The localized surface plasmon resonances in a metallic nanorod are determined using the “electrostatic approximation” and by a finite-difference time-domain numerical solution of Maxwell’s equations. The difference between the two methods is related to the effects of re-radiation, or retardation, which is not included in the electrostatic formulation. It is shown that high-order modes in a metallic nanorod can be modeled by both methods, even beyond the point where the electrostatic method is supposed to fail. This suggests that the simple analytical expressions derived from the electrostatic approximation are valid for describing the large range of resonant modes associated with metallic nanoparticles, including dark modes.
© 2009 OSA
Localized surface plasmon resonances in metallic nanoparticles arise from an interaction between the surface charges and their associated electromagnetic fields. Although Maxwell’s equations provide an accurate description of these resonances, the equations are difficult to solve analytically for particles of geometries other than spheres, ellipsoids or infinite right cylinders. Alternatively, there are many numerical methods that can be used to solve the equations depending on the application, such as the Finite Difference Time Domain method [1,2], Rigorous Coupled Wave Analysis [3–5], the Discrete Dipole Approximation  and the Boundary Element Method [7,8]. While these methods can provide accurate predictions of the properties of plasmonic systems, it is often difficult to extract parameter relationships from the numerical results. In this regard we have been pursuing analytical methods to model the underlying physics of interacting plasmonic systems, even if the methods give only approximate results. Through an understanding of the physics, it is then possible to design advanced systems of interacting plasmonic particles for applications in sensing and metamaterials.
Recently an approximate method was described for solving Maxwell’s equations for plasmonic systems in the case where the nanoparticle is much smaller than the wavelength of light . In this size regime, the electric and magnetic fields decouple and Maxwell’s equations take the same form as in electrostatics. In effect, the nanoparticle experiences a spatially uniform but time varying electric field arising from the applied light field. In this “electrostatic approximation” it is a simple task to calculate the self-sustained surface-charge distributions, or eigenfunctions of the nanoparticle resonances, and obtain the eigenvalues that determine the resonant frequencies. The electrostatic theory has been extended to take account of the interactions between nanoparticles in an ensemble , providing a simple means by which complex nanoparticle systems can be designed to produce the required optical properties. A key aspect of this theory is that it yields relatively simple analytical expressions describing the near-field interactions between optically resonant nanoparticles of arbitrary shape. Such expressions enable us to identify the important terms that affect the resonances of coupled systems and that control the intensity and polarization of light scattering. For example, we were able to use the method to design a plasmonic circuit , the equivalent of the Wheatstone bridge in electronics, which should enable the optical detection of single molecules, such as proteins, or provide a direct measure of phase shifts between two plasmonic nanoparticles. This circuit and similar subwavelength three-particle systems involve the formation of “dark modes” that interact weakly with the incident light [12–14]. The dark modes are associated with quadrupole and higher-order resonances that have zero dipole moments and therefore are not able to be excited directly by plane wave illumination. These three-particle systems can be analyzed algebraically using the electrostatic coupling theory .
Although we use the electrostatic method to obtain an understanding of the interactions between nanoparticles and to help us design plasmonic systems, its accuracy has been questioned, particularly for higher-order resonance modes [16,17]. The main issue relates to retardation, which is the effect of the phase difference between the fields propagating from two different regions of the nanoparticle and it is related to the effects of re-radiation. Such effects are important for coupling in large nanoparticle chains [18,19]. Traditionally, the electrostatic approximation has been derived from the first-order expansion terms for the Mie solution of light scattering from a spherical particle . In this approach, higher-order modes are not represented at all which gives the impression that electrostatic theory is only applicable to dipole modes. However, the eigenmode approach  represents a more complete solution of the problem and does enable a calculation of all of these modes, within the limit where nanoparticle sizes are much less than the wavelength of light.
Since we wish to use the analytical expressions derived from the electrostatic theory for applications involving both dipole modes and higher-order resonance modes, it is important to determine whether or not the theory can model higher-order modes and at what point would we expect the errors to become large. In this paper we compare the results of electrostatic calculations of the resonant modes of a nanoparticle rod with the results of the finite difference time domain (FDTD) method that includes the effects of retardation. The advantage of the electrostatic approximation is that it provides all of the resonant modes of a nanoparticle or an ensemble of nanoparticles, within the limitation of the numerical grid used to represent the nanoparticle surface. Moreover, the resonances are determined by the eigenvalues and the electric permittivity of the nanoparticle according to a simple relation. This means that the eigenvalues need only be evaluated once and the resonances can then be determined for any material. As discussed above, it is possible then to deduce how one nanoparticle couples to another. A limitation with purely numerical techniques such as FDTD is that there is no guarantee that all the resonant modes will be found.
