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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 8 — Apr. 13, 2009
  • pp: 6101–6117
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Fundamental modal properties of SRR metamaterials and metamaterial based waveguiding structures

Rui Yang, Yongjun Xie, Xiaodong Yang, Rui Wang, and Botao Chen  »View Author Affiliations


Optics Express, Vol. 17, Issue 8, pp. 6101-6117 (2009)
http://dx.doi.org/10.1364/OE.17.006101


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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

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Soviet Phys. Uspekhi. 10, 509–514 (1968). [CrossRef]

]–[6

6. R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag. 51, 1516–1529 (2003). [CrossRef]

].

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

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]

], [8

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. 100, 024507 (2006). [CrossRef]

]. 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

9. V. V. Varadan and A. R. Tellakula, “Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. 100, 034910 (2006). [CrossRef]

], [10

10. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2493–2495 (2004). [CrossRef]

]. Several analytical models are employed to examine the resonant property in the SRR transmission spectra [11

11. P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002). [CrossRef]

]–[14

14. P. Markos and C. M. Soukoulis, “Numerical studies of left handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622 (2002). [CrossRef]

]. Careful investigation has been carried out for different SRR parameters [15

15. K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonance for different split ring resonator parameters and designs,” New J. Phys. 7, 168 (2005). [CrossRef]

], [16

16. R. Marqués, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside- couple split ring resonators for metamaterials design-theory and experiments,” IEEE Trans. Microwave Theory Tech. 51, 2572–2581 (2003).

]. 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

17. A. Alú and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers,” IEEE Trans. Microw. Theory Tech. 52, 199–210 (2004). [CrossRef]

]–[24

24. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. 48, 2557–2560 (2006). [CrossRef]

]. The absence of fundamental mode and sign-varying energy flux in the negative refractive index waveguide are revealed [20

20. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative refractive index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

]. Rectangular waveguide filled with anisotropic single negative metamaterials are shown to support backward-wave propagation [21

21. S. Hrabar, J. Bartolic, and Z. Sipus, “Miniaturization of rectangular waveguide using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag. 53, 110–119 (2005). [CrossRef]

]. Moreover, Results for isotropic double negative metamaterial H waveguides are reported, including backward propagation, mode bifurcation and coupling effects [22

22. A. L. Topa, C. R. Paiva, and A. M. Barbosa, “Novel propagation features of double negative H-guides and H-guide couplers,” Microwave Opt. Technol. Lett. 47, 185–190 (2005). [CrossRef]

]. The use of single negative metamaterials as the embedding medium for nonradiative dielectric waveguides is examined [23

23. P. Yang, D. Lee, and K. Wu, “Nonradiative dielectric waveguide embedded in metamaterial with negative permittivity or permeability,” Microwave Opt. Technol. Lett. 45, 207–210 (2005). [CrossRef]

]. Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides is obtained [24

24. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. 48, 2557–2560 (2006). [CrossRef]

]. 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.

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

25. C. Krowne, “Electromagnetic theorems for complex. anisotropic media,” IEEE Trans. Antennas Propag. 32, 1224–1230 (1984). [CrossRef]

]

D=ε0(ε¯·E+Z0κ¯·H)
(1a)
B=μ0(1Z0κ¯T·E+μ¯·H)
(1b)

with Z0=μ0/ε0, where ε̄ and μ̄ are the relative electric permittivity and relative magnetic permeability tensors, and κ̄ is the magnetoelectric coupling dimensionless tensor.

Fig. 1. The SRR unit

For axes fixed to the SRR as shown in Fig. 1, only certain components of ε̄, μ̄ and κ̄ 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

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]

], [8

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. 100, 024507 (2006). [CrossRef]

].

εxx=1,εyy=a+bω2(ω02ω2),εzz=a
(2a)
μxx=1+cω2(ω02ω2),μyy=1,μzz=1
(2b)
κyx==idω0ω(ω02ω2)
(2c)

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

i'×h=ε¯·E+κ¯·h
(3a)
i'×E=κ¯T·E+μ¯·h
(3b)

where ∇'=∇/k 0.

