## Parametric frequency conversion of short optical pulses controlled by a CW background

Optics Express, Vol. 15, Issue 19, pp. 12246-12251 (2007)

http://dx.doi.org/10.1364/OE.15.012246

Acrobat PDF (628 KB)

### Abstract

We predict that parametric sum-frequency generation of an ultra-short pulse may result from the mixing of an ultra-short optical pulse with a quasi-continuous wave control. We analytically show that the intensity, time duration and group velocity of the generated idler pulse may be controlled in a stable manner by adjusting the intensity level of the background pump.

© 2007 Optical Society of America

## 1. Introduction

1. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. **74**, 1–18 (2003). [CrossRef]

2. J.A. Armstrong, S.S. Jha, and N.S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. **QE-6**, 123–129 (1970) [CrossRef]

4. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A **41**, 5645–5649 (1990) [CrossRef] [PubMed]

5. A. Stabinis, G. Valiulis, and E.A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. **86**, 301–306 (1991). [CrossRef]

6. Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. **17**, 1459–1461 (1992). [CrossRef] [PubMed]

7. C.Y. Chien, G. Korn, J.S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. **20**, 353–355 (1995). [CrossRef] [PubMed]

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. **21**, 1582–1584 (1996). [CrossRef] [PubMed]

9. E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt. Soc. Am. B **14**, 1472–1479 (1997). [CrossRef]

10. E. Ibragimov, A. Struthers, D.J. Kaup, J.D. Khaydarov, and K.D. Singer, “Three-wave interaction solitons in optical parametric amplification,” Phys. Rev. E **59**, 6122–6137 (1999). [CrossRef]

11. E. Ibragimov, A. Struthers, and D.J. Kaup, “Soliton pulse compression in the theory of optical parametric amplification,” Opt. Commun. **152**, 101–107 (1998). [CrossRef]

12. S. Fournier, R.L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, “Solitonlike pulse shortening in a femtosecond parametric amplifier,” Opt. Lett. **23**, 627–629 (1998). [CrossRef]

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. **21**, 1582–1584 (1996). [CrossRef] [PubMed]

15. K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Japan **34**, 796–800 (1973). [CrossRef]

16. Y. Ohsawa and K. Nozaki, “Propagation of solitary pulses in interactions of plasma waves. II,” J. Phys. Soc. Japan **36**, 591–595 (1974). [CrossRef]

17. F. Calogero and A. Degasperis, “Novel solution of the system describing the resonant interaction of three waves,” Physica D **200**, 242–256 (2005). [CrossRef]

18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. **97**, 093901 (2006). [CrossRef] [PubMed]

19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E **74**, 065602(R) (2006). [CrossRef]

20. A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. **23**, 1808–1810 (1998). [CrossRef]

21. G.M. Gale, M. Cavallari, T.J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. **20**, 1562–1564 (1995). [CrossRef] [PubMed]

## 2. Three-wave-interaction equations

*τ*=

*t/t*is an arbitrary time parameter; ξ=

_{0}, t_{0}*z/z0, z0*is an unit space-propagation parameter.

*E*are the slowly varying electric field envelopes of the waves at frequencies

_{j}*ω*are the refractive indexes,

_{j}, n_{j}*χ*

^{(2)}is the quadratic nonlinear susceptibility,

*δ*=

_{j}*z*0/(

*v*) with

_{j}t0*v*the linear group velocities, and

_{j}*j*=1,2,3. We assume that the group velocity v

_{3}of the wave with the highest frequency (ω

_{3}=ω

_{1}+ω

_{2}) lies between the group velocities of the other waves, i.e. v

_{1}>v

_{3}>v

_{2}. With no loss of generality, we shall write the Eqs. (1) in a coordinate system such that

*δ*

_{1}=0, which implies 0<

*δ*

_{3}<

*δ*

_{2}. Eqs. (1) exhibit the conserved quantities

*U*

_{1},

*U*

_{2}and 2

*U*

_{3}represent the energies at the frequenciesω

_{1}, ω

_{2}and ω

_{3}.

