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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 17916–17927
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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.

© 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

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

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

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

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

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

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

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

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
τtot(x,t)=n0eπR2u(x,t)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)=12τtot(x,t)φ(x,t)
(2)
can be calculated (see e.g. [10

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)=0lτtot(x,t)2ε0πR2[R2+(xx)2|xx|]dx.
Inserting this formula into Eq. (2), the electric potential energy density of the nanorod reads
𝒱(x,t)=n02e2πR34ε0u(x,t)x0lu(x,t)xg(|xx|)dx,
(3)
where the function
g(|xx|)1+(xxR)2|xx|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 Rλ 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(|xx′|). Note the function takes on significant values only when |xx′| < 10R.

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
uj(x,t)=Ajsin(kjx)sin(Ωjt),j=1,2,,
(15)
where
kj=jπl
(16)
is the angular wave number, and Ωj = vkj is the angular eigenfrequency of electron gas free oscillations. From Eq. (13) it is obvious that Ωj depends on the plasma frequency ωp and the aspect ratio l/R of the metallic nanorod:
Ωj=jπ2ωpRlln(ϑlR).
(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 Ωj, i.e. the incident wavelength roughly equals Λj = 2πcj. From Eq. (17) we get
Λj=2λpjπlR[ln(ϑlR)]12,
(18)
where λp = 2πc/ωp is the plasma wavelength. Recently, utilizing the waveguide theory Novotny [9

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 λeff to an incident wavelength. Considering a half-wave dipole antenna (λeff = 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. εs = 1), Eq. (14) in [9

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+0.22lR)λp.
(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)
Λ1(γ1+γ2lR)λp,
(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 λp 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

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 Eext is parallel with the nanorod main axis (see Fig. 1). Then the wave equation (12) is replaced by
2u(x,t)t2+1τu(x,t)t=v22u(x,t)x2emeEext(t),
(21)
where
Eext(t)=Emextexp(iωt)
(22)
with Emext 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)=2ljqj(t)sin(kjx),
(23)
where kj is given by Eq. (16). After inserting Eq. (23) into Eq. (21) and considering that functions {sin(kix)} form an orthogonal system over the interval (0, l), it is found that qj(t) obeys the differential equation
q¨j+1τq˙j+Ωj2qj=emeEext(t)0lsin(kjx)dx,
(24)
where the angular eigenfrequencies of electron gas free oscillations Ωj = vkj are given by Eq. (17). In the steady state
qj(t)=qm,jexp(iωt),
(25a)
where the amplitudes qm,j are given by
qm,j=eEmextme1Ωj2ω2+iω/τ0lsin(kjx)dx.
(25b)

As kj = /l the integral in Eq. (25b) is proportional to 1 − cos() which means that for even j the amplitudes qm,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)=jum,jsin(jπx/l)exp(iωt),j=1,2,,
(26)
where
um,j=2eEmextjπme1cos(jπ)Ωj2ω2+iω/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,
px(t)=0lxτtot(x,t)dx.
(28)
By using Eqs. (1), (26) and (27) we obtain
px(t)={2n0e2R2lπmej1j2[1cos(jπ)]2Ωj2ω2+iω/τ}Emextexp(iωt),
(29)
where the expression in the parentheses expresses the polarizability of the nanorod. As the term 1 − cos() 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
αj(ω)=8ε0Vj2π2ωp2Ωj2ω2+iω/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 um,j of the j-th mode of the electron displacement and the nanorod polarizability of the corresponding mode can be found
um,j=jαj(ω)Emext2n0eR2l.
(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
Exs(x,y,0,t)=14πε00lτtot(x,t)(xx)[(xx)2+y2]32dx,
(32a)
Eys(x,y,0,t)=14πε00lτtot(x,t)y[(xx)2+y2]32dx,
(32b)
Hzs(x,y,0,t)=14π0lJx(x,t)πR2y[(xx)2+y2]32dx,
(32c)
where τtot is the linear density of the local total charge within the nanorod given by Eq. (1) and Jx = −n0e(∂u/∂t) is the current density. Inserting the electron displacement given by Eq. (26) with the amplitudes expressed by Eq. (31) into Eqs. (32ac) we get
Exs(x,y,0,t)=jEm,x,js(x,y,0;ω)exp(iωt),
(33a)
Eys(x,y,0,t)=jEm,y,js(x,y,0;ω)exp(iωt),
(33b)
Hzs(x,y,0,t)=jHm,z,js(x,y,0;ω)exp(iωt),
(33c)
where the (complex) field amplitudes are
Em,x,js(x,y,0;ω)=j2παj(ω)Emext8ε0l20l(xx)f(|xx|,y)cos(jπx/l)dx,
(34)
Em,y,js(x,y,0;ω)=j2παj(ω)Emexty8ε0l20lf(|xx|,y)cos(jπx/l)dx,
(34b)
Hm,z,js(x,y,0;ω)=jiωαj(ω)Emexty8l0lf(|xx|,y)sin(jπx/l)dx
(34c)
with
f(|xx|,y)[(xx)2+y2]32.
(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

