## Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings

Optics Express, Vol. 14, Issue 20, pp. 9358-9370 (2006)

http://dx.doi.org/10.1364/OE.14.009358

Acrobat PDF (572 KB)

### Abstract

We numerically show that it is possible to achieve adiabatic compression of femtosecond quadratic solitons in aperiodically poled lithium niobate device. Two-colored solitons of the fundamental wavelength of 1560 nm can be adiabatically shaped by using group-velocity matching schemes available in quasi-phase-matching (QPM) devices. We investigate the performance of the adiabatic compression based on two different group-velocity matching schemes: type-I (e: o + o) collinear QPM geometry and type-0 (e: e + e) non-collinear QPM geometry. Two-colored temporal solitons with pulse duration of 35 fs are generated without visible pedestals from 100-fs fundamental pulse. We also show that walking solitons with shorter pulse durations are adiabatically excited under small group-velocity mismatch condition. The walking solitons experience deceleration or acceleration during compression, depending on the sign of the group-velocity-mismatch. The demonstrated adiabatic pulse shaping is useful for generation of shorter pulses with clean temporal profiles, efficient femtosecond second harmonic generation and group-velocity control.

© 2006 Optical Society of America

## 1. Introduction

1. C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to *χ*^{(2)}:*χ*^{(2)} cascading,” J. Opt. Soc. Am. B **11**, 2434–2443 (1994). [CrossRef]

4. A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. **370**, 63–235 (2002). [CrossRef]

5. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. **19**, 1612–1615(1994). [CrossRef] [PubMed]

6. L. Torner, “Stationary solitary waves with second-order nonlinearities,” Opt. Commun. **114**, 136–140 (1995). [CrossRef]

7. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium,” Phys. Rev. Lett. **74**, 5036–5039 (1995). [CrossRef] [PubMed]

8. R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, C. Bosshard, and P. Gunter, “Twodimensional type I quadratic spatial solitons in KNbO_{3} near noncritical phase matching,” Opt. Lett. **27**, 631–633 (2002). [CrossRef]

9. B. Bourliaguet, V. Couderc, A. Barthelemy, G. Ross, P. Smith, D. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. **24**, 1410–1412 (1999). [CrossRef]

10. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic spatial solitons in periodically poled KTiOPO_{4},” Opt. Lett. **28**, 640–642 (2003). [CrossRef] [PubMed]

11. L. Torner, C. Clausen, and M. Fejer, “Adiabatic shaping of quadratic solitons,” Opt. Lett. **23**, 903–905 (1998). [CrossRef]

12. S. Carrasco, J. Torres, L. Torner, and R. Schiek, “Engineerable generation of quadratic solitons in synthetic phase matching,” Opt. Lett. **25**, 1273–1275 (2000). [CrossRef]

14. F. W. Wise, L. Qian, and X. Liu, “Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation,” J. Nonlinear Opt. Phys. Mater. **11**, 317–338 (2002). [CrossRef]

15. X. Liu, L. J. Qian, and F. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade *χ*^{(2)}:*χ*^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

16. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

17. K. Beckwitt, F. Ilday, and F. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. **29**, 763–765 (2004). [CrossRef] [PubMed]

18. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef] [PubMed]

19. S. Ashihara, T. Shimura, K. Kuroda, Nan Ei Yu, S. Kurimura, K. Kitamura, Myoungsik Cha, and Takunori Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. **84**, 1055–1057 (2004). [CrossRef]

*o*+

*o*) in 10-mm MgO-doped periodically poled lithium niobate (PPMgLN) was used to achieve GV matching. The compression mechanism is based on the interplay between amplitude/phase modulation due to cascaded nonlinearities and the material dispersion of the PPMgLN. This non-adiabatic process is inherently accompanied by the large pedestals in the compressed pulses.

_{32}(Type-I: e:

*o*+

*o*) in PPMgLN [20

20. N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO_{3} at the communications band,” Opt. Lett. **27**, 1046–1048 (2002). [CrossRef]

*e*:

*e*+ e) [21

21. S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasiphase-matched gratings,” J. Opt. Soc. Am. B **20**, 853–856 (2003). [CrossRef]

17. K. Beckwitt, F. Ilday, and F. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. **29**, 763–765 (2004). [CrossRef] [PubMed]

15. X. Liu, L. J. Qian, and F. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade *χ*^{(2)}:*χ*^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

18. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef] [PubMed]

_{3}(MgO: LN) geometry (Type-I) are presented in Sec. 3. Numerical results of another GV-mismatch compensation based on the noncollinear QPM (Type-0) geometry are presented in Sec. 4. In Sec.5, we study the tolerance of adiabatic shaping to GV mismatch: the generation of quadratic walking solitons under the small GV mismatch condition is discussed. Here we observe deceleration or acceleration of quadratic walking solitons. Finally, we conclude this paper in Sec. 6.

