Fabrication-conscious neural network based inverse design of single-material variable-index multilayer films

4.1 Error free optimization

Here, we present NN-based optimization results for single material multilayer films with continuously changing refractive index. In the first study, the layer thicknesses are fixed to 30 nm, while their refractive indices can vary within a user-defined range. Our previous work [39] demonstrated that continuously variable refractive indices are achievable with high-density plasma chemical vapor deposition (HDPCVD) grown SiNx by varying the chamber stoichiometry. Based on this experimentally collected data, we set the refractive index range achievable to be between 1.6 and 2.4. The NN-based optimization starts with an initial input of a random multilayer stack. The refractive indices of the stack layers are passed through the online optimizer to generate a new set of modified layer parameters. These parameters are then used to calculate the true optical response with the TMM analytical solver [35, 40]. After the loss is calculated, the gradients are backpropagated through the system, and the online optimizer parameters are updated. Since the matrix-based analytical solver is differentiable, it enables efficient backpropagation of gradients throughout the system, including the solver itself. The optimization results for various optical filtering spectra are given in Figure 3. The insets in the figure represent the refractive index distribution of the optimized multilayer films. For a 200-layer stack with fixed 30-nm thick layers, the optimization results in extremely close conformance with the target transmittance spectra. The largest variation between the target and the optimized device spectra occurs for the 15 nm stopband filter, shown in Figure 3(f), which can be addressed with more layers, higher refractive index ranges, and thickness optimization. To this end, the impact of the layer numbers on the performance has been investigated. Figure 4 shows the effect of the total number of layers on the design performance. For this study, the long pass and the double band stop filter spectra are chosen as the target. For both target spectra, an optimization is performed with 60 and 200-layer optical stacks with fixed layer thicknesses of 30 nm. While the performance of the long pass filters given in Figure 4(a) and (b) does not change much, dual-band stop filters given in Figure 4(c) and (d) show a significant increase in performance as the number of layers increases. Thus, it is safe to conclude that the number of layers needed for high-quality optimization increases with the increasing complexity of the target spectra.

Figure 4:

Transmittance spectra comparison between the target spectra (orange) and the generated optimized optical stack spectra (blue) for a long pass filter with a cut-on at 475 nm for (a) 50, and (b) 200-layer structures. The same plots are shown for a dual stop band filter with stop ranges between 437.5 and 475 nm and 587.5–625 nm for (d) 50, and (e) 200-layer structures.

4.2 Reduction of layer number

The number of layers is yet another parameter in optimizing multilayer films. Usually, the least number of layers needed to satisfy an optimization criterion is not known a priori. Therefore, we implement a layer number reduction scheme into the NN-based optimization architecture. For the layer number reduction, a policy of reduction and performance threshold for stopping criteria is required. As the policy of reduction, removing the last 10 layers of an optimized stack at user-defined intervals is selected in the scope of this specific aim. Different policies could also be implemented, such as combining adjacent layers with similar refractive indices. As the performance threshold, we use mean absolute percentage error (MAPE), which is defined as [41]:

where T ω target and T ω optimized are the target and optimized transmittance spectra. The optimization starts with a large number of layers to guarantee high performance for a given design problem. After optimization converges to a good solution, last 10 layers of the structure are removed, and the remaining structure is reoptimized. This cycle is repeated until the reduced stack can no longer satisfy the performance threshold after a reasonable number of iterations. The impact of the layer reduction is depicted in Figure 5. For the stopping criterion of MAPE 3%, the optimization started with 300 layers and reduced the structure to 130 layers. We will note that the thicknesses are fixed; therefore, the reduced structure becomes thinner. While bandstop performance is somewhat affected with 130 layers, the changes in the performance for the 200-layer structure are negligible compared to the initially optimized structure with 300 layers. In principle, one can use a weighted MAPE or different stopping criterion to prevent performance degradations within desired bands. Using the layer reduction protocol, the total number of layers becomes an optimizable parameter for multilayer optical stack design. This enables optimization with least number of layers possible, allowing an easier and cost-effective fabrication of multilayer optical films.

Figure 5:

Transmittance spectra comparison between optimized stacks with different number of layers. A 300-layer optimized optical stack spectra the (orange) is optimized first. Layer numbers are gradually reduced until the stopping criterion is reached. The transmittance spectra of reduced optical stacks (blue) with (a) 130 layers, and (b) 200 layers are compared to initially optimized structure with 300 layers.

