Layer-multiple-scattering method for photonic structures of general scatterers based on a discrete-dipole approximation/t-matrix point-matching method

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1 V. Yannopapas Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B 631 Layer-multiple-scattering method for photonic structures of general scatterers based on a discrete-dipole approximation/t-matrix point-matching method Vassilios Yannopapas Department of Physics, National Technical University of Athens, GR Athens, Greece (vyannop@mail.ntua.gr) Received October 23, 2013; revised December 5, 2013; accepted December 30, 2013; posted January 23, 2014 (Doc. ID ); published February 27, 2014 We present a hybrid discrete-dipole approximation (DDA)/layer-multiple-scattering (LMS) method for treating photonic structures consisting of general scatterers. The present method is a major extension of the existing LMS formalism and code that combine the merits of both the DDA and LMS technique. Namely, the new hybrid DDA/LMS technique treats, in principle, scatterers of general (nonspherical) shape that might be anisotropic and/ or inhomogeneous, thanks to the DDA component while, at the same time, it incorporates theoretical tools provided by the LMS method, such as the doubling-layer process and the complex frequency band structure that are not met in contemporary electromagnetic solvers. The merging of both techniques is accomplished via a pointmatching module that provides the scattering T-matrix of an arbitrary scatterer. To demonstrate the applicability of the new method, we study the optical properties of 2D and 3D plasmonic lattices of gold nanocubes Optical Society of America OCIS codes: ( ) Multiple scattering; ( ) Metamaterials; ( ) Photonic crystals; ( ) Surface plasmons; ( ) Nanomaterials INTRODUCTION Man-made periodic structures, depending on their functionality and scope, fall within three main categories: photonic crystals, plasmonic materials, and metamaterials. Photonic crystals possess an absolute bandgap, i.e., a frequency region where no electromagnetic (EM) Bloch modes are allowed as a result of destructive interference, allowing for the passive control of light emission and flow within their volume [1]. Plasmonic materials and metamaterials operate both in the subwavelength regime, i.e., around the center of the Brillouin zone. Plasmonic materials are mainly used for boosting the electric field in small volumes and tailoring light absorption [2]. Metamaterials have more unconventional features, such as negative refractive index and/or magnetic permeability with application in optical microscopy, imaging, and cloaking [3]. The fabrication of photonic structures is made either by laser- and electron-beam lithography or by self-assembly. By lithography, one usually realizes planar structures with repeatable units characterized by sharp edges, such as rods, pillars, disks, etc., whereas by self-assembly the repeated units are of spherical or nearly spherical shape. The modeling tools for simulating photonic structures depend on the fabrication technique, i.e., lithography or self-assembly. For lithographic designs purely numerical EM solvers are employed, such as the transfer-matrix method [4], the finite element method (FEM) [5], the finite-difference time-domain (FDTD) method [6], the pseudo-spectral time-domain method (PSTD) [7,8] the Fourier modal method [9], and the discrete-dipole approximation (DDA) [10]. In principle, such methods can be also used for self-assembled structures, such as arrays of spheres requiring, however, huge amounts of memory and computing time. The state-of-the-art technique in simulating two- and three-dimensional arrays of spheres is the layermultiple scattering (LMS) method [11 13]. There, the EM field within a given 2D lattice (plane) of spheres is obtained via the multiple-scattering technique. In between a pair of two consecutive planes of spheres, the EM field is written in a plane wave basis where the multiple-scattering between the planes is fully taken into account. A slab of macroscopic thickness, i.e., containing several hundreds of planes of spheres, can be easily built up via a doubling-layer process [11 13]. By imposing boundary conditions along the growth direction of the crystal slab, one can also calculate the complex frequency band structure [11 13]. In this work, we aim at merging the DDA with the LMS method to a common theoretical framework and computing code (a hybrid DDA/LMS method). By doing so, we combine the merits of both techniques so that the new method is capable of treating photonic structures of general scatterers. Namely, the new hybrid DDA/LMS technique treats, in principle, scatterers of general (nonspherical) shape that