Observation of a strongerthanadiabatic change of light trapped in an ultrafast switched GaAsAlAs microcavity
Abstract
We study the timeresolved reflectivity spectrum of a switched planar GaAsAlAs microcavity. Between 5 and 40 ps after the switching (pump) pulse we observe a strong excess probe reflectivity and a change of the frequency of light trapped in the cavity up to 5 linewidths away from the cavity resonance. This frequency change does not adiabatically follow the fastchanging cavity resonance. The frequency change is attributed to an accumulated phase change due to the timedependent refractive index. An analytical model predicts dynamics in qualitative agreement with the experiments, and points to crucial parameters that control future applications.
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, The Netherlands \affiliationFOM Institute for Atomic and Molecular Physics (AMOLF), Science Park 104, 1098 XG Amsterdam, The Netherlands \affiliationFOM Institute for Atomic and Molecular Physics (AMOLF), Science Park 104, 1098 XG Amsterdam, The Netherlands \affiliationCEA/INAC/SP2M, Nanophysics and Semiconductor Laboratory, 17 rue des Martyrs, 38054 Grenoble Cédex, France \affiliationComplex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, The Netherlands \affiliationCEA/INAC/SP2M, Nanophysics and Semiconductor Laboratory, 17 rue des Martyrs, 38054 Grenoble Cédex, France \affiliationComplex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, The Netherlands
1 Introduction
There is a fast growing interest to control and manipulate information encoded as photons as this promises much broader information bandwidths than with electronic integrated circuits [1]. Therefore it is important to realize timedependent control of photonic systems, in other words, photonic switching. A crucial step is to have control and signal photons interact, such that the signal is phase modulated and its frequency changed to a different value. As the interaction between photons in free space is exceedingly weak [2], one resorts to wellknown methods of nonlinear optics to control light with light [3, 4]. Recently the switching of photonic bandgap crystals and nanophotonic cavities has been advocated in nonlinear optics [5, 6, 7, 8, 9, 10]. These devices permit a judicious tweaking of optical dispersion, offer greatly enhanced field strength in tiny volumes, and have small sizes that make them amenable to largescale integration.
In nanophotonic nonlinear optics [10, 11, 12], a cavity is charged by a probe pulse before a pump pulse arrives. The light in the cavity then adiabatically follows the cavity resonance and exits at the frequency where the cavity has quickly been tuned to. In this paper, we explore conditions where light does not adiabatically follow a singleresonance cavity, after an earlier pump pulse. As a result, the frequency of probe light changes to a value different from the cavity resonance. To the best of our knowledge, such nanophotonic tuning of the light’s frequency has not been observed before.
2 Sample and experimental setup
We have studied a planar microcavity consisting of a GaAs layer with a thickness nm that is sandwiched between two Bragg mirrors made of 12 and 16 pairs of thick layers of nominally pure GaAs and AlAs, see Refs. [13, 14, 15]. A normalincidence reflectivity spectrum reveals the single cavity resonance at eV ( nm) centred in the wide stopband of the Bragg mirrors, see Fig. 1(a). The cavity resonance in Fig. 1(a) has an inhomogeneous linewidth of 1.03 meV, as a result of a finite numerical aperture, and of a slight spatial variation of the resonance frequency. In separate timeresolved experiments, we have determined the average trapping time for photons in the cavity to be fs, corresponding to a quality factor , and a homogeneous cavity linewidth meV [15].
