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Characteristics of band structures in 1D photonic crystals containing alternate left–right handed materials
Solid State Communications 136 (2005) 495–498 www.elsevier.com/locate/ssc
Characteristics of band structures in 1D photonic crystals containing alternate left–right handed materials Zhicheng Yea, Jun Zhengb, Zhaona Wangc, Dahe Liuc,* a
Institute of Semiconductor, Chinese Academy of Sciences, Beijing 100083, China b Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China c Physics Department, Beijing Normal University, Beijing 100875, China Received 27 July 2005; accepted 21 September 2005 by A. Pinczuk Available online 10 October 2005
Abstract This work investigated analytically the band structure of photonic crystals (PCs) with alternate layers of left and right-handed materials in one-dimension. It was found that, under certain conditions, new peculiar band structures not seen in all righthanded material PCs appeared. We transformed the analytic dispersion relation into two cosine terms, and obtained an interesting band structure using the new form of dispersion equation. Conditions for obtaining such peculiar band structure were given. q 2005 Elsevier Ltd. All rights reserved. PACS: 42.70.Qs; 41.20.Jb; 71.20.Tx Keywords: A. Left handed materials; D. Photonic band gaps
Recently, preparation of Left handed material [1] attracted great interest. It has already been made in microwave band though some different opinions exist [2–4]. Fabrication of such kind of material has been achieved in experiments [5] and the results are in good agreement with theoretic suggestions [1]. Since photonic crystals (PCs) without negative index have already aroused so much surprises by its capability of realizing negative refraction [6–8], formation of alternate right/left handed material PC has been investigated intensively and this kind of PC has shown to possess some new characters such as ‘zero-hni stop band’, ‘discrete solution’ and large stop band [9–13]. The ‘zero-hni stop band’ is insensitive to change of scale and incident angle and
* Corresponding author. E-mail address: [email protected] (D. Liu).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.09.027
magnitude of disorder, so the PC with alternate right/left handed material is promising in applications using filter or reflector for a range of wide incident angle. But the condition for obtaining those new band structures has not yet been studied adequately and need further investigations. In the present work, the dispersion relation was expressed in a new way and the conditions for obtaining ‘zero-order stop band’ which is just above zero frequency and ‘discrete solutions’ are derived using this new expression. In addition, how the band structure is related to the dispersion relation was also described and some new band structures were shown. Considering that, a monochromatic plane electromagnetic wave Ey eiðbxCk1 zKutÞ (E-polarization) or Hy eiðbxCk1 zKutÞ (H-polarization) incident on an 1D photonic crystal (Fig. 1). Using Bloch theorem, the analytical dispersion relation [14] in the periodic dielectric structure can be derived as:
496
Z. Ye et al. / Solid State Communications 136 (2005) 495–498 x
1 a ZG 2 0
1(R) 2(L)
1(R)
2(L)
1(R)
2(L)
…
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 32 n1 sin q 31 n22 Kn21 sin2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ; 32 n1 sin q 31 n22 Kn21 sin2 q
and incident light z 1st period
2nd period
3rd period
…
Fig. 1. An 1D PC with alternate right/left-handed material: R(L) represents right(left)-handed material.
