Robust Synthesis
Three ways to find a stable design 1) To obtain a set of solutions of a design problem, provide computational manufacturing experiments simulating real deposition runs and choose solutions providing highest production yields estimations (see for example, article on yield analysis and our paper) 2) To use special algorithms (Robust Synthesis) taking into account stability requirement already in the course of the synthesis process. In Fig. 1 you can see results of the error analysis performed for a conventional and a robust designs of a twoline filter. The robust design (right panel) exhibit much higher stability to thickness errors in index offsets than a conventional design (left panel). 3) To use a deterministic approach taking errors in optical constants into account. In OptiLayer, this approach is realized in the form of Environments manager. 

Fig. 1. Illustrating example: Stability of the spectral characteristics of conventional (left panel) and robust (right panel) twoline filters. 

With Robust synthesis, you can obtain designs which are typically more stable with respect to production errors than standard (conventional) designs. Robust design can be activated by checking Robust Synthesis Enabled check box available through Synthesis > Options > Robust tab (Fig. 2). The new algorithm is based on a simultaneous optimization of a sets of design (design cloud) located in a vicinity of a basic, so called pivotal design. Robust Synthesis options require more computations than analogous standard OptiLayer synthesis options (about M times more, if M is the number of samples of the design cloud). Reasonable M values are 100200. 
The number of the designs in the design cloud M can be specified in The number of samples field on Robust tab. The standard merit function is generalized in the following way: \(GMF=\left\{\sum\limits_{j=0}^M MF^2_j/(M+1)\right\}^{1/2}, \) (Eq. 1) \(MF_j=\left\{\frac 1L \sum\limits_{l=0}^L \left[\frac{S(X^{(j)},\lambda_l)\hat{S}_l}{\Delta S_l}\right]^2\right\}^{1/2} \) (Eq. 2) \(S(X^{(j)},\lambda)\) and \(\hat{S}_l\) are actual and target values of spectral characteristics, \(\Delta S_l\) are target tolerances, \(\{\lambda_l\}, l=1,...,L\) is the wavelength grid. \(X^{(0)}={d_1,...,d_m, n_H,n_L}\) is the pivotal design and \(X^{(j)}={d_1^{(j)},...,d_m^{(j)}, n_H^{(j)},n_L^{(j)}}\) are disturbed designs from the cloud. \(MF_0\) is the standard merit function, \(MF_j\) are merit functions corresponding to designs from the cloud. 
Fig. 2. Activating Robust synthesis option and describing characteristics of the design cloud. 

OptiLayer allows you to specify various expected production errors. In the simplest case, there are errors in layer thicknesses only and no offsets of the optical constants. Errors in layer thicknesses can be specify in the absolute and/or relative scales (Tolerance size panel, Absolute and Relative fields on the Robust tab, see Fig. 2). In the example in Fig. 2, relative errors of 1% are specified. No Drift in Type column specifies absence of offsets in layer refractive indices. 
Absolute errors in layer thicknesses: \(d_i^{(j)}=d_i+\delta_i^{(j)} \;\;\) (Eq. 3) Relative errors in layer thicknesses: \(d_i^{(j)}=d_i+\Delta_i^{(j)}d_i \;\; \) (Eq. 4) 
Please, note that these parameters and the values of the refractive index offsets are not directly corresponding to the deposition process accuracy, yet they are connected with it. These parameters should not be considered literally, they are merely control parameters of the algorithm.  
Fig. 3. Systematic offsets of H and L materials from [0.01; 0.01] and [0.005; 0.005] are specified. 
If you would like to take into account offsets of the refractive indices as well, you need to specify Index Drift level and Type of the offset. If Per Material type is chosen then systematic offsets will take the same value for all layers of the same material. \(n_{H,L}^{(j)}=n_{H,L}+\Sigma_{H,L}^{(j)}, \;\;j=1,...,M\) (Eq. 5) The algorithm with systematic offset is applied when it is assumed that actual refractive indices can be shifted with respect to the nominal ones in the course of the deposition. The systematic errors \(\Sigma_{H,L}^{(j)}\) are random normally distributed errors with zero means and standard deviations \(\Sigma_{H,L}\). 
If Per Layer type is chosen (Fig. 3) then random offsets will take different values for different layers of the same material. \(n_{H,L}^{(j)}=n_{H,L}+\sigma_{H,L,i}^{(j)}, \;\;j=1,...,M\) (Eq. 6) The random errors \(\sigma_{H,L,i}^{(j)}\) are random normally distributed errors with zero means and standard deviations \(\sigma_{H,L}\). 
The algorithm with random offset is applied when it is assumed that actual refractive indices are not stable in the course of the deposition process due to various reasons (for example, instabilities of substrate temperature). 
Example. Twoline Filter: target transmittance is 100% in the ranges 598602 nm and 698702 nm, target transmittance is zero in the ranges 500580 nm, 615693 nm, and 720800 nm (Fig. 4). Layer materials are Nb_{2}O_{5} and SiO_{2}, Suprasil substrate. First, a set of conventional designs was obtained. The structure of one of the conventional designs and its spectral characteristics are shown in Fig. 4. Merit function value (MF) is 2.9.

