## Ultra-broadband pulse evolution in optical parametric oscillators |

Optics Express, Vol. 19, Issue 19, pp. 17979-17984 (2011)

http://dx.doi.org/10.1364/OE.19.017979

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### Abstract

Ultrashort-pulse evolution inside a optical parametric oscillator is described by using a nonlinear-envelope-equation approach, eliminating the assumptions of fixed frequencies and a single χ^{(2)} process associated with conventional solutions based on the three coupled-amplitude equations. By treating the interacting waves as a single propagating field, the experimentally-observed behaviors of singly and doubly-resonant OPOs are predicted across near-octave-spanning bandwidths, including situations where the nonlinear crystal provides simultaneous phasematching for multiple nonlinear processes.

© 2011 OSA

## 1. Introduction

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

^{(2)}susceptibility in non-centrosymmetric media. Despite the assumptions of monochromatic waves implicit in the coupled-amplitude equations, they can be used successfully to model interactions between ultrashort pulses [2

2. M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expression for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. **45**(9), 3996–4005 (1974). [CrossRef]

5. J. E. Schaar, J. S. Pelc, K. L. Vodopyanov, and M. M. Fejer, “Characterization and control of pulse shapes in a doubly resonant synchronously pumped optical parametric oscillator,” Appl. Opt. **49**(24), 4489–4493 (2010). [CrossRef] [PubMed]

^{(2)}nonlinear envelope equation (NEE) [6

6. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**(5), 053841 (2010). [CrossRef]

^{(2)}medium, similar to the octave-spanning supercontinuum models which have been implemented for photonic-crystal fibers [7

7. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**(4), 1135–1184 (2006). [CrossRef]

^{(3)}NEE [8

8. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. **78**(17), 3282–3285 (1997). [CrossRef]

^{(2)}NEE as an analytical technique for studying ultra-broadband χ

^{(2)}interactions is particularly timely because of several parallel experimental observations of ultra-broadband conversion in quasi-phasematched (QPM) interactions, notably in periodically-poled lithium niobate (PPLN) waveguides [9

9. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-photon-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. **32**(17), 2478–2480 (2007). [CrossRef] [PubMed]

10. N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate OPO for mid-infrared frequency comb generation,” Opt. Express **19**(7), 6296–6302 (2011). [CrossRef] [PubMed]

11. S. T. Wong, K. L. Vodopyanov, and R. L. Byer, “Self-phase-locked divide-by-2 optical parametric oscillator as a broadband frequency comb source,” J. Opt. Soc. Am. B **27**(5), 876–882 (2010). [CrossRef]

^{(2)}NEE has been shown to predict the structure of an octave-spanning supercontinuum generated in a PPLN waveguide [6

6. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**(5), 053841 (2010). [CrossRef]

12. M. Conforti, F. Baronio, C. De Angelis, M. Marangoni, and G. Cerullo, “Theory and experiments on multistep parametric processes in nonlinear optics,” J. Opt. Soc. Am. B **28**(4), 892–895 (2011). [CrossRef]

^{(3)}NEE has been validated by direct comparison with Maxwell’s equations for single-cycle optical pulses [8

8. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. **78**(17), 3282–3285 (1997). [CrossRef]

^{(2)}NEE therefore represents a powerful new method for studying ultra-broadband pulse evolution in systems incorporating optical feedback, such as an OPO. To our knowledge, no prior work has been reported in which the χ

^{(2)}NEE is applied in this context, however the rapid progress in degenerate modelocked OPOs – which exploit ultra-broadband χ

^{(2)}interactions for generating mid-infrared frequency-combs [13

13. J. H. Sun, B. J. S. Gale, and D. T. Reid, “Composite frequency comb spanning 0.4-2.4μm from a phase-controlled femtosecond Ti:sapphire laser and synchronously pumped optical parametric oscillator,” Opt. Lett. **32**(11), 1414–1416 (2007). [CrossRef] [PubMed]

14. K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. **36**(12), 2275–2277 (2011). [CrossRef] [PubMed]

^{(2)}NEE, which describes ultra-broadband pulse evolution in an OPO synchronously-pumped by a femtosecond laser. The gain materials used in the simulations are exclusively QPM media, in which the polarity of χ

^{(2)}is modulated along the length of the medium with a period equal to twice the coherence length of the intended χ

^{(2)}process. The longitudinal grating formed in this way can also quasi-phasematch any higher-order process whose coherence length is an odd sub-harmonic of the grating period and, when the magnitude of either the fields or of χ

^{(2)}is sufficiently high, this effect leads to multiple simultaneous nonlinear processes. Effects like this are readily observed when the intensities of the interacting fields are enhanced inside the high-finesse cavity of an OPO. We demonstrate the existence of steady-state and periodic solutions for broadband pulses propagating in an OPO, supported by comparisons with experimental data from a singly-resonant non-degenerate tandem OPO and a doubly-resonant degenerate OPO.

