1. Introduction

The nonresonant interaction of a strong laser field with materials possessing $\chi ^{(2)}$ and $\chi ^{(3)}$ nonlinear optical susceptibilities is widely used for manipulation of the frequency spectrum of the incident light. Numerous applications of nonlinear frequency conversion span from harmonic generation and parametric amplification to supercontinuum generation (SC) of broadband “white” light [1]. Extreme spectral broadening involves a number of coupled physical processes: self-phase modulation (SPM), four-wave mixing (FWM), cross-phase modulation (XPM) and stimulated Raman scattering (SRS), all of which are related to $\chi ^{(3)}$. Characterization and optical control of the third order nonlinearity are important for using these materials as nonlinear optical elements. Furthermore, strong light-semiconductor interactions can be used to manipulate quantum states of matter, considered for quantum computing [2–6].

Nonlinear photonic devices in the long-wavelength infrared (LWIR), from 8 to 15 $\mu$m, are typically designed using semiconductors that are transparent in this range, including CdTe, GaAs, Ge and ZnSe. These materials find application as Pockels cells and electro-optic modulators [7], saturable absorbers [8], and as optical parametric oscillators [9,10]. GaAs, transparent from 1 to 18 $\mu$m, has one of the highest $\chi ^{(2)}$ and $\chi ^{(3)}$ and due to its properties lies in the heart of optoelectronics and integrated technologies based on quantum wells or quantum dots, typically constructed with GaAs or GaAs alloys [11]. The nonlinear optical response of GaAs (band gap $E_{g} = 1.42 \textrm {eV}$) has been studied at resonance near 1 $\mu$m [12], at 2-3 $\mu$m, around two/three photon resonances [13,14], and under extremely nonresonant conditions at 10 $\mu$m (photon energy $\approx E_{g}/12$) with intensities around 1 MW/cm$^{2}$ [15,16]. This material has been successfully applied in bulk form for SC generation [17–19], pulse compression [20], and high-field studies [21,22].

Significant enhancement to the nonlinear optical response has been reported in bulk intrinsic semiconductors for resonant or near-resonant pump configurations [14,23] or doped p- and n-GaAs [24,25]. This enhancement, however, comes at the cost of unavoidable optical losses – for example, in p-GaAs absorption reached 5000 cm$^{-1}$ [25]. Another approach to manipulating the nonlinear optical response of semiconductors is by engineering materials with reduced dimensionality. Systems based on GaAs such as quantum wells [26–28], quantum dots [29], and thin films [30] have demonstrated controllable nonlinearity over a wide range. High nonlinearities of these systems stem from various electron confinement or excitonic effects, but are rooted in resonant processes which also introduce inherent loss. While near-resonant interactions in bulk or structured materials are widely studied and useful in many applications, similar control over the nonlinearity in a bulk semiconductor far away from resonance (and therefore with low loss) would prove valuable for infrared photonics and nonlinear optics.

In this paper, we experimentally demonstrate that the nonlinear optical response of bulk GaAs, excited using infrared photons with energies far below the band gap energy ($E_{g}$/12), can be controlled at laser intensities in the range of 1-10 GW/cm$^{2}$. Specifically, the FWM efficiency of a CO$_{2}$ laser beat-wave in a crystal with negligibly small linear optical loss was found to increase by a factor of 10 by decreasing the beat-frequency. Using a fully microscopic analysis, we show that the observed enhancement of the induced nonlinear response derives from photoexcited free carriers that, once excited at such high fields, are efficiently driven at the beat frequency, emitting light at Stokes and anti-Stokes FWM sideband frequencies. These findings are supported by the observation of the onset of nonlinear absorption at high fields. Numerical solutions of the semiconductor Bloch equations, in agreement with the experimental data, suggest that such a method can be used to control the nonlinearity of an intrinsic semiconductor with the ability to reach almost a 100x increase in FWM efficiency. This opens the way to efficient broadband generation of LWIR radiation pulses and frequency comb generation in bulk, undoped semiconductors.

