- Email: [email protected]
Light transmission along a slab waveguide with a core of anisotropic metamaterial
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 591–595 www.elsevier.de/ijleo
Light transmission along a slab waveguide with a core of anisotropic metamaterial Guoan Zhenga,b, Lixin Ranb, a
Department of Optical Engineering, Zhejiang University, HangZhou 310027, China Department of Information and Electronic Engineering, Zhejiang University, HangZhou 310027, China
b
Received 26 May 2006; accepted 12 December 2006
Abstract We investigated the dispersion property of a slab waveguide with an anisotropic metamaterial core whose permittivity tensor is partially negative. The subwavelength guidance characteristics are presented based on the boundary conditions. The results show that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. It is also found that different orientations of the optical axis of the anisotropic core will lead to different dispersion of the guided modes. If the orientation of the optical axis is properly chosen, the guided modes show a transition from backward wave to a forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infrared frequencies, our result may find some applications in wave trapper, integrated optical and nanophotonic devices. r 2007 Elsevier GmbH. All rights reserved. Keywords: Anisotropic metamaterial; Subwavelength guidance; Slow propagation; Wave trapper
1. Introduction The metamaterial also known as left-handed material [1] has attracted much attention after the first experimental realization at some microwave frequencies [2]. The novel material exhibits simultaneously negative electric permittivity and magnetic permeability values and thus possesses a negative refraction index, as Veselago [1] first predicted theoretically in 1968. Based on the novel property of metamaterial, many ideas and potential applications have been proposed [3–8]. The study of the properties in metamaterial waveguides is a Corresponding author. Tel.: +86 571 87952456; fax: +86 571 88206591. E-mail address: [email protected] (L. Ran).
0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.12.021
large new area, which can be extended to useful applications [6,7]. For conventional waveguides based on total internal reflection, the guiding layers can theoretically be scaled down as much as possible by increasing the difference between the refraction index of the guiding core and its claddings. However, two limitations are associated with the high refraction index guiding core. First, large propagating constant in high refraction index core makes it difficult to couple the wave into the guiding core. Second, one will eventually meet the difficulty on obtaining high refraction index material. For the conventional symmetrical waveguide, there are no cut-off frequencies for the fundamental modes, therefore the size of the guiding core can go to zero theoretically. However, the field confinement is poor for the fundamental modes and the waveguide will
ARTICLE IN PRESS 592
G. Zheng, L. Ran / Optik 119 (2008) 591–595
suffer from large scattering losses while reducing the thickness of the guiding core [9]. In this paper, we investigate the dispersion property of the guided modes in a subwavelength slab waveguide with an anisotropic metamaterial core whose permittivity tensor is partially negative [8]. The result shows that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. Therefore, it is possible to scale down the waveguide to subwavelength size and the field confinement of the high-order modes can be much better than the fundamental modes [10]. It is also found that different orientations of the optical axis of the anisotropic metamaterial core will lead to different dispersions of the guided modes. If the orientation of the optical axis is properly chosen, the guided modes show a transition from the backward wave to the forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infra-red frequencies [11,12], our result may have some applications in wave trapper, integrated optical and nanophotonics device.
Region 3
Air
d
ε1
ε2
Region 2
θ
Air
z Region 1
Fig. 1. Slab waveguide with 1 40 and 2 o0.
optical axis and the z-axis is y (note that the optical axis is in the x–z plane). Consider a TM incidence wave with the magnetic field polarized along y-axis. In order to guide the fields in the core, the fields in the claddings must be evanescent along the x direction. We write the magnetic fields in regions 1–3 as !
!
H 1 ¼ y A1 eax eikz z ;
xo0,
h i ! i r ! H 2 ¼ y A2 eik2x x þ B2 eik2x x eikz z ; !
!
