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Abstract
By generating terahertz pulses in KTP crystals through optical rectification with a pump photon energy varying from below to above the bandgap, we observe a peak of the THz signal at the bandgap energy but also a second one around half the bandgap. This later one is attributed to a two-photon absorption enhanced nonlinearity, which is validated by the similarity of the two-photon absorption coefficient and THz peak amplitude data versus the pump photon energy. A careful analysis of the KTP sample absorption spectral dependence nearby the bandgap demonstrates that KTP is an indirect bandgap crystal, whose absorption below the bandgap involves emission of a phonon related to the symmetric Ti-O stretching vibration, i.e. the ν1 (A1g) mode.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical rectification (OR) of femtosecond laser beams in dielectric crystals is widely used to produce terahertz (THz) waves [1–3], exhibiting large bandwidth [4], as well as huge peak power [5]. The efficiency of the optical-THz conversion is governed by both the nonlinearity of the crystals and the realization of phase-matching, i.e. the group velocity of the laser pulse should be equal to the phase velocity of the THz wave in the crystal, in order to have constructive contributions to the THz signal from all the radiating parts of the crystal. This phase-matching condition is not often achieved in natural crystals. For THz generation it is almost only obtained in organic crystals like DAST or DMTS pumped at a 1.4∼1.7-µm wavelength [5], or in dielectric crystals like ZnTe, but this crystal exhibits a quite moderate nonlinearity [6]. Following the initial suggestion by Armstrong et al. [7] regarding second harmonic generation (SHG), THz generation through OR could be performed in periodic stacks of layers with reversed nonlinearities [8–9]. The thickness of each layer is equal to the coherence length of the THz OR process, which is the thickness of a material at which the THz signal is maximum when phase-matching is not realized. Then the crystal orientation of the next layer is reversed, in such a way the THz field generated in this layer adds to the one generated before (quasi-phase matching), instead of being destructively subtracted as in mono-domain crystals.
Layers with alternative crystallographic orientation can be fabricated in ferroelectric crystals, like LiNbO3 or KTP, by electric field poling [10–11]. Such periodically-poled (PP) crystals have already been used to generate THz waves [12–14]. KTP shows the advantages over LiNbO3 of a better transparency at higher THz frequency and of a much higher damage threshold, even if its involved nonlinear coefficient is weaker: the figure of merit $\eta$ for THz generation through OR, is 1100 and 740 (pm/V)2 respectively for LiNbO3 and KTP [13] (${{\eta = d_{ij}^2} / {{n_{THz}}n_{laser}^2}}$ where ${d_{ij}}$ is the nonlinear OR relevant coefficient, ${n_{THz}}$ and ${n_{laser}}$ are the refractive indices at respectively the THz and laser wavelengths). To enhance this conversion efficiency, it is preferable to excite the crystal nearby a resonance. In KTP, studies are usually focused on the 5.9-THz frequency, at which a polariton is excited, leading to a larger nonlinearity [15]. Another way of benefiting of a larger nonlinearity is to perform OR nearby the KTP bandgap energy. At the vicinity of the bandgap energy, the nonlinearity is strongly enhanced, as known for years for SHG [16–17], and as demonstrated more recently for THz generation through OR in GaSe [18] and ZnTe [19].
Here we report on the efficiency of THz generation in bulk KTP crystals versus the pump laser wavelength, which is tuned from the near IR to the UV range. We study samples of both KTP and rubidium-doped KTP (RKTP), which are either [100], [010], or [001] oriented. As expected, we observe a peak at the bandgap energy (3.55 eV) but also an even more intense signal around half-the-bandgap that originates in a two-photon absorption (TPA) enhanced nonlinearity. To the best of our knowledge, this is the first report of a half-the-bandgap enhanced nonlinear effect on THz generation. Moreover, contrary to what was previously thought based on theoretical considerations, we show that KTP optically behaves as an indirect bandgap crystal.
2. THz generation experiment
KTP and RKTP are orthorhombic crystals, belonging to the mm2 point group, and Pna21 space group. The non-zero elements of their nonlinear second-order tensor are d31, d32, d33, d24, and d15. For SHG, the values of these coefficients are respectively 2.76, 4.74, 18.5, 3.92 and 2.04 pm/V [20]. Because OR is closely related to the electro-optic (EO) effect (assuming the THz frequency Ω is very small as compared to the optical frequency, ω, of the pump), the OR tensor is related to the EO tensor, whose non-null elements are r13=8.8, r23=13.8, r33=35, r42=8.8, and r51=6.9 pm/V [21].
