An optical vortex beam is a structured beam with a spiral phase e−ilϕ in the azimuthal direction ϕ perpendicular to the propagation direction [1–3]. These beams carrying an orbital angular momentum (OAM) of lℏ per photon have opened a route to many promising applications, such as optical communication [4–8], laser ablation [9–12], and micromanipulation [13–15]. These applications can benefit from technologies of optical vortex generation, which can be divided into external modulation outside the cavity [16–20] and direct generation within the cavity [21–23]. Direct generation may face multi-mode superposition, which can be divided into coherent superposition and incoherent superposition [24,25]. With the increase of the peak power of the optical vortex, novel phenomena are emerging. The ultra-intense, ultra-short optical vortex pulse can provide an opportunity to investigate the new phenomena with OAM involved in extreme cases [26–29]. The spiral phase can be introduced to the ultra-short Gaussian mode laser by external modulation through a spiral phase plate [19,28], fork grating [30–34], and spatial light modulator [11,35]. Unfortunately, such methods for ultra-intense cases pose a significant challenge, owing to technical difficulties like optical element damage and the dispersion hindering the synthesis of a high-power broadband optical vortex. As the other way for generation of ultra-short vortices, the direct generation from the mode-locked laser was proved [36,37]. However, this way also cannot generate ultra-short vortices with ultra-high intensity.

Looking back to the development history of ultra-short lasers, we can see that amplification technology plays a very important role in the generation of ultra-short pulses with ultra-high intensity [38,39]. Regenerative amplification can provide a large gain [40], and chirped pulse amplification (CPA) can help to reduce the damage and nonlinear effects [41]. The Ti:sapphire (Ti:S)-based amplifier plays an important role in delivering high peak power [42]. For the ultra-intense, ultra-short optical vortex generation, the blessing of amplification technologies may be indispensable. Compared with conventional ultra-short pulse amplifiers, the input and output of an optical vortex amplifier (OVA) should be both optical vortices with the same topologic charge, and some efforts have been made. A single pass of two Nd-doped yttrium aluminum garnet (Nd:YAG) rods can amplify the optic vortex to several times [43]. A multi-pass amplifier for femtosecond optical vortex pulses in a CPA system with a gain level of 10 was demonstrated [30]. Optical parametric amplification for ultra-short optical vortex pulses generation makes a gain level up to 102 [44–46]. As far as we know, currently, there are no OVAs that can reach a gain level of 106, because these amplifiers cannot make the vortex beam pass through the gain medium for enough times.

In this paper, we develop a high-gain OVA whose gain reaches the 106 level based on regenerative amplification. First, we show that the conventional regenerative amplifier (RA) could not amplify the optical vortex, and the vortex seed will degenerate to a Gaussian beam because of mode competition. Then, to overcome this problem, we pump the RA with the ring-shaped laser for controlling mode competition. Our simulations show that choosing the right pump radius can suppress Gaussian mode generation and increase the gain of the target Laguerre–Gaussian (LG) mode. Therefore, one LG mode can be amplified from a weak vortex seed while maintaining the topological charge (TC) and the spiral phase. Finally, an OVA for the LG0,1(−1) mode has been built experimentally with an amplification gain of up to 1.45×106 for a 2 nJ vortex seed. Based on the OVA, the 51 fs, 1.8 mJ compressed optical vortex has been obtained.

Regenerative amplification has the highest gain among all amplification technologies because the pulse can pass through the gain medium for enough times. It is well known that an RA is an active laser cavity that works as a Q-switched laser operating at the cavity dumping mode with a weak seed instead of oscillation from noise. LGp,l modes are the self-consistent modes of a cylindrically symmetrical cavity, where p is the index of the radial order, and l is the TC. The fundamental mode is the LG0,0, which is also named the Gaussian mode. The seed of the RA should be matched to the most competitive self-consistent mode (usually a Gaussian mode) of the cavity for effective amplification. Traditionally, a Gaussian mode seed is injected into the cavity, and we can obtain an amplified Gaussian output. In spite of the occurrence of mode mismatching, the seed can be still amplified, but it is pulled towards the Gaussian mode. As shown in Fig. 1, a 2 nJ vortex pulse with l=1 evolves into a Gaussian mode during amplification. Unfortunately, the Gaussian mode does not carry any OAM, which means that we cannot amplify vortex pulses directly with a conventional RA. Obviously, the reason for this is the mode competition.