2. Experimental setup

FWM is a third-order nonlinear optical effect which has become a “gold standard” technique for determining the effective nonlinear refractive index $(n_{2,\textrm {eff}})$ of materials [7]. An intense laser pulse comprising two frequencies f$_{1}$ and f$_{2}$ generates a family of Stokes and anti-Stokes sidebands each separated by the beat frequency $\Delta$f = f$_{2}$ - f$_{1}$ inside a material by the process of nonlinear mixing (Fig. 1). The collinear FWM yield of the 1st Stokes sideband at f$_{3}$ = 2f$_{1}$ - f$_{2}$ scales as $E_{\textrm {f}_{3}} \sim n_{2,\textrm {eff}}^{2}E_{\textrm {f}_{1}}^{2}E_{\textrm {f}_{2}}$ where $E_{\textrm {f}}$ is the energy of different frequency components [15,31].

 

Fig. 1. Simplified schematic of the FWM process, where f$_{3}$ = 2f$_{1}$ - f$_{2}$ and f$_{4}$ = 2f$_{2}$ - f$_{1}$.

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In experiment, we study FWM in GaAs using different CO$_{2}$ laser lines. In particular the two different beat-waves used in these measurements are a high frequency beat-wave (HFBW) with the 10P(20) ($\lambda _{1} = 10.59$ $\mu$m) and 10R(16) ($\lambda _{2} = 10.27$ $\mu$m) lines and a low frequency beat-wave (LFBW) with the 10P(20) ($\lambda _{1} = 10.59$ $\mu$m) and 10P(16) ($\lambda _{2} = 10.55$ $\mu$m) lines resulting in $\Delta$f = 872 GHz and 106 GHz, respectively. A simplified schematic of the experimental setup is shown in Fig. 2(a).

 

Fig. 2. (a) The experimental setup. Pulse energy at the interaction point is measured on every shot by energy meters (EM) f$_{1}$ and f$_{2}$. The etalon was used to help filter background FWM signal from optical elements, as shown in (c). Power filtering after the sample is provided by a diffraction grating (DG). BS is beam splitter, SM is scanning monochromator, and HCT is HgCdTe detector. (b) Typical pulse profile with $\tau$ = 200 $\pm$ 20 ps, where the uncertainty represents a standard deviation calculated over 12 pulse profile measurements. (c) (100) GaAs etalon transmission curves measured (solid) and predicted (dashed) for pump and first Stokes wavelengths for the HFBW.

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Linearly polarized 200 ps CO$_{2}$ laser pulses are generated in a master-oscillator power amplifier (MOPA) chain at a repetition rate of 1 Hz, which has been described elsewhere [32]. We used a streak camera to characterize the pulse length; a typical pulse profile for the 10$\mu$m pumps used in this experiment is shown in Fig. 2(b), which had a mean FWHM pulse length of $\tau = 200 \pm 20$ ps.

Before encountering the sample, the 10$\mu$m beam passed through a 700$\mu$m thick (100) GaAs wafer acting as an etalon, as shown in Fig. 2(a). The etalon selectively reflects certain frequencies depending on the beam’s angle of incidence. Figure 2(c) shows transmission curves measured for $\lambda$ = 10.59 $\mu$m and $\lambda$ = 10.27 $\mu$m and extrapolated to $\lambda$ = 10.93 $\mu$m. The purpose of the etalon was to transmit pump frequencies and to filter out FWM sideband light created in the air or in the multiple windows and optics in the MOPA chain before encountering the sample. Tuning the etalon to $11.5^{\circ }$ helped to increase the signal to noise ratio of the HFBW measurements, which was found to be $\geq$ 5 for all HFBW sidebands. Due to the close separation between pump wavelengths in the LFBW case, the etalon proved ineffective and was removed. As a result, these measurements had higher incident intensities and suffered from a worse signal to noise ratio ($\geq$ 2 for all LFBW sidebands measured).

