1. Introduction

In modern optics, beams with special structures such as spatially varying intensity, phase and polarization, are of great interest due to their particular applications [1,2]. One type of the most promising structured beams is that with orbital angular momentum (OAM) [3]. Commonly, there are two kinds of phase-dependences on the azimuthal angle in OAM beams [3,4]. One is called “twisted light” or “vortex light”; the other is angular accelerating vortex beams (AAVBs). Since the radius of a vortex pulse linearly scales with its helicity (i.e., topological charge (TC)), applications that require beam geometry or photon density will be limited to a fixed OAM. While an AAVB has a tailored nonlinear azimuth-dependent phase profile, indicating that it carries local tunable OAM density distributions and exhibits angular acceleration and deceleration upon propagation [3]. Note that the local OAM density describes the OAM distribution of the light field on the reference plane [5]. After numerically integrating it in a limited area, the total OAM of the OAM beams within this area can be obtained. As is known, the different OAM beams with the same total OAM will have different local OAM density distributions [5]. Control of the local or total OAM in a beam is very interesting and important in many applications, as is the spatial distribution or photon density of the beam [4,5]. Therefore, studying the local OAM density distributions of the OAM beams can help us further understand the transfer of OAM in optically induced rotation technology [4,5]. Recently, the studies of OAM beams have extended to the THz region [6,7]. For a THz OAM pulse with an ultra-broad bandwidth (e.g., 0.1–30 THz), it inherently possesses a single-cycle or sub-single-cycle duration and thus shows the strong spatio-temporal coupling (STC) and spectral coupling [6]. These characteristics can advance the OAM beams and facilitate its applications in THz fields such as THz wireless communications, THz super-resolution imaging, particle manipulation, manipulation of chiral matters, detection of astrophysical sources [1,2,7,8]. These growing demands are driving researchers to make great efforts to generate THz OAM beams, which, however, is quite a challenge due to the lack of materials for a desired THz modulation [9].

Some reports demonstrated theoretically and experimentally that the high-order vortex harmonics can be obtained by the frequency up-conversion of twisted light in gas-plasma filaments. The OAM of the high-order harmonics is inherited from the driving vortex laser, and their TCs are a multiple of the harmonic order [10–15]. Even though the THz radiation can be considered as the 0^th order harmonics [16], this kind of the OAM conversion law seems to be not applicable for THz generation via air-filamentation. Very recently, it was found that the local spatial THz phase distributions depend strongly on the frequency of the THz radiation from the air filament [17,18], which may have an ultra-broad bandwidth when pumped by a few-cycle vortex laser field [17] or a two-color vortex laser field [18,19].

In this paper, we find a way to manipulate the local OAM of the ultra-broadband THz pulses in a chirped few-cycle vortex-induced air filament. Two kinds of THz OAM beams can be generated, which have linear azimuth-dependent phases for the THz vortex pulses and nonlinearly azimuth-dependent phase profiles for the THz AAVBs. Their local OAM density distributions have a complicated azimuthal dependence of the intensity and the local wavefront helicity. The properties of the THz OAM beams are determined by the Kerr nonlinearity and the plasma effect in the air filament. Numerical results also show that, when pump intensity is high enough to reach or even exceed the clamping intensity (i.e., the high intensity case), plasmas accumulate greatly, so the contribution of photocurrents overwhelms that of Kerr nonlinearity. In this case, the chirp parameter seems to have little effect on THz generation, where the generated THz beams are AAVBs. By contrast, if the pump works at low intensity level (i.e., the low intensity case) so that plasmas fail to form, Kerr effect plays a dominate role. As a consequence, THz vortex pulses are generated through the nonlinear difference-frequency conversion. As for the medium intensity case, the effects from Kerr nonlinearity and plasma are comparable and competitive. Similar to the high intensity case, THz AAVBs occur but with a smaller phase nonlinearity. The generated THz intensity distributions are closely associated with the chirp parameter. As the absolute value of the chirp parameter increases, their intensities augment remarkably, meanwhile the transverse spatial ring-widths of the THz intensity distributions reinforce.

2. Laser parameters, model and theoretical analysis

In our simulations, the used few-cycle driving vortex pulse is an 800 nm Laguerre-Gaussian (LG) pulse with a TC of l = 1 with the duration of τ₀= 5 fs when the laser pulse is non-chirped (i.e., the chirp parameter is C = 0). The scalar electric field of a conventional twisted light E(r,θ;t) with a TC of l in the temporal domain under cylindrical coordinate system can be expressed as [20]

