RESEARCH ARTICLE

Design of LPWG broad band filter with genetic algorithmoptimization

Girish Semwal & Vipul Rastogi

Received: 7 June 2011 /Accepted: 7 January 2014# The Optical Society of India 2014

Abstract The potential of genetic algorithms (GAs) in designand optimization of long period waveguide grating (LPWG)devices has been explored. To demonstrate the applicability ofGA to LPWG, a wide band spectrum of a sinusoidal LPWGhas been chosen as a target spectrum and the parameters of acorresponding corrugated LPWG have been optimized toachieve this spectrum. The proposed approach should beuseful to design LPWG devices for desired applications.

Keywords Genetic Algorithms (GA) . Long PeriodWaveguideGrating (LPWG) . Optimization .

Broad band filter

Introduction

Long period waveguide grating (LPWG) was proposed toenhance the functionality of long period fiber grating [1, 2].The long period fiber grating (LPFG) has limitations due to itsgeometry and material used in the core and cladding of fiberand hence impose the constraints in manipulating the transmis-sion spectrum. Design of LPWG for different applications inoptical communication were carried out by varying the clad-ding profile without changing the period of grating [3, 4].However, the methods are based on trial and error and are notsuitable for the designing LPWG for pre-defined spectrum.

In the present study, we propose the binary genetic algo-rithm (BGA) for synthesis and optimization of corrugated

waveguide grating parameters to design the broad band filter.The grating parameters such as length, period and corrugationheight were optimized to achieve the pre-defined desiredoutput spectrum.

GA for LPWG

GA is a search and optimization algorithm based on naturalselection of biological evolutions. These algorithms are widelyused to optimize the value of design parameters to find theglobal optimum in the pre-defined range of search space ofvariables. The algorithms have been applied in differentbranches of science and engineering for optimization problems.The GA has two variants: i) binary coded and ii) real coded.Here, we have used binary coded GA (BGA), which works onthe similar procedure of selection, crossover and mutation as inbiological reproduction and generation propagation.

In optoelectronics, BGA has been used for optimization ofcoupling coefficients to design the Fiber Bragg Grating basedband pass filter [5]. In BGA, randomly generated parametersare concatenated in the form of strings called chromosomes.These chromosomes are binary strings of zero and one. Allgenetic operations are applied on these chromosomes toachieve the optimum parameters.

BGA is simplest and fundamental type of GA. It is easy toimplement in an optimization problem. Although, it is binarycoded, the fitness functions are always evaluatedwith real valueof parameters instead of binary string and parameters are to beconverted from their real values to binary and vice-versa. Thisconversion decreases the accuracy in computation. Real codedGA have also been proposed for the optimization of parametersof Bragg grating such as length, grating period, index modula-tion and grating chirp to achieve the desired spectrum [6].

LPWG devices are used in optical communication andsensing technologies. The application specific LPWG devicesare the band rejection filter, broad band filter, band pass filter,

Based on presentation at XXXV OSI symposium held atThiruvananthapuram during 16–18 January 2011.

G. Semwal (*)Instrument Research and Development Establishment (IRDE),Raipur Road, Dehradun, Uttarakhand, Indiae-mail: girishsemwal@rediffmail.com

V. RastogiDepartment of Physics, Indian Institute of Technology Roorkee,Roorkee, Uttarakhand, India

J OptDOI 10.1007/s12596-014-0191-z

wavelength division multiplexers and de-multiplexers. The out-put spectrum of these application specific devices is pre-definedand depends on the parameters of waveguide and grating suchas the length, period, index modulation and corrugation height.To achieve the desired output spectrum, optimization of theseparameters is required. However, as the number of parametersincreases in the devices, its control becomes more complex andsuitable optimization method is desired to achieve the goal. Wehave considered here LPWG based broad band wavelengthrejection filter and have implemented BGA to optimize thecorrugated LPG parameters to obtain the target spectrum.

Procedure

We have considered LPG in a 4-layer waveguide structuresuch as the one proposed in Ref. [1] and shown in Fig. 1.However, instead of sinusoidal grating we have consideredcorrugated grating with corrugation height h. We have select-ed broad rejection band filter of Ref. [1] as the target spectrumto demonstrate the applicability of BGA to the LPWG opti-mization problem. The target spectrum St has been generatedby using the waveguide structure and sinusoidal grating pa-rameters and following the procedure as in Ref. [1]. We havethen applied BGA to optimize the parameters of corrugated

LPG to achieve the same target spectrum. The waveguideparameters, however, have been kept the same.

To calculate the coupling coefficient between the modescoupled through the LPWG, we have solved the scalar waveequation for TE modes by using the suitable boundary condi-tions. The solution gives the propagation constants and themodal fields of the modes. If E0 is the modal field of TE0

mode and Em, m>0 is the modal field of coupled TEm modethen the coupling coefficient can be given by

κ ¼ k0Δn2

2π cμ0

Z

d f −h

d f

E0Emdx ð1Þ

Where c is the velocity of light, Δn2 is the index modula-tion, μ0 is the free space permeability, df is the guiding filmthickness, h is corrugation height and k0 is the free space wavenumber defined by 2π/λ0, λ0 being free space wavelength.

