Crosstalk between three-dimensional plasmonic slot waveguides

Transcription

1 Crosstalk between tree-dimensional plasmonic slot waveguides Georgios Veronis and Sanui Fan Department of Electrical Engineering, Stanford University, Stanford, California Abstract: We investigate in detail te crosstalk between plasmonic slot waveguides. We sow tat te coupling beavior of deep subwavelengt tree-dimensional (3-D) plasmonic slot waveguides is very different from te one of two-dimensional (2-D) metal-dielectric-metal (MDM) plasmonic waveguides. Wile in te 2-D case te coupling occurs only troug te metal, in te 3-D case te coupling occurs primarily troug te dielectric, in wic te evanescent tail is muc larger compared to te one in te metal. Tus, in most cases te coupling between 3-D plasmonic slot waveguides is muc stronger tan te coupling between te corresponding 2-D MDM plasmonic waveguides. Suc strong coupling can be exploited to form directional couplers using plasmonic slot waveguides. On te oter and, wit appropriate design, te crosstalk between 3-D plasmonic slot waveguides can be reduced even below te crosstalk levels of 2-D MDM plasmonic waveguides, witout significantly affecting teir modal size and attenuation lengt. Tus, 3-D plasmonic slot waveguides can be used for ultradense integration of optoelectronic components Optical Society of America OCIS codes: ( ) Guided waves; ( ) Surface plasmons; ( ) Integrated optics devices. References and links 1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelengt optics, Nature 424, (2003). 2. J. Takaara, S. Yamagisi, H. Taki, A. Morimoto, and T. Kobayasi, Guiding of a one-dimensional optical beam wit nanometer diameter, Opt. Lett. 22, (1997). 3. J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, Plasmon polaritons of metallic nanowires for controlling submicron propagation of ligt, Pys. Rev. B 60, (1999). 4. J. R. Krenn, B. Lamprect, H. Ditlbacer, G. Scider, M. Salerno, A. Leitner, and F. R. Aussenegg, Nondiffraction-limited ligt transport by gold nanowires, Europys. Lett. 60, (2002). 5. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, Electromagnetic energy transfer and switcing in nanoparticle cain arrays below te diffraction limit, Pys. Rev. B 62, R16356 R16359 (2000). 6. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requica, Local detection of electromagnetic energy transport below te diffraction limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2, (2003). 7. S. I. Bozevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, Cannel plasmon-polariton guiding by subwavelengt metal grooves, Pys. Rev. Lett. 95, (2005). 8. S. I. Bozevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, Cannel plasmon subwavelengt waveguide components including interferometers and ring resonators, Nature 440, (2006). 9. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, Geometries and materials for subwavelengt surface plasmon modes, J. Opt. Soc. Am. A 21, (2004). 10. K. Tanaka and M. Tanaka, Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide, Appl. Pys. Lett. 82, (2003). (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2129

2 11. F. Kusunoki, T. Yotsuya, J. Takaara, and T. Kobayasi, Propagation properties of guided waves in index-guided two-dimensional optical waveguides, Appl. Pys. Lett. 86, (2005). 12. G. Veronis and S. Fan, Guided subwavelengt plasmonic mode supported by a slot in a tin metal film, Opt. Lett. 30, (2005). 13. L. Liu, Z. Han, and S. He, Novel surface plasmon waveguide for ig integration, Opt. Express 13, (2005). 14. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguci, T. Okamoto, M. Haraguci, and M. Fukui, Two-dimensionally localized modes of a nanoscale gap plasmon waveguide, Appl. Pys. Lett. 87, (2005). 15. E. Feigenbaum and M. Orenstein, Modeling of complementary (Void) plasmon waveguiding, J. Ligtwave Tecnol. 25, (2007). 16. D. M. Pozar, Microwave Engineering, (Wiley, New York, 1998). 17. N. N. Feng, M. L. Brongersma, and L. Dal Negro, Metal-dielectric slot-waveguide structures for te propagation of surface plasmon polaritons at 1.55 μm, IEEE J. Quantum Electron. 43, (2007). 18. G. Veronis and S. Fan, Modes of subwavelengt plasmonic slot waveguides, J. Ligtwave Tecnol. 25, (2007). 19. A. Yariv and P. Ye, Potonics: Optical Electronics in Modern Communications, (Oxford University Press, New York, 2006). 