## Optical frequency comb generation through quasi-phase matched quadratic frequency conversion in a micro-ring resonator |

Optics Express, Vol. 20, Issue 15, pp. 17192-17200 (2012)

http://dx.doi.org/10.1364/OE.20.017192

Acrobat PDF (1175 KB)

### Abstract

We propose optical frequency comb generation in a monolithic micro-ring resonator. Being different from the previously reported nonlinear optical frequency combs, our scheme is based on more efficient quadratic frequency conversion rather than the third-order nonlinearity. To overcome the phase mismatch, a partly poled nonlinear ring is employed. Cascading second harmonic generation and parametric down conversion processes thus are realized through quasi-phase matching (QPM). Coupling equations are used to describe the related nonlinear interactions among different whispering-gallery modes, showing some interesting characteristics that are different from conventional QPM technology.

© 2012 OSA

## 1. Introduction

1. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science **332**(6029), 555–559 (2011). [CrossRef] [PubMed]

3. I. H. Agha, Y. Okawachi, and A. L. Gaeta, “Theoretical and experimental investigation of broadband cascaded four-wave mixing in high-Q microspheres,” Opt. Express **17**(18), 16209–16215 (2009). [CrossRef] [PubMed]

4. 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]

*e.g.*, the four-wave mixing (FWM), two pump photons (

*ω*) generate a pair of signal (

_{p}*ω*) and idler (

_{s}*ω*) photons at the whispering gallery modes (WGMs) [5

_{i}5. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature **450**(7173), 1214–1217 (2007). [CrossRef] [PubMed]

*ω*) generate a second harmonic photon (

_{p}*ω*) then it splits into signal and idler photons cascadingly. However, the involving of

_{sh}*ω*light makes the required phase matching almost impossible due to the material and geometrical dispersions. The generation and splitting of second harmonic photons thus have negligible efficiencies then kill the expected frequency comb generation.

_{sh}6. 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]

7. P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. **35**(1), 23–39 (1963). [CrossRef]

8. Y. L. Lee, B. A. Yu, T. J. Eom, W. Shin, C. Jung, Y. C. Noh, J. Lee, D. K. Ko, and K. Oh, “All-optical AND and NAND gates based on cascaded second-order nonlinear processes in a Ti-diffused periodically poled LiNbO_{3} waveguide,” Opt. Express **14**(7), 2776–2782 (2006). [CrossRef] [PubMed]

9. X. S. Song, Z. Y. Yu, Q. Wang, F. Xu, and Y. Q. Lu, “Polarization independent quasi-phase-matched sum frequency generation for single photon detection,” Opt. Express **19**(1), 380–386 (2011). [CrossRef] [PubMed]

10. Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B **82**(15), 155107 (2010). [CrossRef]

11. A. Rose and D. R. Smith, “Overcoming phase mismatch in nonlinear metamaterials,” Opt. Mater. Express **1**(7), 1232–1243 (2011). [CrossRef]

12. Y. Q. Lu, Y. L. Lu, C. C. Xue, J. J. Zheng, X. F. Chen, G. P. Luo, N. B. Ming, B. H. Feng, and X. L. Zhang, “Femtosecond violet light generation by quasi-phase-matched frequency doubling in optical superlattice LiNbO_{3},” Appl. Phys. Lett. **69**(21), 3155–3157 (1996). [CrossRef]

13. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. **104**(15), 153901 (2010). [CrossRef] [PubMed]

13. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. **104**(15), 153901 (2010). [CrossRef] [PubMed]

## 2. Model and theory

*τ*. As an approximation, the build-up factor is 4/

*τ*for such an “add-drop” system [17]. We treat the nonlinear effects and the coupling between the waveguide and the ring resonator as perturbations. Therefore the ring’s WGMs will not change with the light intensity. To solve the Maxwell equations, the electric field of TE modes (the electric field is parallel to the z axis), are written as,where

^{2}*a*represents the amplitude of WGM

_{n}*n*circulating in the ring;

*f*corresponds to the field distribution of a WGM, which also can be written as

_{n}*φ*is the angle in cylindrical coordinate; corresponds to field distribution along radial direction so it is a function of the radius

*r*. Because the photons of WGMs are mainly constrained in a ring, which is different from the straight waveguides, their momentums would be zero. However, the circular movement of each photon would cause a nonzero angular momentum which is indicated with the mode index

*n*.

