Abstract

We study experimentally and numerically the dynamics of a recently found topological phase transition for discrete quadratic solitons with linearly coupled SH waves. We find that, although no stationary states are excited in the experimental situation, the generic feature of the phase transition of the SH is preserved. By utilizing simulations of the coupled mode equations we identify the complex processes leading to the phase transition involving spatial focusing and the generation of new frequency components. These distinct signatures of the dynamic phase transition are also demonstrated experimentally.

© 2011 OSA

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    [CrossRef]

2011 (2)

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

2010 (3)

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[CrossRef]

2009 (3)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[CrossRef]

F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

M. Heinrich, A. Szameit, F. Dreisow, R. Keil, S. Minardi, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Observation of three-dimensional discrete-continuous x waves in photonic lattices,” Phys. Rev. Lett. 103, 113903 (2009).
[CrossRef] [PubMed]

2008 (2)

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

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]

2007 (1)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (3)

2004 (3)

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimallyengineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. 29, 596–598 (2004).
[CrossRef] [PubMed]

2003 (5)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. 28, 1942–1944 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

C. Conti and S. Trillo, “X waves generated at the second harmonic,” Opt. Lett. 28, 1251–1253 (2003).
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (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 (2002).
[CrossRef] [PubMed]

2001 (2)

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

2000 (2)

A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615 (2000).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

1998 (4)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “Second-harmonic generation and cascaded nonlinearity in titanium-indiffused lithium niobate channel waveguides,” J. Opt. Soc. Am. B 15, 2255–2268 (1998).
[CrossRef]

T. Peschel, U. Peschel, and F. Lederer, “Discrete bright solitary waves in quadratically nonlinear media,” Phys. Rev. E 57, 1127–1133 (1998).
[CrossRef]

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
[CrossRef]

1997 (1)

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

1996 (1)

1995 (1)

C. G. Trevino-Palacios, G. I. Stegeman, M. P. D. Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity dependent mode competition in second harmonic generation in multimode waveguides,” Appl. Phys. Lett. 67, 170–172 (1995).
[CrossRef]

1994 (1)

1993 (1)

1990 (1)

1988 (1)

Aceves, A. B.

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Angelis, C. D.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

A. B. Aceves, C. D. Angelis, S. Trillo, and S. Wabnitz, “Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays,” Opt. Lett. 19, 332–334 (1994).
[CrossRef] [PubMed]

Artigas, D.

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

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]

G. I. Stegeman, M. Sheik-Bahae, E. V. Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

Bache, M.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Baek, Y.

Bakker, H. J.

Baldi, P.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. D. Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity dependent mode competition in second harmonic generation in multimode waveguides,” Appl. Phys. Lett. 67, 170–172 (1995).
[CrossRef]

Bang, O.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615 (2000).
[CrossRef]

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996).
[CrossRef] [PubMed]

Baronio, F.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

Barthelemy, A.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[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 (2002).
[CrossRef] [PubMed]

Büchter, D.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

Burghoff, J.

Carrasco, S.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

Christodoulides, D.

Christodoulides, D. N.

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]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Clausen, C. B.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Clerici, M.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Conti, C.

C. Conti and S. Trillo, “X waves generated at the second harmonic,” Opt. Lett. 28, 1251–1253 (2003).
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (2003).
[CrossRef]

Couderc, V.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

de Boer, M. P.

Delacourt, D.

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F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

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F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
[CrossRef] [PubMed]

T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
[CrossRef] [PubMed]

R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. 30, 1033–1035 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
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[CrossRef]

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T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
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T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 093901 (2002).
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[CrossRef]

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F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

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O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (2003).
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W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

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W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
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W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

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F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
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Rubino, E.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Schiek, R.

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
[CrossRef] [PubMed]

R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. 30, 1033–1035 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimallyengineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. 29, 596–598 (2004).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, and G. I. Stegeman, “Second-harmonic generation and cascaded nonlinearity in titanium-indiffused lithium niobate channel waveguides,” J. Opt. Soc. Am. B 15, 2255–2268 (1998).
[CrossRef]

Schreiber, G.

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]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Setzpfandt, F.

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

F. Setzpfandt, A. Sukhorukov, and T. Pertsch, “Discrete quadratic solitons with competing second-harmonic components” (submitted to Phys. Rev. A).

Sheik-Bahae, M.

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]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Sohler, W.

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. 30, 1033–1035 (2005).
[CrossRef] [PubMed]

T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimallyengineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. 29, 596–598 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615 (2000).
[CrossRef]

Stegeman, G. I.

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]

T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
[CrossRef] [PubMed]

R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. 30, 1033–1035 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimallyengineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. 29, 596–598 (2004).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, and G. I. Stegeman, “Second-harmonic generation and cascaded nonlinearity in titanium-indiffused lithium niobate channel waveguides,” J. Opt. Soc. Am. B 15, 2255–2268 (1998).
[CrossRef]

C. G. Trevino-Palacios, G. I. Stegeman, M. P. D. Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity dependent mode competition in second harmonic generation in multimode waveguides,” Appl. Phys. Lett. 67, 170–172 (1995).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. V. Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

Stryland, E. V.

Suche, H.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

Sukhorukov, A.

F. Setzpfandt, A. Sukhorukov, and T. Pertsch, “Discrete quadratic solitons with competing second-harmonic components” (submitted to Phys. Rev. A).

