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Abstract
Optical vortices with longitudinal orbital angular momentum (OAM) as a powerful tool for particle control, imaging and communication have been greatly developed. We introduce a novel property of broadband terahertz (THz) pulse, represented by frequency-dependent OAM orientation in spatiotemporal domain with transverse and longitudinal OAM projection. We illustrate a frequency-dependent broadband THz spatiotemporal optical vortex (STOV) in plasma-based THz emission driven by cylindrical symmetry broken two-color vortex field. We detect the evolution of OAM by time-delayed 2D electro-optic sampling combined with Fourier transform. This THz optical vortices tunability in spatiotemporal domain expands a new way for studying STOV and plasma-based THz radiation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A light beam carrying orbital angular momentum (OAM) along the propagation direction (longitudinal) with a helical phase at the x-y plane (propagates along the z-direction) is called the vortex beam. Vortex beam has attracted much attention for decades for their various potential applications of particle control [1,2], imaging [3,4], and communication [5–7]. Recently, light beam with transverse OAM was proposed and produced, so-called spatiotemporal optical vortex (STOV) [8,9]. The helical phase of STOV spans in space-time frame instead of the x-y plane and may have arbitrary orientations [10–22].
Frequency conversion greatly expands the spectrum available for optical vortices. Longitudinal OAM and transverse OAM has proven to be conserved in the frequency up-conversion processes of second harmonic generation (SHG) [12,13,23,24] and high harmonic generation (HHG) [11,25–27]. However, there are few studies on the OAM transfer directly via the frequency down-conversion process [28–30], the mechanism of OAM transfer still needs to be studied. Terahertz (THz) radiation generated from intense laser-induced plasma provides an ideal case of frequency down-conversion to THz frequency and has the advantages of without damage threshold limitation, octave-spanning spectrum and controllable polarization.
Plasma-based THz emission usually pumped by two-color Gaussian beam composed of fundamental and second harmonic [31,32]. When spatial phase distribution is introduced in two-color laser field, whether it can produce plasma-based THz vortex emission is an interesting topic. Wang et al. have theoretically found that THz emission from two-color vortex pulses induced plasma does not have OAM, because of the stepwise change of the azimuth phase in transverse plane [30]. Recently, Ivanov et al. experimentally shows the feasibility of plasma-based THz vortex generation in the range of 10-40 THz [29]. The OAM of the THz wave is approximately equal to that of the second harmonic field. However, the spatial phase information of THz field and its evolution in the spatial-time domain is still unclear. Besides, STOV can be produced in air filament by arrested self-focusing collapse of the ultrafast optical pulse, with significant redistribution of phase and energy in plasma channel [33], and thus bring inspiration to the study of plasma-based THz vortex generation.
In this work, we investigated THz optical vortex produced from air plasma driven by two-color vortex laser pulses. We show that the THz beam has a frequency-dependent spatial distribution, which is consistent with the prediction of off-axis phase matching condition. Longitudinal OAM and transverse OAM of THz pulse was detected with its helical phase varying at the azimuthal angle. THz longitudinal OAM is found frequency dependent, presumably owing to the projection of frequency-dependent broadband THz STOV, which is composed of the local THz frequency shift and the diffraction of THz waves.
2. Experiment and data extraction
The experimental set-up is shown in Fig. 1, in which a Ti:sapphire laser delivers linearly polarized pulses with a duration of $40\textrm{fs}$ and pulse energy of $4.0\textrm{mJ}$ at central wavelength of $800\textrm{nm}$ in the repetition rate of $1\textrm{kHz}$. The laser pulse was split into three, (1) the fundamental ($\mathrm{\omega }$) beam to produce a vortex beam, (2) the second harmonic generation (SHG) ($2\mathrm{\omega }$) beam, and (3) the probe beam for electro-optic detection. In beam path (1), the fundamental beam was converted from a Gaussian to a Laguerre Gaussian mode with OAM (${\textrm{l}_\mathrm{\omega }} = 1$) using a radial polarization converter (S-waveplate) inserted between a pair of quarter-waveplates. The $2\mathrm{\omega }$ beam without OAM was produced by a $100{\;\ \mathrm{\mu} \mathrm{m}}$ thick type-I BBO crystal. The two beams co-propagate after a dichroic mirror with parallel polarizations. An air filament was produced near the focus of two-color pulses, from which THz wave was emitted.
