# Perspective on Some Recent and Future Developments in Casimir Interactions
## 3 Nonlinear Optical Response
Assuming the noise has zero mean, one arrives at the following form for the effective dielectric function \(\widetilde{\epsilon}\), describing the physical linear response of Equation (8),  
\[\widetilde{\epsilon}_{ij}(\mathbf{r},\omega)=\varepsilon_{ij}(\mathbf{r}; \omega)+3\int d\,\omega^{\prime}\chi^{(3)}_{ijkl}(\mathbf{r};-\omega,\omega, \ \omega^{\prime},-\omega^{\prime})b\big{(}\,\omega^{\prime})\mathrm{Im}[G_{kl}( \mathbf{r},\ \mathbf{r};\ \omega^{\prime})] \tag{10}\]where \(b(\omega)=\frac{\hbar\omega^{2}}{\pi\varepsilon_{0}c^{2}}\left(1-\exp\left(-\frac{ \hbar\omega}{\mathbf{k}_{0}\overline{\mathbf{I}}}\right)\right)^{-1}\) provides the Bose distribution of frequencies and \(\mathrm{G}_{kl}(\mathbf{r},\ \mathbf{r};\ \omega^{\prime})\) are the components of the retarded dyadic Green's tensor of the corresponding linear system. Note that the dependence on the position \(\mathbf{r}\) is included explicitly in the above equation. The effective permittivity in Equation (10) results from fluctuations in the linear system, quantified by \(b(\omega)\) as well as the imaginary part of the Green's function \(\mathrm{G}_{kl}\), which excite nonlinear responses. The noise in Equation (8) thus enters via the imaginary part of \(\mathrm{G}_{kl}\). Notably, the effective dielectric function carries interesting properties, for example, being nonlocal in the following sense: For two bodies, \(\widetilde{\epsilon}\) is a function of their distance [71]. Because Equation (10) gives the physical linear response, the known formulations for Casimir forces, such as the Lifshitz formula, can be used with \(\widetilde{\epsilon}\) replacing the bare permittivity. By virtue of the fluctuation dissipation theorem, it is the physical linear response, which governs fluctuations. Figure 4 shows as an example the Casimir force for two parallel plates separated by a vacuum gap of width \(d\), at zero temperature. One of the plates is made of a nonlinear material (details given in the figure), while the other plate is a perfect reflector. The graph displays the force while taking the known power law of \(1/d^{4}\) at a larger distance, shows a cross over to a law of \(1/d^{8}\) at smaller distances. This displays the mentioned effect: At a small distance, the distance dependence of the permittivity is notable, thus that the force depends on distance in a two-fold manner: Via confinement of modes, and via the distance dependence of dielectric permittivity \(\widetilde{\epsilon}\). The combination yields the observed power law. Figure 4 also displays that the power law of \(1/d^{8}\) even persists when the bare permittivity approaches unity. This means that an object, which in isolation has no linear response, still feels a Casimir force when close to another object. This analysis shows that the effects of nonlinear optics drastically alter the Casimir effect at small distances, i.e., when the two surfaces strongly interact. Even more drastic effects of nonlinear optics are expected to occur under non-equilibrium conditions, e.g., for objects at different temperatures [71; 73; 74], or for dynamic scenarios discussed below. The inclusion of strong external fields is also a very promising route, and it is expected to allow for a large variety of effects concerning Casimir forces and nonlinear optics (see also the section on DCE below). Last, Casimir forces for mesoscopic bodies with strong even order nonlinear effects, e.g., with large \(\chi^{(2)}\)[11], remain mostly unexplored.