20. J. A. Pereda, A. Vegas, and A. Prieto, An improved compact 2D full-wave FDFD metod for general guided wave structures, Microwave Opt. Tecnol. Lett. 38, (2003). 21. J. Jin, Te Finite Element Metod in Electromagnetics, (Wiley, New York, 2002). 22. S. J. Al-Bader, Optical transmission on metallic wires - fundamental modes, IEEE J. Quantum Electron. 40, (2004). 23. E. D. Palik, Handbook of Optical Constants of Solids, (Academic, New York, 1985). 24. H. J. Hagemann, W. Gudat, and C. Kunz, Optical constants from te far infrared to te x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al 2 O 3, J. Opt. Soc. Am. 65, (1975). 25. G. Veronis and S. Fan, Bends and splitters in metal-dielectric-metal subwavelengt plasmonic waveguides, Appl. Pys. Lett. 87, (2005). 1. Introduction Te capability of guiding ligt at deep subwavelengt scales is of great interest in optoelectronics, in part because suc capability may enable ultradense integration of optoelectronic circuits [1]. Tis prospect for integration as motivated significant recent activities in exploring plasmonic waveguide structures [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Among tese, two-conductor waveguide geometries, wic are te optical analogue of microwave transmission lines [16], are of particular interest because tey support modes at deep subwavelengt scale wit ig group velocity over very wide range of frequencies. As a prominent example of two-conductor waveguide geometries, tree-dimensional (3-D) plasmonic slot waveguides, consisting of a deep subwavelengt slot introduced in a tin metallic film, were recently investigated [12, 13, 14, 17, 18]. To enable ultradense integration, owever, a key consideration is te packing density of optical waveguides and devices. Wen two waveguides are brougt in close proximity, teir modes overlap resulting in coupling and crosstalk between te waveguides [19]. Te crosstalk, in general, becomes stronger as te distance between te waveguides is reduced. Tus, te coupling strengt between two waveguides sets a limit on teir maximum packing density. It is terefore important to investigate te crosstalk between subwavelengt plasmonic slot waveguides. In previous studies, Zia et al. [9] investigated te coupling between two-dimensional (2-D) metaldielectric-metal (MDM) plasmonic waveguides. Tey sowed tat suc waveguides can be put at a distance of 150 nm witout significant crosstalk. In addition, Liu et al. [13] investigated te coupling between 3-D plasmonic slot waveguides formed on te same tin metal film. In tis paper we investigate in detail te crosstalk between plasmonic slot waveguides. We first sow tat for coupled lossy waveguides in general tere is a maximum in te power transfer efficiency from one waveguide to te oter. Tis maximum transfer efficiency is determined by te ratio of te coupling lengt between te two waveguides to teir mean attenuation lengt. (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2130

3 We ten consider te coupling between symmetric plasmonic slot waveguides formed on te same tin metal film. We sow tat te coupling beavior of deep subwavelengt 3-D plasmonic slot waveguides is very different from te one of corresponding 2-D MDM plasmonic waveguides. Wile in te 2-D case te coupling occurs only troug te metal, in te 3-D case te coupling occurs primarily troug te dielectric, in wic te evanescent tail is muc larger compared to te one in te metal. Tus, te coupling between 3-D plasmonic slot waveguides is muc stronger tan te coupling between te corresponding 2-D MDM plasmonic waveguides. Suc a strong coupling can be used to form a directional coupler using slot waveguides. For coupling between asymmetric plasmonic slot waveguides, as well as for vertically-coupled plasmonic slot waveguides wic are formed on parallel tin metal films, we also find tat te coupling troug te dielectric is dominant. On te oter and, we sow tat, by modifying te metal regions between te two slots, te crosstalk between 3-D plasmonic slot waveguides can be reduced even below te crosstalk levels of 2-D MDM plasmonic waveguides. Examples include a structure in wic te metal film separating te two slots as an increased tickness, as well as a structure in wic te metal region separating te two slots is I-saped. We sow tat bot of tese structures greatly reduce te crosstalk between te plasmonic slot waveguides, witout