4.3 Optimization in the presence of systematic growth imperfections

In practice, when fabricating single material continuously changing films, imperfections always occur. These imperfections could be inherent to the nature of multilayer films, such as stress-induced inhomogeneities that lead to shifts in the refractive indices. Additionally, imperfections can be inherent to growth techniques, such as imperfect chamber gas stoichiometry due to switching precursor gas ratios for each layer deposition or slow changes in precursor gasses leading to gradually changing properties instead of clean cut-offs at the layer interfaces. These imperfections will result in refractive index shifts with different dependencies and transitory regions between the layers. Accumulated over many layers, these errors can dramatically reduce the performance of the multilayer stack. Fundamentally, all fabrication imperfections are either systematic or random in their nature. In this section we will discuss systematic errors, and the random errors will follow in the next section. Systematic errors are errors that are replicated with every repeated stack fabrication. Due to their deterministic nature, these errors can be fully compensated for. A variety of systematic errors are simulated during the optimization process to understand (i) the impact of the imperfections on the performance and (ii) efficient ways to compensate for them. Specifically, we modeled a variety of possible systematic errors such as: linear gradient shifts, transitory regions between layers with refractive indices spanning from the bottom to top layer values, and height-dependent refractive index shifts in the system including polynomial, and other higher-order dependencies. As the test bed for this study, a 100-layer optical stack with refractive indices ranging from 1.6 to 2.4 and layer thicknesses ranging from 20 to 120 nm is chosen. Figure 6 depicts the systematic errors applied to each layer of the multilayer stacks. The linearly increasing index shift is implemented by linearly increasing the amount of error in the refractive index at each successive layer. The bottom layers have minimum shifts while top layers have the highest shifts of up to 0.4, which is half of the user-defined range (1.6–2.4) of 0.8. Due to the time needed to change chamber stoichiometry between layers, a transition region between layers is expected. These transitory regions are modeled as 10 nm regions centered between the layer boundaries with a linear refractive index gradient between the interface’s bottom and top layers. Specifically, the 10 nm thick transition region is divided into 1 nm thick sub-layers, with the linear shift implemented as gradual changes in the refractive indices of the sub-layers. This transitory region is applied to all layer interfaces throughout the structure. To generalize further and to demonstrate this method’s capability and versatility higher-order height-dependent functions of refractive index shift were also implemented. These functions introduce an index shift based on the layer height measured from the bottom of the multilayer stack. Specifically, the following height-dependent functions were implemented,

are cubic, trigonometric and hybrid exponential-polynomial height-dependent index shift functions, respectively. Δ n x are the index shifts applied to a specific layer, and x is the distance measured from the bottom of the layer to the bottom of the multilayer stack. Note that the systematic errors will depend on the deposition tool, which is typically unknown and often difficult to measure a priori. The example functions are chosen to demonstrate the error compensation capability of the approach, regardless of the distribution of the systematic error.

Figure 6:

The fabrication imperfections for CVD grown single material multilayer optical stacks. (a) Linear gradient shift, (b) randomly changing shifts in refractive index, (c) graded transition regions between layers, and (d) height dependent functional shift in refractive index.

These errors are implemented individually and in combination to determine the impact on the optimization performance. The results under systematic imperfections are given in Figure 7. A linear gradient shift of indices enlarges the band stop region while the entire spectrum is red-shifted, as shown in Figure 7(a). Transition regions, on the other hand, only redshift the spectra. Figure 7(c) indicates that the combination of these imperfections leads to the linear addition of the degradations introduced by them in the transmittance curve. Height-dependent index shift functions depicted in Figure 7(e)–(f) yield more significant performance degradation, which is more fundamental than a simple red/blue shift. Based on these results, it is safe to conclude that systematic errors can significantly impact the optimized stack performance.

Figure 7:

Multilayer stack optimization under deterministic fabrication imperfections. The blue curve represents the spectra of the original ideally optimized stack without any imperfection introduced. The orange curve is the spectra of the same stack but with fabrication errors added. The black dotted curve is the spectra of the stack after the second stage of optimization to compensate for the fabrication errors is complete. The different plots are for different types of systemic error introduced: (a) linear gradient shift, (b) graded transition regions between layers, (c) linear gradient shift and graded transition regions between layers combined, (d)–(f) polynomial, sinusoidal, and exponential height dependent index shifts, respectively.

A fabrication-in-loop inverse design approach is proposed to counteract the systematic errors in the film growth process effectively. The details of the approach are shown in Figure 8. The proposed method consists of two phases. In the first phase, a multilayer optical stack is optimized with the NN-based inverse design architecture given in Figure 2. The resulting high-performing multilayer film design is then fabricated in the second phase. During the growth process, layer parameters are extracted with in-built metrology tools, which are available in many deposition machines. Specifically, after each layer is grown its properties are extracted with ellipsometry layer-by-layer [14]. Once the imperfect layer data is extracted from experimental measurements, they are integrated into the optimization cycle. To achieve this, the experimentally measured layer parameters are passed through the analytical solver to calculate the expected transmittance spectra. The resulting loss function is then calculated with the imperfect response. This approach allows a gradient-based inverse design method such as the one proposed here to effectively learn the response of a so-called “black box” system, which is the deposition tool in this case. In essence, by using the experimental layer parameters in the optimization cycle, imperfections in the deposition tool can be learned/compensated with a gradient-based optimization method. Importantly, the backpropagation chain is continuous, which is necessary for NN training. Consequently, the NN-based inverse design must learn the imperfections of the machine to produce high-quality stacks in addition to the general EM response of the multilayer films. This cycle of fabrication, measurement, backpropagation, and update continues until a high-performance device is realized experimentally.

