Ultrahigh nonlinear nanoshell plasmonic waveguide with total energy confinement

Dielectric nonlinear waveguides have reached their maximum potential in achieving high nonlinearity due to the limitation of mode confinement beyond the diffraction limit. We theoretically demonstrate that a plasmonic waveguide consisted of a nonlinear subwavelength core coated by a metallic nanoshell can achieve ultrahigh nonlinearity and complete mode confinement. Our results show that the subwavelength nanoshell plasmonic waveguide can possess an ultrahigh Kerr nonlinearity up to 4.1 × 10 4 W −1 m −1 with nearly 100% of the mode energy residing inside the waveguide at λ = 1.55 μm. The optical properties are explored with detailed numerical simulations and are explained in terms of their dispersive properties. ©2011 Optical Society of America OCIS codes: (250.5403) Plasmonics; (190.3270) Nonlinear optics, Kerr effect; (190.4360) Nonlinear optics, devices; (130.4310) Integrated optics, Nonlinear; (250.5300) Photonic integrated circuits. References and links 1. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009). 2. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008). 3. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for alloptical signal processing,” Opt. Express 15(10), 5976–5990 (2007). 4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824– 830 (2003). 5. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189– 193 (2006). 6. A. Moradi, “Plasmon hybridization in metallic nanotubes,” J. Phys. Chem. Solids 69(11), 2936–2938 (2008). 7. 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(7083), 508–511 (2006). 8. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). 9. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008). 10. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87(6), 061106 (2005). 11. J. Li, M. M. Hossain, B. Jia, D. Buso, and M. Gu, “Three-dimensional hybrid photonic crystals merged with localized plasmon resonances,” Opt. Express 18(5), 4491–4498 (2010). 12. M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008). 13. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004). 14. C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17(13), 10757–10766 (2009). 15. J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9(2), 897–902 (2009). 16. W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009). #154978 $15.00 USD Received 20 Sep 2011; revised 28 Oct 2011; accepted 29 Oct 2011; published 8 Nov 2011 (C) 2011 OSA 21 November 2011 / Vol. 19, No. 24 / OPTICS EXPRESS 23800 17. Y. Pu, R. Grange, C. L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010). 18. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). 19. M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. Van Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. De Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). 20. H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44(9A), 6541–6545 (2005). 21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). 22. G. P. Agrawal, “Nonlinear Fiber Optics, 4th ed,” (Academic press, San Diego, 2007). 23. S. Afshar V and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). 24. M. A. Foster, K. D. Moll, and A. L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004). 25. M. D. Turner, T. M. Monro, and S. Afshar V, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering,” Opt. Express 17(14), 11565– 11581 (2009). 26. S. Afshar V, W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, “Small core optical waveguides are more nonlinear than expected: experimental confirmation,” Opt. Lett. 34(22), 3577–3579 (2009). 27. A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” (Chapman & Hall, London, New York, 1983). 28. I. W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15(3), 1135–1146 (2007). 29. D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). 30. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).


Introduction
Recent advancement in the state-of-the-art nanofabrication facilities have led to the development of optical waveguides with subwavelength features and inhomogeneous crosssections, providing enhanced optical nonlinearities and small effective mode areas [1][2][3].Such nonlinear optical waveguides are emerging as vital components for all-optical high speed signal processing [1].The Kerr nonlinear coefficient within a waveguide is determined by the mode confinement and on the bulk nonlinear index coefficient n 2 .To greatly enhance the Kerr nonlinearity, a dramatic reduction of the effective mode area is required.A major problem of current dielectric nonlinear dielectric waveguides is the inability of confining the mode beyond the fundamental diffraction limit.On the other hand, plasmonic waveguides provide mode confinement beyond the diffraction limit; however, they are not explored for potential applications in nonlinear optics.Here, we propose a novel concept for an ultrahigh nonlinear nanoshell plasmonic waveguides with total energy confinement.We show that the silicon-core nanoshell plasmonic waveguide operating at the free-space wavelength 1.55 µm, can possess an ultrahigh Kerr nonlinearity up to 4.1×10 4 W −1 m −1 with nearly 100% of the mode energy residing inside the waveguide.Our results show that this new class of nanoshell plasmonic waveguides could lead to the realization of truly ultra-compact, high-density, integrated nanophotonic devices.

