Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity
Alfredo de Rossi, Michele Lauritano, Sylvain Combrié, Quynh Vy Tran, Chad Husko
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photoniccrystal nanocavity
Alfredo de Rossi, ∗ Michele Lauritano, † Sylvain Combri´e, Quynh Vy Tran, and Chad Husko ‡ Thales Research and Technology, route d´epartementale 128, 91767 Palaiseau, France (Dated: November 9, 2018)We investigate the nonlinear response of GaAs-based photonic crystal cavities at time scaleswhich are much faster than the typical thermal relaxation rate in photonic devices. We demonstratea strong interplay between thermal and carrier induced nonlinear effects. We have introduced adynamical model entailing two thermal relaxation constants which is in very good agreement withexperiments. These results will be very important for Photonic Crystal-based nonlinear devicesintended to deal with practical high repetition rate optical signals.
PACS numbers: 42.70.Qs,42.65.Pc
I. INTRODUCTION
With both a small modal volume and large qualityfactor, optical microcavities exhibit a greatly enhancedlight-matter interaction and a strong non-linear opticalresponse[1, 2, 3, 4]. An emerging class of optical micro-cavities is based on air-clad two-dimensional (2D) Pho-tonic crystals (PCs). A Q-factor greater than 10 wasachieved with this technology [5, 6, 7] which also al-lows small modal volumes ( ≈ ( λ/n ) or 10 − m . More-over, 2D PC technology is a planar technology whichis particularly suited for the fabrication of photonic cir-cuits. Therefore, the move towards all-optical process-ing, long considered impractical, has changed dramati-cally and new possibilities have been opened [8]. Im-pressive experimental demonstrations of low-energy ( f J )optical bistability and all-optical switching [9, 10, 11],wavelength conversion [12], optomechanical effects [13]and dynamical control of the cavity lifetime [14] are re-cent noteworthy achievements. Most of these resultscome from a specific technology, i.e. silicon-based air-clad PCs where the nonlinear process involved is two-photon absorption (TPA) followed by plasma-inducedand thermally-induced refractive index change. The op-tical power required is quite small ( ≈ f J ) and canbe very fast (70 ps by ion implant, as the carrier re-combination time result much shorter in these nanostruc-tured devices than in bulk silicon [15, 16]. It has beenpredicted that, owing to strong light-matter interaction,nonlinear properties can be engineered by introducingnanoparticles[17].The desire for an even faster response time motivatesthe research on alternative materials with a strong opticalKerr effect. Much progress has been made in processingchalcogenide crystals with high-Q PC microcavities havebeen demonstrated recently [18]. III-V semiconductorsare also good candidates for optical switching as they ∗ Electronic address: [email protected] † Also at Engineering Department, University of Ferrara, Italy. ‡ Also at Columbia University, New York, U.S.A. have two very attractive features, compared with silicon.First, the optimization of the Kerr effect with respect toTPA [19, 20]. Self phase modulation due to Kerr effecthas been demonstrated in 2D PCs recently[21, 22]. Anadditional feature of III-V semiconductors is the possibil-ity of exploiting a strong nonlinear effect related to ab-sorption saturation in active structures, such as QuantumWells (QWs) and Quantum Dots (QDs). Fast nonlineardynamics [23], leading to bistability [24] and excitabil-ity [25], has been observed in InP-based 2D PCs withInAsP QWs which are designed to operate with band-edge modes coupled to off-plane free space beams. Lowpower and very fast (2 ps) switching (15 ps for a com-plete on/off cycle) has been demonstrated with a Sym-metric Mach Zehnder-type all-optical switch made withInAs/AlGaAs quantum dots [26].In this paper we focus on the dynamics of the pro-cesses initiated by TPA in GaAs PCs. Although, ideally,all optical switching requires the Kerr effect, in prac-tice TPA, followed by a carrier induced index change,is still a very attractive approach, because of the rela-tive simplicity of the technology. In particular, there isno active material, e.g. QDs, or phase matching condi-tion required. Moreover, compared to silicon, the TPAcoefficient in GaAs is tenfold higher [27] and the non-radiative carrier lifetime in patterned structures can bevery short (8 ps) [28]. A fast and strong nonlinear re-sponse is therefore expected in GaAs PC nanocavities.Very recently, we have demonstrated ultra-fast (6 ps re-covery time) and low power ( ≈
