## Abstract

An analytical model of the response of a free-electron gas within the nanorod to the incident electromagnetic wave is developed to investigate the optical antenna problem. Examining longitudinal oscillations of the free-electron gas along the antenna nanorod a simple formula for antenna resonance wavelengths proving a linear scaling is derived. Then the nanorod polarizability and scattered fields are evaluated. Particularly, the near-field amplitudes are expressed in a closed analytical form and the shift between near-field and far-field intensity peaks is deduced.

© 2012 OSA

## 1. Introduction

The study of nano-plasmonic antennas represents a rapidly growing field of both experimental and theoretical research (see e.g. recent reviews [1–4]). Different numerical simulation techniques for calculation of scattered electromagnetic fields from antennas have been developed and successfully applied to a large variety of systems. However, these simulations generally do not provide a direct insight into physical processes taking place in antennas and make the interpretation of the results rather difficult. Further, if the one dimension of the antennas prevails to the others (typical for nanorods), approaches based on the numerical solution of the Maxwell equations with proper boundary conditions are cumbersome, especially when expressing the local electromagnetic field in the vicinity of such a body. For instance, in case of the finite element method, inconvenient shapes of elements with large angles inherently causing well-known convergence problems have to be generally chosen to match the mesh to the antenna shape properly. Several attempts to describe the processes in plasmonic antennas more analytically have been made (e.g. [5]). Recently, an analytical model based on the RF antenna theory and describing plasmonic antenna resonances was reported in [6] where, in contrast to the RF regime, a constant volume current was assumed.

In the present paper we have developed an analytical approach to the optical antenna problem based on a study of the response of a free-electron gas within a metallic nanorod to the incident electromagnetic wave (Fig. 1). The radiation penetrates into the metal and excites longitudinal oscillations of the free-electron gas along the nanorod. Supposing the nanorod radius is smaller than the skin depth of the metal, which is about 20 nm for frequencies much less than the plasma frequency of the metal [7], the electron longitudinal displacement may be considered approximately uniform over the nanorod cross-section similarly to [6, 8]. First, in Sec. 2 free oscillations of an electron gas under the quasistatic approximation are examined and the phase velocity of electron density waves travelling along the nanorod is derived. This velocity determines nanorod resonance wavelengths, which depend on the plasma frequency and aspect ratio *l/R* of the nanorod, where *l* and *R* are its length and radius, respectively. The resulting resonance wavelength formula is compared with that derived by Novotny [9] who used the results of the waveguide theory. Next, in Sec. 3 the oscillations of an electron gas driven by an external electromagnetic wave are studied and the nanorod polarizability which is crucial for the calculation of scattered fields is derived. Using this polarizability the near-field scattered from the nanorod is evaluated in Sec. 4. Finally, in Sec. 5 the shift between the near-field and the far-field intensity peaks is deduced.

## 2. Calculation of nanorod resonance wavelengths

Providing that electrons move against a background of heavy positive ions the local total charge distribution along the nanorod is changing. As the nanorod radius is considered very small in comparison with the wavelength of the incident wave and thus comparable with the skin depth of the metal, electron oscillations along the nanorod are assumed uniform over the nanorod cross-section and so only the electron displacement *u*(*x*, *t*) will be investigated. The linear density of the local total charge within the nanorod can be expressed by the electron displacement *u*(*x*, *t*) as

*n*

_{0}is the electron concentration in the metal,

*e*is the elementary charge and

*R*is the radius of the circular-shaped nanorod. Now the electric potential

*φ*(

*x*,

*t*) due to the charge density given by Eq. (1), and then the linear density of electric potential energy of the nanorod from the relation can be calculated (see e.g. [10]). Integrating along the nanorod yields

*x*′ contributes to the potential energy density at the point

*x*. Figure 2 reveals that only the charges located at distances shorter than ten nanorod radii

*R*from the point

*x*affects the electric potential energy density at this point significantly. Thus, assuming that

*R*≪

*λ*the field retardation in the nanorod was not considered. Furthermore, the repulsive forces due to Pauli’s exclusion principle in the electron gas within the nanorod are negligible in comparison with the electric forces induced by an external electromagnetic wave (see appendix).

