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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13626–13638
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High-frequency self-induced oscillations in a silicon nanocavity

Nicolas Cazier, Xavier Checoury, Laurent-Daniel Haret, and Philippe Boucaud  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13626-13638 (2013)
http://dx.doi.org/10.1364/OE.21.013626


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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

The membrane photonic crystal nanocavity studied here is a modulated-width waveguide cavity following the design in [17

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] .

]. Light was injected in the cavity and collected with lensed fibers through ridge-type access waveguides suspended by nano-tethers as in [18

18. Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun. 283, 4387–4391 (2010) [CrossRef] .

] (see Fig. 1(a)). A direct coupling between cavity and access waveguides is implemented as illustrated in [19

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] .

]. The measured coupling loss from the lensed fiber to each access waveguide is 8 dB and the input and output powers given afterward are those at the entrance and exit of the photonic crystal. A continuous tunable external cavity laser is modulated by a Mach-Zehnder modulator to generate 10-ns duration pulses with a repetion rate of 100 kHz in order to minimize thermal effects. A 6-GHz bandwidth InGaAs photodiode is used to detected the output signal. In the present experiment, the peak resonance of the studied cavity is at 1585.638 nm, its quality factor Q is around 130000 and the transmission at the resonance is Tmax =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] .

] and in appendix 2, self-pulsing can be expected for operating points on the high-energy branch that characterizes the output power. The occurrence of spontaneous oscillations is very dependent on the carrier recombination time and on a fine tuning of different parameters that can be chosen based on the modeling of the silicon nanocavity.

Fig. 1 (a) (top): schematic view of the photonic crystal with the suspended access waveguides and nanotethers. The input light pulse intensity and the oscillating output are also drawn schematically. Bottom: scanning electron microscope view of the entrance of the photonic crystal waveguide (left) and view of an access waveguide (right). (b) Simulated steady-state output power of the cavity as a function of the input power for different detunings of the pump. For the curve at a detuning of −20 pm, the zone of bistability is between P1 = 0.284 mW and P2 = 0.549 mW. Self-pulsation can be observed for input powers larger than P3 = 1.5 mW, above the zone of bistability. The measurement reported in Fig. 3 (b) was performed for an input power P4 = 2 mW.

The equations describing the temporal behavior of the cavity are [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] .

, 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] .

, 21

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] .

]:
dAdt=A2τ+iΔωAγFCANVeffAγTPA|A|2A2h¯ω+Pinτin
(1)
dNdt=Nτfc+12γTPA|A|4(h¯ω)2
(2)
where 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/(2τ) represents the damping of the electrical field due to intrinsic linear losses, τ = Q/ω0, 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 Pin/τin, τin=2τ/Tmax being the photon injection time and Tmax the transmission maximum at low input power. The interplay between the second term, iΔωA, and the third term, γFCANVeffA, is mainly responsible for the oscillations of the cavity: Δω = ωω0 represents the detuning between the input laser frequency ω/(2π) and the cavity resonance frequency ω0/(2π). γFCANVeffA, where γFCA is a complex number, represents the effect of free-carrier induced absorption and dispersion, and Veff 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 γFCA = 5.83 × 10−14 + 1.7 × 10−12i m3 s−1 using three-dimensional finite difference in time domain (3D-FDTD) simulation.

The fourth term in Eq. (1) represents the effect of two-photon absorption. For the power considered here, this term has a negligible effect on the cavity field decay but its counterpart in Eq. (2), 12γTPA|A|4(h¯ω)2, is crucial. Indeed, it represents the unique source term in the equation for free-carriers, −1/τfc being the effective decay rate of free carriers. From 3D-FDTD, we deduced γTPA = 1.63 × 104 s−1 for the PhC cavity.

For the input power we used experimentally (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

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] .

] is negligible as compared to TPA. For the same reason, we did not take into account the dispersion created by Kerr effect since it is negligible when compared to the dispersion created by free-carriers [24

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] .

