Interacting plasmonic nanostructures beyond the quasi-static limit : a “ circuit ” model

The interaction between individual plasmonic nanoparticles plays a crucial role in tuning and shaping the surface plasmon resonances of a composite structure. Here, we demonstrate that the detailed character of the coupling between plasmonic structures can be captured by a modified “circuit” model. This approach is generally applicable and, as an example here, is applied to a dolmen-like nanostructure consisting of a vertically placed gold monomer slab and two horizontally placed dimer slabs. By utilizing the full-wave eigenmode expansion method (EEM), we extract the eigenmodes and eigenvalues for these constituting elements and reduce their electromagnetic interaction to the structures’ mode interactions. Using the reaction concept, we further summarize the mode interactions within a “coupling” matrix. When the driving voltage source imposed by the incident light is identified, an equivalent circuit model can be constructed. Within this model, hybridization of the plasmonic modes in the constituting nanostructure elements is discussed. The proposed circuit model allows the reuse of powerful circuit analysis techniques in the context of plasmonic structures. As an example, we derive an equivalent of Thévenin’s theorem in circuit theory for nanostructures. Applying the equivalent Thévenin’s theorem, the well-known Fano resonance is easily explained. ©2013 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (310.6628) Subwavelength structures, nanostructures; (310.6805) Theory and design. References and links 1. K. Aslan, J. R. Lakowicz, and C. D. Geddes, “Plasmon light scattering in biology and medicine: new sensing approaches, visions and perspectives,” Curr. Opin. Chem. Biol. 9(5), 538–544 (2005). 2. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). 3. L. R. 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Introduction
Localized surface plasmon resonances (LSPR) in nanoparticles have found many applications spanning from the realm of life sciences, including biomedicine, bio-sensing, cancer treatment [1][2][3][4], to the conventional research domain of electrical engineering, such as the design of nanocircuits, nanofilters and nanoantennas [5][6][7][8].Modern nanofabrication techniques have allowed plasmonic properties to be a explored in a variety of topologies, such as rods [9,10], stars [11,12], split-rings [13], and rings [14,15], to name a few.With the nanoparticles closely spaced, the surface plasmons, i.e. the collective oscillations of conduction electrons, start to interact and, as a result, many exotic line features can be observed in the spectrum of the composite nanostructure, such as the splitting of the resonant frequencies [16] and spectral transparency known as plasmon induced transparency [17].In order to predict the effects of mutual coupling between nanoparticles, several methods have been proposed on the ground of modal analysis (or Eigenmode Analysis).For example, by assuming the size of constituent nanoparticles much smaller than the wavelengths of the incident light, a dipole-interaction model is conceived and applied to the spectral splitting of nanoparticle pairs [18].In the same spirit, the coupling between small nanoparticles is integrated into an optical lumped nanocircuit theory [19].By considering the influence of higher order resonances, the coupling between surface plasmon modes is modeled by hybridization theory, similar to molecular hybridization theory where the atomic orbitals mix and form new molecular orbitals.Further, based on the nanoparticles' electrostatic resonances [20], a methodology of systematically designing and analyzing the optical properties of an ensemble of nanoparticles is presented in [21,22].However, all these models are constructed within the quasi-static limit.It is known that for nanoparticles with larger dimensions, the retardation effects, i.e. the phase changes of the excitation field over the nanoparticle volume, cannot be neglected and a rigorous electrodynamic model is required.With the electromagnetic response of a particle expressed by a set of eigenmodes of scattering fields and internal fields, such size effects can be captured by the Mie theory for spheres [23] and the geometrical Mie theory for arbitrary particles without sharp edges [24,25].
In this work, in contrast to pursue the field eigenmodes which span the whole space, we especially focus on the current eigenmodes supported by arbitrary nanoparticles.We perform a full-wave Eigenmode Expansion Method [26] implemented in the framework of a Volumetric Method of Moments (V-MoM) Solution [27][28][29][30].Utilizing a dolmen-like nanostructure (dimensions, materials and depth profile are shown in Fig. 1), we illustrate that in contrast to the electrostatic modal analysis [20] where the material contribution is the only factor affecting the resonances, in a full-wave eigenmode analysis the radiation, as the result of retardation effects, comes into the picture and plays a crucial role in determining the resonances of surface plasmon modes.Further, we extract the eigenmodes and eigenvalues of the dolmen's constituent structures: a monomer (single vertical gold slab with dimensions W1, L1 as shown in Fig. 1) and dimer (two parallel, horizontally aligned gold slabs with dimensions W2, L2 as shown in Fig. 1), and reduce the electromagnetic interaction between the monomer and the dimer to their eigenmodes' interaction.By making use of the reaction concept [31], the eigenmodes' interaction is simplified to an equivalent circuit model, based on which the hybridization of surface plasmon modes is discussed.Last but not least, the proposed circuit model enables the reuse of powerful circuit analysis techniques in the context of plasmonic applications.As an example, we develop an equivalent of Thevenin's theorem  [32] for composite nanostructures and use it to explain the well-known Fano interference [33][34][35][36][37][38][39] in the dolmen structure.