Fig. 1. Experimental setup for the generation and detection of THz beams. The pulse energies for each path are (1) $\mathrm{\omega }$ pulse: $1.5\textrm{mJ}$, (2) $2\mathrm{\omega }$ pulse: $0.2\textrm{mJ}$, (3) probe pulse: $50{\;\ \mathrm{\mu} \mathrm{J}}$. QWP, quarter-waveplate; SWP, S-waveplate; L1-L2, L5-L6, focusing lens for pump and probe laser pulses; L3-L4, TPX Lenses for THz wave ($\textrm{f} = \textrm{}150,\textrm{}200\textrm{mm}$ respectively); DM, Dichroic mirror for $\mathrm{\omega }$ and $2\mathrm{\omega }$ pulses; BBO, Beta Barium Borate crystal; P1-P2, Glan polarizer; ZnTe, zinc telluride crystals. A1-A2, aperture. M, Dielectric mirror. (Insets) upper: the fringe pattern of the vortex beam and a Gaussian beam, indicates the topological charge of the former is $\textrm{l} = \textrm{}1$; lower, typical intensity distributions at focus of the $\mathrm{\omega }$ and $2\mathrm{\omega }$ beam.
THz electric field was fully reconstructed in three dimensions using a delayed probe beam. The THz waveform was detected through a 2D electric-optical sampling method (two-dimensional time-domain spectroscopy, 2D-TDS), in which THz-induced polarization degradation of a linearly polarized probe beam through a $1\textrm{mm}$ thick ZnTe crystal was measured [34,35]. A chopper worked at the frequency of $4\textrm{Hz}$ was placed in THz generation beam path to improve the signal-to-noise ratio. The modulated 2D probe beam profile was then captured by a charge-coupled device camera (CCD), and the background beam profiles without THz field modulation are also detected in sequence. The 2D distribution of temporal THz waveforms ${\textrm{E}_{\textrm{THz}}}({x,y,t} )$ were extracted by extinction detection [36] using modulated probe beam profile and background profile data (see Appendix A for detailed data extraction).
3. Results and discussion
The experimentally measured 2D distribution of THz intensity at the x-y plane is depicted in Fig. 2(a). It can be seen that the THz intensity presents a crescent shape. This intensity pattern suggests the presence of OAM and the central dark area corresponds to the zero intensity of phase singularity. THz waveforms on the green circle are extracted and plotted in Fig. 2(b). The time-domain waveform changes as the function of azimuthal angle, and indicates 2 times inversion of waveform polarity at the angles of $0.2\mathrm{\pi }$ and $0.9\mathrm{\pi }$, respectively. Because of the spectrum of THz wave exceeds one octave, further analysis is performed in frequency domain.
Fig. 2. (a) THz integral intensity at x-y plane; (b) THz waveform on the green circle in Fig. 2(a) along azimuthal angle. The intensity distribution of (c) $1\textrm{THz}$ and (d) $2.2\textrm{THz}$ in experiment.
The images of THz amplitude at $1\textrm{THz}$ and $2.2\textrm{THz}$ are extracted from the Fourier transformed THz waveforms, as shown in Fig. 2(c) and (d). The radius of intensity distribution decreases and their pattern varies from crescent shape to near closed-ring shape with a dark area breaking the ring intensity distribution as indicated by yellow arrows in Figs. 2(c)-(d).
Figure 3(a) and (b) show the phase distribution for frequency of $1\textrm{THz}$ and $2.2\textrm{THz}$ at x-y plane. Obviously, THz waves possess helical phase distribution with a singularity near the center and near uniformly changed phases along azimuthal angle outside. To the best of our knowledge, this is the first experimental observation of directly generated THz helical phase produced by plasma-based frequency down-conversion. Interestingly, two more singularities with alternate signs appear in the lower left of Fig. 3(b). Helical phase distribution indicates THz wave packet has a longitudinal OAM. In the experiment, clear helical phase distribution at x-y plane can be extracted for frequency in the range of $0.9 - 2.5\textrm{THz}$.