H 3 ¼ y A3 eax eikz z ;
(2a) 0pxpd,
x4d,
We first consider an electrically uniaxial media with a scalar permeability m ¼ 1 and a permittivity tensor 0 1 1 0 0 B C (1) ¼ @ 0 1 0 A, 0 0 2 which is given in the principle coordinates (x–y–z system). The optical axis is along z-direction with dielectric constant e2. For a conventional uniaxial media, e1 and e2 are both positive. However, e2 can be negative and obey the frequency dependence of the wellknown Drude model (note that e1 is still a positive constant). In GHz frequencies, the idea to achieve 2 o0 can be realized in a composite of periodically arranged metallic thin wires aligned along the optical axis [13,14], and in optical, near- and mid-infrared frequencies, it can be, respectively, achieved by various techniques described in Ref. [11]. As an example, for the composite of 10% of SiC nano-spheroids with an aspect ratio of 1/2, aligned with their shorter axis along the optical axis and embedded in quartz, we can obtain 2 ¼ 2:7þ 6 104 i, 1 ¼ 1:6 þ 1 105 i for a wavelength of CO2 laser of 12 mm. A slab waveguide with a partially negative permittivity core is shown in Fig. 1. The thickness of the core is d and the cladding layers are air. The angle between the
(2c)
respectively, where a represents the attenuation factor. Based on Ampere’s law, the electric displacement D is kz a A1 eax eikz z ; D1z ¼ A1 eax eikz z , oi o i r kz A2 eik2x x þ B2 eik2x x eikz z , D2x ¼ o i r 1 A2 ki2x eik2x x þ B2 kr2x eik2x x eikz z , D2z ¼ o
D1x ¼
2. Determination of the guided modes
(2b)
(3a)
ð3bÞ
kz a A3 eax eikz z ; D3z ¼ A3 eax eikz z . (3c) oi o The dispersion relations of the claddings and the core are
D3x ¼
k2z a2 ¼ ðo=cÞ2 ,
(4a)
k22x ð1 cos2 y þ 2 sin2 yÞ þ k2z ð2 cos2 y þ 1 sin2 yÞ þ 2 sin y cos yk2x kz ð2 1 Þ 2 1 ðo=cÞ2 ¼ 0,
ð4bÞ
respectively, where c is the speed of light in the air, and ki2x , kr2x are the solutions of Eq. (4b) for a given kz . The permittivity tensor of the core can be expressed as 1 0 1 cos2 y þ 2 sin2 y 0 sin y cos yð1 2 Þ C B 1 0 ¼ @0 A. 2 2 sin y cos yð1 2 Þ 0 1 sin y þ 2 cos y (5) Applying D ¼ E, we get the z component of E-field as 2 1 1 cos y sin2 y E 2z ¼ D2x sin y cos y þ þ D2z . 2 1 2 1 (6)
ARTICLE IN PRESS G. Zheng, L. Ran / Optik 119 (2008) 591–595
To find the guidance condition, we impose the boundary conditions at x ¼ 0 and d, namely, A1 ¼ A2 þ B2 , i
tion example we give before. The dispersion relation curves for different orientations of the optical axis (i.e. different ys) are shown in Fig. 2(a–d). The thickness d in our numerical calculation is 0:1lp ðlp ¼ 2pc=op Þ which means that the size of the guiding core is 10 times smaller than the working wavelength. The wave number in the propagating direction kz is normalized by kp ðkp ¼ op =cÞ. The range of kz in Fig. 2 is from 0 to 5kp only. In fact, we are not interested in the large wave number because it is difficult to couple wave into the waveguide with a large wave number [9]. The wave number in the transverse direction (i.e. ki2x and kr2x ) can be real or imaginary; the former case corresponds to the oscillating mode and the later case corresponds to the surface mode. The curves in Fig. 2(a–d) are oscillating modes; no surface mode solution is found in these cases. As a comparison, we plot the dispersion curve for isotropic plasmonic core (i.e. 1 ¼ 2 ¼ 1 ðop =oÞ2 ) in Fig. 2(e), in which only surface mode solution can be found, i.e., ki2x and kr2x are imaginary numbers. The result in Fig. 2(e) is the same as that in Ref. [17], therefore Fig. 2(e) can be used as a verification of our theoretical result. There are two curves in Fig. 2(a) with y ¼ 0 . The top one stands for the fundamental TM0 mode and the other one stands for the TM1 mode. These two modes are both forward wave whose group velocity and phase velocity are at the same direction. In Fig. 2(d), the backward (with the opposite directions of group velocity and phase velocity) and forward modes coexist in some frequency regions, so we can use a prism to separate these two types of modes [15]. Some interesting phenomena occur when y ¼ 40 as shown in Fig. 2(b): as the frequency increases, the dispersion curves show a transition from backward modes to forward modes. At the frequency of o ¼ 0:55op , the slope of the dispersion
(7a) i
A2 eik2x d þ B2 eik2x d ¼ A3 ead , (7b) 1 1 ðA2 þ B2 Þkz sin y cos y 2 1 2 cos y sin2 y a þ ðki2x A2 þ kr2x B2 Þ ¼ A1 , ð7cÞ 2 1 i0 i r 1 1 A2 eik2x x þ B2 eik2x x kz sin y cos y 2 1 2 i r cos y sin2 y þ ki2x A2 eik2x x þ kr2x B2 eik2x x 2 1 a ¼ A3 ead , ð7dÞ i0 where we normalize A2 ¼ 1, and A1 , B2 , A3 can be solved using Eqs. (7a)–(7c). For a given kz , we get ki2x and kr2x from Eq. (4b). If ki2x and kr2x satisfy Eq. (7d), the guided mode can exist in the present waveguide. Substituting A1 , B2 , A3 into Eq. (7d), we finally obtain the guidance condition, from which we may start our analysis for different guided modes.
