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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15859–15868
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Optimizing pump-probe switching ruled by free-carrier dispersion

S. Malaguti, G. Bellanca, and S. Trillo  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15859-15868 (2013)
http://dx.doi.org/10.1364/OE.21.015859


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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

Nanocavities with high Q/V ratio allows for strong enhancement of nonlinear behavior which has been exploited for achieving ultra-fast switching in different platforms involving, e.g., defects in photonic crystal (PhC) membranes [1

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

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] .

] or microring resonators [8

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

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] .

]. Such structures, made either in silicon or III–V semiconductors, have demonstrated the viability of free-carrier dispersion (FCD) induced by two-photon absorption (TPA) as a dominant nonlinear effect [12

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

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] .

]. In particular, such mechanism permits pump-probe operations, where a carrier-plasma density induced by TPA of a high-intensity pump coupled via a waveguide, causes the refractive index change inside the cavity and, as a consequence, the wavelength resonant tuning of the probe signal [15

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

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] .

]. However, switching could be quite demanding in terms of energy consumption, as long as the dynamical pump-probe interaction is not properly designed. Moreover, it has been predicted that the non-instantaneous response of the carriers determines the onset of a FCD-driven instability, namely self-pulsing (SP) [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] .

]. This is an widespread phenomenon first predicted for Kerr-like nonlinearities [19

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] .

] and recently investigated theoretically and experimentally in different nano-structures [20

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

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] .

]. In this paper, we give criteria for optimizing bistable pump-probe operations both in terms of minimizing the required switching power and avoiding SP, which could impact the switching dynamics, spoiling the simple bistable features. This is provided by suitable choice of the detunings of the pump-probe pair for a given time constant of carrier dynamics.

2. Coupled-mode equations and lossless dynamics

We point out that Eqs. (1)(3) can be explicitly derived from the following equations for a single envelope at the resonant reference frequency ω0 (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] .

] with zero detuning)
at=inaaα|a|2aγna+P;nt=|a|4nτ.
(4)
By inserting in Eqs. (4) the two-frequency ansatz a = ap exp(−pt) + as exp(−st) and P=Ppexp(iδpt)+Psexp(iδst) and grouping terms with the same frequency, we arrive at Eqs. (1)(3). We recall that two approximations are essentially involved in this step. The first involves approximating |a|4 ≃ |ap|4 in the rate equation for the carrier density, which amounts to assume that the carriers are essentially generated by the pump (|ap|2 ≫ |as|2). The second one involves neglecting the last term in the following expansion of the nonlinear loss term from the first of Eqs. (4): α|a|2aα(|ap|2apexp(iδpt)+2|ap|2asexp(iδst)+ap2as*exp[i(2δpδs)t]). The latter term is indeed responsible for four-wave mixing, i.e. the generation of a new idler frequency 2δpδs 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

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] .

], though only at the expense of adopting a self-consistent model more complicated than Eqs. (1)(3). In the regime considered here, where the probe beam is brought on resonance by the nonlinear effect, the four-wave mixing frequency turns out to be sufficiently detuned from the resonance, and hence is expected to be negligible (this is especially true in the optimal conditions defined in the analysis reported below). In this case, Eqs. (1)(3), where only the incoherent nonlinear coupling between the probe and pump beams is retained (as also done in [14

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] .

]), constitute a satisfactory approximation which allows us to successfully tackle the optimization problem. Note that we retain a factor of two in the cross-TPA nonlinear loss term in Eq. (2) which ultimately stems from the non-degeneracy of the frequencies involved in the underlying nonlinear susceptibility term.

In order to understand the effect of FCD over the pump-probe operation, let us first analyze the dynamics ruled by Eqs. (1)(3) in the absence of nonlinear losses and FCA (lossless limit) [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] .

]. When the loss coefficients α and γ are set to zero, the system reduces to
ajt=i(δj+n)ajaj+Pj,j=p,s;nt=|ap|4nτ.
(5)
The corresponding steady-state (d/dt = 0), once expressed in terms of probe efficiency ηs = Es/Ps, i.e. the cavity energy stored per unit pump power, reads as
Pp=Ep[1+(δp+τEp2)2];ηs=11+(δs+τEp2)2,
(6)
where Ep,s ≡ |ap,s|2 are henceforth the intra-cavity energies. Figure 1 shows the pump intra-cavity energy Ep and the switching efficiency ηs as functions of the input pump power Pp. Bistability occurs simultaneously for the pump and probe whenever δp<δpc5/2, featuring three different levels of energy for the same input power Pp, when the latter is in the range PpPpPp+. The values Pp±=Pp(Eb) can be easily calculated through the first of Eqs. (6) from the branch-point energies
Eb±=3δp±4δp255τ,
(7)
which correspond to the knees (branch points) of the bistable response [see Fig. 1]. Figure 1 clearly shows that the bistability of the pump (black curve) drives the probe response (red curve), so that stable (lower and upper) branches of the probe efficiency correspond to stable branches of the pump, whereas the negative-slope (unstable and inaccessible) branch for the probe (dot-dashed in Fig. 1) arises from the similar branch for the pump.

