## Abstract

We give a geometrical theory of resonances in Maxwell’s equations that generalizes the Mie formulae for spheres to *all* scattering channels of any dielectric or metallic particle without sharp edges. We show that the electromagnetic response of a particle is given by a set of modes of internal and scattered fields that are coupled pairwise on the surface of the particle and reveal that resonances in nanoparticles and excess noise in macroscopic cavities have the same origin. We give examples of two types of optical resonances: those in which a single pair of internal and scattered modes become strongly aligned in the sense defined in this paper, and those resulting from constructive interference of many pairs of weakly aligned modes, an effect relevant for sensing. This approach calculates resonances for every significant mode of particles, demonstrating that modes can be either bright or dark depending on the incident field. Using this extra mode information we then outline how excitation can be optimized. Finally, we apply this theory to gold particles with shapes often used in experiments, demonstrating effects including a Fano-like resonance.

© 2011 OSA

## 1. Introduction

The interaction of light with wavelength sized particles has been intensely investigated for more than a century, continuing to provide interesting and surprising results. This coupling is essential in single molecule spectroscopy and single photon processes [1, 2], while interference between different scattering channels of particles produces classical analogues of quantum processes [3, 4], and carefully designed particles underpin the physical realization of metamaterials [5–7]. The basis all of these effects is that the particle-light interaction, which depends on the composition and shape of the particle and on the properties the incident light, can become very strong around resonances. For particles much larger than the wavelength of light, resonances are described by closed orbits of light rays [8] inside the particle. This geometric approach becomes less and less effective as the size of the particle decreases, eventually requiring the solution of Maxwell’s equations. Mie-type solutions [9, 10], based on symmetry and coordinate separability, provide analytical description of resonances for a few specific shapes of particle. For spheres, internal and scattered fields are expanded by electric and magnetic multipoles and each multipole in one field is coupled exclusively to the corresponding multipole in the other field. Resonances are independent of the incident field and occur when the coefficients of one pair of multipoles reaches a maximum for the particular values of the particle radius, permittivity and susceptibility.

For particles of arbitrary shape, resonances are always (to the best of our knowledge) defined implicitly by maxima in properties such as the far field extinction or scattering efficiency spectra. This is because, unlike the sphere, there is no decomposition of the internal and scattered fields into partial waves that are pairwise coupled. Hence monitoring of calculated specific properties of the scattered field are instead used empirically to define resonances. While there are several methods that can find spectra and their maxima [11], this approach to resonances is unsatisfying because it depends on the incident field; further it fails to recognize the consequences of resonances associated with fields which are strong only in one region of space, and finally requires an *a priori* choice of the property which being monitored to determine the resonance. For example a resonance associated with a strong surface field, that itself is not efficient at transporting energy to infinity, would not appear as an obvious feature in any far field efficiency spectra used to define resonance; nevertheless such resonances can be extremely important in near field applications or through interference with other channels which themselves are able to transport energy into the far field.

Here we introduce a hermitian operator to define field expansions and resonances for any particle, where Mie’s treatment of the sphere is a special case of this more general theory. We use the mathematical framework of the angles between subspaces of functions [12, 13] to reveal the geometrical nature of resonances in Maxwell’s equations. This allows us to predict resonances of particles independently of the particular incident field, obtain the near and far features of particular modes and choose incident fields to optimize their excitation.

## 2. Theory

The theory we develop applies to general metallic and dielectric particles without sharp edges in cases where the interaction between light and matter inside the particle is described by a local macroscopic permittivity and susceptibility. For metallic particles, this means that the free propagation length of carriers is smaller than the skin depth and all the characteristic lengths of the particle. The tangential components of electric and magnetic fields *E,H* are continuous on passing through the particle boundaries, and the energy scattered by a particle flows towards infinity. The interaction of the particle with an incident field is determined by finding appropriate solutions of the Maxwell’s equations in the internal and the external media that satisfy these boundary conditions. We use [14] six component vectors *F* = [*E*,*H*]* ^{T}* for electromagnetic fields. Their projections,

*f*, onto the boundary of the particle are surface fields each with four components, two electric and two magnetic, that form a space ℋ where scalar products are defined in terms of overlap integrals on the surface of the particle, $f\cdot g={\int}_{S}{f}_{j}^{*}{g}_{j}ds$, where the index

