Abstract

Using a combination of numerical and analytical techniques we demonstrate that a metal stripe surrounded by the active and passive dielectrics supports propagation of stable spatial surface-plasmon solitons. Our analytical methods include the multiple scale reduction of the Maxwell’s equations to the coupled Ginzburg-Landau system, and the soliton perturbation theory developed in the framework of the latter.

© 2011 Optical Society of America

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  1. D. K. Gramotnev, and S. I. Bozhevolnyi, "Plasmonics beyond the diffraction limit," Nat. Photonics 4, 83-91 (2010).
    [CrossRef]
  2. E. Feigenbaum, and M. Orenstein, "Plasmon-solitons," Opt. Lett. 32, 674-676 (2007).
    [CrossRef] [PubMed]
  3. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, "Self-focusing and spatial plasmon-polariton solitons," Opt. Express 17, 21732-21737 (2009).
    [CrossRef] [PubMed]
  4. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
    [CrossRef] [PubMed]
  5. A. Marini, A. V. Gorbach, and D. V. Skryabin, "Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures," Opt. Lett. 35, 3532-3534 (2010).
    [CrossRef] [PubMed]
  6. N. N. Ahmediev, and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 2007).
  7. A. Marini, and D. V. Skryabin, "Ginzburg-landau equation bound to the metal-dielectric interface and transverse nonlinear optics with amplified plasmon polaritons," Phys. Rev. A 81, 033850 (2010).
    [CrossRef]
  8. B. A. Malomed, "Evolution of nonsoliton and quasiclassical wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations," Physica D 29, 155-172 (1987).
    [CrossRef]
  9. S. Fauve, and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
    [CrossRef] [PubMed]
  10. B. A. Malomed, and H. G. Winful, "Stable solitons in two-component active systems," Phys. Rev. E 53, 5365 (1996).
    [CrossRef]
  11. J. Atai, and B. A. Malomed, "Stability and interactions of solitons in two-component active systems," Phys. Rev. E 54, 4371 (1996).
    [CrossRef]
  12. W. J. Firth, and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," Eur. Phys. J. D 59, 13-21 (2010).
    [CrossRef]
  13. D. J. Bergman, and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402 (2003).
    [CrossRef] [PubMed]
  14. M. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express 12, 4072-4079 (2004).
    [CrossRef] [PubMed]
  15. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, "Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium," Opt. Express 16, 1385 (2008).
    [CrossRef] [PubMed]
  16. P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, "Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length," Opt. Lett. 35, 1197-1199 (2010).
    [CrossRef] [PubMed]
  17. M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
    [CrossRef] [PubMed]
  18. D. V. Skryabin, A. Gorbach, and A. Marini, "Surface induced nonlinearity enhancement of TM-modes in planar subwavelength waveguides," J. Opt. Soc. Am. B 28, 109-114 (2011).
    [CrossRef]

2011

2010

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, "Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length," Opt. Lett. 35, 1197-1199 (2010).
[CrossRef] [PubMed]

W. J. Firth, and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," Eur. Phys. J. D 59, 13-21 (2010).
[CrossRef]

D. K. Gramotnev, and S. I. Bozhevolnyi, "Plasmonics beyond the diffraction limit," Nat. Photonics 4, 83-91 (2010).
[CrossRef]

A. Marini, A. V. Gorbach, and D. V. Skryabin, "Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures," Opt. Lett. 35, 3532-3534 (2010).
[CrossRef] [PubMed]

A. Marini, and D. V. Skryabin, "Ginzburg-landau equation bound to the metal-dielectric interface and transverse nonlinear optics with amplified plasmon polaritons," Phys. Rev. A 81, 033850 (2010).
[CrossRef]

2009

2008

2007

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

E. Feigenbaum, and M. Orenstein, "Plasmon-solitons," Opt. Lett. 32, 674-676 (2007).
[CrossRef] [PubMed]

2004

2003

D. J. Bergman, and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

1996

B. A. Malomed, and H. G. Winful, "Stable solitons in two-component active systems," Phys. Rev. E 53, 5365 (1996).
[CrossRef]

J. Atai, and B. A. Malomed, "Stability and interactions of solitons in two-component active systems," Phys. Rev. E 54, 4371 (1996).
[CrossRef]

1990

S. Fauve, and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

1987

B. A. Malomed, "Evolution of nonsoliton and quasiclassical wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations," Physica D 29, 155-172 (1987).
[CrossRef]

Adegoke, J. A.

