Abstract

We find some useful and effective approximations of the dispersion relation of the fundamental surface plasmon wave propagating around a curved dielectric-metal interface. These estimations are valid even with a very small curvature radius, which is the case of interest for nano-optics applications, and are validated through the numerical solution of the characteristic equation of the curved interface.

© 2011 Optical Society of America

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References

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  1. S. A. Maier, Plasmonics: Fundementals and Applications (Springer, 2007).
  2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [CrossRef] [PubMed]
  3. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
    [CrossRef]
  4. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
    [CrossRef] [PubMed]
  5. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
    [CrossRef] [PubMed]
  6. A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
    [CrossRef]
  7. V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
    [CrossRef]
  8. A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
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  9. J.-W. Liaw and P.-T. Wu, “Dispersion relation of surface plasmon wave propagating along a curved metal-dielectric interface,” Opt. Express 16, 4945–4951 (2008).
    [CrossRef] [PubMed]
  10. K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
    [CrossRef]
  11. B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
    [CrossRef]
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2010 (1)

2009 (1)

B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
[CrossRef]

2008 (2)

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

J.-W. Liaw and P.-T. Wu, “Dispersion relation of surface plasmon wave propagating along a curved metal-dielectric interface,” Opt. Express 16, 4945–4951 (2008).
[CrossRef] [PubMed]

2007 (2)

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
[CrossRef]

2006 (3)

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

2005 (1)

2001 (1)

2000 (1)

1976 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[CrossRef]

Alú, A.

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Aussenegg, F. R.

Berglund, W.

Bozhevolnyi, S. I.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

Carniel, F.

Chen, L.

B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
[CrossRef]

Costa, R.

Deutsch, M.

K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
[CrossRef]

Dikken, D. J.

Ditlbacher, H.

Drezet, A.

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[CrossRef]

Engheta, N.

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Garcia-Vidal, F. J.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

Gopinath, A.

Guo, B.

B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
[CrossRef]

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Hasegawa, K.

K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
[CrossRef]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Hohenau, A.

Krenn, J. R.

Kuipers, L.

Leitner, A.

Liaw, J.-W.

Maier, S. A.

S. A. Maier, Plasmonics: Fundementals and Applications (Springer, 2007).

Marcuse, D.

Markel, V. A.

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

Martinelli, M.

Martin-Moreno, L.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

Melloni, A.

Moreno, E.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

Nockel, J. U.

K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
[CrossRef]

Oosten, D. V.

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Rodrigo, S. G.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[CrossRef] [PubMed]

Sarychev, A. K.

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

Song, G.

B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
[CrossRef]

Spasenovi, M.

Steinberger, B.

Stepanov, A. L.

Verhagen, E.

Wu, P.-T.

IEEE J. Quantum Electron. (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

IEEE Trans. Nanotechnol. (1)

B. Guo, G. Song, and L. Chen, “Resonant enhanced wave filter and waveguide via surface plasmons,” IEEE Trans. Nanotechnol. 8, 408–411 (2009).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[CrossRef]

Phys. Rev. A (1)

K. Hasegawa, J. U. Nockel, and M. Deutsch, “Curvature-induced radiation of surface plasmon polaritons propagating around bends,” Phys. Rev. A 75, 063816 (2007).
[CrossRef]

Phys. Rev. B (2)

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Science (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Other (1)

S. A. Maier, Plasmonics: Fundementals and Applications (Springer, 2007).

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Figures (8)

Fig. 1
Fig. 1

The system considered in this paper consists of two homogeneous and isotropic regions S 1 and S 2 divided by a curved interface of radius R on the y z plane. The cylindrical coordinates are centered in O.

Fig. 2
Fig. 2

The Poynting theorem discussed in Section 5 is applied at the volume V enclosed by A B C . The arc B C subtends an angle δ θ and A B ¯ = L . The points A, B, and C have, respectively, cylindrical coordinates ( r = 0 , θ = 0 ), ( r = L , θ = 0 ), and ( r = L , θ = δ θ ). The black arrows indicate the outward versors normal to the surfaces A B , A C , and B C .