Here the samples are excited under normal incidence by a single laser beam. It follows, from the symmetry of the nonlinear tensor, that no THz signal is radiated by the [001] samples, while the nonlinear polarization at frequency Ω in the [100] and [010] samples writes:
${I_\omega }$ is the intensity of the pump laser beam, ψ is the polarization angle of the laser E-field, ${\chi _{ij}}$ is an element of the nonlinear susceptibility tensor using conventional contracted index notation, and x, y, z is the crystal coordinate frame (see Fig. 1), corresponding to the crystal axes a, b, c, respectively. In the present study, the laser polarization is horizontal (i.e. along the y-axis for the [100] orientation or along the x-axis for the [010] case) and the detected THz signal polarization is aligned along the z axis ($\psi = 0$ in (1) and (2)). Therefore the detected THz field is proportional to ${\chi _{32}}$ for the [100] sample and to ${\chi _{31}}$ for the [010] one.
Fig. 1. Left: general orientation of the laser and THz beam polarizations; center: present experimental parameters in the [100] crystal case; right: idem for [010] case.
When pumped above the bandgap, photo-carriers are generated in the crystals. However, they do not contribute to the THz generation, as they move from and perpendicularly to the crystal surface, and, moreover, they absorb the THz wave generated in the bulk. Therefore, at any pump wavelength, only the response of excited bound electrons is a source of THz radiation. All the studied KTP and RKTP samples used here are 0.3-mm thick and double-side polished. Our experimental setup is the same as the one described in [19]. The samples served as THz emitter in a common THz time-domain setup and the time-resolved record is performed with an EO detection system that includes a 1.52-mm thick [111]-cut ZnTe crystal. The KTP and RKTP samples are excited by the 50-fs pulses delivered by an OPA (Topas, Light Conversion, pumped with an amplified laser system, Libra from Coherent) whose wavelength is tuned from 310 nm to 800 nm (i.e. from 4 eV to 1.55 eV). At the crystal, the pump spot diameter was 2 mm.
The average power was 10 mW in the ranges 1.55-2.73 eV and 3.40-4.00 eV (peak power density 6.37 GW/cm2). Around 413 nm (3 eV), the output power of the OPA was too weak to allow us to record any THz signal. The experiment was performed in dehydrated air (8% humidity) at room temperature.
3. THz generation results
3.1 THz waveform analysis
Figure 2 shows the THz waveforms generated by the [100] KTP sample and recorded for different photon pump energies whose values are listed on the right side. For the sake of legibility, each curve has been shifted vertically. For excitation below the bandgap, i.e. in the spectral region of transparency of KTP, THz generation occurs only at the two sample surfaces, due to the lack of phase-matching [21]. The first peak at time $t = 0$ps is produced by the exit interface, while the second one at about $t = 1.8$ps comes from the entrance interface. Because the THz wave propagates slower than the optical wave in KTP, the entrance interface peak is detected after the exit one. The delay ${\tau _1}$ between the two peaks is given by:
where d is the sample thickness, ${n_{G\Omega }}$ and ${n_{G\omega }}$ are the group indices at THz and optical frequencies, and c is the velocity of light in vacuum. The group index at THz frequencies is determined by looking at the THz rebound occurring around 9.5 ps (see for example the 3.65-eV waveform). This is the peak generated at the entrance face that is reflected by the exit face back inside the crystal, and then reflected by the entrance face towards the receiver. It is hence detected with a delay ${\tau _2}$, as compared to the second peak of the waveform at $t = 1.8$ps, equal to ${{{\tau _2} = 2{n_{G\Omega }}d} / c}$. The measured value of ${\tau _2}$, averaged over all the waveforms, is equal to 7.54 ps, leading to ${n_{G\Omega }} = 3.77$.Fig. 2. THz waveforms from [100] KTP crystal recorded for different pump photon energies. The thicker curve is recorded for a pump photon energy (3.54 eV) corresponding nearly to the one of the KTP bandgap (3.55 eV).
