The document discusses photonic crystals and their simulation. It notes that photonic crystals show promise for integrated optics but design and fabrication are challenging. Efficient simulation tools are needed to model very low loss integrated photonic circuits. The document describes using an Alternating Direction Implicit Finite Difference Time Domain method to relax the stability limit for simulating photonic crystals with billions of grid points on high performance computing architectures. It provides examples of benchmarking the parallel simulation code.
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http://www.iqol.uwaterloo.ca
http://oxfordplasma.de
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Si
• Crystal – periodic arrangement of
atoms/molecules
• Propagating electrons “see” periodic potential
due to atoms/molecules.
• Conduction properties dictated by crystal
geometry.
• Crystal lattice introduces energy bandgap (Eg)
freq [c/a]
• Optical analogy is photonic crystal
• Periodic potential due to lattice of dielectric
material.
• Propagation of photons controlled by dielectric
contrast and (r/a) ratio.
Band Gap
• Can engineer a photonic bandgap
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• Photonic crystals/PBM shown great deal of promise for
true integrated optics.
• Waveguides with small bends possible making compact
integrated photonic circuits (IPCs) achievable.
• Design/fabrication is challenging
• Efficient simulation tools needed to realize very low
loss IPCs
rods
http://www/photonics.tfp.uni-karlsruhe.de/research.html
http://pages.ief.u-psud.fr
bend splitter resonator cavity
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y
x
2D slice εr = 12.0
z
• Specific PC geometry requires many grid
cells (~107 cells) to resolve even a limited
number of periods.
• Memory intensive computations.
• Modeling of 3D PCB structures dictates
vital need for parallel HPC architectures
with optimized domain decomposition.
εr = 1.0 a
Simple example:
370 x 520 x 50 (~107 grid points)
4.5Gb RAM needed
(Target SPAWAR 3D structure)
Realistic grids > 109 points
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In isotropic medium, Maxwell’s curl
equations are:
∂H
∇ × E = −µ
∂t
∂E
∇× H = ε +J
∂t
K.S. Yee, IEEE Trans. Antennas
Propagat., 14(302) 1966
∂H x ∂Ez ∂E y ∂Ex ∂H z ∂H y
−µ = − ε = − − Jx,
∂t ∂y ∂z ∂t ∂y ∂z
Direct explicit solution of Maxwell’s equations
∂H y ∂Ex ∂Ez ∂E y ∂H x ∂H z (i.e. no matrix inversion required).
−µ = − ε = − − Jy,
∂t ∂z ∂x ∂t ∂z ∂x 2nd order accurate.
∂H z ∂Ex ∂E y ∂Ez ∂H y ∂H x Complete “full-wave” method without
−µ = − ε = − − Jz.
∂t ∂y ∂x ∂t ∂x ∂y approximation (i.e. no pre-selection of output
modes or solution form necessary.)
Easy to parallelize.
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• Introduces “artificial” anisotropic
electric/magnetic* conductivities
within domain boundaries allowing for
absorption/attenuation waves.
• Employs a numerical “split-field”
approach allowing perfect
(theoretical) transmission into
absorbing layer (regardless of
frequency, polarization, or angle of
incidence).
• Perfect electric conductor (PEC)
surrounds PML ABC
• Technique “simulates” effect of J. P. Bérenger, IEEE Trans. Antennas Propagat., 44(110) 1996.
outward propagation of EM waves
to infinity.
Ex = Exy + Exz
∂Exy ∂ ( H zx + H zy )
∂Ex ∂H z ∂H y H y = H yz + H yx ε + σ y Exy =
ε = − − Jx, H = H + H ∂t ∂y
∂t ∂y ∂z z zx zy
∂Exz ∂ ( H yz + H yx )
ε + σ z Exz = −
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• Stability limit, called the CFL 1
Δt FDTD ≤
2 2 2
criterion limits maximum timestep ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞
for solution of PDEs on a finite υ max ⎜ ⎟ +⎜
⎜ Δy ⎟ + ⎜ Δz ⎟
⎟
⎝ Δx ⎠ ⎝ ⎠ ⎝ ⎠
grid.
R. Courant, et al. , IBM Journal , 215(1967).
• For example, a uniform grid of
1nm in Si (εr=12) results in: Alternate-Direction Implicit Approach
• Timestep split into (2) sub-iterations
12 • E-fields updated implicitly along
Δt FDTD ≤
2 specific directions.
⎛ 1 ⎞
(3 × 108 m ) 3 ⎜ ⎟ • H-fields updated explicitly throughout.
s −9
⎝ 1× 10 m ⎠
y
z
RELAXES
Δt FDTD ≤ 6.7 × 10−17 s ⇒ 0.067 fs CFL LIMIT
Long simulation times ! T. Namiki, IEEE MTT 47(10), 2003
(1999).
F. Zheng, et. al, Microwave Guided
Nanostructures Research Group x Wave Lett., 9(11), 441 (1999).
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z
PML
y
Air
PC Slab
Air
PML 370 x 520 x 50 grid
x
~107 grid points
Si slab (εr =12.0)
(PML completely surrounds simulated structure)
cylinders (εr =6.0)
50 Si cylinders
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10. 370 x 520 x 50 grid
Source plane
DISTRIBUTION STATEMENT A: Cleared for public releases; Gaussian pulse ~107 grid points
tw = 15 ps Si slab (εr =12.0)
distribution is unlimited.
50 cylinders (εr =6.0)
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11. Source plane 370 x 520 x 50 grid
Bipolar pulse ~107 grid points
DISTRIBUTION STATEMENT A: Cleared for public releases;
distribution is unlimited. Si slab (εr =12.0)
44 cylinders (εr =6.0)
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• When cylinders are built in domain, each
grid cell is divided into 9 subcells
• The dielectric contribution of each 1/9 of a
grid cell is computed for those subcells
completely within the cylinder radius.
• Results in smoothing around the stair
-cased edges of cylinders
y
x
z
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Task 0
Setup
Parallel Region
Initial Scatter
Task 1 Task N
BC's BC's BC's
calc H field calc H field calc H field
calc E field calc E field calc E field
Communication – plane exchange
Output & Finish
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• Both 1D and 3D decompositions
have been implemented within the
MPI framework of the simulator
• In order to reduce the computation
steps, redundant calculations at
boundary regions were employed
[Hanawa et al., IEEE Trans. on Mag, 43(4),
1545 (2007)]
Speedup of SPAWAR vs. ASU code
• Initial 1D decomposition resulted in good
scaling for long crystal geometry
Interprocessor boundary
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• Speedup for increasingly larger
domains.
Interprocessor boundary
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• 3D decomposition worked best
for more general geometries and
particularly for large problem
domains
• This is the default decomposition
in the code delivered to DoD user
community.
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Benchmarking of parallel ADI-FDTD code. (reduced simulation times)
Demonstration of 3-layer 3D PCG structures and circuits.
J. S. Rodgers, “Quasi-3D photonic crystals for nanophotonics,”
Proceedings of SPIE, vol. 5732, Quantum Sensing and Nanophotonic
Devices II, Manijeh Razeghi, Gail J. Brown, Editors, March 2005, pp.
Nanostructures Research Group 511-519.
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