The term structured laser beam (SLB) generally refers to beams with spatially structured amplitude, phase, or polarization. Solving the scalar paraxial Helmholtz equation in Cartesian or cylindrical coordinates, one can derive the Hermite–Gaussian (HGn,m) and Laguerre–Gaussian (LGp,l) modes, respectively [1]. These eigenmodes, except for the lowest-order TEM00 mode, represent themselves SLBs capable of synthesizing other more complex SLBs by coherent or incoherent superposition. Utilizing the longitudinal electric-field component of a focused HGn,m beam, acceleration of particles is possible [2] and the trapping force in electron acceleration can be even higher compared to the fundamental mode [3]. For LGp,l modes with a topological charge of l≠0, a singularity point, i.e., optical vortex, appears, associated with vanishing amplitude and undefined phase. Such vortices with zero central intensity carry orbital angular momentum of lℏ per photon [4] and enable LGp,l beams to exert additional torques and forces on particles in optical trapping and manipulation [5], expand transmission capacity by exploiting a new degree of freedom in optical communications [6], break the diffraction resolution barrier in fluorescence light microscopy in combination with the stimulated emission depletion technique [7], and involve multi-dimensional states in quantum entanglement [8]. Femtosecond pulse optical vortices exhibiting higher critical power for self-focusing collapse compared to a conventional flattop beam [9], are attractive for some novel applications such as generation of attosecond vortices through high-order harmonic generation (HHG) [10], control of light filamentation for transporting and manipulating microwave radiation in air [11], and fabrication of three-dimensional chiral microstructures [12]. So far, the pulse durations of pulsed optical vortices in the 2-μm spectral region are much longer than 100 fs, thus limiting some potential applications related not only to the specific advantages due to the driving wavelength, e.g.,  in HHG or organic material processing, but also to the availability of better nonlinear materials for frequency conversion to yet longer (mid-IR) wavelengths.

Common methods used for generation of femtosecond optical vortices are based on transformation of a pre-existing ultrashort-pulse TEM00 mode by conventional phase modulation elements, including computer-generated holograms (CGHs) [13,14], spiral phase plates (SPPs) [15,16], and spatial light modulators (SLMs) [17]. Usually CGHs introduce extensive angular dispersion which inevitably results in spatial intensity distortion and temporal pulse broadening, so that complex 4f [13,18] or 2f–2f [14] compressors with additional diffraction gratings are required at the expense of conversion efficiency. SPPs do not produce spatial chirp but introduce topological-charge dispersion due to the high wavelength sensitivity [15], and SLMs generally suffer from low laser damage threshold and high cost [17]. Most importantly, all these methods are not suitable for creating ultrashort-pulse optical vortices in the 2-μm spectral range either because of the lack of suitable materials or bandwidth limitations. Alternatively, HGn,m modes can be transformed into LGp,l modes through rephasing their decomposed terms using “π/2 converters,” e.g.,  single [19] or double cylindrical lenses [20], or two spherical-concave mirrors [21], which ensure a broadband operation range, high mode purity, and high laser damage threshold at lower cost.

Femtosecond pulse HGn,m modes can be directly generated from a mode-locked solid-state laser, as first observed in a Kerr-lens mode-locked Ti:sapphire laser as early as 1991, but in the form of a “mixed” spatial mode [22]. In 2009, pure high-order HG0,m modes were generated from a picosecond self-mode-locked Nd-laser and subsequently converted to LG01 modes [23], and recently the pulse duration near 1 μm was shortened to the 200-fs range employing an Yb-glass laser [24]. However, in the 2-μm spectral range, the pulse durations of lasers with HG01 and thus LG01 modes demonstrated so far were only in the sub-picosecond range and the spectral extent was correspondingly narrower [25,26].

Thus, the state of the art of ultrashort-pulse SLBs in the 2-μm spectral range and their promising applications motivated us to extend the studies to the femtosecond temporal regime. Here, we demonstrate a tabletop laser source that produces ultrashort-pulse HG10 and LG0,m modes both with pulse durations in the 100 fs range. The advantages of such a laser system over the traditional methods for generation of ultrashort-pulse SLBs are experimentally confirmed by characterizing the spatial beam pattern as well as temporal and spectral features. This work paves the route toward few-cycle pulse generation of optical vortices in the 2-μm spectral range. The generated pulsed optical vortices with such a short duration and broad optical spectrum will enable applications such as organic materials machining, novel molecular spectroscopy, and optical vortex infrared supercontinuum generation.

Subsequently, the emitted HG10 laser beam of the mode-locked laser was externally converted into LG0,±1 modes by a simple single-cylindrical-lens (SCL) converter [19]. A lens of f=100mm was used to create a beam waist with a Rayleigh range (ZR) of ∼35mm, followed by a cylindrical lens with fc=35mm, placed exactly at ZR behind the waist (see Section 3.C). The cylindrical lens was rotated by ±45° in the XY-plane to introduce a total Gouy-phase difference of ΔΨ between the X- and Y-directions, where ΔΨ can be expressed as [29] ΔΨ=±[π/2+arctan(Z/ZR−1)]∓[arctan(Z/ZR+1)].(1)

If Z≫ZR, the phase difference will tend to ±π/2, thus converting the femtosecond HG10 to LG0,±1 modes. A CCD camera placed roughly 50 cm behind the cylindrical lens was used to record the LG beam patterns.