2. Electrostatic approximation and the resonances of a nanorod
In the electrostatic method described by Mayergoyz et al , the localized surface plasmon resonances are found from the solution of an integral eigenvalue problem that solves for the self-consistent surface-charge or surface-dipole distributions of mode j of particle p. The resonant frequency of a particle mode j is then related to the eigenvalue , the electric permittivity of the background medium and the frequency-dependent permittivity of the particle according toEq. (1). One of the interesting aspects of the electrostatic method is that it is scale independent, provided that the dimension d of the nanoparticle is much smaller than the wavelength of the radiation in the embedding medium. That is, we require thatEq. (2) is satisfied, the resonances should be independent of the size of the nanoparticle. The amplitude of a resonance excited by an applied electric field is given by Eq. (3).
The solution of the electrostatic eigenvalue problem involves a sum over the surface of the nanoparticle which, for numerical computation, is represented by a finite mesh. Here we consider a nanorod consisting of a cylindrical body 1.5 units long and 0.8 units in diameter with hemispherical ends 0.4 units high, giving a total rod length of 2.3 units. The surface of the rod was tessellated with 1856 triangles and the eigenvalues calculated using the method discussed by Mayergoyz et al . The first ten modes of the rod are shown in Fig. 1 .
The first three modes are longitudinal modes whereas the next four modes are transverse. The transverse modes come in symmetric and anti-symmetric pairs. Since these modes have eigenvalues that are close to one another, their respective resonances are clustered together forming a resonant band. For these modes the resonant peaks are not well separated. The following three modes, 8 to 10, are again longitudinal modes.
It is clear that many of these modes have a zero dipole moment, such as modes 2, 4, 5, 8 and 10. This means that they cannot be excited by a plane wave incident light field, as discussed above. This is problematic for the FDTD method discussed in the next section and it highlights the value of the eigenvalue approach.
3. Resonances of the nanorod
3.1 Electrostatic resonances
To determine the resonant frequency of the nanorod, we assume that the rod is in a dielectric medium with a permittivity, which is approximately the same as water. It is convenient to represent the permittivity of the nanorod using the Drude formula that allows us to choose an appropriate damping factor. This is useful for the FDTD simulation discussed below. For the numerical simulation we useTable 1 , rounded to three significant figures.
3.2 Finite Difference Time Domain model
To compare the resonances obtained using the electrostatic approximation with those found by solving Maxwell’s equations including retardation, we have used an in-house FDTD numerical model based on the work of Taflove . The model uses a three-dimensional grid of points with the electric and magnetic fields placed a half step away from the centre of each point, following the Yee grid formalism . The grid consists of 75×75×75 points to represent the space around the nanorod, with the rod occupying a region 23 grid points in diameter and 67 grid points in length, including the hemispherical ends. The permittivity of the metal of the nanorod is represented by a second-order differential equation in time for the polarization . The coefficients of the differential equation are derived from the Drude model, Eq. (4). A small damping factor was used to promote strong resonances but to ensure some decay, particularly for non-radiating modes. The length scale is set independently of the number of grid points so that the same model can be run multiple times but with different scales to study the effects of size on the resonances.
The model simulates an electromagnetic field incident on the nanoparticle to excite the oscillations. However, a plane wave will only excite a nanoparticle mode if it has a dipole moment . Since many of the modes do not have a dipole moment, this becomes problematic. The simulation was configured so that the excitation source was a rectangular aperture slightly wider than the diameter of the nanorod and about the length of the hemispherical end. The aperture was placed near one end of the rod and the incident electromagnetic wave was emitted through this aperture. The incident wave was a Gaussian-modulated sinusoidal pulse with a centre wavelength at 800 nm and a standard deviation (width) of the Gaussian of 200 nm. This enables us to apply a short “numerical” electromagnetic pulse to the top of the nanorod in the hope that it will excite a large number of modes. The x, y, and z components of the electric field were sampled just below the nanorod, at the end opposite to that of the excitation pulse. Data were sampled as functions of time and then Fourier transformed to obtain the spectra. This process was repeated for four different scales corresponding to nanorod lengths of 90 nm, 230 nm, 460 nm and 690 nm. The resonances were identified by the order of the peak positions in the spectra.
The results of the first two simulations are shown in Fig. 2 . For the 90 nm nanorod we find that there is a good agreement between the excitation resonances calculated using the FDTD method and the resonant modes as determined by the electrostatic approximation. In particular, the locations of the higher-order modes are in agreement. The condition Eq. (2) for the validity of the electrostatic approximation is for the 90 nm rod based on nm for the fundamental resonance. If we use nm which is the resonance of mode 2, we find that so that we would not expect this mode to be accurately reproduced. However, Fig. 2 shows that the modes agree well, certainly within the 5% error we might expect for the FDTD model.