*j*labels the components,

*f*

^{*}is the complex conjugate of

*f*and we sum over repeated indexes. In this formalism the boundary conditions become

*f*

^{0}, of the incident field,

*F*

^{0}(

*x*), onto the surface is equal to the difference between the projections of the internal and scattered fields,

*f*and

^{i}*f*. This suggests that an incident field with small tangent components can excite large internal and scattered surface fields provided that these two fields closely match. This happens when the “angle” between these two fields, and therefore their difference, is small. The angles in question can be rigorously defined as the angles between standing and outgoing waves that are solutions of the Maxwell’s equations for the internal and external media respectively, and that form two subspaces of ℋ. For each particle, these angles and the associated waves characterize completely the particle’s electromagnetic response, which can be determined with arbitrary precision from

^{s}*any*complete set of solutions of the Maxwell equations for the internal and external media.

There exist several sets of exact solutions of the Maxwell equations that are linearly independent and complete [15, 16] on surfaces without sharp edges [17]. We choose two sets of electric and magnetic multipoles,
${\left\{{\tilde{i}}_{n}\right\}}_{n=1}^{\infty}$ for internal fields and
${\left\{{\tilde{s}}_{n}\right\}}_{n=1}^{\infty}$ for scattered fields, centered at different positions within the particle [19, 20]. Any function in ℋ can be approximated to arbitrary precision by a sufficiently large, but finite, number of multipoles [21]; that is,
${\left\{{\tilde{i}}_{n}\right\}}_{n=1}^{\infty}\cup {\left\{{\tilde{s}}_{n}\right\}}_{n=1}^{\infty}$ is complete and no function in this set is the closure of the linear combinations of all the remaining functions. We remark that both the internal and scattered fields exist in real metallic and dielectric particles and fulfill the boundary conditions in Eq. (1) for the electric and magnetic tangential components. For this reason the interaction of light with these particles is determined by surface fields *f* with four components, and completeness in the space ℋ of the surface fields *f* is provided by the union of internal and scattered fields and not by either the scattered or the internal field separately. This point is illustrated by the spherical particles considered in Mie theory, where both internal and scattered modes are necessary to form a complete basis.

One can show [21] that the coefficients of the internal and scattered fields, $\left\{{\tilde{a}}_{n}^{i},{\tilde{a}}_{n}^{s}\right\}$, that minimize the discrepancy between an incident field and the expansion of internal and scattered fields, $\left|{f}^{0}+{\sum}_{n=1}^{N}{\tilde{a}}_{n}^{s}{\tilde{s}}_{n}-{\tilde{a}}_{n}^{i}{\tilde{i}}_{n}\right|$, are the solutions of

The first step is to find orthogonal modes for the scattering and internal fields: for any number of multipoles, *N*, we achieve this through the matrix decomposition

*Q*,

^{i}*Q*are invertible matrices that can be found through SVD or QR decomposition [22] and

^{s}*U*,

^{i}*U*are unitary matrices whose columns are the orthogonal internal and scattering modes respectively. Scalar products between internal and scattering modes form a matrix with decomposition where

^{s}*C*a diagonal matrix with positive elements, and

*V*,

^{i}*V*are unitary matrices acting on the internal and scattered fields, respectively. These identities enable us to simplify the Gram matrix through the transformation

^{s}*U*and Σ =

^{i}V^{i}*U*are matrices whose columns are formed by the so called principal internal and scattering modes {

^{s}V^{s}*i*} and {

_{n}*s*}, which are one of the main tools in this theory.

_{n}**a**

*,*

^{i}**a**

*are the coefficients of the principal modes in the field’s expansion. The most important part of our theory is that, because matrix*

^{s}*C*is diagonal, principal modes are coupled pairwise, i.e., each mode is orthogonal to all but at most one function in the other space. This is the essential feature of the multipoles used in Mie’s theory for spheres. The positive diagonal elements of

*C*define the principal angles,

*ξ*, between

_{n}*s*and

_{n}*i*as follows The terms on the right-hand side of Eq. (8) are the principal cosines [12]: cos(

_{n}*ξ*) and sin (

_{n}*ξ*) are the statistical correlation [23] and the orthogonal distance between

_{n}*s*and

_{n}*i*.

