides and H waveguides by uniaxial bianisotropic SRR metamaterials. Both longitudinal-section magnetic and longitudinal-section electric modes are capable of propagating very slowly due to metamaterial bianisotropic effects. Particularly, the abnormal falling behavior of some higher-order modes, eventually leading to the leakage, may appear when metamaterials are double negative. Fortunately, for other modes, leakage can be reduced due to the magnetoelectric coupling. When the metamaterials are of single negative parameters, leakage elimination can be achieved.

©2009 Optical Society of America

## 1. Introduction

Artificially structured metamaterials have recently garnered considerable attention for their peculiar electromagnetic properties. Especially, great of interest has been devoted to split ring resonator (SRR) which composes the essential part of left-handed metamaterials [1]–[6].

Inherently bianisotropic, SRR metamaterials are made of the isotropic media with two concentric rings separated by a gap, both having splits at opposite sides. As a result, besides the electric and magnetic coupling, the incident field also induces the magnetoelectric coupling [7], [8]. Thus this kind of artificial magnetic media needs a carefully control of the SRR orientation relative to the incident wave as well as the SRR design, otherwise, the electromagnetic response is significantly more complicated [9], [10]. Several analytical models are employed to examine the resonant property in the SRR transmission spectra [11]–[14]. Careful investigation has been carried out for different SRR parameters [15], [16]. However, the presented literatures mainly concentrate on single SRR orientation and experimental as well as simulation results, yet neglecting the rationale behind the work. Therefore, ambiguity still exists in the better understanding of SRR’s resonance behaviors due to the vacancy of uniform theory that are valid and rigorous in the all six distinct SRR orientations relative to electromagnetic field polarization and propagation directions.

In addition, waveguiding structures based on metamaterial media have recently been considered by several research groups showing how the presence of one or both negative constitutive parameters may give rise to unexpected and interesting propagation properties [17]–[24]. The absence of fundamental mode and sign-varying energy flux in the negative refractive index waveguide are revealed [20]. Rectangular waveguide filled with anisotropic single negative metamaterials are shown to support backward-wave propagation [21]. Moreover, Results for isotropic double negative metamaterial H waveguides are reported, including backward propagation, mode bifurcation and coupling effects [22]. The use of single negative metamaterials as the embedding medium for nonradiative dielectric waveguides is examined [23]. Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides is obtained [24]. However, the presented literatures almost focus on the negative effects of both permittivity and permeability to the metamaterial based waveguides, whereas magnetoelectric coupling of the bianisotropic effects may lead to more dramatically unexpected features in the waveguiding structures.

With these considerations, fundamental modal properties of SRR metamaterials and metamaterial based waveguiding structures are studied in this paper. In Section 2, physical insights into the resonance band gap of SRR metamaterials are illustrated. In Section 3, parallel plate waveguides and rectangular waveguides with bianisotropic SRR metamaterials are presented. Dramatically different propagation features of transverse magnetic (TM) and transverse electric (TE) waves are revealed. Nonradiative dielectric (NRD) waveguides and H waveguides containing uniaxial bianisotropic SRR metamaterials are demonstrated for the unique propagation characteristics, providing valuable applications in practice. Finally, in Section 5, conclusions are drawn on various results of this study.

## 2. Physical characteristics of the resonance band gap of SRR metamaterials

To account for the magnetoelectric coupling in Maxwell’s equations, SRR metamaterials can be described by the constitutive relations [25]

with ${Z}_{0}=\sqrt{{\mu}_{0}/{\epsilon}_{0}}$, where *ε*̄ and *μ*̄ are the relative electric permittivity and relative magnetic permeability tensors, and *κ*̄ is the magnetoelectric coupling dimensionless tensor.

For axes fixed to the SRR as shown in Fig. 1, only certain components of *ε*̄, *μ*̄ and *κ*̄ tensors are of significance without losses. It should be noted that the aim of Fig. 1 is to illustrate the relative orientation of the SRR unit, and an array of such ring unit will compose the metamaterials. Therefore, the whole theoretical analysis conducted in this paper is based on the macroscopical effects of metamaterials where the coupling of different rings has been counted [7], [8].

*ω*
_{0} is the resonance frequency, and *a*, *b*, *c*, *d* are related to the geometry of the SRR. For other SRR orientations, the *ε*, *μ*̄ and *κ*̄ tensors just need a coordinates transformation.

Introducing a normalized magnetic field *h* = *Z*
_{0}
*H*, from Maxwell’s curl equation for source free regions together with Eq.(1) and Eq. (2), one may write

where ∇'=∇/*k*
_{0}.

