Finite-Difference Time-Domain Study of Guided Modes in Nano-plasmonic Waveguides

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1 Finite-Difference Time-Domain Stud of Guided Modes in Nano-plasmonic Waveguides Yan Zhao Student Member IEEE and Yang Hao Senior Member IEEE arxiv:cond-mat/6858v cond-mat.other Jun 7 Abstract A conformal dispersive finite-difference timedomain (FDTD) method is developed for the stud of onedimensional (-D) plasmonic waveguides formed b an arra of periodic infinite-long silver clinders at optical frequencies. The curved surfaces of circular and elliptical inclusions are modelled in orthogonal FDTD grid using effective permittivities (EPs) and the material frequenc dispersion is taken into account using an auiliar differential equation (ADE) method. The proposed FDTD method does not introduce numerical instabilit but it requires a fourth-order discretisation procedure. To the authors knowledge it is the first time that the modelling of curved structures using a conformal scheme is combined with the dispersive FDTD method. The dispersion diagrams obtained using EPs and staircase approimations are compared with those from the frequenc domain embedding method. It is shown that the dispersion diagram can be modified b adding additional elements or changing geometr of inclusions. Numerical simulations of plasmonic waveguides formed b seven elements show that row(s) of silver nanoscale clinders can guide the propagation of light due to the coupling of surface plasmons. I. INTRODUCTION It is well known that photonic crstals (PCs) offer unique opportunities to control the flow of light. The basic idea is to design periodic dielectric structures that have a bandgap for a particular frequenc range. Periodic dielectric rods with removed one or several rows of elements can be used as waveguiding devices when operating at bandgap frequencies. A lot of effort has been made to obtain a complete and wider bandgap. It has been shown that a triangular lattice of air holes in a dielectric background has a complete bandgap for TE (transverse electri mode while a square lattice of dielectric rods in air has a bandgap for TM (transverse magneti mode. The devices operating in the bandgap frequencies are not the onl option to guide the flow of light. Another waveguiding mechanism is the total internal reflection (TIR) in one-dimensional (-D) periodic dielectric rods 3. It is analsed in 3 that a single row of dielectric rods or air holes supports waveguiding modes and therefore can be also used as waveguide. In the design of such waveguides consisting of several rows of dielectric rods with various spacings is proposed. Recentl a new method for guiding electromagnetic waves in structures whose dimensions are below the diffraction limit has been proposed. The structures are termed as plasmonic waveguides which have an operation of principle based on near-field interactions between closel spaced noble metal nanoparticles (spacing λ) that can be efficientl ecited at their surface plasmon frequenc. The guiding principle relies on coupled plasmon modes set up b near-field dipole interactions that lead to coherent propagation of energ along the arra. Analogous structures as waveguides in microwave regime include periodic metallic clinders to support propagating waves 5 arra of flat dipoles which support guided waves 6 and Yagi-Uda antennas 7 8 etc. However although these structures can be scaled to optical frequencies with appropriate material properties their dimensions are limited b the so-called diffraction limit λ/(n). On the other hand plasmonic waveguides emplo the localisation of electromagnetic fields near metal surfaces to confine and guide light in regions much smaller than the free space wavelength and can effectivel overcome the diffraction limit. Previous analsis of plasmonic structures includes the plasmon propagation along metal stripes wires or grooves in metal 9 and the coupling between plasmons on metal particles in order to guide energ 5 6 etc. Such subwavelength structures can also find their applications e.g. efficient absorbers and electricall small receiving antennas at microwave frequencies. Recentl composite materials containing randoml distributed electricall conductive material and non-electricall conductive material have been designed 7. The are noted to ehibit plasma-like responses at frequencies well below plasma frequencies of the bulk material. The finite-difference time-domain (FDTD) method 8 is seen as the most popular