1. Introduction

Strong terahertz pulses with electric and magnetic fields exceeding 1 MV/cm and 0.3 T, respectively, are of great interest nowadays for a variety of applications including compact electron acceleration [1–3], post acceleration of laser-driven ions [4], molecular orientation and alignment [5–7], ultrafast control of magnetic order in matter [8, 9], nonlinear terahertz spectroscopy [10], and terahertz streaking techniques [11, 12]. Presently, the most efficient table-top sources of pulsed terahertz radiation are based on optical rectification (OR) of femtosecond laser pulses in crystals with high second-order nonlinearity, such as lithium niobate (LN) and organic crystals (DAST, DSTMS, OH1, HMQ-TMS).

Record high strengths of (focused) terahertz fields have been obtained with organic crystals: 1 MV/cm (0.3 T) with DAST [13], 6 MV/cm (2 T) [14] and 42 MV/cm (14 T) [15] with large-size partitioned DSTMS, 62 MV/cm (20.7 T) with OH1 and 83 MV/cm (27.7 T) with DSTMS [16]. The spectrum of near single-cycle terahertz pulses generated in organic crystals is typically centered in the 2-10 THz range. However, for such applications, as particle acceleration, molecular orientation, and terahertz streaking techniques terahertz pulses with a lower central frequency are optimal.

The frequency range below 2 THz is better accessed with another nonlinear material, LN. To overcome a large velocity mismatch between optical and terahertz waves in LN, pump pulses with a tilted intensity front are used for noncollinear velocity matching [17]. Presently, 1 MV/cm (focused) electric field strength [18] is the highest reported value achieved by this method on the sub-THz range.

Two-, or more generally, multiphoton absorption of the pump is recognized as an essential detrimental factor for terahertz generation at high optical intensities. Multiphoton absorption leads not only to the depletion of the pump pulse but also to the generation of free carriers that absorb terahertz waves [19–21]. In particular, free carrier generation (FCG) is considered as the main effect that limits optical-to-terahertz conversion efficiency of OR in semiconductor materials, such as ZnTe pumped by a Ti:sapphire laser (with the central wavelength $\lambda \approx 0.8\mu$m) [19, 20, 22, 23]. The detrimental effect of FCG on the tilted-pulse-front terahertz generation in LN was demonstrated experimentally [24] and simulated numerically [25]. Suppression of FCG by using pump wavelengths beyond the two-photon or even three-photon absorption edge was proposed [26–28] and demonstrated [29–31] as a potential way for efficient terahertz generation both in semiconductors and in LN.

Recently, however, it was shown that FCG can give rise to an unexpected positive effect, namely, the generation of quasistatic precursors propagating ahead of the pump laser pulses [32]. The electric and magnetic fields in a precursor can be comparable to or even exceed the fields in the main terahertz pulse copropagating the pump pulse. The precursor is generated by a surge of an electric current, which is produced by optically created carriers. The mechanism of the effect is somewhat similar to the terahertz generation by photoconductive antennas, where the optically created carriers are accelerated by an externally applied electric field. In the case of OR, however, the carriers are driven by the intrinsic electric field copropagating the nonlinear polarization induced by the laser pulse in the crystal. Moreover, in photoconductive antennas terahertz radiation is emitted by an immobile point-like time-varying dipole, whereas the precursors are generated by a moving current plane.

