Modal interference in spiky nanoshells

Simon P. Hastings; Zhaoxia Qian; Pattanawit Swanglap; Ying Fang; Nader Engheta; So-Jung Park; Stephan Link; Zahra Fakhraai

doi:10.1364/OE.23.011290

1. Introduction

Plasmonic nanoparticles and nanostructures enable manipulation and enhancement of the electric field of light in the optical frequency range for various applications [1–5]. The Localized Surface Plasmon Resonance (LSPR) usually refers to dipolar resonance of charge oscillations on a nanoparticle. Depending on the shape of the nanoparticle [6] or nanostructure [7], higher order resonances can also be excited either as bright [8, 9] or dark modes [10]. Recently, attention has been focused on the role of modal interferences in further enhancing the localization of the electric field [11, 12]. Interference effects such as Fano-like resonances [13] can couple dark and bright resonances, and allow the far-field detection of dark modes [12, 14, 15].

When two oscillators overlap both spatially and spectrally, both constructive and destructive interferences can be observed [16]. Destructive interference can result in reduced scattering cross sections and generate strongly localized near-fields [17], while constructive interference can result in super-radiant modes with enhanced scattering [12]. In spherically symmetric objects, such as gold nanoparticles and nanoshells, multipole modes are orthogonal and interference can not be observed in the total scattering cross section [13, 18]. In order to design modal interferences, asymmetry can be introduced in the design of the nanoparticle or the nanostructure. Examples include dipolar resonances that are shifted from the center of the object in nanoclusters [19] or fanoshells [14], and structures that are not spherically symmetric such as nanorods [16], nanorice [20], and nanobelts [20].

In non-spherically symmetric nanoparticles, a generalized form of Mie theory called the T-matrix method [21, 22] can be used to estimate the coupling between multipole modes and their effects on both directional and total scattering cross sections. The main advantage of the T-matrix method is that, once determined, it allows rapid computation of scattering and extinction cross sections of a particle in any orientation. However, it is generally challenging to determine the T-matrix for complex shapes. In particular, shapes with sharp corners [23] and high aspect ratios [24] have been shown to be numerically difficult to solve with surface integration methods such as the point matching [25] and Extended Boundary Condition (EBCM) methods [22]. Volume integration methods, including the discrete dipole approximation [26,27] and the finite-difference frequency-domain method [28], have been modified to determine the T-matrix for more complex geometries. However, a separate calculation must be performed for each frequency of interest. It has been previously suggested that finite-difference time-domain (FDTD) simulations could be used to calculate the T-matrix in cases where a broad frequency window is of interest [29]. Here, we have developed an algorithm which uses FDTD simulations to calculate the T-matrix simultaneously for a broad spectral range for complex structures. The FDTD method is robust to the sharp geometry used in this simulation, and in general is able to handle inhomogeneous and anisotropic materials [29]. In this approach, the object is illuminated with a broadband plane-wave from many different directions. By decomposing the incident and scattered fields from each illumination into vector spherical harmonic wave functions [21], an over-determined equation for the T-matrix is produced. The most probable T-matrix is then calculated using the pseudo-inverse matrix based on a least square method. Using this method we ran 28 FDTD simulations and were able to calculate the T-matrix at 75 wavelengths between 550 nm and 850 nm.

Using this method, the T-matrix for model spiky gold nanoshells were calculated in order to investigate modal interference in these nanoparticles. Spiky nanoshells are composed of randomly sized sharp gold spikes decorating the surface of a polystyrene core [30], and have tunable dipolar resonances in the visible and near-IR range [30–32]. It has been recently demonstrated that the disorder in these nanoshells generates spectrally and spatially broad dark quadrupole modes that enable highly efficient and reproducible Quadrupole Enhanced Raman Scattering(QERS) on a single particle level [33]. Here we demonstrate that single particle far-field backscattering measurements of these nanoshells show interference patterns [33]. T-matrix calculations allow us to investigate the origin of these interferences. We show that both dipole→dipole and quadrupole→dipole (i.e., incident quadrupole waves scattered as dipole waves) mode mixing contributes to these interference patterns with a coupling strength of about 5% between different modes at the dipole resonance frequency. Furthermore, mode mixing is not only observed in backscattering but is also present in the total scattering cross section. Comparisons between a disordered nanoshell and a more ordered spiky nanoshell model show that the disorder contributes strongly to the modal interference and allows efficient coupling of the dipole and the dark quadrupole modes, which can contribute to the broadening of quadrupoles and lead to strong QERS enhancements.

2. Methods

2.1. Synthesis and dark field microscopy

As described in our previous publications [30–33], spiky nanoshells were produced using a surfactant-assisted seed growth method. The nanoshells used in this study have a polystyrene core with a diameter of 95 nm, with spikes estimated to be less than 68 nm in length. The morphology of the nanoshells was studied using scanning electron miscroscopy (SEM) (Quanta 600 FEG Mark II operating at 15 kV). Substantial disorder in the size, length and shape of the spikes are evident in the SEM images of these nanoshells shown in the inset of Fig. 1(a). Nanoshells were drop cast onto clean glass slides with an evaporated indexed gold pattern and characterized for single particle backscattering measurements as described in detail in our previous publication [33]. In order to perform single particle surface enhanced Raman spectroscopy (SERS) experiments, as described previously [33], the spiky nanoshells were immersed in a solution of 4-mercaptobenzoic acid (0.0002 g/ml) over night, cleaned with ethanol, and allowed to dry under nitrogen. The combination of backscattering and SEM imaging at low magnification were performed to ensure that only single particle spectra are reported. Single particle SERS measurements were performed first, followed by dark-field backscattering measurements and then SEM measurements at last to ensure that only isolated nanoparticles were used and that SEM damage to the nanoshells was prevented. In this work, only the dark-field backscattering data are presented.

Fig. 1 (a) SEM images of the nanoparticles used in this study. (b)–(f) Examples of single particle backscattering spectra. (b) and (c) show a pronounced double peak, while (d)–(f) show single backscattering peaks.