cosðKTÞ Z cosðk1 t1 Þcosðk2 t2 Þ 1 1 K g C sinðk1 t1 Þsinðk2 t2 Þ 2 g cosðKTÞ Z cosðk1 t1 Þcosðk2 t2 Þ 1 1 K g 0 C 0 sinðk1 t1 Þsinðk2 t2 Þ 2 g
(1)
(2)
Eqs. (1) and (2) represent the relations of E-polarization and H-polarization, respectively. ki (iZ1, 2 representing different layers in one period) corresponds to the wave number of the light propagating in layer i along z axis pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and obey ki ZG ðu2 3i mi =c2 ÞKb2 (c is the speed of light in vacuum, 3i and mi are the permittivity and permeability of the material in corresponding layers, respectively. b is the wave number across (along) x-axis. When 3i and mi are all negative, ki is negative too). ti is the thickness of the corresponding layer. T is the lattice constant of the photonic crystal. K is Bloch wave number, and gZk1m2/ (k2m1), g 0 Zk132/(k231). For simplicity, some transformations of the above equations are made. According to the pffiffiffiffiffiffiffiffi definition of refractive index, ni ZG 3i mi (the sign ‘K’ represents left-handed material) and assuming that b in the first layer is k0n1sin q (k0Zu/c, is the wave number in vacuum), we have: 1 Ca k T cos 0 2pp1 cosðKTÞ Z 2 2p 1Ka k T cos 0 2pp2 (3) C 2 2p 1 C a0 k0 T 2pp1 cosðKTÞ Z cos 2p 2 1Ka 0 k T cos 0 2pp2 C 2 2p Here, respectively, a and a 0 are a ZG
1 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! m2 n1 cos q m1 n22 Kn21 sin2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C m2 n1 cos q m1 n22 Kn21 sin2 q
(4)
p1 Z
t1 t n cos qG 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q;
p2 Z
t1 t n cos qH 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q:
(the sign ‘C’of G and the sign ‘K’ of H in the above equations represent the case of PC with all right handed material and the cases of PC with alternate right/left-handed material, respectively) k0T/2p is the normalized frequency. Using the above equations, the difference between the two kinds of PCs can be seen obviously. As an example, let us consider the case of E-polarization expressed by Eq. (3). The right hand side of Eq. (3) can be taken as the sum of the two terms: ðð1C aÞ=2Þcosððk0 T=2pÞ2pp1 Þ and ðð1KaÞ=2Þcosððk0 T=2pÞ2pp2 Þ. a is not less than 1, jð1C aÞ=2jKjð1KaÞ=2jZ 1. For the PC with all right-handed material, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t p1 Z 1 n1 cos q C 2 n22 Kn21 sin2 q T T and p2 Z
t1 t n cos qK 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q:
Actually, j1/p1j and j1/p2j represent the periods of the two terms when the normalized frequency k0T/2p is taken as an independent variable. Thus, the right-hand side of Eq. (3) can be expressed as a term with larger amplitude and shorter period plus another term with smaller amplitude and longer period. On the contrary, for the PC laid by alternate right/lefthanded materials, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t p1 Z 1 n1 cos qK 2 n22 Kn21 sin2 q ; T T and p2 Z
t1 t n cos q C 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q :
It means that the term with larger amplitude has longer period. And it is obvious that, in either cases mentioned above, the band structures will be periodic which should satisfy 1 1 (5) m Z n Z p; m; n Z 1; 2; 3;/ p1 p2 Here p is the period of band structures. The physical picture of the band structures can be shown clearly by Eq. (3). The following two significant characteristics of the band structures should be addressed:
Z. Ye et al. / Solid State Communications 136 (2005) 495–498
1. Because of the joint effect of the two cosine terms, the band structures are periodic, and their profiles are determined mainly by the term that changes more quickly (i.e. with shorter period). Furthermore, the value of the cosine function is %1, so, for the PCs laid by alternate right/left-handed materials, the profiles of the band structures are mainly determined by the cosine term with smaller amplitude, but for the PCs with all right handed material the profiles of band structures are mainly determined by the cosine term with larger amplitude. Therefore, the band structures of PCs laid by alternate right/left-handed materials will possess abundant profiles, if parameters are chosen properly, the two cosine terms will have appropriate ratio of period and amplitude to make some new band structures appear, such as the band similar to the case of doping impurity into the materials, or the band with ellipse or wave like profile. In Fig. 2, the band structures of four PCs are shown, they have same parameters: 31Z2.25, 32ZK1, m1Z1, m2ZK1, like SiO2/LHM, but with different values of normalized thickness (t2/T) of 0.500, 0.580, 0.588 and 0.700, respectively. The abscissa of the figures is the normalized Blochwave number (KT/2p), the ordinate is the normalized frequency (k0T/2p). In Fig. 2(a) and (d), there are no particular and new profiles in the band structures, they just like that of PCs made by all right-handed material,
497
but, the structures are periodic as predicted by Eq. (3). However, for the PCs laid by alternate right/left handed materials new profiles like wave of the band structures appear in Fig. 2(b) and (c). They all have ‘zero-order’ band gaps, and have inflexion in the profiles caused by the fast changing cosine term with smaller amplitude as mentioned above. Furthermore, the circular and semi circular band gaps could be seen. For example, in Fig. 2(b), there are circular shape bands centering at 0.9, 19.8 and 20.7 and the semi cases circular ones are at 9.5 and 10.3. Comparing Fig. 2(b) with (c), it is easy to see that, the position and the width of the two kinds of circle bands can be adjusted by changing the ratio between layer thickness of the right and the left handed materials. 2. From Eq. (3), the condition for obtaining ‘zero-order stop-band’ can be derived easily. The solution at uZ0 (corresponding to KZ0) always exists. When the value of the term ð1C a=2Þcosððk0 T=2pÞ2pp1 ÞC ð1Ka=2Þcosð ðk0 T=2pÞ2pp2 Þ increases with the increase of u, ‘zeroorder stop-band’ will appear. Taking (k0T/2p) as the independent parameter and having the first order approximation, the condition for the existence of ‘zeroorder stop-band’ can be expressed as: rffiffiffiffiffiffiffiffiffiffiffiffi 1 Ca jp j jp2 jO (6) aK1 1 It is obvious that Eq. (6) is invalid for the PC with all right handed material. If other parameters are known, the range of the normalized thickness (t2/T) can be determined in which ‘zero-order stop-band’ will exist. For example, if the above example is chosen, by substituting the parameters into Eq. (6) and the case of normal incidence is considered, the ‘zero-order stopband’ will exist in the range of 0.5000!t2/T!0.6923. In order to check the validity of Eq. (6), the band structures were calculated by Eq. (3) and the range of relative
Fig. 2. Band structures of four different PCs laid by alternate right/left handed materials.
