Derivation and characterization of dispersion of defect modes in photonic band gap from stacks of positive and negative index materials

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1 Physics Letters A ) erivation and characterization of dispersion of defect modes in photonic band gap from stacks of positive and negative index materials Y.H. Chen, G.Q. Liang, J.W. ong, H.Z. Wang State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan Sun Yat-Sen) University, Guangzhou , People s Republic of China Received 20 July 2005; received in revised form 19 October 2005; accepted 19 October 2005 Available online 26 October 2005 Communicated by R. Wu Abstract An expression is derived to predict the incident angle dependence of the frequency of defect modes inside the zero- n gap of one-dimensional photonic crystals stacking with positive and negative index materials. From this expression, the dependence of the dispersion type of the defect mode on both the refractive index and the wave impedance of the component layers of the photonic crystal are given. For situations with a given value of permeability, it is found that, as the adjacent layers of the defect are low refractive index materials, only positive or negative type of angular dispersion exists; as the adjacent layers of the defect are high refractive index materials, all three types of angular dispersion, including the near-zero dispersion, appear with the refractive index of the defect changes. The dispersion relation diagrams of such photonic crystals are presented, the simulated results agree with the theoretical prediction Elsevier B.V. All rights reserved. PACS: Qs; Jb; Ci Keywords: Photonic crystal; Film; egative index material; ispersion 1. Introduction Since the initial prediction of Yablonovitch 1 and John 2, photonic crystals PCs) have attracted a great deal of interests for their unique electromagnetic properties and potential applications in optoelectronics and optical communications 3, 4. One-dimensional 1) PCs with defect layers have been used for filters, including single channel filters 5 and multiple channel filters 6. However, for conventional filters consisting of positive index materials PIMs), the frequencies of transmission peak will blue shift as the incident angle increases. It makes these filters inefficient in situations of multi-direction incidence. Recently, negative refractive index materials s) with both negative permittivity and negative permeability, which were first suggested theoretically by Veselago 7, have been * Corresponding author. address: stswhz@zsu.edu.cn H.Z. Wang). realized It is then demonstrated that photonic crystals, which are composed of alternating layers of PIM and, have a new type of photonic band gap denoted as zero- n gap) with weak dependence on incident angle 11,12. This new type of gap arises when the volume averaged effective refractive index n) equals zero. A PIM impurity was also introduced in this zero- n gap 13, in which the defect mode is insensitive to the incident angle and polarization. However, the design of controllable defect modes requires predictive formulas for the frequency dependence of the defect modes on physical parameters of PCs, and theory is lacking for predicting whether the near-zero dispersion exists or not. So the applications of such defect modes, such as omnidirectional filters and narrow frequency sharp angular filters 14, are restricted. In this Letter, we derive a simple expression to predict the frequency shift of defect modes inside the zero- n gap as incident angle changes by using the defect mode resonant condition of several 1 defective PC structures with quarter-wave stacks consisted alternately by PIM and. Unlike in most previous studies, we also take into account the magnetic properties /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.physleta