A 7 mm thick anti-reflection coated semi-insulating GaAs slab with a resistivity > 3 $\times 10^{8} \Omega \cdot$cm and a linear absorption coefficient $\leq$ 0.01 cm$^{-1}$ at 10 $\mu$m was used. Electric field polarization was always parallel to the [111] axis of the crystal, and the energy ratio between the two pumps was $E_{\textrm {f}_{1}}/E_{\textrm {f}_{2}} \leq 2$. The electric fields of the two spectrally close pump wavelengths interfere and create a beat-wave with almost double the peak intensity while maintaining the same energy fluence.

The normally incident beam was focused just in front of the GaAs sample (FWHM $0.82 \pm 0.02$ mm) to minimize the self-focusing effect, producing peak intensities in the range of 1-10 GW/cm$^{2}$. On a multishot basis, we began to observe surface damage to our sample at intensities $\geq$ 20 GW/cm$^{2}$. The Rayleigh length of the beam was 20x longer than the sample, so the beam could be treated as a plane wave with a constant area during the entire interaction.

After the sample, the pumps and sidebands were dispersed by a diffraction grating, providing power filtering, before being sent to a scanning monochromator. The beam containing the sideband of interest was analyzed by the monochromator and its energy was detected by a cryogenically cooled HgCdTe energy detector. Every measurement also included a background data set, where the sample was removed from the beam line. The background sideband light included FWM light generated in the laser system as well as scattered light.

3. Experimental results

Figure 3(a) shows measurements of the 1st Stokes sideband energy generated in the GaAs sample for the HFBW as a function of laser intensity. Figure 3(b) shows the same measurement for the LFBW, where the absolute FWM yield is $\sim$40x larger than for the HFBW case. This corresponds to a 10x increase in sideband generation efficiency for comparable intensities. It should be noted that at low intensities (300 ns CO$_{2}$ laser pulses at 1 MW/cm$^{2}$) measurements were made using this GaAs sample with a FWM technique and the results were found to match the measurements in [15] within the experimental uncertainty.

 

Fig. 3. (a) 1st Stokes sideband data measured in GaAs for the HFBW and its linear fit. To enable comparison the data is multiplied by 10. (b) 1st Stokes sideband data for the LFBW and its linear fit. The background for these measurements is represented by the value of the data at zero laser intensity. (c, d) FWM sidebands measured for the HFBW and LFBW respectively, integrated over a spectral region with width defined by our approximately 10nm spectral resolution. Effective intensity is the intensity corresponding to $(E_{1}^{2}E_{2})^{1/3}$ assuming $E_{1}/E_{2} = 2$.

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In Figs. 3(c) and 3(d) we show the experimental FWM sideband measurements for the HFBW and LFBW, respectively, including every sideband measurable above the noise. The LFBW interaction produced more sidebands on both Stokes and anti-Stokes sides than the HFBW, and also had higher efficiency across all sidebands, confirming the increase of FWM yield demonstrated by Figs. 3(a) and 3(b).

We also measured transmission through the sample as a function of intensity (Fig. 4) using a single wavelength CO$_{2}$ laser pulse at $\lambda = 10.59$ $\mu$m. We find that the transmission begins to diverge from lossless propagation (red dashed line) for input intensities greater than 2 GW/cm$^{2}$. The nonlinear absorption data is fit using the equation [33,34] $I_{\textrm {out}} = I_{\textrm {in}}\textrm {exp}\left (-\alpha _{\textrm {eff}}L\right )$ where $\alpha _{\textrm {eff}}$ is the effective absorption coefficient; the best fitting form was found to be $\alpha _{\textrm {eff}} = \alpha _{\textrm {NL}}I_{\textrm {in}}$, which gave $\alpha _{\textrm {NL}} = 0.08 \pm 0.02$ cm/GW. The uncertainty here arises from the uncertainty in the fit of this curve. While the absorption we measure here is not the same as standard two-photon absorption (2PA), we can benchmark the strength of the absorption we observe against 2PA. Resonant or near resonant excitation in GaAs and other semiconductors [13,35] give 2PA coefficients ranging from 2.0 cm/GW to 23 cm/GW. These values are much higher than the coefficient we report here, confirming the rather small optical losses in nonresonant laser-semiconductor interactions.

 

Fig. 4. Nonlinear absorption measured at $\lambda$ = 10.59 $\mu$m in GaAs. The dashed line denotes lossless propagation and the solid curves denotes the fit to the data.