might be anisotropic and/or inhomogeneous, thanks to the DDA component while, at the same time, it incorporates theoretical tools provided by the LMS method, such as the doubling-layer process and the complex frequency band structure, which are not met in contemporary EM solvers. The proposed hybrid DDA/LMS method modularizes the computational process into rapidly converging programming units which addresses the shortcomings of the existing simulators. Currently, the LMS method has been extended so as to treat nonspherical scatterers with axial symmetry [14,15], anisotropic spherical scatterers [16,17] as well as clusters of spherical scatterers [18]. The extension of the LMS to axisymmetric scatterers [14,15] has been achieved by the boundary element method /14/ $15.00/ Optical Society of America

2 632 J. Opt. Soc. Am. B / Vol. 31, No. 3 / March 2014 V. Yannopapas (BEM), which is faster than the DDA in treating homogeneous, isotropic, nonspherical scatterers [19]. However, it lacks the generality of the DDA, which treats anisotropic and inhomogeneous scatterers on equal footing. Originally, the DDA had been mainly applied to individual low-index particles [10,20 22] and 2D arrays of such [23]. However, the rapid increase in computing power, allowed for the study of individual high-index scatterers such as silicon [24,25] and metallic nanoparticles [26] as well as their corresponding 2D arrays [27 29]. Extension of the DDA to 3D arrays is still lacking due to the expected computational overhead. From this point of view, the hybrid DDA/LMS proposed here can be seen as an extension of the DDA to 3D crystals although it possesses much more functionalities than a mere latticetype DDA. 2. THEORY A. Outline of the Hybrid DDA/LMS Method The DDA is among the well-established methods for solving EM scattering from an individual or multiple scatterers (targets) of general shape and moderate size. The central idea of the DDA is the replacement of the actual scatterer by a set of point dipoles interacting with each other and an incident EM field, giving rise to a linear system of equations that provides the polarizations at each point dipole. Due to its conceptual simplicity and transparency, the DDA is very efficient in treating homogeneous or inhomogeneous scatterers with elaborate shape and/or anisotropic EM response. The LMS method is a rigorous theory for solving the problem of scattering of EM waves by a slab containing a finite number of 2D planes of spherical particles and/or homogeneous slabs. The LMS method simulates an actual experimental setup since it provides the transmission, reflection and/or absorption of light incident (with arbitrary angle and polarization) on a finite slab made from several planes of spheres and/or homogeneous slabs. A central quantity in the LMS method is the so-called scattering T-matrix. The latter connects the Fourier coefficients in the spherical-wave expansion of an EM wave scattered off of a finite object with those of the incident wave. The aim of the present paper is to generalize the LMS method along the above lines with hope of superseding the current state-of-start in electrodynamic simulations of metamaterials and photonic crystals. The generalization can be realized by the combination of the DDA and the LMS method within the framework of the point-matching method. According to this (see Fig. 1), the first step is the application of the DDA, wherein the actual scatterer [Fig. 1(a)] is substituted by a finite grid of interacting point dipoles [Fig. 1(b)], which is illuminated by an externally incident EM field. Solution of the corresponding DDA linear system of equations provides the dipole moments at each grid point, as well as the scattered electric field at a given point in space. The scattered electric field calculated by DDA is matched to that written in terms of vector spherical harmonics. The matching of the fields obtained by both methods takes place at points around the spherical scatterer, e.g., on a spherical surface surrounding the scatterer [Fig. 1(c)]. Having ensured the convergence in the calculation of the T-matrix, the latter is embedded into the LMS programming code [Fig. 1(d)], which provides the EM response of the metamaterial containing the scatterers in question [Fig. 1(e)]. B. Discrete-Dipole Approximation Next, we study the optical response of a generally anisotropic scatterer of arbitrary shape using the DDA [10,20 22]. The scattering object is considered as an array of point dipoles (i 1; ;N), each of which is located at the position r i and corresponds to a dipole moment P i and a (positiondependent) polarizability tensor ~α i. The above quantities are connected by where E i is the electric field at ith dipole, P i ~α i E i ; (1) E