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. (34ac) 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 ly the function f(|xx′|, y) in integrands in Eqs. (34ac) 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(jπx/l)cos(jπx/l)+jπlsin(jπx/l)(xx),
(36)
sin(jπx/l)sin(jπx/l)jπlcos(jπx/l)(xx).
(37)
In this approximation the near-field amplitudes given by Eqs. (34ac) can be evaluated analytically:
Em,x,js(x,y,0;ω)j2παj(ω)Emext8ε0l3[β2(x,y)cos(jπx/l)+jπβ3(x,y)sin(jπx/l)],
(38a)
Em,y,js(x,y,0;ω)j2παj(ω)Emext8ε0l3[lyβ1(x,y)cos(jπx/l)+jπylβ2(x,y)sin(jπx/l)],
(38b)
Hm,z,js(x,y,0;ω)jiωαj(ω)Emext8l2[lyβ1(x,y)sin(jπx/l)jπylβ2(x,y)cos(jπx/l)]
(38c)
where we have introduced
β1(x,y)y20lf(|xx|,y)dx=lx(lx)2+y2+xx2+y2,
(39a)
β2(x,y)l0l(xx)f(|xx|,y)dx=l(lx2)+y2lx2+y2,
(39b)
β3(x,y)0l(xx)2f(|xx|,y)dx=xl(lx2)+y2xx2+y2+ln[lx+(lx)2+y2x+x2+y2].
(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 px whereas in the radiation zone to the second derivative of the dipole momentum xω2px, where ω is the angular frequency of the external wave [12

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
|Enear,js(ω)|21(Ωj2ω2)2+ω2/τ2and|Efar,js(ω)|2ω4(Ωj2ω2)2+ω2/τ2.
(40)
By analyzing these functions we immediately get the peak positions for the near-field and far-field
ωnear,j2=Ωj212τ2andωfar,j2=Ωj2(112τ2Ωj2)1Ωj2+12τ2,
(41)
respectively. Thus the frequency shift Δωjωfar,jωnear,j is approximately

Δωj12Ωjτ2.
(42)

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

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 Δωjj ≃ (Ωjτ)−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

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

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 ωp 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

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

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

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.

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

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

]
wrepUV=π43h¯210me(3n)53.
(43)
Considering Pauli’s repulsion Eq. (3) is to be modified as follows:
𝒱=n02e2πR34ε0u(x,t)x0lu(x,t)xg(|xx|)dx+π43h¯210me(3n0)53(1u(x,t)x)53πR2,
(44)
where in the last term for the linear repulsive energy density the local electron concentration has been expressed by nn0 (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
2u(x,t)t2=[1+vF23v2(1u(x,t)x)13]v22u(x,t)x2,
(45)
where vF = (/me)(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 Emext is roughly 104 V/m. Inserting this value into Eq. (27) and putting ω ≃ Ωj ∼ 1014 s−1 and τ ∼ 10−14 s, the electron displacement amplitudes um,j are of the order 10−1 pm. Therefore, max(∂u/∂x) = (/l)um,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 vF2/(3v2)(1u/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

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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]

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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]

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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

  1. N. Berkovich, P. Ginsburg, and M. Orenstein, “Nano-plasmonic antennas in the near infrared regime,” J. Phys.: Condens. Matter24, 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. Photonics5, 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. Express16, 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).

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