Nanoshell plasmonic waveguide beyond the diffraction limit with complete mode confinement
Confining the mode energy totally within the waveguide cross-section becomes a great challenge when approaching the subwavelength regime.Due to the fundamental diffraction limit, the mode energy cannot be confined to dielectric waveguides with subwavelength core sizes.Taking as an example, a silicon waveguide, embedded in air, depicted in Figs.1(a) and 1(b), operating at a free-space wavelength of 1.55 µm, we find that a reduction of the core width from 200 nm to 100 nm results in the dramatic loss of the mode energy confined within the waveguide from 77.4% to 0.22%.On the other hand, plasmonics provides an unmatchable ability to confine fields far beyond the diffraction limit [4][5][6].Surface plasmon polaritons (SPP) can facilitate guiding light with concentrated electromagnetic (EM) energies at the metal-dielectric interfaces [4,7,8].However, confining the mode energy completely within the waveguides physical region is very important for realizing compact integrated nanophotonic components with zero-crosstalk, which has not been addressed in both the current dielectric and plasmonic waveguide designs [1][2][3][8][9][10].
In this work we study the modal characteristics of a plasmonic waveguide comprising of a nonlinear dielectric core coated by a nanoscale metallic shell as depicted in Fig. 1(c).It is demonstrated that such nanoshell plasmonic waveguide possesses three unique features.Firstly, it creates a subwavelength cross sectional confinement of the plasmon mode with an effective mode area as small as 0.0196 µm −2 .Secondly, nearly 100% of the total mode energy can be confined within the subwavelength waveguide.Thirdly, the vectorial nature of the electromagnetic (EM) fields within the cylindrical subwavelength waveguide results in significant increase of the optical Kerr nonlinearity and reaches up to 41251.05W−1 m −1 .Our results show experimentally realizable [11,12] geometrical alterations of the plasmonic waveguide can dramatically change the distribution of the fields from the induced dispersion and yield ultrahigh Kerr nonlinearity.The ability of combining the total mode energy confinement and the ultrahigh nonlinearity in a single nanophotonic component allows the nonlinear nanoshell subwavelength plasmonic waveguide to be potentially used in nonlinear optical switching [13,14], plasmonic modulation [15,16], higher order nonlinear signal generation [17], micro-fluidic sensing, and nano-lasers [18,19] etc.Our geometry of interest shown in Fig. 1(c) is a metallodielectric plasmonic waveguide where a cylindrical nonlinear dielectric core is embedded within a silver metallic nanoshell [11,12].The core has a cross sectional width w, height h, an aspect ratio s (where s is defined as h/w) and the nonlinear core is considered as silicon with a bulk refractive index of 3.48 and a nonlinear coefficient n 2 of 14.5×10 −18 m 2 /W [20].The silver metallic shell was considered to have a complex refractive index n + ik of 0.145263+11.3587i[21] and has a thickness of d.To illustrate the fundamental modal behavior of our plasmon waveguide we characterized the nonlinearity, the mode confinement and the effective mode index for a variety of d, w, and s.The eigenmodes were extracted numerically with the finite element based eigenmode solver (COMSOL Multiphysics) within the plasmonic waveguide at the telecommunications freespace wavelength of 1.55 µm.Figure 1(c) demonstrates the mode energy distribution for the plasmonic waveguide with the same core size as in Fig. 1(b) but with a 50 nm silver coating.The mode energy is tightly confined within the dielectric core of the waveguides.In comparison, the state-of-the-art silicon nanowire waveguides [2,3] and hybrid plasmonic waveguides [8] do not possess such total confinement.
The dependence of the mode confinement on the nanoshell thickness d is further illustrated by the normalized electric field shown in Figs.1(d)-1(f).For a metallic shell with thickness < 50 nm the coupling between the plasmons of the inner and outer surfaces of the metallic shell becomes stronger and results in hybridization of the plasmon modes causing the fields to spread outside of the nanoshell as depicted in Figs.1(d) and 1(e).On the other hand, Fig. 1(f) demonstrates that for metallic shell thickness ≥ 50 nm, the coupling between the plasmons of the two surfaces becomes almost negligible and the mode is strictly confined within the dielectric core.

