Terahertz (THz) radiation has promising applications in diverse fields such as sensing, imaging, spectroscopy, and strong-field physics. However, its applications are still constrained by the absence of reliable high-intensity sources^[1–3]. High-power THz radiation generated by two-color femtosecond laser filamentation has attracted extensive attention due to its unique advantages, including ultrabroad bandwidth, ultrahigh peak intensity, and ultrashort duration^[4,5]. However, the main challenge in this field lies in the significant enhancement of the output THz pulse energy. In recent years, most of the advances have been tied to improving the THz radiation conversion efficiency^[5,6]. The improvement methods are classified into two types: modulation of the driving laser and modulation of the filament’s intrinsic characteristics.

In the early research work, a transition–Cherenkov electromagnetic emission theory has been used to describe THz radiation produced by single-color laser filamentation. This theoretical model describes the influence of the incident laser parameters (pulse energy, pulse duration, and intrinsic phase) and the filament intrinsic parameters (filament length and filament diameter) on THz radiation. The filament length is the intrinsic characteristic of the filament, which determines the THz radiation intensity and the emission angle. The THz radiation conversion efficiency depends on the optimal filament length. Both shorter and longer plasma filaments are not conducive to increasing conversion efficiency^[7–9].

The theoretical understanding of THz radiation produced by two-color laser filament is more complicated. It has been demonstrated experimentally that the THz conversion efficiency of the two-color femtosecond laser filament is determined by multiple laser parameters, including the two-color laser peak power, polarization, and spatiotemporal walk-off^[4,8]. The two-color laser peak power determines the ponderomotive forces that influence the photocurrent intensity^[11]. As the filament intensity is limited by the clamping effect, the THz conversion efficiency cannot be enhanced unrestrictedly by solely increasing the two-color laser peak power^[12]. The polarization modulation that keeps the polarization coincidence of the fundamental wave (FW) and the second harmonic (SH) is also crucial since the THz conversion efficiency obtains the maximum when the polarization of the FW and the SH is parallel^[13,14]. The spatiotemporal walk-off effect also influences the THz conversion efficiency^[10]. During the generation of the two-color laser in β-BBO, the Poynting vector direction of the FW and the SH in the β-BBO has a spatial walk-off, and the group velocity of the FW and the SH in the β-BBO has a temporal walk-off.

The above analysis mainly describes the basic characteristics of the two-color laser field^[5,13–16] and its impact on THz radiation. From another perspective, many research groups have carried out studies on the impact of the intrinsic characteristics of filaments on the intensity and the spatial distribution characteristics of THz radiation. The intensity and directivity of the THz radiation can be improved by extending the filament length^[10,17–19]. However, these works ignored that the spatiotemporal walk-off of the two-color laser was changed when modulating the focal length. When extending the focal length, the propagation distance of the two-color laser is increased, as the two-color laser has different group velocities in the air. We precisely controlled physical parameters such as polarization status and the spatiotemporal walk-off of the two-color laser by a dual-wavelength wave plate and α-BBO. We also demonstrate a quantitative model between the filament length and THz intensity. The laser filament can be seen as a gain channel for THz generation, and the modulation of the filament characteristics (especially the filament length) can further improve the THz conversion efficiency.

In this Letter, we study the relationship between the filament’s intrinsic characteristics (filament length, filament diameter, and plasma density in the filament) and THz radiation intensity by excluding the influence of polarization, peak intensity, and spatiotemporal walk-off between the FW and the SH waves. The THz radiation intensity was found to increase gradually by extending the focal length. This phenomenon is related to the elongation of the filament length but independent of the filament diameter and the evolution of electron density in the filament. A numerical simulation has been developed to describe THz generation in which the whole THz radiation from the filament is considered as an accumulation of the THz waves emitted from each cross-section of the optical filament.

