Abstract

We demonstrate that an interplay between diffraction and defocusing nonlinearity can support stable self-similar plasmonic waves with a parabolic profile. Simplicity of a parabolic shape combined with the corresponding parabolic spatial phase distribution creates opportunities for controllable manipulation of plasmons through a combined action of diffraction and nonlinearity.

© 2013 Optical Society of America

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References

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  1. M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer-Verlag, 2007).
  2. S. I. Bozhevolnyi, Plasmonic Nanoguides and Circuits (Pan Stanford, 2009).
  3. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
    [CrossRef]
  4. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
    [CrossRef]
  5. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
    [CrossRef]
  6. H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
    [CrossRef]
  7. D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
    [CrossRef]
  8. C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
    [CrossRef]
  9. S. I. Bozhevolnyi and K. V. Nerkararyan, Opt. Lett. 35, 541 (2010).
    [CrossRef]
  10. M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012).
    [CrossRef]
  11. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 17, 21732 (2009).
    [CrossRef]
  12. A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
    [CrossRef]
  13. A. I. Latkin, S. K. Turitsyn, and A. A. Sysoliatin, Opt. Lett. 32, 331 (2007).
    [CrossRef]
  14. A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
    [CrossRef]
  15. A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
    [CrossRef]
  16. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
    [CrossRef]
  17. S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
    [CrossRef]
  18. V. I. Kruglov and J. D. Harvey, J Opt. Soc. Am. B 23, 2541 (2006).
    [CrossRef]
  19. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
    [CrossRef]
  20. S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
    [CrossRef]
  21. W. Walasik, Y. Kartashov, and G. Renversez, “Toward low power-plasmon soliton in planar nonlinear structures,” presented at EOS Annual Meeting, Tom 6 Nonlinear Photonics 5895, Aberdeen, Scotland, 2010.

2012 (2)

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012).
[CrossRef]

2011 (2)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

2010 (4)

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

S. I. Bozhevolnyi and K. V. Nerkararyan, Opt. Lett. 35, 541 (2010).
[CrossRef]

2009 (1)

2008 (1)

S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
[CrossRef]

2007 (4)

A. I. Latkin, S. K. Turitsyn, and A. A. Sysoliatin, Opt. Lett. 32, 331 (2007).
[CrossRef]

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

2006 (1)

V. I. Kruglov and J. D. Harvey, J Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

2004 (1)

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

2002 (1)

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Albrecht, M.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Boscolo, S.

S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
[CrossRef]

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi and K. V. Nerkararyan, Opt. Lett. 35, 541 (2010).
[CrossRef]

S. I. Bozhevolnyi, Plasmonic Nanoguides and Circuits (Pan Stanford, 2009).

Brongersma, M. L.

M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer-Verlag, 2007).

Capasso, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Cluzel, B.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Davoyan, A. R.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef]

de Fornel, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Dellinger, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Dudley, J. M.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Elsaesser, T.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Felderer, K.

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Finot, C.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

Frey, H. G.

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

Genevet, P.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Gramotnev, D. K.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

Guckenberger, R.

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

Harvey, J. D.

V. I. Kruglov and J. D. Harvey, J Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Janunts, N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Kartashov, Y.

W. Walasik, Y. Kartashov, and G. Renversez, “Toward low power-plasmon soliton in planar nonlinear structures,” presented at EOS Annual Meeting, Tom 6 Nonlinear Photonics 5895, Aberdeen, Scotland, 2010.

Kauranen, M.

M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012).
[CrossRef]

Kik, P. G.

M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer-Verlag, 2007).

Kivshar, Y. S.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef]

Klein, A. E.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov and J. D. Harvey, J Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Latkin, A.

S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
[CrossRef]

Latkin, A. I.

Lienau, C.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Lin, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

Marini, A.

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

Millot, G.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

Minovich, A.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Neacsu, C. C.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Nerkararyan, K. V.

Neshev, D. N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Nijhof, J. H. B.

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Novokshenov, V. Y.