Considering forward plane wave propagation of the form exp(-*iβz*'), where *β* = *k _{z}*/

*k*

_{0}is the normalized longitudinal wave number, one has

where ∂* _{x}*' stands for ∂/∂

*x*' and ∂

*y*' for ∂/∂

*y*',

*x*'=

*k*

_{0}

*x*,

*y*'=

*k*

_{0}

*y*,

*z*'=

*k*

_{0}

*z*. After substituting Eq. (2) into Eq. (4), and apply the condition that

*E*=

_{z}*h*= 0 for TEM waves, one obtains the following relations

_{z}this is exactly the case shown in Fig. 2(a), where magnetic field **H** is perpendicular to the SRR plane, and incident **E** is parallel to the gapbearing sides of SRR. The normalized wave number of these TEM waves is given by

Ref. [7] concluded the same results by considering the bianisotropy role in SRR metamaterials. At given frequency *ω*, only those modes having *μ _{xx}ε_{yy}* -

*κ*

^{2}>0 will propagate. Those modes with

*μ*-

_{xx}ε_{yy}*κ*

^{2}< 0 will lead to an imaginary

*β*, meaning that all field components will decay exponentially away from the source of excitation. Since

*κ*

^{2}> 0, this SRR orientation will achieve the transmission stop band when the constitutive parameters are single negative, including

*ε*>0 and

_{yy}*μ*<0 case, as well as

_{xx}*ε*<0 and

_{yy}*μ*>0 case. And also when the constitutive parameters are double negative or double positive with the condition |

_{xx}*μ*|<|

_{xx}ε_{yy}*κ*

^{2}|, the transmission stop band will also occur.

Similarly, one obtains

for the case shown in Fig. 2(b), where incident **E** is perpendicular to the SRR plane, and magnetic field **H** is parallel to the gapbearing sides of SRR. The normalized wave number of these TEM waves is given by

which indicates that metamaterials with this SRR orientation has nothing to do with TEM waves of such electromagnetic field polarization and propagation direction. Meanwhile, there is no transmission stop band.

Through the similar analysis, metamaterials with the six SRR orientations can be re-categorized into three groups according to Maxwell’s equations. The ones shown in Fig. 2(a) and Fig. 2(b) are one group, so do those in Fig. 2(c) and Fig. 2(d), as well as those in Fig. 2(e) and Fig. 2(f). The wave numbers for the other four cases are listed in Table 1. The case in Fig. 2(c) has been studied in Ref. [10], where the authors identified the SRR with its outer ring at low frequencies, and illustrated the simulated currents to explain the resonance phenomenon. Here we can see it more clearer that *ε _{yy}* becomes less than zero when frequency

*ω*is larger than the resonance frequency

*ω*

_{0}, leading to the imaginary wave number

*β*, thus achieve the resonance stop band. Also Fig. 2(e) case has the chance to resonance when

*μ*< 0, and there is no resonance stop band for the Fig. 2(d) and Fig. 2(f) case.

_{xx}Since the resonance of SRR is often manifested by a dip in the transmitted (*S*
_{21}) curves, let’s see the *S* parameters for the SRR metamaterial planar slab with complex impedance $Z=\sqrt{\mu /\epsilon}$, complex wave number *β* determined by the six cases above, and thickness *z*' as the distance between the calibration planes

where $\Gamma =\frac{Z-1}{Z+1}$ and *T* = exp(-*iβz*'). Γ is the plane wave reflection coefficient at interface between free space and SRR metamaterials. Since it is related to the reflected power at the interface, we require |Γ| ≤ 1. Therefore, it is easy to see that *S*
_{21} decreases while *T* gets smaller. When *β* becomes pure imaginary, all the field components will decay exponentially from the source of excitation, leading to the dip in transmitted (*S*
_{21}) curves.

So far, three re-categorized groups of metamaterials with six SRR orientations for TEM waves are examined. Every group has one case to achieve resonance due to the imaginary wave number, providing an alternative physical explanation for SRR resonance band gaps.

## 3. Propagation features of waveguides structures with SRR metamaterials

#### 3.1 Parallel Plate Waveguides and Rectangular Waveguides

Geometry of parallel plate waveguide and rectangular waveguides filled with SRR metamaterials are shown in Fig. 3. The strip width *W* in Fig. 3(a) is assumed to be much greater than the separation *l* between the two plates, so that fringing fields and any *x* -variation can be ignored. And it is standard convention to have the longest side of the rectangular waveguide along the *x* -axis, so that *u* > *v* in Fig. 3(b).