].

Classically, this system of differential equations can be solved by first looking for steady state solutions and then by linearizing for small variations around these steady-state solutions (see Refs. [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] .

, 13

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

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] .

] and appendix 2 Solving the equations: steady state and small perturbations). Steady-state solutions of the different variables of the system can be expressed as a function of the energy 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 αr ± iΩ 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 αr > 0, corresponding to an equivalent gain and their period is 2π/Ω.

The set of parameters where self-pulsing occurs is illustrated in Fig. 2. Figure 2(a) shows that, for a cavity with Q=130000, non-attenuated oscillations can only be obtained for a free-carrier lifetime smaller than 0.4 ns and a cavity effective volume smaller than 5.5 μm3. Figure 2(b) shows the dependence of the oscillation period for a free-carrier lifetime of 0.2 ns and an effective volume Veff = 5.25 μm3 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).

Fig. 2 (a) Period of the oscillations as a function of the cavity effective volume Veff and the free-carrier lifetime, for a cavity with a quality factor Q=130000, a detuning of −20 pm and an input power P4 = 2 mW. The oscillations are non-attenuated in the zone of positive gain (inside the region defined by the black thick line). (b) Period of the oscillations as a function of the quality factor and input power for a free-carrier lifetime of 0.2 ns and an effective volume Veff = 5.25 μm3.

The experiments were performed on the same cavity before and after a nitric acid surface treatment used to modify the carrier recombination time [25

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).

] and the quality factor. Nitric acid treatment introduces new surface states that modify the surface recombination velocity of the free-carriers and thus decrease the effective lifetime of the carriers. The silicon is also slightly etched by the nitric acid. The geometry of the PhC is thus modified and the Q factor is slightly increased. The experimental results are presented in Fig. 3(a) for a cavity with a quality factor of 90000 and various detunings. In Fig. 3(a), one observes self-induced oscillations with a period of 0.3 ns, for a laser wavelength detuned from −40 pm to −20 pm. The amplitude of the high-frequency oscillations is strongly damped with a characteristic decay time equal to αr1710ps and the oscillations at 3.18 GHz disappear before the end of the 10-ns pulse. This characteristic time measured at a high input power corresponds to an increase of the photon lifetime in the cavity of one order of magnitude as compared to the photon lifetime 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] .

] using nonlinearities different from the one considered here.

Fig. 3 Experimental (a and b) and simulated (c and d) output power as a function of time for different detunings between the laser wavelength and the resonance wavelength of the cavity. The input optical pulses have a length of 10 ns. Two types of cavities have been investigated. The first cavity has a quality factor of 9 × 104 and a resonance wavelength of 1590 nm. After the cavity had been immersed in a 3:1 HNO3:H2O2 solution, the quality factor of the cavity increased, rising from 9 × 104 to 1.3 × 105, and the resonance wavelength shifted to 1585.638 nm. Measurements (a) before and (b) after the nitric acid treatment. Corresponding modelings, (c) before nitric acid treatment (Q =9×104 and τfc = 0.3 ns) and (d) after the nitric acid treatment (Q = 1.3 × 105 and τfc = 0.2 ns). In Fig. 3(a), a parasitic extrinsic slow modulation of the signal is observed (period around 2 ns). This parasitic effect due to the measurement set-up was significantly suppressed for the subsequent measurements of the cavity [Fig. 3(b)].

The situation is strikingly different for the same cavity but this time with a Q factor of 130000 (i.e. after a nitric acid treatment) as shown in Fig. 3(b) for a detuning of −20 pm. Non-attenuated oscillations with a significant contrast are observed until the end of the pulse for a laser wavelength detuned from −30 pm to −10 pm. The oscillation frequency is 2.8 GHz, which is five times higher than the frequency of the oscillations observed in microring resonators (0.5 GHz) [16

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] .