Figure 3 presents the experimental delay ${\tau _1}$, extracted from Fig. 2 together with the curve calculated with (3). Here there are no adjustable parameters. ${n_{G\Omega }}$ is the value determined above, and ${n_{G\omega }}$ is calculated from data published by Kato and Takaoka [22]. Above the bandgap, the laser beam is completely absorbed before reaching the second face of the sample. Therefore, no THz signal is generated at this face, and thus we see on Fig. 2 only the peak produced at the entrance face of the crystal. Also the weak oscillations observed below the bandgap in the time window 3∼7 ps (see for example curve 2.07-eV in Fig. 2) disappear above the bandgap, because the laser beam does no more propagate in the material (theses oscillations are due to both the dispersion of the refractive index in the THz range and to the lack of phase-matching [23]).
Fig. 3. Delay time τ2 between the 2 first pulses of the recorded THz waveforms of Fig. 1. The continuous line is calculated using expression (3).
Selected spectra of the THz waveforms are depicted in Fig. 4. For the sake of legibility, we chose only 3 spectra corresponding to a laser pump energy below the bandgap (2.26 eV), at the bandgap (3.54 eV) and above the bandgap (3.65 eV). Below the bandgap, the spectrum shows oscillations due to the two peaks in the waveform. Typically, the spectrum spreads up to 2.5 THz, and its maximum is 26 dB over the noise level. Above the bandgap, the spectrum does not exhibit the oscillations, and it is narrower (2 THz maximum), but with the same dynamic range (26 dB). The THz bandwidth shrinkage is clearly explained by the Fourier transform of a single pulse (above bandgap) as compared to the one of a double bipolar pulse (below bandgap). At the bandgap, the spectrum width is smaller than 2 THz, and its dynamic range is lowest (20 dB).
Fig. 4. Spectra (modulus) of the waveforms plotted in Fig. 1 for pump photon energies 2.26, 3.54 and 3.65 eV. The 3.54 and 3.65-eV curves are vertically shifted for the sake of legibility.
3.2 THz signal versus pump photon energy
Apart from the phase-mismatch oscillations below the bandgap excitation, the THz spectra recorded at different pump photon energies are rather similar. This allows us to plot the magnitude of the peaks of the waveforms depicted in Fig. 2 without worrying about dispersion effects at THz frequencies. The data are shown versus the pump photon energy in Fig. 5 for the [100] (x-cut) and [010] (y-cut) samples. For both samples, we depict the maximum values of the first peak at $t = 0$ps (entrance face) and of the second one at $t = 1.8$ps (exit face).
Fig. 5. Peak values of the THz waveforms (Fig. 2) versus the pump photon energy. The left plot is for the x-cut crystal, the right one for the y-cut one. The full (open) circles are signal generated at the entrance (exit) face. In the upper plot, crosses are the entrance face data multiplied by 3.7. The vertical continuous and dotted lines indicate the bandgap and half-the-bandgap energies respectively. The dashed curves are a guide to the eyes.
The signal from the entrance face of the [100] crystal shows a peak around the bandgap energy at ∼3.55 eV, and a second but weaker bump around 2 eV, i.e. just above the half-bandgap energy (1.77 eV). As expected, the signal from the exit face is detected only for photon energies below the bandgap, i.e. in the region of transparency of KTP. It exhibits also a big bump around 2 eV, about 3.7 times stronger than observed from the entrance face. Whatever the generating crystal face, these bumps are about 1-eV broad, and when multiplied by 3.7, the exit data superimpose nicely with the entrance ones. The records are almost similar for the y-cut sample, but the signals are slightly weaker, and mostly the entrance signal is larger than the exit one, which is opposite to what was observed with the x-cut sample. Around 3 eV, our setup sensitivity is not high enough to allow us to record a THz signal that is quite weak. Indeed, it is known that the OR THz generation efficiency decreases when increasing the pump photon energy below the bandgap, due to both the increases of the refractive index and absorption coefficient (see for example Fig. 3 in our previous publication [19]). Here a similar behavior occurs as validated by calculation, but this tendency is hidden by the TPA resonant feature.