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The existence of nonlinear absorption in our semi-insulating sample is experimental evidence for the generation of nonequilibrium carrier distributions in GaAs, allowing for the possibility of both interband and intraband contributions to the overall nonlinearity. We have estimated a lower bound on the density of free carriers using the Keldysh theory [36]. For a beat-wave with peak intensity 5 GW/cm$^{2}$ (Keldysh parameter $\approx$ 1.5), this theory predicts an electron density around 10$^{13}$ cm$^{-3}$ which is far below the critical density for 10 $\mu$m radiation. Therefore the measured optical loss cannot be fully explained by the generation of free carriers via photoexcitation alone and we resort to a full microscopic description of the laser-semiconductor interaction.

4. Quantum mechanical model

We model the interaction of CO$_{2}$ laser beat-waves with a slab of intrinsic GaAs using the semiconductor-Bloch equations (SBEs) [37,38], which fully describe the coupled dynamics of the interband polarization as well as the intraband currents created by accelerating the carrier distributions among the bands. As the SBEs are not based on a perturbative expansion, they implicitly account for nonlinear effects of all orders and the resulting wave-mixing processes. They have been successfully used to explain strong field phenomena including high-harmonic generation [39–42], high-order sideband generation [43], or Wannier-Stark localization in GaAs [44], as well as optical excitations and nonlinear effects in GaAs [45] and other material systems [46]. In the following we summarize the most important steps relevant to our analysis, while extended discussions may be found in [40,41].

The microscopic carrier and polarization dynamics directly map the electronic band structure and the dipole-couplings between the individual bands [47,48]. Consequently, an accurate description of the band dispersion and the dipole-matrix elements is required throughout the whole Brillouin zone (BZ). We, therefore, compute the energy bands and optical dipole matrix-elements using density functional theory (DFT) [49] as implemented in the Vienna ab-initio simulation package (VASP) [50,51]. The Heyd-Scuseria-Ernzerhof (HSE) hybrid functional [52] is used, including spin-orbit coupling [53]. Following the description in [54], we adjust a screening parameter in the HSE functional such that the effective masses and the energetic band separation at high symmetry points of the BZ are accurately reproduced. Our computations are performed in a $\Gamma$-centered 6x6x6 Monkhorst-Pack [55] k-point mesh and the energies are converged until an energy difference of 10$^{-9}$ eV is reached. The optical dipole-matrix elements can be accessed via the linear optics routine as described in [56].

These calculations of the electronic band structure in GaAs find that in the [111] direction, the dipole between heavy hole (HH) and conduction band (CB) vanishes, leaving only a coupling between the CB and light hole band (LH). These two bands (shown in the inset of Fig. 5) are used in our calculations. In the completely filled valence bands of an intrinsic semiconductor, carriers are immobile, preventing charge transport, while, at the same time no charge carriers are available in the empty conduction bands. Only once interband excitations create vacancies in the valence band, charge transport becomes possible.

 

Fig. 5. Simulation results modeling the efficiency of the 1st Stokes FWM sideband are shown in the blue and red lines for the HFBW and LFBW, respectively. Experimental data for each beat-wave is indicated by dots of the same color. Inset: GaAs band structure in the [111] direction used in calculations.

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The dynamical evolution of interband polarization and intraband currents, as well as their mutual interplay is described by the SBEs. Taking into account the CB, as well as the LH band of GaAs, they read [39]:

The Coulomb interaction between charge carriers leads to a renormalization of the bandgap and the formation of excitonic resonances below the bandgap. In bulk GaAs the excitonic binding energy is on the order of 4 meV. While these excitonic features dominate the optical properties for resonant optical excitations, they become less significant for strongly off-resonant excitations with high field strengths. In such situations, the Coulomb interaction typically manifests as an effective dephasing of the polarization [57,58]. We have carefully checked that Coulombic mean field interactions do not modify the qualitative results of our computations. Yet, the Coulomb interaction gives rise to higher order scattering processes, which we include as a phenomenological carrier relaxation and an effective polarization damping. We therefore add a phenomenological dephasing on a timescale defined by $T_{2}$ in Eqs. (1) and (2). In our calculations we use $T_{2} = 300$ fs as a representative value for this nonresonant interaction, however varying $T_{2}$ did not qualitatively change the results.