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Dark-field backscattering measurements were performed with an integration period of two seconds using a 150 grooves/mm grating centered on 800 nm. A 515 nm long-pass filter was used to remove second order reflections. Background and dark counts were removed from the single particle scattering spectra, and the spectrum of the lamp and the spectral response of other optical components were calibrated with a white light reflectivity standard (Labsphere SRS-99-010). Measurements were performed using a Princeton Instruments Acton SP2150i spectrograph. In total, 71 single particle spectra were collected over three sets of measurements, all from the same synthesis batch. Some examples of the single particle spectra are shown in Fig. 1(b)–1(f). A characteristic double peak was observed in about 10% of the single particle backscattering spectra (Fig. 1(b) and 1(c)).

2.2. Finite difference time domain (FDTD) simulations

FDTD simulations were performed using Lumerical Solutions inc.’s FDTD package (versions 8.6 and 8.7) as previously described [31–33]. The simulation boundary conditions were set to Perfectly Matched Layers (PML), and a Total-Field Scattered-Field (TFSF) source was used to inject a plane wave into the simulation. In these simulations, the TFSF source surrounded the scattering object, creating a cubic volume. Inside the TFSF source volume, a plane wave is incident upon the scatterer. At the boundaries of this volume, the TFSF source subtracts the original wave such that the scattered component of the light escapes the volume, but the original wave does not. Outside the TFSF source, only the scattered fields are present. The fields in a cubic volume are collected and used to decompose the scattered wave into its components. This cubic monitor must be at least $\sqrt{3}$ times larger on each side than the source in order to define a spherical surface for the vector spherical harmonic decomposition which does not include any of the source region. In order to illuminate the structure from different angles, the spiky nanoshell itself was rotated. After data collection, this rotation was reversed to the original coordinate system such that the light source has rotated in the opposite direction compared to the particle rotation.

As described in our recent work [33], disordered nanoshells were modeled as a polystyrene core of radius 48 nm decorated with 60 rounded cones. The material properties of gold originated from the CRC Handbook [34], while an ellipsometry measurement was used to determine the optical properties of polystyrene, as in previous work [32,33].The height, tip diameter, and cone angles were randomized with ranges 50 nm – 65 nm, 2 nm – 6 nm, and 30° – 75°, respectively.

In total, eleven randomized spiky nanoshells were produced and modeled using FDTD simulations. All models showed similar spectral features as shown by their total cross section in Fig. 2. Similar to the experimental data, double and multiple peak pattern are also observed in some of the simulated total scattering spectra. One particle was chosen for closer inspection (shown in the inset of Fig. 3(b)). The simulated backscattering spectrum of this particle is compared with a typical experimental backscattering spectrum with a double peak in Fig. 3. A total of 28 simulations were performed on this particular model with varying model orientations with regards to the incident field. These simulations were used to solve the T-matrix. In order to compare the data with more ordered structures, ten simulations of various orientation of a simple gold nanosphere, 23 simulations of a gold ellipsoid, and 28 simulations of various orientations of an ordered spiky nanoshell with 60 identical spikes were also produced and their corresponding T-matrices calculated. The ordered spiky nanoshell used cones with 57 nm length, 4 nm tip radius, and 47 degree cone angle.

Fig. 2 Cross sections of ten random modeled spiky nanoshells with a polystyrene core of radius 48 nm covered with 60 spiky cones. The height, tip diameter and the cone angles were randomly varied between 50 nm and 65 nm, between 2 nm and 6 nm, and between 30 and 75 degrees, respectively. In two of the ten simulations a prominent double peak is observed, although all of them have smaller features which are due to modal interference. Due to the large number of degrees of freedom, a wide range of behavior is in principle possible for this type of geometry, and this data set only captures some of this variation. This data set was previously used in [33], and additional structural details and optical properties in the near-field can be found there.

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Fig. 3 (a) Experimentally measured backscattering spectrum of a single spiky nanoshell with a characteristic double peak feature. The inset shows an SEM image of a typical spiky nanoshell (note that this spectrum does not correspond to this particle). (b) The backscattering cross section of a model disordered spiky nanoshell (inset) simulated using FDTD. In this orientation, the simulated spectrum also shows a double peak feature.

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2.3. T-matrix formalism

In Mie theory, the total scattering cross section and the differential backscattering cross section are described by the following equations [35]:

σ_{scatter} = \frac{λ^{2}}{2 π} \sum_{n = 1}^{\infty} (2 n + 1) ({| a_{n} |}^{2} + {| b_{n} |}^{2})

{(\frac{d σ}{d Ω})}_{bs} = \frac{λ^{2}}{16 π^{2}} {| \sum_{n = 1}^{\infty} (2 n + 1) {(- 1)}^{n} [a_{n} - b_{n}] |}^{2}

Where a_n and b_n are the scattering coefficients for the electric and magnetic modes of order n. As can be seen in Eq. (1), every term in the total scattering cross section is positive-definite. As a result, interference between different terms (i.e. modes) is only possible in directional scattering, such as in the differential backscattering cross section, (dσ/dΩ)_bs, in Eq. (2). T-Matrix theory, a more general form of Mie theory, allows modal interference in structures that are not spherically symmetric. Here, the incident electric field, Eⁱ(r⃗), and the scattered electric field, E^s(r⃗), are expanded as infinite sums of incoming and outgoing vector spherical harmonic wave functions (VSHWF), M, N, RgM and RgN [21]:

E^{i} (\vec{r}) = \sum_{m, n} [a_{m n}^{i (M)} {RgM}_{m n} (\vec{r}) + a_{m n}^{i (N)} {RgN}_{m n} (\vec{r})]

E^{s} (\vec{r}) = \sum_{m, n} [a_{m n}^{s (M)} M_{m n} (\vec{r}) + a_{m n}^{s (N)} N_{m n} (\vec{r})]