significantly affecting teir modal size and attenuation lengt. Tus, wit appropriate design, plasmonic slot waveguides can be used for ultradense integration of optoelectronic components. Te remainder of tis paper is organized as follows. In Section 2 we provide a general analysis of te coupling between lossy waveguides. In Section 3 we describe te simulation metods used for te analysis of te coupled plasmonic waveguides. In Section 4 we investigate various cases of coupling between plasmonic slot waveguides. In Section 5 we propose and analyze structures wic reduce te crosstalk between plasmonic slot waveguides. Finally, our conclusions are summarized in Section Coupling between lossy waveguides Since a key goal in tis paper is to evaluate crosstalk and to develop ways to suppress it, we will be considering te coupler formed between two identical waveguides suc tat pasematcing is automatically satisfied. Tis corresponds to te strongest coupling [19], and terefore te worst-case scenario for crosstalk. Tus, we start from a single-mode waveguide structure, and consider a coupler structure consisting of two suc waveguides, wic satisfies te mirror symmetry relation ε r (x,y) =ε r ( x,y), were ε r is te dielectric function. We assume tat te waveguide is uniform along te z direction. Suc a coupler supports two eigenmodes wit eiter symmetric (E s (x,y)) or antisymmetric (E a (x,y)) electric field distribution wit respect to te y axis. Due to te mirror symmetry of te structure, te two modes are ortogonal, i.e., 1 2 Re[E ν H μ ẑ]ds = δ νμ, ν, μ = s,a, were H s (x,y) (H a (x,y)) is te magnetic field of te symmetric (antisymmetric) mode. Te total electric field is given by E(x,y,z)=c s (z)e s (x,y)+c a (z)e a (x,y), (1) were te modal field amplitudes c s (z), c a (z) are given by c s (z) =c s (0)exp( γ s z), c a (z) = c a (0)exp( γ a z), and te modal propagation constants γ s, γ a are in general complex, i.e., γ s = α s + iβ s, γ a = α a + iβ a. Te properties of te eigenmodes of te coupler completely determine te transfer beavior between te waveguides. To see tis, we define te electric field distributions E l (x,y) E s (x,y)/ 2 + E a (x,y)/ 2, (2) (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2131

4 E r (x,y) E s (x,y)/ 2 E a (x,y)/ 2, (3) wic are igly concentrated eiter on te left (E l (x,y)) or on te rigt (E r (x,y)) waveguide. Note tat te left and rigt field distributions in Eqs. (2) and (3) are also ortogonal to eac oter. For weakly-coupled waveguides tese field distributions are almost identical to te field distribution of an isolated waveguide. Te total electric field in Eq. (1) can ten also be written as E(x,y,z)=c l (z)e l (x,y)+c r (z)e r (x,y). (4) Using Eqs. (1)-(4) we obtain [ ] [ 1 ] cl (z) [ ] cs (z) = c r (z) 1 2 1, (5) c a (z) 2 and c l (z) 2 + c r (z) 2 = c s (z) 2 + c a (z) 2. We now consider te case were only E l (x,y), wic is igly concentrated on te left waveguide, is excited at z = 0, i.e. c l (0) =1, c r (0) =0. Using Eq. (5), te power coupled into te rigt field distribution after a distance L is evaluated to be c r (L) 2 = 1 2 [exp( γ sl) exp( γ a L)] 2. (6) For lossless waveguides, i.e. γ s = iβ s, γ a = iβ a, using Eq. (6), we obtain c r (L) 2 = sin 2 ( β s β a L)=sin 2 ( π L ), 2 2 L c were we defined te coupling lengt as L c π/ β s β a. (7) We observe tat, in te case of coupled lossless waveguides, power is transferred completely from te left to te rigt waveguide for a distance L equal to te coupling lengt L c. Tus, in order to limit te crosstalk between two adjacent lossless waveguides, we must ave L L c. We now consider coupled lossy waveguides. Using Eq. (6) we obtain c r (L) 2 = exp( 2 L L p ) 1 2 [exp( γ L) exp(γ L)] 2, (8) were L p 2/(α s + α a ) is te mean attenuation lengt, and γ =(α s α a )/2 + i(β s β a )/2. Eq. (8) suggests tat, unlike lossless waveguides, for coupled lossy waveguides complete transfer of power from one waveguide to te oter is not possible, since loss is always present during te transfer process. Instead, for coupled lossy waveguides tere is a maximum in power coupled from one waveguide to te oter: p max max L c r(l) 2. (9) As it turned out, for te structures considered in tis paper α s α a /2 β s β a /2, so tat γ i(β s β a )/2. (10) (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2132

