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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 8, APRIL 15, 2007
Multispecies ADI-FDTD Algorithm for Nanoscale Three-Dimensional Photonic Metallic Structures Kyung-Young Jung, Student Member, IEEE, and Fernando L. Teixeira, Senior Member, IEEE
Abstract—The unconditionally stable alternating-directionimplicit finite-difference time-domain (ADI-FDTD) method is extended to model multispecies dispersive media for simulation of nanoscale three-dimensional metallic structures based on optical plasmon resonances. Examples involving modeling of Au nanoparticles show that the proposed ADI-FDTD yields improved computational performance versus standard FDTD in highly refined grids and for moderate Courant numbers. Index Terms—Alternating-direction-implicit (ADI) method, dispersive media, finite-difference time-domain (FDTD) method.
I. INTRODUCTION UBWAVELENGTH integrated photonic devices are of fundamental importance for miniaturization of optical components and better integration with complementary metal–oxide–semiconductor-scale electronic devices. In particular, sub-100-nm photonic devices can be constructed by exploiting plasmon resonances [1], [2]. Numerical modeling of plasmon devices can be done by means of the finite-difference time-domain (FDTD) method [3]. However, FDTD is limited by the Courant stability criterion, which sets a bound of the maximum time step allowed in the time-domain update that depends on the spatial cell size. This criterion impacts the time required to simulate a given structure and becomes particularly taxing in the case of plasmon devices because the geometrical features are much smaller than the wavelength of operation, which necessitates highly refined grids. The alternating-direction-implicit (ADI)-FDTD method [4]–[8] is an attractive alternative to standard FDTD in highly refined grids due to its unconditional stability with moderate computational overhead. Due to the frequency dispersive response of the complex permittivity (conductivity) of metals at optical frequencies, multispecies dispersive models are required for many metals [9]. ADI-FDTD has been previously applied for dispersive media [10], but to date this has been limited to single-species models. In this work, we show how ADI-FDTD can be extended towards modeling of multispecies dispersion in a systematic fashion via auxiliary differential equations (ADEs) for equivalent polarization and current terms. We illustrate the algorithm to calculate localized plasmon resonances in Au nanoparticles described by a Drude–Lorentz dispersion model.
II. FORMULATION time convention. In the visible range, the We assume the dielectric response of gold can be described by a Drude–Lorentz model [9] as (1) and are pole resonances, and are damping where is a weighting coefficient for the Lorentz coefficients, and term. , Ampere’s law reads as With the above
S
(2) By introducing equivalent Drude current [11] and Lorentz terms in Maxwell equations, and applying an polarization inverse Fourier transformation, we obtain the following four time-domain governing equations:
(3) (4) (5) (6) where Maxwell curl equations are displayed in (3) and (4) and two ADEs in (5) and (6). Note that it is possible to choose alternative auxiliary variable pairs, for example, and and , or and , but the above choice leads to the most memory efficient formulation. Equations (3)–(6) are amenable for implementation in either FDTD or ADI-FDTD. In ADI-FDTD, the update at each time step is divided into two substeps. For the first substep , the discretized equations for the and update are written as
Manuscript received September 22, 2006; revised December 15, 2006. This work was supported by the Air Force Office of Scientific Research (AFOSR) under MURI Grant FA 9550-04-1-0359, by the National Science Foundation (NSF) under Grant ECS-0347502, and by OSC under Grant PAS-0061 and Grant PAS-0110. The authors are with the ElectroScience Laboratory and Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LPT.2007.894282 1041-1135/$25.00 © 2007 IEEE
(7)
JUNG AND TEIXEIRA: MULTISPECIES ADI-FDTD ALGORITHM FOR NANOSCALE STRUCTURES
Fig. 1. Spectra of a Au nanosphere (r spatial cell size h nm.
