# Two-dimensional phase-space picture of the photonic crystal Fano laser

## TL;DR

This paper simplifies the complex dynamics of a photonic crystal Fano laser to a 2D phase-space model, revealing new insights into its stability and operation.

## Contribution

It reduces a 5D dynamic model to a 2D model, providing a clearer understanding of the laser's stability and the origin of instabilities.

## Key findings

- Laser operation is confined to two degrees-of-freedom after initial transition.
- The 1D model reduces to a 2D model after the exceptional point.
- A new unstable periodic orbit explains the origin of instability.

## Abstract

The recently realized photonic crystal Fano laser constitutes the first demonstration of passive pulse generation in nanolasers [Nat. Photonics $\boldsymbol{11}$, 81-84 (2017)]. We show that the laser operation is confined to only two degrees-of-freedom after the initial transition stage. We show that the original 5D dynamic model can be reduced to a 1D model in a narrow region of the parameter space and it evolves into a 2D model after the exceptional point, where the eigenvalues transition from being purely to a complex conjugate pair. The 2D reduced model allows us to establish an effective band structure for the eigenvalue problem of the stability matrix to explain the laser dynamics. The reduced model is used to associate a previously unknown origin of instability with a new unstable periodic orbit separating the stable steady-state from the stable periodic orbit.