In 1989, Coullet et al. proposed the existence of optical vortices in laser cavity modes [1]. A few years later, Allen and co-workers demonstrated that Laguerre–Gaussian (LG) spatial amplitude distributions with an azimuthal index different from zero, which constitute one type of an optical vortex, carry orbital angular momentum (OAM) [2]. Since then, optical vortices have emerged as powerful tools across a wide range of applications including, but not limited to, optical trapping and manipulation, optical communications, high-dimensional quantum cryptography, laser micromachining, and microscopy [3]. Moreover ultrashort optical vortices with high energy have been proposed for diverse applications such as hollow plasma drilling [4] or acceleration of attosecond electron slices [5]. In particular, it has been demonstrated that high harmonic generation driven by ultrashort IR fields carrying a topological charge can produce extreme ultraviolet light with OAM [6–8]. These novel laser fields can lead, among other interesting effects, to spatially dependent modified selection rules during single photon ionization [9,10] and enable OAM-induced x ray dichroism [11].

An optical vortex is characterized by a point of zero intensity and an azimuthal phase variation that integrates to $2\pi m$ on a closed path around that point, for a nonzero integer $m$. While vortices can appear randomly in speckle patterns, the term vortex beam describes a beam with a well-defined complex amplitude pattern and a single vortex in the center. This form of phase variation means that the wavefront is helical. $m$ is called the topological charge, and in the context of LG modes, it is indicated by the azimuthal index of the mode. A variety of methods for generating vortex beams have been explored, most notably, a combination of cylindrical lenses [2], spiral phase plates [12], liquid-crystal-based spatial light modulators [13], and holographic plates [14]. In particular, a pair of holographic plates has been utilized to generate femtosecond vortices [15,16]. These approaches rely on imparting a specific spatial phase at a particular target wavelength. Since the spatial spiral phase varies for different wavelengths, implementation of these methods with ultra-broadband sources supporting few-cycle pulses leads to what is known as topological charge dispersion, i.e., different contents of topological charges for different wavelengths of the spectrum. Topological charge dispersion leads, among other things, to a non-preserving spatial distribution during propagation and complex spatiotemporal structures [17].

A number of methods have been proposed and implemented in order to address the topological charge dispersion challenge. Bock et al. employed a diffractive–refractive element to obtain few-cycle vortices at a specific plane from Ti:sapphire oscillator pulses [18,19]. Tokizane et al. combined achromatic wave plates and a segmented half-wave plate for broadband vortex beam generation [20]. The same group further extended this approach, amplifying a broadband vortex beam produced in such way in a two-stage optical parametric amplifier [21]. Among these methods, Atencia et al. developed a compound holographic optical element to generate achromatic vortices [22], while Swartzlander designed achromatic spiral phase plates from two different glasses, providing a perfect topological charge at two wavelengths, akin to achromatic lenses [23].

With the notable exception of Naik et al., who introduced an elegant approach based on a Sagnac interferometer to generate achromatic vortices [24], these techniques often prove complex or challenging to implement, and/or incompatible with the generation of high-energy ultra-broadband vortex pulses. An alternative to generating a wideband optical vortex directly (i.e., transforming a field without OAM) is to use an optical parametric chirped pulse amplifier (OPCPA) [25] to transfer the OAM of a narrowband pump beam to a wideband idler beam. The phase-sensitive nature of the parametric amplification process implies that the idler produced during the amplification of a weak signal wave carries the phase difference between the pump and signal waves. This also holds for helical wavefronts, so the difference in topological charge between the pump and the signal will be transferred to the idler. The concept of transferring the topological charge from the pump to the idler was originally demonstrated in an optical parametric amplifier (OPA) with continuous-wave sources [26] and later implemented in pulsed narrowband optical parametric oscillators where the selective transfer of the topological charge of the pump to the signal or idler was demonstrated [27,28]. Camper et al. demonstrated the generation of a high-energy femtosecond optical vortex in the idler of a noncollinear OPA pumped by a vortex beam [29]. This OPA featured relatively long pulses and consequently their analysis did not address topological charge dispersion. In this work, it is shown that this concept can be extended to ultra-broadband sources supporting few-cycle pulses. The topological charge content of the wideband idler generated in an OPCPA pumped by a narrowband vortex is analyzed, and it is shown that, under certain conditions, the topological charge dispersion can be minimized over the entire bandwidth.

