## Abstract

The dynamical properties of an InP photonic crystal nanocavity are experimentally investigated using pump-probe techniques and compared to simulations based on coupled-mode theory. Excellent agreement between experimental results and simulations is obtained when employing a rate equation model containing three time constants, that we interpret as the effects of fast carrier diffusion from an initially localized carrier distribution and the slower effects of surface recombination and bulk recombination. The variation of the time constants with parameters characterizing the nanocavity structure is investigated. The model is further extended to evaluate the importance of the fast and slow carrier relaxation processes in relation to patterning effects in the device, as exemplified by the case of all-optical wavelength conversion.

©2013 Optical Society of America

## 1. Introduction

During the past decades, increasing attention has been devoted to developing technologies for optical signal processing and on- and inter-chip optical interconnects [1, 2]. Integrated, ultra-compact and low-energy all-optical switches are promising for increasing the integration density and bandwidth as well as decreasing the energy consumption of future communication networks.

Various kinds of structures have been explored for optical switching such as semiconductor optical amplifiers [3], optical fibers exploiting parametric processes [4], micro-ring cavities [5], photonic crystal (PhC) waveguide Mach-Zehnder interferometers [6], silicon-organic hybrid slot waveguides [7], and photonic crystal cavities [8–10]. Among these, two dimensional PhC membrane structures enabling the realization of waveguides with tailored dispersion as well as nanocavities are promising candidates for high-density integrated photonics. Featuring a high quality factor (*Q*-factor) and a small mode volume (*V*), the field intensity can be significantly enhanced inside the cavity (scaling as *Q*/*V*) and therefore non-linear signal processing becomes feasible at very low pulse energies [11, 12].

PhC nanocavity switches have been demonstrated experimentally based on carrier effects using different materials. An effective carrier lifetime of about 100 ps was obtained using a silicon structure [8], while a decay time of only 15 ps was realized using a GaAs structure [9]. A PhC switch based on quaternary InGaAsP material, where carriers are generated using a combination of two-photon and linear absorption, showed ultra-low energy operation and a short switching time window of 20 ps with a relaxation process comprising several characteristic time scales [10]. From previous reports, the switching speed is suggested to be limited by the free carrier relaxation process [13]. Though some strategies have been implemented to increase the carrier recovery rate and thereby the switching speed [8, 14–16], few studies have been devoted to investigating the processes governing the dynamics of the switch. Establishing detailed dynamical models is, however, required for further optimization of the performance of the switch, including the investigation of different configurations and applications.

In this article, the dynamics of InP PhC nanocavity structures are investigated experimentally using short-pulse pump-probe measurements. The results are compared with numerical simulations based on coupled mode theory (CMT) [17] and rate equations for the dynamics of the carrier density governing the cavity properties. As a result, we establish an effective numerical model that consistently accounts for the experimental observations.

The paper is structured as follows: In section 2, we describe the design and fabrication of the investigated InP-based PhC nanocavity structures. Then, in section 3, dynamical switching characteristics measured using pump-probe techniques are reported. A dynamical model based on CMT is introduced in section 4 and, in section 5, the different time constants governing the carrier relaxation and their dependence on material parameters are characterized. Section 6 presents fits of the measured dynamics based on the developed model and, in section 7, the model is applied to evaluate the performance of all-optical wavelength conversion. Our conclusions are summarized in section 8.

## 2. Design and fabrication of a PhC nanocavity structure

As shown in the scanning electron microscope (SEM) images in Figs. 1(a)–1(c), the investigated PhC
structure is an InP air-suspended membrane consisting of a defect cavity of the H0 type [18] formed by shifting two neighboring air holes
0.2*a* in opposite directions, where *a* is the lattice constant.
The cavity is coupled to an input and output waveguide in the Γ-K direction. The
waveguides are standard W1 waveguides (defined as the removal of one row of air holes), but with
the two innermost arrays of holes shifted towards the waveguide center by 0.1*a*
to make the waveguide fundamental mode overlap in frequency with the cavity mode. Compared with
the PhC structures in [9, 19], our structure is realized using a single hole size, which is advantageous from a
fabrication point of view.

