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Abstract
We present an experimental study on pressure-dependent terahertz generation from two-color femtosecond laser filamentation in various gases. Contrary to short-focusing geometry, we find that long filamentation yields higher terahertz energy at lower gas pressures in most gases. This counter-intuitive phenomenon occurs due to multiple peculiar properties associated with filamentation. In practice, filamentation in low-pressure argon provides a maximum laser-to-terahertz conversion efficiency of 0.1%, about 10 times higher than in atmospheric air. In addition, our pressure-dependent study reveals an anticorrelation between terahertz output energy and local plasma fluorescence brightness. This determines the absolute phase difference between two-color laser fields for maximal terahertz generation, as well as verifies the microscopic mechanism of terahertz generation in two-color laser mixing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-power terahertz (THz) sources are indispensable tools to explore strong THz-driven phenomena [1] such as self-phase modulation in semiconductors [2], self-transparency in graphene [3], electron-hole recollisions in quantum wells [4], second and high harmonic generation [5], and electron acceleration [6]. Among many THz sources, femtosecond two-color laser mixing in gases has attracted much interest due to its capability of producing intense, broadband terahertz (THz) radiation [7–36]. In this scheme, a femtosecond laser pulse is mixed and co-focused with its second harmonic pulse to ionize a gaseous medium at the focus. Under the right phase condition between the two-color laser fields, bound electrons can be tunnel-ionized to form asymmetric charge separation (or current) along the laser polarization. This plasma current occurs on the time scale of the laser pulse duration, emitting broadband THz radiation in the far field [11,12]. Compared to other laser-based THz sources including optical rectification in lithium niobates [37–39] or organic crystals [40], two-color laser mixing provides extremely large spectral bandwidths (0.1~200 THz) [21,33]. It can also deliver high peak power (12 MW) [34] and strong field strengths (>16 MV/cm) [36] at kHz repetition rates.
In two-color laser mixing, short-focusing with focal length of 50-200 mm is commonly adopted for THz generation [7–16,18,21–34]. In this setting, however, THz output does not always scale with input laser energy [28,29,35]. Instead, it often saturates or even decreases with modest laser energy [28]. This can happen when the plasma created by the rising part of a laser pulse becomes dense enough to diffract the rest of its energy, prohibiting further enhancement of laser intensity and plasma generation. This plasma-induced defocusing prevents efficient laser energy coupling to the plasma for THz generation. This defocusing effect can be mitigated by increasing the plasma length (or volume) in the transverse direction by cylindrical focusing [35] or in the longitudinal beam direction by weak focusing [36].
Longitudinal weak focusing naturally prompts filamentation, in which the laser intensity is maintained high over a long distance due to a dynamic balance between plasma defocusing and Kerr-induced self-focusing [41–44]. This occurs when the laser power exceeds the critical power for self-focusing Pcr = 3.77λ2/(8πn0n2) for a Gaussian beam [41–44], where λ is the laser wavelength, n0 is the linear refractive index of the medium, and n2 is the nonlinear refractive index. The balance results in a plasma filament ranging from several centimeters to meters. In this long filamentation, an off-axis phase matching condition is favorably satisfied for THz generation, in which the output THz energy increases with growing plasma filament length [25]. In addition, gases with lower ionization potential energy and/or at higher pressures are usually adopted in an attempt to produce more tunnel-ionized electrons and thus stronger THz radiation [12]. Previous studies, however, were mostly limited to short-focusing geometry (f = 100-200 mm) [12,18,19], and no pressure-dependent experiment has been performed in the long-length filamentation regime.
In this paper, we present a comprehensive study of THz generation from long filamentation at a broad range of sub-atmospheric pressures for various gas species. We find that two-color laser filamentation at low pressures (~0.1 atm) dramatically enhances THz radiation in most gases. In addition, low-pressure settings provide reduced dephasing between two-color laser pulses, therefore allowing us to accurately determine an optimal phase for maximal THz generation.
2. Experimental setup
A schematic of our experimental setup is shown in Fig. 1. Laser pulses of <8 mJ, 40 fs, and 800 nm at a repetition rate of 1 kHz are focused by a long-focal-length lens (f = 1 m) into a gas tube of 1.75 m length. A type-I beta barium borate (BBO) crystal of 100 µm thickness is stacked with a thin (45 µm) dual wavelength (800 nm and 400 nm) half-wave plate and placed inside the gas tube to generate the second harmonic (2ω) pulses polarized parallel with the fundamental frequency (ω) pulses [31,34]. The focused ω and 2ω pulses ionize the gas inside the tube, generating a long plasma filament (see the inset to Fig. 1) and simultaneously emitting THz radiation in the forward direction.
