Abstract

We present a first-principles derivation of the variational equations describing the dynamics of the interaction of a spatial soliton and a surface plasmon polariton (SPP) propagating along a metal/dielectric interface. The variational ansatz is based on the existence of solutions exhibiting differentiated and spatially resolvable localized soliton and SPP components. These solutions, referred to as soliplasmons, can be physically understood as bound states of a soliton and an SPP, which dispersion relations intersect, allowing resonant interaction between them [Phys. Rev. A 79, 041803 (2009)]. The existence of soliplasmon states and their interesting nonlinear resonant behavior has been validated already by full-vector simulations of the nonlinear Maxwell’s equations, as reported in [Opt. Lett. 37, 4221 (2012)]. Here, we provide the theoretical analysis of the nonlinear oscillator model introduced in our previous work and present its rigorous derivation. We also provide some extensions of the model to improve its applicability.

© 2013 Optical Society of America

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References

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  1. K. Y. Bliokh, Y. P. Bliokh, and A. Ferrando, “Resonant plasmon–soliton interaction,” Phys. Rev. A 79, 041803 (2009).
    [CrossRef]
  2. C. Milián, D. E. Ceballos-Herrera, D. V. Skryabin, and A. Ferrando, “Soliton–plasmon resonances as Maxwell nonlinear bound states,” Opt. Lett. 37, 4221–4223 (2012).
    [CrossRef]
  3. V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).
  4. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
    [CrossRef]
  5. D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
    [CrossRef]
  6. J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
    [CrossRef]
  7. W. Walasik, V. Nazabal, M. Chauvet, Y. Kartashov, and G. Renversez, “Low-power plasmon–soliton in realistic nonlinear planar structures,” Opt. Lett. 37, 4579–4581 (2012).
    [CrossRef]
  8. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).
  9. L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys. 78, 1199–1202 (2010).
    [CrossRef]
  10. Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
    [CrossRef]
  11. Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
    [CrossRef]
  12. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2008).
  13. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef]
  14. Under the usual consideration that χiiii(3)=3χjkjk(3) and χjkjk(3)=χjjkk(3)=χjkkj(3) (where i, j, k run over the Cartesian coordinates x, y, z), χ(3)≡χxxxx(3)=ε0cn2/2 and χ¯(3)=χ(3)/2.
  15. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon–polariton solitons,” Opt. Express 17, 21732–21737 (2009).
    [CrossRef]
  16. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  17. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  18. Recall that Es=Cfs and H¯y=k0cβnp−1E¯npx, so that the plasmonic projection gives ∫ℝH¯y0|Es|2Enpx∼∫E¯npx(0)Enpxfs2∼O(e−2κsa), whereas the soliton one yields ∫ℝf¯s(0)Enpxfs2∼O(e−(2κs+κs0)a). Both are negligible in the weak coupling approximation. An analogous argument holds for the Es2Enpx* term.
  19. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  20. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
    [CrossRef]
  21. This exchange of energy is visible in the figures presented in [2] showing the propagation of a perturbed soliplasmon field along the surface, in which the flux of the Poynting vector is represented. One should remark at this point that these simulations are the result of solving the full nonlinear vector Maxwell’s equations (1) numerically.
  22. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
    [CrossRef]
  23. E. Feigenbaum and M. Orenstein, “Plasmon–soliton,” Opt. Lett. 32, 674–676 (2007).
    [CrossRef]
  24. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
    [CrossRef]
  25. D. V. Skryabin, A. V. 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]
  26. A. Marini, D. V. Skryabin, and B. Malomed, “Stable spatial plasmon solitons in a dielectric–metal–dielectric geometry with gain and loss,” Opt. Express 19, 6616–6622 (2011).
    [CrossRef]
  27. 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]
  28. C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
    [CrossRef]

2013 (1)

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
[CrossRef]

2012 (2)

2011 (4)

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
[CrossRef]

