Example 5. Designing a beamsplitter maintaining optical flatness after coatings (OIC Design Contest 2013)
Many imaging systems use a dichroic beamsplitter to separate nearinfrared (NIR) and midinfrared (MIR) spectral bands. To maintain image quality, the beam separation must occur at front surface. The contest problem OIC 2013 was to transmit NIR radiation in the range 1.22.4 μm and reflect MIR radiation in the range 35μm. The backside must have an antireflection coating (AR) to minimize ghost images. The spectral requirements for the filter are shown in Fig. 1. Reflectance is calculated per surface and transmittance is calculated through both coatings assuming incoherent addition of the reflected intensity from each surface which includes multiple reflections. 
In addition to meeting the spectral requirements (Fig. 1), the coatings must not degrade the surface flatness of the substrate. The deflection due to coating stress for front \(\delta_1\) and for back \(\delta_2\) sides will be calculated and the net deflection \(\delta\), will be the difference between these two values: \(\delta=\delta_1\delta_2\le 0.01\) waves at 0.633 μm (1) Also, the deflection for the front surface alone must meet the requirement: \(10.00\le\delta_1\le 0.00\) waves at 0.633 μm (2) Four layer materials, L, V, Z, and S, can be used in the design. Refractive indices and stresses of the materials as well as substrate parameters are shown in Figs. 2 and 3. Formulas for \(\delta\) calculation are presented below. 
Fig. 1. Summary of target spectral characteristics (left) and sketch of dichroic beamsplitter to separate the NIR and MIR spectral ranges. 

The average stress of the coating \(\sigma_A\) can be modeled using Stoney's formula: \[\sigma_A=\frac{\delta E}{3(1\nu)\Sigma}\left(\frac{d}{R}\right)^2\qquad (3) \] where the constants \(E\) and \(\nu\) are Young's modulus and Poisson's ratio of the substrate, respectively; \(R\) is the radius of the substrate, \(d\) is the substrate thickness; \(\Sigma\) is total thickness of the coating. On the other hand, the average coating stress \(\sigma_A\) is approximated by: \[\sigma_A=\frac{\Sigma_L\cdot \sigma_L+\Sigma_S\cdot \sigma_S+\Sigma_V\cdot \sigma_V+\Sigma_Z\cdot \sigma_Z}{\Sigma} \qquad (4)\] where \(\sigma_L, \sigma_S, \sigma_V, \sigma_Z\) are stresses in layer materials (Fig. 2) and \(\Sigma_L, \Sigma_S, \Sigma_V, \Sigma_Z\) are total thicknesses of layer materials in the coating. It follows from Eqs. (3) and (4) that deflection of a surface \(\delta\) can be estimated as: \[\delta=\frac{\Sigma_L\cdot\sigma_L+\Sigma_S\cdot\sigma_S+\Sigma_V\cdot\sigma_V+\Sigma_Z\cdot\sigma_Z}{E}\cdot 3(1\nu)\left(\frac{R}{d}\right)^2\; (5)\] 
It follows from Eq. (4) that \[\delta=\alpha_L\Sigma_L+\alpha_S\Sigma_S+\alpha_V\Sigma_V+\alpha_Z\Sigma_Z \qquad (6)\] For the parameters given in Figs. 2 and 3, one can obtain: \[\delta=1.6839\Sigma_L2.5259\Sigma_S0.4210\Sigma_V0.2526\Sigma_Z \qquad (7) \] Such a linear combination can be specified in Thickness/Stress target in OptiLayer. Important: target values in Thickness/Stress target are measured in absolute units normalized to the reference wavelength. For example, the parameter \(\alpha_L\) of the deflection in waves at 0.633 μm is calculated as follows: \[\alpha_L=\frac{200\cdot 3(10.17)}{7.3\cdot 10^4}\cdot\left(\frac{12.5}{1}\right)^2\cdot\frac{1}{0.633}=1.6839\]

Fig. 2. Coating materials: refractive indices and stresses in separate layers. 
Fig. 3. Substrate properties. 
Fig. 4. Specification of stack in the case of dichroic beamsplitter. 
On the contrary to Examples 1, 2, 3, and 4, this contest problem can be solved using stack structure shown in Fig. 4. The stack structure is required due to two reasons: 1. Both coatings should be optimized simultaneously. 2. The frontside reflectance is calculated per surface, i.e. the reflection from the back side goes away from the optical path. For this reason, Wedged type must be specified in Type column (Fig. 4).

Thickness/Stress target is specified as it is shown in Fig. 5. Two targets correspond to requirement (1) and (2), respectively. In both cases range qualifier (R) is used. The number of coatings is two that corresponds to the beamsplitter and AR. Material coefficients are taken from Eq. (7). Range target to achieve spectral requirements (Fig. 1) can be specified in a standard way. 
Fig. 5. Thickness/Stress target for the dichroic beamsplitter. 
Fig. 6. Spectral characteristics of the winning design solution. 
Without the Thickness/Stress target option, good design solutions can be obtained with large design experience and additional tricks. As starting designs for the beamsplitter and AR coating, onelayer coatings of 1 μm thickness can be taken. Spectral characteristics of the winning design are shown in Fig. 6. Total thicknesses of the 37layer beamsplitter and of the 67layer AR coatings are 5.9 an 10.9 μm, respectively. The beamsplitter includes layers of three materials (L, S, and Z) and the AR design contains layers of all four materials (Fig. 2) 
Using the new Thickness/Stress target allows one to obtain a better solution to the same problem without tricks and without large experience. A design that is better than the winning design (Fig. 6) is shown in Fig. 7. Advantages of the design in comparison with the winning design:

Fig. 7. Design solution obtained using new Thickness/Stress target. 
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 4. Alignment of stresses caused by front and backside coatings 