Optical Thermodynamics Beyond the Weak Nonlinearity Limit

Abstract

Optical thermodynamics has recently emerged as a theoretical framework describing a Rayleigh-Jeans (RJ) modal power distribution of multimoded nonlinear photonic circuits. However, its applicability is constrained to systems exhibiting weak nonlinear mode-mode interactions. Here, by employing a Transfer Integral Operator, we circumvent this limitation and establish a steady-state interacting RJ modal distribution -- referred to as non-ideal RJ (NIRJ) -- with renormalized temperature and optical chemical potential. This also builds a natural bridge with earlier work on grand-canonical statistical-mechanical formulations of discrete nonlinear systems. The theory derives the optical analogue of the compressibility factor, which controls the transition from an ideal, non-interacting equation of state (EoS) to a van der Waals-like interacting EoS.

Figures (4)

• Figure 1: An initial preparation (magenta) evolves toward a final state according to the strength of its nonlinear mode-mode interactions. When these interactions are weak, the system converges to the IRJ distribution predicted by ideal OT (red line), but when the interactions are nonnegligible, it converges to an NIRJ distribution (blue) that deviates from the IRJ. These interactions can be conceptualized through a wave picture (bottom panel), where the modes are exchanging power, or through a particle picture (upper panel), where the different modes undergo billiard-like collisions.
• Figure 2: Scaling theory for the difference between the non-ideal equation of state and the OT equation of state. Different colors here represent different TIO chemical potentials, ranging from $\mu=-4.25$ to $\mu=-400.002$, while different shapes represent different nonlinear strengths (squares represent $\chi=0.1$, circles represent $\chi=1$, and diamonds represent $\chi=10$). In all cases, the equation of state is shown to fall along a single universal curve interpolating between 1/2 (dashed line), representing the strongly nonlinear/equipartition limit, and 1 (dotted line), representing the linear limit in which ideal OT is applicable. We also show a theoretical evaluation of $f(z)$ (solid line). While this equation was evaluated for the case of $z\rightarrow 0$, it is shown to be applicable for the full range of $z$.
• Figure 3: (a) The equilibrium state of a strongly nonlinear system, with total power $a=5.614$. Initial conditions shown as black circles. The IRJ calculated according to the initial linear energy (green line, $\beta=0.18$, $\mu_{\rm eff}=-2.227$) differs dramatically from the distribution at $t=10^5$ (red circles), which is instead predicted by the NIRJ (blue line, $\beta=0.013$, $\mu_{\rm eff}=-13.5$) calculated using the asymptotic linear energy. Inset shows a weakly nonlinear system with $a=0.01$. The IRJ (green line,$\beta = 25.5$ and $\mu_{\rm L}=-4.4$) calculated using $h_L(t=0)=-0.00489$ matches the distribution at $t=10^7$ (red circles). (b) The linear energy of the strongly nonlinear system shown in (a) evolves from its initial value $h_L(t=0)=-7$ (green line) to the TIO prediction for the asymptotic linear energy $h_L^\infty=-0.8361$ (blue line). Inset shows the linear energy of the weakly nonlinear system shown in (a), demonstrating that $h_L=-0.00489$ throughout the evolution.
• Figure 4: We report the numerical evaluation of $\mu-\mu_{eff}$ as the temperature is varied for chemical potentials $\mu=-4.25$ (black circles), $\mu=-20.05$ (purple circles), and $\mu=-40.025$ (red crosses). In all cases, the shift in the chemical potential falls along the line $\mu-\mu_{eff}=2 h_{NL}/a$.