Subwavelength guiding of channel plasmon polaritons in a semiconductor at terahertz frequencies Fangming Zhu,1,* Xiaoer Li,2 and Linfang Shen3 1
School of Information Science and Engineering, Hangzhou Normal University, Hangzhou 310036, China
2
Department of Information Science and Electronic Engineering, Zhejiang University, Zhejiang Province, Hangzhou 310027, China 3
The Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China *Corresponding author: [email protected] Received 9 April 2014; revised 25 July 2014; accepted 30 July 2014; posted 5 August 2014 (Doc. ID 209284); published 4 September 2014
We present a numerical investigation of terahertz channel plasmon polaritons (CPPs) propagating in a semiconductor InSb. It is shown that these CPPs can simultaneously exhibit subwavelength field confinement and relatively long propagation length. Moreover, single-mode propagation is available for terahertz CPPs in a certain frequency range. © 2014 Optical Society of America OCIS codes: (230.7380) Waveguides, channeled; (240.6680) Surface plasmons; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.53.005896
1. Introduction
Surface plasmon polaritons (SPPs) are electromagnetic waves coupled to free electron oscillations at the interface between metal and a dielectric [1]. These waves propagate along the interface, decaying evanescently away from it. The unique properties of 2D nature under certain conditions make them quite suitable for applications in subwavelength optics [2]. Most research on SPPs has been done on optical and near-infrared frequencies with metal by manipulating the SPPs at the nanoscale [3]. However, not only metals support electron plasmon oscillations, semiconductors also can support SPPs. Free carriers can be excited either thermally or by doping, which results in a plasma frequency typically in the terahertz domain. THz plasmonics with InSb structures has been proposed and investigated in various systems such as hole arrays [4], gratings [5,6], slits [7], slot cavities [8], and plasmonic antennas [9]. InSb behaves at THz frequencies as a metal at visible 1559-128X/14/265896-05$15.00/0 © 2014 Optical Society of America 5896
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frequencies, i.e., it has a negative real component of permittivity. This behavior can be tuned by modifying the number of excited electrons into the conduction band, i.e., the free carrier density. For example, the permittivity of InSb becomes less negative with smaller jεj when the temperature decreases, due to the decrease in the free carrier density, which leads to lower plasma frequency. Moreover, as the temperature decreases, the electron mobility in InSb significantly increases; then its scattering frequency decreases. As a result, in a lower temperature, SPPs in a semiconductor can not only be more effectively confined but also propagate over a longer distance. The properties of terahertz SPPs guiding in a semiconductor surface (3D system) have received even less attention. In contrast, various schemes for guiding visible SPPs have been proposed; among them, channel plasmon polaritons (CPPs) by triangular-shaped (V-shaped) grooves [10–15] exhibit superior features such as relatively low propagation loss under subwavelength field confinement, efficient transmission through sharp bends, and compatibility with the planar technology. In this paper, we present a numerical investigation of
CPPs guided by V-shaped grooves carved in InSb, showing that CPPs on a semiconductor surface can be tightly localized and simultaneously possess a relatively long propagation length at terahertz frequencies. 2. Numerical Analysis
The structure studied here is a V-shaped groove on an InSb surface, as illustrated in Fig. 1 (see the inset). The geometric parameters characterizing this structure are groove angle θ and groove depth h. The dielectric function of InSb semiconductor in the terahertz frequency range is approximately characterized by a Drude model: εω εlattice −
ω2p : ωω iγ
(1)
For InSb, the lattice permittivity εlattice is taken to be 15.6, and Drude parameters are ωp 46 × 1012 rad∕s and γ 0.3 × 1012 rad∕s [16,17], resulting in the permittivity of ε −37.88 2.55i at f 1.0 THz. It should be noted that Drude parameters may vary considerably between semiconductor samples, owing to variations in impurity density. To understand the CPP properties guiding in the proposed structure, we first analyze the mode effective index for the fundamental mode of CPPs. The CPP effective refractive index is defined as N eff λ∕2πReβ, where β is the propagation constant, and λ is the wavelength in the air. Among the parameters of width, depth, and angle of the V groove, the angle is the main parameter that may largely influence the mode index for a relatively large V groove [18]. The depth/width of the V groove influences the propagation properties only near the cutoff, when the CPP mode begins to be hybridized with a wedge mode at the groove edges. So different groove angles of θ 15°, 20°, and 25° are analyzed, while the a
θ
h
θ θ θ
o o
o
Fig. 1. Effective index of the fundamental CPP mode versus frequency for different groove angles θ 25°, 20°, and 15°. The groove depth is kept at h 750 μm. Inset shows the schematic of the V-shaped groove in InSb.