The device fabrication process is as follows: The photonic crystal pattern is first defined in positive resist (ZEP520A) by electron-beam lithography and then transferred to a 200 nm Si_{3}N_{4} hard mask by CHF_{3}/O_{2} reactive-ion etching (RIE). The resist is removed and the pattern is transferred to a 340 nm InP layer by a cyclic CH_{4}/H_{2} and O_{2} RIE. After removing the Si_{3}N_{4} layer by hydrofluoric acid, the membrane structure is finally formed with diluted H_{2}SO_{4}/H_{2}O_{2} to selectively wet-etch a 1 μm In_{0.53}Ga_{0.47}As sacrificial layer located below the membrane [20].

Figure 1(d) plots the measured linear transmission spectrum of one of the fabricated devices for which *a* = 425 nm, the hole radius is *r* = 100 nm and the membrane thickness is *h* = 340 nm. Both the input and output facets of the 800 μm long device are equipped with mode adapters, shown in Fig. 1(c), to facilitate the out coupling [21]. As illustrated in Fig. 1(d), the device has a resonant wavelength at 1545.2 nm. The insertion loss including both coupling and propagation loss is about −17 dB at resonance. The inset in Fig. 1(d) is the calculated magnetic field intensity corresponding to the excitation of the resonant cavity mode, illustrating that the cavity mode couples well to the waveguide mode. The unloaded *Q*-factor estimated using three dimensional finite-difference time-domain (FDTD) simulations is ~2.5 × 10^{4} and the measured loaded *Q*-factor is 4100.

## 3. Pump-probe setup

A homodyne pump-probe setup, as illustrated in Fig. 2, is used to measure the temporal switching dynamics. The separation of pump and probe beams is accomplished by detuning the signals and selecting the probe using an optical filter. Compared to the heterodyne technique [19], homodyne detection requires spectral separation of pump and probe signals to avoid overlap, but on the other hand allows simple and direct measurement of important properties such as switching contrast and time window.

A laser source emitting short pulses (full-width at half-maximum, FWHM, of 1.5 ps) at a repetition rate of 1.25 GHz (corresponding to a period longer than the free carrier relaxation time) is employed for the pump-probe measurement. Pump and probe signals are generated by bandpass filtering separate parts of the source spectrum using a multiport optical processor (WaveShaper 4000S). The probe signal is then chopped at 100 kHz using a Mach-Zehnder modulator (MZM) and delayed with respect to the pump using a delay line (DL). A counter-propagating scheme is used to avoid pump-probe interactions in the PhC waveguides. Using polarization controllers, the states of polarization of both pump and probe signals are adjusted to TE polarization [22] and are coupled into and out of the PhC waveguides using lensed single mode fibers. A band-pass filter is used to select the probe signal, while blocking the reflected pump. The transmitted probe signal is then detected with a photodiode, amplified, and monitored with a lock-in amplifier. Pump and probe signals are monitored using power meters and an optical spectrum analyzer, while an infrared camera (Xeva-1.7-320) is mounted on a microscope to image the far-field scattered light from the top of the sample [20]. The probe power is kept sufficiently weak so as not to cause any nonlinear effects in the sample. Typically, the average probe input power is set 7~12 dB lower than the average input pump power and the detuning between the probe and pump is larger than the pump spectral FWHM. For smaller probe power or detuning, interference effects are observed around zero delay.

Figure 3 shows an example of a measured pump-probe signal.
The device shows a resonance at 1545.2 nm with a measured loaded *Q*-factor
of 4100 corresponding to a photon lifetime of 3 ps. The switching energy is estimated by
subtracting the coupling loss of ~7.5 dB from the pump power at the input fiber. The coupling
loss (in dB) is obtained as
(*α*_{Insertion}-*α*_{Cavity})/2
for continuous wave excitation, *α*_{Insertion} being the device
insertion loss and *α*_{Cavity} the cavity power transmission at
resonance, which depends on the ratio between the total and in-plane *Q*-factor
of the cavity. The probe signal is detuned from the cavity resonance to shorter wavelengths by 1
nm (2.6 cavity linewidths), while the center wavelength of the pump is fixed to 1545 nm. The
pump and probe pulses are approximately 6 and 12 ps wide, respectively. The inset in Fig. 3 illustrates that the output spectrum of the pump
expands towards shorter wavelengths with increasing pump energy, which is consistent with a
decrease of the refractive index due to band-filling and free carrier dispersion [23] of carriers excited by two-photon absorption (TPA). From
Fig. 3, a switching contrast of 7.8 dB and a FWHM of 18
ps are obtained with a switching energy as low as 85 fJ. This low-energy and fast optical
switching is achieved mainly due to the high *Q*-factor and small mode volume of
the optimized H0 cavity structure. In the following, we analyze the different contributions to
the switching dynamics observed in Fig. 3, in particular
emphasizing the time constants governing the relaxation process.