Ambati, M.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Atai, J.

J. Atai, and B. A. Malomed, "Stability and interactions of solitons in two-component active systems," Phys. Rev. E 54, 4371 (1996).
[CrossRef]

Bahoura, M.

Bartal, G.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman, and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Bolger, P. M.

Bozhevolnyi, S. I.

D. K. Gramotnev, and S. I. Bozhevolnyi, "Plasmonics beyond the diffraction limit," Nat. Photonics 4, 83-91 (2010).
[CrossRef]

Davoyan, A. R.

Dickson, W.

Fainman, Y.

Fauve, S.

S. Fauve, and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Feigenbaum, E.

Firth, W. J.

W. J. Firth, and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," Eur. Phys. J. D 59, 13-21 (2010).
[CrossRef]

Genov, D. A.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Gorbach, A.

Gorbach, A. V.

Gramotnev, D. K.

D. K. Gramotnev, and S. I. Bozhevolnyi, "Plasmonics beyond the diffraction limit," Nat. Photonics 4, 83-91 (2010).
[CrossRef]

Hickey, S. G.

Kivshar, Y. S.

Krasavin, A. V.

Liebscher, L.

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Malomed, B. A.

J. Atai, and B. A. Malomed, "Stability and interactions of solitons in two-component active systems," Phys. Rev. E 54, 4371 (1996).
[CrossRef]

B. A. Malomed, and H. G. Winful, "Stable solitons in two-component active systems," Phys. Rev. E 53, 5365 (1996).
[CrossRef]

B. A. Malomed, "Evolution of nonsoliton and quasiclassical wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations," Physica D 29, 155-172 (1987).
[CrossRef]

Marini, A.

Mayy, M.

Nam, S. H.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Nezhad, M. P.

Noginov, M. A.

Orenstein, M.

Paulau, P. V.

W. J. Firth, and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," Eur. Phys. J. D 59, 13-21 (2010).
[CrossRef]

Podolskiy, V. A.

Reynolds, K.

Ritzo, B. A.

Shadrivov, I. V.

Skryabin, D. V.

Stockman, M. I.

D. J. Bergman, and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Tetz, K.

Thual, O.

S. Fauve, and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Ulin-Avila, E.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Winful, H. G.

B. A. Malomed, and H. G. Winful, "Stable solitons in two-component active systems," Phys. Rev. E 53, 5365 (1996).
[CrossRef]

Zayats, A. V.

Zhang, X.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Zhu, G.

Eur. Phys. J. D

W. J. Firth, and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," Eur. Phys. J. D 59, 13-21 (2010).
[CrossRef]

J. Opt. Soc. Am. B

Nano Lett.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of stimulated emission of surface plasmon polaritons," Nano Lett. 8, 3998-4001 (2008).
[CrossRef] [PubMed]

Nat. Photonics

D. K. Gramotnev, and S. I. Bozhevolnyi, "Plasmonics beyond the diffraction limit," Nat. Photonics 4, 83-91 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

A. Marini, and D. V. Skryabin, "Ginzburg-landau equation bound to the metal-dielectric interface and transverse nonlinear optics with amplified plasmon polaritons," Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Phys. Rev. E

B. A. Malomed, and H. G. Winful, "Stable solitons in two-component active systems," Phys. Rev. E 53, 5365 (1996).
[CrossRef]

J. Atai, and B. A. Malomed, "Stability and interactions of solitons in two-component active systems," Phys. Rev. E 54, 4371 (1996).
[CrossRef]

Phys. Rev. Lett.