Fig. 3. ‘Zero-order stop-band’ width with respect to relative thickness of left- handed material. The ordinate represents normalized band width; the abscissa is the relative thickness of left-handed material. The largest bandwidth just corresponds to p1Z0 at t2/TZ0.6.
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Z. Ye et al. / Solid State Communications 136 (2005) 495–498
phase of the two adjacent layers are reversed, there is only a series of discrete solutions when p2Z2mp (k1t1C jk2jt2Z2mp). Under this condition, the width of the ‘zero-order stop-band’ will reach its maximum (the peak in Fig. 3).
Fig. 4. Dispersion relationship (a) and band structure (b) for the relative thickness case of 0.5625.
thickness for ‘zero-order stop-band’ was plotted in Fig. 3. It can be seen clearly that the two results are in good agreement. Under the above condition and periodic property of the band structures, there will be discrete solutions in the band structures corresponding to uZ0. Using Eq. (5) the relative thickness t2/T that supports discrete solutions is easy to get. When all the parameters are substituted into Eq. (5), it is transformed into: 1 1 Z p; m Z n 1:5K2:5ðt2 =TÞ 1:5K0:5ðt2 =TÞ
Finally, all the above discussion can be summarized as follows. The band structures of both the two kinds of PCs are periodic with the value satisfying Eq. (5), and the profile of the band structures is mainly determined by the term that changes more quickly. ‘zero-order stop-band’ will appear when Eq. (6) is satisfied, and the maximum of the width of ‘zero-order stop-band’ will be obtained when p1Z0. Under the above conditions, the discrete solution will be found in each period. Because the band structures with higher frequencies are more sensitive to disorders, the refractive index and the thickness of every layer should be chosen carefully so that discrete solutions with low frequency could be obtained.
Acknowledgement The authors would like to thank National Natural Science Foundation of China for financial support (grant No. 60277014). Also, we thank Prof Xiangdong Zhang and Peide Han for helpful discussions.
m; n Z 1; 2; 3;/ Here we select the solution 0.5625 and substitute all the parameters into Eq. (3). Then the dispersion relation is cosðKTÞ Z
4:25 1:25 cosðx2p !0:09375ÞK ðx2p 3 3 !1:21875Þ
(7)
Here x is the normalized frequency. The periods of the first and second terms are 1/0.09375 and 1/1.21875, respectively, so the total period is 13 !ð1=1:21875ÞZ 1=0:09375. The discrete points appeared at the middle between the two band gaps every other period, which is 1/0.09375. In Fig. 4, the discrete solutions at frequencies 10.67 and 21.32 approximately correspond to zero frequency predicted by Eq. (7). While it happened that this structure has other discrete solutions at 5.33 and 16.67 with the same period, and this can be obtained using the same analysis as above. Although, these values are approximate, we can see that from Fig. 4(a) they do exit. 3. From Eq. (3), for the PCs laid by alternate right/left handed materials, when p1Z0 (k1t1Kjk2jt2Z0), i.e. the
References [1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S.S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz, Appl. Phys. Lett. 78 (2001) 489. [4] J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Phys. Rev. Lett. 76 (1996) 4773. [5] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. [6] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [7] J.D. Joannopoulous, R.D. Meade, J.N. Winn, Photonic Crystals: Modeling the Flow of Light, first ed, Princeton University Press, Princeton, NJ, 1995. [8] H. Kosaka, T. Kawashima, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) R10096. [9] I.S. Nefedov, S.A. Tretyakov, Phys. Rev. E 66 (2002) 036611. [10] H. Jiang, H. Chen, H. Li, Y. Zhang, S. Zhu, Appl. Phys. Lett. 83 (2003) 5386. [11] J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett. 90 (2003) 083901. [12] L. Wu, S. He, L. Shen, Phys. Rev. B 67 (2003) 235103. [13] D. Bria, et al., Phys. Rev. E 69 (2004) 066613. [14] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984.
