1. Introduction

Electromagnetic scattering from nanometer-scale objects is currently being investigated both experimentally and theoretically for the understanding and prediction of novel near-field effects. This is particularly relevant due to the current research on nanomaterials and the use of nanometer-scale optical probes like near-field scanning optical microscopy (NSOM). Typically, near-field optical studies are performed on noble-metal nanostructures where resonances from surface plasmons play an important role in near-field coupling and propagation effects. Previous studies have focused on surface-enhanced Raman scattering (SERS)[1, 2], anomalous transmission through nanowire gratings[3] and plasmonics[4–7]. In contrast, little attention has been focused on other potentially interesting nanometer-scale optical effects. In this paper, we examine the magneto-optical scattering from nanometer-scale structures using a discrete dipole approximation (DDA). In the simplest form, the approximation replaces individual nanoparticles with a single, electromagnetic dipole radiator. The optical response of a collection of particles is calculated self consistently by calculating the response of each dipole to the incident field plus the scattered field from all the other dipoles.[8, 9] Used in this way, the approximation is often called the coupled dipole approximation (CDA). In addition, the approximation is used as a finite element computation technique where a complicated, non spherical object (target) is discretized into an array of interacting dipole radiators[10, 11]. The electric field, or alternatively, the scattering matrix is then calculated by summing the contribution from each dipole element.

Earlier, we reported on the magneto-optical response (Faraday rotation) from collections of Fe₃O₄ (magnetite) nanoparticles[12]. We found that the spectral Faraday rotation was dependent on the nanoparticle spacing. We were able to qualitatively model this behavior using the DDA formulated in the left-right circular polarization basis corresponding to the normal propagating modes for a magneto-optical medium. Here, we extend those calculations with a more general formulation of the DDA using a non-diagonal dielectric tensor appropriate to gyrotropic media and applicable to arbitrary magnetization direction. In addition, we state a general expression for effective Faraday rotation in terms of scattering matrix elements derived from the calculated polarizabilities. We demonstrate the use of this model by calculating the magneto-optical response of extended nanowire structures and nanoparticle assemblies containing two types of nanoparticles, one a magnetic oxide and one a noble metal.

2. Theory

2.1 Basic equations

In the DDA approximation, either the spherical nanoparticle, with diameter 2a≪λ, or a similarly small element of an extended structure with dimensions on the orderλ is represented by an oscillating dipole with polarizability tensor α j which is determined by the material properties. The dipole moment of the element located at r _j is P _j =αj E _loc,j where the local field E _loc,j is the incident field E _inc,j plus the field radiated by all the other dipoles,

The incident field is given by E _inc,j=E ₀ exp[i κ·r _j -ωt] where ω is the frequency of radiation and κ is the wavevector with magnitude κ=ω/c.

Substituting the expression for the dipole moment and rearranging yields,

where the radiation from a dipole has been written E _dp,k=A _jk P _k . Using the well-known expression for radiation of a dipole located at position _k r evaluated at point _j r [13, 14],

where n _jk is a unit vector pointing from r_k to r_j , and r_jk =|r _j -r _k |. Equations (2) and (3) are simplified by defining A _jj =α1- _j so that the equations can be written as a single matrix equation[14]

For N dipole elements, Ã is a 3N×3N matrix which describes the interactions between dipoles, _inc Ẽ is a 3N×1 vector, and P̃ is a 3N×1 vector of unknown dipole moments. Notice that A _jk , defined in Eq. (3), is itself a 3×3 matrix which becomes the (j, k) element of Ã.