## 2. Adiabatic soliton compression in QPM grating

22. S. Chernikov and P. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. Am. B **8**, 1633–1641(1991). [CrossRef]

23. L. Torner and G. Stegeman, “Soliton evolution in quasi-phase-matched second-harmonic generation,” J. Opt. Soc. Am. B **14**, 3127–3133 (1997). [CrossRef]

*Δk(z)*=

*k*-2

_{2}*k*

_{1}-2

*π*/Λ(

*z*), is determined by the local domain reversal period Λ(

*z*) and the FF and SH wave numbers,

*k*

_{1}and

*k*

_{2}. Here, we consider a chirped QPM grating, where the local effective wave-vector mismatch longitudinally changes and approaches the phase matching condition. FF and SH pulses will dynamically adapt themselves to the soliton solutions for the local wave-vector mismatch. The local wave-vector mismatch should change slowly enough that pulses can adiabatically evolve to proper solitons at each position. Finally, both FF and SH pulses adiabatically evolve into soliton pulses with shorter pulse durations.

## 3. Soliton compression based on collinear QPM geometry

*o*+

*o*) occurs at the fundamental wavelength of 1560 nm [20

20. N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO_{3} at the communications band,” Opt. Lett. **27**, 1046–1048 (2002). [CrossRef]

*z*) used in our simulation varies from 19.98 µm to 20.40 µm (Δ

*k*: 8.15 mm

^{–1}to 0.41 mm

^{–1}) with a simple linear chirp (see Fig. 2). The grating length is 100 mm, which corresponds to 3.3 times the dispersion length of a 100-fs pulse (FF). The coupled wave equations governing the propagation of the FF and SH waves under the slowly varying envelope approximation can be generalized as [24]:

*ρ*=

_{i}(z)*ω*

_{i}d_{32}

*d*(

*z*)/

*cn*and

_{i}*σ*=3

_{i}*ω*

_{i}*χ*

^{(3)}/8

*cn*.

_{i}*E*(

_{i}*z, t*) denotes the amplitude of the electric field, and the subscripts 1 and 2 correspond to FF and SH pulses, respectively. Time

*t*is measured in a frame of reference along with FF propagation. Group velocity dispersion

*k*″=

_{i}*d*

^{2}

*k*

_{i}/d^{2}

*ω*and the material wave-vector mismatch

*Δk*

_{0}=

*k*

_{2}- 2

*k*

_{1}are derived from Sellmeier’s equation for MgO: LN [25

25. D. Zelmon, D. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. magnesium oxide doped lithium niobate,” J. Opt. Soc. Am. B **14**, 3319–3322 (1997). [CrossRef]

*d(z)*is the normalized distribution of second-order nonlinearity along the aperiodic QPM grating. The input FF is a transform-limited pulse with a full-width at half-maximum (FWHM) duration of 100 fs (peak intensity: 20 GW/cm

^{2}) at the wavelength of 1560 nm. Numerical simulation is carried out to solve Eq. (1) with a symmetric split-step beam-propagation method (BPM).

11. L. Torner, C. Clausen, and M. Fejer, “Adiabatic shaping of quadratic solitons,” Opt. Lett. **23**, 903–905 (1998). [CrossRef]

^{2}), as shown in Fig. 3. If we neglect the cubic nonlinear term, the compressed pulse duration becomes 25 fs. This means that the cubic nonlinearity counteracts the cascading self-defocusing nonlinearity and reduces the net nonlinearity. This trend becomes enhanced at higher intensity, because cascaded nonlinearity saturates with intensity. In fact, even if we increase the input peak intensity to be much more than 20 GW/cm

^{2}, the compressed pulse duration decreases only down to 30 fs. Therefore, the shortest pulse duration is ~30 fs, mainly limited by the competition between cascaded second-order and cubic nonlinearity.