The incident electric and magnetic mode wave functions (RgN and RgM respectively) have been regularized at the origin using spherical Bessel functions while the outgoing N and M wave functions employ spherical Hankel functions of the first kind. The coefficients $a_{m n}^{i (N)}$ , $a_{m n}^{i (M)}$ , $a_{m n}^{s (N)}$ , and $a_{m n}^{s (M)}$ define the strength of each electric and magnetic mode of the incoming and scattered light, respectively. Here n indicates the order of the mode (n = 1 implies a dipole, n = 2 a quadrupole, etc.), while m indicates the specific submode and is an integer between −n and n. The T-matrix maps the coefficients of the incident field to those of the scattered field:

a^{s} = T a^{i}

Here, a^s and aⁱ represent all the electric and magnetic modes in that order. In this study the T-matrix was truncated to include dipole and quadrupole terms for both electric and magnetic modes as the octupole and higher order modes are substantially weaker [33]. Therefore, the T-matrix has 16 rows, where we have chosen to order the modes as electric and then magnetic. We also assumed that mode mixing can only occur between different electrical components, T^(EE), or between different magnetic components, T^(MM), but that inter-mixing of magnetic and electric modes (termed here as T^(ME) and T^(EM)) are negligible, i.e., the T-matrix is block diagonal. This assumption is justified as mixing between the electric and magnetic modes requires a strongly chiral structure which is not the case for the spiky nanoshells. Additionally, the magnetic modes are broad and generally very weak in this structure (as shown later in Fig. 7). We therefore removed the T^(ME) and T^(EM) mixing contributions in all following equations. The total scattering cross section is then given by [21]:

\begin{array}{l} σ_{scatter, \hat{β}} (ϕ_{i}, θ_{i}) = \frac{16 π^{2}}{k^{2}} \sum_{n, m} {{| \sum_{m^{'} n^{'}} i^{n^{'}} {(- 1)}^{m^{'}} γ_{- m^{'} n^{'}} T_{m n m^{'} n^{'}}^{(M M)} {\bar{C}}_{- m^{'}, n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β} |}^{2} \\ + {| \sum_{m^{'}, n^{'}} i^{n^{'}} {(- 1)}^{m^{'}} γ_{- m^{'} n^{'}} T_{m n m^{'} n^{'}}^{(E E)} \frac{{\bar{B}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i})}{i} \cdot \hat{β} |}^{2}} \end{array}

Here γ is a normalizing constant, C̄_mn and B̄_mn are vector spherical harmonics [21], β̂ is the polarization vector, and θ_i and ϕ_i indicate the angle of incidence of the incident plane wave. Calculations of C̄_mn and B̄_mn were carried out using the Optical Tweezer Toolbox [36]. Additional details are provided in the appendix. Similarly, the differential backscattering cross section, the intensity of light reflected directly backwards, can be calculated as:

\begin{array}{l} \frac{d σ_{bs, β} (ϕ_{i}, θ_{i})}{d Ω} = \frac{16 π^{2}}{k^{2}} | \sum_{m n m^{'} n^{'}} {(- 1)}^{m^{'} + n} i^{n^{'} - n - 1} γ_{m n} γ_{- m^{'} n^{'}} { \\ {{\bar{C}}_{m n} (θ_{i}, ϕ_{i}) T_{m n m^{'} n^{'}}^{(M M)} [{\bar{C}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}] - {\bar{B}}_{m n} (θ_{i}, ϕ_{i}) T_{m n m^{'} n^{'}}^{(E E)} [{\bar{B}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}]} |}^{2} \end{array}

To calculate the T-matrix a least squares method (LSM) was developed. In this approach, a single particle was simulated in many different orientations with regards to the incident electric field. Both the incident electric and magnetic field, and the scattered electric and magnetic field can be extracted from the FDTD simulation using the built-in total-field scattered field (TFSF) source in FDTD Solutions (more details in the appendix). The incident and the scattered electric fields were expanded in VSHWFs to obtain a set of aⁱ and a^s vectors. Using these vectors, a linear algebra problem was constructed to solve the T-matrix:

A^{s} = T A^{i}

Where A^s and Aⁱ are rectangular matrices composed of sets of matching a^s and aⁱ vectors. The T-matrix was then calculated using the Moore-Penrose least square method [37]. Note that this method is directly fitting incident fields to scattered fields, rather than comparing cross sections. Additionally, this method varies aⁱ by changing the properties of the incident plane wave, and as a result the Aⁱ matrix is dense while other methods of finding the T-matrix are able to isolate single incident components for excitation. More details of the method, including the least squares inversion procedure used to invert Eq. (8) are provided in the appendix. The method was validated by computing the T-matrix and cross sections of a gold nanosphere and a gold ellipsoid as shown in Fig. 4 and Fig. 5. This validation is further discussed in the appendix.

Fig. 4 The T-matrix of a gold sphere with radius 52 nm. (a) The T-matrix at 476 nm, where each position denotes the absolute value of that T-matrix entry. The strongest terms are diagonal terms for the dipole modes. Due to the spherical symmetry, all modes have equal values (although this was not enforced at any point). The matrix is nearly diagonal as expected. (b)–(d) The scattering and extinction cross sections and backscattering differential cross section from FDTD simulations(circles), compared with the values predicted by the T-matrix (solid line), and the Mie theory prediction (crosses).

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Fig. 5 The T-matrix of a gold ellipsoid with values of radii of 52 nm, 40 nm and 40 nm. (a) The T-matrix at 541 nm, near the main plasmon peak of the structure. Each position denotes the absolute value of that T-matrix entry. The strongest terms are diagonal terms for the dipole modes, but the asymmetry creates strong off-diagonal mixing terms between the dipoles. (b) The calculated scattering cross section for 3 orientations of ellipsoid: 0 degrees, 45 degrees and 90 degrees (open symbols) as compared with the corresponding spectra directly calculated using FDTD. Here at 0 degrees the electric field is perpendicular to the long axis of the ellipsoid and the propagation is in the direction of the long axis (as schematically shown in the inset of (b)) and at 90 degrees the electric field is parallel to the long axis of the ellipsoid and the propagation in the direction normal to the long axis. The orientations plotted here were not included in the fitting data set for the T-matrix evaluation.

