This disclosure relates to systems and methods for optimizing fabrication of optical devices.
Optical devices such as metasurfaces may benefit from optimization. Metasurfaces are wavefront shaping devices with subwavelength-scale structures which have been demonstrated for a broad range of applications in sensing, imaging, and information processing. Inverse design using topology optimization has proven to be highly effective for designing metasurfaces with optimized performance, which uses adjoint based gradient computations to iteratively update the dielectric distribution of the device to achieve better performance. A challenge in the inverse design of metasurfaces is to not only optimize device performance but also guarantee device fabricability, for example, enforcing control on the minimum feature size of the pattern so that the fabrication requirement will not go beyond the ability of lithography and etching. Although existing strategies to restrict the minimum feature size such as applying filters and employing analytical penalty functions are able to tackle the problem to some extent, they all lack the ability to strictly avoid the existence of tiny features.
Various embodiments are directed to a method for optimization of photonic devices. The method receives a set of unconstrained latent variables, maps the set of unconstrained latent variables to a constrained space to generate a constrained device, and calculates the permittivity across each element of the constrained device. The method further determines a permittivity-constrained variable gradient based at least partially on the permittivity across each element and optimizes the set of unconstrained latent variables by at least partially using the permittivity-constrained variable gradient.
In various other embodiments, mapping the unconstrained space to a constrained space includes reparameterization of the unconstrained latent variables.
In still various other embodiments, the reparameterization of the unconstrained latent variables includes solving for a set of constrained width space {wi} through the following equations:
where the set of unconstrained latent variables are {ui} and wherein wmin is a minimum width of each feature of the constrained width space, M is the total number of features, and Σi=1M wi=L.
In still various other embodiments, calculating the permittivity across each element of the constrained device includes performing a gray-space relaxation of the set of constrained width space to determine a function of refractive index across each element in the set of constrained width space.
In still various other embodiments, the gray-space relaxation comprises solving for
where x is the position across the width of a certain element, where x=0 is at the middle of the element, and T controls the binarization of the pattern.
In still various other embodiments, the method further includes determining an efficiency-constrained variable gradient of the constrained device based on the permittivity-constrained variable gradient.
In still various other embodiments, the efficiency-constrained variable gradient is further based on an efficiency-permittivity gradient.
In still various other embodiments, the efficiency-constrained variable gradient is calculated using the following equation:
where
is the permittivity-constrained width gradient and
is the efficiency-permittivity gradient.
In still various other embodiments, an efficiency-latent variable gradient is calculated at least partially using a latent variable gradient of the latent unconstrained variables in relation to constrained device calculated using the following equations:
where the set of unconstrained latent variables are {ui} and wherein wmin is a minimum width of each feature of the constrained width space, M is the total number of features, and Σi=1M wi=L.
In still various other embodiments, the efficiency grating in relation to the unconstrained latent variables is calculated at least partially using the following equation:
where
is the constrained width-latent variables gradient and
is the efficiency-constrained width gradient.
In still various other embodiments, the efficiency-permittivity gradient is calculated using an adjoint variable method.
In still various other embodiments, the adjoint variable method includes forward and adjoint simulation.
In still various other embodiments, the method of further includes using a generator to generate the set of unconstrained latent variables; and iteratively updating the generator at least partially based on the permittivity-constrained variable gradient.
In still various other embodiments, the unconstrained latent variables comprise unconstrained latent widths and mapping the set of unconstrained latent variables comprises mapping the unconstrained latent widths to the constrained space to generate constrained widths. The method further includes receiving a set of unconstrained latent permittivities and mapping the set of unconstrained latent permittivities to a constrained space to generate constrained average permittivities of the constrained device. Calculating the permittivity profile across each element of the constrained device is further based on the constrained widths and the constrained average permittivities of the constrained device.
In still various other embodiments, the method further includes determining a permittivity-constrained average permittivity gradient of the constrained device and optimizing the set of unconstrained latent permittivities at least partially based on the permittivity-constrained average permittivity gradient of the constrained device.
In still various other embodiments, the method further includes binarizing the permittivity across each element of the constrained device, where determining a permittivity-constrained variable gradient is based on the binarized permittivity across each element.
