## High-frequency self-induced oscillations in a silicon nanocavity |

Optics Express, Vol. 21, Issue 11, pp. 13626-13638 (2013)

http://dx.doi.org/10.1364/OE.21.013626

Acrobat PDF (1441 KB)

### Abstract

We show that self-induced oscillations at frequencies above GHz and with a high spectral purity can be obtained in a silicon photonic crystal nanocavity under optical pumping. This self-pulsing results from the interplay between the nonlinear response of the cavity and the photon cavity lifetime. We provide a model to analyze the mechanisms governing the onset of self-pulsing, the amplitudes of both fundamental and harmonic oscillations and their dependences versus input power and oscillation frequency. Theoretically, oscillations at frequencies higher than 50 GHz could be achieved in this system.

© 2013 osa

*λ/n*)

^{3}and their high quality factor Q [1

1. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003) [CrossRef] .

3. Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express **17**, 18093–18102 (2009) [CrossRef] [PubMed] .

4. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. **3**, 211–219 (2004) [CrossRef] .

5. P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express **13**, 801–820 (2005) [CrossRef] [PubMed] .

9. J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express **18**, 15484–15497 (2010) [CrossRef] [PubMed] .

10. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics **1**, 319–330 (2007) [CrossRef] .

11. S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A **83**, 051802 (2011) [CrossRef] .

12. K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. **48**, 617–620 (1982) [CrossRef] .

11. S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A **83**, 051802 (2011) [CrossRef] .

13. S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express **20**, 7454–7468 (2012) [CrossRef] [PubMed] .

14. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express **14**, 817–831 (2006) [CrossRef] [PubMed] .

15. M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A **85**, 031803– (2012) [CrossRef] .

16. M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A **85**, 053819 (2012) [CrossRef] .

17. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006) [CrossRef] .

18. Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 10^{6} ultra-high Q silicon photonic crystal cavities,” Opt. Commun. **283**, 4387–4391 (2010) [CrossRef] .

19. Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett. **36**, 1749–1751 (2011) [CrossRef] [PubMed] .

*Q*is around 130000 and the transmission at the resonance is

*T*

_{max}=41% between the entrance and exit of the photonic crystal. Figure 1(b) shows the calculated output power as a function of the input power for different detunings between the laser and the cavity resonance. We can clearly observe the cavity nonlinear behavior as the power is increased. Bistability is obtained when the pump has a negative detuning as compared to the resonance wavelength, i.e. when the laser wavelength is shorter than the cavity resonance wavelength. As explained in [11

11. S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A **83**, 051802 (2011) [CrossRef] .

**83**, 051802 (2011) [CrossRef] .

20. T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express **14**, 377–386 (2006) [CrossRef] [PubMed] .

21. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express **15**, 16604–16644 (2007) [CrossRef] [PubMed] .

*A*is the complex amplitude of the electrical field of the confined mode in the cavity (with

*E*= |

*A*|

^{2}the mode energy) and

*N*is the number of free-carriers. The first term on the right-hand side of Eq. (1), −

*A*/(2

*τ*) represents the damping of the electrical field due to intrinsic linear losses,

*τ*=

*Q/ω*

_{0}, being the photon lifetime in the cavity. This field decay is compensated by the continuous injection of light into the cavity that is taken into account by the last term

*T*the transmission maximum at low input power. The interplay between the second term,

_{max}*i*Δ

*ωA*, and the third term,

*ω*=

*ω*−

*ω*

_{0}represents the detuning between the input laser frequency

*ω*/(2

*π*) and the cavity resonance frequency

*ω*

_{0}/(2

*π*).

*γ*is a complex number, represents the effect of free-carrier induced absorption and dispersion, and

^{FCA}*V*is the cavity effective volume over which the photo-generated free-carriers spread (see appendix 1 Free-carrier dispersion, absorption and generation). Here, we calculated

_{eff}*γ*= 5.83 × 10

^{FCA}^{−14}+ 1.7 × 10

^{−12}

*i*m

^{3}s

^{−1}using three-dimensional finite difference in time domain (3D-FDTD) simulation.

