An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas

At optical frequencies metals behave as an electron plasma and conventional antenna designs need modifications when transferred to this regime. In contrast to antenna theory and to the effective wavelength picture, the position and width of the dipolar resonance of a rectangular cuboidal plasmonic nanoantenna scales nonlinearly with its length, width and height, as shown in this paper directly by analytical formulae. Moreover we show that the quality factor calculated for different sizes varies significantly with size, in contrast to the quasi-static approximation which predicts invariance. We present analytical expressions that provide a tool for direct and precise calculation of the dipolar plasmon resonance which can be applied to the antenna design process. These expressions enable both physical insight and the quick exploration of a wide range of parameters to tailor the plasmon resonance response or scattering by nanoparticles, for either metals or dielectrics, for numerous promising applications in optical sensor, photovoltaic and light emitting device design.

This picture is more effective in the infra-red regime [33] but it fails to provide an accurate solution in the interesting range of optical frequencies, where structures are usually in the range of 100 nm or less. Moreover, under the quasi-static approximation, theory predicts that the Q factor is determined solely by the complex dielectric function of the metal and is independent of the nanostructure form or dielectric environment [34]. We will show that this is not the case for rectangular cuboidal nanoantennas usually used, due mainly to radiation losses, which are neglected in the quasi-static approximation.
In this paper, we provide general analytical formulae for the calculation of the scattering properties of a rectangular cuboid (which, from now on, will be simply referred as cuboid) inside a medium, in the assumption that the field inside the cuboid is constant, corrected for the charges at the vertices. From these formulae, we consider the scaling of the position and width and of the value of the peak of the extinction efficiency of the resonance with the length, width and height of the cuboid. The results improve upon those given by the effective wavelength model for metal structures smaller than 100 nm and illustrate the scaling and behaviour of the Q factor of the resonance, highlighting its change with the nanostructure geometry. The results have been compared for a wide range of geometrical parameters against finite element method (FEM) simulations using COMSOL, obtaining a very good agreement.

Formalism
Consider a rectangular cuboid of volume V placed with its centre at the origin, as shown in figure 1, with length L a = 2a, width L b = 2b and height L c = 2c. The cuboid is composed of a material with a complex, wavelength dependent, dielectric function and is surrounded by a background material with a dielectric constant B . The incident electromagnetic field is a plane wave E(r) = E 0 e ik B z , where, in all the following, it will be assumed the time varying dependence of the field is given by e −iωt . The electric field E 0 is along the x direction and the wave-vector in the background is k B = √ B k 0 = √ B ω/c. A first analytical expression is obtained using the Meier-Wokaun approach [35]. Later we improve the result by means of the Green function formalism. This formalism [36,37] has been used in order to obtain an expression for the scattered field in the far field under the assumption that the electric field inside the cuboid is constant and is given by the field at the central point previously obtained with the Meier-Wokaun approach. The field at the central point is calculated by considering the effects of depolarization in the volume and of the charges induced at the vertices. We can consider this as the zero order of a modal expansion and this can be considered as an educated guess confirmed a posteriori. It is important to point out that we model only the dipolar order and, secondly, that using the experimental dielectric functions of metals instead of the Drude formula (particularly gold), the resonances higher than the dipolar are strongly suppressed by the losses, making this assumption a good approximation. Moreover this approximation is reasonable considering the small scale of the nanoparticles and the fact that gives the exact solution for a small sphere [35]. A full derivation of the resulting equations is provided in the supporting information.
1.1.1. Electric field inside the nanoantenna. Starting from the Meier-Wokaun [35] approach for a sphere, the electric field inside a cuboidal nanoparticle can be approximated by considering the necessary corrections due to the induced charges (see the appendix). The electric field inside  (2) is also shown in dark. (b) Comparison between the extinction cross sections for cubes of side lengths L = 6, 30 and 80 nm calculated using our theory (solid black line) and the ones obtained by FEM simulations (dotted red line). Results obtained using the dipolar approximation (dashed blue line), which works best for small nanoantennas, are also shown. The smaller peaks appearing between 450 and 500 nm in the numerical simulation curves are higher order modes.
the nanoantenna may then be expressed as where E 0 is the incident field, is the solid angle subtended by the side perpendicular to the polarization axis of the cuboid (the x-axis in this case), i.e. the side L b -L c (see figure 1(a)), which gives the singular contribution of the dyadic [37], and is expressed by β is the dynamic geometrical depolarization factor, defined by equation (A.3) in the appendix section and with full derivation in the supplementary data (available from stacks.iop.org/NJP/15/063013/mmedia). For a cube, it is expressed by β cube ≈ 12.6937a 2 . δ is a term that takes into account the polarization charges at the planar ends of the cuboid orthogonal to the x direction, and is expressed by Lastly, the term 16 3 ik 3 B abc constitutes the radiative correction to the field. Note that equation (1) takes into account the effect of depolarization from all dipole moments surrounding the centre [35].