D. V. Skryabin, A. V. 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]

A. Marini, D. V. Skryabin, and B. Malomed, “Stable spatial plasmon solitons in a dielectric–metal–dielectric geometry with gain and loss,” Opt. Express 19, 6616–6622 (2011).
[CrossRef]

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

2010 (3)

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

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]

L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys. 78, 1199–1202 (2010).
[CrossRef]

2009 (2)

K. Y. Bliokh, Y. P. Bliokh, and A. Ferrando, “Resonant plasmon–soliton interaction,” Phys. Rev. A 79, 041803 (2009).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon–polariton solitons,” Opt. Express 17, 21732–21737 (2009).
[CrossRef]

2007 (2)

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

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

2005 (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

1985 (2)

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

1984 (1)

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[CrossRef]

1983 (1)

1980 (1)

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).

Agranovich, V. M.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).

Agrawal, G. P.

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2008).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Babichenko, V. S.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).

Bliokh, K. Y.

K. Y. Bliokh, Y. P. Bliokh, and A. Ferrando, “Resonant plasmon–soliton interaction,” Phys. Rev. A 79, 041803 (2009).
[CrossRef]

Bliokh, Y. P.

K. Y. Bliokh, Y. P. Bliokh, and A. Ferrando, “Resonant plasmon–soliton interaction,” Phys. Rev. A 79, 041803 (2009).
[CrossRef]

Ceballos-Herrera, D. E.

Chauvet, M.

Chernyak, V. Ya.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Davoyan, A. R.

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Eksioglu, Y.

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
[CrossRef]

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
[CrossRef]

Fedyanin, V. K.

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[CrossRef]

Feigenbaum, E.

Ferrando, A.

Gorbach, A. V.

Gordon, J. P.

Güven, K.

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
[CrossRef]

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
[CrossRef]

Hu, B.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

Kartashov, Y.

Kivshar, Y. S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Maier, S. A.

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

Malomed, B.

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

Marini, A.

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Mihalache, D.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[CrossRef]

Milián, C.

C. Milián, D. E. Ceballos-Herrera, D. V. Skryabin, and A. Ferrando, “Soliton–plasmon resonances as Maxwell nonlinear bound states,” Opt. Lett. 37, 4221–4223 (2012).
[CrossRef]

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

Müstecaplioglu, O. E.

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
[CrossRef]

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
[CrossRef]

Nazabal, V.

Nazmitdinov, R. G.

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[CrossRef]

Novotny, L.

L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys. 78, 1199–1202 (2010).
[CrossRef]

Orenstein, M.

Panoiu, N. C.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Renversez, G.

Seaton, C. T.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

Shadrivov, I. V.

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Skryabin, D. V.

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

Walasik, W.

Wallis, R. F.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

Ye, F.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Am. J. Phys. (1)

L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys. 78, 1199–1202 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

J. Appl. Phys. (2)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” JETP Lett. 32, 512–515 (1980).

Opt. Express (2)

Opt. Lett. (4)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Phys. Rev. A (4)

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]

K. Y. Bliokh, Y. P. Bliokh, and A. Ferrando, “Resonant plasmon–soliton interaction,” Phys. Rev. A 79, 041803 (2009).
[CrossRef]

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dynamical analysis of a weakly coupled nonlinear dielectric waveguide: surface-plasmon model as another type of Josephson junction,” Phys. Rev. A 84, 033805 (2011).
[CrossRef]

Y. Ekşioğlu, O. E. Müstecaplioğlu, and K. Güven, “Dissipative Josephson junction of an optical soliton and a surface plasmon,” Phys. Rev. A 87, 023823 (2013).
[CrossRef]

Phys. Rev. Lett. (1)

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef]

Phys. Scr. (1)

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “Polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[CrossRef]

Rep. Prog. Phys. (1)

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface–plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Other (8)

This exchange of energy is visible in the figures presented in [2] showing the propagation of a perturbed soliplasmon field along the surface, in which the flux of the Poynting vector is represented. One should remark at this point that these simulations are the result of solving the full nonlinear vector Maxwell’s equations (1) numerically.

