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Abstract
The nonresonant nonlinear optical response of bulk tellurium (Te) is studied using 220 fs 10 µm laser pulses with photon energy roughly three times smaller than the band gap energy. The Kerr nonlinearity is found to be extremely large (n2,eff = 3.0-6.0 × 10−12 cm2/W), nearly 100 times larger than that of GaAs, depending on crystal orientation. Multiphoton absorption is observed at intensities >109 W/cm2 indicating the importance of free carriers to the overall nonlinear optical response. The large values of the nonlinear susceptibilities of Te open up possibilities of designing thin film elements for mid- and long-wavelength infrared nonlinear photonics.
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1. Introduction
As ultrafast laser sources extend further to longer wavelengths, the long-wavelength infrared (LWIR, 8-14 µm) spectral range is becoming a frontier in ultrafast science [1]. This has triggered a need to develop nonlinear photonics elements in this spectral range.
Of these materials, tellurium (Te) stands out as an elemental narrow-gap (Eg = 0.33 eV) semiconductor possessing unusual optical, electrical, and magnetic properties due to its asymmetric chiral crystal structure. Several magneto-electric and magneto-optic effects in bulk Te have been discovered, including the photogalvanic effect [2], current induced magnetization [3], and kinetic Faraday effect (current induced optical activity) [4]. Recent work has focused on the potential of crystalline Te as a topological material exhibiting more exotic electrical and spin properties [5–12]. In addition, one- and two-dimensional Te structures have been isolated over a relatively large scale [10,13,14], showing photoelectric responses that could have an impact on engineering nano-scale electron transport devices. Some advantageous nonlinear optical properties of tellurium nanostructures have been demonstrated, including a large second order nonlinearity in Te nanowires pumped with a 1.55 µm laser [15] as well as mode-locking and saturable absorption of laser wavelengths between 0.8-2.8 µm in Te nanocrystals [16].
Despite the intense recent study into material properties, the nonlinear optical response of Te has been largely ignored. Te possesses a number of remarkable optical characteristics, including an ultrawide transparency region that extends from mid-to-far-infrared (4-32 µm) as well as large refractive index and birefringence (no = 4.8, ne = 6.2 at 10 µm) [17,18]. This anisotropy allows for straightforward phase-matching of second harmonic generation (SHG) – performing SHG of 10.6 µm CO2 laser pulses uncovered an extremely large second order optical nonlinearity (deff = χ(2)/2 = 600 pm/V) [19], the largest known in this spectral range.
On the other hand, the third order optical nonlinearity $\chi $(3) of bulk Te in the LWIR transparency region, which determines the effective nonlinear refractive index (n2,eff), has not been reported to the best of our knowledge. There are several indications that the nonresonant n2,eff of Te should be extremely large, including refractive index and band gap scaling of n2 [20], as well as observation of strong self-focusing of 10.6 µm radiation inside a bulk Te crystal at a very low intensity [21].
In this article, we report on detailed characterization of the nonlinear optical response of bulk Te in the LWIR at intensities between 0.1-10 GW/cm2 using ultrafast laser pulses. We report on an extremely large nonlinear refractive index far surpassing those measured in other common LWIR nonlinear materials, such as GaAs. In addition, we measure significant spectral broadening induced by self-phase modulation. Finally, we observe significant nonlinear absorption at intensities above 1 GW/cm2, indicating the importance of free carriers to the overall nonlinear optical response at high intensity. An anisotropic nonlinear response is found by comparing the response in primary crystal orientations. Analytical models and numerical propagation calculations are used to validate our findings and assign values to n2,eff and nonlinear absorption coefficients. These findings reveal Te as a material with immense potential in LWIR nonlinear optics.
2. Tellurium
Tellurium crystal is trigonal in structure, with helical chains rotating along the crystal growth axis c. These 1-dimensional chains are arranged in a hexagonal array, where each chain is bonded to neighboring chains via van der Waals interactions. The bond asymmetry, strong covalent bonds parallel to c and weaker bonds perpendicular to c, leads to a uniaxial crystal with large birefringence: the ordinary orientation (electric field of light E ⊥ c) has refractive index no = 4.8 at 10 µm, whereas the extraordinary orientation (E // c) has refractive index ne = 6.2. Te along with its chalcogen neighbor Se are the simplest examples of chiral crystals, which has motivated intense study into their unique responses to external fields.