numerical technique especiall because of its fleibilit in handling material dispersion as well as arbitrar shaped inclusions. In 9 the optical pulse propagation below the diffraction limit is shown using the FDTD method. Also with the FDTD method the waveguide formed b several rows of silver nanorods arranged in heagonal is studied. Despite these eamples of appling the FDTD method for the plasmonic structures the accurac of modelling has not been proven et. When modelling curved structures unless using etremel fine meshes due to the nature of orthogonal and staggered grid of conventional FDTD often modifications need to be applied in order to improve the numerical accurac such as the treatment of interfaces between different materials even for planar structures and the improved conformal algorithms using structured meshes for curved surfaces. In addition to the modifications at material interfaces the material frequenc dispersion has also to be taken into account in FDTD modelling 3 5. However modelling dispersive materials with curved surfaces still remains to be a challenging topic due to the compleit in algorithm as well as the introduction of numerical instabilit. An alternative wa to solve this problem is based on the idea of effective permittivities (EPs) 6 8 in the underling Cartesian coordinate sstem

2 .8.6 E H z E Fig.. Comparison of the filling ratio for E component in FDTD modelling of a circular clinder using ( staircase approimations and ( a conformal scheme. The radius of circular clinder is ten cells. and the dispersive FDTD scheme can be therefore modified accordingl without affecting the stabilit of algorithm. In this paper we first propose a novel conformal dispersive FDTD algorithm combining the EPs together with an auiliar differential equation (ADE) method 8 then appl the developed method to the modelling of plasmonic waveguides formed b an arra of circular or elliptical shaped silver clinders at optical frequencies. Numerical FDTD simulation results are verified b a frequenc domain embedding method 9. To the authors knowledge it is the first time that a conformal scheme is combined with the dispersive FDTD method for the modelling of nano-plasmonic waveguides. II. CONFORMAL DISPERSIVE FDTD METHOD USING EFFECTIVE PERMITTIVITIES Conventionall staircase approimations are often used to model curved electromagnetic structures in an orthogonal FDTD domain. Figure ( shows an eample laout of an infinite-long clinder in the free space represented in a twodimensional (-D) orthogonal FDTD domain. The approimated shape introduces spurious numerical resonant modes which do not eist in actual structures. On the other hand using the concept of filling ratio which is defined as the ratio of the area of material ε to the area of a particular FDTD cell the curvature can be properl represented in FDTD domain as shown in Fig. ( where different levels of darkness indicate different filling ratios of material ε. The accurac of modelling can be significantl improved compared with staircase approimations as will be shown in a later section. According to 8 the EP in a general form is given b ε eff = ε ( n ) + ε n () where n is the projection of the unit normal vector n along the field component as shown in Fig. ε and ε are parallel and perpendicular permittivities to the material interface respectivel and defined as ε = fε + ( f)ε () ε = f/ε + ( f)/ε (3) where f is the filling ratio of material ε in a certain FDTD cell. In this paper we consider the inclusions as silver clinders which at optical frequencies can be modelled using the Drude. - f n n f Fig.. Laout of a quarter circular inclusion in orthogonal FDTD grid for E component. The radius of circular clinder is three cells. dispersion model ε (ω) = ε ( ω p ω jωγ ) () where ω p is the plasma frequenc and γ is the collision frequenc. At the frequencies below the plasma frequenc the real part of permittivit is negative. In this paper we assume that the silver clinders are embedded in the free space (ε = ε ). In order to take into account the frequenc dispersion of the material the electric flu densit D is introduced into standard FDTD updating equations. At each time step D is updated directl from H and E can be calculated from D through the following steps. Substitute () and (3) into () and using the epressions for ε and ε () the constitutive relation in the frequenc domain reads {ω γjω 3 γ + ( f)ω p ω + γ( f)ω p jω}d = ω γjω 3 (γ + ω p)ω + γω pjω +f( f)( n )ω p ε E. (5) Using the inverse Fourier transformation i.e. jω / t we obtain the constitutive relation in the time domain as { } 3 + γ t t 3 + γ + ( f)ωp t + γ( f)ω p D t = 3 + γ t t 3 + (γ + ωp) t + γω p t +f( f)( n )ω p ε E. (6) The FDTD simulation domain is represented b an equall spaced three-dimensional (3-D) grid with periods and z along - - and z-directions respectivel. The time step is t. For discretisation of (6) we use the central finite difference operators in time (δ t ) and the central average operator with respect to time (µ t ): t δ t ( t) 3 t 3 δ3 t ( t) 3 µ t t δ t ( t) µ t t δ t t µ3 t µ t (7)

