Inverse design in photonics by topology optimization: tutorial

Rasmus E. Christiansen; Ole Sigmund

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Inverse design in photonics by topology optimization: tutorial

Rasmus E. Christiansen, Ole Sigmund

arXiv:2008.11816v4Aug 23, 2020physics.opticsphysics.comp-ph

TL;DR

The paper addresses how to design nano-photonic structures with TopOpt while obtaining numerically sound, physically realizable, and geometrically robust designs. It presents a systematic tutorial and applies the approach to cylindrical metalenses and a demultiplexer, showing how auxiliary methods improve designs.

Problem
Designing a monochromatic silicon metalens requires an optimization formulation that avoids pixel-scale variation and tiny features that impair numerical accuracy, manufacturability, and mechanical stability.
Method
The tutorial derives a continuous constrained TopOpt problem and combines density-based adjoint optimization with filtering, thresholding, material interpolation, and multi-state formulations.
Results
TopOpt produced physically realizable and geometrically robust photonic structures, with examples including cylindrical metalenses and an optical demultiplexer.
Takeaways & Limitations
Applying established auxiliary tools during formulation enables systematic photonic TopOpt designs that better satisfy numerical-modelling and fabrication requirements.
Takeaways & Limitations
Because the photonic design problem is non-convex, TopOpt cannot guarantee global optimality and may guarantee only local optimality.

Abstract

from arXiv · show

Topology optimization methods for inverse design of nano-photonic systems have recently become extremely popular and are presented in various forms and under various names. Approaches comprise gradient and non-gradient based algorithms combined with more or less systematic ways to improve convergence, discreteness of solutions and satisfaction of manufacturing constraints. We here provide a tutorial for the systematic and efficient design of nano-photonic structures by Topology Optimization (TopOpt). The implementation is based on the advanced and systematic approaches developed in TopOpt for structural optimization during the last three decades. The tutorial presents a step-by-step guide for deriving the continuous constrained optimization problem forming the foundation of the Topology Optimization method, using a cylindrical metalens design problem as an example. It demonstrates the effect and necessity of applying a number of auxiliary tools in the design process in order to ensure good numerical modelling practice and to achieve physically realisable designs. Application examples also include an optical demultiplexer.

I. INTRODUCTION

The paper introduces density-based Topology Optimization as an inverse-design tool for photonic structures and presents a tutorial for formulating and solving constrained design problems. It emphasizes adjoint sensitivities, fabrication constraints, and reproducible COMSOL implementations.

I. INTRODUCTION: The tutorial uses COMSOL Multiphysics models to support replication of the presented results and provide starting points for advanced photonics applications.The workflow proceeds from defining spatial, temporal, and physical models to deriving and extending a continuous constrained optimization problem.
I. INTRODUCTION: Density-based Topology Optimization uses point-by-point material variation to produce highly optimized structures for specialized physical purposes.The method is applicable across most areas of physics and offers very large design freedom.
I. INTRODUCTION: A central practical challenge is limiting TopOpt’s design freedom so that optimized structures conform to fabrication constraints.The introduction frames manufacturing compatibility as a major consideration when applying the method to photonics.
I. INTRODUCTION: Adjoint sensitivity analysis is indispensable for inverse-design tools with many degrees of freedom and enables efficient gradient computations.The paper distinguishes topology optimization, related photonic inverse-design approaches, and objective-first methods primarily by problem formulation and solution strategy.
III. SPACE: The paper assumes a Cartesian coordinate system and a spatially limited modeling domain with interior ΩI and boundary Γ.These definitions establish the computational domain used for numerical modeling.

IV. TIME

The tutorial defines time-harmonic electromagnetic modeling and derives a continuous constrained optimization problem for a cylindrical silicon metalens. The formulation combines field equations, material interpolation, design constraints, and gradient-based solution methods.