Key to the implementation of the DDA is the connection between the dielectric properties of the material given by the dielectric tensor ε and the polarizability of the individual elements, α _j . The optical interactions with a material which exhibits magneto-optical effects (a gyrotropic material) are described by its dielectric tensor,

where Q≡ε_xy is the magneto-optical Voigt parameter and m_x, m_y and m_z are the direction cosines of the magnetization vector. The induced (uniform) polarization of a dielectric sphere valid for arbitrary anisotropy is given by[15, 16],

where 1 is the 3×3 identity matrix and a is the radius of the sphere This allows the identification of the polarizability as[15]

where (ε+21)^-1 is understood as matrix inversion. Draine et al. have established corrections to the basic formula which include a radiation reaction correction for finite wavelength[17]. From Eq. (7) a Clausius-Mossotti formula can be derived by summing the contributions from individual spherical polarizable elements[16]. Corrections which insure that an infinite lattice of polarizable points has the same dispersion relation as the macroscopic material are also given by Draine[17]. A note of caution: Eq. (7) is exact for infinite wavelength only for an isolated sphere or a dipole on an infinite lattice. For scatters with high aspect ratios (or otherwise arbitrary shapes) Eq. (7) should be modified to account for local field corrections[18]. Here, we wish to highlight the differences between the implementation of the DDA for isotropic and gyrotropic media, so we use the most basic form of the polarizability, Eq. (7).

In our magneto-optical experiments to date we have used the polar geometry in which the magnetic field is parallel (or antiparallel) to the direction of light propagation, in other words, m_z =1 and m_x =m_y =0 in Eq. (5). The equations are valid for arbitrary magnetization; the present implementation of the DDA will accommodate non-zero m_x or m_y without difficulty. We point out that the connection between magnetic field and optical field is through the magnetization-dependent dielectric tensor. In fact, at optical frequencies, the magnetic permeability is equal to unity and the effects of the magnetic field are characterized by the dielectric tensor through its dependence on the magnetization (see, for example, Ref. [19]).

2.2 Scattering matrix elements

The macroscopic magneto-optical effects, Kerr rotation for reflection and Faraday rotation for transmission, are defined in terms of Fresnel coefficients[20]. These familiar Fresnel equations are approximations which apply to optical interactions with surfaces larger than many square wavelengths[21]. More generally, and perhaps more fundamentally, optical properties are manifestations of absorption and reradiation of light from individual elements of the material. For this reason, a slightly different definition is needed for magneto-optical effects arising from materials whose dimensions are on the order of or smaller than an optical wavelength. We define the complex Faraday rotation angle as the difference in the scattering extinction coefficients for left and right circularly polarized light. The Faraday rotation, θ and ellipticity, η (per unit length), are given by

The extinction coefficients are calculated for a nanostructure (or collection of nanostructures) and the observable Faraday rotation is interpreted as being the result of a slab of a given thickness containing a given volume density of nanostructures.

In general, the calculation of the extinction coefficients requires the calculation of dipole moments from Eq. (4) for two orthogonal polarizations of incident electric field. The calculation of the scattering matrix elements using the DDA is given explicitly by Draine[14]. In our case, we take the both the incident and scattered fields to be the ẑ direction (forward scattering), which matches our experimental geometry. Using Draine’s notation, the scattering matrix elements are

where ê is a unit vector for the polarization and m and m′ denote the incident and scattered fields, respectively. ${\mathbf{P}}_j^{\mathit{\left(}m\mathit{\right)}}$ is the calculated dipole moment of the j ^th particle for incident polarization ê _m . Since both the incident and scattered fields are in the ẑ direction, the polarization unit vectors are simply x̂ and ŷ. Basically, we calculate the dipole moments for incident x polarization and evaluate the scattering into the y polarization and show that this is equivalent to the difference in extinction coefficients for left and right circularly polarized light. The Faraday rotation is calculated from the f_xy matrix element,

Substituting for f_xy from Eq. (9),

We now transform into the left-right circular polarization basis to show that this is indeed equivalent to Eq. (8). The usual transformation from (x,y,z) to (r,l,z), where r and l stand for right and left respectively, is given by ê _r =1/√2(êx-iê _y ) and êl=1/√2(ê _x +iê _y ). In this basis the x polarization is written

and the y unit vector is

Substituting this into Eq. (11) yields,

which is equivalent to difference in extinction coefficients for left and right circularly polarized light, C_ext,l-C_ext,r .