## 3. Soliton-based parametric sum-frequency conversion

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. **21**, 1582–1584 (1996). [CrossRef] [PubMed]

*A*

_{1}and

*A*

_{2}with frequencies ω

_{1}and ω

_{2}propagate with speeds v

_{1}and v

_{2}. Whenever the faster pulse overtakes the slower one, an idler pulse

*A*

_{3}at the SF ω

_{1}+ω

_{2}is generated and propagates with the linear speed v

_{3}. Depending on the time widths and intensities of the input pulses, the duration of the SF pulse is reduced with respect to the input pulse widths. Correspondingly, the SF pulse peak intensity grows larger than the input pulse intensities. Figure 1 shows that, eventually, the SF idler pulse decays back into the two original isolated pulses at frequencies ω

_{1}and ω

_{2}. Note that the shapes, intensities and widths of the input pulses are left unchanged in spite of their interaction. As shown in Ref. [8

**21**, 1582–1584 (1996). [CrossRef] [PubMed]

_{1}. Initially, the two pulses propagate uncoupled; as soon as the faster pulse starts to overlap in time with the slower quasi-CW control, their nonlinear mixing generates a short SF idler pulse. The sum-frequency process displayed in Fig. 2 can be analytically explained and explored in terms of stable TWRIS solutions [18

18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. **97**, 093901 (2006). [CrossRef] [PubMed]

_{2}is a stable single component TWRIS (6) with parameters

*p*>0,

*k,q,a*=0. The background control in the interaction region can be modeled with

*A*

_{1}(τ)=

*Ce*. When this faster pulse, pre-delayed with respect to the slower quasi-CW pump at frequency ω

^{-iγτ}_{1}, overtakes the background (at τ=0, in Fig. 2), their collision leads to the generation of a short idler pulse at the SF ω

_{3}. Additionally, a dip appears in the quasi CW-control; whereas the intensity, duration and propagation speed of the input wave at frequency ω

_{2}are modified. Indeed, the signal-pump interaction generates a new stable TWRIS (6), with parameters

*p*̄,

*k*̄,

*q*̄,

*a*̄, moving with the locked nonlinear velocity v̄=z

_{0}/(

*t*δ ¯ ), where

_{0}*p*̄,

*k*̄,

*q*̄,

*a*̄ of the generated TWRIS from the corresponding parameters of the input single wave TWRIS and the complex amplitude of the pump control. This can be achieved by supposing that the input TWRIS adiabatically (i.e., without emission of radiation) reshapes into a new TWRI simulton after its collision with the quasi-CW pump at a given point in time (say, at τ=0). Under this basic hypothesis, the conservative nature of the three-wave interaction permits us to suppose that: i) the energy

*U*

_{23}(4) of the input TWRI soliton is conserved in the generated TWRI simulton; ii) the phase of the ω

_{2}frequency components of the input TWRI soliton and of the generated TWRI simulton is continuous across their time interface (i.e., at τ=0); iii) the amplitude and phase of the control pump coincide with the corresponding values of the asymptotic plateau of the generated TWRI simulton component at frequency ω

_{1}. By imposing the above three conditions, after some straightforward calculations we obtain the following relations that relate the parameters of the incident and of the transmitted TWRIS

_{2}is described by Eqs. (6) with

*p*=1.3,

*k*=0,

*q*=0,

*a*=0, and the background control amplitude with

*C*=1.7,γ=0. After the collision with the CW background, the above equations predict that the generated TWRIS is again described by Eqs. (6), with

*p*̄=1.3,

*k*̄=0,

*q*̄=0, and

*a*̄=1.2. The accuracy of this prediction is well confirmed by its comparison with the numerical solutions of the TWRI Eqs. (1). Indeed, Fig. 3 compares the numerical with the analytical evolutions (along the crystal length ξ) of the energy, the pulse duration and the velocity of the idler and signal pulses which correspond to the case shown in Fig. 2. We performed further extensive numerical simulations, which confirmed the general validity of the above described adiabatic transition model for TWRIS generation upon collision with a CW background.

*C*| in the range [0,

*p*

19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E **74**, 065602(R) (2006). [CrossRef]

*eee*interaction of three-waves with carrier wavelengths of

*λ*

_{1}=1.55

*µm*, λ2=3.4

*µm*,

*λ*

_{3}=1.064

*µm*in a 2

*cm*(8

*cm*) long periodically poled bulk Lithium Niobate crystal with 28

*µm*periodicity. In this case, the parametric mixing of a 100

*fs*(1

*ps*) incident pulse with a quasi–CW (say, with a 3

*ps*(30

*ps*) time duration) control pulse leads to the generation of an ultrashort sum-frequency pulse of approximately the same time width of the incident short pulse whenever the field intensities of the two input pulses are of the order of a few hundreds of

*MW/cm*(or a few

^{2}*MW/cm*, respectively).