Considering forward plane wave propagation of the form exp(-iβz'), where β = kz/k 0 is the normalized longitudinal wave number, one has

'=x'x̂+y'ŷẑ
(4)

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 Ez = hz = 0 for TEM waves, one obtains the following relations

(+β)hx=εyyEy
(5a)
(β)Ey=μxxhx
(5b)

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

β2=μxxεyyκ2
(6)

Ref. [7

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]

] concluded the same results by considering the bianisotropy role in SRR metamaterials. At given frequency ω, only those modes having μxxεyy - κ 2>0 will propagate. Those modes with μxxεyy - 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 εyy>0 and μxx <0 case, as well as εyy <0 and μxx>0 case. And also when the constitutive parameters are double negative or double positive with the condition |μxxεyy|<|κ 2|, the transmission stop band will also occur.

Similarly, one obtains

βhy=εxxEx
(7a)
βEx=μyyhy
(7b)

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

β2=εxxμyy=1
(8)

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.

Fig. 2. Six orientations of SRR relative to different electromagnetic field polarization and propagation direction of the incident TEM field

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

10. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2493–2495 (2004). [CrossRef]

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

Table 1. Wave numbers for the SRR metamaterials shown in Fig. 2(c) ~ Fig. 2(f).

table-icon
View This Table

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=μ/ε, complex wave number β determined by the six cases above, and thickness z' as the distance between the calibration planes

S11=Γ(1T2)1Γ2T2,S21=T(1Γ2)1Γ2T2
(9)

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).

Fig. 3. Geometry of parallel plate waveguide and rectangular waveguide filled with SRR metamaterials

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:

(εxxβ2εxxμyyx'2+εyyβ2μxxεyy+κ2y'2εzz)Ez
=(ββ2εxxμyy+ββ2μxxεyy+κ2)x'y'2hz
(10a)
(μxxβ2μxxεyy+κ2x'2+μyyβ2εxxμyyy'2μzz)hz
=(ββ2εxxμyy+ββ2μxxεyy+κ2)x'y'2Ez
(10b)

3.1.1 Non-cutoff Frequency Modes

For the parallel plate waveguides, TM waves are characterized by hz = 0 and a nonzero Ez field which satisfies the reduced wave Eq. (10a), with ∂x'=0,

[y'2+εzzεyy(μxxεyyκ2β2)]Ez=0
(11)

where

kz2=β2k02=(μxxεyyκ2)k02εyyεzzkc2
(12)

and kc=nπl, (n = 0, 1, 2⋯) is the cutoff wave number constrained to discrete values. /

Fig. 4. Metamaterial waveguide with different SRR orientations

Observe that for n = 0, the TM0 mode is actually identical to the TEM mode shown in Fig. 2(a), therefore, this TM0 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 εyy < 0 and μxxεyy - 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.

Fig. 5. The none cutoff frequency TM modes in parallel plate waveguide with SRR metamaterials εyy = -3, εzz = 1, μxx = -1, κ=1, l = 8 mm

For the rectangular waveguides, one can see that if κ ≠ 0, decoupling of Ez and hz occurs only when ∂x'≡0 or ∂y'≡0, therefore we only consider TEmn 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 TEm0 modes from Eq. (10b)

(x'2β2μxxεyy+κ2μxxμzz)hz=0
(13)

where

kz2=β2k02=(μxxεyyκ2)k02μxxμzzkc2
(14)

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 Ez = 0 and a nonzero hz field which satisfies the reduced wave Eq. (13b), with ∂x'=0. Through the similar derivation, one can obtain

fc=nc2l(n1)
(15)

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 fc=nc2l, 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 TE0n modes in Fig. 4(a) and Fig. 4(c) achieve the cutoff frequency of fc=nc2v, TEm0 modes in Fig. 4(b) obtains the cutoff frequency of fc=nc2u, 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.

Fig. 6. Configuration of nonradiative dielectric waveguide and H waveguide with SRR metamaterials
Fig. 7. Spatial orientation of SRRs in the host isotropic medium

The relative orientation of these two ensembles is in Fig. 7 and the ε̄, μ̄ and κ̄ tensors in Eq. (1) have the following uniaxial form [26

26. S. A. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML),” J. Electromagn. Wave Applic. 12, 821–837 (1998). [CrossRef]

]

ε¯=[ε1000ε2000ε2],μ¯=[μ1000μ2000μ2],κ¯=i[00000κ0κ0]
(16)

where

ε1a,ε21+ςa(ω02ω2)+bω2(ω02ω2)
(17a)
μ11,μ2ξ+(ω02ω2)+cω2(ω02ω2)
(17b)
κdω0ω(ω02ω2)
(17c)

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.