## 4. Model based on coupled mode theory

To analyze the switching dynamics, we employ coupled mode theory based on an extension of the model presented in [17, 19, 22, 24–27]. The dynamic variables in the model are the field amplitudes inside the cavity, the free-carrier density and the temperature of the membrane. Within first order perturbation theory, the dynamical equations for the cavity modes can be formulated as

*a*

_{p}_{,}

*(*

_{s}*t*) is the amplitude of the slowly varying envelope of the cavity mode excited by the pump and signal (probe), respectively, normalized such that |

*a*

_{p}_{,}

*(*

_{s}*t*)|

^{2}represent the corresponding energies in the cavity. The corresponding fields are

*A*

_{p}_{,}

*(*

_{s}*t*) =

*a*

_{p}_{,}

*(*

_{s}*t*)exp(-

*jω*),

_{p,s}t*ω*

_{p}_{,}

*being the center angular frequencies of the input pump and signal, which are represented as ${S}_{p,s}^{i}(t)$ = ${s}_{p,s}^{i}(t)\mathrm{exp}(-j{\omega}_{p,s}t)$, with $|{s}_{p,s}^{i}{|}^{2}$being the powers of the pump and signal in the input waveguide. Furthermore,*

_{s}*ω*is the cold cavity resonance frequency and $\kappa =\sqrt{{\gamma}_{in}/2}$ is the coupling coefficient between the cavity and waveguide mode. The total loss rate $\gamma $ for the cavity has two contributions, $\gamma ={\gamma}_{in}+{\gamma}_{v}$, where ${\gamma}_{in}$ accounts for coupling to the waveguides and ${\gamma}_{v}$ represents coupling to all other modes but the relevant waveguide modes, mainly vertical emission. The loss rates can also be related to the cavity in-plane and vertical (unloaded) quality factors,

_{o}*Q*and

_{in}*Q*, as ${\gamma}_{in}={\omega}_{o}/{Q}_{in}$ and ${\gamma}_{v}={\omega}_{o}/{Q}_{v}$. When the cavity is excited close to its resonance, a highly localized intensity may build up, leading to a change of the local refractive index due to Kerr effects, free-carrier dispersion (FCD), band filling, as well as thermal effects. The change of the refractive index results in a complex time-dependent change, $\Delta {\omega}_{p,s}(t)$, of the cavity resonance frequency, as described by Eqs. (3) and (4). The real part of $\Delta {\omega}_{p,s}(t)$accounts for the resonance frequency shift while the imaginary part accounts for absorption due to TPA and free-carrier absorption (FCA) [28]. Assuming the signal to be much weaker than the pump, we only account for cavity resonance shifts induced by the pump. Furthermore, we restrict attention to the case of below-bandgap excitation, where carriers are generated by TPA and linear absorption can be neglected. In Eqs. (3) and (4), ${K}_{Kerr}={\omega}_{o}c{n}_{2}/({n}^{2}{V}_{TPA})$ and ${K}_{TPA}={\beta}_{TPA}{c}^{2}/(2{n}^{2}{V}_{TPA})$ with

_{v}*c*being the light speed in vacuum,

*n*the background material refractive index, ${n}_{2}$ its Kerr coefficient, and ${\beta}_{TPA}$ the TPA coefficient. The effective TPA mode volume ${V}_{TPA}$ used to characterize the mode averaged local TPA rate was introduced by Johnson et al. in [24], as integrals of the optical cavity mode. In principle, these integrals diverge because of the leaky nature of the cavity modes [29]. For high-

