## Adiabatic and diabatic process of sum frequency conversion |

Optics Express, Vol. 18, Issue 19, pp. 20428-20438 (2010)

http://dx.doi.org/10.1364/OE.18.020428

Acrobat PDF (1748 KB)

### Abstract

Based on the dressed state formalism, we obtain the adiabatic criterion of the sum frequency conversion. We show that this constraint restricts the energy conversion between the two dressed fields, which are superpositions of the signal field and the sum frequency field. We also show that the evolution of the populations of the dressed fields, which in turn describes the conversion of light photons from the seed frequency to the sum frequency during propagation through the nonlinear crystal. Take the quasiphased matched (QPM) scheme as an example, we calculate the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We demonstrate that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffraction of a light field. We finally show that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds.

© 2010 OSA

## 1. Introduction

1. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. **100**(18), 183601 (2008). [CrossRef] [PubMed]

2. M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. **30**(1), 34–35 (1994). [CrossRef]

5. M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature **432**(7015), 374–376 (2004). [CrossRef] [PubMed]

6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. **22**(17), 1341–1343 (1997). [CrossRef] [PubMed]

8. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. **8**(2), 180–198 (2007). [CrossRef]

9. G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B **18**(4), 534–539 (2001). [CrossRef]

7. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B **17**(2), 304–318 (2000). [CrossRef]

10. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B **25**(4), 463–480 (2008). [CrossRef]

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A **78**(6), 063821 (2008). [CrossRef]

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**(15), 12731–12740 (2009). [CrossRef] [PubMed]

17. J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. **72**(1), 56–59 (1994). [CrossRef] [PubMed]

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A **78**(6), 063821 (2008). [CrossRef]

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**(15), 12731–12740 (2009). [CrossRef] [PubMed]

## 2. Propagation equation and its adiabatic solution during the sum frequecy

12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A **78**(6), 063821 (2008). [CrossRef]

13. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**(15), 12731–12740 (2009). [CrossRef] [PubMed]

*z*is the propagation distance of the light field in the crystal.

*c*is the speed of light in vacuum.

*z*, while

*z*. The eigenvalues or the

*G*are

*R*. Borrowing the name of the dressed-pulse fields [17

17. J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. **72**(1), 56–59 (1994). [CrossRef] [PubMed]

*θ*to

*z*. Equation (3) is a general expression for the sum frequency conversion. When the adiabatic condition (3) holds, Eq. (2) becomes Equation (4) means that there is no energy conversion between the dressed fields

**17**(15), 12731–12740 (2009). [CrossRef] [PubMed]

*z*is very slowly.

*q*is real and independent of the position

*z*, i.e.,

*z*, one can easily show the adiabatic condition is satisfied well. Under the adiabatic criterion, the solution of Eq. (4) is

*q*completely governs the energy conversion between the signal field and the sum frequency field. For

*q*or decrease that of

## 3. **Diabatic processes and sum frequency conversion**

18. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. **127**(6), 1918–1939 (1962). [CrossRef]

7. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B **17**(2), 304–318 (2000). [CrossRef]

### 3.1. Energy conversion evolving with the propagation distance

*z*axis into two regions, with one being called as diabatic region (I) and another one as adiabatic region (

*l*being the length of the crystal. The intersection in Fig. 3(a) shows that the dressed fields are in the diabatic region (I), i.e., they are not adiabatic. At the diabatic poin

*n*being the integer, thus

*m*is also a integer. Thus, Eq. (8) becomes Equation (9) indicates that the energy of the dressed fields meet the maximal coherence at the diabatic point, and under this condition the energy is transferred from the signal field to the smu frequency field completely. For different frequency variation

15. L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. **204**(1-6), 413–423 (2002). [CrossRef]

### 3.2. Energy transfer evolving with the frequency of the signal field

**17**(15), 12731–12740 (2009). [CrossRef] [PubMed]

*l*. Figures 5(a) and 5(b) shows the energy evolution of the sum frequency and the left side of the inequality (7)

6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. **22**(17), 1341–1343 (1997). [CrossRef] [PubMed]

8. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. **8**(2), 180–198 (2007). [CrossRef]

**78**(6), 063821 (2008). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. |

2. | M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. |

3. | K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phasematched second-harmonic generation,” IEEE J. Quantum Electron. |

4. | H. Guo, S. H. Tang, Y. Qin, and Y. Y. Zhu, “Nonlinear frequency conversion with quasi-phase-mismatch effect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature |

6. | M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. |

7. | G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B |

8. | D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. |

9. | G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B |

10. | M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B |

11. | L. D. Allen, and J. H. Eberly, |

12. | H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A |

13. | H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express |

14. | M. Shapiro, and P. Brumer, |

15. | L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. |

16. | A. Massiah, |

17. | J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. |

18. | J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(230.4320) Optical devices : Nonlinear optical devices

(140.3613) Lasers and laser optics : Lasers, upconversion

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 24, 2010

Revised Manuscript: July 22, 2010

Manuscript Accepted: July 25, 2010

Published: September 10, 2010

**Citation**

Ren Liqing, Li Yongfang, Li Baihong, Wang Lei, and Wang Zhaohua, "Adiabatic and diabatic process of sum frequency conversion," Opt. Express **18**, 20428-20438 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20428

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### References

- M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100(18), 183601 (2008). [CrossRef] [PubMed]
- M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30(1), 34–35 (1994). [CrossRef]
- K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phasematched second-harmonic generation,” IEEE J. Quantum Electron. 30(7), 1596–1604 (1994). [CrossRef]
- H. Guo, S. H. Tang, Y. Qin, and Y. Y. Zhu, “Nonlinear frequency conversion with quasi-phase-mismatch effect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 066615 (2005). [CrossRef] [PubMed]
- M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004). [CrossRef] [PubMed]
- M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22(17), 1341–1343 (1997). [CrossRef] [PubMed]
- G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17(2), 304–318 (2000). [CrossRef]
- D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8(2), 180–198 (2007). [CrossRef]
- G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18(4), 534–539 (2001). [CrossRef]
- M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B 25(4), 463–480 (2008). [CrossRef]
- L. D. Allen, and J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975)
- H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]
- H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009). [CrossRef] [PubMed]
- M. Shapiro, and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003)
- L. P. Yatsenko, N. V. Vitanov, B. W. Shore, T. Rickes, and K. Bergmann, “Creation of coherent superpositions using Stark-chirped rapid adiabatic passage,” Opt. Commun. 204(1-6), 413–423 (2002). [CrossRef]
- A. Massiah, Quantum Mechanics (North Holland, Amsterdam, 1962), Vol. II.
- J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72(1), 56–59 (1994). [CrossRef] [PubMed]
- J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

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