According to Fig. 2, metamaterial waveguide with different SRR orientations are illustrated in Fig. 4. For Fig. 4(a), one can express the following coupled equations for the longitudinal fields:

$$\phantom{\rule{2.5em}{0ex}}=\left(\frac{-\beta}{{\beta}^{2}-{\epsilon}_{\mathrm{xx}}{\mu}_{\mathrm{yy}}}+\frac{\beta -\mathrm{i\kappa}}{{\beta}^{2}-{\mu}_{\mathrm{xx}}{\epsilon}_{\mathrm{yy}}+{\kappa}^{2}}\right){\partial}_{x\text{'}y\text{'}}^{2}{h}_{z}$$

$$\phantom{\rule{2.5em}{0ex}}=\left(\frac{-\beta}{{\beta}^{2}-{\epsilon}_{\mathrm{xx}}{\mu}_{\mathrm{yy}}}+\frac{\beta -\mathrm{i\kappa}}{{\beta}^{2}-{\mu}_{\mathrm{xx}}{\epsilon}_{\mathrm{yy}}+{\kappa}^{2}}\right){\partial}_{x\text{'}y\text{'}}^{2}{E}_{z}$$

### 3.1.1 Non-cutoff Frequency Modes

For the parallel plate waveguides, TM waves are characterized by *h _{z}* = 0 and a nonzero

*E*field which satisfies the reduced wave Eq. (10a), with ∂

_{z}*x*'=0,

where

and ${k}_{c}=\frac{n\pi}{l}$, (*n* = 0, 1, 2⋯) is the cutoff wave number constrained to discrete values. /

Observe that for *n* = 0, the TM_{0} mode is actually identical to the TEM mode shown in Fig. 2(a), therefore, this TM_{0} mode has a cutoff phenomenon while SRR resonance. However, from (12) one knows that TM mode has the chance to propagate with no cutoff frequency when *ε _{yy}* < 0 and

*μ*-

_{xx}ε_{yy}*κ*

^{2}> 0, as shown in Fig. 5. Similar results hold true for the TE modes in Fig. 4(b), and TM modes in Fig. 4(c), which are corresponding to the resonance cases in the Fig. 2.

For the rectangular waveguides, one can see that if *κ* ≠ 0, decoupling of *E _{z}* and

*h*occurs only when ∂

_{z}_{x'}≡0 or ∂

_{y'}≡0, therefore we only consider TE

_{mn}modes with

*m*= 0 or

*n*= 0, since neither

*m*nor

*n*can be zero for TM modes in a rectangular waveguide. When ∂

_{y'}≡0, one has the following decoupled equation and boundary condition for TE

_{m0}modes from Eq. (10b)

where

Akin to the modes in the parallel plate waveguide, none-cutoff frequency modes also exist under certain condition.

### 3.1.2 Enhanced Bandwidth of Single Mode Operation

For the parallel plate waveguides, the TE modes in Fig. 4(a), characterized by *E _{z}* = 0 and a nonzero

*h*field which satisfies the reduced wave Eq. (13b), with ∂

_{z}*x*'=0. Through the similar derivation, one can obtain

The TM modes in Fig. 4(b), and TE modes in Fig. 4(c) corresponding to the non-resonance cases in the Fig. 2 achieve the identical cutoff frequencies ${f}_{c}=\frac{\mathrm{nc}}{2l}$, which is the maximum value for ordinary TM and TE waves, promising a bandwidth enhancement for single-mode operation in material containing waveguide.

For the rectangular waveguides, one can see that TE_{0n} modes in Fig. 4(a) and Fig. 4(c) achieve the cutoff frequency of ${f}_{c}=\frac{\mathrm{nc}}{2v}$, TE_{m0} modes in Fig. 4(b) obtains the cutoff frequency of ${f}_{c}=\frac{\mathrm{nc}}{2u}$, which are equal to the ones of air containing rectangular waveguide, promising a bandwidth enhancement for single-mode operation in material containing waveguide.

#### 3.2 Nonradiative Dielectric Waveguides and H Waveguides

Consider the particular case of SRR metamaterials where two sets of SRR microstructures with different orientations are included in nonradiative dielectric waveguides and H waveguides as shown in Fig. 6.