If chirping a few-cycle pulse, the temporal distribution can be expressed as

Here, r and θ are the radial and azimuthal coordinates, respectively. A(r) is the radial amplitude in transverse plane. While U₀, l, ω₀, τ₀, C, φ_CE0 and i are the peak amplitude, TC, central frequency, initial pulse duration, chirp parameter, initial carrier-envelop (CE) phase, and imaginary unit, respectively. φ_CE0 is set as zero. The air is supposed to only consists of nitrogen (N₂) gas under the standard atmospheric pressure. Via comparing the simulation results of l = 1 and −1, we find that the variance rules of the THz temporal and transverse spatial distributions for the three intensity cases are similar except for a sign difference. Specifically, the TC sign of the pump affects the intensity maxima direction, the TC sign of the THz radiation and thereby the direction of the positive or negative electric field waveform stripes. We take the high intensity cases with C = 1 at l = 1 and −1 as examples for comparison, shown in Fig. S1 in Supplement 1. For instance, the THz intensity maxima at l = 1 are at the direction of θ≈3π/4 and 7π/4, while those of l=−1 are at θ≈π/4 and 5π/4 [comparing Fig. S1(a) and (e) in Supplement 1]. The corresponding THz local phase curves increase or decrease along θ with almost the same nonlinearity [comparing Fig. S1(b) and (f) in Supplement 1]. Accordingly, the positive or negative electric field peaks in the image exhibit downward or upward inclined crooked stripes [comparing Fig. S1(c) and (g) in Supplement 1]. To avoid repetition, it is appropriate to choose l = 1. It is noted that Gouy phase shift has been included in our simulation and given by φ_Gouy=-(|l|+1)arctan(z/z_R), where z_R =πw₀²/λ₀² is the Rayleigh length and w₀ is the beam waist of the fundamental field. Additionally, its initial CE phase with θ increases monotonically due to the spatial phase (exp(jlθ)) imparted onto the waveforms [17], namely, φ_CE=θ, indicating the strong STC. From a practical point of view, chirped beams present a quadratic temporal dependence in their phases. Namely, the pulse duration τ depends on C via the relation of τ²=τ₀²(1 + C²) [21], where τ₀= 5 fs is the duration when C = 0. C can substantially increase the pulse duration. The pulse bandwidth satisfies (Δω)²= (1 + C²)/τ₀². If C < 0, the laser pulse is down-chirped, otherwise is up-chirped. The above few-cycle vortex field (Eqs. (1) and (2)) is employed as the initial pump field in our numerical calculation.

The physical mechanisms in air-filamentation for both fields (i.e., a two-color field or a few-cycle field) are similar, as well as the simulation methods and simulation results. Specifically, the THz emission from an air filament can be interpreted by the four-wave mixing (FWM) or photocurrent (PC) models [17,22,23]. In the FWM model, the process of ω+ω′-ω′′→ω_THz for the few-cycle field or ω+ω′−2ω→ω_THz for the two-color field occurs; while in the PC model, the THz emission can be interpreted as a result of asymmetric ionization-induced quasi-DC currents produced by an asymmetric pump laser field like a two-color field or a few-cycle field. Thus, it can be simplified as ${E_{\textrm{THz}}} \propto {\partial _t}J$. As for a few-cycle field, these three frequency components within a broad bandwidth possess the same CE phase [22]. Following the expression of the CE phase in the difference-frequency conversion process described in Ref. [24,25], the azimuthal-dependent phase of the generated ultra-broadband THz pulse is given by φ_THz=φ_CE+φ_CE-φ_CE=θ for the few-cycle vortex pump field. Accordingly, the THz field amplitude can be expressed as $u_{\textrm{THz}}^{\textrm{Kerr}} \propto a\textrm{exp} ({i\theta + i{\varphi_a}} )$, where a is the amplitude, and φ_a is the phase shift. In addition, the THz generation mechanism in the FWM-picture pumped by a few-cycle pulse becomes formally equivalent with self-phase modulation (SPM) [26]. Whereas for the PC model, few-cycle pulses produce directly a spatial asymmetry in an air filament, generating directional quasi-DC plasma currents. The CE phase-dependent waveform asymmetry determines the THz emission [27–29]. Moreover, the analytical and simulation results demonstrated that the spatial THz phase distributions depend strongly on the THz frequency. The lower the THz frequency is, the stronger the phase nonlinearity is [17,18]. Accordingly, for the high-frequency THz component (for example, ∼30 THz), its phase varies nonlinearly with CE phase, which indicates that the THz radiation is AAVBs in this case. According to Ref. [10], the THz field can be given by $u_{\textrm{THz}}^{\textrm{plas}} \propto b\textrm{exp} [{i\theta + i\alpha \cos ({2\theta + {\varphi_0}} )+ i{\varphi_b}} ]$ with an amplitude of b. α is an adjustable parameter that determines the magnitude of the acceleration, while φ₀ and φ_b are the phase shift parameters. Since the photo-ionization occurs only around the laser peaks, the transverse spatial ring-widths of the THz intensity distributions from the PC model are much narrower than those by the FWM model, which could be regarded as a characteristic to distinguish plasma effect from Kerr nonlinearity.

