1. Introduction

Strong terahertz (THz) electromagnetic pulse generation is of great current interest due to many potential applications such as rapid THz imaging, intense THz excitation of semiconductors and their nanostructures, as well as nonlinear THz spectroscopy [1]. While intense THz fields exceeding MV/cm can be obtained from large facility sources such as free electron lasers and synchrotron-based sources [1], there is a growing demand for high-power tabletop-scale THz sources. In particular, laser-plasma-based schemes have drawn attention as potential powerful and scalable THz sources. For example, THz radiation by ponderomotive electron acceleration [2] and transition radiation from laser-accelerated electrons crossing a plasma-vacuum boundary [3] have been demonstrated.

Recently, intense THz generation was observed upon mixing the fundamental and its second harmonic laser fields in air [4–7]. The underlying mechanism was initially interpreted as a four-wave difference frequency (FWDF) mixing parametric process in ionized air plasmas produced by the laser fields themselves [4–7]. However, the third order nonlinearity originating from either/both bound electrons of ions (χ ⁽³⁾ _ions) or/and free electrons (χ ⁽³⁾ _{free-electrons}) due to ponderomotive or thermal effects [8] is too small to explain the observed THz field strength [5]. Nonetheless, a recent experiment performed by Xie et al. [7] reported a THz amplitude scaling result (E _THz ∝ χ ⁽³⁾ E _2ω E_ω E_ω supporting FWDF mixing as the principal mechanism for THz generation. In this study, however, the THz scaling was studied over a limited pump energy range, and the origin of χ ⁽³⁾ itself in the air plasma was not addressed.

A photoionization-induced polarization model has also been proposed to explain such THz emission, considering the generation of free elections via tunneling ionization and the resulting polarization transient in the laser field, especially for the few-cycle laser-air interaction [9]. However, in this report, the electron motion is considered only to within a fraction of a laser cycle, as the electron scattering time was extremely underestimated (t _sc ≈ 1 fs) for atmospheric-pressure air [9]. If the plasma polarization changes on the sub-femtosecond timescale, the expected radiation falls mostly in the ultraviolet, and not in the THz range. As a consequence, this model fails to provide a proper mechanism for the THz emission, even for the few-cycle laser-air interaction, making extrapolation to the more complex two-color field case impossible.

In this paper, we develop a photocurrent model to explain the observed THz emission from air plasmas in two-color laser fields, and provide experimental evidence to question the χ ⁽³⁾ _ions and χ ⁽³⁾ _{free-electrons} based FWDF THz generation mechanisms. In our model, a nonvanishing transverse plasma current J _⊥ = eN_eυ_e, where e is the electron charge, N_e is the electron density, and υ_e is the electron velocity, can be produced when the bound electrons are stripped off by an asymmetric laser field, such as a mixed two-color field with the proper relative phase. This photocurrent surge, occurring on the timescale of the photoionization, can produce an electromagnetic pulse at THz frequencies. We note that asymmetric photocurrents were previously observed in gases [10], semiconductors [11], and semiconductor quantum well structures [12], irradiated by mixed two-color laser fields.

2. Theoretical background and simulations

We first show that a nonzero photocurrent can be produced in an alternating laser electric field even if the average field vanishes <E_L> = 0. The laser field containing the fundamental and its second harmonic with the same linear polarization can be expressed as

where E ₁ and E ₂ are the amplitudes of the fundamental (ω) and the second harmonic (2ω) fields, and θ is the relative phase between the ω and 2ω fields. We assume that the bound electrons of atoms under study are librated at a phase ϕ via photoionization. Figure 1(a) shows a combined electric field of ω (λ = 800 nm) and 2ω (λ = 400 nm) lasers with intensities of I_ω= 10¹⁵ W/cm² and I _2ω = 2 × 10¹⁴ W/cm² (assuming 20% efficiency of frequency doubling), respectively, for a relative phase of θ = 0 and π/2. Figure 1(b) shows the trajectories of electrons born at various phases of ϕ = -9π/10, -π/10, π/10, and 9π/10 with respect to the fundamental laser field (for both θ = 0 and π/2). Once bound electrons are liberated via ionization, the electron trajectories and velocities are calculated by classical mechanics. This method is analogous to the model developed to explain high harmonic generation (HHG) and nonsequential ionization in a semi-classical manner [13]. We further simplify our model by including a non-relativistic treatment, thereby ignoring the laser magnetic field, justifiable for our laser intensity regime. Assuming that the initial electron velocity is zero, the drift velocity υ_d of electrons born at ϕ laser phase is given by υ_d = eE ₁ sinϕ/(m_eω) + eE ₂ sin(2ϕ + θ)/(2m_eω), where m_e is the electron mass.

