Curved filament with large bent angle and controllable propagation behavior has always been great expectation and challenge due to its novelty and complexity. The unique properties of curved filaments make it possible to achieve many applications in micro-fabrication, spectroscopy and meteorology. Here we realize experimentally and theoretically control on helical filaments induced by twisted beams in CS2. The results show that helical filaments exhibit a robust pattern and high rotation rate. Specific intensity pattern of the twisted beam confines the filaments in fixed relative position and the azimuthal energy flux drives the rotating of the filamentation pattern. In addition, we demonstrated that the global orbital angular momentum (OAM) of twisted beams is still conservative to be zero, but local OAMs exhibit distinct variation during nonlinear propagation. Our idea has its significance which realizes the construction of helical filaments with flexibility and controllability and then facilitates to push the development of related researches.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Femtosecond (fs) filamentation in transparent media has always been a hot research subject since it was observed for the first time by Braun et al. in 1995 . Special physical phenomena generated during filamentation, including conical emission of broadening super-continuum spectrum  and THz wave , long-range propagation of IR pluses , critical laser intensity for filamentation  and self-compression of pulses , have aroused great interests for researchers to investigate the physical mechanism and potential applications of filaments.
In general, filaments can be generated by using ultrashort laser pulses with cylindrically symmetric spatial profiles like Gaussian  or flat-top  profiles, which finally lead to straight filaments. However, it was demonstrated recently that curved filaments, which extended along bent trajectories, possess novel properties and unique applications [8–12]. Curved waveguides could be fabricated based on the curved refractive index changes induced by the curved filaments. The combination of straight and curved waveguides could be constructed as directional coupler, which was proved to possess accurate and controllable splitting ratio . Moreover, the supercontinuum radiation generated from the curved filaments was demonstrated to be angularly resolved in the far field, which provides a way to directly draw the far-field spectral map and is able to further reveal the nonlinear evolution of the laser pulses . Guiding high electric current along specific paths is always a challenge for protecting against lightning strikes. Curved plasma channels produced by curved filaments act as bent wires, which provide discharged channels for large current surge and are able to bypass obstacles . Besides, helical filaments have been predicted to be a promising approach to generate and guide radio- and micro-waves , as it is possible to construct a hollow plasma tube .
Generally, previous methods to generate curved filaments can be summarized from two main kinds. The first one is to induce the curved filaments by using the self-accelerating beams, like Airy beams  and corrected accelerating parabolic beams . These self-accelerating beams exhibit approximate diffraction-free properties and the main lobes of the self-accelerating beams tend to transversely self-bending during propagation in a finite region . However, the bent angles are suppressed to be small as the self-accelerating beams are inherently subjected to the paraxial limit. Nonparaxial accelerating beam, which has been demonstrated to preserve the shape during nonlinear propagation in colloidal suspensions of polystyrene nanoparticles , may be one of the solutions to generate filament with large bent angle. Another way to overcome this limit is to confine the shift of filamentation in the azimuthal direction, so optical vortices are a good candidate. Different from the propagation behavior in free space, the ring-like structure of optical vortex will break into multiple filamentation in self-focusing nonlinear media due to the azimuthal perturbation . More importantly, it has been proved both in numerical simulations  and experimental results  that the resulting pattern of filaments exhibits a rotation in the direction determined by the chirality of the vortex during propagation. However, it is hard to ensure the stability of the number and relative localizations of filaments. These unstabilities can be solved by introducing the intensity peaks  or the specific polarization structures . Inspired by these works, researchers try to use specific helical beams to guide the filaments. Dual helical filaments in air with a rotation rate of 0.0875°/mm has been achieved .
