Besides spin angular momentum (SAM), orbital angular momentum (OAM) is another intriguing characteristic of light^[1–3]. The beam carrying OAM is the so-called vortex beam. Due to the helical wavefront in longitudinal propagation direction, the vortex beam represents vanishing amplitude and an uncertainty phase around the center. The OAM of the vortex beam is lℏ per photon. They exhibit additional torques and forces that can be applied to particle manipulation^[4], add a new degree of freedom to expand information capacity in optical communication^[5], and break the diffraction resolution limitation in stimulated emission depletion microscopy^[6]. Due to the ultra-intense power, stringent demands for femtosecond vortices have been put forward because of the boom of special applications, such as controlling light filamentation^[7], generating extreme UV vortex by high-order harmonic generation^[8,9], and manipulating microwave radiation in air^[10]. The commonly used method to obtain a femtosecond vortex beam is mode conversion from the TEM00 mode to a Laguerre–Gaussian (LG) beam by phase modulation elements, including spatial light modulators^[11], computer-generated holograms^[12], spiral phase plates^[13,14], and metasurfaces^[15,16]. Usually, angular dispersion introduced by computer-generated holograms inescapably leads to distortion of the intensity distribution in the space domain, as well as broadening of the pulse in the time domain^[12,17]. Spiral phase plates are apt to disperse a topological charge due to the significant dependence on wavelength^[18]. Spatial light modulators and metasurfaces are hampered by a low damage threshold and a significant fabrication cost^[19,20]. Especially, metasurfaces mainly depend on high-cost and time-consuming electron beam lithography or focused ion beam etching^[20].

Compared with mode conversion, femtosecond vortex lasers generated from the resonator are more advantageous. Because the system is more simplified, the output beam quality is excellent, and conversion efficiency is higher due to the absence of phase modulation elements^[21–24]. In 1 µm regime, Yb3+-doped laser crystals own wide emission bandwidth, and thus are favorable for generating femtosecond vortex laser. Based on the off-axis pumping method, by controlling the Gouy phase difference between HG10 and HG01 modes, a 360 fs LG0,1 vortex laser has been achieved from a semiconductor saturable absorber mirror (SESAM) mode-locked Yb:phosphate laser^[25]. Utilizing a defect spot mirror, a 298 fs LG0,1 vortex beam has been reported in a SESAM mode-locked Yb:KYW laser^[26]. Using translation-based off-axis pumping and the angle-based noncollinear pumping techniques, 1st–30th order femtosecond Hermitian–Gaussian (HG) modes have been obtained in a Yb:KGW oscillator and then converted to a femtosecond vortex using a cylindrical lenses mode converter^[27]. The previous studies on ultrafast vortex lasers are almost focused on single mode. However, more complicated beam optical vortex arrays (OVAs) have received less attention. OVAs possess diverse transverse patterns and multiple phase singularities^[28,29], and they are potentially applicable in Bose–Einstein condensate, manipulating multiple microparticles and 3D displays^[30–32]. Recently, transverse mode locking of different LG modes has been found to be utilized to form OVAs^[29,33,34]. Unfortunately, it was usually used to produce CW OVAs rather than femtosecond OVAs. It adds more difficulty to generate femtosecond OVAs because the realization of femtosecond OVAs refers to both transverse mode locking in space and longitudinal mode locking in the time domain.

In our work, we obtain a femtosecond vortex array in a Yb:KGW resonator. A defect spot mirror combined with the off-axis pumping method is utilized to generate three different beams: TEM00, LG0,1, and two-vortex array (TVA). The three beams can be switched with each other by controlling the parameters of the resonator. To the best of our knowledge, this is the first time to realize a mode-locked vortex array with femtosecond pulses from a resonator.

The schematic diagram of the mode-switchable femtosecond vortex laser is demonstrated in Fig. 1. A 976 nm laser diode (LD) is designed as the pump beam. It is then coupled via a fiber with a diameter of 105 µm, and it is subsequently focused into the laser crystal through an optical coupling system with a 1:2 ratio. The laser crystal is an Np-cut Yb:KGW with a dopant concentration of 3% and a size of 4mm×4mm×8mm. Both end facets of Yb:KGW are coated with an antireflection film (R<1%) within the wavelength range at 1030–1080 nm and 976 nm.

A Z-type laser cavity is composed of an input mirror (IM), a high reflection (HR) mirror, an output coupler (OC), and an SESAM. The IM is a flat mirror with high transmissivity at 976 nm (T>99%) on both sides and high reflectance at 1030–1080 nm (R>99%) on the right side. There are different defect points (diameters: 30–250 µm) on the right side of the IM. The HR mirror has a radius of curvature of 500 mm and is coated with high reflectivity (R>99%) at 1030–1080 nm. A concave mirror (radius: 200 mm) with a transmissivity of 9% within the wavelength range of 1030–1080 nm is utilized as OC. A commercial SESAM (Batop Gmbh, SAM-1040-1-1 ps) is used to initiate the mode-locked operation. The length of the cavity is around 1.545 m, with the calculated mode radii on the laser crystal and SESAM being 136 and 60 µm, respectively.

