# Nonlinear Topological Photonics: Capturing Nonlinear Dynamics and Optical Thermodynamics

## TL;DR

This paper reviews advances in nonlinear topological photonics, focusing on new methods to study and characterize nonlinear optical systems and their dynamic behaviors.

## Contribution

The paper introduces data-driven approaches and new theoretical frameworks for analyzing nonlinear topological photonic systems.

## Key findings

- Nonlinear topological systems can exhibit diverse photon phases and functionalities.
- Optical thermodynamics emerges as a new paradigm using nonlinear multimode waveguides.
- The spectral localizer and pseudospectrum methods improve understanding of nonlinear topological effects.

## Abstract

Combining multiple
optical resonators or engineering dispersion
of complex media has provided an effective method for demonstrating
topological physics controlling photons in unprecedented ways such
as unidirectional light propagation and spatially localized modes
between an interface or on a corner. Further, adding nonlinear responses
to those topological photonic systems has enabled achieving diverse
phases of photons in both space and time, allowing for more functionalities
in photonic devices that provide a new playground for studying dynamic
features of nonlinear topological systems. However, most methods for
describing nonlinear topological photonic systems rely on linear topological
theories, making it challenging to accurately characterize the topology
of nonlinear systems. Thus, substantial efforts have focused on rigorously
describing nonlinear topological phases and developing effective tools
to analyze nonlinear topological effects. Meanwhile, coupled multimode
optical waveguides with nonlinear dynamic responses provide an excellent
platform for the statistical description of photons, opening a new
paradigm called “optical thermodynamics”. This review
will introduce the basic concepts of nonlinear topological photonics
and the recent development of theoretical approaches focusing on data-driven
approaches for creating phase diagrams as well as the spectral localizer
framework and the pseudospectrum method for understanding optical
nonlinearities in topological systems. In addition, the new concept
of optical thermodynamics will be introduced with some recent theoretical
works.