Fig. 4. Spectral transmittance and structure of a conventional 31layer design solution. 
Assume, that the expected level of errors in layer thicknesses is 1% and there are systematic offsets of refractive indices, 0.01 of Nb_{2}O_{5} and 0.005 of SiO_{2}. A robust solution can be found with settings shown in Fig. 3 (Per Material). It means that \(\Sigma_{H}=0.01\) and \(\Sigma_{L}=0.005.\) As a starting design, a single layer can be used. Gradual Evolution can be used for synthesis. As a result, a 27layer solution (Fig. 5) is obtained. The merit function value is 10.3 that is larger than in the case of the conventional design (Fig. 4). It is a typical situation for the robust synthesis: robust design solutions approximate target specifications a little bit worse than the conventional designs. The reason is fundamental: additional stability requirements are taken into account in the course of the merit function optimization, i.e. the generalized merit function containing multiple terms (Eq. 2) is optimized instead of the standard merit function. 
Fig. 5. Spectral transmittance and structure of the robust 27layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 systematic offsets in high and lowrefractive indices, respectively. 
Fig. 6. Spectral transmittance and profile of the robust 29layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 random offsets in high and lowrefractive indices, respectively. 
If the expected level of errors in layer thicknesses is 1% and there are random offsets of refractive indices, 0.01 of Nb_{2}O_{5} and 0.005 of SiO_{2}. A robust solution can be found with settings shown in Fig. 3 (Per Layer). It means that \(\sigma_{H}=0.01\) and \(\sigma_{L}=0.005.\) As a result, a 29layer solution (Fig. 6) is synthesized. The merit function value is 6.5. As expected it is bigger than in the case of the conventional design (Fig. 4). 
Important! Merit function (MF) displayed on the bottom of the evaluation window (Fig. 7 and 8) is calculated in two different ways. If the robust option is disabled, the merit function is calculated in the standard way. If the robust option is enabled, MF is calculated via Eqs. 1 and 2. In Fig. 7 and 8 we can see MF values of conventional and robust twoline filters, respectively. Standard MF is smaller at the conventional design. At the same time, the generalized merit function GMF is smaller in the case of the robust solution. 

Fig. 7. Calculation of the merit function when robust option is disabled/enabled (conventional design). 
Fig. 8. Calculation of the merit function when robust option is disabled/enabled (robust design). 
After obtaining the robust design or a series of the robust designs, several important questions arise:

There are no simple answers to these questions. However, there are recommendations, which are typically help you to evaluate the designs stability (of course, with respect to the monitoring technique in use). Computational experiments simulating the deposition process can help you to evaluate stability of your design solution. 1) In the case of nonoptical monitoring technique (quartz crystal or time monitoring), statistical error analysis it recommended (see below). 2) In the case of broadband monitoring (BBM), computational manufacturing with BBM are recommended (without witness chips or with witness chips). 3) In the case monochromatic monitoring, simulations without witness chips or indirect monitoring can be used. In the course of the error analysis, it is reasonable to specify the same levels of the errors in layer thicknesses and offsets of refractive indices. 
In the case of statistical error analysis, designs with imposed errors in layer parameters are generated and spectral characteristics of the disturbed designs are calculated. Fig. 9. Activating statistical error analysis. 

In the example above (TwoLine Filter), the level of errors can be equal to 1% (Rel. RMS (%) column). If in the course of the robust synthesis Tolerance size was specified in absolute values, then that value should be specified in Rel.RMS(%) column. If in the course of the robust synthesis refractive index offsets were specified Per Material, then Per Material Errors box should be checked and the values of offsets should be specified in the RMS column on the Refractive Index tab (Fig. 10). This is, however, just a general recommendation. Of course, other reasonable error levels/settings can be used in the course of the statistical analysis. 
Fig. 10. Specifying errors in layer thicknesses and refractive index offsets in the course of the statistical error analysis. 
Fig. 11. Results of the statistical error analysis of the conventional 31layer conventional design (Fig. 4). Spectral characteristics of the disturbed design degrade significantly especially around the high transmission zone at 700 nm. 
Fig. 12. Results of the statistical error analysis of the robust 27layer design (Fig. 5). It is seen that the design is more stable. 
The numerical measure E(dMF) helps to evaluate the averaged stability of the design solution. This value is displayed on the bottom of the Error Analysis window. The value E(dMF) is expected deviation of the spectral characteristics of the theoretical design averaged by the number of the wavelength and the number of the statistical tests (The number of tests field on the Error Analysis Setup window, Fig. 9). 
Fig. 13. Comparison of the E(dMF) values of the conventional 31layer and 27layer robust designs. 
Important notes:


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