# Optical Frequency Combs in Quadratically Nonlinear Resonators

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## Abstract

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## 1. Introduction

## 2. Intracavity Second Harmonic Generation

## 3. Time-Domain Model for Quadratic Combs

## 4. Combs in Optical Parametric Oscillators

## 5. Single Envelope Equation

## 6. Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

cw | Continuous wave |

DFG | Difference frequency generation |

FF | Fundamental frequency |

FFT | Fast Fourier transform |

FSR | Free spectral range |

FWM | Four-wave mixing |

GVD | Group velocity dispersion |

MI | Modulation instability |

OFC | Optical frequency comb |

OPO | Optical parametric oscillator |

SH | Second harmonic |

SHG | Second harmonic generation |

TH | Third harmonic |

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**Figure 1.**Singly resonant cavity second harmonic generation (SHG). (

**a**) Experimental setup: periodically poled lithium niobate crystal (PPLN), piezoelectric actuator (PZT), photodiode (PD), dichroic mirror (DM). The output beams are detected and processed by radio-frequency (RF) analyzers, while optical spectral analysis is performed by an optical spectrum analyzer in the infrared range and a confocal Fabry-Pèrot interferometer (CFP) in the visible range. (

**b**) Schematic representation of the first steps leading to the formation of a dual optical frequency comb in cavity-enhanced second-harmonic generation: (left) second-harmonic generation with cascaded nondegenerate optical parametric oscillator (OPO) gives rise to two subharmonic sidebands, which in turn (right) lead to successive, multiple second-harmonic and sum-frequency generations. Adapted with permission from [48]. Copyrighted by the American Physical Society.

**Figure 2.**Optical spectral power around the fundamental mode for (

**a**) 170 mW, (

**b**) 2 W, and (

**c**) 9 W of input powers. Adapted with permission from [48]. Copyrighted by the American Physical Society.

**Figure 3.**Optical spectra for phase-mismatched singly resonant cavity SHG. (

**a**–

**d**) Positive phase mismatch; (

**e**–

**h**) negative phase mismatch. Intermodal beat notes corresponding to (

**i**) comb spectrum of Figure 2c, (

**j**) comb spectrum in panel (

**g**).

**Figure 6.**Optical frequency combs (OFC) in a degenerate OPO. (

**a**,

**b**) show the MI gain as a function of the normalized cavity detuning $\Delta ={\delta}_{1}/{\alpha}_{1}$, for the constant solution and the zero solution, respectively. (

**c**) MI gain profiles as a function of the temporal walk-off. Adapted with permission from [54]. Copyrighted by the American Physical Society.

**Figure 7.**OFC in a degenerate OPO. Scheme of the experimental setup: beam splitter (BS), electro-optic phase modulator (EOM), periodically poled lithium niobate crystal (PPLN), piezoelectric actuator (PZT), photodiode (PD). Adapted with permission from [54]. Copyrighted by the American Physical Society.

**Figure 8.**(

**a**–

**c**) Experimental OPO optical spectra for detunings $\Delta =-0.30,0.00,0.30$, respectively. (

**d**–

**f**) Corresponding numerically calculated spectra. From [54]. Copyrighted by the American Physical Society.

**Figure 9.**Numerical simulation of the single-envelope map when SHG and OPO processes are simultaneously phase matched in a lithium niobate microresonator pumped by 100 mW of cw power at 1850 nm. Reprinted with permission from [55] © The Optical Society.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ricciardi, I.; Mosca, S.; Parisi, M.; Leo, F.; Hansson, T.; Erkintalo, M.; Maddaloni, P.; De Natale, P.; Wabnitz, S.; De Rosa, M.
Optical Frequency Combs in Quadratically Nonlinear Resonators. *Micromachines* **2020**, *11*, 230.
https://doi.org/10.3390/mi11020230

**AMA Style**

Ricciardi I, Mosca S, Parisi M, Leo F, Hansson T, Erkintalo M, Maddaloni P, De Natale P, Wabnitz S, De Rosa M.
Optical Frequency Combs in Quadratically Nonlinear Resonators. *Micromachines*. 2020; 11(2):230.
https://doi.org/10.3390/mi11020230

**Chicago/Turabian Style**

Ricciardi, Iolanda, Simona Mosca, Maria Parisi, François Leo, Tobias Hansson, Miro Erkintalo, Pasquale Maddaloni, Paolo De Natale, Stefan Wabnitz, and Maurizio De Rosa.
2020. "Optical Frequency Combs in Quadratically Nonlinear Resonators" *Micromachines* 11, no. 2: 230.
https://doi.org/10.3390/mi11020230