NANO LETTERS
Subwavelength Nonlinear Plasmonic Nanowire
2004 Vol. 4, No. 12 2427-2430
N.-C. Panoiu* and R. M. Osgood, Jr. Department of Applied Physics and Applied Mathematics, Columbia UniVersity, New York, New York 10027 Received September 17, 2004; Revised Manuscript Received October 21, 2004
ABSTRACT We investigate numerically, by means of the finite-difference time-domain method, the propagation characteristics of surface plasmon-polariton (SPP) modes excited in an optical nanowire consisting of a chain of Ag spheres embedded in dielectric shells made of materials with optical Kerr nonlinearity. It is demonstrated that, in the linear limit, the nanowire supports two SPP modes, a transverse and a longitudinal one, separated by ∆λ ) 20 nm. Furthermore, the dependence of the transmission characteristics of these SPP modes, on both the pulse peak power and Kerr coefficient of the dielectric shell, is investigated. Nonlinear optical phenomena, such as power dependence of mode frequency, switching, or optical limiting, are observed and discussed.
Recently, several approaches to subwavelength dimension passive structures for light guiding have been proposed and demonstrated. In one case, generally termed “plasmonic” devices, one makes use of the unique optical properties of metallic materials to provide strong optical confinement and eliminate the limitations imposed by strong diffractive effects. For example, cylindrical metallic waveguides with subwavelength transverse dimensions,1 plasmon waveguides,2 or optical waveguides consisting of chains of resonantly coupled metallic nanoparticles3-6 have been demonstrated. The nanoparticle chains have also been used in more complex device types including combinations to form bends and beam splitters7 or mirrors and interferometers.8 The optical properties of these nanoparticle chains can be further improved if their basic building blocks have a more complex geometry, e.g., ellipsoidal particles5,9 or nanorings,10 or by combining in the same nanostructure materials with different optical properties. One such example is a metallodielectric nanoshell, which consists either of a metallic core surrounded by a dielectric shell11 or, in the opposite case, a dielectric core embedded in a metallic shell.12,13 The frequency of SPP resonances excited in these metallodielectric nanostructures depends sensibly on the material parameters of the core and shell, as well as their relative dimensions,14,15 and therefore, since experimentally the geometry of these nanostructures can be easily controlled and changed, their optical properties are highly tunable. Additionally, the coupling between the SPP resonances depends on the interparticle distance,16 so that the optical properties of nanoparticle chains can also be tuned by changing the distance between adjacent particles in the chain. * Corresponding author. E-mail:
[email protected]. 10.1021/nl048477m CCC: $27.50 Published on Web 11/10/2004
© 2004 American Chemical Society
Despite this work, there have been only limited efforts to explore the questions regarding the nonlinear response of these plasmonic light-guiding structures. In this connection, it is known that the basic material blocks of the nanoparticle chains are a particular efficient medium for nonlinear optical elements. For example, metal nanoparticles have been shown to exhibit strong SPP field enhancement to increase the efficiency of Raman scattering,17 second-harmonic generation,18 and third-order nonlinear processes.19,20 In the present paper, we demonstrate that a nanoparticle chain can be used to form a subwavelength nonlinear optical element, which exhibits optical “limiting” and frequency shifting. Further we show that the use of dielectric/metal nanoparticles leads to lower loss than for pure metal nanospheres. In particular, we use metallodielectric nanoparticle chains to demonstrate that one can achieve dynamic tunability of the optical properties of these nanostructures by incorporating nonlinear (Kerr) optical material in the structure of the metallodielectric nanoparticles that form the chain. Specifically, we consider a nanoparticle chain made of metallodielectric nanoshells, which contain a metallic core (Ag in our case) embedded in a shell made of a dielectric material with Kerr nonlinearity (Figure 1). The geometrical parameters of the nanostructure are R ) 25 nm, radius of the metallic spheres, t ) 10 nm, thickness of the dielectric shell, and D ) 70 nm, center-to-center distance between adjacent spheres, whereas for the material parameters we chose ns ) 1.5, the refractive index of the dielectric shell, and nb ) 1, the refractive index of the background. These values are similar to those used in recent theoretical3 and experimental5 studies, thus we can readily compare our results to those reported in these studies. The nonlinear properties of the nanostructure
Figure 1. Schematics of the optical waveguide: R ) 25 nm is the radius of the metallic spheres, t ) 10 nm is the thickness of the dielectric shell, and D ) 70 nm is the center-to-center distance between adjacent spheres. The sketch at the beginning of the chain depicts the excitation beam.
