1. R. Freund, H. Men, C. Nguyen, P. Parrilo and J. Peraire
Design of Photonic Crystals with Multiple and Combined Band Gaps, plus Fabrication- Robust Design
R. Freund, H. Men, C. Nguyen, P. Parrilo and J. Peraire
MIT
AFOSR, April 2011
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2. Wave Propagation in Periodic Media
For most l, beam(s) propagate through crystal without scattering (scattering cancels coherently).
But for some l (~ 2a), no light can propagate: a band gap
scattering
( from S.G. Johnson )
Rayleigh
1987 – Yablonovitch & John
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3. S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000)
Photonic Crystals
3D Crystals
Band Gap: Objective
S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000)
Applications
By introducing “imperfections”
one can develop:
Waveguides
Hyperlens
Resonant cavities
Switches
Splitters …
Mangan, et al., OFC 2004 PDP24
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4. The Optimal Design Problem for Photonic Crystals
A photonic crystal with optimized 7th band gap.
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5. The Optimal Design Problem for Photonic Crystals
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6. The Optimal Design Problem for Photonic Crystals
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7. The Optimal Design Problem for Photonic Crystals?
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8. Previous Work
There are some approaches proposed for solving the band gap optimization problem:
Cox and Dobson (2000) first considered the mathematical optimization of the band gap problem and proposed a projected generalized gradient ascent method.
Sigmund and Jensen (2003) combined topology optimization with the method of moving asymptotes (Svanberg (1987)).
Kao, Osher, and Yablonovith (2005) used “the level set” method with a generalized gradient ascent method.
However, these earlier proposals are gradient-based methods and use eigenvalues as explicit functions. They suffer from the low regularity of the problem due to eigenvalue multiplicity.
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9. Our Approach
Replace original eigenvalue formulation by a subspace method, and
Convert the subspace problem to a convex semidefinite program (SDP) via semi-definite inclusion and linearization.
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10. First Step: Standard Discretizaton
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11. Optimization Formulation: P0
Typically,
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12. Parameter Dependence
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13. Change of Decision Variables
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14. Reformulating the problem: P1
We demonstrate the reformulation in the TE case,
This reformulation is exact, but non-convex and large-scale.
We use a subspace method to reduce the problem size.
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15. Subspace Method
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16. Subspace Method, continued
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17. Subspace Method, continued
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18. Linear Fractional SDP :
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19. Solution Procedure
The SDP optimization problems can be solved efficiently using modern interior-point methods [Toh et al. (1999)].
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20. Results: Optimal Structures
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21. Results: Computation Time
All computation performed on a Linux PC with Dual Core AMD Opteron 270, 2.0 GHz. The eigenvalue problem is solved by using the ARPACK software.
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22. Mesh Adaptivity : Refinement Criteria
Interface elements
2:1 rule
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23. Mesh Adaptivity : Solution Procedure
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24. Results: Optimization Process
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25. Results: Computation Time
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26. Multiple Band Gap Optimization
Multiple band gap optimization is a natural extension of the previous single band gap optimization that seeks to maximize the minimum of weighted gap-midgap ratios:
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27. Multiple Band Gap Optimization
Even with other things being the same (e.g., discretization, change of variables, subspace construction), the multiple band gap optimization problem cannot be reformulated as a linear fractional SDP. We start with:
where
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28. Multiple Band Gap Optimization
This linearizes to:
where
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29. Solution Procedure
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30. Sample Computational Results: 2nd and 5th TM Band Gaps
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31. Sample Computational Results: 2nd and 5th TM Band Gaps
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32. Optimized Structures TE polarization, Triangular lattice,
1st and 3rd band gaps
TEM polarizations,
Square lattice,
Complete band gaps,
9th and 12th band gaps
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33. Results: Pareto Frontiers of 1st and 3rd TE Band Gaps
TE polarization
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34. Results: Computational Time
Square lattice
Execution Time (minutes)
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35. Future work Incorporate robust fabrication issues, see next slides….
Design of 3-dimensional photonic crystals,
Incorporate robust fabrication issues, see next slides….
Metamaterial designs for cloaking devices, superlenses, and shockwave mitigation.
Band-gap design optimization for other wave propagation problems, e.g., acoustic, elastic …
Non-periodic structures, e.g., photonic crystal fibers.
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36. Need for Fabrication Robustness
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37. Constructing Fabricable Solutions
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38. Constructing Fabricable Solutions, continued
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39. Quality of User-Constructed Solution
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40. Quality of User-Constructed Solution, continued
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41. Fabrication Robustness Paradigm
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42. Fabrication Robustness Paradigm, continued
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43. Basic Results
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44. Fabrication Robustness: Basic Model
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45. Computable FR Problems via Special Structure
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46. Computable FR Problems via Special Structure, continued
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47. Computable FR Problems via Special Structure, continued
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