The electronic band structure of bulk Te has correspondingly been studied both theoretically and experimentally [22–25], with a recent DFT calculation [26] reproducing experimental observables such as band gap energies, structural parameters, and band-edge absorption coefficients. Experiments and calculations both show that the band gap between the upper valence band (VB1) and the conduction band is Eg = 0.33 eV, but the effective absorption edge is blue shifted by approximately 10-20 meV for E // c [27]. This has been associated with symmetry selection rules and vanishing transition dipole moments at the exact band edge in the E // c orientation. A schematic of this band structure is shown in Fig. 1(a), with two primary conduction bands degenerate at the band edge (H point in momentum space) and two valence bands separated by 0.11 eV. In p-doped samples, a strong intervalence band absorption feature has been observed at 11 µm [18]. This absorption only exists in the E // c orientation, as the transition is forbidden for the perpendicular polarization. In reality, the uppermost valence band has a characteristic “camelback” shape with a depth on the order of 1 meV due to spin-orbit interaction [22].
Fig. 1. (a) Simplified band structure at band gap, showing two uppermost valence bands and two lowermost conduction bands. (b) Normalized transmission spectrum (unpolarized) measured for the sample used in experiments, showing broad MIR transmission. Inset shows detail of the band edge, with a blue-shifted band gap for the E // c orientation.
The Te crystal we use in our experiments is 5 mm long, with transverse dimensions 10 × 10 mm, grown with the Czochralski method (Princeton Scientific Corp.). It is cut at 90° from the crystal growth axis with input surface orientation (10$\bar{1}$0). In this orientation, the electric field of a linearly polarized laser beam can be easily oriented either perpendicular (E ⊥ c) or parallel (E // c) to the growth axis c simply by rotating the crystal 90° about the cut face.
In both orientations we study, linearly polarized light propagates perpendicular to the helical atomic chains. While certain physical effects [2,28] depend on the sense of rotation around the crystal axis (i.e., left-handed or right-handed), these are present when the propagation vector is parallel to the atomic chains. Because of this the handedness of our sample, which is unknown, should not play a significant role in our measurements.
Linear transmission of this crystal was measured with an FTIR spectrometer as shown in Fig. 1(b). The spectrometer’s light source is unpolarized, so this measurement is a combination of transmission for E ⊥ c and E // c. Data is corrected to account for the effect of Fresnel reflection. While Te is not transparent in the near infrared, it has a nearly fully transparent MIR transmission window, followed by extended transparency up to 25 µm. The spectrometer only extended to 25 µm, but a different sample from the same manufacturer was previously measured to have transmission on the ∼15% level up to 30 µm.
Two features are of particular note here. First, we observe dips in the linear transmission in the band between 8-12 µm, at 9.3 µm and 10.9 µm (Fig. 1(b)). This is manifestation of intervalence band hole absorption due to equilibrium hole populations with E // c. Dynamics of these transitions in tellurium have been well-studied in the linear regime [29–31]. Based on our transmission data, we estimate a peak absorption coefficient α1 = 1.9 cm-1 at 10.9 µm. Comparing to previous studies [18], this correlates with an equilibrium hole concentration of <1 × 1015 cm-3.
Second, zooming in on the band edge (inset of Fig. 1(b)), we observe a double-humped transmission dependence. This correlates well with previous calculations and measurements of the orientation dependent band gap described above. Comparing to those results, we can assign a band gap of 0.33 eV to the E ⊥ c direction matching literature data, and a slightly blue-shifted band gap of approximately 0.342 eV.
3. Experimental methods
To study the nonlinear optical response of tellurium, we pump the crystal nonresonantly with a ∼220 fs laser pulse with a central wavelength of 10.3 µm. These laser pulses are generated using a commercial Ti:Sapphire laser-based optical parametric oscillator (OPA) overviewed in the following section (further details are included in Supplement 1).
3.1 Laser source
Figure 2 gives a general schematic of the commercial laser system that produces ultrafast LWIR laser pulses. The Ti:Sapphire master oscillator generates broadband pulses centered around 798 nm. This output is split, with half of the energy seeding a picosecond Ti:Sapphire CPA system. To obtain picosecond pulses, the stretcher is modified to mask a significant amount of energy, passing only a small bandwidth centered at 800 nm required for a 1 picosecond pulse (Δλ ≈ 1.5 nm). After compression, the amplifier outputs >8 mJ pulses at a repetition rate of 1 kHz. A measured multi-shot autocorrelation trace is presented in Fig. 2(a), showing 1 ps pulse length.
Fig. 2. General schematic of the laser system used in the present experiments. Insets show details of: (a) autocorrelation function of the 800 nm pump pulse, with a width corresponding to a FWHM pulse length of 1 ps; (b) autocorrelation function of the 1480 nm signal pulse, with a width corresponding to a FWHM pulse length of 520 fs (c) LWIR beam after ∼1 m propagation after generation in GaSe nonlinear crystal.