In consideration that both Kerr nonlinearity (i.e., FWM mechanism) and plasma effect (i.e., PC mechanism) contribute to the ultra-broadband THz radiation during laser filamentation [30,31], we use the unidirectional pulse propagation equation (UPPE) [32,33] for a comprehensive study of THz generation, which is a general method to describe the propagation dynamics for both the two-color field [34] and the few-cycle field [35]. The CE phase in the few-cycle field plays a similar role as the phase difference in the two-color field [22,27–29]. Generally speaking, the THz radiation excited by the few-cycle field has similar transverse spatial distributions and evolution trends as that excited by the two-color field. So it is reasonable to compare the generated THz patterns produced by the few-cycle field with that produced by the two-color field in Ref. [18]. As reported in Ref. [30], both Kerr nonlinearity in neutrals and photocurrents in plasma contribute to THz radiation during laser filamentation. The Kerr self-focusing is usually active over a very short optical path before photocurrents take over in THz generation. Furthermore, the Kerr nonlinearity contributes to the high-frequency THz components (i.e., tens of THz). While the plasma current devotes to the low-frequency THz components (i.e., usually several THz), whose center frequency is located at around the plasma frequency. The plasma frequency given by ω_p= [e²N_e/(ɛ₀m_e)]^1/2 is strongly dependent on the electron density, thereby on the pump intensity. Note that we do not consider the Raman-delayed nonlinearities induced by ro-vibrational transitions of air molecules [30,36].

The spectral-domain UPPE can be expressed as [18,19,34,37]

Here, $\hat{E}$ is the spectrum of pump field with its wave number k(ω)=n(ω)ω/c, where n(ω) is the frequency (ω)-dependent refractive index and c is the light speed in vacuum. µ₀ and n₀ stand for the vacuum permeability and the refractive index of the laser pump, respectively. The nonlinearity term ${\hat{F}_{\textrm{NL}}}\textrm{ = }{\hat{P}_{\textrm{NL}}} + {{i{{\hat{J}}_e}} / \omega } + {{i{{\hat{J}}_{\textrm{loss}}}} / \omega }$ includes the third-order Kerr nonlinear polarization ${\hat{P}_{\textrm{NL}}}$, the current density ${\hat{J}_e}$ and the ionization energy loss ${\hat{J}_{\textrm{loss}}}$. For clarity, we have dropped the (x, y, z, ω) dependence of $({\hat{E},{{\hat{F}}_{\textrm{NL}}},{{\hat{P}}_{\textrm{NL}}},{{\hat{J}}_e},{{\hat{J}}_{\textrm{loss}}}} )$ and the (x, y, z, t) dependence of (E, F_NL, P_NL, J_e, J_loss, N_e). P_NL= 2ɛ₀n₀n₂|E|²E, J_e= (e²/m_e)[(v_c+iω)/(v_c²+ω²)]N_eE, J_loss= R₁(N_at-N_e)U_i/E, where ɛ₀ = 8.85×10⁻¹² F/m and n₂ = 3×10⁻¹⁹ cm²/W are the dielectric constant in vacuum and the nonlinear refractive index of the laser, respectively. Definitely, we use 60×60×4096 grids for the spatial parameters x, y and time t with the steps of Δx=Δy ≈ 23.8µm and Δt ≈ 0.167fs, respectively. Usually, the size of a plasma channel is determined by pulse energy, focusing condition, spatial structure of the pump, etc [22]. Typically, for a TEM₀₀ Gaussian pump pulse in practical experiments, the air-plasma channel has a diameter of around 100µm [22]. While for an annular beam with a phase singularity on its axis (i.e., a vortex beam), a tubular-shaped plasma channel is formed [38,39]. Compared with the TEM₀₀ Gaussian pump laser, LG spatial profile has a larger divergence, thus has larger focused transverse area under the same condition since it increases with TC. Particularly, in our simulation, the beam waist w₀ is set as about 200µm, the radius of intensity maximum is ρ=(l/2)^1/2w₀≈ 141µm≈ 6Δx. The ionization occurs in the vicinity of the maximal intensity. The spatial distribution of electron density [Fig. S9(a) in Supplement 1] keeps ring-shape with narrow ring width for ∼1 mm plasma channel. According to the corresponding cross-section profile along X axis at y = 0 [Fig. S9(c) in Supplement 1], even though there are about 6 to 7 sampling points on the width of the ring, the spatial distribution of electron density is well resolved by interpolating the colormap index or true color value across the line or face. An adaptive step size along z direction is down to ∼1µm.

Usually, micro-plasmas (less than 100µm), elongated plasmas (several hundred of micrometers to several millimeters) and longer filaments (centimeter-scale) are the three popular situations studied in THz generation. For the THz waves produced by micro-plasmas, propagation effects are not taken into consideration [40]. While for elongated plasmas or longer filaments, propagation effects cannot be ignored [18,19,22]. In this situation, the UPPE is the most common method to perform a comprehensive numerical simulation. However, upon the propagation in the longer filaments with a high pump intensity, the vortex pump field may undergo spatio-temporal instabilities, which may influence the THz transverse spatial distributions [18]. Upon comprehensive consideration, first we would like to check the THz OAM characteristic in a short filament with a length of ∼1.0 mm, in which the propagation effects take effects but the spatio-temporal instabilities are unobvious. Here, in our simulation, the ultra-broad THz window of ν<30 THz is selected for a direct analog to practical THz measurements [18].

The transient plasma dynamics are described by the free electron density N_e [19,41], i.e.,

It is noteworthy that, when the pump intensity reaches or even exceeds the clamping intensity, the contribution of plasma effect overwhelms that of Kerr nonlinearity in THz generation [26,40]. Otherwise, if the pump intensity is much lower than clamping intensity, the Kerr nonlinearity is more evident than the plasma effect in THz generation [22,26,42]. Of course, the THz generated from the former reaches much higher levels than that from the latter due to the presence of plasma currents.

