Optical Kinetic Theory of Nonlinear Multimode Photonic Networks

Recent experimental developments in multimode nonlinear photonic circuits (MMNPCs), have motivated the development of an optical thermodynamic theory that describes the equilibrium properties of an initial beam excitation. However, a nonequilibrium transport theory for these systems, when they are in contact with thermal reservoirs, is still terra incognita. Here, by combining Landauer and kinematics formalisms we develop a universal one-parameter scaling theory that describes the whole transport behavior from the ballistic to the diffusive regime, including both positive and negative optical temperature scenarios. We also derive a photonic version of the Wiedemann-Franz law that connects the thermal and power conductivities. Our work paves the way toward a fundamental understanding of the transport properties of MMNPCs and may be useful for the design of all-optical cooling protocols.

Figure 1

The two photonic reservoirs ($L/R$) consist of a large number of coupled nonlinear waveguides. We initially inject two beams at each of the reservoirs with different internal energy and power. At the steady state, the power distribution is given by a RJ thermal state with predetermined temperatures $T_{L/R}$ and chemical potentials ${\mu }_{L/R}$. After reaching a thermal state, the two reservoirs are coupled by an optical junction that transports heat and power currents. For intermediate, but large, length scales these currents acquire a quasi-steady-state $j_h$, $j_a$.

Figure 2

Normalized power current $j_a/j_a^{(\chi )}$ versus the scaling parameter $\lambda =N/l_T$ for a 1D junction of (transverse) size $N$. The different values of the current $j_a$ for a variety of parameters $(\chi ,J,\overline{T},\overline{\mu })$ are shown in the inset. The black dashed line is the interpolating function (11), while the solid line indicates a $1/N$ behavior and is drawn to guide the eye.

Figure 3

The ratio of thermal to power conductivity $\varkappa /\sigma$ versus temperature (black dots) for different values of nonlinearities $\chi =0$, 0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1.0, and junction sizes $N=150$, 300, 600. The theoretical prediction (12) (red dashed line) indicates a $\propto 1/T$ behavior, signifying a separation of relaxation processes between thermal and power transport. In the insets, we show the difference of power (upper inset) and energy (lower inset) densities between the left and right reservoirs for three different reservoir sizes $M$. As $M$ increases, the two reservoirs maintain for longer time their initial energy and power densities, allowing us to extract the (quasi)-steady-state values for the thermal and power currents.