The eigenvalue problem for a nanoscatterer
In this section, the eigenvalue problem is formulated for a general electromagnetic scattering problem and specifically illustrated for monomer, dimer and dolmen structures.The eigenvalues and eigenmodes of the dolmen structure are numerically extracted and their relation to the plasmonic response is discussed.Throughout the rest of this work, the time dependency i t e ω − is considered.The angular frequency of the oscillation is denoted by ω .
Note that the substrate effect is always considered in the analysis.Consider the interaction of electromagnetic waves with the dolmen nanostructure sitting on top of a glass substrate.At all space points, the total electric field In Eq. ( 1), the incident field ( ) ,ω inc E r is the unperturbed field, i.e. the sum of the incoming plane wave and the reflected plane wave without the presence of nanostructures.
Specifically, in the monomer volume (V M ), the total field ( ) ,ω tot E r , due to the volume equivalence principle, is related with the total electric current ( ) ,ω M J r (including both conduction and displacement current), The scattered field In Eqs. , ;ω MD G r r' take the substrate effects into account and describe the self-coupling of the monomer and the mutualcoupling between the dimer and the monomer.The integration in the first term on the right hand side of Eq. ( 3) is carried out over the monomer nanoparticle (V M ), while in the second term this integration is over the volume of the dimer nanoparticle (V D ) excluding the air gap in between.
The functional relations MM C , DD C , MD C and DM C are defined as, For the functional relations MM C and DD C , the associated eigenvalue problems can be found, The subscripts of ( ) λ ω in Eq. ( 12) ( ( ) λ ω in Eq. ( 13)) denote the order of an eigenmode and its corresponding eigenvalue, while their superscripts represent which structure, e.g. the monomer or the dimer, the eigenmode and eigenvalue Moreover, when the volumes of the monomer (V M ) and dimer nanoparticles (V D ), i.e. the dolmen structure (V), are considered as a whole, Eqs. ( 6) and ( 7) can be summarized in a single equation, The functional relation in Eq. ( 15) is defined over the volume of dolmen V, ( Similar to Eqs. ( 12) and ( 13), the associated eigenvalue problem can be defined and the eigenmodes form a complex orthogonal set, , As in [26], we assume that the eigenmodes ( ) ,ω n J r in Eq. ( 17) form the basis for the Hilbert space.Thus, the solution to Eq. ( 15) can be represented by the linear combination of eigenmodes, ( . Since the operators defined in Eqs. ( 8) and ( 9) are not Hermitian but complex symmetric operators, the spectral theorem [40] does not apply.That is, there is no guarantee that solutions to the eigenvalue problems in Eqs. ( 12) and ( 13) always exist.Moreover, even in the case that Eqs. ( 12) and ( 13) do exist, in general, the root system [40,41] forms the basis, i.e. the complete set which spans the domain of the integral operator in Eqs. ( 6) and (7), rather than the complex orthonormal set in Eq. ( 14).Therefore, within the framework of operator theory there could be a problem with mathematical rigorousness.However, in practice the integral operators with infinite dimensions in Eqs. ( 8) and ( 9) are always reduced to finite dimensions by, e.g., the Method of Moments algorithm [42], whereby an N x N matrix is obtained.Under such circumstances, if this matrix is diagonalizable and has distinct eigenvalues, an associated linearly independent set spanning the vector space can be found.Specifics on the mathematical background of this method can be found in [40] and [41].By using the complex orthonormal relation in Eq. ( 17), the coupling coefficient ( ) In order to illustrate how the eigenmodes and eigenvalues determine the response of the dolmen structure, we extract its first five eigenmodes (indicated with Ln), the corresponding eigenvalues, and their coupling to polarized incident light (See Fig. 2).In contrast to the L2 and L4 modes, the charges of the L1, L3 and L5 modes are mainly separated in the Y direction (see surface charge density distributions in inset Figs.2(a) and 2(e)).Thus, the L1, L3 and L5 modes are better coupled with Y polarized light (See Figs.2(c) and 2(g)).As found in [43], an eigenvalues' imaginary part is related with the difference between the power stored in electric field and magnetic field generated by the corresponding eigenmode.Similar to an RLC circuit, the resonance of an eigenmode is reached whenever the electrically stored power balances the magnetically stored one.Thus, at the wavelength where the imaginary part of their eigenvalue crosses zero, the coupling of the L1, L3, and L5 modes to Y polarized light tends to approach a maximum (See the grey dashed vertical lines in Figs.2(b) and 2(c)).The resonances of the L1, L3 and L5 modes are determined by further considering the effects of the eigenvalues' real part (as denoted by the arrows in Fig. 2(c)): the larger the eigenvalues' real part is, the more the resonance shifts from the wavelength where the imaginary part of the eigenvalue crosses zero.Note that this real part is related to the power lost via material dissipation and radiation by the eigenmodes' scattered fields [43].Since the plasmonic material constituting the dolmen is always the same (gold), the difference in the eigenvalues' real part is attributed to the radiative loss.Consequently, from comparing the real part of the eigenvalues of L2 and L4 for example, we can conclude that the L2 mode is more radiative than the L4 mode.The same can be concluded for L1, L3 and L5, where now L5 is the most radiative one, L1 is less radiative, and L3 is the least radiative.This can also be observed from the experimental extinction spectra (Figs.2(d) and 2(h)).Since L3 and L4 are less radiative, i.e. their net dipolar moments are relatively small, they are weakly coupled with the incident field.As a result, their plasmonic responses are overshadowed by the response of the brighter modes.

