Hindawi Publishing Corporation Advances in OptoElectronics Volume 2007, Article ID 46861, 7 pages doi:10.1155/2007/46861 Research Article Anisotropic Left-Handed and μ-Negative Slab Waveguides: Physics and Device Applications Hamidreza Salehi, Sujeet K. Chaudhuri, and Raafat R. Mansour Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G5 Received 25 April 2007; Revised 27 June 2007; Accepted 1 October 2007 Recommended by Stefan A. Maier We study the properties of various anisotropic left-handed slab waveguides. The analysis is extended to anisotropic μ-negative slab waveguides. The possible existence of the plasmon modes in various anisotropic slab waveguide conﬁgurations is discussed. An FDTD program is developed to investigate the potential device applications of these anisotropic structures. A new signal detector and a two-channel harmonic separator multiplexer are designed employing the μ-negative slab waveguide. Copyright © 2007 Hamidreza Salehi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION In this paper, we study the guided mode properties of the anisotropic LHSWs, with the aim of ﬁnding a structure con- The concept of a material with simultaneously negative per- ﬁguration that supports the plasmon modes. Potential device mittivity and permeability, ﬁrst studied by Veselago in 1968 applications of anisotropic LHSWs, that employ ﬁeld local- [1], has attracted physics and engineering communities in ization of the plasmon modes, are explored. recent years. These structures, which exhibit extraordinary To study various device applications of LHSWs, a properties not generally found in nature, are referred to as ﬁnite-diﬀerence time-domain (FDTD) program is devel- negative index materials (NIMs) or left-handed materials oped to numerically study diﬀerent devices that incorporate (LHMs). anisotropic slab waveguides. Berenger’s perfectly matched Diﬀerent research groups have tried to artiﬁcially realize layer (BPML) is chosen to truncate the FDTD grid [10]. The these structures, employing subwavelength periodic struc- Drude dispersion formulation is used to model the permit- tures [2–6]. With the possibility of physically realizing nega- tivity and permeability of the left-handed media. Diﬀerent tive index materials, it is important to investigate the unusual proposed device applications of LHSW such as a new signal properties of these composite structures and their possible detector and a two-channel harmonic separator multiplexer device applications in the microwave and optical frequency are examined by using the developed FDTD code. ranges. The unusual properties of left-handed slab waveg- uides (LHSWs), where a left-handed layer is sandwiched be- 2. ANISOTROPIC LEFT-HANDED SLAB WAVEGUIDES tween positive index layers, have been studied in the litera- ture [7–9]. The analysis of the left-handed slab waveguides (LHSWs) in The existence of plasmon modes in LHSWs can be uti- [7–9] indicated that the structure can support guided modes lized in various device applications, provided these structures with imaginary transverse wavenumbers, referred to as plas- can be realized. The LHSW structure requires negative per- mon modes. The energy of these plasmon modes is concen- meability in both normal and transverse directions, and neg- trated at the interfaces of the left-handed core and the di- ative primitivity in the transverse direction. From a realiza- electric cladding. This characteristic of the plasmon modes tion point of view, it is practically easier to realize an LHM, suggests that an LHSW can have potential applications in sig- that only requires negative permeability in one direction. nal detection devices. In an isotropic LHSW, the left-handed This is due to polarization dependency of split-ring resonator layer exhibits negative permeability in both x and z direc- structures, which