1. Introduction

Driven by the rapid development of ultrafast laser sources, increasing attention has been paid to nonlinear optical processes in semiconductors. Among those phenomena, two-photon absorption (2PA) and its wavelength dependence are particularly well investigated. However, a majority of the studies has been restricted to degenerate configurations where, e.g., the strength of the 2PA is analyzed in a single-color z-scan experiment. In contrast, non-degenerate 2PA is a process where the absorption of an optical field is triggered by a field at a different wavelength. Non-degenerate 2PA has been proposed for applications such as sensitive detection and imaging of mid-infrared radiation [1,2] as well as all-optical switching in Si waveguides [3]. For direct gap semiconductors various experiments employed driving fields close to degeneracy [4], others $\omega /2\omega$ configurations [5]. Also, situations with widely non-degenerate fields were investigated and showed a massive enhancement of the 2PA [1,6]. For indirect gap semiconductors, only very few experiments are available. As an example, Zhang et al. [7] investigate 2PA in Si waveguides close to degeneracy.

However, a systematic investigation of the 2PA coefficient $\beta (\omega _1,\omega _2)$, where the non-degeneracy parameter $\omega _1/\omega _2$ is tuned away from unity while the transition energy $\hbar \omega _1+\hbar \omega _2$ is kept constant, is still missing even for the widely used semiconductors GaAs and Si. Theoretical studies [8–10] suggest characteristic scaling laws for the non-degenerate 2PA coefficient as a function of the parameter $\omega _1/\omega _2$. For transitions across the indirect bandgap of Si, such a model does not exist to date. In this article, we close this gap in literature and provide experimental data for the non-degenerate 2PA coefficient $\beta (\omega _1,\omega _2)$ in the range $1.04 \leq \omega _1/\omega _2 \leq 1.88$ in both bulk GaAs and Si at a constant transition energy of $\hbar \omega _1+\hbar \omega _2=1.57\,$eV. For the two materials, we find a marked increase when $\omega _1/\omega _2$ is tuned away from degeneracy. The results in GaAs compare well with the theoretical models. In addition, we provide results on the crystallographic and polarization anisotropy of 2PA in both materials.

2. Experimental details

The experimental setup is based on a Ti:sapphire regenerative amplifier laser system (Coherent RegA 9000). It provides optical pulses with a central photon energy of 1.57 eV, a pulse energy of 8 $\mathrm{\mu}$J and a pulse duration of $\approx 60\,$fs at a repetition rate of 250 kHz. A major fraction of these pulses is fed into an optical parametric amplifier (Coherent OPA 9800/9850). It emits signal and idler pulses with a fixed sum energy of $\hbar \omega _1+\hbar \omega _2=1.57\,$eV. The corresponding tuning ranges are $\lambda _{\textrm {sig}} = (1210 - 1550)\,$nm and $\lambda _{\textrm {idl}} = (2276 - 1611)\,$nm. Thus, the OPA allows us to systematically investigate non-degeneracy parameters $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$ from 1.04 to 1.88. The pulse lengths and energies depend on the actual wavelength configuration and range from $(\tau _{\textrm {sig}},\tau _{\textrm {idl}})=(90-300)\,$fs (full width at half maximum) and $E_{\textrm {sig}}\leq 140\,$nJ ($E_{\textrm {idl}}\leq 40\,$nJ), with generally a decrease in $\tau _{\textrm {sig}}$ and an increase in $\tau _{\textrm {idl}}$ observed with increasing frequency ratio $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$. Both beams are linearly polarized.

The experimental setup is shown in Fig. 1. Essentially, we measure the strength of the 2PA in a pump-probe fashion with temporally overlapping signal and idler pulses. For that purpose, the idler pulse passes a motorized linear translation stage to adjust the time delay $\tau _{\textrm {d}}$ between the two pulses. Afterwards, the signal and idler beams are superimposed on the sample. An angle of 30$^\circ$ between the two beams allows us to easily separate them after the sample. Note that the angle within the GaAs and Si sample ($\approx 8^\circ$) is even smaller due to refraction and, thus, signal and idler directions can be assumed to be parallel inside the sample. A broadband $\lambda$/2-waveplate allows the polarization of the idler pulse to be rotated by 90$^\circ$. In this way, we can compare 2PA measurements where the signal and idler pulses are polarized either parallel or perpendicular to each other. In the following, these two configurations will be referred to as co-polarized ($\parallel$) and cross-polarized ($\perp$). The idler pulse is blocked after the sample. The detection of the signal transmittance is performed with the lock-in technique referenced to a modulation of the idler beam with a mechanical chopper. This detection scheme is designed to measure 2PA whereby one photon from each beam triggers the optical transition.