groove depth is fixed at h 750 μm. We perform a rigorous numerical calculation of the CPP mode using the full-vector finite element method. To obtain better convergence of the calculated fields, the tip and edges of the V-shaped semiconductor groove are rounded by the curvature radius of r 10 and R 100 μm, respectively. Figure 1 shows the mode effective index as a function of frequency (f ) for the InSb V grooves with three groove angles. The effective index of a flat SPP mode is also shown in the figure for comparison. As seen in Fig. 1, the CPP effective index increases when the frequency grows for each groove angle. It is known that, for a CPP mode, a larger effective index corresponds to a smaller penetration depth in the dielectric (air), so the larger mode index means a stronger field confinement. When the frequency decreases closest to the cut-off frequency, the CPP effective index approaches that of air, indicating the tendency of the CPP mode field to extend gradually outside the groove. Furthermore, for a certain frequency, the CPP effective index increases with the decrease of the groove angle. This means that the CPP mode is more tightly confined in the structure with a smaller groove angle. The cut-off frequency also can be easily read out from Fig. 1. For the groove angle of θ 25°, the cut-off frequency is 0.5 THz, for θ 20°, the cutoff is 0.45 THz, and, for θ 15°, it becomes 0.4 THz. This shows that the groove angle also affects the cut-off frequency of the CPP mode. The cut-off frequency shifts to lower frequencies when the groove angle becomes smaller, indicating a larger frequency band can be achieved for the smaller groove angle. To examine the influence of absorption of a semiconductor at terahertz frequencies on the performance of CPPs, we calculated the propagation length of the fundamental CPP mode as a function of frequency for different groove angles. Figure 2(a) shows the normalized propagation length (L∕λ) versus frequency (f ) for the three different groove angles in Fig. 1. The propagation length for CPPs is determined by L 1∕2∕Imβ, where β is the propagation constant of CPPs. We find that the propagation length decreases as the frequency increases for each groove angle. For a certain frequency, the propagation length also decreases with the decrease of the groove angle. For example, the CPP mode has the propagation length of L 31λ for θ 25°, L 24λ for θ 20°, and L 18λ for θ 15° at f 1.0 THz (λ 300 μm). For the SPP (or CPP) mode, it is physically common that the propagation length decreases as the field confinement increases, and the latter corresponds to the increase of mode index, so the results shown in Fig. 2(a) agree well with those in Fig. 1. To well illustrate the field confinement of the CPP mode, here we introduce the definition of effective mode area Am (the third type A3 in [19]). It is defined as a minimum area in which a set proportion, η (usually η 0.5 is used), of a mode’s energy resides. Unlike the commonly used method of evaluating the mode area based on the region in which the mode 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
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θ=25 o θ=20 o θ=15 o
(b)
Am/A0
θ=25 o θ=20 o θ=15 o
(a)
Fig. 2. Normalized propagation length (a) and normalized effective mode area (b) versus frequency for different groove angles θ 25°, 20°, and 15°. The groove depth is h 750 μm for all three cases.