3. Dispersion relation of the guided modes
0.8
0.8
0.7
0.7
0.7
0.6
1
2 3 kz/kp
ω/ωp
(a)
4
5
0.4
0.4 0
θ = 0°
1
2 3 kz/kp
(b)
0.8
0.8
0.7
0.7
0.6 0.5 0.4
4
5
0
1
θ = 40°
ω/ωp
0
0.6 0.5
0.5
0.5 0.4
ω/ωp
0.8 ω/ωp
ω/ωp
Following Refs. [11,13], we assume 2 obey the Drude model as below: o 2 p , (8) 2 ¼ 1 o where op is the plasmas frequency, and simply assume 1 ¼ 1:6 of an insulator in accordance with the fabrica-
0.6
593
2
(c)
3 4 kz/kp
5
θ = 60°
0.6 0.5
0
1
(d)
2 3 kz/kp θ = 90°
4
5
0.4
0
(e)
1
2 3 kz/kp
4
5
1 = 2 = 1 — (ω p / ω)2
Fig. 2. The dispersion curves with (a) y ¼ 0 , (b) y ¼ 40 , (c) y ¼ 60 , (d) y ¼ 90 and (e) 1 ¼ 2 ¼ 1 ðop =oÞ2 .
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G. Zheng, L. Ran / Optik 119 (2008) 591–595
curves which stand for the group velocity approach zero. Such slow propagation of light can increase the efficiency of nonlinear processes, which are at the root of all-optical interaction [16]. In order to study the transition behaviour shown in Fig. 2(b), we plot the magnetic field distribution with o ¼ 0:54op and o ¼ 0:57op in Fig. 3(a) and (b), in which the former case corresponds to a backward mode and the later case corresponds to a forward mode. To study the propagation behaviours of these two cases, we plot the z component of the time-averaged Poynting vector S z ðS z ¼ RefðE H n Þ ! z gÞ in Fig. 4 (note that S z is not continuous at the boundaries). From Fig. 4, one can see that the energy flows of the propagating modes here are strongly confined in the waveguide. In other words, the high-order modes can propagate in the present waveguide with the thickness of the guiding core 10 times smaller than the working wavelength and with better field confinement than conventional fundamental mode. As shown in Fig. 4(a), the energy flows in the core and the claddings R are contraþ1 directional and the total energy flux 1 S z dx is negative, which means a backward propagation [15], mean while in Fig. 4(b) the energy flux is positive, implying a forward one.
4. Conclusion The dispersion property of a subwavelength slab waveguide with an anisotropic metamaterial core is studied. The result shows that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. Therefore, it is possible to scale down waveguide to subwavelength size and the field confinement of the high order modes can be much better than the fundamental modes. It is also found that if the orientation of the optical axis is properly chosen, the guided modes show a transition from the backward wave to the forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. The anomalous properties of such guided modes are discussed by calculating the time-averaged Poynting vector across the slab. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infra-red frequencies, our result may have some applications in wave trapper, integrated optical and nanophotonic device.