x'2hy+ε2ε1y'2hy=(ε2μ2κ2ε2ε1k'z2)hy
(18)

where kz' = kz/k 0 = β z-z is the complex normalized longitudinal wave number. All other field components can be expressed as

hz=i1k'zy'hy
(19a)
Ex=1k'zε1(y'2hyk'z2hy)
(19b)
Ey=1k'zε2(y'x'hyκy'hy)
(19c)
Ez=i1ε2(x'hyκhy)
(19d)

Express hy as a product of two separate variable functions in the form

hy=f(x')g(y')exp(jk'zz')
(20)

such that

x'2f(x')+k'x2f(x')=0
(21a)
y'2g(y')+k'y2g(y')=0
(21b)

where kx'=kx/k 0 and ky' =ky/k 0 are the complex normalized transverse wave numbers. Substituting back into Eq. (18), the normalized wave numbers can be expressed

k'x12+ε2ε1(k'y2+k'z2)=ε2μ2κ2
(22a)
k'x02+k'y2+k'z2=1
(22b)

One has kx' = k x1' = β x1 - x1 for |x'| < q', while for |x'| > q', one should take kx' = k x0' = β x0 - x0. Apply the boundary conditions on the perfectly electric conductor planes to other field components, one can write

g(y')=Gsin(k'yy')(n=1,2,3,)
(23)

where G is the amplitude constant, and k'y=nπS' with s'=k 0 s. The n index gives the number of half waves along y. And

f(x')={F1exp[k'x0(x'+q')]x'<q'F2[cos(k'x1x')+Rsin(k'x1x')]q'<x<q'F3exp[k'x0(x'q')]q'<x'
(24)

[k'x1cot(k'x1q')+k'x0ε2][k'x1tan(k'x1q')k'x0ε2]+κ2=0
(25)

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

[k'x1cot(k'x1q')+k'x0μ2][k'x1tan(k'x1q')k'x0μ2]+κ2=0
(26)

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

k'x12+μ2μ1(k'y2+k'z2)=ε2μ2κ2
(27)

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 |βz| 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 kz' = 0.

κmax=ε2μ2ε2ε1k'y2k'x12
(28)

Since the guide wavelength defined as λg=2πβz becomes smaller when longitudinal wave number increases, the corresponding phase velocity v = 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.

Fig. 8. Relationship of |βz| and magnetoelectric coupling κ in nonradiative dielectric waveguide with SRR metamaterials (a) ε 1 = 1, ε 2 = 3, μ 2 = 2.5, f = 35 GHz, s = 0.4λ 0, q = 0.6λ 0 (b) ε 1 = 1, ε 2 = -3, μ 2 = -2.5, f = 35 GHz, s = 0.4λ0, q = 0.6λ 0

In Fig. 9(b), one can see that |βz| 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 kz' = 0 can be obtained,

κmin=ε2μ2+ε2ε1k'y2k'x12
(29)

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.

Fig. 9. Variation of the |βz| with frequency f for the dominant LSM01 (conventional NRD waveguide with εr=4, μr=1, s = 4 mm, q = 5 mm; double negative metamaterial NRD waveguide with ε 1 = 1, ε 2 = -4 , μ 2 = -1, κ = 5 and 10, s = 4 mm, q = 5 mm)

Let’s further consider the power flow of LSM01 modes in the nonradiative dielectric waveguide with SRR metamaterials. The time-average power passing a transverse cross-section of the nonradiative dielectric waveguide is

P01=12Rex'=l'l'y'=0s'E×h*·ẑdy'dx'
=±12Rex'=l'l'y'=0s'Exhy*dy'dx'
=±F22G2s'l'2ε1[1βz(λ02s)+βz]
(30)
Fig. 10. Energy flow of LSM01 mode varied with longitudinal wave number in the nonradiative dielectric with SRR metamaterials

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 = 0.4λ 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 |βz| becomes larger than λ0/2s=1.118. What is meant by this is that the increasing |βz| 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 |βz| 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 |βz|, and the total energy will still be conserved since the electromagnetic wave propagates much more slowly.

