Propagation modes and single-guiding-mode conditions of one-dimensional silver nanowires based surface plasmon polariton (SPP) waveguides versus the operating wavelength (500-2000 nm) are investigated. For silver nanowires immersed in a SiO2 matrix, both short-range SPP (SRSPP)-like modes and long-range SPP (LRSPP)-like modes can be guided. However, only the LRSPP-like modes have cutoff radii. For silver nanowires on a SiO2 substrate, the LRSPP-like modes cannot be supported due to asymmetry. While for the SRSPP-like guiding mode, it has a cutoff radius for wavelength longer than 615 nm. For wavelength shorter than 615 nm, there is no cutoff radius for the guiding modes due to the appearance of the interface modes and thus the single-guiding-mode operation is always satisfied.
©2013 Optical Society of America
The surface plasmon polariton (SPP) structures provide a potential solution to key components in future high-capacity electronic and photonic circuits, since they can spatially confine the light well beyond the diffraction limit. This distinctive feature dramatically enhances the density of state near the metal surface and can thereby be used for enhanced light-matter interactions. Thus far, the guiding of SPPs has been demonstrated in a number of strategies including metal-insulator-metal structures, channel plasmonic waveguides, ridge plasmonic waveguides, long-range plasmonic waveguides, dielectric-loaded plasmonic waveguides, hybrid plasmonic waveguides, and metallic nanowires . Among these structures, one-dimensional (1D) silver nanowires, which can be routinely chemically synthesized with high crystallinity and atomically smooth surfaces, have emerged as an appealing candidate for nanoscale confinement, guiding and sensing .
For 1D silver nanowires in a dielectric matrix, present investigations include: (1) plasmon propagation at 632 nm for 65 nm radius nanowire  and at 740-855 nm for 35/40 nm radius nanowire ; (2) SPP interaction with quantum dots at 655 nm for 51 nm radius nanowire . For 1D silver nanowires on a dielectric substrate, present investigations have experimentally revealed moderate propagation lengths at visible (400-750 nm) and short near-infrared (NIR) range (750-1000 nm) [6–16] for SPP propagation. These include: (1) plasmon propagation at wavelength 633 nm for 110 nm radius nanowires , 532/633/980 nm for 130 nm radius nanowires , 830 nm for 50 nm radius nanowires , 780 nm for 75 nm radius nanowires , 633 nm for 375 nm radius curved nanowires and 785 nm for 290/328/375 nm radius curved nanowires ; (2) direct photonic-plasmonic coupling at 488/532/650 nm for 160 nm radius nanowires  and 532/650/980 nm for 50 nm radius nanowires ; (3) plasmon resonator at 785 nm for 60 nm radius nanowires ; (4) SPP interaction with single-photon emitters such as quantum dots at 655 nm for 150 nm radius nanowires with a 15 nm SiO2 coating  and 710 nm for 50 nm radius nanowires with a 25 nm SiO2 coating . Theoretically, it has been shown that coupling of plasmons to silicon substrate with a SiO2 surface spacer layer can severely dampen the propagation length at 633 nm wavelength . The effect of the air gap between the nanowire and the substrate on the guiding plasmon mode at 637 nm has been studied in . In , Wang et al. have studied the guiding plasmon mode at 633 nm and they mainly focused on the effect of substrate (substrate refractive index ranges from 1.37 to 2.7) on the guiding plasmon mode for different nanowire radii.
All the aforementioned reports haven’t studied the mode behavior and the single-guiding-mode conditions versus the operating wavelength, especially at near infrared wavelength between 1000 and 2000 nm . In order to be compatible with current mature optical communication technology, the potential use of SPPs in future high-capacity electronic and photonic circuits should be operated in the NIR range, especially around 1310 and 1550 nm. Besides, the mode distributions unveiling the underlying physics of all the supported modes for most of the present experimental research have yet to be achieved. Whether the propagation modes demonstrated in the experiments are truly guiding modes and whether the waveguides are single-guiding-mode operated are unexplored. Therefore, it is of vital importance to unveil the mode behavior and the single-guiding-mode conditions versus the operating wavelength.