Fig. 1 Experimental setup for THz generation from two-color laser filamentation inside a long gas tube. The emitted THz radiation is refocused by an off-axis parabolic mirror onto a pyroelectric detector for energy measurement and an uncooled microbolometer focal plane array for imaging. Synchronized probe pulses at variable delays are used to measure THz waveforms via electro-optic sampling with a thin GaP crystal. The inset shows a long plasma filament in argon captured by optical side imaging.
A silicon (Si) exit window is used to block the laser beams but transmit THz radiation. The transmitted THz beam is focused by an off-axis parabolic (OAP) mirror of focal length f = 50.8 mm onto a pyro-electric (Gentec-EO Inc., QS9-THZ-BL) or thermopile (Gentec-EO Inc., THz-12-3S-VP) detector for energy/power measurement, or a room-temperature microbolometer focal plane array (FPA) sensor (FLIR, tau2) for THz beam profiling [36,45]. A longpass filter is placed in front of the pyroelectric detector to block high-frequency components at >40 THz. The waveform of emitted THz radiation is characterized by an electro-optic sampling (EOS) technique using a 100-µm thick GaP crystal. A complementary metal-oxide-semiconductor (CMOS) camera is used to capture optical fluorescence emitted from the plasma filament through the transparent tube. Test gas species include helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), nitrogen (N2), oxygen (O2), carbon dioxide (CO2), and room air. Except Kr and Xe, the gas is slowly fed into the tube at a rate of ~0.5 atm/hour while the THz pyroelectric signal, pressure, and filament fluorescence images automatically taken at every 10 seconds.
3. Result and discussion
3.1 THz generation from Ar
First, argon is tested for THz generation. Figure 2(a) shows the output THz energy measured as a function of increasing Ar pressure at laser energy of 8 mJ. It shows that the output yield exhibits periodic oscillations over the entire pressure range. These oscillations result from dephasing between two-color pulses (800 nm and 400 nm) under varying gas pressures (or densities). As THz generation is sensitive to the relative phase between two-color pulses [11], a continuous change in pressure induces a periodic modulation in the THz output. Here the dephasing length ld, the distance over which THz radiation emitted along the filament has the same polarity [25], is given by
where λ is the wavelength at 2ω, and nω and n2ω is the index of refraction at ω and 2ω, respectively. The oscillation period in pressure, Posc, is then related to the dephasing length ld as Posc [atm] = (ld,1 atm/d)⋅1 atm, where ld,1 atm is the dephasing length at 1 atm, and d = 0.55 m is the distance between the BBO and the center of the filament. For argon, ld,1 atm = 0.03 m (with no plasma contribution), and the expected oscillation period is Posc = 0.054 atm. Compared to the previous work on THz generation in a short-length gas cell [12,18,19], our large d yields a much faster oscillation period.Fig. 2 (a) THz output energy as a function of argon pressure inside the gas tube at input laser energy of 8 mJ. (b) THz output signal as a function of laser input energy for argon (squares) and atmospheric room air (triangles) both at 0.08 atm. The inset shows a refocused THz beam profile captured by the microbolometer FPA.
In addition to the oscillations, the output THz energy gradually decreases with increasing pressure. One obvious effect is a group-velocity walk-off between the two-color pulses. As the pressure increases, the temporal separation between the two pulses increases, prohibiting efficient THz generation. We note that a walk-off can also occur in the transverse direction. It is recently known that a high-repetition-rate (kHz) train of filamenting pulses can be self-steered by a gas density gradient created by successive laser heating and resulting buoyant motion of the heated gas [46]. At a 1-kHz repletion rate, this buoyance-induced self-steering can occur even with sub-mJ of laser energy [46]. This density gradient can separate two-color laser pulses (ω and 2ω) in the transverse direction, where the 2ω pulse (400 nm) refracts more than the ω pulse (800 nm). This transverse separation is visually observed at relatively high gas pressures when the two pulses hit the exit window made of high-density polyethylene (HDPE) instead of Si. Thus, both longitudinal and transverse beam walk-offs contribute to the decreasing THz signal with pressure. In addition to this effect, a more fundamental aspect as to why THz generation can be enhanced at low pressures is related to the nature of filamentation and discussed further in the next section.