Mode characteristics of the nonlinear nanoshell plasmonic waveguide
To illustrate the fundamental modal behavior of our plasmonic waveguide the mode confinement is characterized for a variety of w with fixed values of s and d to 3 and 50 nm, respectively.Due to the ellipsoidal symmetry of the waveguide structure the single fundamental mode is x polarized and thus the strong discontinuity of the normal component of the electric fields at the metal-dielectric interface creates an ultrahigh mode confinement.The plasmon mode contains 97.7% of its modal energy within the dielectric core and 99.9% of its modal energy is confined within the waveguide including the metallic shell.It has been demonstrated that the definition for the effective nonlinearity with weak guidance approximation [22] fails for today's state of art subwavelength inhomogeneous cross sections i.e. for strong guidance nonlinear optical waveguides.The subwavelength plasmonic waveguide holds its vectorial nature by having non-negligible longitudinal components of the modal fields which is depicted in Fig. 2.This requires that a full vectorial model should be adopted to accurately predict the Kerr optical nonlinearity.A number of formulations have been developed to correctly predict the effective nonlinearity in such small core waveguides [3,[23][24][25].A generalized full vectorial-based nonlinear Schrodinger equation (VNSE) was developed to accurately predict the pulse propagation within subwavelength and inhomogeneous nonlinear waveguides [23] and experimental results [26] have already confirmed the added accuracy for correctly predicting the Kerr nonlinearity over conventional models.According to the vectorial model, the Kerr optical nonlinear coefficient of a singlemode highly birefringent waveguide is given by [23] where and .
where E and H are the electric and magnetic vector fields of the propagating mode and n(x,y) and n 2 (x,y) are the linear and nonlinear refractive index distributions with respect to the waveguide cross-sections.The effective mode area A eff represents a statistical measure for the mode energy density distribution.The effective mode area also possesses its full vector nature as opposed to the scalar field definition [22].The statistical definition for A eff provides the complete electromagnetic (EM) mode area compared to its counterparts [8] which are completely inaccurate to express the nonlinearity within a waveguide.The effective nonlinear refractive index coefficient 2 n is weighted with respect to the modal field distributions over the inhomogeneous waveguide cross-sections.Equation ( 3) is directly used here to calculate the effective nonlinearity.In all our considered waveguide geometries the strength of the electric fields within the metallic regions are much less than that in the dielectric core, and as the effective Kerr nonlinearity measures on the 4th power strength of the electric fields we can neglect any nonlinear contribution of the metallic portions of the waveguides.It is clear from Fig. 3(a) that the mode possesses an ultra-small effective mode area due to the plasmonic confinement of the system.For a core width of 100 nm, A eff of the nanoshell plasmonic waveguide is 0.0196 µm −2 , which, to the best of our knowledge, is the smallest effective mode area (based on the statistical measure) so far for optical waveguides at wavelength 1.55 µm.The inset of Fig. 3(a) shows the propagation lengths of the plasmon mode is less than that of the SPP modes (~24.5µm at the wavelength of 1.55 µm) at a flat silver-silicon interface.These higher losses arise from the strong localization of the fields to the metallic nanostructure.However, the high subwavelength localization and total energy confinement of the nanoshell plasmon waveguide make them unique candidates for photonic integration on the nanoscale.Further decrease of the core width leads to smaller effective areas and larger nonlinearity; however, this also greatly increases the propagation loss.Here we have restricted the results to a regime where the propagation length is of practical use.
Even though the nonlinearity γ is inversely proportional to A eff , surprisingly, our results show that γ increases at a much faster rate than the decreasing rate of A eff .The reason behind this enhanced increasing rate is that the effective nonlinear index coefficient 2 n of the nanoshell plasmonic waveguide emerges much larger than the bulk 2 n , which has not been reported before.Figures 3(a) and 3(b) show that as the core width decreases from 140 nm to 100 nm [11], A eff decreases 0.52 times, while 2 n increases 1.37 times (from 1.82 to 2.506 times the bulk silicon n 2 ).Thus, the combined effect enhances the optical nonlinearity γ to 2.64 times and reaches a highly enhanced value of 7514W −1 m −1 .Undoubtedly, 2 n plays an intriguing role to stimulate the γ value of the nanoshell plasmonic waveguide which could demonstrate an ultrahigh optical nonlinearity.It should also be noted that the nearly total energy confinement of the plasmonic waveguide holds regardless the variation of the waveguide core cross-section sizes shown in Fig. (3).We now turn our attention to the dependence of this nonlinear enhancement on geometrical properties via the variation of the waveguide aspect ratio s for a constant cross-sectional area of the nonlinear core.Here we restrict our results to modes with an effective mode index n eff >1 where the plasmon mode wavevector k mode >k 0 (k mode →n eff k 0 [27]).Figure 4(a) shows that n n increases from 2.51 to 15.03 as the aspect ratio decreases from 3 to 1.333.Once again, the ultra-small but almost constant values of A eff clearly confirm that it is not A eff that plays the dominating role to significantly enhance the nonlinearity rather it is 2 n of the waveguide core.Figure 4(b) shows that γ reaches an ultrahigh value of 41251.05W−1 m −1 for an aspect ratio of 1.333.For a fair comparison of the enhancement of γ we compare with other highly nonlinear optical waveguides with the same waveguide core material.Our results show that this ultrahigh value of γ in our plasmonic waveguide is 25 times higher than that of the stateof-the-art nonlinear silicon-on-insulator slot waveguide [3].More importantly, at this ultrahigh nonlinearity the mode contains 99.1% of energy within the entire waveguide region.It is the strong lateral localization of the nanoshell plasmonic waveguide that provides almost all the mode energy confined within the tiny silicon core with immense nonlinearity.In contrast, other geometries such as the hybrid plasmonic waveguide [8] contain only 15 to 20% of the mode energy within the low index dielectric region for the mode confinement.As a result, our nanoshell plasmonic waveguide geometry is more suitable for ultrahigh nonlinear interactions with nearly total energy confinement as well as minimal cross-talk at a subwavelength scale.