150 fJ) modulation inGaAs Photonic crystal cavities[29]. We have investigatedoptical bistability in high-Q ( Q = 0 . × ) PC cavi-ties on GaAs and reported an ultra-low threshold power( µW range) [30, 31, 32]. After the submission of thismanuscript, memory operation have been demonstratedin a InGaAsP PC cavity[33]In these experiments, only the slow ( µs ) regime hasbeen explored, which is dominated by the thermally in-duced index change. This is also due to the high ther-mal resistance of the membrane structures compared to,for instance, micro-disks [34] or membranes bonded to a SiO cladding [24] and the lower thermal conductivity ofGaAs with respect to silicon.In this paper we investigate the response of PC micro-cavities at a much faster modulation rate (up to 50 MHz),which is well beyond the typical thermal relaxation timeof photonic devices. Under these conditions, the analysisof nonlinear responses, such as bistability, cannot be ex-plained with static models. First of all, moving to fastertime-scales modifies the relative influence of thermal andcarrier plasma effects. While at very fast time scales (ps)the dynamics tend to be controlled by the carrier lifetime,we will show that in the range between 1 - 100 ns the dy-namics results from the interplay of fast thermal effectsand carrier plasma index shift. We introduce a modelthat incorporates two thermal relaxation constants. Thisis necessary to explain this dynamics and the fact that,despite the high thermal resistance of PCs microcavities,there are thermal effects which develop in less than 10ns. Section II is devoted to experiments made on a PCmicrocavity coupled to a waveguide. In section III wewill introduce our multi-scale model. Discussion of theresults forms the body of section IV. II. EXPERIMENTAL SETUP AND SAMPLEDESCRIPTION
The photonic crystal structure studied here is the well-known optimized three missing holes (L3) PC microcav-ity [35] in an air slab structure (thickness is 265 nm)based on a triangular lattice (period a = 400 nm ) of holeswith radius r = 0 . a . The holes at the cavity edge wereshifted by s = 0 . a . The cavity is side-coupled (thespacing is 3 rows) to a 1 mm long line-defect waveguidealong the Γ K direction, with width W = 1 . √ a (seefigure 1). The fabrication process and the detailed linearcharacterization of similar structures are described else-where [36]. The loaded and intrinsic Q-factors, 7,000 and30,000, respectively, are estimated from measurementsusing the procedure discussed in a previous paper [37].The cavity resonant wavelength is 1567 nm. The charac-terisation setup consist of a tunable external cavity semi-conductor laser (Tunics) with a relative accuracy of 1 pm and narrow linewidth ( << FIG. 1: (Color online) Top: SEM image of the sample, alsoshowing a line-defect photonic crystal waveguide with sidecoupled L A. Sinusoidal modulation
At a low modulation rate (kHz) it has been shown byseveral groups[9, 10, 11, 31, 32] that bistability occurs ata fairly low optical power ( µW range, coupled into thewaveguide). The dominant nonlinear effect is thermalinduced index change because of heating resulting fromcarriers generated by TPA. Since this effect leads to a red-shift of the cavity frequency, bistability is observed whenthe initial detuning ∆ λ | t =0 = λ laser − λ cavity | t =0 ) is pos-itive, where λ cavity | t =0 is the cold cavity resonant wave-length. This is shown in fig. 2. Typically, the detuningis set between √ / λ F W HM = 220pm in our case. If the modulation