In the standard scheme of OR with collinear propagation of the optical and terahertz pulses, the quasistatic precursors can be generated only in (subluminal) materials, such as semiconductors ZnTe and GaP pumped by Ti:sapphire laser, with $n_g>\sqrt{{\epsilon }_0}$, where n_g is the optical group refractive index and ε₀ is the low-frequency dielectric permittivity [32]. Physically, condition $n_g>\sqrt{{\epsilon }_0}$ means that the low frequency waves forming the precursor propagate faster than the pump optical pulse. In ferroelectric materials, such as LN, which are advantageous over semiconductors due to higher values of the nonlinear coefficient and optical damage threshold, n_g is more than two times smaller than $\sqrt{{\epsilon }_0}$. To extend the precursor generation to the materials with $n_g<\sqrt{{\epsilon }_0}$, we propose to use pumping by tilted-pulse-front laser pulses. Indeed, pumping by tilted-pulse-front pulses of a sufficiently large transverse size is equivalent to pumping by ordinary nontilted pulses but in a virtual medium with an effective optical group refractive index $n_g^{\text{eff}}=n_g/\cos \alpha$, where α is the tilt angle [33]. By choosing $\alpha >\stackrel{-1}{\cos }(n_g/\sqrt{{\epsilon }_0})$, condition $n_g^{\text{eff}}>\sqrt{{\epsilon }_0}$ can be fulfilled.

In general, tilted-pulse-front excitation of quasistatic precursors in materials with $n_g<\sqrt{{\epsilon }_0}$, such as LN pumped at $\lambda \approx 0.8$-1 μm or ZnTe pumped at $\lambda >0.84\mu$m, can be more versatile as compared to the standard collinear excitation in ZnTe and GaP at $\lambda \approx 0.7$-0.8 μm [32]. First of all, pumping at the wavelengths beyond the two- or three-photon absorption edge allows one to increase the optical pump intensity and, therefore, the precursor’s magnitude avoiding, at the same time, excessive pump depletion. Additionally, varying the tilt angle allows to change parameter $n_g^{\text{eff}}$. This makes it possible to control the duration and magnitude of the precursor.

In the present paper, we explore potentials of tilted-pulse-front excitation of strong quasistatic precursors in three practically interesting cases, namely, LN pumped at $\lambda =0.8\mu$m and 1.05 μm and ZnTe pumped at $\lambda =1.7\mu$m. In particular, we demonstrate that precursors with at least two orders of magnitude stronger fields, as compared to the collinear geometry [32, 34], can be generated in these cases.

2. Model and approach

We assume that a tilted-pulse-front laser pulse propagates in an electro-optic crystal of thickness L_c in the normal to the phase fronts direction with group velocity $c/n_g$ [Fig. 1(a)]. The intensity front of the pulse (pulse front) is tilted at angle α to the phase fronts and parallel to the crystal boundaries. The projection of the group velocity on the z direction perpendicular to the pulse front is $V=c/n_g^{\text{eff}}=\left(c/n_g\right)\cos \alpha$. Figure 1(b) shows the dependence $n_g^{\text{eff}}\left(\alpha \right)$ for LN and ZnTe.

 

Fig. 1 (a) Geometry of the problem. A high-intensity tilted-pulse-front laser pulse propagates in an electro-optic (EO) crystal and generates free-carriers due to multiphoton absorption. (b) The effective optical group refractive index $n_g^{\text{eff}}$ as a function of the tilt angle α for LN pumped at $\lambda =0.8\mu$m and 1.05 μm and ZnTe pumped at $\lambda =1.7\mu$m. Horizontal lines depict $\sqrt{{\epsilon }_0}$ for the materials.

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The configuration with the pump pulse front parallel to the crystal boundaries is known to be advantageous for the generation of terahertz beams of a high quality, i.e., with a homogeneous field distribution across the beam [28]. The same conclusion holds evidently valid for the generation of quasistatic precursors as well. In ZnTe pumped at 1.7 μm, the configuration in Fig. 1(a) was recently implemented by engraving a grating structure on the entrance surface of a ZnTe crystal [30]. In LN, with a high ε₀, an implementation of such configuration (the contact grating scheme [35]) is much more difficult technologically [36]. The problem can be mitigated by using a combination of the conventional tilted-pulse-front setup and the contact grating scheme [37]. Another possibility is using a stair-step echelon structure on the entrance surface of the crystal [38, 39].