Similar but weaker signals were recorded with the [010] KTP sample can be explained by the fact that in this sample, OR involves the ${d_{31}}$ nonlinear coefficient instead of ${d_{32}}$ for the [100] sample. ${d_{31}}$ is smaller than ${d_{32}}$ if these coefficients behave as the electrooptic ones [21]. This value was confirmed by polarimetric studies, not presented here, in which the polarization angle ψ of the laser (see Eqs. (1) and (2)) was varied. Let us also notice that we did not observe any difference between signals generated in KTP and RKTP.
The THz signals generated at the entrance and exit faces encounter different losses in the KTP material. The entrance signal ${S_{ent}}$ suffers THz loss through the sample, while the exit signal ${S_{exit}}$ is generated by the laser beam that has been attenuated through the sample. Thus we write:
Io is the laser beam intensity. The factor ½ in the exponent of the expression of ${S_{ent}}$ is due to the fact that ${S_{ent}}$ is proportional to the detected THz field, while in the expression of ${S_{exit}}$, one takes into account the attenuation of the laser intensity, so the factor disappears. The experimental ratio R is equal to 3.7 for the x-cut crystal and to 0.65 for the y-cut one. Let us notice that the exit signal is larger than the entrance one for x-cut crystal, while it is the opposite for the y-cut crystal. This is explained by the difference of THz absorption in these two crystals. As the measured THz absorption coefficient, averaged over the range 0.2-2 THz, is equal to 45 cm−1 and 11 cm−1 for respectively the x-cut and y-cut crystals, this leads, using (4), to ${\alpha _{laser}} \approx 10$cm−1. This is larger than the known value of absorption:${\alpha _{o,laser}} = 0.01 \sim 0.03$cm−1 [24].
Finally, from the above explanation, we can now qualitatively comment on the strong decrease of the THz signal generated at the entrance face when pumped around the bandgap wavelength (see Fig. 2). Below the bandgap, i.e. in the range of transparency of KTP, the laser pump beam is weakly absorbed and the photo-carrier density in the crystal is negligible. Thus the THz field generated at the entrance face suffers almost no free carrier absorption and therefore the detected THz signal is strong. Above the bandgap, the laser beam is absorbed within a few microns at the entrance face of the crystal. The THz field, generated in this photo-excited layer, is not attenuated by free carriers inside the crystal, as the laser beam does not propagate in the crystal. Again, the detected THz signal is strong. Near the bandgap, the laser beam penetrates the whole crystal, and creates free carriers all along the crystal thickness. As the THz beam propagates slower than the laser beam, the THz signal generated at the entrance face is attenuated by these already-excited carriers all along the crystal thickness. Thus the THz beam suffers a strong free carrier attenuation and the detected THz signal is weak.
4. Two-photon absorption in KTP
A possible explanation of the discrepancy between the absorption ${\alpha _{laser}}$ at the laser frequency determined here and the published values [24] is linked to a two-photon absorption (TPA) phenomenon, which leads to a nonlinear increase of the absorption coefficient:
Here, with ${\alpha _{laser}} \approx 10$cm−1 and I = 6.4 GW/cm2, we obtain $\beta = 1.56$cm/GW at 2 eV. In KTP, a TPA coefficient value $\beta = 0.1$cm/GW was initially determined by DeSalvo et al. [25]. Later, Maslov et al. [26] reported a 10-times larger value, i.e. $\beta \sim 1$cm/GW at 532 nm when the laser beam is polarized perpendicularly to the crystal z-axis.
Because of this big discrepancy, we decided to measure the TPA coefficient in KTP versus the pump photon energy, in view of validating the value we determined from the analysis of the THz signal. We used the same setup as described above for the THz generation study, but now a power-meter is located just after the THz emitting crystal and it serves to measure the coefficient of transmission of this crystal. The impinging OPA beam power is adjusted with a variable neutral optical density filter (Thorlab).
As expected (see Fig. 6), the transmitted peak power decreases linearly with the power density of the exciting OPA beam whatever the pump photon energy was, which is the signature of a TPA phenomenon. The slope of the fitting linear plot is equal to the TPA coefficient β.
Fig. 6. Transmission coefficient of the x-cut KTP crystal versus the peak power density of the OPA beam, for 3 different values of the pump photon energy. The lines are linear fits of the experimental data.