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3. Results

Using the method described above, the T-matrix of the disordered-spiky nanoshell model shown in Fig. 3 was calculated at 75 different wavelengths, spaced between λ = 550 nm and λ = 850 nm. Figure 6(a) shows the T-matrix at a wavelength of λ = 696 nm, which is the wavelength of the dip in the scattering cross section shown in Fig. 3(b). Figure 6(b) shows the same T-matrix, with the diagonal terms removed to highlight the contribution of the off-diagonal terms to the scattering cross section. At this frequency, each of the electric dipole→dipole and the electric quadrupole→dipole modes are observed to contribute up to about 5% to the scattering coefficient of the dipole mode. Figure 7 shows the cross sections associated only with the diagonal terms of the T-Matrix for this model nanoshell. As expected, the diagonal terms, in particular the electric dipoles (Fig. 7(a)), are the strongest contributors to the scattering cross section at all wavelengths [32]. Both the diagonal magnetic dipoles and the diagonal electric quadrupoles are predominantly dark modes at this frequency (Fig. 7(b) and 7(c)) and have at least two orders of magnitude smaller scattering cross sections at all wavelengths. Furthermore, the magnetic dipoles are non-resonant and only contribute a broad background (Fig. 7(c)).

Fig. 6 (a) The T-matrix at λ = 696 nm, the wavelength of the maximum dip in the backscattering peak. Here the T^(EE) quadrant is shown in the top left corner while T^(MM) is in the bottom right. (b) Off-diagonal terms of the same T-matrix, normalized to the T-matrix entry with the highest absolute value. Each position in both (a) and (b) denotes the absolute value of that T-matrix entry.

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Fig. 7 The cross sections associated with only the diagonal terms of the full T-matrix for the orientation and particle shown in Fig. 3(b). (a) The diagonal electric dipole cross section. (b) The diagonal electric quadrupole cross section, exhibiting substantial breadth but greatly suppressed scattering. (c) The diagonal magnetic dipole cross section exhibits no clear peak and is nearly two orders of magnitude weaker than the electric dipole. The magnetic dipole contributes only a background to the particle cross section.

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4. Discussion

The T-matrix shown in Fig. 6 indicates that the values of different diagonal modes of the same order multipole are not equal. This modal splitting implies that the scattering cross section is orientation dependent. Furthermore, the off-diagonal coupling strengths are also not the same at different orientations, suggesting that the interference patterns observed both experimentally and in simulations are also orientation dependent. This effect is demonstrated in Fig. 8, where the interference feature appears and disappears in the total scattering cross section, depending on the angle of incidence. This suggests that although the interference patterns were observed in ten percent of the nanoshells examined experimentally, they may in fact be present in many more of them when viewed at different orientations. This behavior is a consequence of broken symmetry in these nanoparticles. In spherically symmetric objects (Eq. (1)), the total scattering cross section can not show interference effects or be orientation dependent. The broken symmetry implies that the true eigenmodes of this structure are not the vector spherical harmonic wave functions, but rather slightly perturbed mixtures of them. When these modes are viewed in the far-field, the dipole waveform still dominates, but is perturbed by the dipole→dipole and quadrupole→dipole mode mixing.

Fig. 8 (a) Total scattering cross section, σ_scatter, of the model spiky nanoshell of Fig. 3(b) with the same orientation as in Fig. 3(b). (b) Total scattering cross section, σ_scatter, of the same nanoshell at a different orientation with regards to the incident field. At this orientation the interference pattern is less evident.

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In order to further highlight the effects of the mode mixing in generating the interference patterns, we can calculate the T-matrix assuming that it must be diagonal (i.e., all off-diagonal terms which are responsible for mode mixing are zero). For this non-mixing T-matrix, the maximum likelihood solution using standard Gaussian probabilities results in the following solution:

T_{n m n m} = \frac{\sum_{i} a_{n m, i}^{s} a_{n m, i}^{i}}{\sum_{i} {(a_{n m, i}^{i})}^{2}}

Figure 9 shows a comparison between the non-mixing T-matrix and a full T-matrix solution in predicting the total scattering and backscattering cross section spectra of the model spiky nanoshell. While the non-mixing T-matrix correctly predicts the approximate shape of each curve, it is unable to match the simulated data near apparent interference patterns. Only with the addition of the off-diagonal (mode mixing) terms, is the T-matrix solution able to model the amplitude of the scattering cross section near the interference features, both at λ = 695 nm where a dip is observed in the spectrum, and at λ ≈ 630 nm where a plateau is observed near the dark quadrupole resonance of this structure as shown in Fig. 7(b). Therefore, even though each off-diagonal term contributes at most 5% to the scattered coefficients, they are crucial in properly describing the scattering spectra of spiky nanoshells. Additionally, we can show that mode mixing between the electric dipole modes alone is also not enough to describe the observed interferences and electric quadrupole→dipole mixing is crucial in correctly predicting the spectral features. In order to do this, the T-matrix solution was calculated by only allowing dipole→dipole mode mixing and assuming that the dipole→quadrupole, quadrupole→dipole, and quadrupole→quadrupole mode mixing terms are zero. Figure 10 shows a comparison between the three T-matrix solutions discussed above in predicting the extinction cross section. Neither the non-mixing T-matrix, nor the T-matrix with electrical dipole mode mixing are able to predict the extinction cross section, which includes information about the relative phases of these modes. The extinction cross section is only correctly modeled by including off-diagonal electric terms in the full T-matrix solution. In particular, the interference observed close to the dipole resonance peak, λ = 695 nm, is only explainable with the introduction of quadrupole→dipole interference. Figure 11 shows further comparisons between these three solutions in predicting the total scattering cross section and the extinction cross sections at another orientation of the particle with regards to the incident electric field. We also note that the non-mixing T-matrix solution shown in Fig. 9 gives rise to interference-like patterns, which do not represent real interferences. Therefore the observation of such patterns in a hypothetical nanoparticle structure is not enough to predict a modal interference without knowing the coupling strength or a full T-matrix solution for that structure.

Fig. 9 (a) Total scattering cross section calculated using FDTD (red circles), compared with the total scattering cross section spectra calculated using the full T-matrix solution (black solid lines) and a non-mixing T-matrix solution (gray solid lines). (b) Backscattering cross section calculated using FDTD (purple circles), compared with the total scattering cross section spectra calculated using the full T-matrix solution (black solid lines) and a non-mixing T-matrix solution (gray solid lines). The inset squares indicate the elements of the T-matrix which can be non-zero for each model.

