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In this work, we optimize the structure of the photonic crystal fibers by using genetic algorithms to provide strong light confinement in fiber and small half diffraction angle of output beam. Furthermore, this article shows the potentials of this study, such as optimizing three purposes at the same time and the arbitrary structure design is achieved. We report two optimized results obtained by different optimization conditions. The results show that the half diffraction angle of the output beam of the photonic crystal fibers can be reduced.
© 2014 Optical Society of America
1. Introduction
Optical fibers are the active region of the fiber laser system. In the case of conventional index guiding optical fibers, the fundamental mode in the optical fibers is a Gaussian distribution. The far-field pattern and the half diffraction angle can be estimated by the properties of the Gaussian beam [1]. Recently, nano-structured optical fibers such as photonic crystal fibers (PCFs) have been intensively applied in the application of fiber lasers [2]. The structure of the PCFs consists of periodically arranged air holes and the air-hole region can regarded as the cladding region of optical fibers. The profile of the transverse mode inside the PCFs is different from the Gaussian distribution, especially in the air-hole region. This results in the fact that the far-field pattern and the half diffraction angle of the PCFs is different from the conventional optical fibers. To obtain the output beam with less diffraction, it has been studied that the Bessel distribution can be the analytical solution of the Helmholtz equation in the cylindrical coordinate system [3, 4]. The hollow fibers have also been used to generate the beam with narrower half diffraction angle [5]. Up to now, for the optical mode of PCFs, the analytical solution of the Helmholtz equation to obtain the beam with narrower half diffraction angle has not been proposed.
Genetic algorithms (GAs) [6] method is an optimization method based on the natural selection and genes. It is can efficiently optimize parameters and in the most article are only optimizing one purpose. In 2004, L. Sanchis et al. consider a hexagonal lattice of Si dielectric cylinders structure [7]. The authors optimized a spot size converter having a very low F number with binary-coded GA method. The value of genes as 1 means presence and 0 is absence of a cylinder at a given position. In 2005, A. Håkansson et al. use GAs method to study the relation between the location and the radius of the cylinder defect [8]. In this article, they want to optimize the transmission efficiency. In 2003, L. Shen et al. took advantage of the GAs method to design a structure with the maximal absolute band gap. However, in order to avoid the obtained structure from impractical design, the structure size is 10 × 10 grids [9]. They also add an initial structure in GA process for reducing simulation time. Thus, according to the result of L. Shen, the added initial structure will influence the final structure. The procedure of the structural design in [9] is more like topology optimization [10, 11].
In this study, we adopt the GAs method to optimize the PCFs and want to find a PCF structure having three properties: 1. The optimized PCF can allow only one mode propagating inside the structure (single-mode operation). 2. The optimized PCF has strong field confinement. 3. The output beam of the optimized PCF has small diffraction angle. In the fiber or waveguide optics, the confinement ratio is an important factor. The stronger confinement ratio can prevent the light leaked from the structure. However, the stronger confinement ratio usually make larger diffraction angle. In Fourier optics, the wider light spot size of output beam (near field) can get a narrower spot size in far field. Thus, confinement ratio and diffraction angle are conflicting and it is difficult to find a good solution for optimizing the two factors. The single mode operation is another important factor. If a fiber allows several modes propagating inside the structure, it is difficult to predict the field pattern of the output beam. The diffraction angle and the confinement ratio will become unstable.
Two optimization initial conditions are used. The initial condition of the first GAs case is the PCF with air-holes arranged in triangular lattice. One air hole at the center of the fiber is missing to form a defect. The light is confined around the central defect region. The optical field confined in PCFs can be obtained through the full-vectorial finite different frequency domain method (FDFD method) [12]. The fundamental mode of the common step-index fiber is distributed like a central lobe of the Bessel function. We calculate the results of the far field profile by fast Fourier transform (FFT) on the output beam and also can calculate the diffraction angle in terms of full width at half maximum (FWHM) at the far field. In the first GAs case, the radii of the air holes are the parameters to be varied to reduce the half-diffraction angle of the output beam by calculating the far-field field distribution. In the second GAs case, the initial condition is that the PCFs have randomly distributed index. The parameter which to be varied is the refractive index of all the grids in the calculation region. This kind of optimization is similar to the topology optimization which has been used for lightweight construction and for optimizing two-dimensional photonic crystal structure [10, 11].