are determined by the Kerr coefficient n2, which in our calculations will be a free parameter. Finally, we assume that dispersion properties of the metallic core are described by the Drude dielectric function
[
(ω) ) 0 1 -
ωp2 ω(ω+iγ)
]
(1)
where 0 is the permittivity of free space and ωp ) 13.7 × 1015 rad/s and γ ) 2.7 × 1013 rad/s are the plasma and damping frequencies,21 respectively. Note that eq 1 assumes that interband transition effects and dephasing due to electron surface scattering can be neglected. However, the latter can be included22 by simply replacing γfΓ ) γ + AVF/2R, where A ∼ 1 is a theory-dependent constant and VF is the Fermi velocity. Also, the dependence of the dielectric function of the metallic core on the intensity of light is neglected, a good approximation for the values of optical power considered here. Before studying the guiding characteristics of the nanoparticle chain, we briefly discuss the linear optical properties of an isolated metallodielectric nanoshell. To this end, using the quasistatic approximation, which is valid if the size of the particle is much smaller than the wavelength, we have calculated the extinction and absorption cross-sections of the nanoshell. The results, presented in Figure 2, show an SPP resonance at λr ) 285.4 nm (ωr ) 6.6 × 1015 rad/s). Note that the absorption cross-section is much smaller than the extinction cross section, which means that only a small fraction of the incident light is absorbed by the metallic core. Furthermore, also shown in Figure 2 is the transverse, spatial plot of the ratio between the local field and the excitation field, calculated at the resonant wavelength λr. One important effect illustrated in this figure is the strong field enhancement near the surface of the metallic core, which suggests that the nonlinear optical response of the nanoshell is strongly enhanced at wavelengths close to the SPP resonance. Experimental evidence of this effect has been recently observed in studies of the third-order nonlinear optical susceptibilities of silica-capped Au nanoparticle films.23 To investigate linear and nonlinear optical properties of a chain made of nanoshells we employed the full 3D, finite2428
Figure 2. Extinction and absorption cross-sections of an isolated nanoshell vs wavelength (upper panel) and the field enhancement, calculated at the SPP resonance (lower panel).
difference time-domain (FDTD) numerical method. This algorithm consists of discretizing Maxwell equations on a 3D-grid and then, starting from a given initial condition, marching the resulting iterative relations in time. Upon choosing a suitably refined computational grid, the corresponding numerical solution gives an accurate representation of the dynamics of the electromagnetic field. Throughout our simulations, we considered a chain made of N ) 11 nanoshells, oriented along the z-axis, with the computational domain being a box of dimensions 150 × 150 × 800 nm3 surrounded by a 25 nm-thick perfectly matched layer (PML).24 Both the computational domain and the PML were covered by a 2 nm × 2 nm × 2 nm uniformly coarse grid. Furthermore, we assumed that the metal dispersion is described by the Drude dielectric function given by the eq 1 whereas the dielectric shell has an instantaneous nonlinear Kerr response. Dispersive and nonlinear optical effects can be rigorously incorporated in the FDTD algorithm.24 Finally, because of the heavy computational demand imposed by 3D FDTD simulations, we used a parallel implementation of the FDTD algorithm, which has been run on a computer cluster with 18 Pentium 4 processors at 2.8 GHz. We started our analysis by investigating the linear optical properties, i.e., n2 ) 0 of the nanoparticle chain. This nanostructure supports two propagating SPP modes, one longitudinal, which corresponds to an induced dipole along the chain axis, and one transverse, with the induced dipole perpendicular to the chain axis. To find these modes we proceeded as follows. An electromagnetic beam with a Gaussian transverse profile with its width comparable to the transverse area of the nanoshell, polarized along the z-axis, is excited along the y-axis. Due to its close proximity to the Nano Lett., Vol. 4, No. 12, 2004
Figure 3. Power spectral density of linear longitudinal (a) and transverse (b) SPP modes, calculated after propagation of one (s), two (- - -), and three (- ‚ - ‚ -) interparticle distances. In insets, mode power decay along the particle chain axis.
