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We proposed a three-dimensional model to simulate terahertz generation from LiNbO3 crystal under intense laser excition (up to ~50 mJ/cm2). The impact of three-photon absorption, which leads to free carrier generation and free carrier saturation (when pump fluence above ~10 mJ/cm2) on terahertz generation was investigated. And further with this model, we stated the optimized experimental conditions (incident postion, beam diameter, and pulse duration, etc) for maximum generation efficiency in commonly-used tilted-pulse-front scheme. Red shift of spectrum, spatial distribution “splitting” effects of emitted THz beam, and primilary experimental verification under intense laser excitation are given.
© 2015 Optical Society of America
1. Introduction
Optical rectification (OR) of laser pulses has emerged as the most powerful way to generate high-peak-power terahertz (THz) pulses [1–5 ] and resulted in the highest THz pulse energy to date [3]. Within this category OR with tilted-pulse-front in LiNbO3 (LN) crystal is of particular interest due to its compatibility with pumping by widely available 800 nm and 1 μm laser sources. This approach had produced 0.4 mJ THz pulses by increasing the pump fluence up to 186 mJ/cm2 [6], and the optical-to-THz conversion efficiency at cryogenic temperature had exceeded 3.8% [7]. Such high energy THz pulses are attractive for many applications such as investigation of nonlinear charge dynamics [8], nonlinear interaction with nanostructured materials [9], molecular alignment [10], and high harmonic generation [11], etc.
Experimental studies in the literature have indicated that THz conversion efficiency with this approach goes saturated when pump fluence exceeds certain threshold [12,13 ]. The origin of this saturation is still under discussion and some assumptions (which however are quiet controversial) were proposed, such as self-phase modulation (SPM) [12], free-carried absorption (FCA) induced by multi-photon absorption [13], and cascading effects in conjunction with group velocity dispersion due to angular dispersion (GVD-AD) [14,15 ]. For example, in [12], SPM was proposed to be the principal reason for the saturation rather than FCA. While in [14] and [15], cascading effects in conjunction with GVD-AD was stated to be the stronger limitation than SPM for THz generation in LN crystal. However, their statement were carried out under 1-μm excitation wavelengths which are reasonable to exclude the impact of FCA (induced by weak four-photon absorption under 1-μm laser excitation), and the pump intensity was relatively low to avoid the influence of FCA.
While 800-nm laser are more commonly used, in this case, three-photon absorption (3PA) leads to considerable impact of FCA on THz generation. Recent experimental results depict a strong correlation between pump photon energy distributions on generated THz radiation, which is evidence of the detrimental role played by FCA arising from 3PA [16]. Also in [6], the change of THz power exponent with increasing pump energy in LN crystal was observed, which was suggested to be explained by the saturation of FCA at higher pump intensities, since similar behavior was observed in ZnTe crystal [17]. Some one-dimensional theoretical models were developed to analyze the effect of FCA in LN crystal [13,18 ]. And in a recently work [19], Fülöp et al modeled THz generation from LN and suggested FCA (which was not included in their model) may be the main factor which makes their simulation results different from experimental data. More detailed model considering FCA is required to optimize conversion efficiency and properties of THz pulse output from LN crystal under femtosecond laser excitation with much higher pump fluence.
In this paper, focusing on the impact of FCA induced by 3PA on the THz generation from LN crystal under intense 800-nm femtosecond laser excitation at room temperature, we proposed a three-dimensional (3-D) numerical model simultaneously accounting for gauss spatial and temporal distribution pump beam. Detailed model and parameters are shown in section 2. With the new model, the optimized parameters (pump fluence, beam position and diameter, pump pulse duration, etc) for tilted-pulse-front scheme experiment under intense laser (up to ~10 mJ/cm2) are given in section 3, which is quite different from that without FCA effect under low pump fluence. By further extending pump fluence over 10 mJ/cm2, we find free carrier absorption goes saturated and the THz generation increases again. Red shift of spectrum and beam “splitting” effects of the emitted THz beam under the 3PA effect are found, which along with preliminary experiement results are shown in section 4.
