Transient electromagnetic interactions on plasmonic nanostructures are analyzed by solving the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) surface integral equation (SIE). Equivalent (unknown) electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, are expanded using Rao-Wilton-Glisson and polynomial basis functions in space and time, respectively. Inserting this expansion into the PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations that is solved for the current expansion coefficients by a marching on-in-time (MOT) scheme. The resulting MOT-PMCHWT-SIE solver calls for computation of additional convolutions between the temporal basis function and the plasmonic medium’s permittivity and Green function. This computation is carried out with almost no additional cost and without changing the computational complexity of the solver. Time-domain samples of the permittivity and the Green function required by these convolutions are obtained from their frequency-domain samples using a fast relaxed vector fitting algorithm. Numerical results demonstrate the accuracy and applicability of the proposed MOT-PMCHWT solver.
© 2016 Optical Society of America
Metallic nanostructures support surface plasmon modes upon illumination by electromagnetic fields at optical frequencies . These modes significantly enhance scattering from nanostructures in the near- and far-field regions , enabling their use in various applications, such as the design of nanoantennas for biochemical sensing , subwavelength waveguides for telecommunication , and ring resonators for integrated compact optical components . Characteristics of these plasmon modes, such as resonance frequency, propagation distance, and amplitude decay rate, depend on the nanostructure’s material properties, shape, and size as well as the material properties of the background medium where the nanostructure resides . Therefore, simulation tools, which are capable of characterizing plasmonic field interactions on arbitrarily shaped geometries with certain material properties, have to be utilized to design a plasmonic nanostructure with desired field patterns in the near- and far-field regions.
Simulation tools implemented to analyze plasmonic field interactions make use of finite-difference time-domain (FDTD) schemes [5–7], finite element method (FEM) , discrete dipole approximation (DDA) , or surface integral equation (SIE) solvers [10–14]. Among these methods, FDTD is the most frequently used one due to its simplicity in formulation and implementation. However, FDTD (like any other differential equation-based solver such as FEM) requires volumetric discretizations of the plasmonic nanostructure and the surrounding medium. This becomes inefficient considering the fact that exponentially decaying behavior of the plasmonic fields could only be accurately captured using very small discretization elements, especially in the vicinity of metal/dielectric interfaces. Also, the unbounded surrounding medium has to be truncated into a (bounded) computational domain . This is achieved by using absorbing boundary conditions, which approximate field behavior at infinity. This approximation introduces significant errors, especially while analyzing resonant phenomena requiring long durations of simulations . Additionally, FDTD suffers from numerical dispersion introduced by finite difference approximations used for discretizing time and space derivatives. This produces additional error in the solution for electrically large objects and possibly leads to unphysical effects .
On the contrary, SIE solvers [10–14] only discretize the surfaces of the nanostructures (the dimension of the discretization domain is reduced to two from three—in comparison with FDTD and FEM) and the solution automatically satisfies the radiation condition (the field behavior at infinity is accurately captured without the need for approximate absorbing boundary conditions). However, the use of SIE solvers for characterizing plasmonic field interactions has been limited to the frequency domain. Consequently, these solvers cannot be used when material properties are functions of fields, such Kerr nonlinear materials [18,19], and they have to be executed many times (one execution per frequency) to obtain broadband results, which are typically needed for designing plasmonic devices/systems expected to operate over the support of multiple resonances.
These disadvantages can be overcome through formulation and implementation of SIE solvers for characterizing transient plasmonic field interactions. Time-domain SIE (TD-SIE) solvers have long been used for analyzing electromagnetic field interactions on perfect electrically conducting scatterers and dielectric objects with nondispersive material properties [20–31]. However, despite their advantages, TD-SIE solvers have not been used for analyzing scattering from objects with dispersive material properties, with only one exception , which makes use of finite-difference delay modeling (FDDM) or convolution quadrature (CQ) techniques . This can be attributed to the following challenges: (i) For dispersive media, the dielectric permittivity does not have a closed-form expression in the time domain unless one of the simplified models, such as Drude or Drude–Lorentz , is used. Consequently, the time-domain Green function also does not have a closed-form expression. (ii) Because both the permittivity and the Green function are dispersive, scattered field computations require evaluation of additional convolutions between the permittivity, the Green function, and the equivalent currents. Schemes for discretizing these additional convolutions have never been formulated or implemented. (iii) The dispersive Green function has an infinite tail in the time domain, which significantly increases the computational cost of the convolutions mentioned in (ii) [26–30]. But this cost can be significantly reduced by using the recently developed plane-wave time-domain (PWTD) method  or fast Fourier transform (FFT)-based schemes [27–30,34].