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Fig. 10 The extinction cross section at the same orientations as the data shown in Fig. 9 of the paper. (a) The T-matrix solution for a strictly diagonal (non-mixing) T-matrix. (b) The T-matrix solution where mode mixing is allowed only between the electric dipole modes, but is diagonal in all other modes. (c) The T-matrix solution for the block-diagonal matrix with mode mixing between all electrical modes and between all magnetic modes. It is evident that quadrupole→dipole mixing is required to fully capture the behavior of the system. In each case the line represents the values calculated using the T-matrix and the circles represent the simulated cross section.

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Fig. 11 The scattering (a)–(c) and extinction (d)–(f) cross sections for a different orientation than the one shown in Fig. 9 in the paper. (a,d) show the solutions for a strictly diagonal (non-mixing) T-matrix model. (b,e) show the T-matrix solution when mode mixing is allowed between the electric dipole modes, but the matrix is diagonal in all others modes. (c,f) The full T-matrix solution with block-diagonal mode mixing. It is evident that quadrupole→dipole mode mixing is required to fully capture the behavior of the system. From the results above, we can see that without this mixing, it is not possible to account for the phase or orientation dependence of the modal mixing. In each case the line represents the model values and the circles represent the simulated cross section.

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As discussed above, the off-diagonal T-matrix terms are critical in explaining interference patterns observed in spiky nanoshells. We can understand the exact origin of this effect by examining Eq. (6) in more detail. When the components inside the absolute value terms in Eq. (6) have differing complex phases, it is possible to have destructive interference, while in phase components will result in constructive interfere. Near the dipole resonance, these phases can change rapidly with frequency, resulting in sharp features in the total scattering cross section spectrum. The detailed role of the off-diagonal terms in the observed interference patterns can be examined by calculating the magnitudes of different terms in the sum in Eq. (6) for the full T-matrix solution. The sum over the diagonal components is given by:

\begin{array}{l} σ_{scatter, β}^{Diagonal} (ϕ_{i}, θ_{i}) = \frac{16 π^{2}}{k^{2}} \sum_{n, m} {{| i^{n} {(- 1)}^{m} γ_{- m n} T_{m n m n}^{(M M)} {\bar{C}}_{- m n} (θ_{i}, ϕ_{i}) \cdot \hat{β} |}^{2} \\ + {| i^{n} {(- 1)}^{m} γ_{- m n} T_{m n m n}^{(E E)} \frac{{\bar{B}}_{- m n} (θ_{i}, ϕ_{i})}{i} \cdot \hat{β} |}^{2}} \end{array}

A residual function can be obtained by subtracting the sum of the contributions of all diagonal terms from the total scattering cross section.

Residual = σ_{scatter, β} (ϕ_{i}, θ_{i}) - σ_{scatter, β}^{Diagonal} (ϕ_{i}, θ_{i})

The residual is shown as a function of wavelength in Fig. 12(a). Similarly, the contributions of various off-diagonal terms in the total scattering cross section can be estimated by evaluating the magnitudes of those terms only. These are defined for the electric dipole→dipole and electric quadrupole→dipole mixing as:

σ_{scatter, β}^{dipole, dipole} (ϕ_{i}, θ_{i}) = \frac{16 π^{2}}{k^{2}} \sum_{n = 1, m} {| \sum_{n^{'} = 1, m^{'} \neq m} i {(- 1)}^{m^{'}} γ_{- m^{'} 1} T_{m 1 m^{'} 1}^{(E E)} \frac{{\bar{B}}_{- m^{'} 1} (θ_{i}, ϕ_{i})}{i} \cdot \hat{β} |}^{2}

σ_{scatter, β}^{quad, dipole} (ϕ_{i}, θ_{i}) = \frac{16 π^{2}}{k^{2}} \sum_{n = 1, m} {| \sum_{n^{'} = 2, m^{'}} {(- 1)}^{m^{'} + 1} γ_{- m^{'} 2} T_{m 1 m^{'} 2}^{(E E)} \frac{{\bar{B}}_{- m^{'} 2} (θ_{i}, ϕ_{i})}{i} \cdot \hat{β} |}^{2}

Fig. 12 (a) The residual function of the scattering cross section as described in Eq. (11). This plot is shown normalized to the total cross section in Fig. 13. (b) The magnitude of the off diagonal terms corresponding to mode mixing between dipole→dipole (dashed purple) and quadrupole→dipole(solid black) modes as described in Eq. (12) and Eq. (13).

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Fig. 13 The residual function of the scattering cross section as described in Eq. (11), normalized by the total scattering cross section.

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The contribution of each type of mode mixing is shown in Fig. 12(b). Interestingly, the residual function shown in Fig. 12(a) shows both constructive and destructive interferences. Despite the fact that the quadrupoles themselves scatter very weakly and are dark modes (as shown in Fig. 7(b)), they produce about half of the interference effects observed in the total scattering cross section. The dipole→dipole and quadrupole→dipole mode mixing together produce a total scattering cross section change of just under 20% of the total cross section, while the dipole→quadrupole and quadrupole→quadrupole modes produce almost no change at all. From Eq. (6), changes to the scattered dipole are magnified by the squaring operation due to the existing scattered dipole, while the quadrupole modes are not. This allows the quadrupole modes to couple with the bright far-field scattering mode and therefore be observed in the far-field as interferences. This effect is similar to the Fano-like resonances observed in previous studies [12–14]. However, we note that the complexity and heterogeneity of this structure requires the full T-matrix solution and a simple coupling between one electric quadrupole and one electric dipole is not enough to fully explain the extent and complexity of the effect. When observed at a different orientation, the apparent coupling strength may vary between the dipole→dipole and quadrupole→dipole modes (both bright), and between dipole→quadrupole and quadrupole→dipole modes (both dark).

To further examine the role of the disorder in these observations, the T-matrix of a more ordered spiky nanoshell was also calculated. This ordered structure is very similar to the disordered structure, but all of the spikes decorating the polystyrene cores have the same height, length and tip angles. As shown in Fig. 14, the T-matrix for the ordered model is predominantly diagonal with off-diagonal terms at least an order of magnitude smaller than those observed in the disordered nanoshell. No interference pattern is observed in the total scattering or backscattering spectra in the ordered nanoshell. This indicates the additional asymmetry caused by disorder is required to achieve substantial mode mixing in these structures.