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2008).

Under the usual consideration that χiiii(3)=3χjkjk(3) and χjkjk(3)=χjjkk(3)=χjkkj(3) (where i, j, k run over the Cartesian coordinates x, y, z), χ(3)≡χxxxx(3)=ε0cn2/2 and χ¯(3)=χ(3)/2.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Recall that Es=Cfs and H¯y=k0cβnp−1E¯npx, so that the plasmonic projection gives ∫ℝH¯y0|Es|2Enpx∼∫E¯npx(0)Enpxfs2∼O(e−2κsa), whereas the soliton one yields ∫ℝf¯s(0)Enpxfs2∼O(e−(2κs+κs0)a). Both are negligible in the weak coupling approximation. An analogous argument holds for the Es2Enpx* term.

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

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Figures (3)

Fig. 1.
Fig. 1.

Typical initial soliplasmon profiles for quasi-stationary solutions whose propagation is presented in [2]: (a)-(b) antisymmetric-like solutions or π-soliplasmons; (c)-(d) symmetric-like solutions or 0-soliplasmons. They are numerically computed from Eq. (1) for the metal/Kerr interface considered in the aforementioned reference (λ0=1.5μm, εm=82, εK=2.09, n2=2.6×1020m2/W). Full (dashed) lines plot the x(z) components of the normalized field Eχ(3)E, and the shaded regions at x<0 represent the metal.

Fig. 2.
Fig. 2.

Parallel illumination of a metal/dielectric interface from a Kerr medium. εm,d,K are the linear dielectric constants of the metal (M), linear dielectric (D), and Kerr (K) regions, respectively. n2 is the nonlinear index of the Kerr material.

Fig. 3.
Fig. 3.

Parallel illumination of a metal/Kerr interface from the Kerr medium.

Equations (118)