The emission intensity is created by the macroscopic polarization $P(t) = \sum _{\textbf {k}} d_{\textbf {k}} p_{\textbf {k}} +c.c.$ and the currents $J(t) = \sum _{\textbf {k}} j_{\textbf {k}}f_{\textbf {k}}$, with $j_{\textbf {k}} = \frac {e}{\hbar }\nabla _{\textbf {k}}\epsilon _{\textbf {k}}$. As $j_{\textbf {k}}$ is an odd function in $\textbf {k}$, any $f_{\textbf {k}}$ distribution, which is not symmetric with respect to the $\Gamma$-point, contributes to the currents. In order to prevent unrealistically high currents, we therefore add a relaxation of the carrier distribution to a symmetric distribution via $\Gamma _{\textbf {k}} = -\frac {\hbar }{2\tau }(f_{\textbf {k}} - f_{\textbf {-k}})$ with $\tau = T_{2}/2$, following Ref. [41]. The emitted intensity is then defined by the Fourier transforms of the polarization and current sources respectively via $I_{\textrm {out}}(\omega ) \propto |\omega P(\omega ) + iJ(\omega )|^{2}$ at frequency $\omega$.

The electric field $E(t)$ is defined by the sum of the electric fields of both pump pulses, i.e. the laser beat-wave field. Because carriers and polarization are excited dominantly along the direction of the linearly polarized pump pulses, we perform effectively one-dimensional simulations in reciprocal space. One-dimensional simulations have been successfully applied to describe strong field excitations [40–42].

We solve the SBEs in the time domain for an initially unexcited GaAs sample interacting with a 200 ps long beat-wave field. The ratio of the pump intensities is always taken to be $I_{1}/I_{2} = 2$, matching the energy ratio in experiments. Due to the beat-wave character of the pump setup, this is extremely challenging because two different timescales are involved: As a result of the extremely off-resonant excitation, the numerical time steps need to be small enough to directly resolve the oscillations of the underlying electric field wave. Consequently, the rotating-wave approximation, where only the envelope of the pump is considered, is not applicable here. Therefore, we use $5\times 10^{-2}$ fs as time steps, while we end up with $28\times 10^{6}$ time points in the integration, due to the 200 ps long pump pulses. At the same time, we need to sample the reciprocal space along the full L-$\Gamma$-L line, i.e., Eqs. (1) and (2) have to be solved for overall 1600 k-points.

5. Results of the simulations and discussion

We characterize the nonlinear response by calculating the yield of the 1st Stokes sideband as a function of input intensity (I$_{in}$) for the two beat-waves. The results of these calculations for the LFBW and the HFBW are shown by the solid lines in Fig. 5. Both beat-waves exhibit a similar trend, except that the sideband generation efficiency is significantly increased for the LFBW. Specifically, with increasing beat-wave intensity the sideband efficiency grows rapidly by several orders of magnitude up until 1 GW/cm$^{2}$, at which point it stagnates and exhibits a slowly increasing oscillatory behavior, signature of a highly nonlinear regime. In this regime our calculations indicate that considerable transient carrier populations are created, such that carrier populations and interband polarizations strongly modulate each other and Rabi-flopping occurs. In detail, the computed sideband intensity is sensitive to the actual shape of the beat-wave field, leading to a modulation of the sideband efficiency with input intensity. In the experiment, the intensity of the Stokes sideband is integrated over multiple laser shots, washing out the oscillations. Therefore, a similar averaging is applied in the computations.

The experimental data were separated into different intensity bins with their mean values and standard deviations represented by the points and error bars on Fig. 5. There is reasonable agreement between experiment and simulations — the LFBW yields are consistently higher than the HFBW yields.