This pulse pumps a tunable OPA which generates broadband signal and idler seed and amplifies them to combined total energy of 2 mJ. Signal pulse length of approximately 500 fs is measured via multi-shot autocorrelation (Fig. 2(b), λsig = 1480 nm). Signal and idler are mixed non-collinearly in a large-aperture GaSe crystal. Here, a wavelength-tunable phase matched difference frequency generation (DFG) process occurs; for the purposes of this work, we used DFG pulses centered at 10.3 µm as our pump. Up to 20 µJ pulse energy was produced at this wavelength at a repetition rate of 1 kHz. This pump beam had a very nearly Gaussian transverse profile (Fig. 2(c)), and is linearly polarized.
In addition to the picosecond amplifier, the other half of the master oscillator output is sent to a femtosecond Ti:Sapphire CPA system, which amplifies pulses as short as 34 fs above 5 mJ. This NIR pulse is fully characterized with a SPIDER. The MIR and NIR arms of this laser system are synchronized via an external delay stage and seed pulse selector that supports arbitrary temporal delays between the NIR pulse and DFG pulse with jitter on the order of 1 fs. This setup is ideal for pump-probe and cross-correlation experiments.
We also fully characterize our MIR pump pulse with a sum-frequency generation (SFG) XFROG. The principle of the SFG XFROG is shown in Fig. 3. The fully characterized NIR gating pulse is combined noncollinearly with the MIR pulse in a thin AgGaS2 crystal, phase matched for SFG with angle-tuning. A retrieval algorithm is performed on the time-resolved SFG spectrum (XFROG trace, Fig. 3(a-b)) and the MIR pulse shape and phase is recovered Fig. 3(c).
Fig. 3. 10.3 µm pulse characterization with XFROG. The NIR gating pulse is characterized using SPIDER, results of which are shown in the inset of (a). The gate pulse is scanned through the pump pulse, producing time-resolved SFG measured by a VIS spectrometer. (b) Measured XFROG trace. (c) Retrieved XFROG trace. (c) Measured pump pulse profile (FWHM $\tau $= 220 fs) and phase.
The MIR pump pulses used in these experiments were measured to have pulse width of $\tau $ = 220 fs. The pulse is slightly asymmetric in time, but a Gaussian pulse with FWHM 220 fs is a close approximation to the measured pulse shape. Careful spectral measurements using a scanning monochromator show bandwidth of 850 nm for a TBP of 0.53.
3.2 Experimental setup
To determine the nonlinear optical properties of Te, we use the z-scan method [32] and self-phase modulation. In both cases, the 10.3 µm beam emerging from the GaSe DFG crystal is allowed to propagate for ∼2 m, expanding to a centimeter scale size. The beam is focused with a 50 cm focal length curved metallic mirror (Fig. 4(a)).
Fig. 4. (a) Simple experimental schematic. Interchangeable diagnostics are shown on the right. (b) Pump beam radius in the horizontal and vertical planes after focusing with a 50 cm focal length curved mirror. (c) Calculated intensity profile, I(z), fit with standard Gaussian optics to find w0 = 260 µm.
The 10.3 µm beam is nearly Gaussian throughout the focal region, measured using a pyroelectric IR array with pixel pitch 80 µm. Where the beam waist is < 500 µm, it is magnified by a factor of 5 to provide sufficient resolution. The beam radius is calculated using the second moment of the intensity distribution [33]. As show in Fig. 4(b), the beam radius in the focus (w0) for both the horizontal and vertical planes is between 250-380 µm, with an ellipticity of w0x/w0y = 0.7. Despite the ellipticity, the overall intensity profile can be fit well using an effective spot size of 260 µm (Fig. 4(c)), which corresponds to a Rayleigh length zR = 2 cm. Using this fit, the peak intensity in focus for a 220 fs pulse containing the maximum 20 µJ energy is 56 GW/cm2. A series of calibrated reflective broadband attenuators are used to reduce peak intensity as necessary. The attenuators’ substrates are 1 mm thick ZnSe, which introduce negligible dispersive pulse broadening.
For z-scan measurements the sample is mounted on a translational stage such that it can interact with converging and diverging parts of the beam, as well as a scan of intensity. When a beam with w0 ≈ 3-4 mm is propagated through the sample, signs of total internal reflection and beam distortion are observed. The beam waist is maintained below 2 mm throughout the experiments to eliminate these effects. Closed aperture Z-scan to measure n2,eff is performed by placing a small aperture (rap = 0.35 mm) approximately 40 cm from the focus. After the aperture, an HgCdTe (HCT) detector with a 1 × 1 mm chip measures the transmission through the aperture as a function of sample position. In this geometry, the aperture linear transmission is low, ∼0.5%.