energy drops to e−2 of its peak value, Am is quite little sensitive to the local field confinement with rapid field variations. Furthermore, it is also geometry independent to the measure of confinement owing to the generic feature of η to any field distribution. It is defined by solving the following minimization problem: Z Am min f rdA; f r A∞ Z s:t: f r − ηWrdA 0; A∞
(2)
where f r is the shape function that encompasses a fraction η of the mode’s energy and η 0.5 is used in our study. Assuming that the optimum f r is a step function bounded by a constant energy density contour, W 0 , of the mode, Am can be found out by iteratively solving the minimization problem of Eq. (2) using:
5898
f r 0;
if
Wr < W 0 ;
f r 1;
if
Wr ≥ W 0 ;
(3)
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
where W 0 corresponds to the contour containing η of the mode’s energy. Figure 2(b) shows the normalized effective mode area Am ∕A0 versus the frequency for the InSb V grooves with three different groove angles, where A0 λ∕22 is the diffraction limited area of vacuum. It can be seen that the normalized effective mode area generally decreases as the frequency increases for each groove angle, indicating the field confinement increasing with the frequency. As this variation of Am ∕A0 with the frequency is similar to the normalized propagation length (L∕λ) in Fig. 2(a), there exists a trade-off between the field confinement and propagation loss. However, it is slightly surprising that, at low frequencies close to the cutoff, the value of Am ∕A0 increases with the frequency. This should be attributed to the worse field confinement, and, in this situation, the field energy is no longer concentrated within the groove, and its large portion is distributed around the groove edges. This can be clearly seen from the contour line of W 0 in Fig. 3 (for the case of f 0.6 THz), where the modal field pattern is presented. For a certain frequency, the normalized effective mode area decreases with the decrease of the groove angle, indicating that better field confinement can be achieved with a smaller groove angle. For f 1.0 THz (λ 300 μm), the normalized effective mode area is Am ∕A0 1.12 for the groove angle of θ 25°, it is drastically decreased to Am ∕A0 0.64 for θ 20°, and it reaches to Am ∕A0 0.40 for θ 15°, showing that the groove angle influences the field confinement greatly. Figure 3 shows the electric-field (E) amplitude of the CPP mode, where the dotted line marks the contour line of W 0, which contains η 0.5 of the mode’s energy. The parameters of the structure are as follows: θ 20°, h 750 μm, and a 264 μm. From Fig. 3, it is obvious that the field confinement enhances as the frequency increases. For f 0.6 THz, the E-field extends outside the groove, and obviously the CPP mode is hybridized with wedge plasmon polaritons (WPPs) running along the groove edges. The calculated normalized effective mode area is Am ∕A0 0.92 for this frequency. So, at lower frequencies, CPPs lose their confinement character. As with the frequencies increased, the field confinement also increases, which makes the modal field well localized within the groove. Evidently, only in this situation, the defined mode area can well reflect 0.6 THz
0.8 THz
1.0 THz
1.2 THz
Fig. 3. Distribution of the E-field amplitude of the fundamental CPP mode at different frequencies of f 0.6, 0.8, 1.0, and 1.2 THz. The dotted line marks the contour line of the critical energy density W 0. The groove angle is θ 20°, and the groove depth h 750 μm.
the field confinement. Our numerical calculation shows that Am ∕A0 0.97 for f 0.8 THz, Am ∕A0 0.64 for f 1.0 THz, and Am ∕A0 0.46 for f 1.2 THz. As expected, the normalized effective mode area decreases with the frequency, verifying the field confinement increases with the frequency. This phenomenon is more clearly indicated by the contour line of W 0 (white dotted line) in Fig. 3. It can be seen that, for f 0.6 THz, the contour line of W 0 is broken into several pieces, which clearly indicates the excitation of WPPs at the groove edge and its hybridization with the CPP field. At f 1.2 THz, the CPP mode is deeply subwavelength confined, and, in such situation, it still has a propagation length of L 15λ, indicating the V groove in a semiconductor is a good choice for a terahertz plasmonic waveguide. To vividly demonstrate the guiding properties of the V groove in InSb, we conduct FDTD simulation for the wave transmission along the groove structure with h 750 μm and θ 20°. The waveguide length is taken to be 5000 μm (equals to 10λ at f 0.6 THz). In the simulation, the waveguide is excited using an input port with a pulse of the TE fundamental mode. The spatial variation of the E-field amplitudes associated with CPPs for different frequencies is plotted in Figs. 4(a)–4(d), representing the vertical cutting slice (halving the structure). It can be seen that, for f 0.6 THz, CPPs propagate along the V groove with a looser field confinement. For f 0.8 THz, CPPs propagate with much better field confinement, verifying the field confinement improvement with the increased frequency. When the frequency increases to f 1.0 THz, CPPs almost propagate along the bottom of the V groove, showing the field being highly confined to the groove bottom. For f 1.2 THz, it exhibits the strongest confinement, but the field attenuates significantly over the propagation distance.
