1. Introduction

Terahertz (THz) radiations are subject to great research interests in view of its widely spanned applications such as remote-sensing, chemical spectroscopy, biomedical diagnostics and threat detection [1-3]. Air is of particular interest for THz generation medium for the simplicity of the required setup and the absence of optical damage [4-9]. However, due to the water vapor absorption, THz waves suffer from great loss during their propagation in the atmosphere. THz generation through laser filamentation process has demonstrated a great potential to avoid this strong attenuation [6-9]. By simply manipulating the position of a filament, THz pulses can be emmited at a remotely located target [10]. In fact, this idea has been proven by C. D’Amico et al. recently [6]. The detected THz wave in their work was radially polarized and confined in a forward cone. They interpreted it as a transition-Cherenkov radiation from space charge moving at light velocity in the wake of a femtosecond laser filament.

In this paper, we also have investigated the THz radiation from a femtosecond laser filament in air. The experimental evidences imply that the polarization of the observed THz pulse differed from the previous reports. It consisted of elliptically polarized component. An intuitive explanation about the generation mechanism of elliptically polarized THz pulse has also been proposed. Two orthogonally polarized THz wave could be generated by either four-wave rectification [8] or second order rectification due to the spatial asymmetry of filament plasma. Taking into account the optical birefringence brought forth by femtosecond laser pulse [11], the output THz pulse from the filament could be elliptically polarized depending on the amplitude and phase difference of two orthogonal polarization components.

2. Experimental setup

In our experiment, a 1 kHz, 800 nm, 45 fs Ti-sapphire laser beam was split into two paths. One was used as the pump for the THz wave generation, and the second as the probe for the electro-optic sampling (EOS) setup used to diagnose the THz pulses. Figure 1 schematically illustrates the experimental configuration. The pump beam had an energy of 1.15 mJ/pulse and was focused in ambient air by a plano-convex lens having a focal length of 50 cm. The focused laser beam created a 3-cm long filament. The emitted THz pulse from the filament was collimated by a parabolic mirror (PM1) with a 4-mm diameter hole in its center. Most of the fundamental pulse energy passed through this hole. A 5-mm thick Teflon plate having a diameter of 75 mm was used after the first mirror to block the residual fundamental light. The collimated THz pulse was focused by another parabolic mirror (PM2) into a 1-mm-thick <110> oriented ZnTe crystal (Zomega Terahertz Corporation). The [0,0,1] axis (Z axis) of the ZnTe crystal was oriented horizontally throughout the experiment. The diameter and focal length of these two parabolic mirrors were 5 cm and 10 cm, respectively. The diagnostic of the THz pulse was realized by a standard time-resolved EOS method [12]. In this case, the probe beam was combined with the THz beam by a pellicle beam-splitter (PBS, R=45% for 800 nm) and the two beams propagated collinearly through the ZnTe crystal.

 

Fig. 1. Schematic of the experimental setup. Time delay of the probe beam could be varied by a delay line which is not shown here.

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3. Results and discussions

Figure 2 shows a typical THz electric field waveform obtained in our experiment. The corresponding THz spectrum is given in the inset of Fig. 2. Note that the oscillations after the first cycle of the THz signal in Fig. 2 result from the strong water vapor absorption in air. Figure 3 presents the peak-to-peak amplitude of the THz electric field (solid squares) at different polarization orientation of the pump beam. In Fig.3, θ is defined as the angle of the polarization of the pump beam with respect to the Z axis of the ZnTe crystal. The variation of θ was experimentally achieved by rotating a zero-order half-wave plate (HWP) before the convex lens, while the polarization of probe beam was fixed horizontally. As shown in Fig. 3, the maximum signal was obtained when θ=0°, while the minimum signal was observed when θ=90°. Based on this result, we could deduce that the observed THz pulse can not be totally attributed to the transition-Cherenkov radiation from the axial current excited by the ponderomotive force of a short laser pulse which is independent on the polarization orientation of the driving laser pulses [6].

 

Fig. 2. Typical experimental THz electric field waveform and the corresponding THz spectrum in the inset.