We use a one-dimensional approximation, which works rather well for large-aperture laser pulses utilized in tilted-pulse-front schemes [28]. The optical intensity $I\left(z,t\right)$ in the crystal is assumed high enough to produce n-photon ionization with the density of free carriers [40]

By assuming the Gaussian shape of the laser pulse at the crystal boundary, i.e., $I\left(0,t\right)=I_{0}f\left(t\right)$ with $f\left(t\right)=\exp (-t^2/{\tau }^2)$, we write the intensity at an arbitrary point z inside the crystal ($0<z<L_c$) as $I\left(z,\xi \right)=I_{0}f\left(\xi \right)D\left(z,\xi \right)$, where $\xi =t-z/V$ and factor

The nonlinear polarization induced by the laser pulse in the crystal via nonlinear optical rectification can be written as

To find the electric (E_x) and magnetic (B_y) fields generated by the moving nonlinear polarization ${\mathbf{P}}^{\text{NL}}$, we use the Maxwell equations

To characterize the LN crystal at the terahertz frequencies, we use the parameters of 0.68 mol% Mg-doped stoichiometric LN [27, 47, 48]: ${\epsilon }_0=24.4$, ${\epsilon }_{\infty }=10$, ${\omega }_{\text{TO}}/\left(2\pi \right)=7.44$ THz, and ${\nu }_{\text{ph}}/\left(2\pi \right)=1.3$ THz. In the optical range, we use $n_p=2.16$, $n_g=2.23$, and ${\beta }_3=1\times {10}^{-3}$ cm³/GW² at 0.8 μm [47, 49], and $n_p=2.15$, $n_g=2.2$, and ${\beta }_4=3\times {10}^{-6}$ cm⁵/GW³ at 1.05 μm [50, 51]. The nonlinear coefficient of LN is $d_{\text{eff}}=166$ pm/V [52], $d_{\text{eff}}\left[\text{cm}/\text{cgse}\right]=d_{\text{eff}}\left[\text{pm}/\mathrm{V}\right]\cdot 3\times {10}^{-8}/\left(4\pi \right)$.

For ZnTe pumped at 1.7 μm, we use following parameters [31, 52, 53]: ${\epsilon }_0=10.02$, ${\epsilon }_{\infty }=7.44$, ${\omega }_{\text{TO}}/\left(2\pi \right)=5.32$ THz, ${\nu }_{\text{ph}}/\left(2\pi \right)=0.1382$ THz, $n_p=2.727$, $n_g=2.788$, ${\beta }_4=4\times {10}^{-5}$ cm⁵/GW³, and $d_{\text{eff}}=68.5$ pm/V.

For optically generated free carriers, we use a typical collision rate ${\nu }_{\text{fc}}/\left(2\pi \right)=10$ THz [54].

3. Stationary regime

At first, we neglect the pump depletion by putting $D\left(z,\xi \right)=1$ and consider the stationary solution of Eqs. (4)-(7), which depends only on ξ. We write the derivatives as $\partial /\partial z=\partial /\partial \xi$ and $\partial /\partial t=-V\partial /\partial \xi$, obtain $B_y\left(\xi \right)=n_g{E}_x\left(\xi \right)$ from Eq. (4), and integrate the remaining equations numerically.

Figure 2 shows the stationary solutions $E_x\left(\xi \right)$ excited in LN by a pump pulse with $n_g^{\text{eff}}=5.18$ ($\alpha \approx {64.50}^{\circ }$ for $\lambda =0.8\mu$m and $\alpha \approx {64.87}^{\circ }$ for $\lambda =1.05\mu$m) [Fig. 2(a)] and in ZnTe by a pulse with $n_g^{\text{eff}}=3.32$ ($\alpha \approx {33}^{\circ }$) [Fig. 2(b)]. The plasma density behind the laser pulse is set equal to $f_p={\omega }_p\left(\xi \to \infty \right)/\left(2\pi \right)=1$ THz. According to Eq. (1), the plasma of such density is generated if the pump parameters are related by the equation $I_0^n\left[\text{GW}/{\text{cm}}^2\right]\times \tau \left[\text{fs}\right]\approx 7.3\times {10}^6$, $2.8\times {10}^9$, and $1.3\times {10}^8$ for LN at 0.8 μm, LN at 1.05 μm, and ZnTe at 1.7 μm, respectively. For example, for τ = 300 fs we obtain, respectively, $I_0\approx 29$, 55, and 26 GW/cm².