The TD-SIE solvers [31,32], which make use of FDDM or CQ techniques , formulate and discretize the SIEs in the Laplace domain, approximate the Laplace domain parameter in the -domain using a finite-difference scheme, and finally convert the resulting system into the time domain through the inverse -transform. Since the SIE is constructed in the Laplace domain, these TD-SIE solvers do not face the challenges (i) and (ii) listed above. But they still require a fast method to efficiently compute (discretized) temporal convolutions after conversion into the time domain. This could similarly be remedied by FFT-based schemes [27–30,34].
In this work, a different approach is used. The TD-SIE is constructed directly in the time domain and temporal samples of the dispersive Green function and permittivity are obtained using separate inverse Fourier transforms applied to their frequency-domain representations. More specifically, the well-known frequency-domain Poggio-Miller-Chan-Harrington-Wu-Tsai surface integral equation (PMCHWT-SIE) [35,36] is extended in the time domain to account for scatterers with dispersive material properties, more specifically plasmonic structures. The time-domain PMCHWT-SIE (TD-PMCHWT-SIE) is solved using the well-known marching on-in-time (MOT) scheme [20–30]. The MOT scheme expands the electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, using Rao-Wilton-Glisson (RWG)  and polynomial basis functions [21,28] in space and time, respectively. Inserting this expansion into the TD-PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations that is solved for the current expansion coefficients by marching in time. The additional convolutions are converted into a double discrete summation, where the inner summation corresponds to the discretized convolution of the Green function and temporal basis function  [addressing the challenge (ii) above]. The temporal samples of the time-domain dielectric permittivity and Green function, which are required by this MOT discretization procedure, are obtained numerically from their frequency-domain samples [addressing the challenge (i) above]. This is achieved by representing the frequency-domain Green function and permittivity in terms of a summation of weighted rational functions. The weighting coefficients are found by applying the fast relaxed vector fitting (FRVF) scheme to the frequency-domain samples [39–41]. Time-domain functions are then obtained by analytically computing the inverse Fourier transform of the summation. The resulting MOT-PMCHWT solver is used in characterizing transient field interactions on several different geometries. Accuracy of the solver is demonstrated via comparison of the results with those obtained by frequency-domain solvers or analytical solutions obtained using the Mie series expansion.
A. Time-Domain PMCHWT-SIE
Let denote the volume of the plasmonic nanostructure with time-dependent permittivity and constant permeability . Assume that the nanostructure resides in a nondispersive unbounded background medium denoted by . Constant permittivity and permeability in are represented by and , respectively. Let denote the surface enclosing . An electromagnetic wave with electric and magnetic field intensities, and , excites the nanostructure. It is assumed that and , and , and they are essentially band limited to . Using the surface equivalence principle  and the boundary conditions on , the time derivative of the TD-PMCHWT-SIE is constructed as 3)–(6), the integral operators , , and are given by
B. Time-Domain Permittivity and Inverse Permittivity
Time-domain permittivity and inverse permittivity of the plasmonic nanostructure, and , are obtained by inverse Fourier transforming their frequency-domain counterparts, and , respectively:43] and . Using the FRVF scheme [39–41] (see Appendix A) on and , one can approximate and as 12) and (13) yields
C. Time-Domain Green Function
The integral operators , , and present in Eqs. (3)–(6) and given by Eqs. (7)–(9) require and its derivative with respect to , [due to the presence of the curl operation in Eq. (9)], to be known. For (in the background medium), since and are simply constants, and . Here, is the speed of light in the background medium and is the derivative of the Dirac delta function with respect to its argument. On the other hand, for , general analytical expressions for and are not immediately available due to the dependence of on time. This can be explained by the fact that, for the dispersive , the inverse Fourier transform of the frequency-domain Green function , may not exist in closed-form. Note that, here, is the complex wave number in . To overcome this problem and obtain expressions for and , the FRVF scheme is used in the same way as it is used to obtain the expressions of and in Section 2.B. However, a naive application of the FRVF scheme to the samples of and does not yield accurate representations for and . Singular term (corresponding to a Dirac delta function in time) and phase , (corresponding to a delay in time) are first extracted from . Then, the FRVF scheme (see Appendix A) is applied to the samples of the remaining function at , to yield16), one can write 17) yields A) to the samples of the remaining function at , yield 19), one can write 20) yields 18) with respect to , but this is not possible since the coefficients , , , and depend on and are obtained numerically using the FRVF (i.e., they do not have closed-form expressions). Also, the causality of and is ensured by enforcing and during the execution of the FRVF scheme.
D. Discretization and MOT Scheme37], are the shifted Lagrange interpolation functions [21,28], is the time step size, and are the unknown coefficients, and and are the total numbers of spatial basis functions on and the time steps, respectively. It should be mentioned here that each one of is defined on a pair of triangular patches with support and the Lagrange interpolation functions are piece-wise continuous polynomial functions defined on support , where is the order of . Inserting Eqs. (22) and (23) into Eqs. (1) and (2) and testing the resulting equations with , , yield a system of equations: 30)–(35), are computed using the schemes described in Section 2.E. Once , , and are constructed, unknown vectors , , are obtained recursively by time marching, as described next [20–30]. First, at time is found by solving Eq. (24) with right-hand side (). are then used to compute the scattered fields at time , which are added to the tested incident fields to yield the right-hand side of Eq. (24). is found by solving Eq. (24) with this right-hand side (). Then, and are used to compute the scattered fields at time , which together with form the right-hand side and permit the computation of , and so on. The computational cost of this time-marching scheme is dominated by that of computing the scattered fields, i.e., the discrete summation on the right-hand side of Eq. (24) (see Section 2.F).
E. Computation of the Matrix Entries
1. Contributions from the Background Medium ()
As described in Section 2.C, the Green function in the background medium, i.e., for , . Inserting this in the expressions of , , and in Eqs. (30)–(32) and using the definitions of integral operators , , and in Eqs. (7)–(9), one can obtain37), . The surface integrals present in Eqs. (36) and (37) are computed using numerical quadrature  with proper singularity treatment . One can also use semi-analytical expressions provided in [25,48–50].
2. Contributions from the Plasmonic Medium ()
These require computation of two types of convolutions.
Type-1 Convolutions: Type-1 refers to convolutions of with and and the convolution of with . Let , , and denote the following temporal convolutions:38)–(40), one can rewrite the expressions of , , and , which are needed to compute , , and :
The surface integrals in Eqs. (41)–(43) are computed using numerical quadrature . For every quadrature point pair, , , and have to be computed. Their explicit expressions are obtained by inserting Eqs. (18) and (21) into Eqs. (41)–(43):44)–(46). For these computations, terms involving and are treated carefully using the proper singularity extraction schemes . The boundaries of the time integrals are determined using the support of , , and and the unit step function . It should also be noted here that these integrals are computed using closed-form expressions since , , and consist of polynomial functions .
Type-2 Convolutions: Type-2 refers to the convolutions of with and . Note the presence of the extra temporal convolution (in comparison with type-1 convolutions). Let and denote the following temporal convolutions:47) and (48), one can rewrite the expressions of and , which are needed to compute and : 49) and (50) are computed using numerical quadrature . But this computation can be done very efficiently by making use of type-1 convolutions that are already computed using the same quadrature points. In other words, one can find a way to represent the inner products and (type-2 convolutions) in terms of the inner products and (type-1 convolutions). This is described next. Note that convolutions and do not depend on space but only on the material properties. Let and be discretized using the temporal basis function as 51) and (52) are tested at times , where is an integer. Using the fact that only when and otherwise, and inserting the expressions for and from Eqs. (14) and (15) into the resulting equations, one can obtain 53) and (54) are determined using the supports of and the unit step function . These integrals are computed using closed-form expressions since consists of polynomial functions, respectively . It should also be noted here and for . Inserting Eqs. (51) and (52) into Eqs. (47) and (48), respectively, yields 55) and (56) into Eqs. (49) and (50), respectively, yields 57) and (58) show that and (type-2 convolutions) can be easily computed using and (type-1 convolutions).
F. Computational Complexity
Finally, the computational complexity of the MOT scheme given in Eq. (24) scales with . This cost can be reduced to using the PWTD method  or using FFT-based schemes [27–30,34]. In such cases, the discrete summations in Eqs. (57) and (58) should not be precomputed into the MOT matrices but should be incorporated into time marching through the use of and . This computation can be done very efficiently using a blocked FFT scheme without impacting the computational complexity of the accelerated time marching . The use of these accelerated MOT schemes reduce the computational complexity of the proposed solver to essentially that of FDTD schemes.
