Optical rectification of ultrashort laser pulses in LiNbO3 by tilted-pulse-front excitation is a powerful way to generate near single-cycle terahertz (THz) pulses. Calculations were carried out to optimize the output THz peak electric field strength. The results predict peak electric field strengths on the MV/cm level in the 0.3–1.5 THz frequency range by using optimal pump pulse duration of about 500 fs, optimal crystal length, and cryogenic temperatures for reducing THz absorption in LiNbO3. The THz electric field strength can be increased further to tens of MV/cm by focusing. Using optimal conditions together with the contact grating technique THz pulses with 100 MV/cm focused electric field strength and energies on the tens-of-mJ scale are feasible.
©2011 Optical Society of America
Optical rectification (OR) of femtosecond laser pulses is an efficient method to generate ultrashort THz pulses. The highest THz pulse energies and field strengths in the few-THz frequency range were achieved by using LiNbO3 (LN) as the nonlinear medium having very large effective nonlinear coefficient [1–4]. This requires tilting the pump pulse front relative to its wavefront in order to satisfy the phase matching condition required for efficient THz generation . Pumped by multi-mJ Ti:sapphire lasers this technique allowed to generate ultrashort THz pulses on the 10-µJ energy, and up to the 1-MV/cm electric field strength scale [1,6]. Such THz pulses enabled, for example, the investigation of THz-induced nonlinear optical phenomena directly in the time-domain  as well as time-resolved studies of ultrafast carrier dynamics in semiconductors by THz pump—THz probe measurements [8–10]. Scaling the THz energy up to 50 µJ has been recently demonstrated by using LN and the tilted-pulse-front pumping (TPFP) technique driven by medium-scale Ti:sapphire amplifier systems with pulse energies up to 120 mJ [2,4]. Ultrashort pulses at higher frequencies (several tens of THz, where the conversion efficiency can be higher) with electric field strengths up to 10 MV/cm and 100 MV/cm could be achieved by optical parametric amplification  and difference-frequency generation  in GaSe, respectively. However, due to the material properties of GaSe, this technique is not easily scalable.
Ultrashort THz pulses with energies in the millijoule range and electric field strengths at 100-MV/cm levels, exceeding by far what is presently available, are required by promising new applications. These include investigation of material properties and processes under the influence of extremely high quasi-static fields, particle acceleration by electromagnetic waves, and THz-assisted attosecond pulse generation [13,14]. Some of these applications will require large-scale ultrashort-pulse laser facilities, such as ELI , as the pump source.
A conventional TPFP setup consists of a femtosecond pump laser, an optical grating, an imaging lens or telescope, and the nonlinear material [1–3]. The disadvantage of this setup is that beam distortions caused by imaging errors limit the useful pump spot size [16,17] and, consequently, the THz energy. A simpler scheme was proposed by omitting the imaging optics and bringing the grating directly in contact with the crystal . The advantage of such a contact-grating setup is that, by eliminating imaging errors, larger pumped areas can be efficiently used resulting in higher THz energies and better beam quality.
In this work we show that OR of femtosecond pulses in LN by using the TPFP technique with a contact grating is a promising candidate to reach the high THz field strengths and pulse energies required by the applications mentioned above. The effect of varying the Fourier-limited (FL) pump pulse duration and the crystal temperature (to minimize its THz absorption) will be investigated in detail by numerical calculations and optimal conditions will be given. It will be shown that using FL pump pulse durations longer than the commonly used ~100 fs can allow for larger pump-to-THz energy conversion efficiencies by utilizing longer crystal lengths for THz generation. This is in contrast to simply pre-chirping a short FL pump pulse , which merely allows to achieve the shortest average pulse duration inside the nonlinear crystal.
2. Theoretical model
The applied theoretical model, described in detail in , takes into account (i) the variation of the pump pulse duration (and therefore of the pump intensity) with the propagation distance due to material and angular dispersions [17–19], (ii) the non-collinear propagation of pump and THz beams and (iii) the absorption in the THz range due to the complex dielectric function (determined by phonon resonances). All kinds of nonlinear effects but OR were neglected . The wave equation with the nonlinear polarization was solved in the spectral domain [17,20]. The temporal shape of the THz field was obtained by Fourier transformation.
In the calculations Gaussian pump pulses with a peak intensity of 40 GW/cm2 were used, which is about half of the intensity where the onset of free-carrier absorption was experimentally observed in LN . This is a practical choice, giving close-to-maximum THz yield, since free-carrier absorption limits the useful pump intensity . For the pump wavelength 1064 nm was chosen (the exact choice does not significantly influence the results). The FL pump pulse duration was varied between 50 fs and 1 ps. The pump pulse duration changes as it propagates through the LN crystal  due to the angular dispersion of its spectral components, related to the pulse front tilt . This effect is more pronounced in case of shorter pulse durations. In order to ensure the shortest possible average pump pulse duration within the crystal, and hence the highest pump intensity, the pump pulses were assumed to be pre-chirped such that the FL pulse duration values are reached at the center of the crystal (see Fig. 2 of ). The assumed 40 GW/cm2 peak intensity and a constant pump beam diameter implied a corresponding increase of the pump pulse energy with increasing pulse duration. At each value of the pump pulse duration the electric field strength of the output THz radiation was maximized by choosing optimal crystal length and phase matching THz frequency. The phase matching frequency was iteratively fitted to the central frequency of the generated THz spectrum. For the shortest pump pulse durations the crystal length was set to give maximal THz peak electric field strength at the crystal output. For longer pump pulses, where this choice would have resulted in crystal lengths exceeding 10 mm, the crystal length was set to 10 mm. Due to the large pulse-front-tilt angle (63°), and the associated walk-off effect, longer crystals are impractical even with cm-scale pump beam diameters. The crystal lengths used in the calculations are listed in Table 1 . Throughout the paper, no imaging distortions in the pulse-front-tilting setup are included in the model. Therefore it corresponds either to a pulse-front-tilting setup with optimized imaging conditions (up to about 1 cm pump spot size)  or to a contact-grating device .
In order to avoid photorefraction  and reduce THz absorption  stoichiometric LN (sLN) doped with 0.7 mol% Mg was assumed as the nonlinear medium. LN has significant absorption in the THz range at room temperature (Fig. 1 ). Since decreasing the temperature decreases absorption in the THz range , low temperature cases (100 K and 10 K) were also investigated in the calculations. Since the refractive index of sLN in the THz range is large (n ≈5.0 at 1 THz), the Fresnel-loss at the output surface of the crystal is significant (about 45%). It was also taken into account in the calculations.
3. Results and discussion
3.1. Optimization for the electric field strength
The calculated THz spectra belonging to different pump pulse durations at 100 K are shown in Fig. 2(a) . As it is obvious from Fig. 2(a), the peak spectral intensity increases and the position of the spectral intensity peak shifts towards lower frequencies with increasing pump pulse duration. This behavior can be observed for all temperatures, as it is shown for the central frequencies in Fig. 2(b). The central THz frequency varies between 1.5 and 0.27 THz for the investigated pump pulse duration range.
