Adiabatic and nonadiabatic nanofocusing of plasmons in tapered gap plasmon waveguides is analyzed using the finite-difference time-domain algorithm. Optimal adaptors between two different subwavelength waveguides and conditions for maximal local field enhancement are determined, investigated, and explained on the basis of dissipative and reflective losses in the taper. Nanofocusing of plasmons into a gap of similar to 1 nm width with more than 20 times increase in the plasmon energy density is demonstrated in a silver-vacuum taper of similar to 1 mu m long. Comparison with the approximate theory based on the geometrical optics approximation is conducted

Gramotnev, D K

Pile, DFP

eScholarship - University of California

UC BerkeleyUC Berkeley Previously Published WorksTitleAdiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguidesPermalinkhttps://escholarship.org/uc/item/0866c6tzJournalApplied Physics Letters, 89(4)ISSN0003-6951AuthorsPile, DFPGramotnev, D KPublication Date2006-07-01 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaAPPLIED PHYSICS LETTERS 89, 041111 2006Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gapplasmon waveguidesD. F. P. PileaDepartment of Optical Science and Technology, Faculty of Engineering, The University of Tokushima,Minamijosanjima 2-1, Tokushima 770-8506, JapanD. K. GramotnevbApplied Optics Program, School of Physical and Chemical Sciences, Queensland Universityof Technology, G.P.O. Box 2434, Brisbane QLD 4001, AustraliaReceived 12 March 2006; accepted 4 June 2006; published online 25 July 2006Adiabatic and nonadiabatic nanofocusing of plasmons in tapered gap plasmon waveguides isanalyzed using the finite-difference time-domain algorithm. Optimal adaptors between two differentsubwavelength waveguides and conditions for maximal local field enhancement are determined,investigated, and explained on the basis of dissipative and reflective losses in the taper.Nanofocusing of plasmons into a gap of 1 nm width with more than 20 times increase in theplasmon energy density is demonstrated in a silver-vacuum taper of 1 m long. Comparison withthe approximate theory based on the geometrical optics approximation is conducted. © 2006American Institute of Physics. DOI: 10.1063/1.2236219Localized plasmons in metallic nanostructures offerunique opportunities for nanoscale miniaturization and inte-gration of optical waveguides, devices and circuits,1–7 high-resolution near-field optical microscopy8 and lithography,9surface enhanced Raman scattering allowing single-moleculedetection,10–14 etc. Therefore, the ability to effectively deliverelectromagnetic radiation to the nanoscale is one of the ma-jor directions of research in nano-optics.This problem is usually solved by means of nanofocus-ing of light in metallic nanostructures.15–19 For example,nanofocusing of strongly localized plasmons by sharp metal-lic tips and grooves was considered in the geometrical opticsi.e., adiabatic approximation GOA, demonstrating thepossibility of effective localization focusing of the stronglyenhanced plasmon field into nanoscale regions with dimen-sions as small as several nanometers.18,19However, GOA is applicable only if the plasmon wavevector does not change significantly within one wavelengthin the structure, i.e., at sufficiently small taper angles .18,19At these angles, the plasmon does not experience noticeablereflections in the focusing structure, and its local dispersionand field distribution can approximately be determined atevery distance from the tip of the groove/metal cone, assum-ing that the taper angle is zero.18,19While the approximate methods are straightforward,physically intuitive, and yield important conclusions aboutplasmon nanofocusing, small taper angles are difficult to fab-ricate, and they result in relatively long structures/devices.This may lead to significant dissipative losses in the metalstructure. Increasing taper angle decreases distances the plas-mon travels along the taper during nanofocusing, but mayalso lead to breaching the condition of the adiabaticregime.18,19 This may result in noticeable reflections of theplasmon as it propagates towards the tip of the groove/metalcone. Therefore, on the one hand, increasing taper angleshould lead to decreasing dissipation and increasing localaPresent address: 5130 Etcheverry Hall, NSF Nanoscale Science and Engi-neering Center, University of California, Berkeley, CA 94720-1740; elec-tronic mail: d.pile@berkeley.edubElectronic mail: d.gramotnev@qut.edu.au0003-6951/2006/894/041111/3/$23.00 89, 04111Downloaded 30 Jul 2006 to 128.32.14.230. Redistribution subject to field enhancement efficiency of nanofocusing. On the otherhand, increasing taper angle should result in increasing re-flections of the plasmon. This should lead to increased reflec-tive losses of the plasmon energy, decreasing local field en-hancement and efficiency of nanofocusing. The competitionof these opposing mechanisms may result in an optimal taperangle for fixed