Kerr and free-carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three-photon edge of GaAs

Our structure consists of a GaAs $\lambda$ thick layer (277 nm thick) sandwiched between two Bragg stacks consisting of 12 and 16 pairs of $\lambda/4$ thick layers of nominally pure GaAs or AlAs. The sample is grown with molecular beam epitaxy at 550⁰C to optimize the optical quality Gerard1996 . For experiments outside the present scope the sample was doped with $10^{10}$cm^-2 InGaAs/GaAs quantum dots, which hardly influence our experiment noteQD .

Our setup consists of two independently tunable optical parametric amplifiers (OPA, Topas), that are the sources of the pump and probe beams. The setup has been described in Euser:07a therefore we present only a brief outline here. The OPAs have pulse durations $\tau_{\rm{P}}=140\pm 10$ fs. The pump beam has a much larger Gaussian focus of 113 $\mu$m full width at half maximum than the probe beam (28 $\mu$m), ensuring that only the central flat part of the pump focus is probed. A versatile measurement scheme was developed to subtract the pump background from the probe signal, and to compensate for possible pulse-to-pulse variations in the output of our laser Euser:07a . Separately, continuous-wave (cw) reflectivity was measured with a broad band white light setup with a resolution of $\sim 0.2$ nm Thijssen1999 .

In this work we varied the pump wavelength from 2000 nm to 2400 nm in steps of 200 nm. The probe wavelength was varied independently of the pump wavelength between 1150 nm and 1650 nm, with steps of 5 nm. Measured data was corrected for dispersion in the system.

Figure 1C shows the measured linear reflectivity spectrum of our sample and a transfer matrix (TM) calculation BornWolf . The transfer matrix calculation including the dispersion of GaAs Blakemore1982 and AlAs Fern1971 reproduces the experimental resonance, stopband, and Fabry- Pérot fringes. The only free parameters in the model were the thicknesses of the GaAs (dGaAs = 69.2 nm) and AlAs (dAlAs = 81.0 nm), which agree to the measured values.

The reflectivity spectrum shows the stopband with cavity resonance and Fabry-Pérot fringes. The spectrum can be divided into two spectral regions, namely ’Absorptive’ (right, densely hatched) and ’Dispersive’ (left, hatched), referring to the expected behavior if pump and probe coincide.

The origin of the absorption is explained in figure 1A and 1B, which show schematic energy diagrams at coincidence of pump and probe. The regions are separated at a wavelength which corresponds to the bandgap at a pump wavelength of 2400 nm. In the densely hatched spectral region, the pump and probe photon will be absorbed since the sum of the photon energies is higher than the bandgap of GaAs. In the sparsely hatched (dispersive) spectral region however, the sum of pump and probe photon energies is not larger than the bandgap of GaAs. Therefore absorption is low in this region and behavior at coincidence is therefore mainly dispersive. Only the real part of the refractive index n’ changes.

The origin of the dispersion is the optical Kerr effect. There will be a change in the real part of the refractive index (n’), due to the pump field. The edge of the hatched regions in figure 1C will shift to the red at a pump wavelength of 2000 nm, indicating that we expect absorption in a large part of the spectrum. The behavior at coincidence is therefore mainly absorptive, meaning a change in the imaginary part of the refractive index, (n”).

We focus in this work on the spectral region indicated by the black rectangle. In this region we expect to observe effects of the optical Kerr effect but only little absorption at 2400 nm pump wavelength, while we expect mainly absorption at 2000 nm pump wavelength.

The two plots in figure 2 show the differential reflectivity measured at 2000 nm, $I_{pump}$ = 90 $GW/cm^{2}$ (A) and 2400 nm, $I_{pump}$ = 95 $GW/cm^{2}$ (C) pump wavelength as a function of probe wavelength and delay. Cross sections of figure 2A and C are given in figure 2B and D respectively.

A fringe pattern is visible in figure 2A and C for coincidence of pump and probe and for positive delay. Figure 2B and D show cross sections of 2A and C. The sign difference of differential reflectivity between pump and probe overlap and positive delay is apparent and is a result from the fringe patterns at coincidence and positive delay, which are spectrally shifted with respect to each other. This indicates that the switch mechanism at coincidence is different from the free-carrier mechanism at positive delay.