3 3 where the operators δ t and µ t are defined as in 3: δ t F n F n+ F n (8) µ t F n F n+ + F n. (9) Here F represents field components and m m m z are indices corresponding to a certain discretisation point in the FDTD domain. The discretised Eq. (6) reads { δ t ( t) + γ δ3 t +γ( f)ω p ( t) 3 µ t + γ + ( f)ωp δt δ t t µ3 t δt } D = +(γ + ωp) ( t) µ t + γωp +f( f)( n )ω pµ t δ t ( t) + γ δ3 t δ t t µ3 t ( t) µ t ( t) 3 µ t ε E. () Note that in the above equations we have kept all terms to be the fourth-order to guarantee numerical stabilit. Equation () can be written as D n+ D n + 6D n D n + D n 3 ( t) + γ Dn+ D n + D n D n 3 ( t) 3 + γ + ( f)ωp D n+ D n + D n 3 ( t) + γ( f)ωp D n+ + D n D n D n 3 E n+ E n + 6E n E n + E n 3 = ε ( t) + ε γ En+ E n + E n E n 3 ( t) 3 ( + ε γ + ωp ) E n+ E n + E n 3 ( t) + ε γωp E n+ + E n E n E n 3 + ε f( f)( n )ωp ( E n+ + E n + 6E n 6 + E n + E n 3). () The indices m m and m z are omitted from () since E and D are located at same locations. Solving for E n+ then the updating equation for E in FDTD iterations reads E n+ = b D n+ + b D n + b D n + b 3 D n + b D n 3 (a E n + a E n + a 3 E n + a E n 3 )/a () with the coefficients given b a = ε ( t) + γ ( t) 3 + γ + ωp ( t) + γω p + f( f)( n )ω p 6 a = ε ( t) γ ( t) 3 + γω p t + f( f)( n )ωp 6 a = ε ( t) γ + ωp ( t) + 3f( f)( n )ωp 8 a 3 = ε ( t) + γ ( t) 3 γω p t + f( f)( n )ωp a = ε ( t) b = ( t) + b = ( t) γ ( t) 3 + γ + ωp ( t) γω p + f( f)( n )ω p 6 γ ( t) 3 + γ + ( f)ωp ( t) γ ( t) 3 + γ( f)ω p t b = 6 ( t) γ + ( f)ω p ( t) b 3 = ( t) + b = ( t) γ ( t) 3 γ( f)ω p t γ ( t) 3 + γ + ( f)ωp ( t) + γ( f)ω p γ( f)ω p The computations of H and D are performed using Yee s standard updating equations in the free space. Note that if the plasma frequenc is equal to zero (ω p = ) then () reduces to the updating equation in the free space i.e. E = D/ε. III. FDTD CALCULATION OF DISPERSION DIAGRAM Appling the Bloch s periodic boundar conditions (PBCs) 3 36 FDTD method can be used to model periodic structures and calculate their dispersion diagrams For an periodic structures the field at an time should satisf the Bloch theor i.e. E(d + = E(d)e jka H(d + = H(d)e jka (3) where d is the distance vector of an location in the computation domain k is the wave vector and a is the lattice vector along the direction of periodicit. When updating the fields at the boundar of the computation domain using FDTD the required fields outside the computation domain can be calculated using known field values inside the domain through Eq. (3). Although instead of using real values in conventional FDTD computations the calculation of dispersion diagrams requires comple field values since onl one unit cell is modelled the computation load is not significantl increased. First we appl the developed conformal dispersive FDTD method to calculate the dispersion diagram for -D plasmonic waveguides formed b an arra of periodic infinite-long (along z-direction) circular silver clinders. As shown in Fig. 3 the -D simulation domain (-) with TE modes (therefore onl E E and H z are non-zero fields) is truncated using Bloch s PBCs in -direction and Berenger s perfectl matched laers (PMLs) 39 in -direction. The Berenger s PMLs have ecellent performance for absorbing propagating waves 39

4 Bloch periodic boundar condition perfectl matched laer free space r a silver clinder perfectl matched laer Normalised intensit (a. u.) FDTD with staircase ( =.5 m) -9 FDTD with staircase ( =. 5 m) -9 FDTD with EP ( =.5 m) Embedding -9 Bloch periodic boundar condition Fig. 3. The laout of the -D FDTD computation domain for calculating dispersion diagram for -D periodic structures. The inclusion has a circular cross-section with radius r and the period of the -D infinite structure is a. however for evanescent waves field shows growing behaviour inside PMLs. Since the waves radiated b point or line sources consist of both propagating and evanescent components etra space (tpicall a quarter wavelength at the frequenc of interest) between PMLs and the circular inclusion is added to allow the evanescent waves to deca before reaching the PMLs. The radius of silver clinders is r =.5 8 m and the period is a = m. The plasma and collision frequencies are ω p = rad/s and γ = 3. 