Recent Research on Design Optimization of
Wave Propagation in Metamaterials:
Implementation-Robust Design and
Applications
Han Men, Ngoc Cuong Nguyen, Joel Saa-Seoane,
Robert Freund, Jaime Peraire
AFOSR Optimization and Discrete Mathematics
Program Review,
April, 2012
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Photonic Crystal (PC) Band Gap (BG) Optimization
•
Forward problem: given a photonic crystal with dielectric property ε(r), we can
find its eigenmodes λ’s by solving the governing Maxwell’s equations cast in an
eigenvalue problem.
•
Band gap: a range of prohibitive frequencies over all k’s. The gap-midgap ratio for
the mth band gap is defined as:
•
Inverse problem: find an ε(r), such that its band gap is maximized.
?
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SDP Formulation of BG Optimization – P0
Typically,
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Parameter Dependence
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Change of Decision Variables
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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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Subspace Method
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Subspace Method, continued
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Subspace Method, continued
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Linear Fractional SDP :
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Solution Procedure
The SDP optimization problems can be solved efficiently using modern
interior-point methods [Toh et al. (1999)].
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Need for Fabrication/Implementation Robust Design
Consider the optimized PC designs:
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Implementation Robustness Paradigm
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Implementation Robustness Paradigm, continued
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Basic Results
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Basic Results: continued
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Basic Results: Computation
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Computable IR Problems via Special Structure
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Computable IR Problems via Special Structure,
continued
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Computable IR Problems via Special Structure,
continued
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Implementation Robustness and Photonic
Crystal Design
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Computable IR Problems via Special Structure,
continued
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Implementation Robustness Issues when
Modeling Eigenvalues
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SDP Formulation for Bandgap Optimization
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LP Relaxation of the SDP
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LP Relaxation of the SDP, continued
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LP Relaxation of the SDP, continued
linear fractional SDP
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linear fractional program
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BG optimization: from Nominal to FR
Nominal
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FR
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FR Optimization: Solution Algorithm
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Gradient of Parametric LP
T. Gal, J. Nedoma, Multiparametric Linear Programming, 1972.
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Some results - 1
TE 2nd gap, square lattice,
δ = ν = 5%
Original:
Gap = 65.5%
Worst case:
Gap = 21.9%
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Fab robust:
Gap = 37.2%
5% Manual modification:
Gap = 36.1%
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Some results - 2
TE 1st gap, TM 2nd gap, triangular lattice,
δ = ν = 2%
Original:
Gap = 33.1%
Worst case:
Gap = 23.5%
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Fab robust:
Gap = 29.2%
Rand perturb:
Gap = 24.9%
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Some results - 3
TE 1st gap, triangular lattice,
δ = ν = 10%
Original:
Gap = 96.8%
Worst case:
Gap = 44.5%
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Fab robust:
Gap = 80.7%
Rand perturb:
Gap = 49.8%
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Bandwidth optimization of Single-polarization Single-mode
Photonic Crystal Fiber
Applications of photonic crystals based on index guiding
1
2
1. Index-guiding photonic fiber (PCF). [T. A. Birks et al., Opt. Lett. 22, 961, 1997.]
2. Photonic crystal slab. [S. Assefa et al, Appl Phys Lett 85 25, 2004.]
Design problem
Guided Mode
Unguided Mode
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1. Maximize the dimensionless quantity: bandwidth
2. Enforce the following two constraints
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Single-polarization Single-mode Photonic Crystal Fiber:
Optimal structures
Initial structure
UNDAMENTAL
SFECOND
MODE MODE
After cladding optimization
Seconded mode is attenuated
Fundamental guided mode is localized
After core + cladding optimization
Confinement is enhanced.
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Bandgap optimization of 3D photonic crystal
Applications of photonic crystals based on band gaps
1. Hollow-core photonic band gap fiber. [Mangan, et al., OFC 2004 PDP24.]
2. Photonic crystal switch. [Beggs, et al., IEEE Photonics Tech. Lett. 21, 2009.]
3. Photonic crystal splitter. [Communication Photonics Group, ETH.]
Joannopoulos et al, 2007
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Binary optimization of Metamaterials
• Metamaterials are artificially engineered materials that
have a very special property due to its structure.
• We focus on wave phenomena. Applications:
1. Shock Dispersion 4. Superlensing
2. Negative Poisson 5. Water front control
6. Nonlinear earplugs
3. Cloaking
3
1
5
Spadoni et al.
2
4
Light
Fluid Dynamics
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Elasticity
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Acoustics
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Conclusions
• Solved the photonic crystal bandgap optimization with an LP relaxation of SDP
formulation, enhanced by delayed constraint generation
• Developed a framework for Implementation Robustness, and extended it to
the Fabrication Robustness of photonic 2D crystal bandgap optimization.
• The FR optimal structures provide a starting point to modify within a given
allowance and fabricate an easier structure, while knowing the worst case
scenario (or a lower bound of the bandgap).
• We have extended our methodology to PCFs and working on full 3D
• We are looking at new metamaterial design applications
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