In 1993, Hamster firstly observed sub-picosecond (ps) terahertz (THz) pulses from gas-plasma produced by single-color femtosecond (fs) laser.^[1] Since then, more plasma-based THz generation schemes, such as external DC-bias, two-color fields and few-cycle pulses, have been reported.^[2–4] Thanks to the development of laser pulses with higher power and shorter pulse-width, the THz peak electric field strength has exceeded few MV/cm and the bandwidth has reached up to 200 THz.^[5,6] Thus gas plasma-based THz generation schemes are widely used in the laboratory today, and even commercial products have appeared.

Among these mentioned methods, the two-color experimental scheme, firstly reported by Cook et al. in 2000,^[7] i.e., using a superposition of both fundamental and second-harmonic (SH) pulse fields to generate the plasma, has attracted the most research interest.^[8–10] Its physical mechanism was initially treated as a four-wave mixing process. However, subsequent measured results overturned this intuitive explanation.^[3,11] Using the so-called transient photocurrent model, Kim et al. pointed out that coherent THz waves originate from a net electron current surge generated by an asymmetric two-color laser field,^[9,11] which is now widely accepted. Thus, based on this explanation, researchers have dedicated their efforts to finding various asymmetric ultrashort laser fields, in some cases with additional static electric or magnetic fields, to manipulate the emitted THz fields. For example, multiple-color fs-lasers can boost the THz conversion efficiency significantly to obtain higher intensity THz pulses.^[12,13] Kim et al. demonstrated that the polarization of the THz wave is affected by successive polarization rotation of the local THz plasma sources and the velocity mismatch between the pump laser and the generated THz wave.^[14] Appling a helical electric field along a plasma region, Lu et al. obtained an elliptically polarized THz wave.^[15] Furthermore, the latest research suggests that polarization of the THz wave can be tuned by adjusting the time delays or intensity ratio in the input three-pulse configuration.^[16] In 2015, Wang et al. reported that the magnetic plasma offers an approach to emit circularly or elliptically polarized THz radiation.^[17] However, the results of their two-dimensional (2D) PIC simulations demonstrated that the generated circularly or elliptically polarized THz radiation is narrowband.

In this paper, we utilize the transient photocurrent (PC) model to theoretically investigate the mechanism of polarized THz pulse generation when a static magnetic field is imposed on the gas plasma along the propagation direction of the two-color laser. The simulation results demonstrate that the linearly polarized THz radiation under weak external B-field will be gradually turned to an elliptically or circularly polarized one when the magnetic field strength B is higher than 10 T. Moreover, the generated elliptically or circularly polarized THz wave is broadband rather than narrowband in Ref. [17], and its intensity can be enhanced. The physical mechanism is attributed to that the asymmetric acceleration process of the ionized electrons is changed from one-dimensional to two-dimensional due to the presence of the magnetic field.

It is noted that magnetic fields with tens of tesla, in the form of ms-pulsed field can be easily obtained.^[18,19] Applying destructive methods, even hundreds of tesla, ns-pulsed and thousands of tesla, ns-pulsed B-fields are already available.^[20–22] Compared with two-color fs-lasers and ps-THz pulses, these magnetic fields can be treated as static fields. Besides these traditional methods, Santos et al. proposed an enlightening method to generate kilo-tesla, ns-pulsed quasi-static B-fields by employing intense laser-driven capacitor-coil targets.^[23,24] According to their reports, the B-field strength can be tuned by the laser energy and the B-field direction is perpendicular to the cross section of the hollow coil. By using this method, an all-optic scheme can be proposed that focuses the two-color laser on the coil center to generate a magnetized gas plasma.

As shown in Fig. 1, when two-color linearly polarized (along the x direction) fs-laser pulses are focused in a gas medium, a plasma will be generated around the focal point, where a static magnetic field is imposed on along the z-direction. Then the freed electrons form an asymmetric ionization current whose direction is deviated from the polarization direction of the fs-laser due to the Lorentz electromagnetic force.

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of THz generation in a photoinduced plasma using two-color laser with a static magnetic field imposed on the plasma.

