Nonlinear optical eigenmodes : perturbative approach for classical fields and single photons

In linear optics, the concept of a mode or eigenmode is well established. Often these modes correspond to a set of fields that are mutually orthogonal with intensity profiles that are invariant as they propagate through a given optical system. More generally, using an eigenmode decomposition, one can define a set of orthogonal modes with respect to an optical measure given that is linear in the intensity of the fields or Hermitian in the fields themselves. However, if the intensity of the light is sufficiently large, the dipole response of an optical medium includes nonlinear terms that cause the eigenmode decomposition to break down. In this work, we introduce the eigenmode decomposition in the presence of these nonlinear source terms by introducing small perturbation fields whose interaction is mediated by some high-intensity background field. Unlike the eigenmodes of linear optics, these novel modes correspond to a set of orthogonal fields that are, in general, distributed across multiple wavelengths. Here, we study the definition and interaction of these eigenmodes for classical electromagnetic fields and multiphoton fields. In the context of classical fields, with our eigenmodes established, we highlight the influence of the high-intensity background field on the symmetry of the eigenmodes. At the multiphoton level, we show that the description of multiphoton fields is simplified by using the propagation eigenmodes while remaining equivalent to the standard approach."Finally, I would like to acknowledge the funding provided by the Engineering and Physical Sciences Research Council (EPSRC), U.K. [grant ID: 1793529]" -- Acknowledgement