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High Performance Computing The Physics… F Solve Maxwell Equations with periodic boundary conditions F Numerical method based on Finite Element Method
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Finite Element Modelling of Photonic Crystals
Ben HiettJ Generowicz, M Molinari, D Beckett,
KS Thomas, GJ Parker and SJ Cox
HighPerformance Computing
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Photonic Crystals Photonic Crystals: the presence
of ‘photonic band gaps’ Huge potential in a range of
applications. Highly efficient narrow-band
lasers, integrated optical circuits, high-speed optical communication
networks. Hence a need for accurate and
efficient modelling.
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The Physics… Solve Maxwell Equations with periodic
boundary conditions
Numerical method based on Finite Element Method
0)(
)()()(
1 2
rH
rHrHr c
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Advantages of FEM
Simple and intuitive representation of the photonic crystal structure
Accurate expression of the sharp discontinuities in dielectric constant
0
)sin()(n
nn nxAxf
12
1
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Advantages of FEM Adaptive mesh refinement
allows improvement of solution in specific areas of high relative error
The global matrices comprising the eigenvalue problem are SPARSE. Hence the method scales (almost) linearly in terms of computation and memory requirements
Visualisation of a Global Matrix (non-zero elements
highlighted)
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Domain DiscretisationDomain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
Unit cell Periodically tiled unit cells
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Interpolation Function
Linear, quadratic, cubic… Trade off between
computation/memory cost and solution accuracy
Domain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
x
y
y
1
2
3 1y 2y
3y
x
y
1
2
3
3y
1y 2y
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Elemental EquationsDomain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
0)(
)()()(
1 2
rH
rHrHr c
BkA λ)(
dxiiA ljjl )()( kk
dxxB ljjl )(
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Matrix AssemblyDomain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
1
23
Global matrix assembly via local to global node mapping of elemental matrices
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Solution of the Eigensystem
Computationally Expensive (~95%) Needs to be efficient Sub-space iterative technique Only compute eigenvalues of
interest (lowest) Exploit similarity of adjacent
solutions Search a larger sub-space to
improve convergence
Domain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
BkA λ)(
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VisualisationDomain Discretisation
Selection of Interpolation Function
Derivation of the Elemental Equations
Matrix Assembly
Solution of the Eigensystem
Visualisation
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Convergence Does the method work
in practice? Converges to analytic
solution for free space Verified against other
structures in the literature
Compared with experimentMesh Granularity (no. of elements)
Erro
r
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Triangular Lattice Spectra
Filling fraction = 80%
Rod dielectric constant = 12.25
Unit Cell for
calculation
Periodic repeat of unit cell
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Computational Efficiency Can get more accurate solution by
Using a finer mesh Using a higher order interpolation function
A compromise is necessary
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Which is Best ?
First order is cheap but inaccurate
Higher order gives better accuracy for same compute time
Increa
sed Accu
racy
Reduc
ed Tim
e
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Interesting geometries Exhibit large photonic
band gaps Substrate material can
have low dielectric constant
Practical to manufacture 12-fold symmetric
quasicrystals…
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Quasicrystal Spectra
Filling fraction = 28% Rod dielectric constant = 12.25
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Quasicrystal Spectra (2)Filling fraction = 28%
Rod dielectric constant = 4.0804
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Quasicrystal with Defect
Filling fraction = 28% Rod dielectric constant = 12.25
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Conclusion Finite Elements have advantages
simple and intuitive crystal representation Dielectric discontinuities are modelled accurately Resultant eigenvalue problem is SPARSE
Can use FEM modelling to tune photonic crystal properties
Also have fully 3D extension
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Further Information
E-mail: [email protected]
Web: www.photonics.n3.net
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