Fig. 3. Helical phase distribution for THz frequencies of (a) $1\textrm{THz}$, (b) $2.2\textrm{THz}$; (c) THz intensity distribution circles and phase singularity trajectory as the function of THz frequency; (d) longitudinal OAM evolved with THz frequency.
To validate the observation, we measured the THz beam profile in the range of 0.1-10 THz by using a THz camera (IRXCAM-THz-384, INO) in the focal plane, as shown in Fig. 8(a) in Appendix A. The THz intensity distribution shown the crescent shape. The THz spot was separated by a tilted focusing lens, as shown in Fig. 8(b). The results indicate that generated THz radiation indeed has OAM. This result is similar to the THz beam recorded in Ref. [29], and proved the reliability of our 2D-TDS measurements.
The THz beam profile in Fig. 2 and Fig. 8 are similar to that of a fractional vortex beam [37], the dark area located on opened ring-shaped THz profile is called “structured darkness” which is caused by the depletion of exciting evanescent waves when their exists a step change in spiral phase.
Furthermore, we look into the spatial phase distribution of the 2D THz waves. Figures 3(a)-(b) display the clear signature of phase singularity and smoothed spiral phase, which indicates the observation of broadband THz vortex. Interestingly, the phase singularity is not fixed in transverse space. In Fig. 3(c) we trace out the trajectory of phase singularity and the vortex radius (distance between phase singularity and the peak of the beam profile) for different THz frequencies, and reveal the phase singularity converge at higher THz frequency.
According to Ref. [37], the intrinsic value of longitudinal OAM in the experimental results is defined as the slope of helical phase variation in the azimuth range of opened ring-shaped intensity distribution (for a detailed discussion about longitudinal OAM extraction see Appendix B). Longitudinal OAM of $1\textrm{THz}$ is $0.52$ and $2.2\textrm{THz}$ is $0.7$, which is apparent frequency dependence. The frequency resolved longitudinal OAM is shown in Fig. 3(d). The longitudinal OAM increased from $0.4$ to $0.8$ as the THz frequency increases from $0.9\textrm{THz}$ to $2.5\textrm{THz}$. In order to reflect the trend of longitudinal OAM changes with THz frequency, an exponential fitting curve is plotted in Fig. 3(d). Clearly, the longitudinal OAM increases with THz frequency and tends to converge to be a constant for frequency larger than 2 THz.
In order to clarify the mechanism of THz generation in air filaments driven by vortex two-color field, we used the photocurrent model [38–40] to simulate the far field THz radiation considering the off-axis phase matching condition [41]. In the longitudinal direction, owing to the dispersion of the two-color laser pulses and Gouy phase shift, the relative phase of the two-color laser field changes over $\mathrm{\pi }$ when the filament exceeds the “dephasing length”. In transverse plane, the phase difference varies linearly from $0$ to $4\mathrm{\;\ \pi }$ along azimuthal angle because of the helical phase. Local THz polarity reverses whenever the change in relative phase reaches $\mathrm{\;\ \pi }$. Because of diffraction, THz waves of opposite polarities cancel out in the forward direction but constructively interfere at a certain off-axis angle, forming a conical radiation (see Appendix C for details of simulation).
Our simulation shows that far field THz intensity distribution displays a closed-ring shape and time-domain waveform polarity with reverse 4 times along azimuthal angle, as shown in Fig. 4(a) and (b), respectively. The far-field THz distributions of $1\textrm{THz}$ and $2.2\textrm{THz}$ by simulations are shown in Fig. 4(c) and (d). The ring radius decreases with the increased frequency, which is consistent with the results in Fig. 2(c) and (d). Hence, the off-axis phase matching dominantly shaped the far field THz intensity distribution. Although, the opened-ring shape THz profile recorded experimentally is not well reproduced.
