Open Access
Add to BibSonomyAbstract
Contribution of plasma oscillation to the broadband terahertz (THz) emission is revealed by interacting two-color (ω/2ω) laser pulses with a supersonic jet of nitrogen molecules. Temporal and spectral shifts of THz waves are observed as the plasma density varies. The former owes to the changing refractive index of the THz waves, and the latter correlates to the varying plasma frequency. Simulation of considering photocurrents, plasma oscillation and decaying plasma density explains the broadband THz spectrum and the varying THz spectrum. Plasma oscillation only contributes to THz waves at low plasma density owing to negligible plasma absorption. At the longer medium or higher density, the combining effects of plasma oscillation and absorption results in the observed low-frequency broadband THz spectra.
© 2016 Optical Society of America
1. Introduction
Broadband ultrashort terahertz (THz) pulses are generated from the laser induced plasma by two-color pulses, and it is under intensive studies owing to the potential to produce high intensity THz field and achieve the off-site detection in air [1–8]. From the plasma filament pumped by the two-color laser pulses, the directional photocurrents generate the enhanced supercontinuum at THz wavelength range with the conversion efficiency two orders larger than that from single color intense laser field [9,10]. The low-frequency broadband spectrum is determined by the constructive interference of the attosecond electron bursts whose timing is controllable by detuning the frequencies of the two-color pulses [11,12]. Based on the transient photocurrent model, schemes of controlling the THz polarization and spectrum have been proposed [12–16].
As the ultrashort THz pulses are produced from the ionizing gaseous medium, the surrounding plasma is expected to play an important role in the generation mechanism. The plasma oscillation has been considered as the origin of THz spectrum in previous theoretical studies on the generation of strong THz fields [17,18]. Recently Debayle et al. [19] proposed that the plasma current oscillation is in competition with the photocurrent mechanism and it can contribute to the THz spectrum when the length of medium is less than the skin depth of plasma. Also, the residual plasma oscillation after the interaction of the pumping laser pulses, instead of the photocurrents when the laser is on, was proposed as a source of the low-frequency THz emission [20–22]. However, experimental investigations showed no evidence of plasma oscillation in play, possibly hindered by the complex dynamics of laser propagation in filaments, and it remains controversial on how the laser plasma impacts the broadband THz generation.
We experimentally investigate the THz generation from a jet of nitrogen (N2) molecules pumped with the two-color laser pulses. The molecular beam in vacuum provides a medium of limited length to observe the THz waves under the conditions of varied plasma density. We observed significant frequency shift at the low plasma density that were never shown in air filament, and we found that the changed THz frequency can only correlate to the varied plasma frequencies. The observation suggests that the plasma oscillation contributes to THz spectrum only at low plasma density owing to negligible plasma absorption when the medium length is less than the plasma depth. Our results also show that as the plasma density further increases, the spectrum of THz approaches to a constant central frequency, indicating a more dominant role of the absorption of plasma. In simulation we consider the decay rate of plasma density as an amendment to the semi-analytical one-dimensional model [19] to explain the effect of plasma oscillation.
2. Experimental implementation and results
Experimentally we prepare a short length of nitrogen (N2) molecules using the molecular beam method [5,23]. A beam of N2 molecules is produced from supersonic expansion of a nozzle (General Valve Series 9, orifice diameter of 0.5 mm) in a high vacuum chamber (10−4 Pa). The jet is mounted on a motorized stage and its position is adjustable with a minimum step size of 0.05 mm along the directions of laser propagation (z-axis) and jet expansion (y-axis). Laser pulses are delivered along the z direction from a commercial Ti: sapphire laser amplifier (Coherent Elite-HP-USX) with a duration of 40 fs and a repetition rate of 1 kHz at the center wavelength of 800 nm. As shown schematically in Fig. 1, a portion of the laser pulses (1.2 mJ) passes through a focusing lens (f = 300 mm), a 200 μm type-I beta barium borate (β-BBO) crystal, a calcite for compensating the group delay of the two color pulses, and a zero-order λ/2 (400nm) waveplate for rotating the polarization of the generated 400 nm pulses to be parallel with the 800 nm pulses. The phase delay of the two-color pulses is controlled with a pair of fused silica wedges and the THz yield modulates as the phase delay changes [24]. The two-color pulses focus onto the molecular beam at 0.5 mm downstream of the orifice, where the interaction length is estimated to be Lm = 0.7 mm from the fluorescence of the nitrogen cations in the plasma. The intensity of 800 nm laser field is estimated to be 1.9 × 1014 Wcm−2 based on the measured pulse energy, duration and size of focus, and the peak intensity of 400 nm pulses (conversion efficiency of 20%) is estimated as 9.0 × 1013 Wcm−2. Under these intensities, only single ionization of N2 molecules is dominant according to the calculation of the Ammosov - Delone - Krainov (ADK) theory, even though the laser field intensity is crucial for THz mechanism especially under the condition of multiple ionizations [25]. The left portion of the laser pulses is used as the probe pulses for the measurement of time-domain THz waveforms using the electro-optic (EO) sampling method [26].