At constant round-trip GDD, the pulse duration depends on the product of the intracavity pulse energy (EP) and the self-phase modulation (SPM) coefficient (δL), where δL is inversely proportional to the mode area on the ceramic. By translating M2 from the first to the second femtosecond regime, the estimated mode area reduced roughly 2 times, thus leading to an enhancement of the SPM effect and further shortening of the pulse duration, very similar to previous observations in such mode-locked lasers [33,34]. The mode-area change can be also estimated from Eq. (2) using the measured output power and pulse duration in both cases. The result (a factor of 2.4) is rather close and the deviation can be attributed to nonlinear spatial effects such as self-focusing. Note that the beam radius on the SESAM increased only slightly from 120 to 140 μm when moving from the first to the second mode-locking region, which is a further indication that SPM is the primary mechanism responsible for the pulse shortening, while the SESAM, as a slow saturable absorber, initiates and stabilizes the mode-locking process. Finally, laser operation in the fundamental Gaussian mode was confirmed by the measured beam pattern, as seen in the inset of Fig. 2(b), a typical TEM00 mode with negligible astigmatism.

Subsequently, translating M2 to reach the second mode-locking region, the beam pattern maintained the HG10 mode but exhibited asymmetric intensity distribution of the two lobes [see the inset of Fig. 2(c)]. The higher laser intensity of the lobe that better overlapped with the pump spot enhanced the SPM effect, thus leading to stronger nonlinear astigmatism but also enhanced spectral broadening. In this case, the spectral FWHM was 42 nm with a central wavelength at 2070 nm [see the solid green line in Fig. 2(a)]. By fitting the interferometric autocorrelation trace envelopes [see Fig. 2(c)] using a sech2 function for the pulse shape, a pulse duration of 109 fs was obtained (TBP = 0.320). The exact 8:1 peak-to-background ratio and the perfect fitting indicate chirp-free pulse generation [36]. Mode-locking of yet higher-order HGn,0 modes through increasing the pump offset was not achieved because of the inadequate spatial intensity distribution and the limited pump power.

As shown in Fig. 3(a), the HG10 output beam of the mode-locked laser was transformed to LG0,±1 modes by a simple SCL converter. The middle and bottom rows in Fig. 3 show the measured HG10 and LG0,±1 intensity patterns in the first and second femtosecond regimes, respectively. The imperfect circularity and the nonuniform spatial intensity distribution of the transformed LG0,±1 modes [see Figs. 3(c), 3(d), 3(f), and 3(g)] were inherited from the HG10 mode. Nevertheless, the LG0,±1 modes exhibited clean doughnut intensity profile without other high-order or mixed transverse modes. After passing through the converter, ∼20% of the power was lost due to the uncoated cylindrical lens, resulting in average powers of 40 and 18 mW for the above-mentioned two regimes, respectively.

In conclusion, structured laser beams including the HG10 and LG0,±1 modes were generated near 2 μm in the femtosecond regime employing a mode-locked solid-state laser and a simple SCL converter. No spatial chirp or topological-charge dispersion was introduced in contrast to the commonly used methods for producing ultrashort-pulse vortex beams [13–18]. The almost chirp-free pulses with smooth and broad optical spectra are evidence for the high reliability of the oscillator and the converter. On one hand, this work demonstrates the shortest high-order transverse mode pulses directly generated by a mode-locked solid-state laser, and on the other it shows the first ∼100fs optical vortices in the 2-μm spectral range. It confirms the unique capability of such laser systems to generate ultrashort-pulse SLBs with broad optical spectrum but without unwanted chirp. We believe the present achievement paves the way for generating few-cycle pulse optical vortices at 2 μm, e.g.,  through enhancing the SPM effect since 55 fs (eight optical cycles) has been already achieved in the fundamental Gaussian mode in the present cavity configuration. Finally, the stable femtosecond laser vortices obtained near 2 μm can be employed to generate optical vortex infrared supercontinuum, as a seed for straightforward power scaling using novel single crystal fibers and HHG generation, special microstructuring of transparent materials, and mid-IR vortex generation through nonlinear frequency downconversion.

Acknowledgment. We thank Dr. J. Zhang from Key Laboratory of Transparent and Opto-Functional Inorganic Materials, Shanghai Institute of Ceramics, China, for the Tm:LuYO3 ceramic, and M. Guina from Reflektron Ltd., Tampere, Finland, for the SESAM used in the present study. Y. Zhao acknowledges financial support from the Alexander von Humboldt Foundation through a Humboldt fellowship.