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Photonic band gap from a stack of single-negative materials
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Journal of Magnetism and Magnetic Materials 301 (2006) 371–377 www.elsevier.com/locate/jmmm
Photonic band gap from a stack of single-negative materials L. Gao, C.J. Tang, S.M. Wang Department of Physics, Suzhou University, Suzhou 215006, China Received 15 April 2005; received in revised form 3 July 2005 Available online 11 August 2005
Abstract We investigate the transmission gap of the multilayer structure consisting of alternate permittivity-negative material layer and permeability-negative material layer. It is found that for both normal and oblique incidences, there exists a transmission gap which is invariant with a change of scale length. In view of the fact that the layers are electrically thin, the effective permittivity eeff and permeability meff are introduced to study the transmission property of electromagnetic waves. Numerical results show that the transmission gap can be well explained by the total reflection (eeff meff eair mair sin2 yo0, where y is the incident angle). To one’s interest, the edges of the gap can be accurately determined by eeff meff eair mair sin2 y ¼ 0. In addition, we show that zero effective phase shift gap (zero-feff gap) and zero volume averaged refractive index gap (zero-¯n gap) can also be understood in a similar way. r 2005 Elsevier B.V. All rights reserved. Keywords: Photonic band gap; Single-negative material; Effective medium approximation
1. Introduction Single-negative material (SNM) is the medium of which the relative permittivity e (permeability m) is negative while the relative permeability (permittivity) is positive. For instance, the electromagnetic response of metals in the visible region and near ultraviolet is dominated by the negative permittivity. However, for ordinary metals, at GHz frequencies, the imaginary part of the dielectric constant is about a thousand times bigger than the real part, and hence the dielectric permittivity is Corresponding author. Tel./fax: +86 51265112597.
E-mail address: [email protected] (L. Gao).
essentially imaginary. Fortunately, Pendry et al. demonstrated that a very simple metallic microstructure comprising a regular array of thin wires exhibit negative permittivity at GHz frequencies [1]. Later, metamaterials with effective negative permeability in a particular frequency range have also been obtained by ultilizing split ring resonators [2]. After that, substantial progress has been made to investigate the electromagnetic properties of composite structure consisting of and epsilonnegative media (ENM) and mu-negative media (MNM). Scientists [3,4] adopted ENM and MNM to provide an easy way to construct left-handed material (LHM) [5]. The resonance, tunneling
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.07.010
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L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
and transparency of pairing an ENM slab with MNM slab was also systematically investigated [6]. More recently, the transmission properties of the one-dimensional photonic crystals consisting of alternate ENM layer and MNM layer were investigated [7,8]. A new type of photonic gap called as zero effective phase shift gap (zero-feff gap) was predicted [6]. Zero-feff gap was found to distinguish itself from the Bragg gap in that it is invariant with a change of scale length and insensitive to thickness fluctuation [7]. Moreover, the width of zero-feff gap is only dependent on the thickness ratio of ENM and MNM layers [7]. In Ref. [8], the authors showed a peculiar transmission band gap which is almost insensitive to the different polarizations and incident directions of the incident electromagnetic waves. In other words, the transmission band gap is found to be omnidirectional. The band gap is similar to another omnidirectional transmission band gap (called as zero-¯n gap) existing in the layered structure containing LHMs [9,10]. In this paper, we investigate the transmission gap of the multilayer structure consisting of alternate ENM and MNM layer. We would like to mention that the constitutive parameters used in Ref. [7] are quite special, since the relative permittivity (permeability) of ENM is equal to the relative permeability (permittivity) of MNM [7]. Here, ENM and MNM with more general constitutive parameters [11] are used to form a layered structure. With the help of transfer matrix method [12,13], we study the transmission properties of the layered structure. It is found that, for both normal and oblique incidences, there exists a transmission gap which is invariant with a change of scaling. Since SNM are generally realized in microwave frequency region, their layers are often pffiffiffiffiffi electrically thin, i.e., kd ¼ emod=c51. As a consequence, we adopt effective medium approximation by introducing the effective permittivity eeff and permeability meff to study the wave propagation in this multilayer structure. Numerical results show that the existence of the transmission band gap results from the total reflection (eeff meff eair mair sin2 yp0, where y is the incident angle). For normal incidence, zero effective phase gap (zero-feff gap) [7] and zero average
refractive index (zero-¯n gap) [9] were, respectively, predicted from a stack of alternate ENM and MNM [7] and a stack of alternate ordinary material and LHM [9]. To further verify the validity of our theory, we take one step forward to investigate the properties of these two gaps. Again, these gaps can be well determined by the total reflection condition eeff meff p0. To one’s interest, the edges of gaps are accurately described by eeff meff ¼ 0.