*τ*being the effective decay rate of free carriers. From 3D-FDTD, we deduced

_{fc}*γ*= 1.63 × 10

^{TPA}^{4}s

^{−1}for the PhC cavity.

*P*= 2 mW), the linear absorption that exists in structured silicon [22

22. T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett. **96**, 101103 (2010) [CrossRef] .

23. L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express **18**, 23965–23972 (2010) [CrossRef] [PubMed] .

24. A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A **84**, 053816 (2011) [CrossRef] .

**83**, 051802 (2011) [CrossRef] .

13. S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express **20**, 7454–7468 (2012) [CrossRef] [PubMed] .

16. M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A **85**, 053819 (2012) [CrossRef] .

*E*= |

*A*|

^{2}that appears as a root of a five-degree polynomial. As mentioned before, self-pulsing only happens for a high value of

*E*, i.e. on the upper energy-branch and beyond the bistability zone. The eigenvalues of the matrix associated to this linearized system around the steady-state solutions characterize the cavity dynamics. Self-induced oscillations occur when the system is characterized by a Hopf bifurcation which happens when a pair of conjugate eigenvalues

*α*±

_{r}*i*Ω goes from the left half complex plane to the right one as a parameter of the system is varied, i.e. when the real part of these eigenvalues becomes positive. The oscillations are non-attenuated for

*α*> 0, corresponding to an equivalent gain and their period is 2

_{r}*π/*Ω.

*μm*

^{3}. Figure 2(b) shows the dependence of the oscillation period for a free-carrier lifetime of 0.2 ns and an effective volume

*V*= 5.25

_{eff}*μm*

^{3}and indicates that, in this case, self-sustained oscillations can only be obtained for a quality factor higher than 110000. In the latter case, when we plot the cavity energy vs. the number of free carriers, the energy follows a stable trajectory around an equilibrium point which is characteristic of periodic oscillations (not shown).

*τ*=75 ps given by the Q factor at low power. This effective lifetime increase is a consequence of free-carrier induced nonlinearities that, in these experimental conditions, are not yet sufficient to generate self-sustained oscillations. Similar photon lifetime increases have been observed in active materials [26

26. P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett. **109**, 113903 (2012) [CrossRef] [PubMed] .

16. M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A **85**, 053819 (2012) [CrossRef] .

*V*≃ 5.25

_{eff}*μm*

^{3}could generate non-damped self-sustained oscillations with periods around 0.2–0.3 ns (see Fig. 2(a)). This value is in close agreement with a simple estimate that takes into account the diffusion length of the free-carriers in the slab (see appendix 1). A few values of the free-carrier lifetime were then tested to get the closest fits to the oscillation frequency. It was obtained for a free-carrier lifetime

*τ*= 0.3 ns before the nitric acid surface treatment [Fig. 3(c)], and for a free-carrier lifetime

_{fc}*τ*= 0.2 ns after the nitric acid surface treatment [Fig. 3(d)]. Thermal dispersion is included in the equations (see appendix 4 thermal effects), to reproduce the increase of the output power with time as observed in the measurements but has no impact on the oscillation frequency.

_{fc}23. L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express **18**, 23965–23972 (2010) [CrossRef] [PubMed] .

27. S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London) **437**, 389–392 (2005) [CrossRef] .

28. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express **18**, 3582–3591 (2010) [CrossRef] [PubMed] .