The system under study is the two-stage noncollinear OPCPA schematically shown in Fig. 1. The first stage had Gaussian beams for both the pump and the seed, and the second stage was pumped by an LG mode. The 3D numerical simulations of the parametric processes were performed utilizing the code Sisyfos (Simulation System For Optical Science) [30–32]. The code allows simulating second-order frequency mixing processes including the effects of pump depletion, back-conversion, walk-off, dispersion, diffraction, parasitic effects, and thermal effects. The coupled-wave equations in three dimensions are solved in Fourier space. This code has been utilized before to simulate OPCPAs under realistic conditions [33–35].

 

Fig. 1. Proposed experimental scheme. The beam profiles illustrate Gaussian or Laguerre–Gaussian (LG) modes. The prism ($P$) illustrates an optical element introduced to achieve pulse-front matching. The LG mode of the output idler is also indicated. $M$ represents a set of mirrors to transport/re-size the amplified signal into the second amplification stage.

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All the simulations presented in this work are based on noncollinear type-I phase-matching in BBO crystals, specifically with a noncollinear angle $\alpha = 2.5 ^{\circ }$. The first stage was under the walk-off compensation (WOC) geometry ($\theta _{signal} = 0.4702$ rad, $\theta _{pump} = 0.4265$ rad, with $\theta$ the angle between the crystal axis and the $k$-vector of the signal or the pump). The WOC geometry minimizes space–time couplings in the amplified signal but suffers from parasitic second harmonic of the signal and the idler [36] (taken into account in the simulations). Meanwhile, the second stage was under tangential phase-matching or non-walk-off compensation geometry ($\theta _{signal} = 0.3829$ rad, $\theta _{pump} = 0.4265$ rad), which is better suited for large beams for which the lateral walk-off between the beams is small compared to the beam sizes. The pulse parameters employed herein are similar to a near-infrared OPCPA system developed at the Max Born Institute [37–39]. A measured pulse spectrum from a Ti:sapphire oscillator was utilized to build the input seed pulses, stretched by applying positive group delay dispersion (800 fs$^2$) and energy of 1 nJ. The pump pulses were assumed to have a Gaussian temporal shape with a Fourier transform-limited duration of 1 ps, centered at 515 nm.

It was assumed that the pump beam is transformed into an LG mode by means of a spiral phase plate, which are nowadays commercially available for a target wavelength of 515 nm. Spiral phase plates provide high damage threshold and are easy to implement experimentally. For the simulations shown in this work, it was assumed that the spatial distribution of the pump field is described by an LG mode with a topological charge of $m=1$ and no topological charge dispersion. It is important to remark that it was also assumed that in the second stage, pulse-front matching was implemented [40] (illustrated by the prism ($P$) in Fig. 1). The results can degrade substantially without pulse-front matching (see Supplement 1).

In the first OPCPA stage, the pump pulse energy was 0.4 mJ and the peak intensity incident on the crystal was 97 GW/cm$^2$. The pump and signal beams in the first OPCPA stage had matching sizes (700 µm diameter $1/e^{2}$). The signal pulses were amplified in a 1.5-mm-thick crystal to 14.9 µJ. These amplified signal pulses were utilized to seed the second amplification stage, pumped by an LG mode with a topological charge of $m=1$, pulse energy of 5 mJ, and peak intensity of 50 GW/cm$^2$ incident on a 1.2-mm-thick BBO crystal. It should be pointed out that the following results are almost independent of the value of the topological charge, for low values of $|m|$. A detailed discussion is provided in Supplement 1. In this case, the signal beam diameter was 4.8 mm, approximately 20${\% }$ larger than the pump. At the output of this second OPCPA stage, the idler carries the same topological charge as the pump. This is illustrated in Fig. 2 and Fig. 3 showing frequency-resolved beam profiles and spatial phases at selected wavelengths. The beam profiles show a clear annular shape with the amplitude vanishing in the center. The noncollinear geometry and spatial walk-off cause an uneven spatial intensity distribution (left/right asymmetry), especially toward the edges of the spectrum. The phase profiles show in all cases illustrated in Fig. 3 the characteristic azimuthal phase variation. In order to quantify the topological charge dispersion, the topological charge content was calculated by expanding the spatial part of the laser electric field in helical harmonics [41,42] for each frequency component:

 

Fig. 2. Frequency-resolved idler beam profiles at selected wavelengths of the spectrum. The color scale indicates the normalized intensity.

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Fig. 3. Frequency-resolved idler spatial phase profiles at selected wavelengths of the spectrum. The color scale indicates phase in radians.