The output laser beam is divided into two distinct branches. One branch is used for detecting laser pulses. The effective pulse width is measured by an autocorrelator (APE, pulse Check USB IR). The pulse trains are captured by a digital oscilloscope (LeCroy, HDO4104A) coupled with an InGaAs detector (EOT, ET-3000). Additionally, the laser spectrum is analyzed using an optical spectrum analyzer. To simultaneously monitor the intensity distribution and interference pattern of the vortex laser, another branch is incident in a charge-coupled device (CCD) camera (Dataray, S-WCDLCM-C-UV).

The TEM00 mode is first obtained under a threshold of about 4.7 W [Fig. 2(a)]. After experiencing a Q-switch mode-locked state, its CW mode-locked operation is initiated when the pump power grows up to 7.8 W, and the maximum output power is 383 mW when the pump power reaches 13.8 W. The IM is then calibrated to ensure that the center of the defect point aligns precisely with the pump beam. A defect point with a diameter of 50 µm is determined to be the best size in the experiment. Due to the mode confine of the defect spot, the LG mode starts to oscillate when the pump power exceeds the threshold 5.6 W, as shown in Fig. 2(b). The LG mode switches from Q-switch to CW mode-locked state as the pump power is higher than 9.4 W. The maximum output power of the CW mode-locking laser is up to 340 mW, corresponding to a pulse energy of 3.52 nJ. When the pump power exceeds 11.1 W, the mode-locked LG mode can be switched to TVA by slightly rotating the laser crystal (0.1°) to attain off-axis pumping condition.

In order to determine the OAM, the vortex laser is reflected by a plane concave mirror with 100 mm curvature of radius, interfered with a spherical wave. Figures 3(a1)–3(c1) demonstrate the experimental intensities of TEM00, LG mode, and TVA. The TVA is characterized by two dark spaces in intensity distribution or two singularities in the phase distribution. The interference pattern in Fig. 3(b2) is clockwise, which indicates that the topological charge of the LG mode is l=1. Similarly, the interference pattern of TVA in Fig. 3(c2) has bifurcated upward and downward fringes, representing the topological charges [−1,1]. In order to demonstrate the interference patterns more clearly, the dotted lines are added as auxiliary lines in Figs. 3(b2)–3(c2).

The stability of the mode-locked laser is proved by the pulse train diagrams at different time scales. To realize a stable mode-locking operation, the cavity length is changed by adjusting the position of SESAM. A stable pulse train of TEM00 could be observed when the cavity length is 1.545 m. The pulse trains of TEM00 mode in different time scales (100 ns and 2 ms) are demonstrated in Fig. 4(a1) and the inset, respectively. By moving the defect point to the optical axis and adjusting the cavity length to 1.551 m, a stable pulse sequence of LG0,1 mode is achieved under a pump power of 8.9 W, as shown in Fig. 4(b1). At the same cavity length but higher pump power of 11.1 W, the LG0,1 mode is switched to TVA by off-axis pumping; and the pulse sequence is shown in Fig. 4(c1).

The autocorrelation traces of the mode-locked TEM00, LG0,1, and TVA are measured by the autocorrelator. Assuming a sech2 pulse shape, the pulse durations of TEM00, LG0,1, and TVA are 416, 476, and 520 fs respectively, as shown in Figs. 4(a2)–4(c2). The first reason for the pulse width broadening with the increase of mode order is mainly due to the decrease of phase velocities^[35–37]. According to Ref. [35], although in vacuum, the time delays for l≠0 relative to l=0 Gaussian pulses are on the order of a few femtoseconds. The second reason is intermode dispersion, which increases as higher-order modes oscillate in the cavity, which is similar to the dispersion in the fiber, leading to pulse broadening in mode locking^[38]. Although a femtosecond LG laser has been demonstrated in previous work, to the best of our knowledge, this is the first time to realize a femtosecond vortex array and LG mode can be switched to TVA by an off-axis pumping scheme. It is a flexible, miniaturized, and cost-saving route to attain a mode-switchable femtosecond vortex laser. The central wavelengths of the mode-locked TEM00, LG0,1, and TVA in Fig. 5(a) are 1049, 1047, and 1046 nm, respectively. This indicates the wavelength is blueshifted with the increase of the number of phase singularities. The full widths at half-maximum of TEM00, LG0,1, and TVA are 3.29, 3.38, and 3.67 nm, and their time-bandwidth products are calculated to be 0.375, 0.439, and 0.523, respectively. This indicates that the time-bandwidth product of the TEM00 mode is close to the Fourier transform limit. However, due to the decrease in phase velocities and the existence of mode dispersion, that of the LG mode and TVA is larger. The radio frequency (RF) of the TVA’s pulse sequences is analyzed and the signal-to-noise ratio is 60 dB as shown in Fig. 5(b). The fundamental pulse repetition rate is 96.73 MHz.

