Unidirectional operation criterion in monolithic nonplanar ring oscillators.

Monolithic nonplanar ring oscillators (NPROs) under an applied magnetic field can operate unidirectional single-frequency lasing due to the loss differences among its four eigenpolarizations, where the minimum was empirically estimated to be 0.01%. However, this value has never been verified because the applied magnetic field is not uniformly distributed, making it hard to resolve both theoretically and experimentally. Here, we propose a method to resolve the applied magnetic field through an NPRO by combining finite-element analysis and experimental verification. By introducing the non-uniform magnetic field information to the eigenpolarization theory, the loss differences can be calculated by path integration along the optical path in the NPRO. The critical point, where the bidirectional lasing is emerging, is identified by the relative amplitude noise (RAN) of the laser and by the beating signal between the clockwise (CW) and counterclockwise (CCW) lasing. With this method, we determine that unidirectional operation is possible with loss differences as low as 0.0001% and 0.0003%, corresponding to two different NPRO designs with out-of-plane angles of 90° and 45°, respectively, which increases the precision of the loss differences for unidirectional single-frequency lasing by more than one order of magnitude. Our findings will greatly facilitate NPRO laser design with lowered magnetic field intensity requirements.