massachusetts institute of technology
DESCRIPTION
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. MASSACHUSETTS INSTITUTE OF TECHNOLOGY. 1887 - Rayleigh. 1987 – Yablonovitch & John. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/1.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AFOSR, April 2011
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
![Page 2: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/2.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Wave Propagation in Periodic Media
scattering
For most , beam(s) propagate through crystal without scattering (scattering cancels coherently).
But for some (~ 2a), no light can propagate: a band gap
( from S.G. Johnson )
1887 - Rayleigh 1987 – Yablonovitch & John
![Page 3: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/3.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Photonic Crystals
S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000)
3D Crystals
By introducing “imperfections”one can develop:
• Waveguides• Hyperlens• Resonant cavities• Switches• Splitters• …
Applications
Mangan, et al., OFC 2004 PDP24
Band Gap: Objective
![Page 4: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/4.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The Optimal Design Problem for Photonic Crystals
A photonic crystal with optimized 7th band gap.
![Page 5: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/5.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The Optimal Design Problem for Photonic Crystals
![Page 6: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/6.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The Optimal Design Problem for Photonic Crystals
![Page 7: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/7.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The Optimal Design Problem for Photonic Crystals
?
![Page 8: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/8.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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.
![Page 9: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/9.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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.
![Page 10: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/10.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
First Step: Standard Discretizaton
![Page 11: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/11.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Optimization Formulation: P0
Typically,
![Page 12: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/12.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Parameter Dependence
![Page 13: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/13.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Change of Decision Variables
![Page 14: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/14.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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.
![Page 15: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/15.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Subspace Method
![Page 16: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/16.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Subspace Method, continued
![Page 17: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/17.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Subspace Method, continued
![Page 18: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/18.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Linear Fractional SDP :
![Page 19: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/19.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Solution Procedure
The SDP optimization problems can be solved efficiently using modern interior-point methods [Toh et al. (1999)].
![Page 20: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/20.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Results: Optimal Structures
![Page 21: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/21.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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.
![Page 22: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/22.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Mesh Adaptivity : Refinement Criteria
• Interface elements
• 2:1 rule
![Page 23: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/23.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Mesh Adaptivity : Solution Procedure
![Page 24: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/24.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Results: Optimization Process
![Page 25: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/25.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Results: Computation Time
![Page 26: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/26.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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:
![Page 27: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/27.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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
![Page 28: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/28.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Multiple Band Gap Optimization
This linearizes to:
where
![Page 29: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/29.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Solution Procedure
![Page 30: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/30.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Sample Computational Results:2nd and 5th TM Band Gaps
![Page 31: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/31.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Sample Computational Results:2nd and 5th TM Band Gaps
![Page 32: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/32.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Optimized Structures
• TE polarization,• Triangular lattice, • 1st and 3rd band gaps
• TEM polarizations,• Square lattice, • Complete band gaps,• 9th and 12th band gaps
![Page 33: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/33.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Results: Pareto Frontiers of 1st and 3rd TE Band Gaps
• TE polarization
![Page 34: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/34.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Results: Computational Time
Square lattice
Execution Time (minutes)
![Page 35: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/35.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Future work
• 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.
![Page 36: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/36.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Need for Fabrication Robustness
![Page 37: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/37.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Constructing Fabricable Solutions
![Page 38: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/38.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Constructing Fabricable Solutions, continued
![Page 39: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/39.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Quality of User-Constructed Solution
![Page 40: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/40.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Quality of User-Constructed Solution, continued
![Page 41: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/41.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fabrication Robustness Paradigm
![Page 42: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/42.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fabrication Robustness Paradigm, continued
![Page 43: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/43.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Basic Results
![Page 44: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/44.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fabrication Robustness: Basic Model
![Page 45: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/45.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Computable FR Problems via Special Structure
![Page 46: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/46.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Computable FR Problems via Special Structure, continued
![Page 47: MASSACHUSETTS INSTITUTE OF TECHNOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062422/5681329a550346895d9934fe/html5/thumbnails/47.jpg)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Computable FR Problems via Special Structure, continued