The calculation proceeds as follows. For a given geometry, a lattice is established with individual polarizabilities given by Eq. (7); values of the dielectric tensor are taken from the literature. Equation (4) is solved with incident x-polarized electric field which yields P ^(x) j. The Faraday rotation is calculated using Eqs. (11). Calculations involving few dipoles (<30) were performed using Mathematica[22]. Calculations involving more than 30 dipoles (up to several hundred) were performed using MATLAB using the available complex-conjugate gradient (CCG) matrix solver[23].

3. Results

We first apply the DDA calculations to magnetite nanowires. The model for the nanowires is similar to that used in Ref. [10] and is shown in Fig. 1. The diagonal and off diagonal components of the dielectric tensor are taken from the literature[24, 25]. The Faraday rotation is calculated for nanowires with 2 nm×2 nm cross section and lengths from 2 nm to 100 nm.

 

Fig. 1. Model for the magnetite nanowires. The spheres represent the individual discrete dipole elements. The cross section is 2×2 elements with a length L varied from 2 to 100 elements. In all cases the element grid spacing d=1.0 nm.

Download Full Size | PDF

The results of these calculations are shown in Fig. 2 for incident x polarization and nanowires aligned on the x axis and z axis. The magneto-optics calculations assume that the externally-applied magnetic field is large enough to saturate the magnetization of the material. In order to highlight differences in nanowire orientation, we assume that the magnetization is in the z direction in both Fig. 2 (a) and (b).

The absolute Faraday rotation is dependent on the volume of each scatterer as well as the number density of scatterers and thickness of the sample. We have normalized the calculations in Fig. 2 so that the Faraday rotation at the positive peak (~500 nm) is unity. The spectrum of the 2×2×2 element particle is the same as a single nanoparticle. Little change in the spectral Faraday rotation is noted for nanowires when the nanowire is aligned with polarization of the incident field. However, the Faraday rotation spectrum changes dramatically when the nanowire is aligned in the direction of light propagation (z direction). The overall effect is a blue shift of the spectrum with increasing nanowire length.

 

Fig. 2. Calculations of the Faraday rotation of a magnetite nanowire. The long axis of the nanowire is aligned with the (a) x axis and (b) z axis. In both cases, the magnetic field is applied in the z direction.

Download Full Size | PDF

We also performed calculations on a noble-metal/magnetic nanoparticle system to illustrate possible near-field effects. There are some experimental reports of enhancements in Faraday and Kerr effect for magnetic materials physically located near noble-metal nanoparticles [26–28] or thin films[29–31]. The electromagnetic field is concentrated near the noble metal surface when the optical frequency is resonant with surface plasmon. It is theorized that this is the origin of the observed enhancements and may be considered to be the magneto-optical counterpart of surface-enhanced Raman scattering[29, 32].

Specifically, we calculate the response of assemblies of silver and cobalt ferrite (CoFe₂O₄) nanoparticles separated by a few nanometers. This system was chosen because of the recent experimental report of magneto-optic enhancements in coupled silver-cobalt ferrite nanoparticles[27]. However, complete spectral magneto-optical data is not available, only Faraday rotation at selected laser wavelengths.

For these calculations, each particle is represented by a single dipole. Dielectric tensor data is taken from the literature for CoFe₂O₄.[33, 34] and silver[35] We used the experimental values for the particle radii, 7 nm for CoFe₂O₄ and 3 nm for silver[27]. Our results are shown in Fig. 3 for two-particle systems. A background dielectric constant of 1.7 was chosen so that the plasmon peak of silver was shifted to ~421 nm as the data in Ref. [27] indicate. In our calculations, the presence of the near-by noble metal particle results in enhancements in the Faraday rotation near the plasmon peak and is dependent on the particle geometry.