 

Fig. 2 The stationary oscillograms $E_x\left(\xi \right)$ (normalized to P₀) for (a) LN pumped by a laser pulse with $n_g^{\text{eff}}=5.18$ and (b) ZnTe pumped by a pulse with $n_g^{\text{eff}}=3.32$. In both cases, τ = 300 fs and f_p = 1 THz. In (a), there is a slight difference between the excitation at 0.8 μm (solid) and 1.05 μm (dashed). The curves for f_p = 0 (no FCG) are shown for reference.

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If FCG is neglected, i.e., f_p = 0 and $J_{\text{fc}}=0$ in Eq. (5), the electric field $E_x\left(\xi \right)$ in both materials (Fig. 2) is a unipolar (nearly Gaussian) pulse of the near field of the nonlinear source [33]. No phase-matched wave is generated behind the laser pulse due to its rather long duration τ and a substantial detuning of the tilt angle α from the critical value $\stackrel{-1}{\cos }(n_g/\sqrt{{\epsilon }_0})$ [27]. In the presence of FCG, the near-filed pulses are strongly attenuated and a negative dc component (dc precursor) appears ahead of the laser pulse (Fig. 2). A distinction between the oscillograms generated in LN by the laser pulses with different λ (0.8 and 1.05 μm), but the same $n_g^{\text{eff}}$, is practically indiscernible [Fig. 2(a)] and will be neglected in further analysis.

Figure 3 shows the dependence of the dc precursor’s magnitude E_dc on the parameters f_p and $n_g^{\text{eff}}$ for a fixed τ. In both materials, E_dc decreases monotonically with detuning $n_g^{\text{eff}}$ from $\sqrt{{\epsilon }_0}$ to larger values ($E_{\text{dc}}=0$ for $n_g^{\text{eff}}<\sqrt{{\epsilon }_0}$). The dependence on f_p demonstrates at first a rapid increase, then a smooth maximum, and finally a slight decrease to a stationary value.

According to our numerical analysis, if phonon dispersion and absorption are neglected, E_dc depends on the product $f_p\tau$, rather than on f_p and τ separately. This agrees with the analytical solution for a rectangular laser pulse [32]. With dispersion and absorption included, one can still treat E_dc as a function of $f_p\tau$ with good precision. This allows to easily extend the results in Fig. 3 to other τ.

 

Fig. 3 The dc precursor’s magnitude E_dc (normalized to P₀) in (a) LN and (b) ZnTe as a function of f_p and $n_g^{\text{eff}}$ for τ = 300 fs. The upper panels show the ratio of the precursor’s length L_dc to the crystal thickness L_c as a function of $n_g^{\text{eff}}$.

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If a dc precursor is generated in a crystal of a finite thickness L_c, parameter $n_g^{\text{eff}}$ defines not only the magnitude of the precursor but also its length L_dc (in free space behind the crystal) as

Equation (8) implies that the precursor’s forefront propagates with velocity $c/\sqrt{{\epsilon }_0}$ in the crystal.

In Fig. 3, the upper panels show $L_{\text{dc}}/L_c$ as a function of $n_g^{\text{eff}}$. According to Fig. 3, there is a tradeoff between the precursor’s magnitude and length in the choice of $n_g^{\text{eff}}$. In particular, approaching $n_g^{\text{eff}}$ to $\sqrt{{\epsilon }_0}$ increases the precursor’s magnitude but at the expense of its shorter length. For a reasonable crystal thickness of a few mm [34], the precursor’s length larger than a fraction of a mm, $L_{\text{dc}}/L_c\ge 0.1$, can be obtained for $n_g^{\text{eff}}\ge 5.05$ in LN or $n_g^{\text{eff}}\ge 3.25$ in ZnTe. This restricts the precursor’s magnitude by $E_{\text{dc}}\le 20P_0$ both in LN and ZnTe. For example, for $I_0=$ 40 GW/cm², it gives $E_{\text{dc}}\le 370$ kV/cm for LN and $E_{\text{dc}}\le 120$ kV/cm for ZnTe.