3. Numerical simulation and discussion

The chirp of the pump has an important influence on the nonlinear optical phenomena when the pump laser propagates through a nonlinear medium [37,43–48]. It has been reported that chirp effects may result in the shifts and broadening of the spectra in high-order harmonic generation [46,47]. The typical characteristics for a few-cycle vortex laser field with and without chirp are shown in Fig. 1. It is found that the normalized time-integrated intensity distribution for chirping a few-cycle vortex pulse also presents a doughnut-shape pattern [Fig. 1(a)]. Figure 1(b) shows the corresponding temporal waveforms at different C (i.e., C = 0, −1, −2, 1, 2) at point A (i.e., θ=0, r = w₀(l/2)^1/2). Only the cycle around the pulse peak seems to be independent of C. The distinct temporal waveforms indicate the variance of instantaneous frequencies. The corresponding spectra [Fig. 1(c)] calculated by Fourier transformation show that, as the |C| increases, the spectral width broadens while the spectral peak intensity decreases. Chirping pulses retard the Kerr self-focusing since their input power varies as P = P₀/(1 + C²)^1/2 [49,50], where P₀ is the input power when C = 0. The normalized time-frequency distributions for the different C are shown in Fig. 1(d)-(h) with the Wigner-Ville distributions (WVDs) [51], where the white dash lines are the temporal traces of time-dependent instantaneous frequencies. Note that the WVD is one of the popular time-frequency analysis methods, which concerns the analysis of the signals with the time-varying frequency contents. Such signals are well represented by a time-frequency distribution, which is intended to show how the energy of a signal is distributed over the two-dimensional time-frequency space instead of only one (time or frequency) [51]. One can see that when the laser pulse is non-chirped (i.e., C = 0), the instantaneous frequency is constant [Fig. 1(d)]. It decreases linearly across the temporal waveform when the pulse is down-chirped, or C < 0 [Fig. 1(e) and (f)], while increases linearly when up-chirped, i.e., C > 0 [Fig. 1(g) and (h)]. In addition, the bigger the |C| is, the larger the slope of time-dependent instantaneous frequency curve is. This chirp parameter can cause strong STC in the few-cycle vortex field, which could be found in the variation of rotation speed of the transverse amplitude distribution. The rotation speed increases with the instantaneous frequencies. For an extremely short pulse duration, STC effects caused by pulse travel-time and angular geometry can lead to highly complex propagation effects and related limitations of TC spectrum [52,53]. Accordingly, it can be expected that C has a significant influence on the spatial structure of the generated THz radiation via the frequency down-conversion process during air-filamentation. In the following calculation, the characteristics of the generated THz OAM pulses are presented at different values of C including positive (shown in the main text) and negative (shown in Supplement 1) ones.

 

Fig. 1. Representative characteristics of the 800 nm few-cycle vortex laser field with l = 1. (a) Normalized time-integrated intensity distribution; (b) temporal waveforms at point A with different C; (c) corresponding spectra; (d)-(h) WVDs at C= 0, −1, −2, 1 and 2, respectively.

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When the pump peak intensity reaches or even exceeds the clamping intensity [13,31,54,55], referred to as the high intensity case, the Kerr nonlinearity fades greatly away, and photocurrents play a leading role in THz generation. Note that the clamping intensity in air seems to be about 30 to 80 TW/cm² [13,33]. The ionized electron density is on the order of 10¹⁷ cm⁻³, corresponding to the plasma frequency of several THz. At present, the quasi-DC currents serve as the THz source, i.e., ${E_{\textrm{THz}}} \propto {\partial _t}J$ [17]. This expression determines that the THz local phase curve is strongly dependent on the frequencies [17,18]. Even though the chirp parameter affects the rise time of the electron density and the oscillation of the photocurrent, the THz amplitude still scales as ${E_{\textrm{THz}}} \propto \cos ({{\varphi_{\textrm{CE}}}} )= \cos ({\theta + {\varphi_{CE0}} + \Delta \varphi } )$. It is generally accepted that the THz amplitude is maximal with φ_CE=π/2 and minimal with φ_CE= 0 [22]. Therefore, the THz pulses with two-petal intensity patterns and almost smoothed jump phase curves are obtained. In contrast, when the peak pump intensity level is much lower than the clamping intensity, which is regarded as the low intensity case, it is found that the electron density is on the order of 10¹⁰cm⁻³ (corresponding to the plasma frequency of the order of 10⁻³ THz), so the plasma effect is negligible, and the Kerr nonlinearity becomes dominated. In this situation, the FWM process of ω+ω′-ω′′→ω_THz occurs. Based on the expression of the CE phase (φ_THz=φ_CE+φ_CE-φ_CE=θ) [24,25] and the OAM conservation law (1ħ + 1ħ - 1ħ = 1ħ) [17], the THz vortex pulse with the TC of +1 can be generated. So the phase curve shows a monotonous increase. However, for the case with comparable Kerr nonlinearity and plasma effect, which can be called the medium intensity case, both effects make parallel contributions to the THz radiation. Via analyzing simulation results at different intensities, it is found that, for |C|=1, the range for “the low intensity case” and “the high intensity case” are less than ∼28 TW/cm² and bigger than ∼53 TW/cm², respectively. Accordingly, the range of “the medium intensity case” is between the two values above. For the bigger |C|, the value will shift to be lower.

