## Spectral pulse transformations and phase transitions in quadratic nonlinear waveguide arrays |

Optics Express, Vol. 19, Issue 23, pp. 23188-23201 (2011)

http://dx.doi.org/10.1364/OE.19.023188

Acrobat PDF (1305 KB)

### 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

## 1. Introduction

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

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

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

4. 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]

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

7. 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]

10. 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]

11. 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]

12. 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]

13. 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]

14. 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]

15. 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]

16. 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]

17. 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]

18. 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]

15. 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]

19. S. Droulias, K. Hizanidis, J. Meier, and D. Christodoulides, “X - waves in nonlinear normally dispersive waveguide arrays,” Opt. Express **13**, 1827–1832 (2005). [CrossRef] [PubMed]

21. 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]

22. 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]

23. 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]

*χ*

^{(2)}-material showed the excitation of X-waves at the SH wavelengths from an input FW beam [24

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

26. 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]

27. 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]

18. 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]

18. 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]

## 2. Spatial cw soliton phase transition with two SH modes

**105**, 233905 (2010). [CrossRef]

28. 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]

_{QPM}is the grating period and we consider the first-order QPM process only. The

*j*denotes the different nonlinear interactions. The most efficient interaction for a pair of SH and FW modes is achieved at wavelengths where the respective Δ

*β*= 0, which can be controlled by the quasi-phasematching period Λ

_{j}_{QPM}and the device temperature. A main result in [18

**105**, 233905 (2010). [CrossRef]

_{QPM}= 16.803

*μ*m is employed to achieve efficient nonlinear interaction between FW00 and SH02 at an FW wavelength of approximately 1500 nm. We plot the simulated phase mismatch Δ

*β*

_{1}between the two modes vs. the FW wavelength in Fig. 1(b). However, in the same wavelength range the phase mismatch Δ

*β*

_{2}between FW00 and SH10 mode [see mode profile in Fig. 1(a)] becomes small [29

29. 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]

30. 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]

*U*,

_{n}*V*and

_{n}*W*of the FW00, SH02 and SH10 modes in the

_{n}*n*th waveguide are coupled linearly with the coupling strengths

*c*,

_{U}*c*,

_{V}*c*to the same modes in the neighboring waveguides. Additionally the FW component is coupled with the nonlinear coupling strengths

_{W}*γ*

_{1}and

*γ*

_{2}to the SH02 and SH10 modes at the same waveguide where the efficiency of the interaction is controlled by the phase mismatches Δ

*β*

_{1}and Δ

*β*

_{2}. As can be seen in Fig. 1(b), the difference between the phase mismatches is constant with Δ

*β*

_{1}– Δ

*β*

_{2}= 5

*π*/cm. The linear propagation of the waves in the WGA is determined by the dependence of the longitudinal wavenumber

*k*on the transverse wavenumber

_{m}*κ*and the propagation constant of the single waveguide

_{m}*m*= {

*U,V,W*} again denotes the band, and the

*κ*are normalized to the array pitch. Fig. 1(d) shows a scheme of the three bands governing the propagation of the modes taken into account here.

_{m}29. 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]

*z*is the propagation distance and

*ω*the mean frequency of the FW wave. The physical coupling constants of the used sample are

_{U}*c*=

_{U}*c*= 80/m and

_{V}*c*= 16/m. These coupling constants are used for the theoretical analysis as well and are considered constant in the investigated wavelength range around an FW wavelength of 1500 nm. The power

_{W}*P*= Σ

*(|*

_{n}*U*|

_{n}^{2}+ |

*V*|

_{n}^{2}+ |

*W*|

_{n}^{2}) is conserved during the propagation. Similar to [18

**105**, 233905 (2010). [CrossRef]

*β*. This ansatz is plugged into Eqs. 3 and the ensuing system of nonlinear equations is solved numerically for propagation constants

*β*< 0 where the stationary solutions show the phase transition [18

**105**, 233905 (2010). [CrossRef]

**105**, 233905 (2010). [CrossRef]

## 3. Pulse dynamics near the phase transition: numerical studies

*τ*is the time relative to the frame moving with the FW group velocity. The parameters

*δ*and

_{V}*δ*are the differences in the inverse group velocities between the FW mode and the SH02 and SH10 modes, respectively. Finally,

_{W}*D*accounts for the group velocity dispersion of the mode labeled by the index

_{m}*m*. Eqs. (4) can be integrated numerically by using the standard split-step-algorithm [34]. We account for dispersion only up to the second order, because the shortest temporal features which we observe in our simulations (presented below) have a length of about 2 ps. However, for much shorter pulses, higher order dispersion may play an important role [35

35. 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]