In experiment only the forward THz radiation is measured while the backward emission is not detectable, possibly because of the weak signal level [18,19] especially under the jet condition where the signal to noise ratio is about 2 orders lower than in air. For the EO sampling, both a ZnTe crystal of 100 μm thickness and a GaP crystal of 200 μm are used, allowing the cutoff of the measured bandwidth to be ∼8 THz. The response function of the crystals are calculated to correct the measured THz waves. We only detect the transverse THz polarization with the longitudinal field excluded because the THz radiation is expected to be generated mainly from the transverse currents under the laser intensity of a couple of 1014 Wcm−2 [27].
By altering the stagnation pressure of the jet, we achieve the experimental condition where all the experimental parameters remain constant except for the number density of molecules at the focus. Figure 2(a) shows that the obtained THz waveforms vary significantly as the pressure changes from 0 to 1.0 bar. The THz waves at 0.2 bar and 0.7 bar are retrieved and the Wigner transform maps are plotted in Fig. 2(b) and 2(c) respectively. One can see that the center position of the spectra shifts from 0.8 THz to 1.2 THz while the temporal envelope from 5.2 ps to 5.0 ps. It is noted that, when using the EO sampling crystal of GaP (detection range 0.1 ~8 THz), the main body of THz spectrum lies within low frequency range of 0 ~3 THz under the varied experimental conditions.
Fig. 2 (a) Normalized THz waveforms as a function of stagnation pressure of molecular jet (100 μm ZnTe crystal), and the time-frequency analysis of THz waves at 0.2 bar (b) and 0.7 bar (c).
3. Discussions
The temporal shift of the THz envelope can be attributed to the variation of the refractive index for the pumping laser pulses and the generated THz waves in the plasma, which is mainly determined by the varied gas densities. The temporal positions of the THz waves shown in Fig. 2(a) are retrieved and plotted in Fig. 3(b), indicating an overall shift of ∼0.2 ps. It is known that the frequency-dependent refractive index of the N2 molecules composes of the two contributions: of the neutral molecules and of the plasma [28]
where δngas(ω) is the refractive index drift caused by neutral molecules and δnplasma(ω) by plasma. Here the refractive index is a complex number, n = n1 + i⋅n2, which composes of a real part for phase delay and an imaginary part for absorption [29,30]. As the density of N2 changes, the varied real part (n1) results in a phase delay, and the varied absorption (n2) results in a changed amplitude. We calculate the δngas for THz waves by extrapolating the dispersion formula [28,31]. For the plasma drift, we use nplasma(ω) = (1-ωp2/ω2)1/2 for the pumping laser pulses and nplasma(ω) = (1-ωp2/(ω2 + ivω))1/2 for the THz because the considered wavelength is close to the collision frequency [9]. Under the experimental conditions, the density of N2 molecules and the plasma frequency are calculated by the empirical formula of the free jet expansion [23] and the ADK theory for ionization rate [32], respectively.Fig. 3 (a) Calculated refractive index for 800 nm (black line) and THz waves (1 THz, purple line) at varied stagnation pressures; (b) the center positions (red solid circles) of the temporal envelope of THz waves and simulated temporal advance (blue solid line) of the THz pulses (~1 THz); (c) the THz frequencies (red solid circles) and the calculated 0.7 fp (blue solid line). Note that for the pressure less than 0.7 bar (green dotted line), Lskin > 0.3Lm, the THz frequencies is sensitive to the plasma environment.