For the high intensity case, it seems that C has almost no effect on the THz intensity and phase distributions. When C = 1, the peak intensity of the initial pump is about 70.7 TW/cm², and the generated THz AAVBs are shown in the first row in Fig. 2, implying the narrow two-petal intensity patterns [Fig. 2(a)], the nonlinear phase curves [Fig. 2(b)], the waveforms vs θ [Fig. 2(c)], and the four typical waveforms [Fig. 2(d)] are almost independent of C. Note that the phase curve presents the local phase variance with the azimuthal angle θ at r_max (=w₀(l/2)^1/2). According to the discrete phase values in the Fig. 2(b) for example, φ_THz(2π)-φ_THz(0) 1.0π-(−0.99π) ≈ 2π, the THz phase vs θ varies nonlinearly with a period of 2π. The 2π-periodic phases are satisfied for all the cases [see the second column in Fig. 2 and 3]. The corresponding results for C = 1.5 and 2 are present in the supplementary materials [Fig. S2 in Supplement 1].

 

Fig. 2. THz spatial and temporal characteristics for the high and low intensity cases at C = 1. (a) and (e) Spatial intensity patterns; (b) and (f) local phase curves with θ at r_max; (c) and (g) corresponding THz waveforms vs θ; (d) and (h) typical time-dependent THz waveforms for the four selected points marked by A, B, C and D in panel (c), (g), respectively.

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Fig. 3. THz spatial and temporal characteristics for the medium intensity case at C = 1, 1.5, 2. (a), (e), and (i) Spatial intensity patterns; (b), (f), and (j) local phase curves with θ at r_max; (c), (g), and (k) corresponding THz waveforms vs θ; (d), (h), and (l) typical time-dependent THz waveforms for the four selected points marked by A, B, C and D in panel (c), (g), and (k), respectively.

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In the second row of Fig. 2, a well-defined THz twisted pulse is obtained when C= 1 and the peak intensity of the initial pump is about 14.1 TW/cm², corresponding to the low intensity case. The intensity is doughnut shaped with a zero central intensity [Fig. 2(e)], similar to that of the vortex pump field. Its phase increases linearly vs θ with the slope of ∼1 [Fig. 2(f)]. The waveforms at r_max vary continuously with θ. The positive or negative electric field peaks with almost the same peak amplitude in the image exhibit downward inclined stripes [Fig. 2(g)]. The four typical waveforms at θ ≈ 0, π/2, π, 3π/2 (marked with A, B, C and D, respectively in Fig. 2(g)) plotted in Fig. 2(h) have totally different phases. When C increases, the nonlinearity of the THz phase curve with θ occurs very slightly. However, the THz intensities are azimuthal-modulated, and the modulation depth is proportional to C. This is because the plasma effect still plays a non-negligible and visible role on the final THz intensity pattern, even though the Kerr effect is dominant at the low intensity level. The results for C = 1.5 and 2 are presented in the supplementary materials [Fig. S3 in Supplement 1].

The THz generation for the high and low intensity cases are dominated by the leading plasma effect and the dominant Kerr nonlinearity, respectively. However, as for the medium intensity case, the THz intensity patterns are determined by C [compared with Fig. 3(a), (e) and (i)]. For C = 1 with the peak intensity of the initial pump at ∼35.4 TW/cm², a narrow ring-shaped pattern with an azimuthal-modulated intensity distribution appears, shown in Fig. 3(a). Since the THz radiation produced by the plasma effect has a narrower intensity band and stronger phase nonlinearity than those produced by the Kerr effect, it can be concluded that the plasma effect dominates over the Kerr effect at C= 1. As C increases, the transverse spatial ring-width of intensity increases [Fig. 3(e) and (i)], so the Kerr effect dominates over the plasma effect at a larger C. On the whole, the phase at different C [see Fig. 3(b), (f) and (j)] grows nonlinearly with θ, indicating that the generated THz pulses are AAVBs. Their local OAM density distributions may vary with positions. Their phases with θ can be mapped onto the corresponding waveforms, shown in the third and fourth columns of Fig. 3. Furthermore, the THz electric field amplitude is enhanced dramatically with C.