i E 0 i X j i A ij P j ; (2) which is the sum of the directly incident field E 0 i as well as the field scattered by all the other dipoles j i and it is incident on the ith dipole [second term of Eq. (2)]. The interaction matrix A ij is given from A ij expikr ij r ij k 2 ˆr ij ˆr ij 1 3 ikr ij 1 r 2 ij 3ˆr ij ˆr ij 1 3 ; i j; where 1 3 is the 3 3 unit matrix, r ij r i r j, ˆr ij r ij jr ij j. By combining Eqs. (1) (3), we obtain a linear system of equations, i.e., X N j1 (3) A ij P j E 0 i ; (4) where the diagonal elements of the interaction matrix are essentially the inverse of the polarizability tensor of each dipole, i.e., A ii ~α i 1 : (5) For an anisotropic sphere characterized by a dielectric tensor ~ϵ s and is immersed within an isotropic host of dielectric constant ϵ h, the polarizability tensor of the sphere is given by the Clausius Mossoti formula for anisotropic spheres; that is, Fig. 1. Modeling steps in the hybrid DDA/LMS method. ~α i V s 3ϵ h 4π ~ϵ s ϵ h 1 3 ~ϵ s 2ϵ h : (6) Equation (4) is preferentially solved by conjugate-gradienttype solvers for fast convergence [30]. Having determined the dipole moment P i at each point dipole, one can calculate

3 V. Yannopapas Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B 633 quantities such as the scattering, extinction, and absorption cross sections [10,20 22]. The scattered field E r p at a given point r p in space is the sum of the (secondary) field emitted by each dipole into which the actual scatterer is discretized, i.e., E r p X i A pi P i ; (7) where A pi is provided by Eq. (3) and the point in space does not coincide with one of the dipoles (r p r i ). The polarization vectors P i at each point dipole are provided by Eq. (4). C. Multipole Expansion of the EM Field Let us consider a harmonic EM wave, of angular frequency ω, which is described by its electric-field component: ~Er;tReEr exp iωt: (8) In a homogeneous medium characterized by a dielectric function ϵωϵ 0 and a magnetic permeability μωμ 0, where ϵ 0, μ 0 are the electric permittivity and magnetic permeability of vacuum, Maxwell equations imply that Er satisfies a vector Helmholtz equation, subject to the condition E 0, with p a wave number q ω c, where c 1 p μϵμ 0 ϵ 0 c 0 μϵ is the velocity of light in the medium. The spherical-wave expansion of Er is given by [31] Er X X l l1 m l a H lm f lqrx lm ˆra E i lm q f lqrx lm ˆr ; where a P lm (P E, H) are coefficients to be determined. X lm ˆr are the so-called vector spherical harmonics [31] and f l may be any linear combination of the spherical Bessel function, j l, and the spherical Hankel function, h l. The corresponding magnetic induction, Br, can be readily obtained from Er;t using Maxwell s equations, p ϵμ Br c 0 X X l l1 m l (9) a E lm f lqrx lm ˆr a H i lm q f lqrx lm ˆr ; and we shall now write it down explicitly in what follows. (10) D. Scattering by a Single Scatterer In this subsection, we present a brief summary of the solution to the problem of EM scattering from a single scatterer. We will make use of the compact notation of Ref. [18] for the eigenfunctions and the angular-momentum indices, which allow for easier computer coding. We consider a sphere of radius S, with its center at the origin of coordinates, and assume that its electric permittivity ϵ s and/or magnetic permeability μ s are different from those, ϵ h. μ h of the surrounding homogeneous medium. An EM plane wave incident on this scatterer is described, respectively, by Eq. (9) with f l j l (since the plane wave is finite everywhere), and appropriate coefficients a 0 L, where L denotes collectively the indices Plm. That is, where E 0 r X a 0 L J Lr; (11) L J Elm r i j q l q h rx lm ˆr; J Hlm r j l q h rx lm ˆr; h (12) p and q h ϵ h μ h ω c 0. The coefficients a 0 L depend on the amplitude, polarization and propagation direction of the incident EM plane wave [11 13,31]. Similarly, the wave that is scattered from the sphere is described by Eq. (9) with f l h l, which has the asymptotic form appropriate to an outgoing spherical wave: h l i l expiq h r iq h r as r, and appropriate expansion coefficients a L. Namely, where E r X a L H Lr; (13) L H Elm r i h l q q h rx lm ˆr; h H Hlm r h l q h rx lm ˆr: (14) By applying the requirement that the tangential components of E and H be continuous at the surface of the scatterer, we obtain a relation between the expansion coefficients of the incident and the scattered field, as follows: a L X T LL 0a 0 L0; (15) L 0 where T LL 0 are the elements of the so-called scattering transition T-matrix [32,33]. Eq. (15) is valid for any shape of scatterer; for spherically symmetric scatterers each spherical wave scatters independently of all others, which leads to a transition T-matrix that does not depend on m and