Computational Physics Applied to Photonic Devices

TL;DR

The paper surveys how computational physics enables the study of nonlinear photonic devices, from lasers to Kerr resonators and optical parametric oscillators, by solving model ODEs and PDEs that exhibit bifurcations, chaos, and pattern formation. It highlights a spectrum of phenomena—relaxation oscillations, bistability, cavity solitons, Turing patterns, and frequency combs—demonstrated through small-scale numerical simulations on conventional hardware. The work connects fundamental dynamical systems theory to practical photonic technologies, detailing numerical methods (e.g., Runge–Kutta and split-step Fourier) and illustrating how simulations guide experiments in ultrafast communications, metrology, and quantum photonics. Overall, it argues that accessible computational photonics provides predictive insights and a versatile toolkit for designing and understanding nonlinear optical devices.

Abstract

We all know that the first laser device was realised by Theodore Maiman at Hughes Labs in 1960. Less known is that the very first computer simulations of the relaxation oscillations displayed by Maiman's laser were also performed in 1960 on a digital IBM 704 computer. The reason is that lasers and almost all photonic devices are described by nonlinear equations that are more often than not impossible to be solved analytically, i.e. on a piece of paper. Since then the development and applications of lasers and photonic devices has progressed hand in hand with computer simulations and numerical programming. In this review we introduce and numerically solve the model equations for a variety of devices, lasers, lasers with modulated parameters, lasers with injection, Kerr resonators, saturable absorbers and optical parametric oscillators. By using computer simulations we demonstrate stability and instability of nonlinear solutions in these photonic devices via pitchfork, saddle-node, Hopf and Turing bifurcations; bistability, nonlinear oscillations, deterministic chaos, Turing patterns, conservative solitons; bright, dark and grey cavity solitons; frequency combs, spatial disorder, spatio-temporal chaos, defect mediated turbulence and even rogue waves. There has been a one-to-one correspondence between computer simulations of all these nonlinear features and laboratory experiments with applications in ultrafast optical communications, optical memories, neural networks, frequency standards, optical clocks, future GPS, astronomy and quantum technologies. All of this has been made possible by 'novel insights into spatio-temporal dynamics of lasers, nonlinear and quantum optical systems, achieved through the development and application of powerful techniques for small-scale computing' (2011 Occhialini Medal and Prize of the Institute of Physics and Societa' Italiana di Fisica).

Figures (41)

• Figure 1: (Color online) (a) Bifurcation diagram of a transcritical bifurcation. (b) Bifurcation diagram of a supercritical pitchfork bifurcation. Solid (dashed) lines correspond to stable (unstable) stationary states.
• Figure 2: (Color online) (a) Bifurcation diagram of a saddle-node bifurcation. (b) Bifurcation diagram of an Andronov-Hopf bifurcation. The plane perpendicular to the figure is the $(Re(z), Im(z))$ plane. For $\mu<0$ the trajectory spirals towards the stationary state $z=0$, for $\mu>0$ the trajectory spirals out of the stationary state $z=0$ and relaxes onto the limit cycle (blue oval curve) corresponding to a periodic oscillation. Solid (dashed) lines correspond to stable (unstable) stationary states.
• Figure 3: (Color online) (a) Limit cycle oscillation of Eqs. (\ref{['FPDT']}) for $\gamma=0.752$, and $\alpha=\beta=0.1$. (b) Same as (a) but for $\gamma=0.768$. Period doubled orbit.
• Figure 4: (Color online) (a) Dispersion relation of the SHE, Eq. (\ref{['lambdaSHE']}), for different values of the control parameter $\epsilon$. (b) Stable Turing pattern of the SHE (\ref{['SHE']}) for $\epsilon=0.01$. The horizontal line is the unstable HSS $u_0=0$.
• Figure 5: (Color online) (a) Time evolution of the $X$ variable of the Lorenz equations Eqs. (\ref{['Lorenz']}) for $\sigma=10$, $\rho=28$, and $\beta=8/3$ for two trajectories initially displaced from each other by less than $5x 10^{-4}$. (b) Chaotic oscillations of Eqs. (\ref{['Lorenz']}) in a $(X,Z)$ projection plane.