 

Fig. 1. Experimental setup for the measurement of non-degenerate 2PA.

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For the quantitative analysis of the two-photon absorption coefficient $\beta$, a precise knowledge of the spot diameters in the sample plane is important. For that purpose, we use the knife-edge-method with a titanium plate mounted in the sample plane acting as a knife. Furthermore, the calculation of $\beta$ requires the pulse durations. To measure the pulse durations of the idler, we use a commercial autocorrelator (APE Mini TPA) that can be equipped with different detector units depending on the actual wavelength of the pulse. The signal pulses could also be temporally characterized here. However, the pulse durations of the signal are easily determined by the data analysis described below.

3. Sample characterization

We now turn towards the characterization of the samples. GaAs is a direct semiconductor with a bandgap of $E_{\textrm {g, GaAs}}=1.42\,$eV at room temperature [11]. Si is an indirect semiconductor with a bandgap of $E_{\textrm {g, Si}}=1.12\,$eV at room temperature [11]. Its direct bandgap of 3.4 eV [11] is much larger. As previously described, our experiment uses pulses with $\hbar \omega _1+\hbar \omega _2=1.57\,$eV. This configuration is well suited for a comparison of non-degenerate 2PA for direct and indirect optical transitions. The excess energy above the bandgap is $\Delta E_{\textrm {GaAs}}=150\,$meV for GaAs and $\Delta E_{\textrm {Si (indirect)}}=450\,$meV for Si.

We investigate nominally undoped GaAs and Si specimens with (100) and (110) orientation. The sample thicknesses are 51 $\mathrm{\mu}$m (100) and 52 $\mathrm{\mu}$m (110) for the GaAs and 454 $\mathrm{\mu}$m (100) and 455 $\mathrm{\mu}$m (110) for the Si specimens. For Si we expected the amplitudes of the 2PA signals to be smaller than the GaAs ones, hence we chose thicker samples for Si. The thicknesses are achieved by mechanical grinding of commercial wafers. The orientation of the light polarizations relative to the crystallographic axes is shown in Fig. 2 for all samples.

 

Fig. 2. Orientation of the signal and idler polarizations with respect to the crystallographic axes for the (100) and (110) GaAs and Si samples.

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4. Results

Figure 3 shows exemplary pump-probe transients for a co- and cross-polarized measurement at a wavelength combination of $\lambda _{\textrm {sig}}=1350\,$nm and $\lambda _{\textrm {idl}}=1904\,$nm for the (100) oriented GaAs and Si sample. The signal and idler pulses have Gaussian temporal pulse shapes and the measured idler pulse lengths at the full width at half maximum are $\tau _{\textrm {idl}}=103\,$fs for the GaAs and $\tau _{\textrm {idl}}=160\,$fs for the Si sample. The difference of $\tau _{\textrm {idl}}$ between the two samples is due to the high dependence of $\tau _{\textrm {idl}}$ on the adjustment of the RegA system and the OPA which has been changed between the measurements. To calculate the 2PA coefficient $\beta$ we use the theoretical approach for the normalized signal transmittance of Negres et al. [12]. In short, the results of Negres et al. establish an analytical formula for the shape of the pump-probe transients for our experimental configuration. It assumes pulses with a Gaussian temporal envelope and takes into account the well-known group velocity mismatch of signal and idler pulses. While the approach of Negres et al. in principle takes linear absorption into account as well, we set this linear absorption to zero. This assumption is valid because linear absorption across the band gap is suppressed by the chosen photon energies ($<E_{\textrm {g}}$). Free-carrier absorption can be excluded as the pump-probe traces do not show transient absorption beyond the temporal overlap of pump and probe pulse. This also indicates that the idler pulse does not create a significant amount of carriers, thus there is no significant degenerate 2PA of the idler pulse. Significant degenerate 2PA of the signal pulse can be excluded due to its weak intensity. A fit of this model to the experimental transients reveals the 2PA absorption coefficient $\beta$ as well as the duration of the signal pulse $\tau _{\textrm {sig}}$. The two examples for such a fit, see Fig. 3, show excellent agreement between the model and the measurement.