1 =1
= 15 nm) surrounded by SiO
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with the
(12) where
(8)
and . The left-hand side of (12) forms a tridiagonal matrix, and the associated linear system can complexity. The other electric field be easily solved with components are obtained similarly. In summary, the first substep explicitly of the proposed ADI-FDTD consists of 1) update exfrom (10), 2) update implicitly from (12), 3) update plicitly from (9), and 4) update explicitly from (7). An anal. ogous procedure is employed for the second substep The above recovers the standard ADI-FDTD formulation (for dielectric regions) by a proper choice of material coefficients. III. NUMERICAL EXAMPLES
where the subscript refers to spatial grid indexing and the superscript refers to the time step. The discretized equations for auxiliary variables can be obtained by using central differencing in time, in a standard fashion (9) (10) where
(11) or . From (8), we see that and cannot be updated explicitly. By substituting (7) and (9) into (8), the following implicit update is obtained:
To validate the above formulation, we analyze the surface plasmon resonance of an isolated Au nanosphere with radius nm, in SiO host. In the 500- to 1000-nm range, the following parameters [9] give the best fit to the Johnson and Christy bulk dielectric Au data [12]: THz, THz, THz, THz, and . The refraction index of SiO is set to 1.444, being virtually constant in those wavelengths. The incident field is a -polarized plane wave. The time-domain excitation is a Gaussian modulated sine wave having half power bandwidth (HPBW) at 500–1000 nm. The simulation time window is 30 fs. The Courant number (CN) is , where is the speed of light in vacuum. given by All simulations were performed on a Pentium IV cluster based on a distributed/shared memory hybrid system. nm, First, we employ a grid with spatial cell size corresponding to 500 points per (vacuum) wavelength at the cells is used in this case. smallest wavelength. A grid with Fig. 1 shows the spectra of the Au nanosphere using both standard FDTD and the proposed ADI-FDTD at a distance away from the sphere. The spectral response is obtained by Fourier transforming the time-domain field data (with a Blackman window to remove aliasing), followed by a normalization by the excitation pulse spectrum. Very good agreement is observed between ADI-FDTD and FDTD on this mesh. Table I summarizes the normalized central processing unit (CPU) time, memory requirement, and the surface plasmon
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 8, APRIL 15, 2007
TABLE I COMPARISON BETWEEN ADI-FDTD AND FDTD
in the coarse grid, one has , and hence, CN the accuracy starts to be compromised for CN beyond that. Finally, we consider a plasmon waveguide based on a linear chain of Au nanospheres surrounded by SiO host. The radius of each nanosphere is 15 nm and the intercenter distance is 45 nm. The chain is excited by a dipole with transverse polarization with respect to the chain axis. The time-domain excitation is again a Gaussian modulated sine wave with HPBW at 500–1000 nm. The grid has 390 90 90 cells. Fig. 3 shows amplitude along the chain calculated by ADIa snapshot of , showing electromagnetic energy transFDTD with CN via near-field coupling port below the diffraction limit between closely spaced nanoparticles. IV. CONCLUSION
Fig. 2. Spectra of a Au nanosphere (r nm. with the spatial cell size h
1 =2
= 15 nm) surrounded by silicon dioxide
The ADI-FDTD method has been extended to multispecies dispersive media for the analysis of optical plasmon resonances in nanoscale metallic structures. Numerical results show that the proposed ADI-FDTD is advantageous in highly refined grids and becomes computationally more efficient than FDTD for moderate size CN. The present ADI-FDTD formulation has been illustrated for a two-species Drude–Lorentz dispersive media, but it can be easily applied to arbitrary multispecies dispersive media as well. By using ADEs (instead of recursive convolution), auxiliary variables associated with Lorentz dispersion terms remain real-valued and the formulation can be further extended towards nonlinear media. REFERENCES
Fig. 3. Snapshot of E amplitude along plasmon waveguide excited by a dipole pulse with a y -directed polarization indicated by arrow.
resonance wavelengths for various CN. The ADI-FDTD requires 7.8% more memory than the FDTD method for the same , the CPU time of ADI-FDTD is reduced problem. For CN to about 42% of FDTD, while the resonance wavelength (rel, the relative error in ative) error is below 0.8%. For CN the resonance wavelength is still within 1.3%. In this case, the CPU time required by ADI-FDTD is about 32% of FDTD. nm and Next, we consider a coarser grid with cells. Fig. 2 shows the spectra of the same structure using both FDTD and the proposed ADI-FDTD. For a large CN number, ADI-FDTD results start to deviate from FDTD results, since the becomes more pronounced truncation error due to a larger and [13]. The errors in the resonance wavelength for CN are about 2% and 4.3%, respectively. These coarse CN grid results serve to illustrate the point made previously, that the ADI-FDTD is more suited for very fine grids, where increase in CN produces limited truncation errors. We stress that, even though there is no stability limit, the CN (or, equivalently, ) in ADI-FDTD is still limited by accuracy, . In the case of dispersive media, this implies that for a given should be at least about of the smallest the maximum characteristic time constant of the media, which may include resonant frequencies and relaxation times. In the example con. Using sidered, the smallest characteristic time is
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