Thermo-optical dynamics in an optically pumped Photonic Crystal nano-cavity

Linear and non-linear thermo-optical dynamical reg imes were investigated in a photonic crystal cavity. First, w e have measured the thermal relaxation time in an InP-based nano-cavity with quantum dots in the presence of optical pumping. The experimental m ethod presented here allows one to obtain the dynamics of temperature in a nanocavity. It is based on reflectivity measurements of a cw probe beam cou pled through an adiabatically tapered fiber. Characteristic times o f 1.0±0.2 μs and 0.9 ±0.2 μs for the heating and the cooling processes were o btained. Finally, thermal dynamics were also investigated in a thermo-optical bistable regime. Switch-on/off times of 2 μs and 4 μs respectively w ere measured, which could be explained in terms of a simple non-linear dynamical representation. 2008 Optical Society of America OCIS codes: (190.4390) Nonlinear optics, integrated optics; (1 90. 450) Bistability; (130.3120) Integrated optics devices; (060.1810) Co uplers, switches, and multiplexers. References and links 1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu and A. Imamog lu, “Quantum nature of a strongly coupled single quantu m dot–cavity system,” Nature 445, 896 (2007); S. Laurent, S. Varoutsis, L. Le Gratiet, A. Lemaître, I. Sagnes, F. Raineri, A. Levenson, I. Robert-Phili p and I. Abram, “Indistinguishable single photons from a sin gle quantum dot in two-dimensional Photonic Crystal cavity,” Appl. Phys. Lett. 87, 163107 (2005). 2. M. Notomi, T. Tanabe, A. Shinya, E. Kuramochi, H. T aniyama, S. Mitsugi, M. 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Mc Nab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65 (2005). 8. T. J. Johnson, M. Borselli, and O. Painter, “Self-i nduced optical modulation of the transmission throu gh a high-Q silicon microdisk resonator,” Opt. Express 14, 817-831 (2006). 9. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical th ermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742 (2004). 10. A. M. Yacomotti, P. Monnier, F. Raineri, B. Ben Bak ir, C. Seassal, R. Raj, and J. A. Levenson, “Fast Thermo-Optical Excitability in a Two-Dimensional Ph otonic Crystal,” Phys. Rev. Lett. 97, 143904 (2006). 11. M. T. Tinker and J-B. Lee, “Thermal and optical sim ulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” O ptics Express 18, 7174-7187 (2005). 12. G. Tessier, G. Jerosolimski, S. Hole, D. Fournier, and C. Filloy, “Measuring and predicting the thermoreflectance sensitivity as a function of wave length on encapsulated materials,” Rev. Sci. Instru m. 74 (1), 495 (2003). (C) 2009 OSA 14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 17118 #111641 $15.00 USD Received 21 May 2009; revised 10 Jul 2009; accepted 10 Jul 2009; published 11 Sep 2009 13. L. Pottier, “Micrometer scale visualization of ther mal waves by photoreflectance microscopy,” Appl. Ph ys. Lett. 64, 1618-1619 (1994). 14. Y. Akahane, T. Asano, B.-S. Song and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944 (2003). 15. A. Michon, R. Hostein, G. Patriarche, N. Gogneau, G . Beaudoin, A. Beveratos, I. Robert−Philip, S. Laur ent, S. Sauvage, P. Boucaud, I. Sagnes, “Metal organic v apor phase epitaxy of InAsP/InP(001) quantum dots f or 1.55 μm applications: Growth, structural, and optical pro perties,” J. Appl. Phys. 104, 043504 (2008). 16. I. Hwang, S. Kim, J. Yang, S. Kim, S. Lee, and Y. L ee, “Curved-microfiber photon coupling for photonic crystal light emitter,” Appl. Phys. Lett. 87, 131107 (2005) 17. P. E. Barclay, K. Srinivasan, M. Borselli, and O. P ainter, “Probing the dispersive and spatial propert ies of photonic crystal waveguides via highly efficient co upling from fiber tapers,” Appl. Phys. Lett. 85, (2004) 18. C. Grillet, C. Smith, D. Freeman, S. Madden, B. L-D avies, E. C. Magi, D. J. Moss and B. J. Eggleton, “Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowir es,” Opt. Express 14, 1070 (2006). 19. See for example B. Maes, P. Bienstman, and R. Baets , “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778 (2005). 20. In Section 5 we study nonlinear thermo-optical effe cts which are shown to appear for a signal power gr ate than ~1 mW (see transmission curves as a function o f input power in Fig. 4c). 21. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuram ochi, and T. Tanabe, “Optical bistable switching ac tion of Si high-Q photonic-crystal nanocavities,” Opt. E xpress 13, 2678-2687 (2005). 22. P. Barclay, K. Srinivasan, and O. Painter, “Nonline ar response of silicon photonic crystal microresona t rs excited via an integrated waveguide and fiber taper ,” Opt. Express 13, 801-820 (2005). 23. Convection within the air gap can be neglected sinc e the Rayleigh number for an air gap of thickness δ between two rigid walls, for a few degrees of tempe rature increment, is Ra δ=g ∆T δ/aνT~10, whereas the onset for convection is Ra δ~2 10. See for example J. Taine and J. P. Petit, Heat transfert (Prentice-Hall, 1993). 24. F. G. Della Corte, G. Cocorullo, M. Iodice, and I. Rendina, “Temperature dependence of the thermo-opti c coefficient of InP, GaAs, and SiC from room tempera ture to 600 K at the wavelength of 1.5 μm,” Appl. Phys. Lett. 77, 1614 (2000). 25. F. Raineri, G. Vecchi, A. M. Yacomotti, C. Seassal, P. Viktorovitch, R. Raj and A. Levenson, “Doubly resonant photonic crystal for efficient laser opera tion: Pumping and lasing at low group velocity phot onic modes,” Appl. Phys. Lett. 86, 011116 (2005). 26. The thermal relaxation time for a PhC membrane on o xide can be easily calculated under the hypothesis of 1D vertical heat flow through the oxide layer to th e substrate. For instance, for a 250 nm-thick Si me mbrane (κSi=1.5 W/cm K, αSi=0.9 cm/s) in contact with a 1 μm-thick SiO 2 layer (κSiO2=0.013 W/cm K, αSiO2=0.006 cm/s), a numerical simulation of the 1D heat equation gives tth=950 ns. 27. A. R. A. Chalcraft, S. Lam, D. O'Brien, T. F. Kraus s, M. Sahin, D. Szymanski, D. Sanvitto, R. Oulton, M. S. Skolnick, A. M. Fox, and D. M. Whittaker, “Mode str ucture of the L3 photonic crystal cavity,” Appl. Ph ys. Lett. 90, 241117 (2007). 28. We use the estimated τth from the experimental results, τth~110 ns (see Section 6.1). This is an approximation since τth depends on the geometry of the hot spot which, in the resonant case, is given by the cavity volume, different from the pumped region given by the surfa ce illumination. 29. C. Sauvan, P. Lalanne and J.P. Hugonin, “Slow -wave effect and mode -profile matching in Photonic Crystal microcavities,” Phys. Rev. B 71, 165118 (2005).


Introduction
Active Photonic Crystal nanocavities have recently received much attention due to their importance for quantum optics and quantum electrodynamics [1], as well as for dynamical and nonlinear control of light in photonic circuits [2].In particular, in the context of active semiconductor devices, silicon and III-V-based nanocavities have been shown to exhibit interesting optically induced gain/absorption properties leading to ultrafast switching and bistability, among others [3][4][5][6].
Photon emission and/or carrier induced nonlinearities are obtained through the excitation of electrons in semiconductor conduction bands.Electron-hole nonradiative recombination processes play a fundamental role in the relaxation of excited electronic populations.These are mediated by phonons which become source of heat.While heating turns out to be an unwanted effect in most photonic devices because of detrimental thermal loading (specially in photonic crystal suspended membranes), it can also be used as a mechanism for fast switching (up to 10 MHz-bandwidth) as long as the dimensions of optical cavities are small enough [7][8][9].Moreover, in the context of novel non-linear dynamical mechanisms relying on multiple time-scale processes, it has been demonstrated that the so-called thermo-optical excitable dynamics may lead to repetition rates as high as 1 GHz [10] It is worth pointing out that, in all the above situations, an insight into dynamics of heat dissipation is of central importance since it provides information about the characteristic time scales to take into account when pumping the sample in order to avoid heating up the material, i.e. by modulating the optical pump faster than the thermal relaxation time.In particular, quasi cw light injection for nonlinear operation or laser emission requires the pump pulse to be longer than the carrier recombination lifetime, but shorter than the thermal time.For instance, we have recently demonstrated optical bistability in a 2D PhC following this approach [3].