18. A. A. Savchenkov, A. B. Matsko, M. Mohageg, D. V. Strekalov, and L. Maleki, “Parametric oscillations in a whispering gallery resonator,” Opt. Lett. **32**(2), 157–159 (2007). [CrossRef] [PubMed]

*ω*and

_{m}*ω*to mode

_{n}*ω*.and, for the reversed process,

_{p}*i.e.*, difference frequency generation (DFG) or OPG, the equation would be:and, if the pump wavelength matches a WGM, the power of pump light in coupling waveguide would be coupled into the ring, so intensity of the corresponding WGM follows:where

*γ*is the WGM decay rate that is dominated by the loss. In our configuration, both the ring-to-bus coupling loss and the round-trip bending loss may be taken into account. Smaller ring gives rise to higher bending loss. When the radius is around 12 μm and τ is 0.01, the major loss would be the coupling loss that is 10

_{n}^{9}times higher than the bending one. In this case, the decay rate is given by

*ωτ*, meaning only the coupling loss is considered.

^{2}/2n*P*corresponds to the pump power in the input bus waveguide;

*ε*represents the vacuum permittivity.

_{0}*κ*is the coupling coefficient between three WGMs with frequencies of

_{pnm}*ω*and

_{m}, ω_{n}*ω*

_{p}_{,}which could be expressed as:

*κ*are the dominating factors. Typically, if the coupling coefficients are equal to zero, the couplings among corresponding waves would also be zero and the nonlinear effect disappears. To realize a remarkable nonlinear effect, the coupling coefficient should be maximized. The larger coupling coefficients, the lower pump light power is required.

_{pnm}13. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. **104**(15), 153901 (2010). [CrossRef] [PubMed]

*χ*would be a constant in the whole ring. The angle part of integration in Eq. (6) is

^{(2)}4. 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]

*h(φ)*is defined as:

*θ*is the open angle of the reversely poled domain. In the cases that Δ

*= odd*, to maximize the overlap integration and the nonlinear coupling coefficient

*κ*, the ring should be half poled, which means

_{pnm}*θ = π*. The angular part of overlap integration is obtained as

*= 2*, to maximize the overlap integration,

^{n}× odd*θ = π/2*should be satisfied. In this formulation, odd means an arbitrary odd number, then the integration gives

^{n}10. Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B **82**(15), 155107 (2010). [CrossRef]

11. A. Rose and D. R. Smith, “Overcoming phase mismatch in nonlinear metamaterials,” Opt. Mater. Express **1**(7), 1232–1243 (2011). [CrossRef]

## 3. Simulation results and discussions

*θ*= π, the crystal is equally divided into two domains. To simulate the SHG in the ring, we assume the coupling coefficient between the input waveguide and the ring resonator is the same for both FW and SHW.

19. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n(e), in congruent lithium niobate,” Opt. Lett. **22**(20), 1553–1555 (1997). [CrossRef] [PubMed]

^{3}~10

^{4}, which are quite achievable. On the other hand, the ratio of internal efficiency to external efficiency is

*τ*

^{2}/2. This is because the proportion of power coupled into the ring to the input pump power is

*τ*

^{2}/2. Due to the nature of nonlinear effect, the SHG efficiencies all rise with the increase of pump power. They gradually saturate when the internal efficiency approaches 1. And, in this case, the external efficiency is close to

*τ*

^{2}/2 as shown with the black solid curve in Fig. 2(b). For

*τ*= 0.1,

*τ*

^{2}/2 gives the highest limit of 5% for external efficiency.

*τ*, the internal SHG efficiency simply reduces when

*τ*increases. But for the external SHG efficiency, things are different. Although the internal efficiency still decreases, the internal to external efficiency ratio

*τ*

^{2}/2 increases at the same time. That makes the external efficiency reach a maximum at a given

*τ*value, as shown in Fig. 2(c). However, the efficiency peaks emerge at different

*τ*values for different pump powers. This feature should be considered when design such a nonlinear ring system.