^{2}## 4. Conclusions

## Acknowledgements

## References and links

1. | G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. |

2. | J.A. Armstrong, S.S. Jha, and N.S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. |

3. | S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992). |

4. | Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A |

5. | A. Stabinis, G. Valiulis, and E.A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. |

6. | Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. |

7. | C.Y. Chien, G. Korn, J.S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. |

8. | E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. |

9. | E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt. Soc. Am. B |

10. | E. Ibragimov, A. Struthers, D.J. Kaup, J.D. Khaydarov, and K.D. Singer, “Three-wave interaction solitons in optical parametric amplification,” Phys. Rev. E |

11. | E. Ibragimov, A. Struthers, and D.J. Kaup, “Soliton pulse compression in the theory of optical parametric amplification,” Opt. Commun. |

12. | S. Fournier, R.L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, “Solitonlike pulse shortening in a femtosecond parametric amplifier,” Opt. Lett. |

13. | V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP Lett. , |

14. | D. J. Kaup, “The three-wave interaction — a nondispersive phenomenon,” Stud. Appl. Math. |

15. | K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Japan |

16. | Y. Ohsawa and K. Nozaki, “Propagation of solitary pulses in interactions of plasma waves. II,” J. Phys. Soc. Japan |

17. | F. Calogero and A. Degasperis, “Novel solution of the system describing the resonant interaction of three waves,” Physica D |

18. | A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. |

19. | M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E |

20. | A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. |

21. | G.M. Gale, M. Cavallari, T.J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 19, 2007

Revised Manuscript: July 16, 2007

Manuscript Accepted: July 22, 2007

Published: September 11, 2007

**Citation**

Matteo Conforti, Fabio Baronio, Antonio Degasperis, and Stefan Wabnitz, "Parametric frequency conversion of short optical pulses controlled by a CW background," Opt. Express **15**, 12246-12251 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12246

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### References

- G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003). [CrossRef]
- J. A. Armstrong, S. S. Jha, and N. S. Shiren, "Some effects of group-velocity dispersion on parametric interactions," IEEE J. Quantum Electron. QE-6, 123-129 (1970). [CrossRef]
- S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992).
- Y. Wang and R. Dragila, "Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion," Phys. Rev. A 41, 5645-5649 (1990) [CrossRef] [PubMed]
- A. Stabinis, G. Valiulis and E. A. Ibragimov, "Effective sum frequency pulse compression in nonlinear crystals," Opt. Commun. 86, 301-306 (1991). [CrossRef]
- Y. Wang and B. Luther-Davies, "Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses," Opt. Lett. 17, 1459-1461 (1992). [CrossRef] [PubMed]
- C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou and R. S. Craxton, "Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses," Opt. Lett. 20, 353-355 (1995). [CrossRef] [PubMed]
- E. Ibragimov and A. Struthers, "Second-harmonic pulse compression in the soliton regime," Opt. Lett. 21, 1582-1584 (1996). [CrossRef] [PubMed]
- E. Ibragimov and A. Struthers, "Three-wave soliton interaction of ultrashort pulses in quadratic media," J. Opt. Soc. Am. B 14, 1472-1479 (1997). [CrossRef]
- E. Ibragimov, A. Struthers, D. J. Kaup, J. D. Khaydarov, and K. D. Singer, "Three-wave interaction solitons in optical parametric amplification," Phys. Rev. E 59, 6122-6137 (1999). [CrossRef]
- E. Ibragimov, A. Struthers, and D. J. Kaup, "Soliton pulse compression in the theory of optical parametric amplification," Opt. Commun. 152, 101-107 (1998). [CrossRef]
- S. Fournier, R. L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, "Solitonlike pulse shortening in a femtosecond parametric amplifier," Opt. Lett. 23, 627-629 (1998). [CrossRef]
- V. E. Zakharov and S. V. Manakov, "Resonant interaction of wave packets in nonlinear media," Sov. Phys. JETP Lett. 18, 243-245 (1973).
- D. J. Kaup, "The three-wave interaction - a nondispersive phenomenon," Stud. Appl. Math. 55, 9-44 (1976).
- K. Nozaki and T. Taniuti, "Propagation of solitary pulses in interactions of plasma waves," J. Phys. Soc. Japan 34, 796-800 (1973). [CrossRef]
- Y. Ohsawa and K. Nozaki, "Propagation of solitary pulses in interactions of plasma waves. II," J. Phys. Soc. Japan 36, 591-595 (1974). [CrossRef]
- F. Calogero and A. Degasperis, "Novel solution of the system describing the resonant interaction of three waves," Physica D 200, 242-256 (2005). [CrossRef]
- A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, "Stable control of pulse speed in parametric three-wave solitons," Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, "Inelastic scattering and interactions of three-wave parametric solitons," Phys. Rev. E 74, 065602(R) (2006). [CrossRef]
- A. Picozzi and M. Haelterman, "Spontaneous formation of symbiotic solitary waves in a backward quasi-phasematched parametric oscillator," Opt. Lett. 23, 1808-1810 (1998). [CrossRef]
- G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache,"Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator," Opt. Lett. 20, 1562-1564 (1995). [CrossRef] [PubMed]

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