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Abstract
Terahertz (THz) generation via optical rectification (OR) of near-infrared femtosecond pulses in DSTMS is systematically studied using a quasi-3D theoretical model, which takes into account cascaded OR, three-photon absorption (3PA) of the near-infrared radiation, and material dispersion/absorption properties. The simulation results and the comparison with experimental data for pump pulses with the center wavelength of 1.4 µm indicate that the 3PA process is one of the main limiting factors for THz generation in DSTMS at high pump fluences. The THz conversion efficiency is reduced further by the enhanced group velocity dispersion effect caused by the spectral broadening due to the cascaded OR. We predict that for broadband pump pulses with a duration of 30 fs, the THz conversion efficiency can be enhanced by a factor of 1.5 by using a positive pre-chirping that partially suppresses the cascaded OR and the 3PA effects.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Intense terahertz (THz) transients provide a novel platform to control material and light properties on ultrafast timescale [1] and to construct compact charge particle accelerators [2,3]. Due to stable carrier envelop phase and natural synchronization with laser-driven ultrafast probes, optical rectification (OR) of femtosecond (fs) pulses in nonlinear crystals (NCs) offers one of the most wide-spread methods to generate single-cycle THz pulses with high electric field. Recently, mJ-level THz pulse energy has been obtained via OR in different NCs (for example, LiNbO3 [4] or 4-N,N-dimethylamino-4’-N’-methyl-stilbazolium 2,4,6- trimethylbenzene-sulfonate (DSTMS) [5]) driven by table-top high-energy laser systems. The corresponding peak electric field reaches tens of MV/cm, opening up entirely new avenues in THz science by triggering nonlinear THz responses of materials.
Compared with LiNbO3 – one of the most widely used NCs – organic THz emitters, such as DAST [6], OH1 [7], HMQ-TMS [8] and OHP-CBS [9] offer several advantages: (i) they possess the largest nonlinear coefficients among NCs used for OR, resulting in high optical-to-THz conversion efficiency at room temperature; (ii) phase matching can be achieved in collinear geometry, the generated THz radiation is naturally collimated, enabling excellent THz focusing properties for achieving ultra-high field strengths; (iii) low absorption in the THz range due to the relative weakness and sharpness of phonon resonances enables the generation of very broadband THz pulses. Owing to broadband phase matching condition with a minor phonon absorption, DSTMS can provide octave-spanning, or even multi-octave THz spectra [10], showing promising potential in generating THz pulses with ultra-high electric field [5,11–13]. Since the quest for higher field strength is incessant, upscaling of the THz pulse energy calls for increasing pump fluence. However, the THz conversion efficiency of DSTMS saturates at high pump fluences [12] and the key factors leading to this limitation remained unclear until now.
In this work, we introduce a quasi-3D model of THz generation in DSTMS that takes into account both linear (material dispersion and absorption) and nonlinear (cascaded OR and three-photon absorption) optical effects. The model is tested and compared with experimental results for the most relevant pumping wavelength of 1.4 ${\mathrm{\mu}}$m, where the phase-matching and, correspondingly, the THz conversion efficiency are optimal. The experimental and the simulation results identify three-photon absorption (3PA) as one of the main effects responsible for the saturation of the THz conversion efficiency. Furthermore, we demonstrate that owing to the spectral broadening caused by the cascaded OR, the enhanced GVD effect is detrimental to THz generation even at moderate pump fluences. Our simulations predict that a more efficient THz generation can be obtained using optimized positive chirp of the pump pulses, which leads to the suppression of 3PA and to spectral broadening of the femtosecond pulses. These findings are crucial for achieving an ultimate performance of DSTMS for THz generation and for the design of efficient THz sources driven by high-power fs laser systems using other highly nonlinear optical materials.
