## Dependences of the group velocity for femtosecond pulses in MgO-doped PPLN crystal |

Optics Express, Vol. 19, Issue 6, pp. 5213-5218 (2011)

http://dx.doi.org/10.1364/OE.19.005213

Acrobat PDF (994 KB)

### Abstract

Theoretical investigation on the group velocity control of ultrafast pulses through quadratic cascading nonlinear interaction is presented. The dependences of the fractional time delay as well as the quality factor of the delayed femtosecond pulse on the peak intensity, group velocity mismatch, wave-vector mismatch and the pulse duration are examined. The results may help to understand to what extent some optical operation parameters could have played a role in controlling the ultrashort pulses. We also predict the maximum achievable pulse delay or advancement efficiency without large distortions. A compact solid medium integrating multiple functions including slowing light, wavelength conversion or broadcasting on a single chip, may bring significant practicality and high integration applications at optical communication band.

© 2011 OSA

## 1. Introduction

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature **397**(6720), 594–598 (1999). [CrossRef]

4. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express **14**(25), 12022–12027 (2006). [CrossRef] [PubMed]

5. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News **17**(4), 18–23 (2006). [CrossRef]

5. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News **17**(4), 18–23 (2006). [CrossRef]

6. D. J. Gauthier, “Optical communications - Solitons go slow,” Nat. Photonics **1**(2), 92–93 (2007). [CrossRef]

7. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. **2**(11), 775–780 (2006). [CrossRef]

^{(2)}cascading interactions in quasi-phase-match (QPM) gratings [8

8. 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**(4), 534–536 (2006). [CrossRef] [PubMed]

^{(2)}cascading interactions was investigated in detail. Variation of the fractional delay as well as the delayed pulse quality in QPM gratings with different optical parameters was analyzed. An evaluation factor was also proposed to assess the performance of the delay controlling scheme at the end of our analysis. By this mean, the optimum conditions for best delay performance were found under different situations. In addition, it should be mentioned that in our previous research [9

9. M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. **35**(16), 2672–2674 (2010). [CrossRef] [PubMed]

10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express **16**(10), 6957–6962 (2008). [CrossRef] [PubMed]

11. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. **78**(14), 2752–2755 (1997). [CrossRef]

12. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

## 2. Theoretical model

*5*mol% MgO: PPLN crystal along y direction (see inset of Fig. 1 ). The signal pulse which is called the FF pulse (central wavelength at ω

_{FF}) generates the second harmonic (SH) wave first, then the SH wave will be converted back to the FF due to wave-vector mismatching. When the energy is converted to the SH field, it propagates with the group velocity of the SH field. As a result, the signal experiences deceleration or acceleration since it is dragged by the slower or faster SH pulse depending on the sign of the slight group-velocity mismatch (GVM) and energy exchange between the signal and the SH. We choose

*5mol%*MgO:PPLN crystal to be the representative medium due to its unique material dispersion [9

9. M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. **35**(16), 2672–2674 (2010). [CrossRef] [PubMed]

10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express **16**(10), 6957–6962 (2008). [CrossRef] [PubMed]

13. N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. **82**(20), 3388–3390 (2003). [CrossRef]

10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express **16**(10), 6957–6962 (2008). [CrossRef] [PubMed]

_{31}is used. The coupled wave equations governing the propagation of the FF and SH waves under the slowly varying envelope approximation can be generalized as [14]:

*ρ*(the positive and negative sign depend on the distribution of second-order nonlinearity along the QPM grating, and

_{i}(z) = ± ω_{i}d_{31}/cn_{i}*i = 1,2*) and

*σ*. The

_{i}= 3ω_{i}χ^{(3)}/8cn_{i}*χ*terms are the cubic nonlinear terms include self phase modulation (SPM) and cross phase modulation (XPM). Since the input intensity is quite high, they cannot be neglected in our simulation. And they may counteract the cascading quadratic nonlinearity and reduce the net nonlinearity which will contribute to the construction of slow light soliton.