The hybridization between monomer and dimer eigenmodes: a "circuit" model
To illustrate the interaction between the monomer and the dimer, a simple case is considered, where the current ( ) induced by the Y polarized incident light is taken as a linear combination of the monomer's dipolar mode (See the inset in the rightmost column of Fig. 3(a)) and the dimer's quadrupolar mode which consists of two out-of-phase oscillating dipoles (See the inset in the leftmost column of Fig. 3 , , ., In Eq. ( 20), to the induced current ( ) ,ω J r flowing in the dolmen structure.Substituting Eq. ( 20) into Eqs.( 6) and ( 7), we can find that ', ', , , , ', ', , , .
Since the impedance matrix of a T-Network [32] is a 2 by 2 matrix such as the matrix on the left hand side of Eq. ( 23), we can conceive an equivalent circuit model (See Fig. 3(c)) for the interaction between the monomer's dipolar mode and the dimer's quadrupolar mode.The electromagnetic effects (e.g.self-coupling, mutual coupling, etc.) of the monomer's and dimer's fundamental modes are summarized within two coupled circuit loops, i.e. the "monomer" loop and the "dimer" loop.
The self-coupling of an eigenmode, i.e. the reaction [31] of a current eigenmode to the electric fields generated by itself, is described by ( ) MM c ω for the "monomer" loop and by ( ) DD c ω for the "dimer" loop (See the left and right graphs in Fig. 3(a)).Noticing the eigenvalue problems defined in Eqs. ( 12) and ( 13) and the orthogonality of eigenmodes in Eq. ( 14), we can find that   Eq. ( 16), we can discretize the monomer (V M ) and the dimer (V D ) by hexahedral blocks and associate two adjacent blocks with a presumed local current distribution f(r) of unit amplitude, called basis function.By further applying a testing procedure, we can recast, for example, Eqs. ( 10) and ( 11) into a matrix form [29], , , ' , ' , .
In Eqs. ( 28) and ( 29), g(r) is called testing function.The subscripts denote which nanostructure (monomer or dimer) the basis and testing functions belong to, while the superscripts emphasize the order of the basis and testing function.When the testing functions are identical with the basis functions, the testing procedure is called "Galerkin" testing [42].Though Galerkin testing is accurate and preserves the reciprocal symmetry of Maxwell's equations, it requires eight-fold integrals (a six-fold integral can be seen in Eqs. ( 28) and ( 29) and the extra two-fold comes from the evaluation of the Green's function).To ease the computational efforts, instead we pursue the Galerkin testing procedure in the z direction and a well-known simpler razor-blade testing procedure (which saves a two-fold integration) in the horizontal plane [29].As a consequence, the testing functions are different from the basis functions and therefore the symmetry of the resultant matrices is weakly disturbed, finally leading to the noticeable discrepancy in Fig. 3 Since the matrix on the left hand side of ( 23) describes the coupling between eigenmodes, we call this matrix hereafter the "coupling matrix".Extra attention ought to be paid to the fact that although the quantities defined in Eqs. ( 24)-( 27) have the units of Ohms, they do not conform to the conventional definition of impedance in circuit theory, since Eqs.( 24)-( 27) are obtained by applying the definition of reaction [31] rather than power [44].
On the right hand of Eq. ( 23), the coupling between the Y polarized incident field and the current eigenmodes is modeled as the "voltage" sources driving the loops, ( , .
Note that since the net dipolar moment of the quadrupole mode is zero, it is decoupled from the normally incident Y polarized light and the equivalent source driving the "dimer" loop is shorted-circuited, i.e.
By solving Eq. ( 32), the eigenvectors and eigenvalues of the whole structure can be obtained.Remembering Eq. ( 20), the components of each eigenvector, i.e.
( ) ( ) j ω , are the weights of the contributions from the eigenmodes of the constituent monomer and dimer structures to the eigenmode of the whole dolmen structure.Here, the monomer's dipolar mode and the dimer's quadrupolar mode hybridize into the bonding and anti-bonding modes.In the circuit model, the bonding mode indicates the currents in the monomer's loop and the dimer's loop flowing against each other; the anti-bonding mode is equivalent to the currents in the monomer's loop and the dimer's loop flowing along with each other.
The eigenvalues of the hybridized modes can be found as (See the middle graph in Fig. 3(a)), (  33) provides a good estimation for the eigenvalues of the L1 and L3 dolmen modes.Furthermore, the resonant wavelengths of the hybridized modes are found at the wavelengths where the imaginary part of their eigenvalues crosses zero, around 850 nm for the bonding mode and around 700 nm for the anti-bonding mode.It can be seen from Fig. 3(d) that at these wavelengths the whole equivalent circuit is at resonance. .
The input impedance to the dimer port ( ) D c ω can be derived by calculating the ratio of the voltage across the dimer port to the imposed current ( ) .
The "monomer" loop's current ( ) , is the sum of the currents introduced by two independent sources, .
It can be seen from Figs. 4(a3) and 4(b3) that around 800 nm then ( ) ω ω − due to the mutual coupling from the "dimer" loop.As a result, the voltage drop across the "monomer" loop is approximately zero, the net current minimizes, correspondingly yielding a Fano dip in Fig. 3(d).
By substituting Eq. ( 41) into the second row of Eq. ( 23), we can recover the Thévenin equivalent of the circuit, Physically, Thévenin's theorem selects the dimer's quadrupolar mode as a "port" interfacing the system, which is composed of the dolmen structure and the incident wave, to the "outside" world.The tuning of the dimer's quadupolar mode by, for example, loading the dimer with a different plasmonic material which is equivalent to add an inductive load at the dimer port in the circuit model, can be conducted without considering the detailed electromagnetic properties inside the "port".
The current flowing in the "dimer" loop Combining Eqs. ( 36), (37), and (40) with Eq. ( 42), Eq. ( 42) can be simplified as,   (2) 1 D j ω due to the excitation of the quadrupolar mode are plotted in (b3).As in Fig. 3, the monomer loop and its associated quantities are always denoted by the cyan color, while the dimer loop is specified by the pink color.