are widely used to realize left-handed com- tions, that is, μ < 0and μ < 0. In practice,itisrela- xx zz posites. tively diﬃcult to realize a left-handed composite that exhibits 2 Advances in OptoElectronics yy 2 2 zz zz xx where p = ω μ − (μ /μ )β > 0. The solutions of this μ > 0 2 2 2 2 diﬀerential equation are in the form of A exp (±jpx), which ε > 0 x = d indicates that such a structure cannot support guided modes xx with imaginary wavenumbers; that is, (3) does not have any solutions in the form of A exp (±hx). To overcome this prob- yy zz μ lem, the electric permittivity of the left-handed layer, ,is 2 2 chosen to be positive. This transforms the structure into a yy μ-negative structure, which will be studied in the following x = 0 section. μ > 0 z ε > 0 3. ANISOTROPIC μ-NEGATIVE SLAB WAVEGUIDE Figure 1: Anisotropic μ-negative slab waveguide (SW) of thickness The wave equation of the anisotropic μ-negative medium yy zz xx d sandwiched between two dielectric layers. with μ < 0, μ > 0, and > 0 can have plasmon-type so- 2 2 yy 2 2 zz xx 2 zz lutions if (μ /μ ) β <ω μ . The solutions of the wave 2 2 2 equation negative permeability in two directions using the existing structures, specially at optical frequencies [11–17]. ∂ E (x) − h E (x) = 0, (4) In this section, to comply with the practical limitations of y ∂x realizing an LHSW, the possibility of realizing an anisotropic yy 2 2 μ-negative slab waveguide that can support guided modes 2 2 zz zz xx where h = ω |μ |− (ξβ) > 0and ξ =−μ /μ ,are in 2 2 2 with imaginary transverse wavenumber is investigated. The the form of E (0 <x <d) = A exp (±hx), revealing that this anisotropic μ-negative slab waveguide (SW) is considerably structure can support plasmon modes. Solving the isotropic simpler to fabricate since the composite requires to exhibit wave equation in the cladding and satisfying the electric and negative permeability only in one direction. The schematic magnetic boundary conditions at x = 0and x = d show of an anisotropic μ-negative slab waveguide is shown in that the propagation constant of the even plasmon mode, Figure 1. For the TE polarized waves, with the electric ﬁeld Plasmon TE , is found from the intersection of even in the y direction and the magnetic ﬁelds in the x and z di- rections, Maxwell equations are given by μ u v =− u tanh , zz μ 2 ∂H ∂H x z yy (5) − = jω E , ∂z ∂x d yy 2 2 2 2 2 zz ξ v + u = 4π μ − μ ξ , 2 1 2 1 ∂E xx = jωμ H , (1) ∂z yy 2 zz 2 ∂E y 2 where u = hd, v = qd, h = ω |μ |− ξ β ,and q = zz 2 2 =−jωμ H . ∂x β − ω μ . The propagation constant of the odd plasmon The wave equation in the cladding layer that surrounds the Plasmon mode, TE , is similarly found from the intersection of odd left-handed core remains unchanged because the cladding still consists of isotropic dielectric material. By combining μ u v =− u coth , (1), the wave equation in the anisotropic μ-negative layer of zz μ 2 the slab waveguide is (6) 2 yy 2 2 2 2 zz ξ v + u = 4π μ − μ ξ . 2 1 zz 2 1 ∂ E (x) μ y yy λ 2 zz 2 + ω μ − β E (x) = 0, (2) 2 xx ∂x μ The graphical solution of the plasmon modes, depicted in where β is the propagation constant in the z direction. For Figure 2, indicates that the anisotropic μ-negative SW always the anisotropic LHSW to support the plasmon modes, the supports the even plasmon mode while the odd mode is ex- derivative of the electric ﬁeld with respect to x, ∂E (x)/∂x, cited if the operating frequency is above a certain cutoﬀ fre- should be discontinuous at x = 0and x = d. The bound- quency. As the operating frequency increases, the ﬁeld con- ary conditions at x = 0and x = d require that H (x = centration of plasmon modes at the interfaces of the left- − + − + 0 ) = H (x = 0 ), and H (x = d ) = H (x = d ), where handed and dielectric materials, the core-cladding interface, z z z zz H = (j/ωμ )(∂E (x)/∂x). The