5 We found tat in all cases considered in tis paper, te approximation of Eq. (10) results in less tan 1% error in te calculated maximum transfer power p max. Inserting Eq. (10) into Eq. (8) we obtain c r (L) 2 exp( 2 L )sin 2 ( π L ), L p 2 L c and p max exp( 2xarctan(x 1 )) 1 + x 2, x = 2L c /(πl p ). (11) Terefore, under te approximation of Eq. (10), te maximum transfer power p max is only a function of L c /L p (Fig. 1). Wen te coupling lengt L c is muc smaller tan te mean attenuation lengt L p, te maximum transfer power approaces 1, similar to te lossless case. In te opposite limit, wen te coupling lengt L c is muc larger tan te mean attenuation lengt L p, te maximum transfer power approaces 0 (Fig. 1). In tis case, tere is almost no power coupled from one waveguide to te oter, irrespective of L. In addition, since tere is almost no coupling between te waveguides, te power in eac waveguide attenuates wit te attenuation lengt of an individual plasmonic slot waveguide. Maximum Transfer Power L c / L p Fig. 1. Maximum transfer power p max as a function of L c /L p, under te approximation γ i(β s β a )/2 (see Section 2). Here L c is te coupling lengt, and L p is te mean attenuation lengt. In all cases considered in tis paper, tis approximation results in less tan 1% error in te calculated p max. We would like to empasize tat te above analysis is exact, as long as te propagation constants of te eigenmodes γ s and γ a are known. Terefore, in tis paper we will employ numerical metods to evaluate γ s and γ a directly, and from tese we will determine te coupling beavior. 3. Simulation metod We use a full-vectorial finite-difference frequency-domain (FDFD) mode solver [20, 12] to calculate te symmetric (E s (x,y), H s (x,y)), and te antisymmetric (E a (x,y), H a (x,y)) eigenmodes of coupled waveguides (Sec. 2), and te corresponding modal propagation constants γ s and γ a at a given wavelengt λ 0. For waveguiding structures wic are uniform in te z direction, if an exp( γz) dependence is assumed for all field components, Maxwell s equations reduce to two (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2133

6 coupled equations for te transverse magnetic field components H x and H y [20]. Tese equations are discretized on a non-uniform ortogonal grid resulting in a sparse matrix eigenvalue problem of te form A = γ 2, wic is solved using iterative sparse eigenvalue tecniques [21]. Te discretization sceme is based on Yee s lattice [20]. To calculate te bound eigenmodes of te coupled waveguides, we ensure tat te size of te computational domain is large enoug so tat te fields are negligibly small at its boundaries [22], wile for leaky modes we use perfectly matced layer absorbing boundary conditions [21]. An important feature of tis formulation is te absence of spurious modes [22]. In addition, te frequency-domain mode solver allows us to directly use experimental data for te frequency-dependent dielectric constant of metals [23, 24], including bot te real and imaginary parts, wit no approximation. Once te modal propagation constants γ s and γ a are calculated, te coupling lengt L c between te two plasmonic slot waveguides is calculated using Eq. (7), and te maximum transfer power p max using Eqs. (8) and (9). 4. Coupling between plasmonic slot waveguides Using te analytic teory in Section 2 and te numerical metods in Section 3, we consider coupling between deep subwavelengt plasmonic slot waveguides. We focus on te optical communication wavelengt (λ 0 =1.55 μm), were subwavelengt plasmonic slot waveguides ave attenuation lengts of tens of micrometers [12, 18]. Our reference structure is a symmetric plasmonic slot waveguide [18] consisting of a slot of widt w =50 nm in a silver film of tickness =50 nm embedded in silica (n s =1.44). We first consider te coupling between two suc symmetric plasmonic slot waveguides wic are formed on te same tin silver film (Fig. 2(b)). In general, te coupling lengt L c decreases as te distance D between te slots decreases (Fig. 2(e)). Te presence of te left slot imposes a dielectric perturbation on te propagation of te bound mode of te rigt plasmonic slot waveguide and vice versa. It can be sown tat, for weak coupling, te coupling lengt L c between two waveguides is inversely proportional to te overlap integral of teir field profiles in te perturbed region [19]. In te case of plasmonic slot waveguides formed on te same