To address these issues, here we investigate the propagation modes and single-guiding-mode conditions of silver nanowires in a SiO2 matrix and on a SiO2 substrate versus the operating wavelength from visible to NIR (500-2000 nm). The paper is organized as follows. Section 2 is a description of the structure and the simulation. Section 3 investigates the propagation modes and single-guiding-mode condition of silver nanowires immersed in a SiO2 matrix. Section 4 investigates the propagation modes of silver nanowires on a SiO2 substrate. Section 4.1 investigates the propagation modes above 615 nm wavelength, where the guiding modes can be cutoff (i.e., turning into leaking modes) and Section 4.2 investigates the propagation modes below 615 nm wavelength, where no cutoff radii for the guiding modes exist. Section 5 is the conclusion.
2. Structure and simulation
A 1D silver nanowire with a radius of r is shown in Fig. 1 . The substrate is modeled as a uniform SiO2 with a refractive index n0 = 1.45. The cladding is assumed to be SiO2 (n1 = 1.45) in Section 3 where the nanowires are immersed in a SiO2 matrix, and air (n1 = 1) where the nanowires are located on the substrate in Section 4. The permittivity data of silver from Johnson-Christy in the range of 500 to 2000 nm, which is generally considered superior in relation to chemically synthesized nanowires, is used in the simulation [20, 21]. The considered nanowire radii are below 400 nm, which can be chemically synthesized. A ðnite element method is used to investigate the plasmon propagation modes supported by the waveguide.
The simulations are performed using the commercial finite element-based software Comsol Multiphysics 4.2a with RF module. A triangular mesh is built with a maximum element size of 1/20 of the nanowire diameter for the nanowire and the total number of mesh elements in the simulated area is usually of the order of 104-105. The relative tolerance is 1.0−6 in successive eigenvalues during the iterative eigenvalue computation process, which provides a good convergence of the effect mode index.
As we all know that a metal slab immersed in a dielectric medium can support two kinds of modes: SRSPP mode and LRSPP mode . The SRSPP (LRSPP) mode exhibits several features: (1) the electric field component perpendicular to the slab is anti-symmetric (symmetric) with respect to the metal slab; (2) the mode has no cut-off (cut-off) for the SRSPP (LRSPP) when the slab is thin enough; (3) the propagation loss is comparatively large (small) owing to a considerably portion of energy confined in the metal (dielectric); (4) the mode area is comparatively small (big). We use these four features to draw the analogies between the guiding modes of the nanowires and the SRSPP (LRSPP) modes. Because the wire circumference plays a complicated role in mediating the coupling between the “interfaces” of the nanowire across the center, we use SRSPP-like (LRSPP-like) mode to characterize the guiding modes.
3. Propagation Modes of silver nanowires in a SiO2 matrix
In this Section, we consider the case that the silver nanowires are immersed in a SiO2 matrix. Take a 100 nm-radius silver nanowire at 1000 nm wavelength as an example, two kinds of guiding mode can be supported by this waveguide and their mode fields are provided in Fig. 2 .
- (1) 1st-order mode (termed a SRSPP-like mode). The Ex component and Ey component are perfectly anti-symmetric with respect to the nanowire center.
- (2) 2nd-order mode (termed a LRSPP-like mode). The Ex component and Ey component are perfectly symmetric with respect to the nanowire center.
The effective refractive index neff, the propagation length δ and the mode areas Aeff versus wavelength for different nanowire radii are respectively given in Figs. 3(a) , 3(c) and 3(d), and the cutoff radius versus wavelength is plotted in Fig. 3(b).
- (1) Effective refractive index neff. The effective refractive index neff decreases with an increasing wavelength. The 1st-order mode has no cut-off radius. However, for the 2nd-order mode, the effective refractive index neff decreases with decreasing nanowire radii, in opposite to the 1st-order mode. Therefore, it is cutoff at a small nanowire radius.
- (2) Cutoff radius. There is no cutoff for the 1st-order mode. For the 2nd-order mode, the effective refractive index neff decreases with wavelengths and thus it is cutoff at a long wavelength. Therefore, the single-guiding-mode condition can be satisfied when the 2nd-order mode is cutoff.