Ultrahigh nonlinearity with geometrical alteration
It can also be noticed from the inset of Fig. 4b that the propagation length decreases with a reduction of aspect ratios, giving a trade-off between the ultrahigh nonlinearity and the propagation length.As a figure of merit, we plot the product of the nonlinearity and the propagation length in the inset of Fig. 4b which shows a gain of a factor of two for decreasing aspect ratios.Further decrease of the aspect ratio of the nonlinear core leads to even greater enhancement of the Kerr nonlinearity; however, in this regime, the propagation length of the mode becomes less than the wavelength of operation.Thus we have not included these results here.To understand further the physical reason for the dramatic enhancement of the optical nonlinearity of the nanoshell plasmon waveguide we inspect the variation of the magnitude of the modal fields.The effective nonlinear coefficient 2 n is inversely proportional to the integral of square of the power flow along the propagation direction as can be followed form Eq. ( 3).The denominator in Eq. ( 3) determines the enhancement of the 2 n and can be written as ( )

Origin of the ultrahigh nonlinearity
For the nanoshell plasmon waveguide the mode is x polarized and thus the modal fields E x and H y are strongly dominant over the E y and H x fields respectively.Thus the second term inside the integral of the right hand side of the above equation becomes the multiplication of the weak modal fields and thus can be ignored.So, Eq. ( 4) can be re-written as ( ) The z component of the power flow relies on the lateral components E x and H y .Figure 5(a) depicts the magnitudes of the modal fields for the changing aspect ratios i.e. for the varying wavevector.The field components ( )

∫ ∫
) for each value of the aspect ratios.The same normalization method was followed for the magnetic field components.The E x component is dominant over the other components of the electric fields over the whole aspect ratio range.On the other hand, both the H y and H z components contribute for the high aspect ratio.However, as the aspect ratio decreases, H z becomes dominant over H y , as can be found in Fig. 5(a) for the aspect ratio of 1.333.The reduced magnitude of H y drastically weakens the energy flow along the propagation direction and effectively yields a much higher value of 2 n for the low aspect ratio.The increase of H z component of the magnetic field for low aspect ratios is reasonable from the variation of the effective mode index n eff shown in the inset of Fig. 4(a).This feature states that the n eff value of the plasmon mode reduces for decreasing aspect ratios, indicating the reduction of the plasmon mode wavevector, i.e. it approaches the stationary limit (i.e.k mode →0) when the aspect ratio is reduced.
In the limit of k mode →0, the mode becomes stationary and the current density flows only along the azimuthal direction within the metallic nanoshell cross section.This rotational current flow in the two dimensions causes the magnetic fields to be directed purely along the propagation direction.Figure 5(a) evidently demonstrates the growing dominance of H z over H y for reducing plasmon wave vector (i.e. for approaching to the stationary limit) and illustrates the enhancement of the optical nonlinearity.Figures 5(b) and 5(c) show the distributions of the magnetic field components H y and H z for the lowest aspect ratio 1.333 and depict the dominance of H z over H y .Fig. 6.Dispersion relation of the plasmon mode approaching of the stationary limit at 1.55 µm for d = 50 nm and varying s while the nonlinear core cross-sectional area was kept same as in Fig. 3.An alternative explanation of the enhancement of the effective nonlinear coefficient 2 n can be derived by analyzing the dispersive properties of the nanoshell plasmonic waveguide for varying aspect ratios.We investigate the dispersive properties of the nanoshell plasmon waveguide by using the eigenmodes and analyze the relation with the group velocity [28].Figure 6 depicts the dispersion of the plasmon modes for a range of aspect ratios.Interestingly, for the transition from the high to low aspect ratios, where the plasmon modes approach the stationary limit, the dispersion curves flatten out.As pointed out at Fig. 6, at the wavelength of 1.55 µm the dispersion curve is almost flat for the low aspect ratio of 1.333 which indicates a very small group velocity in contrast to the high group velocity for the high aspect ratio of 3. The square of the group index (n g →c/v g ) of the nanoshell plasmon modes over the varying values of the aspect ratio, shown in Fig. 4(b), exactly replicates the trend of the optical Kerr nonlinearity, illustrating the role of the reduction of the group velocity for the ultrahigh nonlinearity.This large group index greatly prolongs the nonlinear interaction between light and matter i.e. induces the enhanced nonlinear index coefficient 2 n of the system and hence, yields the ultrahigh Kerr nonlinearity.