In order to calculate the peak electric field strength of the THz pulses their temporal shapes were calculated from the spectra by Fourier transformation. Examples are shown in Fig. 3(a) . The calculated peak electric field strength of the THz pulses in air immediately after the output surface of the crystal is shown in Fig. 3(b) for different temperatures as a function of the FL pump pulse duration (τ). The encircled cross indicates the experimental conditions belonging to the values of 100 fs and 300 K. Similar parameters were used in many experiments pumped by amplified Ti:sapphire laser systems [1,2,24,25]. For such experimental conditions our calculations give 240 kV/cm for the peak of the electric field strength (Fig. 3(b)). This exceeds only by about a factor of two the value of 110 kV/cm at the output of the crystal obtained experimentally from the measured peak THz intensity and output spot size . Possible reasons for the difference are the shorter than 100 fs FL pump pulse duration in experiments and imaging errors in the pulse-front-tilting setup . The calculated value for the peak of the THz spectrum is 1.1 THz (Fig. 2(b)), which is similar to the values observed in experiments. This approximate agreement indicates that the present calculation method gives realistic predictions for the order of magnitude of the THz output in real experiments.Increasing the pump pulse duration from the commonly used 100 fs results in significant increase in the THz peak electric field. As shown in Fig. 3(b), by choosing the optimal pump pulse duration of 600 fs for room temperature the THz peak electric field strength can be increased by a factor of more than four, resulting in the extremely high value of 1.0 MV/cm at the output of the crystal. The position of the corresponding spectral peak is reduced to 0.4 THz (Fig. 2(b)). The reason of the increase in field strength is twofold: (i) The longer pump pulse causes a shift of the THz spectrum to lower frequencies, which results in reduced absorption within the crystal (α = 5.9 cm−1 at 0.4 THz instead of α = 18 cm−1 at 1.1 THz, see Fig. 1). (ii) The longer pump pulse also allows a longer THz generation length  (Table 1).
Even higher electric field strength can be reached at lower temperatures. At temperatures of 100 and 10 K the maxima are located at 500 fs (Fig. 3(b)) with field strength values of 2.3 MV/cm and 2.8 MV/cm, respectively, corresponding to an enhancement of about one order of magnitude as compared to 300 K and 100 fs. The reason for this increase is clearly the reduced THz absorption at cryogenic temperatures (Fig. 1). In cases of optimal pump pulse duration the central THz frequency has a value of 0.40 THz at 300 K, 0.64 THz at 100 K and 0.67 THz at 10 K temperatures as it is shown in Fig. 2(b).
Figure 3(a) shows the time-dependent electric field for the optimal pump pulse durations at 300 K and 10 K temperatures. For comparison, the THz pulse shape is also shown for 100 fs and 300 K closest to recent experimental parameters, as mentioned above. It can be seen that a 12-fold increase in the peak electric field strength can be reached by cooling down the crystal to 10 K temperature and using optimal (500 fs) pump pulses, as compared to 100 fs and 300 K.
3.2. Extremely high energies and field strengths from large-area sources
The calculated optical-to-THz energy conversion efficiencies are shown in Fig. 4(a) versus the FL pump pulse duration. The behavior is very similar to the electric field results shown in Fig. 3(b), but the maxima are slightly shifted towards shorter pulse durations. For example, the maxima are located at 400 fs for 10 and 100 K temperatures (instead of 500 fs, as in Fig. 3(b)). As expected, the difference between the efficiency curves belonging to different temperatures are even more pronounced than between the electric field curves of Fig. 3(b). At room temperature the calculated efficiency increases from 0.31% for 100 fs to 2.0% for 500 fs. Cooling the crystal to 100 K gives 8.9% efficiency for 400 fs, while at 10 K the efficiency gets as high as almost 13% for 400 fs.
The TPFP technique is scalable to higher THz energies by increasing the pump spot size and energy. In order to fully exploit the scalability of the TPFP technique to extremely high THz pulse energies and field strengths it will be necessary to use a very large interaction area. This will be enabled by the contact-grating technique , which introduces no imaging errors, and can be pumped by high-energy laser pulses.
Our calculations indicate that it is feasible to scale the THz pulse energy to the tens-of-mJ level by using the contact-grating scheme with LN. Figure 4(b) shows the calculated THz pulse energies obtained by using the efficiencies given in Fig. 4(a) and a pump beam diameter of 5 cm, which is a feasible beam size for use with the contact grating. According to Fig. 4(b), an output THz energy as high as 23 mJ can be obtained by pumping a 10-mm thick LN crystal in a contact-grating setup with pulses of 500 fs duration, 40 GW/cm2 peak intensity, and 5 cm beam diameter (about 200 mJ pump pulse energy) at 10 K temperature. As compared to present experimental status  this corresponds to an increase of more than 400 times in THz pulse energy, which will open up the field for various new applications. The THz electric field strength at the output of the crystal is 2.8 MV/cm in this case, as shown in Fig. 3(b).