At a pump wavelength of 2000 nm we expect a change in n”, since we are in the absorptive regime. At 2400 nm pump wavelength we expect a change in n’ and a negligible change in n”, since we are in the dispersive region (see figure 1).

The differential reflectivity at positive delay is caused by excited free carriers. Figure 3 shows the power dependence of differential reflectivity ($\Delta R/R$) at positive delay.

There is a linear relation between the differential reflectivity and the power cubed, as shown in figure 3. We therefore conclude that the differential reflectivity is caused by free carriers generated by a three photon absorption process.

The fringe pattern in figure 2A and 2C results from Fabry-Pérot interferences within the stack forming the optical microcavity. A change in n’ modifies the optical thickness of the GaAs layers; it induces a spectral shift of the Fabry-Pérot fringes, which results in a periodic differential reflectivity. A change in n” decreases the modulation depth of the Fabry-Pérot fringes.

The differential reflectivity due to a change in the real part of the refractive index depends on measured intensities as follows:

Where $\frac{\Delta R}{R}$ is the differential reflectivity, $\tau$ is the delay between pump and probe, $\omega_{pr}$ is the probe frequency, $\omega_{pu}$ is the pump frequency, $I_{unpu}$ is the reflectance measured if the structure is unpumped and $I_{pu}$ is the reflectance measured if the structure is pumped.

In case of a change in the real part of the refractive index, the fringe pattern shifts spectrally $\Delta\omega_{pr}=\alpha n^{\prime}$, with $\alpha$ a proportionality constant that depends on the exact structure of the fringe pattern. $I_{pu}$ is then given by:

Which can be written as

which yields a differential reflectivity:

Equation 4 shows that in the case of a change in the real part of the refractive index a large differential reflectivity will be measured where the unpumped reflectivity has a large derivative, which is spectrally between successive maxima and minima of the Fabry-Pérot fringes. This is illustrated in figure 4.

In the case of a change in n” the modulation depth of the fringe pattern is affected, while in the case of a change in n’ the position of the fringe pattern is shifted. For this reason the differential reflectivity has maxima at different positions making it possible to distinguish between a purely dispersive and a purely absorptive regime.

Figure 5 shows a TM calculation of the differential reflectivity spectrum of the planar microcavity. The intensity profile in the structure is homogeneous, leading to a homogeneous refractive index change. The intensity profile is homogenous since we pump at a long wavelength in the three photon absorption regime. The homogeneity length (Euser2005 ) is longer than our sample.

Figure 5 shows a comparison between the measured and calculated differential reflectivity pattern. Comparing the measured and calculated patterns shows that the positions of the peaks and troughs in the differential reflectivity pattern near zero delay, are caused by a change in the imaginary part of the refractive index (n”) in the case of 2000 nm pump and a change in the real part of the refractive index (n’) in the case of 2400 nm pump. We conclude that in the case of 2400 nm pump wavelength the fringe pattern near zero delay in the differential reflectivity originates from a Kerr switch, since the process is ultrafast, instantaneous with the laser pulse and is dispersive. In case of 2000 nm pump wavelength the fringe pattern originates from non-degenerate two photon absorption. Furthermore we see a slight deviation of our model near the blue side of the spectrum, indicating some dispersion in the nonlinear effect. We further conclude from figure 5 that absorption can be neglected at 2400 nm pump wavelength, while dispersion can be neglected at 2000 nm pump wavelength.

By varying the size of the change in n’ we extracted the change in refractive index at each wavelength. The results are plotted together with dispersion curves in figure 6. Figure 6 shows no change in refractive index for a negative delay (A), a positive change for zero delay (B) and a negative change for positive delay (C). Furthermore there is a good agreement between the data and the dispersion of the optical Kerr effect sheikbahae1994 and the dispersion of the free carrier excitation Euser:07a .

The scattered symbols deviating strongly from the theoretical model can be attributed to small differences in shape of the measured and calculated reflectivity spectrum. A slight deviation is amplified by the fitting procedure since we extract the refractive index change by fitting the differential reflectivity at one wavelength position.

Figure 6B shows that the change in refractive index induced by the optical Kerr effect is in the order of 0.1%. This is large enough to switch a cavity with a moderate Q of 1000.