2 Y.H. Chen et al. / Physics Letters A ) of materials that influence the properties of the defect modes. From the expression, the dependence of the dispersion type of the defect modes on both the refractive index and the wave impedance of the component layers of the photonic crystals are given. Based on these theories, we find some special characteristics of the angular dispersion when change the refractive index of the defect layer. Then, the dispersion relation diagrams of such photonic crystals are simulated to confirm the theoretical results. Furthermore, dispersion invariability of defect modes is found in our simulations. 2. Resonant condition of the defect modes inside the zero- n gap for 1 defective PCs consisting of alternate PIM and It is known that, the frequency of the defect mode is determined by two factors, the optical phase thickness of the defect layer and the phase change on reflection from the periodic stacks of the PCs. In the case of conventional PCs, which consist of PIMs, as the incident light changes from normal to oblique incidence, the both factors above reduce simultaneously, and it make the defect modes tend to higher frequency. However, for PCs containing s with the zero- n gap, the changes of the two factors can cancel each other in some conditions, and the frequency of the defect mode remain nearly invariant as the incident angle changes, so the near-zero dispersion emerges and it can also be obtained from the formula description in the following sections). Consider a 1 PC consisting of alternate PIM and with relative permittivities and relative permeabilities ε P,µ P ) and ε,µ ), respectively. The corresponding refractive index and the wave impedance are given by n i ± ε i µ i and Z i µ i /ε i µ i /n i, i P, negative sign for ). Then a defect layer is introduced to compose defective PCs with the forms of P ) s P) s system A) and P ) s P) s system B), where P) represents a layer of PIM) with a refractive index of n n P ) and a geometrical thickness of d d P ), represents a defect layer with a refractive index of n and a geometrical thickness of d, s is the number of periods. Here, we assume each layer is of the same quarter-wave optical thickness, n P d P n d, and the optical thickness of the defect layers is n d n d +n P d P.Li11 has shown that a zero- n gap will appear in this PC. As discussed by Smith 15, the defective PC may be completely described by the defect layer and two effective interfaces 1 and 2, and we consider 1 as interface with the layers to the left of the PC, and 2 as interface with the layers to the right of the PC. Then the properties of the PC may be deduced by summing the reflected beams between 1 and 2, and the transmittance of the PC is given by Tν,θ e ) T 1 ν, θ e )T 2 ν, θ e ) 1 R 1 ν, θ e )R 2 ν, θ e ) R 1 ν, θ e )R 2 ν, θ e ) sin δ), 1) where T 1, T 2, R 1, R 2 are the transmittance and reflectance from interfaces 1 and 2 at frequency ν, θ e is the external incident angle, δ is the phase change between the forward and backward waves in the defect layer. The transmittance peaks of the defect modes appear when the resonant condition δ ±φ 1 ν, θ e ) ± φ 2 ν, θ e ) + 4πνn d cos θ 2mπ is satisfied. Here m is an integer, φ 1 and φ 2 are the phase changes on reflection from interfaces 1 and 2, respectively, θ is the refractive angle in the defect layer. As waves are incident upon 1 or 2 from the defect layer, signs are applied to the first two terms in the middle of Eq. 2) 16 when both the defect layer and its adjacent layer are the same type of material, PIMs or s, with the form of P PIM P or ; while + signs are applied when the defect layer and its adjacent layers are different types of material. For symmetrical defective PCs, φ 1 φ 2 φ. 3. erivation of the frequency shift of the defect modes as incident angle changes Then we deduce the phase change φ from these quarter-wave multilayers. The matrix of a PC stack layer without absorption can be simplified 14 as M i Gi ±jz i ±j/z i G i at oblique incidence for small θ e, where G i 1 4 πsin θ e)/n i 2, and + fori P and fori, respectively. The product of the impedance matrices of individual layers may be represented: M m11 jm 12 M i. 4) jm i 21 m 22 Suppose wave from the defect layer incident to four types of PC stacks listed in Table 1, as the refractive indices of most materials are not close to zero, M can be evaluated by using the simplified form Eq. 3) and omitting higher-order terms in G i, and is also listed in Table 1. It has shown that in the limit Zr s ZH i /Zi L)s > 15 the layer with Zi H is of higher wave impedance than that with Zi L ) 15,17, the phase change φ attains limiting value for the case of the incident angle scanning from 0toθ e : φν, θ e ) φν,θ e ) φ, 0) ) ν ν0 kπ + J sin 2 θ e ) ν kπ + J sin 2 θ e, where k and J are parameters described the phase change, is the frequency of the defect mode for normal incidence, ν is the frequency shift. For situation I that wave from the defect layer incident to the adjacent layer with Z H i 15: J I 2Z m 21 /m 11, k I gz / Zi H + Zi L ), 2) 3) 5) 6) 7)