**83**, 051802 (2011) [CrossRef] .

*P*

_{out}= |

*A*|

^{2}/

*τ*

_{in}, is 26 dB below the 2.8 GHz signal [29], as obtained after a Fourier transform of the measured and simulated signal. To explain this behavior, we solved Eqs. (1) and (2) in the harmonic regime (see appendix 5 solving the equations - harmonic analysis) where a solution for the stored energy in the cavity is approximated by

*E*≃

*E*

_{0}+ 2

*E*

_{1}cos(Ω

*t*+

*φ*

_{1}) + 2

*E*

_{2}cos(2Ω

*t*+

*φ*

_{2})). An excellent agreement is obtained between the analytical and simulated spectra [Fig. 4(b)] that confirms the validity of the method. From the analytical expressions of the energy in the cavity, we get the ratio of the second harmonic energy to the first one as:

*E*

_{2}/

*E*

_{1}=

*C*

_{0}×

*E*

_{1}/

*E*

_{0}, where

*C*

_{0}is a function of the frequency Ω, the average energy

*E*

_{0}and the average number of free carriers. For oscillations triggered by an optical input power higher than 2 mW, the expressions of Ω, and

*C*

_{0}, reduce to: and where

*C*

_{0}decreases as the frequency increases. This behavior is confirmed in Fig. 4(c) and (d) that respectively represent

*E*

_{1}/

*E*

_{0}and

*E*

_{2}/

*E*

_{1}calculated with the complete analytical expression as a function of the input power and the detuning. Moreover, the amplitude of the harmonics is significantly smaller than the one observed in the microdisk resonator in [16

**85**, 053819 (2012) [CrossRef] .

*E*

_{2}/

*E*

_{1}= −14.6 dB and

*E*

_{1}/

*E*

_{0}= −6 dB, because the ratio

*E*

_{1}/

*E*

_{0}and

*C*

_{0}are higher in the microdisk, this last fact being explained by the higher frequency of the oscillations in the PhC microcavity. This higher oscillation frequency is a direct consequence of the smaller effective volume of the PhC cavity and of the lower quality factor which causes the system to react faster as seen in Eq. (3).

## 1. Free-carrier dispersion, absorption and generation

*γ*can be written as:

^{FCA}*R*represents the fraction of the optical mode in the silicon part of the cavity calculated by a three-dimensional finite difference in time domain (FDTD) simulation (

_{eff}*R*= 1.1).

_{eff}*σ*= 1.45 × 10

_{r}^{−21}

*m*

^{2}and

*σ*= −5.3 × 10

_{i}^{−27}

*m*

^{3}are respectively the free-carrier absorption and dispersion in silicon given at

*λ*= 2

_{r}*πc/ω*= 1550 nm [21

_{r}21. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express **15**, 16604–16644 (2007) [CrossRef] [PubMed] .

*γ*, can be expressed as [20

^{TPA}20. T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express **14**, 377–386 (2006) [CrossRef] [PubMed] .

*β*= 8.4 × 10

^{−12}m/W [30

30. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. **82**, 2954–2956 (2003) [CrossRef] .

*n*= 3.48 is the silicon refractive index [31], and

*V*= 4.8341 × 10

_{TPA}^{−19}

*m*

^{3}is the TPA volume calculated by three-dimensional finite-difference in time domain modeling (FDTD).

## 2. Solving the equations: steady state and small perturbations

*A*= |

*A*|

*e*and obtained two real equations from the first complex equation, which gave us a system of three real equations:

^{iφ}**83**, 051802 (2011) [CrossRef] .

*A*

_{0}|,

*φ*

_{0},

*N*

_{0}) to these equations, we get: and |

*A*

_{0}|

^{2}is one of the roots of the five-degree polynomial: Linearizing these equations for small perturbations around these steady-state solutions gives:

*A*| only and determine the linearized system behaviour. Self-induced oscillations are possible when the matrix

*M*is characterized by a Hopf bifurcation, which happens when

*M*has a pair of complex conjugate eigenvalues

*α*±

_{r}*i*Ω which crosses the imaginary axis into the right half complex plane, i.e. the real part of the eigenvalue becomes positive as a parameter of the system is varied. These oscillations are non-attenuated for

*α*> 0, and their period is

_{r}*T*= 2

*π/*Ω.

