Evolutionary StrateMicro-t

Carlos Henrique da Silva Santos, IEESão Paulo Federal Institute of Education,

Technology (IFSP), campus ItapetiItapetininga-SP, Brasil

carlos.santos@cefetsp.br

Abstract—This work presents the integrationmethod and artificial intelligence approachoptimize non-intuitive micro-to-nano couplersterm is related to the columns arrangementscouplers, which best optimized solution coupleimportant advantage is the possibility to fabricUnicamp facilities.

Photonic Couplers, Micro to Nano, Optimiza

I. INTRODUCTION The two main global challenges in telecom

are the development of smallest and fastedevices. These requirements are popultelecommunications, computers and entertain

The efforts to develop new technologies optical based devices, especially, based on phhave been motivated for the possibility to adimensional requirements. However, a lcomplex Physics principles and fabricatiorequired. In some cases, the fabrication paccording to device design [1].

In order to attempt the requirements presents the integration between a computaapproach, called evolutionary strategy, and electromagnetism using a bi-dimensional finito model a photonic micro-to-nano coupdimensions compared to wavelength and fabrication limitations of the Unicamp faciliti

The optimization methodology, basicadevice modeled in a bi-dimensional space,triangular mesh to be processed in a 2Dencapsulated as the objective function. parameters are randomly generated by algorithm, attempting a pre-defined range of and its output values are adopted to determinanalyzed device is.

This work presents in the second semodeling specifications and, at third, a brabout evolutionary strategy algorithms. Tpresents optimization efficiency and final devices, which respectively results.

egy Algorithm Appliedo-Nano Coupler Devic

EE Member Science and

ininga

Marcos S. Gonçalves, HuSenior MeUniversity

Campinamarcos@ft.unicamp.br,

n of finite element hes to model and s. The non-intuitive s into the photonic ed 73.94%. Another cate this device into

atio

mmunications area est communication larly applied in

nment ways.

and techniques for hotonic principles, attempt speed and large number of on techniques are process is limited

above, this work ational intelligence the computational

ite element method pler with reduced

attempts current ies.

ally, considers a , discretized by a

D FEM, which is The FEM input the evolutionary acceptable values, e how efficient the

ection the device riefly introduction he fourth section device optimized

II. MICRO-TO-NAN

The photonic crystal-based were designed using a powerfElement Method (2D-FEM) proposed in [3]. Physics anawere focused on the possibilitUnicamp facilities, considering

The photonic micro-to-nano(Figure 1) was discretized in elements in computational doinput micro waveguide that external purple and internal reand they are arranged in a elements. This bi-dimensioninternal and external columnrespectively.

Figure 1. Bi-dimensional representanano c

The refractive index of the coupler substrate is n=1.45 anindex is 3.5.

III. EVOLUTIONARY

d to Optimize ces

ugo E. Hernández Figueroa, ember, IEEE

y of Campinas as-SP, Brasil hugo@dmo.fee.unicamp.br

NO PHOTONIC COUPLER couplers presented in this work

ful frequency domain 2D Finite code [2], based on couplers

lysis and device optimizations ty to fabricate these devices in

g their limitations. o coupler presented in this work

a triangular mesh with 36337 omain enclosed by PML. The operates in �=1.55 μm. The

ed cylinders are called columns rectangular matrix of 12x17

nal photonic coupler presents ns with 0.2 μm and 0.3 μm,

ation of the modeled photonic micro-to-coupler.

input waveguide is n=1.98, the nd output waveguide refractive

Y STRATEGY ALGORITHM

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The Evolutionary Strategy (ES) is a computational intelligent approach based on evolutionary algorithm, which is subclass into the biologically inspired algorithms [4]. It is applied in combinatorial optimization, such proposed in this work.

The ES algorithms are represented by � parents that are recombined in a group of � individuals to generate � offspring. These operations occurs along some iterations, also called generations, where in each generation the number of individuals in the population varies from � individuals to (�+ �) individuals, and after the selection procedure the population is reduced to � individuals again.

