Engineering isospectrality in multidimensional photonic systems: Multidimensional quasi-isospectrality in photonics.

Selective manipulation of energy levels plays an essential role in realizing multichannel wave devices. One of the representative examples is to utilize the concept of quasi-isospectrality: a family of wave systems with an almost identical spectrum except for a part of energy levels. Most approaches toward quasi-isospectrality have employed analytical methods based on symmetry or tridiagonalization, such as supersymmetry, Householder, or Lanczos transformations. Although such analytical approaches provide deterministic and stable designs based on operator factorizations, the mathematical strictness in the factorizations, at the same time, hinders isospectral engineering in a given multidimension. Here we develop the semi-analytical method for engineering isospectrality in multidimensional photonic systems. The method provides the systematic perturbation for the target energy level shifts by decomposing the allowed form of system changes into the perturbation basis. We demonstrate the isospectrality of lower-, higher-, and random-order states while imposing the designed shifts on the other states. The stability analysis shows that the accuracy of the method is determined by the ranges of isospectral state numbers and perturbation strength. The systematic, free-form, and multidimensional natures of the proposed method show great potential for the platform-transparent design of multichannel devices.