As one of the most representative families of structured light, high-order Hermite?Gaussian (HG) mode lasers have garnered increasing interest because of their importance in cutting-edge applications, including laser communication, gravity wave detection, and quantum optics. The most commonly used technique for generating HG-mode lasers is off-axis pumping, which is based on different sizes (intensity distributions) of different orders of HG modes. Because the HG beams have their maximum intensities at the outermost peaks with the pump beam deviating from the axis of the laser cavity, the high-order modes with maxima close to the pump spot have higher overlap with the pump beam than the other modes, thereby dominating the lasing. Using this simple and efficient approach, researchers have demonstrated certain very high-order HG modes with at least one mode index of greater than 100. However, generating two-dimensional (2D) HG-mode beams with two non-zero mode indices remains challenging. When the pump beam deviates from the cavity axis in two orthogonal directions, the lasers operate in a tilted one-dimensional (1D) mode rather than in two 2D modes. Therefore, the generation of 2D HG mode to date has usually relied on complex precision pump shaping, with the HG_25,27 based on a modulated pump beam demonstrating the highest order with an optical efficiency of only 0.2%. In this study, we propose a simple method to generate well-controllable very-high-order 2D HG modes via an off-axis pump by confining the eigenmodes of the cavity employing astigmatism.

As mentioned above, 2D pump displacement does not excite the 2D HG modes but tilts the 1D modes. This is because of the cylindrical symmetry of the laser cavity. A cylindrically symmetric cavity can support both 2D and tilted 1D modes. In other words, the 2D pump displacement in a cylindrically symmetric system is equivalent to the 1D pump displacement after coordinate transformation, as shown in Fig.1. Therefore, generating 2D HG modes via 2D off-axis pumping requires breaking the symmetry by employing astigmatism. The laser cavity is depicted in Fig. 2. The cavity of 1064 nm Nd∶YVO₄ laser is a folded cavity consisting of a flat total reflector M1, a concave folding mirror M2 with a 200 mm radius of curvature, and a flat output coupler with a transmittance of T=3% at a laser wavelength of 1064 nm. The full fold angle was 2θ=20°. The pump beam delivered from a fiber-coupled diode laser at 878.6 nm was focused onto a 0.5% (atomic fraction) doped, 5 mm×5 mm×8 mm Nd∶YVO₄ crystal with a spot radius of 100 μm. Because concave folding mirror M2 has different effective focal powers in the tangential and sagittal planes, that is, astigmatism, the cylindrical symmetry is broken. In this system, the eigensolution of the Helmholtz equation is confined to the HG modes along the directions of the two astigmatic symmetric axes. The tilted 1D HG modes are no longer the eigenmodes of the cavity, in that, they do not self-reproduce after a round trip in the cavity and experience losses. In this case, a simple 2D pump displacement excites the 2D HG modes.

When the position of the pump spot deviates from the cavity axis, the oscillating mode of the laser changes from the fundamental mode to higher order HG modes, and the 2D HG modes are obtained by displacing the pump in both x and y directions, as expected. The relationship between the mode order and pump position aligns well with the theoretical calculations based on the overlap integral. The highest order HG_214,216 is obtained with the pump spot displaced from the cavity axis by 1.9 mm in both x and y directions, as shown in Fig. 3, which is an order of magnitude higher than the previously demonstrated 2D HG modes. The laser threshold is 36 mW for the fundamental mode and gradually increases with the mode order as determined by the pump displacement. The threshold and slope efficiency of the HG_214,216 mode laser are 1.74 W and 13.4%, respectively.

In this study, an ultra-high-order 2D HG-mode laser output is realized for the first time. By introducing astigmatism to confine the eigenmodes of the laser cavity, 2D HG modes are obtained by simple off-axis pumping, and the HG_214,216 achieves its highest-order mode with a watt-level pump power. The well-controllable HG-mode laser also provides a basis for generating Laguerre?Gaussian (LG_p,l) mode vortices with controllable angular and radial indices through astigmatic mode conversion, significantly expanding the degrees of freedom in structured light field manipulation.