discontinuity of ∂E (x)/∂x increases and the propagation constants of even and odd z y y requirement along with the continuity of H at the interfaces modes converge towards an identical value. The electric ﬁeld indicates that for a left-handed slab waveguide to support distribution of both even and odd plasmon modes of the yy zz zz xx plasmon modes, μ needstobenegative. Thewaveequation anisotropic μ-negative SW with μ < 0, μ > 0, and > 0, 2 2 2 yy zz zz xx is depicted in Figures 3 and 4,respectively. With μ =−1, of an LHSW with μ < 0, μ > 0, and < 0is 2 2 2 2 yy xx μ = 1, and = 2.25, the SW also supports guided modes ∂ E (x) with real transverse wavenumbers similar to a conventional + p E (x) = 0, (3) ∂x slab waveguide. The propagation constants of these guided Hamidreza Salehi et al. 3 Even plasmon Anisotropic mode μ-negative layer 0.8 0.6 Odd plasmon mode 0.4 0.2 Even plasmon modes 0 0 −1 −0.50 0.51 1.52 0 12345 6 7 8 x(λ) u = hd Figure 3: Normalized electric ﬁeld distribution of the even plasmon Figure 2: Mode diagram of the anisotropic μ-negative slab waveg- zz yy zz xx modes of the anisotropic μ-negative slab waveguide with μ =−1, uide (LHSW) with μ =−1, μ = 1, and = 2.25 (h = 2 2 2 yy xx μ = 1, and = 2.25. yy 2 2 2 2 2 zz 2 2 ω |μ |− ξ β ,and q = β − ω μ ). 2 2 1 yy 2 2 zz xx 2 zz modes have (μ /μ ) β >ω μ , resulting in the follow- 2 2 2 Anisotropic Odd plasmon 0.8 ing wave equation in the anisotropic μ-negative layer: μ-negative layer mode 0.6 ∂ E (x) 0.4 + p E (x) = 0, (7) ∂x 0.2 yy 2 2 2 2 zz zz xx where p = (ξβ) − ω |μ | > 0and ξ =−(μ /μ ). 2 2 2 −0.2 The solutions of (7) are in the form of E (0 <x <d) = −0.4 A exp (±jpx), indicating that the structure also supports −0.6 guided modes with real transverse wavenumbers. Similarly, solving the isotropic wave equation in the cladding and satis- −0.8 fying the electric and magnetic boundary conditions at x = 0 −1 −1 −0.50 0.51 1.52 and x = d show that the propagation constants of the odd guided modes with real transverse wavenumbers, TE ,are x(λ) 2n+1 found from the intersections of Figure 4: Normalized electric ﬁeld distribution of the odd plasmon zz modes of the anisotropic μ-negative slab waveguide with μ =−1, μ yy xx μ = 1, and = 2.25. v = u tan , 2 zz μ 2 (8) 2 2 yy 2 2 2 zz u − ξ v = 4π μ ξ − μ , 1 2 TE TE TE TE where u = pd and v = qd. The propagation constants of 1 2 3 4 the even guided modes, TE , are similarly found from the 2n intersections of μ u v =− u cot , zz v 10 μ 2 (9) yy 2 2 2 2 2 zz u − ξ v = 4π μ ξ − μ . 1 2 d = λ 1 2 TE The graphical solutions for modes with real transverse num- ber, depicted in Figure 5, indicate that the anisotropic μ- negative SW supports inﬁnite number of discrete guided 0 5 10 15 20 modes with real transverse wavenumbers at any operating u frequencies. The excitation of inﬁnite guided modes with real Figure 5: Graphical solution for modes with real transverse transverse wavenumbers can be troublesome in the design of 2 yy zz wavenumbers, v = qd, u = pd, p = (ξβ) − ω |μ |,and 2 2 any potential device applications that employ the ﬁeld con- q = β − ω μ . ﬁnement property of the plasmon modes. 1 v = qd E (normalized) E (normalized) y z(λ/40) 4 Advances in OptoElectronics 2.5 1.5 0.5 −0.5 Anisotropic μ-negative −2 −1 slab waveguide 0 −1.5 −2 Dielectric waveguide −2.5 The energy carried by plasmon Figure 7: FDTD simulations of the structure in Figure 6, showing modes is focused at the the power density distribution of the propagating guided waves in interface of the LH SW the dielectric slab waveguide and the anisotropic μ-negative slab waveguide (SW) of thickness λ. 2.5 Figure 6: Structure of the proposed signal detector with dielec- tric waveguide thickness of 0.5 λ and anisotropic μ-negative SW of thickness λ . 