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Equation (4) offers an insight into the topological charges forming the spatial profile at a particular frequency component. Figure 4(a) shows the distribution of topological charges calculated at selected wavelengths over the idler spectrum. It is observed that the topological charge of the pump has been transferred to the idler with remarkable purity. In addition, Fig. 4(c) shows the idler spectrum (red) and illustrates the proportion of topological charge $m=1$ across the entire idler spectrum (dashed blue). For comparison, Fig. 4(b) shows the distribution of topological charges in the case in which a fused silica spiral phase plate designed for 1600 nm is utilized to produce a vortex beam with topological charge $m=1$ for an initial Gaussian spatial distribution. In addition, the dashed green line in Fig. 4(c) illustrates the proportion of topological charge $m=1$ across the spectrum in this case. Converting a broadband OPCPA seed into a vortex beam using a spiral phase plate (before amplification) is an approach that has been reported in the literature [43–47], albeit for the case of pulse bandwidths supporting multi-cycle pulses, where the use of a spiral phase plate generates a rather limited topological charge dispersion. However, the effect of saturation over the topological charge dispersion has been observed [46,47].

 

Fig. 4. Topological charge dispersion in the idler. (a) Relative topological charge content ($P_m$ coefficients) at selected wavelengths across the idler spectrum. (b) Relative topological charge content assuming a spiral phase plate (SPP) has been utilized to convert the broadband Gaussian field into a Laguerre–Gaussian. (c) Idler spectrum (red), relative proportion of topological charge $m=1$ across the spectrum for the optical vortex generated during parametric amplification (dashed blue) or utilizing an SPP (dashed green line).

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Under the amplification conditions previously described, the signal field was further amplified to an energy of 1.02 mJ, and the resulting idler had a pulse energy of 0.59 mJ. This corresponds to a total extraction efficiency (considering both the signal and the idler) of 32.2 ${\% }$ and represents a reasonable compromise between efficient energy extraction and minimization of space–time couplings that degrade the peak intensity of the amplified pulses [35]. The spectrum of Fig. 4(c) supports a Fourier transform-limited pulse duration of 8.5 fs, corresponding to approximately 1.67 optical cycles at the carrier frequency of 195.9 THz (1531 nm). The angular dispersion of the idler can be compensated with established methods [40,48] producing high-energy, sub-2 cycle optical pulses in the short-wave infrared with a clean topological charge content after proper dispersion compensation (see Supplement 1 for a detailed analysis on the compressibility of the idler and the pulse duration at different positions across the beam profile). Alternatively, a controlled angular dispersion could be introduced in the signal beam seeding the second stage to obtain an angular-dispersion-free idler [49].

The scenario proposed here represents just one example of many different variations that can be implemented around the same idea. Other approaches include seeding a last OPCPA stage with the idler (with the appropriate angular dispersion) from a previous stage and pumped by an LG mode to obtain an angular-dispersion-free signal with controlled topological charge, combining optical vortices in both pump and seed fields (signal or idler) to control the topological charge of the third field generated during amplification, or designing a multistage, high-energy OPCPA system in which the last amplification stage is pumped by an optical vortex to scale up the pulse energy as already proposed in [29] for a femtosecond OPA. Similar to the first proposed alternative scenario above, an ultra-broadband signal with a clean topological charge can be generated at low or moderate pulse energy and be further amplified in successive OPCPA stages as shown in [43–47], either with Gaussian or flattop pump beams. Given the strong effects of Gaussian pumping on saturation-induced spatiotemporal couplings [35], flattop beam pumps would enable higher extraction efficiencies while maintaining a low topological charge dispersion [46,47]. Additionally, similar scenarios based on sum frequency generation [50] can be implemented. Furthermore, nonlinear propagation in thin glass plates could be utilized to push the pulse duration toward the single-cycle limit [51,52]. Generating few- and near-single-cycle signal and idler pulses with controlled topological charge can be exploited to generate XUV attosecond pulses with controlled OAM [7,8] or to expand the toolbox of ultrafast two-color pump–probe spectroscopic techniques. For example, the optical vortex encodes all the possible values of carrier–envelope phase (CEP) along the azimuthal coordinate of the beam, and therefore, this can be exploited in spatially resolved measurements [53] to record all values of CEP without the need of performing a scan.

In conclusion, this study demonstrates the remarkable capability to transfer the topological charge of a narrowband, high-energy laser pulse with exceptional fidelity to broadband idler pulses during optical parametric chirped pulse amplification. The key element to minimize the topological charge dispersion is the mitigation of space–time couplings through the implementation of pulse-front matching of seed and pump fields. The proposed approach opens the door for the generation of high-energy few-cycle optical vortices with fine control over the topological charge. Follow-up work will focus on the experimental demonstration of the proposed idea.