2
Magnitude of H(x) in A/m
Magnitude of H(x) in A/m
Acknowledgments
1.5 1 0.5 0 -1
0
1
2
This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 60531020, and in part by ZJNSF under Grant R105253.
2 1.5 1
References
0.5 0 -1
0
1
x/d
x/d
(a)
(b)
2
Fig. 3. The distribution of magnetic field across the slab waveguide with y ¼ 40 and kz ¼ kp : (a) o ¼ 0:54op , (b) o ¼ 0:57op .
x 10-7
-7 3 x 10
Sz in W/m2
Sz in W/m2
2 0 -2
-4 -1
0
1
2
2
1
0 -1
0
1
x/d
x/d
(a)
(b)
2
Fig. 4. The z-component of the Poynting vector Sz across the slab waveguide with y ¼ 40 and kz ¼ kp : (a) o ¼ 0:54op , (b) o ¼ 0:57op .
[1] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of epsilon and miu, Sov. Phy. Usp. 10 (1968) 509–514. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000) 4184. [3] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (18) (2000) 3966–3969. [4] L. Wu, S.L. He, H. Chen, On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials, Optics Express 11 (11) (2003) 1283–1290. [5] N. Engheta, An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability, IEEE Antennas Wireless Propagat. Lett. 1 (1) (2002) 10–13. [6] I.V. Shadrivov, A.A. Sukhorukov, Y.S. Kivshar, Guided modes in negative refractive index waveguides, Phys. Rev. E 67 (2003) 057602. [7] B.-l. Wu, T.M. Grzegorczyk, Y. Zhang, J.A. Kong, Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability, J. Appl. Phys. 93 (2003) 9386–9388.
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[8] D.R. Smith, D. Schuring, Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors, Phys. Rev. Lett. 90 (2003) 077405. [9] L.M. Tong, R.R. Gattass, J.R. Aschom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, Subwavelength-diameter silica wires for low-loss optical wave guiding, Nature (London) 426 (2003) 816. [10] K.Q. De, C. Gang, Nanoscale optical waveguides with negative dielectric claddings, Phys. Rev. B 71 (2005) 153107. [11] V.A. Podolskiv, E.E. Narimanov, Strongly anisotropic waveguide as a nonmagnetic left-handed system, Phys. Rev. B 71 (2005) 201101. [12] O. Levy, D. Stroud, Maxwell Garnett theory for mixtures of anisotropic inclusions: application to conducting polymers, Phys. Rev. B 56 (1997) 8035.
595
[13] J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Extremely low frequency plasmas in metallic mesostructures, Phys. Rev. Lett. 76 (1996) 4773. [14] C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah, M. Tanielian, Experimental verification and simulation of negative index of refraction using Snell’s law, Phys. Rev. Lett. 90 (2003) 107401. [15] J. He, S. He, Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate, IEEE Microwave Wireless Comp. Lett. 16 (2006). [16] M. Soljactc, S.G. Johnson, S. Fan, M. Ibanescu, E. Ippen, J.D. Joannopoulos, Photonic-crystal slow-light enhancement of nonlinear phase sensitivity, J. Opt. Soc. Am. B 19 (9) (2002) 2052–2059. [17] E.N. Econnomou, Surface plasmons in thin films, Phys. Rev. 182 (1969) 539–554.
Optik
Optics
Optik 119 (2008) 591–595 www.elsevier.de/ijleo
Light transmission along a slab waveguide with a core of anisotropic metamaterial Guoan Zhenga,b, Lixin Ranb, a
Department of Optical Engineering, Zhejiang University, HangZhou 310027, China Department of Information and Electronic Engineering, Zhejiang University, HangZhou 310027, China
b
Received 26 May 2006; accepted 12 December 2006
Abstract We investigated the dispersion property of a slab waveguide with an anisotropic metamaterial core whose permittivity tensor is partially negative. The subwavelength guidance characteristics are presented based on the boundary conditions. The results show that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. It is also found that different orientations of the optical axis of the anisotropic core will lead to different dispersion of the guided modes. If the orientation of the optical axis is properly chosen, the guided modes show a transition from backward wave to a forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infrared frequencies, our result may find some applications in wave trapper, integrated optical and nanophotonic devices. r 2007 Elsevier GmbH. All rights reserved. Keywords: Anisotropic metamaterial; Subwavelength guidance; Slow propagation; Wave trapper
1. Introduction The metamaterial also known as left-handed material [1] has attracted much attention after the first experimental realization at some microwave frequencies [2]. The novel material exhibits simultaneously negative electric permittivity and magnetic permeability values and thus possesses a negative refraction index, as Veselago [1] first predicted theoretically in 1968. Based on the novel property of metamaterial, many ideas and potential applications have been proposed [3–8]. The study of the properties in metamaterial waveguides is a Corresponding author. Tel.: +86 571 87952456; fax: +86 571 88206591. E-mail address: [email protected] (L. Ran).