 

Fig. 1. (a). Laser fields with the fundamental and second harmonic with a relative phase θ = 0 and π/2. (b) Electron trajectories born at various phases of ϕ = -9π/10, -π/10, π/10, and 9π/10. (c) Drift electron velocity versus ϕ(solid line), overlaid with the laser field (dashed line).

Download Full Size | PDF

As shown in Fig. 1(b), with even-function symmetry (θ = 0), the drift velocity cancels for electrons born at opposite laser field slopes (for example, at ϕ = -π/10 and π/10, ϕ = -9π/10 and 9π/10, correspondingly). With θ = π/2, however, there is a non-vanishing drift velocity in the positive direction. The drift velocity for all ϕ is plotted in Fig. 1(c) for θ = 0 and π/2, along with the corresponding laser field (dotted line). Since the ionization occurs near the peak of the laser field, only the shadowed area of the electron drift velocity contributes to the overall electron current. In contrast to the θ = 0 case, there is a net electron current with θ = π/2, and this, with a time-varying electron density over the entire laser field envelope, produces a current surge which emits THz radiation.

To quantify the resulting THz yield and spectrum, we performed simulations taking into account photoionization and subsequent electron motion using a temporal Gaussian two-color laser field. For the intensity regime of interest (> 10¹⁵ W/cm²), the Keldysh parameter [14] is $\gamma =\sqrt{\frac{U_i}{\left({2U}_P\right)}}=0.36<1$, where U_i = 15.576 eV is the ionization potential energy of N₂ (the primary constituent gas of atmospheric air), and U_p = 59.7 eV is the laser ponderomotive potential energy at I = 10¹⁵ W/cm². The modified Keldysh parameter γ for small diatomic molecules such as N₂ can be even smaller [15, 16]. For γ < 1, tunneling ionization becomes the dominant ionization route and we therefore use the Ammosov–Delone–Krainov (ADK) tunneling ionization rate [17] in our model. Although, the ADK model has been used primarily for noble gases, it also works well for structureless atomic-like molecules such as neutral N₂ [18]. A field-ionized gas can undergo further ionization through electron-ion and electron-neutral collisional process. For an initially singly ionized plasma of atmospheric gas density, the electron-ion collision rate is ν_ei ~ 10¹² s^-1 under our condition [16], which corresponds to a time between collisions of ~1 ps—a time much longer than the pulse length of ~200 fs used in the experiment. The electron-neutral impact ionization rate is also similarly small compared to the ADK rate [16]. Hence, collisional processes during the laser pulse can be ignored for atmospheric pressure gases, but in general they tend to wash out the coherent electron motion on a picosecond time scale, ultimately terminating the photocurrent.

Results of our simulations are shown in Fig. 2 for Gaussian-enveloped ω and 2ω fields with 50 fs (FWHM) pulsewidths, at peak intensities of I_ω = 10¹⁵ W/cm² and I _2ω = 2×10¹⁴ W/cm², and with θ = π/2.

 

Fig. 2. (a). Electron density of atmospheric nitrogen gas irradiated by a combined ω and 2ω laser field with 50 fs pulsewidths and peak intensities of I_ω = 10¹⁵ W/cm² and I _2ω = 2×10¹⁴ W/cm², with a relative phase θ = π/2. (b) Time-varying electron current. (c) Radiation spectrum obtained from the transient electron current. The inset shows an expected THz waveform obtained with 10 THz low-pass filtering.

Download Full Size | PDF

Figure 2(a) shows a time-varying electron density, overlaid with the combined laser field. Here, N₂ gas is ionized up to N₂ ²⁺ with the characteristic stepwise ionization twice per laser cycle. The resulting transverse electron current, J _⊥(t) = ∫^t _t0 eυ_e (t,t′)N_e(t′)dt′, where υ_e(t, t′) is the velocity of electrons born at t=t′ with the electron density of N_e(t′), was computed and is plotted in Fig. 2(b). In this computation, the contribution from electron displacements is ignored because the maximal displacement during the interaction is much smaller than the THz radiation wavelength, and thus is considered a point source. Moreover, transport and space charge effects are excluded for simplicity. In Figure 2(b), it is seen that in addition to its oscillatory feature, a non-zero DC electron current appears after the laser pulse is gone, giving rise to THz emission in the far field. The calculated power spectrum of radiation induced by the transient of current, dJ _⊥(t)/dt, is shown in Fig. 2(c), with θ = π/2, two-color (solid line) and single-color ω (dashed line) fields. Also shown is THz emission with a >20 THz bandwidth (FWHM) (indicated in the box), which is enhanced as compared to the single-color case. The corresponding THz waveform with low frequency filtering (<10 THz) is shown in the inset of Fig. 2(c).