Fig. 1 Typical stationary responses [Eqs. (6)] for pump (Ep vs. Pp, black curve) and probe (ηs vs. Pp, red curve). We highlight the bistable jumps (arrows) occurring at branch point energies Eb±, and the critical pump point [ Ppc, Epc, Eq. (7)] for highest probe efficiency ηs = 1. Dot-dashed portions stand for unstable branches. Here δp = −3, δs = −4, and τ = 1.

In our analysis, we focus on the pump-probe switch-on regime. This means that the probe is initially blue-detuned from the cavity resonance so that the probe is in a low-state corresponding to Es/Ps|Pp=0=(1+δs2)10.5 (less than 50% of the input is stored in the cavity), which requires to operate with δs < −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 Maximal Probe Efficiency (MPE) ηs = 1. By solving s/dEp = 0 from the second of Eqs. (6), we find the following expressions for the critical pump energy Epc and the corresponding input power Ppc,
(Epc)2=δsτ;Ppc=|δs|τ[1+(δpδs)2].
(8)
These critical values identify the point on the pump bistable response such that the injected probe power Ps is switched into the cavity with MPE (ηs = 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 δs, this occurs at the following values of the pump detuning (obtained by imposing Epc=Eb±)
δp±=3δs±4δs21.
(9)
The two values in Eq. (9) along with the threshold value δpc5/2, define three different regimes: (i) δp+δp<δpc; (ii) δpδpδp+; (iii) δpδp, which correspond to the critical value in Eq. (8) lying on the upper (UB), negative-slope branch, or lower branch (LB) of the three-fold pump response, respectively, yet always maintaining the MPE condition ηs = 1. On the other hand, another important figure of merit of the probe switch-on is the 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 δs, when the condition of MPE is achieved, the CR is calculated to be 1+δs21/(ηs|Ep=0), as can be easily obtained from the ratio between the MPE (ηs = 1) and the value of ηs obtained from the second of Eqs. (6) for Ep = 0. Therefore, optimization of the pump-probe switching requires first to choose the value of probe detuning δs accordingly with the desired CR, and then to choose the pump detuning δp that ensures the MPE with lowest possible pump power.

In order to work out the latter criterium we have summarized the results given through Eqs. (8) and (9) in Fig. 2, which shows a level map of the critical pump power Ppc in the parameter plane (δp, δs) with fixed τ, along with samples of pump-probe responses. In Fig. 2(a), the bistable region lying on the left of the vertical dashed line δp=δpc, is divided by the curves δp(δs) [solid red] and δp+(δs) [solid green] into three domains labeled UB, LB, and NSB, according to the name of the branch of the pump response where the critical condition in Eq. (8) falls.

Fig. 2 (a) Level plot of power Ppc (dB units) from Eq. (8) in the plane of detunings (δp, δs). In the bistable region (left of the vertical dashed line δp=δpc), the curves labelled δp+ (green) and δp (red) delimit the domains corresponding to the three branches of the stationary response shown in Fig. 1 (LB/UB stand for the lower/upper branch, NSB stands for the negative-slope branch). The optimum (MPE ηs = 1 with minimal Pp) is achieved for pump detunings δpu (yellow dot-dashed curve). (b)–(e) Corresponding steady-state pump (black) and probe efficiency (red) responses for fixed probe detuning δs = −3 (CR = 10) and increasing values of |δp|. The dots indicate the optimum operation points (the red and black dots give MPE ηs = 1 and corresponding critical values Pp=Ppc, Ep=Epc, respectively): (b) Pump detuning values δp = −1.2, −1.6, −2.1 (Pc = 7.3, 5.1, 3.1) in the UB region; (c) Optimum operation at δp=δpu=2.2 (minimal Ppc=2.8); (d)–(e) MPE at δp+=3.1 and δp=15, respectively. Here τ = 1.