In this article, helical filaments in CS2 with the robust pattern and the high rotation rate have been demonstrated by constructing a kind of twisted beam with the aid of spatial light modulator (SLM). The twisted beam, which is also called radially self-accelerating helical beam, has specific intensity distribution in the azimuthal direction and rotates around its optical axis during propagation. Generally, such a kind of beam can be constructed by collinear coherent superposition of optical vortex beams, such as Laguerre-Gaussian beams [22–28] or Bessel-Gaussian beams [29–33]. This superposition leads to the generated beams carrying variable local obital angular momentum (OAM) although it follows the conservation of global OAM . As eigen-solutions of the Helmholtz equation, the Laguerre-Gaussian and Bessel-Gaussian beams exhibit stable ring-like profiles during propagation. In particular, for the Bessel-Gaussian beams, due to their non-diffraction characteristic, most of the energy is concentrated in the central region in a long propagation region, so it is suitable for constructing long-range filaments . Herein two equal-amplitude Bessel-Gaussian beams with opposite topological charges and different radial-mode indices, which are generated from the adjustable two-dimensional (2D) computer-generated hologram, are collinearly superposed to construct the twisted beam. The generated beam exhibits petaloid intensity pattern and rotates around its optical axis with an uniform rotation rate during propagation.
2. Generation of twisted beams
To generate the twisted beam in experiment, we use a flexible setup in a common path interferometric configuration with the aid of a 4f system composed of paired lenses (L1 and L2) , as shown in Fig. 1. A phase-only SLM (P-SLM) with 1920 × 1080 pixels (each pixel has a 8 × 8 μm2 size) displays the designed holographic grating (HG) and locates at the input plane of 4f system. After passing through such a 2D HG, the input linearly polarized femtosecond Gaussian beam is divided into four equi-amplitude parts carrying the space-variant phase δ(r, ϕ). The spatial filter (SF) is placed at the Fourier plane of the 4f system and only allows two +1st-order diffraction beams to pass through. These two orders are then recombined in common path by a Ronchi grating (RG) located at the output plane of the 4f system. The period of HG is needed to match with RG. When 35-fs laser pulses pass through the system in Fig. 1, the measurement result shows that the pulse duration of the generated twisted beam is broadened to 63 fs.
The complex amplitude transmission function of HG displayed on P-SLM is 36] in the propagation region Zq, which can be estimated geometrically as  36, 37], and the intensity distribution can be expressed as 38], we have 38] and Rq = 2πlqz/kr0. Equation (7) is valid when the radial coordinate r < Rq, r < r0 − Rq and propagation region z ∈ (mZq/2πlq, Zq) . This modified equation exhibits very good approximation to the Bessel wave especially for the central rings  and the modification will be gradually inapparent as the propagation distance increases.
We can find from Eqs. (6) and (7) that: (i) The intensity distribution of this superposed optical field exhibits an azimuthal periodic oscilation, like a ‘petal’ structure with the 2m-fold rotational (C2m) symmetry; (ii) The rotation angle θ of the intensity pattern, as a function of the propagation distance z, can be written as
Clearly, we can raise the rotation rate by increasing the sum of l1 + l2 or the difference of l1 − l2 for two radial spatial frequencies (radial indices), or by using the lower order Bessel beams (i.e., the smaller topological charge m). However, |l1 − l2| is restricted to be small to avoid the separation of central rings of the two Bessel-Gaussian beams. Since the radial index l is not restricted to be an integer, the rotation rate can be adjusted continuously and flexibly within a certain range.
3. Nonlinear propagation of twisted beams
To theoretically explore the nonlinear propagation behaviors of twisted beams in the Kerr medium CS2, which has a relatively high third-order Kerr nonlinear coefficient, the nonlinear wave equation should be used. As is well known, the physical mechanisms of suppressing the nonlinear self-focusing are quite complex, such as plasma defocusing, multi-photon absorption, diffraction, dispersion, and high-order Kerr effect. In different situations, however, specific mechanism(s) could be dominant while others can be ignored. In our case of CS2 under the incidence of ∼63-fs laser pulses, the fifth-order Kerr effect is dominant to resist the self-focusing [39,40] and then the collapse compared with the plasma defocusing when the filamentation is driven by the short-pulse laser (< 70 fs) . The generated filaments carry the lower power density (∼1011 W/cm2) than the breakdown threshold (∼1013 W/cm2) . In particular, it has been demonstrated the fact that the two-photon absorption in CS2 is the dominant multi-photon absorption as the energy loss . Under the slowly varying amplitude approximation and the assumption that the temporal profile of the pulse remains unchanged, a simplified model used here can be expressed as [18,41,42]43], and ψ represents the space-variant complex amplitude normalized by the total field as
On the right side of Eq. (10), the first term denotes diffraction of the field, the second and third terms stand for the contributions from the third-order Kerr nonlinearity and fifth-order nonlinearity [44,45] and their relative relationship is described by the parameter ε. The last term stands for the two-photon absorption as the propagation loss, where β represents the normalized dimensionless two-photon absorption coefficient. We estimate the value of β to be β ∼ 107 from the two-photon absorption coefficient α ∼ 4.5 × 10−13 cm/W of CS2 for 800 nm fs laser pulses [40,46].