Characteristics of band structures in 1D photonic crystals containing alternate left–right handed materials Zhicheng Yea, Jun Zhengb, Zhaona Wangc, Dahe Liuc,* a
Institute of Semiconductor, Chinese Academy of Sciences, Beijing 100083, China b Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China c Physics Department, Beijing Normal University, Beijing 100875, China Received 27 July 2005; accepted 21 September 2005 by A. Pinczuk Available online 10 October 2005
Abstract This work investigated analytically the band structure of photonic crystals (PCs) with alternate layers of left and right-handed materials in one-dimension. It was found that, under certain conditions, new peculiar band structures not seen in all righthanded material PCs appeared. We transformed the analytic dispersion relation into two cosine terms, and obtained an interesting band structure using the new form of dispersion equation. Conditions for obtaining such peculiar band structure were given. q 2005 Elsevier Ltd. All rights reserved. PACS: 42.70.Qs; 41.20.Jb; 71.20.Tx Keywords: A. Left handed materials; D. Photonic band gaps
Recently, preparation of Left handed material [1] attracted great interest. It has already been made in microwave band though some different opinions exist [2–4]. Fabrication of such kind of material has been achieved in experiments [5] and the results are in good agreement with theoretic suggestions [1]. Since photonic crystals (PCs) without negative index have already aroused so much surprises by its capability of realizing negative refraction [6–8], formation of alternate right/left handed material PC has been investigated intensively and this kind of PC has shown to possess some new characters such as ‘zero-hni stop band’, ‘discrete solution’ and large stop band [9–13]. The ‘zero-hni stop band’ is insensitive to change of scale and incident angle and
* Corresponding author. E-mail address: [email protected] (D. Liu).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.09.027
magnitude of disorder, so the PC with alternate right/left handed material is promising in applications using filter or reflector for a range of wide incident angle. But the condition for obtaining those new band structures has not yet been studied adequately and need further investigations. In the present work, the dispersion relation was expressed in a new way and the conditions for obtaining ‘zero-order stop band’ which is just above zero frequency and ‘discrete solutions’ are derived using this new expression. In addition, how the band structure is related to the dispersion relation was also described and some new band structures were shown. Considering that, a monochromatic plane electromagnetic wave Ey eiðbxCk1 zKutÞ (E-polarization) or Hy eiðbxCk1 zKutÞ (H-polarization) incident on an 1D photonic crystal (Fig. 1). Using Bloch theorem, the analytical dispersion relation [14] in the periodic dielectric structure can be derived as:
496
Z. Ye et al. / Solid State Communications 136 (2005) 495–498 x
1 a ZG 2 0
1(R) 2(L)
1(R)
2(L)
1(R)
2(L)
…
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 32 n1 sin q 31 n22 Kn21 sin2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ; 32 n1 sin q 31 n22 Kn21 sin2 q
and incident light z 1st period
2nd period
3rd period
…
Fig. 1. An 1D PC with alternate right/left-handed material: R(L) represents right(left)-handed material.