frequency is increasedwhile the detuning and the average signal power are keptconstant, the bistable behavior disappears. For instance,with a detuning δλ ≈ pm ≈ . λ F W HM and a mod-ulation frequency of 15 kHz, bistability is observed in thissample at 80 µW . The power coupled into the waveguideis estimated considering the amplified laser power andthe input objective coupling factor (here ∼ -10 dB) as inour previous work [31].When the modulation frequency is increased to 1 MHz,the bistable effect and all traces of nonlinear distortionof the transmitted signal disappear. This could be ex-plained in terms of the slow response of the thermo- FIG. 2: (Color online) Low modulation frequency experiment.Oscilloscope traces (a) showing the typical bistable responseof a PC microcavity at low modulation frequency (15 kHz).The average input power in the PC waveguide is about 80 µ W.The bistable cycle is also sketched and the transitions 1-4 arealso marked. Detunings are +0.4, +1.5 and +1.55∆ λ F WHM respectively. Scheme representing a cavity with a negativedetuning b) and going to resonance due to nonlinearly inducedred-shift c). optical effect. An important question arises whether thecarrier plasma effect, much faster but weaker, will thentake over. To answer that, the signal power and themodulation frequency were increased further and the de-tuning (still positive) was swept continuously from zeroto about +700 pm, with the laser always on. This de-tuning value is larger than in low frequency experiments.A strong nonlinear distortion of the transmitted signalis then observed. The onset of the nonlinear behav-ior is simultaneous to the observation of a bright spotthrough the IR camera, thus indicating that the reso-nant frequency of the cavity is red-shifted under a sinu-soidal modulation pump. In particular, fig. 3 shows atypical output signal when the modulation frequency isincreased to 50 MHz and the average power coupled inthe waveguide is about 2.5 mW. The thermo-optic effect,driving the bistability at low modulation frequency, in-tegrates over time. This explains ultra-low power levels.At higher modulation frequencies the power required toobserve nonlinear effects increases. The nonlinear behav-ior is observed with a positive detuning from 600 pm and670 pm ( ≈ λ F W HM ). We don’t believe that theobserved nonlinear distortion corresponds to a bistablebehavior. Indeed, a fast nonlinear mechanism (plasmainduced index change) should be dominate all other non-linearities. That is probably not the case, as the short lifetime of carriers in GaAs compared to silicon dras-tically reduces the strength of this effect, compared tothermo-optic effects.
FIG. 3: (Color online) High modulation frequency experi-ment: oscilloscope traces of the transmitted signal with si-nusoidal modulation (50 MHz) for different detunings. Theaverage power is about 2.5 mW.
The nonlinear distortion observed in figure 3 is quitegeneral and can be reproduced at different modulationfrequencies in the 10 - 100 MHz range. The nonlineartransitions are not sharp, compared to the modulationperiod (response time is a few ns). This time is howevermuch faster than what could be attributed to a ther-mal effect. Additionally, the period, T = 20 ns, is muchshorter than what is estimated to be the thermal recoverytime in air slab PC cavities [9]. We claim that the fastercarrier plasma effect plays a key role at the ns time scale,while the thermal effect accumulates to strongly redshiftthe cavity.This mechanism is represented in Fig. 4. Initially (a),the cavity is at room temperature and laser is detunednegatively (∆ λ <
FIG. 4: (Color online) Scheme representing the interplay ofthermal and carrier plasma effect and how they are relatedto the detuning. The system is at rest (cavity at its coldfrequency) and the laser is switched on (a). The laser wave-length is increased but heating keeps the cavity red detuned(b). When the cavity temperature is high enough, the meanresonant wavelength get closer to the laser and the nonlineardistortion is apparent (c).
