6th Workshop on Numerical Methods for Optical Nanostructures,
ETH Zürich, July 5-7, Zürich Switzeland, 2010
Numerical Structural Optimization in Microoptics and
Nanophotonics Daniel Erni1, Thorsten Liebig1, and Jürg Fröhlich2 1 General and Theoretical Electrical Engineering (ATE),
Faculty of Engineering, University of Duisburg-Essen, and CeNIDE – Center for Nanointegration Duisburg-Essen,
D-47048 Duisburg, Germany
E-Mail: [email protected]
Web: www.ate.uni-due.de
2 Laboratory for Electromagnetic Fields and
Microwave Electronics, ETH Zürich
CH-8092 Zürich, Switzerland
E-Mail: [email protected]
Abstract –The design of advanced functional devices and systems is often based on technical specifications that are either represented as complicated, let alone, contradicting tradeoff relations or situated at the very limit of the physically possible. In either case classical engineering approaches will render inappropriate and have to be reformulated as an inverse problem ready to be solved using numerical optimization. Hence, the question regarding the feasibility of solving such inverse problems with numerical structural optimization is discussed along various examples in the realm of microoptics and nanophotonics.
We focus on population-based design approaches as supported by biological-inspired search heuristics like evolutionary algorithms where a finite population of potential solutions is numerically iterated according to specific genetic reproduction rules, undergoing a kind of artificial evolution. Besides the intended solution, population-based optimization algorithms are apt to deliver structural and temporal information during evolution that can be further exploited in order to provide measures for either refining or accelerating the global search behavior. In an interlude we will further speculate whether physical quantities intrinsic to the device are adequate to be nested into such global search heuristics in order to improve the optimization process.
Besides the success assigned to computer guided engineering schemes, there is a hidden epistemological problem [1] – and thus mostly ignored – regarding the counterintuitive morphology of the best performing outcomes. Here in particular we will address the question whether a formal postprocessing of such findings could provide a measure to reconcile the peculiar outcomes with current engineering expertise.
[1] Jürg Fröhlich and Daniel Erni, "Postprocessing – making technical artifacts more intelligible," accepted contribution for EASST Conference 2010 (EASST 010), ‘Practicing Science and Technology, Performing the Social’, The European Association for the Study of Science and Technology, Sept. 2-4, University of Trento, Track 8: Probing Technoscience, 2010.
1
Numerical Structural Optimization in
Microoptics and Nanophotonics
6th Workshop
«Numerical Methods for Optical Nanostructures», July 5 – 7, ETH Zürich
Daniel Erni, Thorsten Liebig
General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, and
CeNIDE – Center for Nanointegration Duisburg-Essen, University of Duisburg-Essen,
D-47048 Duisburg
Jürg Fröhlich
Laboratory for Electromagnetic Fields and
Microwave Electronics, ETH Zürich, CH-8092 Zürich
-1/27-
What is Numerical Structural Optimization?
«function defines form»
Search for the optimal struc-
ture (shape) according to given specifications.
This is an inverse problem.
Structure: Building block,
device, or system.
The question of the optimal optimizer cannot be answered:
«No free lunch theorem» (Santa Fe Institute).
Note: Biological evolution is
more adaptation than progress (i.e. optimization).
Scopimera (sand bubbler crab), Cyprus.
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2
Why numerical structural optimization?
Example: «Nanoantenna»
Cu-filled CNTs,
nanorobotic pot
welding (ETH Zürich)
3D-FEM
Simulation
X. Cui, D. Erni, L. Dong, and W. Zhang, NANOMETA 2009,
Jan. 5-8, pp. 30, TUE4f.74, Seefeld, Austria, 2009.
Exploiting physical
mechanisms at their limits.
Beating the diffraction limit
while exploiting material
dispersion ⇒ Plasmonics.
Strong local interactions
between field and shape.
Contradictory tradeoffs and
multiple objectives.
Multiple parameters
(e.g. 200).
Epistemology.
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Agenda
Numerical structural optimization:
The spot-size converter as an
introductory example for using
a breeder evolutionary algorithm.
Computer guided design examples:
Dense light bending and photonic
crystal demultiplexer.
«Natural» or intrinsic search strategies.
A remark on epistemology: What can
be deduced from optimal solutions?
Conclusion
What are we going to look at today?
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3
Optical
spot-size converter
in SiO2/SiON
The Problem Setting
Forward Solver
• fast 3D EM simulators
Optimizer
• search heursitics
• structure parametrization
Response
Numerical Structural Optimization I
(1) Vision:
Solving the inverse 3D problem!