2. Modeling
We develop a three-dimensional model which simultaneously accounts for (i) the gauss spatial and temporal distribution of the pump beam, (ii) crystal geometry, (iii) the absorption at THz frequencies due to the complex dielectric function and to free carries generated by pump absorption, (iv) the variation of the pump pulse duration with the propagation distance due to material and angular dispersion, (v) phase mismatch.
We first modified the 1-D equation from [19] for the Fourier component ETHz of the THz field into a 3-D equation, which are further used to establish our 3D model for THz generation from LN crystal under intense laser excitation:
where (x, y, z) and (x’, y’, z’) are the pump and THz pulse coordinates, respectively (see Fig. 1 ), ε0 and c are the permittivity and light speed of free space, respectively, n(Ω) is the refractive index at the THz angular frequency Ω, PNL is the nonlinear polarization calculated by using a gauss spatial and temporal pump pulse electric field as discussed below, Δk is the wave-vector mismatch. The intensity absorption coefficient in THz range α(Ω) can be expressed as:
Fig. 1 Geometry of 3-D spatial model for THz generation and the coordinates used. γ is the required pulse front tilt angle, r0 is the pump beam waist radius, d is the distance between the center of the pump beam and the tip of the crystal, Lp = d tanγ is the pump beam propagation length.
The first term αε is determined by the complex dielectric function of the material, while the second term αfc, called FCA coefficient, is determined by free carries generated by pump absorption, and can be calculated using the Drude-model [19]:
where ε∞ is the high-frequency dielectric constant, τsc is the electron scattering time, and the plasma frequency ωp can be calculated by using the density of free carriers Nfc: and e and meff are the electron charge and effective mass, respectively. I is the time-averaged pump pulse intensity over pulse duration, λ 0 is the pump central wavelength, α 0, β 2, and γ 3 are the linear, two-, and three-photon absorption coefficients, respectively. Tz = T 0[(1 + Cβgz/T 0 2)2 + (βgz/T 0 2)2]1/2 is the pump pulse duration, where βg is group-velocity dispersion parameter which can be calculated as βg = -λ 0 2 D/(2πc), and D = λ/c[nλ(dε/dλ)2 -d 2 nλ/dλ 2] is the dispersion parameter, where nλ is the refractive index at pump wavelength, (dε/dλ) and d 2 nλ/dλ 2 are the the angular dispersion LN crystal dispersion, respectively [20]. C is the chirp parameter, T 0 = τ(1 + C 2)1/2 is the initial pump pulse duration, and τ is the Fourier transform-limited (TL) pulse duration.The nonlinear polarization PNL in Eq. (1) can be expressed as:
where, deff is the nonlinear optical coefficient for optical rectification (OR), ω is the optical pump frequencies, Ep is the pump pulse electric field. The pump pulse with gauss temporal distribution at z = 0 can be expressed as:In Eq. (7), S(x,y) is the spatial distribution coefficient of the pump beam, it can be expressed as S(x,y) = exp[-(x 2 + y 2)/r 0 2] for gauss spatial distribution.The Fourier component of pump pulse electric field with propagation coordinate z can be calculated as:
The substitution of Eq. (8) into Eq. (6) yields:
For calculating the spatial distribution and total energy of the THz pulse emitted from LN crystal, we decomposed the LN crystal (geometry is shown in Fig. 1) into tiny cells with equal sizes (dx’, dy’, dz’), and solved Eq. (1) numerically for each cell with each THz angular frequency. The output THz energy WTHz was obtained by summing up the electric field of THz pulse output from each cell of the LN crystal. And the final THz conversion efficiency is given by η = WTHz/Wp, where Wp is the total energy of the input pump beam.
Please note that in typical experiments with tilted pulse-front scheme, an optical imaging element is needed to tilt the pump pulse wave front and image the wave front into LN crystal. The imaging errors under those experiments, which has been revealed by Pálfalvi et al in [21], have influence on the THz generation efficiency. The imaging errors caused by imaging optics lead to relative group delays between different wavelengths, and change the pump pulse duration and phase matching condition, resulting decrease of the THz generation efficiency. As demonstrated in [21], with a two-lens telescope optics imaging system we are able to reduce imaging errors and further decrease this bad effect, and also with the contact grating scheme there is no imaging optics and thus the influence of imaging errors is elimilatated. Avoiding image errors can increase THz generation efficiency in experiments. Hence, for simplicity and to focus on the impact of FCA on THz generation under intense laser excitation, we didn’t included the image errors in our model and nor in the follow simulations.