2. Model and theory Let us consider the multilayer structure consisting of two kinds of SNM. One kind of SNM is ENM with the relative permittivity and permeability in the form of ea ¼ 1
o2p ; oðo þ i=tÞ
ma ¼ 1,
(1)
where op is the plasma frequency, and t is a relaxation time. Such a form of permittivity is representative of a variety of metal composites, whose permittivity is negative with small imaginary part in the visible region and near ultraviolet, and is essentially imaginary at low frequencies. However, Pentry et al. demonstrated that a very simple metallic microstructure comprising a regular array of thin wires exhibits negative e even at GHz frequencies, described by Eq. (1). The other kind of SNM is MNM with the relative permittivity and permeability expressed as b ¼ 1;
mb ¼ 1
F o2 , o2 o20 þ ioG
(2)
where o0 stands for the magnetic resonance, G is the dissipation factor, and F is the fractional area of the unit cell occupied by the interior of the split ring [14]. For simplicity, in our analysis, both SNM and MNM are assumed lossless, that is, t ¼ 0 and G ¼ 0. Actually, such an assumption has been widely used in Refs. [6–8]. Here, we set the parameters to be op ¼ 10 GHz, o0 ¼ 4 GHz and F ¼ 0:56 [11]. Note that both a and mb will be negative simultaneously when o is in the range (4 and 6 GHz).
ARTICLE IN PRESS L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
It is well known that the dispersion relation for the plane electromagnetic waves propagating in an infinite one-dimensional photonic crystal can be obtained by Bloch-Floquent theorem. For transverse-electric (TE) polarized waves (transversemagnetic (TM) polarized waves can be discussed in a similar way), the dispersion relation is expressed as [15] cosðqdÞ ¼ cosðkaz d a Þ cosðkbz d a Þ 1 kaz =ma kbz =mb þ 2 kbz =mb kaz =ma sinðkaz d a Þ sinðkbz d b Þ,
layer structure is electrically thin rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 2 ðe m sin yÞ 51 d a a a c2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! o2 2 and d b ðeb mb sin yÞ51 . c2 In this case, Based on using transfer matrix method [12,13], we can derive effective medium approximation, in which the effective permittivity eeff and the effective permeability meff of the layered structure are given by [4]
ð3Þ
where q is the Bloch wave vector along the axis of photonic crystals, d d a þ d b is the period, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kjz ¼ ðo2 =c2 Þðej mj sin2 yÞ (y is the incident
meff ¼ eeff
angle, j ¼ a; b). For a given y, if kaz =ma is equal to kbz =mb , Eq. (3) will be simplified as cosðqdÞ ¼ cosðkaz d a kbz d b Þ.
(4)
As both kaz and kbz are imaginary numbers for SNM, we have cosðkaz d a kbz d b ÞX1.
