*5.3. Considerations Regarding the E*ff*ect of the Nonlinear Film Properties on Elastic Scattering*

Albeit not the main scope of this paper, we would like to briefly consider the role of light scattering and the potential impact of nonlinear effects on light scattering properties. We focus here on elastic scattering processes, because with conventional laser coatings, elastic scattering is four orders of magnitude stronger than inelastic scattering. Nevertheless, one must keep in mind that the excitation of nonlinear optical processes may also result in an increase in inelastic scatter contributions, but this discussion is outside the scope of this paper.

For most interference coatings, the interface roughness is the dominating source of light scattering. The angle-resolved scattering (ARS) can be calculated using multilayer vector perturbation (VPT) theories (given in detail, for instance, in References [35–37] and summarized briefly in Reference [38]). The most relevant factors influencing light scattering properties are (i) interface roughness, (ii) the cross-correlation properties of the roughness of different interfaces, (iii) the field strengths at the interfaces, and (iv) the interference properties not only for the specular fields but also for the scattered fields (the latter of course being linked to wavelengths and incident angles). Defects and contamination on the coatings can be considered to be additional sources of light scattering, which we excluded from this discussion.

As a result of VPT, the angle-resolved scattering distribution (the scattered intensity) of a multilayer can be calculated as

$$ARS(\theta\_s) = \frac{16\pi^2}{\lambda^4} \sum\_{i=0}^{Z} \sum\_{j=0}^{Z} F\_i F\_j^\* PSD\_{ij}(f) \, , \tag{7}$$

where *Z* is the number of layers (of course *Z* = 1 for a single film), and *Fi* and *Fj* are optical factors containing information about the optical properties of the perfectly smooth multilayer (design, dielectric functions, etc.) and the conditions of illumination and detection (angles, polarization, etc.). The roughness factor *PSDij* comprises the power spectral density functions of all interfaces (for *i* = *j*) and their cross-correlation properties (for *i j*). The total scatter can be calculated by integrating Equation (7).

It is important to go back into the full treatment of the VPTs to assess the influence of nonlinear effects. The main approach of VPT is that the unperturbed specular fields are used to calculate the fields at the interfaces. Depending on the roughness structure, these fields drive surface currents, which produce scattered waves. These scattered waves then propagate through the coating and interfere. Therefore, we expect an influence of nonlinear effects on the scattering properties in two ways: (i) the field-induced change of the dielectric function leads to a modification of the field distribution inside the coating and hence modified fields at the interfaces, and (ii) the modified dielectric properties change the propagation and interference properties of the scattered waves. It has been demonstrated that even small changes in the field distribution, e.g., caused by small wavelength shifts, can easily lead to an enhancement of the scatter losses by an order of magnitude [38]. Simply put, if for whatever reason the observed specular field deviates from the expected values by several percentage points, a substantial change in the scatter losses can be expected.

Although not trivial, we believe that taking nonlinear effects into account in scatter modeling is straightforward. One precondition is that the scattering effects are still so weak that they do not influence the determination of the dielectric function, including nonlinear effects using the approaches described in this paper. We should then be able to use these new parameters to calculate the scattered fields. Further studies will show if this pragmatic approach is justified. We also believe that for interference coatings, once we have taken nonlinear effects into account in the design, we are also able to bring the light scattering properties back to the level we would expect for a similar coating designed for low fluences, or we even have the possibility of making use of nonlinear effects for scatter loss reduction.