A way to further increase the electric field strength of the THz output is the use of optimized THz imaging optics behind the crystal. For example, by using an optical system consisting of two parabolic mirrors with focal lengths of 50 cm and 8 cm, and typical commercially available aperture sizes, the electric field strength can be scaled to the 10 MV/cm level without any significant frequency cut-off. In this case we have assumed 1 cm THz beam diameter (intensity 1/e2-value) at the output of the LN crystal, which yields a THz beam diameter of 1.6 mm in the image plane. The contact-grating setup will allow to use very large pump beam cross sections with very good output THz beam focusability. Assuming an output THz beam diameter of 5 cm, and using a single parabolic mirror with focal length of 5 cm for focusing, a spot size of 0.57 mm can be reached in the focal plane at the central frequency of 0.67 THz (Fig. 2(b)). Even though in this setup the shape of the THz signal will be distorted because of diffraction , a peak electric field strength of about 100 MV/cm can be reached.
Numerical calculations were performed in order to maximize the electric field strength of THz pulses generated by tilted-pulse-front excitation in LN, motivated by various applications. It was shown that the FL pump pulse duration is a key experimental parameter in the THz generating process. According to the calculations the THz peak electric field strength can be increased by more than a factor of four to the MV/cm level directly at the crystal output by using 600 fs pump pulses instead of the commonly used 100 fs. The importance of the absorption of LN in the THz range was also discussed. The calculations predict about one order of magnitude increase in the THz peak electric field strength when the crystal is cooled to 10 K (thereby reducing its absorption) and 500 fs pump pulses are used, as compared to 100 fs pumping at room temperature. The electric field strength can easily be increased to the 10 MV/cm level by imaging. Using such optimized pump pulse duration, and cooling the LN crystal, in combination with the contact-grating technique will allow to generate THz pulses with focused peak electric field strength on the 100 MV/cm level and tens-of-mJ energy driven by efficient sub-joule class diode-pumped solid-state lasers.
The extremely high pump-to-THz energy efficiency values predicted by the calculations correspond to pump-to-THz photon conversion efficiencies exceeding 100%. We note that internal photon conversion efficiencies well above 100% are caused by cascade effects  and were indicated by recent experiments [1,2]. We note that the influence of the THz field on the pump pulse, which can cause an increase of the THz generation efficiency [28,29], can be significant in case of the very large efficiency values at optimal pump pulse durations at cryogenic temperatures. A more accurate numerical study requires to take these effects into account. We also note that at very high THz fields the effective nonlinearity of LN might change (future experiments should clarify the sign and magnitude of this behavior), which can lead to a significant decrease or increase of the THz generation efficiency. In summary, three main factors are contributing to the predicted increase in the THz yield: (i) longer FL pump pulses, (ii) cooling of the LN crystal, and (iii) large pump spot size and energy, where the latter is not influenced by the uncertainties in the model.
Our preliminary experiments (with a prototype contact grating of sinusoidal profile fabricated on ZnTe by laser ablation and pumped at 1.7 μm wavelength) indicate the working of the contact-grating scheme. Further work is needed to optimize the diffraction efficiency of the grating. Very high diffraction efficiencies can be achieved by using binary gratings [30,31] instead of sinusoidal gratings. Furthermore, our very recent experiments with 1.5 ps, 50 mJ pump pulses delivered about 125 μJ THz pulse energies (0.25% efficiency) at room temperature. This result supports the expectation of a significantly increased THz output for optimal pump pulse duration, as forecast by the calculations.
The predicted development of highly optimized sources for ultra-intense THz pulses will open up new applications in THz-assisted attosecond pulse generation, as well as in particle acceleration and manipulation by electromagnetic waves.
The authors acknowledge the fabrication of the grating structure on ZnTe to Cs. Vass and B. Hopp from Department of Optics and Quantum Electronics, University of Szeged. Financial support from Hungarian Scientific Research Fund (OTKA), grant numbers 76101 and 78262, and from “Science, Please! Research Team on Innovation” (SROP-4.2.2/08/1/2008-0011) is acknowledged.
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