_{n}From the general theory of angles between subspaces [13] and the definition above, the angles *ξ* are invariant under unitary transformation of the multipoles and they completely characterize the geometry of the subspaces of the internal and scattered solutions in ℋ. This geometry is induced by the particular scattering particle through the surface integrals of the scalar products; its relevance to scattering and resonances has not been previously realized. The importance of the principal cosines is twofold: Theoretically they provide analytic equations for the coefficients of the internal and scattered principal modes, generalizing the Mie formulae and clarifying the nature of all scattering channels of a particle. Numerically, they allow us to reduce large matrices to their sub-blocks and eliminate the need for numerical inversion to determination of the mode coefficients. For spherical particles, each pair of modes corresponds to a pair of electric or magnetic multipoles of Mie theory. For non-spherical particles, principal modes are instead combinations of different multipoles (although in some cases there can be dominant contributions from a specific multipole).

The interaction of particles with light can now be interpreted in terms of eigenvalues and orthogonal eigenvectors, ${w}_{n}^{\pm}=\left({i}_{n}\pm {s}_{n}\right)/\sqrt{2}$, of the hermitian operator in Eq. (7); providing interesting analogies between the electromagnetic response of a classical particles with the response of atoms or molecules. However, away from the surface one measures either internal or scattered fields, so we transform the eigenfunctions, { ${w}_{n}^{\pm}$}, to find the coefficients of the principal modes:

*i*′

*=*

_{n}*i*– cos(

_{n}*ξ*)

_{n}*s*,

_{n}*s*′

*=*

_{n}*s*– cos(

_{n}*ξ*)

_{n}*i*are bi-orthogonal to

_{n}*i*,

_{n}*s*(

_{n}*i*′

*·*

_{n}*s*=

_{n}*s*′

*·*

_{n}*i*= 0) with

_{n}*i*′

*·*

_{n}*i*=

_{n}*s*′

*·*

_{n}*s*= sin

_{n}^{2}(

*ξ*). Both the principal or the bi-orthogonal modes fully specify the response of the particle at any point outside and inside the particle. This is shown by recasting the expansions of internal and scattered field as

_{n}*G*(

_{S}*x*,

*s*) is the surface Green’s function [14,21] of the particle and 𝒯

*(*

^{i}*x*) (𝒯

*(*

^{s}*x*)) is 1 inside (outside) the particle and null elsewhere. In practice, because the principal modes are combinations of known solutions of the Maxwell’s equations, propagation of the fields away from the surface (

*I*(

*x*) and

*S*(

*x*)) is performed for Eq. (11) by evaluation of Bessel or Hankel functions and vector spherical harmonics (all at a very low computational cost). Eq. (11) shows that the convergence of principal modes and principal angles as

*N*→ ∞ is a consequence of the convergence of the surface Green’s function [21] for any complete set of solutions of the Maxwell equations. This convergence can be monitored by the surface residual $\left|{f}^{0}+{\sum}_{n=1}^{N}{a}_{n}^{s}{s}_{n}-{a}_{n}^{i}{i}_{n}\right|$, which provides an upper bound for the maximum error of scattered and internal fields that decreases with the distance from the surface [16, 24]. Furthermore, the form of Eq. (9) remains unchanged as

*N*→ ∞, even if

*θ*,

_{n}*i*,

_{n}*s*change [25].

_{n}The modal decomposition in Eq. (11) has several unique advantages. The left-hand terms in the scalar products depend exclusively on the particle, so this allows us to strongly compress the description of scattering by identifying the modes that are coupled to a given field and discarding the others, and also to optimize an incident field in order to excite a specific principal mode. We remark that
${a}_{n}^{i}$,
${a}_{n}^{s}$ in Eqs. (9) and (10) are found by projecting the incident field *f*^{0} onto non-orthogonal vectors, *i _{n}* and

*s*, while sin (

_{n}*ξ*) is defined as the Peterman factor [26] that gives the order of magnitude of transient gain and excess noise in unstable cavity modes. Therefore the presence of strongly aligned vectors with sin(

