Response of plasmonic resonant nanorods: an analytical approach to optical antennas

doi:10.1364/OE.20.017916

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Response of plasmonic resonant nanorods: an analytical approach to optical antennas

Radek Kalousek, Petr Dub, Lukáš Břínek, and Tomáš Šikola »View Author Affiliations

Optics Express, Vol. 20, Issue 16, pp. 17916-17927 (2012)
http://dx.doi.org/10.1364/OE.20.017916

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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.

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

N. Berkovich, P. Ginsburg, and M. Orenstein, “Nano-plasmonic antennas in the near infrared regime,” J. Phys.: Condens. Matter 24, 073202 (2012). [CrossRef]

–4

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]

]). 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

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16, 16529–16537 (2008). [CrossRef] [PubMed]

]). Recently, an analytical model based on the RF antenna theory and describing plasmonic antenna resonances was reported in [6

J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]

] 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

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]

], the electron longitudinal displacement may be considered approximately uniform over the nanorod cross-section similarly to [6

J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]

, 8

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]

]. 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

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

] 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.

Fig. 1 The nanorod illuminated by an external electromagnetic plane wave. For simplicity, the nanorod of a circular cross-section of the radius R being much smaller than the nanorod length l is considered. The incident wavelength λ much longer than R is assumed.

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

τ_{tot} (x, t) = n_{0} e π R^{2} \frac{\partial u (x, t)}{\partial x},

(1)

where n₀ 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

𝒱 (x, t) = \frac{1}{2} τ_{tot} (x, t) φ (x, t)

(2)

can be calculated (see e.g. [10

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , 4th ed. (Elsevier, Butterworth-Heinemann, 2010).

]). Integrating along the nanorod yields

φ (x, t) = \int_{0}^{l} \frac{τ_{tot} (x^{'}, t)}{2 ε_{0} π R^{2}} [\sqrt{R^{2} + {(x - x^{'})}^{2}} - | x - x^{'} |] d x^{'} .

Inserting this formula into Eq. (2), the electric potential energy density of the nanorod reads

𝒱 (x, t) = \frac{n_{0}^{2} e^{2} π R^{3}}{4 ε_{0}} \frac{\partial u (x, t)}{\partial x} \int_{0}^{l} \frac{\partial u (x^{'}, t)}{\partial x^{'}} g (| x - x^{'} |) d x^{'},

(3)

where the function

g (| x - x^{'} |) \equiv \sqrt{1 + {(\frac{x - x^{'}}{R})}^{2}} - \frac{| x - x^{'} |}{R}

(4)

describes how the charge at the point 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).

Fig. 2 Plot of the function g(|x − x′|). Note the function takes on significant values only when |x − x′| < 10R.

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

𝒯 (x, t) = \frac{1}{2} m_{e} n_{0} π R^{2} {[\frac{\partial u (x, t)}{\partial t}]}^{2},

(5)

where 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

\frac{\partial}{\partial t} (\frac{\partial ℒ}{\partial u_{t}}) + \frac{\partial}{\partial x} (\frac{\partial ℒ}{\partial u_{x}}) = \frac{\partial ℒ}{\partial u},

(6)

where

ℒ = 𝒯 - 𝒱

(7)

is the linear density of Lagrangian of the system and 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

\frac{\partial^{2} u (x, t)}{\partial t^{2}} = \frac{ω_{p}^{2} R}{4} \frac{\partial}{\partial x} \int_{0}^{l} \frac{\partial u (x^{'}, t)}{\partial x^{'}} g (| x - x^{'} |) d x^{'},

(8)

where ω_p = [n₀e²/(ε₀m_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

\begin{matrix} \frac{\partial^{2} u (x, t)}{\partial t^{2}} = \frac{ω_{p}^{2} R}{4} \frac{\partial}{\partial x} \int_{0}^{l} [\frac{\partial u (x, t)}{\partial x} + \frac{\partial^{2} u (x, t)}{\partial x^{2}} (x^{'} - x) \\ + \frac{1}{2} \frac{\partial^{3} u (x, t)}{\partial x^{3}} {(x^{'} - x)}^{2} + \dots] g (| x - x^{'} |) d x^{'} . \end{matrix}

(9)

Retaining the first term in the bracket only we obtain

\frac{\partial^{2} u (x, t)}{\partial t^{2}} = \frac{ω_{p}^{2} R}{4} \frac{\partial}{\partial x} [\frac{\partial u (x, t)}{\partial x} \int_{0}^{l} g (| x - x^{'} |) d x^{'}] .