Timeresolved pumpprobe spectroscopy was performed on the cavity in reflection geometry [16]. Two independently tunable optical parametric amplifiers generate pump and probe pulses with fs duration. The pump pulses are incident obliquely to avoid scattering into the detection path (Fig. 2). The pump frequency was not resonant with the cavity and tuned to 0.72 eV ( nm) to switch the refractive index in all GaAs layers (bandgap 1.42 eV) by exciting free carriers by twophoton absorption [17]; the AlAs is hardly excited as its bandgap is much larger (2.2 eV). Also, by setting the pump to the twophoton regime, we evade nondegenerate twophoton absorption at coincidence [18]. Our observations of the change in frequency were reproduced in runs with pump frequencies below 0.62 eV in the threephoton regime of GaAs. To ensure that the microcavity is spatially homogeneously switched [17], we focussed the normalincident probe beam to a much smaller spot (28 m Gaussian diameter) than the pump spot (113 m.) The probe fluence of mJ.cm energy per pulse was kept well below the pump fluence of mJ.cm to prevent inadvertent pumping by the probe pulses. The probe pulses have a relative spectral bandwidth of (1.260 to 1.295 eV) that is much broader than the cavity’s bandwidth of as shown in the unreferenced reflectance spectrum in Fig. 1(b). Thus, we probe the dynamic cavity resonance by resolving the transient reflectivity with a highresolution (0.12 meV) spectrometer at each pumpprobe delay . The stepsize in the delay in Figs. 3 was ps between and ps, and ps henceforth. We used a PI/Acton sp2558 monochromator and a 1024 channels OMAV InGaAs diode array. Due to inadvertent misalignment a slant occurred in the raw spectra that was corrected with a straight line and assuming constant reflectivity in the stopband as in Fig. 1. Spectra are averaged over probe pulses alternatingly reflected from a pumped and an unpumped cavity; the reproducibility of the unswitched resonance confirms that the cavity is unaffected by the experiment. The unswitched resonance also provides an insitu reference for the switched events.
3 Timeresolved transient reflectivity
The cavity’s timedependent resonance is presented in a photon energy versus pumpprobe time delay ()diagram shown in Fig. 2. These data consist of many transient reflectivity spectra while stepping the pumpprobe delay. The transient reflectivity is the timeintegrated signal [16] that exits the cavity as a result of a short probe pulse incident at each (and should not be confused with the instantaneous reflectivity at [19]). The feature at 1.278 eV independent of time delay is the unswitched cavity resonance. When the cavity is optically switched with a short pump pulse, free carriers are excited leading to an ultrafast reduction of the refractive index. As a result, the cavity resonance frequency rapidly increases within 3 ps from to 1.290 eV, while the cavity linewidth hardly changes [13]. This resonance shift corresponds to as much as 14.5 linewidths , a signal of a strongly switched cavity. Subsequently, the carriers relax and the refractive index increases to its unswitched value in about 100 ps [13].
To investigate the spectral features in more detail, we plot in Figure 3(a) transient reflectivity spectra at select time delays. The trough at 1.278 eV in all spectra is the resonance of the unswitched cavity obtained from the events in absence of a pump. The trough of the switched cavity resonance ( eV at 8 ps) is asymmetric due to the timeintegrating nature of the experiment and the dynamics of the cavity resonance: photons exiting from the cavity are recorded while the resonance is shifting. Strikingly, the transient reflectivity exceeds to the red of the switched resonance, in particular near 1.283 eV at 8 ps delay (and at 18 ps delay and 1.282 eV) [19]. The excess transient reflectivity corresponds to frequency shifted probe photons that enter the cavity at a delay ps when the dynamic cavity resonance is still centred at 1.2876 eV. The frequency of the trapped probe photons has changed by as much as 4.6 meV or 5.4 linewidths to 1.283 eV when they exit at a later time .
We make four salient observations: a.) The frequency of frequencychanged photons is substantially different from that of the cavity resonance at : During the trapping time , the cavity shifts by , which is around 0.4 meV, and thus much smaller than the frequency change of the trapped light (4.6 meV at ps). We therefore conclude that the frequency of the trapped photons does not adiabatically follow the cavity resonance. b.) Since the cavity is not resonant with the pump beam, the pump pulses have long vanished at ps. Therefore, the change in the frequency of light is not caused by the temporal overlap with a pump pulse, in contrast to (nanophotonic) nonlinear optics [4, 10, 11, 12]. c.) Since the frequencyshifting occurs long after the pump pulse, the quality factor of the cavity does not degrade during the trapping time . Quite the opposite, actually increases slightly during , in contrast to other switching schemes [10, 11, 12]. d.) The frequencyshifting of the trapped photons seems to correlate with the rate of change of the cavity resonance that characterizes the rate of change of the refractive index: At short probe delays, the rate of change of the resonance is of the order of the characteristic ratio . At long probe delays ps, where , no frequencyshifted photons are observed. In summary, the observations for the frequency change of light in a strongly and ultrafast switched cavity differs fundamentally from previous frequency shifting as studied in, e.g., Refs. [4, 9, 10, 11, 12].