In the case of self-induced heating phenomena in micro and nano optical cavities (i.e., when the heat is produced by optical excitation of the resonant mode in the cavity) the characteristic heat dissipation times strongly depend on the cavity size.As it has been pointed out in [9], these scale with the characteristic cavity length, i.e. small cavities dissipate heat faster.The equilibrium temperature, in turn, scales with the inverse cavity length, mainly due to the reduced heat capacity of small cavities.Fast thermal processes are thus compatible with high thermal loading, showing the importance of thermal measurements in small cavities, such as photonic crystal nanocavities.
Thermal relaxation often involves several time scales.This is especially the case for complex geometries such as a air-bridge photonic crystal slabs, where thermal dynamical measurements are particularly relevant since: i) the heat sink is a complex process given by the actual three-dimensional structure; ii) if air holes are present, full calculations -for instance by means of finite element methods-can be rather cumbersome.Recently, the thermal response in photonic crystal devices has been simulated using a finite-elements method [11].In this study, the device concept was a thermo-optical switch based on a W1 photonic crystal waveguide on silicon.Heating was produced by a pair of electrodes close to the W1.Such a device reaches a stationary temperature of 420° C after 20 µs.Heating and cooling times of 5.6 µs and 3.5 µs at 90% and 10% of the final temperature, respectively, were found.
From the experimental point of view, while thermal loading can be estimated through measurements of thermally induced refractive index change, the thermal relaxation times cannot be obtained straightforwardly; so far, this has been done through parameter fitting from non-linear dynamical models [8,10].This requires a complex set of equations coupling several variables, therefore the fitted relaxation time becomes model-dependent.In this work we present a novel method to directly measure the thermal time out of a photonic crystal nanocavity, based on time-resolved reflectivity within a tapered fiber-assisted optical coupling scheme.
Available thermal characterization techniques in semiconductor microchips have mainly been developed for microelectronics and concern both stationary profiles and dynamics.Stationary temperature profile measurement is a standard technique using infrared CCD cameras that allows quantification of thermal loading averaged over time [12].Dynamical measurements of thermal diffusion coefficients on the microscale is also an available technique based on visualization of thermal waves by photoreflectance microscopy [13].The technique we propose here focuses on thermal dynamics averaged in the optical volume of a resonator; in the case of a photonic crystal nanocavity, the spatial resolution is given by the cavity dimensions, of the order of one micrometer.In Section 2 we describe the photonic crystal sample and we report on the main optical features through photoluminescence experiments.Section 3 is devoted to the description of the taper-assisted coupling method for probing the PhC nanocavity.In Section 4 the thermal dynamical measurements are shown.In Section 5 we illustrate the thermo-optical dynamical phenomena in the context of bistable switches.Section 6 gives a discussion of the results as well as a comparison with thermal properties of other nano and microphotonic systems reported in the literature.Finally, the conclusions are given in Section 7.

Sample description and characterization
The sample is a 10 µm × 50 µm photonic crystal of air holes (triangular lattice with period a=465 nm, holes radius=0.3a) in an InP suspended membrane (air gap size=1.16  and 1b), with a nanocavity in the center.The cavity is a modified L3 [14], where the two holes closing the cavity are shifted away by 0.15a (see.Fig. 1c).The 262 nm-thick (λ/2n) InP membrane incorporates a central single layer of self-assembled InAsP/InP quantum dots (QDs), whose density is 1.5×10 10 cm −2 and whose luminescence at 300 K is centered around 1.55 µm with a 145-nm inhomogeneous broadening [15].The whole structure, incorporating a GaInAs sacrificial layer under the InP, is grown by metalorganic chemical vapor deposition (MOCVD).
The QD photoemission is used to identify the cavity mode under incoherent pump @ 800 nm using a 100 fs-pulses, 80 MHz-repetition rate Ti:Sa laser.The resonant mode filters the QD luminescence, giving a spectral peak at 1490 nm; no laser effect was observed for these cavities.Due to their relatively large emission spectrum, and almost flat emission over the cavity tuning range, QDs are well suited as an active material in the framework of our study since their absorption/gain curves are less sensitive to temperature induced spectral shifts with respect to quantum wells.As a result, it will be a good approximation to neglect the effects of thermally induced electronic band shifts on the cavity mode.Thus, only refractive index change with temperature will be taken into account.

Taper-assisted coupling scheme
Once the optical mode is identified by looking at the photoluminescence, the cavity mode is probed resonantly through a curved tapered fiber (waist=2 µm along 500 µm, radius of curvature=500 µm), which can be approached to the cavity by means of a nanopositioning stage.The curvature of the taper reduces the optical coupling to the substrate outside the photonic crystal membrane, thus decreasing optical losses.The incident polarization angle with respect to the sample plane can be changed by means of a fiber polarization controller.