(2) Approach:
Global, e.g. biological inspired search heursitics
e.g. (a population-based)
Breeder Evolutionary Algorithm
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Example: «Spot-size converter»
Numerical Structural Optimization II
(1) Parametrization of the structure:
Bijective
mapping
Phenotype (converter structure)
Genotype (chromosome) Fitness
(2) Genetic operators:
Selection
• Select two well performing
genotypes 2 parents
Crossover
• Exchange chromosome
segments amongst 2 parents
Mutation
• Coarse random perturbation
of resulting 2 chromosomes
• Reproduction 2 children
Computer
simulation
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4
Numerical Structural Optimization III
Example: «Spot-size converter»
(3) Forward solver (3D-BPM):
Broadening of the mode profile in order to improve the coupling to a
single mode fiber.
(4) Fitness evaluation:
Mode
overlap
F
z
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-8/27-
Population of n
evaluated individuals Forward solver
Selection
Crossover
Mutation
Better than
worst ? Replace
worst
yes
no
Fitness
Optimization loop
(N iterations)
Search heuristics Generate random population
of n genotypes n structures
Initialization
Numerical Structural Optimization IV
Example: «Spot-size converter»
(5) Breeder Evolutionary Algorithm:
2 parents
2 children
5
Improvement of fitness
A single computed
solution (individual)
Numerical Structural Optimization V
Example: «Spot-size converter»
(6) «Evolution»:
• There are always
bad solutions.
• Around 3 minutes
simulation time per individual.
• Overall waiting time:
27 days !
• This was an
«academic» run.
-9/27-
Example: «Spot-size converter»
BPM simulation Measurement
Realization
M. M. Spühler, D. Erni, et al., J. Lightwave Technol.,
vol. 16, no. 9, pp. 1680-1685, September, 1998.
• 3D-BPM & Breeder Evolutionary Algorithm (EA)
• Improvement: 3.7 dB 1.3 dB (@ 1550 nm)
• The fabricated structure even performed better than the simulated converter !
• Shortest converter at that time (1998).
SiON
SiO2
SiO2
Structural Optimization VI
(7) Optimal solution:
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6
pattern #1:
pattern #2:
Numerical Structural Optimization VII
Analysis of the population
(8) Post-processing via pattern correlation:
«How does the final population look like?»
(B) Number of competing «patterns»:
(A) Final population:
Good candidate for
a state variable ! Initialization phase
Evolution phase
Terminal phase
Termination
-11/27-
Numerical Structural Optimization VIII
Analysis of the converter structures
(9) «Rediscovering» the working principle:
(A) Well performing
converters:
(B) Conclusions:
«continuous» section in-plane mode expansion
«intermittent» section out-of-plane mode expansion
These are epistemological statements !
-12/27-
7
Dense Light Bending I
Photonic wires
Rib waveguide
2D-MMP:
T = 6%
Simulation: X. Cui Fabrication: F. Robin (ETH Zürich)
2D-MMP:
T = 99%
Photonic wire
Strong horizontal
light guiding.
conventional
light guiding.
X. Cui, Ch. Hafner et al., Opt. Expr., 14(10), pp. 4351, 2006.
X. Cui, Ch. Hafner, F. Robin, D. Erni, et al., Proc. SPIE vol.
6617, pp. 66170D-1-11, June 2007.
5 m
5 m
1550nm
1550nm
InGaAsP/InP
T < – 4dB
Via Evolution Strategies (ES)
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Dense Light Bending II
Theory: The achromatic photonic crystal bend
Spectral response:
J. Smajic, Ch. Hafner, and D. Erni, Opt. Express,
vol. 11, no. 12, pp. 1378-1384, June 16, 2003.
Rod-type photonic
crystal structure.
Sensitivity analysis
based optimization.
Flattened spectra.
Switching behavior
for r = –30%.
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8
Dense Light Bending III
Reality: Wrestling around with simple device designs
Optimal design of a PhC waveguide bend
• Set-up of a reliable (lossy) 2D model (FEM)
for the hole-type PhC waveguide bend.
• Parameter optimization of the bending area in 2D.
• Verification along a 3D model (FDTD).
• Fabrication in InP/InGaAsP
• End-fire characterization.
• Transmission: –8dB –3dB , bandwidth doubled.
modeling
end-fire
spectra
P. Strasser, D. Erni, et al., J. Opt. Soc. Am. A.,
vol. 25, no. 1, pp. 67, Jan. 2008.
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-16/27-
Designing Complex Devices I
Example: «4-channel demultiplexer» (2) 2D-Modell
(1) Hierarchical
approach:
• Cavities
• Cavity access
• Sections (~20 h)
• Demultiplexer
wavelength [ m]
Tra
nsm
issio
n [
%]
17 m
(3) MB-PE
K. Rauscher, P. Strasser, D. Erni, F. Robin, unpublished., 2006.
9
-17/27-
Designing Complex Devices II
Extended Optimizer Scheme
Forward Solver
• fast 3D EM field solver
Predictor (structure)
• reduced (2D) models
• behavioral models
Optimizer
• search heuristics
• structure parametrization
Response
Interpolation
• model-based
parameter estimation
On acceleration strategies Intrinsic strategies ?
