1. Introduction

Enhancement of the terahertz (THz) radiation is one of the primary aims for the THz generation from gas [1,2] or liquid [3–5] plasmas, which is beneficial to the THz imaging and light-matter interaction [6,7], such as electron-hole re-collisions [8], high harmonic generation [9], electron acceleration [10], and ultrafast material dynamics [11,12]. To excite a gas plasma with a two-color laser field is one of the effective methods to create broadband [2,13–15] and strong [16] THz waves. It is reported that the plasma-based THz fields pumped by 0.8 $\mu$m laser pulses can reach to 8 MV/cm [17,18]. Using a longer wavelength can further improve the THz conversion efficiency [19–22]. The THz field amplitudes can even reach to 100 M/cm with high conversion efficiency by using mid-infrared laser pulses [23].

As the high-power laser pulses apply for THz radiation in an air filament, the output THz power may not achieve the expectation with the laser energy [24]. It even decreases with a continuous increase of laser power [25,26]. Expanding the plasma volumes have been reported to boost the radiated THz power by focusing loosely for a long filament [16,25,27], by one-dimensional focusing with a cylindrical lens [24], and by some special configurations, e.g. parallel [28–30] or cascade multi-filaments [31–33], and so on.

So, to make clear the underlying physics of THz radiation in a long filament excited by a two-color laser field is significant but interesting for developing a strong and broadband THz source. Generally, it is explained by the photocurrent (PC) model [34] or the four-wave-mixing (FWM) model [2]. Both of them work in a short filament without considering the propagation effect. The linear-dipole array (LDA) model was proposed to interpret the mechanism of THz radiation in a long filament [35] for controlling THz polarization [36]. Recently, an effort has been made for the phase evolution of THz radiation in an air filament [37] by applying a travelling direct-current electric field in order to understand the mechanism of the THz radiation.

In this paper, to understand the way of THz radiation in a long filament, we present a simple method to study experimentally the waveform evolution of the THz radiation. We consider a long filament to be the superposition of a series of short filaments, thus perform this work by measuring the THz temporal waveforms from the short filaments excited by the two-color laser fields with different longitudinal positions. We find that the temporal shifts of THz waveforms vary linearly with the longitudinal positions, which allows us to simulate the THz radiation in a long filament by superimposing the position-dependent THz fields from the short filaments. Our simulations have successfully explained the process of the polarization-controlled THz radiation in a long filament excited by a two-color laser field including a circularly polarized laser field and its linearly polarized second-harmonic. Our simulation also works well for the efficient THz generation and its polarization excited by circularly polarized two-color laser fields with considering the propagation effect.

2. Experimental setup

The experimental setup is illustrated in Fig. 1(a). A Ti:sapphire laser system capable of delivering an 1 kHz-3 mJ-100 fs-800 nm pulse chain which is split into the pump and the probe arms with a beam splitter. The pump passes a lens with a focal length of 125 mm and then a 150-$\mu$m-thick $\beta$-barium borate (BBO) crystal (type-I). After BBO, some of the pump is converted into its SH (400 nm). Next to the BBO, a thin ( 45 $\mu$m) dual wavelength plate (DWP) ($\lambda$/2 for 800 nm and $\lambda$ for 400 nm) is used in order to make the polarization of the rest fundamental wave (FW) to be parallel to that of the SH. The rest 800 nm pulse and its SH form the two-color laser field which is focused to excite a short air filament around focal point of the lens. As is shown, the lens, the BBO crystal and DWP are all integrated on a linear translation stage (LTS). Moving the LTS along x-axis allows us to adjust continuously the focal position. More importantly, it ensures the filament position has a constant relative phase between the FW and SH in the focal region. A 2 mm-thick silicon wafer is used after the focal region to block the rest two-color laser field but is transparent for THz radiation. The THz waves are focused by a pair of parabolic mirrors onto a <110>-cut zinc telluride (ZnTe) working as an electro-optical (EO) crystal to modulate the probe beam. In the probe arm, the probe is converted into circular polarization with a quarter-wave plate (QWP). Then the probe passes through a convex lens for the spatial profile matching between the probe and the THz wave on the ZnTe crystal. The THz temporal signal is finally characterized by a balanced detector.

 

Fig. 1. (a) Schematic experimental setup. DWP: zero-order dual-wavelength wave plate (1/2 wave for 800 nm and full wave for 400 nm waves); HWP: half-wave plate of fundamental wave; QWP: quarter-wave plate of fundamental wave; BBO: $\beta$-barium borate crystal; BS: beam splitter; PM: parabolic mirror; BD: balanced detector; LTS: linear translation stage; DS: delay stage; P: polarizer; WP: Wollaston prism; (b) The measured THz electric field and (c) its spectrum and phase.

