$\mathrm{Im}\{\chi^{(3)}\}$ spectra of 110-cut GaAs, GaP, and Si near the two-photon absorption band edge

Fig. 1 shows the pump-probe modulation spectroscopy (PPMS) experimental setup. The light source is a Light Conversion TOPAS-C optical parametric amplifier (tuning range $240<\lambda<2600$ nm) pumped by a 1 kHz Coherent Libra HE USP titanium-doped sapphire regenerative amplifier (nominally $\lambda=800$ nm, $\varepsilon_{P}=4.0$ mJ, $\tau_{FWHM}<50$ fs). Spectral filters attenuate undesired frequency components. A spatial filter consisting of two plano-convex $f=100$ mm lenses and a pinhole (of diameter selected based on beam parameters) homogenizes and shapes the transverse beam profile to approximate a Gaussian. These filters were configured for each excitation wavelength. A linear thin-film polarizer ensures definite polarization. An achromatic half-wave plate (HWP) controls probe polarization relative to the pump. Plano-convex lenses L1 and L2 (with same focal length $f=250$ or 500 mm, depending on beam parameters) focus pump and probe beams to a common spot on the sample. Temporal and spatial overlap was optimized by monitoring two-beam second-harmonic generation (SHG) from a barium borate crystal in the sample position. The optical chopper (OC) was sychronized to the fifth subharmonic of the laser repetition rate ($f_{mod}=200$ Hz) where spectral noise was relatively low, and could be configured to modulate only the pump beam (PPMS experiment) or both beams (probe absorption calibration). The outputs of all photodiode detectors (Thorlabs DET100A for 325 - 1100 nm, Thorlabs PDA30G for 1200 - 2000 nm) were integrated and held for each pulse until the next trigger by Stanford Research Systems SR250 gated integrators. The probe and pump integrated signals were the inputs for two Stanford Research Systems SR830 DSP lock-in amplifiers referenced to the optical chopper TTL output. A Tektronix DPO 2014 oscilloscope recorded the outputs of the lock-in amplifiers, which were digitally recorded by a computer. The computer controlled pump-probe time delay, sample Z-scan to optimize pump-probe spatial overlap, and sample rotation.

The pulse duration was measured by performing a SHG second-order autocorrelation. Pump power was measured with a Coherent FieldMate Si photodiode (PD) head (650 - 1100nm) or with a reference PD calibrated with a thermal head (1200 - 2000 nm). In order to reduce sensitivity on overlap position and geometric effects, the probe radius was kept $5\times$ smaller than the pumpDiels and Rudolph (1996) (the beam waists were not at the sample plane) which necessitated attenuating the probe pulse energy with neutral density filters $\sim 250\times$ lower than the pump to keep the fluence ratio $F_{2}/F_{1}>10$. Pump and probe beams impinged on each sample at $\approx 7.5^{\circ}$ from the surface normal, and at $\approx 15^{\circ}$ with respect to each other. The latter angle became $\approx 5^{\circ}$ (i.e. approximately collinear) inside each sample after taking refraction into account. The beam radii at the overlap position were measured using an automated knife-edge technique, and measurements at $Z=\pm 10$ and $\pm 20$ mm from the overlap were used to measure the beam angles and Rayleigh ranges.

The samples were semi-insulating undoped n-type GaAs (110) with resistivity $\rho=1.9$ – $4.4$ E7 $\Omega$ cm, intrinsic carrier concentration $N_{c}=3.2$ – $11$ E7 cm^-3, carrier mobility $\mu=3.7$ – $4.4$ E3 cm² V^-1 s^-1, etch pitch density $EPD<6000$ cm^-2, and thickness $L=0.538\pm 0.005$ mm; undoped GaP (110) with $L=0.536\pm 0.005$ mm; and undoped Si (110) with $\rho=1$ – $10$ $\Omega$ cm and $L=0.485\pm 0.005$ mm. The spatial and temporal overlap of the beams was fine-tuned for each sample using 2PA prior to all data acquisition scans. The linear optical properties of the samples were measured using a J.A. Wollam M2000 variable-angle spectroscopic ellipsometer (VASE) and are reported in the Supplementary Material. Linear optical properties for $\lambda>1690$ nm were obtained from literature.Skauli et al. (2003); Li (1980)

The 2PA coefficient was determined by performing a time delay scan and fitting the normalized probe transmission to Eq. 7. The fitting algorithm compared the variation in the fitting parameter $\beta_{12}^{\parallel,\perp}I_{2}^{0}L$ for the $N=1$ and $N=2$ cases, and if the variation was below a threshold of $1\%$, then the result for the $N=1$ case was reported; if the variation was larger, the process was repeated, comparing successively higher-order terms. The normalized change in probe transmission was calculated by

where $V_{1}$ is the probe lock-in amplitude during the time delay scan and $\overline{V}_{1}^{cal}$ is the mean lock-in amplitude corresponding to $100\%$ absorption when the probe is modulated by the optical chopper during a calibration scan. After fitting, $\Delta T_{1}(\Delta t)/T_{1}$ was adjusted by the vertical offset parameter which removes the effect of any scattered pump light. When multiple scans were performed at a single excitation energy, the mean value of $\beta_{12}$ was reported and the uncertainty was conservatively quantified by the maximum of the set including the standard error of the mean, the propagated uncertainty of the mean, and the propagated uncertainty in the individual measurements.