B. Low duty cycle excitation
The cavity is excited with low duty cycle, square pulses(duration ∆t = 20 ns, period T=2 µ s), so that the cav-ity has time to cool down before the arrival of the nextpulse. We refer to this case as a “non-heating pulse”because the average temperature of the cavity remainsclose to room temperature. Fig. 5 reports the input andthe transmitted pulse depending at various wavelengthdetunings. The input power in the pulse is estimated tobe on the order of 10 mW. A more accurate determina-tion of this value was complicated by the very low fillingfactor and the use of the EDFA, which is not designedto operate in these conditions. This lead to some fluc-tuation in the level of the amplified pulse. Furthermore,very low duty cycle means that the the power containedin the pulse is a fraction of the amplified spontaneousemission(ASE), which means that it is difficult to deter-mine the exact value of the measurement of the averagepower. Let us first consider a negative detuning: e.g.∆ λ ≈ − . λ F W HM (fig. 5a). The leading edge of theoutput pulse rises with almost linear slope and is about5 ns long. For zero or moderate positive detuning (e.g.∆ λ between 0 and ≈ ∆ λ F W HM ), the output pulse typ-ically has a two step shape with the leading part of thepulse having a lower level (fig. 5b and c). The transi-tion between the two levels is sharp (a few ns) and movesfrom the leading edge to the trailing edge of the pulse asthe detuning is increased. If detuning is increased fur-ther, the pulse takes on an almost triangular shape with a sharp leading edge and a long trailing edge (figure 5 d)before losing any signature of nonlinear distortion.
FIG. 5: (Color online) Experiment with “non heating” pulses.Input (dashed) and output (solid) pulses as a function of thedetuning. (a) ∆ λ = − . λ F WHM , (b) +0 . λ F WHM , (c)+0 . λ F WHM and (d) +1 . λ F WHM . The pulse period is2 µs . Interestingly, the initial leading edge of the outputpulse is very similar in the four cases considered in fig.5, i.e. we don’t see low transmittance associated to theon-resonance case, as long as the power is high. We seethis behavior at much lower power only. We believe thatthis is due to a fast carrier effect which detunes the cavityfaster than our detection apparatus is able to measure (1ns).It is also important to note that no signature of aplasma-induced bistable state is observed when the de-tuning is negative. We think that this is because of fastheating, which dominates the plasma effect before thepulse is extinguished. These features are well explainedby our model.
III. NONLINEAR DYNAMICAL MODEL
The coupled-mode model developed here is based onprevious optical microcavity literature. Particular struc-tures that have been investigated include: microdisks [38]and photonic crystal cavities [39, 40, 41]. The dynamicalvariables are: the optical energy inside the cavity | a ( t ) | ,the free carrier density N ( t ) and the cavity and mem-brane temperature T ( t ). The model considers a singlemode [47] cavity (resonance is at ω and unloaded qual-ity factor Q ), which is side-coupled to a waveguide (theloaded Q-factor is Q ). The modulation of the transmis-sion observed in fig.1 is related to the Fabry-Perot reso-nance due to the finite reflectivity at the waveguide endfacets. This spatially extended resonance has no impacton the nonlinear response (other than merely modulatingspectrally the coupling into the cavity), since the associ-ated field intensity is orders of magniture smaller than inthe cavity. The field in the cavity follows the equation: ∂a ( t ) ∂t = ( iω − iω L − Γ tot a ( t ) + r Γ c P in ( t ) (1)where ω L is the laser frequency, Γ c = ω ( Q − − Q − )gives the cavity to waveguide coupling strength, Γ tot isthe inverse (instantaneous) cavity lifetime, P in ( t ) is thepower in the waveguide and ω = ω +∆ ω NL is the instan-taneous cavity frequency. Following [39, 40], the nonlin-ear change of the cavity frequency is given by:∆ ω NL = − ωn eff ∆ n = − ωn eff [ n I cn V Kerr | a ( t ) | dndT ∆ T ( t ) + dndN N ( t )] (2)Here n is the