*Q*cavities, the divergence as a function of integration domain is relatively small, and one can get an estimate of the mode volume by integrating across a finite domain. In the present approach, we use an approximation to the cavity mode obtained from FDTD as the scattered field from the cavity with no waveguides and evaluate the integrals in a volume of about 10 × 8 × 1.5 μm

^{3}. In addition, we neglect the optical confinements [24] because they are found to be almost unity since nearly all the light field is confined in the membrane area. The expansion coefficient${K}_{Car}$ has contributions from free-carrier dispersion and band filling, both resulting in a reduction of the refractive index for increasing carrier density. Bandgap shrinkage causes a positive index change, and may be important for small carrier densities, leading to a reduction of ${K}_{Car}$ [23]. ${K}_{FCA}=c\sigma /2n$, where $\sigma $ is the absorption cross-section for the carrier density

*N*(

_{c}*t*). Wave mixing (parametric) effects between pump and probe also contribute to the probe field, giving the factor of two difference between Eqs. (3) and (4) for the instantaneous Kerr and TPA effects, while the corresponding effect mediated by carrier density oscillations is neglected, assuming the detuning between the pump and probe to be much larger than the inverse of the carrier lifetime. For thermal dispersion, ${K}_{th}={\omega}_{o}{n}_{T}/n$ in which

*n*is the temperature dependence of refractive index. The output signal amplitude of the probe can be obtained aswhere ${\tau}_{d}$ is the time delay of the probe pulse with respect to the pump, as controlled by the delay line. The probe transmission as a function of delay is calculated as the output probe pulse energy normalized by its value at large negative delay

_{T}The mode averaged temperature difference $\Delta T(t)$ between the PhC cavity and its environment, induced by heating due to the pump pulse, relaxes in several steps [30], but can be approximated by a single decay rate when compared to the faster carrier relaxation

In Eq. (7), ${P}_{abs}(t)=c\sigma {N}_{c}(t){\left|{a}_{p}(t)\right|}^{2}/n+\eta {\beta}_{TPA}{c}^{2}{\left|{a}_{p}(t)\right|}^{4}/({n}^{2}{V}_{TPA})$ is the absorbed light power, adding thermal energy to the cavity with the first and second term originating from FCA and TPA, respectively. Notice that only a fraction $\eta $ of the energy of the two photons involved in the TPA process is converted into thermal energy (kinetic energy of the electrons). The remaining part is used to overcome the semiconductor bandgap energy*E*. We estimate $\eta $ = $(2\hslash {\omega}_{p}-{E}_{g})/2\hslash {\omega}_{p}$ » 17%.

_{g}*C*is the thermal capacitance of the PhC membrane structure which is a product of material density, heat capacity and cavity volume [27]. ${\gamma}_{th}$ is the thermal relaxation rate, which can be estimated using a finite-element method (FEM) [30, 31].

_{InP}## 5. Carrier dynamics

#### 5.1 Carrier diffusion simulations

The carrier dynamics in the PhC cavity is first investigated through FEM simulations. We consider an effective two-dimensional system, where the dynamics on the small length scale corresponding to the membrane thickness is neglected by the consideration of an effective carrier lifetime. This approach was shown to account for experimental results [13, 32]. The spatial and temporal variation of the carrier density *N*(*x*,*y*,*t*) is governed by the following equation:

*τ*= (1/

_{e}*τ*2

_{b}+*S*/

*h*)

^{−1}is the effective carrier lifetime, with contributions due to bulk recombination, with lifetime

*τ*

_{b}_{,}and surface recombination,

*S*being the PhC surface recombination velocity and

*h*the PhC membrane thickness.

*D*is the material ambipolar diffusion coefficient. Normally, if the imbalances between the electron and hole densities as well as their fluxes are small, the ambipolar approximation can well describe the carrier transportations. At the PhC surface, Neumann boundary conditions are applied ${D}_{a}\nabla N(x,y,t)=-SN(x,y,t)$.