• Physics finds shape...
«Natural» Search Strategies I
Ab initio synthesis of an optical microcavity
(1) Problem setting:
(top view) injected
light field Representation:
90 90 array of
dielectric material
pixels (white means low refractive index
and black a high index).
Fitness function
Quality factor of the cavity.
Degree of localization
(i.e. maximal intensity per group of pixels
that are arranged within an square region).
Evolutionary Algorithm
is used here for a nearly unconstrained search, i.e.
there is a large number of degrees of freedom.
Prof. Michal Lipson, Cornell University, A. Gondarenko et al., Phys. Rev. Lett.,
96, 143904 (2006).
-18/27-
10
«Natural» Search Strategies II
Ab initio synthesis of an optical microcavity
after 1 iteration
(2) Emergent resonator topology:
after 600 iterations after 700 iterations after 5000 iterations
In the case of a maximally unconstrained search;
what is the quality of such emergent patterns? Could it be valued as a «natural» outcome?
-19/27-
-20/27-
«Natural» Search Strategies III
Optically induced forces
Radiation pressure virtually
deforms the particle
Rigorously: Evaluation of
the Maxwell stress tensor
at the particle boundary.
11
«Natural» Search Strategies IV
The auto-generated Bragg reflector
-21/27-
«Natural» Search Strategies V
Radiation pressure «molds» photonic crystal bend
-22/27-
12
-23/27-
«Natural» Search Strategies VI
High-Q resonator pill
Prerequisite: Forces indicate the directions to
which a system has to be distorted in order to minimize the system energy.
Idea: Invert the directions to
access the energy maximum: «natural» search strategy.
Increase of the quality factor Q by 38% irrespective the
order of the whispering gallery mode.
r = 10
r = 1 m = 1649 nm
T. Liebig, D. Erni, OWTNM 2008,
June 13-14, Eindhoven,
The Netherlands, 2008.
Best converter
topology
Best multi-
section laser
diode
Best planar
microcavity
access
Best photonic
wire bend
Best omnidirectional
monopole antenna
Force-induced
particle shape relaxation
On Epistemology
Making technical artifacts
more intelligible
Jürg Fröhlich and Daniel Erni, EASST Conference 2010 (EASST 010),
The European Association for the Study of Science and Technology,
Sept. 2-4,University of Trento, Track 8: Probing Technoscience, 2010.
Best performing solutions look alien or
at least counterintuitive with respect to traditional engineering expertise.
Best performing solutions are thus barely intelligible.
Best performing solutions have to be
rediscovered (in a scientific way).
Postprocessing:
Population based numerical structural optimization offers formal modes of
knowledge acquisition (cf. pp. 11-12).
-24/27-
13
Conclusions
Numerical structural optimization is the only mean for
designing advanced functional nanophotonics devices at their physical limit.
The involved optimization methodologies are always highly context dependent, requiring correspondingly
experienced designers.
A timely and robust solution of a true 3D inverse problem is still lacking.
Population-based search strategies (e.g. evolutionary algorithms) offer in addition formal modes of knowledge
acquisition regarding the underlying mechanisms.
It‘s fun because one gets always surprised by the optimizer.
-25/27-
Thanks.
Further Informations:
www.ate.uni-due.de
Check the site
on «Publications»
-26/27-
14
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(1) D. Erni, M. M. Spühler, and J. Fröhlich, "A generalized evolutionary optimization
procedure applied to waveguide mode treatment in non-periodic optical
structures," 8th European Conf. on Integrated Optics ECIO'97, April 2-4, Stockholm, Sweden, pp. 218-221, 1997.
(2) D. Erni, M. M. Spühler, and J. Fröhlich, "Evolutionary optimization of non-periodic
coupled-cavity semiconductor laser diodes," Optical and Quantum Electronics
(OQE), Special Issue: The 1997 International Workshop on Optical Waveguide
Theory and Numerical Modelling, vol. 30, no. 5/6, pp. 287-303, May 1998.
(3) M. M. Spühler, D. Erni and J. Fröhlich, "An evolutionary optimization procedure
applied to the synthesis of integrated spot-size converters," Optical and Quantum
Electronics (OQE), Special Issue: The 1997 International Workshop on Optical
Waveguide Theory and Numerical Modelling, vol. 30, no. 5/6, pp. 305-321, May
1998.