Next the linear density of the kinetic energy of the electron gas within the nanorod will be evaluated. As the amplitudes of free electron oscillations are small (see appendix) the changes of the concentration of the electron gas along the nanorod may be omitted so that

*m*

_{e}is the electron mass. Note that when deriving Eq. (5) much slower motion of electrons than the vacuum velocity of light was considered.

The equation of motion for the electron displacement *u*(*x*, *t*) is consequently obtained from the Euler-Lagrange equation

*u*≡

_{t}*∂u*(

*x*,

*t*)/

*∂t*and

*u*≡

_{x}*∂u*(

*x*,

*t*)/

*∂x*. After inserting Eqs. (3) and (5) into Eq. (7) the Euler-Lagrange equation (6) gives

*ω*

_{p}= [

*n*

_{0}

*e*

^{2}/(

*ε*

_{0}

*m*

_{e})]

^{1/2}is the plasma frequency related to the nanorod metal. As the function

*g*(|

*x*−

*x*′|) given by Eq. (4) takes on significant values only for

*x*′ in the close neighborhood of the point

*x*, the function

*∂u*(

*x*′,

*t*)/

*∂x*′ will be expanded in a Taylor series about the point

*x*. Then the integro-differential equation (8) becomes

*g*(|

*x*−

*x*′|) is significantly nonzero only when |

*x*−

*x*′| < 10

*R*(see Fig. 2), its value does not practically depend on

*x*except when the point

*x*is near the nanorod ends. Approaching the nanorod end the integral ${\int}_{0}^{l}g\left(\left|x-{x}^{\prime}\right|\right)\text{d}{x}^{\prime}$ decreases for

*x*= 0 or

*l*to its half value inside the nanorod. Thus for

*l*≫

*R*the integral can be approximated by its mean value

*ϑ*= 2e

^{−1/2}= 1.213.... Under this approximation Eq. (10) yields the wave equation

*ω*

_{p}is typically 10

^{16}s

^{−1}and the term [ln(

*ϑl/R*)]

^{1/2}is in the order of magnitude 10

^{0}for a wide range of the nanorod aspect ratios 10

^{1}<

*l/R*< 10

^{4}. This indicates the nanorod radius

*R*should be less than 10

^{2}nm in our model in order to keep the phase velocity

*v*below the vacuum velocity of light

*c*. Otherwise our model considering the local charge density uniform over the nanorod cross-section is no longer valid, and a finite penetration depth of electromagnetic field into metals becomes important. Thus, the amplitude of longitudinal charge-density waves decreases with the depth of penetration into a metallic antenna nanorod and the number of free carriers participating in the antenna response to the external electromagnetic wave is no longer proportional to

*R*, as assumed in our model. Hence, the velocity

*v*will not exceed

*c*.

The solutions of the wave equation (12) satisfying the Dirichlet boundary conditions

are represented by standing waves*=*

_{j}*vk*is the angular eigenfrequency of electron gas free oscillations. From Eq. (13) it is obvious that Ω

_{j}*depends on the plasma frequency*

_{j}*ω*

_{p}and the aspect ratio

*l/R*of the metallic nanorod:

If the nanorod is illuminated by an external electromagnetic wave polarized along the nanorod (Fig. 1), the system comes into resonance when the frequency of the incident wave is approximately equal to Ω* _{j}*, i.e. the incident wavelength roughly equals Λ

*= 2*

_{j}*πc*/Ω

*. From Eq. (17) we get*

_{j}*λ*

_{p}= 2

*πc*/

*ω*

_{p}is the plasma wavelength. Recently, utilizing the waveguide theory Novotny [9] derived a linear scaling rule relating the effective nanorod wavelength

*λ*

_{eff}to an incident wavelength. Considering a half-wave dipole antenna (

*λ*

_{eff}= 2

*l*) made of a metal in which only the free-electron gas contributes to its dielectric function (i.e.