*β*

_{1}= −4

*π*/cm, corresponding to an FW wavelength 2.2 nm below the FW00-SH02 phasematching wavelength. A phase difference of

*π*is introduced between adjacent waveguides to fulfill the conditions found for the FW part of the stationary solutions. The output of the WGA after 71 mm of propagation for three different powers is plotted in Fig. 3 ( Media 1), where the amplitude in each panel is normalized to the respective maximum. Fig. 3(a,b) for an input peak power of 2 kW correspond to the realization of a wide spatial soliton similar to [15

15. 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]

36. J. T. Manassah, “Effects of velocity dispersion on a generated second harmonic signal,” Appl. Opt. **27**, 4365–4367 (1988). [CrossRef] [PubMed]

38. 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]

*π*corresponding to a staggered wave. However, due to the large spatial width of the FW field the SH is unstaggered and concentrated at transverse wavenumbers of 0 and 2

*π*. The wavelength scale of the SH components is given in units of the corresponding FW wavelength.

20. 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]

*κ*

_{1}=

*π*. However, the staggered SH is only generated at the phasematching wavelength of the FW and the SH02 component. This also leads to the strong asymmetry of the FW spectrum. As can be seen for all simulated powers, the SH10 does not participate in the generation of staggered SH. This is due to the larger mismatch and weaker linear inter-waveguide coupling of the SH10, which leads to its much higher threshold power for the phase transition [18

**105**, 233905 (2010). [CrossRef]

*γ*

_{2}= 0.

*π*. This happens with the largest efficiency at the phasematching wavelength, leading to the asymmetric shape in the FW spectrum. However, due to the strong localization of the FW, SH intensity is generated for a large range of transverse wavenumbers. Since the phasematching condition is only fulfilled for one longitudinal wavenumber, the SH is generated along isolines of it’s dispersion relation. For the SH these isolines follow a cosine as defined in Eq. (2). SH states existing at the isolines of the dispersion relation were referred to as SH X-waves in [24

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

26. 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]

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

## 4. Experimental measurements

*μ*m width for 8.5 h at a temperature of 1060 °C [28

28. 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]

*μ*m results in linear coupling strengths of

*c*=

_{U}*c*= 80/m for the FW and SH02 modes and

_{V}*c*= 16/m for the SH10 mode. The FW coupling constants are obtained by measuring the Green’s function of the waveguide arrays [12

_{W}12. 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]

39. 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]

*μ*m is imaged on the front facet of the sample. To achieve the necessary phase difference of

*π*between adjacent waveguides the input beam is tilted by off-axis illumination of the coupling objective. This results in an excitation of the waveguides with a spatial FWHM of 4 waveguides. However, not only the FW00 mode but also the 2nd FW mode FW01 is weakly excited. The FWHM length of the pulses is 5.3 ps and the amplifier provides peak powers up to ≈ 12kW with a repetition rate of 5 kHz. To prevent photorefraction by the generated SH radiation the sample is heated to a temperature of 220 °C. At this sample temperature the experimentally used FW wavelength of 1499 nm corresponds to a mismatch between FW00 and SH02 mode of Δ

*β*

_{1}= −4

*π*/cm. This is the same mismatch as used for the simulations in Sec.3. To measure the output of the WGA we simultaneously image the output facet onto an InGaAs- and a CCD-camera to record the spatial profiles of the FW and SH output. To obtain the spatial spectrum of the SH we employ an additional lens in a 2-f configuration to generate a Fourier transform of the SH output. This is recorded with an additional CCD-camera.

40. 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]

*π*[see Fig. 6(c)]. The discrepancies between simulation and measurement may be induced by the waveguide inhomogeneities and the SH induced photorefractive effects. We also note that the rectangular QPM grating [visible in Fig. 5] can fascilitate higher-order cascaded interactions which can act as effective cubic nonlinearity [41

41. 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]

42. 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]

**105**, 233905 (2010). [CrossRef]

*π*, no staggered SH is generated during the propagation. Fig. 7(b) shows results for an input peak power of 2.5 kW, which is slightly larger than power of strongest focusing of the FW component [see Fig. 6(a)]. Here SH is not only detected at 749.5 nm but also at wavelengths of approximately 750.8 nm and 752.3 nm. These wavelengths are ≈ 0.5nm larger than the expected phasematching wavelengths to the SH02 and SH10 modes which we measured independently with a low power cw FW input beam. However, it was shown before in a single waveguide that, due to cascading, the phase-matching wavelengths shift to higher wavelengths with increasing input power [29

29. 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]