Figure 8:

Network architecture for fabrication-in-loop NN-based inverse design of single-material multilayer optical stacks with continuously changing refractive index. (a) The optimization starts under ideal conditions and iterates until convergence. Later, fabrication-related imperfections are added to the ideally produced stack to simulate the imperfections from deposition tools. Modified stack response is calculated and the loss is backpropagated to update NN parameters. This procedure simulates the practical fabrication, (b) case. After ideal optimization, optimized parameters are used to grow the desired multilayer film. The imperfectly grown film will be measured and experimentally retrieved layer parameters are integrated back into the optimization cycle to calculate the imperfect response and loss function. This allows direct integration of imperfections related to the growth process into the optimization loop without disrupting the backpropagation chain. Simulated errors replace the film growth and ellipsometry in the optimization process for the experimental realization of the multilayer films.

In the scope of this study, the experimental growth and measurement process is simulated with systematic fabrication imperfections, as shown in Figure 8(a). After convergence to a good solution in the first phase, the selected error function is applied, and the refractive indices of individual layers are modified accordingly in the second phase. The modified stack response is calculated with the analytical solver, and the gradients are backpropagated after loss function calculation. This procedure, shown in Figure 8(a), simulates the practical fabrication cycle, as shown in Figure 8(b). We note that the introduced errors are not static, meaning that the error that depends on the layer properties, such as index and thickness, are changed every iteration with different optimized stacks.

The results of the compensation for fabrication imperfections are shown in Figure 7 (black dotted line). In the second phase of the optimization, starting from the modified stack with an imperfect response (solid orange line), the NN-based inverse design scheme learns to compensate for fabrication errors to a high degree. As a result, it produces a stack specifically modified to offset the impact of the fabrication errors for all the types of systematic error studied. Analyzing this procedure, the results indicate that the systematic errors in a deposition device can be fully accounted for in the optimization loop to realize high-performance multilayer stacks. The Supplementary Figures S2–S7 shows the loss track and modified device performance in intermediate steps. These results indicate that depending on the complexity of the device, from few tens up to a hundred cycles of optimization can yield multilayer stacks with high performance. The number of iterations also depends on the severity of the errors in the system, the learning rate, and the neural network architecture, which can be optimized for a given design problem. We would like to note that the NN-based inverse design proposes a modified stack, which offsets the imperfections introduced by the deposition tool. In an error free fabrication process, the multilayer film grown with error-compensated layer parameters will perform poorly as the NN is trained to change the layer parameters with respect to the fabrication imperfections. Naturally, the multilayer stack can be optimized with fabrication in the loop from the beginning as shown in Figure 8(b). However, this has the effect of increasing the number of optimization iterations with time consuming experimental growths required as indicated by results shown in Figures S2-S7 in the Supplementary text. This unsurprising result stems from the fact that the NN is learning the EM response of the multilayer film stack in addition to the behavior of the fabrication process in this case. Thus, our two-stage fabrication-in-loop inverse design method minimizes the number of optimization iterations with time-consuming and expensive experimental growths required.

4.4 Optimization in the presence of random growth imperfections

Besides systematic errors, random and unrepeatable errors could be present in the deposition of multilayer films. These errors change from one fabrication process to another and are characterized statistically. Since truly random effects cannot be compensated, they pose a challenge in producing high-quality multilayer films. As pointed out in the introduction, robustness schemes such as including refractive index and layer thickness sensitivities in the loss term of the multilayer film optimization have been proposed to alleviate the impact of the random imperfections [15], [16], [17], [18], [19, 42, 43]. However, these methods have shown limited improvements against minor imperfections where random thickness changes are predicted to be less than a nanometer. This section extends the fabrication-conscious optimization approach to include the case of stochastic imperfections. Following the previous section, the test bed for this study is a 100-layer optical stack with layers having refractive index and layers thicknesses ranging from 1.6 to 2.4 and 20–120 nm, respectively. Following the architecture outlined in Figure 8, the first stage of the optimization produces a high-performing multilayer stack assuming perfect fabrication. In the second stage, simulated random errors are added to the ideal layer parameters, and the EM response is calculated with the imperfect stack, as shown in Figure 8(a). The details of the backpropagation with random imperfections in the optimization loop are given in the Supplementary materials. At every iteration, a different set of random perturbations to both index and thickness is added to the ideal stack parameters. The optimization cycle with random imperfections is ended manually after 700 iterations. Rigorously, we implement the random fabrication errors as additive noise to the learnable base of multilayer stack parameters which can be expressed as:

The imperfect layer parameters t_imp, n_imp are obtained by adding random error functions t_rand, n_rand to the ideal layer parameter t_id, n_id. U 0 , 1 is a random variable with a uniform distribution between 0 and 1, and γ and μ are the user-defined error rate range and the mean of the error distribution, respectively. The results of the optimization with stochastic fabrication imperfections are given in Figure 9. To allow for a fair comparison between multilayer stacks optimized taking into account random imperfections during optimization (black dotted line) and those optimized assuming no imperfections (blue solid line), the EM response of multilayer stacks obtained with both methods have been calculated for 1000 different sets of random perturbations. Figure 9(a)–(h) depict the highest and the lowest performing samples from their respective sets according to the calculated mean squared loss for the given target spectra (solid orange line). For a mean error rate (μ) of 0 and error rate range (γ) of 1 percent, multilayer stack optimized with and without random perturbations perform quite similarly and overall performance is not affected for the best and worst cases as seen in Figure 9(a) and (e), respectively. For μ = 1% and γ = ±1%, a red shift in the spectra is observed for the ideally optimized stack, while the stack optimized with random perturbations compensates for this effect, as shown in Figure 9(b) and (f). For μ = 0% and γ = ±5%, the impact of the random error is more significant as shown in Figure 9(c) and (g). Both stacks perform similarly for best and worst cases with minor improvements in the bandpass region for the fabrication-conscious design. Similar to Figure 9(b) and (f), the case of random error with non-zero mean, μ = 5% and γ = ±5% shown in Figure 9(d) and (h) indicates a redshift for the ideally optimized stack with an even greater impact on the performance. Regardless of the mean of the random error, optimizations with random imperfections perform similarly for the same error ranges. We thus conclude that the proposed NN-based inverse design method can effectively learn the random error’s mean and compensate for it. Unsurprisingly, random imperfections are not fully compensated, as seen in Figure 9, where the impact of the stochastic imperfections remains even for a multilayer stack optimized with fabrication-conscious NN-based inverse design.

Figure 9:

Multilayer stack optimization in the presence of random fabrication imperfections. Results after random error is added for both ideally optimized stacks (blue solid line) and stacks optimized considering random imperfections during the optimization process (black dotted line). The orange solid line is the target spectra. The transmittance of the multilayer stacks produced by both methods is tested with 1000 different sets random imperfections. The transmittance curves for the highest and the lowest performing stacks from their respective sets of results are shown in the top (a)–(d) and bottom (e)–(h) rows, respectively. The mean and the range of the uniform distribution used to create random imperfections are (a and e) μ = 0% and γ = ±1%, (b) and (f) μ = 1% and γ = ±1%, (c) and (g) μ = 0% and γ = ±5%, and (d and h) μ = 5% and γ = ±5%, respectively.

Finally, the impact of combined random and systematic errors is investigated, and the results are shown in Figure 10. For the bandstop filter design, 2 sets of systematic and random errors are implemented. The mean of the stochastic error rate for all the results shown in Figure 10 is 0 percent (μ = 0%). Linear grading and transitory regions are applied with γ = ±1%, and γ = ±5% random shifts to each layer. The results are shown in Figure 10(a) and (b), respectively. The results for height-dependent sinusoidal function with a γ = ±1%, and γ = ±5% random shifts are shown in Figure 10(c) and (d). Looking at the figures and related loss tracks in Figures S2–S7 of the Supplementary, it is clear that the systematic errors are learned and compensated for even in the presence of random errors. Comparing the results with corresponding random error rates between Figures 9 and 10, it is clear that full compensation of the deterministic errors is achievable even in the presence of random imperfections. Starting with both random and deterministic imperfections (solid orange line), a multilayer stack optimized using the fabrication-conscious approach outlined earlier can reach performance only limited by the random imperfections (black dotted line). This approach is most advantageous when systematic errors are dominant in the growth process. Essentially, regardless of the nature of the systematic error, this fabrication-in-loop optimization scheme can produce high-performance films even in the presence of stochastic deposition errors.

Figure 10:

Multilayer stack optimization in the presence of both deterministic and random fabrication imperfections. (a) and (b) Linear gradient shift and graded transition regions between layers combined with γ = ±1% and γ = ±5% random index shifts, respectively (c) and (d) sinusoidal height dependent index shifts (2) with γ = ±1% and γ = ±5% random index shifts, respectively. The solid blue curve represents the ideally optimized spectra without any error. The orange curve represents the ideally optimized spectra after both the error types are introduced. The black dotted curve is the result of optimization to compensate for the errors.