da db ma þ mb , d d
da db da 1 db 1 2 ea þ eb sin y ¼ þ d d d m d mb ! a 1 þ sin2 y d , db a d ma þ d mb
ð6Þ
for TE polarization and eeff ¼
(5)
Except for kaz d a ¼ kbz d b , Eq. (3) has no real solution for Bloch wave vector q. As a result, a transmission gap is opened. In what follows, we will present the numerical results. The transmission coefficients of a multilayer with 200 periods are calculated, as shown in Fig. 1. For normal incidence (see Fig. 1(a)), it is shown that there exists a transmission gap appearing near 5:4 GHz. If the lattice period is changed but the thickness ratio is kept fixed, the position and width of the transmission gap are almost invariant. In other words, the transmission gap is invariant upon a change of scale length. Actually, such a conclusion can also be found for oblique incidence. However, the position and width of the transmission gap for TE wave (see Fig. 1(b)) are quite different from those of the transmission gap for TM wave (see Fig. 1(c)). Therefore, the transmission gap is sensitive to the polarization of the incident electromagnetic waves. It is well known that SNM is usually realized in the microwave frequency region [1,2]. In this regard, the thickness of every layer in the multi-
373
meff
da db ea þ eb , d d
da db da 1 db 1 2 m þ m sin y ¼ þ d a d b d ea d eb ! 1 þ sin2 y d db a d ea þ d eb
ð7Þ
for TM polarization. Eqs. (6) and (7) indicate that the multilayer structure is anisotropic in essence because eeff and meff depend on the incident angle y. Now, we shall display that the transmission gap given in Fig. 1 can be explained simply by total reflection. The total reflection condition is eeff meff eair mair sin2 yo0. In Fig. 2(a) and (b), we plotted the transmission coefficients and eeff meff eair mair sin2 y as a function of the incident frequency o, respectively. Comparing Fig. 2(a) with Fig. 2(b), we find that for both TE and TM waves, the transmission gap just lies in the frequency region where eeff meff eair mair sin2 y is smaller than zero. And the edges of the transmission gap are determined well by the condition eeff meff eair mair sin2 y ¼ 0. Therefore, in the layered structure consisting of two kinds of SNM
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0.0 4.0
4.5
5.0 ω (GHz)
5.5
6.0 4.0
4.5
5.0 ω (GHz)
5.5
6.0
Fig. 1. Transmission coefficients as a function of incident frequency o for (a). normal incidence y ¼ 0, (b) TE polarization with y ¼ 20 , and (c). TM polarization with y ¼ 20 . Solid line: d a ¼ 10 mm, d b ¼ 20 mm; dashed line: d a ¼ 15 mm, d b ¼ 30 mm; and dotted line: d a ¼ 20 mm, d b ¼ 40 mm.
with general physical parameters, there still exists a transmission gap, which can be explained simply as total reflection. Actually, in the transmission gap, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 o2 2 2 ðe m sin yÞ ðe m sin yÞ and d d a a b b a b c2 c2 are found to be so small that the introduction of the effective permittivity eeff and permeability meff is quite reasonable. According to our numerical calculations, the parameters used in Refs. [7,9] are also found to satisfy the condition that the layers are electrically
thin, especially in the frequency range of the zerofeff gap and zero-¯n gap. Hence, we would like to investigate zero-feff gap and zero-¯n gap as reported in Refs. [7,9]. For normal incidence, both eeff and meff are dependent only on the ratio of d a and d b , and the total reflection condition is simplified as eeff meff o0. The transmission coefficients (solid line in Fig. 2 in Ref. [7]) and eeff meff as a function of frequency are shown, respectively, in Fig. 3(a) and (b). From Fig. 3, it is evident that the frequency region of zero-feff band gap is completely consistent with the one in which eeff meff o0. To one’s interest, the left and right edges of zerofeff gap are well determined by eeff ¼ ðd a =dÞea þ
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L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
0.6
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0.2
375
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µeff εeff −µair εair sin2θ
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0.0
-0.5 0.00 (b)
-1.0
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-0.02 4.0
4.5
5.0 ω (GHz)
5.5
6.0
Fig. 2. Transmission coefficients and eeff meff eair mair sin2 y as a function of o for y ¼ 20 .
ðd b =dÞeb ¼ 0 and meff ¼ ðd a =dÞma þ ðd b =dÞmb ¼ 0. As a result, both the position and the width of zero-feff gap are only dependent on the ratio of d a and d b . Therefore, zero-feff gap can be controlled flexibly by the adjustment of the thickness ratio of layers. Due to this property, the application of the transmission gap becomes very convenient. In the end, the transmission coefficients (solid line in Fig. 2(c) in Ref. [9]) and eeff meff are plotted as a function of frequency in Fig. 4. Again, zero-¯n gap region just appears in the frequency range in which the total reflection condition eeff meff o0 is satisfied.
0.4
0.6
0.8 1.0 f (GHz)
1.2
1.4
Fig. 3. Transmission coefficients and eeff meff for zero-feff gap in Ref. [7].
And, like zero-feff gap, the left and right edges of zero-¯n gap are determined by eeff meff ¼ 0 too. In addition, the weak dependence of the two transmission gaps to thickness fluctuation can be understood easily within effective medium approximation [7,9] in virtue of the invariance of eeff and meff .