1.5 4. DEVICE APPLICATIONS One of the potential device applications of the anisotropic μ-negative SW is in the design of a new signal detector struc- 0.5 ture. To explain the concept of the proposed signal detector, assume a dielectric slab waveguide that is connected to an anisotropic μ-negative SW as shown in Figure 6. The dielec- 0 50 100 150 200 tric slab waveguide is 0.5λ wide and has a dielectric con- x(λ/20) stant of 2.25. The anisotropic μ-negativeSWwithwidthof TE mode in dielectric slab waveguide zz yy yy 0 λ , μ =−1, μ =−1, and = 2.25 is placed in the mid- Plasmon modes in left-handed slab waveguide dle of the dielectric slab waveguide, where the power den- sity of the TE mode is maximum. The structure is designed Figure 8: FDTD simulations show the power density distribution of to capture the incoming energy of the TE mode and con- the propagating guided waves in the dielectric slab waveguide and anisotropic μ-negative slab waveguide (SW) of thickness λ ,asde- ﬁne it at the interface of the anisotropic μ-negative SW. The picted in Figure 6. plasmon modes of the anisotropic μ-negative SW are then excited and travel along the structure. In a conventional sig- nal detector, the signal detection is performed in the center of the dielectric slab waveguide, where the power density of ω is deﬁned as the plasma resonant frequency, and the the TE mode is maximum, whereas in the proposed struc- medium exhibits negative permittivity and permeability for ture, the signal detection is performed at the interface of the frequencies below ω< ω . The structure is truncated by anisotropic μ-negative SW, where the power density of the using the split-ﬁeld perfectly matched layer (PML), which plasmon modes is maximum. The proposed structure local- can be conveniently combined with the Drude dispersion izes the incoming signal at the detector, which in turn in- model of the permittivity and permeability in the LHM. The creases the sensitivity of the detector. FDTD simulations shown in Figure 7 indicate that the pro- The proposed signal detector structure, shown in posed structure locally increases the power density of the Figure 6, is simulated using the FDTD program developed guided wave at the core-cladding interface of the anisotropic at the University of Waterloo to simulate various left-handed μ-negative SW. The steady-state ﬁeld distributions of the and anisotropic media. The Drude dispersion formulation is guided TE mode of the dielectric slab waveguide and plas- chosen to model the LHM. The permittivity and permeabil- mon modes of the μ-negative SW, depicted in Figure 8, ity of the negative medium are modeled by indicate that the proposed signal detector can detect sig- nals that are more than 3 dB weaker in comparison with their traditional counterparts. The anisotropic μ-negative = 1 − , r 0 SW also supports undesirable guided modes with real trans- (10) verse wavenumber. A simple overlap integral suggests that more than 95% of the incident power is coupled to the even μ = μ μ 1 − . r 0 and odd plasmon modes as indicated in Figures 9 and 10. x(λ/40) S (normalized) E (normalized) y Hamidreza Salehi et al. 5 3.5 2.5 11.52 2.53 00.02 0.04 0.06 0.08 0.1 Left-handed slab thickness (λ) w/λ Plasmon TE even Figure 11: Eﬀective width gain of the proposed signal detector as a Plasmon TE odd function of the probe eﬀective width, w, indicating that the sensi- tivity of the left-handed detector decreases as w increases. Figure 9: Percentage of the incident power coupled to the plasmon modes of the μ-negative slab waveguide (SW). 