^{2}/2. In addition, the efficiency shows complicate relationship with the coupling coefficient τ. If the pump power is high enough, bigger coupling coefficient gives rise to larger external SHG efficiency, while in lower pump power cases, the internal SHG efficiency would not reaches a maximum, the external SHG efficiency will not simply rise with the increase of τ. The QPM in a nonlinear ring really has many interesting properties that worth studying.

1. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science **332**(6029), 555–559 (2011). [CrossRef] [PubMed]

4. 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]

*ω*and generate a pair of modes at

_{p}*ω*and

_{m}*ω*with 2

_{n}*ω*. The SHG process could be described as the system absorbs two photons with the frequency

_{p}= ω_{sh}= ω_{m}+ ω_{n}*ω*and generate the photon at second harmonic mode with frequency

_{p}*ω*= 2

_{sh}*ω*. Then the OPG process could be described as the system absorbs the photon at the SH mode and generates a photon pair with different frequencies

_{p}*ω*and

_{m}*ω*. Due to the dispersion of the ring, this condition could not be severely satisfied. However, if the line shapes of the ring resonances are considered, then the generated SHW and frequency down-converted signal and idler waves can be made to fit within a specific mode-spacing. Furthermore, if there is a frequency spacing mismatch between the WGMs and the signals generated via SHG and spontaneous down-conversion, there should be a finite bandwidth over which the generated comb can fit in the ring’s resonances. So as an approximation, the modes that satisfy the condition Δ

_{n}*ω*>

_{sh}*ω*are selected. Δ

_{sh}– ω_{m}– ω_{n}*ω*represents the full width of the maximum of the SH mode. It depends on the Q-factor and the coupling condition of our system. We set τ at 0.01 and Δ

_{sh}*ω*= 0.1 THz for the simulation. In order to include more comb modes, the wavelength of pump changes to 1556.7 nm, thus a SHW at 778.35 nm is obtained. In this case, there would be 22 WGMs included in the calculation, including the pump mode, the SH mode and 20 modes with the resonating wavelength around the pump mode. The mode number of these modes are p ± 1, 2, 3, ..., 10. Here ‘p’ indicates the mode number of pump light that is equal to 111. The mode number of SHW WGM is 234. As a consequence, Δ = 12 and the open angle of the reversely poled domain is set at

_{sh}*π/4*. If error occurs for the open angle, the nonlinear conversion efficiency may be affected. The tolerance could be obtained through calculating the angular part integration of Eq. (6). With a 5° offset, the coupling coefficient κ

_{pnm}would only be reduced by 1.5%. The pump power is set at 10 MW/cm.

2. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics **4**(1), 37–40 (2010). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science |

2. | J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics |

3. | I. H. Agha, Y. Okawachi, and A. L. Gaeta, “Theoretical and experimental investigation of broadband cascaded four-wave mixing in high-Q microspheres,” Opt. Express |

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

5. | P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature |

6. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

7. | P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. |

8. | Y. L. Lee, B. A. Yu, T. J. Eom, W. Shin, C. Jung, Y. C. Noh, J. Lee, D. K. Ko, and K. Oh, “All-optical AND and NAND gates based on cascaded second-order nonlinear processes in a Ti-diffused periodically poled LiNbO |

9. | X. S. Song, Z. Y. Yu, Q. Wang, F. Xu, and Y. Q. Lu, “Polarization independent quasi-phase-matched sum frequency generation for single photon detection,” Opt. Express |

10. | Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B |

11. | A. Rose and D. R. Smith, “Overcoming phase mismatch in nonlinear metamaterials,” Opt. Mater. Express |

12. | Y. Q. Lu, Y. L. Lu, C. C. Xue, J. J. Zheng, X. F. Chen, G. P. Luo, N. B. Ming, B. H. Feng, and X. L. Zhang, “Femtosecond violet light generation by quasi-phase-matched frequency doubling in optical superlattice LiNbO |