2. Simulations
As shown in Fig. 1(a), the air holes in silica arranged in triangular lattice with a missing air hole are chosen as a control group. The period (Λ) and the radius(r) of the air holes are 2.3μm, 0.5μm, respectively [12]. The background material of the PCF is silica (n = 1.45). In this case, the most of electromagnetic field is confined inside the central defect region [Figs. 1(b) and 1(c)]. Therefore, we set that the core region is the central regular hexagon area as illustrated in Fig. 1(a) to calculate the optical field confinement ratio of the PCF. The Confinement ratio is obtained by calculating the field intensity inside the central regular hexagon divided by the total field intensity. The length of one side of the hexagon is Ʌ. Total size of calculation domain is 16.1 μm × 14.1 μm. The grid size of the FDFD is 0.1 μm both in X and Y directions, respectively. Zero value boundary condition is adopted surrounding the edges of simulation domain. The wavelength of the propagating electromagnetic wave is 1.5 μm. In the simulated result, only the fundamental mode is found and confined in the core of the PCF structure. Figures 1(b) and 1(c) show the Hx and Hy fields of the Y-polarized and X-polarized fundamental mode, respectively. In this simulation, the three parameters: V parameter, diffraction angle and field confinement ratio of PCF are calculated from the optical fields. V parameter which is used to control the PCF in single-mode operation is given by Eq. (1) [13].
Fig. 1 (a) initial PCF structure, 1(b). The fundamental mode of Hx field, 1(c) The fundamental mode of Hy field.
Equation (1) is an approximation formula. Where k is wave vector, na is the refractive index of air, n0 is the refractive index of background material (silica) and F is the air filling fraction. Normalized frequency is also called V parameter or V number. The smaller V parameter can provide fewer modes propagating in the fiber. However, the calculation of the V parameter of the PCF is more complicated. Although the V parameter in our optimized structure may be slightly different to Eq. (1), this formula still can provide us to judge the tendency of the existing mode number. The field confinement ratio is defined to be the ratio of the field in the core region to the total field in calculation region. The small confinement can cause a lot of power losses in optical fiber. The far-field pattern is calculated by FFT of the optical field. The half-diffraction angle is obtained by calculating the FWHM. For the structure described above, the V parameter, field confinement ratio and half-diffraction angle are 4, 64.31% and 27.1°, respectively.
In the first GAs case, only the radius of the air holes is a variable to reduce the half diffraction angle of the output beam. For reducing simulation time and maintaining the symmetry of the optical mode of the PCF, the radii of the air holes in the first quadrant of the calculation region as shown in Fig. 1(a) are duplicated in the other three quadrants. The flowchart of GAs optimization is shown in Fig. 2. The optimized parameters are considered as a chromosome. It is a serial number composed of all variables in the optimization issue. In our first case, the variables are radius of the 13 air holes in the first quadrant. Each radius is varied in the range from 0.2 to 1.1 μm and divided into 16 portions. It can be coded with 4 bits and each bit can be 0 or 1. For example, radius 0.2 μm is “0000”, radius 0.26 μm is “0001” and radius 1.1 μm is “1111”. The chromosome can be composed of the radii. (e.g., the radii [0.2, 0.26, 1.1, 0.2, 0.26, 1.1, 0.2, 0.26, 1.1, 0.2, 0.26, 1.1, 0.2] μm can form a chromosome [0000, 0001, 1111, 0000, 0001, 1111, 0000, 0001, 1111, 0000, 0001, 1111, 0000].) The chromosome in this case has 52 bits and it represents a kind of structural arraignment of the PCFs. In addition, the initial population is a group of chromosomes. In this case, the population is set as 120. There are 120 chromosomes in one generation and it represent 120 kinds of PCFs. Each chromosome can be completed the same or opposite. All chromosomes in the initial generation are generated by random values. The larger populations can find a better solution, however, it also wastes much time during the simulation. Each chromosome can calculate a cost value which can judge the suitability of the chromosomes. The cost value can be formed with our requirements and consisted of several optimization goals. The cost value in here include the V parameter, field confinement ratio and half diffraction angle. It is represented in Eq. (2)
Fv, Fr, Fd represent the value of the V parameter, field confinement ratio and half diffraction angle, respectively. Wv, Wr, Wd represent the weighting factor employed in Fv, Fr, Fd. The value of the weighting factor can be decided by many ways (e.g., trial and error, linear scaling). For instance, before starting the process of GAs, we randomly choose a PCF. The V parameter, field confinement ratio and half diffraction angle are 3.36, 0.64 and 25.55. Then, we hope that the influence of each factor is the same. The weighting can be set as 0.3, 1.56, and 0.04, respectively. The value in each part of cost value is equal to 1, and the result means that the influence of these three factors are the same. After the GAs process, if the optimized results are not good enough, we can modify the weighting factors in the new starting of the GAs process. We can use this way for several times of modifications and can find a suitable solution.
