## Optimizing pump-probe switching ruled by free-carrier dispersion |

Optics Express, Vol. 21, Issue 13, pp. 15859-15868 (2013)

http://dx.doi.org/10.1364/OE.21.015859

Acrobat PDF (1378 KB)

### Abstract

We address theoretically and numerically pump-probe switching in a nonlinear semiconductor nanocavity where tuning is achieved via a dominant mechanism of free-carrier plasma dispersion. By using coupled-mode approach we give a set of guidelines to optimize the switching performances both in terms of avoiding self-pulsation and keeping switching power to the minimum, ending up by showing that such devices can achieve high-performances with relatively low-power consumption.

© 2013 OSA

## 1. Introduction

1. 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. **30**, 2575–2577 (2005) [CrossRef] [PubMed] .

7. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics **6**, 248–252 (2012) [CrossRef] .

8. Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A **84**, 043847 (2011) [CrossRef] .

11. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. **35**, 55–57 (2009) [CrossRef] .

12. A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett. **10**, 1506–1511 (2010) [CrossRef] [PubMed] .

14. C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. **24**, 1433–1439 (2006) [CrossRef] .

15. A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A **79**, 043818 (2009) [CrossRef] .

16. C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. **94**, 021111 (2009) [CrossRef] .

17. 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(R)(2011) [CrossRef] .

19. 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] .

20. B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A **80**, 033805 (2009) [CrossRef] .

26. X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep. **3**, 01436 (2013) [CrossRef] .

## 2. Coupled-mode equations and lossless dynamics

*ω*

_{0}(the model in [17

17. 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(R)(2011) [CrossRef] .

*a*=

*a*exp(−

_{p}*iδ*) +

_{p}t*a*exp(−

_{s}*iδ*) and

_{s}t*a*|

^{4}≃ |

*a*|

_{p}^{4}in the rate equation for the carrier density, which amounts to assume that the carriers are essentially generated by the pump (|

*a*|

_{p}^{2}≫ |

*a*|

_{s}^{2}). The second one involves neglecting the last term in the following expansion of the nonlinear loss term from the first of Eqs. (4):

*δ*−

_{p}*δ*which turns out to be the specular image of the probe frequency with respect to the pump. Such frequency and the consequent modulation impressed on the carriers could be accounted for [27

_{s}27. Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A **85**, 063824 (2012) [CrossRef] .

14. C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. **24**, 1433–1439 (2006) [CrossRef] .

17. 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(R)(2011) [CrossRef] .

*α*and

*γ*are set to zero, the system reduces to The corresponding steady-state (

*d/dt*= 0), once expressed in terms of probe efficiency

*η*=

_{s}*E*, i.e. the cavity energy stored per unit pump power, reads as where

_{s}/P_{s}*E*≡ |

_{p,s}*a*|

_{p,s}^{2}are henceforth the intra-cavity energies. Figure 1 shows the pump intra-cavity energy

*E*and the switching efficiency

_{p}*η*as functions of the input pump power

_{s}*P*. Bistability occurs simultaneously for the pump and probe whenever

_{p}*P*, when the latter is in the range

_{p}*δ*< −1. Then, the injection of the pump lowers the refractive index, shifting the cavity resonance and inducing the probe beam to jump on a high-state. In particular, our aim is to investigate the conditions to achieve the switching with

_{s}*Maximal Probe Efficiency*(MPE)

*η*= 1. By solving

_{s}*dη*= 0 from the second of Eqs. (6), we find the following expressions for the critical pump energy

_{s}/dE_{p}*P*is switched into the cavity with MPE (

_{s}*η*= 1). However since the pump input-output response is multi-valued, in order to understand where the critical operation point is located on the bistable curve, it is important to identify the conditions under which the critical pump energy coincides with the branching points of the response. For a fixed probe detuning

_{s}*δ*, this occurs at the following values of the pump detuning (obtained by imposing

_{s}*η*= 1. On the other hand, another important figure of merit of the probe switch-on is the

_{s}*Contrast Ratio*(CR) defined as the ratio between the probe energies stored in the resonator in the presence and absence of the pump injection, respectively. For a given probe detuning

*δ*, when the condition of MPE is achieved, the CR is calculated to be

_{s}*η*= 1) and the value of

_{s}*η*obtained from the second of Eqs. (6) for

_{s}*E*= 0. Therefore, optimization of the pump-probe switching requires first to choose the value of probe detuning

_{p}*δ*accordingly with the desired CR, and then to choose the pump detuning

_{s}*δ*that ensures the MPE with lowest possible pump power.