For a plasmonic waveguide, single-mode operation is important, so we further numerically investigate the modal characteristics of CPPs in the V groove. As a typical example, the groove parameters are h 750 μm and θ 20°. Figure 5 shows the effective index for the two lowest-order CPP modes. The fundamental mode (solid line with circles) has a cutoff at f c ≈ 0.45 THz, and the second-order mode (solid line with triangles) has a cutoff at f c ≈ 0.8 THz, so the single-mode propagation is available for CPPs in the terahertz frequency range of 0.45–0.8 THz. The modal shapes of the two modes are shown in the insets for f 1.0 THz, and the fundamental one exhibits much better confinement, as expected. Finally, we give a short discussion about the effect of the curvature radius of tip and edges of the groove on the CPP mode. The tip size may greatly influence the properties of the CPP mode at high frequencies with tight field confinement, due to the energy being concentrated there. The mode index and the propagation length for different radii of rounded tip r 5, 10, and 15 μm are calculated for f 1.0 THz, while the curvature radius of the groove edges is fixed to be R 100 μm. For r 5 μm, we find that N eff 1.064 and L 17.7λ. For r 10 μm, N eff 1.055 and L 23.5λ, and for r 15 μm, N eff 1.050 and L 27.8λ. Clearly, as r increases, the mode index decreases while the propagation length increases, indicating the field confinement decreases with increasing r. The curvature radius of edges might have influence on the CPP mode at lower frequencies closest to the cut-off frequency, when the field extends outside the groove and is hybridized with the edge WPPs. We have calculated the mode index and propagation length for different curvature radii R 50 and R 100 μm and found that N eff and L are almost the same for both cases, showing that the edge curvature radius has little effect on the property of the CPP mode.
λ=500 µm
CPP 1st
CPP 2 nd
st nd
(a)
0.6 THz
0.8 THz
(b)
(c)
1.0 THz
1.2 THz
(d)
Fig. 4. Spatial variation of the E-field amplitudes associated with the CPP transmission for different frequencies: (a) 0.6 THz, (b) 0.8 THz, (c) 1 THz, and (d) 1.2 THz. Geometric parameters are as follows: θ 20°, h 750 μm, and the waveguide length is 5000 μm.
Fig. 5. Effective index as a function of frequency for the first and second mode of CPPs. Insets show the E-field amplitude of the two CPP modes at f 1.0 THz. The parameters of the V groove are the same as in Fig. 3. 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
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3. Conclusion
We have presented a numerical study of CPP propagation along a semiconductor InSb surface at terahertz frequencies. Our numerical simulations have shown that CPPs sustained by an InSb V groove can be guided on a subwavelength scale. These CPPs have relatively long propagation length under subwavelength confinement, while single-mode propagation is still available in the terahertz range. Our results show that CPPs have promising applications in the THz regime. This work was supported by the National Natural Science Foundation of China under grant no. 61372005. References 1. H. Raether, Surface Plasmons (Springer, 1988). 2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 3. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nanooptics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). 4. J. Gómez Rivas, C. Janke, P. Haring-Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express 13, 847–859 (2005). 5. J. Gómez Rivas, M. Kuttge, H. Kurz, P. Haring-Bolivar, and J. A. Sánchez-Gil, “Low-frequency active surface plasmon optics on semiconductors,” Appl. Phys. Lett. 88, 082106 (2006). 6. R. Parthasarathy, A. Bykhovski, B. Gelmont, T. Globus, N. Swami, and D. Woolard, “Enhanced coupling of subterahertz radiation with semiconductor periodic slot arrays,” Phys. Rev. Lett. 98, 153906 (2007). 7. T. H. Isaac, J. Gómez Rivas, J. R. Sambles, W. L. Barnes, and E. Hendry, “Surface plasmon mediated transmission of subwavelength slits at THz frequencies,” Phys. Rev. B 77, 113411 (2008).