 

Fig. 3. Exemplary pump-probe transients for a co- and cross-polarized 2PA measurement at $\lambda _{\textrm {sig}}=1350\,$nm and $\lambda _{\textrm {idl}}=1904\,$nm for the (100) oriented (a) GaAs and (b) Si sample. The blue (red) data correspond to co- (cross-) polarized configurations. Solid lines represent the corresponding fits according to the model by Negres et al. [12].

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For this specific wavelength combination, the fits to the GaAs data represent curves with $\beta _\parallel =22.24\,_{-1.66}^{+1.85}\,$cm/GW with $\tau _{\textrm {sig}}=(204.0\pm 2.3)\,$fs and $\beta _\perp = 7.19\,_{-0.55}^{+0.61}\,$cm/GW with $\tau _{\textrm {sig}}=(205.6\pm 3.2)\,$fs. The corresponding values for Si read $\beta _\parallel =0.97\,_{-0.07}^{+0.08}\,$cm/GW with $\tau _{\textrm {sig}}=(255.4\pm 4.5)\,$fs and $\beta _\perp =0.46\,_{-0.03}^{+0.04}\,$cm/GW with $\tau _{\textrm {sig}}=(261.0\pm 4.6)\,$fs. The temporal widths are specified as the full width at half maximum. The spot sizes are $A_{\textrm {sig}}=2000\,\mu \textrm {m}^2$ and $A_{\textrm {idl}}=34700\,\mu \textrm {m}^2$ for GaAs and $A_{\textrm {sig}}=6300\,\mu {\textrm {m}}^2$ and $A_{\textrm {idl}}=27800\,\mu {\textrm {m}}^2$ for Si. In general, the spot sizes are adjusted that the idler spot is, at least, four times larger than the signal spot. Therefore, the intensity of the idler pulse across the signal spot is practically identical to its peak intensity. The idler peak intensity is an input parameter for the fitting procedure, for the measurement examples these are $I_{\textrm {idl}}^0=0.15\,{\textrm {GW}}/{\textrm {cm}}^2$ for GaAs and $I_{\textrm {idl}}^0=0.12\,{\textrm {GW}}/{\textrm {cm}}^2$ for Si. The errors for $\beta$ are estimated by varying the input parameters within the experimental inaccuracies. Since all pump-probe data sets for each sample look rather comparable and the analysis follows analogously the above procedure, we now restrict the presentation to the values for the two-photon absorption coefficient $\beta$ and its dependence on $\omega _1/\omega _2$ as well as the beam polarization and crystallographic orientation.

In Fig. 4 the extracted 2PA coefficients $\beta$ in the measurement range of $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}=1.04-1.88$ are summarized for GaAs. Some data points at certain frequency ratios appear twice as we repeated part of the measurements at a later stage to test for consistency. Most remarkably, both $\beta _\parallel$ and $\beta _\perp$ increase by more than a factor of 2 over this range. This holds true for both crystal orientations. Note that the scatter and the error bars of the experimental data also increase with $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$ because the noise of the optical parametric amplifier increases for widely non-degenerate operation. When comparing the two crystallographic orientations, we find different magnitudes for $\beta _\parallel$ and $\beta _\perp$ at the same frequency combinations. For the (100) orientation the $\beta _\parallel$ values are generally higher than for the (110) orientation. The opposite trend is seen for $\beta _\perp$. These findings reflect the pronounced anisotropy of the third-order optical nonlinearity of GaAs.