Various embodiments are directed to a method for optimization of photonic devices, the method including receiving a set of variables including a set of refractive indices including a set of corresponding thicknesses, calculating a reflection gradient with respect to the set of refractive indices and their set of corresponding thicknesses, and optimizing the set of variables at least partially using the reflection gradient to obtain an optimized device including a set of optimized refractive indices with corresponding optimized thicknesses.
In various other embodiments, the method further includes calculating the reflection spectrum of the set of variables, where calculating the reflection gradient includes calculating a partial differential of the reflection spectrum.
In still various other embodiments, calculating the reflection spectrum comprises using a transfer matrix method.
In still various other embodiments, the method further includes: using a generator to generate the set of variables; using the set of variables to calculate a corresponding reflection gradient; and iteratively updating the generator at least partially using the reflection gradient.
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
The description will be more fully understood with reference to the following figures and data graphs, which are presented as various embodiment of the disclosure and should not be construed as a complete recitation of the scope of the disclosure.
Nanophotonic devices may be capable of manipulating and guiding electromagnetic waves propagating in free space and on chip, and they have a broad range of applications in imaging, sensing, and optical communications. Among the most effective methods to design these devices is gradient-based topology optimization, which has been used to realize metagratings, metalenses, and on-chip photonic devices that utilize complex electromagnetic wave dynamics to achieve exceptional performance. Topology optimization is performed by discretizing the device structure into small voxels, initializing each voxel with grayscale dielectric permittivity values c, and then iteratively modifying these permittivity values in a manner that improves a figure of merit (FoM). These modifications are based on gradient terms
that can be computed for each pixel using the adjoint variables method or autodifferentiation. Topology optimization can be performed in the context of local optimization, in which local gradients are directly used to perform gradient descent on grayscale device structures, or global optimization with global topology optimization networks (GLOnets), in which local gradient calculations are combined with generative neural network training to perform global population-based optimization.
A critical concern in the inverse deign methods for optical devices may be guaranteeing device fabricability, making it desirable to incorporate practical experimental constraints such as the minimum feature size or robustness to fabrication imperfections. Existing techniques such as applying threshold filters and adding penalties to the figure of merit may be able to push the output devices towards the desired constrained design space. However, they may lack the ability to enforce hard constraints such as avoiding the appearance of tiny features smaller than the minimum specified feature size.
Further, a critical issue concerning topology-optimized devices is the practical implementation of hard geometric constraints imposed by experimental considerations. A particularly important geometric constraint is minimum feature size (MFS), which arises due to limitations in lithography patterning resolution and etching aspect ratio. The imposition of MFS constraints also enables proper base patterns to be defined and used in algorithms that enforce fabrication robustness, in which base patterns are codesigned with their geometrically eroded and dilated forms. Without these constraints, topology-optimized devices often possess complex geometric shapes with very small feature sizes, making them difficult if not impossible to experimentally fabricate in a reliable manner.
Current methods to impose MFS constraints fall in one of three classes. The first is to set the device voxel dimensions or spacing between features to match the desired MFS. While this method works, it adds significant granularity to the device design space, limiting the final device performance. The second method is to use regularization terms that penalize the FoM when the MFS constraint is violated. While this technique will generally push devices toward regions of the design space that satisfy the desired constraints, it does not guarantee their enforcement. A third method is to optimize the device in an unconstrained manner and then use threshold filters to incorporate constraints. This method can be applied during the iterative optimization process or after unconstrained optimization is performed, and while it has the potential to work well, it only works when optimized devices in the unconstrained space locally map onto high performance optima in the constrained space.
Various embodiments of this disclosure relate enforcing strict geometric constraints within an inverse optimizer, while maintaining the fine pixel-level granularity of the physical design space, by reparameterizing the physical design space itself.
Further, various embodiments of this disclosure relate to a method that accurately imposes constraints through reparametrization. In some embodiments, the method may impose hard constraints in the optimization process through reparametrizing the problem. The pixelated device pattern is reparametrized by a new set of parameters where the constraints in the old design space are naturally eliminated, and optimization is performed in this new design space instead. This method may be used to design metasurfaces such as metagratings including silicon nanoridges with strict restrictions on minimum feature size.
