Optical vortices in the Laguerre-Gaussian (LG) mode that have a unique hollow intensity profile and non-zero orbital angular momentum are highly significant for various applications. The LG mode laser can be generated using external-cavity devices, such as holograms or cylindrical lens pairs, to transform a Hermite-Gaussian beam into a LG beam, or using the intracavity, where the intracavity components are utilized to preferentially oscillate the certain high-order modes within a laser resonator. In comparison to the external-cavity approaches, intracavity approaches typically yield superior power handling, beam quality, and conversion efficiency. However, there are very few demonstrations regarding high-order LG mode laser oscillations with angular indices (m) beyond 30. The main challenge is that the beam patterns of the very-high-order mode lasers become highly complex, which makes it difficult to fabricate mode-selecting elements with a sufficient accuracy to precisely manage the loss and gain of a certain mode. In this study, we demonstrate the generation of an ultra-high-order LG mode output based on mode selection enabled by intracavity spherical aberration (SA). By calculating the focusing behavior of the high-order LG mode beam and the SA of the intracavity lens, the relationship between the angular indices m of the high-order LG_0,±m vortex laser and the cavity parameters is determined. In the experiment, the ultra-high-order LG_0,±m vortex lasers with tunable angular indices m of up to 280 are obtained with an end-pumped Nd∶YVO₄ laser at a wavelength of 1064 nm, under an incident diode pump power of only 2.06 W.

The experimental arrangement of the laser that generates the ultra-high-order LG mode output is depicted in Fig. 1. Two lenses, L1 and L2, with focal lengths of f₁=150 mm and f₂=51.8 mm, respectively, are inserted into the cavity of an end-pumped Nd∶YVO₄ laser to collimate the beam and introduce SA for mode selection. The laser is pumped by a fiber-coupled diode laser at 878.6 nm, with a pump beam radius of approximately 120 μm at the input facet of the a-cut Nd∶YVO₄ crystal and a Rayleigh length of approximately 0.9 mm. The crystal is located near the total reflector M1, while the distances between the crystal and lens L1 (d₁) and between lenses L1 and L2 (d₂) are 155 mm and 20 mm, respectively. The plano-concave input mirror M1 with a small radius of curvature of 50 mm generates a small beam waist near it, enabling the beam to expand significantly when it reached the lenses, thus enhancing the SA and resultant mode selection capability. The output coupler M2 is a flat mirror with a transmittance of 10% at 1064 nm. The beam waist position of the LG beam behind the focusing lens L2 can be obtained using Eq. (1). Considering that the beam arriving at lens L2 is well-collimated by lens L1, the relationship can be simplified as indicated in Eq. (3). Because the output coupler M2 is a flat mirror, the oscillating beam should have its waist exactly on the mirror surface. The defocused modes (with the beam waist deviated from M2) suffered a loss larger than that of the "on-focus" mode with the beam waist on M2. The spherical lens with SA is used as L2, and the focal length is not a constant but varies with the incident beam height. Therefore, mode selection can be achieved by adjusting the location of M2 within a small range to have different orders of modes focused on it. Moving the output coupler M2 toward lens L2 will result in modes with a larger m and vice versa.

Figure 4 presents certain typical beam patterns recorded during the experiment. With an incident pump power of 2.06 W, the lowest order propagation-invariant single-mode LG_0,±m mode laser is LG_0,±38, which is obtained at distance between M2 and L2 of d₃=51.48 mm, and the highest order is LG_0,±280, which is obtained at d₃=48.91 mm. The beams are petal-like because both the +m and -m components have a similar intensity, and the mode order can be determined by counting the surrounding dark bars. The high-order LG mode optical vortices demonstrate good mode purity and stability. Figure 5 presents the theoretical relationship of the mode order and the d₃ obtained using Eq. (3), as well as the experimental results, which sufficiently match the theoretical results. The slope efficiency of the laser decreases with the mode order owing to the increasing SA-induced cavity loss and decreasing mode matching.

In summary, ultra-high-order LG mode vortex beams with selective angular indicesare obtained by utilizing the SA of an off-the-shelf spherical lens in the laser cavity. By calculating the focusing behavior of the high-order LG mode beam and the SA of the intracavity lens, the relationship between the angular indices m of the high-order LG_0,±m vortex laser and the cavity parameters is determined. In the experiment, an ultra-high-order LG_0,±m vortex laser with tunable angular indices m of up to 280 is obtained with an end-pumped Nd∶YVO₄ laser at a wavelength of 1064 nm, under an incident diode pump power of only 2.06 W. The ultra-high-order LG_0,±m vortex laser exhibits good stability in terms of power and the transverse mode. The mode evolution in the experiment sufficiently matches with that in the theoretical model. This study provides theoretical and experimental references for the generation of ultra-high-order LG mode vortex lasers. By increasing the pump power or pump overlap to enhance the laser gain, arbitrary high-order modes can be expected using this method.