*Δk*, the compressed pulses have shorter (longer) pulse duration with larger (smaller) pedestal. The effective wave-vector mismatch

*Δk*is set to be 0.35 mm

^{–1}for periodic QPM grating, which gives the same compression factor as the adiabatic scheme. Then we can compare the performance as a compressor by the difference in pulse quality. Here self-defocusing nonlinearity is induced by the cascaded processes. The optimum propagation length (10 mm) for the maximum compression factor is taken after the initial compress stage, which is denoted by a black arrow in Fig. 3(d). After initial compress stage, FF and SH have same dynamic behavior. From the results in Figs. 3(a)–3(b), the output FF and SH pulses are compressed by a factor of about 3, but the pulse quality is degraded: the main spike is accompanied by a broad pedestal in both FF and SH pulses. From Fig. 3(d), FF and SH pulses experience several stretch and compression phases during the initial compression stage.

*Δk*, but they are very close to the hyperbolic-secant or Gaussian functions. In simple terms, a sech

^{2}(

*t*) function is used to fit the central part of the FF and SH pulses at each position. In this paper, the term quality factor is defined as the fractional amount of energy carried by the central spike of the FF and SH pulses, normalized by the launched energy. Figure 4 shows the quality factor as a function of propagation distance in the linearly chirped QPM grating (solid line). We can see that the quality factor is around 1 during propagation. This means that two-color (FF and SH) solitons show strong robustness and adapt to slight variations of

*Δk*by reshaping their profiles. We note that the quality factor is larger than 1 at several regions due to mathematical artifacts, i.e., an intensity profile having a super-Gaussian shape will produce a quality factor larger than 1. Figs. 5(a)–5(b) show the evolutions of FF and SH pulses in the linearly chirped 100-mm QPM grating. No obvious energy leakage is visible. In contrast, the quality factor becomes as low as 0.6 in the periodic QPM grating (dash line in Fig. 4). The main lobe of FF and SH splits into a multi-peaked structure and a large part of the launched energy flows away in the form of a broad continuum, as shown in Figs. 5(c)–5(d).

## 4. Soliton compression in noncollinear QPM geometry

27. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. **25**, 2464–2468 (1989). [CrossRef]

*χ*

^{(2)}temporal solitons in BBO crystal [28

28. P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. **81**, 570–573 (1998). [CrossRef]

21. S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasiphase-matched gratings,” J. Opt. Soc. Am. B **20**, 853–856 (2003). [CrossRef]

29. N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, “Group-velocity-matched noncollinear second-harmonic generation in quasi-phase matching,” J. Opt. Soc. Am. B **22**, 1283–1289 (2005). [CrossRef]

30. A. Schober, M. Charbonneau-Lefort, and M. Fejer, “Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion,” J. Opt. Soc. Am. B **22**, 1699–1713 (2005). [CrossRef]

31. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. **22**, 1553–1555 (1997). [CrossRef]

*β*depends on the QPM grating vector

*=2π/Λ, which is always longer than grating vector in collinear QPM grating, i.e. Λ < Λ*

**K**_{0}. That is, the walk-off angle

*β*(or the QPM period Λ) is a free parameter in the noncollinear QPM geometry. We express this freedom by a parameter

*R*=Λ/Λ

_{0}(0<

*R*<1), where Λ

_{0}is the QPM period of the collinear geometry. Once R is chosen, (1) the propagation angle

*α*with respect to the grating vector

*and the walk-off angle*

**K***β*are determined to satisfy the noncollinear QPM condition, and (2) the pulse-front tilt angle

*ρ*is determined for GV mismatch compensation. For example, when R is 0.3 (Λ=5.61 µm), the achromatic QPM condition at the fundamental wavelength of 1560 nm requires

*α, β*, and

*ρ*to be 74.3°, 3.5° and 36.5°, respectively. FF is required to have a suitable pulse-front tilt before entering the QPM grating and the suitable spectral angular dispersion is necessary to compensate for the pulse-front tilt of output FF and SH, respectively.