3 448 Y.H. Chen et al. / Physics Letters A ) Table 1 The product M and the expressions of J for different types of PC stacks Types of quarterwave stacks M Z P >Z P ) s Z r ) x j Z P G x 1 P s0 Z 1 r + Z G x 1 s0 Z r) x 2s j GP x 1 Z P s0 Z r Z s0 1 Z r ) x 2s 1 Z r ) x Z >Z P P ) s Z r ) x j Z P G x 1 P s0 Z r + Z G x 1 s0 Z 1 r ) x 2s j GP x 1 Z P s0 1 Z r Z s0 Z r ) x 2s 1 Z r ) x Z P <Z P ) s 1 Z r ) x j G P Z x 1 P s0 Z r + G Z x 1 s0 1 Z r ) x 2s j GP x 1 Z P s0 Z 1 r Z s0 Z r ) x 2s Z r ) x Z <Z P P ) s 1 Z r ) x j G P Z x 1 P s0 1 Z r + G Z x 1 s0 Z r) x 2s j GP x 1 Z P s0 Z r Z s0 Z 1 r ) x 2s Z r ) x J π 2 π 2 Z 1 Zr 2 1) n 2 Z + Z ) P n 2 P Z2 Z 1 Zr 2 1) n 2 P Z P Z ) n 2 Z2 P π 1 Z 2 Z Zr 2 1) n 2 + Z2 ) n 2 P Z P π 1 ZP 2 Z Zr 2 1) n 2 Z2 ) P P n 2 Z and for situation II that incident to the adjacent layer with Z L i : J II 2m 12 /m 22 Z, k II Zi H Zi L /gz Z H i + Zi L ), where g equals +1 or 1, respectively, corresponds to PIM or defect. Then combining product M in Table 1 and Eqs. 6) and 8), we get the expressions of parameter J listed in the fourth column of Table 1. Substituting Eq. 5) into Eq. 2) and assuming that θ e is small, one has δν, θ e ) δν,θ e ) δ, 0) ±2 φν,θ e ) φ, 0) + 2gπcos φ 1) ± k j π ν ) + 2J j sin 2 θ e gπ 10) n 2 sin 2 θ e, where j I, II. For the mth order of defect mode, δν,θ e ) δ, 0) 2mπ, Eq.10) can be written as ±2J j gπ ) n 2 sin 2 θ e, ±2k j π ν 8) 9) 11) where the sign + or is decided from Eq. 2). Positive or negative type of dispersion of the defect mode corresponds to positive or negative value of ν. When ν 0, the near-zero dispersion emerges. 4. ispersion properties of the defect modes inside the zero- n gap From Eq. 11), one finds that the frequency shift of the defect mode is determined by parameters of k j, J j and n. We can obtain such parameters from both the refractive index and the impedance of the component layers of the photonic crystals, and demonstrate the dispersion types of the defect modes inside the zero- n gap from the frequency shift. For situations when µ i 1 Z i 1/ n i ), some special characteristics of the dispersion of the defect modes are found. We first consider system A with the form of P ) s P) s when the adjacent layers of the defect are low refractive index materials n P < n, Z P >Z ). For a PIM defect, it is given by Table 1 and Eq. 7) that J A I π 2 1 n n /n P ) 2 1 n n 2 + n 2 n 2 p n )>0, ka 1 P I n 1/n P +1/n P ) > 0. By substituted J A I and ki A into Eq. 11), it can be obtained that ν/ ) 2JI A π/n 2 ) sin2 θ e / ki Aπ), as n changes, ν is always positive. Similarly, for a defect, it can be obtained that JI A > 0 and ki A < 0, Eq. 11) should be written as ν/ 2JI A +π/n 2 ) sin2 θ e /ki Aπ).So ν is always negative as n changes. Then we consider system A when the adjacent layers of the defect are high refractive index materials n P > n,z P < Z ). For a PIM defect, it is given by Table 1 and Eq. 9) that JII A π n 2 n P /n ) n 2 n + n P )<0, and k n 2 P n2 II A n /n P n ) 1/n P +1/ n ) > 0. By inserting J A II and kii A into Eq. 11), it can be obtained that ν/ ) 2JII A + π/n2 ) sin2 θ e /kii Aπ).As n increases, ν changes from positive value to zero to negative value. n can be found to satisfied the condition that ν 0. Similarly, for a defect, it is obtained that JII A < 0, ka II < 0, and ν/ ) 2JII A π/n2 ) sin2 θ e / kii Aπ).As n changes, ν can also be positive, negative or zero, and the three types of dispersion also appear. Similarly, we also analyze the dispersion properties of the defect modes in system B with the form of P ) s P) s and the results are listed in Table Simulations In order to simulate the dispersion properties of the defect modes and prove the theoretical results, the dispersion relation diagrams for these 1 defective PCs with the zero- n gap are given in this section. Let a wave be incident from vacuum onto the PC. The dispersion relation ωκ) of a perfect 1 PC have been obtain as follow 18, 2 cos κd Tr T P ω), 12) where T P ω) is a unitary 2 2 transfer matrix links the electric fields of waves propagating forward and backward in the same layer, and κ is the Bloch wave number. Solutions of the infinite system can be propagating or evanescent, corresponding to the real or imaginary Bloch wave numbers, respectively. The band

PIM Positive egative n <n P P ) s P ) s P) s PIM egative Positive n P >n P ) s P ) s P) s PIM Positive egative n >n P P ) s P ) s P) s PIM ear-zero Fig. 1.