2. Experiment
The optical pump pulses are delivered from an optical parametric amplifier, which is driven by a Ti:sapphire femtosecond amplifier at a repetition rate of 1 kHz. The pulse energy of the optical pump can reach as high as 181.0 ${\mathrm{\mu}}$J. The beam size of the optical pump is around 2 mm FWHM, and the corresponding maximum peak fluence is 4.0 mJ/cm2. Its spectrum, which is measured by a fiber-coupled spectrometer (Avantes Corp.) equipped with an InGaAs detector array, is centered at 1428 nm (209.9 THz) with a FWHM bandwidth of 90 nm. A commercial <001>-cut DSTMS crystal with thickness of 0.54 mm was utilized as a THz emitter. To validate the developed theoretical model, all the experimental parameters shown above were utilized in simulations.
For the comparison with simulation results, the transmitted spectrum of the optical pump was measured at various peak fluences (0.4 mJ/cm2, 2.2 mJ/cm2, 4.0 mJ/cm2), and the THz field emitted from DSTMS was detected by a 400 ${\mathrm{\mu}}$m thick GaP crystal, which was gated by a femtosecond probe at 800 nm with 35 fs FWHM. The average power of the generated THz pulses as a function of pump fluence was characterized by a calibrated thin-film pyroelectric detector (SLT sensors, THz 20).
3. Quasi-3D theoretical model
In order to reveal the main limiting factors of THz generation, we introduce a model with slowly varying envelope and plane wave approximations. The transverse beam profile of the collimated optical pump was assumed as a Gaussian function of radial distance, R, shown in Fig. 1(b) and spatial phase evolution (such as self-focusing effect) is neglected in the model for simplification. As shown in Fig. 1(a), the spatially dependent evolution of the THz electric field, ${E_{THz}}{\rm{(\varOmega ,\;r,\;z)}}$, at angular frequency ${\rm{\varOmega }}$ can be described as [14],
Fig. 1. Spatial dependent spectra and time-domain pulse shape evolution of optical pump for optical rectification process in DSTMS. (a) scheme of optical rectification in DSTMS pumped by intense femtosecond pulses; (b) transverse Gaussian distribution for optical pump with a FWHM of 2 mm at pulse energy of 181.3 ${\rm{\mu J}}$, the corresponding fluence at the center is 4 mJ/cm2; spectra (c) and pulse (d) re-shaping evolution along to propagation of optical pump in DSTMS at different radial distance (labeled A, B, C, D in (b)).
In the case of optical pump pulse evolution in DSTMS, both linear effects (linear absorption and group velocity dispersion (GVD)) and high-order nonlinear effects (the cascaded OR and the three-photon absorption (3PA)) are considered to reveal the actual physics situations for THz generation via OR process. The corresponding nonlinear wave equation for evolution of the optical pump pulses in the frequency domain is [14],
The first term on the right-hand of Eq. (2) accounts for the absorption with the coefficient $\alpha ({\omega ,R,z} )= {\alpha _0}(\omega )+ \gamma {I^2}({\omega ,R,z} )$ where ${\alpha _0}$, $\gamma $ are the linear absorption [16] and 3PA coefficients of DSTMS [17]. The second term corresponds to the GVD effect, ${\beta _2}$ is the material dispersion coefficient of DSTMS, ω0 is the center angular frequency of the optical pump pulses. The third and fourth terms represent the down-conversion and up-conversion of spectral components in optical pump pulses due to cascaded OR process. Here, the self-phase modulation (SPM) and stimulated Raman scattering (SRS) are neglected, since the intrinsic nonlinear refractive index of DSTMS is very small at a wavelength near 1.4 ${\mathrm{\mu}}$m [17], which has the optimized phase matching condition at this wavelength range. We utilize a 4th order Runge-Kutta method to numerically solve the above equations with a spatial resolution of 10 ${\mathrm{\mu}}$m in the z-axis and 200 ${\mathrm{\mu}}$m along the transverse radial directions.