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3. Experimental results

Figure 1(b) shows the measured THz electric field by pump-probe method. Figure 1(c) presents the corresponding spectrum (the solid red line) and phase ($\phi _f$) obtained by Fourier transformation (the solid blue line). The orange dashed line is the linear fit of the phase as a function of frequency f. The fitted phase can be described as $\phi _f = \alpha \cdot 2 \pi f+\beta$, where $2\pi \alpha$ and $\beta$ are the slope and vertical intercept of the $\phi _f$, respectively [37]. The range of f is between 0 and 2.5 THz when linearly fitting the Fourier transformed phase here. Our calculation indicates that the THz spectra less than 1 THz are suppressed due to electron-neutral collisions, which agrees well with the PC theory.

The linearly polarized two-color laser field in the focal region is expressed as $E_1(t) = f_{FW}(t) \textrm {cos}(\omega t) + f_{SH}(t) \textrm {cos}(2\omega t + \phi _L)$, where $f_{FW}(t)$ and $f_{SH}(t)$ are the amplitudes of the FW and the SH, and $\phi _L$ is the relative phase of the two-color field, respectively. $\phi _L$ includes two parts: $\phi _0$ and $\phi _d$, and $\phi _L = \phi _0 + \phi _d$. Here, $\phi _0$, a constant, is the relative phase of two-color field at the instant the SH is generated. And $\phi _d$ is the dispersion-induced relative phase related to the propagation distance $L_p$ when the SH walks up to the filamentation. Obviously, $\phi _d$ is also a constant for a given $L_p$ as moving the LTS.

To check how the propagation effect works, the filament position is scanned along x via moving the LTS. Figure 2(a) shows a sequence of the THz waveforms with a range of x from 0 to 12 mm. The temporal shifts of the THz field are indicated by the dashed black and blue lines. As the dashed blue line shown in Fig. 2(a), the THz field has a linearly temporal shift with the increase of the location x. Figure 2(b) records how the delay $\tau$ at positive peak of the THz field depends on longitudinally position x. The $\tau$ exhibits a linear relationship with x: 15 fs per 1 mm of the movement.

 

Fig. 2. (a) Measured waveforms of the THz radiations when the stage is located from x = 0 to x 12 mm. And (b) the corresponding temporal shift. The extracted (c) $\alpha$ and (d) $\beta$ as a function of x and their linear fitting results.

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To further estimate the evolution of THz waveform, we apply a Fourier transform to the sequence THz fields and fit the phases with $\phi _f = \alpha \cdot 2 \pi f+\beta$ as we did in Fig. 1(c). According to the previous report [36], $\alpha$ represents the time delay between the probe pulse and the THz field, while $\beta$ is the carrier-envelope phase of the THz field. Figures 2(c) and (d) depict the $\alpha$ and $\beta$ as a function of x, where the orange and gray lines, denote the linear fit of $\alpha$ and $\beta$ with $d\alpha /dx =20$ fs and $d\beta /dx=0.017\pi$, respectively.

Next, considering the clamping intensity, the longitudinal intensities of the laser fields along a long filament can be further explored. For this purpose, we focus on the filament length dependent THz radiations excited by circularly polarized two-color laser, where the intensity of THz radiation from each slice of filament will not be affected by the Gouy phase shift. The measured setup is shown in Fig. 3(a). Here, the filament length is 22 mm. A metal iris with a clear aperture of 2 mm is installed with its axis along the filaments to block the forward-propagating THz signals before the iris. Therefore, the filament length of the THz radiation is effectively controlled. The THz radiation is focused by a pair of parabolic mirrors and collected by a Golay cell. The measured THz powers and their linear fit are depicted in Fig. 3(b). Interestingly, the THz powers increase linearly with the filament length. This implies the intensities of laser fields in a long filament are almost uniform.

 

Fig. 3. (a) The experimental setup of circularly polarized two-color laser field generation. b-BBO: type I $\beta$-BBO crystal; a-BBO: $\alpha$-BBO crystal; lens: a lens with a focus length of 30 cm; AQWP: achromatic quarter-wave plate. (b) The experimentally measured relationship between the energies of THz radiations and filament length. (c) The relative phase $\phi _L(x)$ as a function of x.