3. Experiment

The above hypotheses are tested by performing THz emission experiments using an amplified Ti:sapphire laser system delivering 815 nm, 50 fs, 25 mJ pulses at a 10 Hz repetition rate. The pulsewidth was stretched to ~200 fs with a beam diameter of ~8 mm to minimize self-focusing in the beam path, and most of the laser’s energy was directed to f = 150 mm lens, followed by a 100-μm thick BBO frequency-doubling crystal (Type I) placed near the focus. Here, the crystal ordinary axis for ω was detuned to ~50° with respect to the p-polarized ω field, producing a p-polarization component of 2ω field, but with a somewhat compromised frequency doubling efficiency (maximum ~10%). Both ω and 2ω pulses were focused into air (591 torr), with the relative phase θ controlled by adjusting the separation of the BBO crystal and the laser focal spot [5]. The peak ω-field intensity was estimated to be ~10¹⁶ W/cm², with a 60 μm (FWHM) spot size. The resulting THz field was measured using an electro-optic sampling technique [19] employing a 1-mm thick ZnTe <110> crystal. Although the THz was detectable either with p- and s-polarization, the strongest THz signal was observed with p-polarization, consistent with previous measurements [5, 7]. Figure 3(a) shows a typical THz waveform resulting from a pump-probe scan. The corresponding THz spectrum is plotted in the inset of Fig. 3(a). The oscillations in Fig. 3(a) are due to strong water absorption in air.

 

Fig. 3. (a). Typical experimental THz waveform and the corresponding THz spectrum in the inset. (b) THz yield vs. BBO-to-plasma distance d with two different BBO angles φ = 28° and 34°.

Download Full Size | PDF

To verify the validity of our photocurrent model, we studied the phase dependence θ of THz radiation yield. Figure 3(b) shows the THz intensity versus the BBO-plasma distance d (see the inset) for two different BBO angles φ = 28° and 34° with respect to the BBO ordinary axis of ω. The least squares fits of data (solid lines) are also shown in Fig. 3(b) and extrapolated toward d = 0, where near d = 0 the direct measurement is impossible because of laser-induced damage to the BBO crystal. It is clear that the THz yield exhibits an oscillatory behavior due to air dispersion for ω and 2ω. Here the phase at the plasma front can be expressed as θ = ω(n_ω-n _2ω)d/c + θ ₀, where c is the speed of light in vacuum, n_ω and n _2ω is the refractive index of air at ω and 2ω frequency, respectively, and θ ₀ is the ω-2ω phase difference right after the BBO crystal. With careful angular tuning (±0.2° around the phase-matching angle), the dephasing length can be as large as 1 mm. Hence, with our 100-μm crystal thickness, the ω and 2ω fields can be considered as phase-matched (θ ₀ ≈ 0) directly after the crystal and, even for nonzero φ cases, there are always in-phase ω and 2ω components projected along the incoming p-polarized ω field. We note that the relative phase slippage in the plasma is given by Δθ = (3π/4)(l/λ)(N_e/N_c), where l ≈ 7 mm is the plasma length and N_c = m_eω/(4πe ²) = 1.7 × 10²¹ cm^-3 is the critical density at λ = 815 nm. With N_e ~ 1.24 × 10¹⁶ cm^-3, as the upper-limit electron density obtained from the critical density at 1 THz, the estimated relative phase slippage is Δθ ≈ 0.3 rad, which still produces a phase-matching condition in the plasma. With θ ≈ 0 (or d ≈ θ ₀ ≈ 0), the extrapolated THz yield approaches zero, consistent with the photocurrent model. However, this is in sharp contrast with polarization-based FWDF mixing which predicts the maximum THz yield at θ = 0, as the rectified term from the third order polarization is proportional to E ² _ω(t)E _2ω(t)cosθ [4–7]. We note that sinθ dependence can be obtained from FWDF mixing with an assumption of E_ω(t) = sinωt and E _2θ(t) = sin(2ωt + θ) [5]. In either case, THz radiation from polarization-based FWDF mixing becomes maximal when the peaks of ω and 2ω fields overlap in time. In contrast, the photocurrent model predicts that the maximal THz yield occurs with a θ = π/2 phase slippage (or sinθ dependence) between the ω and 2ω fields.

As a complementary experiment looking for THz contribution from χ ⁽³⁾ _ions and χ ⁽³⁾ _{free-electrons}, we examined the response in a preformed plasma. The preformed plasma was produced prior to the main ω and 2ω pulse arrival by reprogramming the regenerative amplifier Pockels cell switch-out timing, conveniently generating a contrast-controllable prepulse (with respect to the main pulse) of 9.6 ns. Figure 4 shows the THz emission yield for five cases (A, B, C, D, and E) of first to second pulse intensity contrast ratio.