Based on the PC model, the radiation field from plasma is proportional to the rate of change of such asymmetric ionization current, written as^[9]

In our simulation, the fundamental laser has a Gaussian formation, center wavelength λ = 800 nm, full width at half maximum (FWHM) T_FWHM = 50 fs, and focal beam radius w₀ = 10 μm. According to the experimental scheme, a 0.1 mm thick type-I β barium borate (BBO) crystal is used to generate the second harmonic (SH) field. The peak intensities of the fundamental and the SH laser pulses after the BBO crystal are I_ω = 1.29 × 10¹⁴ W/cm² and I_2ω = 0.45 × 10¹⁴ W/cm², respectively. We assume that the fundamental and the SH laser pulses have the same polarization direction, which can be precisely controlled by a dual-band waveplate. The gas is nitrogen and its density is assumed to be 2.4 × 10¹⁹ cm⁻³. The strength of the magnetic field B ∈ [0,1000 T].

Firstly, we obtain the THz E-field amplitudes along the x and y directions as the functions of time and magnetic field B, as depicted in Figs. 2(a) and 2(b), respectively. From these two figures, we extract some THz waveforms (E_x and E_y) corresponding to different B-values (see Figs. 2(c) and 2(d)). When B < 10 T, E_x is a single cycle pulse, while E_y has no significant value compared to E_x. Therefore, the B-field at this scale has little effect on the generation of THz radiation, which agrees with the conclusion in Ref. [17]. When B increases from 10 T to 500 T, the values of E_y are prominent and continually enhanced (see Fig. 2(d)), while the peak values of E_x are reduced gradually (see Fig. 2(c)). It is worth noting that both E_x and E_y are still single cycle pulses with the increase of B.

Fig. 2.

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	Fig. 2. (color online) (a) Calculated THz field amplitude along the x axis as a function of the time interval and the external magnetic field strength. (b) Calculated THz field amplitude along the y axis as a function of the time interval and the external magnetic field strength. (c) (d) Time-varying THz waveforms along the x and y axes in case of different magnetic field strengths.

We plot the radiation spectra along two directions in the THz gap in Figs. 3(a) and 3(b) to investigate their characteristics. Figure 3(a) reveals that the B-field has no significant effect on the spectral range along the x direction, but induces a slight enhancement to the spectral amplitude. Figure 3(b) indicates that the spectrum along the y direction undergoes a process from scratch with increasing B. When B = 10 T, both spectral range and spectral amplitude along the y direction are much smaller than those along the x direction. With increasing B, the effective spectral range along the y direction is gradually enlarged. Three other different B-values (200 T, 350 T, and 500 T) correspond to the effective spectral ranges of 17 THz, 28 THz, and 40 THz, respectively. As a consequence, our simulations reveal that the THz radiation generated along both directions are broadband in this scheme and the effective spectral range along the y direction can be tuned by varying B.

Fig. 3.

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	Fig. 3. (color online) (a) and (b) Radiation spectra along the x and y axes in case of different magnetic field strengths. (c) and (d) Phase difference and amplitude ratio between x and y components in frequency domain with different magnetic field strengths.

Figures 3(c) and 3(d) plot the phase difference φ_x − φ_y and the amplitude radio E_x/E_y between the x and y components in frequency domain with different B-field strengths, which are main characteristics of the polarized beams. When B = 10 T, the phase difference increases almost linearly with the frequency and the amplitude radio is greater than 1. Hence, the THz radiation is still nearly linearly polarized with B = 10 T. With increasing B, the curves of phase difference become smooth and gradually approximate the value of −π/2, see Fig. 3(c). In the meantime, the regions of the amplitude radio curves close to value 1 increase from several THz to tens of THz, as shown in Fig. 3(d). Summing up the above, the THz radiations change from linearly polarized to nearly broadband circularly polarized with the increase of B.

Next we plot the peak values of E_x and E_y versus B-field in Fig. 4(a). It is clear that the peak THz electric field along the x direction decreases consistently, while that along the y direction increases firstly and slightly decreases finally with increasing B. They become equal at 325 T. Figure 4(b) plots the THz pulse energy versus B, the maximum of the THz pulse energy is obtained at 410 T. Figure 4(c) plots the phase difference φ_x − φ_y and the amplitude radio E_x/E_y between the x and y components in frequency domain with B = 325 T. We can find that the phase difference φ_x − φ_y is around −π/2 for 0–20 THz and the amplitude radio E_x/E_y is approximately equal to 1 for 5–18 THz. Therefore, the THz radiation can be considered as a nearly broadband circularly polarized radiation when B = 325 T.