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The Gouy phase shift in plasma can be calculated. In an air plasma, the frequency-dependent refractive index for the excited beam can be expressed as $n(\omega )=1+\delta n_{plasma}(\omega )$, where $\delta n_{plasma}(\omega )=(1-\omega _p^2/\omega ^2)^{1/2}$ [38] with the plasma frequency $\omega _p = (n_ee^2/m_e\varepsilon _0)^{1/2}$.Here, $\varepsilon _0$ is vacuum permittivity, while $e$, $m_e$, and $n_e$ are electron charge, mass, and density, respectively. $n_e$ is set as $1.74\times 10^{17} cm^{-3}$. In a long filament, $\phi _L(x)=\phi _0+\phi _d +\phi _{plasma}(x)$, where $\phi _{plasma}(x) = 2\pi \textit {x}[n(2\omega )-n(\omega )]/\lambda _{800nm}$. x is the longitudinal position of the plasma ranging from 0 to l (the filament length). We calculate the $\phi _{plasma}(x)$ as a function of x, as displayed in Fig. 3(c). It shows a linear relationship between the $\phi _{plasma}(x)$ and x and suggests that the $\phi _L(x)$ would have $\sim 2\pi$ phase shift while the excited beam travels 22 mm in the filament.

Here, we would replicate the process of THz radiation from a long plasma by the two-color laser fields composed of the circularly polarized FW and linearly polarized SH. We extract the measured waveform of THz field at x = 0, labeled as $E_{THz}(t_0)$, so the THz field at x shall be $\boldsymbol{f}_{THz} (x)=E_{THz} (t_0+\tau _x ) e^{i\theta _x} [\textrm {cos}\phi _L (x)\hat {e}_y+\textrm {sin}\phi _L (x)\hat {e}_z]$, where $\tau _x=x d\alpha /dx$, $\theta = x d\beta /dx$, and $\phi _L (0)=0$. Many reports indicated the THz radiation in a long filament is the coherent longitudinal superposition of THz waves from each slice of the filament [35–37,39,40], so the far-field THz radiation $\textbf {E}_{THz} (t)= \int _0^l\boldsymbol{f}_{THz} (x)dx$, where l is the filament length. Our calculation results are presented in Fig. 4. We can see, while increasing the filament length, the polarization of the output THz field can be controlled cyclically from linear to elliptical, circular, which agrees with our experiments and theoretical results [36]. It means that we have successfully characterized the generation of THz radiation in a long filament, which gives us a further understanding of the LDA theory.

 

Fig. 4. The calculation results of polarization-dependent THz radiation at different filament length.

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4. Discussions

The results above suggest the waveform evolution of THz field from a long filament can be simulated by the coherent superposition of those from a series of short filaments. Here, we would discuss the enhancement and polarization of the THz radiation by a co-rotation circularly polarized two-color laser field from a long filament.

It is well known that a circularly polarized two-color laser field can enhance the THz radiation comparing with a linearly polarized two-color laser field from a short filament [41]. Our previous work confirmed it can strengthen the THz radiation from a long air-filament, too [42]. The enhancement of THz radiation from a short filament is explained by the PC theory, which depends deeply on the laser pump intensity [41]. However, the enhancement can be suppressed due to the intensity clamping. Additionally, the free electron trajectories ionized by the circular two-color laser fields are similar to those by a two-color laser field composed of circular FW and linear SH, so a circular two-color field may be better to generate the polarization-controllable THz field.

To investigate the THz radiation from a long filament with a circularly polarized two-color laser field, we firstly calculate the THz radiation in a slice of plasma with the PC theory. In the simulation, the peak intensities of FW and its SH in a two-color laser field with linear polarization are set to $1.2\times 10^{14}$ ${\rm W}/cm^2$ [43] and $0.18\times 10^{14}$ ${\rm W}/cm^2$ in the filament region, respectively. Distinctly, the peak intensities of FW and SH with circular polarization are only $0.6\times 10^{14}$ ${\rm W}/cm^2$ and $0.09\times 10^{14}$ ${\rm W}/cm^2$, respectively. Here, we compare the THz intensities from three kinds of laser fields. For convenience, the three kinds of fields, i.e., linearly polarized two-color laser fields, two-color laser fields composed by circularly polarized FW and linearly polarized SH, and co-rotation circularly polarized two-color laser fields, are labelled as $FW_\textrm {L}SH_\textrm {L}$, $FW_\textrm {C}SH_\textrm {L}$, and $FW_\textrm {C}SH_\textrm {C}$, respectively.

According to the static tunnelling ionization theory [44], the ionization rate can be expressed by $w(t)=4\omega _a (U_{O_2}/U_H )^{5/2} [E_a/E_L(t)]exp[-2E_a/3E_L (t) \cdot U_{O_2}/U_H ^{3/2}]$ with $\omega _a = 4.134\times 10^{16} s^{-1}$. Here, $E_a$ and $E_L$ are the amplitudes of the atomic field and the laser field, while $U_H$ and $U_{O_2}$ are ionization potentials of hydrogen atoms and oxygen molecules, respectively. The local current (LC) approximation [45] without propagation effects shall be described by $\partial \boldsymbol{J} (t)/\partial t=q^2/m\cdot \rho (t) \boldsymbol{E} (t)- \boldsymbol{J} (t)/\tau _c$ and $\partial \rho (t)/\partial t=w(t)[\rho _0-\rho (t)]$, where $\rho _0$ is the neutral molecule density, $q$ and $m$ are the charge and mass of the electron, and $\tau _c$ is the current decay time due to collisions.