Conclusion
In summary, we propose an approach to characterize the interaction between plasmonic nanostructures by an equivalent circuit model.The eigenvalue problem for a composite nanoscatterer is first illustrated with a dolmen structure.The effects of the eigenmodes and eigenvalues on the plasmonic response are discussed.By further examining the eigenmodes of the dolmen's composite structures, the monomer and dimer, we boil down the interaction between plasmonic nanostructures to the interaction between their eigenmodes.As an example, the interaction of the monomer's dipolar mode with the dimer's quadrupolar mode is emphasized and summarized in a coupling matrix.Furthermore, an equivalent circuit model is conceived, based on which the plasmonic mode hybridization is elaborated.Last but not least, an equivalent of Thevenin's theorem in circuit theory for plasmonic nanostructures is derived.From this perspective, the well-known Fano dip in the dolmen's spectral response is explained.
It is worth noticing that although the discussion is confined here to the context of two linearly coupled circuits, i.e. a 2 x 2 network, the proposed circuit model can be readily generalized to an N x N network to include the electromagnetic interactions between multiple nanostructures supporting multiple eigenmodes.In this way, the proposed model allows the reuse of many well-established circuit analysis techniques [32] and therefore promises a substantial simplification in future analyses and design of coupled nanoscale plasmonic systems.