3. Conclusion and discussion In conclusion, we have investigated the transmission gap of the multilayer structure consisting
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376
0.1
duced to investigate the transmission gap. We find that the transmission gap is well explained by the total reflection condition eeff meff eair mair sin2 yo0. Furthermore, We discuss zero-feff gap and zero-¯n gap reported in previous works. We numerically show that the band edges of two gaps are described by a simple formula eeff meff ¼ 0. Here, some comments are in order. Since the properties of the photonic gaps are compact and robust against length scaling and disorder, it would be of great interest to investigate the guided wave in layered waveguide made of SNMs. Initial results show that there exists the omnidirectional guided mode, when ðd a =dÞea þ ðd b =dÞeb ¼ 0 or ðd a =dÞma þ ðd b =dÞmb ¼ 0 is satisfied. In addition, bright and dark gap solitons in LHM were studied more recently [16]. Therefore, we would like to take one step forward to investigate the bistability and soliton in one-dimensional photonic crystals containing SNMs.
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Acknowledgements
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0.0
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0.2
This work was supported by the National Natural Science Foundation of China under Grant no. 10204017 (L.G.) and the Natural Science of Jiangsu Province under Grant no. BK2002038 (L.G.).
-0.1
(b)
-0.2 1.5
2.0
2.5 f (GHz)
3.0
3.5
Fig. 4. Transmission coefficients and eeff meff for zero-¯n gap in Ref. [9].
of alternate two kinds of SNMs. It is shown that there exists a transmission gap which is invariant with a change of scale length. And, the transmission gap is found to be dependent on the polarization of the incident electromagnetic waves, leading to a polarization-dependent transmission gap. Since SNM is usually realized in the microwave frequency region, the thickness of every layer in the multilayer structure is electrically thin. As a result, the effective permittivity eeff and permeability meff of the multilayer structure are intro-
References [1] J.B. Pentry, A.J. Holden, W.J. Stewart, I. Youngs, Phys. Rev. Lett. 76 (1996) 4773. [2] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theor. Tech. 47 (1999) 2075. [3] D.R. Fredkin, A. Ron, Appl. Phys. Lett. 81 (2002) 1753. [4] A. Lakhtakia, C.M. Krowne, Optik 114 (2003) 305. [5] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [6] A. Alu`, N. Engheta, IEEE Trans. Antennas. Propagation 51 (2003) 2558. [7] H.T. Jiang, H. Chen, H.Q. Li, Y.W. Zhang, J. Zi, S.Y. Zhu, Phys. Rev. E. 69 (2004) 066607. [8] L.G. Wang, H. Chen, S.Y. Zhu, Phys. Rev. B. 70 (2004) 245102. [9] J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett. 90 (2003) 083901. [10] H.T. Jiang, H. Chen, H.Q. Li, Y.W. Zhang, S.Y. Zhu, Appl. Phys. Lett. 83 (2003) 5386. [11] R. Ruppin, J. Phys.: Condens. Matter 13 (2001) 1811.
ARTICLE IN PRESS L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377 [12] Z.M. Zhang, C.J. Fu, Appl. Phys. Lett. 80 (2002) 1097. [13] L. Gao, C.J. Tang, Phys. Lett. A 322 (2004) 390. [14] D.R. Smith, W. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184.
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[15] P. Sheng, R.S. Stepleman, P.N. Sanda, Phys. Rev. B 26 (1982) 2907. [16] G. D’Aguanno, N. Mattiucci, M. Scalora, M.J. Bloemer, Phys. Rev. Lett. 93 (2004) 213902.
Journal of Magnetism and Magnetic Materials 301 (2006) 371–377 www.elsevier.com/locate/jmmm
Photonic band gap from a stack of single-negative materials L. Gao, C.J. Tang, S.M. Wang Department of Physics, Suzhou University, Suzhou 215006, China Received 15 April 2005; received in revised form 3 July 2005 Available online 11 August 2005
Abstract We investigate the transmission gap of the multilayer structure consisting of alternate permittivity-negative material layer and permeability-negative material layer. It is found that for both normal and oblique incidences, there exists a transmission gap which is invariant with a change of scale length. In view of the fact that the layers are electrically thin, the effective permittivity eeff and permeability meff are introduced to study the transmission property of electromagnetic waves. Numerical results show that the transmission gap can be well explained by the total reflection (eeff meff eair mair sin2 yo0, where y is the incident angle). To one’s interest, the edges of the gap can be accurately determined by eeff meff eair mair sin2 y ¼ 0. In addition, we show that zero effective phase shift gap (zero-feff gap) and zero volume averaged refractive index gap (zero-¯n gap) can also be understood in a similar way. r 2005 Elsevier B.V. All rights reserved. Keywords: Photonic band gap; Single-negative material; Effective medium approximation
1. Introduction Single-negative material (SNM) is the medium of which the relative permittivity e (permeability m) is negative while the relative permeability (permittivity) is positive. For instance, the electromagnetic response of metals in the visible region and near ultraviolet is dominated by the negative permittivity. However, for ordinary metals, at GHz frequencies, the imaginary part of the dielectric constant is about a thousand times bigger than the real part, and hence the dielectric permittivity is Corresponding author. Tel./fax: +86 51265112597.