*π*for both SH modes. The power of the staggered components is just too weak to be detected in the integrated measurements shown above. Finally we show a measurement result for a power of 9 kW in Fig. 7(c). Here we now see strong staggered SH components generated at both SH modes, where the maxima are shifted further towards longer wavelengths. To emphasize that the main contribution to the staggered SH indeed is generated at the phasematching wavelengths we integrate the SH intensity of Fig. 7(c) over spectral domains centered at the (shifted) phasematching wavelengths with a spectral width corresponding to the spectral width of the input pulse. The central wavelengths are indicated by the dotted white lines in Fig. 7(c). The normalized result is plotted in Fig. 7(d). It shows that the SH intensity maximum at the phasematching wavelengths of 751 nm for the SH20 and 752.3 nm for the SH10 is at a wavenumber of

*π*. In contrast, the maximum intensity is at 0,2

*π*for the SH generated at 749.5 nm, corresponding to half of the input FW wavelength.

## 5. Conclusion

**105**, 233905 (2010). [CrossRef]

**105**, 233905 (2010). [CrossRef]

**28**, 1251–1253 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B |

3. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

4. | T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

5. | S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. |

6. | C. Denz, S. Flach, and Y. S. Kivshar, eds., |

7. | 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. |

8. | O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. |

9. | R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. |

10. | 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. |

11. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

12. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

13. | 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. | 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 |

15. | R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. |

16. | G. I. Stegeman, M. Sheik-Bahae, E. V. Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. |

17. | 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 |

18. | 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. |

19. | S. Droulias, K. Hizanidis, J. Meier, and D. Christodoulides, “X - waves in nonlinear normally dispersive waveguide arrays,” Opt. Express |

20. | Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete |

21. | 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 |

22. | S. Carrasco, J. P. Torres, D. Artigas, and L. Torner, “Generation of multicolor spatial solitons with pulsed light,” Opt. Commun. |

23. | 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. |

24. | C. Conti and S. Trillo, “X waves generated at the second harmonic,” Opt. Lett. |

25. | 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 |

26. | 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 |

27. | R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized discrete quadratic solitons,” Opt. Lett. |

28. | 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 |

29. | 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. |

30. | F. Setzpfandt, D. N. Neshev, R. Schiek, F. Lederer, A. Tünnermann, and T. Pertsch, “Competing nonlinearities in quadratic nonlinear waveguide arrays,” Opt. Lett. |

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

32. | T. Peschel, U. Peschel, and F. Lederer, “Discrete bright solitary waves in quadratically nonlinear media,” Phys. Rev. E |

33. | A. A. Sukhorukov, Y. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E |

34. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

35. | 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 |

36. | J. T. Manassah, “Effects of velocity dispersion on a generated second harmonic signal,” Appl. Opt. |

37. | 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. |

38. | 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. |

39. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express |

40. | 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. |

41. | C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. |

42. | 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. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Waveguide Arrays

**History**

Original Manuscript: September 7, 2011

Revised Manuscript: October 16, 2011

Manuscript Accepted: October 20, 2011

Published: November 1, 2011

**Virtual Issues**

Nonlinear Optics (2011) *Optical Materials Express*

**Citation**

Frank Setzpfandt, Andrey A. Sukhorukov, Dragomir N. Neshev, Roland Schiek, Alexander S. Solntsev, Raimund Ricken, Yoohong Min, Wolfgang Sohler, Yuri S. Kivshar, and Thomas Pertsch, "Spectral pulse transformations and phase transitions in quadratic nonlinear waveguide arrays," Opt. Express **19**, 23188-23201 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23188

Sort: Year | Journal | Reset

### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003). [CrossRef] [PubMed]
- A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B43, 163001 (2010). [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]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev.3, 243–261 (2009). [CrossRef]
- C. Denz, S. Flach, and Y. S. Kivshar, eds., Nonlinearities in Periodic Structures and Metamaterials (Springer, 2010).
- 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]
- 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]
- 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]
- 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. Express14, 6055–6062 (2006). [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,” Nature422, 147–150 (2003). [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]
- 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]
- 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. B15, 2255–2268 (1998). [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]
- S. Droulias, K. Hizanidis, J. Meier, and D. Christodoulides, “X - waves in nonlinear normally dispersive waveguide arrays,” Opt. Express13, 1827–1832 (2005). [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]
- 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]
- 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]
- 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. E68, 026610 (2003). [CrossRef]
- 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. A83, 043834 (2011). [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]
- 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. News19, 24–31 (2008). [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]
- 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).
- T. Peschel, U. Peschel, and F. Lederer, “Discrete bright solitary waves in quadratically nonlinear media,” Phys. Rev. E57, 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. E63, 016615 (2000). [CrossRef]
- W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).
- M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010). [CrossRef]
- J. T. Manassah, “Effects of velocity dispersion on a generated second harmonic signal,” Appl. Opt.27, 4365–4367 (1988). [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]
- S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.