To assess nonlinear absorption (NLA), the full beam is collected in the far field with a short focal length off-axis parabolic (OAP) mirror and focused onto a pyroelectric array. This array has a linear response throughout the MIR spectral region, acting as a total energy detector with a dynamic range corresponding to 15 bits. Total energy is measured in this way as the sample is scanned through the focal region. Beam collection is necessary as strong self-focusing is observed, which dramatically affects the beam size and spatial distribution in the far field.
As another test of nonlinearity, we measure spectral broadening due to self-phase modulation with a fixed sample position and beam size, w = 0.95 mm with peak intensity in air of 2.6 GW/cm2. The beam emerging from the sample is focused with a short focal length OAP onto the entrance slit of a scanning monochromator. The light is dispersed with a 100 grooves/mm diffraction grating blazed at 9 µm with known spectral efficiency. The 550 mm focal length of the spectrometer gives spectral resolution at 10.3 µm of ∼5 nm. After the exit slit, the light is focused with another short focal length OAP onto the 1 × 1 mm chip of a cryogenically cooled HCT detector to measure spectral energy density. The pump spectrum is measured in the same setup by removing the sample.
In all the above high-field measurements, no optical damage is observed on the polished Te surfaces after extended exposure to laser radiation. Different locations on the surface were used for experiments, showing no discernable difference in optical quality. We use semi-insulating GaAs (L = 7 mm, [111] orientation) as a control sample. This allows us to evaluate the validity of our experimental methods, as the Kerr nonlinearity of GaAs is well studied [34]. All experiments were performed at room temperature.
4. Experimental results
4.1 Closed aperture Z-scan
The closed aperture Z-scan measurements were made in GaAs as a calibration and both orientations of Te as specified above. These results are shown in Fig. 5, which specify the peak intensity at focus for each measurement. The full power beam was attenuated to stay below the onset of significant nonlinear absorption. To reduce the effect of shot-to-shot energy fluctuations, the signal measured by the HCT detector after the aperture is averaged over 16 individual shots to comprise one data point. The total data set at each z position consists of 150 data points; shown are the mean value with error bars representing the standard deviation of this data set.
Fig. 5. Closed aperture z-scan results in (a) 7 mm GaAs, (b) 5 mm Te, E ⊥ c orientation, (c) 5 mm Te, E // c orientation.
In all cases, relative transmission through the aperture is reduced when the sample is placed in an intense converging beam (z < 0) and increased when the sample is placed in an intense diverging beam (z > 0). This dependence gives the characteristic z-scan shape corresponding to positive n2.
The z-scan technique is a sensitive measure of nonlinear phase shift, $\mathrm{\Delta }{\mathrm{\Phi }_0} = {k_0}{n_{2,\textrm{eff}}}{I_0}L,$ where k0 is the vacuum central wavenumber of the laser, I0 is the peak intensity, and L is the sample length. In extracting this quantity, and thus n2,eff, the thin sample approximation is often used [32]. This states that the length of the sample is much shorter than the characteristic length over which the beam size changes, $L \ll {z_R}/\mathrm{\Delta }{\mathrm{\Phi }_0}$, where $\mathrm{\Delta }{\mathrm{\Phi }_0} $ is the peak on-axis nonlinear phase shift in the focus. When this is satisfied, the beam and intensity can be considered constant throughout the length of the sample, simplifying analysis tremendously.
In the special case where linear transmission through the aperture is small (S ≈ 0, as in our experimental setup), and $\mathrm{\Delta }{\mathrm{\Phi }_0} \le \mathrm{\pi }$, the transmission difference between the peak and valley of the Z-scan trace, ΔTp-v is linear with the nonlinear phase shift, satisfying the relationship [32]
Calculations of nonlinear phase shift and n2,eff using this formula are summarized in Table 1.
Table 1. Measured and calculated parameters of the closed aperture Z-scan experiment under the thin sample approximation.a
The experiment in GaAs satisfies the thin sample approximation, and calculated n2,eff matches the literature value extremely well, providing confidence in the calibration of our method. Error bars are calculated using standard uncertainty propagation, taking into account shot-to-shot fluctuation and uncertainty in the pump intensity measurement.