Fig. 3
Fig. 3

Dispersion functions ω ( γ ) of the first four TM-SPW propagating around a curved air-silver interface of radius R = 100 nm . The black bold line represents the fundamental mode (FM). The thin dashed line is the light line ω = k 0 n 1 .

Fig. 4
Fig. 4

Real part of the dispersion function of the fundamental SPW propagating around a curved air-silver interface of radius R = 100 nm . The three diagrams represent the function in different frequency ranges (respectively, 0 < ω < 3 · 10 15 , 3 · 10 15 < ω < 6 · 10 15 , and 6 · 10 15 < ω < 8 · 10 15 ). The gray line is the dispersion function of the air-silver straight interface; the black line represents the numerical results obtained by means of Eq. (3); the dotted line represents the estimation obtained by means of the A E ; and the circles represent the estimation obtained by means of the P E . The dashed line is the light line.

Fig. 5
Fig. 5

Same as Fig. 4 but the imaginary part is represented. Also, estimation obtained by means of the Poy E is shown (triangles).

Fig. 6
Fig. 6

Same as Fig. 4 but with R = 600 nm .

Fig. 7
Fig. 7

Same as Fig. 5 but with R = 600 nm .

Fig. 8
Fig. 8

The dispersion functions of the first four TM-SPW propagating around a curved silver-air interface of radius R = 100 nm . The black bold line represents the FM. The thin dashed line is the light line ω = k 0 n 1 . Im ( γ / R ) 0 when ω = 0 even in the case of the fundamental mode (in this example γ / R = i · 2.50 · 10 6 ).

Equations (26)