], notably because the quality factor of our cavity is about ten times lower than for the microring resonator, causing the cavity to react faster. This clearly demonstrates that self-sustained self-pulsing at GHz frequencies can be achieved in silicon photonic crystal nanocavities.

We have compared the experimental results with those obtained by the modeling described above. The only adjustable parameters were the free-carrier lifetime and the effective volume of the cavity. Only an effective volume close to Veff ≃ 5.25 μm3 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 τfc = 0.3 ns before the nitric acid surface treatment [Fig. 3(c)], and for a free-carrier lifetime τfc = 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.

If we consider that the transmitted signal is detected by a photodetector with a 1 A/W response and a 50 Ohm impedance circuitry, the radio-frequency power delivered by the nanocavity at 2.8 GHz is 20 nW. The radio-frequency power could also be collected directly by electrical contacts in close proximity of the photonic crystal thus eliminating the need for a fast photodetector [23

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] .

]. These performances compare favorably with those achieved with spin-transfer nano-oscillators where a single spin-torque nano-oscillator emits a microwave power up to one nW [27

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] .

]. Moreover, ultra-short free-carrier lifetime as small as 12 ps has been recently demonstrated in silicon nanowaveguides using a reverse bias and a p-i-n junction [28

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] .

]. If we combine this lifetime value with a Q=10000 cavity, operation at 50 GHz can be expected for an input power of 50 mW with the advantage of generating the RF signal directly at the junction electrical outputs. In III–V system, the same lifetime parameters lead to a 100 GHz operating frequency because of the difference of nonlinear coefficients [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] .

]. As high-frequency operation requires low-Q cavities, self-pulsation could be easily observed with cavities on silicon-on-insulator with a further advantage of a better thermal management. Low-Q cavities can also exhibit lower cavity and effective volumes which are also an advantage for the onset of self-pulsing.

A remarkable property of the oscillations in PhC microcavity is their high spectral purity. As can be seen in Fig. 4(a), the amplitude of the second harmonic of Pout = |A|2/τin, is 26 dB below the 2.8 GHz signal [29

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.

], 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 EE0 + 2E1 cos(Ωt +φ1) + 2E2 cos(2Ωt +φ2)). 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: E2/E1 = C0 × E1/E0, where C0 is a function of the frequency Ω, the average energy E0 and the average number of free carriers. For oscillations triggered by an optical input power higher than 2 mW, the expressions of Ω, and C0, reduce to:
Ω~τfcγTPAγiFCA2(h¯ω)2Veff×Tmax(Pin2τ)25Δω5
(3)
and
C0~1Ωτfc×((1+ΔωΩ)3γTPAτfcE04h¯ω)
(4)
where γiFCA=Im(γFCA). These equations show that, counter intuitively, the signal is more sinusoidal at high injection power than at low power, because the frequency of the oscillations increases with the input power and C0 decreases as the frequency increases. This behavior is confirmed in Fig. 4(c) and (d) that respectively represent E1/E0 and E2/E1 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

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] .

], for which we have found that E2/E1 = −14.6 dB and E1/E0 = −6 dB, because the ratio E1/E0 and C0 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).

Fig. 4. : (a) Measured and simulated spectra of the oscillations with thermal effects for a detuning of −20 pm. (b) Comparison between the simulated and the analytically calculated spectra of the cavity with a continuous input power and without the thermal effects. (c) Ratio of the fundamental harmonic amplitude E1 to the average energy in the cavity E0 as a function of the input power and the detuning (analytical expression). (d) Ratio of the second harmonic amplitude E2 to the fundamental amplitude E1(analytical expression).

In conclusion, we have experimentally demonstrated that self-pulsing can occur at GHz frequencies in photonic crystal nanocavities. This self-pulsing effect differs significantly from the one created by thermal effects, which is limited to MHz frequencies. The operation frequency is here controlled by the carrier lifetime and photon lifetime in the cavity. Self-pulsing in photonic crystal cavities presents the advantage of simplicity to realize ultra-compact microwave sources with high spectral purity. As shown by the model we have developed, the remarkable spectral purity of the self-oscillations in PhC microcavities mainly results from the small volume of the PhC cavities and cannot be observed in larger cavities such as micro-disks. These microwave oscillators on an optical carrier are intrinsically compatible with optical delay lines based on dispersion engineering in photonic crystal waveguides, their combination allowing one the design of more complex architectures for microwave photonics.