In some embodiments, via a sequence of mathematical transformations, the method uses variables in an unconstrained latent space to reparametrize the original constrained device space such that the constraints are naturally imposed through the reparameterization process. Adjoint based optimization may then be performed with this newly introduced unconstrained variable space where performance gradients are first calculated regarding the original constrained device space and then backpropagated to the unconstrained latent space.
Photonic devices including metasurfaces such as high efficiency metagratings including silicon nanoridges may be fabricated with accurate control on the minimum feature size are designed. Silicon nanoridges may deflect normally incident light to the +1 diffraction order.
At all stages of optimization, an individual metagrating period may be defined by a permittivity profile ε=ε(X), which denotes the material distribution at each voxel located at x within an individual grating period. ε(x) is normalized so that 0 represents air and 1 represents silicon. The metagrating period has a width L that is subdivided into N=256 voxels, so that both x and E are vectors with dimension N. Our FoM is deflection efficiency, which is defined as the electromagnetic power deflected into the +1 diffraction order normalized to the total incident beam power. The deflection efficiency is denoted as Eff=Eff(c) and calculated with a rigorous coupled-wave analysis (RCWA) electromagnetic solver such as Reticolo.
The objective is to find a permittivity profile that maximizes the deflection efficiency, which can be cast as the following optimization:
maximize Eff(ε)
subject to ε∈{0,1}N
To define feature size constraints on the pixelized pattern ε(x), the pixelized pattern may be transformed to a vector of discrete width values, w, with which MFS constraints can be readily defined and incorporated. For devices of fixed topology, each element in w represents ridge widths and air gap values which may generalize to devices of varying topology. By reframing the pixelized patterns to a vector of geometric values, the original optimization problem may be reformulated for devices of fixed topology to be:
wmin denotes the MFS, wi denotes the width of each structural feature, which can be a silicon ridge or air gap, and M denotes the total number of silicon and air features. The total device width should be equal to the grating period L.
By first transforming the pixelated pattern into a pattern represented by the ridge width and ridge separation values, the minimum feature size can be described as wi≥wmin with i=1, 2, . . . , M where wmin denotes the minimum feature size, wi denotes the width of each bar (silicon ridge or air gap), and M denotes the total number of bars.
The reparametrized inverse design process may be used to design high efficiency metagratings with accurate control on the minimum feature size.
The reparameterization process of mapping {ui} into n(x) is elaborated in the lower panel of
As illustrated in the leftmost box on the lower panel of
The transformation of u to w may be defined as w=F(u, wmin). In this manner, MFS constraints on the metagrating pattern, have been imposed onto the width vector w.
Through the transformations described in the leftmost box on the lower panel of
With a physical device in hand,
may be evaluated using the adjoint variables method, in which forward and adjoint simulations are performed using the RCWA simulator. This gradient term may be then used to calculate
using back-propagation, which may be based on the chain rule. Backpropagation can be used in the algorithm because the entire reparameterization process can be described as a continuous and differentiable computation graph. In practice, backpropagation may be performed by programming the algorithm in PyTorch and using built-in autodifferentiation packages. Finally, u may be updated using the adaptive moment estimation (Adam) algorithm, which may be a variant of gradient descent, and the entire process is iteratively repeated.
A metagrating may be designed that deflects normally incident TM-polarized light at a wavelength of 850 nm to 65 degrees.
In some embodiments, the method may be used to enforce fabrication constraints through reparameterization. In the new reparameterized design space, those constraints may be naturally eliminated, and fabrication optimization may be achieved. The device designed by the method may be able to achieve extremely high performance and strictly satisfy fabrication constraints. Implementations of this design scheme may include variations in the topology, enrichment in constraints, and diversification of structural geometries.
