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Omnidirectional resonance modes in photonic crystal 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) materials. 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 insensitive to incident angle. Moreover, as the periods of the heterostructure increases, the resonance transmission modes will split and be located symmetrical on both sides of the midfrequency of the zero-eff gap. © 2006 Optical Society of America OCIS codes: 260.0260, 310.6860, 120.2440, 350.2460.
1. INTRODUCTION Photonic crystals (PCs) have attracted extensive interest for their unique electromagnetic properties and potential applications in optoelectronics and optical communications.1,2 It has been proven that a photonic bandgap (PBG) is formed as the result of the interference of the Bragg scattering in a periodical dielectric structure. If the PBGs of the constituent PCs are aligned properly, heterostructures will form, leading to the generation of multiple resonance modes.3–5 However, in such structures consisting of positive-index materials (PIMs), the frequencies of resonance modes will blueshift as the incident angle increases, making these structures inefficient in application at situations of multidirectional incidence. Recently, negative-index materials (NIMs), with both negative permittivity and negative permeability , have been realized.6–8 It is demonstrated that PCs, which are composed of alternating layers of PIMs and NIMs, possess a PBG corresponding to a zero-averaged refractive ¯ gap).9 As the incident angle inindex (denoted as zero-n creases, the frequency shift of the defect modes inside the ¯ gap is small.10 But such frequency shifts should zero-n not be ignored when compared to the low frequencies of the defect mode, and the defect may be sensitive to the incident angle when it shifts to higher frequency.11 In addition to the NIMs, other metamaterials called singlenegative (SNG) materials, including the -negative (MNG) materials and -negative (ENG) materials, deserve special attention.12,13 It is then found that stacking alternating layers of MNG and ENG materials leads to another type of PBG with zero effective phase (denoted as zero-eff gap).14 A defect mode inside the zero-eff gap is insensitive to incident angle.15 However, there is a lack of structure that can produce a number of resonance modes with weak incident angle dependence, and the number 0740-3224/06/102237-4/$15.00
and corresponding frequencies of these resonance modes can be adjusted by changing the structural parameters. In this paper, 1D periodic PC heterostructures consisting of alternating MNG and ENG materials are demonstrated. Such structures can generate resonance transmission modes with weak dependence on incident angle. As the period of the heterostructure increases, the resonance modes will split and be symmetrical in both sides of the center of the zero-eff gap. In addition, the fields tend to be strongly localized at the interfaces between the two sub-PCs of the heterostructures.
2. COMPUTATIONAL MODEL Suppose that 1 = a,
1 = a −
2 mp
2
共1兲
in MNG materials and 2 = b −
2 ep
2
,
2 = b
共2兲
in ENG materials, where ep, mp are, respectively, the magnetic plasma frequency and the electronic plasma frequency. These kinds of dispersion for 1 and 2 may be realized in special metamaterials.16 In Eqs. (1) and (2), is the angular frequency measured in gigahertz. In the following calculation, we choose a = b = 1, a = b = 3, and mp = ep = 10 GHz. A zero-eff gap14 will be found in 1D PCs constituted by a periodic repetition of MNG and ENG layers with the thickness of dN and dN, respectively. In Fig. 1, we show the dependence of the band gaps on the ratio of the thicknesses of the two SNG layers 共dN / dN兲 at normal incidence under the lattice constant dN © 2006 Optical Society of America
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Fig. 1. Dependence of the PBGs on the ratio of the thickness of the two SNG materials under dN = 12 mm.
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cordingly, A cannot sustain the propagation of EM waves with frequency located in its PBG, so the EM waves will be localized in B. Like electrons and phonons in a semiconductor, the confinement of photons will also lead to the quantization of frequencies.3 The photons at these frequencies can pass through the structure by tunnelling, so several sharp resonance modes appear in the zero-eff gap (the midfrequency of the gap is ⬃0.8 GHz). Similarly, a number of resonance modes are also found in the Bragg gap (the midfrequency of the gap is ⬃5 GHz). It can be seen from Fig. 3 that, when the incident angle increases from 0° to 60°, the resonance modes inside the zero-eff gap remain nearly invariant, whereas the resonance modes inside the Bragg gap change quickly. Because these two kinds of gaps are locate at different frequency regions, for comparison we use a normalized frequency shift ⌬ / normal to denote the change of the resonance modes, where ⌬ = 兩-normal兩; normal is the frequency of the resonance mode at normal incidence. Figure 4 gives ⌬ / normal as a function of the incident angle for TE and TM polarizations. It can be seen from Fig. 4 that, as the incident angle increases, the resonance modes inside the zero-eff gap shift very slowly 共⌬ / normal ⬍ 0.01兲,
Fig. 2. Schematic of a 1D periodic PC heterostructure composed of alternating MNG and ENG materials.