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We first verified if the observed THz pulse was linearly polarized. Recalling that by using EOS method to detect THz wave, the change of the signal as a function of the angle between the THz electric field polarization and the [0, 0, 1] axis of the ZnTe crystal obeys the following rule: [13]:

Where ϕ and α correspond to the angles of the THz wave polarization and the probe beam polarization with respect to the [0,0,1] axis of the ZnTe crystal, respectively, while E_THz denotes the magnitude of the THz electric field which is assumed to be independent of θ in the calculations as all the experimental conditions except for the pump beam polarization were kept constant. Since α=0° in our experiment, Eq. (1) can be further shortened as:

 

Fig. 3. Solid squares (right axis): measured peak-to-peak amplitude of THz electric field as a function of θ defined as the angle of the pump pulse polarization with respect to the [0,0,1] axis of the ZnTe crystal); red solid line (left axis): calculated orientation dependent EOS signal of a linearly polarized THz wave according to Eq. (2) (θ=ϕ, ϕ being the angle between the THz wave polarization and the [0,0,1] axis of ZnTe crystal); green dashed line (left label): the same as the red solid line but θ=ϕ+90°, i.e. polarization of the THz wave is perpendicular to that of the pump beam.

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We first assumed that the polarization of the generated THz pulse was the same as the pump beam, namely, θ=ϕ. For this case, the calculated EOS signal as a function of θ is indicated in Fig. 3 by the red curve (left axis). Immediately, one can find that it does not fit the experimental results. Furthermore, we calculated the case where the polarization of THz pulse is perpendicular to that of the pump beam (θ=ϕ+90°). The result curve is shown in Fig. 3. by the green line. The calculated signal peaks at θ=0° and has a minimum at θ=180°. Clearly, it does not fit the experimental results either. The assumption of linear polarization failed to reproduce the experiment results. Hence, the discrepancy between the calculated and experimental results ruled out the possibility that the THz polarization is linear, which could be related to ponderomotive THz emission with the present of pre-formed plasma [4, 5]. There is therefore only one choice left for the polarization of the THz pulse generated by the filament in our experiment, and it is elliptical polarization

However, it is more difficult to directly compare the evolution of the θ dependence EOS signal of an elliptically polarized THz pulse with the theoretical prediction compare with the linear polarization case as in Fig. 3. Nevertheless, we know that the energy of the electric field recorded by the EOS setup is proportional to the square of the field peak-peak amplitude, which has been readily given in Fig. 3 as a function of θ. On the other hand, according to Eq. (2), when the incident THz wave is elliptically polarized, the EOS setup only response to the projection of the THz electric field on the vertical axis since the Z axis of the ZnTe crystal is oriented horizontally. It is equivalent to calculate the transmission through a polarizer whose optical axis is along the vertical direction in the experiment. Therefore, for elliptically polarized THz wave, the energy of the recorded electric field by EOS setup would have the same analytic solution as the transmittance through the equivalent polarizer, which yields:

where e represents the ratio of the magnitudes of the minor axis to the major axis of the polarization ellipse, and β corresponds to the angle between the major axis and the optical axis of the virtual polarizer (i.e. vertical direction in our case). When θ varies, β is simply given by β=β ₀+θ.β ₀ denotes the angle between the major axis and the vertical direction when θ=0. Based on the above arguments, we first squared the measured peak-to-peak amplitudes of the EOS waveforms. The outcome is depicted in Fig. 4 as the solid squares. Afterwards, we optimized the parameters of Eq. (3) in order to fit the measured curve. The red solid curve in Fig. 4 was obtained with the parameters: e=0.25 and β ₀=11.5°. Thus, we have confirmed that an elliptically polarized THz pulse has been observed in our experiment accompanying the occurrence of femtosecond laser filamentation in air.

 

Fig. 4. Solid squares: normalized energy of the electric field recorded by the EOS setup; red solid line: calculated corresponding energy of an elliptically polarized THz wave according to Eq. (3) with e=0.25 and β ₀=11.5o.