*ε*

_{∞}= 1 in the Drude formula) and surrounded by vacuum (i.e.

*ε*

_{s}= 1), Eq. (14) in [9] derived for a very thin nanorod yields the incident wavelength corresponding to the first mode of the antenna To compare our result [Eq. (18)] with that obtained by Novotny [Eq. (19)] the function

*ψ*(

*ξ*) =

*ξ*[ln(

*ϑξ*)]

^{−1/2}, with

*ξ*=

*l/R*being the nanorod aspect ratio, was expanded in a Taylor series about a point

*ξ*

_{0}. As the second- and higher-order derivatives of the function

*ψ*(

*ξ*) are much smaller than the first one for 20 <

*ξ*

_{0}< 150, our result takes the same form as Eq. (19)

*γ*

_{1}and

*γ*

_{2}depend on

*ξ*

_{0}. For 20 <

*ξ*

_{0}< 150 the coefficient

*γ*

_{1}ranges from 1.1 to 4.0 while the coefficient

*γ*

_{2}decreases from 0.30 to 0.25. Hence, it can be concluded that both distinct approaches provide nearly the same value of the first resonance wavelength for very thin nanorods of radii up to 10 nm. When the radius of the nanorod exceeds the penetration depth the charge distribution may be no more considered uniform over the nanorod cross-section. This leads to a decrease of the number of free carriers participating in the antenna response which results in the modification of the value of the plasma wavelength. Then the plasma wavelength

*λ*

_{p}depending on the material only should be replaced by its effective value depending on

*R*. Thus the resonance wavelength of the nanorod becomes dependent both on the aspect ratio and the thickness of the nanorod reported e.g. in [8].

## 3. Nanorod polarizability

In this section the longitudinal oscillations of the electron gas driven by an external electromagnetic plane will be studied. The wave incidents perpendicularly to the nanorod while its electric field **E**^{ext} is parallel with the nanorod main axis (see Fig. 1). Then the wave equation (12) is replaced by

*ω*being the amplitude and the angular frequency of the external wave, respectively. In equation (21) the damping term (1/

*τ*)(

*∂u/∂t*) accounting for Joule’s heat produced within the metallic nanorod has been introduced. The symbol

*τ*represents the electron relaxation time. For simplicity, the radiation damping has been neglected in this equation.

The electron displacement *u*(*x*, *t*) fulfilling the boundary conditions given by Eq. (14) can be expressed as

*k*is given by Eq. (16). After inserting Eq. (23) into Eq. (21) and considering that functions {sin(

_{j}*k*)} form an orthogonal system over the interval (0,

_{i}x*l*), it is found that

*q*(

_{j}*t*) obeys the differential equation

*=*

_{j}*vk*are given by Eq. (17). In the steady state where the amplitudes

_{j}*q*

_{m,j}are given by

As *k _{j}* =

*jπ*/

*l*the integral in Eq. (25b) is proportional to 1 − cos(

*jπ*) which means that for even

*j*the amplitudes

*q*

_{m,j}are equal to zero. This has a direct physical explanation. As the external electromagnetic wave impinges on the nanorod under the right angle, the driving force is constant along the nanorod at the same moment and can thus excite only the modes symmetrical with respect to the nanorod center.

After inserting Eqs. (25a,b) into Eq. (23) the following expression for the electron displacement is obtained

*jπ*) in Eq. (29) is equal to zero for

*j*= 2, 4, 6,..., even modes are not excited. On the other hand, if

*j*= 1, 3, 5,..., this term is equal to 2 and the polarizabilities of odd modes read

*V*=

*πR*

^{2}

*l*being the nanorod volume. The term

*j*

^{2}in the denominator indicates that the higher mode is excited, the smaller amplitude of its dipole momentum is induced.

