Example 4. Alignment of stresses caused by front and backside coatings
The new stress/thickness target allows you to align stress in front and backside 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) on the backside of the substrate, which will compensate this stress. Assume that we need to compensate stress of a 15layer TiO_{2}/SiO_{2} quarterwave mirror (QWM). The QWM central wavelength is 1030 nm. Total thickness is 2126 nm, total thicknesses of SiO_{2} and TiO_{2} layers are 1226 nm and 900 nm, respectively. The highreflectance 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 backside coatings. 
According to the wellknown Stoney's formula, the stress caused by a multilayer is: \[ \sigma_A=\frac{\delta E}{3 (1\nu) (\Sigma_H+\Sigma_L)}\cdot\left(\frac{d}{R}\right)^2 \qquad (1)\] where \(\delta\) is surface deflection caused by stress; \(E\) and \(\nu\) are Young's modulus and Poisson's ratio of the substrate, respectively; \(d\) and \(R\) are substrate thickness and diameter, respectively. On the other hand, the average coating stress \(\sigma_A\) can be approximated by: \[\sigma_A=\frac{(\Sigma_H\sigma_H+\Sigma_L\sigma_L)}{(\Sigma_H+\Sigma_L)} \qquad (2)\] where \(\sigma_H\) and \(\sigma_L\) are stress in separate H and L thinfilm materials. It follows from Eqs. (2) and (3) that: \[ \delta=\frac{(\Sigma_H\sigma_H+\Sigma_L\sigma_L)\cdot 3(1\nu)}{E} \cdot\left(\frac{R}{d}\right)^2 \qquad (3) \] Evidently, the deflection \(\delta\) can be represented as a linear combination of total thicknesses of H and L layer materials: \[\delta=\alpha_1\Sigma_H+\beta_1\Sigma_L\] 
Actually, the stress can be compensated by alignment of total thicknesses of layer materials as it was done in Example 3. However, it may happen that due to some reasons, the backside coating should be produced from other materials. In this case, the approach demonstrated in Example 3 does not work. The parameters \(E\) and \(\nu\) for Stoney's formula (1) can be found in the literature. Stresses \(\sigma_H\) and \(\sigma_L\) can be found in the literature as well or measured (for example, with the help of a profiler or an interferometer). In our example, we take \(\sigma_{TiO_2}=540\) MPa, \(\sigma_{SiO_2}=1100\) MPa, \(E=70000\) MPa, \(\nu=0.25\). Assume that substrate diameter is 25 mm and substrate thickness is 1 mm. Using Eq. (3), we represent the deflection caused by the QWM is: \[ \delta=4.284\cdot\Sigma_{TiO_2}8.73\cdot\Sigma_{SiO_2} \] To compensate stress caused by the frontside coating (QWM), a backside AR coating can be designed. The deflection sign of the AR coating is opposite. 
Fig. 2. Specification of the Thickness/Stress target. 
To minimize stresses in double sided coatings (QWM and AR), the difference of deflections caused by stresses in front and backside coatings is to be minimized. The corresponding Thickness/Stress target should be specified as it is shown in Fig. 2. In comparison with Example 3, the number of thickness/stress targets is equal to 1. The target value is zero since we minimize the difference of deflections. 
An important design hint: QWM must be loaded as a backside coating and substrate back side must be disabled. As a result, a 12layer AR operating in the range 9301150 nm can be obtained (Figs. 3 and 4). Total thickness of the AR design is 2003 nm, total thicknesses of SiO_{2} and TiO_{2} layers are 1344 nm and 659 nm, respectively. 

Fig. 3. Reflectance of the 12layer compensating AR (blue) and a doublesided coating with the QWM on the front side and the AR on the back side (red). 
Fig. 4. Refractive index profile of the compensating AR. 
Go to Example 1. Optimization of total thickness of one coating material Go to Example 2. Minimization of the total coating thickness Go to Example 3. Reducing of stress using alignment of thicknesses of materials on front and back sides Go to Example 5. Designing a beamsplitter maintaining optical flatness after coatings (OIC Design Contest 2013) Auxiliary information for stresstargets specification You may be also interested in reading the following articles: 