Effective-medium models and experiments for extraordinary ...

20 downloads 0 Views 339KB Size Report
electric plasma with eff=1− c. 2/ 2.4 Recently, experiments and simulations5,6 revealed unusual EM wave transmissions through a waveguide loaded with split ...
APPLIED PHYSICS LETTERS 92, 041122 共2008兲

Effective-medium models and experiments for extraordinary transmission in metamaterial-loaded waveguides Hao Xu,1,3 Zhiyu Wang,2 Jiaming Hao,3 Jiajie Dai,3 Lixin Ran,2 J. A. Kong,2 and L. Zhou3,a兲 1

Surface Physics Laboratory (State Key Laboratory), Fudan University, Shanghai 200433, People’s Republic of China 2 The Electromagnetic Academy, Zhejiang University, Hangzhou 310027, People’s Republic of China 3 Physics Department, Fudan University, Shanghai 200433, People’s Republic of China

共Received 15 December 2007; accepted 14 January 2008; published online 1 February 2008兲 We show that a metallic waveguide behaves as an electric 共magnetic兲 plasma for transverse-electric 共transverse-magnetic兲 polarized electromagnetic 共EM兲 waves at frequencies below cutoff value. Inserting anisotropic resonance structures of either electric or magnetic type into a waveguide, we find extraordinary transmissions of EM waves with different polarizations through the waveguide at frequencies well below the waveguide’s cutoff value, following two different mechanisms. Microwave experiments, in excellent agreements with finite-different-time-domain simulations, are performed to demonstrate all theoretical predictions. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2840163兴 Electromagnetic 共EM兲 wave tunneling through classically opaque medium has drawn much attention recently.1–3 A metallic waveguide is opaque for EM waves at frequencies below its cutoff value ␻c, and it is usually characterized as an electric plasma with ␧eff = 1 − ␻2c / ␻2.4 Recently, experiments and simulations5,6 revealed unusual EM wave transmissions through a waveguide loaded with split ring resonators 共SRRs兲.7 The physics is that a SRR possesses a negative ␮ at some frequencies, so that the system becomes transparent since both ␧eff and ␮eff are ⬍0.5,6 Such loaded-waveguide systems attracted extensive attention recently8–11 since they provide an alternative approach to build negative index metamaterials.12 Here, we show that the physics behind such resonanceinduced transparencies is much richer than the simple picture presented.5,6 We found that while a waveguide can indeed be viewed an electric plasma4–6 for transverse-electric 共TE兲 waves, it must be viewed as a magnetic plasma for transverse-magnetic 共TM兲 waves. We performed experiments and finite-difference-time-domain 共FDTD兲 simulations13 to study the transmissions of EM waves through waveguides loaded with different resonance structures. Our results unambiguously verified the above pictures. Consider a metallic waveguide of a rectangular crosssection a ⫻ b, filled uniformly with a uniaxial medium with in in in in in J J ␧r = diag关␧in 储 , ␧ 储 , ␧ z 兴 and ␮ r = diag关 ␮ 储 , ␮ 储 , ␮ z 兴. Here, z denotes the wave propagation direction. Fixing the transverse wave vector as kx = m␲ / a , ky = n␲ / b, we solved the Maxwell J r−1 · 共k ⫻ E兲兴 = −共␻ / c兲2J equations, k ⫻ 关␮ ␧r · E, to get the dispersion relations and the wave forms of EM waves inside the waveguide. For a TE mode, the dispersion relation is found as

a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected].

in 2 in in kz2 = 共␻/c兲2␧in 储 ␮ 储 − 共 ␻ c/c兲 ␮ 储 / ␮ z ,

共1兲

S = E2储 zˆkz/共␻␮0␮in 储 兲.