wavelength and parameters of the media incontact.Recently there has been significant interest in using gapplasmon waveguides for subwavelength nano-optical appli-cations with subwavelength localization provided in onedimension20,21 or even both22–24 dimensions normal to thedirection of propagation. Therefore, in this letter, we conductrigorous numerical analysis of nanofocusing of plasmons intapered gap waveguides in both the adiabatic and nonadia-batic regimes and determine the optimum conditions formaximally effective focusing with the strongest local fieldenhancement. An optimal adaptor between two subwave-length plasmonic waveguides is also suggested and analyzed.Numerical confirmation of the previously developed ap-proximate method,19 its applicability conditions, and accu-racy are presented and discussed.The analyzed structure is shown in Fig. 1. A vacuum gapbetween two silver halfspaces is tapered from an initial gapwidth wi=vac/2=316.4 nm where vac=632.8 nm is thevacuum wavelength for the He–Ne laser to a final gap widthwf =vac/400=1.512 nm with the taper angle . The final gapwidth is taken as the minimal width for which the approxi-mation of continuous electrodynamics and the local Drudemodel for the determination of the metal permittivity are stillapproximately valid.19,25,26 The structure is assumed to beuniform and infinite in the z direction the system of coordi-nates is presented in Fig. 1.A metallic gap can support two different types ofcoupled plasmon modes—with the symmetric and antisym-metric distributions of the magnetic field across the gap.However, only the symmetric gap plasmon can exist at anarbitrarily small gap width. Its wave number and localizationincrease with decreasing gap width, while the wave numberand effective permittivity of the antisymmetric gap plas-© 2006 American Institute of Physics1-1AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp041111-2 D. F. P. Pile and D. K. Gramotnev Appl. Phys. Lett. 89, 041111 2006mon reduces to zero at a critical gap width and then becomesimaginary below the critical width. Therefore, only sym-metric gap plasmons can experience effectivenanofocusing.19A symmetric gap plasmon with the wave vector qgpw,where w is the varying width of the gap, propagates in thedirection of the taper i.e., along the x axis—Fig. 1. This isequivalent to the geometry of normal incidence of the plas-mon focused by a V groove.19 In this case, we can use thetwo-dimensional 2D finite-difference time-domain FDTDalgorithm for the analysis of nanofocusing. If the incidentplasmon propagates at a nonzero angle with respect to thedirection of the taper, then three-dimensional FDTD shouldbe used instead, which will require substantial computationalresources and will not allow analysis of nanofocusing withinsignificant propagation distances. Therefore, the analysis inthis letter is limited to the case of normal plasmon incidenceFig. 1.The 2D FDTD algorithm is based on the local Drudemodel to allow for the negative permittivity of the metal.27First-order Mur-type boundary conditions are used at theedges of the computational window.28 The dimensions of thecomputation grid cells are x=vac/120 and y=vac/4,000.In the FDTD simulations, the symmetric gap plasmonwas excited using a line source of the electromagnetic field,parallel to the z axis, with TM polarization magnetic fieldalong the z axis, located near the edge of the computationalwindow in the wide section of the gap with the width wi. Asthe gap plasmon propagates into the taper, its wave numberqgp increases together with increasing localization of theplasmons in the y direction. This means effective focusing ofthe incident plasmon into the region of the width that is ofthe order of wf far beyond the diffraction limit. The taperednanogap Fig. 1 is analogous to a cylindrical lens, thoughwith much tighter focusing of electromagnetic energy to avery narrow region with a possibility of a significant en-hancement of the local field see also Ref. 19. In addition,the considered structure Fig. 1 represents an adaptor be-tween two subwavelength gap plasmon waveguides ofstrongly different widths. The analysis and optimization ofsuch an adaptor are important for the development of plas-monic circuits, interconnectors, and nano-optical devices.Figure 2a presents the typical FDTD x ,y dependenceof the square of the normalized magnitude of the magneticfield H2 / Hi02 in the gap plasmon, where Hi0 is the initialamplitude of the magnetic field in the plasmon at the begin-ning of the taper i.e., at x=−L in Fig. 1. The dissipation inFIG. 