IV. TIME: The electromagnetic model assumes time-harmonic behavior and disregards transient effects by representing time dependence with eiωt.The spatially dependent fields are separated from their time-harmonic factors.
VI. OPTIMIZATION PROBLEM: The baseline design problem seeks a cylindrical silicon metalens that focuses normally incident TM-polarized light monochromatically.In the two-dimensional model, material invariance out of plane turns the three-dimensional focal line into a focal point.
VI. OPTIMIZATION PROBLEM: TopOpt formulates structural design as maximizing a figure of merit Φ(ξ) subject to equality and inequality constraints on the continuous design field ξ(r).The design field takes values in [0,1], while the constraints measure physical and design requirements.
VI. OPTIMIZATION PROBLEM: The metalens figure of merit is based on electromagnetic field intensity at the focal point for fixed input power and TM-polarized illumination.Alternative objectives include focal-spot powerflow or field-profile differences over the focal plane.
VI. OPTIMIZATION PROBLEM: A continuous design field interpolates material parameters between air and silicon, with ξ = 0 corresponding to silicon and ξ = 1 to air.The interpolation introduces an equality constraint and bounds on ξ.
VI. OPTIMIZATION PROBLEM: TopOpt solves the continuous constrained problem with gradient-based algorithms that require sensitivities of the objective and constraints with respect to ξ.Finite-difference sensitivities would require repeated perturbed solves for each design variable at every iteration.

A. A note on adjoint sensitivity analysis

Adjoint sensitivity analysis can be derived before or after numerical discretization, with a trade-off between implementation simplicity and exactness for the discretized model.

A. A note on adjoint sensitivity analysis: Adjoint sensitivities may be derived analytically before discretization or directly for the discretized numerical system.Analytical derivations can be simpler to implement, whereas discretized adjoint gradients are exact for the numerical model.

B. A note on optimality

Optimization results for non-convex design problems should not be described as globally optimal because mathematical optimization methods cannot guarantee global optimality.

B. A note on optimality: No global-, artificial-intelligence-, or gradient-based optimization method guarantees global optimality for a non-convex problem.At most, an optimization method can guarantee local optimality, for example through satisfaction of KKT conditions.

C. A note on the design uniqueness

TopOpt can produce multiple qualitatively different photonic designs with similar figures of merit, so outcomes may depend strongly on the initial design field. The tutorial supports reproducibility and model setup through COMSOL examples and explicitly defined design parameters.

C. A note on the design uniqueness: Different initial design fields can lead TopOpt to qualitatively different geometries with similar figures of merit.The final geometry may therefore vary substantially even when the FOM changes little.
Reproducibility: The tutorial provides COMSOL models intended to reproduce the presented results and serve as starting points for advanced photonics applications.The models reproduce data used to create Figs. 3–8 when executed without modification.
Model problem: The metalens example defines a designable silicon-and-air region above a silicon slab for focusing TM-polarized plane waves at a specified focal point.The focal distance and design-region width determine the lens numerical aperture.
Model problem: The model domain uses a PML on three sides and first-order scattering boundary conditions, with a Gaussian-localized TM plane wave introduced at the lower boundary.These boundary-condition choices are made for simplicity.
Parameter specification: The examples organize geometry, physical, material, discretization, and optimization-manipulation parameters across Tables I–III.Table III includes the initial design field, filter strength, threshold, filter radius, and inner-iteration count.

A. Case 1: Naive approach

The naive metalens optimization permits intermediate material values that improve the optimized design but are impractical to fabricate. Thresholding produces a physical silicon–air design, yet sharply reduces performance; artificial attenuation instead preserves performance after thresholding, though tiny features remain.

Naive optimization: The naive optimization solves for a continuous design field and produces a metalens containing intermediate material values.The field is obtained from eq. (14), with black and white representing silicon and air; gray regions arise during optimization.
Naive optimization: Intermediate values enable precise local manipulation of wavelength and phase, but they are non-physical or impractical to realize.Fabrication would require removing intermediate values or approximating them with deep sub-wavelength perforations.
Naive optimization: The figure of merit falls from Φ ≈18.2 [V2/m2] to Φ ≈4.7 [V2/m2] after thresholding ξ(r) at 0.5.The thresholded design also transmits significantly less energy and focuses less power at the focal point.
Artificial attenuation: Artificial attenuation penalizes intermediate design values by adding imaginary loss to the relative-permittivity interpolation.Intermediate values in regions with non-zero electric field then propagate less energy, which is detrimental to maximizing the FOM.
Artificial attenuation: With attenuation, thresholding at 0.5 yields an almost identical binary design and preserves near-identical performance, with Φ ≈17.84 for both designs.The electric fields look nearly identical, and focal-plane power does not drop after thresholding.
Artificial attenuation: The attenuated design remains difficult to model and manufacture because it contains pixel-scale variation, tiny features, and narrow regions that may be mechanically unstable.The tutorial introduces further methods for amending tiny and single-pixel features.