## 2. Model

6. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**(5), 053841 (2010). [CrossRef]

*R*, where

*R*is the peak cavity reflectivity. The phase of the spectral filter determines the delay,

*T*, which is added to the resonant pulse every roundtrip of the cavity. This delay is relative to the centre of the comoving frame, so it is not generally equal to the delay between the pump and signal pulses. The spectral filter can be modified to include the group-delay dispersion characteristics of additional cavity elements and is applied to the field leaving the nonlinear medium by simply multiplying the field:

*I*is the pulse intensity averaged across its beam radius and

*n*is the refractive index at the pump wavelength.

## 3. Experimental validation

16. K. A. Tillman, D. T. Reid, D. Artigas, and T. Y. Jiang, “Idler-resonant femtosecond tandem optical parametric oscillator tuning from 2.1 µm to 4.2 µm,” J. Opt. Soc. Am. B **21**(8), 1551–1558 (2004). [CrossRef]

^{(2)}processes. An insight into the origin of these outputs is given by Fig. 1(b) which presents a logarithmic plot of the spectral evolution of the field in the OPO crystal once steady-state has been reached. In the first section of the crystal the pump pulse (a) can be seen converting into a signal pulse at 1.32 µm (b) and an idler pulse at 2.35 µm (c). In the second section of the crystal, difference frequency mixing between the signal and idler pulses leads to a mid-infrared pulse at 3.0 µm (d), which interacts with the pump to create a near-infrared pulse at 1.17 µm (e). The grating period of the second section is nearly phasematched for second-harmonic generation of 3.0 µm, and consequently a second-harmonic pulse can be seen at 1.5 µm (f), which mixes with the intense pump pulse to produce a weak 2-µm output (g). A direct comparison of the output-coupled mid-infrared spectrum recorded from the OPO and that predicted by the simulation is shown in Fig. 2 , and reveals a number of similarities. The sharp features predicted by the simulation are not resolved by the low-resolution mid-infrared spectrometer used in the experiment, however the bandwidths, positions and shapes of the spectra are very comparable, giving a high degree of confidence in the accuracy of the simulation. The solid line in the simulation result in Fig. 2 shows the exact spectrum obtained from the model, while the dashed line shows this spectrum after filtering (by convolution) with a 35-nm FWHM Gaussian band-pass filter. The measured spectrum in Fig. 3 was recorded with a spectrometer whose best resolution was 17 nm, however to improve the signal level we used it without entrance or exit slits, which is realistically expected to increase this value to a figure similar to that applied numerically.

10. N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate OPO for mid-infrared frequency comb generation,” Opt. Express **19**(7), 6296–6302 (2011). [CrossRef] [PubMed]

10. N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate OPO for mid-infrared frequency comb generation,” Opt. Express **19**(7), 6296–6302 (2011). [CrossRef] [PubMed]

17. D. T. Reid, J. M. Dudley, M. Ebrahimzadeh, and W. Sibbett, “Soliton formation in a femtosecond optical parametric oscillator,” Opt. Lett. **19**(11), 825–827 (1994). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgements

## References and links

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

2. | M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expression for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. |

3. | E. C. Cheung and J. M. Liu, “Theory of a synchronously pumped optical parametric oscillator in steady-state operation,” J. Opt. Soc. Am. B |

4. | B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO |

5. | J. E. Schaar, J. S. Pelc, K. L. Vodopyanov, and M. M. Fejer, “Characterization and control of pulse shapes in a doubly resonant synchronously pumped optical parametric oscillator,” Appl. Opt. |

6. | M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A |

7. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

8. | T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. |

9. | C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-photon-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. |