The variations of the refractive indexes for 800 nm pump pulses and the THz waves (1 THz) are calculated and shown in Fig. 3(a), respectively. One can see that the change of the refractive index for 800 nm is 4 orders lower than that for the THz waves. Calculation also indicates that the refractive index drift caused by plasma δnplasma is 5 orders larger than by neutral molecules δngas, so the velocity change of THz waves by plasma dominates the temporal envelope position. Figure 3(b) also shows the calculated time shift of the THz waves by Δt = Ln/c with L the interaction length of 0.7 mm, which is consistent with the experimental observation. During the laser pulse of tens of femtoseconds, electrons are tunneling ionized and the time scale of THz emission extends to picoseconds. So the THz radiation is inevitably affected by surrounding plasma environment. Since the temporal advance of the THz waves is sensitively related to the plasma density, the phenomenon offers a means of measuring the plasma density in situ accurately.
With the varying plasma density, the underlying physics of the spectral shift can be analyzed. We consider the two possible mechanisms of the frequency shift during the generation of transient photocurrents: one is the spatially varied phase difference of the two-color laser pulses, φ(z), in the plasma; and the other is originated from the detuning of spectrum of the driving laser pulses. For the former case, the dispersion within the short length of plasma increases the phase difference of the two color pulses, which can modify not only the generated THz spectrum but also the yield [6,9]. We calculate the increased phase difference δφ = φ(0) -φ(Lm) = 0.0086 π for 0.2 bar and δφ = 0.038 π for 1 bar, which are both too little to modulate the THz yield.
Secondly, the detuning of the two-color laser pulse spectrum in plasma may change the timing of the attosecond electron bursts and then modify the generated THz frequency through interference [12,16]. We calculated the shifted spectrum of the driving two-color laser pulses with the varying refractive index induced by laser plasma. At the stagnation pressure of 0.2 bar and 0.7 bar, the frequency shift for 800 nm is 0.1 THz and for 400 nm 0.05 THz with which the THz spectrum can be evaluated by photocurrent model. When we neglect the propagation effects in the jet gas, the free electron density ne(t) and transient current J(t) are governed
where w is the ionization rate calculated by ADK theory, n0 is the initial density of the jet molecules. The generated THz spectrum is determined by ETHz (ω) ∝ ωJ(ω). The electron-ion collisional rate was determined by [9]. For the stagnation pressure of 0.2 bar to 0.7 bar, the rate v in jet molecules was calculated to be 1.9 ∼6.5 × 1011 Hz.As a result, the calculated frequency shift of THz is less than 0.01 THz, much less than the experimental results of 0.4 THz as shown in Fig. 3(b). Based on the shifted spectrum of laser pulses, we also calculate the THz spectrum using four-wave mixing model [15,33], and the THz frequency shifts ~0.0045 THz, which is close to the simulation result of photocurrents model, but has a one-order difference with the observed THz frequency shift.
The discrepancy of the above simulation with the experimental observation can be explained the contribution of plasma frequency to THz spectrum, since the small-size beam and low stagnation pressure of molecules provides a medium length shorter than the skin depth of plasma and permits the transmittance of plasma frequency, as proposed by [19]. For a uniformly distributed plasma in a cylinder volume, the resonance frequency is fc = fp / = 0.707fp. The peak frequency of THz spectrum is approximately equal to the plasma resonance frequency, namely fTHz = fc [21]. We plot the observed central frequencies and the calculated frequencies of fc in Fig. 3(c), and the two curves match well. The consistence indicates that the plasma oscillation plays a major role in the obtained THz spectra.