4 Y.H. Chen et al. / Physics Letters A ) Table 2 The dispersion types for different PC structures in situations when µ i 1 Types of stacks PC structures Type of dispersion n P <n P ) s P ) s P) s PIM Positive egative n <n P P ) s P ) s P) s PIM egative Positive n P >n P ) s P ) s P) s PIM Positive egative n >n P P ) s P ) s P) s PIM ear-zero Fig. 1. ispersion relations of defect modes in PC structure of P ) s P) s. The parameters are n P 1.5, n 3 with a thickness ratio of d P /d 3/1.5. The sign circlet, plus and dot represent the defect modes corresponding to the defect layers with a) n 1.5, 3, 5, and b) n 1.5, 3, 5, respectively. structure of the perfect 1 PC can be obtained from the solutions of Eq. 12). When the defect is introduced, an additional real Bloch wave can be obtained as sinhiκ d) TrT P ω)t Q ω) 1 2 TrT P ω) TrT Q ω) TrT Q, ω) 13) where T Q ω) describes the relation between the amplitudes of the waves in the defect layer and its adjacent layers. The solution of Eq. 13) yields the dispersion relation of the defect modes. The dispersion relations of the defect modes can be calculated from Eqs. 12) and 13). The dispersion relation diagrams of system A in situations of n P < n are shown in Fig. 1, in which we let µ P µ µ 1, n P 1.5, n 3. The gray areas represent the regions of propagating states, whereas the white areas represent regions containing evanescent states. The two oblique solid lines are light line corresponding to the incidence with an angle of 90. The sign circlet, plus and dot denote the defect modes corresponding to the defect layers with a) n 1.5, 3, 5, and b) n 1.5, 3, 5, respectively. In Fig. 1a), the dispersions of the defect modes are always positive type, in accordance with the theoretical prediction. It is also noted from Fig. 1a) that, with increasing n, the angle dependence of the defect mode tends to be weakened. This phenomena can also be understood from Eq. 11) and Table 1. AsJI A 1/ n ) and ν 2JI A + π/n 2 ) sin2 θ e, ν is in inverse proportion to n. Therefore, for a larger n, ν is smaller and then the angle dependence of the defect mode is weaker. It can be Fig. 2. ispersion relations of defect modes in structure of P ) s P) s with n P 3, n 1.5 andd P /d 1.5/3. The sign circlet, plus and dot represent the defect modes corresponding to the defect layers with n 1.5, 2.7, 4, respectively. seen from Fig. 1b) that, the defect modes possess only negative type of dispersion when the defect layers are s, which is the same as the theoretical prediction. Fig. 2 shows the dispersion relations of the defect modes of system A in situations of n P > n, in which we let µ P µ µ 1, n P 3, n 1.5 and we get n 2.72 from Eq. 11) for the condition of ν 0. As shown in Fig. 2, in frequency range from πc/a to πc/a,the dispersion of the defect mode changes from positive to nearzero to negative type as n increases from 1.5 to 2.72 to 4 and is the same as theoretical prediction.

5 450 Y.H. Chen et al. / Physics Letters A ) Moreover, in our calculation, it is found that the dispersion relations of system A and B are the same when both Zi A Zi B and n A i n B i is satisfied. Zi A, ZB i, na i, nb i are the impedance and the refractive index of the ith layer in system A or B, respectively), this phenomenon can also be understood by the theories in Sections 2 and 3. FromEqs.2), 7), 9) and 11), the frequency shift for system A can be written as ν A 2Jj A + π ) sin 2 θ n 2 e kj Aπ, 14) where j I, II. Similarly, for system B we have ν B 2Jj B π ) sin 2 θ n 2 e kj Bπ. 15) When Zi B Zi A and n B i n A i, it can be obtained from Table 1, Eqs.7) and 11) that, Jj B Jj A, kb j ka j,so ν B ν A and the dispersion of the defect modes in system A and B are the same. According to such dispersion invariability between system A and B, the dispersion relations of system B can be easily obtained from that of system A. It is noted that all that have been made are dispersive. Then, we suppose that the effective permittivity and permeability in the are given by Fig. 3. The transmission spectra of a defective PC with structure of P ) s P) s at θ e 0 and θ e 30 for a) TE waves and b) TM waves. Structure parameters: n P 3, d P 10 mm, ε ν) 1 ωep 2 /2πν)2, µ ν) 1 ωmp 2 /2πν)2, ω ep 28.3 GHz, ω mp 22.2 GHz, d 