Reflectivity measurements are well suited to probe the cavity resonance since they naturally avoid the uncoupled light background.In order to measure the reflection upon evanescent interaction between the taper and the cavity, a fibered 50% coupler is used.The coupling scheme, depicted in Fig. 1c, is similar to other approaches already reported for coupling light resonantly into W1 waveguides and cavities [16][17][18].When the taper is positioned at the center of the defect in contact with the cavity, and broadband femtosecond pulses are injected into the fiber (120 fs pulses @ 1492 nm, linearly polarized), a reflected signal is detected showing a narrow peak (Fig. 2a, black line).This peak disappears when the taper is moved away from the cavity.The polarization angle is set to optimize the reflected signal.
In order to estimate the coupling efficiency, the transmission spectrum has been measured.This also shows a resonant feature at the same wavelength (Fig. 2b).We estimate the coupling efficiency using a simple model based on mode coupled theory [19].Let τ c be the taper-cavity coupling time and τ 0 the cavity photon lifetime (defined for the field amplitude), the latter including the intrinsic cavity lifetime and other additional taper-induced cavity losses, such that ω/2Q loaded =1/τ 0 +1/τ c .The ratio of the -in resonance-transmitted power to the input power reads where 2 t f is the transmitted power in the vicinity of the cavity.The power measured at the fiber output is P t =5.7 nW.The transmitted power out of resonance is P i ~11nw.Assuming which gives τ 0 /τ c ≈0.39 using Eq. ( 1).We then define the coupling efficiency (η c ) as the probability that a photon inside the cavity couples to the fiber (in either direction, hence P c ∝2/τ c ) with respect to the photon emission probability throughout all the coupling channels (P tot ∝2/τ c +1/τ 0 ), namely η c ≡P c /P tot =2/(2+τ c /τ 0 ).We point out that for an ideal coupling τ c <<τ 0 which gives η c →1.In our case we obtain a coupling efficiency of η c =44%.Let us stress that this high coupling efficiency is of great interest in many applications, like efficient single photon sources, or light extraction of cavity based nano-lasers.

Thermal relaxation dynamics
First, the probe power injected into the taper was set to a low level (<1 mW) in order to prevent any self-induced thermal or electronic effect [20].In order to produce heat, the cavity is optically pumped by the surface using a modulated beam at 800 nm focused down to a 3.2 µm-diameter spot (@1/e 2 of the intensity).This wavelength is mainly absorbed in the InP (bandgap wavelength ~ 900nm @ 300 K).Non-radiative carrier relaxation processes are the central mechanisms producing heat in the membrane.Thermal effects increase the refractive index, shifting the resonance to a longer wavelength.Figure 2a  Thermal dynamics is measured as follows.The wavelength of a tunable laser is set at a given detuning ∆λ 0 with respect to the resonance of the unpumped cavity (∆λ 0 ≡λ c -λ p , where λ c is the wavelength at the center of the resonance and λ p the probe wavelength), probing the cavity-induced reflection.Heating up the cavity results in a thermally induced wavelength shift, producing a change in the reflected intensity of the probe light.A high sensitivity fibercoupled avalanche photodiode (APD) with a 1.44 GHz-bandwidth is used to temporally resolve the change of the reflected signal.Traces are recorded on a 1 GHz -bandwidth oscilloscope.Upon 10 µs width-50 µs period square pump pulse excitation, the reflected signal exhibits either a drop-out followed by a recovery for ∆λ 0 ≥0 (i.e.blue detuning), or intensity peaks for ∆λ 0 <0 (red detuning).Both situations are directly related to the thermal dynamics of the resonance towards the equilibrium states.For a nearly resonant probe (∆λ 0 =0, Fig. 3a), the signal decreases in the presence of the pump beam due to thermally-induced shift of the resonance.For red detuning, instead, the resonance "passes through" the injection wavelength leading to a maximum of intensity in the heating process, followed by a maximum in the cooling process (Figs.3b-3d).In order to extract the characteristic thermal time from the time evolution of the reflected probe, the spectral shape of the resonance must be taken into account.The resonance can be fitted with a Lorentzian function.Considering the time dependence of the center of the resonance λ c (t) -which contains the refractive index dependence with temperature-, the timedependent reflectivity R(t) is thus modelled as R(t)=1/[1+(λ c -λ p ) 2 /(γ/2) 2 ], where γ is the FWHM of the resonance.The resonance width γ has been measured from the reflectivity signals as a function of detuning, and gives γ=0.33 nm, corresponding to a quality factor of Q loaded ≅4520.The time dependent wavelength shift becomes ∆λ(t) ≡λ c (t)-λ p =±[R(t) -1 -1] 1/2 γ/2, where the two roots indicate blue or red shift of the probe with respect to the cavity resonance.We point out that the kinetics of the center of the resonance also contains an ultrafast process related to carrier-induced index variation.By means of an ultrafast pump and probe technique we have verified that such processes decrease the refractive index blue-shifting the resonance in a time shorter than 1 ns, eventually being followed by a thermally driven red-shift.Therefore, they lead to short downward peaks for red detuning, and short upward peaks for blue detuning.We point out that such effects remain small in our measurements (see arrow in Fig. 3c).