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8. J. Li and K. J. Webb, “Terahertz field enhancement in doped semiconductor slot cavities,” J. Appl. Phys. 106, 124901 (2009). 9. V. Giannini, A. Berrier, S. A. Maier, J. A. Sánchez-Gil, and J. Gómez Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010). 10. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069– 1071 (2004). 11. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002). 12. D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30, 1186–1188 (2005). 13. 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 440, 508–511 (2006). 14. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. MartinMoreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006). 15. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005). 16. S. C. Howells and L. A. Schlie, “Transient terahertz reflection spectroscopy of undoped InSb from 0.1 to 1.1 THz,” Appl. Phys. Lett. 69, 550–552 (1996). 17. T. H. Isaac, W. L. Barnes, and E. Hendry, “Determining the terahertz optical properties of subwavelength films using semiconductor surface plasmons,” Appl. Phys. Lett. 93, 241115 (2008). 18. S. I. Bozhevolnyi and K. V. Nerkararyan, “Analytic description of channel plasmon polaritons,” Opt. Lett. 34, 2039–2041 (2009). 19. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10, 105018 (2008).
School of Information Science and Engineering, Hangzhou Normal University, Hangzhou 310036, China
2
Department of Information Science and Electronic Engineering, Zhejiang University, Zhejiang Province, Hangzhou 310027, China 3
The Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China *Corresponding author: [email protected] Received 9 April 2014; revised 25 July 2014; accepted 30 July 2014; posted 5 August 2014 (Doc. ID 209284); published 4 September 2014
We present a numerical investigation of terahertz channel plasmon polaritons (CPPs) propagating in a semiconductor InSb. It is shown that these CPPs can simultaneously exhibit subwavelength field confinement and relatively long propagation length. Moreover, single-mode propagation is available for terahertz CPPs in a certain frequency range. © 2014 Optical Society of America OCIS codes: (230.7380) Waveguides, channeled; (240.6680) Surface plasmons; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.53.005896
1. Introduction
Surface plasmon polaritons (SPPs) are electromagnetic waves coupled to free electron oscillations at the interface between metal and a dielectric [1]. These waves propagate along the interface, decaying evanescently away from it. The unique properties of 2D nature under certain conditions make them quite suitable for applications in subwavelength optics [2]. Most research on SPPs has been done on optical and near-infrared frequencies with metal by manipulating the SPPs at the nanoscale [3]. However, not only metals support electron plasmon oscillations, semiconductors also can support SPPs. Free carriers can be excited either thermally or by doping, which results in a plasma frequency typically in the terahertz domain. THz plasmonics with InSb structures has been proposed and investigated in various systems such as hole arrays [4], gratings [5,6], slits [7], slot cavities [8], and plasmonic antennas [9]. InSb behaves at THz frequencies as a metal at visible 1559-128X/14/265896-05$15.00/0 © 2014 Optical Society of America 5896
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
frequencies, i.e., it has a negative real component of permittivity. This behavior can be tuned by modifying the number of excited electrons into the conduction band, i.e., the free carrier density. For example, the permittivity of InSb becomes less negative with smaller jεj when the temperature decreases, due to the decrease in the free carrier density, which leads to lower plasma frequency. Moreover, as the temperature decreases, the electron mobility in InSb significantly increases; then its scattering frequency decreases. As a result, in a lower temperature, SPPs in a semiconductor can not only be more effectively confined but also propagate over a longer distance. The properties of terahertz SPPs guiding in a semiconductor surface (3D system) have received even less attention. In contrast, various schemes for guiding visible SPPs have been proposed; among them, channel plasmon polaritons (CPPs) by triangular-shaped (V-shaped) grooves [10–15] exhibit superior features such as relatively low propagation loss under subwavelength field confinement, efficient transmission through sharp bends, and compatibility with the planar technology. In this paper, we present a numerical investigation of
CPPs guided by V-shaped grooves carved in InSb, showing that CPPs on a semiconductor surface can be tightly localized and simultaneously possess a relatively long propagation length at terahertz frequencies. 2. Numerical Analysis
The structure studied here is a V-shaped groove on an InSb surface, as illustrated in Fig. 1 (see the inset). The geometric parameters characterizing this structure are groove angle θ and groove depth h. The dielectric function of InSb semiconductor in the terahertz frequency range is approximately characterized by a Drude model: εω εlattice −
ω2p : ωω iγ
(1)
For InSb, the lattice permittivity εlattice is taken to be 15.6, and Drude parameters are ωp 46 × 1012 rad∕s and γ 0.3 × 1012 rad∕s [16,17], resulting in the permittivity of ε −37.88 2.55i at f 1.0 THz. It should be noted that Drude parameters may vary considerably between semiconductor samples, owing to variations in impurity density. To understand the CPP properties guiding in the proposed structure, we first analyze the mode effective index for the fundamental mode of CPPs. The CPP effective refractive index is defined as N eff λ∕2πReβ, where β is the propagation constant, and λ is the wavelength in the air. Among the parameters of width, depth, and angle of the V groove, the angle is the main parameter that may largely influence the mode index for a relatively large V groove [18]. The depth/width of the V groove influences the propagation properties only near the cutoff, when the CPP mode begins to be hybridized with a wedge mode at the groove edges. So different groove angles of θ 15°, 20°, and 25° are analyzed, while the a
θ
h
θ θ θ
o o
o
Fig. 1. Effective index of the fundamental CPP mode versus frequency for different groove angles θ 25°, 20°, and 15°. The groove depth is kept at h 750 μm. Inset shows the schematic of the V-shaped groove in InSb.