 

Fig. 3. Calculated magneto-optical response of a silver-CoFe₂O₄ particle pair as a function of interparticle spacing. The optical wave is propagating in the z direction and has linear polarization in the x direction. The calculations are performed for particles aligned on the (a) z axis and (b) y axis.

Download Full Size | PDF

 

Fig. 4. Calculated (a) optical extinction and (b) magneto-optical response of a silver-CoFe₂O₄ particle nanoparticle assembly. The 3×3×3 array of nanoparticles are arranged in a rock-salt structure.

Download Full Size | PDF

Figure 4 shows the optical extinction and magneto-optical response of a three-dimensional binary array: CoFe₂O₄ and silver arranged in a rock-salt structure. Binary assemblies of noble metal and magnetic particles have recently been demonstrated. These assemblies can have a variety of mesoscopic arrangements isostructural with rock salt, Cu₃Au and AlB₂, for example[36]. The optical extinction spectrum is dependent, of course, on interparticle spacing, even splitting into two peaks at the smallest separation. Previous calculations have shown line broadening, splitting and even line narrowing in regular arrays of noble metal nanoparticles[8]. In our calculations, the primary effect in the magneto-optical spectrum is at or near the surface plasmon wavelength of the noble-metal particles and results in an increase in the magnitude of the Faraday rotation of up to a factor of five. The spectrum is also dependent on the particle arrangement. For example, in the calculations presented here, the ferrite particles are located on the unit cube corners; slightly different spectra are obtained if the positions of the ferrite and silver particles are interchanged.

4. Discussion

These calculations provide a tool for modeling the magneto-optical scattering from non-spherical structures as well as provide a framework for the basic, qualitative understanding of the near-field coupling effects on magneto-optical scattering. Incidentally, none of these effects are predicted by any effective-medium type model or any mean-field theory which does not include near-field interactions. The calculations for the nanowires predict shape and orientation effects. The effects are most evident for the wires aligned with the z axis. This suggests that the effects are due to an anisotropy induced by the wire. The incident x-polarized optical wave consists of equal parts left- and right-circular polarizations. For bulk materials or nanowires aligned in the x-y plane, these are the normal modes of propagation. However, for the wires aligned with the z axis, these modes no longer propagate without change of polarization state; there is a depolarization of the light. Note that magneto-optical effects arise due differences in the real or imaginary parts of the index of refraction for circularly polarized light Anisotropy in the x-y plane could effect linear but not circular birefringence or dichroism.

For the binary nanoparticle arrangements, the presence of the noble-metal and subsequent concentration of the electromagnetic field perturbs the polarization of the magnetic nanoparticle and, hence, the reradiation of the optical field. As would be expected, the most profound changes are observed near resonances of the silver particle (or particle arrangement). We predict that such effects would be observed for particle spacings on the order of 20 nm or less. In Ref. [27], the silver and cobalt ferrite particles were attached forming dimers (two-particle assemblies). The optical absorption shows a peak at ~421 nm, but the primary differences between the measured Faraday rotation for cobalt ferrite alone and cobalt ferrite with silver particles attached occurs at much longer wavelengths. Other reports of magneto-optical enhancements resulting from composites with noble-metal nanoparticles contain Faraday rotation data at even fewer wavelengths. Without more complete spectral Faraday rotation, we can only speculate on the absence of an observable effect near the plasmon resonance.

5. Conclusion

We have developed a model for magneto-optical scattering from nanometer-scale structures based on a modification of the discrete dipole approximation. The model can be used as finite element method for the calculation of the magneto-optical response of non-spherical nanostructures with dimensions approaching the wavelength of light. The model can also be used to calculate magneto-optical scattering from collections of nanoparticles, where each nanoparticle is represented by a radiating dipole. The particle collections need not be homogeneous; in fact, we apply the model to binary nanoparticle systems which consist of a noble-metal and magnetic oxide nanoparticles. We calculate enhancements of the Faraday rotation near the plasmon resonances of the metal nanoparticles. These enhancements are evident for interparticle spacings on the order of 20 nm or less.