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  • Open Access

Scalar product for the radiation of resonant modes

Maria Paszkiewicz-Idzik1,2,*, Lukas Rebholz2,*, Carsten Rockstuhl3,2, and Ivan Fernandez-Corbaton3,†

  • *These authors contributed equally to this work.
  • †Contact author: ivan.fernandez-corbaton@kit.edu
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Phys. Rev. A 112, 013518 – Published 16 July, 2025

DOI: https://doi.org/10.1103/1rhp-ts2q

Abstract

We introduce the conformally invariant scalar product, originally devised for radiation fields, to the study of the modes of optical resonators. This scalar product allows one to normalize and compare resonant modes using their corresponding radiation fields. Such fields are polychromatic fields free of divergences, which are determined from the complex frequencies and the modal fields on the surface of the resonator. The scalar product is expressed as surface integrals involving the modal fields, multiplied by closed-form factors incorporating the complex frequencies. In a practical application, we study the modes of disk-shaped whispering gallery resonators and show that the proposed scalar product accurately predicts the geometry-dependent anticrossings between modes.

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Phys. Rev. A 112, 013518– Published 16 July, 2025
  • Received 20 March 2025
  • Accepted 24 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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