3. Results
3.1 Results of optimized parameters for tilted-pulse-front scheme
We first investigate the influence of different factors on the THz generation with the effect of 3PA with pump fluence 0 - 10 mJ/cm2 at room temperature.
With the new 3D model described above, we carried out calculation to analyze the influence of 3PA on THz generation from LN crystal, and further stated the optimized experimental conditions for maximum generation efficiency. The coefficients of laser and materials used in our simulation are lists in Table 1 , which are experimental data taken from published literatures. Please note that those data (especially material’s parameters) are variable according to the LN crystal used in experiments, and one can use our proposed 3D model and simply adjust the coefficients used in their experiment to interpret their experiment observations.
Table 1. Coefficients of laser and materials used in our simulation.
In order to address the influence of FCA under intense laser excitation on THz generation, with Eqs. (3)-(5) we quantitatively show in Fig. 2 free carrier density Nfc and FCA coefficient αfc under several typical experiment situations. Femtosecond laser with Fourier TL pulse duration τ of 50 fs [27], 120 fs [12], 200 fs [12], 500 fs [28], and 700 fs [28] are the most commonly used laser excitation source for pulsed THz generation.
Fig. 2 (a) Free carrier density Nfc as a function of pump fluence, and (b) FCA coefficient αfc under 200-fs laser excitation (as an example) as a function of THz frequency under different pump fluence (FCA coefficient under the rest pump fluence and pulse duration used in our simulation are able to be calculated with Eq. (5)).
With quantitatively derived relation between FCA and laser excitation, we get the THz conversion efficiency (shown in Fig. 3 ) as a function of pump energy (fluence) for several typical experiment situation (different Fourier TL pump pulse duration τ and incident position d with a constant pump beam waist radius of 3 mm). As can be seen, the THz conversion efficiency saturates and then decreases when pump fluence goes higher than a threshold. And interesting, the threshold varies with the incident position and pump pulse duration. This discovery has been confirmed by a previous experiment [12].
It’s simple to understand the variation of optimal pump energy (with maximum THz generation efficiency) under different incident positions and with differennt pulse durations. When incident position is increased (from 1.5 mm to 4.5 mm in Fig. 3), pump beam propagation length Lp (shown in Fig. 1) increase proportionally, which leads to excitation pump pulse Tz dispersion (longer), time-averaged pump pulse intensity over pulse duration I decreasing, and consequently free carrier density induced by 3PA decreasing. The change of pump pulse duration results in similar effect on free carrier generation. Additionally, the pump beam waist radius will also affect the pump beam intensity and the free carrier generation.
This discover means to keep THz generation with maximum efficiency we need to investigate the THz conversion efficiency under different experimental conditions, and find out the optimal parameters for maximum generation efficiency to guide futher experiments. We found that the THz conversion efficiency η = f(τ,r0,d,Wp) is a function of pump pulse duration τ, pump beam waist radius r 0, incident position d, and pump energy Wp. With our proposed 3D model, we revealed the optimal parameters with maximum THz consersion dfficiency for several experiment situations (pump pulse duration τ, pump beam waist radius r 0, incident position d), which are shown in Fig. 4 .
Fig. 4 (a) Optimal incident position d (blue solid lines) and pump fluence threshold (PFT) (red dash lines) for the peak THz conversion efficiency (in Fig. 3) under different pump beam waist radius r 0, and (b) the corresponding peak THz conversion efficiency as a function of pump beam waist radius r 0 with optimal d and PFT.