E-mail address: [email protected] (L. Gao).
essentially imaginary. Fortunately, Pendry et al. demonstrated that a very simple metallic microstructure comprising a regular array of thin wires exhibit negative permittivity at GHz frequencies [1]. Later, metamaterials with effective negative permeability in a particular frequency range have also been obtained by ultilizing split ring resonators [2]. After that, substantial progress has been made to investigate the electromagnetic properties of composite structure consisting of and epsilonnegative media (ENM) and mu-negative media (MNM). Scientists [3,4] adopted ENM and MNM to provide an easy way to construct left-handed material (LHM) [5]. The resonance, tunneling
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.07.010
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L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
and transparency of pairing an ENM slab with MNM slab was also systematically investigated [6]. More recently, the transmission properties of the one-dimensional photonic crystals consisting of alternate ENM layer and MNM layer were investigated [7,8]. A new type of photonic gap called as zero effective phase shift gap (zero-feff gap) was predicted [6]. Zero-feff gap was found to distinguish itself from the Bragg gap in that it is invariant with a change of scale length and insensitive to thickness fluctuation [7]. Moreover, the width of zero-feff gap is only dependent on the thickness ratio of ENM and MNM layers [7]. In Ref. [8], the authors showed a peculiar transmission band gap which is almost insensitive to the different polarizations and incident directions of the incident electromagnetic waves. In other words, the transmission band gap is found to be omnidirectional. The band gap is similar to another omnidirectional transmission band gap (called as zero-¯n gap) existing in the layered structure containing LHMs [9,10]. In this paper, we investigate the transmission gap of the multilayer structure consisting of alternate ENM and MNM layer. We would like to mention that the constitutive parameters used in Ref. [7] are quite special, since the relative permittivity (permeability) of ENM is equal to the relative permeability (permittivity) of MNM [7]. Here, ENM and MNM with more general constitutive parameters [11] are used to form a layered structure. With the help of transfer matrix method [12,13], we study the transmission properties of the layered structure. It is found that, for both normal and oblique incidences, there exists a transmission gap which is invariant with a change of scaling. Since SNM are generally realized in microwave frequency region, their layers are often pffiffiffiffiffi electrically thin, i.e., kd ¼ emod=c51. As a consequence, we adopt effective medium approximation by introducing the effective permittivity eeff and permeability meff to study the wave propagation in this multilayer structure. Numerical results show that the existence of the transmission band gap results from the total reflection (eeff meff eair mair sin2 yp0, where y is the incident angle). For normal incidence, zero effective phase gap (zero-feff gap) [7] and zero average
refractive index (zero-¯n gap) [9] were, respectively, predicted from a stack of alternate ENM and MNM [7] and a stack of alternate ordinary material and LHM [9]. To further verify the validity of our theory, we take one step forward to investigate the properties of these two gaps. Again, these gaps can be well determined by the total reflection condition eeff meff p0. To one’s interest, the edges of gaps are accurately described by eeff meff ¼ 0.
2. Model and theory Let us consider the multilayer structure consisting of two kinds of SNM. One kind of SNM is ENM with the relative permittivity and permeability in the form of ea ¼ 1
o2p ; oðo þ i=tÞ
ma ¼ 1,
(1)
where op is the plasma frequency, and t is a relaxation time. Such a form of permittivity is representative of a variety of metal composites, whose permittivity is negative with small imaginary part in the visible region and near ultraviolet, and is essentially imaginary at low frequencies. However, Pentry et al. demonstrated that a very simple metallic microstructure comprising a regular array of thin wires exhibits negative e even at GHz frequencies, described by Eq. (1). The other kind of SNM is MNM with the relative permittivity and permeability expressed as b ¼ 1;
mb ¼ 1
F o2 , o2 o20 þ ioG
(2)
where o0 stands for the magnetic resonance, G is the dissipation factor, and F is the fractional area of the unit cell occupied by the interior of the split ring [14]. For simplicity, in our analysis, both SNM and MNM are assumed lossless, that is, t ¼ 0 and G ¼ 0. Actually, such an assumption has been widely used in Refs. [6–8]. Here, we set the parameters to be op ¼ 10 GHz, o0 ¼ 4 GHz and F ¼ 0:56 [11]. Note that both a and mb will be negative simultaneously when o is in the range (4 and 6 GHz).
