# Perspective on Some Recent and Future Developments in Casimir Interactions
## 3 Nonlinear Optical Response  
The pioneering experiments by [PERSON] et al., demonstrating second harmonic generation, have opened up the field of nonlinear optics [62]. This area of investigation deals with light-matter interactions involving media whose dielectric and magnetic response properties are nonlinear functions of the electric and magnetic fields. The discoveries of new materials, in bulk, and at the nanoscale, have created novel opportunities for fundamental studies and applications of nonlinear optics. Recent works have shown that Weyl semimetals and transition metal dichalcogenide layers, for example, can experience much enhanced nonlinear optical response, which in turn can give rise to photogalvanic effects, photocurrents, and nonlinear Hall effect [11; 63; 64; 12]. On the other hand, although electromagnetic fluctuations in nonlinear classical systems and quantum field theories have been studied in the past, the concept of fluctuational electrodynamics in the context of Casimir interactions has been explored only in limited situations. These include some general questions of noise [65; 66], forces for small particles with nonlinear polarizability [67; 68; 69], or some limited scenarios of nonlinear parallel plates [70; 71; 72; 73]. There have also been studies of thermal radiation in classical nonlinear Langevin models [74; 75]. The approach of references [70; 71] for fluctuational electrodynamics including nonlinear optics starts from the nonlinear Helmholtz equation for the vector electric field \(\mathbf{E}\), including stochastic sources \(\mathbf{J}\),  
\[\
abla\times\
abla\times\mathbf{E}-\frac{\omega^{2}}{c^{2}}\mathbf{E}-\frac{ \omega^{2}}{c^{2}}\mathbf{P}=i\omega\mathbf{J}. \tag{8}\]  
The optical response of the involved materials enters via the displacement field \(\mathbf{P}\), which is a nonlinear function of the electric field, and it is determined by the material properties. While the linear response is expressed in terms of the permittivity \(\varepsilon\), nonlinear terms are expressed by higher-order susceptibilities, and, with [PERSON] summation convention, the \(i\) component of the displacement field is,  
\[P_{i}=\big{(}\varepsilon_{ij}-\delta_{ij}\big{)}E_{j}+\int d\omega_{1}\,d \omega_{2}d\omega_{3}\delta(\omega-\omega_{\sigma})\chi^{(3)}_{ijkl}(-\omega_ {\sigma},\omega_{1},\omega_{2},\omega_{3})E_{j}(\omega_{1})E_{k}(\omega_{2})E_ {l}(\omega_{3})+\ldots \tag{9}\]  
We restrict the following discussion to symmetric materials, for which the polarization field is an odd function of the electric field, thus that in the series expansion of Equation (9), the dots represent fifth-order terms and so on. The nonlinear (third-order) term in Equation (9), expressed via the third-order susceptibility \(\chi^{(3)}\), couples different frequency modes, and, in general, different directions of the electric field. Equations (8) and (9) pose the mathematical difficulty of being nonlinear, which can be addressed in a perturbative manner: The results reviewed here are valid in leading order of \(\chi^{(3)}\). Another task is the treatment of the stochastic sources \(\mathbf{J}\) in Equation (8). Following references [70; 71], one starts by computing the linear response of Equation (8). The presence of \(\chi^{(3)}\) in combination with the noise render it nontrivial.