Figure 2(b) shows that the output THz energy increases with input laser for both Ar and room air at 0.08 atm, without exhibiting any saturation up to 8 mJ of laser energy. In the case of Ar, the maximum THz energy estimated immediately after filamentation is 9.6 µJ at input laser energy of 8 mJ. Here the pyroelectric detector is calibrated at wavelengths of 532 nm, 650 nm, and 800 nm, which all provides a similar responsivity of 2.5 × 104 V/W (or 40 nJ/V per pulse at 1 kHz) at a chopping rate of 13 Hz [36]. The detector has a black organic coating and provides a flat response over a wide range of frequencies (7.5–3000 THz). The maximum pyroelectric voltage observed is 3 V with HDPE (16.7%), longpass (25%), and Si (30%) filters, with each transmission characterized with the emitted THz radiation. At below ~7.5 THz, the detector’s sensitivity gradually drops with decreasing frequency, reaching about ~10% of its flat value at 1 THz. Because of this, 9.6 µJ is somewhat underestimated when THz radiation below 7.5 THz is included. This yields a conversion efficiency exceeding 1.2 × 10−3, which is more than 10 times greater than typical values (~10−4) achieved with short-length focusing in air [34]. The efficiency (~0.1%) is comparable to those achieved with longer wavelength lasers to drive filamentation in atmospheric air [32].
Figure 3(a) shows a typical THz waveform measured by EOS when 8 mJ of laser energy is focused into 0.05 atm Ar. The corresponding spectrum is obtained by Fourier transformation and shown in Fig. 3(b) (blue line). This measurement provides mostly low-frequency radiation (<10 THz) due to material absorption and dispersion at high THz frequencies in the GaP crystal. A complementary spectral measurement is conducted by using THz bandpass filters [47] to characterize high frequency radiation at >10 THz. This method shows a much broader spectrum (red line) compared to the EOS result (blue line). Here the detection range is limited to 30 THz due to filter availability. With tight refocusing, the emitted THz radiation is confined into a spot size of <50 µm in full width at half maximum (FWHM) as shown in the inset to Fig. 2(b). This provides field strengths of 30 MV/cm at the focus, calculated from the estimated THz energy (9.6 μJ), beam spot size (<50 μm in FWHM), and waveform (in Fig. 3(a)) using Eq. (1) in Ref [36].
Fig. 3 (a) THz waveform measured by electro-optic sampling with a 100-µm thick GaP. (b) THz spectrum obtained from (a) via Fourier transformation (blue solid blue) and independently characterized by pyro-electric detection using various THz bandpass filters (red solid line with dots).
3.2 THz generation from various gases
Pressure-dependent THz generation is tested with other gases including He, Ne, N2, room air, O2, CO2, Kr, and Xe. The results are shown in Fig. 4. Similar to Ar, most gases exhibit dephasing-induced modulations in THz output yields. Among the tested gases, He is the least dispersive and thus displays the slowest oscillation period. Xe is the most dispersive and expected to yield the fastest oscillations although those are not clearly pronounced due to our coarse measurements made for Kr and Xe. The measured oscillations periods in pressure, Posc, are tabulated in Table 1 along with their expected values obtained from Eq. (1).
Fig. 4 Relative THz output energy as a function of gas pressure when ionized by 8 mJ laser pulses with the gas tube filled with helium, neon, nitrogen, room air, oxygen, carbon dioxide, krypton, or xenon.
Table 1. Measured and predicted pressure periods (Posc) in THz generation for various gases when irradiated by 800 nm, 8 mJ, 40 fs laser pulses with the BBO-to-focus length of d = 0.55 ma
Aside from the local oscillations, most gases yield decreasing THz output with increasing pressure except He, Kr, and Xe. In the case of He, the THz signal is overwhelmed by its local dephasing effect, which makes it difficult to extract its long-term pressure-dependent behavior from our limited pressure range (<0.8 atm). Moreover, unlike other gases, the critical power for self-focusing in He (268 GW at 1 atm [48]) is larger than the laser power (200 GW), and thus no filamentation in He is expected under our experimental condition. In the case of Kr and Xe, strong filamentation occurs even at low pressures, producing a large amount of white light (supercontinuum) and also weakly ablating the exit window and producing floating particles inside the gas tube. This yielded some artifacts in the THz and fluorescence signals. For instance, the THz signal for Kr initially decreases with pressure like other gases but starts to increase at ~0.1 atm due to strong white light generation and then decreases at ~0.4 atm possibly due to ablation. The Xe signal also exhibits a similar trend although the scan was terminated at 0.45 atm to avoid further damage on the exit window.