ARTICLE IN PRESS L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
It is well known that the dispersion relation for the plane electromagnetic waves propagating in an infinite one-dimensional photonic crystal can be obtained by Bloch-Floquent theorem. For transverse-electric (TE) polarized waves (transversemagnetic (TM) polarized waves can be discussed in a similar way), the dispersion relation is expressed as [15] cosðqdÞ ¼ cosðkaz d a Þ cosðkbz d a Þ 1 kaz =ma kbz =mb þ 2 kbz =mb kaz =ma sinðkaz d a Þ sinðkbz d b Þ,
layer structure is electrically thin rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 2 ðe m sin yÞ 51 d a a a c2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! o2 2 and d b ðeb mb sin yÞ51 . c2 In this case, Based on using transfer matrix method [12,13], we can derive effective medium approximation, in which the effective permittivity eeff and the effective permeability meff of the layered structure are given by [4]
ð3Þ
where q is the Bloch wave vector along the axis of photonic crystals, d d a þ d b is the period, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kjz ¼ ðo2 =c2 Þðej mj sin2 yÞ (y is the incident
meff ¼ eeff
angle, j ¼ a; b). For a given y, if kaz =ma is equal to kbz =mb , Eq. (3) will be simplified as cosðqdÞ ¼ cosðkaz d a kbz d b Þ.
(4)
As both kaz and kbz are imaginary numbers for SNM, we have cosðkaz d a kbz d b ÞX1.
da db ma þ mb , d d
da db da 1 db 1 2 ea þ eb sin y ¼ þ d d d m d mb ! a 1 þ sin2 y d , db a d ma þ d mb
ð6Þ
for TE polarization and eeff ¼
(5)
Except for kaz d a ¼ kbz d b , Eq. (3) has no real solution for Bloch wave vector q. As a result, a transmission gap is opened. In what follows, we will present the numerical results. The transmission coefficients of a multilayer with 200 periods are calculated, as shown in Fig. 1. For normal incidence (see Fig. 1(a)), it is shown that there exists a transmission gap appearing near 5:4 GHz. If the lattice period is changed but the thickness ratio is kept fixed, the position and width of the transmission gap are almost invariant. In other words, the transmission gap is invariant upon a change of scale length. Actually, such a conclusion can also be found for oblique incidence. However, the position and width of the transmission gap for TE wave (see Fig. 1(b)) are quite different from those of the transmission gap for TM wave (see Fig. 1(c)). Therefore, the transmission gap is sensitive to the polarization of the incident electromagnetic waves. It is well known that SNM is usually realized in the microwave frequency region [1,2]. In this regard, the thickness of every layer in the multi-
373
meff
da db ea þ eb , d d
da db da 1 db 1 2 m þ m sin y ¼ þ d a d b d ea d eb ! 1 þ sin2 y d db a d ea þ d eb
ð7Þ
for TM polarization. Eqs. (6) and (7) indicate that the multilayer structure is anisotropic in essence because eeff and meff depend on the incident angle y. Now, we shall display that the transmission gap given in Fig. 1 can be explained simply by total reflection. The total reflection condition is eeff meff eair mair sin2 yo0. In Fig. 2(a) and (b), we plotted the transmission coefficients and eeff meff eair mair sin2 y as a function of the incident frequency o, respectively. Comparing Fig. 2(a) with Fig. 2(b), we find that for both TE and TM waves, the transmission gap just lies in the frequency region where eeff meff eair mair sin2 y is smaller than zero. And the edges of the transmission gap are determined well by the condition eeff meff eair mair sin2 y ¼ 0. Therefore, in the layered structure consisting of two kinds of SNM
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0.0 4.0
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5.0 ω (GHz)
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5.0 ω (GHz)
5.5
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Fig. 1. Transmission coefficients as a function of incident frequency o for (a). normal incidence y ¼ 0, (b) TE polarization with y ¼ 20 , and (c). TM polarization with y ¼ 20 . Solid line: d a ¼ 10 mm, d b ¼ 20 mm; dashed line: d a ¼ 15 mm, d b ¼ 30 mm; and dotted line: d a ¼ 20 mm, d b ¼ 40 mm.
