Efficient FDTD algorithms for dispersive Drude-critical points media based on bilinear z-transform
step Δt via the following relations c0p = 2a1p Dt + a0p (Dt)2 , c1p = 2a0p (Dt)2 c2p = a0p (Dt) − 2a1p Dt, d0p = 4 + 2b1p Dt + b0p (Dt)
K.P. Prokopidis and D.C. Zografopoulos Finite-difference time-domain (FDTD) schemes based on the bilinear z-transform are introduced for modelling time-domain wave propagation in dispersive Drude-critical points media. The accuracy and efficiency of the proposed technique are verified by comparisons with other FDTD algorithms and analytical solutions.
Introduction: During recent years, the numerical simulation of electromagnetic wave propagation in metals in the visible (VIS) and infrared (IR) spectrum has received considerable attention. One of the most successful and well established approaches to study optical wave interaction with metals is the finite-difference time-domain (FDTD) method [1]. In particular when applications involving plasmonic devices are considered [2], the accurate modelling of the dispersion of metals over a wide spectrum is essential. The commonly used Drude and DrudeLorentz models often result in being unsatisfactory or demand many terms for accurate fitting, thus increasing memory requirements and hindering the efficient broadband simulation of practical devices. An analytical dispersive model including a single Drude term and two critical point pairs (DCP) has been recently introduced for the modelling of gold [3] and thereafter extended to other metals [4,5]. The DCP model has been proven capable of describing the dielectric dispersion of metals in the VIS/IR spectral region more accurately than the widely used Drude-Lorentz media. FDTD implementations of the DCP model have been thus far based on the recursive convolution (RC) [4] and the trapezoidal RC (TRC) [6] techniques. Although these provide accurate results, they involve complex arithmetics and they cannot be easily combined with perfectly matched layers (PMLs). Here, we propose a set of efficient FDTD algorithms based on the bilinear z-transform, which involve purely real numbers and show reduced memory requirements. Furthermore, contrary to the implementations of [4,6], in the proposed schemes the dielectric flux density D is explicitly present, thus rendering them compatible with the use of material-independent PMLs [7] and allowing for the study of anisotropic materials. FDTD formulation: The relative dielectric permittivity ɛr(ω) of the DCP model, assuming e jωt time dependence and N critical point pairs, is described by
v2D v( j g − v) N Ap Vp e jwp Ap Vp e−jwp + + Vp − v + jGp p=1 Vp + v − jGp
1r ( v ) = 11 +
(1)
(4b)
d1p = 2b0p (Dt)2 − 8, d2p = 4 − 2b1p Dt + b0p (Dt)2
(4c)
1r ( v ) = 11 +
M p=1
a1p jv + a0p b2p ( jv)2 + b1p jv + b0p
(2)
as an explicit function of jω. One can readily verify that for the Drude term a0p = ω2D, a1p = 0, b0p = 0, b1p = γ, and b2p = 1, while for the critical point pair a1p = − 2ApΩpsinϕp, a0p = 2ApΩp(Ωpcosϕp − Γpsinϕp), b1p = 2Γp, b0p = Ω2p + Γ2p, and b2p = 1. It is stressed that the proposed general expression incorporates all commonly used dispersion models, i.e. Debye, Drude, Lorentz and static conductivity. We opt for the use of the bilinear z-transform (BT), defined as jv Dt2 (1 − z−1 )/(1 + z−1 ), in order to implement a discrete scheme. The BT is widely used in the field of signal processing, such as in analogue-to-digital filter conversion [8]. In the context of FDTD studies, the BT has been already successfully implemented for the modelling of dispersive media [9,10], providing very efficient frequency approximation owing to the small introduced phase error [9]. After the BT, ɛr(ω) reads −1
−2
M c +c z + c2p z 0p 1p 1r (z) = 11 + −1 + d z−2 2p p=1 d0p + d1p z