(10)

Further, as the function g(|x − x′|) is significantly nonzero only when |x − x′| < 10R (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 (| x - x^{'} |) d x^{'}

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

\int_{0}^{l} g (| x - x^{'} |) d x^{'} ≃ \frac{1}{l} \int_{0}^{l} \int_{0}^{l} g (| x - x^{'} |) d x d x^{'} ≃ R ln (ϑ \frac{l}{R}),

(11)

where ϑ = 2e^−1/2 = 1.213.... Under this approximation Eq. (10) yields the wave equation

\frac{\partial^{2} u (x, t)}{\partial t^{2}} = v^{2} \frac{\partial^{2} u (x, t)}{\partial x^{2}}

(12)

with

v = \frac{ω_{p} R}{2} \sqrt{ln (ϑ \frac{l}{R})}

(13)

being the phase velocity of longitudinal free-electron density waves in the nanorod. For noble metals, such as Au or Ag, the plasma frequency ω_p is typically 10¹⁶ s⁻¹ and the term [ln(ϑl/R)]^1/2 is in the order of magnitude 10⁰ for a wide range of the nanorod aspect ratios 10¹ < l/R < 10⁴. This indicates the nanorod radius R should be less than 10² 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

u (0, t) = u (l, t) = 0

(14)

are represented by standing waves

u_{j} (x, t) = A_{j} sin (k_{j} x) sin (Ω_{j} t), j = 1, 2, \dots,

(15)

where

k_{j} = \frac{j π}{l}

(16)

is the angular wave number, and Ω_j = vk_j is the angular eigenfrequency of electron gas free oscillations. From Eq. (13) it is obvious that Ω_j depends on the plasma frequency ω_p and the aspect ratio l/R of the metallic nanorod:

Ω_{j} = \frac{j π}{2} ω_{p} \frac{R}{l} \sqrt{ln (ϑ \frac{l}{R})} .

(17)

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_j = 2πc/Ω_j. From Eq. (17) we get

Λ_{j} = \frac{2 λ_{p}}{j π} \frac{l}{R} {[ln (ϑ \frac{l}{R})]}^{- \frac{1}{2}},

(18)

where λ_p = 2πc/ω_p is the plasma wavelength. Recently, utilizing the waveguide theory Novotny [9

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

] derived a linear scaling rule relating the effective nanorod wavelength λ_eff to an incident wavelength. Considering a half-wave dipole antenna (λ_eff = 2l) 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

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

] derived for a very thin nanorod yields the incident wavelength corresponding to the first mode of the antenna

λ = (2.76 + 0.22 \frac{l}{R}) λ_{p} .

(19)

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 ξ₀. As the second- and higher-order derivatives of the function ψ(ξ) are much smaller than the first one for 20 < ξ₀ < 150, our result takes the same form as Eq. (19)

Λ_{1} ≃ (γ_{1} + γ_{2} \frac{l}{R}) λ_{p},

(20)

where the coefficients γ₁ and γ₂ depend on ξ₀. For 20 < ξ₀ < 150 the coefficient γ₁ ranges from 1.1 to 4.0 while the coefficient γ₂ 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

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]

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

\frac{\partial^{2} u (x, t)}{\partial t^{2}} + \frac{1}{τ} \frac{\partial u (x, t)}{\partial t} = v^{2} \frac{\partial^{2} u (x, t)}{\partial x^{2}} - \frac{e}{m_{e}} E^{ext} (t),

(21)

where

E^{ext} (t) = E_{m}^{ext} exp (i ω t)

(22)

with

E_{m}^{ext}

and 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

u (x, t) = \frac{2}{l} \sum_{j} q_{j} (t) sin (k_{j} x),

(23)

where k_j is given by Eq. (16). After inserting Eq. (23) into Eq. (21) and considering that functions {sin(k_ix)} form an orthogonal system over the interval (0, l), it is found that q_j(t) obeys the differential equation