with general physical parameters, there still exists a transmission gap, which can be explained simply as total reflection. Actually, in the transmission gap, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 o2 2 2 ðe m sin yÞ ðe m sin yÞ and d d a a b b a b c2 c2 are found to be so small that the introduction of the effective permittivity eeff and permeability meff is quite reasonable. According to our numerical calculations, the parameters used in Refs. [7,9] are also found to satisfy the condition that the layers are electrically
thin, especially in the frequency range of the zerofeff gap and zero-¯n gap. Hence, we would like to investigate zero-feff gap and zero-¯n gap as reported in Refs. [7,9]. For normal incidence, both eeff and meff are dependent only on the ratio of d a and d b , and the total reflection condition is simplified as eeff meff o0. The transmission coefficients (solid line in Fig. 2 in Ref. [7]) and eeff meff as a function of frequency are shown, respectively, in Fig. 3(a) and (b). From Fig. 3, it is evident that the frequency region of zero-feff band gap is completely consistent with the one in which eeff meff o0. To one’s interest, the left and right edges of zerofeff gap are well determined by eeff ¼ ðd a =dÞea þ
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L. Gao et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 371–377
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Fig. 2. Transmission coefficients and eeff meff eair mair sin2 y as a function of o for y ¼ 20 .
ðd b =dÞeb ¼ 0 and meff ¼ ðd a =dÞma þ ðd b =dÞmb ¼ 0. As a result, both the position and the width of zero-feff gap are only dependent on the ratio of d a and d b . Therefore, zero-feff gap can be controlled flexibly by the adjustment of the thickness ratio of layers. Due to this property, the application of the transmission gap becomes very convenient. In the end, the transmission coefficients (solid line in Fig. 2(c) in Ref. [9]) and eeff meff are plotted as a function of frequency in Fig. 4. Again, zero-¯n gap region just appears in the frequency range in which the total reflection condition eeff meff o0 is satisfied.
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Fig. 3. Transmission coefficients and eeff meff for zero-feff gap in Ref. [7].
And, like zero-feff gap, the left and right edges of zero-¯n gap are determined by eeff meff ¼ 0 too. In addition, the weak dependence of the two transmission gaps to thickness fluctuation can be understood easily within effective medium approximation [7,9] in virtue of the invariance of eeff and meff .
3. Conclusion and discussion In conclusion, we have investigated the transmission gap of the multilayer structure consisting
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duced to investigate the transmission gap. We find that the transmission gap is well explained by the total reflection condition eeff meff eair mair sin2 yo0. Furthermore, We discuss zero-feff gap and zero-¯n gap reported in previous works. We numerically show that the band edges of two gaps are described by a simple formula eeff meff ¼ 0. Here, some comments are in order. Since the properties of the photonic gaps are compact and robust against length scaling and disorder, it would be of great interest to investigate the guided wave in layered waveguide made of SNMs. Initial results show that there exists the omnidirectional guided mode, when ðd a =dÞea þ ðd b =dÞeb ¼ 0 or ðd a =dÞma þ ðd b =dÞmb ¼ 0 is satisfied. In addition, bright and dark gap solitons in LHM were studied more recently [16]. Therefore, we would like to take one step forward to investigate the bistability and soliton in one-dimensional photonic crystals containing SNMs.
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Acknowledgements
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This work was supported by the National Natural Science Foundation of China under Grant no. 10204017 (L.G.) and the Natural Science of Jiangsu Province under Grant no. BK2002038 (L.G.).
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Fig. 4. Transmission coefficients and eeff meff for zero-¯n gap in Ref. [9].
of alternate two kinds of SNMs. It is shown that there exists a transmission gap which is invariant with a change of scale length. And, the transmission gap is found to be dependent on the polarization of the incident electromagnetic waves, leading to a polarization-dependent transmission gap. Since SNM is usually realized in the microwave frequency region, the thickness of every layer in the multilayer structure is electrically thin. As a result, the effective permittivity eeff and permeability meff of the multilayer structure are intro-
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