1. Introduction

Single, or few-mode resonant cavities are likely to be critical components in all-optical information processing systems. Microfabricated cavities that confine photons in three dimensions and that can be coupled efficiently to single-channel waveguides are of particular interest due to their miniature scale, and the low optical power required to effect the nonlinear response needed for all-optical control of signals carried by the waveguides. There have been dramatic recent advances (in both disk and planar photonic crystal (PPC) geometries) optimizing Q-factors, minimizing mode volumes, and coupling to single channel waveguides [1, 2, 3, 4, 5, 6]. These advances have enabled researchers to show that the cavity resonant frequency and Q-values [7] can be substantially influenced by using low power optical excitation to inject free carriers into the cavities, thus modifying their refractive indices (both real and imaginary parts) via the Drude effect.

While much of the literature has focussed on the use of free-carrier injection to change the resonant reflectivity characteristics of the waveguide-coupled microcavity, thereby effecting an all-optical switch [8, 9, 10] or bistability [5, 11, 12, 13], recent work has highlighted the fact that the same effect can be used to shift the emission frequency of a populated microcavity — i.e. one that is storing electromagnetic energy [14, 15]. In the present paper, we show experimental results and simulations of the ultrafast optical perturbation of a high-Q, PPC-based microcavity mode that demonstrate the inextricable connection between microcavity-based switching and frequency-shifting processes. In particular, we show how both the frequency shift and Q-spoilage associated with rapid free-carrier injection influences the dynamic polarization properties within the microcavity.

2. Experiment

To fully characterize the cavity dynamics that occur when free carriers are injected on time-scales much faster than the cavity lifetime, we perform a pump-probe experiment where the pump (used to inject free carriers) consists of a 130 fs pulse above the bandgap of the host silicon, and the probe diagnostic involves resonant scattering of a second 130 fs pulse, tuned in the vicinity of the dynamically perturbed resonance. The resonant scattering spectrum in the vicinity of the original and perturbed cavity frequencies is time-integrated, and therefore reveals the complete power spectrum of electromagnetic energy emitted from the cavity as a function of the relative arrival times of the probe, which injects energy into the cavity, and the pump, which dynamically alters the refractive index and damping of the cavity.

 

Fig. 1. Optical set-up of the pump-probe resonant scattering experiment. The two beams propagate colinearly after beamsplitter 1 (BS1), with a relative delay controlled by the delay line on the pump beam path. The resonantly scattered radiation is collected in reflection and detected in the cross polarization by a Fourier transform (FT) spectrometer. A scanning electron microscope image of the L3-microcavity is shown in the inset. The scalebar denotes 1 µm.

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The microcavity, which is fabricated in a free-standing silicon membrane, consists of three missing holes in an otherwise uniform hexagonal photonic crystal [16], which has a lattice spacing of 465 nm, an air hole radius of 175 nm, and a thickness of 196 nm. It is fabricated using standard electron beam lithography, reactive ion etching, and wet chemical etching techniques, leaving a free-standing planar photonic crystal microcavity separated from the supporting Si substrate by a 1.1 µm air gap. The cavity mode of interest has a quality factor of Q ₀=35,000 at a frequency of 6307.9 cm^-1. This mode is not the fundamental mode of the cavity, but rather a quasi-localized mode within the continuum of states, near the low energy edge of the photonic bandgap [17]. Many such modes are typically observed in both resonant scattering experiments and FDTD simulations, but their experimental spectra vary from sample to sample and are difficult to correlate to the simulations. Translation of the sample by about 1 micron completely eliminates the resonant scattering feature, confirming that the mode is tightly confined to the microcavity. Tight spatial confinement and a high quality factor are the two prerequisite properties for this investigation, making this mode an attractive candidate for study.

The probe pulses, obtained as the “signal” beam from an optical parametric oscillator (OPO), consist of an 80 MHz train of 130 fs pulses at ~1550 nm and an average power of 0.5 mW (6 pJ per pulse) at the sample. The 200 pJ pump pulses are derived from the Ti-sapphire 810 nm laser which drives the parametric down-conversion process inside the OPO. The relative timing of the pump and probe pulses is controlled at the sub-picosecond level by an optical delay line on the pump beam path, as shown in Fig. 1. The optical delay line position corresponding to zero delay between the pump and the probe pulses was found by inserting a beta-barium borate (BBO) crystal at the sample plane and detecting the sum-frequency radiation between the pump and the probe beams on a photodiode detector.