Notably, we observe that the THz central frequency varies linearly at low stagnation pressures and the converges to a constant central frequency at around 0.7 bar as pressure becomes higher, where the plasma skin depth is close to the medium length, Lskin = c/[ω Im(n)] = 0.3 Lm. Under the experimental condition of the changing gas pressures, the saturation of electron density can be ruled out [34]. It has been proposed that the plasma oscillation current can mainly contribute to the THz spectrum when plasma skin depth is larger than the medium length, and the THz peak frequency increases with plasma frequency for Lskin > Lm [19]. The local electric field δE emitted by the plasma is expressed as
where. Here ωcf the calculated 2πfc under the final electron density (nef) after gas ionization, EL the laser electric field, and τ the laser pulse width, so the THz field oscillates at plasma resonance frequency of fc. Notably, indicating the contribution of the photoionization currents in the semi-analytical model. When Lskin < Lm at higher plasma density, the photocurrent contribution becomes more dominant, but the THz spectra ought to show shifting frequency because of the presentable contribution of the plasma frequency, shown in Fig.7 of [19]. However, our experimental result indicates otherwise for Lskin < 0.3 Lm.The shifting of THz spectrum to a constant central frequency can be understood by introducing a decay rate of plasma density into the semi-analytical one-dimensional model [19]. In the laser plasma generated from the molecular jet, the time-evolution of electron density can be expressed as [35,36]
where β is the decay coefficient used to characterize the electron-ion recombination. The decay rate β has been measured to be β = 3.1 × 10−12 m3/s in the previous study of [37]. The ratio between Lskin and Lm affects the THz spectrum mainly by absorption during transmission in plasma. The absorption function of THz spectrum by plasma [9] iswhere z is the position of THz generation and Lskin = c/(ω⋅n2). The THz spectrum is then obtained by integrating the THz emission along the plasma spatial dimension [38]where, ω is the beam radius at different position, , θ(z) is the phase at the exit of the plasma. The simulation result of ETHz (ω) is shown in Fig. 4, and it is consistent with experimental results regarding the turning to a constant central frequency.Fig. 4 (a) Normalized THz spectrum as a function of stagnation pressure; (b) simulated THz spectrum.
The longer skin depth for low plasma density leads to negligible absorption according to Eq. (6). Hence the contribution of plasma oscillation can be observed experimentally in Fig. 3(c) for low pressure. On the contrary, the shorter skin depth for high plasma density leads to the generated δE strongly absorbed in the plasma. So the peak frequency of THz no longer change under the varied dense plasma density, but with decay of plasma density after ionization, the skin depth will change longer until that the skin depth become close to the length of the plasma, in which δE with the plasma oscillation characteristics is reappear. So the transmitted plasma resonance frequency under high plasma density is directly related to the medium length but not the plasma density and keep invariant.
Experimentally we also found that the amplitude of the THz waves from the jet molecules decreases 10 times as the driving laser field changes from two color pulses to one color pulses, shown in Fig. 5(a). The last item G in Eq. (4) indicates a non-zero DC electron current density after the laser pulse. One can see that two color pulses can effectively increase G, as shown in Fig. 5(c), and the THz yield from plasma oscillation also increases effectively as shown in Fig. 5(b). It explains well why the amplitude of the THz waves by one color driving pulses decreases much even through it is contributed from plasma oscillation.
Fig. 5 (a) Measured terahertz electric field waveform for two-color pulse (red line) and one-color pulse (blue line), respectively. (b) Simulated unfiltered local δE(t) for two-color pulse (red line) and one-color pulse (red line). (c) δE(t) is enlarged in time scale of −0.02 ps~0.04 ps of (b).
There is some inconsistency at larger plasma density in our simulation result comparing to the experiments shown in Fig. 3(b). We propose two possible reasons that may improve the model. Firstly, the increase of plasma density may lead to the off-axis phase matching condition of THz emission and reduce the transmittance of plasma in forward direction, and the photocurrent mechanism becomes dominant [8,15] (conical THz radiation has been found in related studies based on photocurrent in long plasma filament [39,40]). Secondly, under the higher stagnation pressure, the pumping pulses and THz waves may experiences more complex propagation effects such as plasma defocusing and diffraction, with which one may improve the accuracy of simulation.
4. Conclusion
In conclusion, the contribution of plasma oscillation is identified for the generation of broadband THz pulses in ultrashort two-color laser plasma in experiments. Temporal and spectral shifts of THz waves are observed as the plasma density varies at low stagnation pressure. The former owes to the changing refractive index of the THz waves, and the latter correlates to the varying plasma frequency. Simulation of considering photocurrents, plasma oscillation and decaying plasma density explains the broadband THz spectrum and the varying THz spectrum. Plasma oscillation only contributes to THz waves at low plasma density owing to negligible plasma absorption. At the longer medium or higher density, the combining effects of plasma oscillation and absorption results in the observed low-frequency broadband THz spectrum. Our study provides significant guidance for the tunable terahertz spectrum radiation based on plasma oscillation.
Funding
National Natural Science Foundation of China (11274326, 61521093, 61405222, 11134010 and 11127901); Shanghai Yangfan Program (14YF1406200).