Simulated nonlinear propagating behaviors of twisted beams with l1 = 6 and l2 = 10 are shown in Fig. 2, in which (a) and (b) correspond to m = 1 and m = 2, respectively. In our simulations, the input power P is taken as P = 4Pc, ε = 10−4, and β = 107. The three-dimensional graphs in Fig. 2 show the nonlinear propagating behaviors of the twisted beams within the propagation region ζ ∈ [0, 1], where three two-dimensional profiles show the intensity distributions at ζ = 0, ζ = 0.5 and ζ = 1, respectively. Clearly, the intensity convergence and clamping appear at the intensity peaks of the twisted beam and the resulting filamentation patterns exhibit the C2m symmetry. The filamentation pattern rotates as the propagation distance increases, which will ultimately build a helical filamentation trajectory in three-dimensional space. In the region around the central filamentation, there exhibit the Archimedes spiral structures. We have confirmed that the rotation angle (θ) and the rotation rate (θ̇) of the intensity pattern for the twisted beam are the same for both the linear and nonlinear propagation. That is to say, the θ and the θ̇ are independent of nonlinearity, while the nonlinearity is dominant to produce the filamentation.
4. Experimental study and results
To observe the propagating process of filaments, we need to capture the transverse profiles at different propagation distance without changing the position of input plane. So we propose to use a wedge quartz cell filled with CS2. The wedge cell had a wedge angle of α = 18°. The experimental setup is shown in Fig. 3(a). The femtosecond laser system used is an ultrafast Ti:sapphire laser system (Coherent Inc.), which delivers linearly polarized laser pulses at a central wavelength of λ = 800 nm with a pulse duration of 35 fs and a repetition rate of 1 kHz. The twisted beam generated from the 4f system shown in Fig. 1, which has a radius of r0 = 1.1 mm, a pulse duration of ∼63 fs and a pulse energy of 9.13 μJ, is incident normally into the wedge quartz cell filled with CS2. The filaments will be terminated on the output plane of the quartz cell because of the very low nonlinearity of the quartz glass. The intensity patterns on the output plane are then imaged on a detector (Beamview, Coherent Inc.) by an achromatic lens with a focal length of 60 mm. To avoid the influence of environmental disturbance, we take 750 images and make an averaging to get the final result. The exposure time for each image is 5 ms. An infrared cut-off filter with a cut-off wavelength of ∼750 nm is inserted between the lens and the detector, to block the original femtosecond laser at the central wavelength of 800 nm, while to allow only the supercontinuum radiation (shorter than 750 nm) induced by the filaments to pass through. The broadened spectra associated with the helical filaments are recorded by a spectrometer (Ocean Optics USB4000, covering a spectrum region from 200 nm to 1100 nm). Moving the wedge quartz cell with a distance of d in the +x direction [Fig. 3(a)] will increase an available propagation region of Δz = d/tan α for the filaments along the z direction inside the quartz cell. Correspondingly, the whole image system should be moved a corresponding distance of Δz in the +z direction to obtain clear images.