C. Case 3: Filter and threshold

Case 3 combines spatial filtering with thresholding to produce nearly discrete, fabrication-friendly metalens designs while preserving focusing performance. The resulting geometry has larger features and avoids rapid pixel-scale variations.

Filtering: The smoothing filter controls spatial design-field variations and reduces single-pixel features by introducing a weak lengthscale.It is implemented through an auxiliary PDE in the design domain, with the original design field as input.
Filtering: Increasing the filter radius rf controls the size of features appearing in the filtered design field.
Thresholding: Thresholding uses a smoothed Heaviside approximation to recover a nearly discrete design field.
Thresholding: The threshold sharpness β ranges from weak thresholding at β = 1 to an effectively binary field as β approaches infinity.A continuation scheme increases β during optimization, allowing the design to develop gradually before becoming nearly black and white.
Results: Thresholding at 0.5 produces an almost identical silicon-and-air design with fewer and larger features and no single-pixel features.
Results: Φ ≈17.64 is obtained for both optimized and thresholded designs, only ≈1% below the unfiltered case.The electric-field intensity and focal-plane power flow are near identical, while the geometry is significantly simplified.
Results: The filtered design is easier to fabricate and avoids rapid pixel-by-pixel material variations that may jeopardize numerical convergence and precision.

D. Case 4: Multiple state-equation optimization

Case 4 reformulates the figure of merit to optimize focusing at three wavelengths simultaneously. Compared with single-wavelength optimization, this yields better average broadband performance while accepting lower performance at the central wavelength.

Formulation: The multi-objective formulation combines several figures of merit into one objective, with the example using a p-norm.
Formulation: The metalens targets λ ∈ {500 nm, 550 nm, 600 nm} across a 100 nm band instead of optimizing only one wavelength.
Formulation: The formulation gives greatest weight to the lowest Φi, while the required state equations can in principle be solved in parallel.
Evaluation: The resulting design is evaluated through electric-field intensity at the three targeted wavelengths and FOM values over 480 nm to 620 nm.
Results: The three-wavelength design performs best when averaged over the target wavelengths and the full interval, although the single-wavelength design performs best at its optimized wavelength.Broadband improvement is obtained by accepting a performance drop at the central wavelength.

IX. MODEL PROBLEM: THE DEMULTIPLEXER

The demultiplexer example applies TopOpt to a compact silicon-and-air photonic device that routes one input wavelength-dependently to two output waveguides. The formulation also demonstrates robustness to near-uniform fabrication variations.

Model problem: The optically small demultiplexer has a device footprint of approximately 2.4λ^2 and directs light to different output waveguides according to wavelength.
Model problem: The model domain contains air, silicon input and output waveguides, and a central designable region whose material distribution is optimized for output power flow.
Model problem: A Gaussian-enveloped, TE-polarized plane wave is introduced through the input waveguide, with boundary conditions used to truncate the model domain.
Robustness: Double filtering applies the filter-and-threshold procedure twice to optimize robustness toward near-uniform geometric perturbations.
Robustness: The design targets near-uniform erosion or dilation of ±8 nm, approximating over- or under-exposure during electron-beam lithography.
Evaluation: Transmittance is reported both relative to an identically excited reference waveguide and relative to the input-waveguide power.
Results: The optimized design exhibits approximately identical power flow through the relevant output waveguide across six cases, matching the FOM preference.Individual case weights can be changed to target larger or smaller transmittance for selected cases.

