We establish a one-to-one mapping between the local phase slip and the spatial position near the focus by scanning a thin jet along the propagation direction of laser beams. The measurement shows that the optimal phase of terahertz can be utilized to characterize in situ the spatially dependent relative phase of the two-color field. We also investigate the role of the Gouy phase shift on terahertz generation from two-color laser-induced plasma. The result is of critical importance for phase-dependent applications of two-color laser-field, including high-order harmonic and terahertz generation.
© 2016 Optical Society of America
Since it was first demonstrated by Cook et. al.  in 2000, terahertz (THz) radiation from two-color laser-induced plasma has been extensively studied [2–8], due to its broad-bandwidth, easy implementation and high efficiency. Several models, including four-wave mixing  and transient photo-current [9, 10], have been proposed to explain the underlying physical mechanism. In this intense THz generation scheme, the relative phase between the fundamental and second-harmonic laser fields, as a phase-gating, is extremely critical for THz generation [11–14]. The question of determining the value of the optimal phase, which maximizes THz radiation, has to be addressed. This problem was solved very recently by a synchronous measurement of high-order harmonics and THz generation [15–17].
However, THz pulses are usually generated from air plasma in the laser focus. As is well known, the Gouy phase shift (GPS) is introduced  while the electromagnetic wave passes through its focus. This phase shift has important consequences in the optical range of the electromagnetic spectrum. It also provides us an empirically adjustable quantity in some phase-dependent researches such as controlling the chemical reactions , reshaping the electromagnetic pulses [20,21], optimizing THz detection efficiency  and measurement precision , and so on. In principle, the GPS ϕGouy  of a continuous Gaussian wave, is related to the wavelength λ and beam waist w0 by25]. Indeed, the details of the phase change in the focus depend on the spatial profile of the laser beam and the focusing geometry. The GPS of complex laser beams and ultrashort pulses cannot be easily calculated based on Eq. (1). Efforts are still being made to precisely measure and control the spatial variation of the GPS in the whole focal region. The so-called stereo-ATI (above-threshold-ionization) scheme is used to determine the evolution of the carrier-envelope phase in the focus of few-cycle laser pulses . The ion imaging is performed to characterize in situ the GPS of an annular beam for further high-precision attosecond measurements .
In this paper, we demonstrate the first experimental determination of the evolution of the relative phase in the focus of the two-color laser pulses by means of THz generation. This is important because this method can be applied to complex beam profiles and realized simultaneously with strong field experiment. By scanning the thin gas jet through the focus, we demonstrate directly that THz generation from long gas plasma is the superposition of that from sliced gas medium. We also investigate the phase averaging effect on THz generation in air plasma.
2. Experimental details
Figure 1 shows the experimental setup. 25 fs laser pulses with the energy of 1.6 mJ centered at 790 nm from a 1 kHz Ti:sapphire system (Femtopower Compact PRO; Femtolasers Produktion GmbH) are split into two arms as the pump and probe beams. The pump beam is frequency doubled by a type-I β-barium borate (β-BBO), after which temporally separated two color lasers with orthogonal polarizations are thus formed. To make their polarization parallel, a special dual-wavelength waveplate (DWP) is employed, which introduces a phase retardation of λ/2 for fundamental, and λ for the second-harmonic. A wire grid polarizer (WGP) after the DWP further eliminates the residual fundamental laser fields with different polarization. The α-BBO is used to make the two-color field temporally overlap with each other. A pair of fused-silica wedges are adopted to finely control the relative phase between the two-color laser field. The two color laser pulses are focused by a spherical mirror (SM) with a focal length of 200 mm onto a gas jet to generate THz. The jet with 50 μm inner diameter is placed in a vacuum chamber pumped down to 10−5 mbar. The generated THz waves are focused by a hole-drilled off-axis parabolic mirror (PM) onto a 1 mm thick (110)-cut ZnTe crystal, which is shined on by the delayed probe beam for electro-optic sampling (EOS) detection. The two-color laser beams after the gas jet propagate through another β-BBO crystal to frequency double the residual fundamental laser and the resulted second harmonic will interfere with the second harmonic components in the two-color field.
The interference of second-harmonics measured by Detector 2 is used to monitor the shift of the relative phase between two-color fields . Varying the relative phase between fundamental and second-harmonic fields by inserting one of the wedges, the modulation of second-harmonic is recorded at a high vacuum (∼ 10−5mbar) as shown with red squares in Fig. 2(a). When a gas jet with 50 μm diameter and 3 bar backing pressure is inserted into the focus, the measured intensity interference is shown as blue dots in Fig. 2(a). There is no significant difference between the two cases. Therefore, the dispersion introduced by the thin gas jet can be neglected.