The spatially dependent evolution of optical spectra in DSTMS has been simulated for a peak fluence of 4 mJ/cm2 at the center. The simulated spectral re-shaping evolution in DSTMS at a different radial distance is depicted in Fig. 1(c). Large frequency down-shift and spectral broadening were observed close to the center of pump beam, resulting in pulse duration broadening and multi-pulse splitting combined with the GVD effect, as shown in Fig. 1(d), whereas the spectrum and pulse shape nearly remain the same across the whole thickness of DSTMS at the edge of the optical beam. These results indicate that significant spectral re-shaping is attributed to a strong cascaded effect during THz generation from OR process in DSTMS, when pumped by intense femtosecond pulses [14].
For comparison with the experimental data, the simulated spectra for different radial positions are summed up (integrated) over the complete pump spot. Thus, our model takes into account not only the propagation along the z-axis, but also the radial intensity distribution of the near-infrared and the THz beams. Therefore, effectively the model can be called quasi-3D.
4. Validation of model: comparison to experimental results
The spatially averaged transmitted optical spectra, $ |(\omega )\propto {2\pi} \int_0^\infty |E_{op} (\omega ,R,z)|^2 RdR$, were simulated with optical pump fluence ranging from 0.1 to 4.0 mJ/cm2 (the corresponding pulse energy is from 4.5 to 181.3 ${\rm{\mu J}}$). As Fig. 2(a) shows, significant center wavelength red-shift and spectral broadening were observed with increasing fluence. However, a saturation of the red-shift occurs at a fluence of higher than 1 mJ/cm2. The center wavelength red-shift is a direct consequence of cascaded OR processes and proportional to THz transient strengths. This suggests that THz conversion efficiency suffers from saturation at high pump fluence levels. In contrast, the spectral broadening monotonically increases in the simulated fluence range since the stronger cascaded OR effects emerge at the edge area of the beam cross-section at a higher fluence level. In addition to the red-shift, an increasing amount of blue-shift was observed, which is induced via THz-plus-optical sum frequency generation (SFG) process and it is accounted for by the fourth term in the right-hand side of Eq. (2).
Fig. 2. (a) Simulated pump fluence dependent output spectra of pump pulses from DSTMS; (b-d) comparison between experimental and simulated spectra at peak fluence of 0.4 mJ/cm2, 2.2 mJ/cm2 and 4.0 mJ/cm2. The simulated spectra are multiplied by the spectrometer response function [dash line in the panel (b)]. The dot line indicates the initial center wavelength of pump pulses.
Figure 2(b-c) depict simulated spectra corrected by the spectrometer response function in comparison to the experimental results. One can see that the model qualitatively reproduces the experimentally observed red-shift and the spectral broadening. The absence of the fringes in the recorded spectra can be a consequence of the surface roughness and imperfections in the DSTMS crystal leading to the phase front distortion and suppressing the interference. Other quantitative discrepancies between simulated and experimental spectral shapes may be caused by the following reason: the actual cross-section of the OPA beam is non-Gaussian.
Besides optical pump spectra, the simulated THz spectra are verified by experiments as well. As Fig. 3(a) shows, the cut-off frequency of the simulated THz spectrum extends up to 15 THz. In order to compare the simulation with the experimental results, we multiply it by the response function of the GaP detector. This function, shown as the dashed line in Fig. 3(a), has been calculated using equations from [18] with additional averaging over the spectrum of the electro-optic sampling pulse and a factor that takes into account the diffraction effect for the focusing on the detector crystal [19]. The product of the simulated THz field emitted by DSTMS and the GaP detector response function is shown as a red area in Fig. 3(b). The simulated THz spectrum agrees excellently with the experimental results.
Fig. 3. (a) Simulated THz field at pump fluence of 0.4 mJ/cm2 (red solid line) and the response function of 400 um GaP detector (grey dashed line); (b) Simulated THz spectrum calculated as a product of calculated THz field emitted by DSTMS and the experimental GaP detector response function (red area). The blue area shows Fourier transform of the THz pulse measured by the GaP detector.