*, while the propagation angle α is same with that (74.3°) in the periodic noncollinear QPM grating. The value of grating vector*

**K***is determined by the grating period along the coordinate*

**K***z*(

_{s}*z*‖

_{s}*), which has same period with the grating period along the line*

**K***z*=y sin(α)+

_{s}*z*cos(α) in (

*y, z*) coordinates. That is, grating vector

*along this line is determined by*

**K***(*

**K***z*). Therefore the grating vector

_{s}*(*

**K***y, z*) can be expressed as two components [

*K*

_{l}(

*z*),

_{s}*K*(

_{p}*z*)] in (

_{s}*y, z*) coordinates. The first-order Fourier component of the nonlinear coefficient distribution is expressed as

*d*(

*z, y*)=2/

*πd*

_{33}

*exp*[

*iK*·

_{p}*z*+

*iKl*·

*y*]. The wave-vector diagram of noncollinear SHG is shown in Fig. 6(b). The effective wave-vector mismatch Δ

*(*

**k***z*)=

*k*-2

_{2z}*k*

_{1}-

*K*is parallel to the

_{p}*z*direction, where

*k*

_{2z}and

*k*represent the

_{2y}*z*and

*y*components of

*k*

_{2}. Δ

*k*(

*z*) is designed to vary linearly from -40 mm

^{–1}to 5 mm

^{–1}. Strictly speaking, GV mismatch compensation becomes imperfect as we introduce effective wave-vector mismatch. Such degradation of the GV-mismatch compensation, however, is too small to affect the nonlinear propagation, and therefore is not taken into account here.

29. N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, “Group-velocity-matched noncollinear second-harmonic generation in quasi-phase matching,” J. Opt. Soc. Am. B **22**, 1283–1289 (2005). [CrossRef]

*L*=∂

_{i}*+[(2*

_{z}*k*)

_{i}^{-1}∂2

*yy*-(

*k**

*/2)∂2*

_{i}*],*

_{tt}*ρ*=2

_{i}*ωid*

_{33}/

*πcn*,

_{i}*δ*=1/v

_{g1}-1/v

_{g2}is temporal walk-off,

*v*

_{g1}and

*v*

_{g2}are the group velocities of the FF and SH pulses, and

*η*=

*k*. The standard (2 + 1)D (temporal, spatial plus propagation) coupled-wave equations are solved by using the 2D BPM method.

_{2y}/k_{2z}^{2}. The grating length

*L*is set to be 3 mm, which is 3 times the effective dispersion length of the 100-fs duration pulse (FF). The effective GVD induced by the pulse-front tilting is given by [32]:

*c*is the velocity of light in vacuum and dα/dλ is the spectral angular dispersion. The effective GVD is usually one order of magnitude larger than the intrinsic material GVD. Different from the type-I case, the self-focusing nonlinearity works for soliton formation, because the net dispersion is anomalous: the net dispersion is mainly governed by the anomalous dispersion induced by pulse-front tilt. Therefore the cubic nonlinearity supports the cascading (self-focusing) nonlinearity to work for compression. In principle, the compression factor increases with intensity. For example, the compressed pulse duration becomes 25 fs when the input peak intensity is 40 GW/cm

^{2}. However, the material damage threshold limits the highest input intensity.

*d*

_{33}used in the type-0 phase matching geometry makes the cascaded nonlinearity more efficient for soliton compression.

*D*=

*βL*. The aperture length

*D*is 180 µm after 3 mm propagation, which is much less than the 500 µm (FWHM) beam width. It is pointed out that spatial walk-off in the noncollinear interaction has less influence on the pulse compression if large-diameter beams are used. This means that spatial walk-off does not degrade the interaction between FF and SH pulses. To verify this point, we investigate the temporal behaviors without considering the spatial walk-off. The equations for this case are similar to Eq. (1) except that the GVD coefficients of FF and SH are modified by the effective anomalous GVD, which is induced by the pulse-front tilting. The normalized energy ratio and pulse duration of the FF and SH pulses are calculated by using a 1D BPM method (red lines), as shown in Figs. 7(a)–7(b) for comparison. The agreement between the results for 1D and 2D simulations is clearly shown, which demonstrates that spatial walk-off has little influence on the pulse compression.

*d*

_{33}(Type-0: e:

*e*+

*e*) can be used in this scheme; (2) Shorter gratings become available in noncollinear QPM grating due to the induced effective GVD from pulse-front tilting. Since GV-mismatching compensation in other wavelengths can be achieved in non-collinear QPM gratings, adiabatic soliton compression can be applied to a broad FF wavelength. This approach shows an efficient way to achieve adiabatic soliton compression in the bulk

*χ*

^{(2)}media.