_{a}Equation (8) is solved subject to an initial carrier distribution, generated by the pump pulse via TPA, which is taken to be proportional to the square of the light intensity with a distribution following the cavity field profile *ε*(*x,y*)|*E*(*x,y*)|^{4}. For use in CMT, we do not need the full spatial variation of the carrier density, but rather a mode averaged carrier density, ${N}_{c}(t)$, which can be expressed as

In the simulations, we use the same parameters as in the experiments, i.e. the investigated structure is a PhC H0 cavity as the fabricated one with *a* = 425 nm, *r* = 100 nm, *h* = 340 nm and the cavity hole shift is 0.2*a*. We use *D _{a}* = 10 cm

^{2}/s for InP corresponding to a mobility of 5400 cm

^{2}/(V⋅s) for the electrons and 200 cm

^{2}/(V⋅s) for the holes.

Figure 4 shows the calculated temporal variations of the
effective carrier density for different parameter values. It is observed that the relaxation process contains both fast and slow components. Figure 4(a) illustrates the dependence of the carrier density
on its initial spatial profile. The red curve corresponds to an initial profile ∝
*ε*(*x,y*)|*E*(*x,y*)|^{4},
while the others correspond to a profile determined by the overlap of two identical Gaussian
distributions with FWHM values of *R _{I}* and their maxima placed at the
two primary antinodes of the cavity field

*ε*(

*x,y*)|

*E*(

*x,y*)|

^{4}. From Fig. 4(a), where surface and carrier bulk recombination are not included, we find that the initial cavity mode distribution affects only the fast carrier decay rate but not the slow one, for which the carrier spatial distribution has become uniform, as seen in the insets of Fig. 4(a). Figure 4(b) illustrates the dependence on the ambipolar diffusion coefficient

*D*, when the initial carrier profile is fixed (∝

_{a}*ε*(

*x,y*)|

*E*(

*x,y*)|

^{4}), and the surface and bulk carrier decay rates are set to zero. It is found that

*D*affects both the fast and slow carrier relaxation rates. Figure 4(c) shows the result of varying the surface recombination rate

_{a}*S.*The initial carrier profile is again fixed (∝

*ε*(

*x,y*)|

*E*(

*x,y*)|

^{4}),

*D*= 10 cm

_{a}^{2}/s and the bulk carrier decay rate is set to zero. It is seen that surface recombination affects the slow carrier relaxation tail significantly but has negligible influence on the fast relaxation. Figure 4(d) illustrates the dependence of the dynamics on the bulk carrier recombination rate for fixed surface recombination rate

*S =*2 × 10

^{4}cm/s. For InP,

*τ*has a typical value spanning from tens to several hundreds of nanoseconds [33]. As

_{b}*τ*decreases, the slow relaxation rate increases slightly while the fast relaxation rate remains almost the same. These results show that the carrier relaxation dynamics is mainly determined by carrier diffusion and surface recombination.

_{b}#### 5.2 Carrier rate equations

Based on the FEM simulation results, we formulate a phenomenological rate equation model for the carrier dynamics that can readily be combined with the CMT for the optical fields to describe the switching dynamics. Unlike most previous work [24–27, 30], which employ only one carrier rate equation with a single decay rate, we find it necessary to use three coupled carrier rate equations containing three characteristic time constants to accurately describe the carrier dynamics:

*N*(

_{c}*t*) is the effective carrier density in the region defined by the cavity resonant mode profile, cf. Eq. (9). The carrier density

*N*(

_{c}*t*) couples with a short time constant

*τ*

_{1}to a density

*N*

_{2}(t) defined as the density in a larger neighboring area, thus mainly accounting for the initial fast decay of

*N*(

_{c}*t*) due to fast diffusion. The density

*N*

_{2}(t) subsequently couples with time constant

*τ*

_{2}to

*N*

_{3}(t), being the density in a yet larger neighboring region as in the insets of Fig. 4(a). The need to introduce several carrier densities arises due to the non-exponential relaxation implied by the diffusion process. In addition, all the carrier populations are subject to bulk and surface recombination with a time constant

*τ*

_{3}. As we shall see later,

*τ*

_{3}may actually also have a small contribution from (slow) carrier diffusion, when the total recombination rate is small. In Eqs. (11) and (12),

*R*

_{12}and

*R*

_{23}represent the ratios between the carrier volume of

*N*(

_{c}*t*) and

*N*

_{2}(t), and between the carrier volume of

*N*

_{2}(

*t*) and

*N*

_{3}(t), respectively, and reflect the conservation of carriers when they are exchanged between the different volumes. Furthermore, $G(t)={\beta}_{TPA}{c}^{2}{\left|{a}_{p}(t)\right|}^{4}/(2\hslash {\omega}_{p}{n}^{2}{V}_{FCA}^{2})$ is the carrier generation term due to TPA, with the effective FCA mode volume ${V}_{FCA}$ characterizing the mode averaged free carrier density generated by TPA [24].