References and links
1. K. Reimann, “Terahertz radiation: A table-top source of strong pulses,” Nat. Photonics 2(10), 596–597 (2008). [CrossRef]
2. S. I. Mitryukovskiy, Y. Liu, B. Prade, A. Houard, and A. Mysyrowicz, “Coherent synthesis of terahertz radiation from femtosecond laser filaments in air,” Appl. Phys. Lett. 102(22), 221107 (2013). [CrossRef]
3. T. I. Oh, Y. S. You, N. Jhajj, E. W. Rosenthal, H. M. Milchberg, and K. Y. Kim, “Intense terahertz generation in two-color laser filamentation: energy scaling with terawatt laser systems,” New J. Phys. 15(7), 075002 (2013). [CrossRef]
4. T. I. Oh, Y. J. Yoo, Y. S. You, and K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105(4), 041103 (2014). [CrossRef]
5. N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102(9), 093001 (2009). [CrossRef] [PubMed]
6. M. Li, W. Li, Y. Shi, P. Lu, H. Pan, and H. Zeng, “Verification of the physical mechanism of THz generation by dual-color ultrashort laser pulses,” Appl. Phys. Lett. 101(16), 161104 (2012). [CrossRef]
7. P. G. D. Martínez, I. Babushkin, L. Bergé, S. Skupin, E. Cabrera-Granado, C. Köhler, U. Morgner, A. Husakou, and J. Herrmann, “Boosting terahertz generation in laser-field ionized gases using a sawtooth wave shape,” Phys. Rev. Lett. 114(18), 183901 (2015). [CrossRef] [PubMed]
8. V. A. Andreeva, O. G. Kosareva, N. A. Panov, D. E. Shipilo, P. M. Solyankin, M. N. Esaulkov, P. González de Alaiza Martínez, A. P. Shkurinov, V. A. Makarov, L. Bergé, and S. L. Chin, “Ultrabroad terahertz spectrum generation from an air-based filament plasma,” Phys. Rev. Lett. 116(6), 063902 (2016). [CrossRef] [PubMed]
9. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15(8), 4577–4584 (2007). [CrossRef] [PubMed]
10. T.-J. Wang, S. Yuan, Y. Chen, J.-F. Daigle, C. Marceau, F. Théberge, M. Châteauneuf, J. Dubois, and S. L. Chin, “Toward remote high energy terahertz generation,” Appl. Phys. Lett. 97, 111108 (2010). [CrossRef]
11. L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D numerical simulations of THz generation by two-color laser filaments,” Phys. Rev. Lett. 110(7), 073901 (2013). [CrossRef] [PubMed]
12. I. Babushkin, S. Skupin, A. Husakou, C. Köhler, E. Cabrera-Granado, L. Bergé, and J. Herrmann, “Tailoring terahertz radiation by controlling tunnel photoionization events in gases,” New J. Phys. 13(12), 123029 (2011). [CrossRef]
13. H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009). [CrossRef] [PubMed]
14. X. Lu and X.-C. Zhang, “Generation of elliptically polarized terahertz waves from laser-induced plasma with double helix electrodes,” Phys. Rev. Lett. 108(12), 123903 (2012). [CrossRef] [PubMed]
15. A. V. Borodin, N. A. Panov, O. G. Kosareva, V. A. Andreeva, M. N. Esaulkov, V. A. Makarov, A. P. Shkurinov, S. L. Chin, and X.-C. Zhang, “Transformation of terahertz spectra emitted from dual-frequency femtosecond pulse interaction in gases,” Opt. Lett. 38(11), 1906–1908 (2013). [CrossRef] [PubMed]
16. T. Balčiūnas, D. Lorenc, M. Ivanov, O. Smirnova, A. M. Zheltikov, D. Dietze, K. Unterrainer, T. Rathje, G. G. Paulus, A. Baltuška, and S. Haessler, “CEP-stable tunable THz-emission originating from laser-waveform-controlled sub-cycle plasma-electron bursts,” Opt. Express 23(12), 15278–15289 (2015). [CrossRef] [PubMed]
17. Z.-M. Sheng, K. Mima, J. Zhang, and H. Sanuki, “Emission of electromagnetic pulses from laser wakefields through linear mode conversion,” Phys. Rev. Lett. 94(9), 095003 (2005). [CrossRef] [PubMed]