_{n}*ξ*) << 1 is at the origin of large surface fields in nanoparticles as well as large transient gain and excess noise [27] in macroscopic unstable cavities and dissipative systems governed by non-hermitian operators [28]. The physical origin of this correspondence is that both macroscopic cavities and scattering particles are open systems in which both internal and external modes are necessary to provide a full description of the interaction with the environment: reducing the theory to one set of modes implies loss of information. The mathematical counterpart to this is that the separate internal and scattered fields are not modes of a hermitian operator and their optimal excitation is provided by the corresponding bi-orthogonal modes, even in the case of non-absorbing particles. These bi-orthogonal modes are surface fields that are either totally reflected or totally absorbed. The physical realizations of these surface fields may be challenging, and will be studied elsewhere, but we can use them to find and optimize an incident field able to couple to the principal modes. For well aligned mode pairs (small

_{n}*ξ*), the coefficient

_{n}*a*,

^{i}*a*are of the same order, but this is not the case for weakly aligned pairs, which can have qualitatively different absorption and differential scattering cross sections (DSCS). Moreover, Eqs. (9) and (11) show that modes can be “dark” for

^{s}*specific*incident fields but couple well to other incident fields.

To fully characterize the interaction of particles with electromagnetic fields, we need also to evaluate the ability of principal modes to transport energy. This is determined by the integral of the Poynting vector of each mode over the surface of the particle,

*n̂*is the outward (inward) pointing normal to the surface for scattered (internal) modes and ${E}_{n}^{s/i}$, ${H}_{n}^{s/i}$ are the electric and magnetic components of the principal mode with $\left|{a}_{n}^{i/s}\right|=1$. Large values of this “intrinsic” mode flux, ${\Phi}_{n}^{s}$, corresponds to radiative modes that are very effective at transporting energy from the particle surface to infinity, while small values correspond to modes that are mainly confined near the surface and can only transport energy effectively close to the surface, but not into the far field region. Analogously, large (small) values of ${\Phi}_{n}^{i}$ correspond to strongly (weakly) absorbing modes.

^{s/i}We now show that this approach gives us an unambiguous way to generalize the definition of Mie resonances to any particle without sharp edges, while also being computationally efficient. Principal modes and scalar products are functions of the frequency dependent permittivity and susceptibility of the particle, and therefore the principal angles *ξ _{n}* change with the frequency of incident light. Internal and scattered coefficients diverge when the denominators of Eqs. (9) and (10) vanish. This happens when a pair of normalized internal and scattering modes are parallel. For a sphere the angular dependence of internal and scattered modes can be factored out and the condition

*i*=

_{n}*s*can be recast in terms of the amplitude of the electric and magnetic components giving the usual Mie resonance condition, which can be interpreted geometrically in terms of alignment between internal and scattered modes. For spheres, the condition

_{n}*i*=

_{n}*s*occurs at complex wavelengths; for real wavelengths, resonances correspond to minima of the principal angles. This is also generally true for any smooth particle because the linear independence and completeness of the principal modes makes perfect alignment impossible. So, as with spherical particles [29], actual resonances correspond to minimum angles (

_{n}*ξ*≠ 0) of pairs in ℋ, which are also minima of the eigenvalues of the hermitian operator in Eq. (7).

_{n}We also need to consider the energy transported by resonances. A resonant pair can have very different ${\Phi}_{n}^{s}$ and ${\Phi}_{n}^{i}$, which means that it can be strong or weak depending on whether it is observed in the near or far field, or if scattering or absorption are measured. Furthermore, one important difference between spherical and non-spherical structures is that the total flux of energy scattered or absorbed (integrals of the Pointing vectors over all directions) is given by the sum of principal mode contributions plus interference terms between modes, which are absent for spherical particles. This is due to the orthogonality of the multipoles used in Mie theory being equivalent to orthogonality of the modes’ fluxes. Hence for non-spherical particles, efficiencies can have strong peaks caused by constructive interference within a group of modes, as well as sharp asymmetric features resulting from Fano-like interference between broad and narrow resonances. In the following, we give examples of both of these phenomena. For all particles presented here we use a fitted dielectric function [30, 31] for gold.