Appendices:

1. Free-carrier dispersion, absorption and generation

The complex coefficient appearing in the third term of eq. (1), γFCA can be written as:
γFCA=γrFCA+iγiFCA=c2nReff(σri2ωcσi)×(ωrω)2
(5)

The real and imaginary parts are responsible for absorption and dispersion respectively. 1/Reff 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 (Reff = 1.1). σr = 1.45 × 10−21m2 and σi = −5.3 × 10−27m3 are respectively the free-carrier absorption and dispersion in silicon given at λr = 2πc/ωr = 1550 nm [21

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] .

].

The coefficient appearing in the fourth term of eq. (1), γTPA, can be expressed as [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] .

]: γTPA=(βh¯ω(cn)2)/VTPA, where 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] .

] is the TPA coefficient in bulk silicon, n = 3.48 is the silicon refractive index [31

31. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data 9, 98 (1980).

], and VTPA = 4.8341 × 10−19m3 is the TPA volume calculated by three-dimensional finite-difference in time domain modeling (FDTD).

2. Solving the equations: steady state and small perturbations

To solve the cavity equations, we wrote A = |A|e and obtained two real equations from the first complex equation, which gave us a system of three real equations:
d|A|dt=|A|2τγrFCA|A|NVeffγTPA|A|32h¯ω+cos(φ)Pinτin
(6)
dφdt=ΔωγiFCANVeffsin(φ)|A|Pinτin
(7)
dNdt=Nτfc+12γTPA|A|4(h¯ω)2
(8)

This system only differs from the one used in Ref. [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] .

], in the fact that it is non-normalized and completely real. Looking for steady-state solutions (|A0|, φ0, N0) to these equations, we get:
N0=τfcγTPA|A0|42(h¯ω)2,
(9)
tan(φ0)=ΔωτfcγTPAγiFCA|A0|42(h¯ω)2Veff12τ+γTPA|A0|22h¯ω+τfcγTPAγrFCA|A0|42(h¯ω)2Veff,
(10)
and |A0|2 is one of the roots of the five-degree polynomial:
(|A0|2τ+γTPA|A0|32h¯ω+τfcγTPAγrFCA|A0|52(h¯ω)2Veff)2+(Δω|A0|τfcγTPAγiFCA|A0|52(h¯ω)2Veff)2Pinτin=0
(11)
Linearizing these equations for small perturbations around these steady-state solutions gives:
ddt(δ|A|δφδN)=M(δ|A|δφδN),
(12)
M=(12τγrFCAN0Veff3γTPA|A0|22h¯ωsin(φ0)PinτinγrFCA|A0|Veffsin(φ0)|A0|2Pinτincos(φ0)|A0|PinτinγiFCAVeff2γTPA|A0|3(h¯ω)201τfc)
(13)

The eigenvalues of this matrix can be calculated as a function of |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 αr ± iΩ 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 αr > 0, and their period is T = 2π/Ω.

In Eq. (11), for input power in the range considered here, typically Pin > 2 mW in the case of the PhC cavity, from Eq. (11), we have τfcγTPAγiFCA|A0|52(h¯ω)2Veff~Pinτin+Δω|A0| because γiFCAγrFCA. Note that the detuning Δω 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:
|A0|~2(h¯ω)2VeffτfcγTPAγiFCA(Pinτin)1/25×(1Δω5×2(h¯ω)2VeffτfcγTPAγiFCA(τinPin)25)1
(14)
. Neglecting small terms in the matrix M, we have Ω|A0|Pin/τin and
Ω~τfcγTPAγiFCA2(h¯ω)2Veff×Tmax(Pin2τ)25Δω5.
(15)

3. Estimating the cavity effective volume

The cavity effective volume for the free-carriers found experimentally (Veff ≃ 5.25 μm3) 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] .