Dipolar approximation.
The above expression for the internal electric field can be used together with the dipolar expressions for the scattering and absorption cross sections: and respectively, along with the polarizability α, obtained from the dipole moment p = 0 B αE 0 , and defined as to obtain very simple expressions for the cross sections, with the extinction cross section defined as σ ext = σ sca + σ abs .

Far-field scattering.
Although the field inside the nanoantenna allows us to derive simple dipolar formulae for the scattering, absorption and extinction cross sections, a more accurate result is obtained by now considering scattering in the far field using the Green function formalism (see the appendix), since in this case we do not assume that the scatterer is a point dipole. Starting with equation (A.6) and considering the field inside the nanoantenna given by equation (1), the scattering cross section is obtained as where = − B . A derivation is shown in the appendix section. The extinction cross section is also obtained from the scattered field by means of the optical theorem using equation (A.9) where is the real part. The absorption cross section is calculated by using the expression σ abs = σ ext − σ sca . In order to compare with simulation results, we consider the case of gold (Au) nanoantennas. The dielectric function of the Au has been expressed using Drude model, which fits well to the experimental dielectric function but neglects the interband transition region. Note that this is chosen only for convenience during the comparison. In fact, the model works with an arbitrary dielectric function, , and therefore also with any experimental dielectric function. In particular, we have simulated gold using the following expression: where ∞ = 10.7026, ω P is the plasma frequency (ω P = 1.3748 × 10 16 Hz) and γ is the collision frequency (γ = 1.1738 × 10 14 Hz). We will show that the analytic formulae agree very well with numerical calculations over a wide range of values, from just a few to ≈100 nm.   (4) and (5), dashed blue line) and the ones obtained by FEM simulations (dotted red line). The dimensions of either L a , L b , L c which are not indicated in the inset on each figure are equal to 60 nm. The smaller peaks appearing between 450 and 500 nm in the numerical simulation curves are higher order modes.

Results and discussion
The geometry of the scattering problem is shown in figure 1(a), where the electric field is incident in the x direction. Figure 1(b) shows the comparison between the extinction cross section for cubes of different side lengths L calculated using our approach (equation (8), solid black line), our dipolar approximation (equations (4) and (5), dashed blue line) and that obtained by numerical FEM simulations using COMSOL (dotted red line). The plasmon resonance appears as a single main peak that red-shifts and widens with increasing the size of the particle. Smaller peaks appear between 450 and 500 nm in the simulations curves due to higher order modes. Figure 2 shows the comparison for cuboids of different side lengths between our theory (equation (8), solid black line), our dipolar approximation (equations (4) and (5), dashed blue line) and the same obtained by FEM simulations (dotted red line). To avoid numerical errors due to the discontinuity at the surface in the FEM simulations, all sharp corners and edges of the cuboid have been slightly smoothed by spherical or cylindrical surfaces of radius R = L/10 where L is the shortest side length. The total scattering cross sections were obtained by integrating the scattered power flux over an enclosing spherical surface outside the cuboid, while the absorption cross sections were determined by integrating the Ohmic heating within the cuboid. Each numerical simulation curve shows a strong dipolar resonance, as well as several weak higher order resonant modes.
As shown in figures 1 and 2, theory and FEM simulations are generally in good agreement. Both expressions (8), (4) and (5) give good information about the plasmonic resonance. In particular, equation (8)  and very well its width and strength for particles of only few nm to more than 100 nm. We can also see that our model works better for larger particles, while, for smaller ones, we notice a small blue-shift of the resonance of approximately 20 nm. This is due to the fact that the constant field approximation does not exactly capture the excited modes inside the cuboid.
The good agreement of these analytical results with complex simulations means that the analytic expression provides a simple, quick, yet accurate method of evaluating the effect of varying size parameters on the plasmon resonance. For example, one can study the behaviour of the resonance when varying the cube side length. Interestingly, one discovers that the plasmon resonance peak shifts nonlinearly with the length, as shown very clearly by figure 3, which shows the extinction efficiency, σ ext /L 2 , for cubes of different side length L as a function of the wavelength. As expected, the resonance red-shifts and broadens when increasing the length of the structure, due to the increases in the effective wavelength and radiative losses, respectively. From figure 3 one can see the existence of two scaling regimes, one nonlinear for side lengths smaller than approximately 80 nm and one linear for side lengths larger than that.
It is also possible to check quickly the tuning of the plasmon resonance to a desired position by changing any of the three dimensions of the cuboid. In particular, as shown in figure 4, the plasmon resonance position scales nonlinearly when changing the length, width and height of the cuboid, in contrast to the predictions of the effective wavelength model which show a linear scaling with the length. In particular, the squeezing of the width and height causes a significant red-shift of the resonance.
These results are important to establish design rules for transferring antenna technology to the optical regime. This technique can then be extended to complex designs composed, for instance, of many antenna rods such as Yagi-Uda antennas.
Another interesting parameter to characterize a plasmon resonance is the quality (Q) factor, which is inversely proportional to the width of the resonance. Figure 5  Quasi-static theory is always above the values from our theory, because it neglects radiation losses. Results are calculated using equation (8).
important to tune the plasmon resonance to a specified region and to have a resonance with the highest Q factor achievable in order to have the greatest sensitivity. Quasi-static theory predicts that with losses occurring only in the metal part of the nanostructure, the Q factor may be  expressed as where and are the real and imaginary parts of , respectively, and we have used the definition of from the Drude model (equation (9)). Note that equation (10) neglects losses due to radiation, which effectively increase the loss coefficient , making the Q factor smaller. Figure 5(b) shows the comparison between the results predicted by equation (10) and the values calculated using our formalism from Q = ω/ ω, where ω is the FWHM of the plasmon resonance, for cuboids of different side lengths. The quasi-static theory always produces values above those from our theory because it neglects radiation losses, and, in the worst case, can be off by up to a factor of 2 compared to the value from our theory.
Another important parameter for sensing is the figure of merit (FOM) [9], which is defined as FOM = ∂λ ∂n / λ, where λ is the resonance width and ∂λ ∂n is the shift of the resonance wavelength upon a change of the refractive index of the surrounding medium n. Clearly, for a known background refractive index, upon a shift ∂n of its value which causes a shift of the peak of the plasmon resonance ∂λ, equation (8) makes it possible to obtain ∂λ ∂n and the width of the resonance λ. In this way we have a direct and useful method to obtain the FOM.
Finally, we consider how the maximum of the extinction efficiency, σ ext /(L a L b ), shown in figure 6 varies with changing length, width and height. This provides important information on how to increase the extinction efficiency at resonance for the same volume of material. Changing the length L a , the extinction efficiency monotonically increases, in some cases to very large values (over 40), whereas changing the width L b has the inverse effect, since we obtain large values only for small widths. Changing the height L c has a small mixed effect with an increase or decrease for smaller height L c depending of the values of the length and width. This can be used for the fabrication of strongly scattering nanoantennas, with low absorption, since it shows the marginal contribution of a change in size of each dimension.

