### Example 3. Reducing of stress using alignment of thicknesses of materials on front and back sides

 The new stress/thickness target allows you to align total thicknesses of layer materials in front- and back-side coatings. Some complicated coatings may consist of dozens of layers. Thin films layers have mechanical stresses. After summing up stresses from all layers, the resulting stress is high enough to bend even a relatively thick substrate. To avoid this effect it may be useful to  deposit an antireflection coating (AR) at the back-side of the substrate, which together with suppression of unwanted reflections from the back side will also compensate this stress. Assume that we need to compensate stress of a 15-layer TiO2/SiO2 quarter-wave mirror (QWM). The QWM central wavelength is 1030 nm. Total thickness is 2126 nm, total thicknesses of SiO2 and TiO2 layers are 1226 nm and 900 nm, respectively. The high-reflectance zone of this QWM is from 930 nm to 1150 nm. The AR should operate in the same range. Fig. 1. General schematic of a substrate coated by a front- and back-side coatings. Fig. 2. Thickness/Stress target specification in the case when a compensating AR coating is to be designed. As stress depends linearly on the total thicknesses of coating materials (see Example 4 as well), it is reasonable to align total thicknesses of thin-film materials in front and back sides. In this case, Thickness/Stress target can be specified as it is shown in Fig. 2. Actually, it corresponds to the following optimization equations: $|\alpha_1\Sigma_{SiO_2}^{(1)}-\beta_1\Sigma_{SiO_2}^{(2)}|\rightarrow \min$ $|\alpha_2\Sigma_{TiO_2}^{(1)}-\beta_2\Sigma_{TiO_2}^{(2)}|\rightarrow \min$ The indices (1) and (2) corresponds to the front- and back-side coatings. An important design hint: QWM must be loaded as a back-side coating and substrate back side must be disabled. As a result, a 12-layer AR operating in the range 930-1150 nm can be obtained (Figs. 3 and 4). Total thickness of the AR design is 2126 nm, total thicknesses of SiO2 and TiO2 layers are 1226 nm and 900 nm, respectively. These thicknesses are equal to the thicknesses of SiO2 and TiO2 layers in the QWM design. Fig. 3. Reflectance of the 12-layer compensating AR (red) and a double-sided coating with the QWM on the front side and the AR on the back side. Fig. 4. Refractive index profile of the compensating AR. You can read more about AR compensating coatings for dispersive mirror in the publication. Auxiliary information for stress-targets specification