For both orientations of Te, the calculated nonlinear phase shift is very large, approximately $\pi $ radians even at relatively low intensities < 1 GW/cm2. Qualitative observations of the beam size after the crystal support this, with strong self-focusing causing the beam to evolve into a tight spot only a few centimeters after the sample at the highest intensities. For this experimental setup, the thin sample approximation [32] is not satisfied for either orientation of Te. Further attenuating the beam or focusing more loosely could help to reduce nonlinear phase shift, but these were both impractical solutions given our setup. A thick sample Z-scan theory exists [35], but it requires $\mathrm{\Delta }{\mathrm{\Phi }_0} \ll \pi $, which is not applicable here. In breaking the thin sample approximation, we should expect a departure from linearity in Eq. (1); in this way, n2,eff calculated from Eq. (1) gives only a lower limit of n2,eff in Te.
We calculate extremely large n2,eff for both orientations, approaching 100x larger than that in GaAs. An effective length ${L_{\textrm{e}ff}} = ({1 - exp ({ - {\mathrm{\alpha }_1}L} )} )/{\mathrm{\alpha }_1}$ = 3.5 mm is used for the E // c orientation to account for linear absorption [32]. n2,eff in the E // c orientation is ∼2x larger than in the E ⊥ c orientation, a significant anisotropy. This anisotropy correlates with the crystal symmetry and birefringence. Further, an additional dipole moment is accessible in the E // c orientation, that between the higher lying valence bands, which may help boost n2,eff.
Clearly, compared to other nonresonant materials in the LWIR, bulk Te in both orientations has among the largest known n2,eff. It is on the order or approaching extremely large third order nonlinearities measured in organic solutions or semiconductors in the near-infrared, where nonlinear effects are naturally more efficient.
4.2 Self-phase modulation
In order to establish a more reliable value of n2,eff in Te, we fix the position of the sample and measure the spectral broadening incurred by self-phase modulation (SPM) using the spectrometer. A slightly different focusing geometry is used giving zR = 4.5 cm, so the beam size stays essentially constant over the 5 mm sample length. In this position, the beam radius is 980 µm. Taking into account Fresnel reflection, the peak intensity inside the crystal for SPM measurements is 1.5 GW/cm2 for E ⊥ c and 1.2 GW/cm2 for E // c. The results of these spectral measurements are shown in Fig. 6(a) and 6(b), respectively.
Fig. 6. Measurements of spectral broadening in (a) Te, E ⊥ c and (b) Te, E // c caused by self-phase modulation.
The spectral region shown was scanned in 50 nm steps where each displayed data point is averaged in a similar way as described above. A moving average over a 200 nm window is displayed in the solid line to guide the eye. Each curve is normalized such that the total integrated spectral energy is the same, for the sake of comparison. In addition, corrections are made for the spectral dependence of relative grating efficiency, detector sensitivity, and spectral resolution.
The input pump spectrum is close to Gaussian, and has a FWHM bandwidth of 0.85 µm. The spectrum after propagation through 5 mm of Te E ⊥ c has been broadened to 1.38 µm, a bandwidth increase of 1.6x. Similarly, for E // c, the spectrum is broadened to 1.82 µm, a 2.1x increase. The anisotropy of the spectral broadening matches that seen in the Z-scan measurements, where E // c has a larger interaction despite slightly lower intensity, supporting the conclusion that n2,eff (//) > n2,eff (⊥). As before, the same measurement was carried out in the control 7 mm GaAs at similar intensities, and no spectral broadening was observed for these experimental conditions.
In principle, the amount of spectral broadening should depend on n2,eff. In the plane wave approximation, the instantaneous frequency is
We also observe a slight blue shift in the central wavelength in both orientations including an asymmetry in the SPM favoring anti-Stokes frequency shifts, more pronounced for E // c. There are two possible explanations for this asymmetry. First, the linear absorption in E // c orientation is slightly stronger on the red side of the pump pulse spectrum due to the absorption feature peaked at 10.9 µm – this could lead to an apparent blue-shift. However, this does not explain a blue-shift in the E ⊥ c case, where linear absorption is much lower. Another explanation is the slight asymmetry of the pump pulse in time, with a faster falling edge than rising edge (Fig. 3(d)). The derivative in Eq. (2) shows that the leading edge gives new lower frequency components, and the opposite for the falling edge. Thus, with our measured pulse profile, we can expect asymmetric SPM favoring the blue. This has been observed before, with a similar resulting spectral shape [38].
4.3 Open aperture Z-scan
Throughout measurements described in the previous sections, we observed signs of nonlinear absorption in Te as we remove attenuation or scan the sample through the focal region at high intensity. We use the open aperture z-scan as described in Section 3.2 to quantify the NLA. At each z position, several shots were summed, and the signals (counts) corresponding to transverse energy distribution measured by the 2D pyroelectric array were integrated over the entire beam. Several of these measurements were averaged at each z position and normalized to the total counts when the sample was far from the focus to find normalized transmittance. These results are shown in Fig. 7(a) for Te E ⊥ c and 7(b) for E // c.