Conclusions
In summary, we have derived analytical expressions for the extinction, scattering, and absorption cross sections of a rectangular cuboid, showing a nonlinear scaling of the plasmon resonance position, width and extinction efficiency. This enables one to precisely downscale antenna designs to optical frequencies by enabling a convenient calculation of the dipolar plasmon resonance. Moreover, the formalism can be applied to the design of strongly scattering nanoantennas with low absorption and to the analysis of operational parameters such as the Q factor and FOM. In fact, we have shown that the Q factor changes significantly with the dimensions of the cuboid, in contrast to the quasi-static approximation, which predicts invariance with size. This enables the simultaneous tailoring of the Q factor of the plasmon resonance and of the FOM for sensing applications.
which can be expressed analytically with a rather long expression, given in the supporting information. The third term can be integrated in a simple way to give 16 3 ik 3 B abc, which takes into account the radiative correction to the field. In this way we obtained By considering also the effect of polarization charges at the planar ends of the cuboid and orthogonal to the x direction, another term in the expression of E dep,x appears. Using equation P · n = σ where σ is the surface charge at the planar ends and n is the external normal vector, we obtain that the charge at the surfaces, which we consider concentrated at each vertex, is given by q = P x bc for each vertex, where q is positive in the x = a and negative in the x = −a planar surfaces. The contribution to the field along x at the centre of the cuboid given by the charges at the vertices is where we have taken into account the projection of the electric field along x generated by the charges. Using the expression of equations (A.4) and (A.1) and by defining δ as equation (3), we are left with the expression of equation (1), which gives a corrected expression that takes into account the induced charges at the surface of the cuboid.

A.2. Far-field
Consider the expression of the scattered field given by the Green formulation, where ↔ GFF is the Green function in the far field: where R = r − r , R = |r − r | and = − B , were is the dielectric function of our antenna and B that of the surrounding medium.
Given that the structure is symmetric in the x − z and y − z planes, it can be shown that the electric field fulfils the following symmetry relationships: which are valid for each z = constant plane. Our main assumption is that the field inside the cuboid is given by equation (1). With this assumption and by doing the approximations of 1/R ≈ 1/r , R ≈ r − r cos γ , where γ is the angle between r and r , and 1 − RR R 2 ≈ 1 − rr r 2 since the higher order terms generate contributions that go to zero faster than 1/r 2 which are negligible in the far field, one obtains From the scattered field, equations (7) and (8) give the scattering and extinction cross sections.