It is common practice to associate a dispersive medium in the discrete domain with an equivalent digital filter whose transfer function is the inverse of the dielectric permittivity of the medium. In dispersive FDTD implementations, the electric field is derived from the constitutive relation, while the dielectric displacement through Faraday’s law. The constitutive relation in the medium is expressed as D(z) = ɛ0ɛr(z)E(z), which describes a digital filter in the z-domain. After substituting ɛr(z) and defining an auxiliary variable Fp(z), the following relation is yielded D(z) = 10 11 E(z) + 10
(3)
where the coefficients are related to the medium parameters and the time
M
Fp (z)
(5)
p=1
where Fp (z) =
c0p + c1p z−1 + c2p z−2 E(z) d0p + d1p z−1 + d2p z−2
(6)
Equation (6) is switched into the time domain by means of the relation z−m F(z) ↔ F n−m , where n refers to the time instant nΔt, as follows = Fn+1 p
1 (c0p En+1 + c1p En d0p
(7)
+ c2p En−1 − d1p Fn − d2p Fn−1 ) After turning (5) to the time domain and substituting (7) into (5) the following update equation of the electric field is obtained En+1 =
1 1 0 11 + 10
M
c0p p=1 d0p
[Dn+1 − 10
M 1 p=1 d0p
(8)
(c1p En + c2p En−1 − d1p Fnp − d2p Fn−1 p )] The variables Fp are then updated through (7). This scheme requires two previous values of the E-field and 2M = 2N + 2 values for the Fp, translating into a total of 2N + 4 additional auxiliary values. Memory efficient implementation: In this Section optimised BT algorithms that reduce FDTD memory requirements are presented. To start with, one can observe that (3) describes a set of digital filters in parallel. After some algebraic manipulations, ɛr(z) can be expressed as a transfer function with denominator and numerator of Kth-order, K K fk z−k / gk z−k (9) 1r (z) = 1 + k=1
with parameters defined as in [4]. The Drude term can be included in a generalised expression involving M = N + 1 terms given by
(4a) 2
2
k=0
where K = 2M. It is straightforward to prove that the previous transfer function is equivalent to the single difference equation En+1 = −
K
fk En+1−k +
k=1
K g k n+1−k D k=0 10
(10)
Equation (10) describes an IIR filter, which can be implemented using the Transposed Direct Form II (TDF-II) [8]. This scheme requires K = 2N + 2 additional auxiliary variables and, in terms of memory requirements, it is equivalent to the RC [4] and TRC [6] methods. Memory demands can be further reduced via a modified version of the TDF-II as follows g0 n+1 g1 n D + D 10 10
(11a)
= W nk+1 − fk+1 En W n+1 k gk+1 n + D , k = 1, 2, . . . , K − 2 10
(11b)
En+1 = −f1 En + W n1 +
W n+1 K−1 =
gK n D − fK En 10
(11c)
This implementation requires only K − 1 = 2N + 1 additional auxiliary variables for each component and per FDTD cell, that is one less than the RC and TRC methods.
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Numerical results: To validate the proposed FDTD schemes, we consider two examples where analytical solutions are available. First, we simulate light transmission through a thin slab of gold modelled with a Drude term and two critical points [5]. The spatial cell was Δz = 1 nm and the time step Δt = 0.3Δz/c, where c is the speed of light in vacuum. In Fig. 1, the analytically computed transmission coefficient and the relative errors for the RC, TRC and the proposed BT method are displayed for a slab thickness of 20 nm. The BT method is shown to be of equal accuracy with the TRC one, while it outperforms the RC implementation.
BT TRC RC
0.16 transmission: relative error, %
transmission
0.18
0.14 0.12 0.10
0.5 0.4 0.3 0.2 0.1 0 200 400 600 800 1000 λ, nm
Acknowledgment: This work was supported in part by the EU Marie-Curie grant ALLOPLASM (FP7-PEOPLE-2010-IEF-273528). © The Institution of Engineering and Technology 2013 16 January 2013 doi: 10.1049/el.2013.0198 One or more of the Figures in this Letter are available in colour online.
