Abstract

We study optical beams that are supported at the surface of a medium with a linear index potential and by a piecewise linear wedge-type potential. In the linear limit the modes are described by Airy functions. In the nonlinear regime we find families of solutions that bifurcate from the linear modes and study their stability for both self-focusing and self-defocusing Kerr nonlinearity. The total power of such nonlinear waves is finite without the need for apodization.

© 2012 Optical Society of America

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  1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. M. A. Bandres, Opt. Lett. 33, 1678 (2008).
    [CrossRef]
  4. J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
    [CrossRef]
  5. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
    [CrossRef]
  6. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
    [CrossRef]
  7. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
    [CrossRef]
  8. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [CrossRef]
  9. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
    [CrossRef]
  10. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
    [CrossRef]
  11. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
    [CrossRef]
  12. J. Giannini and R. Joseph, Phys. Lett. A 141, 417 (1989).
    [CrossRef]
  13. T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
    [CrossRef]
  14. R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
    [CrossRef]
  15. I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
    [CrossRef]
  16. Y. Fattal, A. Rudnick, and D. M. Marom, Opt. Express 19, 17298 (2011).
    [CrossRef]
  17. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, Opt. Lett. 36, 1164 (2011).
    [CrossRef]
  18. N. K. Efremidis, Opt. Lett. 36, 3006 (2011).
    [CrossRef]
  19. Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, Opt. Lett. 36, 3230 (2011).
    [CrossRef]
  20. O. Valleé and M. Soares, Airy Functions and Applications to Physics (World Scientific, 2004).
  21. D. M. Greenberger, Am. J. Phys. 48, 256 (1980).
    [CrossRef]

2011 (8)

2010 (4)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef]

2009 (2)

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

2008 (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

M. A. Bandres, Opt. Lett. 33, 1678 (2008).
[CrossRef]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

1989 (1)

J. Giannini and R. Joseph, Phys. Lett. A 141, 417 (1989).
[CrossRef]

1980 (1)

D. M. Greenberger, Am. J. Phys. 48, 256 (1980).
[CrossRef]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

Ayelet, G.-P.

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Chen, R.-P.

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

Chen, Z.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

Chremmos, I.

Christodoulides, D. N.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Chu, X.-X.

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Efremidis, N. K.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

Fattal, Y.

Giannini, J.

J. Giannini and R. Joseph, Phys. Lett. A 141, 417 (1989).
[CrossRef]

Greenberger, D. M.

D. M. Greenberger, Am. J. Phys. 48, 256 (1980).
[CrossRef]

Hu, Y.

Joseph, R.

J. Giannini and R. Joseph, Phys. Lett. A 141, 417 (1989).
[CrossRef]

Kaminer, I.

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

Kivshar, Y. S.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Liu, S.

Liu, W.

Lou, C.

Marom, D. M.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Mills, M. S.

Miroshnichenko, A. E.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Neshev, D. N.

Papazoglou, D. G.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Prakash, J.

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

Rudnick, A.

Segev, M.

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

Shadrivov, I. V.

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Soares, M.

O. Valleé and M. Soares, Airy Functions and Applications to Physics (World Scientific, 2004).

Song, D.

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Tzortzakis, S.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Valleé, O.

O. Valleé and M. Soares, Airy Functions and Applications to Physics (World Scientific, 2004).

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

Wang, H.

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

Ye, Z.

Yin, C.-F.

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

Zhang, P.

Zhang, Z.

Zhao, J.

Am. J. Phys. (1)

D. M. Greenberger, Am. J. Phys. 48, 256 (1980).
[CrossRef]

Nat. Photon. (3)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, G.-P. Ayelet, and A. Arie, Nat. Photon. 3, 395 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (9)

Phys. Lett. A (1)

J. Giannini and R. Joseph, Phys. Lett. A 141, 417 (1989).
[CrossRef]

Phys. Rev. A (1)

R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, Phys. Rev. A 82, 043832 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Other (1)

O. Valleé and M. Soares, Airy Functions and Applications to Physics (World Scientific, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

Linear mode profiles for (a) a wedge potential with d1=1, d2=4, (b) linear potential at the surface of a semi-infinite medium with d2=1. In both cases, the first six modes are depicted.

Fig. 2.
Fig. 2.

Typical amplitude profiles of nonlinear waves supported at the surface of a Kerr nonlinear medium. In (a), (b), and (c) SF; and in (d), (e), and (f) SDF amplitude profiles belonging to the families 1, 4, and 8 are shown, respectively.

Fig. 3.
Fig. 3.

(a) Power P (solid curve) and growth rate G (dashed–dotted curve) as a function of the propagation constant k ofthe fourth family of surface waves in the presence of SDFnonlinearity. Corresponding dynamics for (b) k3.42, (c) k3.94 [(d) shows details of the same simulation], (e) k6.54, and (f) k10.1.

Fig. 4.
Fig. 4.

Typical amplitude profiles supported by a wedge index potential in the case of (a)–(d) families 1–4 in a SF medium, and (e)–(f) families 3 and 4 in a SDF medium. (g) Unstable and (h) stable dynamics of beams belonging to the fourth SDF family.

Equations (8)

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

iψz+(1/2)ψxxV(x)u+γ|ψ|2ψ=0.
V(x)={d1x2x<0d2x2x>0,
ψj(x,z)=AjAi[σj|dj|1/3(xcj)]ei(djcj/2)z,
c1d1=c2d2,
A1Ai(ξ1)=A2Ai(ξ2),
σ1|d1|1/3A1Ai(ξ1)=σ2|d2|1/3A2Ai(ξ2),
ku+(1/2)u+V(x)u+γu3=0.
ψ(x,z)=[u(x)+A(x,z)+iB(x,z)]eikz,

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