1. Introduction

Fibre gratings are one of most important developments in the field of optical fibre technology. Their invention was nominated one of the milestone events in the history of optical communications. They are very simple, low cost, distributed reflectors whose reflection and transmission spectra are wavelength-dependent and can be accurately adjusted by proper design. Gratings are written in fibres making use of the photosensitivity of certain types of materials.

Fibre gratings are classified as either short-period (also called Bragg gratings) or long-period gratings. Long-period gratings have periods that are much higher than the wavelength, ranging from a few hundred μm to few millimetres. Short-period gratings have periods that are comparable to the wavelength, typically around 0.5μm [1, 2, 3].

The synthesis methods for fibre gratings can be classified into two categories: The inverse scattering methods [4]; and the optimisation methods [5]. The inverse scattering methods directly compute the required grating profile from the grating reflection or transmission spectra by solving the mathematical inverse problem while the optimisation method directly minimise the difference between the synthesised and targeted spectra. The optimisation can be performed by several optimisation algorithms such as Simulated Annealing [6], Tabu Search [7], Genetic Algorithm (GA) [5, 8] and recently by Particle Swarm Optimisation [9].

In this paper we propose a novel acceleration technique to compute the objective function in GA algorithm based on the stochastic variation of the wavelength samples. The technique will be applied to the problem of fibre grating sinthesys. The results will show that our technique can improve GA performance. The outline of this paper is as follows: Section 2 the theoretical approach and the synthesis algorithm are presented, Section 3 contains a short description of Genetic Algorithms and the proposed acceleration technique. Section 4 contains numerical results and comparisons, and finally the main conclusions are drawn in Section 5.

2. Theory

In our work the well-known transfer matrix method is applied to solve the couple mode equations and to obtain the spectral response of the fibre grating. In this approach, the grating is divided into uniform sections, each section is represented by a 2x2 matrix. By multiplying these matrices, a global matrix that describes the whole grating is obtained. We will describe the approach developed by Yamada et al. [10].

We assume that the refractive index, inside a ith uniform section from a Bragg grating, can be described by

Where n_eff is the effective index of fibre core, Δn_i is the refractive index amplitude modulation, and Λ_i, is the section grating period.

A nonuniform fibre grating of length L is divided into M uniform gratings, i. e. section, as illustratted in Fig. 1. The propagation through each uniform section i is described by a matrix F_i defined such that

The matrix F_i for one section is defined by [1]

here $j=\sqrt{-1}$, R_i and S_i are the slowly varying amplitudes of the fundamental mode travelling in the +z and -z directions, respectively. γ_i, is the coupling coefficient defined as ${\gamma }_i\equiv \sqrt{{\kappa }_i^2-{\hat{\sigma }}_i^2}$, where ${\hat{\sigma }}_i=\pi \left(\frac{2n_{\mathit{eff}}}{\lambda }-\frac{1}{{\Lambda }_i}\right),$ κ_i = $\frac{\pi }{\lambda }$Δn_i, and λ is the wavelength, and Δz_i is the section thickness.

 

Fig. 1. The transfer matrix method applied to obtain the spectral characteristics of a fibre grating.

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Once all the matrices for the individual sections are known, we find the output amplitudes from

The reflection coefficient of the entire grating is defined as ρ = S ₀/R ₀ and the reflectivity as Γ = |ρ|² [1, 2]. The main drawback of this method is that M may not be made arbitrarily large, since the coupled-mode theory approximations are not valid when uniform grating section is only few grating periods long [10]. Thus, it requires Δz >> Λ.

3. Grating design using genetic algorithm

A genetic algorithm is a search procedure based on the mechanics of natural selection and natural genetics. The basic structure of a genetic algorithm consists of: a set of chromosomes, a mutation function, and a crossover operator.

A chromosome in the genetic algorithm context is an encoding representation of a solution. The aim of the genetic algorithm is to produce near-optimal solutions by letting a population of random solutions – the chromosomes, be manipulated by a sequence of unary and binary transformations – mutation and crossover, respectively. The whole process is governed by a biased selection scheme which leads the set of solutions towards high-quality solutions.

The objective of the crossover operator is to implicitly combine good properties of two different chromosomes chosen by the selection operator, and therefore hopefully transfer these properties to their offsprings. Whereas the mutation is to push the search process towards a more wide search area by randomly slightly changing a chromosome encoding so as to increase the chances of finding good solutions.

The process of performing the crossover, the mutation and selecting the survivor elements of the population is then repeated until the stopping criteria are met [11].

In the following section we will show how we have mapped the fibre grating design problem into the genetic algorithm structures.

3.1. Encoding the problem into a chromosome

In general, the design of fibre gratings consists on finding the right values for a set of physical parameters given the desired reflection or transmission spectra. The grating parameters usually are the grating length, L, the number of sections, M, the section length, Δz, the section period, Λ, and the modulation amplitude Δn.

Before starting any operation, it is necessary to define the parameters bounds and their representation on the genetic algorithm. For sake of simplicity, we have decided to use only Δn and Λ from M sections, i. e., 2M optimisation parameters. Figure 2 shows the encoding chromosome and individuals.

 

Fig. 2. The encoded chromosome: a float-point vector of amplitude modulations and section periods.

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3.1.1. The objective function

Genetic algorithm uses the concept of fitness to represent how good is a particular solution. In the fibre grating syntheses problem it is customary to use the inverse of the mean squared error between the computed and the target reflectivity as the Fitness function [12], i.e.,

Where Γ(λ_j), Γ_target (λ_j) are the computed and the target reflectivity, respectively, and δΓ_j is the tolerance for the j^th wavelength. In the Standard Sampling (SS) technique, the λ_j values are uniformly distributed over S = S_SS samples between Λ_min and λ_max and are given by

with $\mathrm{\Delta \lambda }=\frac{{\mathit{\lambda }}_{\mathit{max}}-{\mathit{\lambda }}_{\mathit{min}}}{S-1}$.