^{(3)}*E*denotes the amplitude of the electric field,

_{i}(z,t)*n*denotes the refractive index and

_{i}*ω*denotes the angular frequency. The subscripts 1 and 2 correspond to FF and SH pulses, respectively. Time t is measured in a time frame moving with the linear group velocity of the FF pulse. k

_{i}_{i}

^{’}is the inverse group velocity, and

*k*is the group velocity dispersion (GVD);

_{i}^{”}= d^{2}k_{i}/dω^{2}*δ = k*is the GVM,

^{’}_{SH}−k^{’}_{FF}*Δk*is derived from Sellmeier’s equation for MgO:LN [15

_{0}= k_{2}-2k_{1}15. D. E. Zelmon, D. L. 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**(12), 3319–3322 (1997). [CrossRef]

16. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B **11**(12), 2434–2443 (1994). [CrossRef]

17. J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. **29**(7), 757–776 (1997). [CrossRef]

*Δk = k*, and Λ is the poling period. In the case of negligible GVD, The general solution

_{2}-2k_{1}-2π/Λ*E*of Eq. (2) can be written as [18

_{1}(z,t) = I_{1}(z,t)^{1/2}exp(iϕ(z,t))18. F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express **14**(11), 4774–4779 (2006). [CrossRef] [PubMed]

*I*is the square of the amplitude and

_{1}(z,t) = |E_{1}(z,t)|^{2}*ϕ(z,t)*is the phaseIn the Eq. (3) and (4),

*f(t) = I(t,0)*is the initial pulse shape, and

*g(t)*is the initial pulse distribution,

*γ = −2δρ*, and

_{1}ρ_{2}/Δk^{2}*κ = ρ*.

_{1}ρ_{2}/Δk## 3. Results and discussion

*50 fs*pulse (peak intensity:

*50 GW/cm*) at the wavelength of

^{2}*1530 nm*and the temperature is set to be

*20°C*. Numerical simulation is carried out to solve Eq. (1) with a symmetric split-step beam-propagation method (BPM) and the waveforms of the input and output pulses are shown in Fig. 1. Firstly, we investigate the fractional time delay of the output FF pulse as a function of the FF input peak intensity (as is seen in Fig. 2 ). Besides, we also evaluate the quality of the output pulses. The pulse shape of quadratic soliton is 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. It should be noted that, in most of our simulations, when the quality factor drops below 0.75, the pulse will degrade and split into a multi-peaked structure, being no longer tolerable for maintaining pulse integrity. From the inset in Fig. 2, we can see the pulse broadening with no time delay at low intensity case (below 0.

*1 GW/cm*). Group delay control by varying input FF peak intensity can be realized at higher intensity due to the strong nonlinear interaction and dragging between the FF and SH pulses. The rate of time delay increment decreases with the increasing input peak intensity. This is consistent with the theoretical prediction because the quadratic cascading nonlinearity saturates with the increasing input intensity. But we can also find that with the input pulse peak intensity increased, the quality factor decreases, which is due to the strong interaction between the FF and SH pulses that results in a broad pedestal accompanied by the main spike [19

^{2}19. W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express **16**(1), 355–361 (2008). [CrossRef] [PubMed]

*30 fs*where the quality factor is also close to maximum. For it’s the minimum pulse duration that can tolerate the given GVM without the complete walk-off between the FF and SH wave. So in practical case, we can carefully choose the input pulse duration to get a high fractional delay with good pulse quality simultaneously.

*Q*is the quality factor.One can evaluate the delaying performance by DQP. In the practical point of view, since the poling period of the MgO:PPLN crystal is fixed after it is fabricated, we can’t modulate the group velocity mismatch and the wave-vector mismatch independently. It’s more efficient to tune the peak intensity and the duration of the input pulse to obtain the maximum DQP. Figure 5 gives an example to optimize the delaying performance by selecting the specific peak intensity and duration of the input pulse. We can see that the optimum input pulse duration can always be found with different input peak intensity. This can help us to find the optimum conditions for this group velocity control scheme by looking for the largest achievable DQP.