The selection is basically defined according to the use of parents and offspring groups. In this context, two basic recent ES selections are:

(�/�+�)ES: This selection criterion considers both groups (parents and offspring) to be selected. The non-selected individuals are removed from the population, storing best individuals along the generations.

(�/�,�)ES: This selection criterion consider only offspring group of each generation to select � individuals. For this reason, the amount of individuals in offspring group must be greater or equal than in parent group (� � �). That absent the use of memory to store best individuals.

This work used only first version, the (�/�+�)ES. It was chosen by the static scenery and memory requirements necessary in this optimization.

IV. OPTIMIZATIONS AND FINAL DEVICES RESULTS The aim of this work is to provide most efficient micro-to-

nano coupler, considering an initial constraint of 70% coupling relating input and output signal power. The optimization process considers the total of t=216 elements modeled, internal and external columns. In the case of external column being a different material from the coupler substrate, internal column will be set with similar value.

In this case, considering distinct arrangements in the optimization to provide a non-intuitive micro-to-nano coupler, with simple or double probability combination (1) [5],

( )!

!t

ctP

t c=

− (1)

being t the total of elements and c the element arrangement, in this case, with two elements. The amount of possible configurations is close to 2.158×10407 permutations. Considering sequential 2D FEM runtime execution demands 82.77 seconds, to evaluate each combination was necessary 178.618×10407 seconds (~2.067×10404 days) to be performed.

For this reason, an artificial intelligent approach, called Evolutionary Strategy, was adopted to perform the optimization in practical time and to achieve successful results.

In order to optimize this device, the objective function adopted was based on the input and output power from the fundamental mode. In Fig. 1 it is possible to note two yellow

lines in the left and right waveguides, representing the input and output integration analysis, respectively. If the integrals are analytically evolved, the expression of the power (in this case the output power) is given by:

20

04effE wd

Pβ

ωμ= (2)

Where � is the constant phase, E0 is the amplitude of the x component of the electric field, w is the waveguide depth, � angular frequency, μ0 magnetic permeability in free space and deff is waveguide effective width expressed by (3), where d is waveguide width and �s the substrate electric permittivity.

2 20

2eff

s

d dβ ω μ ε

= +−

(3)

Considering the 2D approximation, the depth of the input and output waveguides are the same. Besides, the amplitude of the x component of the electric field in the input waveguide is fixed in 1 V/m. Thus the final objective function was given by (4). Where �out, �in, deffout and deffout are calculated separately in the input and output waveguides.

20out out

in

out effout

in in eff

E dPP d

ββ

= (4)

During the optimization process only the electric field amplitude (E0) is obtained in the simulation. This measurement is made in a point far enough from the coupler where only the fundamental mode is present. This prevents that the non-guided field interfere in the coupling determination.

The sequential processing optimizations, was performed in a laptop with Intel Centrino 2 Processor, 4 GB of RAM and Windows Vista based operating system. The runtime optimization process of these bi-dimensional devices, demands around 270,000 seconds (more than 3 days).

An initial optimization considers columns with refractive index equal to coupler substrate (n=1.45) or free space refractive index (n=1.0). The Fig. 2 shows the optimization convergence curve during 200 generations. For this optimization was adopted �=30 parents combined in �=2 to generate �=20 offspring, under a mutation fee of 9%.

Figure 2. Micro-to-nano coupler optimization considering columns with

n=1.45 or n=1.0.

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The algorithm has successfully convergeoptimized full columns with n=1.0 acefficiency close to 18% and optimized 53.88the final results do not satisfy the initial coconstraint, which determines values higher case, best solution achieves coupling efficieThe final optimized device structure that detn=1.0 is presented in Figure 3 (a). Fig. 3 (bfield distributions in this first optimized devic

(a)

(b) Figure 3. (a)Optimized micro-to-nano coupler structu

field distribution.

Considering the non-attempt of the initiafirst optimization, a second optimization efusing a different refractive index in the columor n=1.98 (SiN4). In this second optimizparameters used to configure the evolutionarmaintained.

This second optimization achieved goodcoupling efficiency results. The coupling eout values higher than 70%, being 73.94% wdistribution different from the first optimizedin Fig. 4.