Our microscopic calculations attribute the observed increase of FWM efficiency to an efficient driving of intraband currents by the laser beat-wave. Although photon energies (hf $\approx$ 0.12 eV) are much smaller than the band gap of 1.42 eV in GaAs, strong electric fields produce a transition probability between an occupied valence and empty conduction state. Once carriers are generated, the beat-wave acts as a bias to accelerate electrons and holes through the Brillouin zone (BZ), causing intraband currents. Especially, when $\Delta$f $\lesssim$ 1 THz, the beat-wave modulation efficiently drives currents with very low frequency bias by accelerating electrons and holes to high-$k$ states and non-parabolic regions of the BZ [38], changing the charge carriers’ contribution to the optical nonlinearity.

The importance of intraband currents in reproducing the experimental results is further demonstrated in Fig. 6. The dashed curve indicates a calculation for the HFBW case with the intraband currents disabled. In this case, only interband transitions contribute to the optical nonlinearity. This calculation gives Stokes sideband efficiency more than an order of magnitude lower than the experimental results in Fig. 5 including both interband and intraband effects. For very low intensities (<100 MW/cm$^{2}$), both computations provide comparable results before the free carrier effects set in and strongly increase FWM efficiency. This analysis strongly suggests that intraband currents are responsible for the significantly increased strength of the nonlinear response of GaAs above 1 GW/cm$^{2}$. Note that similar behavior was observed when pumping doped p-GaAs or n-GaAs with much weaker MW/cm$^{2}$ CO$_{2}$ laser beat-waves but with orders of magnitude higher losses due to the doping [24,25].

 

Fig. 6. Extrapolation of the beat-wave enhancement of the Kerr nonlinearity in simulations. The dashed line indicates simulations with the intraband currents switched off, disallowing current contributions to the FWM process.

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In general, the driving of currents in solids becomes more efficient the longer the wavelength of the driving field because the duration of a half cycle of the electric field (the time before its sign flips) increases, leading to a larger excursion in $k$-space. Therefore, the physical picture presented above suggests inverse scaling of FWM efficiency with the difference frequency $\Delta$f. We test this prediction by extrapolating $\Delta$f to values smaller than those used in experiment. As $\Delta$f goes from 872 GHz (blue curve in Fig. 6), corresponding to a half cycle bias time of 0.57 ps, to 4 GHz and a half cycle bias time of 125 ps (dotted curve in Fig. 6), the sideband efficiency increases by more than an order of magnitude at all intensities. Note that the CO$_{2}$ laser lines provide a convenient method to tune $\Delta$f smoothly between the values in Fig. 6.

The dramatic enhancement of the nonlinearity in bulk GaAs seen both in experiment and theory has temporal limitations imposed on it by the period of the beat frequency chosen - the length of the pulse must be on the order of the beat period or longer for the effect to manifest itself. The relatively long pulses used in this experiment are useful for reaching the smallest beat frequencies provided by the regular band of the CO$_{2}$ laser [59]. To reach beat frequencies smaller than this, simultaneous oscillation on the regular and sequence bands [60] could allow for 10 $\mu$m laser beat-waves with $\Delta$f as small as 4 GHz.

6. Conclusion

To summarize, we have shown that at GW/cm$^{2}$ intensities, much below the dielectric breakdown threshold, the nonresonant nonlinear optical response of GaAs can be enhanced due to a significant free carrier contribution. This contribution results in a change of FWM efficiency and therefore the effective Kerr nonlinearity at high intensities. Nonlinear currents are efficiently driven by the CO$_{2}$ laser beat-wave, and varying the beat frequency gives the ability to control the nonlinearity of a material like GaAs by orders of magnitude. This robust control may prove useful for a variety of applications, including LWIR source development. In a recent numerical study [61], a source was described based on nonlinear optical processes in GaAs which produce a comb of sidebands in the middle infrared.

The fully quantum mechanical model used in this paper considers nonperturbative laser semiconductor interactions from first principles, and therefore fully describes radiation effects of all orders. It gives good agreement with experimental four-wave mixing observables. The nonperturbative physics presented in the paper is general, and ought to be taken into account while describing nonresonant laser-semiconductor interactions at intensities >1 GW/cm$^{2}$.