 

Fig. 4. GaAs 2PA coefficients $\beta$ for co- (blue) and cross-polarized (red) measurements as a function of the frequency ratio $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$ for (a) (100) and (b) (110) crystallographic orientation. Note, the solid lines represent the corresponding fits according to the model by Hannes and Meier [8].

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We now compare our data to theoretical predictions for the non-degenerate 2PA. As displayed in Fig. 4, our experimental data is consistent with the theoretical scaling function of Hannes and Meier [8] for direct band gap semiconductors:

Their approach is based on the semiconductor Bloch equation for a two-band model assuming a k-independent dipole matrix element. The solid lines in Fig. 4 represent the corresponding fit of this scaling behavior for the co- and cross-polarized measurement data. Apart from the scaling proposed in Ref. [8], other theoretical studies by Sheik-Bahae et al. [5] and Aversa et al. [10] propose somewhat different dependencies of $\beta (\omega _1,\omega _2)$ on a variation of $\omega _1/\omega _2$. These predictions are in rather good agreement with our data as well (fit curves not shown). To clearly distinguish between those models, data for a larger range of $\omega _1/\omega _2$ would be required which is inaccessible with our apparatus.

A comparison with previous experimental results is only possible for degenerate 2PA coefficients as no data is available for non-degenerate configurations at similar wavelengths. Fishman et al. report a degenerate co-polarized 2PA coefficient of $\approx 1.5\,$cm/GW [1] and Cirloganu et al. $\approx 10.0\,$cm/GW [13] at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}=1580\,$nm. These values are lower compared to our 2PA coefficients of $\bar {\beta }_{(100),\parallel }=18.1\,$cm/GW and $\bar {\beta }_{(110),\parallel }=12.9\,$cm/GW at our data points closest to degeneracy ($\lambda _{\textrm {sig}}=1550\,$nm, $\lambda _{\textrm {idl}}\approx 1611\,$nm). Only a few studies focus on the anisotropy of the 2PA and further specify the beam polarizations in relation to the crystallographic orientation of the specimens. Dvorak et al. [14] investigate a (001) oriented crystal with signal and idler beam polarization parallel to the [100] axis and extract a 2PA coefficient of $\approx 20.0\,$cm/GW at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}=950\,$nm. Bepko [15] reports a degenerate 2PA coefficient of $\approx 80\,$cm/GW at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}=1060\,$nm for a (110) oriented sample with signal and idler beam polarization parallel to the [1-10] axis. DeSalvo et al. report differences in the degenerate 2PA coefficients at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}=1064\,$nm for (100), (110) and (111) oriented GaAs specimens, varying between 18 cm/GW and 25 cm/GW [16].

We now turn towards the investigation of the 2PA in Si. Figure 5 summarizes the results for both $\beta _\parallel$ and $\beta _\perp$, again comparing (100) and (110) oriented crystals. Remarkably, the Si data shows similar trends as GaAs although in Si the 2PA occurs phonon-assisted across the indirect bandgap. In particular, the 2PA coefficients also increase with higher degrees of non-degeneracy. In our parameter range of $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$, they increase by a factor of $1.8-2.0$ depending on the polarization configuration and the crystallographic orientation. In comparison to GaAs, the absolute values of $\beta$ are smaller as the phonon-assisted 2PA is more unlikely than the 2PA for a direct transition. For the (100) orientation the $\beta _\parallel$ values are generally smaller than for the (110) orientation, for $\beta _\perp$ the values are largely independent of the orientation.

 

Fig. 5. Si 2PA coefficients $\beta$ for co- (blue) and cross-polarized (red) measurements as a function of the frequency ratio $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$ for (a) (100) and (b) (110) crystallographic orientation.

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Our results for the 2PA coefficients, $\beta _{(100),\parallel }=0.9\,$cm/GW and $\beta _{(110),\parallel }= 1.1\,$cm/GW, closest to degeneracy ($\lambda _{\textrm {sig}}=1550\,$nm, $\lambda _{\textrm {idl}}\approx 1611\,$nm) are in good agreement with previous experimental studies of the degenerate 2PA coefficient obtained with the z-scan technique. Bristow et al. [17] report 0.75 GW/cm at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}\approx 1600\,$nm for a (001) oriented sample and Dinu et al. [18] 0.79 cm/GW at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}\approx 1540\,$nm for a (110) oriented sample. Note that in both cases, the relative orientations of the crystallographic axes with respect to the beam polarizations are not specified. Lin et al. [19] report a 2PA coefficient of 0.5 cm/GW at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}\approx 1600\,$nm for a (100) oriented Si specimen with signal and idler pulse parallel to the [010] direction. All these studies focus on co-polarized configurations.