Reparameterized local optimization may be used to design metagratings that deflect normally incident transversemagnetic-polarized light at a wavelength of 850 nm to 65 degrees. In one example, the MFS constraint may be set to 60 nm, the topology is fixed to contain three silicon ridges (M=6), and the device thickness is 335 nm. The refractive index of silicon may be used and only the real part of the index may be used to simplify the design problem. Complex refractive indices can be readily incorporated into the optimization procedure without loss of generality. For the disclosed system, the role of the imaginary part of the index may be small, and its inclusion results in a less than 2% drop in the total deflection efficiency for all of the optimized devices.
spans many voxeis, which translate to large
upon backpropagation. These gradients allow relatively large changes to the device layout to be made each iteration in the early stages of optimization, while the algorithm is broadly searching for a local optimum. It is noted that the initial magnitude of T cannot be too large and must be judiciously chosen because the grayscale and binary design spaces are different, and these differences become more substantial as T increases. Sufficient correlation between these two design spaces is required for gradients within the grayscale design space to reliably improve the device over the course of binarization. As the local optimizer evolves, T is gradually reduced and the device becomes more binarized. At this stage,
becomes more localized to the air-silicon interfaces and the gradients provide fine-tuning of the device layout, in a manner akin to conventional boundary optimization. Upon the completion of optimization, T is decreased to zero, and the device is fully binarized and contains only silicon and air material.
Reparameterization not only is able to enforce robust physical constraints, it also reduces the computational cost for the optimization process because it serves to reduce the dimensionality of the physical design space itself. Through reparameterization, N dimensional (N=256) pixelized device patterns are reduced to a M—1-dimensional latent vector representations, leading to over an order of magnitude reduction in the dimensionality of the design parameters. This results in a physical design space that is dramatically smaller with fewer local optima, yielding a simplified optimization problem that requires less computational cost for both local and global optimization algorithms.
In some embodiments, reparameterization may be applied to GLOnets to enable population-based global optimization of fixed-topology devices with hard geometric constraints.
G
ϕ
:U
M-1(−1,1)→Pϕ(u) (5)
where Pϕ(u) denotes the probability of generating u in the latent device space. The reparameterization process then follows, which maps u to the physical device pattern ε using the same formalism specified for the reparameterized local optimizer.
The next step is the evaluation of the loss function using metrics calculated from the generated devices. The loss function is engineered so that minimizing the loss function maximizes the probability that the neural network generates the optimal latent vector u*, which maps onto the globally optimized device ε*. For reparameterized GLOnets, the loss function is defined to be:
The efficiencies, Eff, and efficiency gradients,
of the physical devices are calculated with forward and adjoint electromagnetic simulations. K is the batch size, and σ is a hyperparameter that biases network training toward devices possessing relatively high efficiencies and large gradients. To minimize the loss function, backpropagation is performed to modify network weights ϕ in the generator. As the mapping functions that link the noise vector, latent vector, and physical device pattern profile are continuous and differentiable, backpropagation is performed via the chain rule in a straightforward manner. Upon the completion of network training, the network generates latent vectors that map to physical devices clustered around the global optimum.
Reparameterized GLOnets are used to optimize metagratings with the same specifications as those designed by local optimization which are discussed in connection with
Over the course of network training, the best generated device and the overall device distribution shift toward higher efficiency regimes (top panel,
The evolution of the device distribution over the course of network training is examined in more detail in
Fabrication Imperfections Enforced with Reparameterization
Devices that satisfy MFS constraints still may not guarantee good experimental performance as the device could be sensitive to other types of fabrication imperfections. Robustness to fabrication imperfections can be readily incorporated into reparameterized GLOnets. In some embodiments robustness may be incorporated in fixed-topology reparameterized GLOnets. In some embodiments, robustness may be incorporated in variable-topology reparameterized GLOnets. Robustness criteria are practically important due to geometric imperfections arising from all experimental fabrication processes. In some cases, geometrically eroded and dilated versions of the devices in the FoM may be used to account for fabrication imperfections.
Eff
avg=0.5×Efforiginal+0.25×Efferoded+0.25×Effdilated (7)
Eff
avg=0.5×Eff(w)+0.25×Eff(we)+0.25×Eff(wd) (11)
Fixed-topology reparameterized GLOnets with robustness may be used to globally optimize a metagrating with a MFS constraint of 30 nm and δw=10 nm. The device may operate with a wavelength of 850 nm and deflection angle of 65 deg, and the topology may be fixed to contain three silicon ridges.