Curves of ∆λ(t) are plotted as a function of time in Figs.3e-3h for the heating and cooling processes.Note the slight jumps in the signals at ∆λ~0, which are due to the discontinuity of the two roots in the inverse of the lorentzian function.On the basis of such curves, the characteristic times have been calculated as the times at which the modulus of the total dynamical wavelength shift (|λ c (t)-λ c (t=0)|=|∆λ(t)-∆λ 0 |) becomes 1/e of its stationary value.As can be seen in Figs.3e-3h, the heating and cooling times for the different detuning values are close to each other, as it is expected.Under the hypothesis that the characteristic times do not depend on the initial detuning, we average out the measured values for different detuning which yields t C =0.9±0.2 µs and t H =1.0±0.2 µs for the cooling and heating processes, respectively.We then conclude that there is no significant difference between cooling and heating times in the weak-probe regime.It is worth pointing out that for the zero-detuning case (Fig. 3e) the cooling time is about a factor 2 smaller compared to the other cases.For zero-detuning, indeed, the reflectivity time trace for the cooling process is mainly affected by the spectral tail at the blue side of the resonance, which is slightly different to the tail at the red side (see the slight asymmetry of the resonance in Fig. 2a), which may explain the substantial deviation for this situation.Therefore, our technique is more accurate for thermally-induced wavelength shifts within the FWHM of the resonance, i.e. far from the spectral tails (|∆λ(t)|< γ/2).

Thermo-optical bistability
In this section we study the thermal dynamics in the context of bistable switches.Thermooptical bistability [21][22] can be obtained through self-induced red shift of the resonance upon (large enough) cw injection from a single beam.Indeed, the thermally-induced refractive index increase also takes place for a nearly optically resonant cw injection, in absence of any incoherent pump.In such a case, bistable operation can be expected provided that: i) the injection wavelength is at the red side of the resonance; ii) the detuning between the injection beam and the cold resonance is larger (in modulus) than ~ 2 / 3γ ; iii) the injection power exceeds a given threshold.As long as the nonlinearity comes from a thermo-optical effect, the switching times should be related to the characteristic thermal relaxation time obtained before.
For injection powers greater than ~1 mW, thermo-optical bistability was observed.Bistable behaviour can be shown by measurements of output power as a function of input power for different detuning, leading to hysteresis cycles.Figure 4a shows the spectral range of wavelength detuning used in this experiment.Importantly, the power sweep in such measurements must be quasi-stationary, i.e. the duration of the power ramp must be much longer than the thermal dissipation time.Injection modulation is thus introduced by a 10 kHz triangular amplitude modulation of the input beam.In such conditions, simultaneous input modulation and optical transmission through the taper were measured as a function of time, the latter using a DC-coupled 1 GHz photodiode (Fig. 4b).For |∆λ 0 |>0.35 nm, hysteresis loops are observed (Fig. 4c).Their size increases for larger detuning-values, up to |∆λ 0 |~0.72 nm where the maximum injection power remains below the bistability threshold.Switching times have been measured as the time widths between the minimum and the maximum of the switch processes.The switch on time is 1.8 µs, whether the switch off time is 4 µs.Both are of the order of the 1 µs characteristic time obtained with the pump and probe technique.Detuning-values with respect to the cavity resonance are, from ∆λA to ∆λK: -0.11, -0.27, -0.36, -0.46, -0.52, -0.56, -0.6, -0.64, -0.66, -0.68 and -0.72 nm.The input power is measured at the fiber taper input.