1. The geometry of the tapered gap between two metallic media.qgpw is the wave vector of the gap plasmon generated by a line sourceparallel to the z axis of a magnetic field component in the z axis locatednear the edge of the computational window in the wide gap section. Thisstructure also represents an adaptor between two gap plasmon waveguidesof the widths wi and wf. The length of the taper L is determined by the taperangle .the silver is assumed to be zero i.e., m=−16.2 with the zeroDownloaded 30 Jul 2006 to 128.32.14.230. Redistribution subject to imaginary part, the taper angle =3.24° corresponding tochange in gap width per wavelength 632.8 nm in vacuumalong the x axis of −35 nm, and the corresponding lengthof the taper L5.6 m. In particular, it can be seen thatsubstantial plasmon localization simultaneously with thestrong local field enhancement is achieved in the structure.For example, an 31.2 times increase of the energy densityin the plasmon at the narrowest part of the taper at x=0 isachieved Fig. 2a. For x0, the amplitude of the gap plas-mon is independent of x because w=wf =const recall thatthe dissipation is assumed to be zero.Using GOA Ref. 19 for the analysis of nanofocusing inthe same silver-vacuum tapered gap gives the increase in thefield intensity of 31.9 times, which is only 2% differentfrom the FDTD result. This is in agreement with the expec-tation that the GOA approach provides good accuracy for theanalysis of plasmon nanofocusing, if the taper angle  issmaller than the critical angle c17°, which determines theapplicability of GOA in the considered structure.19The effect of dissipation in the metal on nanofocusing isillustrated by Fig. 2b, for which we assume that m=−16.2+0.5i. As a result, the energy density increase withinthe taper of 11.8 times has been obtained using FDTDsimulations Fig. 2b, which is significantly below the en-hancement achieved in Fig. 2a. This reduction in the in-crease of the plasmon energy density is due to dissipativelosses in the metal. GOA gives the increase of the plasmonenergy density in the same structure =3.24° with dissipa-tion of 9.0 times. The main reason for the discrepanciesbetween the FDTD and GOA approaches for very small taperangles is due to the accumulation of numerical errors in theFDTD method. This is related to relatively large distances ofplasmon propagation along the sharp taper in the above ex-ample, L5.6 m, and this results in a very large numberof time iterations.Therefore, analysis of plasmon nanofocusing in struc-tures with small taper angles significantly below c1 shouldrather be conducted using the approximate GOA approachthat provides high accuracy of results. FDTD method is in-efficient in this case and may result in noticeable numericalerrors. At the same time, as mentioned above, this situation isassociated with significant dissipative losses for the focusedplasmons, because they have to propagate large distancesalong the taper. Increasing taper angle decreases dissipation,but eventually causes breaching the GOA applicability con-dition, and the numerical approaches become essential. For-tunately, numerical errors associated with large number ofFIG. 2. The typical instantaneous x ,y dependencies of the relative squareof the magnetic field H2 / Hi02 in the plasmon inside the silver-vacuumtaper at the vacuum wavelength vac=632.8 nm He–Ne laser, wi=316.4 nm, wf =1.512 nm, =3.24°, and Hi0 is the initial amplitude of themagnetic field in the plasmon at x=−L. a Dissipation in the metal is as-sumed to be zero: m=−16.2. b Dissipation is nonzero: m=−16.2+0.5i.time iterations reduce with increasing taper angle, whichAIP license or copyright, see http://apl.aip.org/apl/copyright.jsp041111-3 D. F. P. Pile and D. K. Gramotnev Appl. Phys. Lett. 89, 041111 2006makes the FDTD approach increasingly more efficient. Fromthis point of view, FDTD and GOA appear to complementeach other, covering the whole possible range of taperangles.More detailed comparison of the GOA and FDTD ap-proaches is presented in Fig. 3 that shows the dependenciesof the square of the relative plasmon amplitude Hf02 / Hi02at the end of the taper i.e., at x=0—Fig. 1 on angle . Thevalues for the initial and final gap widths are the same for allthe curves in Fig. 3: wi=316.4 nm and wf =1.512 nm. Thismeans that increasing taper angle  results in decreasing itslength L, and this leads to smaller dissipative losses. In theabsence of dissipation, GOA gives no dependence of therelative plasmon intensity on taper angle dashed line in Fig.3. This is because GOA neglects the reflective losses, andthis is correct only for small taper angles c17°.19 Thisis the reason for substantial discrepancies between the FDTDand GOA curves for both the cases with and without dissi-pation at taper angles 10° Fig. 3. On the other hand, if10°, good agreement between the FDTD and GOA re-sults can be seen Fig. 3. This is again in agreement with theapplicability conditions for GOA derived in Ref. 19.Another important aspect that follows from Fig. 3 is thatthe FDTD dependence in the presence of dissipation sug-gests that there exists an optimal taper angle opt13.5° seecircles in Fig. 3 at which the relative plasmon intensity ismaximal: Hopt2 / H0221.8. Note that it is not possible toobtain this optimal angle in GOA—compare circles andsquares in Fig. 3. This is because the optimal taper angle isdetermined by the competition of