It is found that the intensity maxima almost stay at the same direction for high intensity case with different C [see Fig. S2 in Supplement 1]. However, as for the low and medium intensity cases, the THz intensity distributions rotate obviously with an increase of C [see Fig. S3 in Supplement 1 and Fig. 3]. This is related to the carrier phase shift in the plasma filament, i.e., Δφ=Δnω₀L_plas/c, where Δn=Δn_Kerr+Δn_plas is the total refractive index variance. Note that the effect of the Gouy phase shift on THz generation is much smaller than carrier phase shift by refractive index variance, thus it can be negligible [33]. Generally, for the few-cycle Gaussian pulses, the THz field reaches its maximum when the CE phase is 0.5π or 1.5π [22]. On the contrary, reverse state is observed when the CE phase is 0 or π. When the spiral phase (exp(ilθ)) and also the carrier phase shift (Δφ) in the plasma filament are imparted onto the few-cycle field, its CE phase with θ can be expressed by φ_CE=θ+φ_CE0+Δφ with φ_CE0 =0. At the very beginning of the plasma filament, Δφ is zero too. The directions of the THz intensity maxima are along θ =0.5π and 1.5π. The THz intensity distribution is sensitive to θ. That is why the THz intensity patterns are azimuthal-modulated. If the phase has an additional small deviation along θ, namely, φ_CE0 is not zero, it can be predicted that the initial phase can affect the initial THz intensity maxima direction. When the few-cycle vortex pulses with different C propagate in the plasma filament, Δφ varies for different total refractive index variance Δn. Thus the THz intensity maxima do not appear at θ =0.5π and 1.5π. In other words, the THz intensity patterns rotate, which will certainly rotate the local OAM distributions. Based on the formulas of the refractive index variance induced by the plasma effect Δn_plas=-VN_e (in which V = (4/9)e²/(ɛ₀m_eω₀²) is the effective volume of the plasma [13]) and by the Kerr effect Δn_Kerr=+n₂I_pk (where n₂= 4×10⁻¹⁹ cm²/W is the nonlinear refractive index in air, and I_pk is the peak intensity), the calculated values of Δn_plas and Δn_Kerr at different cases are shown in Table 1. For the three intensity cases, Δn_plas stays basically unchanged with an increase of |C|, while Δn_Kerr decreases relatively dramatically. Moreover, for the high intensity case, Δn_plas is larger than Δn_Kerr and the intensity maxima directions are almost the same for different C [see Fig. S2 in Supplement 1]. Whereas for the low and medium intensity cases, Δn_Kerr is larger than Δn_plas and the THz intensity patterns rotates as |C| increases [see Fig. S3 in Supplement 1 and Fig. 3], so we can draw the conclusion that the Kerr effect is the main mechanism determining the direction of the THz intensity maxima. Interestingly, our simulation shows that the phase of the generated THz radiation changes linearly for vortex beams or nonlinearly for AAVBs by 2π as θ varies from 0 to 2π. This indicates all the THz beams have the same TC value of 1 according to the formula $Q = {\left\langle {{{\partial \varphi } / {\partial \theta }}} \right\rangle _\theta }$ . In other words, even though the phase varies locally, the total OAM is conserved.

Table 1. Calculated values of Δn_plas and Δn_Kerr at different cases

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Besides the THz transverse spatial distributions (especially local phase curves) and temporal waveforms mentioned above, it is necessary to calculate their OAM density distributions quantitatively and discuss how they vary. We consider the OAM density for a general complex scalar field, u(x,y,z), which varies with time (in complex notation) as exp (iωt). We briefly outline the approach for defining the OAM density starting from the Poynting vector. We define a vector potential in the Lorentz gauge as a representation of the linearly polarized (in the x direction) laser mode as [20,56]

The angular momentum density in a transverse electromagnetic field is M = r×p, where the linear momentum density is p=ɛ₀E×B. In the paraxial approximation, the time average of the real part of p for any scalar field u is given by [4,20,56]

Here, $\hat{z}$ is the unit vector in the z direction and $\nabla$ is the gradient operator in r and θ. Thus, the azimuthal component of the linear momentum density for an arbitrary scalar field with inclined wavefront described by any azimuthally-dependent phase φ(θ) is [4]

The local OAM density M_z is related to p through M_z= r×p. Accordingly, in the transverse plane, it can be given by [4]

There is complicated dependence of the intensity |u|² and the local angular helicity ${{\partial \varphi } / {\partial \theta }}$. According to Eq. (10), it is necessary to check the azimuthal dependence of the THz intensity (|E_THz|²) and the local THz wavefront helicity (${{\partial {\varphi _{\textrm{THz}}}} / {\partial \theta }}$) for the THz OAM radiation at the above three intensity cases for different C (C = 1, 1.5, 2), shown in Fig. 4. Through an overall observation of Fig. 4, the THz intensity and the local THz wavefront helicity seem to have a similar characteristic for most cases, i.e., inverse oscillation [4].

 

Fig. 4. Azimuthal dependence of |E_THz|² and ${{\partial {\varphi _{\textrm{THz}}}} / {\partial \theta }}$ for the high (first column), medium (second column) and low (third column) intensity cases at different C (C = 1, 1.5, 2).