We finally further elaborate on the polarization anisotropy of 2PA. To this end, we analyze the ratio $\beta _\perp /\beta _\parallel$ for both GaAs and Si. The data are summarized in Fig. 6. The errorbars are calculated according to the error propagation, taking into account the errors of $\beta _\perp$ and $\beta _\parallel$. Most likely the errors for $\beta _\perp /\beta _\parallel$ are overestimated as a part of the inaccuracies cancel out, when computing the ratio, which can also be seen in the fairly low scatter of $\beta _\perp /\beta _\parallel$. Interestingly, the polarization dependence $\beta _\perp /\beta _\parallel$ in GaAs, see Fig. 6(a), also shows a slight increase with increasing $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$ for both sample orientations. This trend has not been observed experimentally prior to our experiments, however a simplified theoretical model [20] has predicted this increase of $\beta _\perp /\beta _\parallel$ whereby $\beta _\perp$ and $\beta _\parallel$ approach each other for widely non-degenerate configurations. The value at degeneracy is dependent on the specific theoretical model, Ref. [20] reports a value of 0.5 for k-independent dipole matrix elements. In addition, the comparison of the two sample orientations shows a more pronounced anisotropy for the (100) crystallographic orientation when compared to the (110) oriented sample. Dvorak et al. report a value of $\beta _\perp /\beta _\parallel \approx 0.41$ [14] at $\lambda _{\textrm {sig}}=\lambda _{\textrm {idl}}=950\,$nm for a (001) oriented crystal with signal and idler beam polarization parallel to the [100] axis. The deviation to our value is possibly related to the different transition energy of $\approx 2.6\,$eV of the experiment in Ref. [14].

 

Fig. 6. Ratio $\beta _\perp /\beta _\parallel$ between co- and cross-polarized 2PA coefficients for (a) GaAs and (b) Si as a function of the frequency ratio $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$. The color indicates the (100) (blue) and (110) (red) crystallographic orientations. The corresponding dotted lines are guides to the eye.

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In Fig. 6(b) we show the ratio of the 2PA coefficient for the co- and cross-polarized configuration for Si. Also in Si, we observe an increase of $\beta _\perp /\beta _\parallel$ with increasing $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}$. It is even more pronounced than in GaAs. For the high deviation of the data point at $\omega _{\textrm {sig}}/\omega _{\textrm {idl}}\approx 1.1$ and the (110) oriented sample, we assume a mistake in the experimental adjustment as this is the only data point with a large deviation from the general trend. The comparison of the two sample orientations shows a higher anisotropy for the (110) crystallographic orientation than for (100). So far, we cannot provide an intuitive picture of why the anisotropy of GaAs and Si shows different results when comparing these two crystallographic orientations.

5. Summary

In summary, we have systematically investigated the non-degenerate two-photon absorption coefficient $\beta (\omega _1,\omega _2)$ for the most important semiconductors GaAs and Si over a range of non-degeneracy parameters while keeping the transition energy $\hbar \omega _1+\hbar \omega _2$ constant. Both GaAs and Si show an enhancement of the two-photon absorption strength with an increasing non-degeneracy parameter regardless of whether the 2PA is phonon-assisted. As expected, the overall strength is higher for GaAs than for Si. The comparison of the data gives rise to a similar scaling behavior for both types of semiconductors. For GaAs, a good agreement with previous theoretical calculations can be shown. Further, polarization and crystallographic anisotropy of the non-degenerate two-photon absorption coefficient $\beta (\omega _1,\omega _2)$ can be observed for both materials. The results contribute to a more detailed understanding of the non-degenerate two-photon absorption in GaAs and Si and therefore to their widely useful applications in nonlinear photonics.