The good agreement between simulation and experiments in frequency down-shift and spectral broadening of transmitted optical spectra and emitted THz field demonstrate that the quasi-3D model can capture the essential physical processes of THz efficiency saturation in DSTMS. A similar model, which involved a delayed χ(3) process, well reproduces the spectral broadening of infrared pulses with higher fluence in DAST crystal as well [20].
5. Analysis of the THz conversion efficiency
To reveal the role of the cascaded OR and the 3PA effects in THz conversion efficiency saturation, we selectively switch-on, and switch-off various high-order nonlinear effects in the simulations. The THz conversion efficiency is calculated as the ratio between the intensities of the generated THz pulse and incident optical pump pulse:
The different cases are as follows: (i) all the nonlinear effects besides the OR are neglected; (ii) Only the cascaded OR effect is involved; (iii) Only the 3PA effect and the OR without cascading are considered; (iv) both cascaded OR and the 3PA effects are included. Material dispersion and linear absorption in the THz and NIR regimes are considered in all the cases. The simulated parameters of optical pump pulses and DSTMS are the same as in Section 2. Figure 4 shows the simulated THz conversion efficiency as a function of pump fluence at various cases mentioned above. For case (i), when only GVD and linear absorption are considered, this approximates as an undepleted pump condition due to low linear absorption (< 2 cm−1) and material dispersion (∼1177 fs2/mm) of DSTMS at a wavelength close to 1.4 ${\mathrm{\mu}}$m [16]. The THz conversion efficiency shows a linear dependence of pump fluence, which is expected for a 2nd order nonlinear effect and also agrees with theoretically predicted THz conversion efficiency via the OR process [21].
Fig. 4. Simulated THz conversion efficiency as a function of pump fluence dependence by switching on and off various high-order nonlinear effects. Material dispersion and linear absorption are considered for all the cases. The lines correspond to the fits according to Eq. (4). The inset shows simulated pump fluence dependence of THz conversion efficiency in the low fluence regime.
In case (ii), with the cascaded OR effect included, the THz conversion efficiency decreases and shows saturation at high fluence level, as circles in Fig. 4 indicate. Since DSTMS possesses high 2nd-order nonlinear susceptibility, the dispersive effects due to the GVD effect are dramatically enhanced by the spectral broadening of the optical spectra via a strong cascaded OR effect. Consequently, the peak power of optical pulses rapidly decreases when propagating within DSTMS, result in THz conversion efficiency reduction at a high fluence level. As the inset of Fig. 4 shows, the cascaded OR effect in combination with the GVD effect is still detrimental to THz generation even at a relatively low fluence level. This trend was also demonstrated in LiNbO3 based on a tilted pulse front scheme when pumped by intense fs pulse at 800 nm [14].
In case (iii), when only the 3PA effect is involved, a stronger THz conversion efficiency saturation was observed. The main impact of the 3PA is that it limits the highest pump fluence for THz generation, since the absorbed intensity of optical pump is proportional to ${{\rm{I}}^{\rm{3}}}{\rm{(\omega )}}$. For relatively low pump fluence, the 3PA effect only slightly degrades THz conversion efficiency, compared to the case (i), as shown in the inset of Fig. 4. It should be noticed that the photoexcitation free-carrier absorption (FCA) of the THz wave was not included in the simulation due to lack of the essential intrinsic material parameters (effective mass of electron meff, scattering time ${\rm{\tau }}$) of DSTMS [22]. Therefore, the 3PA effect would have a stronger effect on saturation of the THz conversion efficiency in the actual OR process.
In case (iv), with the cascaded OR and the 3PA effects, the THz conversion efficiency is reduced further and saturates at lower pump fluence, compared to case (iii).
