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Chen et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. B 2237

Omnidirectional resonance modes in photoniccrystal heterostructures containing single-negative

materials

Y. H. Chen, J. W. Dong, and H. Z. Wang

State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan (Sun Yat-Sen) University,Guangzhou 510275, China

Received March 24, 2006; accepted June 29, 2006; posted July 6, 2006 (Doc. ID 69344)

Multiple omnidirectional resonance modes are generated in the periodic arrangement of photonic crystal (PC)heterostructures with two sub-PCs consisting of single-negative (permittivity- or permeability-negative) ma-terials. The key to designing such heterostructures is that only one of the sub-PCs possesses the zero-�eff gap.It is found that the resonance transmission modes inside the zero-�eff gap of these heterostructures are insen-sitive to incident angle. Moreover, as the periods of the heterostructure increases, the resonance transmissionmodes will split and be located symmetrical on both sides of the midfrequency of the zero-�eff gap. © 2006Optical Society of America

OCIS codes: 260.0260, 310.6860, 120.2440, 350.2460.

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. INTRODUCTIONhotonic crystals (PCs) have attracted extensive interest

or their unique electromagnetic properties and potentialpplications in optoelectronics and opticalommunications.1,2 It has been proven that a photonicandgap (PBG) is formed as the result of the interferencef the Bragg scattering in a periodical dielectric structure.f the PBGs of the constituent PCs are aligned properly,eterostructures will form, leading to the generation ofultiple resonance modes.3–5 However, in such structures

onsisting of positive-index materials (PIMs), the fre-uencies of resonance modes will blueshift as the incidentngle increases, making these structures inefficient in ap-lication at situations of multidirectional incidence.Recently, negative-index materials (NIMs), with both

egative permittivity � and negative permeability �, haveeen realized.6–8 It is demonstrated that PCs, which areomposed of alternating layers of PIMs and NIMs, pos-ess a PBG corresponding to a zero-averaged refractivendex (denoted as zero-n̄ gap).9 As the incident angle in-reases, the frequency shift of the defect modes inside theero-n̄ gap is small.10 But such frequency shifts shouldot be ignored when compared to the low frequencies ofhe defect mode, and the defect may be sensitive to the in-ident angle when it shifts to higher frequency.11 In addi-ion to the NIMs, other metamaterials called single-egative (SNG) materials, including the �-negative

MNG) materials and �-negative (ENG) materials, de-erve special attention.12,13 It is then found that stackinglternating layers of MNG and ENG materials leads tonother type of PBG with zero effective phase (denoted asero-�eff gap).14 A defect mode inside the zero-�eff gap isnsensitive to incident angle.15 However, there is a lack oftructure that can produce a number of resonance modesith weak incident angle dependence, and the number

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nd corresponding frequencies of these resonance modesan be adjusted by changing the structural parameters.

In this paper, 1D periodic PC heterostructures consist-ng of alternating MNG and ENG materials are demon-trated. Such structures can generate resonance trans-ission modes with weak dependence on incident angle.s the period of the heterostructure increases, the reso-ance modes will split and be symmetrical in both sides ofhe center of the zero-�eff gap. In addition, the fields tendo be strongly localized at the interfaces between the twoub-PCs of the heterostructures.

. COMPUTATIONAL MODELuppose that

�1 = �a, �1 = �a −�mp

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n MNG materials and

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n ENG materials, where �ep, �mp are, respectively, theagnetic plasma frequency and the electronic plasma fre-

uency. These kinds of dispersion for �1 and �2 may be re-lized in special metamaterials.16 In Eqs. (1) and (2), � ishe angular frequency measured in gigahertz. In the fol-owing calculation, we choose �a=�b=1, �a=�b=3, andmp=�ep=10 GHz. A zero-�eff gap14 will be found in 1DCs constituted by a periodic repetition of MNG and ENG

ayers with the thickness of dN� and dN�, respectively. Inig. 1, we show the dependence of the band gaps on theatio of the thicknesses of the two SNG layers �dN� /dN��t normal incidence under the lattice constant d

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2238 J. Opt. Soc. Am. B/Vol. 23, No. 10 /October 2006 Chen et al.

12 mm. The gray areas represent the regions of propa-ating states, whereas the white areas represent regionsontaining evanescent states. It can be seen from Fig. 1hat the zero-�eff gap can be widened by enlarging the dif-erence between dN� and dN�. However, when dN�=dN�,he zero-�eff gap is closed. Such properties of the zero-�effap are useful for designing periodic structures with mul-iple resonance modes.

Here we consider a 1D periodic PC heterostructureomposed of two sub-PCs, A and B, arranged periodicallys �AnBm�N. An and Bn are in turn composed of two differ-nt period units, a and b, respectively, where n and m arehe period number of a and b in A and B, respectively.oth a and b consist of a pair of MNG and ENG materi-ls. The thickness of the layers in a and b are d1, d2, d3nd d4, respectively, as shown in Fig. 2. In the followingalculation, we choose d1�d2 in a, and d3=d4, in b. Ac-ording to Fig. 1, the zero-�eff gap exists in A, but not in.