We conducted two experiments in this Letter. First, we study the effects of the two-color laser properties and the filament properties on THz intensity. The experiment consists of a lens (f=300mm) and a β-BBO crystal. Figure 1(a) shows the experimental setup for two-color femtosecond laser filamentation. Laser pulses (45 fs, 4.4 mJ, 800 nm) with horizontal linear polarization were focused by a plano–convex lens of 30 cm focal length. A 107 µm thick β-BBO crystal with type I phase matching was used to generate the SH wave. Both the SH intensity and the polarization of the FW could be controlled by rotating the β-BBO crystal. Figure 1(b) shows that the THz emission from the two-color filament was detected with a Michelson interferometer consisting of a 100 µm thick high-resistance silicon wafer and two flat mirrors: M1 (fixed) and M2 (movable). A parabolic mirror (OPM) focused the output radiation on the Golay cell detector (GC-1 P, Tydex Inc.). As shown in Fig. 1(c), the two-color laser filament was characterized by measuring the side fluorescence of the filament. A 357 nm filter was placed in front of the CCD camera to detect the fluorescence emission over the N2+ band while rejecting the scattered light from the FW wave, the SH wave, and other emissions from nitrogen.

We have further controlled the polarization and compensated the time delay of FW and the SH pulses through a double wavelength wave plate (DWP) and α-BBO crystal in Fig. 1(a) to precisely study the influence of the filament length on THz radiation to quantitatively study the relationship among the THz radiation and the filament length, filament diameter, and electron density to explain the mechanism of the filament length modulation on THz radiation. Three kinds of lenses (f=300, 400, and 500 mm) were used to produce different filament lengths to further explain the principles of filament length modulation. In the experiment, a dual-wavelength wave plate is used to modulate the polarization of the two-color laser to be the same polarization. Three α-BBO crystals were used to compensate for the spatiotemporal walk-off of the two-color laser. We have designed different α-BBO crystals for different experimental conditions (f=300, 400, and 500 mm) in our experiment. The parameters of the three α-BBO crystals used in our experiment are shown in Table 1. (The detailed calculation process can be found in the Supplementary Information.) Vg is the group velocity of the laser beam propagating in the α-BBO. τt is the time compensation parameter of the α-BBO.

Figure 1(d) shows the experimental setup of the time-resolved shadowgraphs, which can be used to estimate the electron density inside the filament by calculating the absorption of the probe laser (400 nm) by the plasma generated by the filament.

In this Letter, we first study the effect of rotating β-BBO crystal on the evolution of THz radiation intensity. As shown in Fig. 2(a), the FW (as the incident wave) can be decomposed into the ordinary wave and extraordinary wave. The ordinary wave component converts to the SH, while the extraordinary wave component does not convert to the SH and propagates with a polarization direction perpendicular to the FW. Therefore, the β-BBO crystal rotation angle determined the angle of the FW polarization and β-BBO crystal optic axis, which dominates the SH efficiency and the polarization status of the FW and the SH. Figure 2 shows the evolution laws of the THz radiation intensity, the filament length, and the SH intensity by rotating the β-BBO angle. In data collection and organization, the SH intensity minimum point is set as the β-BBO rotation angle of 0° position. In Fig. 2(b), it has been demonstrated that, with only one β-BBO crystal to produce two-color laser filamentation, the SH maximum point is misaligned with the THz maximum point. The maximum point (−120°) of THz radiation results from the two-color laser with parallel polarization status, which is consistent with the conventional photocurrent model.

On the other side, in Fig. 2(b), the physical mechanism of the THz radiation increases at −120° and 0° has a significant difference. As mentioned above, THz radiation increases at −120° depending on the polarization status of the FW and the SH laser field. During the process of rotating the β-BBO angle, the polarization status of the two-color laser should be parallel at −120°, −60°, and 0°. However, as is shown in the blue line in Fig. 2(b), the SH intensity reduces even to zero at 0°, and the driving laser transforms close to the single-color laser, which means that the SH intensity and polarization status do not lead to the changing of THz radiation. As shown in Figs. 2(b) and 2(c), in the red box area, the driving laser (single color) energy increases from 4.2 mJ (−20°) to 4.39 mJ (0°), and the filament length increases from 4.2 mm (−20°) to 5.2 mm (0°). The driving laser intensity changes, leading to changes in filament length. Since the total radiated THz signal consists of the accumulation of all points along the filament, this change in the filament affects the THz radiation intensity^[18].

The transient current model focuses on the interaction process between laser and gas molecules at the geometric optical focus. The two-color laser is mixed to ionize gas molecules, and a directional photoelectron current can be produced, which simultaneously emits THz radiation in the far field^[6,18]. However, the THz radiation is generated in the entire region where the filament exists, which is the limitation of the transient current model. To understand this phenomenon, we need to study the relationship between THz radiation and the intrinsic characteristics of the filament.