Among all tested gases, Ar yields the largest THz signal at low pressures. This is because Ar has relatively high ionization potential (see Table 1), and thus ionization occurs at high laser intensities. After ionization, the laser intensity clamps at Icl ≈1.1 × 1014 W/cm2 for example in Ar [51,52], and remains more or less the same along the filament due to the balancing act between plasma defocusing and Kerr-induced self-focusing [53,54]. This high-intensity clamping in Ar results in strong THz radiation compared to other gases. We note that He and Ne have even higher ionization potential but do not yield filamentation at low pressures (<0.5 atm) because of their high critical power Pcr necessary for self-focusing (see Table 1).
3.3 Filamentation at low gas pressures
Previous studies show that femtosecond laser filamentation can occur at low pressures as long as the laser power exceeds the critical power for self-focusing Pcr at the given pressures [55–57]. It is also known that the peak laser intensity inside a filament is clamped, and the clamped intensity is nearly independent of gas pressure [58]. This is because the nonlinear index of refraction (n2) and plasma density (Ne) effects scale together with varying gas density (or pressure) [58]. This results in an interesting outcome. At laser power equal to or greater than the critical power, P = I ⋅ Af ≥ Pcr, the filament cross-sectional area Af increases with decreasing pressure p as [57]
where r is the filament radius. This is because the laser intensity remains constant due to clamping, but the critical Pcr scales inversely with gas pressure as Pcr ∝ 1/p [55–58]. Then the number of free electrons N# within the filament of length l and radius r is given bywhere Ne is the electron density that is linearly proportional to the gas pressure (Ne ∝ p), and the filament length l does not change significantly with pressure in the filamentation regime [56]. Equation (3) shows that the number of total THz-emitting free electrons within the filament is independent of gas pressure. In addition, reduced electron density Ne at low pressures leads to less THz absorption in the forward direction. These two effects make low-pressure operation more favorable for THz generation.To verify this result, single-color laser filamentation is examined for Ar at various pressures. Figure 5(a) shows plasma fluorescence emitted along the laser propagation direction at various Ar pressures when 800 nm, 7 mJ laser pulses are weakly focused (f = 1m) into the gas tube. Here the fluorescence image taken at each pressure is radially integrated and plotted along the laser propagation direction. It shows that the onset of filamentation shifts toward the laser source with increasing gas pressure due to self-focusing [41–44]. More importantly, the integrated fluorescence signal is almost independent of gas pressure. This is clearly observed in Fig. 5(b) showing a total fluorescence signal as a function of Ar pressure. Note that Ar fluorescence arises from electronic transitions in neutral atoms excited by electron collisions, three-body electron-ion recombination, and a broad continuum by Bremsstrahlung emission as a dominant source at high laser power [59]. In general, all processes are seeded by free electrons born during field sensitive tunneling ionization. Although the fluorescence signal is not directly proportional to the electron density or laser intensity, the trend shown in Fig. 5(b) supports that the number of total electrons within the filament does not change significantly with gas pressure. This confirms that low-pressure settings are not disadvantageous in generating free electrons as a fundamental source for THz radiation. At the laser power of 175 GW, the threshold pressure for self-focusing in Ar is estimated to be ~0.05 atm, under which no filamentation is expected. This is consistent with a rapid drop in the fluorescence around 0.05 atm in Fig. 5(b).
Fig. 5 (a) Measured plasma fluorescence variation along the laser propagation direction z at varying Ar pressure when ionized by single-color (800 nm), 7 mJ laser pulses at 1 kHz. (b) Fluorescence integrated from 4.3 cm to 17.8 cm and plotted as a function of Ar pressure.
3.4 Two-color laser filamentation
In the case of two-color laser filamentation, low-pressure operation is more favorable for THz generation because it is as effective as atmospheric processes in producing free electrons and also yields a less walk-off between two-color pulses, potentially generating more free electrons at low pressures. This is clearly shown in Fig. 6(a), which displays fluorescence intensity as a function of the beam propagation distance z and Ar gas pressure. Compared to the single-color case shown in Fig. 5(a), the filament is much brighter at low pressures (<0.4 atm). At >0.4 atm, the filament appears to break into two segments with faint fluorescence emitted in between. This dim light is not well captured by the camera. Another imaging artifact is related to the multiple vertical stripes shown in Fig. 6(a). Their positions and sizes are insensitive to pressure changes and possibly caused by unknown interference effects in our imaging system and/or fluorescent light scattering by dusty particles caught on the inner surfaces of the transparent tube.