Linear regime: thermal dynamics in a 2D PhC membrane
The energy absorbed in the pumped region can be essentially dissipated in two ways.The first mechanism of heat flow corresponds to heat conduction through the membrane (Fig. 5a).In order to obtain an analytical expression for the associated thermal resistance, let us consider a 2D cylindrical heat conductor of radius R with thermal conductivity κ, and a heat source of radius r (Fig. 5b).The thermal resistance for a length d (membrane thickness) between r and R is R plane =ln(R/r)/2πκd~1.7 10 4 K/W, where: R is half of the −smallest, i.e. closest to the heat sink− lateral size of the membrane (R=5µm), r the pump radius (r=1.6µm) and κ=κ eff =0.46 W/cm K, κ eff being an effective thermal conductivity calculated as a geometrical average of the material with holes.The second mechanism of heat flow corresponds to thermal conduction through the air gap [23].This can be estimated as R airgap =δ/κ air πr 2 ~6 10 6 K/W, where δ is the air gap thickness (δ=1.16µm) and κ air =2.4 10 -4 W/cm K. Since R airgap >>R plane , we can neglect heat transfer across the air gap and assume that the suspended membrane behaves as a quasi 2D system for heat diffusion and conduction.Eventually, heat is dissipated out of the membrane through the substrate.Now we derive the order of magnitude of the thermal time scale from a 2D heat equation.An analytical expression for the heat dissipation time in the quasi-stationary limit can be derived: τ th ≈ρC p r 2 ln(R/r)/2κ, where ρ is the density and C p the specific heat capacity in the pumped region.Using typical thermal constants for InP (ρC p =1.5 J/cm 3 K), we obtain τ th ~50ns.In order to compare the calculated τ th with the experimental results, we estimate τ th from the red shift of the resonance.We first calculate the temperature increase from tabulated values of refractive index change per unit temperature (n T, n T /n≈0.63 10 -4 K -1 @ 300 K for InP [24]) in the following way: ∆T=(n/n T )|∆λ th /λ|.Using the measured red shift |∆λ th |=0.3 nm with 165 µW of incident power, we obtain ∆T≈3.2K. Now, the temperature increase of the membrane under continuous pump can be related to τ th through ∆T=η 800 P pump τ th /ρC p Vol, where Vol=πr 2 d (d=membrane thickness) is the volume of the pumped region and η 800 is the fraction of the incident power absorbed at 800 nm (η 800 ≈33% [25]).From the last expression we obtain τ th ≈186 ns.Note that τ th is about one order of magnitude below the measured relaxation times.The difference between the thermal time scale for energy dissipation in the quasi-stationary limit and the relaxation time is related to a generic dynamical feature of heat diffusion in this kind of systems rather than just to the particularities of the photonic crystal.Indeed, such a difference has already been observed in other geometries without air holes such as microdisks [8].In order to investigate the relaxation time t th , let us look for the time dependent solutions of a 2D rectangular membrane of size 2L x ×2L y with thermal diffusivity α and a heat source Q(r), symmetric under reflections x→-x and y→-y (Fig. 5c).Boundary conditions are T(x=L x )=T(y=L y )=0 and ∂T/∂x| x=0 =∂T/∂y| y=0 =0 for symmetrical solutions (T is temperature increase with respect to the substrate temperature).The general solution is a superposition of a particular (stationary) solution T st (r) and the homogeneous solution, T(r,t)=T st (r)+T h (r,t).It can be easily showed that T h can be expanded in Fourier series, ( ) , 0 , cos( ) cos( ) with k xn =(2n+1)π/2L x , k yn =(2n+1)π/2L y and σ nm /α=k xn 2 +k ym 2 .The amplitudes A nm can be obtained from the initial conditions.In the case of the heating process (Q(r)≠0) the initial condition is T (H) (r,t=0)=0, and the coefficients A (H)  nm are calculated as the Fourier transform of -T (H)  st (r).Thus, the Fourier modes relax with characteristic times given by t th,nm =4L x 2 L y 2 /απ 2 [(2m+1 . Note that higher order modes are dissipated faster.Therefore, we can expect the relaxation dynamics to be driven by the lowest order mode provided it is efficiently excited.In our case, since L y >>L x , t th,00 ≈4L x 2 /απ 2 ≈300 ns, meaning that the smallest length governs the relaxation process.This estimation is about a factor 3 below the measured relaxation time.In addition, Eq. 2 also accounts for cooling processes, whose dynamics is denoted by T (C) (r,t).In such a case the heat source is turned off (Q(r)=0) hence T (C)  st (r)=0 and the initial condition reads T (C) (r,t=0)=T (H)  st (r).Therefore A (C)  nm =-A (H) nm , hence T (C) (r,t)=-[T (H) (r,t)-T (H) st (r)], which shows that the relaxation dynamics of the cooling process is the same as for the heating process.