refractive index of the bulk material and n eff the effective refractive index in the cavity, i.e.: n eff = R n ( ~r ) | E ( ~r ) | dV / R | E ( ~r ) | dV , and n I is theKerr coefficient and V Kerr the Kerr nonlinear volume,defined as: V − Kerr = R n I ( ~r ) /n I | E ( ~r ) | n ( ~r ) dV ( R n ( ~r ) | E ( ~r ) | dV ) (3)The refractive index change due to the plasma effect is dn/dN = − ω p / nω N , with ω p = e N/ǫ m ∗ the plasmafrequency. In principle, holes and electrons both con-tribute to this effect, however, given the much smallereffective mass of electrons in GaAs, the contribution ofholes is negligible. The inverse instantaneous photon life-time is: Γ tot = ω Q + Γ T P A + Γ
F CA (4)the first term is the inverse linear cavity lifetime, Γ
T P A and Γ
F CA are the contributions from two photon (TPA)and free carrier absorption (FCA), respectively. TheTPA term is: Γ
T P A = β c /n | a ( t ) | /V T P A with β representing the TPA coefficient in units of m/W and V T P A = V Kerr the nonlinear effective volume[40], whilethe free carrier absorption is proportional to the com-bined free carrier density Γ
F CA = ( σ e + σ h ) N ( t ) c/n . Theevolution of carrier density follows the rate equation: ∂N ( t ) ∂t = − N ( t ) τ N + c /n β V T P A ~ ω V car | a | (5)Here ~ is the reduced Planck’s constant, τ N is the effec-tive carrier lifetime, and V car is the volume in which thecarriers spread and recombine[40]). In this approxima-tion, the population decay is dominated by recombina-tion at the surface, due to the large surface to volume ratio typical of photonic crystal strucures. Carriers areassumed to spread and distribute homogeneously withinthe carrier volume V car , which is assumed to correspondto the region of the membrane delimited by the holesaround the cavity. This approximation is rough, but it isa reasonable choice. FIG. 6: (Color online) Modelling heat diffusion in the PCmembrane. Local cavity temperature T c vs. time calculatedwith a 2D thermal diffusion equation (dots) and fitting (solidthick and thin lines) with the model eq. 6. Inset: enlargedview of the short and the long term behaviour. Inset right: fit-ted τ th,m = 200 ns with a single exponential (dashed line).Fitwith 4 unconstrained parameters (thin line). Constrained fitwith τ th,m = 200 ns τ th,c = 8 . ns (thick solid line, mainplot and inset left). Thermal capacitances for the cavity andthe membrane are 0 .
43 10 − W/K and 3 . − W/K respec-tively.
Because of the high thermal resistance in membranePCs, thermal effects are very important and must bemodelled appropriately. This is, for instance, the scopeof ref.[42]. In contrast with existing literature, we intro-duce a more complicated model for the thermal response.We assume that the heat generated in the cavity hasnot yet diffused to the border of the membrane at thetime scale of interest (1 ns). Therefore, a very impor-tant physical quantity is the thermal capacitance of thecavity C th,c = c v ρV th,c , associated with a small regionof the membrane, with volume V th,c roughly correspond-ing to the cavity volume. Heating and the spreading ofthe heat are modelled by solving the two dimensionalheat diffusion equation. The heat is generated for sometime (10 ns) inside the cavity and then the system is al-lowed to cool down. We assume that the radiative andconvective contributions are negligible; therefore, sincethe membrane is suspended in air, all the heat has toflow through it. The result of this calculation is shownin fig. 6. The cavity temperature T c increases with arate that is governed by C th,c and T c decreases at twoexponential time scales. The fast time scale τ th,c is asso-ciated to a relatively fast transfer of heat from the cavity,where it is generated, to the neighbouring region in themembrane. We associate a second thermal capacitance TABLE I: Physical paramenters used in the dynamical model.Parameter Symbol Value Ref.TPA coefficient β ( cm/GW ) 10 . n I ( cm /W ) 1 . − [27]Loaded Q Q Q V mod . λ/n ) calc.TPA Volume V TPA . λ/n ) calc.Carrier Volume V car V mod Thermo-optic coeff. dn/dT ( K − ) 2 .