0.08
K.P. Prokopidis (Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece)
0.06
E-mail:
[email protected]
0.04
D.C. Zografopoulos (Consiglio Nazionale delle Ricerche, Istituto per la Microelettronica e Microsistemi (CNR-IMM), Roma 00133, Italy)
0.02 0 200
300
400
500
600 λ, nm
700
800
900
References
1000
Fig. 1 Relative error (%) on the transmission coefficient of light through a 20 nm-thick gold slab against wavelength for RC, TRC and proposed BT methods Inset: Analytical solution
As a second example, we consider a metal-insulator-metal (MIM) Ag/ air/Ag waveguide with a spacing of d = 100 nm. The DCP parameters of Ag are given in [5]. A two-dimensional 400 × 120 domain with cell sizes Δx = 10 nm and Δz = 5 nm backed by CPML [7] was employed in order to calculate the modal effective index of the fundamental TM-polarised mode, using a simple averaging technique at the Ag/air interface. Fig. 2 shows the FDTD results directly compared to the analytical solution [2]. The relative error was found to be lower than 0.1% in the whole wavelength range of interest, indicating the accuracy of the proposed scheme for rigorous numerical simulations of plasmonic applications. 1.38 1.36 modal effective index
Conclusion: We have presented novel FDTD schemes based on the bilinear z-transform, for the modelling of generalised critical-point dispersion media, such as noble metals in the VIS/IR spectrum. The proposed algorithm shows equal or better accuracy compared to the TRC and RC techniques, respectively, involving only real arithmetics and allowing for the termination of the computational domain via PMLs. A memory-efficient implementation characterised by a reduced number of auxiliary variables has also been introduced.
EZ at 800 nm
1.34 1.32
Ag
Z
Ag
500 nm
max x
d
1.30
min
1.28
1 Taflove, A., and Hagness, S.C.: ‘Computational electrodynamics: The finite-difference time-domain method’ (Artech House, 2005, 3rd edn) 2 Dionne, J.A., Sweatlock, L.A., Atwater, H.A., and Polman, A.: ‘Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization’, Phys. Rev. B, 2006, 73, p. 035407 3 Etchegoin, P.G., Ru, E.C.L., and Meyer, M.: ‘An analytic model for the optical properties of gold’, J. Chem. Phys., 2006, 125, p. 164705 4 Vial, A.: ‘Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finitedifference time domain method’, J. Opt. A, Pure Appl. Opt., 2007, 9, pp. 745–748 5 Vial, A., Laroche, T., Dridi, M., and Le Cunff, L.: ‘A new model of dispersion for metals leading to a more accurate modeling of plasmonic structures using the FDTD method’, Appl. Phys. A, Mater. Sci. Process., 2011, 103, pp. 849–853 6 Shibayama, J., Watanabe, K., Ando, R., Yamauchi, J., and Nakano, H.: ‘Simple frequency-dependent FDTD algorithm for a Drude-critical points model’. Asia-Pacific Microwave Conf., Yokohama, Japan, 2010, pp. 73–75 7 Roden, J.A., and Gedney, S.D.: ‘Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media’, Microw. Opt. Technol. Lett., 2000, 27, (5), pp. 334–339 8 Proakis, J.G., and Manolakis, D.G.: ‘Digital signal processing: principles, algorithms, and applications’ (Prentice Hall, 1996, 3rd edn) 9 Hulse, C., and Knoesen, A.: ‘Dispersive models for the finite-difference time-domain method: design, analysis, and implementation’, J. Opt. Soc. Am. A, 1994, 11, (6), pp. 1802–1811 10 Pereda, J., Vegas, A., and Prieto, A.: ‘FDTD modeling of wave propagation in dispersive media by using the Mobius transformation technique’, IEEE Trans. Microw. Theory Tech., 2002, 50, (7), pp. 1689–1695
1.26 1.24
analytical
1.22
FDTD
1.20 400
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λ, nm
Fig. 2 Effective index of fundamental TM-polarised mode of Ag/Air/Ag MIM plasmonic waveguide computed with proposed FDTD method and the analytical solution Inset: z-component of electric field at free-space wavelength of 800 nm
ELECTRONICS LETTERS 11th April 2013 Vol. 49 No. 8