metal film, L c will terefore be inversely proportional to te modal field amplitude of te plasmonic slot waveguide on te adjacent slot. Since far from te slot te modal field amplitude decays asymptotically as exp( αρ)/ ρ, were α = Re γ 2 ( 2πn s λ ) 2 [12], te coupling lengt 0 L c will increase wit D as Dexp(αD). We ave indeed verified tat tis is te case for D >200 nm, were te coupling between te waveguides is weak. In Fig. 2(f) we sow te maximum transfer power p max between te two plasmonic slot waveguides as a function of D (Fig. 2(b)). As expected, p max decreases as D increases. For D >0.94 μmweavep max < 10 3, so tat te crosstalk between te two plasmonic waveguides is negligible. As mentioned above, p max is a function of te ratio of te coupling lengt L c to te mean attenuation lengt L p (Eq. (11)). In te weak coupling regime (D >200 nm) te mean attenuation lengt L p of te modes supported by te structure is insensitive to D, since te fraction of te optical power in te metal is not affected by te coupling. Tus, in tis regime p max is solely determined by L c. In te strong coupling regime (D <200 nm) te coupling lengt is in te order of a few microns (Fig. 2(e)), and te maximum transfer power p max approaces 1 (Fig. 2(f)). In tis regime, te structure of Fig. 2(b) can be used as a directional coupler to perform power division, power coupling and switcing [19]. In te strong coupling regime, as D decreases, tere is an increase in te fraction of te modal power in te metal region separating te two slots for te mode wit symmetric electric field distribution (E s (x,y)). Consequently, te attenuation lengt of tis mode decreases significantly as D is decreased. Tis results in decreased mean attenuation (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2134

7 w Silver D Silica (a) Silver Silica w D (b) Coupling Lengt (μ m) Silver Silica (a) w D (c) (c) (b) (d) (e) D (μ m) Maximum Transfer Power Air Silver Silica (a) w (c) D (b) (d) (f) (d) D (μ m) Fig. 2. (a) Scematic of two coupled 2-D MDM plasmonic waveguides. (b) Scematic of two coupled symmetric plasmonic slot waveguides wic are formed on te same tin metal film. (c) Scematic of two vertically-coupled symmetric plasmonic slot waveguides wic are formed on parallel tin metal films. (d) Scematic of two coupled asymmetric plasmonic slot waveguides wic are formed on te same tin metal film. (e) Coupling lengt L c as a function of te distance D between two coupled plasmonic waveguides. Results are sown for 2-D MDM plasmonic waveguides (green line), symmetric plasmonic slot waveguides formed on te same metal film (red line), vertically-coupled symmetric plasmonic slot waveguides formed on parallel metal films (blue line), and asymmetric plasmonic slot waveguides formed on te same metal film (black line). In all cases, te slot widts are w =50 nm, te metal film ticknesses are =50 nm, and te operating wavelengt is λ 0 =1.55 μm. (f) Maximum transfer power p max as a function of te distance D between two coupled plasmonic waveguides. All oter parameters are as in (e). lengt L p and faster decay of te total power in te system of coupled slot waveguides wen compared wit an isolated waveguide. In addition, in tis regime te beavior of te coupling strengt wit respect to te distance D cannot be inferred directly from te field profile of an isolated plasmonic slot waveguide. In te strong coupling regime te coupler can be used for power division. However, we note tat an alternative way to perform power division is to use a splitter. Suc a device is extremely compact and as almost zero splitting loss [25]. Te beavior of coupling in 3-D plasmonic slot waveguides turns out to be completely different from te one of te corresponding 2-D MDM plasmonic waveguides. In Figs. 2(e) and 2(f) we also sow L c and p max, respectively, for a corresponding 2-D MDM plasmonic waveguide coupler wit te same dielectric layer widt w (Fig. 2(a)). We observe tat for te coupled 3-D plasmonic slot waveguides, L c is order of magnitudes smaller, and p max is order of magnitude (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2135