To implement silver nanowires for single-guiding-mode operation at 1310 nm and 1550 nm, the nanowire radii should be smaller than 184 nm and 326 nm, respectively. For typical nanowire radii of 50 nm, 75 nm and 100 nm, the single-guiding-mode operation wavelengths are respectively above 771 nm, 940 nm and 1055 nm, and the silver nanowire waveguides in this case are thereby no longer single-mode guided in visible wavelengths. To excite the 1st-order mode, the incident polarization should be parallel to the nanowire. While for the 2nd-order mode, perpendicular incident polarization is required.
- (3) Propagation length δ and mode area Aeff. The SPP propagation length δ is defined by the distance when the mode power decays to 1/e. The mode area Aeff is defined by the mode’s total energy density per unit length divided by its peak energy density . The 1st-order mode has a considerable portion of energy is confined inside the metal, thus having smaller propagation length and mode area. For the 2nd-order mode, most of the energies confined in the SiO2 layer around the nanowire, thus holding longer propagation length and simultaneously larger mode area.
Therefore, we can term the 1st-order mode a SRSPP-like mode and 2nd-order mode a LRSPP-like mode in terms of their mode symmetries, cut-off behavior, propagation losses and mode areas.
4. Propagation modes of silver nanowires on a SiO2 substrate
4.1 The operating wavelength is above 615 nm
We first analyze the propagation modes of silver nanowires on a SiO2 substrate when the operating wavelength is above 615 nm. Take 1550 nm operating wavelength as an example, two kinds of propagation modes can exist:
- (1) Guiding mode (termed a g-mode). The energy density distribution of the g-mode at r = 30 nm is provided in Fig. 4(a) and the distributions of its two main electric components (Ex and Ey) are provided in Figs. 4(e) and 4(f), respectively. The phases of Ex component and Ey component are anti-symmetric with respect to the nanowire center. The amplitude of Ey component below the y = 0 cutline is larger than that above the y = 0 cutline owing to the substrate effect. This g-mode originates from the strong coupling of the surface plasmons from the two opposite Ag/dielectric interfaces across the center. Therefore, we can term it a SRSPP-like mode (its cut-off behavior, propagation loss and mode area is to be shown later). Note that due to the asymmetric structures with respect to the y = 0 cutline, this coupling only develops the SRSPP-like anti-symmetric mode rather than the LRSPP-like symmetric mode.
- (2) Leaking mode (termed as l-mode). At r = 50 nm, the propagation mode has a refractive index lower than the substrate index n0. Therefore, the g-mode turns into the l-mode and its mode distribution is provided in Fig. 4(b). To clearly reveal the leaking characteristic of the l-mode, its mode distribution is also given in Fig. 4(d) in a different color bar with saturation. In contrast, there is no leaking for the g-mode shown in Fig. 4(c). For a large radius, the coupling of surface plasmons from the two interfaces Ag/Air and Ag/SiO2 gets weak due to limited penetration depths of SPP into the metal. Since the substrate possesses a comparatively large refractive index, a substantial portion of energy is dragged into the substrate rather than around the nanowire and thereby l-mode is formed.
To further explore the characteristics of propagation modes of silver nanowires on a substrate versus operating wavelength, the effective refractive index neff, the propagation length δ, the mode areas Aeff and the energy percentages in air/SiO2 versus wavelength for different nanowire radii are respectively given in Figs. 5(a) , 5(c), 5(d) and 5(e), and the cutoff radius versus wavelength is plot in Fig. 5(b).
- (1) Effective refractive index neff. The effective refractive index neff decreases with an increasing wavelength. This can be interpreted in terms of SPP penetration depth into metal. The SPP penetration depth into metal between Ag/dielectric interface can be represented by (1 + εd/εm)1/2/k0 (εd and εm are respectively the permittivity for dielectric and metal and k0 is related to the wavelength λ0 by k0 = 2π/λ0). As the wavelength increases, the penetration depth decreases and thus the coupling of surface plasmons between the two interfaces Ag/Air and Ag/SiO2 is weakened. Consequently, a substantial portion of energy is dragged into the SiO2 substrate, thus leading to a decrease in the effective refractive index.