In order to overcome the mode competition and realize the amplification of vortex pulses, we need the net gain of the target LG mode carrying target OAM greater than that of other modes, particularly the Gaussian mode. This is also the key to the vortex emission within the cavity, and some methods have been reported, such as a spot-defect mirror [47] and a ring-shaped pump [36,48,49]. Here, we choose the ring-shaped pump for the suppression of the Gaussian mode and excitation of the vortex modes. Our setup is reformed from a commercial 1 kHz 800 nm Ti:S RA (Coherence Inc., Legend-Elite-He). This RA without a beam expander (BE) and convex axicon (CA) is optimized in the Gaussian mode, and all components do not have further adjustment after the optimization except for the pump-related components. As shown in Fig. 2, the ring-shaped pump is realized by using a CA (Thorlabs, AX2505-A) to modulate a 50 ns, 15 mJ, 532 nm beam [50]. Besides, a variable BE is required to adjust the pump radius (the pump radius is inversely proportional to the beam expansion ratio), and the expansion ratio is set at 3.5 for the motivating of the LG0,1(−1) mode.

For an active cavity without seed, its initial signal starts from the noise, and the final output mode is determined by the gains of modes. The differences in these gains can be reflected in the mode probabilities of the output. For our RA, the concerned LG modes in the output mainly include LG0,0, LG0,1, and LG0,−1. The purity ρp,l of LGp,l in the optical field E for a coherent superposition can be expressed as where the inner product ⟨B|A⟩ of two fields A and B is defined as ∬B*(x,y)A(x,y)dxdy. We simulate the laser oscillation from noise with the different radii of the ring-shaped pump or different expansion ratios of BE using the method proposed in Ref. [51]. First, we set the expansion ratio of BE as four. Our simulations start with different noises each time, and we can obtain the different outputs oscillating from the different noises. For an output, we can get the corresponding purities of LG0,1, LG0,−1, and LG0,0 as (ρ0,1,ρ0,−1,ρ0,0). We simulate 1000 oscillations from different noises, and the purities of the 1000 outputs are obtained. As shown in Fig. 3(a), the (ρ0,1,ρ0,−1,ρ0,0) are plotted in a three-dimensional coordinate, while the black star is for the average mode purities or mode probabilities (ρ¯0,1,ρ¯0,−1,ρ¯0,0). As can be seen from Fig. 3(a), LG0,1 and LG0,−1 start to appear, but LG0,0 is not well suppressed. When the expansion ratio is set as three, LG0,0 is well suppressed (ρ0,0→0), as shown in Fig. 3(c). However, LG0,1 and LG0,−1 are not very well motivated because the pump radius is too large. However, if the expansion ratio is 3.5, LG0,1 and LG0,−1 are well motivated, while LG0,0 is well suppressed, as shown in Fig. 3(b). The purities lay near the blue line, which represents ρ0,1+ρ0,−1=1, and the mode probabilities are near the center of the blue line. Because the ring-shaped pump is not selective between LG0,1 and LG0,−1, the gains for these modes are the same, and we cannot select a unique vortex mode from the series of self-consistent LG modes by the ring-shaped pumped cavity. Consequently, the system generates the two modes with the same possibility in the absence of external seeds.

For the RA, the final output depends on not only the gain of each mode, but also the mode and energy of the input seed. We simulate the RA seeding with a vortex pulse with l=1. For different seed energies, our simulation also runs 1000 times, and the mode probabilities are obtained. As shown in Fig. 4, if the seed is weak enough, LG0,1 and LG0,−1 are the two most possible modes to survive with the same possibility of 45%. At the same time, the LG0,0 mode is well suppressed. The LG0,1 mode purity increases with the increase of seed energy, while LG0,−1 mode purity decreases. If the seed energy rises to 0.01 nJ, the purity goes up to 96% for LG0,1 but drops to 0.8% for LG0,−1, while LG0,0 always keeps its purity less than 0.12%. Our simulations also find that, if the TC of the seed changes to −1 instead of 1, the high probability mode will change to LG0,−1 instead of LG0,1.

In our experiments, we first align BE for matching the radius of the ring-shaped pump for the motivation of LG0,1(−1) and the suppression of LG0,0. The expansion ratio of BE is chosen as 3.5 according to our simulation. Figure 5(a) shows the ring-shaped pump at one of the Ti:S surfaces. Correspondingly, the output beam profile from the RA without seeding is shown in Fig. 5(b). The donut-shaped distribution of the output implies that LG0,0 has been successfully suppressed. The Dammann vortex grating can be used to check the TC of the output by measuring whether the pattern on the corresponding diffraction order is solid [52]. Figure 5(c) is the phase structure of our Dammann vortex grating, which is loaded into a spatial light modulator. Figure 5(d) shows the simulated far-field diffraction patterns with parallel illumination. By determining whether the spots on the green box are solid, we can find out whether the beam contains an OAM component with l=0, while the blue and red boxes are for l=+1 and −1. Figure 5(e) presents the measured far-field spots after the Dammann vortex grating from the output of the unseeded RA. A very bright solid spot in the center is an exception, which is from the strong zero-order diffraction. Other solid spots only locate within the blue and red boxes, which means the output beam includes both LG0,1 and LG0,−1 modes.