Fig. 4.

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	Fig. 4. (color online) (a) Peak THz electric fields along two directions versus the static magnetic field strength. (b) THz pulse energy versus the static magnetic field strength. (c) Phase difference and amplitude ratio between the x and y components in frequency domain when the magnetic field B = 325 T.

To further investigate the polarized properties of the THz radiation, we plot the three-dimensional THz electric field at B = 325 T and decompose it into three mutually orthogonal components with their polarization perpendicular to each other, as shown in Fig. 5(a). It is clearly seen that the absolute values of amplitude of the x and y components are almost equal. Thus the THz wave is approximate circularly polarized. Moreover, figures 5(b)–5(i) show the projections of the three-dimensional THz electric fields in the x–y plane, which make the relationship between the polarization of the THz radiation and the B-value more intuitive. The THz radiation is nearly linearly polarized when B < 10 T. With the increase of B, the linear polarization gradually turns into elliptically polarization. When B is up to 325 T, a nearly circularly polarized THz radiation is generated. Therefore, the polarization of the THz radiation is tunable via changing the B-field strength in this scheme.

Fig. 5.

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	Fig. 5. (color online) (a) Temporal evolution of THz electric field with the magnetic field B = 325 T: three-dimensional (black) and projected (green) fields, as well as the x (blue) and y (red) components. (b)–(i) Projections of three-dimensional THz electric fields in x–y plane with different magnetic field strengths.

As mentioned above, the asymmetric ionization of bound electrons and the asymmetric electron acceleration play crucial roles in the whole process of THz generation. With an external B-field imposed on the gas plasma, these asymmetry processes could be more complex. By analyzing the force acting on the ionized electron motion, we will discuss our results. As the direction of the B-field is along the propagation direction of the two-color laser, the freed electrons will be deviated from the polarization direction of the laser due to the Lorentz force. After neglecting an attenuation item caused by the electron–ion collision, the freed electrons subjected to the laser field and static B-field accelerate as

Fig. 6.

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	Fig. 6. (color online) (a) and (b) Electron velocity and forces in the x–y plane at different time.

We decompose the electron velocity and forces in the x and y directions at two different time, as shown in Figs. 6(a) and 6(b). Vectors v and B can be written as v = v_x + v_y = v_x e_x + v_y e_y and B = B_x + B_y = B_x e_x + B_y e_y. Angle α is the angle between the velocity vector and the y axis. Angle β is the angle between the Lorentz force and the y axis, β = α −π/2. No matter which quadrant velocity vector lies in, the acceleration of the electrons can be given by

The electron velocities along the x and y directions are plotted in Figs. 7(a) and 7(b), from which it can be seen that an external B-field of 325 T introduces a y-component to the electron velocity, which complicates the electron motion in the two-dimensional space. In addition, Figure 7(c) indicates that the ionized electrons only oscillate along the direction of the laser polarization without a B-field. In contrast, with an external B-field of 325 T imposed, the electron motion turns to be two-dimensional, and the electron oscillation amplitude along the y direction is approximately one-tenth of that along the x direction, as shown in Figs. 7(c) and 7(d). As a result, the presence of the external B-field changes the ionized electron motion from one-dimensional to two-dimensional, generating elliptically or circularly polarized THz radiations.

Fig. 7.

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	Fig. 7. (color online) (a) and (b) Electron velocities along two orthogonal directions with and without a static magnetic field. (c) and (d) Electron displacements along two orthogonal directions with and without a static magnetic field.

By using the transient PC model, we systematically investigate the THz pulse generation when a static magnetic field is imposed on gas plasma along the propagation direction of the two-color laser. The influence of the static magnetic field is analyzed from the point of view of ionized electron motion. It is found that the static magnetic field introduces a y-component of electron motion, changing the THz radiation from the linearly polarized one to an elliptically or circularly polarized one. Our simulations demonstrate that the polarization of the THz radiation can be tuned by changing the magnetic field strength. If the magnetic field strength is less than 10 T, the THz radiation is still nearly linearly polarized. When the magnetic field strength increases, the generated THz radiation gradually turns to be elliptically or even nearly broadband circularly polarized. Consequently, this scheme offers an extremely significant way to obtain tunable polarized broadband THz radiation.