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 Optical Society of America

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References

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  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).

2012 (2)

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

2011 (2)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

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]

2010 (1)

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]

2009 (1)

2008 (2)

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]

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]

2007 (1)

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

2006 (1)

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]

Aizpurua, J.

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]

Barnard, E. S.

Barnes, W. L.

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]

Berkovich, N.

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

Bharadwaj, P.

Biagioni, P.

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

Brongersma, M. L.

Bryant, G. W.

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]

Chandran, A.

Deutsch, B.

Dorfmüller, J.

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]

García de Abajo, F. J.

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]

Ginsburg, P.

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

Hecht, B.

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

Huang, J. S.

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

Jackson, J. D.

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

Kern, K.

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]

Khunsin, W.

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]

Kittel, C.

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

Landau, L. D.

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

Lifshitz, E. M.

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

Nordlander, P.

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]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

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

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

Orenstein, M.

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

Rockstuhl, E. C.

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]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

Vogelgesang, R.

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]

White, J. S.

Zuloaga, J.

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]

Adv. Opt. Photon. (1)

J. Opt. A: Pure Appl. Opt. (1)

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 (1)

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

Nano Lett. (3)

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 (1)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

Opt. Express (1)

Phys. Rev. Lett. (1)

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

Rep. Prog. Phys. (1)

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

Other (4)

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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Figures (4)

Fig. 1
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.

Fig. 2
Fig. 2

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

Fig. 3
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.

Fig. 4
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.

Equations (57)