groove depth is fixed at h 750 μm. We perform a rigorous numerical calculation of the CPP mode using the full-vector finite element method. To obtain better convergence of the calculated fields, the tip and edges of the V-shaped semiconductor groove are rounded by the curvature radius of r 10 and R 100 μm, respectively. Figure 1 shows the mode effective index as a function of frequency (f ) for the InSb V grooves with three groove angles. The effective index of a flat SPP mode is also shown in the figure for comparison. As seen in Fig. 1, the CPP effective index increases when the frequency grows for each groove angle. It is known that, for a CPP mode, a larger effective index corresponds to a smaller penetration depth in the dielectric (air), so the larger mode index means a stronger field confinement. When the frequency decreases closest to the cut-off frequency, the CPP effective index approaches that of air, indicating the tendency of the CPP mode field to extend gradually outside the groove. Furthermore, for a certain frequency, the CPP effective index increases with the decrease of the groove angle. This means that the CPP mode is more tightly confined in the structure with a smaller groove angle. The cut-off frequency also can be easily read out from Fig. 1. For the groove angle of θ 25°, the cut-off frequency is 0.5 THz, for θ 20°, the cutoff is 0.45 THz, and, for θ 15°, it becomes 0.4 THz. This shows that the groove angle also affects the cut-off frequency of the CPP mode. The cut-off frequency shifts to lower frequencies when the groove angle becomes smaller, indicating a larger frequency band can be achieved for the smaller groove angle. To examine the influence of absorption of a semiconductor at terahertz frequencies on the performance of CPPs, we calculated the propagation length of the fundamental CPP mode as a function of frequency for different groove angles. Figure 2(a) shows the normalized propagation length (L∕λ) versus frequency (f ) for the three different groove angles in Fig. 1. The propagation length for CPPs is determined by L 1∕2∕Imβ, where β is the propagation constant of CPPs. We find that the propagation length decreases as the frequency increases for each groove angle. For a certain frequency, the propagation length also decreases with the decrease of the groove angle. For example, the CPP mode has the propagation length of L 31λ for θ 25°, L 24λ for θ 20°, and L 18λ for θ 15° at f 1.0 THz (λ 300 μm). For the SPP (or CPP) mode, it is physically common that the propagation length decreases as the field confinement increases, and the latter corresponds to the increase of mode index, so the results shown in Fig. 2(a) agree well with those in Fig. 1. To well illustrate the field confinement of the CPP mode, here we introduce the definition of effective mode area Am (the third type A3 in [19]). It is defined as a minimum area in which a set proportion, η (usually η 0.5 is used), of a mode’s energy resides. Unlike the commonly used method of evaluating the mode area based on the region in which the mode 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
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θ=25 o θ=20 o θ=15 o
(b)
Am/A0
θ=25 o θ=20 o θ=15 o
(a)
Fig. 2. Normalized propagation length (a) and normalized effective mode area (b) versus frequency for different groove angles θ 25°, 20°, and 15°. The groove depth is h 750 μm for all three cases.