{\ddot{q}}_{j} + \frac{1}{τ} {\dot{q}}_{j} + Ω_{j}^{2} q_{j} = - \frac{e}{m_{e}} E^{ext} (t) \int_{0}^{l} sin (k_{j} x) d x,

(24)

where the angular eigenfrequencies of electron gas free oscillations Ω_j = vk_j are given by Eq. (17). In the steady state

q_{j} (t) = q_{m, j} exp (i ω t),

(25a)

where the amplitudes q_m,j are given by

q_{m, j} = - \frac{e E_{m}^{ext}}{m_{e}} \frac{1}{Ω_{j}^{2} - ω^{2} + i ω / τ} \int_{0}^{l} sin (k_{j} x) d x .

(25b)

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

u (x, t) = \sum_{j} u_{m, j} sin (j π x / l) exp (i ω t), j = 1, 2, \dots,

(26)

where

u_{m, j} = - \frac{2 e E_{m}^{ext}}{j π m_{e}} \frac{1 - cos (j π)}{Ω_{j}^{2} - ω^{2} + i ω / τ}

(27)

are the amplitudes of excited modes. Having derived the formula for electron displacement it is possible to evaluate the linear density of the local total charge given by Eq. (1) and finally the dipole momentum of the nanorod,

p_{x} (t) = \int_{0}^{l} x τ_{tot} (x, t) d x .

(28)

By using Eqs. (1), (26) and (27) we obtain

p_{x} (t) = {\frac{2 n_{0} e^{2} R^{2} l}{π m_{e}} \sum_{j} \frac{1}{j^{2}} \frac{{[1 - cos (j π)]}^{2}}{Ω_{j}^{2} - ω^{2} + i ω / τ}} E_{m}^{ext} exp (i ω t),

(29)

where the expression in the parentheses expresses the polarizability of the nanorod. As the term 1 − cos(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

α_{j} (ω) = \frac{8 ε_{0} V}{j^{2} π^{2}} \frac{ω_{p}^{2}}{Ω_{j}^{2} - ω^{2} + i ω / τ},

(30)

with V = πR²l being the nanorod volume. The term j² 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

u_{m, j} = - \frac{j α_{j} (ω) E_{m}^{ext}}{2 n_{0} e R^{2} l} .

(31)

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

E_{x}^{s} (x, y, 0, t) = \frac{1}{4 π ε_{0}} \int_{0}^{l} \frac{τ_{tot} (x^{'}, t) (x - x^{'})}{{[{(x - x^{'})}^{2} + y^{2}]}^{\frac{3}{2}}} d x^{'},

(32a)

E_{y}^{s} (x, y, 0, t) = \frac{1}{4 π ε_{0}} \int_{0}^{l} \frac{τ_{tot} (x^{'}, t) y}{{[{(x - x^{'})}^{2} + y^{2}]}^{\frac{3}{2}}} d x^{'},

(32b)

H_{z}^{s} (x, y, 0, t) = \frac{1}{4 π} \int_{0}^{l} \frac{J_{x} (x^{'}, t) π R^{2} y}{{[{(x - x^{'})}^{2} + y^{2}]}^{\frac{3}{2}}} d x^{'},

(32c)

where τ_tot is the linear density of the local total charge within the nanorod given by Eq. (1) and J_x = −n₀e(∂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

E_{x}^{s} (x, y, 0, t) = \sum_{j} E_{m, x, j}^{s} (x, y, 0; ω) exp (i ω t),

(33a)

E_{y}^{s} (x, y, 0, t) = \sum_{j} E_{m, y, j}^{s} (x, y, 0; ω) exp (i ω t),

(33b)

H_{z}^{s} (x, y, 0, t) = \sum_{j} H_{m, z, j}^{s} (x, y, 0; ω) exp (i ω t),

(33c)

where the (complex) field amplitudes are

E_{m, x, j}^{s} (x, y, 0; ω) = - j^{2} \frac{π α_{j} (ω) E_{m}^{ext}}{8 ε_{0} l^{2}} \int_{0}^{l} (x - x^{'}) f (| x - x^{'} |, y) cos (j π x^{'} / l) d x^{'},

(34)