The TPA coefficient β is plotted versus the photon energy on Fig. 7 (open circles) for the x-cut sample. We retrieve the same results as Maslov et al. [26] (grey squares in Fig. 7). In agreement with them, we also obtain the same β values for the y-cut sample. Our experimental β values exhibit a similar behavior versus pump photon energy as the one of the THz peak values (i.e. the 2-eV bump of Fig. 5) that are plotted again for comparison. Note that we plot here only the THz signal generated at the exit face, and thus we divide the measured waveform peak value by the transmission (intensity) of the crystal at the optical frequencies, in order to normalize the THz signals to the pump laser intensity at the exit face.
Fig. 7. TPA nonlinear coefficient β (open circles) versus pump photon energy. Grey squares are values published by Maslov et al. [26]. The THz peak values (Fig. 5), divided by the crystal optical transmission, are plotted again for comparison (black circles, given here in arbitrary units). The continuous line is calculated with the model of Ref. [35]. The dashed line indicates half-the-bandgap.
This similarity between the THz peak curve and the TPA one, versus the pump photon energy, confirms that the bump of the THz signal around half-the-bandgap is definitively related to a two-photon interband carrier excitation, and hence to the related enhanced nonlinearity. We did not find in the literature any theoretical paper that describes qualitatively this effect. However a simple analysis in terms of energy levels diagram (see for example paragraph 3.2.4 in [27]) may help to understand the resonant behavior of THz generation at half the bandgap, and thus its relation to the TPA phenomenon. Let us assume that TPA occurs first through the excitation from the ground state a (top of valence band at energy $E = 0$) of a virtual energy level b at ${{{E_g}} / 2}$, and then from b to the energy level c (bottom of the conduction band at ${E_g}$). The expression of the nonlinear susceptibility writes (equation 3.2.27 in [27] adapted to optical rectification):
Let us now look again at the relative magnitudes of the x-cut and y-cut signals, which are respectively proportional to the ${d_{32}}$ and ${d_{31}}$ nonlinear coefficients. As previously explained, we get:
We measure on Fig. 5 ${{{S_{exit,x}}} / {{S_{exit,y}}}} = 2.0$ at 1.55 eV, where the TPA contribution to nonlinearity is very low, and using ${\alpha _{laser,y}} - {\alpha _{laser,x}} \approx 0.014$[24], we derive ${{{d_{32}}} / {{d_{31}}}} = 1.99$. This is comparable with both the electro-optics coefficients ratio ${{{r_{32}}} / {{r_{13}}}} = 1.57$ and the SHG ones ratio ${{{d_{32}}} / {{d_{31}}}} = 1.72$ published in [20–21]. This value was confirmed by polarimetric studies, not presented here, in which the polarization angle ψ of the laser (see Eqs. (1) and (2)) was varied.
5. Indirect bandgap of KTP
Several theoretical papers have been published on modelling the TPA coefficient β in semiconductors [28–35]. The expression of β for direct bandgap semiconductors, derived from band calculations including exciton effects and band non-parabolicity [30–34], is [35]:
The curve calculated with (9) is shown as a continuous line on Fig. 7. The magnitude, with ${f_2} \approx 0.54$, and the shape of the calculated curve correspond nicely to the experimental data, but the calculated curve is shifted towards higher photon energies as compared to the measured values. A better fit would have been achieved by taking ${E_g} = 3.2$eV instead of ${E_g} = 3.55$eV. Let us notice that such a shift has not been reported [36] for another large bandgap crystal, namely GaN, which is a direct bandgap semiconductor.