10. | N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate OPO for mid-infrared frequency comb generation,” Opt. Express |

11. | S. T. Wong, K. L. Vodopyanov, and R. L. Byer, “Self-phase-locked divide-by-2 optical parametric oscillator as a broadband frequency comb source,” J. Opt. Soc. Am. B |

12. | M. Conforti, F. Baronio, C. De Angelis, M. Marangoni, and G. Cerullo, “Theory and experiments on multistep parametric processes in nonlinear optics,” J. Opt. Soc. Am. B |

13. | J. H. Sun, B. J. S. Gale, and D. T. Reid, “Composite frequency comb spanning 0.4-2.4μm from a phase-controlled femtosecond Ti:sapphire laser and synchronously pumped optical parametric oscillator,” Opt. Lett. |

14. | K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. |

15. | G. P. Agrawal, |

16. | K. A. Tillman, D. T. Reid, D. Artigas, and T. Y. Jiang, “Idler-resonant femtosecond tandem optical parametric oscillator tuning from 2.1 µm to 4.2 µm,” J. Opt. Soc. Am. B |

17. | D. T. Reid, J. M. Dudley, M. Ebrahimzadeh, and W. Sibbett, “Soliton formation in a femtosecond optical parametric oscillator,” Opt. Lett. |

**OCIS Codes**

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 13, 2011

Manuscript Accepted: August 14, 2011

Published: August 29, 2011

**Citation**

Derryck T. Reid, "Ultra-broadband pulse evolution in optical parametric oscillators," Opt. Express **19**, 17979-17984 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-17979

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### References

- J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev.127(6), 1918–1939 (1962). [CrossRef]
- M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expression for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys.45(9), 3996–4005 (1974). [CrossRef]
- E. C. Cheung and J. M. Liu, “Theory of a synchronously pumped optical parametric oscillator in steady-state operation,” J. Opt. Soc. Am. B7(8), 1385–1401 (1990). [CrossRef]
- B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B67(5), 537–544 (1998). [CrossRef]
- J. E. Schaar, J. S. Pelc, K. L. Vodopyanov, and M. M. Fejer, “Characterization and control of pulse shapes in a doubly resonant synchronously pumped optical parametric oscillator,” Appl. Opt.49(24), 4489–4493 (2010). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81(5), 053841 (2010). [CrossRef]
- J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006). [CrossRef]
- T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett.78(17), 3282–3285 (1997). [CrossRef]
- C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-photon-exchanged periodically poled lithium niobate waveguides,” Opt. Lett.32(17), 2478–2480 (2007). [CrossRef] [PubMed]
- N. Leindecker, A. Marandi, R. L. Byer, and K. L. Vodopyanov, “Broadband degenerate OPO for mid-infrared frequency comb generation,” Opt. Express19(7), 6296–6302 (2011). [CrossRef] [PubMed]
- S. T. Wong, K. L. Vodopyanov, and R. L. Byer, “Self-phase-locked divide-by-2 optical parametric oscillator as a broadband frequency comb source,” J. Opt. Soc. Am. B27(5), 876–882 (2010). [CrossRef]
- M. Conforti, F. Baronio, C. De Angelis, M. Marangoni, and G. Cerullo, “Theory and experiments on multistep parametric processes in nonlinear optics,” J. Opt. Soc. Am. B28(4), 892–895 (2011). [CrossRef]
- J. H. Sun, B. J. S. Gale, and D. T. Reid, “Composite frequency comb spanning 0.4-2.4μm from a phase-controlled femtosecond Ti:sapphire laser and synchronously pumped optical parametric oscillator,” Opt. Lett.32(11), 1414–1416 (2007). [CrossRef] [PubMed]
- K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett.36(12), 2275–2277 (2011). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, 2001).
- K. A. Tillman, D. T. Reid, D. Artigas, and T. Y. Jiang, “Idler-resonant femtosecond tandem optical parametric oscillator tuning from 2.1 µm to 4.2 µm,” J. Opt. Soc. Am. B21(8), 1551–1558 (2004). [CrossRef]
- D. T. Reid, J. M. Dudley, M. Ebrahimzadeh, and W. Sibbett, “Soliton formation in a femtosecond optical parametric oscillator,” Opt. Lett.19(11), 825–827 (1994). [CrossRef] [PubMed]

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