3.2. The proposed acceleration technique

It is necessary to compute the Fitness of each individual. It means that the processing time of the entire population is affected by the number of wavelength samples, S. As S increases, the wavelength sampling presented in Eq. (6) may leads to precise values of the reflectivities. However low values of S indicate large wavelength spacing, leading to errors on the Fitness function. We present in this paper a novel acceleration technique based on the stochastic variation of the wavelength samples.

In our technique, the number of wavelength samples is reduced and the samples are chosen stochastically. Basically, a different set of samples is used to evaluate all individuals of the same generation during the selection process. Thus, after N_G generations, it is possible that N_G different set of points has been tested. After the population calculation the best individual is evaluated using the standard sampling technique, i.e. S_SS points, and it is stored. This individual will have samples enough to obtain a reasonable estimation between the target and the calculated curves. It is used as elitist and its fitness is compared with the fitness of the best individual stored from past generation.

In the Reduced Stochastic Sampling (RSS) technique, the j_th wavelength is defined as:

where ξ is a random variable with uniform distribution in [0,1], $\mathrm{\Delta \lambda }=\frac{{\mathit{\lambda }}_{\mathit{max}}-{\mathit{\lambda }}_{\mathit{min}}}{S_{RSS}-1}$, and S_RSS is the number of samples in the RSS technique.

It is possible to estimate how the RSS technique improves de GA performance. For a given population size N_P, traditional GA computes the Fitness function N_P times using S_SS samples. On the other hand, the RSS technique reduces the samples to S_RSS plus the evaluation of the best individuals with S_SS samples. Therefore, one can define a speedup factor f as

4. Numerical examples

To demonstrate the effectiveness of the RSS technique, the design of two fibre Bragg gratings were investigated. All computation performed in this paper have used a AMD Athlon^™ 64 3200+ computer with 1GB of ROM.

The first design consists of a 1 cm long grating with a bandwidth (BW) of 0.4 nm, design wavelength at 1550 nm, Γ = 1 inside the BW and Γ = 0 otherwise. The number of sections was fixed to M = 50. The effective refractive index of the fibre core was 1.45. The imposed constrain was the amplitude modulation 0 ≤ Δn ≤ 8 × 10^-4.

Figure 3 shows the Fitness curves of best invidual as a function of the processing time for both the standard sampling (SS) and the reduced stochastic sampling (RSS). As one can see,with the RSS technique it is possible to get higher values to the Fitness with less computational effort. In Fig. 3 we have chosen S_SS = 500, S_RSS = 20, and N_p = 50, resulting in a speedup factor f = 16.67.

The speedup factor described in Eq. (8) was also tested for others S_RSS values. Figure 4 shows f as a function of the ratio S_RSS/S_SS when N_p = 50 for the previous design. The stars represent the results obtained with the RSS technique. The results show a good agreement between theory and simulations. Figure 5 shows the computed reflectivity spectra obtained with both techniques. The side lobes obtained for the design with the RSS techniques are about 37 dB down. Figure 6 shows the profile of the amplitude modulation Δ_n. As one can see, the results are very close each other.

 

Fig. 3. Fitness for standard sampling (SS) and for reduced stochatic sampling (RSS) as a function of the processing time.

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Fig. 4. Speedup fator as a function of the ratio S_RSS/S_SS when N_P = 50. The stars represent the results obtained with the RSS technique.

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The second design consists of a 1cm long Triangular Fibre Bragg Grating (TFBG) commonly used as a readout device in fibre Bragg grating based sensor applications [13]. The goal of the TFBG filter design is to find the optimum refractive index modulation amplitude, Δn, and section period, Λ, for a linear edge reflectivity spectrum and desired bandwidth.

The triangular spectrum mask with 1.0 nm bandwidth and 100% reflectivity at λ = 1550 nm is shown in Fig. 7. The number of sections was fixed to M = 50 and the effective refractive index of the fibre core was 1.45. The imposed constrain was the amplitude modulation 0 ≤ Δn ≤ 8 × 10^-4. Figure 7 also shows the computed reflectivities for both SS and RSS techniques. Again, the resusts are very close to each other.

Figure 8 shows the Fitness curves of best invidual as a function of the processing time for both techniques. The RSS technique was able to perform better than SS, after 13 minutes of calculations, the RSS Fitness was 52% superior.

Figure 9 shows the amplitude modulation and section period profiles for the best individual using the RSS technique. Similar profiles were found using the SS technique.

 

Fig. 5. Reflectivity curves obtained via standard sampling (SS) and reduced stochastic sampling (RSS). The dotted line represents the SS technique.(a) Reflectivity and (b) phase response.

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Fig. 6. Profile of the perturbation to the effective refractive index. (a) SS and (b) RSS techniques.

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Fig. 7. Target and computed reflectivities using both SS and RSS techniques of a TFBG.

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Fig. 8. Fitness for standard sampling (SS) and for reduced stochatic sampling (RSS) as a function of the processing time.

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Fig. 9. (a) amplitude modulation and (b) section period profiles of the best individual using the RSS technique.

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5. Conclusions

In this paper we have presented a new acceleration technique for the design of fibre gratings using genetic algorithm. Ou technique is based on the stochastic variation of the wavelength samples and it is able to significantly reduce the computation effort. The numerical examples presented show that the RSS technique can be used with a minimum of wavelength samples.

We also have tested both techniques for the design of different grating profiles. In all tests RSS have shown the same performance. In addition, the proposed method is general, and would thus be useful for other inverse problems such as multilayer filters, thin film devices and photonic crystals based devices.