48 10 − [43]Therm. eff. vol.(c) V th,c . ∗ V mod calc.Therm. eff. vol. (m) V th,m ∗ V mod calc.Specific Heat c v ρ ( W/K m − ) 1.84 10 [43]Carrier lifetime τ N τ th,c τ th,m
200 ns calc.Therm. resistance R th = P τ th C th . K/W calc.FCA cross section σ e,h (10 − m ) 9 . λ . µm ) . [44] C th,m to the whole membrane. The second time scale τ th,m takes into account the spreading of the heat overthe rest of the membrane (with temperature T m ) to thebulk semiconductor structure. This picture is substan-tially different from previous models with a single timeconstant τ = C th,c R th that is obtained from the thermalresistance R th , defined as W = R th ∆ T (ratio betweenthe increase of the temperature over the heating powerat steady state). We will show that these two time scalesplay a crucial role in understanding the system responseunder sinusoidal and single pulse excitation. The corre-sponding model entails therefore two auxiliary equations: ∂T c ( t ) ∂t = − T c ( t ) − T m ( t ) τ th,c + 2 ~ ωτ N V car C th,c V th N ( t ) (6) ∂T m ( t ) ∂t = − T m ( t ) − T τ th,m − T ( t ) − T τ th, C th,c C th,m (7)The thermal effective volume of the cavity and of themembrane V th,c V th,m and their thermal relaxation times τ th,c , τ th,m are obtained by fitting the solution of theheat diffusion equation with the solution of the two timeconstant model (eq. 6). If we fit all four parameters (twotime constants and two capacitances) the fit turns out tobe very good in the first 50 ns, yet tends to underestimatethe long time constant and therefore to underestimatethe total thermal resistance. To prevent this, we first fita simple exponential to the long term behaviour, whichgives a time constant of approximately 200ns. Then we fitthe complete curve by constraining τ th,m = 200 ns . Theresult is shown in fig. 6 and confirms that this modelis well adapted to describe the thermal behaviour of PCmicrocavities. The result of this calculation is given intable I. IV. DISCUSSION
Simulations have been carried out with parametersgiven in Table I. All of them are well known or can be cal-culated or measured with reasonable accuracy. The car-rier volume V car is the exception. The impact of carrierdiffusion has been investigated theoretically in a recentpaper [45], providing some hints to explain a fast recov-ery time in Silicon PCs. The carrier lifetime in patternedGaAs (e.g. 2D PCs) is however much shorter than in Sil-icon. Very recently we estimated it to be about 6 ps inour GaAs cavities[29], which is consistent with previous(8 ps) estimates for GaAs PC structures [28].In the limit where the dynamic is much slower than τ N , which the case considered in this work, τ N and V car play the same role and what matters here is the ratio τ N /V car . To show that, let us consider the density of thegenerated carriers : N ( t ) = c /n β V T P A ~ ω τ N V car | a | (8)The blue-shift of the cavity resonance due to the gen-erated carriers is therefore proportional to the instan-taneous power absorbed, P a . The red shift induced byheating is proportional to the absorbed energy and tothe inverse of the thermal capacitance. Then, the rela-tive strength of thermal and carrier effects depends onthe ratio τ N /V car and on the thermal capacitance. Thedynamical behaviour of the system investigated here isbasically determined by these two quantities.Modelling was performed by varying only one param-eter, the carrier volume, and adjusting the input powerand the detuning around reasonable values. All othervalues are well known or calculated precisely.Fig. 7 reports the simulated responses of a square pulseat various detunings. When the initial detuning is nega-tive (e.g. ∆ λ (0) = − pm , fig. 7 a and b), the carriereffect is responsible for the drop of the leading edge ofthe transmitted pulse. The signal coupled in the cavityis strong enough to cause considerable generation of car-riers and the resulting blue-shift tends to tune the cavityinto resonance so that transmission is reduced (the cav-ity is side-coupled). Within a ns time scale, the heatingfollowing carrier recombination red-shifts and thereforedetunes the cavity again. Such fast heating arises fromthe small thermal capacitance of the cavity. After thepulse is extinguished, the cavity is red-shifted by about100 pm with respect to the initial position and recoversits initial state in a