0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.12.021
large new area, which can be extended to useful applications [6,7]. For conventional waveguides based on total internal reflection, the guiding layers can theoretically be scaled down as much as possible by increasing the difference between the refraction index of the guiding core and its claddings. However, two limitations are associated with the high refraction index guiding core. First, large propagating constant in high refraction index core makes it difficult to couple the wave into the guiding core. Second, one will eventually meet the difficulty on obtaining high refraction index material. For the conventional symmetrical waveguide, there are no cut-off frequencies for the fundamental modes, therefore the size of the guiding core can go to zero theoretically. However, the field confinement is poor for the fundamental modes and the waveguide will
ARTICLE IN PRESS 592
G. Zheng, L. Ran / Optik 119 (2008) 591–595
suffer from large scattering losses while reducing the thickness of the guiding core [9]. In this paper, we investigate the dispersion property of the guided modes in a subwavelength slab waveguide with an anisotropic metamaterial core whose permittivity tensor is partially negative [8]. The result shows that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. Therefore, it is possible to scale down the waveguide to subwavelength size and the field confinement of the high-order modes can be much better than the fundamental modes [10]. It is also found that different orientations of the optical axis of the anisotropic metamaterial core will lead to different dispersions of the guided modes. If the orientation of the optical axis is properly chosen, the guided modes show a transition from the backward wave to the forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infra-red frequencies [11,12], our result may have some applications in wave trapper, integrated optical and nanophotonics device.
Region 3
Air
d
ε1
ε2
Region 2
θ
Air
z Region 1
Fig. 1. Slab waveguide with 1 40 and 2 o0.
optical axis and the z-axis is y (note that the optical axis is in the x–z plane). Consider a TM incidence wave with the magnetic field polarized along y-axis. In order to guide the fields in the core, the fields in the claddings must be evanescent along the x direction. We write the magnetic fields in regions 1–3 as !
!
H 1 ¼ y A1 eax eikz z ;
xo0,
h i ! i r ! H 2 ¼ y A2 eik2x x þ B2 eik2x x eikz z ; !
!
H 3 ¼ y A3 eax eikz z ;
(2a) 0pxpd,
x4d,
We first consider an electrically uniaxial media with a scalar permeability m ¼ 1 and a permittivity tensor 0 1 1 0 0 B C (1) ¼ @ 0 1 0 A, 0 0 2 which is given in the principle coordinates (x–y–z system). The optical axis is along z-direction with dielectric constant e2. For a conventional uniaxial media, e1 and e2 are both positive. However, e2 can be negative and obey the frequency dependence of the wellknown Drude model (note that e1 is still a positive constant). In GHz frequencies, the idea to achieve 2 o0 can be realized in a composite of periodically arranged metallic thin wires aligned along the optical axis [13,14], and in optical, near- and mid-infrared frequencies, it can be, respectively, achieved by various techniques described in Ref. [11]. As an example, for the composite of 10% of SiC nano-spheroids with an aspect ratio of 1/2, aligned with their shorter axis along the optical axis and embedded in quartz, we can obtain 2 ¼ 2:7þ 6 104 i, 1 ¼ 1:6 þ 1 105 i for a wavelength of CO2 laser of 12 mm. A slab waveguide with a partially negative permittivity core is shown in Fig. 1. The thickness of the core is d and the cladding layers are air. The angle between the