Fig. 4. (a) Simulated THz integral intensity distribution at x-y plane; (b) THz waveform on the green circle in Fig. 4(a) along azimuthal angle. Simulated intensity distribution for THz frequencies of (c) $1\textrm{THz}$ and (d) $2.2\textrm{THz}$.
We also extracted the spatial phase distribution from simulation, as shown in Fig. 5(a), spiral phase is clearly presented with the extracted longitudinal OAM is nearly $2$. Local THz in x-y plane showing a four-lobe intensity distribution and stepwise variance phase distribution along azimuthal angle (see Fig. 9 in Appendix D), which has been proved no OAM exist [30]. However, the four-lobed profile is rotated along with propagation because of the two-color phase difference evolves along z axis. The spatial phase of far field THz wave varies near-linearly with azimuthal angle, results in the observed longitudinal OAM of the THz wave packet.
Fig. 5. Simulated far field helical phase at the frequency of 1 THz, (a) without, (b) with external disturbances and wavefront defects.
The differences between simulation and experiment mainly lie in the beam profile and the position of phase singularities. In simulation, THz radiation have a constant longitudinal OAM and singularity position with increased THz frequency, which in experimental observation longitudinal OAM varies from $0.4$ to $0.8$ with multiple moving singularities. Since the local THz field inside filament is inherited from the helical phase of the driven fields, the mismatch between experiments and simulation indicates some underlying mechanism may have been overlooked in theoretical model.
Considering the “structured darkness” on intensity distribution in Fig. 2(a), (c)-(d), and multiple phase singularities in Fig. 3(b). In the propagation of a vortex beam with large longitudinal OAM, unavoidable environment perturbations in intensity and distortions in the wavefront will make the helical phase singularity fuzzy and split [42]. As a verification, a small disturbance was introduced in ionization rate calculation which imprint the distortions into local THz wavefront and intensity distribution, results show no significant change in the intensity distribution, as shown in Fig. 4(c). The corresponding phase distribution is displayed in Fig. 5(b). As we can see, the disturbance divides one phase singularity into two with same sign, which is still surrounded by ring shape intensity distribution. However, this still cannot be fully consistent with the experimental observations, for example, the multiple singularities outside the ring-shaped profile displayed in Fig. 3(b). So, the “structured darkness” and non-integer longitudinal OAM are robust against the environmental disturbances in pump laser pulses.
Recent works in STOV demonstrates that the OAM of STOV may have an arbitrary orientation in the spatiotemporal domain [20–22]. Considering the vector property of OAM and off-axis phase matching gives THz wave packets a conical spatiotemporal structure, the spatiotemporal distribution of THz OAM in experiment ought to be considered. A typical feature of STOV is the existence of transverse OAM [8,9]. Different from the time-independent longitudinal OAM, transverse OAM only lives in a Fourier-transform-limited pulse. We believe the non-integer longitudinal OAM is the longitudinal projection of the total OAM which orientation in spatiotemporal domain. Thus, the residual OAM lives in transverse projection.
Due to the broadband spectrum of THz wave, phase distribution for transverse OAM was extracted in x-ω plane at y = 0, thanks to the OAM conservation in Fourier transforms [8]. The spectrum normalized intensity and phase distribution are depicted in Fig. 6(a) and Fig. 6(b), respectively. As we can see the clear helical phase and singularity appear (x = 1.1 mm, f = 0.9 THz, marked by a white arrow) in Fig. 6(b), the corresponding intrinsic transverse OAM is 1.56. In the intensity distribution, two zero-intensity areas are shown in the upper right corner marked by a white circle in Fig. 6(a), which indicates the split singularities caused by propagation. This intensity distribution is similar to the propagated STOV with transverse OAM equals 2 in a recent work [8].
Fig. 6. Transverse OAM observed in the spatial-frequency domain, the spectrum intensity (a) and phase (b) distribution in x-ω plane at y = 0. (c) The THz frequency dependent OAM orientation in STOV wave packet, showing the rotation in space.
