Example 4. Alignment of stresses caused by front- and backside coatings

The new stress/thickness target allows you to align stress 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) on the back-side of the substrate, which will 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.

stress compensation in multilayer

Fig. 1. General schematic of a substrate coated by a front- and back-side coatings.

According to the well-known 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 thin-film 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 back-side 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 front-side coating (QWM), a back-side AR coating can be designed.

The deflection sign of the AR coating is opposite.

deflection of coated substrate

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 back-side 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 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 2003 nm, total thicknesses of SiO2 and TiO2 layers are 1344 nm and 659 nm, respectively.

compensation of stress in coatings

Fig. 3. Reflectance of the 12-layer compensating AR (blue) and a double-sided coating with the QWM on the front side and the AR on the back side (red).

stress compensation

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 stress-targets specification

You may be also interested in reading the following articles:

Design targets

Combined targets

Antireflection coatings

Laser-related coatings