To better illustrate how the MPE and the corresponding critical pump value move on the bistable response, the latter is shown in Figs. 2(b)–2(e) for different (increasing in modulus) values of δp<δpc, and fixed δs = −3. As shown in Fig. 2(b), when δp+δp<δpc, the critical point (8) lies on the UB, and moves together with the MPE condition at lower pump powers for increasing detuning in modulus. Under this regime, the minimum driving pump power needed to yield the MPE condition, is clearly obtained when Pp=Pp+ (i.e., when the cavity is driven at the first knee of the response), which is obtained for δp=δpu=2.2, as illustrated in Fig. 2(c).

Keeping on increasing |δp| 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 Pp after switch-on to the UB, which still requires to drive the cavity above the knee Pp=Pp+. This situation holds up to the limit value δp=δp+=3.1 such that the MPE is obtained exactly at the upper knee of the response as shown in Fig. 2(d).

Further increasing |δp| above the value |δp+| [i.e. entering the region labeled NSB in Fig. 2(a)], makes the critical point (8) to move on the negative-slope branch, thus making the MPE condition unaccessible. This regime holds until |δp| reaches the new limit value |δp|, in correspondence of which the cavity operates on the first knee of the LB [see Fig. 2(e)]. For detunings |δp||δp|, the MPE condition becomes accessible again since the critical point (8) lies on the LB, though the required input powers grow so large to become unpractical, as can be clearly seen in Fig. 2(e).

Therefore we conclude that optimized pump-probe operation which allows to obtain MPE with minimal pump power requires to operate at the pump detuning δp=δpu such that the critical point coincides with the UB energy level [black dot in Fig. 2(c), not to be confused with Eb in Eq. (7)] that corresponds to the knee level of input power Pp=Pp+. This optimum detuning can be found for any choice of the signal detuning δs by imposing that the largest real root Ep=Ep(Pp+) coincides with the critical value Epc=δs/τ. The pump detunings δpu that fulfills this constraint (no simple analytical expression can be found since it involves the roots of a quintic polynomial) for different values of probe detuning δs, leads to the curve of optimal detunings reported (in dot-dashed yellow line) in the parameter plane in Fig. 2(a). The choice of δp on this curve guarantees to reach the MPE condition with minimal required pump power.

It is worth noting that the curves delimiting the different regions in Fig. 2(a) remain unchanged with respect to variations of τ. This means that for fixed δs (fixed CR), by varying τ, the pump power level needed to obtain MPE changes, but the required value δpu is the same in the same cavity, regardless of the specific carrier lifetime.

3. Role of self-pulsing

In Figs. 3(a)–(b) we compare the pump-probe temporal dynamics corresponding to the steady-states displayed in Fig. 2(c) for δp = −2.2 and Fig. 2(d) for δp = −3.1 (with fixed δs = −3, τ = 1). In both cases the up-switching of the pump to the UB, which occurs at peak value Pp (we use Pp = 3, 6 for δp = −2.2, −3.1), always triggers the onset of SP. However, in the optimal case δp=δpu=2.2, when the pump starts to exhibit SP, the probe has nearly reached the MPE condition in up-switching and shows only small spurious oscillations driven by those of the pump beam. Therefore SP does not dramatically deteriorate the probe switching performances. Conversely, the case δp = −3.1 (non-optimal detuning), besides requiring a nearly double power Pp, 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 ηs ∼ 1, which is reached only in a particular instant when the pump experiences down-switching [as expected from Fig. 2(d)].

Fig. 3 Temporal dynamics of pump (solid black) and probe (solid red) energies for δs = −3 and pump detuning: (a)–(c) δp = −2.2 (peak power Pp = 3); (b)–(d) δp = −3.1 (peak power Pp = 6). The left column cases (a)–(b) and right column cases (c)–(d) are relative to τ = 1 and τ = 4, respectively. The (blue) dashed line is the driving pump Pp(t) (Ps is a cw signal).

The SP, however, is greatly affected by the carrier lifetime, and can be inhibited by τ large enough [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] .

]. This is illustrated in Figs. 3(c)–3(d), where we display the dynamics obtained with 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 [δp = −2.2, Fig. 3(c)], the MPE is reached in up-switching and maintained until the pump ramps down, whereas for δp = −3.1 [Fig. 3(d)], MPE is reached again only in down-switching.

In terms of typical dimensional quantities that correspond to pump-probe experiments performed in III–V photonic crystal cavities [15

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

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] .

