Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods
In this paper, we consider the optimal design of photonic crystal structures for two-dimensional square lattices. The mathematical formulation of the bandgap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and th...
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| Format: | Article |
| Language: | en_US |
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Elsevier
2012
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| Online Access: | http://hdl.handle.net/1721.1/69624 https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 https://orcid.org/0000-0003-1132-8477 |
| _version_ | 1811092893753933824 |
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| author | Men, Han Nguyen, Ngoc Cuong Freund, Robert Michael Parrilo, Pablo A. Peraire, Jaime |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Men, Han Nguyen, Ngoc Cuong Freund, Robert Michael Parrilo, Pablo A. Peraire, Jaime |
| author_sort | Men, Han |
| collection | MIT |
| description | In this paper, we consider the optimal design of photonic crystal structures for two-dimensional square lattices. The mathematical formulation of the bandgap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue problems for multiple values of the wave vector parameter. The resulting optimization problem is large-scale and non-convex, with low regularity and non-differentiable objective. By restricting to appropriate eigenspaces, we reduce the large-scale non-convex optimization problem via reparametrization to a sequence of small-scale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. Numerical results are presented for both transverse magnetic (TM) and transverse electric (TE) polarizations at several frequency bands. The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design. |
| first_indexed | 2024-09-23T15:33:19Z |
| format | Article |
| id | mit-1721.1/69624 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:33:19Z |
| publishDate | 2012 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/696242022-10-02T02:53:22Z Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods Men, Han Nguyen, Ngoc Cuong Freund, Robert Michael Parrilo, Pablo A. Peraire, Jaime Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Sloan School of Management Freund, Robert Michael Nguyen, Ngoc Cuong Freund, Robert Michael Parrilo, Pablo A. Peraire, Jaime In this paper, we consider the optimal design of photonic crystal structures for two-dimensional square lattices. The mathematical formulation of the bandgap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue problems for multiple values of the wave vector parameter. The resulting optimization problem is large-scale and non-convex, with low regularity and non-differentiable objective. By restricting to appropriate eigenspaces, we reduce the large-scale non-convex optimization problem via reparametrization to a sequence of small-scale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. Numerical results are presented for both transverse magnetic (TM) and transverse electric (TE) polarizations at several frequency bands. The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design. 2012-03-09T17:12:39Z 2012-03-09T17:12:39Z 2010-01 2010-01 Article http://purl.org/eprint/type/JournalArticle http://hdl.handle.net/1721.1/69624 Men, H. et al. “Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods.” Journal of Computational Physics 229.10 (2010): 3706-3725. https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 https://orcid.org/0000-0003-1132-8477 en_US http://dx.doi.org/10.1016/j.jcp.2010.01.023 Journal of Computational Physics Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Elsevier Prof. Freund via Alex Caracuzzo |
| spellingShingle | Men, Han Nguyen, Ngoc Cuong Freund, Robert Michael Parrilo, Pablo A. Peraire, Jaime Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title | Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title_full | Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title_fullStr | Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title_full_unstemmed | Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title_short | Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods |
| title_sort | bandgap optimization of two dimensional photonic crystals using semidefinite programming and subspace methods |
| url | http://hdl.handle.net/1721.1/69624 https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 https://orcid.org/0000-0003-1132-8477 |
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