decreasing dissipativelosses and increasing reflective losses when  is increasedsee above. It is clear that opt is a function of the wavefrequency and permittivities of the metal and the dielectricfilling the gap. For example, increasing dissipation i.e., in-creasing imaginary part of the permittivity in the metal re-sults in increasing optimal taper angle.A taper with the optimal angle works as an optimal adap-tor between the waveguides. It enables the most effectiveenergy coupling between two gap plasmon waveguides ofdifferent widths. For example, at the optimal taper i.e., at=opt13.5° in the considered structure, highly efficientcoupling of gap plasmons from the 316.4 nm waveguide intothe 1.512 nm nanowaveguide or nanofocusing into this nar-row region with the 2,200% enhancement of the energydensity can be achieved by the tapered structure whoselength is only 1.3 m. Increasing taper angle above optresults in further reduction of the adaptor length and maystill correspond to a significant enhancement of the plasmonFIG. 3. The dependencies of the square of the relative plasmon amplitudeHf02 / Hi02 at the end of the taper i.e., at x=0—Fig. 1 on taper angle ,calculated using FDTD circles and crosses and GOA dashed curve andsquares in the absence of dissipation crosses and dashed line: m=−16.2and with dissipation in silver squares and circles: m=−16.2+0.5i. Theother structural and wave parameters are the same as for Fig. 2.Downloaded 30 Jul 2006 to 128.32.14.230. Redistribution subject to field in the narrow waveguide. For example, if 57.9° theincrease in the energy density in the plasmon at the end ofthe taper at the beginning of the narrow waveguide is850% with the taper length of 0.29 m.In summary, this letter presents the numerical FDTDanalysis of adiabatic and nonadiabatic nanofocusing of plas-mons in tapered nanogaps between two identical metallicmedia. The analysis was compared with the previously de-veloped approximate theory of adiabatic nanofocusing inmetal V grooves,19 providing good agreement of the approxi-mate and rigorous results for taper angles satisfying the ap-plicability condition for the approximate theory.19 At thesame time, it is also demonstrated that in the nonadiabaticregime of nanofocusing, reflections of plasmons in the focus-ing structure play a significant role, resulting in the existenceof an optimal taper angle for achieving the maximal localfield enhancement.The obtained results will be important for effective de-livery of electromagnetic energy to the nanoscale, includingnanooptical devices and waveguides, quantum dots, singlemolecules, etc. In addition, optimal adaptors between differ-ent subwavelength plasmonic waveguides were suggestedand analyzed.The authors gratefully acknowledge support from the Ja-pan Society for the Promotion of Science and the High Per-formance Computing Division at the Queensland Universityof Technology.1S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel,and A. A. G. Requicha, Nat. Mater. 2, 229 2003.2J. R. Krenn, Nat. Mater. 2, 210 2003.3J. Takahara and T. Kobayashi, Opt. Photonics News 15, 55 2004.4D. F. P. Pile and D. K. Gramotnev, Opt. Lett. 29, 1069 2004.5D. F. P. Pile and D. K. Gramotnev, Appl. Phys. Lett. 86, 161101 2005.6D. F. P. Pile and D. K. Gramotnev, Opt. Lett. 30, 1186 2005.7D. K. Gramotnev and D. F. P. Pile, Appl. Phys. Lett. 85, 6323 2004.8S. Kawata, Topics in Applied Physics Springer, Berlin, 2001, Vol. 81.9W. Srituravanich, N. Fang, C. Sun, Qi. Luo, and X. Zhang, Nano Lett. 4,1085 2004.10K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari,and M. S. Feld, Phys. Rev. Lett. 78, 1667 1997.11S. Nie and S. R. Emory, Science 275, 1102 1997.12R. Hillenbrand, T. Taubner, and F. Kellmann, Nature London 418, 1592002.13B. Pettinger, B. Ren, G. Picardi, R. Schuster, and G. Ertl, Phys. Rev. Lett.92, 096101 2004.14T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata,Phys. Rev. Lett. 92, 220801 2004.15Kh. V. Nerkararyan, Phys. Lett. A 237, 103 1997.16A. J. Babadjanyan, N. L. Margaryan, and Kh. V. Nerkararyan, J. Appl.Phys. 87, 3785 2000.17H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H.Yao, Opt. Express 13, 10795 2005.18M. I. Stockman, Phys. Rev. Lett. 93, 137404 2004.19D. K. Gramotnev, J. Appl. Phys. 98, 104302 2005.20H. Miyazaki and Y. Kurokawa, Phys. Rev. Lett. 96, 097401 2006.21J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, Phys. Rev.B 73, 035407 2006.22D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, T.Okamoto, M. Haraguchi, and M. Fukui, Appl. Phys. Lett. 87, 2611142005.23G. Veronis and S. Fan, Opt. Lett. 30, 3359 2005.24L. Liu, Z. Han, and S. He, Opt. Express 13, 6645 2005.25I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev.B 69, 121403R 2004.26A. Liebsch, Phys. Rev. Lett. 54, 67 1985.27D. F. P. Pile, Appl. Phys. B: Lasers Opt. 81, 607 2005.28G. Mur, IEEE Trans. Electromagn. Compat. 40, 100 1998.AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguides

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