13. | J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. |

14. | V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B |

15. | V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. |

16. | Kiyotaka Sasagawa, and Masahiro Tsuchiya, “High efficiency third harmonic generation in PPMgLN disk resonator,” in CLEO 2007 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), Paper CThC6. |

17. | J. Heebner, R. Grover, and T. A. Ibrahim, |

18. | A. A. Savchenkov, A. B. Matsko, M. Mohageg, D. V. Strekalov, and L. Maleki, “Parametric oscillations in a whispering gallery resonator,” Opt. Lett. |

19. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n(e), in congruent lithium niobate,” Opt. Lett. |

**OCIS Codes**

(160.3730) Materials : Lithium niobate

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4975) Nonlinear optics : Parametric processes

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 29, 2012

Revised Manuscript: July 5, 2012

Manuscript Accepted: July 5, 2012

Published: July 12, 2012

**Citation**

Zi-jian Wu, Yang Ming, Fei Xu, and Yan-qing Lu, "Optical frequency comb generation through quasi-phase matched quadratic frequency conversion in a micro-ring resonator," Opt. Express **20**, 17192-17200 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17192

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### References

- T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332(6029), 555–559 (2011). [CrossRef] [PubMed]
- J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010). [CrossRef]
- I. H. Agha, Y. Okawachi, and A. L. Gaeta, “Theoretical and experimental investigation of broadband cascaded four-wave mixing in high-Q microspheres,” Opt. Express17(18), 16209–16215 (2009). [CrossRef] [PubMed]
- S. N. Zhu, Y. Y. Zhu, and N.-B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science278(5339), 843–846 (1997). [CrossRef]
- P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450(7173), 1214–1217 (2007). [CrossRef] [PubMed]
- 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]
- P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys.35(1), 23–39 (1963). [CrossRef]
- Y. L. Lee, B. A. Yu, T. J. Eom, W. Shin, C. Jung, Y. C. Noh, J. Lee, D. K. Ko, and K. Oh, “All-optical AND and NAND gates based on cascaded second-order nonlinear processes in a Ti-diffused periodically poled LiNbO3 waveguide,” Opt. Express14(7), 2776–2782 (2006). [CrossRef] [PubMed]
- X. S. Song, Z. Y. Yu, Q. Wang, F. Xu, and Y. Q. Lu, “Polarization independent quasi-phase-matched sum frequency generation for single photon detection,” Opt. Express19(1), 380–386 (2011). [CrossRef] [PubMed]
- Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B82(15), 155107 (2010). [CrossRef]
- A. Rose and D. R. Smith, “Overcoming phase mismatch in nonlinear metamaterials,” Opt. Mater. Express1(7), 1232–1243 (2011). [CrossRef]
- Y. Q. Lu, Y. L. Lu, C. C. Xue, J. J. Zheng, X. F. Chen, G. P. Luo, N. B. Ming, B. H. Feng, and X. L. Zhang, “Femtosecond violet light generation by quasi-phase-matched frequency doubling in optical superlattice LiNbO3,” Appl. Phys. Lett.69(21), 3155–3157 (1996). [CrossRef]
- J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett.104(15), 153901 (2010). [CrossRef] [PubMed]
- V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B20(6), 1304–1308 (2003). [CrossRef]
- V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett.92(4), 043903 (2004). [CrossRef] [PubMed]
- Kiyotaka Sasagawa, and Masahiro Tsuchiya, “High efficiency third harmonic generation in PPMgLN disk resonator,” in CLEO 2007 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), Paper CThC6.
- J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators (Springer, 2008), Chap. 3.
- A. A. Savchenkov, A. B. Matsko, M. Mohageg, D. V. Strekalov, and L. Maleki, “Parametric oscillations in a whispering gallery resonator,” Opt. Lett.32(2), 157–159 (2007). [CrossRef] [PubMed]
- D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n(e), in congruent lithium niobate,” Opt. Lett.22(20), 1553–1555 (1997). [CrossRef] [PubMed]

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