Let us now take into account the pump depletion by including the factor $D\left(z,\xi \right)$ [Eq. (2)]. To characterize depletion, we introduce the depletion length L_D as the distance z, at which $D\left(L_D,0\right)=0.5$, i.e.,

For the highest peak optical intensities, which are used further, i.e., $I_0=70$ GW/cm² for LN and 40 GW/cm² for ZnTe, L_D is estimated as 1.3 mm for LN at 0.8 μm, 9.6 mm for LN at 1.05 μm, and 7.9 mm for ZnTe at 1.7 μm. We can conclude that for a few mm thick crystals pump depletion will not substantially affect the precursor generation in LN at 1.05 μm and in ZnTe. For LN at 0.8 μm, on the contrary, pump depletion appears to be an essential factor. To reduce its effect one should use lower intensities, $I_0\sim 30-40$ GW/cm², when pumping by a Ti:sapphire laser.

4. FDTD simulation

With the results of the foregoing analysis in hand, we now turn to a simulation study of a dc precursor generation by using an in-house developed finite-difference time-domain (FDTD) code.

Figures 4(a)–4(c) show the snapshots of the electric field E_x generated in LN and ZnTe by the laser pulses with three different I₀ and the same $n_g^{\text{eff}}$ as in Fig. 2. Unlike the stationary solutions in Fig. 2, the precursor in Figs. 4(a)–4(c) looks as a plateau of a finite length [slanted, in Figs. 4(a) and 4(b), or with imposed oscillations, in Fig. 4(c)] propagating ahead of the laser pulse. The magnitude of the precursor is as high as $\sim 100-200$ kV/cm in LN and $\sim 70$ kV/cm in ZnTe. The corresponding precursor lengths exceed 1 mm and 0.5 mm.

 

Fig. 4 (a)-(c) Snapshots of E_x at four moments of time for different I₀ (indicated in the frames). (a) LN, $\lambda =0.8\mu$m, $n_g^{\text{eff}}=5.18$. (b) LN, $\lambda =1.05\mu$m, $n_g^{\text{eff}}=5.18$. (c) ZnTe, $\lambda =1.7\mu$m, $n_g^{\text{eff}}=3.32$. The shaded regions show the crystals. (d) Peak optical intensity I_p as a function of distance into the crystal. In (a)-(d), τ = 300 fs.

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For LN pumped at 0.8 μm [Fig. 4(a)], increasing the pump intensity above 50 GW/cm² adds little to the precursor’s magnitude. This is explained by fast depletion of a high intensity pump: the peak optical intensity $I_p=I_{0}D\left(z,0\right)$ drops from $I_0=70$ GW/cm² to 50 GW/cm² at a short distance of $\approx$0.5 mm from the entrance boundary of the crystal [Fig. 4(d)].

For LN pumped at 1.05 μm, I_p decreases with z significantly slower than for 0.8 μm [Fig. 4(d)]. As a result, the slant of the plateau is less pronounced and the plateau magnitude increases with I₀ even for $I_0>50$ GW/cm² [Fig. 4(b)].

For ZnTe, increasing I₀ above 30 GW/cm² adds little to the precursor’s magnitude [Fig. 4(c)] due to a fast pump depletion [Fig. 4(d)].

Figure 5 confirms the prediction of the stationary analysis (Fig. 3) that approaching $n_g^{\text{eff}}$ to $\sqrt{{\epsilon }_0}$ allows one to enhance the precursor’s magnitude, both in LN and ZnTe, but at the expense of its shorter length. The magnitudes of the precursors in Fig. 5 are as high as $\sim 400$ kV/cm for LN [Fig. 5(a)] and $\sim 150$ kV/cm for ZnTe [Fig. 5(b)]. The precursors’ lengths are $\sim 0.5$ mm and $\sim 0.3$ mm, respectively.

