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

Optics Express, Vol. 20, Issue 16, pp. 17916-17927 (2012)

http://dx.doi.org/10.1364/OE.20.017916

Acrobat PDF (2979 KB)

### 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

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]

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]

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]

*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]

## 2. Calculation of nanorod resonance wavelengths

*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 where

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

*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.

*=*

_{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:

*, i.e. the incident wavelength roughly equals Λ*

_{j}*= 2*

_{j}*πc*/Ω

*. From Eq. (17) we get where*

_{j}*λ*

_{p}= 2

*πc*/

*ω*

_{p}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]

*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

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

*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) where the coefficients

*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

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

**E**

^{ext}is parallel with the nanorod main axis (see Fig. 1). Then the wave equation (12) is replaced by where with

*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.

*u*(

*x*,

*t*) fulfilling the boundary conditions given by Eq. (14) can be expressed as where

*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 where the angular eigenfrequencies of electron gas free oscillations Ω

*=*

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

_{j}*q*

_{m,j}are given by

*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.

*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 with

*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.

## 4. Near electromagnetic field scattered from a nanorod

*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 where the (complex) field amplitudes are with Using the above results the maps of amplitudes of scattered electromagnetic field in the vicinity of a golden nanorod with dimensions

*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

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

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

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

*p*whereas in the radiation zone to the second derivative of the dipole momentum

_{x}*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*By analyzing these functions we immediately get the peak positions for the near-field and far-field respectively. Thus the frequency shift Δ

*ω*≡

_{j}*ω*

_{far,j}−

*ω*

_{near,}

*is approximately*

_{j}*we see that the higher modes give the smaller frequency shifts Δ*

_{j}*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

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]

*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

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]

## 6. Conclusions

*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

**98**, 266802 (2007). [CrossRef] [PubMed]

**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]

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

*n*is [14] Considering Pauli’s repulsion Eq. (3) is to be modified as follows: where in the last term for the linear repulsive energy density the local electron concentration has been expressed by

*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 where

*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

^{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

^{−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

## References and links

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

2. | P. Biagioni, J. S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. |

3. | L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics |

4. | P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. |

5. | E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express |

6. | J. Dorfmüller, R. Vogelgesang, W. Khunsin, E. C. Rockstuhl, and K. Kern, “Plasmonic nanowire antennas: Experiment, simulation, and theory,” Nano Lett. |

7. | W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. |

8. | G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. |

9. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

10. | L. D. Landau and E. M. Lifshitz, |

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

12. | J. D. Jackson, |

13. | J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon system,” Nano Lett. |

14. | C. Kittel, |

**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

Sort: Year | Journal | Reset

### 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).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.