### Robust Synthesis

 Errors in layer thicknesses and instability of the refractive indices of thin film materials are the main reasons of the deviations of experimental spectral characteristics of produced optical coatings from theoretical performances of corresponding multilayer designs. Errors in layer thicknesses are inevitable even in modern deposition plants equipped with high precision monitoring devices. It is known also that optical constants of layer materials may deviate from the nominal constants specified in the corresponding theoretical designs. These deviations are explained by various factors inside the deposition chamber, for example, change of materials in melting pot, unstable substrate temperature, cleaning conditions, etc.   It is known that different solutions of the same design problem exhibit different sensitivities to deposition errors.  Some designs may be sensitive with respect to even small errors. Other solutions are stable to errors of large magnitudes. Stability of the design solutions is tightly interconnected with a monitoring technique used to control the deposition process. For example, the same design solution might be stable in the case of time-monitoring control and sensitive to errors in the cases of broad band or monochromatic monitoring. 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 two-line 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) two-line 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 100-200. 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. Two-line Filter: target transmittance is 100% in the ranges 598-602 nm and 698-702 nm, target transmittance is zero in the ranges 500-580 nm, 615-693 nm, and 720-800 nm (Fig. 4).  Layer materials are Nb2O5 and SiO2, 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 31-layer 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 Nb2O5 and 0.005 of SiO2. 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 27-layer 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 27-layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 systematic offsets in high- and low-refractive indices, respectively. Fig. 6. Spectral transmittance and profile of the robust 29-layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 random offsets in high- and low-refractive indices, respectively. If the expected level of errors in layer thicknesses is 1% and there are random offsets of refractive indices, 0.01 of Nb2O5 and 0.005 of SiO2. 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 29-layer 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 two-line 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:  How to evaluate stability of the obtained robust designs?   How to compare stability of the conventional and robust designs?   What design solution is the most stable to deposition errors? 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 non-optical 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 (Two-Line 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 31-layer 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 27-layer 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 31-layer and 27-layer robust designs. Important notes: The robust algorithm takes not all sources of the deposition errors into account. Influence of some factors should be considered separately. It is recommended to obtain and analyze a series of good design solutions using various design techniques. It can happen that the levels of errors in layer parameters is to high to meet target requirements with the help of the robust algorithm. It is reasonable to stop the computations if either the generalized merit function almost does not decrease or there are very insignificant changes in the pivotal design. It means that a state of dynamic equilibrium has been achieved and at the specified error level it is not reasonable to search for a more complicated design. Almost all OptiLayer optimization algorithms support the robust synthesis. You maybe also interested in the following articles: