Abstract

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

1. INTRODUCTION

Metallic nanostructures support surface plasmon modes upon illumination by electromagnetic fields at optical frequencies [1]. These modes significantly enhance scattering from nanostructures in the near- and far-field regions [1], enabling their use in various applications, such as the design of nanoantennas for biochemical sensing [2], subwavelength waveguides for telecommunication [3], and ring resonators for integrated compact optical components [4]. 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 [1]. 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 [57], finite element method (FEM) [8], discrete dipole approximation (DDA) [9], or surface integral equation (SIE) solvers [1014]. 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 [15]. 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 [16]. 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 [17].

On the contrary, SIE solvers [1014] 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 [2031]. However, despite their advantages, TD-SIE solvers have not been used for analyzing scattering from objects with dispersive material properties, with only one exception [32], which makes use of finite-difference delay modeling (FDDM) or convolution quadrature (CQ) techniques [33]. 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 [19], 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) [2630]. But this cost can be significantly reduced by using the recently developed plane-wave time-domain (PWTD) method [26] or fast Fourier transform (FFT)-based schemes [2730,34].

The TD-SIE solvers [31,32], which make use of FDDM or CQ techniques [33], formulate and discretize the SIEs in the Laplace domain, approximate the Laplace domain parameter s in the z-domain using a finite-difference scheme, and finally convert the resulting system into the time domain through the inverse z-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 [2730,34].

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 [2030]. The MOT scheme expands the electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, using Rao-Wilton-Glisson (RWG) [37] 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 [38] [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 [3941]. 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.

2. FORMULATION

A. Time-Domain PMCHWT-SIE

Let V1 denote the volume of the plasmonic nanostructure with time-dependent permittivity ε1(t) and constant permeability μ1. Assume that the nanostructure resides in a nondispersive unbounded background medium denoted by V0. Constant permittivity and permeability in V0 are represented by ε0 and μ0, respectively. Let S denote the surface enclosing V1. An electromagnetic wave with electric and magnetic field intensities, Einc(r,t) and Hinc(r,t), excites the nanostructure. It is assumed that Einc(r,t)0 and Hinc(r,t)0,   rV1 and t<0, and they are essentially band limited to fmax. Using the surface equivalence principle [42] and the boundary conditions on S, the time derivative of the TD-PMCHWT-SIE is constructed as [25]

n^(r)×tEinc(r,t)=n^(r)×[tE0sca(r,t)tE1sca(r,t)],
n^(r)×tHinc(r,t)=n^(r)×[tH0sca(r,t)tH1sca(r,t)].
Here, n^(r) is the unit normal vector at rS, which is pointing toward V0, and Epsca(r,t) and Hpsca(r,t) are the scattered electric and magnetic field intensities in Vp, p{0,1}. Epsca(r,t) and Hpsca(r,t) are expressed in terms of equivalent electric and magnetic current densities, J(r,t) and M(r,t), which are introduced on S using the surface equivalence principle:
tE0sca(r,t)=L0{μ0J(r,t)}+Q0{ε01J(r,t)}K0{M(r,t)},
tH0sca(r,t)=L0{ε0M(r,t)}+Q0{μ01M(r,t)}+K0{J(r,t)},
tE1sca(r,t)=L1{μ1J(r,t)}Q1{ε¯1(t)*J(r,t)}+K1{M(r,t)},
tH1sca(r,t)=L1{ε1(t)*M(r,t)}Q1{μ11M(r,t)}K1{J(r,t)}.
In Eqs. (3)–(6), the integral operators Lp{·}, Qp{·}, and Kp{·} are given by
Lp{X(r,t)}=Sgp(R,t)*t2X(r,t)ds,
Qp{X(r,t)}=Sgp(R,t)*·X(r,t)ds,
Kp{X(r,t)}=×Sgp(R,t)*tX(r,t)ds.
Here, gp(R,t) is the Green function of the unbounded medium that has the same permittivity and permeability as Vp, R=|rr| is the distance between source and observation points, r and r, “*” denotes temporal convolution, and ε¯1(t) is the time-domain inverse permittivity in V1.

B. Time-Domain Permittivity and Inverse Permittivity

Time-domain permittivity and inverse permittivity of the plasmonic nanostructure, ε1(t) and ε¯1(t), are obtained by inverse Fourier transforming their frequency-domain counterparts, ε1(ω) and 1/ε1(ω), respectively:

ε1(t)=F1{ε1(ω)},
ε¯1(t)=F1{ε¯1(ω)}=F1{1/ε1(ω)}.
Here, it is assumed that frequency samples of ε1(ω) and ε¯1(ω), i.e., ε1(ωi) and ε¯1(ωi), i=1,,NωVF, where ωi2π[fminVF,fmaxVF], are available from tabulated experimental data, such as those provided for silver and gold in [43] and [44]. Using the FRVF scheme [3941] (see Appendix A) on ε1(ωi) and ε¯1(ωi), one can approximate ε1(ω) and ε¯1(ω) as
ε1(ω)ε1m=1NpVFbmϵjω+amϵ,
ε¯1(ω)1ε1m=1NpVFbmϵ¯jω+amϵ¯,
where j=1 is the imaginary number, ε1=1 is the high frequency limit of ε1(ω) as ω, and {amϵ,bmϵ} and {amϵ¯,bmϵ¯} are the pole/residue pairs obtained by applying the FRVF scheme to ε1(ωi) and ε¯1(ωi). Inverse Fourier transforming Eqs. (12) and (13) yields
ε1(t)ε1δ(t)+m=1NpVFbmϵu(t)eamϵt=ε1δ(t)+γ1(t),
ε¯1(t)1ε1δ(t)+m=1NpVFbmϵ¯u(t)eamϵ¯t=1ε1δ(t)+γ¯1(t),
where δ(·) and u(·) are the Dirac delta and the unit step functions, respectively. It should be noted here that the causality of ε1(t) and ε¯1(t) is ensured by enforcing amϵ>0 and amϵ¯>0 during the execution of the FRVF scheme.

C. Time-Domain Green Function

The integral operators Lp{·}, Qp{·}, and Kp{·} present in Eqs. (3)–(6) and given by Eqs. (7)–(9) require gp(R,t) and its derivative with respect to R, Rgp(R,t) [due to the presence of the curl operation in Eq. (9)], to be known. For p=0 (in the background medium), since ε0 and μ0 are simply constants, g0(R,t)=δ(tR/c0)/(4πR) and Rg0(R,t)=δ(tR/c0)/(4πc0R)+δ(tR/c0)/(4πR2) [45]. Here, c0=1/ε0μ0 is the speed of light in the background medium and δ(·) is the derivative of the Dirac delta function with respect to its argument. On the other hand, for p=1, general analytical expressions for g1(R,t) and Rg1(R,t) are not immediately available due to the dependence of ε1(t) on time. This can be explained by the fact that, for the dispersive ε1(ω), the inverse Fourier transform of the frequency-domain Green function g1(R,ω)=ejk1R/(4πR), may not exist in closed-form. Note that, here, k1=ωε1(ω)μ1 is the complex wave number in V1. To overcome this problem and obtain expressions for g1(R,t) and Rg1(R,t), the FRVF scheme is used in the same way as it is used to obtain the expressions of ε1(t) and ε¯1(t) in Section 2.B. However, a naive application of the FRVF scheme to the samples of g1(R,ω) and Rg1(R,ω) does not yield accurate representations for g1(R,t) and Rg1(R,t). Singular term 1/(4πR) (corresponding to a Dirac delta function in time) and phase ejωR/c1, c1=1/ε1μ1 (corresponding to a delay in time) are first extracted from g1(R,ω). Then, the FRVF scheme (see Appendix A) is applied to the samples of the remaining function at ωi2π[fminVF,fmaxVF], i=1,,NωVF to yield

ejωR/c1[g1(R,ω)ejωR/c14πR]=ej(k1RωR/c1)14πRdg(R)+m=1NpVFbmg(R)jω+amg(R).
Here, {amg(R),bmg(R)} are the pole/residue pairs and dg(R) is the constant term (with respect to frequency dependence) generated by the FRVF scheme. Note that amg(R), bmg(R), and dg(R) are functions of R. Using Eq. (16), one can write
g1(R,ω)ejωR/c1[14πR+dg(R)+m=1NpVFbmg(R)jω+amg(R)].
Inverse Fourier transforming Eq. (17) yields
g1(R,t)δ(τ1)4πR+dg(R)δ(τ1)+m=1NpVFbmg(R)u(τ1)eamg(R)τ1,
where τ1=tR/c1 is the delay representing the time retardation due to the finite speed of light. A similar method is used to obtain an expression for Rg1(R,t) from the samples of Rg1(R,ω). Note that Rg1(R,ω)=ejk1R/(4πR)[jk1+1/R]. Extracting the singularity and the phase, and applying the FRVF scheme (see Appendix A) to the samples of the remaining function at ωi2π[fminVF,fmaxVF], i=1,,NωVF yield
ejωR/c1[Rg1(R,ω)+(jωc1R+1R2)ejωR/c14π]=ej(k1RωR/c1)4π(jk1R+1R2)+14πR2+jω4πc1RdgR(R)+jωfgR(R)+m=1NpVFbmgR(R)jω+amgR(R).
Here, {amgR(R),bmgR(R)} are the pole/residue pairs, dgR(R) is the constant term (with respect to frequency dependence), and fgR(R) is the coefficient of the linear term generated by the FRVF scheme. Using Eq. (19), one can write
Rg1(R,ω)ejωR/c1[jω(fgR(R)14πc1R)+dgR(R)14πR2+m=1NpVFbmgR(R)jω+amgR(R)].
Inverse Fourier transforming Eq. (20) yields
Rg1(R,t)δ(τ1)(fgR(R)14πc1R)+δ(τ1)(dgR(R)14πR2)+m=1NpVFbmgR(R)u(τ1)eamgR(R)τ1.
One could assume Rg1(R,t) could be directly obtained by taking the derivative of Eq. (18) with respect to R, but this is not possible since the coefficients amgR(R), bmgR(R), dgR(R), and fgR(R) depend on R and are obtained numerically using the FRVF (i.e., they do not have closed-form expressions). Also, the causality of g1(R,t) and Rg1(R,t) is ensured by enforcing amg(R)>0 and amgR(R)>0 during the execution of the FRVF scheme.

D. Discretization and MOT Scheme

To numerically solve the TD-PMCHWT-SIE in Eqs. (1) and (2), first S is discretized into triangular patches. Then J(r,t) and M(r,t) are expanded in space and time as

J(r,t)=j=1Ntn=1NsJn(jΔt)fn(r)Tj(t)=j=1Ntn=1Ns{I¯jJ}nfn(r)Tj(t),
M(r,t)=j=1Ntn=1NsMn(jΔt)fn(r)Tj(t)=j=1Ntn=1Ns{I¯jM}nfn(r)Tj(t).
Here, fn(r) represent the well-known RWG functions [37], Tj(t)=T(tjΔt) are the shifted Lagrange interpolation functions [21,28], Δt is the time step size, {I¯jJ}n=Jn(jΔt) and {I¯jM}n=Mn(jΔt) are the unknown coefficients, and Ns and Nt are the total numbers of spatial basis functions on S and the time steps, respectively. It should be mentioned here that each one of fn(r) is defined on a pair of triangular patches with support Sn and the Lagrange interpolation functions T(t) are piece-wise continuous polynomial functions defined on support [Δt,dΔt], where d is the order of T(t). Inserting Eqs. (22) and (23) into Eqs. (1) and (2) and testing the resulting equations with n^(r)×fn(r)δ(tjΔt), n=1,,Ns, yield a system of equations:
[Z¯¯0JJZ¯¯0JMZ¯¯0JMZ¯¯0MM][I¯jJI¯jM]=[V¯jJV¯jM]j=1j1[Z¯¯jjJJZ¯¯jjJMZ¯¯jjJMZ¯¯jjMM][I¯jJI¯jM].
Here, vectors V¯jJ and V¯jM of size Ns×1 store the tested incident fields and their elements are given by
{V¯jJ}n=fn(r),tE0inc(r,t)t=jΔt,
{V¯jM}n=fn(r),tH0inc(r,t)t=jΔt.
Note that here fn(r),X(r,t)t=jΔt=Snfn(r)·X(r,t)ds|t=jΔt represents the inner product operation due to spatial testing. The elements of the matrices Z¯¯jjJJ, Z¯¯jjJM, and Z¯¯jjMM of size Ns×Ns are given by
{Z¯¯jjJJ}n,n={Z¯¯jjJJ,0}n,n+{Z¯¯jjJJ,1}n,n,
{Z¯¯jjJM}n,n={Z¯¯jjJM,0}n,n+{Z¯¯jjJM,1}n,n,
{Z¯¯jjMM}n,n={Z¯¯jjMM,0}n,n+{Z¯¯jjMM,1}n,n,
where
{Z¯¯jjJJ,0}n,n=μ0fn(r),L0{fn(r)Tj(t)}t=jΔt+1ε0fn(r),Q0{fn(r)Tj(t)}t=jΔt,
{Z¯¯jjJM,0}n,n=fn(r),K0{fn(r)Tj(t)}t=jΔt,
{Z¯¯jjMM,0}n,n=ε0μ0{Z¯¯jjJJ,0}n,n,
{Z¯¯jjJJ,1}n,n=μ1fn(r),L1{fn(r)Tj(t)}t=jΔt+1ε1fn(r),Q1{fn(r)Tj(t)}t=jΔt+fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt,
{Z¯¯jjJM,1}n,n=fn(r),K1{fn(r)Tj(t)}t=jΔt,
{Z¯¯jjMM,1}n,n=ε1fn(r),L1{fn(r)Tj(t)}t=jΔtfn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt+1μ1fn(r),Q1{fn(r)Tj(t)}t=jΔt.
Here, Z¯¯jjJJ,p, Z¯¯jjJM,p, and Z¯¯jjMM,p, p{0,1}, correspond to contributions from the scattered fields generated in the background medium (p=0) and inside the nanostructure (p=1). Their entries, which are given by Eqs. (30)–(35), are computed using the schemes described in Section 2.E. Once Z¯¯jjJJ, Z¯¯jjJM, and Z¯¯jjMM are constructed, unknown vectors [I¯jJI¯jM]T, j=1,,Nt, are obtained recursively by time marching, as described next [2030]. First, [I¯1JI¯1M]T at time t=Δt is found by solving Eq. (24) with right-hand side [V¯1JV¯1M]T (j=1). [I¯1JI¯1M]T are then used to compute the scattered fields at time t=2Δt, which are added to the tested incident fields [V¯2JV¯2M]T to yield the right-hand side of Eq. (24). [I¯2JI¯2M]T is found by solving Eq. (24) with this right-hand side (j=2). Then, [I¯1JI¯1M]T and [I¯2JI¯2M]T are used to compute the scattered fields at time t=3Δt, which together with [V¯3JV¯3M]T form the right-hand side and permit the computation of [I¯3JI¯3M]T, 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 (p=0)

As described in Section 2.C, the Green function in the background medium, i.e., for p=0, g0(R,t)=δ(tR/c0)/(4πR). Inserting this in the expressions of {Z¯¯jjJJ,0}n,n, {Z¯¯jjJM,0}n,n, and {Z¯¯jjMM,0}n,n in Eqs. (30)–(32) and using the definitions of integral operators L0{·}, Q0{·}, and K0{·} in Eqs. (7)–(9), one can obtain

{Z¯¯jjMM,0}n,n=ε0μ0{Z¯¯jjJJ,0}n,n=ε04πSnfn(r)·Snfn(r)t2Tj(t)|t=jΔtR/c0Rdsds14πμ0Sn·fn(r)Sn·fn(r)Tj(jΔtR/c0)Rdsds,
{Z¯¯jjJM,0}n,n=14πSnfn(r)·SnR^×fn(r)R[t2Tj(t)c0+tTj(t)R]t=jΔtR/c0dsds.
In Eq. (37), R^=(rr)/R. The surface integrals present in Eqs. (36) and (37) are computed using numerical quadrature [46] with proper singularity treatment [47]. One can also use semi-analytical expressions provided in [25,4850].

2. Contributions from the Plasmonic Medium (p=1)

These require computation of two types of convolutions.

Type-1 Convolutions: Type-1 refers to convolutions of g1(R,t) with ·fn(r)Tj(t) and fn(r)t2Tj(t) and the convolution of Rg1(R,t) with fn(r)tTj(t). Let CT(R,t), C(R,t), and CR(R,t) denote the following temporal convolutions:

CT(R,t)=g1(R,t)*t2Tj(t),
C(R,t)=g1(R,t)*Tj(t),
CR(R,t)=Rg1(R,t)*tTj(t).
Using the definitions in Eqs. (38)–(40), one can rewrite the expressions of fn(r),L1{fn(r)Tj(t)}t=jΔt, fn(r),Q1{fn(r)Tj(t)}t=jΔt, and fn(r),K1{fn(r)Tj(t)}t=jΔt, which are needed to compute {Z¯¯jjJJ,1}n,n, {Z¯¯jjJM,1}n,n, and {Z¯¯jjMM,1}n,n:
fn(r),L1{fn(r)Tj(t)}t=jΔt=Snfn(r)·SnCT(R,t)fn(r)dsds|t=jΔt,
fn(r),Q1{fn(r)Tj(t)}t=jΔt=Sn·fn(r)SnC(R,t)·fn(r)dsds|t=jΔt,
fn(r),K1{fn(r)Tj(t)}t=jΔt=Snfn(r)·SnR^×fn(r)CR(R,t)dsds|t=jΔt.

The surface integrals in Eqs. (41)–(43) are computed using numerical quadrature [46]. For every quadrature point pair, CT(R,t), C(R,t), and CR(R,t) have to be computed. Their explicit expressions are obtained by inserting Eqs. (18) and (21) into Eqs. (41)–(43):

CT(R,t)=[dg(R)+14πR]t2T(t)|t=tjΔtR/c1+m=1NpVFbmg(R)Δtmin(tjΔtR/c1,dΔt)u(ttjΔtR/c1)eamg(R)(ttjΔtR/c1)t2T(t)dt,
C(R,t)=[dg(R)+14πR]T(tjΔtR/c1)+m=1NpVFbmg(R)Δtmin(tjΔtR/c1,dΔt)u(ttjΔtR/c1)eamg(R)(ttjΔtR/c1)T(t)dt,
CR(R,t)=[fgR(R)14πRc1]t2T(t)|t=tjΔtR/c1+[dgR(R)14πR2]tT(t)|t=tjΔtR/c1+m=1NpVFbmgR(R)Δtmin(tjΔtR/c1,dΔt)u(ttjΔtR/c1)eamgR(R)(ttjΔtR/c1)tT(t)dt.
For every quadrature point pair, CT(R,t), C(R,t), and CR(R,t) are computed using Eqs. (44)–(46). For these computations, terms involving 1/R and 1/R2 are treated carefully using the proper singularity extraction schemes [47]. The boundaries of the time integrals are determined using the support of T(t), tT(t), and t2T(t) and the unit step function u(ttjΔtR/c1). It should also be noted here that these integrals are computed using closed-form expressions since T(t), tT(t), and t2T(t) consist of polynomial functions [38].

Type-2 Convolutions: Type-2 refers to the convolutions of g1(R,t) with ·fn(r)γ¯1(t)*Tj(t) and fn(r)γ1(t)*t2Tj(t). Note the presence of the extra temporal convolution (in comparison with type-1 convolutions). Let D(R,t) and DT(R,t) denote the following temporal convolutions:

D(R,t)=g1(R,t)*γ¯1(t)*Tj(t),
DT(R,t)=g1(R,t)*γ1(t)*t2Tj(t)=t2g1(R,t)*γ1(t)*Tj(t).
Using the definitions in Eqs. (47) and (48), one can rewrite the expressions of fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt and fn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt, which are needed to compute {Z¯¯jjJJ,1}n,n and {Z¯¯jjMM,1}n,n:
fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt=Sn·fn(r)SnD(R,t)·fn(r)dsds|t=jΔt,
fn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt=Snfn(r)·SnDT(R,t)fn(r)dsds|t=jΔt.
The surface integrals in Eqs. (49) and (50) are computed using numerical quadrature [46]. 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 fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt and fn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt (type-2 convolutions) in terms of the inner products fn(r),Q1{fn(r)Tj(t)}t=jΔt and fn(r),L1{fn(r)Tj(t)}t=jΔt (type-1 convolutions). This is described next. Note that convolutions Fj(t)=γ¯1(t)*Tj(t) and FjT(t)=γ1(t)*Tj(t) do not depend on space but only on the material properties. Let Fj(t) and FjT(t) be discretized using the temporal basis function Tl(t) as
Fj(t)=γ¯1(t)*Tj(t)l=1NtFljTl(t),
FjT(t)=γ1(t)*Tj(t)l=1NtFljTTl(t),
where Flj and FljT are the expansion coefficients. To find them, Eqs. (51) and (52) are tested at times sΔt, where s is an integer. Using the fact that T([sl]Δt)=1 only when s=l and T([sl]Δt)=0 otherwise, and inserting the expressions for γ(t) and γ¯(t) from Eqs. (14) and (15) into the resulting equations, one can obtain
Flj=Fj(lΔt)=γ¯1(t)*Tj(t)|t=lΔt=Δtmin(lj,d)Δtγ¯1([lj]Δtt)T(t)dt=m=1NpVFbmϵ¯Δtmin(lj,d)Δtu([lj]Δtt)eamϵ¯([lj]Δtt)T(t)dt,
FljT=FjT(lΔt)=γ1(t)*Tj(t)|t=lΔt=Δtmin(lj,d)Δtγ1([lj]Δtt)T(t)dt=m=1NpVFbmϵΔtmin(lj,d)Δtu([lj]Δtt)eamϵ([lj]Δtt)T(t)dt.
Note that the boundaries of the time integrals in Eqs. (53) and (54) are determined using the supports of T(t) and the unit step function u([lj]Δtt). These integrals are computed using closed-form expressions since T(t) consists of polynomial functions, respectively [38]. It should also be noted here Flj=0 and FljT=0 for jl+1. Inserting Eqs. (51) and (52) into Eqs. (47) and (48), respectively, yields
D(R,t)=g1(R,t)*Flj(t)l=1NtFljg1(R,t)*Tl(t),
DT(R,t)=t2g1(R,t)*FljT(t)l=1NtFljTt2g1(R,t)*Tl(t)=l=1NtFljTg1(R,t)*t2Tl(t).
Finally, inserting Eqs. (55) and (56) into Eqs. (49) and (50), respectively, yields
fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt=Sn·fn(r)Sn[l=1NtFljg1(R,t)*Tl(t)]·fn(r)dsds|t=jΔt=l=1Ntfn(r),Q1{fn(r)Tl(t)}t=jΔtFlj,
fn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt=Snfn(r)·Sn[l=1NtFljTg1(R,t)*t2Tl(t)]fn(r)dsds|t=jΔt=l=1Ntfn(r),L1{fn(r)Tl(t)}t=jΔtFljT.
Equations (57) and (58) show that fn(r),Q1{γ¯1(t)*fn(r)Tj(t)}t=jΔt and fn(r),L1{γ1(t)*fn(r)Tj(t)}t=jΔt (type-2 convolutions) can be easily computed using fn(r),Q1{fn(r)Tl(t)}t=jΔt and fn(r),L1{fn(r)Tl(t)}t=jΔt (type-1 convolutions).

F. Computational Complexity

In this section, computational complexity of the MOT-PMCHWT-SIE solver is described step by step following the derivation of the matrix entries (Section 2.E) and the MOT scheme (Section 2.D):

  • (i) Computation of {Z¯¯jjJJ,0}n,n, {Z¯¯jjJM,0}n,n, and {Z¯¯jjMM,0}n,n in Eqs. (36) and (37) scales with O(Ns2) because g0(R,t) is simply a Dirac function in time and space (no temporal tail).
  • (ii) Computation of type-1 convolutions in Eqs. (41)–(43) has a complexity of O(NtNs2) because g1(R,t) has a temporal tail.
  • (iii) Convolutions in Eqs. (53) and (54) depend only on the permittivity of the plasmonic medium. Therefore Flj and FljT are computed once and stored in the memory. Cost of computing Flj and FljT scales with O(Nt) because γ1(t) and γ¯1(t) have temporal tails.
  • (iv) Computation of type-2 convolutions in Eqs. (57) and (58) scales with O(NtNs2) as a consequence of (ii) and (iii). As a result, computation of {Z¯¯jjJJ,1}n,n, {Z¯¯jjJM,1}n,n, and {Z¯¯jjMM,1}n,n scales with O(NtNs2). This assumes that the results of discrete summations in Eqs. (57) and (58) are directly incorporated into the matrix elements.

Finally, the computational complexity of the MOT scheme given in Eq. (24) scales with O(Nt2Ns2). This cost can be reduced to O(NtNslog2(Nt)log(Ns)) using the PWTD method [26] or O(NtNs3/2log(Ns)log(NtNs)log(Nt)) using FFT-based schemes [2730,34]. 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 j=1l1FljI¯jJ and j=1l1FljTI¯jM. This computation can be done very efficiently using a blocked FFT scheme without impacting the computational complexity of the accelerated time marching [29]. 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

E0inc(r,t)=p^E0incG(tk^·r/c0),
where E0inc=1  V/m is the amplitude, p^ is the polarization unit vector, k^=x^sinθinccosφinc+y^sinθincsinφinc+z^cosθinc is the unit vector along the direction of propagation defined by the angles θinc and φinc, G(t)=cos(2πf0[tt0])e(tt0)2/2σ2 is a Gaussian pulse with modulation frequency f0, duration σ, and delay t0. In all examples, t0=8σ and σ=3/(2πfbw), where fbw denotes an effective band. This specific selection of σ ensures that 99.998% of the incident energy is within the frequency band [f0fbw,f0+fbw]. The free-space wavelengths at the minimum and maximum frequency of the excitation are λmax=c0/[f0fbw] and λmin=c0/[f0+fbw].

In all examples, the unknown coefficients of the equivalent electric and magnetic current densities, Jn(kΔt) and Mn(kΔt), k=1,,Nt, n=1,,Ns, are computed by the proposed MOT-PMCHWT-SIE solver under this excitation. After the time-domain simulation is completed, the Fourier transforms of Jn(kΔt) and Mn(kΔt) are computed using the discrete time Fourier transform (DTFT). Normalizing the results with the DTFT of the samples of the Gaussian pulse, G(kΔt), yields coefficients of the frequency-domain (i.e., time harmonic) electric and magnetic current densities denoted by Jn(lΔω) and Mn(lΔω), l=1,,Nf, where Δω=2πΔf, Δf is the frequency step, and Nf is the number of frequency samples. This is followed by the computation of the frequency-domain scattering and extinction cross sections, Csca(lΔω) and Cext(lΔω) [51]:

Csca(lΔω)=116π2|E0inc|2Ω|F(r^,lΔω)|2dΩ,
Cext(lΔω)=1k0|E0inc|2Im{E0incp^·F(k^,lΔω)}.
Here, r^=x^sinθcosφ+y^sinθsinφ+z^cosθ is the unit vector along the direction defined by angles θ and φ, dΩ=sinθdθdφ is the differential solid angle, and F(r^,lΔω) represents the scattered electric field pattern in the far field and computed by inserting Jn(lΔω) and Mn(lΔω) into
F(r^,lΔω)=j(lΔω)μ0N(r^,lΔω)+jk0r^×L(r^,lΔω),
where
N(r^,lΔω)=n=1NsJn(lΔω)Snfn(r)ejk0r·r^ds,
L(r^,lΔω)=n=1NsMn(lΔω)Snfn(r)ejk0r·r^ds.
Note that here k0=(lΔω)ε0μ0, it is assumed that r during the derivation of Eqs. (62)–(64), and r^·F(r^,lΔω)=0 is explicitly enforced during the computation of Csca(lΔω). Extinction efficiency of a scatterer, Qext(lΔω), is defined as the ratio of Cext(lΔω) to its geometrical cross section on the plane perpendicular to the direction of propagation of the incident field k^. In all examples presented in this section, Qext(lΔω), Cext(lΔω), and Csca(lΔω) computed by the proposed MOT-PMCHWT-SIE solver are compared to those obtained by the frequency-domain (FD) PMCHWT-SIE solver of [12], which computes Jn(lΔω) and Mn(lΔω) on the same discretization but directly in the frequency domain. Note that FD-PMCHWT-SIE solver has to be executed Nf 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, ε1(ωi), i=1,,NωVF, 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 [43]. It should be noted here that experiments were carried out at only 49 frequency points; therefore, spline interpolation is used to generate samples at ωi2π[fminVF,fmaxVF], i=1,,NωVF, NωVF=1000 required by Eqs. (12) and (13). Here, fminVF=155, and fmaxVF=1595  THz (corresponding to free-space wavelengths of λmaxVF=c0/fminVF=1937 and λminVF=c0/fmaxVF=188  nm), and the number of terms in the expansions NpVF=100. Figures 1 and 2 compare ε1(ω) in Eq. (12) obtained using the FRVF to that provided by the four-term Lorentz model described in [7], for gold and silver, respectively. The figures clearly show that both the FRVF expansion and the Lorentz model generate the real part of ε1(ω) 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 γ1(t) and γ¯1(t), which correspond to the Fourier transform of the summation in Eqs. (12) and (13), i.e., ε1(t) and ε¯1(t) without the Dirac delta term [see Eqs. (14) and (15)].

 figure: Fig. 1.

Fig. 1. (a) Real and (b) imaginary parts of ε1(ω) for gold.

Download Full Size | PPT Slide | PDF

 figure: Fig. 2.

Fig. 2. (a) Real and (b) imaginary parts of ε1(ω) for silver.

Download Full Size | PPT Slide | PDF

 figure: Fig. 3.

Fig. 3. Amplitude of (a) γ1(t) and (b) γ¯1(t) for gold and silver.

Download Full Size | PPT Slide | PDF

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 [43] experimental data. For the FRVF scheme, NpVF=100, NωVF=1000, λminVF=188, and λmaxVF=1937  nm. The excitation parameters are p^=x^, k^=z^, f0=900, fbw=600  THz, λmin=200, and λmax=1000  nm. The currents induced on the sphere surface are discretized using Ns=1926 RWG basis functions and the simulation is executed for Nt=2500 time steps with step size Δt=0.0333  fs. Figures 4(a) and 4(b) plot the amplitudes of J1(kΔt) and M1(kΔt), k=1,,Nt 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 Qext(lΔω), l=1,,Nf, Nf=100, Δf=12  THz, computed by the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers, and those obtained from Mie series solution. Results agree well with each other.

 figure: Fig. 4.

Fig. 4. Amplitude of J1(t) and M1(t) induced on (a) the gold sphere and (b) the silver sphere.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5.

Fig. 5. Qext(ω) computed for (a) the gold sphere and (b) the silver sphere.

Download Full Size | PPT Slide | PDF

C. Gold Rounded Cube

Next, scattering from a gold rounded cube [52] 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 [43] experimental data. For the FRVF scheme, NpVF=20, NωVF=1000, λminVF=292, and λmaxVF=1088  nm. The excitation parameters are p^=x^, k^=z^, f0=685, fbw=315  THz, λmin=300, and λmax=810  nm. The currents induced on the cube surface are discretized using Ns=6543 RWG basis functions and the simulation is executed for Nt=900 time steps with step size Δt=0.05  fs. Figure 6 compares Qext(lΔω), l=1,,Nf, Nf=200, Δf=3.15  THz, obtained using the proposed MOT-PMCHWT-SIE solver, to that computed by the null-field method [52]. Results are in good agreement with each other.

 figure: Fig. 6.

Fig. 6. Cext(ω) computed for the gold rounded cube.

Download Full Size | PPT Slide | PDF

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 [52]. The radius of the fillets on the edges and the corners is 10 nm. The permittivity of gold is obtained using the Johnson–Christy [43] experimental data. For the FRVF scheme, NpVF=20, NωVF=1000, λminVF=292, and λmaxVF=1088  nm. The excitation parameters are p^=x^, k^=z^, f0=685, fbw=315  THz, λmin=300, and λmax=810  nm. The currents induced on the cube surface are discretized using Ns=6543 RWG basis functions and the simulation is executed for Nt=900 time steps with step size Δt=0.05  fs. Figure 7 compares Cext(lΔω), l=1,,Nf, Nf=200, Δf=3.15  THz obtained using the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers to that computed by the DDA [52]. 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 [52] and explained by the fact that the DDA loses accuracy as the permittivity of the scatterer increases [53].

 figure: Fig. 7.

Fig. 7. Cext(ω) computed for the gold rounded triangular prism.

Download Full Size | PPT Slide | PDF

E. Gold-Coated Silica Sphere

Scattering from a silica sphere coated with gold [7] 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 [43] experimental data. For the FRVF scheme, NpVF=100, NωVF=1000, λminVF=188, and λmaxVF=1937  nm. The excitation parameters are p^=x^, k^=z^, f0=940, fbw=560  THz, λmin=200, and λmax=788  nm. The currents induced on the sphere surface are discretized using Ns=2130 RWG basis functions and the simulation is executed for Nt=2000 time steps with step size Δt=0.0333  fs. Figure 8 plots Cext(lΔω), l=1,,Nf, Nf=100, Δf=11.2  THz computed by the MOT-PMCHWT-SIE solver to that obtained from the Mie series solution. Results are in good agreement.

 figure: Fig. 8.

Fig. 8. Qext(ω) computed for the gold shell with silica core.

Download Full Size | PPT Slide | PDF

F. Silver Dimer

In this example, the scatterer is an x-directed dimer consisting of two silver spheres residing on the xy-plane [54]. 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 [44]. For the FRVF scheme, NpVF=100, NωVF=1000, λminVF=196, and λmaxVF=1033  nm. Two simulations are carried out for two polarizations of the incident field, p^=x^ and p^=y^. The remaining excitation parameters are k^=z^, f0=750, fbw=250  THz, λmin=300, and λmax=600  nm. The currents induced on the sphere surfaces are discretized using Ns=1428 RWG basis functions. Both simulations are executed for Nt=2000 time steps with step size Δt=0.0333  fs. Figure 9 compares Cext(lΔω), l=1,,Nf, Nf=100, Δf=5  THz, computed by the MOT-PMCHWT-SIE and FD-PMCHWT-SIE solvers. Results agree well with each other and also with those provided in [54].

 figure: Fig. 9.

Fig. 9. Cext(ω) computed for the silver dimer under two excitations with different polarizations.

Download Full Size | PPT Slide | PDF

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 xy-plane [55]. 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 x-direction. The permittivity of gold is obtained using the Johnson–Christy [43] experimental data. For the FRVF scheme, NpVF=20, NωVF=1000, λminVF=496, and λmaxVF=1216  nm. The excitation parameters are p^=y^, θinc=78 and φinc=0, f0=450, fbw=150  THz, λmin=500, and λmax=1000  nm. The currents induced on the surface of the scatterer are discretized using Ns=4380 RWG basis functions. The simulation is executed for Nt=1500 time steps with step size Δt=0.083  fs. Figure 10 compares Csca(lΔω), l=1,,Nf, Nf=100, Δf=3  THz, obtained using the proposed MOT-PMCHWT-SIE solver, to that computed by the frequency-domain FEM [55]. 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.

 figure: Fig. 10.

Fig. 10. Csca(ω) computed for the gold disk with the nonconcentric cavity.

Download Full Size | PPT Slide | PDF

4. CONCLUSION

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.

APPENDIX A

This section briefly describes the fundamental idea behind the FRVF scheme. The details can be found in [3941]. Let F(ω) represent a function defined in the frequency domain. FRVF approximates F(ω) as a sum of rational functions:

F(ω)jωf+d+m=1Nbmjω+am.
Here, N is termed fitting order, d and f are optional parameters, and am and bm are poles and residues, respectively, while d and f are real or can be forced to be zero, and am and bm are either real or have to come in complex conjugate pairs.

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 S(ω) and S(ω)F(ω) represent two auxiliary functions:

S(ω)=1+m=1NbmSjω+a˜m,
S(ω)F(ω)jωfSF+dSF+m=1NbmSFjω+a˜m.
Here, a˜m are “guessed” poles of S(ω) and S(ω)F(ω) for the given iteration. Multiplying Eq. (A2) with F(ω) and equating the resulting expression to the right-hand side of Eq. (A3), and evaluating the final equation at frequency samples ωi, i=1,,Nsamp, where N<Nsamp, yield an overdetermined system:
A¯¯X¯=Y¯.
Here, [A¯¯]i=[1jωi+a˜11jωi+a˜N1jωiF(ωi)jωi+a˜1F(ωi)jωi+a˜N] represents the ith row of A¯¯, X¯=[b1SFbNSFdSFfSFb1SbNS]T is the unknown vector, and {Y¯}i=F(ωi). Unknown X¯ is obtained by solving Eq. (A4) in the least-squares sense. One can show that zeros of the S(ω) are equal to the poles of F(ω) by rewriting Eqs. (A1)–(A3) in terms of sum of partial fractions [39]. These zeros are equal to the eigenvalues of the matrix D¯¯H¯B¯T, where D¯¯ is a diagonal matrix with entries {D¯¯}m,m=a˜m, {H¯}m=1, and {B¯}m=bmS. 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 ωi, i=1,,Nsamp, yielding an overdetermined system as in Eq. (A4), where now [A¯¯]i=[1jωi+a11jωi+aN1jωi] and X¯=[b1bNdf]T. 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 F(ω) is obtained.

REFERENCES

1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

2. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008). [CrossRef]  

3. D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015). [CrossRef]  

4. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006). [CrossRef]  

5. C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004). [CrossRef]  

6. A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005). [CrossRef]  

7. F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007). [CrossRef]  

8. Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010). [CrossRef]  

9. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]  

10. A. M. Kern and O. J. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009). [CrossRef]  

11. M. Araújo, J. Taboada, J. Rivero, D. Sols, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. 37, 416–418 (2012). [CrossRef]  

12. M. Araújo, J. Taboada, D. Sols, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express 20, 9161–9171 (2012). [CrossRef]  

13. B. Gallinet, A. M. Kern, and O. J. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]  

14. J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011). [CrossRef]  

15. A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

16. K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011). [CrossRef]  

17. P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994). [CrossRef]  

18. J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006). [CrossRef]  

19. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A 15, 487–502 (1998). [CrossRef]  

20. D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992). [CrossRef]  

21. G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997). [CrossRef]  

22. H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007). [CrossRef]  

23. A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999). [CrossRef]  

24. B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003). [CrossRef]  

25. B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009). [CrossRef]  

26. P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005). [CrossRef]  

27. A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004). [CrossRef]  

28. H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005). [CrossRef]  

29. H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010). [CrossRef]  

30. A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002). [CrossRef]  

31. X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010). [CrossRef]  

32. X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011). [CrossRef]  

33. C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988). [CrossRef]  

34. C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995). [CrossRef]  

35. J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991). [CrossRef]  

36. K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003). [CrossRef]  

37. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982). [CrossRef]  

38. I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015). [CrossRef]  

39. B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999). [CrossRef]  

40. B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006). [CrossRef]  

41. D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008). [CrossRef]  

42. R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989). [CrossRef]  

43. P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]  

44. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), Vol. 3.

45. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

46. S. Rao, Time Domain Electromagnetics (Elsevier, 1999).

47. D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984). [CrossRef]  

48. H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007). [CrossRef]  

49. H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011). [CrossRef]  

50. H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011). [CrossRef]  

51. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

52. C. Forestiere, G. Iadarola, G. Rubinacci, A. Tamburrino, L. Dal Negro, and G. Miano, “Surface integral formulations for the design of plasmonic nanostructures,” J. Opt. Soc. Am. A 29, 2314–2327 (2012). [CrossRef]  

53. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]  

54. E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010). [CrossRef]  

55. M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

References

  • View by:
  • |
  • |
  • |

  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  2. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
    [Crossref]
  3. D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
    [Crossref]
  4. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
    [Crossref]
  5. C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
    [Crossref]
  6. A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
    [Crossref]
  7. F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
    [Crossref]
  8. Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
    [Crossref]
  9. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
    [Crossref]
  10. A. M. Kern and O. J. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
    [Crossref]
  11. M. Araújo, J. Taboada, J. Rivero, D. Sols, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. 37, 416–418 (2012).
    [Crossref]
  12. M. Araújo, J. Taboada, D. Sols, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express 20, 9161–9171 (2012).
    [Crossref]
  13. B. Gallinet, A. M. Kern, and O. J. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
    [Crossref]
  14. J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
    [Crossref]
  15. A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).
  16. K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
    [Crossref]
  17. P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).
    [Crossref]
  18. J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
    [Crossref]
  19. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A 15, 487–502 (1998).
    [Crossref]
  20. D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
    [Crossref]
  21. G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
    [Crossref]
  22. H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
    [Crossref]
  23. A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
    [Crossref]
  24. B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
    [Crossref]
  25. B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
    [Crossref]
  26. P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
    [Crossref]
  27. A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
    [Crossref]
  28. H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
    [Crossref]
  29. H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
    [Crossref]
  30. A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
    [Crossref]
  31. X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
    [Crossref]
  32. X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
    [Crossref]
  33. C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988).
    [Crossref]
  34. C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
    [Crossref]
  35. J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
    [Crossref]
  36. K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
    [Crossref]
  37. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
    [Crossref]
  38. I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
    [Crossref]
  39. B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
    [Crossref]
  40. B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006).
    [Crossref]
  41. D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
    [Crossref]
  42. R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989).
    [Crossref]
  43. P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  44. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), Vol. 3.
  45. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  46. S. Rao, Time Domain Electromagnetics (Elsevier, 1999).
  47. D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
    [Crossref]
  48. H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
    [Crossref]
  49. H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
    [Crossref]
  50. H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
    [Crossref]
  51. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  52. C. Forestiere, G. Iadarola, G. Rubinacci, A. Tamburrino, L. Dal Negro, and G. Miano, “Surface integral formulations for the design of plasmonic nanostructures,” J. Opt. Soc. Am. A 29, 2314–2327 (2012).
    [Crossref]
  53. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [Crossref]
  54. E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
    [Crossref]
  55. M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

2015 (2)

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

2012 (3)

2011 (5)

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
[Crossref]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

2010 (5)

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

B. Gallinet, A. M. Kern, and O. J. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
[Crossref]

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

2009 (2)

A. M. Kern and O. J. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[Crossref]

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

2008 (2)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

2007 (3)

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

2006 (3)

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006).
[Crossref]

2005 (3)

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
[Crossref]

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

2004 (2)

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[Crossref]

2003 (3)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

2002 (1)

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

1999 (2)

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

1998 (1)

1997 (1)

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

1995 (1)

C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
[Crossref]

1994 (2)

P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[Crossref]

1992 (1)

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

1991 (1)

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

1989 (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989).
[Crossref]

1988 (1)

C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988).
[Crossref]

1984 (1)

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

1972 (1)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Al-Bundak, O.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

Amin, M.

M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Araújo, M.

Araújo, M. G.

Bagci, H.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Butler, C.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

Chew, W. C.

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

Christy, R.-W.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

Coronado, E. A.

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

Dal Negro, L.

de Aberasturi, D. J.

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

de La Chapelle, M. L.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

De Zutter, D.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

Deschrijver, D.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Dhaene, T.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

Donepudi, K.

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

Draine, B. T.

Drezek, R. A.

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

Duyne, R. P. V.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Encina, E. R.

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

Ergin, A. A.

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Flatau, P. J.

Forestiere, C.

Gallinet, B.

Glisson, A.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Greene, J. H.

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Gustavsen, B.

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006).
[Crossref]

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Hao, F.

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

Harrington, R. F.

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989).
[Crossref]

Hu, Y.

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

Iadarola, G.

Jiang, P.-L.

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
[Crossref]

Jin, J.-M.

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

Johnson, P. B.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Karlsson, A.

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

Kern, A. M.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Laluet, J.-Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Landesa, L.

Liz-Marzán, L. M.

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

Lomakin, V.

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

Lu, M.

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

Lubich, C.

C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
[Crossref]

C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988).
[Crossref]

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Macas, D.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Manara, G.

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

Martin, O. J.

Medgyesi-Mitschang, L.

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

Miano, G.

Michielssen, E.

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
[Crossref]

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Monorchio, A.

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Mrozowski, M.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

Noelck, S. J.

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

Nordlander, P.

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[Crossref]

Obelleiro, F.

Ostermann, A.

C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
[Crossref]

Oubre, C.

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[Crossref]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), Vol. 3.

Pazynin, V.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

Petropoulos, P. G.

P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).
[Crossref]

Putnam, J.

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

Rao, S.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

S. Rao, Time Domain Electromagnetics (Elsevier, 1999).

Rao, S. M.

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

Reggiannini, R.

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

Rikte, S.

Rivero, J.

Rubinacci, G.

Schatz, G. C.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

Schaubert, D.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

Semlyen, A.

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
[Crossref]

Serrano-Montes, A. B.

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Shanker, B.

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

Sirenko, K.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

Sirenko, Y. K.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

Sols, D.

Taboada, J.

Taboada, J. M.

Taflove, A.

Tamburrino, A.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Ulku, H. A.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

Uysal, I. E.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

Vechinski, D.

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

Vial, A.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Wang, X.

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

Weile, D. S.

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

Wilton, D.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Yilmaz, A.

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

Yilmaz, A. E.

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

Yuan, J.

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Zhao, L. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

ACS Nano (1)

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

Adv. Opt. Mater. (1)

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

Chem. Phys. Lett. (1)

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (2)

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

IEEE Microwave Wireless Compon. Lett. (1)

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

IEEE Trans. Antennas Propag. (16)

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).
[Crossref]

IEEE Trans. Electromagn. Compat. (2)

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (1)

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

IEEE Trans. Power Del. (2)

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
[Crossref]

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006).
[Crossref]

IMA J. Numer. Anal. (1)

C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
[Crossref]

J. Electromagn. Waves Appl. (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989).
[Crossref]

J. Opt. Soc. Am. A (6)

J. Phys. Chem. B (2)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[Crossref]

J. Phys. Chem. C (1)

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

Nat. Mater. (1)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Nature (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Numer. Math. (1)

C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (2)

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Proc. IEEE (1)

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
[Crossref]

Prog. Electromagn. Res. (1)

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

Other (7)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), Vol. 3.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

S. Rao, Time Domain Electromagnetics (Elsevier, 1999).

M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1. (a) Real and (b) imaginary parts of ε 1 ( ω ) for gold.
Fig. 2.
Fig. 2. (a) Real and (b) imaginary parts of ε 1 ( ω ) for silver.
Fig. 3.
Fig. 3. Amplitude of (a)  γ 1 ( t ) and (b)  γ ¯ 1 ( t ) for gold and silver.
Fig. 4.
Fig. 4. Amplitude of J 1 ( t ) and M 1 ( t ) induced on (a) the gold sphere and (b) the silver sphere.
Fig. 5.
Fig. 5. Q ext ( ω ) computed for (a) the gold sphere and (b) the silver sphere.
Fig. 6.
Fig. 6. C ext ( ω ) computed for the gold rounded cube.
Fig. 7.
Fig. 7. C ext ( ω ) computed for the gold rounded triangular prism.
Fig. 8.
Fig. 8. Q ext ( ω ) computed for the gold shell with silica core.
Fig. 9.
Fig. 9. C ext ( ω ) computed for the silver dimer under two excitations with different polarizations.
Fig. 10.
Fig. 10. C sca ( ω ) computed for the gold disk with the nonconcentric cavity.

Equations (68)

Equations on this page are rendered with MathJax. Learn more.

n ^ ( r ) × t E inc ( r , t ) = n ^ ( r ) × [ t E 0 sca ( r , t ) t E 1 sca ( r , t ) ] ,
n ^ ( r ) × t H inc ( r , t ) = n ^ ( r ) × [ t H 0 sca ( r , t ) t H 1 sca ( r , t ) ] .
t E 0 sca ( r , t ) = L 0 { μ 0 J ( r , t ) } + Q 0 { ε 0 1 J ( r , t ) } K 0 { M ( r , t ) } ,
t H 0 sca ( r , t ) = L 0 { ε 0 M ( r , t ) } + Q 0 { μ 0 1 M ( r , t ) } + K 0 { J ( r , t ) } ,
t E 1 sca ( r , t ) = L 1 { μ 1 J ( r , t ) } Q 1 { ε ¯ 1 ( t ) * J ( r , t ) } + K 1 { M ( r , t ) } ,
t H 1 sca ( r , t ) = L 1 { ε 1 ( t ) * M ( r , t ) } Q 1 { μ 1 1 M ( r , t ) } K 1 { J ( r , t ) } .
L p { X ( r , t ) } = S g p ( R , t ) * t 2 X ( r , t ) d s ,
Q p { X ( r , t ) } = S g p ( R , t ) * · X ( r , t ) d s ,
K p { X ( r , t ) } = × S g p ( R , t ) * t X ( r , t ) d s .
ε 1 ( t ) = F 1 { ε 1 ( ω ) } ,
ε ¯ 1 ( t ) = F 1 { ε ¯ 1 ( ω ) } = F 1 { 1 / ε 1 ( ω ) } .
ε 1 ( ω ) ε 1 m = 1 N p VF b m ϵ j ω + a m ϵ ,
ε ¯ 1 ( ω ) 1 ε 1 m = 1 N p VF b m ϵ ¯ j ω + a m ϵ ¯ ,
ε 1 ( t ) ε 1 δ ( t ) + m = 1 N p VF b m ϵ u ( t ) e a m ϵ t = ε 1 δ ( t ) + γ 1 ( t ) ,
ε ¯ 1 ( t ) 1 ε 1 δ ( t ) + m = 1 N p VF b m ϵ ¯ u ( t ) e a m ϵ ¯ t = 1 ε 1 δ ( t ) + γ ¯ 1 ( t ) ,
e j ω R / c 1 [ g 1 ( R , ω ) e j ω R / c 1 4 π R ] = e j ( k 1 R ω R / c 1 ) 1 4 π R d g ( R ) + m = 1 N p VF b m g ( R ) j ω + a m g ( R ) .
g 1 ( R , ω ) e j ω R / c 1 [ 1 4 π R + d g ( R ) + m = 1 N p VF b m g ( R ) j ω + a m g ( R ) ] .
g 1 ( R , t ) δ ( τ 1 ) 4 π R + d g ( R ) δ ( τ 1 ) + m = 1 N p VF b m g ( R ) u ( τ 1 ) e a m g ( R ) τ 1 ,
e j ω R / c 1 [ R g 1 ( R , ω ) + ( j ω c 1 R + 1 R 2 ) e j ω R / c 1 4 π ] = e j ( k 1 R ω R / c 1 ) 4 π ( j k 1 R + 1 R 2 ) + 1 4 π R 2 + j ω 4 π c 1 R d g R ( R ) + j ω f g R ( R ) + m = 1 N p VF b m g R ( R ) j ω + a m g R ( R ) .
R g 1 ( R , ω ) e j ω R / c 1 [ j ω ( f g R ( R ) 1 4 π c 1 R ) + d g R ( R ) 1 4 π R 2 + m = 1 N p VF b m g R ( R ) j ω + a m g R ( R ) ] .
R g 1 ( R , t ) δ ( τ 1 ) ( f g R ( R ) 1 4 π c 1 R ) + δ ( τ 1 ) ( d g R ( R ) 1 4 π R 2 ) + m = 1 N p VF b m g R ( R ) u ( τ 1 ) e a m g R ( R ) τ 1 .
J ( r , t ) = j = 1 N t n = 1 N s J n ( j Δ t ) f n ( r ) T j ( t ) = j = 1 N t n = 1 N s { I ¯ j J } n f n ( r ) T j ( t ) ,
M ( r , t ) = j = 1 N t n = 1 N s M n ( j Δ t ) f n ( r ) T j ( t ) = j = 1 N t n = 1 N s { I ¯ j M } n f n ( r ) T j ( t ) .
[ Z ¯ ¯ 0 JJ Z ¯ ¯ 0 JM Z ¯ ¯ 0 JM Z ¯ ¯ 0 MM ] [ I ¯ j J I ¯ j M ] = [ V ¯ j J V ¯ j M ] j = 1 j 1 [ Z ¯ ¯ j j JJ Z ¯ ¯ j j JM Z ¯ ¯ j j JM Z ¯ ¯ j j MM ] [ I ¯ j J I ¯ j M ] .
{ V ¯ j J } n = f n ( r ) , t E 0 inc ( r , t ) t = j Δ t ,
{ V ¯ j M } n = f n ( r ) , t H 0 inc ( r , t ) t = j Δ t .
{ Z ¯ ¯ j j JJ } n , n = { Z ¯ ¯ j j JJ , 0 } n , n + { Z ¯ ¯ j j JJ , 1 } n , n ,
{ Z ¯ ¯ j j JM } n , n = { Z ¯ ¯ j j JM , 0 } n , n + { Z ¯ ¯ j j JM , 1 } n , n ,
{ Z ¯ ¯ j j MM } n , n = { Z ¯ ¯ j j MM , 0 } n , n + { Z ¯ ¯ j j MM , 1 } n , n ,
{ Z ¯ ¯ j j JJ , 0 } n , n = μ 0 f n ( r ) , L 0 { f n ( r ) T j ( t ) } t = j Δ t + 1 ε 0 f n ( r ) , Q 0 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j JM , 0 } n , n = f n ( r ) , K 0 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j MM , 0 } n , n = ε 0 μ 0 { Z ¯ ¯ j j JJ , 0 } n , n ,
{ Z ¯ ¯ j j JJ , 1 } n , n = μ 1 f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t + 1 ε 1 f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t + f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j JM , 1 } n , n = f n ( r ) , K 1 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j MM , 1 } n , n = ε 1 f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t + 1 μ 1 f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t .
{ Z ¯ ¯ j j MM , 0 } n , n = ε 0 μ 0 { Z ¯ ¯ j j JJ , 0 } n , n = ε 0 4 π S n f n ( r ) · S n f n ( r ) t 2 T j ( t ) | t = j Δ t R / c 0 R d s d s 1 4 π μ 0 S n · f n ( r ) S n · f n ( r ) T j ( j Δ t R / c 0 ) R d s d s ,
{ Z ¯ ¯ j j JM , 0 } n , n = 1 4 π S n f n ( r ) · S n R ^ × f n ( r ) R [ t 2 T j ( t ) c 0 + t T j ( t ) R ] t = j Δ t R / c 0 d s d s .
C T ( R , t ) = g 1 ( R , t ) * t 2 T j ( t ) ,
C ( R , t ) = g 1 ( R , t ) * T j ( t ) ,
C R ( R , t ) = R g 1 ( R , t ) * t T j ( t ) .
f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n C T ( R , t ) f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n C ( R , t ) · f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , K 1 { f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n R ^ × f n ( r ) C R ( R , t ) d s d s | t = j Δ t .
C T ( R , t ) = [ d g ( R ) + 1 4 π R ] t 2 T ( t ) | t = t j Δ t R / c 1 + m = 1 N p VF b m g ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g ( R ) ( t t j Δ t R / c 1 ) t 2 T ( t ) d t ,
C ( R , t ) = [ d g ( R ) + 1 4 π R ] T ( t j Δ t R / c 1 ) + m = 1 N p VF b m g ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g ( R ) ( t t j Δ t R / c 1 ) T ( t ) d t ,
C R ( R , t ) = [ f g R ( R ) 1 4 π R c 1 ] t 2 T ( t ) | t = t j Δ t R / c 1 + [ d g R ( R ) 1 4 π R 2 ] t T ( t ) | t = t j Δ t R / c 1 + m = 1 N p VF b m g R ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g R ( R ) ( t t j Δ t R / c 1 ) t T ( t ) d t .
D ( R , t ) = g 1 ( R , t ) * γ ¯ 1 ( t ) * T j ( t ) ,
D T ( R , t ) = g 1 ( R , t ) * γ 1 ( t ) * t 2 T j ( t ) = t 2 g 1 ( R , t ) * γ 1 ( t ) * T j ( t ) .
f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n D ( R , t ) · f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n D T ( R , t ) f n ( r ) d s d s | t = j Δ t .
F j ( t ) = γ ¯ 1 ( t ) * T j ( t ) l = 1 N t F l j T l ( t ) ,
F j T ( t ) = γ 1 ( t ) * T j ( t ) l = 1 N t F l j T T l ( t ) ,
F l j = F j ( l Δ t ) = γ ¯ 1 ( t ) * T j ( t ) | t = l Δ t = Δ t min ( l j , d ) Δ t γ ¯ 1 ( [ l j ] Δ t t ) T ( t ) d t = m = 1 N p VF b m ϵ ¯ Δ t min ( l j , d ) Δ t u ( [ l j ] Δ t t ) e a m ϵ ¯ ( [ l j ] Δ t t ) T ( t ) d t ,
F l j T = F j T ( l Δ t ) = γ 1 ( t ) * T j ( t ) | t = l Δ t = Δ t min ( l j , d ) Δ t γ 1 ( [ l j ] Δ t t ) T ( t ) d t = m = 1 N p VF b m ϵ Δ t min ( l j , d ) Δ t u ( [ l j ] Δ t t ) e a m ϵ ( [ l j ] Δ t t ) T ( t ) d t .
D ( R , t ) = g 1 ( R , t ) * F l j ( t ) l = 1 N t F l j g 1 ( R , t ) * T l ( t ) ,
D T ( R , t ) = t 2 g 1 ( R , t ) * F l j T ( t ) l = 1 N t F l j T t 2 g 1 ( R , t ) * T l ( t ) = l = 1 N t F l j T g 1 ( R , t ) * t 2 T l ( t ) .
f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n [ l = 1 N t F l j g 1 ( R , t ) * T l ( t ) ] · f n ( r ) d s d s | t = j Δ t = l = 1 N t f n ( r ) , Q 1 { f n ( r ) T l ( t ) } t = j Δ t F l j ,
f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n [ l = 1 N t F l j T g 1 ( R , t ) * t 2 T l ( t ) ] f n ( r ) d s d s | t = j Δ t = l = 1 N t f n ( r ) , L 1 { f n ( r ) T l ( t ) } t = j Δ t F l j T .
E 0 inc ( r , t ) = p ^ E 0 inc G ( t k ^ · r / c 0 ) ,
C sca ( l Δ ω ) = 1 16 π 2 | E 0 inc | 2 Ω | F ( r ^ , l Δ ω ) | 2 d Ω ,
C ext ( l Δ ω ) = 1 k 0 | E 0 inc | 2 Im { E 0 inc p ^ · F ( k ^ , l Δ ω ) } .
F ( r ^ , l Δ ω ) = j ( l Δ ω ) μ 0 N ( r ^ , l Δ ω ) + j k 0 r ^ × L ( r ^ , l Δ ω ) ,
N ( r ^ , l Δ ω ) = n = 1 N s J n ( l Δ ω ) S n f n ( r ) e j k 0 r · r ^ d s ,
L ( r ^ , l Δ ω ) = n = 1 N s M n ( l Δ ω ) S n f n ( r ) e j k 0 r · r ^ d s .
F ( ω ) j ω f + d + m = 1 N b m j ω + a m .
S ( ω ) = 1 + m = 1 N b m S j ω + a ˜ m ,
S ( ω ) F ( ω ) j ω f SF + d SF + m = 1 N b m SF j ω + a ˜ m .
A ¯ ¯ X ¯ = Y ¯ .

Metrics