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τ tot ( x , t ) = n 0 e π R 2 u ( x , t ) x ,
𝒱 ( x , t ) = 1 2 τ tot ( x , t ) φ ( x , t )
φ ( x , t ) = 0 l τ tot ( x , t ) 2 ε 0 π R 2 [ R 2 + ( x x ) 2 | x x | ] d x .
𝒱 ( x , t ) = n 0 2 e 2 π R 3 4 ε 0 u ( x , t ) x 0 l u ( x , t ) x g ( | x x | ) d x ,
g ( | x x | ) 1 + ( x x R ) 2 | x x | R
𝒯 ( x , t ) = 1 2 m e n 0 π R 2 [ u ( x , t ) t ] 2 ,
t ( u t ) + x ( u x ) = u ,
= 𝒯 𝒱
2 u ( x , t ) t 2 = ω p 2 R 4 x 0 l u ( x , t ) x g ( | x x | ) d x ,
2 u ( x , t ) t 2 = ω p 2 R 4 x 0 l [ u ( x , t ) x + 2 u ( x , t ) x 2 ( x x ) + 1 2 3 u ( x , t ) x 3 ( x x ) 2 + ] g ( | x x | ) d x .
2 u ( x , t ) t 2 = ω p 2 R 4 x [ u ( x , t ) x 0 l g ( | x x | ) d x ] .
0 l g ( | x x | ) d x 1 l 0 l 0 l g ( | x x | ) d x d x R ln ( ϑ l R ) ,
2 u ( x , t ) t 2 = v 2 2 u ( x , t ) x 2
v = ω p R 2 ln ( ϑ l R )
u ( 0 , t ) = u ( l , t ) = 0
u j ( x , t ) = A j sin ( k j x ) sin ( Ω j t ) , j = 1 , 2 , ,
k j = j π l
Ω j = j π 2 ω p R l ln ( ϑ l R ) .
Λ j = 2 λ p j π l R [ ln ( ϑ l R ) ] 1 2 ,
λ = ( 2.76 + 0.22 l R ) λ p .
Λ 1 ( γ 1 + γ 2 l R ) λ p ,
2 u ( x , t ) t 2 + 1 τ u ( x , t ) t = v 2 2 u ( x , t ) x 2 e m e E ext ( t ) ,
E ext ( t ) = E m ext exp ( i ω t )
u ( x , t ) = 2 l j q j ( t ) sin ( k j x ) ,
q ¨ j + 1 τ q ˙ j + Ω j 2 q j = e m e E ext ( t ) 0 l sin ( k j x ) d x ,
q j ( t ) = q m , j exp ( i ω t ) ,
q m , j = e E m ext m e 1 Ω j 2 ω 2 + i ω / τ 0 l sin ( k j x ) d x .
u ( x , t ) = j u m , j sin ( j π x / l ) exp ( i ω t ) , j = 1 , 2 , ,
u m , j = 2 e E m ext j π m e 1 cos ( j π ) Ω j 2 ω 2 + i ω / τ
p x ( t ) = 0 l x τ tot ( x , t ) d x .
p x ( t ) = { 2 n 0 e 2 R 2 l π m e j 1 j 2 [ 1 cos ( j π ) ] 2 Ω j 2 ω 2 + i ω / τ } E m ext exp ( i ω t ) ,
α j ( ω ) = 8 ε 0 V j 2 π 2 ω p 2 Ω j 2 ω 2 + i ω / τ ,
u m , j = j α j ( ω ) E m ext 2 n 0 e R 2 l .
E x s ( x , y , 0 , t ) = 1 4 π ε 0 0 l τ tot ( x , t ) ( x x ) [ ( x x ) 2 + y 2 ] 3 2 d x ,
E y s ( x , y , 0 , t ) = 1 4 π ε 0 0 l τ tot ( x , t ) y [ ( x x ) 2 + y 2 ] 3 2 d x ,
H z s ( x , y , 0 , t ) = 1 4 π 0 l J x ( x , t ) π R 2 y [ ( x x ) 2 + y 2 ] 3 2 d x ,
E x s ( x , y , 0 , t ) = j E m , x , j s ( x , y , 0 ; ω ) exp ( i ω t ) ,
E y s ( x , y , 0 , t ) = j E m , y , j s ( x , y , 0 ; ω ) exp ( i ω t ) ,
H z s ( x , y , 0 , t ) = j H m , z , j s ( x , y , 0 ; ω ) exp ( i ω t ) ,
E m , x , j s ( x , y , 0 ; ω ) = j 2 π α j ( ω ) E m ext 8 ε 0 l 2 0 l ( x x ) f ( | x x | , y ) cos ( j π x / l ) d x ,
E m , y , j s ( x , y , 0 ; ω ) = j 2 π α j ( ω ) E m ext y 8 ε 0 l 2 0 l f ( | x x | , y ) cos ( j π x / l ) d x ,
H m , z , j s ( x , y , 0 ; ω ) = j i ω α j ( ω ) E m ext y 8 l 0 l f ( | x x | , y ) sin ( j π x / l ) d x
f ( | x x | , y ) [ ( x x ) 2 + y 2 ] 3 2 .
cos ( j π x / l ) cos ( j π x / l ) + j π l sin ( j π x / l ) ( x x ) ,
sin ( j π x / l ) sin ( j π x / l ) j π l cos ( j π x / l ) ( x x ) .
E m , x , j s ( x , y , 0 ; ω ) j 2 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 ) ] ,
E m , y , j s ( x , y , 0 ; ω ) j 2 π α j ( ω ) E m ext 8 ε 0 l 3 [ l y β 1 ( x , y ) cos ( j π x / l ) + j π y l β 2 ( x , y ) sin ( j π x / l ) ] ,
H m , z , j s ( x , y , 0 ; ω ) j i ω α j ( ω ) E m ext 8 l 2 [ l y β 1 ( x , y ) sin ( j π x / l ) j π y l β 2 ( x , y ) cos ( j π x / l ) ]
β 1 ( x , y ) y 2 0 l f ( | x x | , y ) d x = l x ( l x ) 2 + y 2 + x x 2 + y 2 ,
β 2 ( x , y ) l 0 l ( x x ) f ( | x x | , y ) d x = l ( l x 2 ) + y 2 l x 2 + y 2 ,
β 3 ( x , y ) 0 l ( x x ) 2 f ( | x x | , y ) d x = x l ( l x 2 ) + y 2 x x 2 + y 2 + ln [ l x + ( l x ) 2 + y 2 x + x 2 + y 2 ] .
| E near , j s ( ω ) | 2 1 ( Ω j 2 ω 2 ) 2 + ω 2 / τ 2 and | E far , j s ( ω ) | 2 ω 4 ( Ω j 2 ω 2 ) 2 + ω 2 / τ 2 .
ω near , j 2 = Ω j 2 1 2 τ 2 and ω far , j 2 = Ω j 2 ( 1 1 2 τ 2 Ω j 2 ) 1 Ω j 2 + 1 2 τ 2 ,
Δ ω j 1 2 Ω j τ 2 .
w rep U V = π 4 3 h ¯ 2 10 m e ( 3 n ) 5 3 .
𝒱 = n 0 2 e 2 π R 3 4 ε 0 u ( x , t ) x 0 l u ( x , t ) x g ( | x x | ) d x + π 4 3 h ¯ 2 10 m e ( 3 n 0 ) 5 3 ( 1 u ( x , t ) x ) 5 3 π R 2 ,
2 u ( x , t ) t 2 = [ 1 + v F 2 3 v 2 ( 1 u ( x , t ) x ) 1 3 ] v 2 2 u ( x , t ) x 2 ,

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