共2兲

where ␻c = c冑共n␲ / a兲2 + 共m␲ / b兲2 is the cutoff frequency of the waveguide for a given mode.14 Simple calculations show that the energy flux S = E ⫻ H is given by

Let us consider a homogeneous medium with effective per¯ . For a plane EM wave propamittivity ¯␧ and permeability ␮ gating along the z direction 共with a wave-vector kz兲 inside the medium, the dispersion relation and the energy flux are ¯, kz2 = 共␻/c兲2¯␧ · ␮ ¯ 兲 = 兩H兩2zˆkz/共␻␧0¯␧兲. S = 兩E兩2zˆkz/共␻␮0␮

共3兲

Comparison between Eqs. 共1兲 and 共2兲 and Eq. 共3兲 suggests that the waveguide with inclusions can be viewed as an effective medium15 with 2 in ¯␧TE共␻兲 = ␧in 储 − 共 ␻ c/ ␻ 兲 / ␮ z ,

¯ TE共␻兲 = ␮in ␮ 储

共4兲 14

for TE waves. For TM waves, similar calculations that

show

in 2 in in kz2 = 共␻/c兲2␧in 储 ␮ 储 − 共 ␻ c/c兲 ␧ 储 /␧ z ,

S = H2储 zˆkz/共␻␧0␧in 储 兲.

共5兲

Again, comparing Eq. 共5兲 with Eq. 共3兲, we find that for TM waves, the effective parameters15 of the waveguide with inclusion should be ¯␧TM共␻兲 = ␧in 储 , 2 in ¯ TM共␻兲 = ␮in ␮ 储 − 共 ␻ c/ ␻ 兲 /␧ z .

共6兲

Equations 共4兲 and 共6兲 are our key results. Conclusions drawn from analyzing Eqs. 共4兲 and 共6兲 are summarized in Table I. First, for a hollow waveguide, while Eq. 共4兲 shows that for TE modes, it is indeed an electric plasma,4–6 Eq. 共6兲 shows

0003-6951/2008/92共4兲/041122/3/$23.00 92, 041122-1 © 2008 American Institute of Physics Downloaded 26 Feb 2008 to 61.129.42.30. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

041122-2

Appl. Phys. Lett. 92, 041122 共2008兲

Xu et al.

TABLE I. Transparency conditions and effective refraction indices of waveguides filled with inclusions for different polarizations.

Hollow waveguide

TE

TM

Electric plasma WG 共␻兲 = 1 − 共␻c / ␻兲2 ␧eff

Magnetic plasma WG ␮eff 共␻兲 = 1 − 共␻c / ␻兲2

Inclusion E resonance M resonance E resonance M resonance Case 3: Case 4: Transparency Case 1: Case 2: 2 2 ␮in ␧in ␮in conditions ␧in 储 ⬎ 共␻c / ␻兲 储 ⬍0 储 ⬍0 储 ⬎ 共␻c / ␻兲 Refraction index Positive Negative Negative Positive

that for TM modes, it behaves as a magnetic plasma. Second, we find that two possible inclusions, namely, a uniaxial electric resonant material 共UERM兲 or a uniaxial magnetic resonant material 共UMRM兲, can be inserted into a hollow waveguide to make it transparent at frequencies below ␻c. Consider the TE mode as an example. If the waveguide is filled with a UERM, Eq. 共4兲 shows that the whole waveguide 2 ¯ 共␻兲 = 1, indicating and ␮ is described by ¯␧共␻兲 = ␧in 储 − 共␻c / ␻兲 that it becomes transparent when ␧in is positively large 储 enough. The transparent medium possesses a positive refrac¯. tive index characterized by simultaneously positive ¯␧ and ␮ On the other hand, when the inclusion is a UMRM, the efWG ¯ 共␻兲 and ␮ fective parameters are ¯␧共␻兲 = 1 − 共␻c / ␻兲2 = ␧eff in = ␮储 , so that the waveguide becomes transparent when ␮in 储 ⬍ 0, and the system possesses a negative refraction index. The same arguments hold for the TM polarization.16 We performed FDTD simulations to verify all predictions presented in Table I. As shown in Figs. 1共a兲 and 1共b兲, the UERM we designed is a metallic cross 共CRS兲 structure and the UMRM consists of two double-ring SRRs perpendicularly interconnected. We employed FDTD simulations17 to calculate the normal transmission spectra of EM waves through slabs consisting of periodic arrays of either CRS 共periodicity= 11.43 mm in x-y plane兲 or SRR structures 共periodicity= 10 mm in x-y plane兲. By fitting the FDTD transmission spectrum of the CRS array 关solid circles in Fig. 2共b兲兴, we find that the CRS plate can be accurately modeled =1 as a 1.235-mm-thick slab of UERM with ␧CRS 储 + 260/ 共5.522 − f 2兲 + 1300/ 共17.122 − f 2兲, where f is the frequency measured in gigahertz. Similarly, we find that the