 

Fig. 5 (a) The same as in Fig. 4(b) but for $n_g^{\text{eff}}=5.05$. (b) The same as in Fig. 4(c) but for $n_g^{\text{eff}}=3.27$.

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Figure 6 shows the precursor generation by longer, than in Figs. 4 and 5, laser pulses with τ = 600 fs. For LN, increasing τ allows one to enhance the precursor’s magnitude at low pump intensities. For example, for $I_0=30$ GW/cm², the precursor’s magnitude in Fig. 6(a) is two times as large as in Fig. 4(b). The enhancement can be explained by a higher plasma density behind the longer laser pulse and, therefore, a stronger E_dc according to Fig. 3(a). For high pump intensities, increasing τ provides practically no enhancement due to a saturation of the dependence $E_{\text{dc}}\left(f_p\right)$ in Fig. 3(a). For ZnTe, increasing τ removes oscillations from the precursor’s shape and decreases to some extent its magnitude at high pump intensities [Fig. 6(b)]. The decrease is in accord with Fig. 3(b), where the dependence $E_{\text{dc}}\left(f_p\right)$ demonstrates a slight decline at high f_p.

 

Fig. 6 (a) The same as in Fig. 4(b) but for τ = 600 fs. (b) The same as in Fig. 4(c) but for τ = 600 fs.

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5. Tentative proof-of-principle experiment

For proof-of-principle experimental verification of our predictions, one can use the conventional tilted-pulse-front setup, where the tilt of the pump pulse front is introduced by diffracting the pulse off an optical grating and imaging the front into a 63° prism-cut LiNbO₃ crystal by a lens (or two-lens telescope) [26]. To realise, for instance, the situation shown in Fig. 5(a), the LiNbO₃ prism is to be of a cm size and an Yb-doped laser amplifier (1.05 μm central wavelength) with a 500 fs (FWHM) pulse duration and 2.7 mJ pulse energy should be used as a pump. For a 3-mm diameter of the laser beam, the peak optical intensity will be about 50 GW/cm² and, according to Fig. 5(a), the precursor’s magnitude as high as ∼300 kV/cm will be reached. To detect the precursor, we propose to use electro-optic sampling in a (110)-cut GaP crystal placed in the vicinity (at a few mm distance) of the exit face of the LiNbO₃ prism. The precursor emitted from the prism will enter the GaP crystal and propagate toward the probe optical beam introduced into the crystal from the opposite side. After reflection from the crystal boundary faced to the LiNbO₃ prism, the probe pulse will propagate in the same direction with the precursor thus accumulating a polarization change due to the Pockels effect. The change can be measured by a standard ellipsometric scheme.

6. Conclusions

To conclude, tilted-pulse-front excitation scheme is promising for the generation of propagating quasistatic electromagnetic fields (dc precursors) of a high magnitude. In particular, our calculations predict that a 500 fs (FWHM) pulse with the peak intensity of 70 GW/cm² from Yb-doped laser amplifier can generate in a 5 mm thick LiNbO₃ crystal a dc precursor with the electric (magnetic) field as high as ∼0.2-0.4 MV/cm (∼0.1 T) and corresponding length of ∼1-0.5 mm. By using a monolithic structure with a ZnTe crystal, similar to that in Ref. [30], and pump pulses of 500 fs duration and 30 GW/cm² peak intensity at $\lambda =1.7\mu$m, one can generate quasistatic fields as high as ∼70-150 kV/cm (∼0.02-0.05 T) with corresponding lengths of ∼0.5-0.3 mm. Strong quasistatic (subterahertz) fields can be a useful tool for particle acceleration, molecular orientation, ultrafast control of magnetic order in matter, and in terahertz streaking techniques.