3 Hz respectivel in order to closel match the bulk dielectric function of silver. The FDTD cell size is = =.5 9 m with the time step t = /( s (where c is the speed of light in the free space) according to the Courant stabilit criterion 8. Although the stabilit condition for high-order FDTD method is tpicall more strict than the conventional one since the average operator µ t is applied to develop the algorithm we have not found an instabilit for a complete time period of more than time steps used in all simulations. A wideband magnetic line source is placed at an arbitrar location in the free space region of the -D simulation domain in order to ecite all resonant modes of the structure within the frequenc range of interest (normalised frequenc f = ωa/(π.5): g(t) = e ( t t τ ) e jωt () where t is the initial time dela τ defines the pulse width and ω is the centre frequenc of the pulse (f =.5). The magnetic fields at one hundred random locations in the free space region are recorded during simulations transformed into the frequenc domain and combined to etract individual resonant mode corresponding to each local maimum. For each wave vector a total number of time steps are used in our simulations to obtain enough accurate frequenc domain results. In order to demonstrate the advantage of EPs and validate the proposed conformal dispersive FDTD method we have Frequenc (ω a/π Fig.. Comparison of the first resonant frequenc (transverse mode) at wave vector k = π/a calculated using the FDTD method with staircase approimations the FDTD method with EPs and the frequenc domain embedding method. also performed simulations using staircase approimations for the circular clinder as shown in Fig. (. Figure shows the comparison of the first resonant frequenc (transverse mode) at wave vector k = π/a of the plasmonic waveguide calculated using the FDTD method with staircase approimations the FDTD method with EPs and the frequenc domain embedding method. With the same FDTD spatial resolutions the model using EP shows ecellent agreement with the results from the frequenc domain embedding method on the contrar the staircase approimation not onl leads to a shift of the main resonant frequenc but also introduces a spurious numerical resonant mode which does not eist in actual structures. The same effect has also been found for nondispersive dielectric clinders. It is also shown in Fig. that although one ma correct the main resonant frequenc using finer meshes the spurious resonant mode still remains. The problem of frequenc shift and spurious modes become severer when calculating higher guided modes near the flat band region (i.e the region where waves travel at a ver low phase velocit). Even with a refined spatial resolution the staircase approimation fails to provide correct results (not shown). On the other hand using the proposed conformal dispersive FDTD scheme all resonant modes are correctl captured in FDTD simulations as demonstrated b the comparison with the embedding method as shown in Fig. 5. According to previous analsis using the frequenc embedding method the fundamental mode in the modelled plasmonic waveguide is transverse mode and the second guided mode is longitudinal which is also shown b the distribution of electric field intensities in Fig. 6 from our FDTD simulations. The higher guided modes are referred to as plasmon modes. For demonstration of field smmetries and due to the TE mode considered in our simulations we have plotted the distributions of magnetic field corresponding to different resonant modes at wave number k = π/a as marked in Fig. 5 as shown in Fig. 7. Sinusoidal sources for ecitation of certain single