Several conclusions can made from Fig. 4(a): (i) the optimal incident position d of pump beam is nearly proportional to pump beam waist radius r 0, and increases with pump pulse duration increasing; (ii) the pump fluence threshold (optimal pump energy with maximum THz generation efficiency) is nearly inversely proportional to pump beam waist radius r 0, and keep constant of ~3 mJ/cm2 when pump beam waist radius r 0 goes increasing. And from Fig. 4(b), we conclude that (iii) under optimal experimental setup shown in Fig. 4(a), there is an optimal waist radius for maximum THz conversion and 500-fs excitation leads to the highest conversion (~2.5 × 10−3) among the four typical lasers, (iv) and this optimal radius increases with pump pulse duration increasing. For guiding the experiement for higher THz generation under intense laser, to avoid damage to optical elements (grating, mirror, etc) and LN crystal [29], excitation laser with longer Fouier TL pulse duration and larger pump beam is preferred.
We further investigate the maximum THz conversion efficiency η(τ) = max[f(τ,r0,d,Wp)] as a function of pump pulse duration. In Fig. 5 shows the optimal pump beam waist radius and its corresponding maximum THz conversion efficiency as a function of pump pulse duration. As can be seen that the optimal pump beam waist radius is nearly proportional to pump pulse duration, and the optimal pump pulse duration is about 500 fs. The maximal THz conversion efficiency is about ~2.5 × 10−3, which are consistent with previous experiment results [13].
Fig. 5 Optimal pump beam waist radius r 0 (blue line) and corresponding maximal THz conversion efficiency (red line) as a function of pump pulse duration.
3.2 Influence of free carrier saturation
In this subsection, we’ll reveal the influence of saturation of free carrier generation under higher intense laser (from 10 mJ/cm2 to 50 mJ/cm2) on THz generation. Free carrier saturation and its impact on THz generation has been experimentally observed and modeled in ZnTe crystal [17]. As demonstrated previously, FCA reduces THz conversion efficiency, while as shown in [17], by further increasing pump fluence, saturation of FCA occours and consequenctly leading to THz conversion efficiency increasing. Although never modeled in LN crystal, such relation has been observed in [6].
First, we estimate the saturation concentration of free carriers in LN crystal to be ~4 × 1020m−3 by using our model with the parameters from [6] (Please note, the accurate number of carriers density for saturation is adjustable in our model according to the LN crystal used in experiment). With Eq. (3), we’re able to reveal the relation between free carrier density and pump fluence (up to 50 mJ/cm2).
As an example, we only investigated the influence of free carrier saturation when pumping with 800-nm laser and pulse duration of 700-fs. The experiments under laser system with different paramenters has similar behavior and one can get results with our model. Free carrier density as a function of pump fluence with saturation effect is shown in Fig. 6(a) . In Fig. 6(b) (solide lines), we show the dependence of THz generation with pump fluence under free carrier saturation effect. There’re three regimes. In the 1st regime (under pump fluence lower than ~3 mJ/cm2), emitted THz energy increases with increasing pump fluence; in the 2nd regime (under pump fluence from ~3 mJ/cm2 to ~10 mJ/cm2), emitted THz energy decreases with increasing pump fluence; while in the 3rd regime (under pump fluence higher than ~10 mJ/cm2), emitted THz energy goes increasing again with increasing pump fluence. In all of the 3 regimes, THz generation efficiency increases with increasing pump fluence. The behavior in the 2nd regime is due to THz absorption by free carrier, while the havior in the 3rd regime is due to saturation of free carrier generation when further increasing pump fluence.
Fig. 6 (a) Free carrier density as a function of pump fluence with saturation effect, insert: conrresponding FCA coefficient, and (b) THz conversion efficiency and corresponding THz pulse energy as a function of pump fluence: dash lines—simulation results without consideration of free carrier saturation, solid lines— simulation results with consideration of free carrier saturation.
Such relationship between THz generation and excitation energy has been observed and well model in THz generation from ZnTe under intense laser excitation [17]. For comparison, we also show the relation between THz generation with pump fluence without consideration of saturation of free carrier generation. Under the effect of saturation of free carrier generation, we expect mJ-level THz pulses generation by further increasing the pump pulses energy.
4. Spectrum, beam properties and preliminary experimental verification
For many applications, such as THz time-domain spectroscopy and THz imaging, the THz beam properties, such as spectrum, spatial intensity distribution are very important. With our proposed model, we simulated the emitted THz pulse spectrum under intense laser excitation with the effect of FCA induced by 3PA.
