The quantum harmonic oscillator is an indispensable paradigm to understand the concept of quantum-classical correspondence, quantized radiation fields, and quantum optics. The eigenmodes of the two-dimensional (2D) quantum harmonic oscillator can be analytically solved as Hermite–Gaussian (HG) modes with rectangular symmetry or Laguerre–Gaussian (LG) modes with circular symmetry^[1]. Since the paraxial wave equation for the spherical laser cavity is identical to the Schrödinger equation for the 2D harmonic oscillator, the HG and LG eigenmodes play an important role in exploring the laser transverse modes^[2–4]. With the advent of end-pumped configurations, the high-order HG modes^[5–8] and LG modes^[9–15] can be efficiently generated in diode-pumped solid-state lasers. The Ince–Gaussian (IG) modes, another form of eigenfunctions to the paraxial wave equation, have been recently introduced^[16] and been also experimentally observed in stable resonators^[17–19].