. OMNIDIRECTIONAL RESONANCEODES IN PERIODIC HETEROSTRUCTURES

he transmission spectra of �A8B4�2 at the incident anglese=0° and �e=60° for different polarization are shown inig. 3. Here we choose d1=12 mm, d2=6 mm, d3=12 mm,nd d4=12 mm. As mentioned above, the zero-�eff gap ofhe A PC is inside the transmission band of the B PC. Ac-

ig. 1. Dependence of the PBGs on the ratio of the thickness ofhe two SNG materials under dN�=12 mm.

ig. 2. Schematic of a 1D periodic PC heterostructure composedf alternating MNG and ENG materials.

ordingly, A cannot sustain the propagation of EM wavesith frequency located in its PBG, so the EM waves wille localized in B. Like electrons and phonons in a semi-onductor, the confinement of photons will also lead to theuantization of frequencies.3 The photons at these fre-uencies can pass through the structure by tunnelling, soeveral sharp resonance modes appear in the zero-�eff gapthe midfrequency of the gap is �0.8 GHz). Similarly, aumber of resonance modes are also found in the Braggap (the midfrequency of the gap is �5 GHz). It can beeen from Fig. 3 that, when the incident angle increasesrom 0° to 60°, the resonance modes inside the zero-�effap remain nearly invariant, whereas the resonanceodes inside the Bragg gap change quickly.Because these two kinds of gaps are locate at different

requency regions, for comparison we use a normalizedrequency shift �� /�normal to denote the change of theesonance modes, where ��= ��-�normal�; �normal is the fre-uency of the resonance mode at normal incidence. Figuregives �� /�normal as a function of the incident angle forE and TM polarizations. It can be seen from Fig. 4 that,s the incident angle increases, the resonance modes in-ide the zero-�eff gap shift very slowly ��� /�normal�0.01�,

ig. 3. Transmission spectra of structure �A8B4�2 at incidentngle (a) �=0°, (b) �=60° for TE wave, and (c) �=60° for TMave.

ig. 4. Dependence of the resonance transmission peaks on thencident angle for different polarizations. The parameters are theame as those in Fig. 3.

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Chen et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. B 2239

hereas the resonance modes inside the Bragg gap shiftuickly and �� /�normal achieves 0.25 at incident angle 80°.he very weak dependence of the resonance modes in theero-�eff gap may be useful in the designing of multiple-hannel omnidirectional filters.

Next, we study the relation between the resonanceodes inside the zero-�eff gap and the �AB� number. Fig-

re 5 shows the transmission spectra of structuresA8B4�2, �A8B4�3, and �A8B4�4 with d1=12 mm, d2=6 mm,3=20 mm, and d4=20 mm. It can be seen from Fig. 5(a)hat two resonance modes appear at frequencies 0.727nd 0.871 GHz, respectively, and are located in regionshat are lower and higher, respectively, than the centralrequency of the band gap. As the periods of the PC het-rostructure �AB� increases, each resonance mode willplit into two and three, as shown in Figs. 5(b) and 5(c).uch a phenomenon can be explained as follows. Becausehe frequency of incident light is located in the zero-�effap of A but the transmission band of B, B can be consid-red to be a defect of A, the resonance modes appear. It isemonstrated that the zero-� gap of A corresponds to

ig. 5. Calculated transmission spectra of �A8B4�N: (a) for N2, (b) for N=3, and (c) for N=4, respectively.

ig. 6. The electric field distributions in 1D periodic PC hetero-tructures. Corresponding resonant frequencies are (a).871 GHz in �A8B4�2, (b) 0.866 GHz in �A8B4�3, and (c) 0.864 GHzn �A8B4�4, respectively.

eff

he phase-mismatch k1d1�k2d2.14 As a defect B is in-erted, the phase thickness of B can compensate ���k1d1−k2d2� partially so that the phase-match conditionf the whole structure can be satisfied at two symmetricrequencies that are lower and higher than the midfre-uency of the zero-�eff gap and two resonance modes ap-ear. These resonance modes, also known as eigenmodes,orrelated with every defect. When these defects arerought together and arranged alternatively at a certainnterval, previously degenerate eigenmodes will split ow-ng to their coupling with one another. The number ofplit modes is equal to that of the defect layers.17 In ourase, each defect layer corresponds to two degenerateigenmodes, so �AB�N with N-1 defect layers should have�N-1� resonance modes.To understand how resonance modes were generated in

he defect of �AB�N, we calculated the field distribution inhe periodic heterostructures. Figure 6 exhibits the elec-ric field distribution at frequencies of the resonanceodes in �A8B4�2, �A8B4�3, and �A8B4�4, respectively. As

hown in Fig. 6, the electric fields are localized mainly inhe corresponding defect and reach maxima on the edge ofhe defects and at the interfaces from ENG layers to MNGayers.

. CONCLUSIOND periodic photonic heterostructures stacked with alter-ate MNG and ENG materials are proposed to generatemnidirectional resonance modes. The key to designinguch heterostructures is to have only one of the two sub-Cs possess the zero-�eff gap. In contrast with the reso-ance modes inside the Bragg gap, the resonance modes

nside the zero-�eff gap of this periodic heterostructureave very weak dependence on incident angle (normal-

zed frequency shift �� /�normal�0.01). As the periods ofhe heterostructure increase, the resonance transmissionodes will split and will be located symmetrically in both

egions that are higher and lower than the central fre-uency of the zero-�eff gap. The properties of this struc-ure provide such possible applications as multiple-hannel omnidirectional filtering.

CKNOWLEDGMENTShis work is supported by National 973 Project of China

2004CB719804) of China, the National Natural Scienceoundation of China (10274108), and the Natural Scienceoundation of Guangdong Province of China.

H. Z. Wang is the corresponding author and can beeached at stswhz@zsu.edu.cn.

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