The elongated zero intensity area (indicated by a blue arrow curve in Fig. 6(a)) caused by phase singularity indicating the transverse OAM dislocation along THz frequency. According to the varying longitudinal OAM, the OAM of THz STOV shown a frequency-dependent spatiotemporal orientation, as shown in Fig. 6(c). OAM orientation rotated from the near parallel with the x-y plane to the near parallel with z-axis (time axis) with the increased THz frequency indicated by gradient color from red to blue. It’s important to noted that, these vectors shown here aim to describe the variation of OAM orientation, a single THz frequency will not own transverse OAM, that is, broadband in nature. Considering the longitudinal OAM emerged from 0.9 THz, we can infer that the low frequency (0.1-0.8 THz) THz radiation possess transverse OAM.
According to our experiment results the OAM is not conserved between pump field and THz radiation. The phase variation of pump field mainly affects the amplitude of THz field, but cannot fully transfer to the THz field. The spiral phase of the THz field includes two parts: quasi-continuous phase variation and discontinuous phase jump [29]. These two parts can be distinctly separate for lower THz frequencies, and gradually merge with each other for higher THz frequencies. That is the reason why the OAM is not converged in the lower frequency range (0.1-3.0 THz) of our measurement, while approximately conserved for higher THz frequency range (10-40THz) in Ref. [29].
We believe the frequency-dependent THz STOV is inherited from the two-color vortex pump field. The pump intensity distribution in x-y plane is unstable and asymmetric along azimuthal angle, and the beam profile rotated in the frequency of fundamental beam along with pulse propagation. Simulated intensity distribution at x-y plane in t = 0 of pump field is shown in Fig. 7. But the intermediate reason overlooked by photocurrent model in air plasma remains unknown. According to the movement of THz singularity at x-y plane, as shown in Fig. 3(c) the white dot line, maybe the pump intensity dependent frequency shift of local THz emission cause the asymmetry distribution of local THz intensity, thus forming the helical phase in the spatiotemporal domain in far field.
4. Conclusion
In conclusion, we have studied THz vortex generated from two-color field induced air plasma filament. The OAM transfer mechanism in laser plasma was investigated. With the increased THz frequency, the far-field intensity distribution radius is decreased, which is dominated by off-axis phase matching. THz has a near linearly varying helical phase distribution at x-y plane, and its longitudinal OAM increases with THz frequency bringing the varied intensity distribution pattern. The varying longitudinal OAM is due to the longitudinal projection of the frequency-dependent broadband THz STOV. Helical phase distribution at x-ω plane shows the transverse projection of STOV, reflects the correlation of different THz frequencies. The frequency-dependent broadband THz STOV originates from the intensity distribution asymmetry of the vortex two-color field. The unique broadband THz spectrum endows tunability of THz vortex in spatiotemporal domain, which expands a new dimension for the study of plasma-based THz emission. The THz frequency-dependent STOV can provide torque in three-dimensional. This is a powerful tool in live particle control in the spatiotemporal domain such as living cells, nanoparticles, thanks to the low THz photon energy and the frequency-related three-dimensional torque. On the other hand, detected THz STOV can be used to investigate the nonlinear propagation of the optical vortices in plasma channel, such as the intensity and phase evolution and redistribution.
Appendix A: experiment data acquisition and processing
Measured the THz beam profiles in 0.1-10 THz by a THz camera (IRXCAM-THz-384, INO) in the focal plane as shown in Fig. 8. THz intensity distribution shown the crescent shape in Fig. 8(a), and the THz spot was separated by tilted focusing lens indicating the OAM existence, as shown in Fig. 8(b).
In the acquisition of time-delayed 2D electro-optic sampling experimental data, 200 repeated two-dimensional images were collected with a charge-coupled device camera (CCD) under each probe delay. Half of images are background signals and half are terahertz (THz) signals.
Fig. 8. THz intensity distribution before (a) and after (b) tilt the focusing lens, the dark stripe located between two lobes indicates the topological charge is approximately equal to 1.
Background signal (${I_b}$): Only probe beam passes through ZnTe crystal.
THz signal (${I_{out}}$): Probe beam and THz beam pass through ZnTe crystal at the same time.