], where it is calculated as Veff=(2d)×(3a)×h, with d ≃ 1.1 μm the diffusion length of the free-carriers, h the slab thickness, and 3a the distance between the air holes nearest to the center of the cavity in the orthogonal direction (a being the photonic crystal period). Using this formula would give Veff = 0.32 μm3 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] .

], the diffusion of free-carriers in a photonic crystal is not very different from the diffusion of free-carriers in a silicon membrane without holes. Furthermore, it assumes that the free-carriers are all created in the cavity center, but the electric field profile of the cavity resonant mode calculated by FDTD is in fact quite large and is included in a volume approximately equal to (8a)×(23a)×h. If we assume that the free-carriers are created uniformly in this volume, and then diffuse in the rest of the cavity, we can approximate the effective cavity volume for the free-carriers by Veff=(8a+2d)×(23a+2d)×h. This formula gives Veff = 4.06 μm3, which is close to the value of Veff that was deduced from the experimental measurements.

4. Thermal effects

If we take into account the thermal dispersion, the equations used for the modeling become [14

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] .

]:
dAdt=A2τ+iΔωAγFCAANVeffiγTAΔTγTPA|A|2A2h¯ω+Pinτin
(16)
dNdt=Nτfc+12γTPA|A|4(h¯ω)2
(17)
(ρSiCpSiVeffT)dΔTdt=ΔTRT+h¯ω(γTPA|A|4(h¯ω)2+2γrFCA|A|2h¯ωNVeff)
(18)

where ΔT is the temperature difference in the cavity generated by the input power, ρSi = 2.33 g/cm3 is the silicon density, CpSi=0.7J/(g×K) is the thermal capacity of silicon, and γT=ωndndT is the thermal dispersion with dndT=1.86×104K1[14

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] .

]. Two parameters are unknown: the thermal resistance of the cavity RT and the effective cavity volume that depends on the temperature VeffT. To determine these two coefficients, measurements on the cavity after nitric acid treatment were performed again with pulses of 200 ns length. These measurements allowed us to observe thermal oscillations with a frequency around 20 MHz. We used these oscillations to determine approximate values of the thermal resistance and effective volume by comparing measurements and modeling, and found RT ≈ 8 × 103 K/W and VeffT3×1018m3, which were used in the modeling as shown in Fig. 3(c) and 3(d).

5. Solving the equations: harmonic analysis

An approximate analytical expression for the time evolution 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 |A|2A0(2)+2A1(2)cos(Ωt+φA1(2)) with φA1(2) a phase term introduced for generality. We then solved the linear first order differential equation that is obtained for 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: dA/dt(R+F(t))A+Pin/τin where 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, EE0 + 2E1 cos(Ωt +φ1)+ 2E2 cos(2Ωt +φ2)) where E0, E1, E2 depend on A0(2) and A1(2). For consistency, this expression of E must be equal to the first approximation we initially used in Eqs. (1) and (2), i.e. E0=A0(2), E1=A1(2) with E2 negligible. Finally, solving these last two equations gives expressions for E0, E1 and E2.

The detailled calculation is given in the following. According to the experimental results, if we neglect thermal effects, the time evolution of the energy is quasi-sinusoidal. The energy can then be written as:
|A(t)|2A0(2)+2A1(2)cos(Ωt+φA1(2))
(19)
and |A|4=E(t)2((A0(2))2+2(A1(2))2)+4A0(2)A1(2)cos(Ωt+φA1(2)) plus an harmonic term that is neglected. Equation (2) becomes :
dNdtNτfc+12γTPA(h¯ω)2[(A0(2))2+2(A1(2))2+4A0(2)A1(2)cos(Ωt+φA1(2))]
(20)