fraction of µs . Let us now considerthe case of zero detuning (fig. 7c and d). The carrierinduced index change produces an instantaneous (withrespect to the time scale considered here) blue shift thatadds to the initial detuning. Heating, following absorp-tion, tunes the cavity back and transmission decreasesonce again. Thus more carriers are generated and theplasma effect tends to oppose heating but, in the end,the heating dominates. At these modulation rates ther-mal relaxation is not sufficiently fast to dissipate heat FIG. 7: (Color online) Simulation of the response (solid) to asingle pulse (dashed), depending on the detuning ∆ λ (0): -120pm (a,b), 0 pm (c,d), 80 pm (e,f) and 150 pm (e,f). Right:corresponding instantaneous frequency shift ∆ λ NL of the cav-ity resonance (solid line) and carrier plasma (thin-dashed) andthermo-optical contributions (dotted). The laser frequency isalso shown (thin solid). P in = 1.2 mW, 0.9 mW, 1.2 mW and1.0 mW respectively. from the cavity after each duty cycle. Thus heat accu-mulates over time, causing a net red-shift of the cavityresonance. When the instantaneous detuning ∆ λ NL be-comes positive, carrier generation starts to decrease andso does the carrier induced shift. This initiates a positivefeedback that quickly detunes (red) the cavity and makesthe carrier density drop very fast. The delay between thepulse leading edge and this transition is clearly related tothe amount of the initial detuning (fig. 7e and f). Whenthe initial detuning is positive and large enough (fig. 7gand h), then heating tends to tune the cavity into reso-nance, thus reducing the transmission but also generat-ing carriers. The plasma effect will partially oppose thered-shift, thus explaining an almost linear change withtime. The sharp step observed in fig. 5b,c can onlybe explained by the thermal effect overtaking the carrierplasma index change. This step cannot be reproducedfor any choice of free parameters if the plasma inducedindex change is suppressed in the model. This is shownin figure 8. In panel a and b we ran the same simulationas in fig. 7e, but we have suppressed the contribution ofcarriers to the frequency shift of the cavity. In this casethe transmission increases with a linear, but finite, slopeafter the dip, independent of the pulse power. This isnot what we see experimentally. Conversely, when theplasma effect is included, the sharp transition is repro- FIG. 8: (Color online) Simulated response when the contri-bution to the plasma shift from carrier plasma is suppressed(a) and full model (c). The excitation power is 1.2 mW (thickline). The case with P in =1.1 mW and 1.3 mW are also plot(thin lines). The detuning ∆ λ (0) is 80 pm. The correspond-ing instantaneous frequency shifts ∆ λ NL is also plot (b,d) forthe case P in =1.2 mW. duced correctly. There is a physical reason for that: theobserved steep step requires a fast index change with,say, a negative sign, and a slower index change, withopposite sign. A simple explanation is that at first theplasma effect dominates (over a very short time, a fewns) before thermal heating takes over. If a plasma effectis ruled out, there is no way to explain the experimentalresults. A Kerr effect cannot be considered because ithas the same sign as the thermal effect. After the sub-mission of this manuscript, critical slowing down (CSD),implying a transition from a low (off-resonance) to a high(on-resonance) transmission state, has been reported inInP-based PC cavities[46]. This effect also results into astep in the pulse response. We believe however that thisis a different mechanism than what we describe here. Thereason is that our system is side-coupled, thus CSD wouldmanifest as a transition from a high (off-resonance) to alow (on resonance) transmission state. We did not ob-serve that. A possible explanation is the carrier lifetimein GaAs PCs being mush shorter than in Silicon or InPPCs. This implies that the plasma dispersion effect ismuch weaker than the thermal induced index change.We found experimentally that the response of the cav-ity is very sensitive to the power of the pulse, in particularin the case reported in 5e. This behavior is well repro-duced by theory. This means that input power must beset precisely in simulation and it cannot be the same forall the detunings considered here. Indeed, experimen-tally the peak pulse