## 5. Adiabatic soliton compression under small GV mismatch

*χ*

^{(2)}cascaded interactions in QPM gratings [33

33. M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis, and K. Kitamura, “Group-velocity control by quadratic nonlinear interactions,” Opt. Lett. **31**, 534–536 (2006). [CrossRef] [PubMed]

34. S. Carrasco, J. Torres, L. Torner, and F. W. Wise, “Walk-off acceptance for quadratic soliton generation,” Opt.Comm. **191**, 363–370 (2001). [CrossRef]

## 6. Conclusions

## 7. Acknowledgments

## References and links

1. | C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to |

2. | G. I. Stegeman, D. J. Hagan, and L. Torner, “ |

3. | L. Torner and A. Barthelemy, “Quadratic solitons: recent developments,” IEEE J. Quantum Electron. |

4. | A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. |

5. | A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. |

6. | L. Torner, “Stationary solitary waves with second-order nonlinearities,” Opt. Commun. |

7. | W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium,” Phys. Rev. Lett. |

8. | R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, C. Bosshard, and P. Gunter, “Twodimensional type I quadratic spatial solitons in KNbO |

9. | B. Bourliaguet, V. Couderc, A. Barthelemy, G. Ross, P. Smith, D. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. |

10. | H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic spatial solitons in periodically poled KTiOPO |

11. | L. Torner, C. Clausen, and M. Fejer, “Adiabatic shaping of quadratic solitons,” Opt. Lett. |

12. | S. Carrasco, J. Torres, L. Torner, and R. Schiek, “Engineerable generation of quadratic solitons in synthetic phase matching,” Opt. Lett. |

13. | R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, “One-dimensional spatial soliton families in optimally engineered quasi-phase-matched lithium niobate waveguides,” Opt. Lett. |

14. | F. W. Wise, L. Qian, and X. Liu, “Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation,” J. Nonlinear Opt. Phys. Mater. |

15. | X. Liu, L. J. Qian, and F. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade |

16. | S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B |

17. | K. Beckwitt, F. Ilday, and F. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. |

18. | J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. |

19. | S. Ashihara, T. Shimura, K. Kuroda, Nan Ei Yu, S. Kurimura, K. Kitamura, Myoungsik Cha, and Takunori Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. |

20. | N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO |

21. | S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasiphase-matched gratings,” J. Opt. Soc. Am. B |

22. | S. Chernikov and P. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. Am. B |

23. | L. Torner and G. Stegeman, “Soliton evolution in quasi-phase-matched second-harmonic generation,” J. Opt. Soc. Am. B |

24. | G. P. Agrawal, |

25. | D. Zelmon, D. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. magnesium oxide doped lithium niobate,” J. Opt. Soc. Am. B |

26. | G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, “Temporal solitons in |

27. | E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. |

28. | P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. |

29. | N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, “Group-velocity-matched noncollinear second-harmonic generation in quasi-phase matching,” J. Opt. Soc. Am. B |

30. | A. Schober, M. Charbonneau-Lefort, and M. Fejer, “Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion,” J. Opt. Soc. Am. B |

31. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. |

32. | J. C. Diels and W. Rudolph, |

33. | M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis, and K. Kitamura, “Group-velocity control by quadratic nonlinear interactions,” Opt. Lett. |

34. | S. Carrasco, J. Torres, L. Torner, and F. W. Wise, “Walk-off acceptance for quadratic soliton generation,” Opt.Comm. |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.5520) Ultrafast optics : Pulse compression

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 25, 2006

Revised Manuscript: September 11, 2006

Manuscript Accepted: September 12, 2006

Published: October 2, 2006

**Citation**

Xianglong Zeng, Satoshi Ashihara, Nobuhide Fujioka, Tsutomu Shimura, and Kazuo Kuroda, "Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings," Opt. Express **14**, 9358-9370 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9358

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### References

- C. Menyuk, R. Schiek, and L. Torner, "Solitary waves due to Chi (2):Chi (2) cascading," J. Opt. Soc. Am. B 11, 2434-2443 (1994). [CrossRef]
- G. I. Stegeman, D. J. Hagan, and L. Torner, "Chi (2) Cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons," Opt. Quantum. Electron. 28, 1691-1740 (1996). [CrossRef]
- L. Torner and A. Barthelemy, "Quadratic solitons: recent developments," IEEE J. Quantum Electron. 39, 22-30 (2003). [CrossRef]
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