For the 230 nm long nanorod, the electrostatic approximation is much less accurate. For this nanorod, with nm, we find that which is clearly violating the condition given by Eq. (2). The spectra associated with this rod are red-shifted with respect to the shorter nanorod, but note also that the longer wavelength resonances are broader. This is a consequence of radiation damping. The longer is the rod, the stronger the dipole moment and the more efficiently it will radiate. The electrostatic approximation does not include radiation as this requires a coupling between the electric and magnetic fields. For small dimensions, this coupling is weak (and is neglected in the electrostatic approximation) so there is little radiation from the small nanorod. The broadening of the spectral peaks by radiation damping can be included in the electrostatic approximation  which appears as an additional loss term. This can also lead to a small shift in the spectral peak position, depending on the electric permittivity of the nanorod.
The results of four nanorod lengths are summarized in Fig. 3 . It is interesting to note that the lowest-order mode is the least accurately represented as the nanorod length is increased, showing the greatest fractional error. However, the errors in the higher-order modes increase more slowly. For the 230 nm long nanorod, the errors are below 8% for all modes except for mode 1 which has an error of about 18%. There is some fluctuation in the errors of the high-order modes for the longer nanorods which is probably due to the difficulty in identifying the individual modes, since some of them are not easily excited and the resonances can be broad.
For comparison with the results in Fig. 2, we show in Fig. 4 the resonant modes excited by a plane-wave electromagnetic pulse incident on the nanoparticle with the electric field polarized parallel to the long axis. The pulse was exactly the same as used previously except that it was applied to the entire rod instead of through the aperture. It is apparent from these spectra that many of the modes are missing. In particular, mode 2 is absent and in this regard can be considered a dark mode. From Fig. 1 it is clear that this mode has a strong linear-quadrupole moment with no dipole moment. As we discussed in section 2, modes that have no dipole moment cannot be excited by a plane wave when the nanoparticle is much smaller than the wavelength of the incident radiation. This highlights one of the limitations with this numerical method in determining the resonant modes of nanoparticle.
The comparison between the two calculations clearly shows that the electrostatic method is capable of predicting the resonances of many of the modes of a nanorod, including the high-order modes. This is important as it allows us to make use of the analytical formulae associated with this method for predicting the resonant effects arising from interactions between metallic nanoparticles and, in particular, the resonances associated with dark modes.
In terms of the failure of the electrostatic method, it is apparent that the low-order mode is most poorly represented as the size of the nanorod increases. A clue to the origin of this behaviour can be found in the second-order correction formulae as described by Mayergoyz et al . The zero-order term is the electric field of the nanoparticle that arises in direct response to the applied field. There is no magnetic response at this order because the electrons in the metal are driven mainly by the electric field (magnetic fields interact with electric currents). The first-order corrections to the electric field are zero because these corrections only involve the excitation of a magnetic field by the zero-order electric field. The second-order correction represents the excitation of an electric field by the induced magnetic field, which represents the effect of retardation. The electric permittivity at resonance with the second-order correction can be written in the formEq. (5) is that the correction depends on the eigenvalue and is larger when the eigenvalues are smaller (i.e. those close to 1). This means that the lower-order modes are most affected by increases in the size of the nanoparticle, which is consistent with the results in Fig. 3. Moreover, we see that the correction scales with the ratio of the nanoparticle surface area A to the square of the resonance wavelength λ in the embedding medium. For a spherical particle, this gives a scaling with the ratio of the square of the diameter to the square of the wavelength, in agreement with Eq. (2). However, the scaling with the nanorod area involves a product of its length L and its radius R so that we should use, rather than the square of the length which would give the erroneous result as used in Eq. (2). This suggests that nanorods or high aspect ratio nanostructures can be much larger than a corresponding nanosphere before significant errors appear in the electrostatic approximation. Although these conclusions are consistent with our numerical observations, they must be taken as rough trends since the integral ratio also depends on the geometry of the nanoparticle and on the nature of the resonant modes.
In this paper we have modeled the localized surface plasmon resonances of a nanorod using the electrostatic approximation and the FDTD method. It was shown that the electrostatic formulation can accurately predict the resonances of both low and high-order modes, including dark modes, provided that the nanorod is not too large. This is important because we wish to use the simple analytical expressions provided by this method to design optically resonant nanoparticle systems. It was found that the simple criterion for the validity of the electrostatic approximation does not provide an adequate measure of the point of failure of the method. It was shown that the second-order correction to the resonance permittivity involves the eigenvalues of the resonances and should scale as the ratio of the surface area of the nanoparticle to the square of the wavelength of the radiation in the medium. This suggests that the modes of higher aspect ratio nanostructures can be predicted with higher accuracy than lower aspect ratio nanostructures with similar dimensions.
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