first particle in the chain, the beam excites a SPP resonance on the first particle in the chain, with the corresponding dipole along the z-axis, which through resonant near-field interactions couples to SPP resonances on subsequent particles in the chain to form a longitudinal SPP propagating mode. To excite a transVerse SPP mode, the incoming beam is polarized along the x-axis. In both cases, the frequency width of the beam was three times the width of the SPP resonance of an isolated nanoshell. Finally, the values of the electric fields at the middle points between adjacent nanoshels were recorded and then Fourier transformed. Figure 3 summarizes our calculated results pertaining to the linear optical properties of SPP modes supported by the nanoparticle chain. The graph depicts the two calculated SPP modes with frequencies ωl ) 5.98 × 1015 rad/s (longitudinal) and ωt ) 5.63 × 1015 rad/s (transverse), both close to the SPP resonance of an isolated nanoshell. Furthermore, from the variation with the propagation distance of the total power in the two modes along the chain we obtained their power absorption coefficients. Our results show that the longitudinal mode experiences larger losses, as compared to the transverse one, namely Rl ) 10.2 × 106m-1 (3 dB/68 nm) vs Rt ) 7.7 × 106 m-1 (3 dB/90 nm). Note that these absorption coefficients are approximately six times smaller than those in a chain made of metallic spheres with no dielectric shells.5 This loss reduction is due to the fact that the dielectric shells enhance the field in the interparticle space, increasing the coupling between the particles; in addition, the field in the metallic core is reduced, which leads to lower absorption in the metallic particles. An additional loss decrease, by ∼10, can be achieved if particles with elongated shape are used.5,25 Nano Lett., Vol. 4, No. 12, 2004
Figure 4. (a) Power spectral density of the longitudinal SPP mode, calculated after propagation of three interparticle distances. The curves correspond to input peak powers of 0.1 GW/cm2, 0.77 GW/cm2, 6 GW/cm2, 46 GW/cm2, and 360 GW/cm2. In inset, the frequency shift of the SPP mode, at 3 (f) and 10 (O) interparticle distances. The oblique line, a guide to the eye, connects local minima in the spectra. (b) Normalized transmission vs power, calculated at λ ) 335 nm (O), λ ) 345 nm (]), and λ ) 376 nm (f).
When a nonzero Kerr nonlinearity is included in numerical simulations, the optical response of the particle chain becomes dependent on the power coupled into the SPP modes. As previously explained, a beam with peak power P excites a SPP resonance on the first particle in the chain, which then excites a SPP propagating mode; however, when the nonlinear response of the Kerr dielectric is included, the optical properties of the SPP modes depend on both the input power P as well as the Kerr coefficient n2. To investigate this relationship we proceeded as follows (here, for brevity, we present only the results pertaining to the longitudinal mode). First, we fix the Kerr coefficient to n2 ) 1.5 × 10-17 m2/W, which is close to Kerr nonlinearities for GaAs, and varied the peak power P of the excitation beam. The results, presented in Figure 4, show that the frequency of the SPP mode decreases with P, a variation of more than 150 THz being induced when P increases from zero to 46 GW/cm2. Note that the SPP mode frequency shift is induced by the power in the SPP mode and not by the power of the incoming beam. We estimate that the efficiency of power coupling between the incoming beam and the SPP mode is ∼1.6 × 10-4, which means that the frequency shift of 150 THz is induced by an SPP mode power of ∼0.6 mW. Thus, because of the ultra-small transverse area of the SPP mode, the 2429
coupled through near field interactions. We showed that, in the linear limit, this nanostructure supports longitudinal and transverse SPP modes and calculated the absorption coefficients of both modes. We have also demonstrated that under relatively modest excitation powers nonlinear optical response can be observed. The nonlinear optical response of the particle chain includes phenomena such as power dependence of mode frequency, ultralow power switching, and optical limiting. Finally, potential applications of these nanostructures to nanodevices with new functionality were discussed.
Figure 5. Power spectral density of the longitudinal SPP mode, calculated after the SPP modes have propagated three interparticle distances. The curves correspond to Kerr coefficients of n2 ) 0 (- - -) and n2 ) 8 × 10-14 m2/W (s). In inset, the SPP mode frequency shift vs the Kerr coefficient.