4 Model for the frequency change of trapped light
The frequency change of light in a strongly and ultrafast changing cavity can be calculated as follows without assumptions on adiabaticity in the following: Photons from the short probe pulse that enter the cavity are trapped for a while  on average the cavity trapping time  before exiting. The trapped light experiences a changing refractive index, which results in a phase shift that changes the light frequency, whereas the cavity resonance shifts by a different amount. If the probe pulse enters the cavity at delay , each angular frequency component travels a distance (or correspondingly a time ) through the cavity and acquires a phase shift
(1) 
with the refractive index at time , and the refractive index at time . The refractive indices are determined by the pumpprobe time delay and by the dynamics of the refractive index (here: dynamics of relaxing free carriers). The phase shift as a function of propagation length affects the time dependence of both the amplitude and phase profile of the exiting probe light , equal to
(2) 
Here frequency components are defined by the spectrum of the incident probe pulse . The function is the timeevolution kernel (Fourier transform of the Smatrix) that determines the range of cavity propagation lengths and hence the temporal profile of the light exiting from the cavity. Thus, light exiting at later times accumulates more phase shift since the interaction length is longer compared to light that exits early on. The accumulated phase is positive as the refractive index increases while the light exits from the cavity. Since the change in frequency is related to minus the time derivative of the phase (), the increased phase causes the frequency of the light to decrease.
The dynamics of the light field in a multilayer sample is described by a straightforward generalization of eq. 1 that takes into account reflection and transmission coefficients of the 59 interfaces of the GaAs and AlAs layers in the sample. The refractive index of the GaAs layers are taken to be switched by Drude freecarrier dispersion to have the resonance frequency match the experiments (Fig. 2). The time dependent refractive indices in the GaAs layer and those in the GaAs mirrors have been calculated in Ref. [14]. Equation 2 was solved numerically, where was determined by solving Maxwell’s equations for the 12 and 16 GaAs/AlAs Bragg mirrors with the time dependent refractive indices as above. The width of the cavity resonance determined by this exact solution does not account for interface roughness caused by imperfect growth and is therefore slightly narrower (0.74 meV) than the homogeneous linewidth which is gratifying as there are no free parameters. We verified that for no frequency changes occur. While the accumulated phase per cavity roundtrip is small due to the small change in during one roundtrip, the phase accumulates substantially within roundtrips. Fig. 3(b) shows the power spectrum of the waveforms at the same delays as in Fig. 3(a). For better comparison with the experiment we average over switched and unswitched spectra. At ps, we find the unswitched cavity resonance at 1.278 eV, and the switched cavity resonance at 1.287 eV. The switched cavity resonance exhibits a strong asymmetry caused by the timeintegration of the shifting cavity resonance that we also observe in the experiment. The redshift of the trapped light explains the occurrence of transient reflectivity peaks 100% at frequencies below the switched cavity resonance at 1.286 eV and 8 ps in Fig. 3(b). The experimental shifts are larger than in the model, which we tentatively attribute to beyondDrude carrier dynamics in the experiments since the carrier density ( cm) approaches the range where scattering of electrons with other electrons, holes, or phonons become significant [20]. The strongerthanadiabatic response could possibly also be caused by an interaction of the weak probe pulses with the free carrier plasma. The qualitative phenomena of the spectra are however fully captured without such an assumption. Overall, we find a good qualitative agreement between the theoretical and experimental curves.
Fig. 4 shows the excess reflectivity (or intensity of frequency changed light) versus delay. Both the model and the measurements reveal a maximum amplitude at small pumpprobe delays of 5 to 10 ps. We propose that the excess reflectivity is large when the rate of change of the resonance is substantial compared to the cavity’s characteristic ratio . The time dependence of the calculated excess reflectivity agrees well with the measured data, which indicates that the dynamics of the trapped light and of the refractive indices are correctly modeled. With increasing delay the rate of change of the refractive index decreases, hence the phase modulation of the trapped probe light and the excess reflectivity decrease.