E_{m, y, j}^{s} (x, y, 0; ω) = - j^{2} \frac{π α_{j} (ω) E_{m}^{ext} y}{8 ε_{0} l^{2}} \int_{0}^{l} f (| x - x^{'} |, y) cos (j π x^{'} / l) d x^{'},

(34b)

H_{m, z, j}^{s} (x, y, 0; ω) = j \frac{i ω α_{j} (ω) E_{m}^{ext} y}{8 l} \int_{0}^{l} f (| x - x^{'} |, y) sin (j π x^{'} / l) d x^{'}

(34c)

with

f (| x - x^{'} |, y) \equiv {[{(x - x^{'})}^{2} + y^{2}]}^{- \frac{3}{2}} .

(35)

Using the above results the maps of amplitudes of scattered electromagnetic field in the vicinity of a golden nanorod with dimensions l = 1 μm and R = 10 nm have been calculated for the first two modes close to resonance, i.e. for ω = Ω₁ and Ω₃ (Fig. 3a, c). For comparison, the results obtained by 3D finite-difference time-domain (FDTD) simulations [11

F. D. T. D. Solutions (version 7.5.5), from Lumerical Solutions, Inc., http://www.lumerical.com.

] 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.

Fig. 3 The scattered electric and magnetic fields (normalized to the incident electromagnetic wave) in the vicinity of a golden nanorod of the length l = 1.0 μm and diameter R = 10 nm at the resonance wavelengths Λ₁, Λ₃. The nanorod is depicted by the gray rectangle. The material parameters of Au used in calculations were n₀ = 5.0 × 10²⁸ m⁻³ and τ = 3.2 × 10⁻¹⁴ s. The maps of amplitudes of near-field components calculated by Eqs. (34a–c) and by FDTD simulations are shown in (a, c) and (b, d), respectively. Note the different scales in the x and y axis, and the phase shift π/2 between electric and magnetic fields.

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

cos (j π x^{'} / l) ≃ cos (j π x / l) + \frac{j π}{l} sin (j π x / l) (x - x^{'}),

(36)

sin (j π x^{'} / l) ≃ sin (j π x / l) - \frac{j π}{l} cos (j π x / l) (x - x^{'}) .

(37)

In this approximation the near-field amplitudes given by Eqs. (34a–c) can be evaluated analytically:

E_{m, x, j}^{s} (x, y, 0; ω) ≃ - j^{2} \frac{π α_{j} (ω) E_{m}^{ext}}{8 ε_{0} l^{3}} [β_{2} (x, y) cos (j π x / l) + j π β_{3} (x, y) sin (j π x / l)],

(38a)

E_{m, y, j}^{s} (x, y, 0; ω) ≃ - j^{2} \frac{π α_{j} (ω) E_{m}^{ext}}{8 ε_{0} l^{3}} [\frac{l}{y} β_{1} (x, y) cos (j π x / l) + j π \frac{y}{l} β_{2} (x, y) sin (j π x / l)],

(38b)

H_{m, z, j}^{s} (x, y, 0; ω) ≃ j \frac{i ω α_{j} (ω) E_{m}^{ext}}{8 l^{2}} [\frac{l}{y} β_{1} (x, y) sin (j π x / l) - j π \frac{y}{l} β_{2} (x, y) cos (j π x / l)]

(38c)

where we have introduced

β_{1} (x, y) \equiv y^{2} \int_{0}^{l} f (| x - x^{'} |, y) d x^{'} = \frac{l - x}{\sqrt{{(l - x)}^{2} + y^{2}}} + \frac{x}{\sqrt{x^{2} + y^{2}}},

(39a)

β_{2} (x, y) \equiv l \int_{0}^{l} (x - x^{'}) f (| x - x^{'} |, y) d x^{'} = \frac{l}{\sqrt{(l - x^{2}) + y^{2}}} - \frac{l}{\sqrt{x^{2} + y^{2}}},

(39b)

\begin{matrix} β_{3} (x, y) \equiv \int_{0}^{l} {(x - x^{'})}^{2} f (| x - x^{'} |, y) d x^{'} = \frac{x - l}{\sqrt{(l - x^{2}) + y^{2}}} - \frac{x}{\sqrt{x^{2} + y^{2}}} \\ + ln [\frac{l - x + \sqrt{{(l - x)}^{2} + y^{2}}}{- x + \sqrt{x^{2} + y^{2}}}] . \end{matrix}