Solving this linear first order differential equation, we get the following expression for N: N(t) = N0 + 2N1 cos(Ωt + φN1), with N0=τfcγTPA((A0(2))2+2(A1(2))2)2(h¯ω)2τfcγTPA(A0(2))22(h¯ω)2, N1=τfcγTPA2(h¯ω2)2A0(2)A1(2)(1+iΩτfc) and φN1 a phase term. Replacing this expression of N in the Eq. (1), we get the first order, linear, ordinary differential equation :
dAdt(R+F(t))A+Pinτin
(21)
with
R=12τiΔω+γFCAN0Veff+γTPAA0(2)2h¯ω
(22)
F(t)=2γFCAN1Veffcos(Ωt+φN1)+2γTPAA1(2)2h¯ωcos(Ωt+φA1(2))
(23)

For time t ≫ 1/R, the harmonic regime is reached and the solution of this equation is :
A(t)=KeRt0tF(t)dt0teRt+0tF(t)dtdt
(24)
with 0tF(t)dt=α1FCAsin(Ωt+φN1)+αTPAsin(Ωt+φA1(2)), K=Pinτin, αTPA=2ΩγTPAA1(2)2h¯ω and αFCA=2ΩγFCAN1Veff.

In the case of a photonic crystal cavity with an input power of 2 mW and a detuning of −20 pm, as in the experiments, we have αTPA = 0.016 and αFCA = 0.004 + 0.123i and the main contribution to the integral 0tF(t)dt comes from the imaginary part of αFCA. To further simplify the calculations, we, first, only consider this contribution in the evaluation of R and of the integral 0tF(t)dt, i.e. R12τ+i(γFCAN0VeffΔω) and 0tF(t)dtαiFCAsin(θ) with αiFCA=2ΩiγiFCAN1Veff, γiFCA=Im(γFCA) and θ = Ωt + φN1.

The Jacobi-Anger expansion is used to develop e∫F(t) as :
e0tF(t)dteαiFCAsin(θ)=i3nIn(αiFCA)einθ
(25)
where the In are the modified Bessel functions of the first kind. In this expansion, only the terms for 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 I1(αiFCA)(eiθeiθ) in the case of our experimental conditions. As a consequence, we have the following expressions :
0teRt+0tF(t)dtdt=eRt[I0(αiFCA)RiI1(αiFCA)(eiθR+iΩeiθRiΩ)]
(26)
and
A=K(I0(αiFCA)RiI1(αiFCA)(eiθR+iΩeiθRiΩ))eαiFCAsin(θ)
(27)
We then deduced the energy as:
E=|A|2=K2|I0(αiFCA)RiI1(αiFCA)(eiθR+iΩeiθRiΩ)|2
(28)
that can be decomposed as
E=(E0+2E1cos(Ωt+φ1)+2E2cos(2Ωt+φ2))
(29)
with
E0=K2(I0(αiFCA)2|R|2+2|I1(αiFCA)|2(Ω2+|R|2)(|R|2Ω2)2+(2ΩRe(R))2)
(30)
E1=K22ΩIm(R)I0(αiFCA)|I1(αiFCA)|(|R|2Ω2)2+(2ΩRe(R))2
(31)
and
E2=K2|I1(αiFCA)|2(|R|2Ω2)2+(2ΩRe(R))2
(32)

For consistency, this expression of E must be equal the one initially introduced in Eq. (19), i.e. E0=A0(2), E1=A1(2) with E2 negligible. We then get a set of two transcendental equations, the first one in the unknown A0(2) and the second one in the unknown A0(2) and A1(2), A1(2) being one of the harmonic magnitudes we are looking for. The magnitude of the fundamental A1(2) can be determined by solving numerically the equation A1(2)=E1 and the amplitude of the second harmonic can then be calculated by using the expression of E2. A simple expression can be obtained for the ratio of the second harmonic to the first one E2/E1:
E2E1|I1(αiFCA)|22ΩIm(R)I0(αiFCA)|I1(αiFCA)|
(33)

In the conditions of the experiment, |αiFCA|1, I0(αiFCA)~1, I1(αiFCA)~12αiFCA=iγiFCAN1ΩVeff, and we get:
E2E1C0×E1E0
(34)
where
C0=γiFCAN0Veff|R|2Ω2Im(R)1+Ω2τfc2
(35)

C0 depends only on N0 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).

For oscillations triggered by an input optical power higher than 2 mW, the expressions of Ω and R reduce to Ω~γiFCAN0VeffΔω and RiΩ, and the expression of C0 simplify to
C0~γiFCAN0τfcΩ2Veff~1Ωτfc×(1+ΔωΩ)
(36)
C0 will be lower at high frequencies, i.e. the signal will be more sinusoidal.

The same method can be used if the two-photon absorption and the free carrier absorption are taken into account in the calculation of Eq. (24). The expressions of E0, E1, E2 and C0 then become more complicated but they still simplify for an input optical power higher than 2 mW and we get for C0 :
C0~1Ωτfc×((1+ΔωΩ)3γTPAτfcE04h¯ω)
(37)

Numerically, for a detuning of −20 pm, C0 varies between 0.27 and 0.084 for an input power varying from 1.5 mW to 20 mW. For Pin= 2 mW, C0 = 0.24 and E1/E0 = −21.4 dB, which gives us E2/E1 ≃ −33.6 dB, which is very close to the values found by the numerical simulation (see Fig. 4b) : E1/E0 = −21.3 dB and E2/E1 = −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.

In the case of a microring resonator of the kind described in Ref. [16

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] .

], we have according to our simulations αTPA = 0.042 and αFCA = 0.04 + 1.21i for a detuning of −2.5 pm and an input power of 1 mW. Since |αFCA| > 1, we can no longer write eαiFCAsin(θ)~I0(αiFCA)iI1(αiFCA)(eiθeiθ), I0(αiFCA)~1 nor I1(αiFCA)~12αiFCA, and the formulae above are no longer very accurate. However, they can still be used to estimate the importance of the second harmonic. Numerically, for a detuning of −2.5 pm and an input power Pin= 1 mW, similar to the experimental parameters used in Ref. [16

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] .

], C0 = 0.37, E1/E0 = −6 dB and E2/E1 ≃ −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 C0, 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

This work was supported by the Agence Nationale de la Recherche ( ANR) through the project PHLORA (ANR 2010 JCJC 0304 01). This work was also supported by the french RENATECH network and Conseil général de l’Essonne.

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) 425, 944–947 (2003) [CrossRef] .

2.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005) [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] .

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 17, 2944–2953 (2009) [CrossRef] [PubMed] .

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

8.

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82, 041308(R) (2010) [CrossRef] .

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] .

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 × 106 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] .

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] .

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] .

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. 109, 113903 (2012) [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] .

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. 82, 2954–2956 (2003) [CrossRef] .

31.

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data 9, 98 (1980).

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] .

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

  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]
  2. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005). [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. Express17, 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. Express13, 801–820 (2005). [CrossRef] [PubMed]
  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. Express17, 2944–2953 (2009). [CrossRef] [PubMed]
  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.30, 2575–2577 (2005). [CrossRef] [PubMed]
  8. X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010). [CrossRef]
  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. Express18, 15484–15497 (2010). [CrossRef] [PubMed]
  10. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1, 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. A83, 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]
  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. Express20, 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. Express14, 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. A85, 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. A85, 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 × 106 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]
  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. Express14, 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. Express15, 16604–16644 (2007). [CrossRef] [PubMed]
  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. Express18, 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. A84, 053816 (2011). [CrossRef]
  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.109, 113903 (2012). [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. Express18, 3582–3591 (2010). [CrossRef] [PubMed]
  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.82, 2954–2956 (2003). [CrossRef]
  31. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data9, 98 (1980).
  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]

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