Abstract

We numerically demonstrate the formation of the spatiotemporal version of the so-called diffractive resonant radiation generated in waveguide arrays with Kerr nonlinearity when a long pulse is launched into the system. The phase matching condition for the diffractive resonant radiation that we have found earlier for CW beams also works well in the spatiotemporal case. By introducing a linear potential, one can introduce a continuous shift of the central wavenumber of a linear pulse, whereas in the nonlinear case one can demonstrate that the soliton self-wavenumber shift can be compensated by the emission of diffractive resonant radiation, in a very similar fashion as it is done in optical fibers. This work paves the way for designing unique optical devices that generate spectrally broad supercontinua with a controllable directionality by taking advantage of the combined physics of optical fibers and waveguide arrays.

© 2013 OSA

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  7. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett.83, 963–966 (1999).
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  8. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
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  10. S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett.35, 235–237 (2010)
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  11. F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
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  12. S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B104, 453–468 (2011).
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  21. V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
    [CrossRef]
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  29. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).
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    [CrossRef] [PubMed]
  32. D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 17051708 (2003).
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    [CrossRef]

2013 (1)

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett.110, 113903–113903 (2013).
[CrossRef]

2012 (2)

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
[CrossRef]

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

2011 (1)

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B104, 453–468 (2011).
[CrossRef]

2010 (1)

2008 (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

2004 (1)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiationin photonic crystal fibers,” Phys. Rev. E70, 016615–016623 (2004).
[CrossRef]

2003 (5)

P. St and J. Russell, “Photonic Crystal Fibers,” Science299, 358–362 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear optical waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 17051708 (2003).
[CrossRef]

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222, 413–420 (2003).
[CrossRef]

V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
[CrossRef]

2002 (1)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

2001 (3)

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901–203904 (2001).
[CrossRef] [PubMed]

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett.87, 233901–233904 (2001).
[CrossRef] [PubMed]

R. Holzwarth, M. Zimmermann, Th. Udem, T. W. Hánsch, P. Russbüldt, K. Gäbel, R. Poprawe, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser,” Opt. Lett.26, 1376–1378 (2001).
[CrossRef]

1999 (2)

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett.83, 963–966 (1999).
[CrossRef]

1998 (1)

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

1993 (1)

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E47, 2073–2082 (1993).
[CrossRef]

1990 (2)

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B2, 889–900 (1990).
[CrossRef]

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A41, 426–439 (1990).
[CrossRef] [PubMed]

1988 (1)

1986 (2)

1965 (1)

Agrawal, G. P.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222, 413–420 (2003).
[CrossRef]

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic Press, 2008).

G. P. Agrawal, Nonlinear fiber optics, 5th ed. (Academic Press, 2012).

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

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Belyaeva, T. L.

V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
[CrossRef]

Biancalana, F.

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett.110, 113903–113903 (2013).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiationin photonic crystal fibers,” Phys. Rev. E70, 016615–016623 (2004).
[CrossRef]

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” submitted.

Brauer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

Chen, H. H.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A41, 426–439 (1990).
[CrossRef] [PubMed]

Christodoulides, D. N.

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear optical waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett.87, 233901–233904 (2001).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Discrete self- focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988).
[CrossRef] [PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

Corro, G. H.

V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
[CrossRef]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

de Sterke, C. M.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett.83, 963–966 (1999).
[CrossRef]

Dreisow, F.

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
[CrossRef]

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

Efremidis, N.K.

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett.87, 233901–233904 (2001).
[CrossRef] [PubMed]

Gäbel, K.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

Gordon, J. P.

Granados, M. A.

V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
[CrossRef]

Hánsch, T. W.

Herrmann, J.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901–203904 (2001).
[CrossRef] [PubMed]

Holzwarth, R.

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901–203904 (2001).
[CrossRef] [PubMed]

Jones, A. L.

Joseph, R. I.

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

Karpman, V. I.

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E47, 2073–2082 (1993).
[CrossRef]

Keil, R.

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
[CrossRef]

Kivshar, Y. S.

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

Knight, J. C.

Kuehl, H. H.

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B2, 889–900 (1990).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear optical waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Lee, Y. C.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A41, 426–439 (1990).
[CrossRef] [PubMed]

Lenz, G.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an array of curved optical waveguides,” Phys. Rev. Lett.83, 963–966 (1999).
[CrossRef]

Longhi, S.