frequency switching effect described here is achieved by employing an extremely small amount of optical energy. Furthermore, the fact that the observed frequency shift is due to the nonlinear response of the nanoshells is also illustrated by the inset in Figure 4a, which shows the power dependence of the SPP mode frequency shift, calculated at two propagation distances. Thus, due to the losses in the particle chain, the power in the SPP mode decreases with the propagation distance, and therefore the corresponding frequency shift decreases, too. Figure 4a suggests that the nanoparticle chain can be used as an active nanodevice, namely as an optical limiting device. Thus, the transmission spectra calculated for different powers P, normalized by the peak power, show that at certain frequencies (wavelengths), the normalized transmission is strongly dependent on the power P. This effect is illustrated in Figure 4b, which depicts the dependence of the normalized transmission on the power P, computed at three wavelengths. This figure shows that, depending on the operating wavelength, the transmitted power can can be either increased or decreased by tuning the input power P. In a different set of numerical experiments, we fixed the peak power, P ) 100 MW/cm2 and varied the Kerr coefficient n2 from zero to n2 ) 10-12 m2/W. The main results are summarized in Figure 5, which illustrates that as the Kerr coefficient n2 varies, the corresponding frequency of the SPP mode changes too, with a steep variation starting at n2 ∼ 10-15 m2/W; for the peak power P shown here, the frequency changes by about 30 THz. Notice also that unlike the linear regime, when optical nonlinearities are included, upon propagation along the particle chain the spectral profile of the SPP mode develops a series of modulations, a phenomenon which is an expression of self-modulation effects. In conclusion, we studied both linear and nonlinear properties of a particle chain made of metallic spheres embedded into dielectric shells made of Kerr optical material,
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Acknowledgment. This work was supported by the NIST Advanced Technology Program Cooperative Agreement, Grant No 70NANB8H4018 and also in part by the DoD STTR, Grant No FA9550-04-C-0022. References (1) Takahara, J.; Yamagishi, S.; Taki, H.; Morimoto, A.; Kobayashi, T. Opt. Lett. 1997, 22, 475-477. (2) Yatsui, T.; Kourogi, M.; Ohtsu, M. Appl. Phys. Lett. 2001, 79, 45834585. (3) Quinten, M.; Leitner, A.; Krenn, J. R.; Aussenegg, F. R. Opt. Lett. 1998, 23, 1331-1333. (4) Krenn, J. R.; Dereux, A.; Weeber, J. C.; Bourillot, E.; Lacroute, Y.; Goudonnet, J. P.; Schider, G.; Gotschy, W.; Leitner, A.; Aussenegg, F. R.; Girard, C. Phys. ReV. Lett. 1999, 82, 2590-2593. (5) Maier, S. A.; Kik, P. G.; Atwater, H. A. Appl. Phys. Lett. 2002, 81, 1714-1716. (6) Citrin, D. S. Nano Lett. 2004, 4, 1561-1565. (7) Brongersma, M. L.; Hartman, J. W.; Atwater, H. A. Phys. ReV. B 2000, 62, R16356-R16359. (8) Ditlbacher, H.; Krenn, J. R.; Schider, G.; Leitner, A.; Aussenegg, F. R. Appl. Phys. Lett. 2002, 81, 1762-1764. (9) Chen C. J.; Osgood, R. M. Phys. ReV. Lett. 1983, 50, 1705-1708. (10) Aizpurua, J.; Hanarp, P.; Sutherland, D. S.; Kall, M.; Bryant, G. W.; Garcia de Abajo, F. J. Phys. ReV. Lett. 2003, 90, 057401. (11) Liz-Marzan, L. M.; Giersig, M.; Mulvaney, P. Langmuir 1996, 12, 4329-4335. (12) Zhou, H. S.; Honma, I.; Komiyama, H.; Haus, J. W. Phys. ReV. B 1994, 50, 12052-12056. (13) Averitt, R. D.; Sarkar, D.; Halas, N. J. Phys. ReV. Lett. 1997, 78, 4217-4220. (14) Oldenburg, S. J.; Averitt, R. D.; Westcott, S. L.; Halas, N. J. Chem. Phys. Lett. 1998, 288, 243-247. (15) Prodan, E.; Nordlander, P. Nano Lett. 2003, 3, 543-547. (16) Stuart, H. R.; Hall, D. G. Phys. ReV. Lett. 1998, 80, 5663-5666. (17) Nie, S. M.; Emery, S. R. Science 1997, 275, 1102-1106. (18) Antoine, R.; Brevet, P. F.; Girault, H. H.; Bethell, D.; Schiffrin, D. J. J. Chem. Soc., Chem. Commun. 1997, 1901-1902. (19) Ricard, D.; Roussignol, P.; Flytzanis, C. Opt. Lett. 1985, 10, 511513. (20) Magruder, R. H.; Yang, Li; Haglund, R. F.; White, C. W.; Yang, Lina; Dorsinville, R.; Alfano, R. R. Appl. Phys. Lett. 1993, 62, 17301732. (21) Ordal, M. A.; Bell, R. J.; Alexander, R. W.; Long, L. L.; Querry, M. R. Appl. Opt. 1985, 24, 4493-4499. (22) Kreibig U.; Vollmer, M. Optical Properties of Metal Clusters; Springer: Berlin, 1995. (23) Hamanaka, Y.; Fukuta, K.; Nakamura, A.; Liz-Marzan, L. M.; Mulvaney, P. Appl. Phys. Lett. 2004, 84, 4938-4940. (24) Taflove A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method; Artech House: Boston, 2000. (25) Sonnichsen, C.; Franzl, T.; Wilk, T.; von Plessen, G.; Feldmann, J.; Wilson, O.; Mulvaney, P. Phys. ReV. Lett. 2002, 88, 077402.
NL048477M
Nano Lett., Vol. 4, No. 12, 2004