cosðKTÞ Z cosðk1 t1 Þcosðk2 t2 Þ 1 1 K g C sinðk1 t1 Þsinðk2 t2 Þ 2 g cosðKTÞ Z cosðk1 t1 Þcosðk2 t2 Þ 1 1 K g 0 C 0 sinðk1 t1 Þsinðk2 t2 Þ 2 g
(1)
(2)
Eqs. (1) and (2) represent the relations of E-polarization and H-polarization, respectively. ki (iZ1, 2 representing different layers in one period) corresponds to the wave number of the light propagating in layer i along z axis pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and obey ki ZG ðu2 3i mi =c2 ÞKb2 (c is the speed of light in vacuum, 3i and mi are the permittivity and permeability of the material in corresponding layers, respectively. b is the wave number across (along) x-axis. When 3i and mi are all negative, ki is negative too). ti is the thickness of the corresponding layer. T is the lattice constant of the photonic crystal. K is Bloch wave number, and gZk1m2/ (k2m1), g 0 Zk132/(k231). For simplicity, some transformations of the above equations are made. According to the pffiffiffiffiffiffiffiffi definition of refractive index, ni ZG 3i mi (the sign ‘K’ represents left-handed material) and assuming that b in the first layer is k0n1sin q (k0Zu/c, is the wave number in vacuum), we have: 1 Ca k T cos 0 2pp1 cosðKTÞ Z 2 2p 1Ka k T cos 0 2pp2 (3) C 2 2p 1 C a0 k0 T 2pp1 cosðKTÞ Z cos 2p 2 1Ka 0 k T cos 0 2pp2 C 2 2p Here, respectively, a and a 0 are a ZG
1 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! m2 n1 cos q m1 n22 Kn21 sin2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C m2 n1 cos q m1 n22 Kn21 sin2 q
(4)
p1 Z
t1 t n cos qG 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q;
p2 Z
t1 t n cos qH 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q:
(the sign ‘C’of G and the sign ‘K’ of H in the above equations represent the case of PC with all right handed material and the cases of PC with alternate right/left-handed material, respectively) k0T/2p is the normalized frequency. Using the above equations, the difference between the two kinds of PCs can be seen obviously. As an example, let us consider the case of E-polarization expressed by Eq. (3). The right hand side of Eq. (3) can be taken as the sum of the two terms: ðð1C aÞ=2Þcosððk0 T=2pÞ2pp1 Þ and ðð1KaÞ=2Þcosððk0 T=2pÞ2pp2 Þ. a is not less than 1, jð1C aÞ=2jKjð1KaÞ=2jZ 1. For the PC with all right-handed material, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t p1 Z 1 n1 cos q C 2 n22 Kn21 sin2 q T T and p2 Z
t1 t n cos qK 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q:
Actually, j1/p1j and j1/p2j represent the periods of the two terms when the normalized frequency k0T/2p is taken as an independent variable. Thus, the right-hand side of Eq. (3) can be expressed as a term with larger amplitude and shorter period plus another term with smaller amplitude and longer period. On the contrary, for the PC laid by alternate right/lefthanded materials, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t p1 Z 1 n1 cos qK 2 n22 Kn21 sin2 q ; T T and p2 Z
t1 t n cos q C 2 T 1 T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 Kn21 sin2 q :
It means that the term with larger amplitude has longer period. And it is obvious that, in either cases mentioned above, the band structures will be periodic which should satisfy 1 1 (5) m Z n Z p; m; n Z 1; 2; 3;/ p1 p2 Here p is the period of band structures. The physical picture of the band structures can be shown clearly by Eq. (3). The following two significant characteristics of the band structures should be addressed:
Z. Ye et al. / Solid State Communications 136 (2005) 495–498
1. Because of the joint effect of the two cosine terms, the band structures are periodic, and their profiles are determined mainly by the term that changes more quickly (i.e. with shorter period). Furthermore, the value of the cosine function is %1, so, for the PCs laid by alternate right/left-handed materials, the profiles of the band structures are mainly determined by the cosine term with smaller amplitude, but for the PCs with all right handed material the profiles of band structures are mainly determined by the cosine term with larger amplitude. Therefore, the band structures of PCs laid by alternate right/left-handed materials will possess abundant profiles, if parameters are chosen properly, the two cosine terms will have appropriate ratio of period and amplitude to make some new band structures appear, such as the band similar to the case of doping impurity into the materials, or the band with ellipse or wave like profile. In Fig. 2, the band structures of four PCs are shown, they have same parameters: 31Z2.25, 32ZK1, m1Z1, m2ZK1, like SiO2/LHM, but with different values of normalized thickness (t2/T) of 0.500, 0.580, 0.588 and 0.700, respectively. The abscissa of the figures is the normalized Blochwave number (KT/2p), the ordinate is the normalized frequency (k0T/2p). In Fig. 2(a) and (d), there are no particular and new profiles in the band structures, they just like that of PCs made by all right-handed material,
497