*P*

_{in}> 2 mW in the case of the PhC cavity, from Eq. (11), we have

*ω*cannot be neglected since it has still a significant influence on the energy inside the cavity as can be seen on Fig. 1(b). One then gets the approximate expression: . Neglecting small terms in the matrix M, we have

## 3. Estimating the cavity effective volume

*V*≃ 5.25

_{eff}*μm*

^{3}) is ten times larger than the cavity volume used in Ref. [20

20. T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express **14**, 377–386 (2006) [CrossRef] [PubMed] .

*d*≃ 1.1

*μm*the diffusion length of the free-carriers,

*h*the slab thickness, and

*a*being the photonic crystal period). Using this formula would give

*V*= 0.32

_{eff}*μm*

^{3}for the investigated cavity. But this model assumes that the free-carrier diffusion stops at the air holes, which is not the case. According to Ref. [32

32. T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech. **26**, 1396–1403 (2008) [CrossRef] .

*V*= 4.06

_{eff}*μm*

^{3}, which is close to the value of

*V*that was deduced from the experimental measurements.

_{eff}## 4. Thermal effects

14. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express **14**, 817–831 (2006) [CrossRef] [PubMed] .

*T*is the temperature difference in the cavity generated by the input power,

*ρ*= 2.33 g/cm

_{Si}^{3}is the silicon density,

14. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express **14**, 817–831 (2006) [CrossRef] [PubMed] .

*R*and the effective cavity volume that depends on the temperature

_{T}*R*≈ 8 × 10

_{T}^{3}K/W and

## 5. Solving the equations: harmonic analysis

*E*(

*t*) = |

*A*(

*t*)|

^{2}of the energy in the cavity can be obtained when the harmonic regime is reached for a constant input power. The sketch of the method is the following. First, in Eq. (2), we neglected higher order harmonics of |

*A*|

^{2}, the stored energy in the cavity, and replaced it by the approximation

*N*, the number of free carriers. This expression for

*N*and the approximate value of |

*A*|

^{2}are used to transform Eq. (1) in a first order, linear, ordinary differential equation of the form:

*R*is a constant and

*F*(

*t*) is a sum of two sinusoids. Solving this equation and taking the square modulus of its solution, we get for the energy,

*E*≃

*E*

_{0}+ 2

*E*

_{1}cos(Ω

*t*+

*φ*

_{1})+ 2

*E*

_{2}cos(2Ω

*t*+

*φ*

_{2})) where

*E*

_{0},

*E*

_{1},

*E*

_{2}depend on

*E*must be equal to the first approximation we initially used in Eqs. (1) and (2), i.e.

*E*

_{2}negligible. Finally, solving these last two equations gives expressions for

*E*

_{0},

*E*

_{1}and

*E*

_{2}.

*N: N*(

*t*) =

*N*

_{0}+ 2

*N*

_{1}cos(Ω

*t*+

*φ*

_{N}_{1}), with

*φ*

_{N}_{1}a phase term. Replacing this expression of

*N*in the Eq. (1), we get the first order, linear, ordinary differential equation : with

*t*≫ 1/

*R*, the harmonic regime is reached and the solution of this equation is : with

*α*= 0.016 and

^{TPA}*α*= 0.004 + 0.123

^{FCA}*i*and the main contribution to the integral

*α*. To further simplify the calculations, we, first, only consider this contribution in the evaluation of

^{FCA}*R*and of the integral

*θ*= Ω

*t*+

*φ*

_{N}_{1}.

*e*

^{∫F}^{(}

^{t}^{)}as : where the

*I*are the modified Bessel functions of the first kind. In this expansion, only the terms for

_{n}*n*= −1, 0 and 1 are conserved as it can be checked that the influence of other terms on the final values of the fundamental and the second harmonic is negligible as compared to the influence of

*E*must be equal the one initially introduced in Eq. (19), i.e.