(4) M. M. Spühler, B. J. Offrein, G.-L. Bona, R. Germann, I. Massarek and D. Erni, "A
very short planar silica spot-size converter using a non-periodic segmented
waveguide," J. Lightwave Technol., vol. 16, no. 9, pp.1680-1685, Sept. 1998.
(5) M. M. Spühler, D. Erni, "Towards structural optimization of planar integrated
lightwave circuits," Optical and Quantum Electronics (OQE), Special Issue: The 1999 International Workshop on Optical Waveguide Theory and Numerical
Modelling, vol. 32, no. 6/8, pp. 701-718, Aug. 2000.
(6) D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich
and Ch. Hafner, "Applications of evolutionary optimization algorithms in
computational optics," ACES Journal: Special Issue on Genetic Algorithms, vol. 15, no. 2, pp. 43-60, July 2000.
(7) E. Moreno, D. Erni, Ch. Hafner, R. E. Kunz, and R. Vahldieck, "Modeling and
optimization of non-periodic grating couplers," Optical and Quantum Electronics
(OQE), vol. 34, no. 11, pp. 1051-1069, Nov. 2002.
(8) D. Wiesmann, R. Germann, G.-L. Bona, C. David, D. Erni, and H. Jäckel, "Add-drop filters based on apodized surface-corrugated gratings," J. Opt. Soc. Am. B,
vol. 20, no. 3, pp. 417-423, March 2003.
(9) J. Smajic, Ch. Hafner, and D. Erni, "Design and optimization of an achromatic
photonic crystal bend," Opt. Express, vol. 11, no. 12, pp. 1378-1384, June 16,
2003.
(10) J. Smajic, Ch. Hafner, and D. Erni, "Optimization of photonic crystal structures,"
J. Opt. Soc. Am. A, vol. 21, no. 11, pp. 2223-2232. Nov. 2004.
(11) A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bächtold, "In-plane coupling into
circular-grating resonators for all-optical switching," 8th International Conference
on Transparent Optical Networks (ICTON’2006), Special Session on Microresonators and Photonic Molecules, June 18-22, Tu.A1.6, pp. 88-91,
Nottingham, UK, 2006.
(12) X. Cui, Ch. Hafner, F. Robin, D. Erni, K. Tavzarashvili, and R. Vahldieck, "Sharp
trench waveguide bend with photonic crystals: Simulation, fabrication and
characterization," Proc. SPIE vol. 6617, WoP 2007 – World of Photonics Congress, (SPIE Europe Optical Metrology), pp. 66170D-1-11, June 17-21,
Munich, Germany, 2007.
(13) T. Jalali, K. Rauscher, A. Mohammadi, D. Erni, Ch. Hafner, W. Bächtold, and M.
Z. Shoushtari, "Efficient effective permittivity treatment for the two-dimensional
finite difference time-domain simulation of photonic crystals," J. Comput. Theor. Nanosci., vol. 4, no. 3, pp. 644-648, May 2007.
(14) P. Strasser, G. Stark, F. Robin, D. Erni, K. Rauscher, R. Wüest, and H. Jäckel,
"Optimization of a 60° waveguide bend in InP-based 2D planar photonic crystals,"
J. Opt. Soc. Am. A., vol. 25, no. 1, pp. 67-73, Jan. 2008.
(15) T. Liebig, and D. Erni, "Using optically induced forces in numerical structural optimization," XVII Int. Workshop on Optical Waveguide Theory and Numerical
Modeling (OWTNM 2008), June 13-14, pp. 36, PO-14, Eindhoven, The
Netherlands, 2008.
(16) T. Liebig, I. Kemper, and D. Erni, "Iterative strategies for the structural design of
nanophotonic components," 1st Int. Workshop on Theoretical and Computational Nano-Photonics (TaCoNa 2008), Dec. 3-5, pp. 52, Bad Honnef, Germany, 2008.
(17) X. Cui and D. Erni, "Optimization of nanophotonic structures by using genetic
algoritms and evolutionary strategies," 1st Int. Workshop on Theoretical and
Computational Nano-Photonics (TaCoNa 2008), Dec. 3-5, pp. 43, Bad Honnef,
Germany, 2008.
(18) Jürg Fröhlich, Daniel Erni, "Search for the optimum: Engineers challenged by
machines?," Workshop 'Engineering as Technoscience – From Calculation and
Simulation towards Search Heuristics', 16.-17. Juli, Universität Duisburg-Essen,
Gerhard-Mercator-Haus, 2007.
(19) Daniel Erni, Jürg Fröhlich, "Engineering expertise in the context of computer guided design," Workshop 'Engineering as Technoscience – From Calculation
and Simulation towards Search Heuristics', 16.-17. Juli, Universität Duisburg-
Essen, Gerhard-Mercator-Haus, 2007.
Selected Publications