Finally, using Eqs. (27) and (30) the relationship between the complex amplitude *u*_{m,}* _{j}* of the

*j*-th mode of the electron displacement and the nanorod polarizability of the corresponding mode can be found

## 4. Near electromagnetic field scattered from a nanorod

The scattered electromagnetic field in the vicinity of a nanorod illuminated by an external plane electromagnetic wave can be calculated in the quasistatic approximation. Considering the geometry in Fig. 1 and the Coulomb and Biot-Savart laws, the scattered field components read

*τ*

_{tot}is the linear density of the local total charge within the nanorod given by Eq. (1) and

*J*= −

_{x}*n*

_{0}

*e*(

*∂u/∂t*) is the current density. Inserting the electron displacement given by Eq. (26) with the amplitudes expressed by Eq. (31) into Eqs. (32a–c) we get

*l*= 1

*μ*m and

*R*= 10 nm have been calculated for the first two modes close to resonance, i.e. for

*ω*= Ω

_{1}and Ω

_{3}(Fig. 3a, c). For comparison, the results obtained by 3D finite-difference time-domain (FDTD) simulations [11] for the identical antenna are shown in Fig. 3(b, d). A very good qualitative and quantitative agreement between the maps calculated by analytical formulae and by simulations is evident.

In a close proximity of the nanorod where *l* ≫ *y* the function *f*(|*x* − *x*′|, *y*) in integrands in Eqs. (34a–c) has a sharp peak for *x*′ in the neighborhood of *x* and so it is decisive for the value of the integrals. Thus the functions cos(*jπx*′/*l*) and sin(*jπx*′/*l*) can be expanded in a Taylor series about the point *x* and then approximated by the first two terms as

Note that Eqs. (38b,c) approximate the field amplitudes very well for all modes in the whole near-field domain. On the other hand, the two terms in the bracket on the right hand side of Eq. (38a) do not approximate the higher mode amplitudes sufficiently and thus more terms in Taylor’s series in Eq. (36) need to be taken into account.

## 5. Shift of resonance peaks in the near- and far-field spectra

In the nanorod vicinity the electric field is proportional to the dipole momentum *p _{x}* whereas in the radiation zone to the second derivative of the dipole momentum

*p̈*∝

_{x}*ω*

^{2}

*p*, where

_{x}*ω*is the angular frequency of the external wave [12]. This is why the shift between the near-field and the far-field intensity peaks occurs. Considering Eq. (29) we get for particular mode

*j*

*ω*≡

_{j}*ω*

_{far,j}−

*ω*

_{near,}

*is approximately*

_{j}Considering relation (17) for Ω* _{j}* we see that the higher modes give the smaller frequency shifts Δ

*ω*and that the shifts depend upon the nanorod dimensions. In particular, for a very thin nanorod the shifts depend linearly on the aspect ratio according to Eq. (20) and the relation 1/Ω

_{j}*= Λ*

_{j}*/(2*

_{j}*πc*). Furthermore, the shifts depend on the intrinsic damping expressed by 1/

*τ*

^{2}, as also reported in [13]. Naturally, for ideal metals (where

*τ*→ ∞) the shift between the near-field and far-field peaks becomes zero. As the electron relaxation time

*τ*in metals is typically 10

^{−14}s, the relative frequency shift Δ

*ω*/Ω

_{j}*≃ (Ω*

_{j}*)*

_{j}τ^{−2}will be about 0.1 % for the visible light while in the near-infrared it will be about 0.5 %. In Fig. 4 the near-field and far-field spectra around two resonance peaks (

*j*= 1, 3) with the corresponding frequency shifts are shown. The red shift of the near-field intensity peak with respect to the far-field intensity peak was observed for instance in [8] and qualitatively explained in [13] by a simple mass-and-spring model of a localized plasmon system.