_{p}*δ*,

_{p}*δ*) with fixed

_{s}*τ*, along with samples of pump-probe responses. In Fig. 2(a), the bistable region lying on the left of the vertical dashed line

*δ*= −3. As shown in Fig. 2(b), when

_{s}*δ*| results in further shifting the critical point on the UB of the response. However this does not allow to decrease the minimum required pump power, since these points can be reached only through hysteresis by decreasing

_{p}*P*after switch-on to the UB, which still requires to drive the cavity above the knee

_{p}*δ*| above the value

_{p}*δ*| reaches the new limit value

_{p}*δ*by imposing that the largest real root

_{s}*δ*, leads to the curve of optimal detunings reported (in dot-dashed yellow line) in the parameter plane in Fig. 2(a). The choice of

_{s}*δ*on this curve guarantees to reach the MPE condition with minimal required pump power.

_{p}*τ*. This means that for fixed

*δ*(fixed CR), by varying

_{s}*τ*, the pump power level needed to obtain MPE changes, but the required value

## 3. Role of self-pulsing

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

*a*|

_{p}^{2}≫ |

*a*|

_{s}^{2}, SP is induced by the pump and its threshold analysis carried out in [17

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

*P*(qualitatively similar results are obtained also for other waveforms).

_{p}*δ*= −2.2 and Fig. 2(d) for

_{p}*δ*= −3.1 (with fixed

_{p}*δ*= −3,

_{s}*τ*= 1). In both cases the up-switching of the pump to the UB, which occurs at peak value

*P*(we use

_{p}*P*= 3, 6 for

_{p}*δ*= −2.2, −3.1), always triggers the onset of SP. However, in the optimal case

_{p}*δ*= −3.1 (non-optimal detuning), besides requiring a nearly double power

_{p}*P*, leads to a dynamics where strong oscillations are exhibited by both beams. In this case SP heavily affects the dynamics of the probe over the whole signal duration, up to the MPE point

_{p}*η*∼ 1, which is reached only in a particular instant when the pump experiences down-switching [as expected from Fig. 2(d)].

_{s}*τ*large enough [17

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

*τ*= 4, while the detunings are the same as in Figs. 3(a)–3(b). In this case no SP occurs and the pump dynamics shows only strongly damped relaxation oscillations that are characteristic of any bistable system. With optimal detuning [

*δ*= −2.2, Fig. 3(c)], the MPE is reached in up-switching and maintained until the pump ramps down, whereas for

_{p}*δ*= −3.1 [Fig. 3(d)], MPE is reached again only in down-switching.

_{p}15. A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A **79**, 043818 (2009) [CrossRef] .

16. C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. **94**, 021111 (2009) [CrossRef] .

*τ*=

*τ*

_{r}/τ_{0}= 1 and

*τ*= 8 ps amounts to have (at telecom wavelength

_{r}*λ*

_{0}= 1.55

*μ*m)

*Q*≃ 5000. The dynamics in Fig. 3(a)–3(b) can be observed by operating with the following dimensional detunings: Δ

*λ*= −0.465 nm (

_{s}*δ*= −3), and Δ

_{s}*λ*= −0.341 nm (

_{p}*δ*= −2.2) or Δ

_{p}*λ*= −0.481 nm (

_{p}*δ*= −3.1), respectively. The injected powers turns out to be

_{p}*P*= 2 mW and

_{in}*P*= 4 mW, respectively. With the same carrier lifetime

_{in}*τ*= 8 ps,

_{r}*τ*= 4 [Fig. 3(c)–3(d)] can be obtained by lowering the quality factor to

*Q*= 1200, while the detuning values and powers become in this case: Δ

*λ*= −1.937 nm, Δ

_{s}*λ*= −1.421 nm and

_{p}*P*= 17.25 mW (

_{in}*δ*= −2.2), or Δ

_{p}*λ*= −2 nm and

_{p}*P*= 37.73 mW (

_{in}*δ*= −3.1).

_{p}*δ*= −15) that the MPE is obtained at a critical energy coincident with the first knee of the response [

_{p}*δ*|, when the driving power is slightly higher than the value

_{p}*δ*= −15 and

_{p}*P*= 250 (corresponding in real-world units to Δ

_{p}*λ*= −2.31 nm and pump power

_{p}*P*= 168.4 mW), and

_{in}*τ*= 1. As shown, the pump undergoes SP, and its evolution is attracted in phase space towards a stable limit cycle (supercritical Hopf bifurcation). In this case, the onset of SP deteriorates the pump-probe operations by inducing an abrupt switch-off of the probe, which remains in a low state (very far from MPE). Conversely, even a slight decrease of pump power, as shown in Fig. 4(b) for a driving level

*P*= 248.5 (

_{p}*P*= 167.4 mW) leaves the intra-cavity pump on the lower branch, thus allowing for the probe to switch-on and achieve dynamically MPE when the pump reaches its maximum level of intra-cavity energy (yet on the lower branch). This regime, however, is not advantageous due to the large driving power

_{in}*P*required to reach the MPE.