 

Fig. 2. Example of a raw resonant scattering spectrum, showing the sharp mode feature superimposed on the non-resonant laser line-shape. The data of Figs. 3 and 4 are obtained by normalizing the raw spectrum by a similar spectrum for which the mode is temperature-tuned to a different frequency. This normalization procedure removes the Fabry Perot fringes evident in the spectrum and results in an essentially background-free probe of the mode.

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The two beams are recombined at a beamsplitter (BS1), after which they propagate collinearly. A 100× microscope objective is positioned to focus the probe beam onto the microcavity from normal incidence to a spot diameter of 2.5 µm [6]. In this geometry, approximately 2% of the probe beam, or 10 µW, is coupled into the microcavity mode [6], well below the typical threshold for which nonlinear effects due to two-photon absorption are observed [5]. Due to chromatic aberration, the pump beam is not focussed precisely at the plane of the cavity, so it has a spot radius of ~10µm on the sample. Approximately 1% of the pump spot area (containing 2 pJ of energy) overlaps with the mode, and about 3% of this energy is absorbed (since aSi(775 nm)=1.6×10⁵ m^-1), creating a free-carrier density of 5×10¹⁷ cm-¹. The resonantly scattered light reflected from the microcavity is measured in the crossed polarization [18], and detected with a Fourier transform spectrometer. A typical spectrum is shown in Fig. 2, in which the sharp mode feature is visible on top of the broad non-resonantly scattered laser line-shape, which is modulated by a pattern of Fabry Perot fringes associated with the crossed polarizers. To generate the clean series of spectra shown in Figures 3 and 4, both the non-resonant laser line-shape and the Fabry Perot fringes are normalized out by dividing each of the raw spectra by a reference spectrum acquired with the same beam alignment but the mode temperature-tuned out of the spectral window of interest.

Three relevant regimes of interest can be identified according to the relative delay between the pump and the probe beams. Defining t ₀ as the excitation time of the microcavity mode via the probe pulse, t_p as the arrival time of the pump pulse, and τ_m as the lifetime of the mode, these regimes are: t_p<t₀ (I), t₀<t_p<t₀+τ_m (II), and t_p≫t₀+τ_m (III).

 

Fig. 3. Resonant scattering spectra from a microcavity for a wide range of delay times between the pump and the probe beams. The spectra are offset so that their non-resonant backgrounds intercept the vertical axis at the corresponding pump-probe delay, τ=t₀-t_p. The red spectrum in the lower plot shows the bare mode spectrum with the pump off. The bare mode and the two spectra at negative delays are not fully resolved, and so appear broader than the numbers quoted in the text.

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3. Results and discussion

Focussing first on regimes I and III, Fig. 3 shows a series of spectra for different values of the pump-probe delay, defined as τ=t₀-t_p. These spectra are relatively easy to interpret, and demonstrate the intuitive result for canonical all-optical switching behaviour. When the pump beam is blocked, the bare cavity mode spectrum shown in red is obtained, with Q ₀=35,000. With the pump beam unblocked, the mode is blue-shifted for both positive and negative values of the delay, indicating that there is a steady-state background population of free carriers, N_bg, maintained by the 80 MHz train of pump pulses. At negative delays, where this background population is sampled, the 1.2 cm^-1 shift and 40% reduction in the Q-factor (Q(-200 ps)=20,000) with respect to the bare cavity mode are consistent with N_bg=1.6×10¹⁷ cm^-3. At 0 ps, there is a transient blue-shift of the mode to ω₂=6310.9 cm^-1, and subsequently the mode red-shifts as a function of increasing delay. The transient frequency shift, ω₂-ω₁, is 1.8 cm^-1, and the total shift with respect to the bare mode, ω₂-ω₀, is 3.0 cm^-1, or 16 line-widths (G₀) of the bare cavity mode.

The mode position as a function of positive delay essentially corresponds to a map of the free-carrier relaxation rate, since even during the relatively rapid free-carrier decay immediately following injection (with a ~400 ps decay constant), the carrier density is still approximately constant during the lifetime of the cavity mode (τ_c(0ps) ~8 ps at the peak density). Both the 400 ps carrier initial decay constant and the fact that the decay is not a single exponential function are consistent with other experiments on silicon microresonators [5, 9].