5 5.3 d) e) Frequenc (ω a /π c ).5..5 Embedding FDTD with EP Wave vector (π/ Fig. 5. Comparison of dispersion diagrams for an arra of infinite-long (along z-direction) circular silver clinders calculated using the FDTD method with EPs and the frequenc domain embedding method Fig. 6. Normalised total electric field intensities corresponding to ( transverse and ( longitudinal modes at wave number k = π/a as marked in Fig. 5. The structure is infinite along -direction. mode are used and the sources are placed at different locations corresponding to different smmetries of field patterns. All field patterns are plotted after the stead state is reached in simulations. The modes ( ( and (d) in Fig. 7 are even modes (relative to the direction of periodicit of the waveguide i.e. -ais) and ( and (e) are considered as odd modes. The above comparison of the simulation results calculated using the conformal dispersive FDTD method and the embedding method clearl demonstrates the effectiveness of appling the EPs in FDTD modelling. Furthermore in contrast to the embedding method the main advantage of the FDTD method is that arbitrar shaped geometries can be easil modelled. We have applied the conformal dispersive FDTD method to stud the effect of different inclusions on the dispersion diagrams of -D plasmonic waveguides. The geometries considered are two rows of periodic infinite-long (along z-direction) circular silver clinders arranged in square lattice and a single row of periodic infinite-long (along z-direction) elliptical silver clinders. The elliptical element has a ratio of semimajor-tosemiminor ais : where the semiminor ais is equal to the radius of the circular element (5. nm). For the two rows of circular nanorods the spacing between two rows (centreto-centre distance) is 75. nm. The dispersion diagrams for these structures are plotted in Figs. 8 and. Comparing the dispersion diagrams for a single circular element in Fig. 5 and two circular elements in Fig. 8 we can see that the dispersion e) d) Fig. 7. Normalised distributions of magnetic field corresponding to different resonant modes at wave number k = π/a as marked in Fig. 5: ( ( (d): even modes and ( (e): odd modes. The structure is infinite along - direction. (Note that the coordinate has been rotated 9 degrees anti-clockwise from Fig. 3 for better presentation of the figure.) Frequenc (ω a / π c ) Wave vector (π/ Fig. 8. Dispersion diagram for two rows of periodic infinite-long (along z- direction) circular silver clinders arranged in square lattice calculated from conformal dispersive FDTD simulations. diagram has been modified due to the change of inclusion. The strong coupling between two elements introduces additional guided modes to appear in dispersion diagram. Such phenomenon has also been studied for dielectric (non-dispersive) nanorods previousl. The distributions of magnetic field for selected guided modes as marked in Fig. 8 are plotted in Fig. 9. The modes ( ( and (d) are even modes while ( and (e) are odd modes. The dispersion diagram for a single elliptical element as inclusion is shown in Fig.. It can be seen that more guided modes appear which is caused b the change of inclusion s geometrical shape from circular to elliptical. The frequenc corresponding to the lowest mode has been lowered due to the increase of inclusion s volume. The distributions of magnetic field for selected guided modes are plotted in Fig.. The modes ( and (d) are even modes and ( ( and (e) are odd modes respectivel. IV. WAVE PROPAGATION IN PLASMONIC WAVEGUIDES FORMED BY FINITE NUMBER OF ELEMENTS In order to stud wave propagations in plasmonic waveguides formed b a finite number of silver nanorods we have e) d)

6 6 d) e) d) e) Fig. 9. Normalised distributions of magnetic field corresponding to different guided modes as marked in Fig. 8: ( ( (d): even modes and ( (e): odd modes. The structure is infinite along -direction. (Note that the coordinate has been rotated 9 degrees anti-clockwise from Fig. 3 for better presentation of the figure.) Frequenc (ω a / π c ) Wave vector (π/ Fig.. Dispersion diagram for a single row of periodic infinite-long (along z-direction) elliptical silver clinders calculated from conformal dispersive FDTD simulations. replaced PBCs in -direction with PMLs and added additional cells for the free space region to the simulation domain. The number of nanorods under stud is seven. The spacing (pseudo-period) between adjacent elements remains the same as for infinite structures considered in the previous section. For a single mode ecitation we choose the frequenc of corresponding mode from dispersion diagram and ecite sinusoidal sources at different locations with respect to the smmetr of different guided modes at one end of the waveguides. For the plasmonic waveguides formed b different tpes of inclusions we have chosen certain eigen modes: mode Fig. 7( for a single row of circular clinders mode Fig. 9(d) for two rows of circular clinders and mode Fig. (e) for a single row of elliptical clinders. The distributions of magnetic field intensit for different waveguides operating in these guided modes are plotted in Fig.. The field plots are taken after the stead state is reached in simulations. It is clearl seen that single guided modes are coupled into these waveguides but the ecitation of certain modes highl depends on the smmetr of field patterns. The energ that can be coupled e) d) Fig.. Normalised distributions of magnetic field corresponding to different guided modes as marked in Fig. : ( (d): even modes and ( ( (e): odd modes. The structure is infinite along -direction. (Note that the coordinate has been rotated 9 degrees anti-clockwise from Fig. 3 for better presentation of the figure.)