Fluctuational electrodynamics for nonlinear materials in and out of thermal equilibrium

We develop fluctuational electrodynamics for media with nonlinear optical response in and out of thermal equilibrium. Starting from the stochastic nonlinear Helmholtz equation and using the fluctuation dissipation theorem, we obtain perturbatively a deterministic nonlinear Helmholtz equation for the average field, the physical linear response, as well as the fluctuations and Rytov currents. We show that the effects of nonlinear optics, in or out of thermal equilibrium, can be taken into account with an effective dielectric function. We discuss the heat radiation of a planar, nonlinear surface, showing that Kirchhoff's law must be applied carefully. We find that the spectral emissivity of a nonlinear nanosphere can in principle be negative, implying the possibility of heat-flow reversal for specific frequencies.

Figure 1

Spatial dependence of the effective dielectric function inside a single isotropic nonabsorbing plate ($\varepsilon =4$). $z$ is the distance from the surface and ${\lambda }_0^{\prime }=2\pi \frac{c}{{\omega }^{\prime }}$ is the wavelength corresponding to ${\omega }^{\prime }$ in vacuum. Note that while $N_{\mathrm{plate}}^{\mathrm{eq}}\left(\mathbf{r};\omega ,{\omega }^{\prime }\right)$ depends also on $\omega$, the plotted quantity is independent of $\omega$ due to division with ${\chi }^{\left(3\right)}\left(\mathbf{r};-\omega ,\omega ,{\omega }^{\prime },-{\omega }^{\prime }\right)$.

Figure 2

The effective dielectric function inside one of two nonabsorbing parallel plates ($\varepsilon =4$), where $z$ is the distance from the surface and $d$ is the separation between the plates. ${\lambda }_0^{\prime }=2\pi \frac{c}{{\omega }^{\prime }}=600\mathrm{nm}$ is the corresponding wavelength in vacuum.

Figure 3

Same quantity as in Fig. 2 for lossy plates with a dielectric constant $\varepsilon =4+i$ and vacuum wavelength ${\lambda }_0^{\prime }=2\pi \frac{c}{{\omega }^{\prime }}=600\mathrm{nm}$.

Figure 4

The nonequilibrium effective dielectric function inside a semi-infinite plate, where $z$ is the distance from the surface. $\varepsilon =4+i$, ${\lambda }_0^{\prime }=2\pi \frac{c}{{\omega }^{\prime }}=50\mu \mathrm{m}$, and $T_{\mathrm{obj}}=\frac{\hslash {\omega }^{\prime }}{k_{\mathrm{B}}}\approx 287\mathrm{K}$ were fixed while $T_{\mathrm{env}}$ was varied between zero and $2T_{\mathrm{obj}}$.