_{p}## 4. Effect of Losses

*τ*= 1 and

*α*= 0.1, the choice of probe detuning, for instance

*δ*= −3, fixes the contrast ratio to CR= 5.5 from Fig. 5(b). Then the optimum pump detuning remains, with good approximation, unchanged with respect to the lossless case (

_{s}*δ*= −2.2), whereas the MPE obtained with such detuning, decreases to ∼ 0.55 as inferred from Fig. 5(a). The outcome of the numerical integration of Eqs. (1)–(3) reported in Fig. 5(c) confirms such values of MPE and CR. Moreover, comparing with the lossless case in Fig. 3(a), we notice that SP is suppressed. Indeed the TPA losses moves the SP pump energy threshold at a value (

_{p}*E*= 1.9, from the linearized analysis in [17

_{H}**83**, 051802(R)(2011) [CrossRef] .

*E*= 1.65). Therefore we can summarize by saying that the losses on one hand decrease the achievable MPE, and on the other hand stabilize the dynamics against SP even for low values of

*τ*. However the latter statement is true only for the optimum detuning. Non-optimal values of detunings, as shown for instance in Fig. 5(d) for

*δ*= −3.1 [comparable with the lossless case in Fig. 3(b)], still leads to SP, besides reaching a reduced MPE in down-switching.

_{p}## 5. Conclusions

## Acknowledgments

## References and links

1. | 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. |

2. | M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express |

3. | 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 |

4. | E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. |

5. | S. Combrié, N. V. Q. Tran, A. de Rossi, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 |

6. | K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics |

7. | K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics |

8. | Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A |

9. | V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett. |

10. | Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. |

11. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. |

12. | A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett. |

13. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express |

14. | C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. |

15. | A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A |

16. | C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. |

17. | 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 |

18. | A. Rodriguez, M. Soljaĉić, J. D. Joannopoulos, and S. G. Johnson, “ |

19. | K. Ikeda and O. Akimoto, “Instability Leading to Periodic and Chaotic Self-Pulsations in a Bistable Optical Cavity,” Phys. Rev. Lett. |

20. | B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A |

21. | V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A |

22. | 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 |

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

24. | T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A |

25. | T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon. |

26. | X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep. |

27. | Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(190.4360) Nonlinear optics : Nonlinear optics, devices

(160.5298) Materials : Photonic crystals

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 3, 2013

Revised Manuscript: June 5, 2013

Manuscript Accepted: June 7, 2013

Published: June 25, 2013

**Citation**

S. Malaguti, G. Bellanca, and S. Trillo, "Optimizing pump-probe switching ruled by free-carrier dispersion," Opt. Express **21**, 15859-15868 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15859

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### References

- 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.30, 2575–2577 (2005). [CrossRef] [PubMed]
- M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express13, 2678–2687 (2005). [CrossRef] [PubMed]
- 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. Express14, 377–386 (2006). [CrossRef] [PubMed]
- E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007). [CrossRef]
- S. Combrié, N. V. Q. Tran, A. de Rossi, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 μm,” Opt. Lett.33, 1908–1910 (2008). [CrossRef]
- K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010). [CrossRef]
- K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012). [CrossRef]
- Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84, 043847 (2011). [CrossRef]
- V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002). [CrossRef]
- Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett.31, 341–343 (2006). [CrossRef] [PubMed]
- I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett.35, 55–57 (2009). [CrossRef]
- A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010). [CrossRef] [PubMed]
- I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express17, 22124–22137 (2009). [CrossRef] [PubMed]
- C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol.24, 1433–1439 (2006). [CrossRef]
- A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009). [CrossRef]
- C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009). [CrossRef]
- S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011). [CrossRef]
- A. Rodriguez, M. Soljaĉić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express, 15, 7303–7318 (2007). [CrossRef] [PubMed]
- 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]
- B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009). [CrossRef]
- V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011). [CrossRef]
- S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express20, 7454–7468 (2012). [CrossRef] [PubMed]
- M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012). [CrossRef]
- T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012). [CrossRef]
- T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012). [CrossRef]
- X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013). [CrossRef]
- Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012). [CrossRef]

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