The relative orientation of these two ensembles is in Fig. 7 and the *ε*̄, *μ*̄ and *κ*̄ tensors in Eq. (1) have the following uniaxial form [26]

where

*ϛ*, *ξ*, reflect the different changes of *ε* and *μ* components in *ŷŷ* and *ẑẑ* direction. Therefore, *ε*
_{1} and *μ*
_{1} are always positive, whereas *ε*
_{2} and *μ*
_{2} can be negative in certain frequency band.

For the full wave analysis of the nonradiative dielectric waveguides and H waveguides modal properties, the eigenvalue problem is solved in the complex plane so as to examine the leakage property of H waveguides. The LSM modes are characterized by *h _{x}* = 0 with nonzero

*h*as the supporting field which satisfies

_{y}where *k _{z}*' =

*k*/

_{z}*k*

_{0}=

*β*

_{z}-

*iα*is the complex normalized longitudinal wave number. All other field components can be expressed as

_{z}Express *h _{y}* as a product of two separate variable functions in the form

such that

where *k _{x}*'=

*k*/

_{x}*k*

_{0}and

*k*' =

_{y}*k*/

_{y}*k*

_{0}are the complex normalized transverse wave numbers. Substituting back into Eq. (18), the normalized wave numbers can be expressed

One has *k _{x}*' =

*k*

_{x1}' =

*β*

_{x1}-

*iα*

_{x1}for |

*x*'| <

*q*', while for |

*x*'| >

*q*', one should take

*k*' =

_{x}*k*

_{x0}' =

*β*

_{x0}-

*iα*

_{x0}. Apply the boundary conditions on the perfectly electric conductor planes to other field components, one can write

where *G* is the amplitude constant, and ${k\text{'}}_{y}=n\frac{\pi}{S\text{'}}$ with *s*'=*k*
_{0}
*s*. The *n* index gives the number of half waves along *y*. And

Enforcing the continuity conditions at both *x*'=±*q*', the modal equation for the LSM modes can be finally derived

the order of eigensolution of Eq. (25) gives the *m* index (*m* = 0, 1, 2, ⋯) appearing in LSM_{mn} From the similar derivation, the LSE modes can be defined as

and the normalized transverse wavenumber in the slab should be given by

instead of Eq. (22a). Hereafter, we only consider the LSM modes, and the following results hold true for the LSE modes.

#### 3.2.1 Slow Wave Propagation

Figure 8(a) presents the operational diagram for LSM modes, the real part of the longitudinal wave number |*β _{z}*| decreases gradually as the magnetoelectric coupling turns larger in the case that

*ε*

_{2}and

*μ*

_{2}are of positive values. Maximum

*κ*is achieved under the cutoff condition

*k*' = 0.

_{z}Since the guide wavelength defined as ${\lambda}_{g}=\frac{2\pi}{{\beta}_{z}}$ becomes smaller when longitudinal wave number increases, the corresponding phase velocity *v* = *Tλ* of the modes will be much slower. From Eq. (17), one can see that in the frequency that *ω* is far larger than the resonance frequency *ω*
_{0}, both positive *ε*
_{2} and *μ*
_{2} as well as smaller absolute value of *κ* can be obtained, thus slow wave propagation will appear.

In Fig. 9(b), one can see that |*β _{z}*| shows a general upward trend when the magnetoelectric coupling becomes significant in the case that

*ε*

_{2}and

*μ*

_{2}are both negative. Minimum value for

*κ*with the cutoff condition

*k*' = 0 can be obtained,

_{z}From Eq. (17), *ε*
_{2} and *μ*
_{2} have the chance to become negative when *ω* is little larger than the resonance frequency *ω*
_{0}. Meanwhile, the magnetoelectric coupling *κ* has the absolute value which can be infinitely large within this frequency band. Therefore, the guided waves are able to propagate very slowly, and even approach zero velocity.

Besides, we should stress that when the magnetoelectric coupling vanishes, Eq. (25) and Eq. (26) become a product of two elementary modal equations, which is similar to those of conventional nonradiative dielectric waveguides and H waveguides. Fig. 9 shows that the operational diagram for LSM_{01} modes in a conventional nonradiative dielectric waveguide and a nonradiative dielectric waveguide with SRR metamaterials. Since there exists minimum value for magnetoelectric coupling *κ* in the double negative case, we choose *κ* = 5 and 10 to make sure that LSM_{01} propagates. As can be seen within [30 GHz, 31 GHz], the longitudinal wave number of LSM_{01} mode in nonradiative dielectric with SRR metamaterials is always larger than that of the conventional one, thus traveling more slowly, which indicates that nonradiative dielectric waveguide with SRR metamaterials allows more number of wavelength to propagate within the same length, providing feasibility of miniaturization for nonradiative dielectric waveguide.