*E*

_{2}negligible. We then get a set of two transcendental equations, the first one in the unknown

*E*

_{2}. A simple expression can be obtained for the ratio of the second harmonic to the first one

*E*

_{2}/

*E*

_{1}:

*C*

_{0}depends only on

*N*

_{0}and Ω, which are easily calculated from the steady-state solutions and the eigenvalues of the linearized system (see above: solving the equations: steady state and small perturbations).

*R*reduce to

*R*∼

*i*Ω, and the expression of

*C*

_{0}simplify to

*C*

_{0}will be lower at high frequencies, i.e. the signal will be more sinusoidal.

*E*

_{0},

*E*

_{1},

*E*

_{2}and

*C*

_{0}then become more complicated but they still simplify for an input optical power higher than 2 mW and we get for

*C*

_{0}:

*C*

_{0}varies between 0.27 and 0.084 for an input power varying from 1.5 mW to 20 mW. For

*P*= 2 mW,

_{in}*C*

_{0}= 0.24 and

*E*

_{1}/

*E*

_{0}= −21.4 dB, which gives us

*E*

_{2}/

*E*

_{1}≃ −33.6 dB, which is very close to the values found by the numerical simulation (see Fig. 4b) :

*E*

_{1}/

*E*

_{0}= −21.3 dB and

*E*

_{2}/

*E*

_{1}= −34.7 dB. The amplitude of the second harmonic is very low compared to the fundamental, which justifies the approximation of the energy function by a sinusoid we made at the beginning.

**85**, 053819 (2012) [CrossRef] .

*α*= 0.042 and

^{TPA}*α*= 0.04 + 1.21

^{FCA}*i*for a detuning of −2.5 pm and an input power of 1 mW. Since |

*α*| > 1, we can no longer write

^{FCA}*P*= 1 mW, similar to the experimental parameters used in Ref. [16

_{in}**85**, 053819 (2012) [CrossRef] .

*C*

_{0}= 0.37,

*E*

_{1}/

*E*

_{0}= −6 dB and

*E*

_{2}/

*E*

_{1}≃ −14.6 dB according to the formula (37), a value which is close to the value given by a full numerical simulation (− 15 dB). Therefore, we can conclude that the greater nonlinearity of the oscillations in a microring resonator is caused firstly by the higher value of

*C*

_{0}, itself explained by the lower value of the frequency of the oscillations in a microdisk, and secondly by the higher amplitude of those oscillations.

## Acknowledgments

## References and links

1. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) |

2. | B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. |

3. | Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express |

4. | M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

5. | P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express |

6. | C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express |

7. | T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. |

8. | X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B |

9. | J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express |

10. | J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics |

11. | S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A |

12. | K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. |

13. | S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express |

14. | T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express |

15. | M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A |

16. | M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A |

17. | E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. |

18. | Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 10 |

19. | Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett. |

20. | T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express |

21. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express |

22. | T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett. |

23. | L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express |

24. | A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A |

25. | T. J. Johnson and O. Painter, “Passive modification of free carrier lifetime in high-Q silicon-on-insulator optics,” 2009 Conference On Lasers and Electro-optics and Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), 1–5, 72–73 (2009). |

26. | P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett. |

27. | S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London) |

28. | A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express |

29. | If we consider that the transmitted signal is detected by a photodetector, the RF power is proportional to the square of the electric intensity generated by the photodetector, i.e. to the square of the optical intensity. |

30. | M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. |

31. | H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data |

32. | T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech. |

**OCIS Codes**

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(230.4910) Optical devices : Oscillators

(140.3948) Lasers and laser optics : Microcavity devices

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 8, 2013

Revised Manuscript: May 17, 2013

Manuscript Accepted: May 21, 2013

Published: May 30, 2013

**Citation**

Nicolas Cazier, Xavier Checoury, Laurent-Daniel Haret, and Philippe Boucaud, "High-frequency self-induced oscillations in a silicon nanocavity," Opt. Express **21**, 13626-13638 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13626

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### References

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