Fig. 6 (a) Measured normalized plasma fluorescence along the laser propagation direction (z) at various argon pressures. (b) Pressure dependent fluorescence brightness integrated along z (blue solid line) plotted with measured THz output energy (red dotted line). The relative phase between the two is plotted in (ii). (c) Simulated normalized electron density as functions of both z and pressure. (d) Comparison between simulated electron density (blue solid line) and THz energy (red dotted line).
By contrast, the fluorescence that is periodically modulated with pressure is real and revealed to be anti-correlated with far-field THz radiation. This is clearly seen in Fig. 6(b), where the fluorescence intensity is integrated along the laser direction and plotted as a function of pressure with the corresponding THz signal. The relative phase between the signals is extracted and plotted in Fig. 6(b)(ii). It shows that they are nearly out of phase by π. This result unveils a close relation between phase-dependent ionization and THz generation. In two-color laser mixing, the combined laser field can be expressed as
where θ is the relative phase between the fundamental (Eω) and second harmonic (E2ω) fields. In the tunneling regime, the ionization rate is highly nonlinear and strongly enhanced when the two-color wave crests are in phase with θ = 0, but this condition yields minimal THz radiation according to the semi-classical plasma current model [11]. The model also predicts that maximal THz radiation occurs with θ = 0.5 π, and at the same time this phase condition yields less tunneling ionization according to Eq. (4). We note that the plasma current model ignores electron scattering by atomic Coulomb potentials with a strong field approximation [11]. Nonetheless, the experimental result shown in Fig. 6(b) strongly supports the plasma current model as a microscopic mechanism. This contrasts to recent simulation results obtained from solving the time-dependent Schrödinger equation (TDSE) with Coulomb scattering taken into account, in which the optimal phase for THz generation is determined to be θTDSE = 0.8 π [22]. Although the discrepancy is not fully resolved, this experimental result confirms that Coulomb scattering plays little or no role in THz generation under our experimental conditions.Figure 6(c) reproduces the fluorescence image in Fig. 6(a) by simulating two-color Gaussian beam propagation at various gas pressures and calculating the resulting electron densities along the laser propagation direction. In general, it replicates well the periodic modulations caused by the dephasing effect. However, the curved stripes are not clearly observed in the measured fluorescence, particularly at high pressures. This might be due to laser shot-to-shot fluctuations and multi-filament formation that could smear out the modulations at high pressures. Nonetheless, the anti-correlation between the fluorescence and THz signals is clearly reproduced as shown in Fig. 6(d). In addition, the modulation stripe is more or less straight at <0.1 atm, and this indicates that the coherence length for THz generation [25] is very long at low pressures. This is another reason why low-pressure filamentation can yield more THz radiation.
4. Conclusion
We report efficient THz generation from two-color laser filamentation at low gas pressures with maximum conversion efficiency of 10−3, about 10 times greater than typical values obtained in atmospheric air. This high efficiency benefits from (i) an increased plasma filament diameter that can contain as many THz-emitting free electrons as in atmospheric pressure, (ii) reduced longitudinal and transverse beam walk-offs between two-color laser pulses, (iii) a long coherence length for phase matching, and (iv) reduced THz absorption in the plasma, all occurring at low pressures. Also, our combined measurement of THz and plasma fluorescence in low-pressure filamentation reveals an optimal phase of 0.5 π for maximal THz generation, supporting the microscopic plasma current model. As a practical source, low-pressure filamentation in Ar has been demonstrated to provide ~10 μJ of THz energy with a focused spot size of <50 μm in FWHM, potentially producing a field strength in access of 30 MV/cm at 1 kHz repetition rate. Such a strong THz source can be a useful tool in studying THz-driven novel phenomena and nonlinear THz spectroscopy.
Funding
Air Force Office of Scientific Research (FA9550-16-1-0163); National Science Foundation (NSF) (1351455).
Acknowledgment
The authors thank Donghoon Kuk and Zheqiang Zhong for their contributions in building the gas tube used in this experiment.
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