0.8 0.7 Interface B 0.6 0.5 Channel 2 0.4 Anisotropic μ-negative slab waveguide 0.3 Input Channel 1 0.2 0.1 Interface A 11.52 2.53 Left-handed slab thickness (λ) TE TE 1 3 Figure 12: Schematic of the proposed two-channel multiplexer. TE TE 4 2 The anisotropic μ-negative slab waveguide has a thickness of 0.5λ , yy zz xx length of L = 6.25λ , with μ =−1, μ = 1, and = 2.25. The 0 2 2 2 Figure 10: Percentage of the incident power coupled to the guided dielectric waveguides, located in the input, output channel one, and modes of the μ-negative slab waveguide (SW) with real transverse output channel two, have a thickness of 0.25λ with = 2.25. wavenumbers. In the proposed signal detector structure, the signal can be uide supports two guided modes with imaginary transverse detected by a solid state probe, such as a photodiode, that wavenumbers, namely, even and odd plasmon modes. As the is placed at the interface of the μ-negativeSW. Theeﬀective operating frequency increases, the propagation constants of width of the probe changes the sensitivity of the signal de- the even and odd plasmon modes converge towards a com- tector. Increasing the eﬀective width, w, of the photodiode mon value. The diﬀerence between the propagation con- decreases the eﬀective width gain of the proposed structure, stants of the even and odd plasmon modes as a function which is deﬁned as the ratio of the total power detected in of frequency is depicted in Figure 13. If the operating wave- the proposed signal detector to that of a conventional detec- length is greater than the anisotropic μ-negative SW thick- tor. Figure 11 shows the eﬀective width gain of the proposed ness, the even and odd plasmon modes interfere construc- structure as a function of the photodiode eﬀective width. tively at interface A of the structure in Figure 12,and de- Another potential device application of the anisotropic structively at interface B at the input waveguide junction. μ-negative SW is in the design of a two-channel harmonic The structure can be designed to guide the fundamental fre- separator multiplexer. The structure of the proposed two- quency, f , to channel 2 and its harmonics to channel 1. To channel multiplexer is depicted in Figure 12. The operation do so, the propagating wave with fundamental wavelength λ mechanism of the proposed structure is as follows. At any should interfere constructively at interface B of the structure operating frequency, the anisotropic μ-negative slab waveg- in Figure 12, while interfere destructively at interface A at Normalized power (%) Normalized power (%) Gain (dB) 6 Advances in OptoElectronics −0.1 1.5 Channel 2 1 −0.2 Anisotropic μ-negative slab 0.5 waveguide −0.3 Input Channel 1 −0.4 −0.5 −0.5 −1 −1.5 −0.6 −2 −0.7 0.30.40.50.60.70.80.91 Figure 15: Proposed two-channel harmonic separator multiplexer d/λ directing the ﬁrst harmonic of the fundamental frequency to chan- nel 1. Figure 13: Δ propagation constants of even and odd plasmon modes of the μ-negative slab waveguide (SW) as a function of the operating wavelength. port plasmon modes with imaginary transverse wavenum- ber. However, it is conﬁrmed that anisotropic μ-negative slab 1.5 waveguides can support plasmon modes, which localizes the ﬁeld at the interfaces of the waveguide. Two potential device applications of the anisotropic μ-negative slab waveguide are Channel 2 identiﬁed. An FDTD program is developed to analyze the 0.5 Anisotropic μ-negative slab performance of the proposed devices. waveguide Input Channel 1 0 REFERENCES −0.5 [1] V. Veselago, “The electrodynamics of substances with simul- taneously negative values of ε and μ,” Soviet Physics Uspekhi, −1 vol. 10, no. 4, pp. 509–514, 1968. [2] D.R.Smith,W.J.Padilla,D.C.Vier, S. C. Nemat-Nasser, and −1.5 S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Physical Review Letters, vol. 84, Figure 14: Proposed two-channel harmonic separator multiplexer no. 18, pp. 4184–4187, 2000. directing the fundamental frequency to channel 2. [3] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Transactions on Antennas and Propagation, vol. 51, no. 7, pp. 1516–1529, 2003. the output waveguide junction. 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