- (2) Cutoff radius. Beyond 615 nm, there is a cutoff radius for the g-mode. Below 615-nm, there is no cutoff radius and the reason will be analyzed in Section 4.2. Contrary to conventional photonic waveguides, metal nanowires on a substrate can support g-modes for arbitrarily small non-zero radii (r < 32 nm) and the g-modes in metal nanowires cut off at large radii. The cutoff radius decreases drastically with the wavelength below 800 nm and comparatively slowly at long wavelengths.
To implement silver nanowires for single-guiding-mode operation at 1310 nm and 1550 nm, the nanowire radii should be lower than 36.7 nm and 34 nm, respectively. For typical nanowire radii of 50 nm, 75 nm and 100 nm, the single-guiding-mode operation wavelengths are below 832 nm, 663 nm and 630 nm, respectively. When the nanowire radius is above 160 nm, the g-mode will not be supported for any of these wavelengths. Therefore, the experimental demonstrations of SPP propagation modes, including using 130 nm radius nanowires at 980 nm , 290/328/375 nm radius curved nanowires at 785 nm , and 150 nm radius nanowires at 980/1550 nm , are actually l-modes rather than g-modes.
- (3) Propagation length δ, mode area Aeff and energy percentage in air and SiO2 substrate. The l-mode exhibits a larger propagation length and mode area compared with the g-mode. This is because the l-mode has a considerable percentage of energy confined in SiO2 and subsequently a small percentage of energy in air and Ag. For the g-mode, the propagation length δ increases with wavelength because of comparatively low metal losses at long wavelengths. The mode area Aeff also increases with wavelength because the SPPs at long wavelengths have comparatively large penetration depths into dielectric.
4.2 The operating wavelength is below 615 nm
As has been pointed in Section 4.1, there is no cutoff radius for the guiding mode below 615 nm. To unveil the underlying physics, we first analyze the mode evolution versus nanowire radius for a wavelength slightly above 615 nm (620 nm here), for which there is a cutoff radius for the g-mode.
Figure 6 provides the energy density distributions of nanowires at 620 nm wavelength for three radii (60 nm, 160 nm and 320 nm). At r = 60 nm, the propagation mode is g-mode with most of the energies confined around the nanowire. At r = 160 nm, g-mode turns into l-mode with a considerable portion of energy leaked into the substrate. At r = 320 nm, another kind of guiding mode with an effective refractive index above n0 appears. For this guiding mode, all the energies are confined in the Ag/SiO2 interface (including the air gap in between), and it is thus termed as interface mode (termed as i-mode) here. When the nanowire is thick enough, the surface plasmons between the two interfaces Ag/Air and Ag/SiO2 decouple due to the limited penetration depths into metal. Since the substrate processes a large refractive index, the field mainly concentrates between the Ag/SiO2 interface rather than surrounding the nanowire in this case. When r increases further, the neff approaches that of interface mode between infinite Ag/SiO2 interfaces.
Figure 7 provides the effective index neff and propagation length versus radius (40 nm to 400 nm) for these three propagation modes in silver nanowires. At 620 nm wavelength, the effective index of the g-mode decreases first with increasing radii. At 123 nm < r < 229 nm, neff goes down below 1.45 and thus only l-mode can propagate. At r > 229 nm, the i-mode can be guided with neff approaching (εdεm/(1 + εdεm))1/2. Therefore, the l-mode area can be regarded as a transition between the g-mode and the i-mode. At 615 nm wavelength, the largest refractive index for the g-mode and the smallest refractive index for the i-mode are 1.45 for the 160 nm radius nanowire. Below 615 nm wavelength, the g-mode transitions into i-mode gradually and therefore no cutoff radius exists.
Propagation modes and single-guiding-mode conditions of one-dimensional silver nanowire based SPPs waveguides versus the operating wavelength from visible to NIR (500-2000 nm) are extensively investigated. When the metal nanowires are immersed in a SiO2 matrix, both SRSPP-like modes and LRSPP-like modes can be guided. However, only the LRSPP modes have cutoff radii. The nanowire radii should be smaller than 184 nm and 326 nm at 1310 nm and 1550 nm, respectively, to guarantee single-guiding-mode operation.