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Figure 5 focuses the corresponding local THz OAM density (M_z) distributions in the transverse plane. Given by the azimuthal-modulated intensity and nonlinear local phase profile, the local OAM may vary with position and yet be effectively averaged out by motion. Their spatial ring-widths seem to coincide with those of the corresponding THz intensity distributions. Most of their structures are complex uneven rings because the azimuthal-dependent intensity and the corresponding angular helicity are not able to cancel out completely. Remarkably, the M_z is exclusively positive. It can be easily understood, since the z-component of the OAM density is proportional to the θ-component of the Poynting vector [56]. As for the high intensity case, the M_z distributions are almost the same [first column in Fig. 5]. This is due to the similar THz intensity and local angular helicity [first column in Fig. 4]. Only one maximum occurs on the one side, which could be explained by the big difference of the two peak values of the local angular helicity. It is more interesting for the M_z structure of the medium intensity case at C = 1 [Fig. 5(b)], it seems to have two symmetrical saddle points and two maximum points because the azimuthal-dependent intensity and the corresponding angular helicity exactly cancel out [Fig. 4(b)]. As C increases, the M_z structure becomes fully asymmetric. The bigger the C is, the greater the asymmetry shows [compare Fig. 5(e) and (h)]. While for that of the low intensity case at C = 1 [Fig. 5(c)], even though the THz intensity distribution [Fig. 2(e)] in the transverse plane shows a doughnut shape, and the local THz phase curve is linear visibly, the discrete data points of the azimuthal-dependent intensity and local wavefront helicity oscillate slightly. This does not prevent the occurrence of doughnut-shaped OAM density structure [Fig. 5(c)]. As C increases, the variance is much similar to that of the medium intensity case. With regard to the cases at negative C (i.e., C=−1, −1.5, −2), the variance rule is similar to the cases discussed above. To avoid repetition, the detailed discussion will be not be covered here. The simulation results can be seen in the supplementary materials [Fig. S7 and S8 in Supplement 1].

 

Fig. 5. Local THz OAM density (M_z) distributions for the high (first column), medium (second column) and low (third column) intensity cases at different C (C = 1, 1.5 and 2).

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Furthermore, the above simulation results have revealed that the dominate mechanism for THz generation is determined by C. This can be further verified by a numerical analysis of the Kerr and plasma effect terms. Note that the “Kerr effect term” indicates the third-order Kerr nonlinear polarization ${\hat{P}_{\textrm{NL}}}$, and the “plasma effect term” presents the current density ${\hat{J}_e}$ in Eq. (3). They can be expressed as ${\hat{P}_{\textrm{NL}}} = {\cal F}\{{2{\varepsilon_0}{n_0}{n_2}{{|E |}^2}E} \}$ and ${\hat{J}_e} = {\cal F}\{{({{e^2}/{m_e}} )[{{{({{v_c} + i\omega } )} / {({{v_c}^2 + {\omega^2}} )}}} ]{N_e}E} \}$, respectively. The values of the “Kerr effect term” and “plasma effect term” are integral calculation in x, y and ω dimensions, i.e., the diagnostic method is global. It is calculated in spectral domain (Eq. (3)) with the same parameters, and the values are given in arbitrary units. To a certain extent, their values can represent the levels of Kerr effect and plasma effect. The scaling laws of Kerr and plasma terms vs C are shown in Fig. 6(a) and (b), respectively. The value of Kerr effect term increases with |C| for the three intensity cases. The variation of the Kerr effect term is roughly symmetric with respect to C = 0. While the plasma effect term gradually decreases monotonically when C changes from negative to positive. This is because the chirp effects affect the rise time of the electron density and the oscillation of photocurrent due to the dependency of the time-dependent oscillation period, which confirms that a faster oscillation could induce a bigger plasma current. Specifically, for negative C, the oscillation frequency decreases as time increases, which increases the rise time of electron density, thereby generating a bigger plasma current. Whereas for positive C, the reverse is true, i.e, generating a smaller plasma current. Moreover, at the low intensity case, the Kerr effect is a few orders larger than the plasma effect, which indicates the Kerr effect is dominant and the generated THz radiation is vortex beams. In contrast, the Kerr effect is much smaller than the plasma effect at the high intensity case, which shows the plasma effect is dominant and THz AAVBs are obtained. For the medium intensity case, both effects are at the same level, indicating that they play a comparable role in THz generation. Accordingly, the THz electric field can be considered as

 

Fig. 6. (a) Kerr effect term and (b) plasma effect term for the three intensity cases. (c) - (d) Phase data vs fitting curves for the medium intensity case at different C (C = 1, 1.5 and 2).

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Since the THz radiation is sensitive to the CE phase of an intense few-cycle laser field, we then investigate its variation in the air filament, i.e., Δφ=Δnω₀L_plas/c [13]. Note that the clamping intensity is the intensity that the dynamic between the Kerr self-focusing and the plasma defocusing effect is balanced. At the clamping intensity, the peak intensity can be clamped inside the plasma filament with the increase of pump power and the length of the filament increases [57]. When increasing the |C|, the few-cycle pulse spectrum is broadened in the sub-cycle time scale, leading to a decrease of the pump peak intensity. Meanwhile, the increase of |C| also results in the increase of asymmetry of the few-cycle temporal waveforms and so is N_e. Figure 7(a) presents the relationship between N_e and Δn vs the pump peak intensity. Note that in the plot, N_e indicates the maximal electron density at ∼1 mm for the different cases. When |C| increases, the electron density increases slightly for all the cases due to the enhanced asymmetry. Moreover, we calculate the average electron densities of N_{e_ave}= 2.87×10¹⁰, 9.21×10¹⁴, and 2.73×10¹⁷ cm⁻³ for the three intensity cases, respectively, which are the average for the every intensity case at C=−1, −1.5, −2, i.e., N_{e_ave}= [N_e (C=−1)+ N_e (C=−1.5)+ N_e (C=−2)]/3. The total refractive index variance Δn is positive for the low and medium intensity cases, but it is negative for the high intensity case. We take the high intensity case as an example to estimate the value of THz peak amplitude and we estimate that it seems to be a little more than 100 kV/cm based on the pump intensity and the electron density. Additionally, Fig. 7(b) and its inset are the full spectra and the corresponding THz and fundamental components, respectively. As is known, the Kerr nonlinearity for THz generation in a few-cycle field can be taken as SPM, which depends on the pump pulse waveforms and spectra [26]. Chirp effects enhance SPM [see the green circles in Fig. 7(b)]. For the low intensity case and even the medium intensity case with bigger |C|, the THz component of the spectrum is nothing more but the part of the broad pump spectrum. This is because in these cases, the Kerr effect is dominate for THz generation, which can be explained by an increase of SPM with the generated THz field coming from the extreme tail of the “broadened” spectrum [26]. So the THz component contains the vortex properties (i.e., linear local phase profiles) of the initial pump [see the second row in Fig. 2]. However, for the high intensity cases and the medium intensity cases with smaller |C|, the transient electron current results in the THz spectral components [58,59], which causes the generation of THz AAVBs with nonlinear local phase profiles [18]. Namely, the THz radiation from plasma effect inherits part of the properties of the pump field, such as the TC.