power is not constant since, a) theoutput power from the amplifier showed a marked depen-dence on slight change in wavelength, b) the waveguidetransmission is modulated by a strong Fabry-P´erot ef-fect (fig. 1). The fact that the detuning in experiments FIG. 9: (Color online) Simulation of the response of the cavityto a sinusoidal excitation for the following detunings (fromtop to bottom): 440 pm (a,d), 490 pm (b,e) and 500 pm(c,f). Input peak power is 1 mW. The transmitted pulses areshown on the top (a,b,c), the instantaneous frequency shift(solid think curve) and the thermo-optical contribution aloneare shown on the bottom (d,e,f). and in the theory are not the same can be understood byaccepting some error in the measurement of the detun-ing. The reason is that the laser was directly modulatedwhich induced some deviation from the nominal settingof the laser on the order of 100 pm.This relatively fast (few ns rise-time) nonlinear effectreported in fig. 5 is well understood through modelling.The complexity of the model is justified by the number ofdifferent time scales present. An important point of thispaper is to demonstrate that the same model is able toexplain also the response to a sinusoidal excitation. Theresponse is calculated for a time long enough to ensurethat we reach the steady-state regime. The simulationsshown in fig. 9 are carried out in order to mimic exper-imental conditions, that is, the excitation wavelength isdetuned negatively with respect to the cold cavity fre-quency. As the excitation is turned on, the wavelengthis increased adiabatically with respect to the modulationperiod. Indeed it is found that the cavity stays red de-tuned, and is pushed towards longer wavelengths, untilthe excitation power and the heat generated is enough tosustain the detuning.The simulations clearly support the experimental re-sults shown in figure 3 and interpretation in fig. 4. In-deed, if we now compare the measured and calculatedoscilloscope traces we conclude that the dynamics are ac-curately reproduced, as the detunings used in modellingcorrespond to what is measured experimentally.In particular, the response is very well understoodby looking at the instantaneous frequency of the cav-ity, which crosses the laser frequency at different points depending on the detuning. If the detuning ∆ λ is toosmall, the cavity wavelength is always higher than thelaser frequency, as the instantaneous cavity nonlinearshift ∆ λ NL = λ cavity ( t ) − λ cavity (0) due to heating ismuch larger over the whole modulation cycle. When thedetuning ∆ λ increases, ∆ λ NL also increases on average,as the resonance is closer to the laser frequency, but lessthan ∆ λ does. Thus, there exists a combination of pulsepower, modulation frequency and detuning such that ∆ λ and ∆ λ NL are close one to each other. If ∆ λ is increasedfurther, heating is not enough to keep ∆ λ NL close to itand nonlinear distortion is lost. The role of the plasmainduced index change and the interplay with the thermo-optic effect is also clear in figure 7(b,d,f,).It is interesting to note that the plasma effect van-ishes when t < ns and t > ns in figs. 9b,d and f,although minima of the sinusoidal excitation are at t=0and t=20 ns (figs. 9a,c,e). This important point is un-derstood when considering that while the plasma effectis almost instantaneous (within the time scale relevantto this work), the thermo-optic effect follows the timeintegral of the optical power. At time t = 0 the carrierpopulation is negligible and the cavity is cooling down,and therefore is approaching the laser wavelength. As thepower inside the cavity increases, carriers are generatedwith a rate proportional to the power squared. A carrierinduced index change builds up very fast and producesthe difference between the thin and the thick curves. Thepower absorbed produces heating. Instantaneous heat-ing is the time integral of instantaneous absorbed powerand therefore builds up with a delay with respect to theplasma effect. The thermal effect then dominates theplasma effect. As soon as the resonance is crossed againand the detuning becomes comparable to half the cavitylinewidth (100 pm), nonlinear absorption decreases veryquickly and the carrier population declines accordingly.Therefore the carrier effect disappears. The cavity againstarts to cool down and the cycle begins anew. V. CONCLUSION