## Theoretical and experimental investigation of broadband cascaded four-wave mixing in high-*Q* microspheres

Optics Express, Vol. 17, Issue 18, pp. 16209-16215 (2009)

http://dx.doi.org/10.1364/OE.17.016209

Acrobat PDF (238 KB)

### Abstract

We analyze the process of cascaded four-wave mixing in a high-*Q* microcavity and show that under conditions of suitable cavity-mode dispersion, broadband frequency combs can be generated. We experimentally demonstrate broadband, cascaded four-wave mixing parametric oscillation in the anomalous group-velocity dispersion regime of a high-*Q* silica microsphere with an overall bandwidth greater than 200 nm.

© 2009 OSA

## 1. Introduction

1. M. Nakazawa, K. Suzuki, and H. A. Haus, “Modulational instability oscillation in nonlinear dispersive ring cavity,” Phys. Rev. A **38**(10), 5193–5196 (1988). [CrossRef] [PubMed]

*Q*) microcavities, such as the silica microsphere [2

2. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A **76**(4), 043837 (2007). [CrossRef]

3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

*χ*

^{(3)}nonlinear interactions scales with the square of the quality factor [2

2. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A **76**(4), 043837 (2007). [CrossRef]

3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

*Q*silica cavity with nearly 100% efficiency and render the cavity a fiber-in, fiber-out source, which is low-cost, compact, operates at room temperature, and does not require any special or high-power laser sources. Applications for these highly nonlinear microcavities range from low-power parametric oscillators [4

4. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF(_{2}) resonator,” Phys. Rev. Lett. **93**(24), 243905 (2004). [CrossRef]

5. 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]

6. 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]

2. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A **76**(4), 043837 (2007). [CrossRef]

3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

**76**(4), 043837 (2007). [CrossRef]

*Q*microtoroid [6

6. 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]

*Q*cavity is characterized by both narrow spectral features due to the high finesse and a large bandwidth, which makes it ideal for frequency metrology and for optical spectroscopy applications.

*k*,where

_{NL}*k*,

_{s}*k*, and

_{i}*k*are the wavevectors, for the signal, pump and idler waves, respectively,

_{p}*γ*is the nonlinear coupling coefficient, and

*P*is the power inside the medium. Near the point of zero GVD, the wavevector mismatch is small, and the process is phase matched at low pump power. However, since the expression for the gain [8

_{p}8. R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. **18**(7), 1062–1072 (1982). [CrossRef]

*k*vanishes.

_{NL}*k*-vectors on resonance, namely

*m*is the cavity mode number, and

*L*the effective cavity length. This implies that on resonance

*m*about the pump mode, that is,

*n*is the number of orders between the pump and the signal/idler modes. In the strong-pump approximation, the cavity modes are given by

*n*is the effective refractive index (including material and cavity/waveguide dispersion),

_{eff}*n*

_{2}the nonlinear index coefficient, and

*I*the intracavity pump intensity. In order for the four-photon scattering process to occur efficiently at strong pump powers, the conditions

_{p}*n*Δω is such that the signal, idler, and pump are resonant with cavity modes and the nonlinear refractive index is included in the calculation of the mode frequencies. Thus, to predict which FWM signal/idler pairs will oscillate first at a particular input pump power, we calculate which signal/idler modes will be ‘symmetrized’ in the presence of the strong pump circulating in the cavity, i.e., which set of cavity modes does the nonlinear refractive index balance the material/cavity dispersion and lead to frequency matching between the signal/idler and cavity resonances. Since

*n*

_{2}is positive in most materials, this can only occur when the pump frequency is located in the anomalous-GVD regime. In addition, since

*n*is an integer, there exists a unique value of the pump power that maximizes the gain for a specific pair of signal/idler modes. For example, for a fused-silica cavity 480 μm in length, an effective mode area of 10 μm

^{2}, and considering only material dispersion (no waveguide/cavity dispersion), a 17-mW input pump power at 1550 nm will lead to oscillation of the 5th order signal/idler modes, which are separated by 18 nm from the pump wavelength. On the other hand, at 96 mW of input power, oscillation will occur at the 21st order modes, which corresponds to a shift of 72 nm from the pump wavelength. At these high input powers, enough power builds up at the signal/idler pairs for efficient re-mixing with the pump, which leads to cascading of the FWM process [9

9. T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fiber optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. **23**(2), 818–826 (2005). [CrossRef]

## 2. Numerical modeling

*Q*cavity, the cavity resonances and the phase-matching conditions shift due to the nonlinear refractive index. Thus, the fields associated with the FWM process that are initially resonant may no longer be so as the intensity dynamics in the cavity change. Due to the number of field modes (> 100 for a 100 mW pump) being generated, predicting the steady-state frequencies is problematic, and a coupled-amplitude model is not viable [9

9. T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fiber optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. **23**(2), 818–826 (2005). [CrossRef]

*a priori*. We treat all the field modes together as a single field that is launched into a one-dimensional ring cavity - with an effective refractive index that accounts for the cavity-mode dispersion of a silica microsphere [2

**76**(4), 043837 (2007). [CrossRef]

10. I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interactions,” Appl. Phys. B **83**(2), 303–309 (2006). [CrossRef]

^{2}, which was also extracted from the exact solver [10

10. I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interactions,” Appl. Phys. B **83**(2), 303–309 (2006). [CrossRef]

11. R. A. Fisher and W. Bischel, “The role of linear dispersion in plane-wave self-phase modulation,” Appl. Phys. Lett. **23**(12), 661–663 (1973). [CrossRef]