In the above results, s < 0.5λ 0 is assumed throughout which means that the proposed waveguiding structure functions as nonradiative dielectric waveguide. When s increases to s > 0.5λ 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 TM1 or TE1 mode in the air-filled parallel plate region. Consequently, the condition for leakage can be written as βz < kp, where kp is the normalized longitudinal wave number of the TM1 or TE1 satisfying kp2=1(πs')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 |βz| 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),

k'z2=ε1ε2(ε2μ2κ2k'x12)k'y2
(31)

one can conclude that when ε1ε2(ε2μ2κ2)>0, |kz| will demonstrate a general upward trend as f increase, and may have the chance to get a fall while ε1ε2(ε2μ2κ2)<0. The similar trend holds true for |βz|. 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 TM1 mode in air filled parallel plate guide. In the region where the dispersion curve of LSM mode is located below the curve of TM1 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 |βz| = 0 while f becomes larger.

Fig. 11. Abnormal falling behavior of LSM higher modes in the proposed nonradiative dielectric waveguide/H waveguide with double negative parameters (ε 1 = 1, ε 2 = -3, μ 2 = -1, κ = 1, s = 5 mm, q = 5 mm) and TM1 mode in air filled parallel plate guide
Fig. 12. Variation of the |βz| versus frequency f for LSM modes in the proposed nonradiative dielectric waveguide/H waveguide ((a) for double positive metamaterials with ε 1 = 1, ε 2 = 2, μ 2 = 2.5, κ = 0.1, 1.2, and 1.4, s = 5 mm, q = 5 mm (b) for double negative metamaterials with ε 1 =1, ε 2 = -3, μ 2 =-1.5, κ = 2.3, 2.5, and 3, s = 5 mm, q = 5 mm) and TM1 mode in air filled parallel plate guide

Figure 12 shows dispersion curve |βz| versus frequency f for LSM modes in proposed H waveguide under the condition ε1ε2(ε2μ2κ2)>0. It can be seen in Fig. 12(a), |βz| presents an increase when 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, |βz| increases when 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, |βz| of LSM modes become larger as κ increases. Furthermore, the leakage can hardly happen in this case even when κ = 0, since βz > kp can be easily satisfied. However, LSE modes will no longer exist under such condition. On the other hand, when 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.

Fig. 13. Variation of the |βz| versus frequency f for LSM modes in proposed nonradiative dielectric waveguide/H waveguide with single negative metamaterials ε 1=1, ε 2 =-3.5, μ 2 =1.5, κ = 0, 1, and 3, s = 5 mm, q = 5 mm and TM1 mode in air filled parallel plate guide

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

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. 100, 024507 (2006). [CrossRef]

]. 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)

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Soviet Phys. Uspekhi. 10, 509–514 (1968). [CrossRef]

2.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

3.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenonmena,” IEEE Trans. Microwave Theory Tech. 47, pp. 2075–2084 (1999). [CrossRef]

4.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability an permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

5.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

6.

R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag. 51, 1516–1529 (2003). [CrossRef]

7.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]

8.

D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. 100, 024507 (2006). [CrossRef]

9.

V. V. Varadan and A. R. Tellakula, “Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. 100, 034910 (2006). [CrossRef]

10.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2493–2495 (2004). [CrossRef]

11.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002). [CrossRef]

12.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402(1–4), 2004. [CrossRef] [PubMed]

13.

T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, “Ab initio numerical simulation of left handed metamaterials: comparison of calculation and experiments,” J. Appl. Phys. 90, 5419–5424 (2001). [CrossRef]

14.

P. Markos and C. M. Soukoulis, “Numerical studies of left handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622 (2002). [CrossRef]

15.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonance for different split ring resonator parameters and designs,” New J. Phys. 7, 168 (2005). [CrossRef]

16.

R. Marqués, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside- couple split ring resonators for metamaterials design-theory and experiments,” IEEE Trans. Microwave Theory Tech. 51, 2572–2581 (2003).

17.

A. Alú and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers,” IEEE Trans. Microw. Theory Tech. 52, 199–210 (2004). [CrossRef]

18.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386–9388 (2003). [CrossRef]

19.

Y. S. Xu, “A study of waveguides field with anisotropic metamaterials,” Microwave Opt. Technol. Lett. 41, 426–431 (2004). [CrossRef]

20.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative refractive index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

21.

S. Hrabar, J. Bartolic, and Z. Sipus, “Miniaturization of rectangular waveguide using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag. 53, 110–119 (2005). [CrossRef]

22.

A. L. Topa, C. R. Paiva, and A. M. Barbosa, “Novel propagation features of double negative H-guides and H-guide couplers,” Microwave Opt. Technol. Lett. 47, 185–190 (2005). [CrossRef]

23.

