Abstract
<jats:p>Learning-based structured-light generation has primarily been developed for extracavity single-pass systems, where neural networks infer phase masks under a fixed one-way propagation operator. Extending this paradigm to digital lasers, however, is fundamentally more challenging. Unlike the single-pass case, intracavity mode design is an inverse eigenmode problem in which the desired field must be established as the dominant self-reproducing cavity mode. Here, we propose EigenNet, a physics-constrained neural network that solves for the cavity round-trip operator T, enforcing γE = TE to establish the target pattern as the dominant lasing eigenmode E. Given an arbitrary target intensity pattern, EigenNet predicts a pure-phase intracavity hologram in a single forward inference of about 3.5 ms, which is approximately 20× faster than iterative optimization at equivalent fidelity. We validate EigenNet numerically by round-trip optical-field tomography and experimentally in a nondegenerate cavity with a phase-only spatial light modulator, demonstrating the generation and streaming switching of diverse laser beam patterns. These results establish an iteration-free route to real-time programmable laser-mode engineering in learnable resonators.</jats:p>