but, the structures are periodic as predicted by Eq. (3). However, for the PCs laid by alternate right/left handed materials new profiles like wave of the band structures appear in Fig. 2(b) and (c). They all have ‘zero-order’ band gaps, and have inflexion in the profiles caused by the fast changing cosine term with smaller amplitude as mentioned above. Furthermore, the circular and semi circular band gaps could be seen. For example, in Fig. 2(b), there are circular shape bands centering at 0.9, 19.8 and 20.7 and the semi cases circular ones are at 9.5 and 10.3. Comparing Fig. 2(b) with (c), it is easy to see that, the position and the width of the two kinds of circle bands can be adjusted by changing the ratio between layer thickness of the right and the left handed materials. 2. From Eq. (3), the condition for obtaining ‘zero-order stop-band’ can be derived easily. The solution at uZ0 (corresponding to KZ0) always exists. When the value of the term ð1C a=2Þcosððk0 T=2pÞ2pp1 ÞC ð1Ka=2Þcosð ðk0 T=2pÞ2pp2 Þ increases with the increase of u, ‘zeroorder stop-band’ will appear. Taking (k0T/2p) as the independent parameter and having the first order approximation, the condition for the existence of ‘zeroorder stop-band’ can be expressed as: rffiffiffiffiffiffiffiffiffiffiffiffi 1 Ca jp j jp2 jO (6) aK1 1 It is obvious that Eq. (6) is invalid for the PC with all right handed material. If other parameters are known, the range of the normalized thickness (t2/T) can be determined in which ‘zero-order stop-band’ will exist. For example, if the above example is chosen, by substituting the parameters into Eq. (6) and the case of normal incidence is considered, the ‘zero-order stopband’ will exist in the range of 0.5000!t2/T!0.6923. In order to check the validity of Eq. (6), the band structures were calculated by Eq. (3) and the range of relative
Fig. 2. Band structures of four different PCs laid by alternate right/left handed materials.
Fig. 3. ‘Zero-order stop-band’ width with respect to relative thickness of left- handed material. The ordinate represents normalized band width; the abscissa is the relative thickness of left-handed material. The largest bandwidth just corresponds to p1Z0 at t2/TZ0.6.
498
Z. Ye et al. / Solid State Communications 136 (2005) 495–498
phase of the two adjacent layers are reversed, there is only a series of discrete solutions when p2Z2mp (k1t1C jk2jt2Z2mp). Under this condition, the width of the ‘zero-order stop-band’ will reach its maximum (the peak in Fig. 3).
Fig. 4. Dispersion relationship (a) and band structure (b) for the relative thickness case of 0.5625.
thickness for ‘zero-order stop-band’ was plotted in Fig. 3. It can be seen clearly that the two results are in good agreement. Under the above condition and periodic property of the band structures, there will be discrete solutions in the band structures corresponding to uZ0. Using Eq. (5) the relative thickness t2/T that supports discrete solutions is easy to get. When all the parameters are substituted into Eq. (5), it is transformed into: 1 1 Z p; m Z n 1:5K2:5ðt2 =TÞ 1:5K0:5ðt2 =TÞ
Finally, all the above discussion can be summarized as follows. The band structures of both the two kinds of PCs are periodic with the value satisfying Eq. (5), and the profile of the band structures is mainly determined by the term that changes more quickly. ‘zero-order stop-band’ will appear when Eq. (6) is satisfied, and the maximum of the width of ‘zero-order stop-band’ will be obtained when p1Z0. Under the above conditions, the discrete solution will be found in each period. Because the band structures with higher frequencies are more sensitive to disorders, the refractive index and the thickness of every layer should be chosen carefully so that discrete solutions with low frequency could be obtained.
Acknowledgement The authors would like to thank National Natural Science Foundation of China for financial support (grant No. 60277014). Also, we thank Prof Xiangdong Zhang and Peide Han for helpful discussions.
m; n Z 1; 2; 3;/ Here we select the solution 0.5625 and substitute all the parameters into Eq. (3). Then the dispersion relation is cosðKTÞ Z
4:25 1:25 cosðx2p !0:09375ÞK ðx2p 3 3 !1:21875Þ
(7)
Here x is the normalized frequency. The periods of the first and second terms are 1/0.09375 and 1/1.21875, respectively, so the total period is 13 !ð1=1:21875ÞZ 1=0:09375. The discrete points appeared at the middle between the two band gaps every other period, which is 1/0.09375. In Fig. 4, the discrete solutions at frequencies 10.67 and 21.32 approximately correspond to zero frequency predicted by Eq. (7). While it happened that this structure has other discrete solutions at 5.33 and 16.67 with the same period, and this can be obtained using the same analysis as above. Although, these values are approximate, we can see that from Fig. 4(a) they do exit. 3. From Eq. (3), for the PCs laid by alternate right/left handed materials, when p1Z0 (k1t1Kjk2jt2Z0), i.e. the
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