The relative phase can control the THz intensity finely in the linearly-polarized two-color field. The total laser field is expressed as E(t) = Eω cos(ωt) + E2ω cos(2ωt + ϕ), where the Eω and E2ω are the fundamental and the second-harmonic amplitudes respectively, ϕ is the relative phase composed of two controllable parts ϕd and ϕG, ϕ = ϕd − ϕG. The dispersion-induced relative phase ϕd is related to the wedge insertion Δ, ϕd(Δ) = 2ϕω − ϕ2ω = 2ω (nω − n2ω)Δ/c, where nω and n2ω are the refractive indices of the fused silica and c is the speed of light in vacuum, the GPS-induced relative phase ϕG is z-dependent along the propagation of two color lasers, ϕG(z) = 2ϕGouy(ω, z) − ϕGouy(2ω, z). We place the gas jet at the focus of two color lasers (ϕG = 0), and record THz waveform varying the relative phase ϕd. The resolution of the relative phase ϕd is 0.04π according to the period of the second harmonic. The modulation of THz intensity is illustrated in Fig. 2(b). The experimental data (red dots) can be well fitted by a cosine function in the form of , where ITHz is the THz intensity, A is the modulation depth, is the measured optimal phase which maximizes the THz intensity by inserting the optical wedge. At different jet positions z along the laser propagation, , where ϕ0 is the absolute value of the optimal phase which can be calibrated by the intrinsic attochirp of different high-order even harmonics [15, 16].
3. Results and discussion
3.1. GPS in THz generation with two color laser fields
We can establish the mapping between the GPS-induced relative phase ϕG and the jet position, if we calibrate the optimal phase ϕ0 and record the measured optimal phase . Scanning the thin gas jet through the focus with a step of 0.5 mm along the propagation direction, we maximize THz intensity varying the relative phase by inserting one of the optical wedges. Figure 3(a) shows the measured THz yield at optimal phase when the gas jet lies in different positions. Since the stronger laser field can ionize gaseous atoms more efficiently and produce larger photocurrent, the THz intensity peaks at the focus and decreases symmetrically in the Rayleigh region as indicated in Fig. 3(a). This solves the difficulty in finding the exact focus in practical experiment. Figure 3(b) presents the relationship between the measured optimal phase and the jet position, which agrees well with the calculated GPS-induced relative phase ϕG. The results show that the change of the measured optimal phases at different gas jet positions depends mainly on the Gouy phase, which indicates that the absolute optimal phase is constant when gas jet moves. So we can use the measured optimal phase to measure the relative phase slippage of two-color laser. It also opens up the possibility of measuring the relative phase of two-color field by using the optimal phase ϕ0.
In order to further inspect the influence of other factors on the optimal phase, we fix the jet at the focus (ϕG = 0) and record the dispersion-induced relative phase ϕd under different pump intensities and the ratios of two color lasers. The laser intensity is changed from the minimum, which can deliver detectable THz radiation, to the maximum, which is estimated about 1.8 × 1014W/cm2 by the cutoff of the high-order harmonic generation . It is found that the optimal phases of THz remain approximately constant as shown in Fig. 4(a). The ratios of two color pulses are changed by using BBO crystals with various thicknesses while the laser intensity is maximum. Figure 4(b) shows the optimal phases have little shift while the intensity ratios changed from 0.1% to 25%. The additional phase shift caused by the BBO thickness can be compensated by adjusting the insertion of the wedge, which makes sure the modulation of second-harmonic is the same in every measurement. The zero points in both Figs. 4(a) and 4(b) are measured with the low intensity of two-color laser-filed whose ratio is about 5% to eliminate the effect of plasma and produce measurable THz radiation. It is worth noting that the optimal phase of THz is robust against the laser intensities and the intensity ratios of second-harmonic to fundamental pulses, which agrees well with the soft-collision theory [15,17] under the experimental conditions. Figures 4(a) and 4(b) indicate that the optimal phase of THz generation will be a good tool to calibrate and monitor the relative phase of two-color laser-field.
3.2. GPS effect for long gas medium
For long gas medium, the measured THz waveform can be considered as a coherent superposition of THz waveform from different slices along the propagation direction. We move the gas jet along the laser propagation direction at fixed ϕd, and record THz waveforms for all z-positions. Figure 5(a) shows the contour of THz waveforms when the jet lies at the different positions along the Rayleigh range with two specific relative phase delays and . Since the phase changes by π in the propagation through the focus, virtually all possible relative phase are available moving the thin gas jet. There always exists at least one position where the relative phase is optimal for THz yield. The relative phase is set optimum at the focus in the left of Fig. 5(a). The amplitude of THz electric field peaks at the focus and decreases symmetrically due to the GPS and decreased laser intensity when the gas jet moves away from the focus. It’s not easy to decouple the effect of the Gouy phase shift from the laser intensity variation on the THz yield in the whole focal region. In order to manifest the phase gating effect, we compare THz yield in the same laser intensity but at another relative phase delay . The phase-gating can turn off THz radiation at the laser focus completely and the coupling effect between the laser intensity and the GPS reproduces the maximal THz yield in the 1.5 mm as shown in the right of Fig. 5(a).