..6.8 Fig.. Normalised distributions of magnetic field intensit corresponding to different guided modes for seven-element plasmonic waveguides formed b ( a single row of circular nanorods (the corresponding eigen mode is shown in Fig. 7() ( two rows of circular nanorods arranged in square lattice (the corresponding eigen mode is shown in Fig. 9() ( a single row of elliptical nanorods (the corresponding eigen mode is shown in Fig. (e)). (Note that the coordinate has been rotated 9 degrees anti-clockwise from Fig. 3 for better presentation of the figure.) into the waveguides also depends on the matching between source and the plasmonic waveguide. V. CONCLUSIONS In conclusion we have developed a conformal dispersive FDTD method for the modelling of plasmonic waveguides formed b an arra of periodic infinite-long silver clinders at optical frequencies. The conformal scheme is based on effective permittivities and its main advantage is that since conventional orthogonal FDTD grid is maintained in simulations no numerical instabilit is introduced. The material frequenc dispersion is taken into account using an auiliar differential equation method. The comparison of dispersion diagrams for one-dimensional periodic silver clinders calculated using the conformal dispersive FDTD method the conventional

7 7 dispersive FDTD method with staircase approimations and the frequenc domain embedding method demonstrates the accurac of the proposed method. It is shown that b adding additional element or changing the geometr of inclusions the corresponding dispersion diagram can be modified. Numerical simulations of plasmonic waveguides formed b seven elements show that the eigen modes in infinite structures can be ecited but highl depend on the smmetr of field patterns of certain modes. Further work includes the investigation of the effects of different number of elements in plasmonic waveguides on guided modes and the calculation of group velocit of different modes propagating in these waveguides. Although results presented in this paper have been focused at optical frequencies with future advances in microwave plasmonic materials novel applications can be found in the designs of small antenna and efficient absorbers. ACKNOWLEDGMENT The authors would like to thank Mr. N. Giannakis for providing simulation results using the embedding method and thank Dr. Pavel Belov for helpful discussions. The authors would also like to thank reviewers for their valuable comments and suggestions. REFERENCES J. D. Joannopoulos R. D. Meade and J. N. Winn Photonic crstals: Molding the flow of light Princeton U. Press Princeton N.J G. Qiu F. Lin and Y. Li Complete two-dimensional bandgap of photonic crstals of a rectangular Bravais lattice Opt. Commun. vol. 9 pp S. Fan J. Winn A. Deveni J. C. Chen R. D. Meade and J. D. Joannopoulos Guided and defect modes in periodic dielectric waveguides J. Opt. Soc. Am. B vol. pp D. Chigrin A. Lavrinenko and C. Sotomaor Torres Nanopillars photonic crstal waveguides Opt. Epress vol. pp J. Shefer Periodic clinder arras as transmission lines IEEE Trans. Microwave Theor Tech. vol. pp Jan B. A. Munk D. S. Janning J. B. Pror and R. J. Marhefka Scattering from surface waves on finite FSS IEEE Trans. Antennas Propagat. vol. 9 pp Dec.. 7 R. J. Maillou Antenna and wave theories of infinite Yagi-Uda arras IEEE Trans. Antennas Propagat. vol. 3 pp Jul A. D. Yaghjian Scattering-matri analsis of linear periodic arras IEEE Trans. Antennas Propagat. vol. 5 pp. 5-6 Aug.. 9 J. C. Weeber A. Dereu C. Girard J. R. Krenn and J. P. Goudonnet Plasmon polaritons of metallic nanowires for controlling submicron propagation of light Phs. Rev. B vol. 6 pp B. Lamprecht J. R. Krenn G. Schider H. Ditlbacher M. Salerno N. Felidj A. Leitner F. R. Aussenegg and J. C. Weeber Surface plasmon propagation in microscale metal stripes Appl. Phs. Lett. vol. 79 pp