FIG. 2. 共Color online兲 For a 50-mm-long waveguide with cross section of 22.86⫻ 10.16 mm2 filled periodically with ten CRS plates: 共a兲 the calculated ¯␧ and ␮ ¯ vs frequency for TE wave; 共b兲 FDTD transmission spectrum through a CRS array and measured spectrum through a single CRS resonator in a parallel-plate waveguide; 共c兲 FDTD simulated TE-wave transmission spectra through the loaded and hollow waveguides; and 共d兲 measured TEwave transmission spectra through the loaded and hollow waveguides. In experiments, the waveguide is 100-mm-long filled with 18 CRS plates periodically.

SRR structure shown in Fig. 1共b兲 can be viewed as a = 1 + 9 / 共7.562 − f 2兲 3.6-mm-thick slab of UMRM with ␮SRR 储 2 2 + 28/ 共19.34 − f 兲 by fitting the FDTD transmission spectrum shown as the dash-dot lines in Figs. 3共b兲 and 5共b兲. To verify the predictions presented in Table I, we selected four waveguides with geometries specified in Figs. 2–5. In our FDTD simulations, we set the source and receiver antennas on different sides of the waveguide to measure the signals transmitted though the waveguide. For TE 共TM兲 mode, we place the source/receiver antennas parallel to the y共z兲 axis, as shown in Figs. 1共c兲 and 1共d兲. We first calculated the transmission spectra through those hollow waveguides and depicted the spectra as dotted lines in Figs. 2共c兲, 3共b兲, 4共b兲, and 5共b兲, correspondingly. In each spectrum, we find the received signal significantly enhanced 共reduced兲 at frequencies above 共below兲 the cutoff value of the designed

FIG. 1. 共Color online兲 Geometries of the 共a兲 CRS structure printed on a FIG. 3. 共Color online兲 For a 20-mm-long waveguide with cross section of d = 1.2-mm-thick substrate 共with ␧r = 2.2兲 and the 共b兲 SRR structure. Here, 10⫻ 10 mm2 filled with five SRR layers periodically: 共a兲 the calculated ¯␧ l1 = 9, l2 = 7.4, w = 0.2, a = 11.43, b = 0.2, c = 0.4, g = 0.4, and s = 3.6 共all in ¯ vs frequency for TE wave; 共b兲 FDTD simulated TE-wave transmisunits of mm兲. Schematic pictures of FDTD simulations 共experiments兲 to and ␮ calculate 共measure兲 the EM wave transmissions through waveguides filled sion spectra through the loaded and hollow waveguides, and the normal transmission spectrum through a slab of SRR array 共dash-dot line兲. with inclusions for 共c兲 TE polarization and 共d兲 TM polarization. Downloaded 26 Feb 2008 to 61.129.42.30. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

041122-3

Appl. Phys. Lett. 92, 041122 共2008兲

Xu et al.

FIG. 4. 共Color online兲 Same as Fig. 2 except that here the EM wave is TM polarized and the waveguide is filled with eight CRS plates with periodicity 5 mm in FDTD simulations.

waveguide, justifying the simulation scheme we adopted. We then periodically inserted some resonant units into those waveguides and calculated the transmission spectra through such loaded waveguides. The FDTD spectra were shown in Figs. 2共c兲, 3共b兲, 4共b兲, and 5共b兲 as solid lines, corresponding precisely to the four cases presented in Table I. In all situations, we find significantly enhanced transmissions within some frequency windows well below the waveguides’ cutoff frequencies. Qualitatively, the positions of those enhanced transmission bands 共ETBs兲 agree well with the theoretical predictions 共Table I兲. For example, the transparency condition in case 1 dictates the ETB to appear on the left of the resonance frequency, which is indeed the case as shown in Fig. 2共c兲. These intriguing phenomena can be quantitatively understood by using Eqs. 共4兲 and 共6兲. The effective parameters of the inclusions filling the waveguides are calculated by ␧in 储 in = 共1 − p兲 + p␧CRS and ␮in 储 储 ⬅ 1 for cases 1 and 3 and by ␧ 储 SRR ⬅ 1 and ␮in for cases 2 and 4. Here, p is the 储 = 共1 − p兲 + p ␮ 储 volume fraction of the inserted materials in different cases. in ¯ ¯ Setting ␧in 储 and ␮ 储 into Eqs. 共4兲 and 共6兲 to obtain ␧ and ␮ of

