Inverse solving the Schrödinger equation for precision alignment of a microcavity

In paraxial approximation, the electromagnetic eigenmodes inside an optical microresonator can be derived from a Schrödinger-type eigenvalue problem. In this framework, tilting the cavity mirrors introduces a linear term to the potential energy of the system. In our paper, we apply solution strategies for inverse problems to precisely determine and control the relative orientation of two mirrors forming an optical microcavity. Our approach employs the inversion of the Schrödinger equation to reconstruct the effective potential landscape, and thus mirror tilts, from observed mode patterns. We investigate regularization techniques to address the ill-posed nature of inverse problems and to improve the stability of solutions. Our method consistently achieves an angle resolution of order 100 nanoradians per measurement. We consider our method applicable to a wide variety of optical resonators and driving schemes.