Figure 4. Optimized micro-to-nano coupler structureindex equal than substrate (n=1.45) or SiN4

ed, because a non-chieved coupling 8%. Unfortunately, oupling efficiency than 70%. In this ncy close to 55%. tach columns with

b) presents electric ce.

ure and (b) its electric

al constraint in the ffort was initiated mns, being n=1.45 zation process all ry algorithms were

d convergence and efficiency reached with a geometrical d solution, as seen

e and with refractive 4 (n=1.98).

The electric field distributbest coupling than first optimiz(a) and Fig. 5 (b). In thesereflection in the junction betwoutput waveguide. It makes serefractive indexes is considerewith n=1.45 or n=1.98, and twith n=3.5, which is following

(

Figure 5. (a) Three-dimensional andelectric field distribution in

The results achieved in thisto be detached is the high coupminiaturized distance from thwaveguide is another importawhich lower than three waveleother important consideration ris a device with simplified geodifferent fabrication techniquetopic is related to the geometrfabrication restrictions.

Considering the optimizefficiency bandwidth is anothanalyzed in photonic communi

Fig. 6 shows the coupwavelengths of the two optimbandwidth represented in blacefficiency than second optimizAlthough, the first device presin a larger range from �=1.387

The second optimized dcoupling efficiency higher thaμm to �=1.587 μm. It alconsidering an interval to be an

tion presents lower losses and zed structure, as shown in Fig. 5 e figures we can note a high ween the optimized coupler and ense when the high contrast in

ed, being columns and substrate the output waveguide modeled some fabrication specification.

a)

(b)

d (b) bi-dimensional visualization of the n the micro-to-nano coupler.

s second optimization important pling efficiency of 73.94% and a he input waveguide and output ant detail, measuring 4.3 μm,

engths (1.55 μm). There are two related to fabrication. At first, it ometrical arrange, which allows s and technologies. The second rical properties attempting local

zed structures the coupling her important parameter to be cations.

pling efficiency for different mized devices. The first device ck line presents lower coupling zed device presented in red line. sents more homogeneous values μm to �=1.587 μm.

device presents an interesting an 70% in range from �=1.533 lows the fabrication process

nalyzed.

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Figure 6. Coupling efficiency bandwidth for the first (black) and second

(red) optimized coupler.

V. CONCLUSIONS This work presents a computational approach, which

integrates a bi-dimensional Finite Element method and evolutionary strategy algorithm to model and optimize photonic devices, respectively. The coupler obtained is based on optical pixel idea which corresponds to a small column. By changing its index refraction and its diameter it is possible to modify the light propagation.

This computational tool allows the optimization of high efficient miniaturized non-intuitive micro-to-nano coupler devices working on wavelengths close to �=1.55 μm. In addition, the optimization runtime with this computational intelligent approach is practicable for many photonic applications in runtime and optimization convergence. Next step of this work is the fabrication of these efficient photonic coupler devices.

ACKNOWLEDGMENT The authors wish to thanks FAPESP for the CEPID CePOF

and CNPq for the INCT Fotonicom, for the financial support. The authors wish also to thank Paulo Jarschel and Luiz Barea for technical fabrications instructions.

REFERENCES [1] W. Cai, V. Shalaev, “Optical Metamaterials Fundamentals and

Applications”, Springer Science+Business Media, USA,(2010). [2] M. S. Gonçalves, H. E. Hernández-Figueroa, A. C. Bordonalli, “Time

Domain Full-Band Method Using Orthogonal Edge Basis Functions”, IEEE Photonics Technology Letters, v. 18, pp. 52-54, 2006.

[3] C. H. Silva-Santos, M. S. Gonçalves, H. E. Hernández-Figueroa, “Designing Novel Photonic Devices by Bio-Inspired Computing”, IEEE Photonics Technology Letters, v. 22, pp. 1777-1779, 2010.

[4] C. H. Silva-Santos, “Computação Bio-Inspirada e Paralela para a Análise de Estruturas Metamateriais em Microondas e Fotônica”, Ph.D. thesis, School of Electrical and Computer Engineering, University of Campinas, Campinas, Brasil, 2010.

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