Fig. 7. Open aperture z-scan results in (a) Te, E ⊥ c, (b) Te, E // c, (c) GaAs, which shows negligible NLA at the intensities used. (d) log-log fitting to predict the MPA order of the NLA process. The dashed lines correspond to slope fitting of the data.
Measured at several times higher peak intensity than the open aperture z-scan (≈ 5 GW/cm2), the nonlinear absorption at the maximum intensity results in as low as 20% for both Te orientations. The same scan was made in the control GaAs sample, which showed no signs of NLA within the experimental uncertainty (Fig. 7(c)). While the magnitude of the peak absorption is the same for both Te orientations, the intensity dependence appears to be different. As apparent in Fig. 7(d), the onset of NLA in the E // c orientation occurs at higher intensity than E ⊥ c, but the slope is stronger. This may indicate a difference in the underlying physics behind NLA in the two different orientations.
NLA is typically associated with excitation of nonequilibrium free carriers and/or free carrier absorption (FCA). In the nonresonant interaction case, $\hbar \mathrm{\omega } < {E_g}$, multiple photons are required to span the band gap and excite an electron-hole pair. This multiphoton absorption (MPA) process scales with intensity to the Nth power, where N is the number of photons to cross the gap:
When intensity is near-constant over the course of the sample, integrating Eq. (3) shows that the log-log slope of -ΔT is equal to N-1, as typically one MPA order dominates. For high-order MPA, this dependence has been shown to behave in this way even with rather large changes in transmission [39].This slope fitting is shown for both orientations of Te in Fig. 7(d). Only data points with -ΔT < 30% were used in the fitting process, as saturation and change of slope is found beyond that point. The fitted slope for E ⊥ c is 1.3, and it is 2.4 for E // c. Using the band gaps determined in Section 2, the central pump wavelength is nominally in the 3 PA regime for both orientations. This gives a slope close to 2 for both orientations. The slope fitting does not match the prediction, but does support the statement that NLA has a stronger intensity dependence in the E // c orientation.
An analytical theory of the open-aperture z-scan exists for MPA orders 3-5 [40]. Assuming Gaussian beams and pulses in time, analytical expressions for T(z) are derived and can be used to fit MPA coefficients to our experimental data. The fitting of this theory for both E ⊥ c and E // c are shown in Fig. 8. The best-fit is found by minimizing the root mean square error (RMSE) of the analytic curve and the experimental data, with αN as the only free parameter. For E ⊥ c, there is a good fit for pure 3 PA using α3 = 5.6 cm3/GW2. Compared to other common semiconductors, this is a rather large 3 PA coefficient. Note that at wavelengths between 2.3-2.7 µm, α3 has been measured in GaAs and Si to be 0.35 and 0.035 cm3/GW2, respectively [41,42]. However, compared to other narrow band gap semiconductors such as InSb or InAs, where 3 PA coefficients have been found to be as large as α3 = 1-10 × 103 cm3/GW2 [43,44], the NLA is Te is quite low.
Fig. 8. Analytical fitting of theory of z-scan with 3 PA from Ref. [40]. The curves shown are calculated with α3= 5.6 cm3/GW2 and 4.4 cm3/GW2 for E ⊥ c and E // c orientations, respectively. In E // c, 3 PA, 4 PA, and 5 PA theory all produce equally poor fits to the experimental data.
The 3 PA theory slightly overestimates absorption in the region |z| > 2.5zR, and underestimates within 1 Rayleigh length of the focus. This behavior suggests stronger intensity dependence; when 4 PA theory is applied, however, the best-fit (α4 = 15 cm5/GW3) does not improve the data matching a significant amount. In fact, the 3 PA and 4 PA theory curves are so similar that it is difficult to unambiguously assign a single MPA order to this data. Potential causes of the poor matching at high intensities will be discussed in section 5.2.
In the E // c orientation, as can be seen in Fig. 8(b), fitting any MPA order is unsuccessful, again suggesting a difference in physics between the two orientations related to different hole dynamics in the valence band.