## 3. Numerical validation

From the point of view of numerical calculations, this approach is a surface method that relies on the the fact that
${\sum}_{n=1}^{N}{a}_{n}^{i}{I}_{n}$,
${\sum}_{n=1}^{N}{a}_{n}^{s}{S}_{n}$ converge to the exact fields at any point inside and outside the particle [14, 21] as *N*→ ∞. We calculate
$\left|{f}^{0}+{\sum}_{n=1}^{N}{a}_{n}^{s}{s}_{n}-{a}_{n}^{i}{i}_{n}\right|$ to determine the numerical error in the evaluation of the surface fields; also the scattered power both at infinity and on the surface of the particle is evaluated to check that the error in the propagation, i.e. in the evaluation of the special functions, is negligible. Comparison with Mie theory [32] for metallic spheres is shown in Fig. 1 and demonstrates that this method is numerically very accurate for particles with radii between [10^{−2},4] times the wavelength of light. The timings for these calculations are shown in Table 1.

## 4. Resonances in gold nanorods and nanodiscs

We now consider light interaction with experimentally relevant gold particles in vacuum, without considering the possible presence of supporting substrates.

Figure 2 shows calculated near and far field properties of a rounded nanodisc (height 20 nm and diameter 120 nm), illuminated axially by plane-wave light [33]. We find a strong dipole resonance, see Fig. 2(a) and 2(b), at 613 nm: this is compatible with a previous experimental and numerical observation [34] of a dipole resonance around 700 nm for an array of these particles on a substrate.

Figure 3(a) shows the landscape of a group of principal mode angles as a function of the order of the mode’s principal cosines and wavelength of incident light. The modes discussed in this section do not change significantly when the number of sources is increased. The “height” of this landscape is sin^{−1}(*ξ*), i.e. the largest value of Eqs. (9) and (10) for |*f*^{0}| = 1; while the shading of the traces overlaid on top show, for each wavelength, the values of the intrinsic mode fluxes
${\Phi}_{n}^{s}$ normalized into the range [0, 1]. Figure 3(d) shows the same information as Fig. 3(a), but instead for
${\Phi}_{n}^{i}$. These two figures do not depend on the incident field and provide a visualization of the properties of the surface Green’s function and principal modes of this particle. We can see that there is only one mode pair with comparable intrinsic fluxes, Φ* ^{i}* and Φ

*, which has a resonance at 613 nm. Most of the other mode pairs are strongly absorbing and not able to effectively scatter energy away from the particle. However, one poorly aligned mode pair is capable of strongly radiating if excited, but very weakly absorbs. We also clearly identify a subset of absorbing pairs that become resonant at short wavelengths around 525 nm.*

^{s}Figures 3(b) and 3(e) show the same landscape, but with the amplitudes,
$\left|{a}_{n}^{i/s}\right|$, of the internal and scattered principal modes due to the *specific* axial incident field overlaid on top. This illustrates the *excitation paths* of the modes as the wavelength changes. We can see that the short-wavelength absorbing resonance around 525 nm is not effectively excited by the specific incident field used; similarly the weakly aligned mode pair which does not pass through a resonance in this range and is capable of strong scattering but weak absorption shows excitation, but only of its internal mode. In contrast the the internal and scattered amplitudes of the pair which reaches resonance at 613 nm is strongly excited by this field.

Figures 3(c) and 3(f) show the effective flux carried by the modes for the incident field used (
${\Phi}_{n}^{i/s}{\left|{a}^{i/s}\right|}^{2}$) again normalized to [0,1]. By comparing all of these figures, we see that the observed resonance is a Mie-like single principal cosine pair, which reaches maximum alignment (its smallest value of *ξ*) at the resonance. The internal field also contains a second more weakly aligned excited mode, where its counterpart in the scattering field is not excited. This mode does not absorb much energy on its own, as shown in Fig. 3(f), but affects the surface current, which can be found using these fields and the Ohm equation. The internal and scattered near field of the resonant pair, an electric dipole, is shown in Fig. 2(c). Other strongly aligned mode pairs are not excited, i.e. are dark: this is because they rapidly vary at the surface as in Fig. 2(d), so do not couple to the smoothly varying incident field. The appreciable asymmetry of the absorption and scattering efficiencies are explained by the asymmetry in the principal cosine of the resonant mode as a function of wavelength. For non axial incident light, the main peak in Fig. 2(a) becomes smaller, due to a weaker coupling with this resonant mode (with a corresponding decrease in the amplitudes of Figs. 3(b) and 3(e).