Let’s further consider the power flow of LSM_{01} modes in the nonradiative dielectric waveguide with SRR metamaterials. The time-average power passing a transverse cross-section of the nonradiative dielectric waveguide is

$$\phantom{\rule{1.5em}{0ex}}=\pm \frac{1}{2}\mathrm{Re}{\int}_{x\text{'}=-l\text{'}}^{l\text{'}}{\int}_{y\text{'}=0}^{s\text{'}}{E}_{x}{{h}_{y}}^{*}\mathrm{dy}\text{'}\mathrm{dx}\text{'}$$

$$\phantom{\rule{1.5em}{0ex}}=\pm \frac{{F}_{2}^{2}{G}^{2}s\text{'}l\text{'}}{2{\epsilon}_{1}}\left[\frac{1}{{\beta}_{z}}\left(\frac{{\lambda}_{0}}{2s}\right)+{\beta}_{z}\right]$$

For *ε*
_{2} and *μ*
_{2} are both positive, we choose ‘+’, and for the double negative metamaterial case, we choose ‘-’. As we all know, the double negative metamaterials have the negative wave number, which leads to the positive Poynting vector in Eq. (30). With the choice of *s* = 0.4*λ*
_{0}, Fig. 10 shows general trend of power flow varied with absolute value of longitudinal wave number by not taking the constant coefficient into account. It can be seen that *P*
_{01} will increase as |*β _{z}*| becomes larger than $\sqrt{{\lambda}_{0}/2s}=1.118$. What is meant by this is that the increasing |

*β*| will enhance the power flow. From the above analysis, we can discern that metamaterial waveguide with double negative parameters has the chance to achieve infinite large |

_{z}*β*| around the metamaterial resonance frequency. Therefore, inserting metamaterials will strengthen the power flow of the conventional NRD waveguide. It is worth noting that such strengthen trend of power flow is based on the variation of |

_{z}*β*|, and the total energy will still be conserved since the electromagnetic wave propagates much more slowly.

_{z}In the above results, *s* < 0.5*λ*
_{0} is assumed throughout which means that the proposed waveguiding structure functions as nonradiative dielectric waveguide. When *s* increases to *s* > 0.5*λ*
_{0}, the waveguiding structure in Fig. 6 becomes H waveguide, and the results such as slow wave propagation and enhanced energy flow shown in Fig. 8–10 still work.

#### 3.2.2 Abnormal Guidance and Leakage Suppression

Usually, all the nonradiative dielectric waveguide and H waveguide components preserve the vertical symmetry so that a general *n* = 1 dependence may be assumed. Therefore, if the modes can leak power, they must do so in the form of TM_{1} or TE_{1} mode in the air-filled parallel plate region. Consequently, the condition for leakage can be written as *β _{z}* <

*k*, where

_{p}*k*is the normalized longitudinal wave number of the TM

_{p}_{1}or TE

_{1}satisfying ${k}_{p}^{2}=1-{\left(\frac{\pi}{s\text{'}}\right)}^{2}$. Given

*s*= 5 mm, the waveguiding structure in Fig. 2 works as a nonradiative dielectric waveguide when frequency

*f*< 30 GHz. No electromagnetic wave can propagate between the parallel plates because of the cutoff property, thus no leaky wave exits. However, when

*f*> 30 GHz, it functions as an H waveguide, therefore, leakage may come into being.

Considering the propagation features of the nonradiative dielectric waveguide and H waveguide with bianisotropic SRR metamaterials, the |*β _{z}*| of most modes become larger when

*f*increases like the modes in conventional waveguides. However, some higher-order LSM modes in the proposed waveguiding structure may operate dramatically differently. From Eq. (22a),

one can conclude that when $\frac{{\epsilon}_{1}}{{\epsilon}_{2}}\left({\epsilon}_{2}{\mu}_{2}-{\kappa}^{2}\right)>0$, |*k _{z}*| will demonstrate a general upward trend as

*f*increase, and may have the chance to get a fall while $\frac{{\epsilon}_{1}}{{\epsilon}_{2}}\left({\epsilon}_{2}{\mu}_{2}-{\kappa}^{2}\right)<0$. The similar trend holds true for |

*β*|. Fig. 11 presents the abnormal falling behavior of LSM higher-order modes in the proposed nonradiative dielectric waveguide and H waveguide with double negative parameters, and TM

_{z}_{1}mode in air filled parallel plate guide. In the region where the dispersion curve of LSM mode is located below the curve of TM

_{1}mode, one expects the leakage. As can be seen, if the falling behavior continues in the H-guide, leakage will eventually happens. Fortunately, such higher order modes can propagate only when the parameters are both negative. Moreover, lots of them cannot exit in the H-guide region since the cutoff of |

*β*| = 0 while

_{z}*f*becomes larger.