(
function of the material occupying the space volume of the monomer V M .The tensor Green's functions belong to.Due to the reciprocity of the Green's function, both MM C and DD C are complex #199641 -$15.00USD Received 17 Oct 2013; revised 27 Nov 2013; accepted 27 Nov 2013; published 10 Dec 2013 (C) 2013 OSA symmetric (non-Hermitian) operators.Their eigenfunctions a complex orthonormal set,

Fig. 2 .
Fig. 2. Eigenmodes, eigenvalues and plasmonic response of the composite dolmen structure.The first five eigenmodes are extracted for the dolmen structure and their charge density distribution is shown in the insets of (a) for the L1, L3, and L5 modes and (e) for the L2 and L4 modes (blue and red indicate negative and positive charge, resp.).The real (solid) and the imaginary (dash) parts of their eigenvalues are presented in (a,e) and (b,f), resp.The coupling coefficients are shown in (c) and (g).The measured extinction cross sections of the dolmen structure are shown in (d) and (h) for Y and X polarized incident light, resp.
actually the eigenvalues of the monomer's dipolar mode and the dimer's quadrupolar mode, (

ω
describe the mutual coupling, i.e. the reaction[31] of a current eigenmode to the electric fields generated by another current eigenmode, between the monomer's and the dimer's fundamental modes (See Fig.3(b)),

27 )
Due to the reciprocity of Maxwell's equations, is worth noticing that in order to numerically evaluate the functional relations in Eqs.(8)-(11) and in #199641 -$15.00USD Received 17 Oct 2013; revised 27 Nov 2013; accepted 27 Nov 2013; published 10 Dec 2013 (C) 2013 OSA (b) between the imaginary parts of ( ) eigenvalues derived from Eq. (33) (see red and green curves in the middle column of Fig.3(a)) with the eigenvalues directly retrieved from Eq. (17) (see red and green curves in Figs.2(a) and 2(b) and the light grey curves in the middle column of Fig.3(a)), we can summarize that Eq. (

Fig. 3 .
Fig. 3.The equivalent circuit model of the dolmen structure excited by Y-polarized incident light.(a) shows the real (solid) and imaginary (dashed) parts of the self-coupling of the monomer's dipolar mode (cyan, left column), the dolmen's L1 and L3 modes (red and green, resp., middle column), and the dimer's quadrupolar mode (pink, right column).The eigenvalues of the L1 and L3 modes directly extracted as in Eq. (17) (the red and green lines in Figs.2(a) and 2(b)) are shown as grey lines in (a).The charge density distributions of the dimer's quadrupolar mode, the monomer's dipolar mode, and their hybridized modes are shown in the insets of (a).(b) shows the real (solid) and imaginary (dashed) parts of the mutual coupling.Knowing the self-coupling and mutual coupling, as well as the driving voltage source, an equivalent circuit for the dolmen structure can be constructed as shown in (c).The Thévenin's equivalent of the equivalent circuit is also presented in (c).As a result, the current circulating in each loop can be found as in (d).The reference directions of the currents are shown in (c).

1 Me
is to say, in the equivalent circuit (Fig.3(c)) the source voltage imposed by the source field ( ) ω is balanced by the electromotive force ( Fig. 3(c), the Thévenin theorem "seals" the electrical circuit left to the dimer port into a "black box" which contains an open circuit voltage

( ) 1 Dj
ω is thus the ratio of the open circuit voltage ( ) D oce ω to the input impedance to the "dimer" port ( )

1 .ω
case where the resonances of the monomer's dipolar and the dimer's quadrupolar mode are degenerate, the imaginary parts of cross zero at the same wavelengths.As a result, the denominator of Eq. (43) reaches a minimum and accordingly the current in the dimer loop becomes maximum (See Fig.3(d) and the pink curves in Fig.4(b3)).

Fig. 4 .
Fig. 4. Illustration of Thévenin's theorem for coupled nanostructures.In (a1), only the monomer is present and excited by an incoming plane wave.The polarization and propagation direction of the incident plane wave are denoted by the blue and red arrows, resp.In (a2), the equivalent circuit model of (a1) is plotted with the open-circuit voltage explicitly defined.(a3) shows the corresponding the real (solid) and imaginary (dashed) parts of the currents circulating in the monomer's loop ( ) (1) 1 M j

1 D
(b1), the monomer structure is excited by the quadrupolar mode in the dimer structure with an amplitude of ( ) j ω .The equivalent circuit of (b1) is illustrated in (b2).The real (solid) and imaginary (dashed) parts of the currents in the monomer's loop( )