energy drops to e−2 of its peak value, Am is quite little sensitive to the local field confinement with rapid field variations. Furthermore, it is also geometry independent to the measure of confinement owing to the generic feature of η to any field distribution. It is defined by solving the following minimization problem: Z Am min f rdA; f r A∞ Z s:t: f r − ηWrdA 0; A∞
(2)
where f r is the shape function that encompasses a fraction η of the mode’s energy and η 0.5 is used in our study. Assuming that the optimum f r is a step function bounded by a constant energy density contour, W 0 , of the mode, Am can be found out by iteratively solving the minimization problem of Eq. (2) using:
5898
f r 0;
if
Wr < W 0 ;
f r 1;
if
Wr ≥ W 0 ;
(3)
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
where W 0 corresponds to the contour containing η of the mode’s energy. Figure 2(b) shows the normalized effective mode area Am ∕A0 versus the frequency for the InSb V grooves with three different groove angles, where A0 λ∕22 is the diffraction limited area of vacuum. It can be seen that the normalized effective mode area generally decreases as the frequency increases for each groove angle, indicating the field confinement increasing with the frequency. As this variation of Am ∕A0 with the frequency is similar to the normalized propagation length (L∕λ) in Fig. 2(a), there exists a trade-off between the field confinement and propagation loss. However, it is slightly surprising that, at low frequencies close to the cutoff, the value of Am ∕A0 increases with the frequency. This should be attributed to the worse field confinement, and, in this situation, the field energy is no longer concentrated within the groove, and its large portion is distributed around the groove edges. This can be clearly seen from the contour line of W 0 in Fig. 3 (for the case of f 0.6 THz), where the modal field pattern is presented. For a certain frequency, the normalized effective mode area decreases with the decrease of the groove angle, indicating that better field confinement can be achieved with a smaller groove angle. For f 1.0 THz (λ 300 μm), the normalized effective mode area is Am ∕A0 1.12 for the groove angle of θ 25°, it is drastically decreased to Am ∕A0 0.64 for θ 20°, and it reaches to Am ∕A0 0.40 for θ 15°, showing that the groove angle influences the field confinement greatly. Figure 3 shows the electric-field (E) amplitude of the CPP mode, where the dotted line marks the contour line of W 0, which contains η 0.5 of the mode’s energy. The parameters of the structure are as follows: θ 20°, h 750 μm, and a 264 μm. From Fig. 3, it is obvious that the field confinement enhances as the frequency increases. For f 0.6 THz, the E-field extends outside the groove, and obviously the CPP mode is hybridized with wedge plasmon polaritons (WPPs) running along the groove edges. The calculated normalized effective mode area is Am ∕A0 0.92 for this frequency. So, at lower frequencies, CPPs lose their confinement character. As with the frequencies increased, the field confinement also increases, which makes the modal field well localized within the groove. Evidently, only in this situation, the defined mode area can well reflect 0.6 THz
0.8 THz
1.0 THz
1.2 THz
Fig. 3. Distribution of the E-field amplitude of the fundamental CPP mode at different frequencies of f 0.6, 0.8, 1.0, and 1.2 THz. The dotted line marks the contour line of the critical energy density W 0. The groove angle is θ 20°, and the groove depth h 750 μm.
the field confinement. Our numerical calculation shows that Am ∕A0 0.97 for f 0.8 THz, Am ∕A0 0.64 for f 1.0 THz, and Am ∕A0 0.46 for f 1.2 THz. As expected, the normalized effective mode area decreases with the frequency, verifying the field confinement increases with the frequency. This phenomenon is more clearly indicated by the contour line of W 0 (white dotted line) in Fig. 3. It can be seen that, for f 0.6 THz, the contour line of W 0 is broken into several pieces, which clearly indicates the excitation of WPPs at the groove edge and its hybridization with the CPP field. At f 1.2 THz, the CPP mode is deeply subwavelength confined, and, in such situation, it still has a propagation length of L 15λ, indicating the V groove in a semiconductor is a good choice for a terahertz plasmonic waveguide. To vividly demonstrate the guiding properties of the V groove in InSb, we conduct FDTD simulation for the wave transmission along the groove structure with h 750 μm and θ 20°. The waveguide length is taken to be 5000 μm (equals to 10λ at f 0.6 THz). In the simulation, the waveguide is excited using an input port with a pulse of the TE fundamental mode. The spatial variation of the E-field amplitudes associated with CPPs for different frequencies is plotted in Figs. 4(a)–4(d), representing the vertical cutting slice (halving the structure). It can be seen that, for f 0.6 THz, CPPs propagate along the V groove with a looser field confinement. For f 0.8 THz, CPPs propagate with much better field confinement, verifying the field confinement improvement with the increased frequency. When the frequency increases to f 1.0 THz, CPPs almost propagate along the bottom of the V groove, showing the field being highly confined to the groove bottom. For f 1.2 THz, it exhibits the strongest confinement, but the field attenuates significantly over the propagation distance.