12. A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, “Split-step-type angular plane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B **3**(6), 885–890 (1986). [CrossRef]

*E*(pump + noise) enters the cavity through a partially reflecting beam splitter, of reflectivity

_{in}*R*, transmittivity

*T*, that are chosen to satisfy the critical-coupling condition often encountered in tapered-fiber coupling, as well as to match the quality factor of our silica microspheres (2 × 10

^{7}). For the cavity of length

*L*, The intracavity field at position

*z*= 0, iteration

*t*is given by,where

*Q*= 2 × 10

^{7}, the inclusion of 20 signal/idler cascaded pairs, which corresponds to our experimental observations, requires one million points in the FFT array, which sets a severe limit on the speed of the split-step method. In order to run a simulation to steady state, less bandwidth needs to be considered to reduce the size of the FFT arrays and the time for the simulation to be completed. This is achieved with a 17-mW input pump and 90-nm overall bandwidth. Upon reaching steady state, the most important parameter to extract is the separation between the individual peaks [13

13. P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. **101**(5), 053903 (2008). [CrossRef] [PubMed]

*ω*=

_{p}*ω*+

_{s}*ω*), then the individual peaks will no longer be equidistant; the separation between the peaks will be determined solely by the cavity-mode dispersion, which is not flat. In these simulations, the frequency resolution is 13 MHz, which allows for one data point per mode (for

_{i}*Q*= 2 × 10

^{7}, the linewidth is 13 MHz).

*f*between the FWM peaks is detuned by 109 MHz from the cavity FSR due to pump cross-phase modulation, which also indicates that the dynamics are governed by cascaded FWM rather than by the first-order nearly degenerate FWM process [2

**76**(4), 043837 (2007). [CrossRef]

**93**(8), 083904 (2004). [CrossRef] [PubMed]

## 3. Experiment

14. A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B **22**(2), 459–465 (2005). [CrossRef]

15. T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express **12**(20), 4742–4750 (2004). [CrossRef] [PubMed]

*Q*= 2 × 10

^{7}) but small compared to the thermal response (≈10 μs). This allows for power buildup in the cavity mode, which leads to the desired nonlinear response while minimizing the thermal drift due to the average power. The average power can be reduced by decreasing the duty cycle of the pump or by increasing the period. The easiest way to achieve this pulsed operation is to scan the laser about the desired mode of the microsphere. We increase the scan speed to reduce the dwell time in the mode (and hence the instantaneous heating) and increase the scan range to reduce the ”average power” heating. For a silica microsphere with a

*Q*= 2 × 10

^{7}and for a frequency scan of 2.5 GHz at a 1-kHz rate (both within the range of the piezoelectric transducer in the laser cavity), the dwell time in the mode is 2 μs, which is shorter than the thermal response time and longer than the optical buildup time such that the average coupled power is negligible.

**76**(4), 043837 (2007). [CrossRef]

**76**(4), 043837 (2007). [CrossRef]

## 4. Conclusion

*Q*cavity from a basic ring-cavity model that was chosen to mimic the dispersion in a high-

*Q*silica microsphere. A full-field simulation based on the ring-cavity model and a split-step Fourier method was developed and implemented, and it confirms that the observed FWM peaks arise from a cascaded FWM process. Experimental realization of broadband cascaded FWM in a silica microsphere is demonstrated resulting in an overall bandwidth > 200 nm.

## Acknowledgments

## References and links

1. | M. Nakazawa, K. Suzuki, and H. A. Haus, “Modulational instability oscillation in nonlinear dispersive ring cavity,” Phys. Rev. A |

2. | I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A |

3. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. |

4. | A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF( |

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

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

7. | G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001). |

8. | R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. |

9. | T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fiber optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. |

10. | I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interactions,” Appl. Phys. B |

11. | R. A. Fisher and W. Bischel, “The role of linear dispersion in plane-wave self-phase modulation,” Appl. Phys. Lett. |

12. | A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, “Split-step-type angular plane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B |

13. | P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. |

14. | A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B |

15. | T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express |

**OCIS Codes**

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 6, 2009

Revised Manuscript: August 22, 2009

Manuscript Accepted: August 22, 2009

Published: August 27, 2009

**Citation**

Imad H. Agha, Yoshitomo Okawachi, and Alexander L. Gaeta, "Theoretical and experimental investigation of broadband cascaded four-wave mixing in high-Q microspheres," Opt. Express **17**, 16209-16215 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16209

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### References

- M. Nakazawa, K. Suzuki, and H. A. Haus, “Modulational instability oscillation in nonlinear dispersive ring cavity,” Phys. Rev. A 38(10), 5193–5196 (1988). [CrossRef] [PubMed]
- I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A 76(4), 043837 (2007). [CrossRef]
- T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]
- A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF(2) resonator,” Phys. Rev. Lett. 93(24), 243905 (2004). [CrossRef]
- 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]
- 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]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).
- R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]
- T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fiber optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. 23(2), 818–826 (2005). [CrossRef]
- I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interactions,” Appl. Phys. B 83(2), 303–309 (2006). [CrossRef]
- R. A. Fisher and W. Bischel, “The role of linear dispersion in plane-wave self-phase modulation,” Appl. Phys. Lett. 23(12), 661–663 (1973). [CrossRef]
- A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, “Split-step-type angular plane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3(6), 885–890 (1986). [CrossRef]
- P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008). [CrossRef] [PubMed]
- A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B 22(2), 459–465 (2005). [CrossRef]
- T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12(20), 4742–4750 (2004). [CrossRef] [PubMed]

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