To understand the evolution of filaments and THz radiation intensity, we measured the filament length, filament diameter, and plasma density to find the leading factor that influences the THz radiation. In this part, the polarization and the spatiotemporal walk-off of the two-color laser have been compensated. The improved experimental setup aims to eliminate the influence of factors other than the intrinsic characteristics of filaments on THz intensity. (Specific modifications can be found in the experimental setup section.) The filament length and the filament diameter were measured by the side fluorescence imaging. The plasma density was measured by a time-resolved shadowgraph scheme. The time delay between the main pulse and the probe pulse was adjusted by the translation stages, with a delay time of Δt.

Figure 3(a) shows the side fluorescence image of filament produced by lenses with focal lengths of 300, 400, and 500 mm, respectively. The strength of the N2+ fluorescence signal at 337 nm is directly proportional to the total number of N2+ in the B2Σu+. Figure 3(b) shows the fluorescence intensity distribution along the X-Y cross-section of z′ position in Fig. 3(a). Figure 3(c) shows the fluorescence intensity distribution along the Z-direction of the filament. The two-color filament diameters with focal lengths of 300, 400, and 500 mm are 239, 243, and 245 µm, respectively.

Subsequently, the filament length is extracted on the whole length of the filament shown in Fig. 3(c). The two-color filament lengths with focal lengths of 300, 400, and 500 mm are 5.2, 9.3, and 14.5 mm, respectively. Under the focal length modulation, the evolution of the filament diameter is negligible, but the elongation of the filament is obvious. We simultaneously estimated the filament length and compared it with the experiment. The filament length is determined by the FW intensity since the FW intensity far exceeds the SH intensity (the conversion efficiency of the SH is 5%) and the FW intensity far exceeds the critical power. The filament is formed before the linear focus of the lens and quickly vanishes behind the linear focus. The filament length can be written as lfilament=βIFWF21+βIFWF≈βIFWF2=βPFWF2/D2.(1)

Here, β=λ/(4πPc), where Pc is the critical power. IFW is the FW intensity, and the PFW is the FW power. F and D are the focus length and incident beam diameter, respectively^[20]. The comparison between the experimental and simulation results of the filament length is shown in Fig. 4. The filament length simulation method is in good agreement with experimental results when the driving laser peak power far exceeds the critical power.

We analyzed the electron density in the filaments with different focal lengths. The shadowgraphs with different Δt were recorded to retrieve the variation of plasma density inside the filament. The filament shadowgraph was gradually lengthened as Δt changed from 0 to 6000 fs since the two-color pump pulses and the SH probe pulse gradually overlapped. When Δt<0fs, the filament and the probe pulse do not overlap in the time domain. Δt=0fs is the starting overlap point of the filament and the probe pulse. When Δt>6000fs, the filament shadowgraph stopped changing since the temporal coincidence point of the main pulse and the probe pulse had already passed. The plasma density calculation process is in the Supplementary Information^[6]. As shown in Fig. 5, the plasma density of different focusing conditions is 4.93×1016cm−3 (f=300mm), 2.45×1016cm−3 (f=400mm), and 1.12×1016cm−3 (f=500mm), respectively.

Under precise control of the two-color laser polarization and spatiotemporal walk-off conditions, the THz radiation time domain curve and spectrum for different focal lengths are shown in Figs. 6(a)–6(c). According to the photocurrent model, higher electron density in the filament is beneficial for achieving higher THz radiation conversion efficiency. However, this conclusion is contrary to the experimental results shown in Fig. 6. As the focal length changes from 300 to 500 mm, the THz radiation intensity gradually increases, despite the decreasing plasma density in the filament. From Figs. 4–6, only the evolution trend of filament length is consistent with the evolution trend of THz radiation intensity. We can obtain the photocurrent intensity and the THz radiation electric field intensity on the cross-section of the filament by precisely calculating the electron density and the velocity at the cross-section of the filament. Finally, the THz radiation intensity produced by the whole filament length can be calculated by the integral of the THz radiation from each cross-section of the whole filament. The calculating result is shown in Fig. 6(d) (purple line), and the calculating result matches well with the experimental data (orange line).