Fig. 14 The T-matrix calculated for an ordered version of the model spiky nanoshell, with the same number of cones(60), but identical cone sizes and shapes. The cones have 57 nm height, cone angle 47 degrees, and tip radius 4 nm. These represent the average values used in the disordered nanoshells. (a) The full T-matrix solution for this structure at 696 nm. Each position denotes the absolute value of that T-matrix entry. The diagonal dipole modes are the dominant terms and all nearly equal, which indicates that this structure is mostly isotropic in the far-field. (b)The off-axis terms. These terms are a factor of 10 smaller than those observed in the disordered structure and a factor of 200 smaller than the strength of the dipole modes. (c) Scattering cross section of the ordered nanoshell as calculated by FDTD (open symbols) compared with the cross section calculated using the T-matrix (solid line). (d) Extinction cross section of the ordered nanoshell as calculated by FDTD (open symbols) compared with the cross section calculated using the T-matrix (solid line). Both the scattering and the extinction cross sections are independent of direction of illumination within the accuracy of these calculations. This is further evidence that this particle is isotropic in the far-field, while the disordered structure is not.

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The mode mixing discussed above can also strongly affect the electric near-field intensity in these disordered nanoshells, resulting in spectrally broadened quadrupoles and strong Quadrupole Enhanced Raman Scattering (QERS) in these systems [33]. This effect is due to the symmetry of the T-matrix [21]:

T_{- m^{'} n^{'} (- m) n}^{(i j)} = {(- 1)}^{m^{'} + m} T_{m n m^{'} n^{'}}^{(j i)}

where i, j are either electric, E, or magnetic, M, modes. While we did not explicitly enforce this requirement in our method, the least-square method solution shown in Fig. 6 still results in symmetric patterns in the T-matrix. This symmetry means that the quadrupole→dipole mixing shown in Fig. 12(b) as a function of wavelength can also lead to a reciprocal dipole→quadrupole mixing and interference in the dark quadrupole modes. Figure 7 shows that in the far-field the quadrupole→dipole mixing results in the broadening of the quadrupole scattering cross section and spectral overlap between the dipole and quadrupole scattering cross sections. However, in the far-field the quadrupole mode is dark and its scattering cross section is significantly weaker than that of the dipole mode. Because the quadrupole modes scatter very weakly, the quadrupole moment (calculated directly from the polarization current as described in the supporting info and our previous work [33]) is a better indicator of quadrupole excitation than the quadrupole scattering coefficients above. Figure 15 shows that the near-field quadrupole moments are strongly but heterogeneously excited across a broad range of wavelengths due to interferences with the dipole moments. The dipole induced quadrupole broadening dramatically enhances the near-field intensity of the electric field, strengthening the QERS enhancement [33].

Fig. 15 Quadrupole moment broadening for a typical orientation of the disordered particle in Fig. 3(b). For this illumination, light propagated in the ŷ direction and was polarized along ẑ. In the absence of disorder, the only excited quadrupole is in the yz plane, and forms a narrow peak around 630 nm. This data appeared in our previous studies of this particle [33]

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5. Conclusion

In this work we developed a novel method to calculate the T-matrix of arbitrarily-shaped nanoparticles using FDTD simulations and a least square matrix inversion algorithm. This is a general method that can be applied on a wide range of nanostructures to investigate the coupling between various modes and quantify the strength of modal interference in a classical electrodynamical framework. Using this method we demonstrated that modal interferences with complicated spectral shapes can explain both the far-field interference patterns in spiky nanoshells as well as their ability to efficiently enhance the near-field intensity of the electric field and produce strong QERS signals. We showed that in disordered spiky nanoshells, modal interference can be observed in the backscattering cross sections as well as the total scattering cross section and strongly depends on the orientation of the nanoshell with respect to the angle of incidence of light. The modal interference has non-trivial participation of both bright dipole→dipole and otherwise dark quadrupole→dipole modes with both constructive and destructive interferences. The mode coupling and modal interference directly results from the built-in disorder in these nanoshells which is intrinsically robust to error in construction, generating reliable coupling of the dark quadrupole modes to the far-field. In a reciprocal process, the coupling results in the broadening of the quadrupole modes due to the dipole→quadrupole coupling and allows strong enhancement of the near-field intensity.

This work provides a general recipe for analyzing modal interferences of arbitrary nanoparticles using existing commercially available software. With the advent of faster computers, this method should be able to generate a T-matrix for almost any particle, providing an analytical framework for studying modal interference in nanoparticles. Furthermore we demonstrated that in principle the observation of a dip or increase in the backscattering spectrum alone is insufficient to describe the patterns as interferences, as shown in the scattering spectrum of a diagonal-T-matrix solution, which indicates the importance of obtaining the full T-matrix solution to verify the existence of modal interferences in a complicated structure.

A. Scattering formalism

The scattering behavior of a particle can be derived from the scattering dyad [21]:

\begin{array}{l} \overset{̿}{F} (θ, ϕ; θ^{'}, ϕ^{'}) = \frac{4 π}{k} \sum_{n, m, n^{'}, m^{'}} {(- 1)}^{m^{'}} i^{n^{'} - n - 1} \times \\ {[T_{m n m^{'} n^{'}}^{(M M)} γ_{m n} {\bar{C}}_{m n} (θ, ϕ) + T_{m n m^{'} n^{'}}^{(E M)} i γ_{m n} {\bar{B}}_{m n} (θ, ϕ)] \cdot γ_{- m^{'} n^{'}} {\bar{C}}_{- m^{'} n^{'}} (θ^{'}, ϕ^{'}) \\ + [T_{m n m^{'} n^{'}}^{(M E)} γ_{m n} {\bar{C}}_{m n} (θ, ϕ) + T_{m n m^{'} n^{'}}^{(E E)} i γ_{m n} {\bar{B}}_{m n} (θ, ϕ)] \cdot γ_{- m^{'} n^{'}} \frac{{\bar{B}}_{- m^{'} n^{'}} (θ^{'}, ϕ^{'})}{i}} \end{array}