The discrepancy between the experimentally obtained time constants τ th and t th and their theoretical estimations may be due to the strong approximations used in our simple models, in particular the cylindrical symmetry to represent a 2D heat diffusion and conduction in the membrane, the effective conductivity of the photonic crystal as a simple spatial average of thermal conductivities, and the temperature being fixed to the ambient temperature at the end of the membrane (perfect heat sink at r=R in the cylindrical model, and at L x and L y in the rectangular model).Full 3D-finite element numerical simulations, as in Ref. [11], should be carried out in order to better account for thermal dynamics in this system.
The measured relaxation times are comparable to the numerical values obtained for a similar geometry in Ref. [11], where a slight asymmetry in turn-on/off processes has been observed.From the dynamical point of view, an asymmetry may arise if the system is heated during a time shorter compared to the characteristic heating time.In our case, since the pumpon and pump-off times −10 µs and 40 µs respectively− are much longer compared to the relaxation time (~1 µs) we do not expect such an asymmetry.The difference in relaxation times with respect to Ref. [11] may be due to differences in size of the heated regions, membrane size, differences in thermal constants (the thermal conductivity of Si is about a factor 2 greater than the one of InP) and, most importantly, the presence of a 1 µm-thick oxide layer underneath the PhC membrane.An oxide cladding is also present in Ref. [10], where the thermal relaxation time observed for a 5 µm-size Bloch mode in a 2D InP PhC slab on SiO 2 /Si (C) 2009 OSA 14 September 2009 / Vol.17, No. 19 / OPTICS EXPRESS 17126 was of the order of 1 µs as well.In both cases, the lateral dimension of the heated region being quite large, the heat flow can be expected to be essentially driven by the ~1 µmthickness SiO 2 cladding directly to the substrate [26].The Si-based PhC air-bridge membrane reported in [7] also exhibits characteristic times close 1 µs for thermal effects, similar to our InP membrane in terms of lateral size.Finally, with respect to other geometries, we find our results comparable to the times found in silicon microdisks [8].In the case of silica microdisks, in contrast, thermal dynamics lies on the millisecond timescale [9], mainly due to the low thermal diffusivity of silica; indeed, the thermal diffusivity (κ/ρC p ) is about two orders of magnitude smaller compared to silicon.

Nonlinear regime: thermo-optical bistability
In the nonlinear regime, the switching times were measured as the time interval between minimum and maximum of time traces.The obtained switching times are ~2 µs for the switch-on process (from the cold to the hot states), and ~4 µs from the switch-off process (from the hot to the cold states).As expected, both are related to the 1 µs characteristic time measured in the linear regime.Phase space for bistability conditions.Blue, green and red lines: stationary solutions of the intracavity power (P) as a function of the normalized wavelength shift (θ) for: bistable conditions (Pin,input=2 mW, blue), switch on conditions (Pin,input=3.3mW, green) and switch off conditions (Pin,input=1.55mW, red).Black line: stationary solutions of θ.The intersections of the black line with the curves give the steady states: θst +(-) is the stable state for the switch on (off) process.The direction of the dynamical flow is indicated by the arrows.The opposite slopes of dP/dθ at the fixed points mathematically explain the difference in relaxation times (see text).Parameters are: P0=2 mW (calculated with Vc=0.08 µm 3 , τth=186 ns, Γ=0.015, Qloaded=4520, η1500=0.00057 and a=100 cm -1 ), Pin=0.263Pin,input (Pin,input is injected power into the fiber), ξ=16.2 (using vg=c/10, taken from Ref.The difference between switch-on and switch-off times can be explained from simple considerations of dynamical thermo-optical nonlinearities in the following way.The temperature variation ∆T(t) imposes a dynamical thermal red shift |∆λ th (t)/λ|=(n T /n)∆T(t), with ∆T(t)=η 1500 P(t)τ th /ρC p V c , where: P(t) is the intracavity power; η 1500 is the fraction of intracavity power absorbed at 1500 nm, η 1500 ~2ΓaL, with Γ the optical confinement factor in the QDs layer, a the absorption coefficient and L is the cavity length; V c is the cavity volume (V c ~0.76 (λ/n) 3 [27]).Defining the normalized wavelength shift as θ(t)=[λ in -λ c (t)]/(γ/2), where λ in is the injection (rather than "probe" as in the linear case) wavelength, the dynamical equation becomes 0 0 ( ) 1 ( ) ( )

Conclusions
In conclusion, we have developed a method to directly measure the thermal relaxation time out of a photonic nanocavity.This technique relies on reflectivity measurements of a cw probe beam coupled to the cavity through a tapered fiber, becoming sensitive to the thermally induced increase of the refractive index within the cavity.We applied this method to an InPbased nanocavity on a suspended membrane.Wavelength shifts up to 0.3 nm for 165 µW pumping power @ 800 nm were observed.The obtained values for the characteristic thermal times are close to 1 µs, more than a factor 3 compared to a rough estimation of the relaxation time of the fundamental mode in a rectangular 2D membrane, showing the relevance of the experimental measurements.In addition, the thermal relaxation time has been shown to be closely related to switch on/off times within an opto-thermal bistable regime.In a general sense, additional engineering of photonic crystal geometries together with the inclusion of new materials should improve heat sink properties, which can be experimentally tested with the method proposed here.This would allow both increasing the repetition rate of thermal switches and reducing thermal loading for cw device operation.The latter is of central importance for obtaining fully stationary active and nonlinear functionalities such as efficient cw lasers and amplifiers, as well as long memories and logic gates based on cw bistable operation in photonic crystal circuits.