is diagonal in l, i.e., T LL 0 T L δ LL 0 [33]. E. Calculation of the T-matrix via Point-Matching The point-matching (PM) method for calculating the T-matrix of general scatterers was first proposed in Ref. [34]. Here, we restate the DDA-based PM method in the spirit of the LMS formalism [18]. Note in passing that a PM method for the calculation of the T-matrix has also been presented in conjunction with the FDTD method [35]. Moreover, the PM method has been employed for obtaining the T-matrix and the multipole moments for a collection of spherical scatterers [36]. We assume that a single spherical EM wave is incident on an arbitrary scatterer, i.e., E 0 r J L0 r. This means that, in Eq. (15), we set a 0 L δ LL 0 which becomes a L T LL 0 : (16) Eq. (13) then becomes X T LL0 H L r p E r p : (17) L

4 634 J. Opt. Soc. Am. B / Vol. 31, No. 3 / March 2014 V. Yannopapas The matrix elements of the L 0 -column of the T-matrix can be calculated from the above equation, provided that we know the scattered field E at a sufficient number of points r p in space. Equation (17) must be solved for different incident spherical waves J L0 to obtain all the T-matrix columns. Namely, we calculate the scattered field via DDA [Eq. (7)] at several points on a spherical surface surrounding the scatterer, so that Eq. (17) becomes a linear system of equations for the T-matrix elements. However, in practice, we calculate the scattered field E via DDA at a large number of points, so that the unknowns T LL0 are fewer than the system equations. In this case, Eq. (17) is solved by seeking a least-squares solution. The scattering/extinction and absorption cross sections of light scattered off a single scatterer can be calculated in a spherical wave expansion by use of the obtained T-matrix [see Eq. (15) of Ref. 14]. F. Hybrid DDA/LMS Method Having calculated the T-matrix via Eq. (17), we insert it within the existing LMS formalism and computer code (see Fig. 2) which allows us to perform light-transmission or band structure calculations for 2D and 3D periodic arrays (photonic and plasmonic crystals, metamaterials, etc.) of scatterers described by the given T-matrix. In case where the 2D unit cell consists of more than one scatterer, e.g., a cluster of scatterers, one can first calculate the total scattering T-matrix of all the scatterers within the unit cell via real-space multiplescattering and then insert the resulting T-matrix within the LMS code, in the manner of Ref. [18]. The latter possibility renders the hybrid DDA/LMS method a multiscale EM solver, as it allows for the study of hierarchically organized photonic structures, where more than one-length scales are involved, e.g., the period of the photonic structure and the characteristic inter-particle distance within the cluster of scatterers. 3. TEST EXAMPLE We consider a 2D square array with lattice constant a 150 nm consisting of 100 nm gold nanocubes. The dielectric function of gold is taken from experiment [37]. Figure 3 shows the cross sections for linearly polarized light incident normally on a nanocube face. The curves are obtained by discretizing the cubes into dipoles while the cutoff in the angular momentum expansion is taken l max 7. We note that the cross sections already converge for a lower cutoff (l max 4); however, for the periodic photonic structures of Fig. 2. Flowchart of the hybrid DDA/LMS method. Fig. 3. Extinction (solid), scattering (dotted), and absorption (dashed line) cross sections (in arbitrary units) for light incident normally on one of the faces of gold nanocube with 100 nm edge. The curves are calculated by assuming 512 dipoles in the DDA and l max 7 in the angular momentum expansion. The inset shows the relative error in the scattering spectrum when assuming angular momentum cutoffs l max 6 and l max 7 (black line), as well as the DDA convergence, when assuming and point dipoles within the nanocube (gray line). gold nanocubes studied right below, a higher angular momentum cutoff (l max 7) is needed to achieve convergence [11 13]. The computer time needed for the calculation of the cross sections depicted in Fig. 3 was 9.25 h for 101 frequency points on a single core i7-3930k 3.2 GHz while it required about 620 MB of RAM. We note here that a computer time per frequency is meaningless in our case since the iterations needed for solving the DDA Eq. (4) via a conjugategradient-type solver [30] depends critically on frequency [26]. The evident peak in all three curves is attributed to the excitation of the surface plasmon resonance at 540 nm (when defined from the absorption cross section). In the inset of Fig. 3 we show the convergence (spectrum of relative error) of the scattering cross section in terms of the angular momentum expansion (black line) and of the number of dipoles (gray line) into which the nanocube is discretized in the DDA. In Fig. 4 we show the spectra of light transmittance/ reflectance/absorbance from a 2D square array of 100 nm gold nanocubes (those of Fig. 3) with lattice constant a 150 nm. In both Figs. 3 and 4, convergence is achieved for l max 7 in the EM spherical wave expansion and for 37 reciprocal-lattice vectors in the EM plane wave expansion [11 13] for Fig. 4. For these parameters, the relative error in the transmittance averaged over the studied spectrum ( nm) is 1.3%, in the reflectance curve 0.3%, and in the absorbance curve 0.8%. The computer time needed for the calculation of the transmittance/reflectance/absorbance spectra depicted in Fig. 4 was about 1.3 s per frequency on a single core i7-3930k 3.2 GHz while it required around 400 MB of RAM. As the only quantity that can be compared with the case of a single nanocube (Fig. 3) is absorbance, we observe a strong redshift of the surface plasmon peak, now located at 576 nm, which is a result of the interaction of the surface plasmons among the nanocubes of the square lattice [38]. Finally, Fig. 5 studies the case of a 3D plasmonic lattice; namely, a simple cubic lattice of 100 nm gold nanocubes with

5 V. Yannopapas Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B 635 Fig. 4. Transmittance, reflectance, and absorbance for light incident normally on an infinitely periodic 2D square array of 100 nm gold nanocubes with lattice constant a 150 nm. lattice constant of 150 nm, viewed as a succession of layers parallel to the (001) crystallographic plane. Figure 5(a) shows the dispersion curves (band structure) for an infinite crystal and Figs. 5(b) and 5(c) show the reflectance and absorbance spectra from finite slabs of the crystal, respectively. Figure 5(a) depicts both the real Rk z and imaginary Ik z parts of the wavevector k z, i.e., normal to the (001) crystallographic plane; from Rk z and Ik z, the effective-medium properties of the structure can be obtained. In Figs. 5(b) and 5(c), we only depict plasmonic-crystal slabs up to 4 layers as both the reflectance and absorbance spectra saturate very fast with increasing slab thickness. This means that adding more than 4 layers of the crystal does not practically change the depicted spectra. From applications point of view, even a single layer of a lattice of nanocubes may serve as an efficient broadband light absorber thanks to the wide absorbance plateau from 300 to about 500 nm. In the case studies presented here, the nanocubes are separated from each other by 50 nm. As the inter-cube distance, Fig. 5. (a) Dispersion curves along the [001] crystallographic direction of simple cubic crystal of 100 nm gold nanocubes with lattice constant a 150 nm. The solid (broken) line depicts the dispersion of the real (imaginary) part of k z. (b) Reflectance and (c) absorbance of light incident normally on finite slabs of various thicknesses (number of layers) of the simple cubic crystal of (a). i.e., the lattice constant of the square lattice, becomes smaller and the nanocubes approach each other, the number of terms needed in the spherical-wave expansion of the EM field increases rapidly. This is caused by the unnatural choice of a local dielectric function for metallic scatterers when the latter come very close to each other. When a nonlocal dielectric function is taken into account this effect is mitigated and an angular-momentum cutoff l max 15 is sufficient for converged results in the cases under study. However, the DDA/ LMS programming code scales linearly with l max as the time-consuming part is the solution of DDA system, i.e., Eq. (4). As stated above, Eq. (4) must be solved for different spherical waves whose number scales as (2l max 1). So, small inter-particle distances within a single plane of particles does not prohibitively increase the computer time as the corresponding programming code of the hybrid DDA/LMS method scales linearly with l max. A last point on the convergence of the presented method. Since the DDA method is ideally suited for scattering targets of small to medium size compared with the operating wavelength, so does the hybrid DDA/LMS method. For scatterers of large size, time-domain methods (FDTD and PSTD) are superior [39,40] in terms of computer time and memory suggesting that a hybrid FDTD/LMS [41] or PSTD/LMS code is more favorable in this case. 4. CONCLUSION We have presented a novel electromagnetic solver, namely, a hybrid discrete-dipole approximation/layermultiple-scattering method that can treat photonic structures consisting scatterers, which, in general, may be nonspherical, inhomogeneous, and/or anisotropic. 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