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It is necessary to point out that a combination of linear polarized THz wave (i.e. e=0) and a radially polarized THz wave, which would lead to constant EOS signal, could give rise to a result similar to Fig. 4. However, the results presented in Fig. 3 have indicated that neither radially polarized nor linearly polarized THz wave had a significant contribution in the experiment. Therefore, only elliptically polarized THz wave need to be considered in the present case. The depolarization of THz wave was observed in ref. 5 by using 50 mJ, 120 fs laser pulses and with strong external focusing (f=5 cm). The authors of ref. 5 explained this phenomenon by severe laser beam distortion and breakup. Since we did not observe beam breakup and distortion during our experiment, this hypothesis could not explain our observation. Nerveless, it was reported that THz wave could be generated by femtosecond laser pulse in air through the four wave rectification process [8]:

Eq. (4) hints that the optical frequencies ω ₁, ω ₂ and ω ₃ should satisfy ω ₁+ω ₂≅ω ₃. Since we did not use the popular ω-2ω scheme [8, 14-16], it is required that the pump laser spectrum need to be octave-spanned [17]. This condition can be easily fulfilled because in the course of filamentation, the laser spectrum could cover an extremely broad rang, extending from ultraviolet to infrared [18-20]. Hence, THz wave with polarization parallel to the pump polarization (referred to x direction in the following discussion) can be obtained in air via

where Δk=k ₁+k ₂-k ₃ describes the phase matching condition and L is the effective interaction length. On the other hand, the generation of orthogonally polarized THz wave could be realized mathematically through

Note that χ ⁽³⁾ _yxxx=0 for an isotopic material like air. However, X. Xie et al. have observed non-vanished χ ⁽³⁾ _yxxx when plasma is produced in air using femtosecond Ti-sapphire laser pulses. They have attributed it to the spatial asymmetry of laser induced plasma [8]. Accordingly, THz pulse could be generated along both orthogonal axes. Two components would travel at different phase velocities due to the substantial difference in the non-linear refractive indices generated by the femtosecond laser pulse along its polarization axis and the orthogonal axis, respectively [11]. Finally, an elliptically polarized THz pulse could be generated at the end of the filament. The shape of the polarization ellipse depends on both the accumulated phase delay and the amplitudes of the two polarization components.

The argument outlined above is mainly focused on the third order nonlinear optical susceptibility. It is worth mentioning that due to the spatial non-uniformity of the laser induced plasma, second order nonlinear optical process such as second harmonic generation have been observed [21, 22] using longer laser pulses and lower intensities (order of 10¹² W/cm²) than the clamped intensity inside a filament (~5×10¹³W/cm²) [20]. This could provide another alternative hypothesis of the generation mechanism of the elliptically polarized THz pulse. The second order optical rectification could be realized in this case through ω _THz=ω _{800 +THz}-ω ₈₀₀ along both orthogonal axes leading to the detected THz pulse.

4. Conclusion

In conclusion, we have observed elliptically polarized THz emission from a filament induced by intense femtosecond laser pulses. Two orthogonal polarization components may be generated simultaneously through four-wave rectification in view of non-zero susceptibility tensor elements χ ⁽³⁾ _xxxx and χ ⁽³⁾ _yxxx. Though χ ⁽³⁾ _yxxx is vanished in normal air, it could be non-negligible in the presence of plasma [8]. The spatial non-uniformity of plasma might also allow second order optical rectification giving rise to THz wave generation along both the laser polarization and the orthogonal polarization. In this case, further study is under way to clarify the role of the involved nonlinear optical processes. On the other hand, significant birefringence induced by the femtosecond laser pulse will lead to phase delay between the laser pulse polarization axis and the orthogonal axis. This in term produces elliptically polarized THz pulse at the output of the filament. This phase delay is potentially controllable, since the filament length can be easily controlled by varying the pump laser parameters such as pulse energy, chirp and focal length [18-20], realizing controllable polarization of the generated THz wave. This can be potentially helpful to the control of excitons in semiconductor nanostructures [23] and of molecular rotational wave packets [24].