T. Yatsui M. Kourogi and M. Ohtsu Plasmon waveguide for optical far/near-field conversion Appl. Phs. Lett. vol. 79 pp R. Zia M. D. Selker P. B. Catrsse and M. L. Brongersma Geometries and materials for subwavelength surface plasmon modes J. Opt. Soc. Am. A vol. pp R. Charbonneau N. Lahoud G. Mattiussi and P. Berini Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons Opt. Epress vol. 3 pp D. F. P. Pile D. K. Gramotnev Channel plasmon-polariton in a triangular groove on a metal surface Opt. Lett. vol. 9 pp M. Quinten A. Leitner J. R. Krenn and F. R. Aussenegg Electromagnetic energ transport via linear chains of silver nanoparticles Opt. Lett. vol. 3 pp M. L. Brongersma J. W. Hartman and H. A. Atwater Electromagnetic energ transfer and switching in nanoparticle chain arras below the diffraction limit Phs. Rev. B vol. 6 pp T. J. Shepherd C. R. Brewitt-Talor P. Dimond G. Fiter A. Laight P. Lederer P. J. Roberts P. R. Tapster and I. J. Youngs 3D microwave photonic crstals: Novel fabrication and structures Electron. Lett. vol. 3 pp A. Taflove Computational Electrodnamics: The Finite Difference Time Domain Method. Norwood MA: Artech House S. A. Maier P. G. Kik and H. A. Atwater Optical pulse propagation in metal nanoparticle chain waveguides Phs. Rev. B vol. 67 pp W. M. Saj FDTD simulations of D plasmon waveguide on silver nanorods in heagonal lattice Opt. Epress vol. 3 no. 3 pp June 5. K.-P. Hwang and A. C. Cangellaris Effective permittivities for secondorder accurate FDTD equations at dielectric interfaces IEEE Microwave Wirel. Compon. Lett. vol. pp Y. Hao and C. J. Railton Analzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes IEEE Trans. Microwave Theor Tech. vol. 6 pp Jan R. Luebbers F. P. Hunsberger K. Kunz R. Standler and M. Schneider A frequenc-dependent finite-difference time-domain formulation for dispersive materials IEEE Trans. Electromagn. Compat. vol. 3 pp. -7 Aug. 99. O. P. Gandhi B.-Q. Gao and J.-Y. Chen A frequenc-dependent finitedifference time-domain formulation for general dispersive media IEEE Trans. Microwave Theor Tech. vol. pp Apr D. M. Sullivan Frequenc-dependent FDTD methods using Z transforms IEEE Trans. Antennas Propagat. vol. pp. 3-3 Oct N. Kaneda B. Houshmand and T. Itoh FDTD analsis of dielectric resonators with curved surfaces IEEE Trans. Microwave Theor Tech. vol. 5 pp Sep J.-Y Lee and N.-H Mung Locall tensor conformal FDTD method for modeling arbitrar dielectric surfaces Microw. Opt. Tech. Lett. vol. 3 pp. 5-9 Nov A. Mohammadi and M. Agio Contour-path effective permittivities for the two-dimensional finite-difference time-domain method Opt. Epress vol. 3 pp J. E. Inglesfield A method of embedding J. Phs. C: Solid State Phs. vol. pp F. B. Hildebrand Introduction to Numerical Analsis. New York: Mc- Graw-Hill C. T. Chan Q. L. Yu and K. M. Ho Order-N spectral method for electromagnetic waves Phs. Rev. B vol. 5 pp H. Holter and H. Steskal Infinite phased-arra analsis using FDTD periodic boundar conditions-pulse scanning in oblique directions IEEE Trans. Antennas Propagat. vol. 7 pp M. Turner and C. Christodoulou FDTD analsis of phased arra antennas IEEE Trans. Antennas Propagat. vol. 7 pp D. T. Prescott and N. V. Shule Etensions to the. FDTD method for the analsis of innitel periodic arras IEEE Microwaves and Guided Waves Letters vol. pp Oct J. R. Ren O. P. Gandhi L. R. Walker J. Fraschilla and C. R. Boerman Floquet-based FDTD analsis of two-dimensional phased arra antennas IEEE Microwave and Guided Wave Letters vol. pp J. A. Roden S. D. Gedne M. P. Kesler J. G. Malone and P. H. Harms Time-domain analsis of periodic structures at oblique incidence: Orthogonal and nonorthogonal FDTD implementations IEEE trans. Microwave Theor and Techniques vol. 6 pp S. Fan P. R. Villeneuve and J. D. Joannopoulos Large omnidirectional band gaps in metallodielectric photonic crstals Phs. Rev. B vol. 5 pp M. Qiu and S. He A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crstal with dielectric and metallic inclusions J. Appl. Phs. vol. 87 pp J. R. Berenger A perfectl matched laer for the absorption of electromagnetic waves J. Computat. Phs. vol. pp. 85- Oct. 99. P. B. Johnson and R. W. Christ Optical constants of the noble metals Phs. Rev. B vol. 6 pp N. Giannakis J. Inglesfield P. Belov Y. Zhao and Y. Hao Dispersion properties of subwavelength waveguide formed b silver nanorods Photon6 September Manchester UK. W. Song Y. Hao and C. Parini Comparison of nonorthogonal and Yee s FDTD schemes in modelling photonic crstals submitted to Opt. Epress 6.