To check the bandgap energy value of our samples, we measured their transmission in the visible-UV range using a Shimadzu UV-2600 spectrometer that delivers an unpolarized light beam. For both samples, the absorption starts to increase around 3.4∼3.5 eV (see inset of Fig. 8) due to intra-band transitions, and the curves cannot be fitted with direct bandgap absorption expression, but with indirect intra-band absorption, namely [37]:
For such an absorption behavior, it is worthwhile to plot the spectrum of the square-root of $\alpha (\omega )$, which varies linearly with the difference $\hbar \omega - {E_g}$. The results are presented in Fig. 7 for both the x-cut and y-cut samples. Above 3.75 eV, the transmitted beam is too weak to be measured, and thus the observed plateau is an artefact. For the y-cut sample, the measured data are well fitted by expression (10), with ${E_g} = 3.5 \pm 0.05$eV. The x-cut sample follows the same law, but the agreement is not perfect: the experimental data show a pronounced tail below the bandgap, which can be fitted by [37]:
This expression results from the emission of a phonon in the absorption process, whose energy is ${E_{phonon}}$. The role of phonon emission and absorption at the edge of the indirect bandgap is well known in common semiconductors like Si [38] or even large bandgap semiconductors like SrTiO3 [39]. In KTP, an intense phonon line, deduced from IR reflectivity and Raman scattering experiments, occurs at ∼700 cm−1 [40–41], i.e. ${E_{phonon}} \approx 173$meV. It is due to the ν1 (A1g) mode, which corresponds to a symmetric Ti-O stretching vibration. It is efficiently excited when the optical field is aligned along the b or c axes of the crystal, but not when it is aligned along the a axis. With the y-cut sample, the unpolarized light beam excites the phonon only along the c axis, while with the x-cut sample, phonons along both b and c axes are excited, resulting in a stronger phononic contribution to the intra-band absorption. Here (see Fig. 8), the dashed line, calculated with (11) and ${E_{phonon}} = 173$meV fits well the x-cut sample. This dashed line crosses the zero level at ${E_g} - 2{E_{phonon}}$. The above observation of an indirect intra-band absorption was not expected since KTP is known as being a direct bandgap semiconductor [42–44]. However, the calculated band structure of KTP is very complex and its valence and conduction bands in the reciprocal space are almost flat [42–44]. Another explanation for the absorption tail below the bandgap energy for the x-cut sample could be the Urbach rule [45]. However, this Urbach tail appears likely in disordered crystals (structural or composition disorder) or in crystals exhibiting defects, and it should be seen for both x-cut and y-cut samples (except if these crystals were differently grown). As our samples are of high crystallographic quality, this explanation is unlikely. Nevertheless, this should be confirmed by recording the absorption spectrum versus temperature.
Fig. 8. Square-root of the measured coefficient of absorption for KTP (x-cut sample: full circles; y-cut sample: open circles) versus the photon energy. The continuous line is calculated with (10) and Eg=3.5 eV, and the dashed line with (11) and Ephonon=173 meV. The inset shows the measured absorption versus the photon energy on a linear scale diagram.
6. Conclusion
Performing THz generation in KTP and RKTP through OR, with a pump laser beam whose photon energy is tuned from 1.55 to 4 eV, permits us to observe a peak of the THz signal at the bandgap energy but also around half the bandgap. This later signal increase is definitively due to a TPA enhanced nonlinearity, which is validated by the amazing similarity of the TPA coefficient and THz peak amplitude data versus the pump photon energy. Our determination of the TPA coefficient of KTP, over the range 1.55-2.73 eV, is in agreement with and completes data published by Maslov et al. [26]. A careful analysis of the KTP sample absorption spectral dependence nearby the bandgap demonstrates that KTP is an indirect bandgap crystal. Just below the bandgap, absorption involves emission of a phonon, which likely corresponds to a symmetric Ti-O stretching vibration. The indirect bandgap behavior must be validated by complementary works, for example by measuring the near bandgap absorption versus temperature. As well, the TPA enhancement of the nonlinearity merits to be described and explained with a complete theory. However, we can predict that this effect should be more pronounced with crystals exhibiting a large TPA efficiency. The TPA coefficient is known to vary inversely to the third power of the band gap energy (see Eq. (8) [30–35]), and thus GaAs (β = 23 cm/GW) or CdTe (β = 22 cm/GW) [45] should show a strong TPA enhancement of the OR nonlinearity.
Funding
Knut och Alice Wallenbergs Stiftelse; Stiftelsen Olle Engkvist Byggmästare; Stiftelsen för Strategisk Forskning; NATO Science for peace (GS-5396); Agence Nationale de la Recherche (ANR-17-CE24-0031-01).
Acknowledgments
We acknowledge Mr. Gilles de Moor, LEPMI laboratory at University Savoie Mont Blanc, for having performing the UV-visible transmission measurements.
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.
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