(39c)

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 ∝ ω²p_x, where ω is the angular frequency of the external wave [12

J. D. Jackson, Classical Electrodynamics (J. Wiley & Sons, 1999).

]. 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

{| E_{near, j}^{s} (ω) |}^{2} \propto \frac{1}{{(Ω_{j}^{2} - ω^{2})}^{2} + ω^{2} / τ^{2}} and {| E_{far, j}^{s} (ω) |}^{2} \propto \frac{ω^{4}}{{(Ω_{j}^{2} - ω^{2})}^{2} + ω^{2} / τ^{2}} .

(40)

By analyzing these functions we immediately get the peak positions for the near-field and far-field

ω_{near, j}^{2} = Ω_{j}^{2} - \frac{1}{2 τ^{2}} and ω_{far, j}^{2} = Ω_{j}^{2} {(1 - \frac{1}{2 τ^{2} Ω_{j}^{2}})}^{- 1} ≃ Ω_{j}^{2} + \frac{1}{2 τ^{2}},

(41)

respectively. Thus the frequency shift Δω_j ≡ ω_far,j − ω_near,_j is approximately

Δ ω_{j} ≃ \frac{1}{2 Ω_{j} τ^{2}} .

(42)

Considering relation (17) for Ω_j we see that the higher modes give the smaller frequency shifts Δω_j 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/(2πc). Furthermore, the shifts depend on the intrinsic damping expressed by 1/τ², as also reported in [13

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]

]. 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 Δω_j/Ω_j ≃ (Ω_jτ)⁻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

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]

] and qualitatively explained in [13

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]

] by a simple mass-and-spring model of a localized plasmon system.

Fig. 4 The near-field (red) and far-field (blue) spectra around two resonance peaks (left: j = 1, right: j = 3) of the golden nanorod of the length l = 1.0 μm and radius R = 10 nm (n₀ = 5.0 × 10²⁸ m⁻³, τ = 3.2 × 10⁻¹⁴ s). Note the corresponding frequency shifts Δω₁ and Δω₃.

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

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

]. 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

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]

, 13

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]

] 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.

Appendices

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

C. Kittel, Introduction to Solid State Physics , 8th ed. (J. Wiley & Sons, 2005).

]

w_{rep} \equiv \frac{U}{V} = \frac{π^{\frac{4}{3}} {\bar{h}}^{2}}{10 m_{e}} {(3 n)}^{\frac{5}{3}} .

(43)

Considering Pauli’s repulsion Eq. (3) is to be modified as follows:

𝒱^{'} = \frac{n_{0}^{2} e^{2} π R^{3}}{4 ε_{0}} \frac{\partial u (x, t)}{\partial x} \int_{0}^{l} \frac{\partial u (x^{'}, t)}{\partial x^{'}} g (| x - x^{'} |) d x^{'} + \frac{π^{\frac{4}{3}} {\bar{h}}^{2}}{10 m_{e}} {(3 n_{0})}^{\frac{5}{3}} {(1 - \frac{\partial u (x, t)}{\partial x})}^{\frac{5}{3}} π R^{2},

(44)

where in the last term for the linear repulsive energy density the local electron concentration has been expressed by n ≃ n₀ (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

\frac{\partial^{2} u (x, t)}{\partial t^{2}} = [1 + \frac{v_{F}^{2}}{3 v^{2}} {(1 - \frac{\partial u (x, t)}{\partial x})}^{- \frac{1}{3}}] v^{2} \frac{\partial^{2} u (x, t)}{\partial x^{2}},

(45)

where v_F = (h̄/m_e)(3π²n₀)^1/3 is the Fermi velocity and v is given by Eq. (13). For intensities of common light sources (reaching 10⁰ kW/m²) the amplitude of the external electromagnetic wave

E_{m}^{ext}

is roughly 10⁴ V/m. Inserting this value into Eq. (27) and putting ω ≃ Ω_j ∼ 10¹⁴ 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 ∼ 10⁰ μm 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 n₀ ∼ 10²⁸ m⁻³) of the dimensions R ∼ 10¹ nm and l ∼ 10⁰ μm the term

v_{F}^{2} / (3 v^{2}) {(1 - \partial u / \partial x)}^{- 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).