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2E[·E]=k02D(E)=k02εLEk02PNL(E),PNL(E)=χ(3)[E·E*]E+χ¯(3)[E·E]E*,
E=Enp({Ai(z)}i=1N)+Es({Ci(z)}i=1N),
2Enpz2+L0Enp[·Enp]+2Esz2+L0Es=k02PNL(Enp)k02PNL(Es)k02QK(Enp,Es),
QK(Enp,Es)=χ(3){|Enp|2Es+|Es|2Enp+2Re(Enp*Es)[Enp+Es]}+χ¯(3){Enp2Es*+Es2Enp*+2[EnpEs][Enp*+Es*]}.
2Enpz2+L0Enp[·Enp]=k02PNL(Enp)
2Esz2+L0Es=k02PNL(Es).
E¯s(x,z)=uE¯s(x;C)eiβsz,E¯sCsech(γ2k0C[xa]),C,aR+,
βs2=k02[εK+γ2C2],
{t2+k02[εK+γ|E¯s|2]}E¯s=βs2E¯s,
E¯s(x,z)Es(x,z)=C(z)sech(γ2k0|C(z)|[xa]),C(z)=|C(z)|exp[iφs(z)]C.
Es(x,z)=ueiφs(z)E¯s(x;|C(z)|),
2Esz2+L0Es=k02{[χ(3)+χ¯(3)]|Es|2}Es,
2Esz2+{t2+k02[εK+γ|Es|2]}Es=0.
{t2+k02[εK+γ|Es|2]}Es=βs2Es,
2Es(x;C(z))z2+βs2Es(x;C(z))=0.
2z2[C(z)fs(x;|C(z)|)]+βs2[C(z)fs(x;|C(z)|)]=0.
[d2dz2+βs2][C(z)Rdxfs(x,0)fs(x;|C(z)|)]=0.
d2dz2[C(z)Ns(|C(z)|)]d2C(z)dz2Ns(|C(z)|).
Ns[d2dz2C(z)+βs2(|C(z)|)C(z)]=0.
εL(x)=εp(x){εmifx0εdifx>0.
E¯np(x,z)=enp(x)eiβnpz=(enpt(x)enpz(x))eiβnpz,
βnp2enpt+2enptt(iβnpenpz+t·enpt)=k02[εpenpt+χ(3)(enp·enp*)enpt+χ¯(3)(enp·enp)enpt*].
PNLt=[χ(3)(enp·enp*)enpt+χ¯(3)(enp·enp)enpt*]eiβnpz=[γ|enpt|2γ¯enpz2]enpteiβnpz,
PNLz=[γ¯|enpt|2γenpz2]enpzeiβnpz.
Dt=εnpenpteiβnpz;εnpεL+(γ|enpt|2γ¯enpz2),Dz=ε¯npenpzeiβnpz;ε¯npεL+(γ¯|enpt|2γenpz2).
[2E(·E)]0,
·D=0,
t·(εnpenpt)+iβnpε¯npenpz=0enpz=it·(εnpenpt)βnpε¯np.
εnpεL+γ|enpt|2,ε¯npεL+γ¯|enpt|2.
iβnpenpzεnpε¯npt·enpttεnpε¯np·enpt.
(t2+k02εnp)enpt+t(Fnpt·enpt)=βnp2enpt,
E¯npx(x,z)=enpx(x)eiβnpz=Afnp(x;A)eiβnpz,AR+,
enpx(x)=Afnp(x;A,B),enpz(x)=Bgnp(x;A;B).
E¯npx(x,z)Enpx(x,z)=A(z)fnp(x;|A(z)|)A(z)C.
(z2+t2+k02εp)EnpxzxEnpzx(t·Enpt)k02γ|Enpt|2Enpx,
Enpt(x,z)=eiφp(z)[|A(z)|fnp(x;|A(z)|)0]=eiφp(z)[enpx(x;|A(z)|)0]=eiφp(z)enpt(x;|A(z)|),
t·Enpt=eiφp(z)t·enpt=iβnpEnpzFnpt·Enpt.
2Enpxz2z(Enpzx)+iβnpEnpzx+[t2+k02εnp]Enpx+x(Fnpt·Enpt)=0.
2Enpxz2z(Enpzx)+iβnpEnpzx+βnp2Enpx=0.
[d2dz2+βnp2]RH¯y(0)Enpx(z)=(ddziβnp)RH¯y(0)(Enpzx).
