Nonlinear optical thermodynamics from a van der Waals-type equation of state

Abstract

Optical thermodynamic theory provides distinct viewpoint to rich set of optical phenomena in multimode optical systems. However, standard theory ignores nonlinear effect, severely limiting its range of application. Within a mean-field approximation, we develop a nonlinear optical thermodynamic theory, taking into account the renormalization of linear spectrum due to inter-mode interaction, reminiscent of the van der Waals theory of gases. The resultant nonlinear equation of state enables prediction of power localization, as well as cooling/heating in optical Joule-Thomson expansion, thus providing a unified thermodynamic perspective on nonlinear control and transport of optical waves.

Figures (8)

• Figure 1: Thermalization of two subsystems ($R_A$ and $R_B$) with different nonlinear coefficients after turning on their coupling. Parameters: Number of modes $M_A=M_B=100$, nonlinear coefficients $\chi_A=-0.03$, $\chi_B=-0.06$, nearest neighbor coupling $\kappa_A=\kappa_B=1$, subsystem coupling $\kappa_{AB}=0.1$. (a) The two subsystems are initially in their respective equilibrium states with parameters shown in the inset. After turning on the coupling, the composite system evolves to equilibrium, and $R_A$ and $R_B$ reach the same temperature. (b) The chemical potentials ($\mu$) obtained from the linear theory (LOT, pink and cyan lines) are different from each other even when the system reaches equilibrium. In contrast, the chemical potential $\tilde{\mu}$ defined from the nonlinear theory (NOT, red and blue lines) reaches the same value.
• Figure 2: Instability of a 1D homogeneous waveguide array at positive temperatures. (a) Schematic of the waveguide array with $M=50$, $\kappa=1$. The initial input satisfies a R-J distribution (red line). (b) The isothermal power elastic modulus $\kappa^P _T$ as a function of input power density $P/M$ for $\chi=0.1$ at different temperatures. The curve color indicates pressure $\tilde{p}$, normalized independently for each $T$. The black dashed lines are the LOT result. The inset shows the complex-$\psi$ space calculated from the DNSE at $P/M=1.3$, with the color indicating the proportion of optical power $|\psi _m|^2/P$. (c) Similar results as (b), but for $\chi=-0.1$. $\kappa_T^{P}$ becomes negative at high input optical power. (d) Evolution of the optical power $|\psi _m|^2$ along propagating distance $z$ for each waveguide at $P/M=$0.5 (I), 0.8 (II), and 1.3 (III), as marked in (c). One ensemble is selected randomly from numerical simulation of the DNSE for each case. (e) Energy stored in the mean-field and the interaction Hamiltonian, obtained from numerical simulation, during evolution along $z$ at $P/M=$1.3. The inset shows the corresponding equilibrium distribution. (f) Optical entropy as a function of temperature at $P/M=1.3$, with a red arrow indicating evolution from the initial to the final state. The inset shows the complex-$\psi$ space obtained from 500 ensembles.
• Figure 3: Cooling and heating in optical J-T expansion described by the nonlinear optical thermodynamic theory. (a) Schematic of a structure realizing J-T expansion: a single waveguide couples into the center of a 19 $\times$ 19 square waveguide array with $\kappa=1$. (b) The modal occupancies at input $P=8$ for $\chi=\pm 0.5$, obtained by $z$-averaging. (c) During the evolution in case (b), the internal energy $\widetilde{U}$ is conserved. (d) Variation of the optical J–T response coefficient $\eta$ with increasing $M$ for a 1D homogenous waveguide array with $\kappa=1$, starting from $M=30$ with initial condition $T=5$, $P=26.46$. (e) Corresponding temperature evolution for the case in (d). (f) Schematic of the extended 1D waveguide array and its modal occupations for the case $\chi=0.5$ in (e), where the array is expanded from the initial $M=30$ (I) to $M=100$ (II), and then to $M=500$ (III), eventually achieving a negative temperature $T=-1.27$. Lines: theory; Dots: numerics.
• Figure S1: Instability of a uniform 1D waveguide array shown in Fig. \ref{['fig:soliton']}(a) at negative temperature $T=-0.5$. All other parameters are the same as Fig. \ref{['fig:soliton']}. (a) The isothermal power elastic modulus $\kappa^P_T$. (b) Evolution of the optical power $|\psi _m|^2$ along propagating distance $z$ for each waveguide at $P/M=$0.5 (I), 0.8 (II), and 1.3 (III, IV), with one ensemble randomly selected for each case. (c) The corresponding complex-$\psi$ space obtained from 500 ensembles, with the color indicating optical power $|\psi _m|^2$. Unlike positive $T$, when $\chi>0$, the high input power drives the system unstable and leads to localization, whereas for $\chi<0$ the system remains stable. For $\chi=0.1$, we calculated $P/M$ = 0.5 (I), 0.8 (II), and 1.3 (III), shown in Figs. \ref{['fig:fwd_S']}(b) and (c). For case (III), the system shows strong localization independent of the initial phase, whereas for $\chi=-0.1$ (IV), as predicted, the localization disappears.
• Figure S2: The isothermal power elastic modulus $\kappa^P_T$ for (a) square ($M=M_x \times M_y =19 \times 19$), (c) honeycomb ($M=M_x \times M_y =16 \times 23$), (e) irregular ($M=300$) and (f) Lieb array ($M=3\times N_x \times N_y =3\times10 \times 10$) with $\kappa=1$, $\chi=0.1$ at $T=0.5$. For the irregular array, we randomly generate 300 sites within a circular region of radius 15, with the inter-site spacing denoted by $l_{mn}$ and minimum distance $l_{mn}^{\min}=1$. The coupling coefficient is set to $\kappa _{mn}=\exp \left[ -2\left( l_{mn}-l_{mn}^{\min} \right) \right]$. (b, d, f) The complex-$\psi$ space distribution obtained from 500 ensembles for $\kappa^P_T$=0.15 (I), 0.05 (II), -0.05 (III), and -0.15 (IV). (h) Ratio of energy transferred to the nonlinear part $1-\widetilde{U}(z)/\widetilde{U}(0)$ as a function of propagation distance for the Lieb array. All results are consistent with the theoretical predictions. When $\kappa^P_T=0.15$, the system thermalizes as described. When $\kappa^P_T=0.05$, the $\widetilde{U}$ gradually transfers to $H_{\text{int}}$, indicating that nonlinearity starts to compete with diffraction, and progressively localizes the optical power during the evolution. For $\kappa^P_T<0$, the optical power becomes strongly localized.