The corresponding coupled mode equations are given by

dA0

dz¼ jκAme

jδz ð2Þ

dAm

dz¼ −jκA0e

−jδz ð3Þ

δ ¼ β0−βmð Þ−2πΛ

ð4Þ

Fig. 1 Corrugated grating and refractive index profile of a 4-layer planarwaveguide. a A typical LPWG structure. b Corresponding refractiveindex profile of 4-layers planar waveguide

Fig. 2 The generated targeted spectrum St by using the sinusoidal LPG

Fig. 3 Variation of fitness function with number of generations

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Where δ is detuning parameter, A0(z), Am(z) the Amplitudeof core mode and mth cladding mode; β0, βm propagationconstant for core mode and mth cladding modes and Λ gratingperiod.

The coupled mode equations have been solved byRunge–Kutta method to calculate the transmitted powerat a given wavelength and thereby the transmissionspectrum Sc has been computed for some initial valuesascribed to various corrugated LPG parameters, pitch Λ,corrugation depth h, modulation index Δn, and grating

length L. The fitness function has been evaluated usingfollowing relation.

Fitness ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

St−Scð Þ2q

ð5Þ

The selection of best chromosomes (LPWG parameters)for next generation was performed with Boltzmann probabil-ity distribution [7]. This selection method was found suitable

Fig. 4 Convergence of computedspectrum Sc towards the targetspectrum St, (a–h)

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as compared to the tournament selection or roulette wheelselection method. Single point crossover for generating newchromosomes has been carried out. Judicious selection ofnumber of parameters, selection procedure, and crossoverand mutation rate has been carried out for proper convergenceof GA optimization.

The step-by-step procedure for implementation GA isoutlined as follows:

(i) Generate the random values of parameters to be opti-mized for corrugated grating for the given populationsize.

(ii) Compute the output spectrum Sc for complete popula-tion size.

(iii) Compute the fitness function using Eq. (5) with targetspectrum St for complete population size.

(iv) Arrange the fitness function and corresponding param-eters in ascending order.

(v) Use the selection procedure to select best chromo-somes for next generation to generate newchromosomes.

(vi) Use the crossover and mutation process to generatenew chromosomes from best selected chromosomes.

(vii) Continue the step (ii) to (vi) till the problem convergesto its optimum value.

(viii) The optimum value is the tolerance limit of fitnessfunction to be achieved.

Numerical results

We have used the following values of waveguide parametersto carry out numerical simulation:

ns ¼ 1:5; n f ¼ 1:52; ncl ¼ 1:51; nc ¼ 1; d f ¼ 2 mm;

and dcl ¼ 5:5 mm

ð3Þ

The target spectrum has been generated by using a sinu-soidal grating in the guiding film of the waveguide with Λ=191.6 μm,Δn=6×10−4 and L=16mm. The target spectrum isshown in Fig. 2.

To achieve the similar broad band rejection filterfrom corrugated grating, three parameters of corrugatedgratings; L, Λ and h were optimized by BGA. Thepopulation size for BGA was taken as 50 because ofhighly non-linear nature of problem. The selection ratewas chosen to be 0.5 and mutation was 0.3. The GAwas used for 150 generation after that the computedspectrum became constant showing that parameters were

completely optimized. The fitness function as definedby Eq. (2) has been plotted in Fig. 3 with respect to thenumber of generations (iterations). Fitness function ini-tially decreases rapidly for first few generations andthen the rate of decrease becomes slow and finally itbecomes almost constant. The corresponding computedspectra sampled at some intermediate generations and atthe final generation are shown in Fig. 4. One canclearly see the convergence towards the target spectrum.The GA optimization shows that optimized grating pe-riod of corrugated grating is same as the sinusoidalgrating however; the length is different than the sinu-soidal grating. The length and corrugation height bothcontrol the coupling coefficient of grating and hence thetransmission spectrum characteristics of corrugated grat-ing. The optimized length and corrugation height are18 mm and 96 nm respectively.

Conclusion

We have used the binary coded GA to design thecorrugated long period grating for broad band rejectionfilter for predefined target spectrum. The simulationresults show that the GA is an efficient tools to designand optimize the waveguide grating devices. The meth-od is powerful for designing the LPWG to achieveapplication specific complex spectrum for telecommuni-cation and sensing application.

Acknowledgments This work has been partially supported by UK-India education and research initiative(UKIERI) major award.

References

1. V. Rastogi, K.S. Chiang, Long period grating in planar optical wave-guide. Appl. Opt. 41, 6351 (2002)

2. Q. Liu, K.S. Chiang, V. Rastogi, Analysis of corrugated long periodgrating in slab waveguide and their polarization dependence. J.Lightwave Technol. 21, 3399 (2003)

3. M.S. Kwon, S.Y. Shin, Spectral tailoring of uniform long periodwaveguide grating by the cladding thickness control. Opt. Commun.250, 41–47 (2005)

4. Q. Liu, K.S. Chiang, Design of long period waveguide grating filter bycontrol of waveguide cladding profile. J. Lightwave Technol. 24, 3540(2006)

5. J. Skar, K.M. Risvik, A genetic algorithm for the inverse problem insynthesis of fiber gratings. J. Lightwave Technol. 16, 1928 (1998)

6. G. Cormier, R. Boudreau, Real coded genetic algorithm for bragggrating parameter synthesis. J. Opt. Soc. Am. B 18, 1771 (2001)

7. A. Ray, D.C. Srivastava, Non-linear least squares ellipse fitting usingthe genetic algorithm with applications to strain analysis. J. Struct.Geol. 30, 1593 (2008)

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