3. NUMERICAL RESULTS
In this section, accuracy and applicability of the proposed MOT-PMCHWT-SIE solver are demonstrated through its application to the analysis of scattering from several nanostructures. In all examples, it is assumed that the nanostructure is residing in free space and excited by a plane wave with electric field
In all examples, the unknown coefficients of the equivalent electric and magnetic current densities, and , , , are computed by the proposed MOT-PMCHWT-SIE solver under this excitation. After the time-domain simulation is completed, the Fourier transforms of and are computed using the discrete time Fourier transform (DTFT). Normalizing the results with the DTFT of the samples of the Gaussian pulse, , yields coefficients of the frequency-domain (i.e., time harmonic) electric and magnetic current densities denoted by and , , where , is the frequency step, and is the number of frequency samples. This is followed by the computation of the frequency-domain scattering and extinction cross sections, and :62)–(64), and is explicitly enforced during the computation of . Extinction efficiency of a scatterer, , is defined as the ratio of to its geometrical cross section on the plane perpendicular to the direction of propagation of the incident field . In all examples presented in this section, , , and computed by the proposed MOT-PMCHWT-SIE solver are compared to those obtained by the frequency-domain (FD) PMCHWT-SIE solver of , which computes and on the same discretization but directly in the frequency domain. Note that FD-PMCHWT-SIE solver has to be executed times for this operation.
A. Accuracy of FRVF Scheme
In this section, the accuracy of the expansions in Eqs. (12) and (13) is demonstrated. The samples of the scatterer’s permittivity, , , which are used in the FRVF scheme to generate the terms in these expansions, are obtained from the experimental data of Johnson–Christy tabulated for gold and silver in . It should be noted here that experiments were carried out at only 49 frequency points; therefore, spline interpolation is used to generate samples at , , required by Eqs. (12) and (13). Here, , and (corresponding to free-space wavelengths of and ), and the number of terms in the expansions . Figures 1 and 2 compare in Eq. (12) obtained using the FRVF to that provided by the four-term Lorentz model described in , for gold and silver, respectively. The figures clearly show that both the FRVF expansion and the Lorentz model generate the real part of accurately. However, the imaginary part generated by the Lorentz model does not match the experimental data. For the sake of completeness, Figs. 3(a) and 3(b) plot amplitude of and , which correspond to the Fourier transform of the summation in Eqs. (12) and (13), i.e., and without the Dirac delta term [see Eqs. (14) and (15)].
B. Gold and Silver Spheres
In this example, the scatterer is a silver or gold sphere of radius 50 nm. The permittivities of gold and silver are obtained using the Johnson–Christy  experimental data. For the FRVF scheme, , , , and . The excitation parameters are , , , , , and . The currents induced on the sphere surface are discretized using RWG basis functions and the simulation is executed for time steps with step size . Figures 4(a) and 4(b) plot the amplitudes of and , computed during the simulations of the gold and silver spheres, respectively. Figures clearly show the stability of the results. Figures 5(a) and 5(b) compare , , , , computed by the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers, and those obtained from Mie series solution. Results agree well with each other.
C. Gold Rounded Cube
Next, scattering from a gold rounded cube  is analyzed using the proposed MOT-PMCHWT-SIE solver. The dimension of the cube is 200 nm and the radius of fillets on the edges and corners is 20 nm. The permittivity of gold is obtained using the Johnson–Christy  experimental data. For the FRVF scheme, , , , and . The excitation parameters are , , , , , and . The currents induced on the cube surface are discretized using RWG basis functions and the simulation is executed for time steps with step size . Figure 6 compares , , , , obtained using the proposed MOT-PMCHWT-SIE solver, to that computed by the null-field method . Results are in good agreement with each other.
D. Gold Rounded Triangular Prism
In this example, the scatterer is a gold rounded triangular prism of height 40 nm and edge length 200 nm . The radius of the fillets on the edges and the corners is 10 nm. The permittivity of gold is obtained using the Johnson–Christy  experimental data. For the FRVF scheme, , , , and . The excitation parameters are , , , , , and . The currents induced on the cube surface are discretized using RWG basis functions and the simulation is executed for time steps with step size . Figure 7 compares , , , obtained using the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers to that computed by the DDA . Results obtained by the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers match well with each other, but the result computed by the DDA differs from those, especially around the peak values. The same type of mismatch is also observed in  and explained by the fact that the DDA loses accuracy as the permittivity of the scatterer increases .