RH¯y(x,0)xEnpz(x,z)=R(xH¯y(x,0))Enpz(x,z)=ik0cRεnpE¯npz(x,0)Enpz(x,z)=O(Enpz2)0,
(d2dz2+βnp2)RH¯y(0)Enpx(z)+ik02RH¯y(0)ΔlεpEnpx(z)=0.
H¯y(x,0)=k0cβnp(0)E¯npx(x,0)=Kfnp(x;|A(0)|),
Enpx(x,z)=A(z)fnp(x;|A(z)|).
d2dz2[NnpA(z)]+Nnp[βnp2+iλnp]A(z)=0,
NnpRfnp(x;|A(0)|)fnp(x;|A(z)|)
λnp1NnpRfnp(x;|A(0)|)Δlεp(x)fnp(x;|A(z)|).
d2dz2[Nnp(|A(z)|)A(z)]Nnp(|A(z)|)(d2A(z)dz2),
Nnp[d2dz2A(z)+β¯np2(|A(z)|)A(z)]=0,
E(x,z)=Enp(x;A(z))+uEs(x;C(z)).
2Enpxz2z(Enpzx)+iβnpEnpzx+L0Enpx+x(Fnpt·Enpt)+2Esz2+L0Es=k02[γ|Enpt|2Enpx+γ|Es|2Es+QK(Enpx,Es)],
εL(x)={εp(x)xdεKx>d,
εL=εp+Δεp,εL=εK+Δεs,
Δεp(x)={0xdεKεdx>d
Δεs(x)={εmεKx0εdεK0<xd0x>d,
L0=(t2+k02εp)+ΔεpL0p+Δεp,L0=(t2+k02εK)+ΔεsL0s+Δεs,
2Enpxz2z(Enpzx)+iβnpEnpzx+[L0p+k02γ|Enpt|2]Enpx+x(Fnpt·Enpt)eigenvalue equation for NL plasmon+2Esz2+[L0s+k02γ|Es|2]Eseigenvalue equation for soliton=k02(ΔεpEnpx+ΔεsEs)k02QK(Enpx,Es).
[2z2+βnp2]Enpx[ziβnp]xEnpz+[2z2+βs2]Es=k02[ΔεpEnpx+ΔεsEs+QK(Enpx,Es)].
In(a)=RFp(x)fsn(xa),
In(a)enκsa0dpFp(x)enκsxO(eκsa)n,
REnpz(x)fs(xa)eκsa0dpEnpz(x)enκsxO(eκsa)O(Enpz)0,
QK(Enpx,Es)=γ[2|Es|2Enpx+Es2Enpx*+2|Enpx|2Es+Enpx2Es*].
2Enpxz2+βnp2Enpx+2Esz2+βs2Esk02[ΔεpEnpx+ΔεsEs+γ{2|Enpx|2Es+Enpx2Es*}].
Nnp[d2dz2A+βnp2A]+δps[d2dz2C+βs2C]=ΔppAΔpsCΔK[2|A|2C+A2C*],
δpsRfnp(0)fs(z),Δpsk02Rfnp(0)Δεsfs(z),ΔKk02Rγfnp(0)fnp2(z)fs(z).
Δppk02Rfnp(0)Δεpfnp(z)k02(εkεd)de2κpxO(e2κpd)0,
δsp[d2dz2A+βnp2A]+Ns[d2dz2C+βs2C]=ΔssCΔspAΔK[2|A|2C+A2C*],
δspRfs(0)fnp(z),Δspk02Rfs(0)Δεpfnp(z),ΔKk02Rγfs(0)fnp2(z)fs(z).
Δssk02Rfs(0)Δεsfs(z)O(e[κs(0)+κs]a)0,
ΔKO(e[κs(0)+κs]a)0.
O(eκs(z)a)O(eκs(0)a)z.
O(e[κs(0)+κs]a)O(e2κs(0)a)O(e2κsa)0.
[NnpδpsδspNs][(d2dz2+βnp2)A(d2dz2+βs2)C]=[ΔppΔpsΔspΔss][AC][ΔKAC*2ΔK|A|2ΔKAC*2ΔK|A|2][AC].
[NnpδpsδspNs]1,
(d2dz2+Bnp2)A=qpsCqK(2|A|2C+A2C*),(d2dz2+Bs2)C=qspAqK(2|A|2C+A2C*).
Bnp2βnp2(δpsΔpsNsΔppNnpNsδpsδsp),Bs2βs2(δspΔspNnpΔssNnpNsδpsδsp).