8 larger tan te quantities of te corresponding 2-D MDM plasmonic waveguides. For example, a minimum distance of D min =0.16 μm between 2-D MDM plasmonic waveguides is sufficient to maintain negligible crosstalk (p max < 10 3 ), wile for 3-D plasmonic slot waveguides te corresponding minimum distance is D min =0.94 μm (Fig. 2(f)). We empasize tat, wile in te 2-D case te coupling occurs only troug te metal, in te 3-D case te coupling occurs primarily troug te dielectric, in wic te evanescent tail is muc larger compared to te one in te metal. We also note tat, as, te modal propagation constant of a 3-D plasmonic slot waveguide approaces asymptotically te modal propagation constant of te corresponding 2-D MDM plasmonic waveguide ( lim γ 3D ()=γ 2D ) [18]. Similar asymptotic beavior is observed in te coupling lengt L c and te maximum transfer power p max between 3-D plasmonic slot waveguides ( lim L c,3d ()=L c,2d and lim p max,3d () =p max,2d ). However, wile te modal caracteristics of 3-D plasmonic slot waveguides converge fast to te corresponding caracteristics of 2-D MDM waveguides (γ 3D ( = 100nm) γ 2D ) [18], in te case of coupled 3-D plasmonic slot waveguides, te convergence of L c,3d () and p max,3d () to te corresponding 2-D MDM quantities L c,2d and p max,2d is muc slower and does not occur until >1μm. Tis is again due to te fact tat te coupling troug te dielectric is muc stronger tan te coupling troug te metal, so tat only wen te metal film becomes quite tick does te coupling troug te dielectric become negligible. Since te coupling in 3-D plasmonic slot waveguides occurs primarily troug te dielectric, te asymptotic beavior of vertically-coupled structures (Fig. 2(c)) is very similar to tat of orizontally-coupled structures (Fig. 2(b)). Te coupling lengt L c for te vertically-coupled waveguides also increases wit D as Dexp(αD) for D >200 nm (Fig. 2(e)), since far from te slot te modal field amplitude decays asymptotically as exp( αρ)/ ρ in bot te orizontal and vertical directions [12]. In addition, in te weak coupling regime (D >200 nm) te dependence of te maximum transfer power p max on D for te vertically-coupled plasmonic slot waveguides is similar to te one of orizontally-coupled waveguides (Fig. 2(e)). Tere are, on te oter and, some differences between tese two systems. In general, te coupling is stronger for orizontally-coupled plasmonic slot waveguides, since, for a given distance D, te modal field amplitude is larger in te orizontal tan in te vertical direction. In addition, for D <200 nm te modes of te vertically-coupled plasmonic slot waveguides leak into one of te modes of te dielectric-metal-dielectric-metal-dielectric (DMDMD) structure formed by te two parallel metal films. Tus, te vertically-coupled slot waveguides cannot be used as an efficient directional coupler, since te leakage of power to te modes of te corresponding DMDMD structure eliminates te guiding. In Figs. 2(e) and 2(f) we also sow te coupling lengt L c and maximum transfer power p max, respectively, between two asymmetric plasmonic slot waveguides [18] formed on te same tin silver film (Fig. 2(d)). Eac waveguide consists of an air slot in te silver film wic is deposited on silica. Since n s =1.44>1, te field decay rate is larger in air [12]. Tus, in te case of coupled asymmetric plasmonic slot waveguides, te coupling occurs primarily troug te dielectric substrate. In addition, since n s =1.44>1, te effective index of te mode of an asymmetric plasmonic slot waveguide is lower tan te effective index of te mode of te corresponding symmetric plasmonic slot waveguide [18]. Tus, te field decay rate in te far field of te slot (D >500 nm) is smaller in te asymmetric case, resulting in stronger coupling (Figs. 2(e), and 2(f)). Note also tat in te strong coupling regime te mode wit antisymmetric electric field distribution (E a (x,y)) may become leaky into te substrate. More specifically, if te mode of te asymmetric slot waveguide wit widt 2w and eigt is leaky, ten for te coupled asymmetric slot waveguides of widt w and eigt (Fig. 2(d)) tere exists a cutoff distance D cutoff below wic te antisymmetric mode becomes leaky. (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2136