To determine the saturated THz conversion efficiency and corresponding pump fluence for cases (ii-iv), the simulation was fitted by a phenomenological function,
Table 1. Fitted saturated pump fluence, THz efficiency and quantum conversion efficiency
6. Optimizing THz conversion efficiency by pre-chirping optical pump pulses
As demonstrated in other nonlinear crystal THz emitters, pumped by fs pulses sources with narrow duration at different repetition rates, the THz conversion efficiency can be enhanced by pre-chirping pump pulses [26–28]. Here, the THz conversion efficiency as a function of pre-chirp of optical pump was also investigated by our model to present instruction of optimizing THz generation vis OR in DSTMR. The thickness of DSTMS was assumed the same as in Section 2. The Gaussian pulses with three different durations (30 fs, 50 fs, 100 fs FWHM) at 1430 nm were used in the simulation. The pump fluence was 5 mJ/cm2 in all cases. As shown in Fig. 5, in the case of 30 fs pulses, an enhancement by a factor up to 1.5 was obtained for pumping with positively chirped pulses, whereas Fourier-transform limited pulses give the highest THz conversion efficiency when pumped by 100 fs pulses. This is attributed to the suppressed cascaded OR and 3PA effects in DSTMS when pumped by 100 fs pulses with a broader duration (the corresponding peak power decreases by about 3.3 times). Without pre-chirping, a higher THz conversion efficiency was achieved with 100 fs pulses than 30 fs pulses.
Fig. 5. Simulated THz conversion efficiency as a function of pre-chirp for Gaussian pulses with pulse duration of 30 fs (red dots), 50 fs (blue dots) and 100 fs (green dots) FWHM.
The mechanism of the asymmetric behavior is caused by the variation of pulse duration evolution in DSTMS for positively and negatively chirped pulses. Since the material dispersion of DSTMS (∼1177 fs2/mm) is positive at 1430 nm, negatively chirped pulses would become initially narrow within DSTMS, which results in enhanced cascaded OR and 3PA effects in the second half of DSTMS. However, the duration of positively chirped pulses monotonically broadens in DSTMS, suppressing the cascaded OR effect and the 3PA across the whole thickness of DSTMS. As Fig. 5 shows, the optimized negative chirp is almost equal to the total material dispersion of DSTMS (∼ 635 fs2), which means that the shortest duration would be obtained, without considering the high-order nonlinear effects. In addition, the asymmetric behavior has been demonstrated in other OR studies involving intense table-top fs laser sources [28]. These observations demonstrate that THz waves can be efficiently generated via OR in DSTMS, when pumped by positively chirped pulses with broad spectral bandwidth.
7. Conclusion
The saturation of THz generation via optical rectification in DSTMS crystal was systemically studied by a quasi-3D model, which simultaneously accounted for the strong nonlinear interaction between THz and optical field (cascaded OR), 3PA, GVD, and material absorption at THz and optical regimes. The simulation results reveal that the red-shift and spectral broadening of the center wavelength are a direct consequence of the cascaded OR effect, and correlated with the amount of THz generation, which consistently reproduced the measured transmitted optical spectra at different pump fluence levels, resulting in THz decreasing conversion efficiency due to the enhanced dispersive effect. The 3PA effect plays an important role in the limitation of THz generation from DSTMS, when pumped by intense fs pulses at wavelength near 1.4 ${\mathrm{\mu}}$m, as compared to the combination of the cascaded OR and the GVD effects. The good agreement with experimental results in transmitted optical spectra, THz spectra and pump-fluence dependent THz conversion efficiency, validates the developed quasi-3D theoretical model. In addition, the simulation results demonstrate that THz conversion efficiency can be further enhanced by chirp optimization, by suppressing the cascaded OR and the 3PA effects. We believe that these conclusions have important implications for the maximum conversion efficiency that can be achieved from optical rectification in organic nonlinear crystals and are crucial to the design of intense THz sources, pumped by fs pulse sources with extremely high intensities.
Funding
Foundation of President of China Academy of Engineering Physics (YZJJLX2018001); National Natural Science Foundation of China (No.11704358, No.12002326).
Acknowledgments
Jiang Li thanks China Scholarship Council (file no.201804890029) for financial supports.
Disclosures
The authors declare no conflicts of interest.
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