(2c)
respectively, where a represents the attenuation factor. Based on Ampere’s law, the electric displacement D is kz a A1 eax eikz z ; D1z ¼ A1 eax eikz z , oi o i r kz A2 eik2x x þ B2 eik2x x eikz z , D2x ¼ o i r 1 A2 ki2x eik2x x þ B2 kr2x eik2x x eikz z , D2z ¼ o
D1x ¼
2. Determination of the guided modes
(2b)
(3a)
ð3bÞ
kz a A3 eax eikz z ; D3z ¼ A3 eax eikz z . (3c) oi o The dispersion relations of the claddings and the core are
D3x ¼
k2z a2 ¼ ðo=cÞ2 ,
(4a)
k22x ð1 cos2 y þ 2 sin2 yÞ þ k2z ð2 cos2 y þ 1 sin2 yÞ þ 2 sin y cos yk2x kz ð2 1 Þ 2 1 ðo=cÞ2 ¼ 0,
ð4bÞ
respectively, where c is the speed of light in the air, and ki2x , kr2x are the solutions of Eq. (4b) for a given kz . The permittivity tensor of the core can be expressed as 1 0 1 cos2 y þ 2 sin2 y 0 sin y cos yð1 2 Þ C B 1 0 ¼ @0 A. 2 2 sin y cos yð1 2 Þ 0 1 sin y þ 2 cos y (5) Applying D ¼ E, we get the z component of E-field as 2 1 1 cos y sin2 y E 2z ¼ D2x sin y cos y þ þ D2z . 2 1 2 1 (6)
ARTICLE IN PRESS G. Zheng, L. Ran / Optik 119 (2008) 591–595
To find the guidance condition, we impose the boundary conditions at x ¼ 0 and d, namely, A1 ¼ A2 þ B2 , i
tion example we give before. The dispersion relation curves for different orientations of the optical axis (i.e. different ys) are shown in Fig. 2(a–d). The thickness d in our numerical calculation is 0:1lp ðlp ¼ 2pc=op Þ which means that the size of the guiding core is 10 times smaller than the working wavelength. The wave number in the propagating direction kz is normalized by kp ðkp ¼ op =cÞ. The range of kz in Fig. 2 is from 0 to 5kp only. In fact, we are not interested in the large wave number because it is difficult to couple wave into the waveguide with a large wave number [9]. The wave number in the transverse direction (i.e. ki2x and kr2x ) can be real or imaginary; the former case corresponds to the oscillating mode and the later case corresponds to the surface mode. The curves in Fig. 2(a–d) are oscillating modes; no surface mode solution is found in these cases. As a comparison, we plot the dispersion curve for isotropic plasmonic core (i.e. 1 ¼ 2 ¼ 1 ðop =oÞ2 ) in Fig. 2(e), in which only surface mode solution can be found, i.e., ki2x and kr2x are imaginary numbers. The result in Fig. 2(e) is the same as that in Ref. [17], therefore Fig. 2(e) can be used as a verification of our theoretical result. There are two curves in Fig. 2(a) with y ¼ 0 . The top one stands for the fundamental TM0 mode and the other one stands for the TM1 mode. These two modes are both forward wave whose group velocity and phase velocity are at the same direction. In Fig. 2(d), the backward (with the opposite directions of group velocity and phase velocity) and forward modes coexist in some frequency regions, so we can use a prism to separate these two types of modes [15]. Some interesting phenomena occur when y ¼ 40 as shown in Fig. 2(b): as the frequency increases, the dispersion curves show a transition from backward modes to forward modes. At the frequency of o ¼ 0:55op , the slope of the dispersion
(7a) i
A2 eik2x d þ B2 eik2x d ¼ A3 ead , (7b) 1 1 ðA2 þ B2 Þkz sin y cos y 2 1 2 cos y sin2 y a þ ðki2x A2 þ kr2x B2 Þ ¼ A1 , ð7cÞ 2 1 i0 i r 1 1 A2 eik2x x þ B2 eik2x x kz sin y cos y 2 1 2 i r cos y sin2 y þ ki2x A2 eik2x x þ kr2x B2 eik2x x 2 1 a ¼ A3 ead , ð7dÞ i0 where we normalize A2 ¼ 1, and A1 , B2 , A3 can be solved using Eqs. (7a)–(7c). For a given kz , we get ki2x and kr2x from Eq. (4b). If ki2x and kr2x satisfy Eq. (7d), the guided mode can exist in the present waveguide. Substituting A1 , B2 , A3 into Eq. (7d), we finally obtain the guidance condition, from which we may start our analysis for different guided modes.
