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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  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]
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    [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]
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    [CrossRef] [PubMed]
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    [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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  14. W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161–1166(2000).
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2010

2009

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

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

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

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

2001

2000

1976

1975

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

1969

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.

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.

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.

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

Phys. Rev.

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

Phys. Rev. A

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

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.

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

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

Other

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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