Where B̄_−mn and C̄_−mn are vector spherical harmonics [21], θ′ and ϕ′ represent the direction of the incident light, while θ and ϕ represent the direction of scattering and k is the scaler wave number being considered. Here we have chosen to reverse the standard order of the T-matrix so that electric modes are in the top left quadrant of the T-matrix, while magnetic modes are in the bottom right. In this work, cross terms between electric and magnetic modes (labeled T^(EM) and T^(ME)) were assumed to be zero. These terms will be dropped in all subsequent expressions. The factor γ has been used to normalize the waves:

γ_{n m} = \sqrt{\frac{(2 n + 1) (n - m)!}{4 π n (n + 1) (n + m)!}}

By integrating over all outward angles and summing the intensity of vertically and horizontally polarized components, we can calculate the total scattering cross section of a plane wave with incident polarization β̂ [21]:

\begin{array}{l} σ_{scatter \hat{β}} (ϕ_{i}, θ_{i}) = \int_{4 π} d Ω [{| f_{v \hat{β}} (θ, ϕ; θ_{i}, ϕ_{i}) |}^{2} + {| f_{h \hat{β}} (θ, ϕ; θ_{i}, ϕ_{i}) |}^{2}] = \\ \frac{16 π^{2}}{k^{2}} \sum_{n, m} {{| \sum_{m^{'}, n^{'}} i^{n^{'}} {(- 1)}^{m^{'}} γ_{- m^{'} n^{'}} T_{m n m^{'} n^{'}}^{(M M)} {\bar{C}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β} |}^{2} \\ + {| \sum_{m^{'} n^{'}} i^{n^{'}} {(- 1)}^{m^{'}} γ_{- m^{'} n^{'}} T_{m n m^{'} n^{'}}^{(E E)} \frac{{\bar{B}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i})}{i} \cdot \hat{β} |}^{2}} \end{array}

where f_vβ and f_hβ are the horizontal and vertical components of the F̿ dyad acting on the incident polarization. The extinction cross section is given by the optical theorem (neglecting the electric to magnetic mode mixing):

\begin{array}{l} σ_{e β} = \frac{4 π}{k} Im [\hat{β} \cdot \overset{̿}{F} (θ_{i}, ϕ_{i}, θ_{i}, ϕ_{i}) \cdot \hat{β}] = \frac{16 π^{2}}{k^{2}} Im [\sum_{n m m^{'} n^{'}} {(- 1)}^{m^{'}} i^{n^{'} - n - 1} γ_{m n} γ_{- m^{'} n^{'}} { \\ [\hat{β} \cdot {\bar{C}}_{m n} (θ_{i}, ϕ_{i})] T_{m n m^{'} n^{'}}^{(M M)} [{\bar{C}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}] + [\hat{β} \cdot {\bar{B}}_{m n} (θ_{i}, ϕ_{i})] T_{m n m^{'} n^{'}}^{(E E)} [{\bar{B}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}]}] \end{array}

And the differential backscattering cross section is simply the amount of light reflected directly backwards with either polarization (again neglecting electric to magnetic mode mixing):

\begin{array}{l} \frac{d σ_{bs, \hat{β}} (ϕ_{i}, θ_{i})}{d Ω} = {‖ \overset{̿}{F} (π - θ_{i}, π + ϕ_{i}, θ_{i}, ϕ_{i}) \cdot \hat{β} ‖}^{2} = \frac{16 π^{2}}{k^{2}} ‖ \sum_{n m m^{'} n^{'}} {(- 1)}^{m^{'} + n} i^{n^{'} - n - 1} γ_{m n} γ_{- m^{'} n^{'}} { \\ {{\bar{C}}_{m n} (θ_{i}, ϕ_{i}) T_{m n m^{'} n^{'}}^{(M M)} [{\bar{C}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}] - {\bar{B}}_{m n} (θ_{i}, ϕ_{i}) T_{m n m^{'} n^{'}}^{(E E)} [{\bar{B}}_{- m^{'} n^{'}} (θ_{i}, ϕ_{i}) \cdot \hat{β}]} ‖}^{2} \end{array}

B. Calculation of the T-matrix

A simple, effective, and stable approach to solve the T-matrix for a nanostructure is to reformulate the problem as a matrix inversion problem. In particular, since the T-matrix predicts the scattering coefficients at any given angle, sampling of the scattering function at various angles of the incident field can determine the truncated T-matrix, using a pseudo-inverse approach. As more terms are included in the T-matrix, more angles of illumination are required. Here, FDTD simulations were used to produce a set of input and output values, by rotating the structure randomly up to 180 degrees around each axis, which is equivalent to rotating the incident direction of light in the opposite fashion. The matrix inversion problem is then expressed as follows:

\begin{matrix} [\begin{matrix} | & | & | & | \\ a_{s, 1} & a_{s, 2} & \dots & a_{s, N} \\ | & | & | & | \end{matrix}] = T [\begin{matrix} | & | & | & | \\ a_{i, 1} & a_{i, 2} & \dots & a_{i, N} \\ | & | & | & | \end{matrix}] \\ A^{s} = T A^{i} \end{matrix}

Where the n^th column of A^s is the coefficients of the scattered field after exposure at a particular angle with an incident field described by the coefficients in the n^th column of Aⁱ. The elements of the aⁱ and a^s vectors are the coefficients of the spherical harmonics for each field. The far-field scattered light was isolated from the incident field by using the built-in Total Field-Scattered Field (TFSF) source function in the Lumerical software. Once isolated, the radial component of the scattered field can be decomposed using the spherical harmonics in order to determine the scattered coefficients.