## 6. Conclusions

Summarizing, we have developed an analytical method for calculating the scattered electromagnetic field from an ultra-thin metallic nanorod illuminated by an external plane electromagnetic wave. Longitudinal oscillations of the free-electron gas along the nanorod with frequencies from the near-infrared to visible have been studied in the quasistatic approximation. This provides a simple analytic expression for nanorod resonance wavelengths which depends on the plasma frequency *ω*_{p} and the nanorod aspect ratio *l*/*R*. For a broad interval of the aspect ratios 20 < *l*/*R* < 150 of very thin nanorods our results correspond to a linear scaling rule derived in [9]. Knowing the resonance frequencies of a nanorod its polarizability and the scattered near-field have been evaluated. The results are in a very good agreement with FDTD simulations carried out. Furthermore, it has been shown that the near-field amplitudes can be expressed in a closed analytical form. Finally, based on our model the red shift of the near-field spectral peak with respect to the far-field one reported in literature [8, 13] was explained theoretically.

In conclusion, we believe that our analytical approach gives a more straightforward physical insight into mechanisms of the interaction of light with a metallic nanorod. It is obvious that the obtained results can be directly utilized in a study of plasmonic properties of arrays of nanoantennas since the optical properties of the whole system may be expressed using the polarizability of a single nanorod.

## Appendix: Influence of repulsive forces of electron gas on the nanorod response

Let us investigate the influence of repulsive forces among electrons due to Pauli’s exclusion principle on the free-electron gas longitudinal oscillations along the nanorod. The volume density of energy accumulated due to Pauli’s repulsion of electrons in a metal possessing a local electron concentration *n* is [14]

*n*≃

*n*

_{0}(1 −

*∂u*/

*∂x*). By expressing

*𝒱*in Eq. (7) by Eq. (44) instead of Eq. (3) and under the same approximations as adopted in Eqs. (10) and (11) the wave equation (12) is modified as

*v*

_{F}= (

*h̄*/

*m*

_{e})(3

*π*

^{2}

*n*

_{0})

^{1/3}is the Fermi velocity and

*v*is given by Eq. (13). For intensities of common light sources (reaching 10

^{0}kW/m

^{2}) the amplitude of the external electromagnetic wave ${E}_{\text{m}}^{\text{ext}}$ is roughly 10

^{4}V/m. Inserting this value into Eq. (27) and putting

*ω*≃ Ω

*∼ 10*

_{j}^{14}s

^{−1}and

*τ*∼ 10

^{−14}s, the electron displacement amplitudes

*u*

_{m,j}are of the order 10

^{−1}pm. Therefore, max(

*∂u*/

*∂x*) = (

*jπ*/

*l*)

*u*

_{m,j}for

*l*∼ 10

^{0}

*μ*m is of the order 10

^{−7}which is much smaller than 1. Thus (1 −

*∂u*/

*∂x*)

^{−1/3}≃ 1 holds. Furthermore, considering metallic nanorods (where

*n*

_{0}∼ 10

^{28}m

^{−3}) of the dimensions

*R*∼ 10

^{1}nm and

*l*∼ 10

^{0}

*μ*m the term ${v}_{\text{F}}^{2}/\left(3{v}^{2}\right){\left(1-\partial u/\partial x\right)}^{-1/3}$ in Eq. (45) is of the order 10

^{−4}which means that the repulsive forces in the free-electron gas within the nanorod are negligible in comparison with the electric forces induced by an external electromagnetic wave.

## Acknowledgments

The authors thank Prof. B. Lencová for a critical reading of the manuscript and Prof. M. Lenc for inspiring comments. This work was supported by the project GACR ( P102/12/1881), European Regional Development Fund ( CEITEC-CZ.1.05/1.1.00/02.0068), Grant of the Technology agency of the Czech Republic (TACR) No. TE01020233 and by the EU 7th Framework Programme (Contract No. 286154 - SYLICA and 280566 - UnivSEM).

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