The two-color laser field is given as El(t)=Eωcos(ωt+φ0)+E2ωcos[2(ωt+φ0)+θ], where Eω and E2ω are, respectively, the electric field of the FW and the SH, ω and φ0 are, respectively, the angular frequency and the initial phase of the FW, and θ is the phase difference between the FW and the SH while propagating in the filament^[21,23]. The ionization process is simulated using the static tunneling (ST) model, which expresses the relationship between the free electron density in the plasma and the incident electric field, so the density of free electrons at the time of t at the position z along the filament is described as ∂Ne(t)∂t=WST(t)[N0−Ne(t)].(2)

Here, WST(t) is the ionization rate determined by the two-color laser field EL(t), and N0 is the density of the neutral molecules. After the laser pulse arrives at t0, the electron drift velocity is given by v(t0,t)=em∫t0tEl(t′)dt′,(3)where e and m are the electron charge and mass, respectively. The electron current producing the THz radiation is given by J(t)=−e∫t0tv(t0,t)dNe(t).(4)

Here, t0 is the initial ionization time, dNe(t) is the electron density variation within the time range t0 and t0+dt, and v (t0,t) is the ionized electrons instantaneous velocity within the time range t0 and t0+dt. J(t) is the current density of a cross-section along the z-direction of the entire filament^[6]. And, ETHz-crosssection(t)∝dJ(t)dt.(5)

For a long filament, the initial phase difference [θ=ω(n2ω−nω)d/c] between the FW and its SH also influences the THz radiation by the entire filament, where ω is the angular frequency of the FW, nω and n2ω are the refractive indices of the FW and the SH, d is the distance between the β-BBO and the focus, and c is the speed of light in vacuum. In the experiment, we controlled the θ as π/2 by changing d. To get the THz radiation in the far field, we need to consider the generation of the THz signal from each cross-section of the filament, and coherent superposition of this signal should be considered.

In this Letter, z0 and z1 are the starting position and the end position of the filament along the direction (z) of laser transmission, respectively. As the THz wave propagates from z0 to z1, the THz wave phase can be written as Φ(Z1)=Φ(Z0)+kTHzη(z)dz, where kTHz=ωTHz/c is the THz wave vector. η(z)=[(1−nee2)/(ε0meωTHz2)] is the refractive index of the plasma, where ne is the plasma density, e is the electron charge, ε0 is the dielectric constant in vacuum, and me is the electron mass. By introducing the THz wave phase and the spatial coordinates, we can estimate the THz wave at Z1 in the filament. Thus, ETHz(z,t)∝dJ(t)dteiΦ(Z1).(6)

The coherent emission of the THz wave from the whole filament in the far field can be shown as ITHz∝∫τ0τ∫z0zdETHz2(z,t).(7)

Conventionally, for the photocurrent model, THz radiation intensity gradually decreases with the focal length extension. The reason is the decrease of the peak power density inside the filament because the long focal length leads to a larger filament diameter. This phenomenon can also lead to a decrease in the electron density inside the filament. The decreasing of the two-color peak power density and the electron density leads to the attenuation of the THz radiation from the filament. However, this conclusion is contrary to our experimental results (Fig. 4). The reason is due to the photocurrent model only calculating the transient current produced at the cross-section of the focal position. This limitation results in the photocurrent model not being able to describe the effect of the filament length on THz radiation.

In our experiment (Figs. 4–6), polarization status and spatiotemporal walk-off have been adjusted to the optimal status. The peak power of the driving laser exceeds the critical power needed for self-focusing in air, which means that the filament produced by all focal lengths reaches saturation of the peak intensity inside the filament^[22,24,25]. As shown in Figs. 3 and 5, the diameter of the filament gradually increases while the electron density inside the filament gradually decreases. The difference is that the filament length is significantly extended. By characterizing the plasma density, filament diameter, and filament length and precisely calculating the THz radiation along the whole filament length (Fig. 6), we can conclude that the filament length is the core parameter that influences the THz radiation intensity. When the filament length is lengthened, the THz signal increases obviously. The simulation curve shows a good coincidence with the measured THz signal in the experiment.

In summary, it has been shown that THz generation by two-color filament can be effectively modulated by changing the filament length, after precisely controlling the polarization and the spatiotemporal walk-off of the two-color laser. Besides, the filament diameter and the plasma density in the filament are not the dominant factors influencing the THz radiation intensity. This experimental result complements the traditional transient photocurrent model at the intense long filament condition. The simulation in this Letter can support the experimental results, where the THz radiation from the total two-color laser filament is considered as a THz signal accumulation from each filament cross-section. This practical THz modulation mechanism can be used for promising THz sources.