Sukhorukov, A. A.

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615 (2000).
[CrossRef]

Szameit, A.

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[CrossRef]

M. Heinrich, A. Szameit, F. Dreisow, R. Keil, S. Minardi, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Observation of three-dimensional discrete-continuous x waves in photonic lattices,” Phys. Rev. Lett. 103, 113903 (2009).
[CrossRef] [PubMed]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
[CrossRef] [PubMed]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Torner, L.

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

Torres, J. P.

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

Trevino-Palacios, C. G.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. D. Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity dependent mode competition in second harmonic generation in multimode waveguides,” Appl. Phys. Lett. 67, 170–172 (1995).
[CrossRef]

Trillo, S.

Trull, J.

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (2003).
[CrossRef]

Tünnermann, A.

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

M. Heinrich, A. Szameit, F. Dreisow, R. Keil, S. Minardi, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Observation of three-dimensional discrete-continuous x waves in photonic lattices,” Phys. Rev. Lett. 103, 113903 (2009).
[CrossRef] [PubMed]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
[CrossRef] [PubMed]

Valiulis, G.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (2003).
[CrossRef]

van Linden van den Heuvell, H. B.

Vannahme, C.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Vicencio, R. A.

Wabnitz, S.

Wise, F. W.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Zentgraf, 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 (2002).
[CrossRef] [PubMed]

Zhou, B. B.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B: Lasers Opt. (1)

F. Setzpfandt, D. N. Neshev, A. A. Sukhorukov, R. Schiek, R. Ricken, Y. Min, Y. S. Kivshar, W. Sohler, F. Lederer, A. Tünnermann, and T. Pertsch, “Nonlinear dynamics with higher-order modes in lithium niobate waveguide arrays,” Appl. Phys. B: Lasers Opt. 104, 487–493 (2011).
[CrossRef]

Appl. Phys. Lett. (1)

C. G. Trevino-Palacios, G. I. Stegeman, M. P. D. Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity dependent mode competition in second harmonic generation in multimode waveguides,” Appl. Phys. Lett. 67, 170–172 (1995).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

F. Baronio, A. Barthelemy, S. Carrasco, V. Couderc, C. D. Angelis, L. Lefort, Y. Min, P. H. Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B: Quantum Semiclassical Opt. 6, S182–S189 (2004).
[CrossRef]

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

J. Phys. B (1)

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[CrossRef]

Laser Photon. Rev. (1)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[CrossRef]

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. 192, 347–355 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (11)

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
[CrossRef]

A. B. Aceves, C. D. Angelis, S. Trillo, and S. Wabnitz, “Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays,” Opt. Lett. 19, 332–334 (1994).
[CrossRef] [PubMed]

O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996).
[CrossRef] [PubMed]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. 28, 1942–1944 (2003).
[CrossRef] [PubMed]

T. Pertsch, R. Iwanow, R. Schiek, G. I. Stegeman, U. Peschel, F. Lederer, Y. H. Min, and W. Sohler, “Spatial ultrafast switching and frequency conversion in lithium niobate waveguide arrays,” Opt. Lett. 30, 177–179 (2005).
[CrossRef] [PubMed]

G. I. Stegeman, M. Sheik-Bahae, E. V. Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

C. Conti and S. Trillo, “X waves generated at the second harmonic,” Opt. Lett. 28, 1251–1253 (2003).
[CrossRef] [PubMed]

F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. 34, 3589–3591 (2009).
[CrossRef] [PubMed]

R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. 30, 1033–1035 (2005).
[CrossRef] [PubMed]

L. D. Noordam, H. J. Bakker, M. P. de Boer, and H. B. van Linden van den Heuvell, “Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects,” Opt. Lett. 15, 1464–1466 (1990).
[CrossRef] [PubMed]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimallyengineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. 29, 596–598 (2004).
[CrossRef] [PubMed]

Opt. Photon. News (1)

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photon. News 19, 24–31 (2008).
[CrossRef]

Phys. Rep. (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]

Phys. Rev. A (2)

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Phys. Rev. E (3)

T. Peschel, U. Peschel, and F. Lederer, “Discrete bright solitary waves in quadratically nonlinear media,” Phys. Rev. E 57, 1127–1133 (1998).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615 (2000).
[CrossRef]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, “Nonlinear x waves in second-harmonic generation: Experimental results,” Phys. Rev. E 68, 026610 (2003).
[CrossRef]

Phys. Rev. Lett. (8)

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[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 (2002).
[CrossRef] [PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

M. Heinrich, A. Szameit, F. Dreisow, R. Keil, S. Minardi, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Observation of three-dimensional discrete-continuous x waves in photonic lattices,” Phys. Rev. Lett. 103, 113903 (2009).
[CrossRef] [PubMed]

F. Setzpfandt, A. A. Sukhorukov, D. N. Neshev, R. Schiek, Y. S. Kivshar, and T. Pertsch, “Phase transitions of nonlinear waves in quadratic waveguide arrays,” Phys. Rev. Lett. 105, 233905 (2010).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Other (3)

C. Denz, S. Flach, and Y. S. Kivshar, eds., Nonlinearities in Periodic Structures and Metamaterials (Springer, 2010).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

F. Setzpfandt, A. Sukhorukov, and T. Pertsch, “Discrete quadratic solitons with competing second-harmonic components” (submitted to Phys. Rev. A).

Supplementary Material (2)

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