5 Discussion
Previously, ultrafast switching of a nanophotonic cavity has been discussed in the context of the adiabatic theorem in quantum mechanics [21]. In quantum mechanics, adiabatic and nonadiabatic indicate the ability for a highfrequency degree of freedom to follow or not a change in a lowfrequency degree of freedom, such as a 2level spin in an external magnetic field. In the adiabatic case, the change of the field is sufficiently slow for the occupied quantum state to remain in the same state. In the nonadiabatic case, the change is so fast that the occupied state does not follow the field and is (partially) transferred to other states. One might thus naively think that our observation is in a nonadiabatic regime. However, it appears that in a quantum mechanical nonadiabatic transition the phase of the system behaves erratically [22], while the phase is wellbehaved in an adiabatic system. Since the phase of light in our microcavity varies smoothly, as evidenced by the appearance of interference fringes in the transient reflectivity spectrum 3, we conclude that our experiments are in the adiabatic limit in quantum mechanics. Hence we propose to denote the observation that the frequency of light does not adiabatically follow the cavity resonance as notadiabatic to distinguish it from nonadiabatic behavior in quantum mechanics.
If we switch both GaAs and AlAs layers in our model, the transient reflectivity spectra are quite similar to the spectra for only GaAs switching (Fig. 3). Hence, spatially homogeneous or inhomogeneous switching is unimportant to the results. This confirms that our observations are in the adiabatic limit in quantum mechanics, since nonadiabatic dynamics is distinguished from adiabatic dynamics by the fact that the wavefunction changes symmetry, e.g., when spatial inhomogeneity is introduced [8].
Finally we briefly discuss several ways to apply and control our new frequency change mechanism. i. Since the frequency changes away from the cavity resonance, frequency changes are greater than those achievable by adiabatic following. Currently, the trapped photons are redshifted since the nature of the switching mechanism (excited carriers) has an increasing refractive index in time. Blueshifted light is expected to occur if the refractive index decreases as can be realized with phononinduced switching [23] or by electronic Kerr switching [24]. ii. To increase the efficiency of the frequency change, it is desirable to provide faster relaxation times in the order of ps by optimized sample growth [25], since is then even greater compared to . iii. To maximize the ratio of bandwidth that is frequency shifted to the probe bandwidth, the probe’s bandwidth could be significantly decreased. We expect the intracavity intensity, and therefore the transient reflectivity, to only depend on the spectral density of probe light near the instantaneous resonance wavelength, as long as the probe pulses remain much shorter than the cavity lifetime. Irrespective of the detailed freecarrier physics, we therefore expect similar results for pulses with a bandwidth down to 2 meV. iv. This scheme can easily applied in transmission mode. From our model, we find that a GaAs/AlAs microcavity in transmission mode predicts the same frequency shifts as in reflection. This is particularly attractive as the modulation depth varies with respect to 0 and not with respect to . v. We find that the excess amplitude increases substantially by only doubling . A further increase of saturates the excess amplitude. vi. Since the frequencyconversion occurs during the relaxation of the free carriers, will always increase during the conversion process. Higher pump powers are therefore not expected to limit the switching efficiency. vii. Notadiabatic cavity switching is suited to control frequency shifting of quantum states of light as the required strong pump field does not overlap with fragile quantum states. Hence our method opens new avenues in quantum optics and quantum information processing.
6 Conclusions
We have studied frequencyresolved femtosecond pumpprobe reflectivity of a planar GaAsAlAs microcavity. Between 5 and 40 ps after a pump pulse we observe a strong excess probe reflectivity. The frequency of light trapped in the cavity changes by up to 5 linewidths away from the cavity resonance, and does not adiabatically follow the fastchanging cavity resonance. The frequency change is attributed to an accumulated phase change due to the timedependent refractive index. An analytical model predicts dynamics in qualitative agreement with the experiments, and points to crucial parameters that control future applications.
7 Acknowledgments
We thank Bart Husken for help, and Ad Lagendijk, Klaus Boller, Pepijn Pinkse, Shanhui Fan for discussions. This work is part of the research program of FOM that is financially supported by NWO. WLV thanks NWOVici, and Smartmix Memphis.
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[19]
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