One could then simply invert the incident Aⁱ matrix, assuming that sufficiently random sampling has been performed such that it is full rank and square:

T = A^{s} {[A^{i}]}^{- 1}

In this approach, one requires one illumination angle per row of the T-matrix. Since we assume that the electric and magnetic modes are decoupled, half that number of measurements are needed. However, due to numeric instability in the FDTD algorithm and the T-matrix truncation, the inversion of Aⁱ in the above approach is numerically unstable and will result in an inaccurate T-matrix calculation. Instead, by over sampling one can construct an over-determined least squares problem for each row of the T-matrix:

A^{s} = T A^{i}

Where A^s and Aⁱ are now rectangular matrices formed by the column vectors of the a^s and aⁱ coefficients extracted from each simulation. In order to find the most probable T-matrix in a least squares sense, the problem must be slightly reformulated. The standard form for a least squares inversion of this type is:

y = M x

Where vector y and matrix M are known and the problem is over-determined in the sense that M is not square. The minimization problem is then solved by the pseudo-inverse, pinv:

arg min_{x} | M x - y | = pinv (M) y

Equation (22) can be put into this form by taking the transpose:

{(A^{s})}^{T} = {(A^{i})}^{T} {(T)}^{T}

Where superscript T denotes transpose. The nth column of (T)^T and the nth column of (A^s)^T are now in the correct form, so the solution of the nth row of the T-matrix is simply given by:

{(T_{n^{th} row})}^{T} = pinv (A^{i T}) {[A_{n^{th} row}^{s}]}^{T}

A built-in function in Matlab(TM) was used to calculate the pseudo-inverse using the Moore-Penrose method [37]. To accommodate the decoupling of the electric and the magnetic modes, Eq. (26) was employed separately for the T^(EE) and T^(MM) quadrants of the T-matrix, and the T^(EM) and T^(ME) quadrants were assumed to be zero.

In choosing the dimension of the T-matrix in this work, we have assumed that octupoles and higher order modes as well as their coupling to the dipole mode are negligible. In principle there may be contributions from higher order modes, but these modes are weaker, and due to their smaller angular extent are more prone to noise from digitization. Additionally, by including the octupole, 14 new rows need to be added to the T-matrix, dramatically increasing the number of free fit parameters. This can result in over-fitting and instability in the pseudoinverse function. As a structure becomes larger, these modes will scatter more light and will be less prone to over-fitting.

In order to verify the validity of this approach, we calculated the T-matrix for a simple gold sphere with radius 52 nm without applying additional constraints due to symmetry. Figure 4(a), shows that the calculated T-matrix is almost perfectly diagonal, with off diagonal terms several orders of magnitude smaller than the corresponding diagonal terms of the same rank. This T-matrix can be used to calculate the total scattering (Fig. 4(b)), backscattering (Fig. 4(c)) and extinction (Fig. 4(d)) cross sections. These results are in excellent agreement with the corresponding cross sections directly calculated using FDTD and those calculated using Mie theory. The accurate prediction of the extinction and backscattering cross section values demonstrate that the phase of the T-matrix elements are also accurately predicted. It is important to note that this approach allows simultaneous calculation of the T-matrix for a broad spectral range.

To further validate the method, the T-matrix of a gold nano-ellipsoid was also analyzed using this method. The ellipsoid had an aspect ratio of 1.3 with radii values of 52 nm, 40 nm, and 40 nm. The T-matrix was calculated based on 20 simulations at random orientations. The results shown in Fig. 5 indicate that this approach is able to accurately model shapes with strong mode mixing parameters and explicit asymmetry.

In order to show the accuracy of the calculated T-matrix, as well as the fact that the number of orientations chosen for this calculation was more than adequate, for a single orientation we compare a T-matrix based on all orientations and a T-matrix using every other orientation, excluding this one, in fitting the T-matrix. The new T-matrix was then used to calculate the scattering cross section for this excluded orientation, and compared to the old (full) T-matrix cross section. For the orientation shown in Fig. 11, the average change in scattering cross section is 0.6%, while the maximum change is 3.2%, suggesting that sufficient samples have been included and the T-matrix can predict the spectra of orientations that have not been directly used in the calculations.

C. Calculation of the quadrupole moments

While the scattering cross section is useful for studying the far-field, it poorly describes the dark quadrupole modes due to their extremely small scattering. In our previous work [33], the quadrupole moments of a typical orientation of this particle were calculated (see Fig. 15). This allows modal activity to be measured independently of scattering efficiency, although it still depends on the ability of the mode to couple to the incident field. These are calculated using the polarization current by applying the equation [38]:

Q_{α β} = \frac{i}{ω} \int {\vec{J}}_{α} \vec{r^{'}} {\vec{r^{'}}}_{β} + {\vec{J}}_{β} (\vec{r^{'}}) {\vec{r^{'}}}_{α} d V^{'}

Where J⃗_α is the polarization current, ${\vec{r^{'}}}_{β}$ is the position, α and β denote coordinate axis and the integral is over all space. While Fig. 7 shows a distinct decrease in scattering from the quadrupole modes as the wavelength approaches the electric dipole resonance, the quadrupole moments themselves remain strong (see Fig. 15). This is because the quadrupole moments are a description of how well the far-field couples to the quadrupole currents, while the quadrupole scattering (Fig. 7) measures both how well the far-field couples to the quadrupole moment, and how well the dark quadrupoles can scatter that power as a function of wavelength.

Acknowledgments

S.P.H and Z.F. acknowledge C. Li’s helpful contributions to the discussions. This work was supported by the startup funds from the University of Pennsylvania (for Z.F) and partial support from the MRSEC program of the National Science Foundation under award no. DMR-11-20901 at the University of Pennsylvania. S.L. thanks the Robert A. Welch Foundation (Grant C-1664), ACS-PRF ( 50191-DNI6) and NSF ( CHE-0955286) for financial support of this work. N.E. acknowledges partial support from the US Air Force Office of Scientific Research (AFOSR) Grant number FA9550-10-1-0408. P.S. acknowledges support from the Royal Thai Government. S.-J.P. acknowledges the support from the Camille Dreyfus teacher scholar award program.

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Modal interference in spiky nanoshells

Abstract

1. Introduction

2. Methods

2.1. Synthesis and dark field microscopy

2.2. Finite difference time domain (FDTD) simulations

2.3. T-matrix formalism

3. Results

4. Discussion

5. Conclusion

A. Scattering formalism

B. Calculation of the T-matrix

C. Calculation of the quadrupole moments

Acknowledgments

References and links

Cited By

Figures (15)

Equations (27)

Optics Express