References and links

1.	N. Berkovich, P. Ginsburg, and M. Orenstein, “Nano-plasmonic antennas in the near infrared regime,” J. Phys.: Condens. Matter 24, 073202 (2012). [CrossRef]
2.	P. Biagioni, J. S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012). [CrossRef] [PubMed]
3.	L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). [CrossRef]
4.	P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]
5.	E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16, 16529–16537 (2008). [CrossRef] [PubMed]
6.	J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]
7.	W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]
8.	G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]
9.	L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]
10.	L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , 4th ed. (Elsevier, Butterworth-Heinemann, 2010).
11.	F. D. T. D. Solutions (version 7.5.5), from Lumerical Solutions, Inc., http://www.lumerical.com.
12.	J. D. Jackson, Classical Electrodynamics (J. Wiley & Sons, 1999).
13.	J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]
14.	C. Kittel, Introduction to Solid State Physics , 8th ed. (J. Wiley & Sons, 2005).

OCIS Codes
(260.5740) Physical optics : Resonance
(250.5403) Optoelectronics : Plasmonics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: June 8, 2012
Revised Manuscript: July 13, 2012
Manuscript Accepted: July 16, 2012
Published: July 20, 2012

Citation
Radek Kalousek, Petr Dub, Lukáš Břínek, and Tomáš Šikola, "Response of plasmonic resonant nanorods: an analytical approach to optical antennas," Opt. Express 20, 17916-17927 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17916

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References

N. Berkovich, P. Ginsburg, and M. Orenstein, “Nano-plasmonic antennas in the near infrared regime,” J. Phys.: Condens. Matter24, 073202 (2012). [CrossRef]
P. Biagioni, J. S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys.75, 024402 (2012). [CrossRef] [PubMed]
L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5, 83–90 (2011). [CrossRef]
P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon.1, 438–483 (2009). [CrossRef]
E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express16, 16529–16537 (2008). [CrossRef] [PubMed]
J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett.10, 3596–3603 (2010). [CrossRef] [PubMed]
W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt.8, S87–S93 (2006). [CrossRef]
G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett.8, 631–636 (2008). [CrossRef] [PubMed]
L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Elsevier, Butterworth-Heinemann, 2010).
F. D. T. D. Solutions (version 7.5.5), from Lumerical Solutions, Inc., http://www.lumerical.com .
J. D. Jackson, Classical Electrodynamics (J. Wiley & Sons, 1999).
J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett.11, 1280–1283 (2011). [CrossRef] [PubMed]
C. Kittel, Introduction to Solid State Physics, 8th ed. (J. Wiley & Sons, 2005).

Adv. Opt. Photon.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon.1, 438–483 (2009). [CrossRef]

J. Opt. A: Pure Appl. Opt.

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt.8, S87–S93 (2006). [CrossRef]

J. Phys.: Condens. Matter

N. Berkovich, P. Ginsburg, and M. Orenstein, “Nano-plasmonic antennas in the near infrared regime,” J. Phys.: Condens. Matter24, 073202 (2012). [CrossRef]

Nano Lett.

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett.8, 631–636 (2008). [CrossRef] [PubMed]
J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett.10, 3596–3603 (2010). [CrossRef] [PubMed]
J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett.11, 1280–1283 (2011). [CrossRef] [PubMed]

Nat. Photonics

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5, 83–90 (2011). [CrossRef]

Opt. Express

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express16, 16529–16537 (2008). [CrossRef] [PubMed]

Phys. Rev. Lett.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]

Rep. Prog. Phys.

P. Biagioni, J. S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys.75, 024402 (2012). [CrossRef] [PubMed]

Other

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed. (Elsevier, Butterworth-Heinemann, 2010).
F. D. T. D. Solutions (version 7.5.5), from Lumerical Solutions, Inc., http://www.lumerical.com .
J. D. Jackson, Classical Electrodynamics (J. Wiley & Sons, 1999).
C. Kittel, Introduction to Solid State Physics, 8th ed. (J. Wiley & Sons, 2005).

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