In theory, the vortex array is analyzed by transverse mode locking. In the resonator, multiple transverse modes can simultaneously oscillate if the modes satisfy the threshold condition. Under the case that all the modes are in-phase, stationary transverse patterns will be formed by the transverse mode-locked effect^[29,33,34]. Among them, the vortex array is a typical transverse pattern. The vortex array can be described as the superposition of different basic LG or HG modes. The formula that describes LG mode is $$\begin{gathered} LGpl=2p!π(p+|l|)!1ω(z)Lp|l|(2r2ω(z)2)exp[−r2ω(z)2−ikr22R(z)] \\ ⁢exp[−i(2p+|l|+1)tan−1(zz(R))]exp[−ilφ], \end{gathered}$$(1)where ω(z) is the beam width, R(z) is the radius of curvature, z(R) is the Rayleigh length, and Lp|l| is the Laguerre polynomial. The light field of the vortex array is^[39]$$\begin{gathered} E(r,φ)=g1×LGp1,l1exp(iθ1)+g2×LGp2,l2exp(iθ2) \\ +g3×LGp3,l3exp(iθ3), \end{gathered}$$(2)where LGp1,l1, LGp2,l2, and LGp3,l3 are the transverse modes that participate in constituting the vortex array, pj (j=1, 2, 3) is the radial index, and lj (j=1, 2, 3) is the topological charge. g1, g2, g3 are the weights of different LGp,l beams. θ1, θ2, θ3 represent the rotating angles of LGp1,l1, LGp2,l2, LGp3,l3, and they satisfy θ1+θ2+θ3=(2n+1)π, where n is an integer. From Eq. (2), the stationary transverse pattern is related to different LG modes, their weights, and rotating angles. For instance, in 2p+l=0 state, there is only a single TEM00 mode. LG0,±1 are two frequency-degenerated modes. TVA is composed of three modes. The rotating angles of LG0,1, LG0,−1, and LG1,0 are 7π/50, 43π/50, and 0, respectively. The simulation results are demonstrated in Fig. 3. Figures 3(a3)–3(c3) are the intensity distributions of TEM00, LG mode, and TVA, and Figs. 3(a4)–3(c4) are the corresponding interference patterns. The clockwise direction represents a positive topological charge. From Fig. 3(c4), the topological charges of TVA are [−1,1]. Figures 3(a5)–3(c5) show the phase distributions. That around the singularity of LG0,1 is 2π in Fig. 3(b5). In Fig. 3(c5), the phase distribution around the left and right singularities are (2π,2π).

It is noteworthy that the mode selection is realized by defect spot mirror and off-axis pumping. Without a defect spot on the input mirror, the femtosecond TEM00 mode is first obtained in the resonator. Due to the presence of the defect spot, the mode confine results in the LG mode preferential oscillation, whereas the TEM00 mode is restrained owing to mode competition. Combined with off-axis pumping by a rotating laser crystal, the three transverse modes LG1,0, LG0,1, and LG0,−1 exceed the thresholds and oscillate simultaneously in the resonator, so TVA is constructed by these three modes. Therefore, diverse vortex lasers with a femtosecond pulse are achieved in one resonator. Moreover, compared with previous research that reported femtosecond LG modes, we demonstrate a femtosecond TVA laser. TVA possesses an intriguing transverse pattern and multiple phase singularities. Because TVA is the combination of different LG modes, it adds more difficulty to producing femtosecond pulses than a single LG mode. The defect spot mirror combined with off-axis pumping ensures simultaneous oscillation of multiple LG modes, and these LG modes are phase-locked to TVA by transverse mode locking. With the help of nonlinearity induced by elements of the resonator, both longitudinal and transverse mode locking are achieved to produce a femtosecond vortex array. Therefore, we believe that the generation of the vortex array is due to the unique structure of our resonator.

In summary, we demonstrate a mode-switchable femtosecond vortex laser in a Yb:KGW-based resonator. Mode locking of TEM00, LG0,1 and TVA are achieved by aligning the defect spot on the input mirror and utilizing off-axis pumping. Without a defect point, TEM00 mode with a pulse duration of 416 fs and maximum output power of 383 mW is obtained. By utilizing a 50 µm defect spot, the resonator delivers a 340 mW LG0,1 mode. The pulse duration is as short as 476 fs, and the corresponding pulse energy is 3.52 nJ. Under off-axis pumping conditions, the LG0,1 mode can be switched to TVA. The shortest pulse width reaches 520 fs, and the pulse repetition rate is 96.73 MHz. The maximum output power is 401 mW, and the pulse energy is 4.15 nJ. The signal-to-noise ratio of TVA is 60 dB. The femtosecond vortex laser system is an attractive structured light source for ultrafast microprocessing, ultrafast filament optics, and extreme UV OAM beams.