Figure 5(a) shows the calculated ionization probabilities as a function of laser phase $\phi _L$ in different laser fields. As is seen, the ionization probability in $FW_\textrm {L}SH_\textrm {L}$ approaches saturation but only $\sim 45\%$ and $\sim 50\%$ molecules are ionized in $FW_\textrm {C}SH_\textrm {C}$ and $FW_\textrm {C}SH_\textrm {L}$, respectively. Apparently, the high ionization rate brings larger residual current in $FW_\textrm {L}SH_\textrm {L}$. Figure 5(b) presents the simulation results of the generated THz energies with x. Interestingly, the maxima of THz radiations in three kinds of laser fields are close, which suggest the $FW_\textrm {C}SH_\textrm {C}$ and $FW_\textrm {C}SH_\textrm {L}$ can strongly stretch the electron drift comparing with $FW_\textrm {L}SH_\textrm {L}$ as the previous studies [41,46]. As is well known, the THz radiation in $FW_\textrm {L}SH_\textrm {L}$ is dependent on sin$^2\phi _L(x)$. In a long filament, the intensity of THz radiation $\int _0^lP_{THz} (x)dx$ with $FW_\textrm {L}SH_\textrm {L}$ is suppressed by the Gouy phase. As is shown in Fig. 5(b), the intensity with $FW_\textrm {C}SH_\textrm {L}$ is also suffered a bit. As we extract from Fig. 2(b), we estimate the intensity of THz generated from $FW_\textrm {C}SH_\textrm {C}$ can reach to 2.5 times of that from $FW_\textrm {L}SH_\textrm {L}$ in the 22-mm-long filament. Besides, the intensity from $FW_\textrm {C}SH_\textrm {L}$ can also reach to 1.5 times of that from $FW_\textrm {L}SH_\textrm {L}$.

 

Fig. 5. (a) The simulated ionization probabilities of O$_2$ and (b) The energies of THz waves as a function of x with different polarized two-color laser fields.

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Furthermore, we have simulated the 3-dimensional electric field of THz radiation with different filament lengths in $FW_\textrm {C}SH_\textrm {C}$. We take the method similar to that in Fig. 4, but calculate the initial THz waveform at x = 0 by LC model. In the simulation, we assume the THz fields are recorded by a <110> ZnTe EO sensor, so the calculated THz bandwidth is limited to a range from 0 to 3 THz due to the sensor bandwidth. The simulation results are displayed in Fig. 6. As the filament length increases, the polarization of the THz radiation changes cyclically from linear (4 mm) to elliptical (12 mm), circular (18 mm), and elliptical (22 mm) again. Thus, $FW_\textrm {C}SH_\textrm {C}$ can also be used to manipulate the polarization of THz radiation, which is superior than $FW_\textrm {C}SH_\textrm {L}$ for THz power. Assume that the THz radiation in a long plasma is expressed as $\int _0^L E_{THz} \begin {pmatrix} cos(\omega _\textrm{THz}t) \\ cos(\omega _\textrm{THz} t+\phi _l) \end {pmatrix} dl$, where $\phi _l$ is relative phase shift of the THz field in the propagation process. We can see that $\phi _l$ can cause the length-dependent polarization state of the THz radiation. In order to show intuitively the changes in the polarization states with different filament lengths, we measured the $E{min}/E_{max}$ as a function of the filament length in Fig. 6(e), where $E_{min}$ and $E_{max}$ are the electric fields of the minor and major axes in the polarization plane.

 

Fig. 6. (a-d) The simulated 3-dimensional electric trajectories of THz waves with different lengths of the laser filament. (e) The $E_{min}/E_{max}$ with the different filament lengths

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5. Conclusion

In this paper, we have experimentally investigated the propagation effect of THz radiation from a long filament by checking the evolution of THz temporal waveform through shifting the longitudinal position of a short filament. Adding the extra intensity distribution of laser fields in a long filament, we rebuilt successfully the polarization-controllable THz radiation in a long filament by $FW_\textrm {C}SH_\textrm {L}$, which can well match with the LDA theory. It is helpful to understand the mechanism of THz radiation in a long filament. And we discussed how the intensity of THz radiation in $FW_\textrm {C}SH_\textrm {C}$ is stronger than that in $FW_\textrm {L}SH_\textrm {L}$. The simulation results suggest the Gouy phase shift in a long filament suppressed the THz radiation with $FW_\textrm {L}SH_\textrm {L}$. Besides, $FW_\textrm {C}SH_\textrm {C}$ can flexibly control the polarization of THz radiation with higher THz power.