20 mm, n 2.72,d 22 mm and s 8. ε 1 ω2 ep ω 2 1 ω2 ep 2πν) 2, µ 1 ω2 mp ω 2 1 ω2 mp 2πν) 2, 16) 17) where ω ep, ω mp are, respectively, the magnetic plasma frequency and the electronic plasma frequency. In Eqs. 16) and 17), ω ep, ω mp, ω and ν are the frequency measured in GHz. The dispersion can be realized in a composite made of periodically L C loaded transmission lines with good microwave properties In the following calculation, we choose d P 10 mm, n P 3,µ P 1, d 20 mm, ω ep 28.3 GHz, ω mp 22.2 GHz. It can be obtained that the character frequency of the quarter-wave stack is c/λ 0 c/4n P d P / ) Hz 2.5 GHz, and ε ) 2.25, µ ) 1, n ) 1.5. The transmission spectra of the PC with structure of P ) s P) s at the incident angles θ e 0 and θ e 30 is shown in Fig. 3, in which we let n 2.72, µ 1, d 2n P d P /n 22 mm and s 8. It can be seen from Fig. 3 that the frequency of the defect mode at normal incidence is 2.5 GHz), and the defect mode remains nearly invariant as the incident angle changes. The dependence of the defect modes on the incident angles for different n is shown in Fig. 4.AsshowninFig. 4, the dispersion relations of the defect modes change from positive to near-zero then to negative types as n changes from 1.5 to 2.72 and onwards to 4. The unique properties of the defect modes still exist when the dispersion of the is considered. Fig. 4. The dependence of the defect modes on the incident angle for different polarizations. The sign circlet, plus and square represent the defect modes corresponding to the defect layers with n 1.5, 2.72, 4, respectively. The other parameters are the same as those in Fig Conclusion In this Letter, we have derived a simple expression to demonstrate the dispersion properties of the defect modes inside the zero- n gap by using the defect mode resonant condition of 1 defective PC stacking with alternate PIMs and s. It is found from the analysis of this expression that, for situations when the permeabilities µ i 1, as the adjacent layers of the defect are low refractive index materials, only positive or negative type of dispersion exists; as the adjacent layers of the defect are high refractive index materials, all three types of dispersion, including the near-zero dispersion, appear as the refractive index of the defect changes. Then, the dispersion relation diagrams of

6 Y.H. Chen et al. / Physics Letters A ) such defective PCs are presented. The simulated results agree with the theoretical ones. Moreover, dispersion invariability is found in our simulations. The derived expression is useful in the designing of omnidirectional filters and narrow frequency sharp angular filters. Acknowledgements Work is supported by the ational atural Science Foundation of China ), ational CB719804) Project of China and ational AA311022) Project of China, and the atural Science Foundation of Guangdong Province of China. References 1 E. Yablonovitch, Phys. Rev. Lett ) S. John, Phys. Rev. Lett ) J.C. Knight, J. Broeng, T.A. Birks, P.St.J. Russell, Science ) C.M. Bowden, J.P. owling, H.O. Everitt, J. Opt. Soc. Am. B ) Q. Qin, H. Lu, S.. Zhu, C.S. Yuan, Y.Y. Zhu,.B. Ming, Appl. Phys. Lett ) Z.S. Wang, L. Wang, Y.G. Wu, L.Y. Chen, Appl. Phys. Lett ) V.G. Veselago, Sov. Phys. Usp ) J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Phys. Rev. Lett ) J.B. Pendry, A.J. Holden,.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech ) R.A. Shelby,.R. Smith, S. Schultz, Science ) J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett ) J. Li,.G. Zhao, Z.Y. Liu, Phys. Lett. A ) H.T. Jiang, H. Chen, H.Q. Li, Y.W. Zhang, Appl. Phys. Lett ) G.Q. Liang, P. Han, H.Z. Wang, Opt. Lett ) C.R. Pidgeon, S.. Smith, J. Opt. Soc. Am ) H.A. Macleod, Thin-Film Optical Filters, Adam Hilger, Philadelphia, J.S. Seeley, J. Opt. Soc. Am ) Y. Fink, J.. Winn, S. Fan, C. Chen, J. Michel, J.. Joannopoulos, E.L. Thomas, Science ) H. ĕmec, P. Kužel, J. Opt. Soc. Am. B ) H.Y. Sang, Z.Y. Li, B.Y. Gu, Phys. Lett. A ) K.Y. Xu, X.G. Zheng, W.L. She, Appl. Phys. Lett ) G.V. Eleftheriades, A.K. Iyer, P.C. Kremer, IEEE Trans. Microwave Theory Tech ) A. Grbic, G.V. Eleftheriades, J. Appl. Phys ) L. Liu, C. Caloz, C.C. Chang, T. Itoh, J. Appl. Phys ) 5560.

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