 

Fig. 7. (a) Relationship of N_e, Δn vs initial pump peak intensity and (b) full spectra at C = 1, 1.5 and 2 for the medium intensity case. The inset is the corresponding THz and fundamental components.

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Finally, it is necessary to check the propagation dynamics of a longer distance for both the pump laser field (i.e., full field) and the generated THz field. We found that the propagation dynamics are closely relative to the electron density. Note that, for the low and medium intensity cases, when propagating several millimeters, the spatial distribution of electron density keeps ring-shape. While for the high intensity case with high electron density (i.e., the order of ∼10¹⁷cm⁻³), at the beginning of the filament, it performs ring-shape too. But upon further propagation, it becomes more complex configuration [see the comparison in Fig. S9 in Supplement 1]. Besides, the plasma effect is main contribution to THz generation, and the pump pulses would develop spatio-temporal instabilities upon longer propagation. Figure 8 and 9 present its spatio-temporal and transverse spatial distributions of the pump laser field (i.e., E(x,t), E(x,y), I(x,y)) and the generated THz field (i.e., E_THz(x,t), E_THz(x,y), I_THz(x,y)) at C = 1 for different propagation distances (L = 1.0 and 2.5 mm), respectively. The phase singularity is retained during air-filamentation for both the pump laser field and the generated THz field. At the beginning of the air-filament, the pump laser field propagates stably [see the first row in Fig. 8], thereby generating steady THz OAM beams [see the first row in Fig. 9]. Upon further propagation, the vortex pump pulse develops spatio-temporal instabilities, and starts to undergo transverse distortion. The lobes of its amplitude profile begin to bifurcate, as evidenced by Fig. 8(e) and (h). The spatio-temporal instabilities directly influence the transverse spatial and spatio-temporal distributions of the generated THz pulses [see the second and third rows in Fig. 9], thereby their local OAM density distributions [see Fig. 10]. The spatiotemporal instability characteristics at the evolution of the generated ultra-broadband THz OAM radiation have been presented in Ref. [60]. These characteristics are much similar to those presented by the second figure in the last row (THz component at 25 THz) in Fig. 5 in Ref. [18], which has obvious bifurcations. In contrast, for the low intensity cases and even the medium intensity cases with bigger C, when propagating for longer distance (for example, several millimeters), we see almost no changes with the pump laser field (i.e., full field) due to very small electron density. Basically, the characteristics of the generated THz pulses are well kept. We take the medium intensity case at C = 1.5 for different distances as examples for comparison of the slightly differences for the THz spatio-temporal and transverse spatial distributions to show the propagation dynamics, shown in Figs. S10 and S11 in Supplement 1.

 

Fig. 8. Spatio-temporal and transverse spatial distributions of the pump laser field (i.e., full field) for the high intensity case with C = 1 at different propagation distances of L = 1.0, 2.5, and 3.2 mm.

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Fig. 9. Same as Fig. 8, except for the generated THz field.

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Fig. 10. Local THz OAM density (M_z) distributions for the high intensity case with C = 1 at different propagation distances of L = 1.0, 2.5, and 3.2 mm.

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4. Conclusions

This paper presents numerically a method to tune the local THz OAM density distributions without changing its TC from a laser-induced short air filament by chirping a few-cycle vortex laser. The simulation results show that the THz vortex pulses with linear azimuth-dependent phases and the THz AAVBs with nonlinear azimuth-dependent phases can be produced. The geometric intensity, the phase properties and also the local OAM density of the THz OAM beams can be controlled by the chirp parameter. If pumped at the low intensity case, the Kerr effect dominates over the plasma effect, so the generated THz radiation shall be vortex pulses. For the high intensity case, the leading plasma effect dominates. Instead, if the pump intensity is at the medium level, the Kerr nonlinearity and the plasma effect may become comparable and competitive. The THz AAVBs are obtained for both high and medium intensity cases. The work may find its applications in the OAM (de)multiplexing ultra-broadband THz communication, particle manipulation, plasma control optically-driven micro-machines, trapping in-parallel molecular or cell assays, material processing and nonlinear optics [61,62].