For a plasmonic waveguide, single-mode operation is important, so we further numerically investigate the modal characteristics of CPPs in the V groove. As a typical example, the groove parameters are h 750 μm and θ 20°. Figure 5 shows the effective index for the two lowest-order CPP modes. The fundamental mode (solid line with circles) has a cutoff at f c ≈ 0.45 THz, and the second-order mode (solid line with triangles) has a cutoff at f c ≈ 0.8 THz, so the single-mode propagation is available for CPPs in the terahertz frequency range of 0.45–0.8 THz. The modal shapes of the two modes are shown in the insets for f 1.0 THz, and the fundamental one exhibits much better confinement, as expected. Finally, we give a short discussion about the effect of the curvature radius of tip and edges of the groove on the CPP mode. The tip size may greatly influence the properties of the CPP mode at high frequencies with tight field confinement, due to the energy being concentrated there. The mode index and the propagation length for different radii of rounded tip r 5, 10, and 15 μm are calculated for f 1.0 THz, while the curvature radius of the groove edges is fixed to be R 100 μm. For r 5 μm, we find that N eff 1.064 and L 17.7λ. For r 10 μm, N eff 1.055 and L 23.5λ, and for r 15 μm, N eff 1.050 and L 27.8λ. Clearly, as r increases, the mode index decreases while the propagation length increases, indicating the field confinement decreases with increasing r. The curvature radius of edges might have influence on the CPP mode at lower frequencies closest to the cut-off frequency, when the field extends outside the groove and is hybridized with the edge WPPs. We have calculated the mode index and propagation length for different curvature radii R 50 and R 100 μm and found that N eff and L are almost the same for both cases, showing that the edge curvature radius has little effect on the property of the CPP mode.
λ=500 µm
CPP 1st
CPP 2 nd
st nd
(a)
0.6 THz
0.8 THz
(b)
(c)
1.0 THz
1.2 THz
(d)
Fig. 4. Spatial variation of the E-field amplitudes associated with the CPP transmission for different frequencies: (a) 0.6 THz, (b) 0.8 THz, (c) 1 THz, and (d) 1.2 THz. Geometric parameters are as follows: θ 20°, h 750 μm, and the waveguide length is 5000 μm.
Fig. 5. Effective index as a function of frequency for the first and second mode of CPPs. Insets show the E-field amplitude of the two CPP modes at f 1.0 THz. The parameters of the V groove are the same as in Fig. 3. 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
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We have presented a numerical study of CPP propagation along a semiconductor InSb surface at terahertz frequencies. Our numerical simulations have shown that CPPs sustained by an InSb V groove can be guided on a subwavelength scale. These CPPs have relatively long propagation length under subwavelength confinement, while single-mode propagation is still available in the terahertz range. Our results show that CPPs have promising applications in the THz regime. This work was supported by the National Natural Science Foundation of China under grant no. 61372005. References 1. H. Raether, Surface Plasmons (Springer, 1988). 2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 3. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nanooptics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). 4. J. Gómez Rivas, C. Janke, P. Haring-Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express 13, 847–859 (2005). 5. J. Gómez Rivas, M. Kuttge, H. Kurz, P. Haring-Bolivar, and J. A. Sánchez-Gil, “Low-frequency active surface plasmon optics on semiconductors,” Appl. Phys. Lett. 88, 082106 (2006). 6. R. Parthasarathy, A. Bykhovski, B. Gelmont, T. Globus, N. Swami, and D. Woolard, “Enhanced coupling of subterahertz radiation with semiconductor periodic slot arrays,” Phys. Rev. Lett. 98, 153906 (2007). 7. T. H. Isaac, J. Gómez Rivas, J. R. Sambles, W. L. Barnes, and E. Hendry, “Surface plasmon mediated transmission of subwavelength slits at THz frequencies,” Phys. Rev. B 77, 113411 (2008).
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