The values of $\sigma$ and $\eta$ were determined by performing rotation scans of the sample fixed at the spatial overlap position and $\Delta t=0$ in both co-polarized and cross-polarized geometries. Power fluctations can sometimes complicate the signal, and so the magnitude of the pump-induced absorption was corrected according to

and

where $V_{ref}(\theta)$ is a reference voltage monitoring the incident power, $V_{1}(\Delta t\rightarrow-\infty)$ is the lock-in amplitude of the probe at large negative time delay (and thus negligible 2PA, capturing any offset), and $\overline{(V_{1}^{cal}/V_{ref}^{cal})}$ is the mean of the probe lock-in amplitude normalized to the reference amplitude during the calibration scan.

Zincblende semiconductors can exhibit the photorefractive effect which can lead to transient two-beam coupling.Dvorak et al. (1994) Transient energy transfer between beams propagating along the $110$-direction due to the photorefractive effect is antisymmetric about rotations of $R_{Z}(\pi)$, while 2PA is symmetric.Dvorak et al. (1994) Analyzing only the symmetric part removes any contribution from the photorefractive effect. In addition, random fluctuations are also reduced. The symmetric and antisymmetric parts of the change in probe transmission are

where $\Delta T_{1}^{+}(\theta)/T_{1}$ and $\Delta T_{1}^{-}(\theta)/T_{1}$ are the symmetric and antisymmetric parts, respectively, of the change in probe transmission upon rotations of $R_{Z}(\pi)$.

Fig. 2 (a) shows an example of a time delay scan for GaP at $\hbar\omega=1.55$ eV in co-polarized geometry. Baseline probe absorption $\Delta T/T$ returned to zero at both positive and negative delays $|\Delta t|>250$ fs, indicating negligible FCA. We observed the same symmetry in all $\Delta t$ scans presented in this study.

For $|\Delta t|<250$ fs, the width of the measured 2PA response (data points) fits very closely to the calculated response (solid black curve) with the pulse duration independently measured by SHG second-order autocorrelation. We obtained a similar match for most data presented here. For a small number of exceptional cases, we observed a 2PA response slightly wider than the SHG autocorrelation. For these cases, we considered the possiblity of saturation effect and this led to a widening of error bars in reported spectra (see Supplementary Material).

In addition to revealing complete information regarding the $\mathrm{Im}\{\chi^{(3)}(\omega)\}$ tensor structure, the 2-beam PPMS technique offers another advantage over single-beam techniques in its ability to discern anomalous effects from 2PA. Such effects include FCA, transient energy transfer originating from photorefractive phase gratings, nonlinear refractive index phase gratings, 2PA amplitude gratings, free-carrier gratings, and saturable absorption which can couple the two beams.Smirl et al. (1988); Valley and Smirl (1988)

The temporal width of the nonlinear loss was anomalously broadened in GaAs and GaP as the excitation neared their respective band gaps in both polarization geometries. Fig. 2 (b) shows this anomaly in GaP at $\hbar\omega=1.91$ eV (0.85$E_{g}$) and it was also observed in GaAs at $\hbar\omega=1.38$ eV (0.96$E_{g}$). Pump depletion by pump-pump 2PA cannot account for this effect. The observed temporal FWHM in the nonlinear loss was a factor of 2.3 – 3 greater than expected for 2PA alone. Asymmetry in the time delay scans also supports the presence of other effects not accounted for in our model. Both of these features are obvious against the second-order autocorrelation signal expected for 2PA (as measured by SHG) which is superimposed on the data.

A fs-scale time delay-dependent FCA cross-section could produce such an anomaly as free carriers excited predominantly by pump-pump 2PA diffuse through $k$-space in the conduction band. Zero time delay in this experiment was determined by a fit parameter, but the peak could be shifted towards positive time delays as a result of this effect. Another possibility is that a photorefractive index grating could induce transient energy transfer and generate the observed anomaly. Both of these effects are encompassed more generally by any non-instantaneous response or memory effect as the 2PA process approaches resonance. If such an effect is present it can induce a polarization density responsive not only to the instantaneous value of the electric field but also to its history on fs timescales.Diels and Rudolph (1996) Measurements at different crystal orientations, pulse energies, pulse durations, and a nondegenerate two-photon absorption experimentFishman et al. (2011); Hutchings and Stryland (1992); Bolger et al. (1993); Hannes et al. (2019) could reveal more information about the nature of such anomalies.