 

Fig. 4. THz emission yield from a preformed air plasma with a varying double-pump-pulse contrast

Download Full Size | PDF

With ~20% prepulse (case B), the THz yield significantly dropped to ~25% of the signal obtained without the prepulse (case A). This reduction could be a result of the pre-ionized ions and electrons not contributing to the THz emission when the main pulse arrives at the preformed plasma, or possibly simply due to a nonlinear effect associated with the frequency doubling and plasma current generation due to the decreased laser field strength (prepulse removing some energy from the main pulse). However, with the time-reversed intensity profile (case E), where the prepulse of case B acts as a postpulse in case E, the THz emission yield becomes as high as 65% of the prepulse-free signal (case A). This discrepancy, in fact, disputes the possibility that the second scenario (energy removal) causes the THz yield reduction. Moreover, the increased THz yield in case E can be understood since most of the THz emission now comes from the first pulse (prepulse), which is more intense than the second pulse (postpulse). Therefore, in all our trials with the prepulse present, we observe a significant drop in the THz output originating from the second pulse. This can be explained as follows: the free electrons produced by the first pulse produce THz radiation via a photocurrent effect, but later on, when the electron coherent motion is lost and the second pulse arrives, those free electrons simply follow the oscillating laser field of the second pulse without acquiring a net drift current. Therefore, it is the ionizing (not ionized) medium where the THz emission originates. This result contradicts the possibility of FWDF mixing by coupling to pre-ionized ions (χ ⁽³⁾ _ions) or free electrons (χ ⁽³⁾ _{free-electrons}) as an origin for THz generation. This is because χ ⁽³⁾ _ions or χ ⁽³⁾ _{free-electrons} based FWDF should give a similar effect whether the plasma is formed by the rising edge of the pulse itself or by other methods. As a result, the experiment indicates that the THz generation mechanism is closely related to the instantaneous nature of the interaction dynamics. Specifically, the generation mechanism itself is related to tunneling ionization and subsequent coherent electron motion induced by the laser field.

Regarding the long time scale (9.8 ns) between the pre- and main pulse, we note that the preformed plasma can cause a significant plasma defocusing effect to the main pulse for subpicosecond time intervals. This strongly affects the main pulse propagation and the resulting THz generation. In contrast, nanosecond delays can provide a sufficient time for the preplasma to evolve and provide a plasma channel suitable for waveguiding of the main laser pulses [20].

The THz power scalability was also investigated by increasing laser energy to the tens of mJ levels, in contrast to previous reports in the sub-mJ regime [4–7]. Figure 5 shows the peak THz field strength as a function of the incoming optical laser pump energy (before frequency doubling).

 

Fig. 5. Peak THz field strength as a function of the incoming laser energy with ±5% error sizes.

Download Full Size | PDF

The peak THz field amplitude after the air plasma was estimated as ~150 kV/cm with 2-THz bandwidth filtering. Here the THz field was calibrated by applying a known DC voltage across a ZnTe crystal and measuring the resulting optical probe leakage into the detector in the absence of the ω and 2ω pulses. Unlike the previous reports, the THz output strongly saturates with increasing laser energy. This saturation is attributed to strong THz absorption in the long (~7 mm) air plasma. In general, the THz yield scales as E _THz ∝ dJ _⊥/dt ~ eυ_d (dN_e/dt), and thus the THz yield grows with increasing laser field, due to increases in both the electron drift velocities (υ_d) and the ionization rates (dN_e/dt). However, as the plasma electron density increases due to high laser fields and large background gas densities, the THz absorption in the over-dense plasma can also dramatically attenuate the THz yield, impacting future experimental approaches to efficient scaling.

4. Conclusion

In conclusion, we have developed a transient photocurrent model in which optical field ionization and subsequent electron motion in a symmetry-broken laser field are key mechanisms for producing a quasi-DC electron current and simultaneous THz pulse radiation. In general, the directional current and resulting THz generation becomes most efficient when the fundamental and its properly phase-locked second harmonic fields are mixed. With any other two-color wavelength combination, our simulation shows that the resulting quasi-DC current becomes extremely weak, compatible to that obtained with a single-color field. This is due to the fact that the current magnitude and direction varies at each combined laser cycle and the net current summed over an entire set of cycles vanishes unless one of the laser frequencies is an integer multiple of the other one. Our theoretical and experimental investigation provides insight into the THz generation mechanism and will speed further optimization and scalability of tabletop, ultrafast THz sources.