When the metal nanowires are located on a SiO2 substrate, the LRSPP-like modes cannot be supported owing to asymmetry. While for the SRSPP-like guiding mode, it has a cutoff radius for wavelength longer than 615 nm. For instance, to support guiding modes, the nanowire radii should be smaller than 36.7 nm and 34 nm at 1310 nm and 1550 nm, respectively. For larger radii, only leaking modes exist. The leaking modes exhibit longer propagation lengths and larger mode areas compared with the guiding modes. Therefore, the propagation modes demonstrated in some experiments in silver nanowires on a substrate are actually leaking modes rather than guiding modes. For wavelength shorter than 615 nm, there is no cutoff radius for the guiding modes due to the appearance of the interface modes and thus the single-guiding-mode operation is always satisfied. These findings unveil the underlying physics of silver nanowires based SPPs and are thus essential for applying metal nanowires for plasmonic circuit integration, nanoscale confinement, plasmonic sensing, etc.
This work is supported by the National Natural Science Foundation of China (Grant Nos. 61275030, 61205030 and 61235007), the Opened Fund of State Key Laboratory of Advanced Optical Communication Systems and Networks, the Opened Fund of State Key Laboratory on Integrated Optoelectronics, the Fundamental Research Funds for the Central Universities (2012QNA5003), Doctoral Fund of Ministry of Education of China (Grant No 20120101120128), the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR).
References and links
1. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
2. N. P. de Leon, M. D. Lukin, and H. Park, “Quantum plasmonic circuits,” IEEE J. Quantum Electron. 18(6), 1781–1791 (2012). [CrossRef]
3. Z. P. Li, K. Bao, Y. R. Fang, Y. Z. Huang, P. Nordlander, and H. X. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10(5), 1831–1835 (2010). [CrossRef] [PubMed]
4. B. Wild, L. Cao, Y. Sun, B. P. Khanal, E. R. Zubarev, S. K. Gray, N. F. Scherer, and M. Pelton, “Propagation lengths and group velocities of plasmons in chemically synthesized gold and silver nanowires,” ACS Nano 6(1), 472–482 (2012). [CrossRef] [PubMed]
5. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef] [PubMed]
8. A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. N. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]
9. C. H. Dong, X. F. Ren, R. Yang, J. Y. Duan, J. G. Guan, G. C. Guo, and G. P. Guo, “Coupling of light from an optical fiber taper into silver nanowires,” Appl. Phys. Lett. 95(22), 221109 (2009). [CrossRef]
11. X. Guo, M. Qiu, J. M. Bao, B. J. Wiley, Q. Yang, X. N. Zhang, Y. G. Ma, H. K. Yu, and L. M. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009). [CrossRef] [PubMed]
12. R. X. Yan, P. Pausauskie, J. X. Huang, and P. D. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. U.S.A. 106(50), 21045–21050 (2009). [CrossRef] [PubMed]
13. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]
14. L. L. Wang, C. L. Zou, X. F. Ren, A. P. Liu, L. Lv, Y. J. Cai, F. W. Sun, G. C. Guo, and G. P. Guo, “Exciton-plasmon-photon conversion in silver nanowire: polarization dependence,” Appl. Phys. Lett. 99(6), 061103 (2011). [CrossRef]
16. Z. P. Li, K. Bao, Y. R. Fang, Z. Q. Guan, N. J. Halas, P. Nordlander, and H. X. Xu, “Effect of a proximal substrate on plasmon propagation in silver nanowires,” Phys. Rev. B 82(24), 241402 (2010). [CrossRef]
17. C. L. Zou, F. W. Sun, Y. F. Xiao, C. H. Dong, X. D. Chen, J. M. Cui, Q. Gong, Z. F. Han, and G. C. Guo, “Plasmon modes of silver nanowire on a silica substrate,” Appl. Phys. Lett. 97(18), 183102 (2010). [CrossRef]
19. Q. Li, S. Wang, Y. Chen, M. Yan, L. Tong, and M. Qiu, “Experimental demonstration of plasmon propagation, coupling, and splitting in silver nanowire at 1550-nm wavelength,” IEEE J. Quantum Electron. 17(4), 1107–1111 (2011). [CrossRef]
20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
21. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005). [CrossRef]
22. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009). [CrossRef]
23. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008). [CrossRef]