Comparing the above method based on the wedge quartz cell in Fig. 3(a) with the parallel quartz cells of different lengths (of 10.020, 20.395 and 31.143 mm) in Fig. 3(b), our experimental design provides a continuous variation of propagation distance without changing the input plane. Figure 3(c) shows the filamentation results in the three parallel quartz cells and the wedge cell whose input planes locate at the same position. As a function of the propagation distance, the filaments orientation recorded in the parallel quartz cells (squares) are consistent to the linear fitting line of results obtained from the wedge quartz cell (circles), meaning that the oblique output plane in the wedge quartz cell hardly influences the orientation of the filamentation.
Figure 3(d) shows the spectral broadening and Fig. 3(e) further confirms this broadening effect by the supercontinuum spectrum after passing through an infrared cut-off filter. These results and the captured output patterns in Fig. 4 prove clearly the appearance of filamentation.
Figure 4 shows the measured propagation behaviors of filaments generated by the twisted beam with different l and m but fixed pulse energy and radius. The number of filaments is 2m and the pattern keeps the C2m rotational symmetry during propagation. All the images are normalized by its maximum intensity. The white arrows denote the orientation of filamentation patterns, which are parallel to the connection between the diagonal filaments. Clearly, as the propagation distance increases, the filamentation pattern will rotate, as shown from the left to the right in Fig. 4. For instance, an ∼45° rotation can be observed after propagating a distance of 19.65 mm (= 3.275 × 6 mm) without the split or fusion of filaments, as shown in Fig. 4(b).
Furthermore, we explore the stability and controllability of the helical filaments. Figure 5 shows the orientation angle of the filamentation pattern as a function of the propagation distance. We can find from Fig. 5 that the orientation angle of the filamentation pattern varies linearly with the propagation distance. In particular, from Fig. 5(b), which shows the comparison between the results with exchanged radial indices but unchanged topological charge m =1, the rotation direction (or chirality) of the filamentation pattern is determined by the Bessel-Gaussian beam with larger radial index. By using the linear fitting of experimental data, the fitted slopes (θ̇exp) representing the rotation rates are very close to the theoretical predictions (θtheo), and the detailed results are also listed in Table 1.
As is well known, the propagation behavior of an optical field is closely related to the distribution of energy flux, which is usually represented by the Poynting vector. The magnitude and the direction of the Poynting vector indicate the intensity and the local direction of the energy flow, respectively. To explore the mechanism of the generation of helical filaments, it is necessary to investigate the evolution of the Poynting vector during the nonlinear propagation. The time-averaged Poynting vector in the Lorenz gauge needs to be solved Fig. 6 (m = 1 (a) and m = 2 (b)). At the initial position z = 0 (or ζ = 0) in the first column of Fig. 6, the intensity distribution exhibits the spiral structure, but the energy flux is along the radial direction. As the twisted beam propagates, the perfect spiral structures are broken and the azimuthal component of the energy flux (Poynting vector) appears, as shown in the second and third columns of Fig. 6. From the second column of Fig. 6, it can be found that the azimuthal component around the intensity peaks (the filaments) is along the clockwise direction, which is the same as the rotation direction of filaments. The existence of the azimuthal component of energy flux with the increasing of propagation distance drives the continuous rotation of the filamentation pattern. However, when the propagation distance arrives at the maximum propagation region (the third columns of Fig. 6), the energy flux along the anticlockwise becomes dominant, implying the inverse energy transport and resisting the rotation of filamentation pattern.
To further investigate the evolution of this azimuthal energy flux, we calculate the local OAMs. As reported in the previous works, the propagation of the twisted beam in free space exhibits the variation of local OAM but the global OAM still keeps conserved . However, when the high-power twisted beam propagates in the nonlinear Kerr medium, the evolution of OAM is not confirmed. The OAM density Lz can be determined from the relationship Eq. (13), we can now define the local OAMs inside specific areas. The red circle (in the second column of Fig. 6) is set to distinguish the filamentation region from the surrounding, and the radius of the circle rf is defined to encircle all the points with the intensity above the half value of the intensity peaks. The local OAMs inside and outside the circle can be calculated from
Figure 7 shows the dependence of the local OAMs (Lin and Lout) on the propagation distance ζ (or z) within the propagation region of the twisted beam, in which the solid lines correspond to the cases considering the linear and nonlinear effects simultaneously, while the dotted lines correspond to the cases considering the linear effect only for comparison. The perfect cancellation of Lin and Lout implies the global zero OAM and the conservation of OAM for the above two cases. The local OAMs Lin undergoes the evolution from negative value to positive one, and the inversion happens near the maximum propagation region (ζ ∼ 0.9Z), while the local OAMs Lout of the surrounding field exhibits the opposite process. Such an inverse behavior appears for both linear and nonlinear propagation, while the existence of nonlinearity has a little influence. It is also found that the evolution of local OAMs for the same topological charges in different propagation situations exhibits the similarity, but the nonlinearity suppresses the OAM change.