Figure 7 shows the spectrum of emitted THz pulse under several pump fluence with pump pulse of 200 fs. We find that, with pump fluence increasing, the high-frequency component in the emitted THz spectrum is suppressed and the frequency of peak intensity in the emitted THz spectrum shifts to lower frequency (red-shift). Similar results had been observed in ZnTe under intense laser excitation [12]. It’s smiple to understand this behavior with our model, the free carrier density increases with the pump fluence increasing and leads to strong FCA. according to the Drude behavior of FCA, THz aborption by free carrier is differenct among frequencies which was shown previously in Fig. 2(a). For a given pump fluence, there is a absorption peak in absorption (αfc) spectrum, and absorption peak goes blue-shift (higher frequency) with pump fluence increasing. Hence, under higher excitation pump fluence, higher-frequency component in the emitted THz spectrum is absorbed and results in lower-frequency THz emission.
And then, we simulated the spatial intensity distribution of THz pulses emitted from the LN crystal with pump fluence lower and higher than the threshold (the threshold is defined as the pump fluence with maximum THz conversion efficieny in Fig. 3). Figures 8(a) and 8(b) show the simulated spatial intensity distribution of emitted THz beam under 50-fs 800-nm excitation pump fluence of 1.3 mJ/cm2 and 13 mJ/cm2 with pump beam radius of 5mm, respectively. We find that the emitted THz intensity distribution spreads with the pump fluence increasing, and “splits” into two parts. For comparison, we conducted similar experiment under laser system with center wavelength of ~800 nm and pulse duration of ~50 fs. The experimental observation under pump fluence of 2 mJ/cm2 and 16 mJ/cm2 and pump beam radius of 5mm, which were recorded by a microbolometer THz camera, are shown in Figs. 8(c) and 8(d), respectively.
Fig. 8 Simulation and experimental observation of the emitted THz beam spot. x’ and y’ correspond to coordinates in Fig. 1. (a) simulation with pump fluence of 1.3 mJ/cm2, (b) simulation with pump fluence of 13 mJ/cm2, (c) experiment with pump fluence of 2 mJ/cm2, (d) experiment with pump fluence of 16mJ/cm2.
It’s simple to understand the beam splitting effect. From Fig. 3 we conclude that THz generation decreases when pump fluence exceeds the threshold due to FCA. In our simulation and experiment, excitation pump with gauss spatial distribution is used, which means excition pump fluence in the pump beam center is much stronger than that in the pump beam edge. It consequently results in stronger FCA in center of pump beam and weaker THz generation in the center of emitted THz beam.
5. Conclusion
In conclusion, a three-dimensional model was used to investigate the influence of FCA induced by 3PA on terahertz generation from LiNbO3 crystal with tilted-pulse-front under intense 800-nm femtosecond laser excitation. This model simultaneously accounts for (i) the gauss spatial and temporal distribution of the pump beam, (ii) crystal geometry, (iii) the variation of the pump pulse duration with the propagation distance due to material and angular dispersion, (iv) the absorption at THz frequencies due to the complex dielectric function and to free carries generated by pump absorption, (v) phase mismatch. The optimization of tilted-pulse-front scheme with the effect of 3PA which is highly dependent on optical pump conditions was given, which includes pump flucence, beam diameter and postion, pump duration. The optimum pump pulse duration is about 500 fs, and larger pump beam can be adopted when using longer pump pulse duration for higher efficiency. By further increase pump flucenc, we found that FCA goes saturated. And the variations of THz conversion efficiency and pulse energy versus pump pulse energy under saturation effect of FCA were also be discussed, and our results reveal that the THz pulse energy increase with the pump energy higher than 10 mJ/cm2, mJ-level THz pulses generation is achievable under tilted-pulse-front scheme. The red-shift of THz spectrum and intensity distribution “splitting” induced by 3PA under intense laser was also predicted and experimentally verified in this work.
Acknowledgments
The authors would like to thank the sponsors of this work— National Key Basic Research Program of China (No. 2015CB755405), the National Natural Science Foundation of China (Nos. 61205100 and 61427814), and Foundation of President of China Academy of Engineering Physics (No. 201501033).
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