In order to investigate the validity of the 3-time constant model, Fig. 5(a) compares the calculated results using FEM and 3-time constant rate
equations with optimized fitting parameters. The generation term *G*(*t*) in Eq. (10) is set to be zero, the initial carrier
profile is given by
*ε*(*x,y*)|*E*(*x,y*)|^{4}
and we consider *S* = 0 cm/s and 1/*τ _{b}* = 0.
For comparison, a simulation result using 2-time constant rate equations as in [19] is also plotted with optimized fitting parameters. Figure 5(a) suggests that the relaxation curves can be fitted
well using the 3-time constant model, while the 2-time constant model introduces an obvious
kink between the fast and slow decay components.

By fitting the FEM solutions with the 3-time constant model, we obtain the dependence of the time constants *τ*_{1}, *τ*_{2}, *τ*_{3} and the carrier volume ratios *R*_{12}, *R*_{23} on the initial carrier density profile, cf. Figs. 5(b)–5(e). The initial profile is again set as the overlap of two identical Gaussian distributions with a FWHM *R _{I}*. Carrier band-to-band recombination is not considered. As

*R*increases, meaning the initial profile expands,

_{I}*R*

_{12}and

*R*

_{23}increase as illustrated in Fig. 5(e). From Figs. 5(b)–5(d), the fast time constant

*τ*

_{1}(several picoseconds) and

*τ*

_{2}(several tens of picoseconds) increase with

*R*while the slow time constant

_{I}*τ*

_{3}(several hundreds of picoseconds) remains almost the same. This is as expected, since a more localized carrier profile gives faster initial carrier diffusion, while the following slow diffusion is not mode dependent since the broadened carrier distribution behaves similarly to that of a cavity-free PhC, only being affected by the effective diffusion coefficient determined by the material and structure of the PhC.

Figure 6(a) shows the dependence of the time constants
*τ*_{1}, *τ*_{2},
*τ*_{3} and the carrier volume ratios
*R*_{12}, *R*_{23} on the carrier diffusion
coefficient.The initial carrier profile was set proportional to
*ε*(*x,y*)|*E*(*x,y*)|^{4},
*S* = 0 cm/s and 1/*τ _{b}* = 0. As expected, all
time constants decrease when the carrier diffusion coefficient increases. From Fig. 6(c), when the recombination rate is finite, for
example,

*S =*2 × 10

^{4}cm/s and

*τ*

_{b}= 100 ns, which are typical values for InP PhC membrane structures, we find that

*τ*

_{3}only depends weakly on

*D*, implying that it is mainly determined by surface and carrier bulk recombination, which is also the implicit assumption about the form used for the rate equation, which corresponds to the total carrier number decaying with a time constant

_{a}*τ*

_{3}.

Figure 7(a) shows the variation of the time constants
*τ*_{1}, *τ*_{2},
*τ*_{3} with the surface recombination velocity
*S* for 1/*τ _{b}* = 0. While

*τ*

_{1}and

*τ*

_{2}do not vary with

*S*,

*τ*

_{3}decreases significantly from an upper value (the longest time constant) of 290 ps originating from slow carrier diffusion. The tendency is similar when varying the bulk recombination rate 1/

*τ*, cf. Fig. 7(b). The short relaxation times are nearly constant while the slowest vary, albeit much less compared to the dependence on the surface recombination velocity. From the fitting to FEM simulations, we find that the carrier volume ratios depend weakly on

_{b}*S*and

*τ*and remain approximately constant at

_{b}*R*

_{12}= 0.5 and

*R*

_{23}= 0.4. This means that, like the fast time constants

*τ*

_{1}and

*τ*

_{2}, the carrier volumes depend only on the initial carrier profile and the effective diffusion coefficient of the structure. While the modeling of the carrier density dynamics in terms of three carrier densities is an approximate representation of the diffusion and recombination processes in the PhC membrane, the dependence of model parameters on material constants and excitation conditions supports the interpretation of the different carrier densities.

## 6. Comparison between experiment and theory

The CMT model including the dynamics of the cavity field, carrier density and membrane temperature has now been formulated and will be compared to the pump-probe measurements.

Using the physical parameters shown in Table 1, simulations are carried out by solving Eqs. (1)–(7) and Eqs. (10)–(12) numerically, which is readily accomplished using a standard routine for solving ordinary differential equations.

Figure 8 shows the measured (dots) and simulated (lines)
probe transmission versus pump-probe delay for different pump energies and two different probe
detunings δ, corresponding to switch-on and switch-off configurations. The corresponding spectral configurations of pump, probe and cavity are also illustrated.
Two different samples are considered, Figs. 8(a) and
8(b), with similar H0 cavity structures but different
*Q*-factors. In all cases we find excellent agreement between measurements and
simulations using the CMT model employing three carrier rate equations. In all cases the
transmission recovers in two steps, a fast and a slow one, confirming previous findings [9]. We emphasize, however, that it was not possible to obtain
quantitative agreement employing a model considering only two carrier densities and two time
constants.

In Fig. 8(a), all the traces correspond to a fixed set of carrier time constants and carrier volume ratios: *τ*_{1} = 3 ps, *τ*_{2} = 85 ps_{,} τ_{3} = 180 ps, *R*_{12} = 0.5, *R*_{23} = 0.4. For Fig. 8(b) we similarly have: *τ*_{1} = 3 ps, *τ*_{2} = 45 ps_{,} τ_{3} = 175 ps, *R*_{12} = 0.5, *R*_{23} = 0.3. The measurement data from the two structures were fitted independently: The fact that similar values are obtained suggests that these are related to intrinsic structural properties, supporting the appropriateness of the phenomenological model formulated in Section 5. From the comparison between simulations and experiments, it can be concluded that thermal effects are negligible at this low signal repetition rate (the calculated thermal induced resonance red shift is less than 0.02 nm) due to the high thermal conductivity of InP.

For the switch-on case (δ < 0), the switching contrast increases with the pump energy. When the strong pump pulse is injected into the cavity, the probe transmission is first reduced due to the combination of TPA loss and the instantaneous Kerr effect, which red shifts the resonance. However, this reduction is relatively small and therefore not noticeable in the experiments unless very large pump powers are used. Following the Kerr effect, free carriers are generated by TPA, causing strong carrier-induced nonlinearity which dominates over the thermo-optic effect, shifting the cavity resonance towards shorter wavelengths, hence increasing the probe transmission. For the switch-off case (δ > 0), in contrast, the blue-shift of the cavity resonance causes a reduction of the probe transmission. Comparing the results in Figs. 8(a) and 8(b), shows that the switching contrast is larger, the switching energy is smaller and the switching window is shorter for the structure with higher *Q*-factor mainly due to the sharper slope of the resonant line. A very high *Q*-factor can, however, limit the signal operation speed and make it difficult to avoid cross-talk between pump and probe. In Fig. 8(a), a switching contrast of nearly 8 dB can be achieved for switch-on with an energy of 180 fJ. According to the simulations, the absorbed energy due to TPA and FCA is of the order of 20 fJ.

## 7. Patterning effects in all-optical wavelength conversion

While a major part of the transmission induced by the pump recovers within tens of picoseconds, it is well-known that a slow component, albeit small, may lead to detrimental patterning effects [37]. Thus, after verifying our CMT model, we apply it, as an example, to evaluate the role of the carrier dynamics on the performance of the nanocavity for all-optical wavelength conversion [38, 39].