D. J. Bergman, and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

S. Fauve, and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength Discrete Solitons in Nonlinear Metamaterials," Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Physica D

B. A. Malomed, "Evolution of nonsoliton and quasiclassical wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations," Physica D 29, 155-172 (1987).
[CrossRef]

Other

N. N. Ahmediev, and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 2007).

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Figures (2)

Fig. 1
Fig. 1

(a) Dielectric-metal-dielectric structure. (b) The coexistence domain of the stable zero and nonzero spatially homogeneous SPPs in the (g,Δβ)-plane. gth is indicated with the full lines and g0 with the dashed one. Other parameters are β = 1.43, l = 0.0026; fϒ = 3.5 × 10−3(1 + 0.1i); κ = 0.0028 corresponds to w = 98nm. (b) Subcritical dependence of the intensity |R|2 of the spatially homogeneous SPP vs gain g, Δβ = −2.5 × 10−4.

Fig. 2
Fig. 2

(a) Maximum of the soliton intensity, max|ψa|2 vs gain g. The crosses correspond to the Newton-Raphson method, while the full line corresponds to the analytical results. The dotted line correspond to the spatially homogeneous SPPs. The other parameters are as in Fig. 1(c). (b) Dots show the numerically computed soliton profiles |ψp(y)| (blue) and |ψa(y)| (red). Full lines are the soliton profiles predicted by the perturbation theory. The solitons shown correspond to the large amplitude branch. g = 0.0042 and the other parameters as in Fig. 1(c).

Equations (11)

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e a = ( e x a e z a ) = θ ( x + w / 2 ) ( i β q m 1 ) e q m ( x + w / 2 ) + θ ( x w / 2 ) ( i β q d 1 ) e q d ( x + w / 2 ) ,
e p = ( e x p e z p ) = θ ( x w / 2 ) ( i β q d 1 ) e q d ( x w / 2 ) + θ ( x + w / 2 ) ( i β q m 1 ) e q m ( x w / 2 ) ,
E x = [ ψ p ( y , z ) e x p ( x ) + ψ a ( y , z ) e x a ( x ) + δ E x + ] e i β z , E y = [ ϕ p ( y , z ) e y p ( x ) + ϕ a ( y , z ) e y a ( x ) + ] e i β z , E z = [ ψ p ( y , z ) e z p ( x ) + ψ a ( y , z ) e z a ( x ) + δ E z + ] e i β z ,
L ^ = ( β 2 ɛ ˜ i β x i β x x x 2 ɛ ˜ ) , N ^ = ( 2 i β z + y y 2 + δ ɛ ˜ x z 2 x z 2 y y 2 + δ ɛ ˜ ) , ɛ ˜ = ɛ m [ θ ( x + w / 2 ) θ ( x w / 2 ) ] + ɛ d [ θ ( x w / 2 ) + θ ( x w / 2 ) ] , δ ɛ = i ɛ m [ θ ( x + w / 2 ) θ ( x w / 2 ) ] + ( 1 / 2 ) α [ θ ( x w / 2 ) θ ( x w / 2 ) ] + [ i ɛ a + χ 3 | ψ a | 2 | e a | 2 ] θ ( x w / 2 ) , b = ( y ϕ p x e y p + y ϕ a x e y a i β y ϕ p e y p + i β y ϕ a e y a ) .
i z ψ p + ( 1 / 2 β ) y y 2 ψ p + ( i l Δ β ) ψ p + κ ψ a = 0 , i z ψ a + ( 1 / 2 β ) y y 2 ψ a + [ i ( l g ) + Δ β ] ψ a + f ϒ | ψ a | 2 ψ a + κ ψ p = 0 ,
ϒ = k 2 n 2 β 3 ( ɛ d ɛ m ) 2 ɛ d 2 ,
( g th l κ 2 / l ) ( g th 2 l ) 2 = 4 Δ β 2 ( l g th ) .
d N d z + 2 l + d y | ψ p | 2 2 ( g l ) + d y | ψ a | 2 + 2 f ϒ + d y | ψ a | 4 = 0 ,
ψ a = η 1 / ( β f ϒ ) sech ( η y ) e i μ z + O ( κ 2 ) ,
ψ p = 2 κ η β f ϒ { cosh ( η y ) ln [ 2 cosh ( η y ) ] η y sinh ( η y ) } e i μ z + O ( κ 3 ) ,
d η d z + β 2 κ 2 l C η 3 2 ( g l ) η + 4 ϒ 3 β ϒ η 3 = 0 ,

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