Abstract

By introducing symmetry-breaking in geometry, we reveal the existence of thresholdless crescent waves, i.e., nonlinear diffractionless modes pinged to the boundary of a curvature, in an elliptical ring. An effective nonlinear Schrödinger equation along the azimuthal direction is derived by taking the transformation in the curvilinear coordinate of elliptical symmetry, which illustrates the formation of trapping potentials (barriers) along the semi-major (minor) axis. Our results demonstrate an alternative but efficient approach to access optical modes with a variety of inhomogeneous potentials.

© 2013 Optical Society of America

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  1. C. P. Jisha, Y. Y. Lin, T.-D. Lee, and R.-K. Lee, Phys. Rev. Lett. 107, 183902 (2011).
    [CrossRef]
  2. I. E. Tamm, Z. Phys. 76, 849 (1932).
    [CrossRef]
  3. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 32, 2948 (2007).
    [CrossRef]
  4. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
    [CrossRef]
  5. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
    [CrossRef]
  6. S. M. Vukovic, I. V. Shadrivov, and Yu. S. Kivshar, Appl. Phys. Lett. 95, 041902 (2009).
    [CrossRef]
  7. S. H. Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, Opt. Lett. 35, 1847 (2010).
    [CrossRef]
  8. Y. Y. Lin, R.-K. Lee, and Yu. S. Kivshar, Opt. Lett. 34, 2982 (2009).
    [CrossRef]
  9. W. Shockley, Phys. Rev. 56, 317 (1939).
    [CrossRef]
  10. W.-X. Yang, Y. Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, Opt. Lett. 35, 3207 (2010).
    [CrossRef]
  11. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).
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    [CrossRef]
  13. Y. Y. Lin and R.-K. Lee, Opt. Lett. 33, 1377 (2008).
    [CrossRef]
  14. A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
    [CrossRef]
  15. Y. V. Bludov, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 35, 3339 (2010).
    [CrossRef]

2011 (1)

C. P. Jisha, Y. Y. Lin, T.-D. Lee, and R.-K. Lee, Phys. Rev. Lett. 107, 183902 (2011).
[CrossRef]

2010 (3)

2009 (3)

2008 (2)

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Y. Y. Lin and R.-K. Lee, Opt. Lett. 33, 1377 (2008).
[CrossRef]

2007 (1)

2006 (1)

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

2005 (1)

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

1939 (1)

W. Shockley, Phys. Rev. 56, 317 (1939).
[CrossRef]

1932 (1)

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

Bartal, G.

Bludov, Y. V.

Christodoulides, D. N.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Desyatnikov, A. S.

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Dreisow, F.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Garanovich, I. L.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Hache, A.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Heinrich, M.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Jisha, C. P.

C. P. Jisha, Y. Y. Lin, T.-D. Lee, and R.-K. Lee, Phys. Rev. Lett. 107, 183902 (2011).
[CrossRef]

Kartashov, Y. V.

Kivshar, Yu. S.

W.-X. Yang, Y. Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, Opt. Lett. 35, 3207 (2010).
[CrossRef]

S. M. Vukovic, I. V. Shadrivov, and Yu. S. Kivshar, Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Y. Y. Lin, R.-K. Lee, and Yu. S. Kivshar, Opt. Lett. 34, 2982 (2009).
[CrossRef]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

Konotop, V. V.

Lee, R.-K.

Lee, T.-D.

C. P. Jisha, Y. Y. Lin, T.-D. Lee, and R.-K. Lee, Phys. Rev. Lett. 107, 183902 (2011).
[CrossRef]

W.-X. Yang, Y. Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, Opt. Lett. 35, 3207 (2010).
[CrossRef]

Lin, Y. Y.

Makris, K. G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Morandotti, R.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Nam, S. H.

Nolte, S.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Pertsch, T.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Salamo, G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Shadrivov, I. V.

S. M. Vukovic, I. V. Shadrivov, and Yu. S. Kivshar, Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Shockley, W.

W. Shockley, Phys. Rev. 56, 317 (1939).
[CrossRef]

Sorel, M.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Stegeman, G. I.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Sukhorukov, A. A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Suntsov, S.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Szameit, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Tamm, I. E.

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Torner, L.

Tunnermann, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

Ulin-Avila, E.

Vukovic, S. M.

S. M. Vukovic, I. V. Shadrivov, and Yu. S. Kivshar, Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Vysloukh, V. A.

Yang, H.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

Yang, W.-X.

Zhang, X.

Appl. Phys. Lett. (1)

S. M. Vukovic, I. V. Shadrivov, and Yu. S. Kivshar, Appl. Phys. Lett. 95, 041902 (2009).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. (1)

W. Shockley, Phys. Rev. 56, 317 (1939).
[CrossRef]

Phys. Rev. Lett. (4)

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and Yu. S. Kivshar, Phys. Rev. Lett. 101, 203902 (2008).
[CrossRef]

C. P. Jisha, Y. Y. Lin, T.-D. Lee, and R.-K. Lee, Phys. Rev. Lett. 107, 183902 (2011).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Z. Phys. (1)

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Other (1)

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Elliptical potential, V, and (b) corresponding top-view used in the simulations, with the semi-major axis a=1.8, semi-minor axis b=1, and potential depth V0=10.

Fig. 2.
Fig. 2.

Examples of the intensity profile for crescent waves are shown along the (a) semi-minor and (b) semi-major axes, corresponding to the markers A and B in (c), respectively. (c) Formation power P versus propagation constant β for crescent waves along the semi-minor (solid-line) and semi-major (dashed-line) axes are shown in black, while the crescent waves in the inscribed (dashed-line) and circumscribed (solid-line) circles are shown in red, which bifurcate from the corresponding symmetric donut-shaped modes (in blue).

Fig. 3.
Fig. 3.

Required threshold power, Pth, to support crescent waves along the y(solid-line)/x(dashed-line) axis for different ellipticity, a/b.

Fig. 4.
Fig. 4.

Effective diffraction coefficient Deff, effective nonlinearity geff, and effective potential Veff for the reduced quasi-1D nonlinear equation along the azimuthal direction θ, with the formula in Eq. (4). Dashed and dotted–dashed lines in each panel indicate the cases of the inscribed and circumscribed circles, with the radii 1 and 1.8, respectively.

Equations (6)

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iΨz+12(2x2+2y2)Ψ+|Ψ|2Ψ+V(x,y)Ψ=0,
V(r)={V0cos2(πr),12r320,otherwise
2x2+2y2=1h2(|μ|2+|θ|2),
L=121h2(|μM|2|Θ|2+|M|2|θΘ|2)β|MΘ|2+12|MΘ|4+V|MΘ|2.
βΘ=θ{DeffΘθ}+geff|Θ|2Θ+VeffΘ,
Deff=1201h|M|2dμ0h|M|2dμ,geff=0h|M|4dμ0h|M|2dμ,Veff=01h|Mμ|2dμ+20hV|M|2dμ20h|M|2dμ.

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