Given that the final pixelized pattern ε(x) may be generated from the width space {wi}, fabrication imperfections may be defined in terms of pattern erosion (width of Si ridge decreases and width of air gap increases) and dilation (width of Si ridge increases and width of air gap decreases). In some embodiments, an averaged deflection efficiency may be defined as the figure of merit to reflect fabrication robustness where during each iteration the efficiency of multiple devices (e.g. three devices) may be averaged. The multiple devices may include:
In this way, robustness to fabrication imperfections may be introduced to the design space constrained with minimum feature size.
In some embodiments, the optimization techniques may be used to optimize optical devices such as photonic thin film stacks. Thin film stacks have been used in many optical systems.
{n*,t*}=arg min{n,t}Σλ,θ,pol((n,t|λ,θ,pol)−*(λ,θ,pol))2 (5)
The desired reflection spectrum is denoted as *(λ,θ,pol). The objective may be described to determine Equation 6:
O(n,t)=Σλ,θ,pol((n,t|λ,θ,pol)−*(λ,θ,pol))2 (6)
To determine the optimization, a global optimization algorithm may be used. In some embodiments the optimization algorithm may be a transfer matrix method (TMM).
In some embodiments, a generative neural network may be used to determine a distribution of thin film stack configurations.
The gradient of the loss function with respect to the neuron weights may be calculated by propagation, and the generator may then be optimized by gradient descent. After optimization, the distribution of thin film stack configurations may converge to a delta function located at the global optimum.
In practice, the refractive indices of thin film layers may not take arbitrary values and may be chosen from a material database and thus n may be a categorical variable. However, the gradients may not be calculated for categorical variables. To overcome this issue, a new scheme to generate refractive indices is illustrated in
α is a hyperparameter to tune the sharpness of the Softmax function 910. The ith row of probability matrix P 912 is a 1×M vector, which represents the probability distribution over the material database for the ith layer. Then the expected refrative index of each layer given by this probability distribution is calculated by ni(λ)=Σj=1M mj(λ)·Pij, which is a continuous variable with the valid gradient. As such, the gradient with respect to the refractive index ni may be backpropagated to the probability matrix P 912 and then the neurons. In addition, to ensure the material probability distribution of each layer converges to only one material, the hyperparameter a may increase over the course of optimization.
Since the desired reflection spectra usually cover a wide range of wavelengths and incident angles, finding the proper thin film configuration may be complex and challenging. A deep neural network may be beneficial to produce complex distributions in the design space in the early stage of optimization, while the ability to express a relatively simple distribution is also important for smooth convergence at the end.
A 3-layer thin film stack forming a three-layer AR coating for Si was designed to minimize average reflection from silicon over the incident angle range between 0° and 60° and the wavelength range between 400 nm and 1100 nm. The refractive index was between 1.09 and 2.60 and the thicknesses were between 0 nm and 200 nm of each layer.
P0 is the input power and V(A) denotes the eye's sensitivity spectrum. The generative neural network optimized device achieved χ of 18.0 compared with a genetic algorithm optimized device which achieved χ of 14.8. The generative neural network improved the performance by 22%.
Embodiments Including Optimization with Variable Topology
A global optimizer may benefit from the ability to search for the proper device topology as well as the detailed layout for that topology. Optimizers for fixed-topology devices can perform this task by parametrically sweeping across a wide range of topologies, as performed in
with its function defined as:
The right half is defined in a similar way on {tilde over (x)}(i)∈
from the center location to the interface between the ith and the i+1th section:
Thus, a continuous and differentiable mapping function ĥ is defined to transform from the piecewise vectors (w,{circumflex over (ε)}) to the device's permittivity profile ε, such that the gradient can be backpropagated during the optimization process. In some embodiments, a gradient based on the width vector (wi) and the center permittivity vector ({circumflex over (ε)}i) may be solved for. A process such as gradient descent may be used to optimize each of these vectors in order to optimize the device. In some embodiments, these width and permittivity may be solved for simultaneously. In some embodiments, the optimum width of the elements may be solved for followed by the optimum permittivity or vice versa.
