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: daniel.erni@uni-due.de

Web: www.ate.uni-due.de

2 Laboratory for Electromagnetic Fields and

Microwave Electronics, ETH Zürich

CH-8092 Zürich, Switzerland

E-Mail: j.froehlich@ifh.ee.ethz.ch

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

-6/27-

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.

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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

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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 !

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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)

-13/27-

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%.

-14/27-

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).

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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).

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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