9 Silver Silica w D i (a) Coupling Lengt (μ m) (b) i (μ m) Maximum Transfer Power (c) i (μ m) Fig. 3. (a) Scematic of a structure consisting of two coupled plasmonic slot waveguides, in wic te metal film separating te two slots as an increased tickness i. (b) Coupling lengt L c as a function of i (solid line). Te distance between te two plasmonic slot waveguides is D =150 nm. All oter parameters are as in Fig. 2(e). Also sown are te asymptotic value of L c for i (das-dotted line), and L c for two coupled 2-D MDM plasmonic waveguides wit te same w and D (dased line). (c) Maximum transfer power p max as a function of i (solid line). Also sown are te asymptotic value of p max for i (das-dotted line), and p max for two coupled 2-D MDM plasmonic waveguides wit te same w and D (dased line). All oter parameters are as in (b). 5. Reducing crosstalk between plasmonic slot waveguides In te previous section we saw tat te coupling beavior of deep subwavelengt 3-D plasmonic slot waveguides is very different from tat of 2-D MDM plasmonic waveguides, and te minimum distance required for negligible crosstalk is terefore muc larger in te 3-D case. Here we sow tat, wit appropriate design, te crosstalk between deep subwavelengt 3-D plasmonic slot waveguides can be reduced even below te crosstalk levels of 2-D MDM plasmonic waveguides. As a starting point, to reduce te coupling troug te dielectric, we increase te tickness i of te metal film separating te two slots (Fig. 3(a)). We observe tat as i increases, te coupling strengt between te slots decreases, so tat L c increases (Fig. 3(b)), and p max decreases (Fig. 3(c)). As i, bot L c and p max approac constant values. In addition, for i >0.9 μm, we ave L c,3d ( i ) > L c,2d and p max,3d ( i ) < p max,2d, i.e. te crosstalk between 3-D plasmonic slot waveguides is reduced below te crosstalk levels of te corresponding 2-D MDM plasmonic waveguides. Te use of a ticker metal region in te center provides two effects tat elp to suppress te crosstalk. First, te ticker metal film removes part of te dielectric between te slots. Since te field decays muc slower in te dielectric compared to te metal, te dielectric region between te slots sould be most critical to te coupling. Terefore, replacing te dielectric between te slots by metal greatly reduces te crosstalk. Second, te use of a ticker metal film breaks te local symmetry in eac individual slot. As a result, compared wit te mode in an individual slot (Fig. 4(a)), te mode in te coupler is actually pused away from te central metal region (Fig. 4(b)). (Similar modal pattern is also observed in plasmonic strip (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2137

10 waveguides [18].) Consequently, te coupling between 3-D plasmonic slot waveguides can be even weaker compared wit te coupling between te corresponding 2-D MDM structures. Te crosstalk between te waveguides can be furter reduced by using an I-saped central metal region (Fig. 5(a)), wic furter reduces te coupling troug te dielectric. In Figs. 5(b) and 5(c) we sow te coupling lengt L c and maximum transfer power p max, respectively, between te two plasmonic slot waveguides as a function of w i (Fig. 5(a)). We observe tat, for a given tickness i,asw i increases, te coupling strengt between te slots decreases, so tat L c increases (Fig. 5(b)), and p max decreases (Fig. 5(c)). As w i, bot L c and p max approac constant values. In addition, we observe tat te additional metal films do not significantly modify te modal power density profile (Figs. 4(b), 4(c)). We also found tat te presence of te additional metal films modifies te attenuation lengt of te supported modes by only 1%. In oter words, te I-saped metal region furter reduces te crosstalk between te plasmonic slot waveguides, witout significantly affecting teir modal size and attenuation lengt. S max 50 nm (a) 0 (b) (c) Fig. 4. (a) Power density profile of te fundamental mode of a symmetric plasmonic slot waveguide consisting of a slot of widt w =50 nm in a silver film of tickness =50 nm embedded in silica (n s =1.44). (b) Power density profile of te mode wit symmetric electric field distribution (E s (x,y)), supported by te structure of Fig. 3(a) for i =1 μm. All oter parameters are as in Fig. 3(b). Since te coupling between te two plasmonic slot waveguides is negligible for i =1 μm (Figs. 3(b), 3(c)), te power density profiles of te symmetric (E s (x,y)) and antisymmetric (E a (x,y)) modes are almost identical. (c) Power density profile of te mode wit symmetric electric field distribution (E s (x,y)), supported by te structure of Fig. 5(a) for w i =150 nm. All oter parameters are as in Fig. 5(b). Since te coupling between te two plasmonic slot waveguides is negligible for w i =150 nm (Figs. 5(b), 5(c)), te power density profiles of te symmetric (E s (x,y)) and antisymmetric (E a (x,y)) modes are almost identical. We also note tat, even toug te dimensions of te I-saped metal region ( 400 nm) are not deep subwavelengt, te lengtscale of interest is te distance D (pitc) between te slots. Tus, wit appropriate design, plasmonic slot waveguides can be very densely packed (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2138