Figure 12 shows dispersion curve |*β _{z}*| versus frequency

*f*for LSM modes in proposed H waveguide under the condition $\frac{{\epsilon}_{1}}{{\epsilon}_{2}}\left({\epsilon}_{2}{\mu}_{2}-{\kappa}^{2}\right)>0$. It can be seen in Fig. 12(a), |

*β*| presents an increase when

_{z}*f*becomes lager, but experience a decrease as

*κ*turn larger at the same frequency while

*ε*

_{2}and

*μ*

_{2}are positive. Therefore, smaller magnetoelectric coupling

*κ*will reduce the leakage of the propagating modes. To examine further, from Eq. (17) one knows that positive

*ε*

_{2}and

*μ*

_{2}as well as minimum

*κ*can exit together in the frequency bands that are far larger than the SRR resonance frequency

*ω*

_{0}, in which the proposed waveguiding structure may exactly works as a H waveguide, thus such reduced leakage scheme can be fulfilled. On the contrary, Fig. 12(b) shows that under the same frequency, |

*β*| increases when

_{z}*κ*becomes more significant in the case that

*ε*

_{2}and

*μ*

_{2}are both negative. From Eq. (17), one can see that

*ε*

_{2}and

*μ*

_{2}have the chance to become both negative when the frequency is little larger than the SRR resonance frequency

*ω*

_{0}. Meanwhile, the magnetoelectric coupling

*κ*has the absolute value which can be infinitely large within this frequency band. Therefore, the leakage of guided waves is able to be greatly reduced.

For the more special case when *ε*
_{2} and *μ*
_{2} are of single negative value, Fig. 13 illustrates that when *μ*
_{2} > 0 and *ε*
_{2} < 0, |*β _{z}*| of LSM modes become larger as

*κ*increases. Furthermore, the leakage can hardly happen in this case even when

*κ*= 0, since

*β*>

_{z}*k*can be easily satisfied. However, LSE modes will no longer exist under such condition. On the other hand, when

_{p}*μ*

_{2}< 0 and

*ε*

_{2}> 0, only LSE modes can propagate in the proposed nonradiative dielectric waveguide amd H waveguide, and by choosing proper parameters, the leakage can also be eliminated.

## 4. Conclusion

Rigorous full wave analysis of bianisotropic SRR metamaterials has been developed for six SRR orientations relative to TEM wave directions. An alternative physical explanation for SRR resonance band gaps has been gained by revealing the fact that imaginary wave number leads to the SRR resonance. Model equations for TE and TM waves have been derived by considering parallel plate waveguide and rectangular waveguide with SRR metamaterials. Modes propagation with no cutoff frequency and an increase of single mode operation in material containing waveguide has been demonstrated. Different dispersion properties have been imparted to nonradiative dielectric waveguide and H waveguide by employing SRR metamaterials as the embedding medium. It is shown that the guided modes propagate more slowly when magnetoelectric coupling of SRR metamaterials becomes more significant. Zero-speed transmission can even occur within certain frequency band. Corresponding strengthened energy flow in the proposed structures has also been demonstrated. These results in turn make it possible to miniaturize the nonradiative dielectric waveguide and H waveguide. In addition, some abnormal higher-order LSM and LSE modes, eventually leading to the leakage, may come into existence when metamaterials are double negative. Fortunately, for other LSM and LSE modes, leakage can be reduced due to the magnetoelectric coupling. Furthermore, when metamaterials are of single negative parameters, leakage elimination has been achieved.

However, the electromagnetic responses of SRR metamaterials are in fact much more complicated for practical use [8]. Therefore, restrictive conditions, such as the absence of losses, certain direction of the incident wave, are assumed throughout this theoretical analysis. Here we employ different permeabilities and permittivities to discuss the possible performance enhancement by introducing the bianisotropic metamaterial model with the effective medium theory of tensor parameters, providing alternative means of characterizing the SRR metamaterial based waveguiding structures.

## Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Grant No. 60771040) and in part by State Key Laboratory Foundation (Grant No. 9140C0704060804)

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