E. Gold-Coated Silica Sphere
Scattering from a silica sphere coated with gold  is analyzed using the proposed MOT-PMCHWT-SIE solver. The radius of the silica sphere and the thickness of the gold layer are 40 nm and 10 nm, respectively. The relative permittivity of the silica is 2.04 while the permittivity of the gold is obtained using the Johnson–Christy  experimental data. For the FRVF scheme, , , , and . The excitation parameters are , , , , , and . The currents induced on the sphere surface are discretized using RWG basis functions and the simulation is executed for time steps with step size . Figure 8 plots , , , computed by the MOT-PMCHWT-SIE solver to that obtained from the Mie series solution. Results are in good agreement.
F. Silver Dimer
In this example, the scatterer is an -directed dimer consisting of two silver spheres residing on the -plane . The radius of the spheres and the shortest distance between them are 15 nm and 1.5 nm, respectively. The permittivity of silver is obtained using the Palik experimental data . For the FRVF scheme, , , , and . Two simulations are carried out for two polarizations of the incident field, and . The remaining excitation parameters are , , , , and . The currents induced on the sphere surfaces are discretized using RWG basis functions. Both simulations are executed for time steps with step size . Figure 9 compares , , , , computed by the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers. Results agree well with each other and also with those provided in .
G. Gold Disk with a Nonconcentric Cavity
In the last example, the scatterer is a gold disk embedding a nonconcentric cavity and residing on the -plane . The radius and thickness of the disk are 50 nm and 10 nm, respectively. The radius of the circular cavity is 20 nm and its center is offset from the center of the disk by 29 nm in the -direction. The permittivity of gold is obtained using the Johnson–Christy  experimental data. For the FRVF scheme, , , , and . The excitation parameters are , and , , , , and . The currents induced on the surface of the scatterer are discretized using RWG basis functions. The simulation is executed for time steps with step size . Figure 10 compares , , , , obtained using the proposed MOT-PMCHWT-SIE solver, to that computed by the frequency-domain FEM . The results agree with each other reasonably well. The positions of the two main peaks in the results are close enough yet the peak at 660 nm is less pronounced and slightly blueshifted in the result obtained by the MOT-PMCHWT-SIE solver.
An MOT scheme for solving the PMCHWT-SIE enforced on surfaces of plasmonic nanostructures is described. The unknown equivalent electric and magnetic current densities introduced on these surfaces are expanded by RWG and polynomial basis functions in space and time, respectively. Inserting this expansion into the PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations. This system is then solved for the unknown expansion coefficients using the MOT scheme.
The PMCHWT-SIE requires evaluation of additional convolutions involving the plasmonic medium’s permittivity and Green function and the temporal basis function to compute the scattered fields. This convolution is discretized in a way that is fully consistent with the MOT scheme. The computation of the discretized convolution is carried out with almost no additional cost and without changing the computational complexity of the MOT scheme. Time-domain samples of the permittivity and the Green function required by this computation are obtained from their frequency-domain samples using the FRVF algorithm. Frequency samples are generated using tabulated data obtained from experiments.
Numerical results that demonstrate the accuracy of the MOT-PMCHWT-SIE solver are presented. The quantum-corrected version of the same solver, which can accurately be applied to scatterers separated by subnanometer distances, is under development.
This section briefly describes the fundamental idea behind the FRVF scheme. The details can be found in [39–41]. Let represent a function defined in the frequency domain. FRVF approximates as a sum of rational functions:
The FRVF scheme is iteratively executed. Each iteration has two stages: identification of poles and residues of the right-hand side of Eq. (A1). First, poles are identified as described next. Let and represent two auxiliary functions:A2) with and equating the resulting expression to the right-hand side of Eq. (A3), and evaluating the final equation at frequency samples , , where , yield an overdetermined system: A4) in the least-squares sense. One can show that zeros of the are equal to the poles of by rewriting Eqs. (A1)–(A3) in terms of sum of partial fractions . These zeros are equal to the eigenvalues of the matrix , where is a diagonal matrix with entries , , and . This completes the identification of the poles of the right-hand side of Eq. (A1) for the given iteration. Once these poles are known, the residues are identified as described next. Eq. (A1) is evaluated at , , yielding an overdetermined system as in Eq. (A4), where now and . Solving this system in the least-squares sense yields the residues of the right-hand side of Eq. (A1) for the given iteration and completes the iteration. Poles obtained in one iteration are used as an initial guess in the next one and the iterations are repeated until a preset level of accuracy is obtained, i.e., an accurate representation for is obtained.
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