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Pertsch, T.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Pile, D. F. P.

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

Raschke, M. B.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Renversez, G.

W. Walasik, Y. Kartashov, and G. Renversez, “Toward low power-plasmon soliton in planar nonlinear structures,” presented at EOS Annual Meeting, Tom 6 Nonlinear Photonics 5895, Aberdeen, Scotland, 2010.

Richardson, D. J.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

Ropers, C.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Shadrivov, I. V.

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef]

Skryabin, D. V.

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Sysoliatin, A. A.

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

Turitsyn, S. K.

S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
[CrossRef]

A. I. Latkin, S. K. Turitsyn, and A. A. Sysoliatin, Opt. Lett. 32, 331 (2007).
[CrossRef]

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

Vogel, M. W.

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

Walasik, W.

W. Walasik, Y. Kartashov, and G. Renversez, “Toward low power-plasmon soliton in planar nonlinear structures,” presented at EOS Annual Meeting, Tom 6 Nonlinear Photonics 5895, Aberdeen, Scotland, 2010.

Witt, S.

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

Zayats, A. V.

M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012).
[CrossRef]

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

Zharov, A. A.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

IEEE J Quantum Electron. (1)

S. Boscolo, A. Latkin, and S. K. Turitsyn, IEEE J Quantum Electron. 44, 1196 (2008).
[CrossRef]

J Opt. Soc. Am. B (1)

V. I. Kruglov and J. D. Harvey, J Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

Nano Lett. (1)

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, Nano Lett. 7, 2784 (2007).
[CrossRef]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012).
[CrossRef]

Nat. Phys. (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nat. Phys. 3, 597 (2007).
[CrossRef]

Nature Nanotech. (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nature Nanotech. 6, 151 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Phys. Rev. B (1)

D. K. Gramotnev, D. F. P. Pile, M. W. Vogel, and X. Zhang, Phys. Rev. B 75, 035431 (2007).
[CrossRef]

Phys. Rev. Lett. (5)

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef]

Phys. Status Solidi RRL (1)

A. R. Davoyan, I. V. Shadrivov, Y. S. Kivshar, and D. K. Gramotnev, Phys. Status Solidi RRL 4, 277 (2010).
[CrossRef]

Theor. Math. Phys. (1)

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Other (3)

W. Walasik, Y. Kartashov, and G. Renversez, “Toward low power-plasmon soliton in planar nonlinear structures,” presented at EOS Annual Meeting, Tom 6 Nonlinear Photonics 5895, Aberdeen, Scotland, 2010.

M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer-Verlag, 2007).

S. I. Bozhevolnyi, Plasmonic Nanoguides and Circuits (Pan Stanford, 2009).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of a tapered metal–dielectric–metal waveguide. (b) Schematic of the plasmonic beam propagation in linear (dashed curve) and nonlinear (solid curve) regimes.

Fig. 2.
Fig. 2.

Beam cross-sections for (a) untapered and (b) tapered waveguides, respectively, after 20  μm of propagation. Solid curves correspond to numerical results, whereas dashed are our analytic predictions. Solid parabolic curve corresponds to the input beam profile.

Fig. 3.
Fig. 3.

Magnetic field profiles at the metal–dielectric interface for (a) nonlinear untapered (ϕ=0°) and (b) nonlinear tapered (ϕ=1.6°) waveguides. Field profiles are plotted up-till few microns from the taper tip.

Fig. 4.
Fig. 4.

(a) Evolution of the beam amplitude with propagation obtained from the analytical theory for all studied cases. The fields are normalized to the initial amplitude a(0). (b) Evolution of product a2b with propagation for tapered and untapered waveguides, for different input amplitudes.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

2iσAz+2Ay2I+iA(dσdz+Γ)+Nnl|A|2A=0,
2σadadz+2IC+(dσdz+Γ)=0,2σdCdz4IC2Na2b2=0,2σbdbdz+4IC=0,
2iσuz+Ib22uξ2+Nnlna2(ξ2+|u|2)u=0.

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