3. Dispersion relation of the guided modes
0.8
0.8
0.7
0.7
0.7
0.6
1
2 3 kz/kp
ω/ωp
(a)
4
5
0.4
0.4 0
θ = 0°
1
2 3 kz/kp
(b)
0.8
0.8
0.7
0.7
0.6 0.5 0.4
4
5
0
1
θ = 40°
ω/ωp
0
0.6 0.5
0.5
0.5 0.4
ω/ωp
0.8 ω/ωp
ω/ωp
Following Refs. [11,13], we assume 2 obey the Drude model as below: o 2 p , (8) 2 ¼ 1 o where op is the plasmas frequency, and simply assume 1 ¼ 1:6 of an insulator in accordance with the fabrica-
0.6
593
2
(c)
3 4 kz/kp
5
θ = 60°
0.6 0.5
0
1
(d)
2 3 kz/kp θ = 90°
4
5
0.4
0
(e)
1
2 3 kz/kp
4
5
1 = 2 = 1 — (ω p / ω)2
Fig. 2. The dispersion curves with (a) y ¼ 0 , (b) y ¼ 40 , (c) y ¼ 60 , (d) y ¼ 90 and (e) 1 ¼ 2 ¼ 1 ðop =oÞ2 .
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curves which stand for the group velocity approach zero. Such slow propagation of light can increase the efficiency of nonlinear processes, which are at the root of all-optical interaction [16]. In order to study the transition behaviour shown in Fig. 2(b), we plot the magnetic field distribution with o ¼ 0:54op and o ¼ 0:57op in Fig. 3(a) and (b), in which the former case corresponds to a backward mode and the later case corresponds to a forward mode. To study the propagation behaviours of these two cases, we plot the z component of the time-averaged Poynting vector S z ðS z ¼ RefðE H n Þ ! z gÞ in Fig. 4 (note that S z is not continuous at the boundaries). From Fig. 4, one can see that the energy flows of the propagating modes here are strongly confined in the waveguide. In other words, the high-order modes can propagate in the present waveguide with the thickness of the guiding core 10 times smaller than the working wavelength and with better field confinement than conventional fundamental mode. As shown in Fig. 4(a), the energy flows in the core and the claddings R are contraþ1 directional and the total energy flux 1 S z dx is negative, which means a backward propagation [15], mean while in Fig. 4(b) the energy flux is positive, implying a forward one.
4. Conclusion The dispersion property of a subwavelength slab waveguide with an anisotropic metamaterial core is studied. The result shows that, at some specific frequencies, many high-order modes can exist in present waveguide even with the thickness of the guiding core 10 times smaller than the working wavelength. Therefore, it is possible to scale down waveguide to subwavelength size and the field confinement of the high order modes can be much better than the fundamental modes. It is also found that if the orientation of the optical axis is properly chosen, the guided modes show a transition from the backward wave to the forward wave as the frequency increases. During this transition, the group velocity of some guided modes can approach zero. The anomalous properties of such guided modes are discussed by calculating the time-averaged Poynting vector across the slab. Since the anisotropic metamaterial we discuss here can be fabricated in GHz, near- and mid-infra-red frequencies, our result may have some applications in wave trapper, integrated optical and nanophotonic device.
2
Magnitude of H(x) in A/m
Magnitude of H(x) in A/m
Acknowledgments
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This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 60531020, and in part by ZJNSF under Grant R105253.
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References
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1
x/d
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(b)
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Fig. 3. The distribution of magnetic field across the slab waveguide with y ¼ 40 and kz ¼ kp : (a) o ¼ 0:54op , (b) o ¼ 0:57op .
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Fig. 4. The z-component of the Poynting vector Sz across the slab waveguide with y ¼ 40 and kz ¼ kp : (a) o ¼ 0:54op , (b) o ¼ 0:57op .
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