Our vortex seed with l=1 is converted from a Gaussian mode pulse from a mode-locked oscillator with a Q-plate (QP) and two achromatic quarter-wave plates (QW-1 and QW-2), as shown in Fig. 2. The final seed before the incident to the system has the pulse energy of 2 nJ, and we check the TC of the seed by a cylindrical lens [53]. Figure 6(a) is the recorded spatial profile of the seed focused by a cylindrical lens, which features one dark stripe across the beam spot, showing that the seed has a TC of one. The stripe orientation depends on whether the TC is positive or negative. Correspondingly, the amplified output is recorded as Fig. 6(b), which also presents a donut-shaped profile. The corresponding far-field distribution after the Dammann vortex grating is shown in Fig. 6(c). Compared with Fig. 5(e), its solid spots only exist inside the blue box. This means that the output contains LG0,1 mode only and has the same TC as the seed. Then, we rotate simultaneously QW-1 and QW-2 with an angle of 90° to change the TC value of the seed from 1 to −1. Compared with Fig. 6(a), the cylindrical lens focused pattern of the seed changed the orientation of the tilted dark stripe, as shown in Fig. 7(a). This implies that the seed changes to an opposite TC. As shown in Fig. 7(b), the amplified output seeded with l=−1 has a similar profile as that in Figs. 5(b) and 6(b). Compared with Fig. 6(c), Fig. 7(c) confirms further that the output has changed its TC value from +1 to −1, showing that our RA can keep the output maintaining the same TC from the seed. From Fig. 5(e), the rings around the spots are oval and disconnected on the minor axis sides. We infer that they result from the astigmatisms due to the slight misalignment of the Fourier lens after the Dammann vortex grating. The slightly oval rings on both sides of the red and blue boxes in Figs. 6(c) and 7(c) are also caused by the astigmatism.

From Fig. 6(b), we can get different cross-section intensity distributions across the center along with different directions; then, we obtain the average cross section from the orientations, as shown as the black line in Fig. 8(a). The dashed red line is the theoretical fit by LG0,1, which agrees with the average data very well. For the amplified LG01 pulse, the spiral phase distribution is verified by the Dammann vortex grating, and we can determine the phase term as e−ilϕ with l=1. The amplitude term can be obtained from the captured intensity. With these two terms, the mode purity of LG01 from the output is 95% according to Eq. (1). This means that our RA outputs high-quality LG0,1 pulses. Compared with the same RA cavity with the Gaussian pump, our RA with a ring-shaped pump has smaller gain under the same pump power because of the deduction of pump density. Smaller net gain will result in larger amplification pass number N for saturated output and smaller output power. According to our experiment, the N for saturated amplification is 53 for the ring-shaped pump, and it is 27 for the Gaussian pump under the same pump energy of 15 mJ. Before entering the compressor, the output pulse energies are 2.9 mJ for the ring-shaped pump and 4.5 mJ for the Gaussian pump. From this, the gain of our OVA is 1.45×106, and the corresponding output energies are 1.8 mJ and 3.2 mJ after compression. The pulse durations of the outputs have been measured with spectral phase interferometry for direct electric-field reconstruction (SPIDER). The spectral intensity and phase of the amplified LG0,1 pulse are shown in Fig. 8(b). Figure 8(c) shows that the LG0,1 pulse duration is about 51 fs with the black line, which is wider than 45.3 fs of the Gaussian pulse. This is because the larger N implies to induce larger high-order material dispersion, which cannot be compensated by a compressor. However, the red line in Fig. 8(c) reveals that the spectrum of the amplified LG0,1 pulse can support a Fourier-transform-limited pulse with a pulse duration as short as 35 fs.

Summarily, in order to overcome the mode competition problem for the amplification of an optical vortex in RA, we develop a high-gain OVA with the mode-control cavity. The RA is pumped with the ring-shaped laser for controlling mode competition. Our simulation shows that choosing the right radius of the pump can suppress the generation of Gaussian mode and increase the gain of LG0,1(−1) mode. In our experiment, an RA for LG0,1(−1) mode is built. It verifies that, without a seeding, the amplifier has a donut-shaped output containing two opposite OAM states simultaneously. By seeding pulses with the topologic charge of one, the system will output amplified LG0,1 mode pulses with the same topologic charge as the seed. The sign of the output TC can be changed accordingly by changing that of the vortex seed. This shows that our OVA, whose gain is 1.45×106 for 2 nJ seed, can maintain the TC and the spiral phase of the vortex seed during amplification. Finally, a 1.8 mJ, 51 fs optical vortex after compression is obtained.