18. W.-M. Wang, Z.-M. Sheng, H.-C. Wu, M. Chen, C. Li, J. Zhang, and K. Mima, “Strong terahertz pulse generation by chirped laser pulses in tenuous gases,” Opt. Express 16(21), 16999–17006 (2008). [CrossRef] [PubMed]
19. A. Debayle, L. Gremillet, L. Bergé, and Ch. Köhler, “Analytical model for THz emissions induced by laser-gas interaction,” Opt. Express 22(11), 13691–13709 (2014). [CrossRef] [PubMed]
20. V. B. Gildenburg and N. V. Vvedenskii, “Optical-to-THz wave conversion via excitation of plasma oscillations in the tunneling-ionization process,” Phys. Rev. Lett. 98(24), 245002 (2007). [CrossRef] [PubMed]
21. C. S. Liu and V. K. Tripathi, “Tunable terahertz radiation from a tunnel ionized magnetized plasma cylinder,” J. Appl. Phys. 105(1), 013313 (2009). [CrossRef]
22. N. V. Vvedenskii, A. I. Korytin, V. A. Kostin, A. A. Murzanev, A. A. Silaev, and A. N. Stepanov, “Two-color laser-plasma generation of terahertz radiation using a frequency-tunable half harmonic of a femtosecond pulse,” Phys. Rev. Lett. 112(5), 055004 (2014). [CrossRef] [PubMed]
23. G. Scoles, Atomic and Molecular Beam Methods, Vol. 1 (Oxford University Press, 1988).
24. J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009). [CrossRef] [PubMed]
25. A. Debayle, P. G. D. Martínez, L. Gremillet, and L. Bergé, “Nonmonotonic increase in laser-driven THz emissions through multiple ionization events,” Phys. Rev. A 91(4), 041801 (2015). [CrossRef]
26. P. C. M. Planken, H. K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18(3), 313–317 (2001). [CrossRef]
27. P. González de Alaiza Martínez, X. Davoine, A. Debayle, L. Gremillet, and L. Bergé, “Terahertz radiation driven by two-color laser pulses at near-relativistic intensities: Competition between photoionization and wakefield effects,” Sci. Rep. 6, 26743 (2016). [CrossRef] [PubMed]
28. L. A. Johnson, J. P. Palastro, T. M. Antonsen, and K. Y. Kim, “THz generation by optical Cherenkov emission from ionizing two-color laser pulses,” Phys. Rev. A 88(6), 063804 (2013). [CrossRef]
29. J. D. Jackson, Classical Electrodynamics (Berkeley, 1998), Chap. 7.
30. R. P. Feynman, R. B. Leighton, and M. Sands, Feynman Lectures on Physics (Addison–Wesley, 1964), Vol. 2, Chap. 32.
31. E. R. Peck and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am. 56(8), 1059–1063 (1966). [CrossRef]
32. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).
33. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25(16), 1210–1212 (2000). [CrossRef] [PubMed]
34. J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B 91(1), 45–48 (2008). [CrossRef]
35. H. Yang, P. Liu, H. Lu, X. Ge, R. Li, and Z. Xu, “Time-evolution of electron density in plasma measured by high-order harmonic generation,” Opt. Express 20(17), 19449–19454 (2012). [CrossRef] [PubMed]
36. S. Tzortzakis, B. Prade, M. Franco, and A. Mysyrowicz, “Time-evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181(1-3), 123–127 (2000). [CrossRef]
37. R. B. Bryan, R. B. Holt, and O. Oldenberg, “Recombination and afterglow in nitrogen and oxygen,” Phys. Rev. 106(1), 83–86 (1957). [CrossRef]
38. E. Constant, D. Garzella, P. Breger, E. Mevel, C. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82(8), 1668–1671 (1999). [CrossRef]
39. V. Blank, M. D. Thomson, and H. G. Roskos, “Spatio-spectral characteristics of ultra-broadband THz emission from two-colour photoexcited gas plasmas and their impact for nonlinear spectroscopy,” New J. Phys. 15(7), 075023 (2013). [CrossRef]
40. Y. S. You, T. I. Oh, and K. Y. Kim, “Off-axis phase-matched terahertz emission from two-color laser-induced plasma filaments,” Phys. Rev. Lett. 109(18), 183902 (2012). [CrossRef] [PubMed]