The THz waveform detected by EOS contains not only the amplitude information, but also the phase information. Therefore, it is possible to combine a THz wave by a coherent superposition of THz waves measured at all z-positions with the same ϕd. Figure 5(b) shows that the phase dependence of combined THz (middle) is the same as that measured at the focus (left). The same phase dependence is obtained for the low-pressure gas as shown in the right of Fig. 5(b), since the dispersion of the plasma and neutral atoms can be ignored when the pressure of argon in the chamber is about 10−2 mbar. Our experiment promotes to the intense research on spatiotemporal dynamics of THz generation in gases [4, 5, 21].
We consider the THz waveform from a long gas medium as the coherent superposition of many gas slices of a cylindrical source to understand the phase-dependence of THz emission. In the far field, the length of the THz source is much shorter than the propagation distance and the detected THz electric field E⃗(r⃗, Ω) can be expressed as
In low-pressure gas cell, the laser intensity distribution is symmetric about the focus, which leads to the symmetric amplitude of THz, A⃗(−z, Ω) = A⃗(z, Ω). The phase Δϕ(z) introduced only by GPS is antisymmetric, Δϕ(−z) = −Δϕ(z). According to Eq. (2), THz waveforms obtained with the superposition of symmetric slices before and after the focus will have the same phase as that generated from the focus. Therefore, Fig. 5(b) shows the same phase dependence between THz generated from the long gas medium and the gas jet at the focus.
While increasing the pressure in the chamber, the dispersion of plasma and neutral gasous atoms cannot be neglected any more. The dispersion will break the antisymmetric phase distribution induced by GPS gradually, and the symmetric THz amplitude will also be broken down. In this condition, A⃗(−z, Ω) ≠ A⃗(z, Ω), Δϕ(−z) ≠ −Δϕ(z). The integral in the Eq. (2) cannot carry out to be a real number again. When we fill argon in the chamber with a higher pressure (∼ 10mbar), the measured THz waveforms for various relative phase are shown in the Fig. 6(a). The amplitude of THz radiation cannot be reduced to zero as the black line presented in the Fig. 6(a), even if we adjust the relative phase of two-color field carefully. Figure 6(b) shows the modulation of THz intensity detected by the thermal detector, which is characterized with a noticeable background due to the phase slippage near the focus.
The GPS also can change the ellipticity of THz generation. As discussed in [11, 30], when either one of the two color pulses is elliptically or circularly polarized, the polarization of THz wave is very close to linear polarization, and the polarization direction of the THz will rotate while varying the relative phase of two color lasers. In order to display the effect of GPS more intuitively, the linearly-polarized fundamental laser is replaced by the circularly polarized, and its second-harmonic is elliptically polarized with an ellipticity of about 5/11. Here, the ellipticity is defined as e = EMinor_axis/EMajor_axis. The THz waveforms are recorded by a polarization-sensitive air-biased-coherent-detection system , Fig. 7(a) depicts the rotation of THz polarization generated at the focus when the relative phase is changed by the insertion of one glass wedge. Since the GPS leads to the spatial variation of relative phase in the whole focal region, the direction of linearly polarized THz in the plasma varies with the position. In low pressure, the laser intensity is symmetric and the relative phase is antisymmetric with respect to the focus. The coherent superposition of THz waveforms from different plasma slices produces linearly polarized THz radiation according to Eq. (2). Due to the dispersion of the plasma and neutral atoms in high pressure, the antisymmetric phase distribution of GPS is broken and THz waves are not out-of phase from the symmetric plasma slices before and after the focus. This effect is more remarkable in high pressure. Thus, the elliptically polarized THz wave can be produced from the coherent superposition in high pressure. Figure 7(b) shows the increase of THz ellipticity with gas pressure. This also indicates that precise control of the spatial phase distribution can be a simple tool to manipulate the THz ellipticity.
In conclusion, we present the first full and unambiguous investigation on the effect of Gouy phase in air-plasma THz generation. Our results show that the optimal phase of THz is robust enough to measure in situ the value of the relative phase in the phase-sensitive two-color experiments. By means of the THz phase meter, we establish a one-to-one mapping between the Gouy phase slippage and the spatial THz generation in the focus. We find that the phase averaging in air plasma decreases the modulation depth and increases the ellipticity of THz radiation in linearly and circularly polarized two-color field, respectively. These findings contribute to the attempt to characterize the distribution of THz electric field within the plasma filament. We envision that the approach outlined here will serve to characterize the phase variation of more complex beam profile and open up doors for the observation of ultrafast electron dynamics by means of THz generation as well as high-precision attosecond studies.
This work is supported by the National Basic Research Program (973 Program) of China (Grant No. 2013CB922203), and the National Natural Science Fund of China (Grant Nos. 11374366, 11474359, 11574396 and 11404400). Z. L. and D. Z. acknowledge support through the National Natural Science Fund of China (Grant No. 61490694).
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