11 in a planar plasmonic integrated optical circuit wit a pitc of only 150 nm. In addition, te proposed geometries consist of a number of interconnected metal regions. We note tat te current electronic integrated circuit tecnology already uses nanoscale metallic structures as interconnects. It sould terefore be possible to fabricate te proposed geometries wit CMOScompatible fabrication tecniques, using litograpy and etcing processes. t Silver Silica w i w i Coupling Lengt (μ m) (b) w i (μ m) D Maximum Transfer Power (a) 10 3 (c) w i (μ m) Fig. 5. (a) Scematic of a structure consisting of two coupled plasmonic slot waveguides, in wic te metal region separating te two slots is I-saped, and includes two additional metal films of tickness t. (b) Coupling lengt L c as a function of w i (solid line). Te metal film ticknesses are i =400 nm, t =50 nm. All oter parameters are as in Fig. 3(b). Also sown are te asymptotic value of L c for w i (das-dotted line), and L c for two coupled 2-D MDM plasmonic waveguides wit te same w and D (dased line). (c) Maximum transfer power p max as a function of w i (solid line). Also sown are te asymptotic value of p max for w i (das-dotted line), and p max for two coupled 2-D MDM plasmonic waveguides wit te same w and D (dased line). All oter parameters are as in (b). 6. Summary In tis paper we first sowed tat for coupled lossy waveguides in general tere is a maximum in te power transfer efficiency from one waveguide to te oter. Tis maximum transfer efficiency depends on te ratio of te coupling lengt between te two waveguides to teir mean attenuation lengt. We ten investigated in detail te crosstalk between plasmonic slot waveguides. We calculated te eigenmodes of coupled plasmonic waveguides at a given wavelengt using a fullvectorial FDFD mode solver. We first considered te coupling between symmetric 3-D plasmonic slot waveguides formed on te same tin metal film. We sowed tat te coupling beavior of deep subwavelengt 3-D plasmonic slot waveguides is very different from te one of corresponding 2-D MDM plasmonic waveguides. Wile in te 2-D case te coupling occurs only troug te metal, in te 3-D case te coupling occurs primarily troug te dielectric, (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2139

12 in wic te evanescent tail is muc larger compared to te one in te metal. Tus, te coupling between 3-D plasmonic slot waveguides is muc stronger tan te coupling between te corresponding 2-D MDM plasmonic waveguides. Suc a strong coupling can be used to form a directional coupler using slot waveguides, to perform power division, power coupling and switcing. We also considered te coupling between asymmetric plasmonic slot waveguides, as well as vertically-coupled plasmonic slot waveguides wic are formed on parallel tin metal films. In te case of coupled asymmetric plasmonic slot waveguides, te coupling occurs primarily troug te dielectric substrate. In addition, te field decay rate in te far field of te slot is smaller in te asymmetric case, resulting in stronger coupling compared to te symmetric case. Finally, we sowed tat, by modifying te metal regions between te two slots, te crosstalk between 3-D plasmonic slot waveguides can be reduced even below te crosstalk levels of 2-D MDM plasmonic waveguides. Examples include a structure in wic te metal film separating te two slots as an increased tickness, as well as a structure in wic te metal region separating te two slots is I-saped. We sowed tat bot of tese structures greatly reduce te crosstalk between te plasmonic slot waveguides, witout significantly affecting teir modal size and attenuation lengt. Tus, wit appropriate design, plasmonic slot waveguides can be used for ultradense integration of optoelectronic components. Acknowledgments Tis researc was supported by DARPA/MARCO under te Interconnect Focus Center and by AFOSR grant FA (C) 2008 OSA 4 February 2008 / Vol. 16, No. 3 / OPTICS EXPRESS 2140