γ is a tunable hyperparameter that controls the binarization of the permittivity vector {circumflex over (ε)} and is analogous to the hyperparameter T used to control the binarization of the device widths. As with T, γ is initially set to produce grayscale refractive index values and is manually increased in a gradual manner so that the final device possesses binary refractive index values of silicon or air. {circumflex over (M)} specifies the total number of sections and sets the upper limit in the topological complexity of generated devices. For example, if {circumflex over (M)}=10, physical devices with up to five nanoridges can be generated. In some embodiments, the physical device may include less than
nanoridges. For example, it {circumflex over (M)}=10, the physical device may have 4 nanoridges.
The ability for variable-topology reparameterized GLOnets to search across different device topologies is demonstrated for metagratings with the same operating parameters specified in
Variable-topology reparameterized GLOnets were benchmarked with two other methods. The first is local adjoint based optimization, where 200 unconstrained topology optimization iterations are first performed to produce binary devices from random grayscale devices, followed by 50 boundary optimization iterations to refine the binary device layouts and incorporate MFS constraints. The second is the original GLOnets, where a Gaussian filter is used at the network output in an attempt to impose a MFS constraint. The device designs are benchmarked by two criteria, deflection efficiency and the MFS in the physical device.
In the version of reparameterization discussed above in connection with
In some embodiment, the binary devices may be obtained and evaluated at each step during the iterative optimization. Unlike the reparameterization process described in connection with
the grayscale pattern may be binarized.
of the binary pattern may be simulated with an EM solver. Lastly, the gradient evaluated using the binary device may be directly applied onto the grayscale pattern to update the grayscale pattern, and eventually backpropagate onto the latent vector. The gradient may be used to optimize the latent vector through processes such as gradient descent. In some embodiments, the gradient on the binary device
may be directly applied to the grayscale device because the grayscale device is close enough to the binary device under a small T. In this manner, the EM simulation at each step may be performed for the binary device. The hyperparameter T can be set as a constant value throughout the optimization as long as the binary device is similar enough to its corresponding grayscale version.
For purposes of this discussion, cloud services are one or more applications that are executed by one or more server systems to provide data and/or executable applications to devices over a network. The server systems 3210, 3240, and 3270 are shown each having three servers in the internal network. However, the server systems 3210, 3240 and 3270 may include any number of servers and any additional number of server systems may be connected to the network 3260 to provide cloud services. In accordance with various embodiments of this invention, the optimization system that uses systems and methods that optimizes photonic devices while enforcing fabrication constraints in accordance with an embodiment of the invention may be provided by a process being executed on a single server system and/or a group of server systems communicating over network 3260.
Users may use personal devices 3280 and 3220 that connect to the network 3260 to perform processes that optimizes photonic devices while enforcing fabrication constraints in accordance with various embodiments of the invention. In the shown embodiment, the personal devices 3280 are shown as desktop computers that are connected via a conventional “wired” connection to the network 3260. However, the personal device 3280 may be a desktop computer, a laptop computer, a smart television, an entertainment gaming console, or any other device that connects to the network 3260 via a “wired” connection. The mobile device 3220 connects to network 3260 using a wireless connection. A wireless connection is a connection that uses Radio Frequency (RF) signals, Infrared signals, or any other form of wireless signaling to connect to the network 3260. In the example of this figure, the mobile device 3220 is a mobile telephone. However, mobile device 3220 may be a mobile phone, Personal Digital Assistant (PDA), a tablet, a smartphone, or any other type of device that connects to network 3260 via wireless connection without departing from this invention.
As can readily be appreciated the specific computing system used to optimizes photonic devices while enforcing fabrication constraints is largely dependent upon the requirements of a given application and should not be considered as limited to any specific computing system(s) implementation.
The processor 3305 can include (but is not limited to) a processor, microprocessor, controller, or a combination of processors, microprocessor, and/or controllers that performs instructions stored in the memory 3320 to manipulate data stored in the memory. Processor instructions can configure the processor 3305 to perform processes in accordance with certain embodiments of the invention.
Peripherals 3310 can include any of a variety of components for capturing data, such as (but not limited to) cameras, displays, and/or sensors. In a variety of embodiments, peripherals can be used to gather inputs and/or provide outputs. The optimizer element 3300 can utilize network interface 3315 to transmit and receive data over a network based upon the instructions performed by processor 3305. Peripherals and/or network interfaces in accordance with many embodiments of the invention can be used to gather inputs that can be used to optimize photonic devices while enforcing fabrication constraints.