], operating with τ = τr0 = 1 and τr = 8 ps amounts to have (at telecom wavelength λ0 = 1.55 μm) Q ≃ 5000. The dynamics in Fig. 3(a)–3(b) can be observed by operating with the following dimensional detunings: Δλs = −0.465 nm (δs = −3), and Δλp = −0.341 nm (δp = −2.2) or Δλp = −0.481 nm (δp = −3.1), respectively. The injected powers turns out to be Pin = 2 mW and Pin = 4 mW, respectively. With the same carrier lifetime τr = 8 ps, 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: Δλs = −1.937 nm, Δλp = −1.421 nm and Pin = 17.25 mW (δp = −2.2), or Δλp = −2 nm and Pin = 37.73 mW (δp = −3.1).

For completeness, we have also investigated the temporal dynamics corresponding to the case in Fig. 2(e), namely for pump detunings so large (δp = −15) that the MPE is obtained at a critical energy coincident with the first knee of the response [ EcEb]. For such large values of |δp|, when the driving power is slightly higher than the value Pp+ of the knee, the cavity dynamics is ruled by the up-switching of the pump to the upper branch, which then undergoes SP. This is illustrated in Fig. 4(a) for δp = −15 and Pp = 250 (corresponding in real-world units to Δλp = −2.31 nm and pump power Pin = 168.4 mW), and 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 Pp = 248.5 (Pin = 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 Pp required to reach the MPE.

Fig. 4 As in Fig. 3 for δs = −3, δp = −15, and τ = 1, contrasting two dynamical behaviors: (a) pump self-switching (dark curve) towards a SP state [the inset shows the corresponding limit cycle in the phase plane Re(ap) − Im(ap)], obtained with maximum driving power Pp = 250 (slightly larger than knee value Pp+); (b) stable behavior for Pp = 248.5 (slightly lower than Pp+), with MPE reached with pump on the lower branch (note the different vertical scale in the two plots).

4. Effect of Losses

Fig. 5 Effect of TPA: level plots of (a) MPE; (b) CR, in the plane (α, δs) for fixed optimum value of pump detuning ( δpu=2.2) and τ = 1 (labels UB and NSB stand for uppe and negative slope branch, respectively). (c)–(d) Temporal pump-probe dynamics for α = 0.1 and τ = 1, contrasting: (c) stable probe switching for δp = −2.2 (MPE of 0.55 and a CR of 5.5) and (d) SP-dominated dynamics for δp = −3.1.

Based on such maps one can still optimize the pump-probe switching. For instance, with τ = 1 and α = 0.1, the choice of probe detuning, for instance δs = −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 (δp = −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 (EH = 1.9, from the linearized analysis 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] .

]) larger than the value of the jump on the UB (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 δp = −3.1 [comparable with the lossless case in Fig. 3(b)], still leads to SP, besides reaching a reduced MPE in down-switching.

5. Conclusions

In summary, the analysis of CME for pump-probe operation in a semiconductor nanocavity allows us to assess the optimized conditions for pump-induced switching of the probe beam. For a given desired contrast ratio, the optimization in terms of reaching the best probe efficiency with minimum driving pump power requires the proper choice of the pump detuning, while a constraint on the characteristic lifetime of the carriers is necessary to avoid self-pulsing.

Acknowledgments

This work has been supported by the 7th framework of the European Commission through the COPERNICUS (www.copernicusproject.eu) project.

References and links

1.

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2.

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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] .

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I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009) [CrossRef] [PubMed] .

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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] .

18.

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] .

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] .

21.

V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A 83, 043816 (2011) [CrossRef] .

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 20, 7454–7468 (2012) [CrossRef] [PubMed] .

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 85, 031803(R)(2012) [CrossRef] .

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 86, 063808 (2012) [CrossRef] .

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. 6, 554–559 (2012) [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] .

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] .

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

  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]
  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. Express13, 2678–2687 (2005). [CrossRef] [PubMed]
  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. Express14, 377–386 (2006). [CrossRef] [PubMed]
  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.90, 101118 (2007). [CrossRef]
  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 μm,” Opt. Lett.33, 1908–1910 (2008). [CrossRef]
  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. Photonics4, 477–483 (2010). [CrossRef]
  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. Photonics6, 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. A84, 043847 (2011). [CrossRef]
  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.14, 74–76 (2002). [CrossRef]
  10. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett.31, 341–343 (2006). [CrossRef] [PubMed]
  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]
  13. 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]
  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. A79, 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. A83, 051802(R)(2011). [CrossRef]
  18. 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]
  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. A80, 033805 (2009). [CrossRef]
  21. V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011). [CrossRef]
  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. Express20, 7454–7468 (2012). [CrossRef] [PubMed]
  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. A85, 031803(R)(2012). [CrossRef]
  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. A86, 063808 (2012). [CrossRef]
  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.6, 554–559 (2012). [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]
  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. A85, 063824 (2012). [CrossRef]

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