X. BRIEF DISCUSSIONS ON USEFUL TOOLS

The tutorial surveys auxiliary TopOpt tools that improve manufacturability, geometric robustness, connectivity, and fabrication compatibility. It illustrates their use in photonic designs, including a demultiplexer evaluated at two wavelengths.

Demultiplexer example: Figure 8 compares dilated, nominal, and eroded demultiplexer designs using power-flow magnitudes at λ = 1300 nm and λ = 1550 nm.The corresponding designs and wavelength-specific power-flow maps are arranged across panels (a–i).
Material interpolation: Material interpolation can strongly affect both optimized geometry and performance, so selecting an appropriate scheme is important for each problem.For plasmonic nanoparticles, a nonlinear scheme outperformed several alternatives by orders of magnitude in final design performance.
Length-scale control: Minimum-length-scale tools address nanometer-scale features that exceed practical fabrication limits.The double-filter technique supports geometric robustness, while geometric constraints can impose minimum length-scales when robustness is not the design priority.
Demultiplexer example: The demultiplexer’s reported power flow is ≈7.06 · 10^-4 W/m at λ = 1300 nm and ≈7.47 · 10^-4 W/m at λ = 1550 nm.
Connectivity: Connectivity constraints can prevent free-floating members when physical realization requires every structural member to remain connected.A virtual-temperature method is given as one way to impose this constraint.
Fabrication constraints: TopOpt can enforce fabrication-specific dimensional restrictions by mapping the design field and integrating sensitivities along the constrained dimension.This enables two-dimensional blueprints for fabrication processes with little or no out-of-plane variation.
Fabrication constraints: Filtering and thresholding have been used to eliminate proximity-effect correction in electron-beam and optical-projection lithography.The design field can serve directly as exposure dose or as a fabrication mask, while related procedures can also perform optimization-based correction.
Additional tools: Additional techniques include design-variable linking and perturbation-based treatment of random geometric uncertainties.The tutorial points to broader overviews of tools for length-scale control and manufacturability.

XI. CONCLUSION

The tutorial applies TopOpt to cylindrical metalenses and a demultiplexer, showing how iterative formulation improvements produce physically realizable and geometrically robust photonic structures. It concludes with software and methodological guidance, while noting that large-scale three-dimensional modeling remains computationally challenging.

XI. CONCLUSION: The tutorial demonstrates TopOpt for photonic structural design using cylindrical metalenses and an optical demultiplexer.
XI. CONCLUSION: Iteratively adding established methods transformed a naive formulation into designs that are physically realizable and geometrically robust.
XI. CONCLUSION: The examples use two spatial dimensions, while three-dimensional extension is methodologically straightforward but computationally bottlenecked by large-scale electromagnetic state-equation solves.Finite-difference time-domain solvers and overlapping-domain techniques are suggested as possible approaches for some large-scale three-dimensional problems.
XI. CONCLUSION: The supplied COMSOL software is intended as a starting point for adapting TopOpt to other photonics applications.
XI. CONCLUSION: A compact accompanying MATLAB tutorial provides a 200-line implementation and compares TopOpt with a non-gradient genetic-algorithm approach.

Appendix A: COMSOL Model Description

The appendix describes the COMSOL model organization used to set up, optimize, analyze, and visualize the metalens design problem. It identifies the model parameters, design operations, studies, derived values, plots, and practical auxiliary settings.

Model scope: The MetalensCase1.mph model implements the first metalens-design iteration discussed in the paper.
Model parameters: Global Definitions contain parameters such as lens width, target wavelength, and design resolution.
Model setup: The standard 2D component organizes the objective function, design-field operations, plotting operations, material interpolation, probes, mappings, and perfectly matched layers.
Optimization and analysis: The Optimization study defines the optimization method, evaluation limit, optimization type, targeted frequencies, and participating physics interfaces.
Results and visualization: Derived Values compute the figure of merit and power flow, while plots visualize the optimized design, electric-field intensity, focal-plane flow, and analysis results.
Practical settings: Additional parameters are included for practical tasks such as setting up the model geometry.