20. W. J. Lu, Y. P. Chen, X. F. Chen, and Y. Xia, “Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity,” IEEE J. Quantum Electron. **46**(7), 1099–1104 (2010). [CrossRef]

22. M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, “Observation of spectral drift in engineered quadratic nonlinear media,” Appl. Phys. Lett. **93**(2), 021107 (2008). [CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature |

2. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science |

3. | Z. M. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science |

4. | Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express |

5. | R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News |

6. | D. J. Gauthier, “Optical communications - Solitons go slow,” Nat. Photonics |

7. | J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. |

8. | 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. |

9. | M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. |

10. | J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express |

11. | S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. |

12. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science |

13. | N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. |

14. | G. P. Agrawal, |

15. | D. E. Zelmon, D. L. 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 |

16. | C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B |

17. | J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. |

18. | F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express |

19. | W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express |

20. | W. J. Lu, Y. P. Chen, X. F. Chen, and Y. Xia, “Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity,” IEEE J. Quantum Electron. |

21. | M. Conforti, F. Baronio, C. De Angelis, G. Sanna, D. Pierleoni, and P. Bassi, “Spectral shaping of feratosecond pulses in aperiodic quasi-phase-matched gratings,” Opt. Commun. |

22. | M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, “Observation of spectral drift in engineered quadratic nonlinear media,” Appl. Phys. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 2, 2010

Revised Manuscript: January 20, 2011

Manuscript Accepted: January 22, 2011

Published: March 4, 2011

**Citation**

Yu-Ping Chen, Wen-Jie Lu, Yu-Xing Xia, and Xian-Feng Chen, "Dependences of the group velocity for femtosecond pulses in MgO-doped PPLN crystal," Opt. Express **19**, 5213-5218 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5213

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### References

- L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
- M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
- Z. M. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318(5857), 1748–1750 (2007). [CrossRef] [PubMed]
- Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14(25), 12022–12027 (2006). [CrossRef] [PubMed]
- R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News 17(4), 18–23 (2006). [CrossRef]
- D. J. Gauthier, “Optical communications - Solitons go slow,” Nat. Photonics 1(2), 92–93 (2007). [CrossRef]
- J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2(11), 775–780 (2006). [CrossRef]
- 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(4), 534–536 (2006). [CrossRef] [PubMed]
- M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. 35(16), 2672–2674 (2010). [CrossRef] [PubMed]
- J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express 16(10), 6957–6962 (2008). [CrossRef] [PubMed]
- S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]
- N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82(20), 3388–3390 (2003). [CrossRef]
- G. P. Agrawal, Nonlinear fiber optics, 3rd ed., (Academic Press, 2001).
- D. E. Zelmon, D. L. 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(12), 3319–3322 (1997). [CrossRef]
- C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11(12), 2434–2443 (1994). [CrossRef]
- J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]
- F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express 14(11), 4774–4779 (2006). [CrossRef] [PubMed]
- W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express 16(1), 355–361 (2008). [CrossRef] [PubMed]
- W. J. Lu, Y. P. Chen, X. F. Chen, and Y. Xia, “Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity,” IEEE J. Quantum Electron. 46(7), 1099–1104 (2010). [CrossRef]
- M. Conforti, F. Baronio, C. De Angelis, G. Sanna, D. Pierleoni, and P. Bassi, “Spectral shaping of feratosecond pulses in aperiodic quasi-phase-matched gratings,” Opt. Commun. 281, 1693–1697 (2008).
- M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, “Observation of spectral drift in engineered quadratic nonlinear media,” Appl. Phys. Lett. 93(2), 021107 (2008). [CrossRef]

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