In order to get an independent measurement of NLA for both orientations, we measured energy throughput at a fixed z position (w = 0.95 mm) and use a series of calibrated attenuators to vary intensity. For these measurements, the pulse length is ∼500 fs due to slight modifications to the laser system but the peak laser intensity was similar, on the order of 2 GW/cm2. Total energy throughput for each orientation is found using a calibrated power meter and normalized for Fresnel reflection (Fig. 9). Using modeling described in section 5.1, this experimental data matches well with α3 = 10 cm3/GW2 in the E ⊥ c orientation, a value consistent with the analytic 3 PA theory. In the E // c orientation, this method gives α3 ≈ 15 cm3/GW2, a slightly larger NLA coefficient than for E ⊥ c. It should be noted that interaction with longer pulse lengths could increase the contribution of free-carrier effects (Section 5.2). In addition, these measurements show that at low intensities where NLA is negligible, transmission in the E // c orientation is lower than in the E ⊥ c orientation. This supports the attribution of linear absorption features near 10 µm to the E // c orientation (Fig. 1(b)), and is also in line with literature measurements [18]
Fig. 9. Nonlinear absorption measurements in Te with a fixed sample position. Solid lines represent modeling results for the 3 PA coefficients listed in the legend. Dashed line is the case of no nonlinear absorption.
5. Modeling and discussion
5.1 Nonlinear Schrödinger equation modeling
In order to assign numerical values of n2,eff and αN to the nonlinear refraction and absorption processes we observe in experiments, we turn to numerical modeling. Both self-phase modulation and the open-aperture z-scan measurements are modeled using the generalized nonlinear Schrödinger equation (gNLSE), solved in a 2-dimensional cylindrical geometry. Measured experimental parameters of pulse length, beam profile, and intensity are taken into account, and best-fit values of nonlinear coefficients are found by scanning a parameter space of n2,eff and αN. Further details of the model are presented in Supplement 1, Section 1.
For the E ⊥ c orientation, the results of the modeling are shown in Fig. 10 using a pure 3 PA mechanism for NLA. Best-fit parameters are found to be to n2,eff = 3.0 × 10−12 ± 0.5 × 10−12 cm2/W and α3 = 6 ± 1 cm3/GW2. Error bars are assigned based on the discretization of the scanned parameter space. This fit n2,eff value is close to that measured in the closed aperture z-scan experiment, 1.5 × 10−12 cm2/W; however, it is larger, following the prediction that the z-scan method provided only a lower limit.
Fig. 10. (a) Spectral broadening experiment (black) and modeling (red) in the E ⊥ c orientation, with the best-fit parameters. (b) NLA experiment (black) and modeling (red) with the best fit parameters. The best fit when modeling 4 PA is also shown, for comparison.
The spectrum calculated after propagation using these nonlinear parameters is compared to the measured nonlinearly broadened spectrum in Fig. 10(a), where good agreement is achieved. The open aperture z-scan measurements are also reproduced rather well (Fig. 10(b)), especially at lower incident intensities. Numerical calculations nearly exactly overlap the MPA theory presented in Fig. 8, suggesting that certain effects that are not considered in the analytical theory – e.g., dispersion and self-focusing – do not play a dominant role in our measurements.
Due to the strong intensity dependence of 3 PA, a significant portion of the absorption occurs early in the propagation length. For example, given these fit parameters, at an intensity of 2.5 GW/cm2 65% of the total absorption occurs within the first 1 mm of the 5 mm sample.
Modeling is also performed for the E // c orientation. The SPM fitting is nearly independent of the NLA model used – we find an n2,eff value of 5.5 × 10−12 ± 0.5 × 10−12 cm2/W. Similar to the MPA theory, the open aperture z-scan data is not fit well.
5.2 Additional nonlinear absorption mechanisms
Just as with the 3 PA theory, the highest absorption near focus is not well described by the numerical model. Increasing the intensity dependence to I4 (4 PA) does not improve the fitting, as shown in Fig. 9(b). It should be noted that for both 3 PA and 4 PA regimes, the best fit n2,eff is the same, demonstrating the independence and robustness of the spectral broadening model.
There are several potential explanations for the increased absorption at high intensity that are not included in the MPA theory used in gNLSE modeling. The first of these is free carrier absorption (FCA). This can be described by:
assuming a single dominant MPA order N, where σc is the FCA cross section, and nc is the free carrier density, which in this case is assuming that the carrier life time is much longer than the pulse length. Integrating Eq. (5) in time and substituting it into Eq. (4) shows that the FCA term scales with IN + 1 when a significant free carrier density has been excited. This stronger intensity dependence may explain stronger measured absorption at the focus for the E ⊥ c orientation.An FCA cross section of σc = 5 × 10−17 cm2 was estimated in Te in the context of nondegenerate 2 PA during second harmonic generation [45]. In addition, calculations with α3 = 6 cm3/GW2 show that at the start of the sample a peak local free-carrier density on the order 1018 cm-3 is generated. However, when we perform the same calculation as before including FCA, very little change in the peak absorption is found with this value of σc, (36.6% transmission to 34.7% with FCA). Despite the FCA absorption product αFCA = σcnc = 500 cm-1, these peak carrier density exists over only a very short length. It is possible that a much larger σc would give reasonable agreement, but this requires mutual fitting of the MPA and FCA coefficients which is beyond the scope of this work.