Figure 4(a) and 4(b) shows the calculated optical efficiencies and DSCS of a 480 nm long rod with diameter 40 nm, illuminated axially by plane-wave light. There is a strong resonance at 205 nm and a weaker absorption peak at ∼515 nm. The DSCS demonstrates that this particle strongly scatters this incident light forward, particularly at short wavelengths. Figures 5(a) and 5(d) show that most mode pairs are either strongly radiative or absorbing, except for one weakly aligned pair that is both absorbing and radiating. The peak at 205 nm is not a single mode, but instead the excitation amplitudes of the principal modes, Figs. 5(b) and 5(c), show it to be due to constructive interference between a group of several weakly aligned principal mode pairs.

Many of these mode pairs are important mainly in the near field, but collectively they very efficiently extract energy from the incident field. Such multimode resonances are potentially very useful for sensing applications because interference between different scattering channels leads to enhanced sensitivity to perturbations near to the scattering surface. The absorption feature at ∼515 nm is due to a small group of modes, two of which are weakly aligned, both absorbing comparable amounts of energy and becoming more strongly aligned at around this resonance. The scattering mode of the pair with the weakest alignment is instead dominant, as shown in Fig. 5(c), where the resonance at ∼515 nm is enhanced, but in absolute terms this resonance is barely observable in the far field scattering efficiency of Fig. 4(a). Figure 6 shows the same particle illuminated equatorially with an incident light polarization of 45° with respect to its long axis. Analyzing other particles with the same diameter but varying length, we find that the broad feature at around 200–450 nm, containing structures similar to the composite modes of Fig. 4, is insensitive to the particle length. It also shows no clear hot or cold spots, see Fig. 6(c). The sharp resonance at 676 nm shifts with rod length, and its surface field as shown in Fig. 6(d), has the strong nodal local structure of a “waveguide” mode on the long axis, remarkably similar to the experimental results of Ref. [35]. Figure 7 shows only mode pairs that cannot be excited by symmetry for axial incidence; the broad features at short wavelength are multimode resonances similar to the one discussed for axial incidence, and most of the corresponding modes are not shown as a result. Similar to the disc, in Fig. 7 most mode pairs are either absorbing or radiating and the three resonances at around 550–700 nm are all absorbing. The excitation paths in Figs. 6(b) and 6(e) show that only the best aligned of the three resonant pairs is excited for this particular field, this resonance is not visible for axial incident light, but becomes excited as the angle of incidence is rotated towards the equator at 90° (Fig. 6, Media 1); the more weakly aligned mode pair at ∼600 nm becomes excited instead for incident angles of ∼ 50°. Figure 6(d) shows that near the sharp resonance at 676 nm, the energy is transported into the far field by the resonant mode and by a weakly aligned, non-resonant, scattering mode. This leads to a Fano-like asymmetric feature in the total scattering cross-section that is sharper and more asymmetric than the single pair resonance, this is due to the interference between these two modes. On the contrary, absorption being due only to the resonant mode, produces a feature in the absorption cross-section which is symmetric with its peak coinciding with the maximum alignment of the mode pair.

## 5. Conclusion

In summary we provide a theory that generalizes Mie’s formulae to all particles without sharp edges, and reveals the common mechanism behind resonances in nanoparticles and excess noise in macroscopic cavities. This enables a very detailed analysis of the near and far field properties of the electromagnetic response of particles by finding expansions of the internal and scattered fields in terms of modes that are coupled pairwise. A particle can absorb more energy than it scatters or vice-versa, depending on the ability of the scattering and internal modes to transport energy and on their coupling with the incident field. We further show that there are sharp resonances caused by strong alignment between one pair of internal and one scattered mode, and broad resonances due to several pairs of modes having weak, but similar, alignment. We also provide an example of Fano-type resonance in the total scattering cross section of a particle due to interference effects between a resonant and a non-resonant mode, this effect is not possible for spherical particles. We also find that even simple particles that are far smaller than the wavelength of light posses many modes that are dark with respect to incident fields not matching their surface structure. These modes however play an important role for other incident fields, especially when applying near field excitation. This approach can be generalized to assemblies of particles and to complex particles, and is applicable to any process where the interaction between the system and the environment is described by internal and external functions complete at the boundary, including scattering of acoustic or electron waves and coupling to optical cavities.

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