Transverse oscillation in laser cavities

Recent experiments on lasers with large Fresnel numbers have demonstrated a close correspondence between the transverse degrees of freedom (transverse modes) and the two-dimensional harmonic oscillator. Mode patterns that correspond to classical trajectories have been observed. These experiments suggest that such systems may serve as convenient platforms for studying quantum-classical correspondence. But why the paraxial wave equation for the laser leads to two-dimensional harmonic oscillation for the transverse degrees of freedom is yet to be clarified. We use two methods to show that such a correspondence is not a mathematical coincidence: the focusing mirror that stabilizes the laser gives rise to the transverse oscillation, and the reduced equation that governs the transverse degrees of freedom is the Schrödinger equation for the harmonic oscillator. For the fundamental Gaussian mode, the theory remains valid in the weakly nonlinear limit.