= 12 mm. The gray areas represent the regions of propagating states, whereas the white areas represent regions containing evanescent states. It can be seen from Fig. 1 that the zero-eff gap can be widened by enlarging the difference between dN and dN. However, when dN = dN, the zero-eff gap is closed. Such properties of the zero-eff gap are useful for designing periodic structures with multiple resonance modes. Here we consider a 1D periodic PC heterostructure composed of two sub-PCs, A and B, arranged periodically as 共AnBm兲N. An and Bn are in turn composed of two different period units, a and b, respectively, where n and m are the period number of a and b in A and B, respectively. Both a and b consist of a pair of MNG and ENG materials. The thickness of the layers in a and b are d1, d2, d3 and d4, respectively, as shown in Fig. 2. In the following calculation, we choose d1 ⫽ d2 in a, and d3 = d4, in b. According to Fig. 1, the zero-eff gap exists in A, but not in B.
Fig. 3. Transmission spectra of structure 共A8B4兲2 at incident angle (a) = 0°, (b) = 60° for TE wave, and (c) = 60° for TM wave.
3. OMNIDIRECTIONAL RESONANCE MODES IN PERIODIC HETEROSTRUCTURES The transmission spectra of 共A8B4兲2 at the incident angles e = 0° and e = 60° for different polarization are shown in Fig. 3. Here we choose d1 = 12 mm, d2 = 6 mm, d3 = 12 mm, and d4 = 12 mm. As mentioned above, the zero-eff gap of the A PC is inside the transmission band of the B PC. Ac-
Fig. 4. Dependence of the resonance transmission peaks on the incident angle for different polarizations. The parameters are the same as those in Fig. 3.
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Fig. 5. Calculated transmission spectra of 共A8B4兲N: (a) for N = 2, (b) for N = 3, and (c) for N = 4, respectively.
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the phase-mismatch k1d1 ⫽ k2d2.14 As a defect B is inserted, the phase thickness of B can compensate ⌬ = 兩k1d1 − k2d2兩 partially so that the phase-match condition of the whole structure can be satisfied at two symmetric frequencies that are lower and higher than the midfrequency of the zero-eff gap and two resonance modes appear. These resonance modes, also known as eigenmodes, correlated with every defect. When these defects are brought together and arranged alternatively at a certain interval, previously degenerate eigenmodes will split owing to their coupling with one another. The number of split modes is equal to that of the defect layers.17 In our case, each defect layer corresponds to two degenerate eigenmodes, so 共AB兲N with N-1 defect layers should have 2共N-1兲 resonance modes. To understand how resonance modes were generated in the defect of 共AB兲N, we calculated the field distribution in the periodic heterostructures. Figure 6 exhibits the electric field distribution at frequencies of the resonance modes in 共A8B4兲2, 共A8B4兲3, and 共A8B4兲4, respectively. As shown in Fig. 6, the electric fields are localized mainly in the corresponding defect and reach maxima on the edge of the defects and at the interfaces from ENG layers to MNG layers.
4. CONCLUSION
Fig. 6. The electric field distributions in 1D periodic PC heterostructures. Corresponding resonant frequencies are (a) 0.871 GHz in 共A8B4兲2, (b) 0.866 GHz in 共A8B4兲3, and (c) 0.864 GHz in 共A8B4兲4, respectively.
whereas the resonance modes inside the Bragg gap shift quickly and ⌬ / normal achieves 0.25 at incident angle 80°. The very weak dependence of the resonance modes in the zero-eff gap may be useful in the designing of multiplechannel omnidirectional filters. Next, we study the relation between the resonance modes inside the zero-eff gap and the 共AB兲 number. Figure 5 shows the transmission spectra of structures 共A8B4兲2, 共A8B4兲3, and 共A8B4兲4 with d1 = 12 mm, d2 = 6 mm, d3 = 20 mm, and d4 = 20 mm. It can be seen from Fig. 5(a) that two resonance modes appear at frequencies 0.727 and 0.871 GHz, respectively, and are located in regions that are lower and higher, respectively, than the central frequency of the band gap. As the periods of the PC heterostructure 共AB兲 increases, each resonance mode will split into two and three, as shown in Figs. 5(b) and 5(c). Such a phenomenon can be explained as follows. Because the frequency of incident light is located in the zero-eff gap of A but the transmission band of B, B can be considered to be a defect of A, the resonance modes appear. It is demonstrated that the zero-eff gap of A corresponds to
1D periodic photonic heterostructures stacked with alternate MNG and ENG materials are proposed to generate omnidirectional resonance modes. The key to designing such heterostructures is to have only one of the two subPCs possess the zero-eff gap. In contrast with the resonance modes inside the Bragg gap, the resonance modes inside the zero-eff gap of this periodic heterostructure have very weak dependence on incident angle (normalized frequency shift ⌬ / normal ⬍ 0.01). As the periods of the heterostructure increase, the resonance transmission modes will split and will be located symmetrically in both regions that are higher and lower than the central frequency of the zero-eff gap. The properties of this structure provide such possible applications as multiplechannel omnidirectional filtering.
ACKNOWLEDGMENTS This work is supported by National 973 Project of China (2004CB719804) of China, the National Natural Science Foundation of China (10274108), and the Natural Science Foundation of Guangdong Province of China. H. Z. Wang is the corresponding author and can be reached at
[email protected].
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