Different from the previously reported methods producing the bent or the helical filaments, our method exhibits the high rotation rate of the filamentation pattern. The previous works can only lead to the helical filaments in experiments with rotation angular velocities of ∼0.0875°/mm  and ∼0.15°/mm , while we have achieved a great increase up to ∼2.3°/mm. Furthermore, no splitting or fusion of the filaments has been observed in the experiments and simulations, and we have demonstrated that the rotation rate of filamentation pattern is the same as that of twisted beam during the linear propagation. The helical filaments keep the high stability during the whole propagation, which is hard to be achieved in the previous works [11,19]. Based on the experiments and simulations, the helical filaments with the large rotation angle can be predicted from Eq. (8), even in other propagation media such as air. If the laser pulses have the higher power/energy, it is allowed to use the slightly expanded twisted beams (keeping the radial spatial frequencies, i.e. l1 and l2). Thus the twisted beam will extend the propagation region of filaments in the case without significantly reducing the rotation rate. This could be expected when the damage threshold of the P-SLM can be greatly raised or the experimental design can be significantly improved.
In fact, for the helical filaments, the bending effect is determined by the rotation rate. Higher rotation rate means a higher radial acceleration and a more remarkable azimuthal shift during the propagation. The remarkable bending effect achieved by the twisted beam provides the possibility to push the development of related research and benefit to related applications. Specifically, the helical filaments with the robust pattern and high rotation rate are possible to produce a supercontinuum far-field spectral map with an improved angular resolution, which can provide more precise information about the nonlinear pulse evolution . In addition, if the helical filaments with high rotation rate can be achieved, it is possible to produce high-quality tubular plasma channels to guide microwave signal by the helical filaments  instead of cylindrical filament array .
In conclusion, we have demonstrated the helical filaments with stable number, robust pattern and controllable uniform rotation rate, generated by the twisted beams in CS2. Based on the method of collinear coherent superposition of two Bessel-Gaussian beams with opposite topological charges and different radial indices, we produce the twisted beams with centrosymmetric petal intensity patterns and spiral profiles. Such specific intensity patterns confine the fs laser filaments at fixed relative positions obeying C2m symmetry, while the azimuthal energy flux drives the rotation of filaments. A ∼45° rotation angle has been achieved after propagating a distance of 19.65 mm in experiment, corresponding to a high rotation rate of ∼2.3°/mm, which is more than one order of magnitude comparing to the results reported in the previous works [11, 19]. In addition, we discuss the OAM evolution of the high-power twisted beam during the propagation in nonlinear media. The results show that the global OAM of the superposed beam is still conserved to be zero, while the local OAMs within the filamentation region and the background area exhibit the opposite variation. Although the existence of nonlinearity suppresses the changes of local OAMs in the filamentation region, the dynamics of helical filaments still stably inherits the helical propagation behavior of the twisted beam with more converged and clamped intensity in the filamentation region. Furthermore, we predict the larger rotation angle can be achieved if the damage threshold of P-SLM experimental design can be improved.
National key R&D Program of China (2017YFA0303800, 2017YFA0303700); National Natural Science Foundation of China (NSFC) (11534006, 11774183, 11674184); Natural Science Foundation of Tianjin (16JCZDJC31300); 111 Project (B07013).
We acknowledge the support by Collaborative Innovation Center of Extreme Optics.
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