P. Yang, D. Lee, and K. Wu, “Nonradiative dielectric waveguide embedded in metamaterial with negative permittivity or permeability,” Microwave Opt. Technol. Lett. 45, 207–210 (2005). [CrossRef]

24.

P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. 48, 2557–2560 (2006). [CrossRef]

25.

C. Krowne, “Electromagnetic theorems for complex. anisotropic media,” IEEE Trans. Antennas Propag. 32, 1224–1230 (1984). [CrossRef]

26.

S. A. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML),” J. Electromagn. Wave Applic. 12, 821–837 (1998). [CrossRef]

OCIS Codes
(230.7370) Optical devices : Waveguides
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: January 29, 2009
Revised Manuscript: March 5, 2009
Manuscript Accepted: March 5, 2009
Published: March 31, 2009

Citation
Rui Yang, Yongjun Xie, Xiaodong Yang, Rui Wang, and Botao Chen, "Fundamental modal properties of SRR metamaterials and metamaterial based waveguiding structures," Opt. Express 17, 6101-6117 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6101


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and μ," Soviet Phys. Uspekhi. 10, 509-514 (1968). [CrossRef]
  2. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenonmena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability an permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  6. R. W. Ziolkowski, "Design, fabrication and testing of double negative metamaterials," IEEE Trans. Antennas Propag. 51, 1516-1529 (2003). [CrossRef]
  7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and lefthanded metamaterials," Phys. Rev. B 65, 144440 (2002). [CrossRef]
  8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig "Calculation and measurement of bianisotropy in a split ring resonator," J. Appl. Phys. 100, 024507 (2006). [CrossRef]
  9. V. V. Varadan, A. R. Tellakula, "Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100, 034910 (2006). [CrossRef]
  10. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, C. M. Soukoulis, "Electric coupling to the magnetic resonance of split ring resonators," Appl. Phys. Lett. 84, 2493-2495 (2004). [CrossRef]
  11. P. Gay-Balmaz and O. J. F. Martin, "Electromagnetic resonances in individual and coupled split ring resonators," J. Appl. Phys. 92, 2929-2936 (2002). [CrossRef]
  12. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402(1-4), 2004. [CrossRef] [PubMed]
  13. T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, S. Schultz, "Ab initio numerical simulation of left handed metamaterials: comparison of calculation and experiments," J. Appl. Phys. 90, 5419-5424 (2001). [CrossRef]
  14. P. Markos and C. M. Soukoulis, "Numerical studies of left handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002). [CrossRef]
  15. K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, "Investigation of magnetic resonance for different split ring resonator parameters and designs," New J. Phys. 7, 168 (2005). [CrossRef]
  16. R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside- couple split ring resonators for metamaterials design-theory and experiments," IEEE Trans. Microwave Theory Tech. 51, 2572-2581 (2003).
  17. A. Alú and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), doublenegative (DNG), and/or double-positive (DPS) layers," IEEE Trans. Microw. Theory Tech. 52, 199-210 (2004). [CrossRef]
  18. B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003). [CrossRef]
  19. Y. S. Xu, "A study of waveguides field with anisotropic metamaterials," Microwave Opt. Technol. Lett. 41, 426-431 (2004). [CrossRef]
  20. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative refractive index waveguides," Phys. Rev. E 67, 057602 (2003). [CrossRef]
  21. S. Hrabar, J. Bartolic, and Z. Sipus, "Miniaturization of rectangular waveguide using uniaxial negative permeability metamaterial," IEEE Trans. Antennas Propag. 53, 110-119 (2005). [CrossRef]
  22. A. L. Topa, C. R. Paiva, and A. M. Barbosa, "Novel propagation features of double negative H-guides and H-guide couplers," Microwave Opt. Technol. Lett. 47, 185-190 (2005). [CrossRef]
  23. P. Yang, D. Lee, and K. Wu, "Nonradiative dielectric waveguide embedded in metamaterial with negative permittivity or permeability," Microwave Opt. Technol. Lett. 45, 207-210 (2005). [CrossRef]
  24. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, "Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides," Microwave Opt. Technol. Lett. 48, 2557-2560 (2006). [CrossRef]
  25. C. Krowne, "Electromagnetic theorems for complex. anisotropic media," IEEE Trans. Antennas Propag. 32, 1224-1230 (1984). [CrossRef]
  26. S. A. Tretyakov, "Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML)," J. Electromagn. Wave Applic. 12, 821-837 (1998). [CrossRef]

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