Benchmarking Five Global Optimization Approaches for Nano-optical Shape Optimization and Parameter Reconstruction

Numerical optimization is an important tool in the field of computational physics in general and in nano-optics in specific. It has attracted attention with the increase in complexity of structures that can be realized with nowadays nano-fabrication technologies for which a rational design is no longer feasible. Also, numerical resources are available to enable the computational photonic material design and to identify structures that meet predefined optical properties for specific applications. However, the optimization objective function is in general non-convex and its computation remains resource demanding such that the right choice for the optimization method is crucial to obtain excellent results. Here, we benchmark five global optimization methods for three typical nano-optical optimization problems: \removed{downhill simplex optimization, the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, particle swarm optimization, differential evolution, and Bayesian optimization} \added{particle swarm optimization, differential evolution, and Bayesian optimization as well as multi-start versions of downhill simplex optimization and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm}. In the shown examples from the field of shape optimization and parameter reconstruction, Bayesian optimization, mainly known from machine learning applications, obtains significantly better results in a fraction of the run times of the other optimization methods.

[1]  Alexander Y. Piggott,et al.  Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer , 2015, Nature Photonics.

[2]  Robert Magnusson,et al.  Particle swarm optimization and its application to the design of diffraction grating filters. , 2007, Optics letters.

[3]  Carsten Rockstuhl,et al.  Shape design of a reflecting surface using Bayesian Optimization , 2018 .

[4]  Philippe Lalanne,et al.  Near-to-Far Field Transformations for Radiative and Guided Waves , 2016 .

[5]  Carsten Rockstuhl,et al.  Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes , 2017, Other Conferences.

[6]  Peter Monk,et al.  Finite Element Methods for Maxwell's Equations , 2003 .

[7]  A compact and simple photonic nano-antenna for efficient photon extraction from nitrogen vacancy centers in bulk diamond , 2017, 1704.08951.

[8]  E. Bor,et al.  Differential evolution algorithm based photonic structure design: numerical and experimental verification of subwavelength λ/5 focusing of light , 2016, Scientific Reports.

[9]  J. Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[10]  Eric Jones,et al.  SciPy: Open Source Scientific Tools for Python , 2001 .

[11]  J. Fischer,et al.  Three‐dimensional optical laser lithography beyond the diffraction limit , 2013 .

[12]  Yudong Zhang,et al.  A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications , 2015 .

[13]  Seyed Mohammad Mirjalili,et al.  Optical buffer performance enhancement using Particle Swarm Optimization in Ring-Shape-Hole Photonic Crystal Waveguide , 2013 .

[14]  L. Zschiedrich,et al.  Hp-finite element method for simulating light scattering from complex 3D structures , 2015, Advanced Lithography.

[15]  Carl E. Rasmussen,et al.  Manifold Gaussian Processes for regression , 2014, 2016 International Joint Conference on Neural Networks (IJCNN).

[16]  Matthijs Langelaar,et al.  System Robust Optimization of Ring Resonator-Based Optical Filters , 2016, Journal of Lightwave Technology.

[17]  M. E. H. Pedersen,et al.  Tuning & simplifying heuristical optimization , 2010 .

[18]  Ying Li,et al.  Computational metrology and inspection (CMI) in mask inspection, metrology, review, and repair , 2012 .

[19]  P. N. Suganthan,et al.  Differential Evolution: A Survey of the State-of-the-Art , 2011, IEEE Transactions on Evolutionary Computation.

[20]  Neil D. Lawrence,et al.  Batch Bayesian Optimization via Local Penalization , 2015, AISTATS.

[21]  Frank Scholze,et al.  Reconstructing Detailed Line Profiles of Lamellar Gratings from GISAXS Patterns with a Maxwell Solver , 2017, 1704.08032.

[22]  Bernd Bodermann,et al.  Quantifying parameter uncertainties in optical scatterometry using Bayesian inversion , 2017, Optical Metrology.

[23]  Nando de Freitas,et al.  Taking the Human Out of the Loop: A Review of Bayesian Optimization , 2016, Proceedings of the IEEE.

[24]  D. Sculley,et al.  Google Vizier: A Service for Black-Box Optimization , 2017, KDD.

[25]  C. Rockstuhl,et al.  Core-Shell Particles as Building Blocks for Systems with High Duality Symmetry , 2017, Physical Review Applied.

[26]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[27]  Oriol Vinyals,et al.  Qualitatively characterizing neural network optimization problems , 2014, ICLR.

[28]  J. Mørk,et al.  Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides , 2009, 0909.3233.

[29]  M. G. Saber,et al.  Performance Analysis of a Differential Evolution Algorithm in Modeling Parameter Extraction of Optical Material , 2017, Silicon.

[30]  G. Ehret,et al.  Optical dimensional metrology at Physikalisch-Technische Bundesanstalt (PTB) on deep sub-wavelength nanostructured surfaces , 2016 .

[31]  J. Rho,et al.  Fabrication of three-dimensional suspended, interlayered and hierarchical nanostructures by accuracy-improved electron beam lithography overlay , 2017, Scientific Reports.

[32]  L. Zschiedrich,et al.  Numerical optimization of the extraction efficiency of a quantum-dot based single-photon emitter into a single-mode fiber. , 2018, Optics express.

[33]  Alexander Y. Piggott,et al.  Fabrication-constrained nanophotonic inverse design , 2016, Scientific Reports.

[34]  F Schmidt,et al.  Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography , 2015, Nature Communications.

[35]  L. Lauhon,et al.  Evolutionary Design and Prototyping of Single Crystalline Titanium Nitride Lattice Optics , 2017 .

[36]  Lukas Chrostowski,et al.  Silicon Photonics Circuit Design: Methods, Tools and Challenges , 2018 .

[37]  Carl E. Rasmussen,et al.  Derivative Observations in Gaussian Process Models of Dynamic Systems , 2002, NIPS.

[38]  Michael Mrejen,et al.  Plasmonic nanostructure design and characterization via Deep Learning , 2018, Light: Science & Applications.

[39]  Philipp Gutsche,et al.  Chiral scatterers designed by Bayesian optimization , 2017, 1712.07091.

[40]  Dan Dalacu,et al.  Nanowire waveguides launching single photons in a Gaussian mode for ideal fiber coupling. , 2014, Nano letters.

[41]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[42]  Eli Yablonovitch,et al.  Adjoint shape optimization applied to electromagnetic design. , 2013, Optics express.

[43]  Mikio C. Aoi,et al.  Exploiting gradients and Hessians in Bayesian optimization and Bayesian quadrature. , 2017, 1704.00060.

[44]  N. Bonod,et al.  Optimized 2D array of thin silicon pillars for efficient antireflective coatings in the visible spectrum , 2016, Scientific Reports.