qps(NsΔpsδpsΔssNnpNsδpsδsp),qsp(NnpΔspδspΔppNnpNsδpsδsp).
qK(NsΔKδpsΔKNnpNsδpsδsp),qK(NnpΔKδspΔKNnpNsδpsδsp).
Bnp2βnp2,Bs2βs2,
qpsΔpsNnp,qspΔspNs,
qKΔKNnp,qK0.
(d2dz2+βnp2)A=ΔpsNnpCΔKNnp(2|A|2C+A2C*),(d2dz2+βs2)C=ΔspNsA.
βnp2βp2,
ΔKγdfp(0)fp2(z)fs(z)adO(e3κpd)0.
(d2dz2+βp2)A=ΔpsNpC,(d2dz2+βs2)C=ΔspNsA.
E¯px(x)={βpE0k0εmeκmxx0βpE0k0εdeκdxx>0.
fp(x)E¯px(x)E¯px(0)={eκmxx0eκdxx>0.
fs(x)2eκs(ax),x<d,a>d>0.
Δps=k02[(εmεK)0fpfs+(εdεK)0dfpfs]d|κdκs|12eκsak02[εmεKκs+κm+(εdεK)d],
Δsp=k02(εKεd)dfsfp2k02(εKεdκdκs)edκd+(da)κs,
Np=12(1κd+1κm),
Ns=2κs.
A(z)=A˜(z)eik0nKz,C(z)=C˜(z)eik0nKz,
idA˜dz=μpA˜+qC˜,idC˜dz=μsC˜+q¯A˜
μp(βp2k02εK)2k0nK,μs(βs2k02εK)2k0nK=k0γ4nK|C˜|2.
qΔps2k0nKNp=k02nKNpRfpΔεsfs,q¯Δsp2k0nKNs=k02nKNsRfsΔεpfp.
q=k0nKNpeκsa(εmεK)[1κm+κs+(εdεKεmεK)d],q¯=k0nKNseκsa(εKεd)[1κdκsd].
κm=k0εm1εm+εdk0(εm)1/2,κd=k0εd1εm+εdk0εd(εm)1/2,
q¯qκs4κdκm(εdεK)κd(εm)=κsκd(εdεK4εd),
q¯q(εdεK4εd2)(εmγ¯2)1/2.
q¯q101(40γ¯)1/20.5γ¯1/2.
iddz(A˜C˜)=(μpqq¯μs)(A˜C˜),
βnp2=βp2+k02γp|A|2+O(|A|4).
(d2dz2+βnp2)A=ΔpsNnpCΔKNnp(2|A|2C+A2C*),(d2dz2+βs2)C=0.
[d2dz2+β¯np2(A,C)]A=Δ¯ps(|A|,|C|)NnpC[d2dz2+βs2(|C|)]C=0,
β¯np2(A,C)=βp2+k02γp|A|2+ΔK(|C|)NnpAC*Δ¯ps(|A|,|C|)=Δps(|C|)+2ΔK(|C|)|A|2.
Nnp[d2dz2A+βnp2A]+δps[d2dz2C+βs2C]=ΔppAΔpsCΔK[2|A|2C+A2C*]ΓK[2|C|2A+C2A*],
δsp[d2dz2A+βnp2A]+Ns[d2dz2C+βs2C]=ΔssCΔspAΔK[2|A|2C+A2C*]ΓK[2|C|2A+C2A*],
ΓKk02Rγfnp(0)fnp(z)fs2(z),ΓKk02Rγfs(0)fnp(z)fs2(z).
(d2dz2+Bnp2)A=qpsCqK(2|A|2C+A2C*)pK(2|C|2A+C2A*),(d2dz2+Bs2)C=qK(2|A|2C+A2C*)pK(2|C|2A+C2A*),
pK=NsΓKδpsΓKNnpNsδpsδsp,pK=NnpΓKδspΓKNnpNsδpsδsp.
(d2dz2+Bnp2)A=qpsCpK(2|C|2A+C2A*),(d2dz2+Bs2)C=pK(2|C|2A+C2A*).
B¯s2Bs2+pK(2C*A+CA*),
(d2dz2+B¯s2)C=0.
I(B¯s2)=pK|C||A|sin(φpφs)
ΔεpNL(x)=ε(x)εp(x)=ε(x)εK=γ|Es|2ifx>0.
ΔspNLk02Rfs(0)ΔεpNLfnp(z)=k02|C|2Rγfs(0)fs2(z)fnp(z)=ΓK|C|2.
ΔppNLk02Rfnp(0)ΔεpNLfnp(z)=k02|C|2Rγfnp(0)fs2(z)fnp(z)=ΓK|C|2.

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