We turn now to an investigation of frequency conversion in regime II. Figure 4(a) shows a series of spectra corresponding to dynamic perturbations during the ring-down of the microcavity mode. In this regime, the probe beam excites the mode, with frequency ω₁ and width Γ₁ (the equilibrium values in the presence of N _bg), and as the energy in the microcavity decays, the pump pulse arrives and changes the cavity properties by injecting free carriers. Thus, the probe pulse arrives before the pump, as indicated by the negative delay values in Fig. 4. Most of the spectrally broad probe pulse scatters away from the cavity within a few hundred femtoseconds, and so, for all the spectra at negative delay times in Fig. 4, only the energy stored resonantly in the mode remains when the pump pulse perturbation arrives.

At a probe delay of -60 ps, only 3% of the initial energy of the mode at ω₁ remains in the microcavity when the perturbation occurs, and so most of the energy collected by the spectrometer is due to the emission from the unperturbed mode. As the delay approaches 0, a decreasing fraction of the collected energy is from the unperturbed mode: the spectrum near the original mode frequency broadens considerably and radiation is also emitted in the oscillating part of the spectrum towards higher frequency. At -14 ps, the spectral weight is broadly distributed over the range between ω₁ and ω₂, yielding an almost dispersive line-shape. Only at -7 ps is the clear signature of the perturbed cavity mode at ω₂ revealed.

This behaviour is largely reproduced using a phenomenological model to describe the rapid shift of the resonant frequency and lifetime (Q-factor) of a damped harmonic oscillator, as induced by the free-carrier modification to the refractive index, and hence ring-down characteristics, of the cavity. The simple model is summarized by the following equation for the resonant material polarization in the cavity:

where q (t) is the Heaviside step function. The first term describes the evolution between time t=0 and t=t_p of the polarization associated with the fields trapped in the mode. The second term represents the evolution after the pump pulse perturbation at time t_p, with new frequency ${\omega }_2={\omega }_1+\Delta \omega e^{-\left(t-t_p\right)⁄{\tau }_c}$ and line-width ${\Gamma }_2={\Gamma }_1+\Delta \Gamma e^{-\left(t-t_p\right)⁄{\tau }_c}$, where τ_c=400 ps is the free-carrier lifetime. The scaling factor, $e^{-i{\omega }_1{t}_p-{\Gamma }_1{t}_p}$, is necessary to match the amplitude of the oscillator between the two regimes at t=t_p. The Heaviside functions in the equation represent an instantaneous perturbation, which is a reasonable assumption in the classical limit, since the 130 fs pump pulse is more than two orders of magnitude shorter than the mode lifetime.

Time-domain data were numerically generated according to Eq. (1) and then Fourier transformed, and the set of parameters (Δω,Γ₁,Γ₂) were varied to provide the best match between the simulation, as shown by the red spectra in Fig. 4(b), and the experiment. Qualitatively, this simple model captures all the main features of the data. At 0 delay, the frequency shift determined from the best fit is Δω=1.8 cm^-1, and the damping parameter due to free-carrier absorption is Γ₂=2.0Γ₁, which gives Q(0 ps)=10,000. The pump-induced change in the cavity index with respect to the bare cavity is, from perturbation theory, Δn=-nΔω/ω=0.0017. Using this index shift and Eq. (4) in [19], the maximum free-carrier density in the cavity at 0 delay is found to be N=5.2×10¹⁷ cm^-3, which is consistent with experimental estimates based on the pump pulse energy and spot-size. Similarly, the extinction coefficient at this carrier density is calculated to be k=9.5×10^-5 [19]. The effect of the free-carrier absorption on the mode Q-factor is estimated using the three-dimensional finite-difference time-domain (3D-FDTD) method, assuming n and k do not change over the mode lifetime. This predicts that the bare cavity Q-factor of 35,000 is spoiled by a factor of 3.7, which is in good agreement with the experimental ratio of 3.5.

We have investigated the dependence of the simulation results on the finite time of the perturbation. The spectra are actually best fit when a 500 fs linear transition for both ω and Γ is used in place of the step function. This likely reflects both the pulse broadening (pump and probe) beyond 130 fs that occurs during propagation through the 100× microscope objective, and the convolution of the probe pulse shape with the integral of the pump pulse shape (the carrier density is roughly proportional to this integral pulse-shape). The results of these simulations are shown by the blue spectra in Fig. 4(b). As can be seen in the figure, there is actually a very small effect of the non-instantaneous transition, with the main difference revealed in the spectrum at τ=-13 ps, for which there is a reduced spectral weight at the original cavity frequency due to the finite width transition.