Conclusion
We have shown that total mode energy confinement is achievable in a nanoshell nonlinear plasmonic waveguide.We have theoretically demonstrated that an ultrahigh nonlinearity can be achieved up to 41,251 W −1 m −1 due to the significant enhancement of the effective nonlinear index coefficient along with the subwavelength confinement by controlling the geometrical properties of the plasmonic waveguide.The combined properties of the enhanced nonlinearity, the subwavelength mode area and complete mode energy confinement forms an excellent silicon based nanoplasmonic platform for the realization of high-density photonic integration of nonlinear optical switching [13,14], higher order nonlinear signal generation [17] and all optical modulators [29], nano-lasers [19,20] and gain assisted plasmonic propagation [30].

Fig. 1 .
Fig. 1.Concept of the nanoshell plasmonic waveguide and associated total mode confinement.(a, b) Mode energy density distribution of a silicon waveguide with an aspect ratio of 3, calculated numerically for a 200 nm and a 100 nm core width respectively.(c), Mode energy distributions for a nanoshell plasmonic waveguide with a silicon core width of 100 nm and silver nanoshell of thickness 50 nm.(d) The normalized electric field |E| of the plasmonic mode for a 10 nm thick silver shell on the silicon core, showing large leaking of the fields outside of the waveguide.(e) |E| for a 25 nm silver shell, showing less field leakage.(f) |E| for a 50 nm thick silver shell, showing minimal field leakage and leading to nearly total confinement of the fields to the nonlinear core region.

Fig. 2 .
Fig. 2. Vectorial plots of the modal fields of the nanoshell plasmonic waveguide with a core aspect ratio of 3, a core width of 100 nm, and a 50 nm thick silver nanoshell.(a-c) Spatial distributions of the absolute values of the Ex, Ey, Ez, components.(d-f) Spatial distributions of the absolute values of the Hx, Hy, and Hz components.

Fig. 3 .
Fig. 3. Effective mode area, Kerr nonlinearity and effective nonlinear index of the plasmon mode for w=100 nm, d=50 nm and s=3.(a) Calculated Aeff of the plasmonic waveguide for

Fig. 4 .
Fig. 4. Characteristics of the effective nonlinear index, mode area, Kerr nonlinearity and group index for d = 50 nm and varying s while the nonlinear core cross-sectional area was kept same as in Fig. 3. (a) The normalized 2 n and Aeff for decreasing aspect ratios of plasmon waveguide.Inset: the neff versus aspect ratio s.(b) Kerr nonlinearity and square of the group index as function of aspect ratios for the plasmonic waveguide with a silicon core.Inset: The propagation length L and product of γ and L as a figure of merit.

Fig. 5 .
Fig. 5. (a) Magnitude of the modal field components of the plasmon mode for d = 50 nm and varying s while the nonlinear core cross-sectional area was kept same as in Fig. 3. (b, c) Spatial distributions of the Hy, and Hz components of the magnetic fields respectively of the plasmon mode for an aspect ratio of 1.333.