Below the two-photon absorption band edge of GaAs and GaP, nonlinear loss was very small except at high intensities (a factor of 1 – 20 greater than intensities at excitation energies $\hbar\omega>E_{g}/2$). This is consistent with 2PA being energetically disallowed but a transient coherent artifactLebedev et al. (2005) was still observed near zero time delay in both polarization geometries. A transient coherent artifact was also observed in Si near to but slightly above $E_{g}/2$ at $\hbar\omega=0.62$ eV, where the magnitude of 2PA was also quite low. This coherent artifact was asymmetric and notably included an antisymmetric component (about $\Delta t=0$) where $\Delta T_{1}/T_{1}>0$ briefly for positive time delays. While $\mathrm{Im}\{\chi^{(3)}(\omega)\}\rightarrow 0$ rapidly at the two-photon band edge, $\mathrm{Re}\{\chi^{(3)}(\omega)\}$ in general does not. $\mathrm{Re}\{\chi^{(3)}(\omega)\}$ encodes the nonlinear refractive index which is related to $\mathrm{Im}\{\chi^{(3)}(\omega)\}$ by nonlinear Kramers-Kronig relations.Boyd (2008) Below resonance, $\mathrm{Re}\{\chi^{(3)}(\omega)\}\sim\omega^{-3}$ in semiconductors and thus is the dominant third-order nonlinearity at low excitation energies.Boyd (2008)

This coherent artifact shown in Fig. 2 (c) likely arises from a nonlinear refractive index phase grating, which can transfer energy between the probe and pump and produce such antisymmetric time delay profiles.Palfrey and Heinz (1985) This effect is a background effect that likely occurs in all measurements, but is so weak that it is only visible when 2PA is no longer the dominant third-order nonlinearity.

The 2PA spectrum for GaAs measured by PPMS in co-polarized geometry is shown in Fig. 3 along with the fit using the Jones-Reiss model in Eq. 27, published values using mostly single-beam techniques, and ab initio calculations. The best fit of this data to the Jones-Reiss model gives $B=510\pm 210$ cm GW^-1 and a 2PA peak of $\beta^{JR}_{max}=24\pm 10$ cm GW^-1. The 2PA coefficient reported in literature has a large spread, mainly from single-beam experiments clustered around 1053 nm using picosecond lasers.

The 2PA spectrum for GaP measured by PPMS in co-polarized geometry is shown in Fig. 4 along with a fit using a cumulative model for direct and indirect transitions, a fit using the Jones-Reiss model for direct transitions near $E_{0}/2$, and a comparison to published values using single-beam techniques. The cumulative model is defined as using the Garcia-Kalyanaraman model for $E_{g}/2\leq\hbar\omega<E_{0}/2$, and a sum of this function and the Jones-Reiss model above $E_{0}/2$. The best fit to this model gives $B=44\pm 18$ cm GW^-1 and $C=49\pm 35$ cm GW^-1, with 2PA contributions peaking at $\beta^{GK}_{max}=1.1\pm 0.8$ cm GW^-1 for indirect transitions and $\beta^{JR}_{max}=2.1\pm 0.9$ cm GW^-1 for direct transitions.

The 2PA spectrum for Si measured by PPMS in co-polarized geometry is shown in Fig. 5 along with the fit using the Garcia-Kalyanaraman model and a comparison to published values using single-beam techniques. The best fit of this data to the Garcia-Kalyanaraman model gives $C=41\pm 25$ cm GW^-1, peaking at $\beta^{GK}_{max}=0.9\pm 0.6$ cm GW^-1, which is considerably lower than the fit to the data from Bristow, et al.Bristow, Rotenberg, and van Driel (2007) of $C=86$ cm GW^-1, peaking at $\beta^{GK}_{max}=1.93$ cm GW^-1. The measured dispersion differs from the phenomenological model. These results are tabulated in the Supplementary Material. An inset in Fig. 5 shows the degenerate 2PA spectrum predicted by ab initio calculations valid above $E_{0}^{\prime}/2=1.75$ eV and published values using a two-beam technique.

An example of a rotation scan for GaP at $\hbar\omega=1.55$ eV showing symmetric and antisymmetric parts of the normalized $-\Delta T_{1}(\theta)/T_{1}$ in co- and cross-polarized geometries is shown in Fig. 6. The fits of the co- and cross-polarized symmetric data share global parameters $\sigma$ and $\theta_{0}$, while the other parameters are independent. The antisymmetric part is typically $<10\%$ of the magnitude of the symmetric part.

The spectra of anisotropy and dichroism parameters for GaAs, GaP, and Si are reported in Table 2, along with the calculations of the relative amplitudes of $\mathrm{Im}\{\chi^{(3)}_{xxyy}\}$ and $\mathrm{Im}\{\chi^{(3)}_{xyyx}\}$ to $\mathrm{Im}\{\chi^{(3)}_{xxxx}\}$. The values of $\sigma$ from this work are compared to the range of theoretical predictions of $\sigma$ for GaAs and GaP from Table 1 as well as results reported in literature for GaAs and Si. $|\sigma|$ is largest for GaP, which is consistent with theoretical predictions at the two-photon band edge. The values of $\eta$ from this work are compared to a previously reported value in literature for GaAs, and ab initio calculations for anisotropy in GaAs are also shown. See the Supplementary Material for additional information on comparing values between different sources in literature.