Another possible explanation of the discrepancy between the experiment and calculations is more complicated ionization dynamics than a simple MPA model. Using the well-established Keldysh ionization model [46], we find that for the whole range of intensities in the open aperture z-scan measurements (1-5 GW/cm2) the Keldysh parameter varies between 1.8 and 0.8 in the E ⊥ c orientation of Te. This places our study in an intermediate regime, where ionization is a blend between the more commonly understood limits of MPA (γ >> 1) and tunneling (γ << 1) (see details in Supplement 1, section 3). In addition, due to the large ponderomotive energy (${U_p} \propto I{\mathrm{\lambda }^2}$), the Keldysh theory predicts that the effective band gap increases at high intensity – for our parameters, the threshold between 3 PA and 4 PA (Eg,eff = 0.36 eV) occurs at an intensity of 1.38 GW/cm2. These potential complications to the ionization rate may introduce ionization physics which is not captured with simple MPA scaling.
It is also possible that the overall nonlinearity at these long wavelengths is enhanced by free carriers. At lower intensities, the optical nonlinearity n2,eff is dominated by n2,bound. However, if significant carrier densities become excited n2,free can come to dominate, causing a large change in n2,eff. This could result in stronger self-focusing and more nonlinear loss. Furthermore, free holes in Te have been predicted to exhibit a peculiar nonlinear optical response due to the camelback valence band [47].
As discussed in section 4.3, there may be some suppression of NLA in the E // c orientation up to 1 GW/cm2, compared to the E ⊥ c orientation. This could be related to intervalence band transitions, only allowed for this orientation, or interference of different excitation pathways which could lead to reduced (or enhanced) NLA in certain intensity regimes.
Finally, the detailed band structure of Te is not considered in the current gNLSE model. It is possible that single photon or multiphoton transitions exist between conduction or valence bands, adding additional absorption channels after nonequilibrium carriers are generated. Included could be photon driven transitions into higher energy conduction bands, or phonon-assisted intervalley scattering into other regions in the Brillouin zone. For both orientations, but especially E // c, it is clear that there are complicated microscopic dynamics that cannot be fully taken into account by the gNLSE. More complete quantum mechanical modeling taking full account of the structure and symmetries of the energy bands and transition dipole moments is required to understand the full physics of the absorption processes occurring in Te.
We present a summary of the results of our study in Table 2. A figure of merit for all-optical switching devices, $\textrm{FOM} = {\textrm{n}_2}\textrm{I}/\mathrm{\lambda }\mathop \sum \nolimits_{\textrm{N} = 1}^\infty {\mathrm{\alpha }_\textrm{N}}{\textrm{I}^{\textrm{N} - 1}}$ [48], represents the usefulness of the material to provide nonlinear phase shift with low-loss. Due to the multiphoton nature of the absorption, the FOM will depend on intensity. The value given in the table represents the intensity range at which FOM > 1, which is the threshold for the material to be useful. The FOM drops at low intensity for E // c due to linear absorption.
Table 2. Summary of nonlinear optical measurements and modeling in tellurium.a
6. Conclusion
We have presented measurements of the nonlinear optical response, both nonlinear refraction and nonlinear absorption, in single crystal tellurium. The effective nonlinearity is measured to be extremely large, ∼3 × 10−12 cm2/W for E ⊥ c (100x larger than that in GaAs). A factor of nearly 2 difference between the two orientations is measured, where n2,eff(E // c) is the larger of the two. Nonlinear absorption is measured up to 5 GW/cm2 input intensity. While the loss is large over a 5 mm sample, with shorter samples and slightly less intense pulses the figure of merit can be large. The strong nonlinear optical response, in addition to unique electrical and magnetic properties, makes Te an intriguing material for thin film nonlinear photonics devices in the MIR.
Finally, our results reveal interesting nonlinear optical behavior at intensities > 1 GW/cm2, especially in the E // c orientation, that have yet to be fully understood. Calculations have already predicted efficient high-harmonic generation and spectral features which may help uncover the underlying physics of strong-field nonresonant interactions in tellurium [26]. This response warrants future experimental and numerical study.
Funding
Office of Naval Research (MURI N00014-17-1-2705).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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