The extinction detection of two-dimensional CCD images was carried out. Take a point on the plane, its intensity I satisfies the formula for extinction detection as follows [36]:
${I_0}$ is the intensity of probe beam; $\eta $ is the contribution by scattering; ${\varGamma _0}$ is the shift of phase induced by the residue birefringence of ZnTe crystal; $\varGamma $ is the THz induced birefringence contribution and $\varGamma \; \propto {E_{THz}}$. Usually the value of the phase term is small, so we take this approximation:
So,
Solving these equations and we can get this:
In ${I_{out}}(t )$ the minimum value is ${I_0}\eta $, and ${I_b}$ is the baseline value, so we can figure out the ${\varGamma _0}$, finally we can get the $\varGamma (t )$ which is in direct proportion to ${E_{THz}}(t )$.
The above operation was performed on all points in the plane to obtain THz distribution information.
Appendix B: experiment longitudinal OAM extraction
Considering the “structured darkness” on THz intensity distribution at x-y plane for all detected THz frequency. So, we choose the phase in C-shaped intensity distribution azimuth range, and linear fitting was performed, the change rate along azimuthal angle is defined as the intrinsic longitudinal OAM. The results are less than 1, this is similar to the fractional vortex beams. A fractional vortex is a superposition of a series of integer order vortices, but our experiment does not satisfy this condition. So, a topological charge less than one and continuously varying with THz frequency in our results is similar to the time-varying OAM in vortex HHG [27] which contains all intermediate OAM states.
Appendix C: photocurrent model and off-axis phase matching
In the simulation, the far field THz intensity distribution is composed of the coherent superposition of local THz emission in air plasma. And air plasma is driving by vortex two-color laser field.
Vortex two-color laser field was used as the driving field. The wavelength of $\mathrm{\omega }$ field is $800\textrm{nm}$, and beam waist is $50{\;\ \mathrm{\mu} \mathrm{m}}$, and topological charge is $1$. The wavelength of $2\mathrm{\omega }$ field is $400\textrm{nm}$, and beam waist is $50{\;\ \mathrm{\mu} \mathrm{m}}$, and topological charge is $0$. The laser intensity of $\mathrm{\omega }$ field in simulation is $1 \times {10^{14}}\; \textrm{W}/\textrm{c}{\textrm{m}^2}$, and the intensity of $2\mathrm{\omega }$ field is one-fourth of $\mathrm{\omega }$ field. The medium is nitrogen at one atmosphere.
The laser field of $E({z,r,t} )$ is written as a Laguerre–Gaussian beam:
The local THz emission is calculated by photocurrent model with finite lifetime of free electrons in laser plasma, and it given by [38–40]:
Here, ${W_{ADK}}$ is the ionization rate and was calculated based on the ADK theory [44]:
The far field THz emission is then calculated in the frequency domain by [41]:
Appendix D: simulated local THz distribution and phase difference along medium
Local THz shows four-lobe distribution at x-y plane and stepwise variance phase distribution along azimuthal angle, as shown in Fig. 9(a) and Fig. 9(b) respectively. Phase in intensity distribution radius along azimuthal angle is shown in Fig. 9(c), a stepwise variance phase from 0 to $4\mathrm{\pi }$. This four-lobe intensity distribution pattern rotated along z direction, because of the phase difference of two-color field induced by refractive index difference ($\varDelta \textrm{n} \approx 9.1 \times {10^{ - 6}}$) and Gouy phase, as shown in Fig. 9(d). When phase difference varies more than $2\mathrm{\pi }$, the local THz distribution will turn it around. In far field, the superposition of the rotated local THz makes the intensity distribution evolve to a ring shape, at the same time, the phase changes linearly from the stepwise change. The linearly changed helical phase gives the far-filed THz wave packet OAM.
Funding
National Natural Science Foundation of China (12174412, 11874373); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021241); Scientific Instrument Developing Project of the Chinese Academy of Sciences (YJKYYQ20180023); Provincial Natural Science Foundation of Henan (202300410017); Xinxiang University Doctor Initial Research Program (1366020150).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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