Memory 3320 includes an optimization application 3325 and generator 3330. The optimization application 3325 and the generator 3330 in accordance with several embodiments of the invention can be used to optimize photonic devices while enforcing fabrication constraints.
The optimization application 3325 and generator 3330 may be used to perform the methods described above and the methods described below in
In some embodiments, mapping the unconstrained space to a constrained space may include reparameterization of the unconstrained latent variables.
In some embodiments, calculating the permittivity across each element of the constrained device may include performing a gray-space relaxation of the set of constrained width space to determine a function of refractive index across each element in the set of constrained width space.
In some embodiments, the optimization application 3325 may further determine an efficiency-constrained width gradient of the constrained device based on the permittivity-constrained width gradient. The efficiency-constrained width gradient is further based on an efficiency-permittivity efficiency gradient.
In some embodiments, the generator 3330 may generate the set of unconstrained latent variables; and the generator 3330 may be iteratively updated at least partially based on the permittivity-constrained width gradient.
In various embodiments, the optimization application 3325 may receive a set of variables including a set of refractive indices including a set of corresponding thicknesses; calculate a reflection gradient with respect to the set of refractive indices and their set of corresponding thicknesses; optimize the set of variables at least partially using the reflection gradient to obtain an optimized device including a set of optimized refractive indices with corresponding optimized thicknesses.
In some embodiments, the optimization application 3325 may further calculate the reflection spectrum of the set of variables, where calculating the reflection gradient comprises calculating a partial differential of the reflection spectrum. Calculating the reflection spectrum may include using a transfer matrix method.
In some embodiments, the generator 3330 may be used to generate the set of variables; the set of variables may be used to calculate a corresponding reflection gradient; and the generator 3330 may be iteratively update at least partially using the reflection gradient.
Although a specific example of element 3300 is illustrated in this figure, any of a variety of optimizer elements can be utilized to perform processes optimize photonic devices while enforcing fabrication constraints similar to those described herein as appropriate to the requirements of specific applications in accordance with embodiments of the invention.
The optimization method 3400 may further include mapping (3404) the set of unconstrained latent variables to a constrained space to generate a constrained device. The mapping may include reparameterization of the unconstrained latent variables. the reparameterization of the unconstrained latent variables may include solving for a set of constrained width space {wi} through the following equations:
where the set of unconstrained latent variables are {ui} and wherein wmin is a minimum width of each feature of the constrained width space, M is the total number of features, and Σi=1M wi=L.
The optimization method 3400 may further include calculating (3406) the permittivity across each element of the constrained device. Calculating the permittivity across each element of the constrained device may include performing a gray-space relaxation of the set of constrained width space to determine a function of refractive index across each element in the set of constrained width space. The gray-space relaxation may include solving for:
where x is the position across the width of a certain element, where x=0 is at the middle of the element, and T controls the binarization of the pattern.
The optimization method 3400 may further include determining (3408) a permittivity-constrained width gradient based at least partially on the permittivity across each element. The optimization method 3400 may further include optimizing (3410) the set of unconstrained latent variables at least partially using the permittivity-constrained width gradient. The permittivity-constrained width gradient may be at least partially used to calculate the efficiency-constrained width gradient.
In embodiments where a generator is used to generate the set of unconstrained latent variables, the generator may be iteratively updated at least partially based on the permittivity-constrained width gradient. The generator may be used to generate another set of unconstrained latent variables which may be used to update the constrained device.
While the above description contains many specific embodiments of the invention, these should not be construed as limitations on the scope of the invention, but rather as an example of one embodiment thereof. It is therefore to be understood that the present invention may be practiced in ways other than specifically described, without departing from the scope and spirit of the present invention. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents.
This application claims the benefit of and priority under 35 U.S.C. § 119(e) to U.S. Provisional Patent Application Ser. No. 63/023,766 entitled “Multi-objective, Robust Constraints Enforced Global Topology Optimizer for Photonic Devices,” filed May 12, 2020, which is incorporated herein by reference in its entirety for all purposes.
This invention was made with government support under Contract No. DE-AR0001212 awarded by the Department of Energy. The Government has certain rights in the invention.
Number | Date | Country | |
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63023766 | May 2020 | US |