Figure 9 shows the calculated probe transmission in the switch-on case when the pump consists of 5 ps FWHM Gaussian pulses at 10 GHz and 40 GHz repetition rates, respectively. The injected pump signal corresponds to a sequence of consecutive “1’s”, the input probe is a weak continuous wave signal with a detuning of −1.5 nm from the cavity resonance of 1545.2 nm. The simulation results are obtained when the output signals reach steady states. Instead of using pseudorandom binary sequences, patterning effects can also be tested using a periodic signal by investigating the dependence of the extinction ratio on the period [40]. Figure 9(a) shows that, at a repetition rate of 10 GHz, the switch fully recovers to the initial state in-between pump pulses and a large modulation depth can be achieved with a high pump energy. However, at a higher repetition rate of 40 GHz, as shown in Fig. 9(b), the extinction ratio degrades, i.e. patterning effects become serious and the switching performance is deteriorated. In this case, there is a maximum in the modulation depth as a function of pump energy. This fact can be attributed to three factors. First, for very large energies, the cavity resonance has already shifted far away from the pump spectrum due to a large density of accumulated free carriers caused by the slow carrier relaxation. Then, injecting a stronger pump, will not increase the resonance shift, since only the initial part of the pump is coupled to the cavity. Second, at large pump powers, the cavity transmission maximum may shift across the probe wavelength leading to a decrease of the probe transmission. Finally, a larger density of free carriers will cause larger FCA, thus reducing the probe transmission.

In Fig. 9, a transient oscillation of the probe transmission curve is seen, which can be explained as resulting from interference between the probe signal and the cavity mode [41]. When the switching energy is low, the frequency difference between the cavity mode and probe is large, so the beating period is short and the oscillations can be clearly seen. However, as the pump energy increases, the cavity resonance blue shifts closer to the probe and the beating period becomes longer and the oscillations are no longer visible due to the overall transmission decay. These beating-induced oscillations can introduce a decay of energy faster than the cavity photon lifetime and it was recently suggested to use such coherent effects to improve the performance of the switch [25].

From the results in Fig. 9, it can be concluded that the wavelength conversion performance, especially at high repetition rates, is seriously affected by the slow carrier recovery process. To improve the performance, the time constants can be decreased by implementing strategies such as capturing carriers by impurities [8, 42] or quantum wells [14], employing a thinner membrane, sweeping out carriers using electric contacts [16, 43], or doping with n-type carriers to reduce the ambipolar lifetime. The *Q*-factor should be chosen so that the dynamics is not limited by the cavity photon lifetime [41]. Also, other PhC structures may be employed to improve the switching performance, such as multi-mode and multi-port cavity structures employing a low-*Q* mode for the probe [20, 38, 44], and Fano structures [45, 46]. Although the switching speed is one of the most important characteristics of the switch, it should be noted that simply increasing the carrier relaxation rate will increase the energy consumption, since it will reduce the accumulation of carriers during the pump excitation, which will affect the switching contrast as well. This energy-bandwidth trade-off [41] is a major issue for nonlinear optical switches and needs to be further investigated in systems experiments.

## 8. Conclusion

The carrier dynamics and switching characteristics of an InP photonic crystal H0 nanocavity were experimentally investigated using pump-probe measurements. The results were compared to simulations using a model based on coupled-mode theory and a rate equation description of the carrier dynamics. Excellent agreement between measurements and simulations was obtained when employing a model containing three relaxation times, accounting for the joint effects of carrier diffusion, surface recombination and bulk recombination. The dependence of the parameters entering the rate equation model on the physical material parameters for the switch was deduced by numerical simulations of the diffusion equation, confirming the relevance of the suggested rate equation model and its physical interpretation. The model was next applied to analyze the role of the fast and slow carrier relaxation processes on patterning effects in the device, as exemplified by the case of all-optical wavelength conversion. It was shown that although a short switching window may be realized by exploiting fast carrier diffusion effects, the accumulation of carriers due to slow (bulk and surface) recombination in the cavity region may be detrimental at high repetition rates. Further investigations of these limitations and alternative structures for alleviating them are thus needed. The coupled-mode theory model suggested in this paper can be used to analyze various applications and structures, and furthermore has a form that can be easily extended to include additional effects, thus making it useful for exploring the potential of nanocavity switches for systems applications.

## Acknowledgments

The authors acknowledge financial support from Villum Fonden, via the NATEC (NAnophotonics for TErabit Communications) Centre, and the European Union (COPERNICUS project, FP-ICP-249012).

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