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
[CrossRef]

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B104, 453–468 (2011).
[CrossRef]

S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett.35, 235–237 (2010)
[CrossRef] [PubMed]

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” submitted.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 17051708 (2003).
[CrossRef]

Mitschke, F. M.

Mollenauer, L. F.

Nolte, S.

J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
[CrossRef] [PubMed]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
[CrossRef]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998).
[CrossRef]

Poprawe, R.

Russbüldt, P.

Russell, J.

P. St and J. Russell, “Photonic Crystal Fibers,” Science299, 358–362 (2003).
[CrossRef]

Russell, P. St. J.

Santhanam, J.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222, 413–420 (2003).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Serkin, V. N.

V. N. Serkin, T. L. Belyaeva, G. H. Corro, and M. A. Granados, “Stimulated Raman self-scattering of femtosecond pulses. I. Soliton and non-soliton regimes of coherent self-scattering,” Quantum Electron.33, 325 (2003).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear optical waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiationin photonic crystal fibers,” Phys. Rev. E70, 016615–016623 (2004).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 17051708 (2003).
[CrossRef]

St, P.

P. St and J. Russell, “Photonic Crystal Fibers,” Science299, 358–362 (2003).
[CrossRef]

Stegeman, G. I.

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Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett.110, 113903–113903 (2013).
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Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” submitted.

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F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
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J. M. Zeuner, N.K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett.109, 023602–023606 (2012).
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F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett.97, 10008 (2012).
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D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear optical waveguide lattices,” Nature424, 817–823 (2003).
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F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Phys. Rev. A (2)

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A41, 426–439 (1990).
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[CrossRef]

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett.110, 113903–113903 (2013).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901–093904 (2002).

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Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” submitted.

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Figures (2)

Fig. 1
Fig. 1

(a,b) Diffractive resonant radiation in the spatiotemporal case in the (n, z, τ)-space (a) and (κ, z, τ)-space (b). (c) The close view of the uppermost slide in Fig. 1(b) is depicted separately with the black dashed line representing the input wavenumber of the soliton (κ0 = 0.6) and the solid white line showing the predicted DifRR wavenumber calculated by using the phase matching condition with this input soliton wavenumber value. Parameters: c = 1, κ0 = 0.6, A0 = 1.

Fig. 2
Fig. 2

(a,b) PBOs in the linear regime depicted in the (n, z)-plane (a) and (κ, z)-plane (b). Parameters for (a,b): c = 1, κ0 = 0, A0 = 0.8, α = 0.5. (c,d) Anomalous compensation of SSWS in the nonlinear regime depicted in the (n, z)-plane (c) and (κ, z)-plane (d). Parameters for (c,d): c = 1, κ0 = 0, A0 = 0.6, α = 0.01. White straight lines in (b,d) represent the wavenumber calculated by using Eq.(12).

Equations (13)

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i d a n ( z ) d z + c [ a n + 1 ( z ) + a n 1 ( z ) ] + | a n ( z ) | 2 a n ( z ) = 0 ,
κ z ( k x ) = 2 c cos ( k x d ) ,
κ z ( κ ) = κ z ( κ 0 ) + m 1 D m m ! Δ κ m ,
[ i z D 2 2 n 2 + m 3 D m m ! ( i n ) m + | Ψ ( n , z ) | 2 ] Ψ ( n , z ) = 0.
a sol ( z , n ) = A 0 sech = ( n A 0 2 c cos ( κ 0 ) ) exp ( i k sol z ) ,
k lin ( Δ κ ) m 2 D m m ! Δ κ m = 2 c [ cos ( κ ) cos ( κ 0 ) + sin ( κ 0 ) Δ κ ] .
k lin ( Δ κ ) = k sol ,
i z a n + c [ a n + 1 + a n 1 ] + 1 2 τ 2 a n + | a n | 2 a n = 0 ,
( i d d z + α n ) a n + c [ a n + 1 + a n 1 ] + | a n | 2 a n = 0 ,
[ i z + α n D 2 2 n 2 + m 3 D m m ! ( i n ) m + | Ψ | 2 ] Ψ = 0.
κ ( z ) = i 2 E + ( Ψ * n Ψ Ψ n Ψ * ) d n ,
κ ( z ) = κ 0 + α z ,
n 0 ( z ) = n 0 ( 0 ) + c cos ( κ 0 ) ( α z 2 + 2 κ 0 z ) .

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