FIG. 5. 共Color online兲 Same as Fig. 3 except that here the EM wave is TM polarized and the waveguide is 34-mm-long to allow the antennas probing inside.

¯ the loaded waveguides, we plotted the calculated ¯␧ and ␮ versus frequency in Figs. 2共a兲, 3共a兲, 4共a兲, and 5共a兲, corre¯ ⬎0 spondingly. In each case, we find a pass band with ¯␧ · ␮ 共the shaded region兲 below the waveguide’s cutoff value, which coincides perfectly with the ETB shown in the FDTD transmission spectrum. As experimental confirmation on case 2 was available,5,6 we focus on cases 1 and 3 to experimentally verify our predictions. We fabricated the CRS plates and waveguides based on theoretical designs and performed microwave measurements following the scheme depicted in Fig. 1. The source and receiver antennas are both 5 mm monopoles, connected to a vector network analyzer Agilent-8722ES. The measured transmission spectra through a 100-mm-long hollow waveguide and through the waveguide loaded with 18 pieces of CRS plates were depicted in Figs. 2共d兲 and 4共c兲 for TE and TM polarizations, respectively. The spectrum of a single CRS plate placed in a parallel-plate waveguide is depicted in Fig. 2共b兲 to indicate the resonance frequency position. For each polarization, the measured transmission spectrum through the loaded waveguide clearly shows an ETB, whose position coincides well with the FDTD results as well as the theoretical analysis. In short, we demonstrate that a waveguide behaves as an electric 共magnetic兲 plasma for the TE 共TM兲 polarized wave, and inserting resonant materials yields high EM wave transmissions through an opaque waveguide following two mechanisms. This work was supported by the China-973 Program 共Nos. 2004CB719800 and 2006CB921701兲, the NSFC 共Nos. 10504003, 60725417, 60531020, and 60671003兲, FYTEF, PCSIRT, MOE of China 共No. B06011兲, ZJNSF 共No. R105253兲, and the NCET No. 07-0750. 1

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature 共London兲 391, 667 共1998兲. 2 A. Alu and N. Engheta, IEEE Trans. Antennas Propag. 51, 2558 共2003兲. 3 L. Zhou, W. J. Wen, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 94, 243905 共2005兲. 4 J. D. Jackson, Classical Electrodynamics, 3rd ed. 共Wiley, New York, 1999兲. 5 R. Marques, J. Martel, F. Mesa, and F. Medina, Phys. Rev. Lett. 89, 183901 共2002兲. 6 J. D. Baena, L. Jelinek, R. Marqués, and F. Medina, Phys. Rev. B 72, 075116 共2005兲. 7 J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 共1999兲. 8 A. Dechant and M. Okoniewski, Electron. Lett. 42, 4 共2006兲. 9 T. Ueda and M. Tsutsumi, IEEE Trans. Magn. 41, 3532 共2006兲. 10 P. A. Belov and C. R. Simovski, Phys. Rev. E 72, 036618 共2005兲. 11 S. Antipov, L. Spentzouris, W. Liu, W. Cai, and J. G. Power, J. Appl. Phys. 102, 034906 共2007兲. 12 R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 共2001兲. 13 K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 共1966兲. 14 We note that similar dispersion relations were derived in Ref. 11. 15 These effective-medium parameters can also be derived by comparing the impedances of the loaded waveguides and a homogenous medium. 16 Compared with previous works Refs. 5 and 6, our results are in agreement with their general conclusions 共see case 2 in Table I兲, but go beyond their results. 17 CONCERTO 4.0, Vector Fields Limited, England, 2004.

Downloaded 26 Feb 2008 to 61.129.42.30. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp