An Analytical Model for Characteristic Impedance in ... - IEEE Xplore
An Analytical Model for Characteristic Impedance in Nanostrip Plasmonic Waveguides Amir Hosseini, Hamid Nejati, and Yehia Massoud Department of Electrical and Computer Engineering, Rice University, Houston, TX, 77005 [email protected] Abstract— In this paper, we investigate the performance of a plasmonic nanostrip waveguide as a transmission line. We show that a nanostrip waveguide allows for subwavelength single-mode propagation. We present an analytical model of the characteristic impedance, which effectively captures the impacts of metal strip and dielectric film thicknesses as well as plasmonic effects. We show that the characteristic impedance model can be used as a reliable tool to design compact plasmonic components.
Waveguides that support subwavelength optical propagation are essential for enabling integration of nanometer-scale photonic devices. Experimental and theoretical studies have demonstrated the potential of plasmonic waveguides to support optical surface modes, known as Surface Plasmon Polaritons (SPPs), beyond the diffraction limit of light [1], [2]. Several plasmonic structures have been proposed for subwavelength transmission and manipulation of light such as channel waveguides [3], plasmon slot waveguides [4], Bragg reflectors [5], resonators, power divider [6] and micro-cavities [7]. So far, the proposed plasmonic couplers mainly belong to the mach-zehnder couplers class [8]. In mach-zehnder couplers, when implemented in a Metal-Insulator-Metal (MIM) configuration, the coupling occurs through the lossy metallic layer, which results in small coupling coefficients. Note that MIM configuration is necessary for true subwavelength confinement of light. MIM waveguides guide light in subwavelength scales with minimal field decay out of the waveguide physical cross section even for frequencies far from the plasmon resonance. Another coupler structure, branch coupler, is one of the basic components with wide range of circuit and system applications. Common applications are power monitoring and generation of a desired power division in beam-forming networks, realization of balanced circuits, matched attenuators and phase shifters. Branch couplers are also utilized as directional filters in multiplexers [9], [10]. 978-1-4244-1684-4/08/$25.00 ©2008 IEEE
FVH-FDM (d=25nm, t=20nm) FVH-FDM (d=30nm, t=25nm)
2.8 2.6 2.4
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I. I NTRODUCTION
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y 2 1.8 1.6
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Fig. 1. Variation of effective index for the fundamental and second modes versus nanostrip width for nanostrip waveguides of d = 25nm and t = 20nm (dotted line) and d = 30 nm and t = 25 nm (solid line). A schematic of the nanostrip waveguide is shown in the inset. Effective indices are calculated at λ = 1550 nm.
In this paper, we present an analytical model for the characteristic impedance of the nanostrip waveguide shown in the inset of Figure 1. Our model effectively captures the impact of geometrical parameters and plasmonic effects to present a transmission line model for the nanostrip waveguide. We show that the presented model can be utilized to design an optical 3dB branch coupler for maximum coupling and isolation at the telecommunication wavelength (λ = 1550 nm). The FiniteDifference Time-Domain (FDTD) simulation results show equal division of input power between the outputs as well as subwavelength light confinement. II. F UNDAMENTAL M ODE OF P LASMONIC NANOSTRIP WAVEGUIDE
A nanostrip structure, shown in the inset of Figure 1, is a planar transmission line consisting of a
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Fig. 2. Nanostrip waveguide field profiles of the fundamental mode with d = 25 nm, t = 20 nm and w = 60 nm, (a) Hx and (b) Hy . Hx field profile for nanostrip waveguide with d = 25 nm, t = 20 nm and w = 500 nm for (c) the fundamental mode and (d) the second mode. The field profiles are calculated for λ = 1550 nm.
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Fig. 3. Variation of modal size versus nanostrip width at λ = 1550 nm for nanostrip waveguide of d = 25 nm and t = 20 nm (dotted line) and d = 30 nm and t = 25 nm (solid line). The propagation length versus nanostrip width is shown in the inset.
damental mode for large w values. For waveguide and device applications, mono-mode propagation is usually an important constraint to be fulfilled. To ensure mono-mode propagation, we can choose w values much smaller than λ/2nef f,1 . Modal-size, defined as the square root of the area in which the power density is more than 1/e2 times its maximum value, can be used to quantify the light confinement in photonic components. Figure 3 shows the modal size variation for the nanostrip waveguide. Note that for large w where the fringe effects are not dominant, the modal size becomes almost a linear function of w. Figure 3 depicts that the nanostrip waveguide can effectively confine the light in subwavelength dimensions. Finally, the propagation loss does not affect the performance of structures with total length much smaller than the propagation length [1/(2k0 (nef f ))]. Figure 3 inset depicts the variation of nanostrip waveguide propagation length versus metallic strip width (w).
metallic strip with width w and thickness t lying on a dielectric thin film of thickness d. The dielectric film is supported by a metallic layer underneath. We choose all metallic materials to be silver characterized by the experimental data from [11]. Silicon dioxide (n = 1.5) is assumed for the dielectric layer. Effective index variations (nef f = β/k0 ) of the first two modes are shown in Figure 1, where β is the propagation constant and k0 is the vacuum wave-number. Note that a decrease in the dielectric layer thickness or the metallic strip width results in larger nef f values. In addition, for the nanostrip waveguide fundamental mode, the light confinement in the transverse direction increases nef f values compared to those in 2D MIM waveguides. The transverse magnetic field profiles of the nanostrip waveguide fundamental and the second modes are shown in Figure 2. For large enough metallic strip, the nanostrip waveguide supports multiple propagating modes, where the number of III. P LASMONIC NANOSTRIP WAVEGUIDE supported modes (N ) is given by the approximate C HARACTERISTIC I MPEDANCE expression, 2w If we let w in the inset of Figure 1 approach infinnef f,1 , (1) N< λ ity, the resulting structure becomes an MIM wavegwhich predicts a single mode propagation for w < uide. The concept of characteristic impedance (Z0 ) 300 nm. nef f,1 is the effective index of the fun- has been applied to MIM waveguides to improve the 2347
transmission coefficient of the MIM splitter [12]. For an MIM waveguide with a dielectric layer of refractive index n, Z0 associated with the supported transverse-magnetic (TM) mode is given as follows Ey d βM IM d = 2 . (2) Z0,M IM = Hx n ω0 Note that (2) is valid since electromagnetic field penetration into the metal exponentially decays from the dielectric-metal interface, and the field profile inside the dielectric region is almost uniform at wavelengths far from the surface plasmon resonance. In addition, (2) is the general form of the MIM waveguide characteristic impedance derived in [12]. Similarly, we can define characteristic impedance for nanostrip waveguides. Since the fundamental nanostrip mode is hybrid, the character= istic impedance becomes Z0 = V /I ( l E.dl)/( c H.dl), where s and c are an arbitrary path connecting the metal layer to the metal strip and a loop enclosing the metallic strip, respectively. In a discretized computational window in the FVHFDM simulations, averaging on integrals calculated along several different paths improves the accuracy of Z0 calculations. Note that the difference between these nanostrip Z0 and microstrip Z0 increases for smaller w, where the nef f values are larger. This behavior of plasmonic waveguides is fundamentally different from conventional waveguides, where nef f remains constant or decreases as the waveguide dimensions shrink. We can capture the effect of plasmonic effects by incorporating nef f in nanostrip Z0 expression as follows:
values in (3) can be accurately calculated using the Effective Index Method (EIM) [14]. The characteristic impedance is developed by a non-linear least square regression to the FVH-FDM data. A range of parameters (50 nm < w < 500 nm, 20 nm < d < 40 nm and 15 nm < t < 35 nm) is explored for fitting the FVH-FDM data to the proposed nanostrip impedance model at λ = 1550 nm with a maximum error of 8 %. IV. A C OMPACT 3dB B RANCH C OUPLER D ESIGN
A branch coupler is a four-port directional coupler that divides the input port power (1) between two output ports (2 and 3), while the forth port (4) is isolated [see Figure 4(a)]. A branch coupler structure usually consists of two main transmission lines that are shunt-connected by two secondary branch lines, which lead to a symmetrical fourport structure. Branch couplers can be characterized using the scattering matrix elements. These elements determine reflection (S11), coupling (S13 ), directivity (S13 /S14 ), isolation (S14 ) and power split ratio (S13 /S12 ). In the case of 3dB branch couplers |S13 /S12 | = 1 at the central wavelength. Equations (3) and (4) can be used to find the optimum values for the physical dimensions of the branches satisfying the matching and maximum coupling conditions. Matching condition at the input port holds when 1/Z02 =1/Zs2 -1/Zp2 , where the Z0 is the characteristic impedance of the input/output ports, Zs is the series branch characteristic impedance and Zp is the shunt branch characteristic impedance. Assuming Zp = Z0 , we can choose √ Zs be equal to Z0 / 2 to fulfill the matching and maximum coupling conditions at the same time. The coupler structure in Figure 4 is designed for w −1 η wef f ef f a central wavelength of 1550 nm. The input and + Q + R ln +S , Z0 = nef f d d output waveguides are 50 nm wide, and d = 25 nm (3) and t = 20 nm is assumed for whole structure. The where series and parallel branch lines are chosen to be 2d P wef f = w + t 1 + ln( ) . (4) 95 nm and 50 nm wide, which correspond to 17.5 Ω π t and 22.7 Ω characteristic impedances, respectively. In (3) and (4), P = 1.050, Q = 2.959, R = 0.575, The lengths of the series and shunt branches are S = 1.030 and w/d > 1.5 are assumed. While wef f equal to l = λg /4, where λg = λ/nef f is the guiding encapsulates the effects of finite t, nef f accounts wavelength of the supported waveguide mode. We have used FDTD method with perfectly for the fringing fields and asymmetric waveguide structure as well as plasmonic effects. In order to matched layers [15] to model light transmission avoid expensive FVH-FDM simulations, the nef f through the designed coupler structure. Power pro2348
R EFERENCES
1
2
150
[1] R. Zia, J. A. Schuller, and M. L. Brongersma, “ Nearfield characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B, vol. 74, p. 165415, 2006.
z 50
145
[2] J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B, vol. 73, 2006.
x 50
95
4
3
Fig. 4. Top view of the nanostrip 3dB branch line coupler, which is designed and optimized for optical frequency range. all numbers are in nanometer.
[3] D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Appl. Phys. Lett., vol. 30, no. 10, pp. 1186–1188, August 2005. [4] I. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express., vol. 13, pp. 6645–6650, 2005. [5] A. Hosseini and Y. Massoud, “A low-loss metal-insulatormetal plasmonic bragg reflector,” Opt. Express., vol. 14, pp. 11 318–11 323, 2006. [6] S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature, vol. 440, pp. 508–511, 2006. [7] A. Hosseini and Y. Massoud, “Optical range microcavities and filters using multiple dielectric layers in metalinsulator-metal structures,” J. Opt. Soc. Am. A, vol. 24, pp. 221–224, 2007. [8] B. Wang and G. Wang, “Surface plasmon polariton propagation in nanoscale metal gap waveguides,” Opt. Lett., vol. 29, pp. 1992–1994, 2004.
Fig. 5. FDTD simulation results of the nanostrip 3dB branch coupler with power transmission (Pz ) at λ0 = 1550 nm.
[9] C. Pon, “Hybrid-ring directional coupler for arbitrary power division,” IEEE Trans. Microw. Theory Tech., vol. MTT-9, no. 6, p. 529535, 1961. [10] Y. Sung, C.S.Ahn, and Y. Kim, “Branch-line couplers
file (Pz ) of the structure at λ0 = 1550 nm is shown using defected ground structure,” IEICE Trans. Commun., in Figure 5. The incident wave is almost equally vol. E88B, no. 4, pp. 1665–1667, 2005. distributed between the output branches (|S13 /S12 | = [11] P. B. Johnson and R. W. Christy, “Optical Constants of 1.07 at λ0 = 1550 nm) , while the port 4 bears the Noble Metals,” Phys. Rev. B, 1972. negligible output power. [12] G. Veronis and S. Fan, “Bends and splitters in metal-
dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett., vol. 87, no. 13, p. 131102, 2005.
V. C ONCLUSION
[13] J. Hong and M. Lancaster, Microstrip filters for
In this paper, we investigated the performance RF/microwave applications. Wiley New York, 2001. of a plasmonic nanostrip waveguide as a transmis[14] S. Bozhevolnyi, V. Volkov, E. Devaux, and T. Ebbesen, sion line. We presented an analytical model of the “Channel plasmon-polariton guiding by subwavelength characteristic impedance, which effectively captures metal grooves,” Phys. Rev. Lett., vol. 95, p. 046802(4), the impacts of metal strip width and dielectric film 2005. thickness as well as plasmonic effects. We showed [15] C. Rappaport, M. Kilmer, and E. Miller, “Accuracy that the characteristic impedance model can be used considerations in using the pml abc with fdfd helmholtz equation computation,” Inter. J. Num. Model., vol. 13, pp. as a reliable tool to design compact plasmonic 471–482, 2000. components. 2349
Waveguides that support subwavelength optical propagation are essential for enabling integration of nanometer-scale photonic devices. Experimental and theoretical studies have demonstrated the potential of plasmonic waveguides to support optical surface modes, known as Surface Plasmon Polaritons (SPPs), beyond the diffraction limit of light [1], [2]. Several plasmonic structures have been proposed for subwavelength transmission and manipulation of light such as channel waveguides [3], plasmon slot waveguides [4], Bragg reflectors [5], resonators, power divider [6] and micro-cavities [7]. So far, the proposed plasmonic couplers mainly belong to the mach-zehnder couplers class [8]. In mach-zehnder couplers, when implemented in a Metal-Insulator-Metal (MIM) configuration, the coupling occurs through the lossy metallic layer, which results in small coupling coefficients. Note that MIM configuration is necessary for true subwavelength confinement of light. MIM waveguides guide light in subwavelength scales with minimal field decay out of the waveguide physical cross section even for frequencies far from the plasmon resonance. Another coupler structure, branch coupler, is one of the basic components with wide range of circuit and system applications. Common applications are power monitoring and generation of a desired power division in beam-forming networks, realization of balanced circuits, matched attenuators and phase shifters. Branch couplers are also utilized as directional filters in multiplexers [9], [10]. 978-1-4244-1684-4/08/$25.00 ©2008 IEEE
FVH-FDM (d=25nm, t=20nm) FVH-FDM (d=30nm, t=25nm)
2.8 2.6 2.4
Real (Neff)
I. I NTRODUCTION
3
2.2
1st mode
y 2 1.8 1.6
metal strip
x
w
dielectric layer metal film
t
2nd mode
d
1.4
substrate 1.2 1 50
100
150
200
250
300
350
400
450
500
Width (nm)
Fig. 1. Variation of effective index for the fundamental and second modes versus nanostrip width for nanostrip waveguides of d = 25nm and t = 20nm (dotted line) and d = 30 nm and t = 25 nm (solid line). A schematic of the nanostrip waveguide is shown in the inset. Effective indices are calculated at λ = 1550 nm.
In this paper, we present an analytical model for the characteristic impedance of the nanostrip waveguide shown in the inset of Figure 1. Our model effectively captures the impact of geometrical parameters and plasmonic effects to present a transmission line model for the nanostrip waveguide. We show that the presented model can be utilized to design an optical 3dB branch coupler for maximum coupling and isolation at the telecommunication wavelength (λ = 1550 nm). The FiniteDifference Time-Domain (FDTD) simulation results show equal division of input power between the outputs as well as subwavelength light confinement. II. F UNDAMENTAL M ODE OF P LASMONIC NANOSTRIP WAVEGUIDE
A nanostrip structure, shown in the inset of Figure 1, is a planar transmission line consisting of a
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0
0.5
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Fig. 2. Nanostrip waveguide field profiles of the fundamental mode with d = 25 nm, t = 20 nm and w = 60 nm, (a) Hx and (b) Hy . Hx field profile for nanostrip waveguide with d = 25 nm, t = 20 nm and w = 500 nm for (c) the fundamental mode and (d) the second mode. The field profiles are calculated for λ = 1550 nm.
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Fig. 3. Variation of modal size versus nanostrip width at λ = 1550 nm for nanostrip waveguide of d = 25 nm and t = 20 nm (dotted line) and d = 30 nm and t = 25 nm (solid line). The propagation length versus nanostrip width is shown in the inset.
damental mode for large w values. For waveguide and device applications, mono-mode propagation is usually an important constraint to be fulfilled. To ensure mono-mode propagation, we can choose w values much smaller than λ/2nef f,1 . Modal-size, defined as the square root of the area in which the power density is more than 1/e2 times its maximum value, can be used to quantify the light confinement in photonic components. Figure 3 shows the modal size variation for the nanostrip waveguide. Note that for large w where the fringe effects are not dominant, the modal size becomes almost a linear function of w. Figure 3 depicts that the nanostrip waveguide can effectively confine the light in subwavelength dimensions. Finally, the propagation loss does not affect the performance of structures with total length much smaller than the propagation length [1/(2k0 (nef f ))]. Figure 3 inset depicts the variation of nanostrip waveguide propagation length versus metallic strip width (w).
metallic strip with width w and thickness t lying on a dielectric thin film of thickness d. The dielectric film is supported by a metallic layer underneath. We choose all metallic materials to be silver characterized by the experimental data from [11]. Silicon dioxide (n = 1.5) is assumed for the dielectric layer. Effective index variations (nef f = β/k0 ) of the first two modes are shown in Figure 1, where β is the propagation constant and k0 is the vacuum wave-number. Note that a decrease in the dielectric layer thickness or the metallic strip width results in larger nef f values. In addition, for the nanostrip waveguide fundamental mode, the light confinement in the transverse direction increases nef f values compared to those in 2D MIM waveguides. The transverse magnetic field profiles of the nanostrip waveguide fundamental and the second modes are shown in Figure 2. For large enough metallic strip, the nanostrip waveguide supports multiple propagating modes, where the number of III. P LASMONIC NANOSTRIP WAVEGUIDE supported modes (N ) is given by the approximate C HARACTERISTIC I MPEDANCE expression, 2w If we let w in the inset of Figure 1 approach infinnef f,1 , (1) N< λ ity, the resulting structure becomes an MIM wavegwhich predicts a single mode propagation for w < uide. The concept of characteristic impedance (Z0 ) 300 nm. nef f,1 is the effective index of the fun- has been applied to MIM waveguides to improve the 2347
transmission coefficient of the MIM splitter [12]. For an MIM waveguide with a dielectric layer of refractive index n, Z0 associated with the supported transverse-magnetic (TM) mode is given as follows Ey d βM IM d = 2 . (2) Z0,M IM = Hx n ω0 Note that (2) is valid since electromagnetic field penetration into the metal exponentially decays from the dielectric-metal interface, and the field profile inside the dielectric region is almost uniform at wavelengths far from the surface plasmon resonance. In addition, (2) is the general form of the MIM waveguide characteristic impedance derived in [12]. Similarly, we can define characteristic impedance for nanostrip waveguides. Since the fundamental nanostrip mode is hybrid, the character= istic impedance becomes Z0 = V /I ( l E.dl)/( c H.dl), where s and c are an arbitrary path connecting the metal layer to the metal strip and a loop enclosing the metallic strip, respectively. In a discretized computational window in the FVHFDM simulations, averaging on integrals calculated along several different paths improves the accuracy of Z0 calculations. Note that the difference between these nanostrip Z0 and microstrip Z0 increases for smaller w, where the nef f values are larger. This behavior of plasmonic waveguides is fundamentally different from conventional waveguides, where nef f remains constant or decreases as the waveguide dimensions shrink. We can capture the effect of plasmonic effects by incorporating nef f in nanostrip Z0 expression as follows:
values in (3) can be accurately calculated using the Effective Index Method (EIM) [14]. The characteristic impedance is developed by a non-linear least square regression to the FVH-FDM data. A range of parameters (50 nm < w < 500 nm, 20 nm < d < 40 nm and 15 nm < t < 35 nm) is explored for fitting the FVH-FDM data to the proposed nanostrip impedance model at λ = 1550 nm with a maximum error of 8 %. IV. A C OMPACT 3dB B RANCH C OUPLER D ESIGN
A branch coupler is a four-port directional coupler that divides the input port power (1) between two output ports (2 and 3), while the forth port (4) is isolated [see Figure 4(a)]. A branch coupler structure usually consists of two main transmission lines that are shunt-connected by two secondary branch lines, which lead to a symmetrical fourport structure. Branch couplers can be characterized using the scattering matrix elements. These elements determine reflection (S11), coupling (S13 ), directivity (S13 /S14 ), isolation (S14 ) and power split ratio (S13 /S12 ). In the case of 3dB branch couplers |S13 /S12 | = 1 at the central wavelength. Equations (3) and (4) can be used to find the optimum values for the physical dimensions of the branches satisfying the matching and maximum coupling conditions. Matching condition at the input port holds when 1/Z02 =1/Zs2 -1/Zp2 , where the Z0 is the characteristic impedance of the input/output ports, Zs is the series branch characteristic impedance and Zp is the shunt branch characteristic impedance. Assuming Zp = Z0 , we can choose √ Zs be equal to Z0 / 2 to fulfill the matching and maximum coupling conditions at the same time. The coupler structure in Figure 4 is designed for w −1 η wef f ef f a central wavelength of 1550 nm. The input and + Q + R ln +S , Z0 = nef f d d output waveguides are 50 nm wide, and d = 25 nm (3) and t = 20 nm is assumed for whole structure. The where series and parallel branch lines are chosen to be 2d P wef f = w + t 1 + ln( ) . (4) 95 nm and 50 nm wide, which correspond to 17.5 Ω π t and 22.7 Ω characteristic impedances, respectively. In (3) and (4), P = 1.050, Q = 2.959, R = 0.575, The lengths of the series and shunt branches are S = 1.030 and w/d > 1.5 are assumed. While wef f equal to l = λg /4, where λg = λ/nef f is the guiding encapsulates the effects of finite t, nef f accounts wavelength of the supported waveguide mode. We have used FDTD method with perfectly for the fringing fields and asymmetric waveguide structure as well as plasmonic effects. In order to matched layers [15] to model light transmission avoid expensive FVH-FDM simulations, the nef f through the designed coupler structure. Power pro2348
R EFERENCES
1
2
150
[1] R. Zia, J. A. Schuller, and M. L. Brongersma, “ Nearfield characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B, vol. 74, p. 165415, 2006.
z 50
145
[2] J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B, vol. 73, 2006.
x 50
95
4
3
Fig. 4. Top view of the nanostrip 3dB branch line coupler, which is designed and optimized for optical frequency range. all numbers are in nanometer.
[3] D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Appl. Phys. Lett., vol. 30, no. 10, pp. 1186–1188, August 2005. [4] I. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express., vol. 13, pp. 6645–6650, 2005. [5] A. Hosseini and Y. Massoud, “A low-loss metal-insulatormetal plasmonic bragg reflector,” Opt. Express., vol. 14, pp. 11 318–11 323, 2006. [6] S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature, vol. 440, pp. 508–511, 2006. [7] A. Hosseini and Y. Massoud, “Optical range microcavities and filters using multiple dielectric layers in metalinsulator-metal structures,” J. Opt. Soc. Am. A, vol. 24, pp. 221–224, 2007. [8] B. Wang and G. Wang, “Surface plasmon polariton propagation in nanoscale metal gap waveguides,” Opt. Lett., vol. 29, pp. 1992–1994, 2004.
Fig. 5. FDTD simulation results of the nanostrip 3dB branch coupler with power transmission (Pz ) at λ0 = 1550 nm.
[9] C. Pon, “Hybrid-ring directional coupler for arbitrary power division,” IEEE Trans. Microw. Theory Tech., vol. MTT-9, no. 6, p. 529535, 1961. [10] Y. Sung, C.S.Ahn, and Y. Kim, “Branch-line couplers
file (Pz ) of the structure at λ0 = 1550 nm is shown using defected ground structure,” IEICE Trans. Commun., in Figure 5. The incident wave is almost equally vol. E88B, no. 4, pp. 1665–1667, 2005. distributed between the output branches (|S13 /S12 | = [11] P. B. Johnson and R. W. Christy, “Optical Constants of 1.07 at λ0 = 1550 nm) , while the port 4 bears the Noble Metals,” Phys. Rev. B, 1972. negligible output power. [12] G. Veronis and S. Fan, “Bends and splitters in metal-
dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett., vol. 87, no. 13, p. 131102, 2005.
V. C ONCLUSION
[13] J. Hong and M. Lancaster, Microstrip filters for
In this paper, we investigated the performance RF/microwave applications. Wiley New York, 2001. of a plasmonic nanostrip waveguide as a transmis[14] S. Bozhevolnyi, V. Volkov, E. Devaux, and T. Ebbesen, sion line. We presented an analytical model of the “Channel plasmon-polariton guiding by subwavelength characteristic impedance, which effectively captures metal grooves,” Phys. Rev. Lett., vol. 95, p. 046802(4), the impacts of metal strip width and dielectric film 2005. thickness as well as plasmonic effects. We showed [15] C. Rappaport, M. Kilmer, and E. Miller, “Accuracy that the characteristic impedance model can be used considerations in using the pml abc with fdfd helmholtz equation computation,” Inter. J. Num. Model., vol. 13, pp. as a reliable tool to design compact plasmonic 471–482, 2000. components. 2349
