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.

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  1. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [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]
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  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 Schrdinger 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]
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    [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).
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2011 (1)

2010 (5)

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]

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]

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]

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]

2009 (1)

2008 (2)

2007 (2)

E. Feigenbaum and M. Orenstein, “Plasmon-solitons,” Opt. Lett. 32, 674–676 (2007).
[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]

2004 (1)

2003 (1)

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 (2)

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 (1)

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

1987 (1)

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

Adegoke, J. A.

Ahmediev, N. N.

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

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]

Ankiewicz, A.

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

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 Schrdinger 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 (1)

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 (1)

Nano Lett. (1)

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 (1)

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

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (1)

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 (2)

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. (3)

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 (1)

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

Other (1)

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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