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δ r r a ( r ) + δ r a ( r ) / r + ( k 0 2 n i 2 γ 2 / r 2 ) a ( r ) = 0 ( i = 1 , 2 ) .
δ r H γ ( 1 ) ( k 0 n 2 R ) H γ ( 1 ) ( k 0 n 2 R ) = ρ · δ r J γ ( k 0 n 1 R ) J γ ( k 0 n 1 R ) .
δ r H γ ( 1 ) ( x 2 ) H γ ( 1 ) ( x 2 ) = ( x 2 x 1 ) 2 δ r J γ ( x 1 ) J γ ( x 1 ) .
δ r H γ ( 1 ) ( x 2 ) H γ ( 1 ) ( x 2 ) δ r H 0 ( 1 ) ( x 2 ) H 0 ( 1 ) ( x 2 ) = k 0 n 2 ( ( 1 / 2 ) H 1 ( 1 ) ( x 2 ) ( 1 / 2 ) H 1 ( 1 ) ( x 2 ) H 0 ( 1 ) ( x 2 ) ) = k 0 n 2 ( H 1 ( 1 ) ( x 2 ) H 0 ( 1 ) ( x 2 ) ) .
δ r J γ ( x 1 ) J γ ( x 1 ) = k 0 n 1 ( γ x 1 J γ + 1 ( x 1 ) J γ ( x 1 ) ) k 0 n 1 ( γ x 1 ( 1 / Γ ( γ + 2 ) ) ( x 1 / 2 ) γ + 1 ( 1 / Γ ( γ + 1 ) ) ( x 1 / 2 ) γ ) = k 0 n 1 ( γ x 1 x 1 2 ( γ + 1 ) ) .
H 1 ( 1 ) ( x 2 ) H 0 ( 1 ) ( x 2 ) = x 2 x 1 ( γ x 1 x 1 2 ( γ + 1 ) ) .
γ = x 1 2 ( 1 2 H 1 ( 1 ) ( x 2 ) x 2 H 0 ( 1 ) ( x 2 ) ) .
2 ( H x ) δ y ϵ ϵ δ y H x δ z ϵ ϵ δ z H x + k 0 2 ϵ H x = 0.
δ r r h + δ r h r δ r ϵ ϵ δ r h + ( k 0 2 ϵ γ 2 r 2 ) h = 0 ,
δ u u h δ u ϵ ϵ δ u h + ( k 0 2 ϵ e 2 u / R γ 2 R 2 ) h = 0.
δ u u h δ u ϵ ϵ δ u h + ( k 0 2 ϵ γ 2 R 2 ) h = 0.
γ / R = β = k 0 n 1 2 n 2 2 n 1 2 + n 2 2 , h s ( u ) = e k 1 u ( u 0 k 1 = β 2 k 0 2 n 1 2 ) , h s ( u ) = e k 2 u ( u > 0 k 2 = β 2 k 0 2 n 2 2 ) .
δ u ( 1 ϵ δ u h c ) + k 0 2 e 2 u / R h c = λ c ϵ h c ,
δ u ( 1 ϵ δ u h s ) + k 0 2 h s = λ s ϵ h s ,
L c = ϵ ( δ u ( 1 ϵ δ u ) + k 0 2 e 2 u / R ) ,
L s = ϵ ( δ u ( 1 ϵ δ u ) + k 0 2 ) .
f 1 , f 2 = f 1 ( u ) f 2 ( u ) ϵ ( u ) δ u .
δ λ = h s , δ L [ h s ] h s , h s ,
δ λ = 1 R 2 1 / ( 2 T 1 + 2 ) 1 / ( T 1 ) + 1 / ( 2 T 2 2 ) 1 / ( T 2 ) 1 / ( 2 T 1 x 1 2 ) + 1 / ( 2 T 2 x 2 2 ) , T 1 , 2 = x 1 , 2 4 / ( x 1 2 + x 2 2 ) .
ϕ A B = r = 0 R e r h x * 2 δ r = γ c 0 μ 0 2 k 0 ϵ 1 r = 0 R | J γ ( k 0 n 1 r ) | 2 r δ r ,
V 2 | E | 2 δ V = 0.
α R = P Loss ( γ , ϵ 2 ) / P T X ( γ , ϵ 1 ) .
P Loss ( γ , ϵ 2 ) P T X ( γ , ϵ 1 ) P Loss ( Re ( γ ) , ϵ 2 ) + i · α · δ P Loss / δ α P T X ( Re ( γ ) , ϵ 1 ) + i · α · δ P T X / δ α ,
P Loss ( γ , ϵ 2 ) = ( 1 / 2 ) δ θ · R · Re ( e θ ( R + ) h x * ( R + ) ) = Re ( R · δ θ · c 0 μ 0 | J γ ( x 1 ) | 2 · δ r H γ ( 1 ) ( x 2 ) 2 ( H γ ( 1 ) ( x 2 ) ) i k 0 ϵ 2 ) .
r = 0 R e r h x * 2 δ r = Re ( γ ) c 0 μ 0 2 k 0 ϵ 1 r = 0 R | J Re ( γ ) ( k 0 n 1 r ) | 2 r δ r = Re ( γ ) c 0 μ 0 2 k 0 ϵ 1 r = 0 R ( J Re ( γ ) ( k 0 n 1 r ) ) 2 r δ r = C · HPR ( a , b , x 1 2 ) ; C = Γ ( 2 Re ( γ ) ) · γ c 0 μ 0 2 k 0 ϵ 1 · ( x 1 2 2 ) 2 Re ( γ ) ; a = ( Re ( γ ) , 1 / 2 + Re ( γ ) ; b = ( 1 + Re ( γ ) , 1 + Re ( γ ) , 1 + 2 Re ( γ ) ) ;
α R = Re ( | J Re ( γ ) ( x 1 ) | 2 · δ r H Re ( γ ) ( 1 ) ( x 2 ) · ϵ 1 · 2 2 Re ( γ ) 1 ( H Re ( γ ) ( 1 ) ( x 2 ) ) · i ϵ 2 · Re ( γ ) · x 1 4 Re ( γ ) · Γ · HPR ) ,

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