Fig. 1 .
Fig. 1. (a).Sketch of the photonic crystal-membrane sample.(b).SEM cross-section image of a 5µm-width membrane showing the ~ 1µm-thickness air gap.(c).Tapered fiber optical coupling scheme for transmission and reflectivity measurements.A SEM image of the L3 cavity is shown.

Fig. 2 .
Fig. 2. (a).Reflectivity spectrum of the cavity resonance without pump (black line), and with 165µW pump (red line).(b).Simultaneous transmission and reflection.The thick arrow indicates the out-of-resonance transmitted power used to estimate the coupling efficiency (see text).

Fig. 5 .
Fig. 5. (a).Sketch of the sample cross section, showing the direction of the heat flow.(b).Simplified model for cylindrical 2D heat conduction in a membrane; κeff represents the effective thermal conductivity of the photonic crystal membrane (see the definition of the other parameters in the text).(c).Rectangular geometry for the calculation of transient dynamics in a 2D membrane.(d).Phase space for bistability conditions.Blue, green and red lines: stationary solutions of the intracavity power (P) as a function of the normalized wavelength shift (θ) for: bistable conditions (Pin,input=2 mW, blue), switch on conditions (Pin,input=3.3mW, green) and switch off conditions (Pin,input=1.55mW, red).Black line: stationary solutions of θ.The intersections of the black line with the curves give the steady states: θst +(-) is the stable state for the switch on (off) process.The direction of the dynamical flow is indicated by the arrows.The opposite slopes of dP/dθ at the fixed points mathematically explain the difference in relaxation times (see text).Parameters are: P0=2 mW (calculated with Vc=0.08 µm 3 , τth=186 ns, Γ=0.015, Qloaded=4520, η1500=0.00057 and a=100 cm -1 ), Pin=0.263Pin,input (Pin,input is injected power into the fiber), ξ=16.2 (using vg=c/10, taken from Ref.[29]) and θ0=3.4.
Fig. 5. (a).Sketch of the sample cross section, showing the direction of the heat flow.(b).Simplified model for cylindrical 2D heat conduction in a membrane; κeff represents the effective thermal conductivity of the photonic crystal membrane (see the definition of the other parameters in the text).(c).Rectangular geometry for the calculation of transient dynamics in a 2D membrane.(d).Phase space for bistability conditions.Blue, green and red lines: stationary solutions of the intracavity power (P) as a function of the normalized wavelength shift (θ) for: bistable conditions (Pin,input=2 mW, blue), switch on conditions (Pin,input=3.3mW, green) and switch off conditions (Pin,input=1.55mW, red).Black line: stationary solutions of θ.The intersections of the black line with the curves give the steady states: θst +(-) is the stable state for the switch on (off) process.The direction of the dynamical flow is indicated by the arrows.The opposite slopes of dP/dθ at the fixed points mathematically explain the difference in relaxation times (see text).Parameters are: P0=2 mW (calculated with Vc=0.08 µm 3 , τth=186 ns, Γ=0.015, Qloaded=4520, η1500=0.00057 and a=100 cm -1 ), Pin=0.263Pin,input (Pin,input is injected power into the fiber), ξ=16.2 (using vg=c/10, taken from Ref.[29]) and θ0=3.4. .