 

Fig. 4. (a) Experimental spectra of a high-Q mode dynamically perturbed by a population of free carriers. The data are normalized by a reference spectrum of the non-resonantly scattered laser line-shape. (b) Normalized spectra from simulations of a perturbed harmonic oscillator in which the perturbation of ω and G occurs instantaneously (blue traces), as in Eq. 1, and linearly over a 500 fs time width (red traces). The dashed black line in both plots indicates the e ^-1 lifetime of 17 ps of the mode (at ω₁) before the dynamic perturbation.

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The robustness of the oscillations and the broadening of the original cavity resonance peak at large negative delays confirms their interpretation as due to the abrupt (on the time-scale of the original cavity mode decay) termination of the original Lorentzian decay by the pump pulse, which then initiates a second Lorentzian decay that is allowed to fully decay. The broadened original resonance, and the “ringing” in frequency space, with a period inversely proportional to the time delay between the start of the Lorentzian decay at t=0 and its termination at t=τ, is the frequency domain manifestation of an “ideal filter” applied to the decaying field in the time domain. The finite transition time amounts to apodizing this ideal filter, and the effect of the apodization is small when the time-scale is much less than the mode decay constant.

 

Fig. 5. Simulation of a dynamically perturbed mode with no change in damping, and a new mode frequency that does not relax with time. The unperturbed mode frequency and Q are the same as in Fig. 4(b). The dashed red line is explained in the caption to Fig. 4.

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Recently, it was conjectured that this dynamic tuning process would lead to adiabatic frequency conversion when applied to a mode which undergoes a dynamic frequency change with no change in lifetime [14]. By applying our model to this scenario, we generate the spectra shown in Fig. 5. In the absence of a change in the Q-factor, there is a smooth transition of spectral weight between one frequency and the other, which looks intuitively like adiabatic frequency shifting, as claimed in [14]. This is in marked contrast to the case when the Q-factor is spoiled by the perturbation, as in our experiment, and the contrast is greater for higher-Q modes. Our actual results (experiment and model) show that the spectral signature of the perturbed mode is not revealed until close to 0 delay, complicating the intuitive adiabatic shift interpretation. This analysis demonstrates that dynamic changes in the Q-factor play a significant role in determining the nature of the frequency transition. In either case, with or without the modified Q factor, there is clearly an eventual transfer of emitted radiation from the old to the new cavity mode frequency, but the interpretation of the transition regime is subtle.

Thus we have shown that the classical spectra from a dynamically perturbed microcavity mode can be robustly modelled as a classical damped oscillator which undergoes a rapid (on the time-scale of the mode lifetimes) change in frequency and lifetime. Finally, we comment on the possible implications of this cavity perturbation scheme on the quantum optical properties of the emitted radiation. When external changes to the frequency of a harmonic oscillator are made suddenly on the time-scale of the mode period (as opposed to its lifetime), it is well known that squeezed states can be generated [20, 21]. When considering a resonant mode of a microcavity excited by a strong laser field, it is appropriate to represent the mode (oscillator) using coherent states. These states are squeezed by a sudden frequency change, whereas a more gradual change has no effect on the variances of the the oscillator quadratures [20]. These two limiting cases are distinguished by the relation of the perturbation rate, γ, to the oscillator frequency, ω. The change can lead to squeezing if 2πγ/ω≫1. In the context of the present pump-probe experiment, the oscillator period is ~5 fs, whereas the free-carrier perturbation occurs on the time-scale of the pump pulse, which is >130 fs. Realizing a perturbation fast enough to generate squeezed states would be very challenging to implement at optical frequencies, but might be applicable with ultra-short laser pulses (<10 fs) and mid-IR microcavity modes (which would have longer periods).

4. Summary

In summary, we have used an ultrafast pump-probe experiment to reveal the total power spectrum of radiation emitted by a populated optical microcavity that is dynamically perturbed by optically injecting free carriers on sub-picosecond time-scales. This work has shown the connection between frequency conversion and optical switching processes. It reveals that both the change in cavity frequency and the cavity Q-factor on time-scales much faster than the cavity mode lifetime give rise to spectral structure that is richer than the intuitive adiabatic transfer of energy from one mode to another.