Then, choosing several couples of chromosomes produce offspring by use of exchanging genes, and the choosing rule is roulette wheel. Roulette-wheel selection can be treated as quasi-random selection. However, the better chromosome has a larger chance to be selected. The selected chromosomes exchange parts of genes (bits) and the proportion of the exchanged genes (bits) is called crossover rate. Thus, we set the crossover rate as 80% in our simulation. The higher crossover rate can speed up new the chromosomes of offspring generation, but it will slow down the speed of convergence. Several offspring having largest cost value is excluded directly from the new generation. Furthermore, some bits (gene) are changed directly by mutation and the mutation rate in this work is set as 0.01. Also, the higher mutation rate can speed up new chromosomes into the offspring generation but it will slow down the speed of convergence. And another situation must be considered in GA method. In order to choose few numbers of chromosomes having best performance in the parent generation, we can directly allow these chromosome into offspring generation. This parameter is called on elite number. It can prevent the elite eliminated from the crossover.
The best cost value is reduced with the evolution of the generations in GAs process. Then GAs is stopped when the cost value is small enough. The chromosome having minimum cost value is the best parameters, and it can be treated as the solution of the optimization issue.
The main parameters adopted in GAs process are total variables, total populations, elites and crossover probability, the values are set to be 52, 120, 2, and 80%, respectively. Figure 3(a) is the optimized fiber structure after the first GAs optimization. The optimized radii from r1 to r13 are 0.31μm, 0.65μm, 0.26μm, 0.48μm, 0.76μm, 0.26μm 0.88μm, 0.31μm, 0.37μm, 0.2μm, 0.2μm, 0.2μm and 0.2μm, respectively. In this figure, the radii of the air holes which are close to the core region are enlarged, and others are reduced. Figure 3(b) shows the fundamental modes in Hx field and only one confined mode is found in the fiber. The mode can also be observed to be strongly confined in the core region as Fig. 1(b). Figure 3(c) shows the normalized far-field pattern in X-axis of the initial structure (blue solid line) and that of the first GAs case (black dash line). The width of the central lobe of the far-field pattern is reduced for the first GAs case indicating the reduction of the half diffraction angle. The mean FWHM of the far-field pattern is decreased around 5.7%. After the GAs optimization, the V parameters, field confinement ratio, and half diffraction angle, are 3.36, 64.4%, 25.6°, respectively. Comparing with the initial structure [Fig. 1], the field confinement ratio is maintained to be around 64% and the half diffraction angle is decreased 5.7%.
Fig. 3 (a) the optimized structure after the 1st GAs case, (b) The fundamental mode of Hx field, (c) the comparison of the normalized field intensity between the initial and the 1st GAs cases in X axis
In the second GAs case, the refractive index of each grid is the parameter to be chosen in binary. The value of the refractive index is unity or 1.45. The method is very similar to the topology optimization, but no initial shape of the structure is required. The size of the simulation domain is chosen as 30 grids × 30 grids in the first quadrant, and the grid size in both X and Y axes is set to be 0.23 μm. The number of variables is 900. The total populations, elites and crossover probability for the GAs optimization are 1600, 2, and 80%, respectively.
In order to avoid the optimized structure becoming unrealistic for manufacture. A structural parameter (floating factor) adds to the cost value. In order to understand the amount of floating structure, a method is used for calculation in this simulation. We set a value as 2 to represent silica (material) and 0 to represent air (empty). And one-grid layer is added to the edges of the simulation domain and is set as 1. Then, we choose a 3 × 3 grids domain and sweeping the square both in X and Y direction of the 61 × 61 grids in each scanning round from the first 3 × 3 grids domain to the last. In the beginning, if the center grid of the 3 × 3 square is nonzero (value 1 or 2) and at least one value of the grid is 1, we set all values 2 become 1. After sweeping X and Y axes, it has to sweep again until all values are not changed. Finally, the amount of the value 2 are floating factor.
The results of the second GAs case are shown in Fig. 4. Figure 4(a) is the optimized structure. The white and dark color represents an air hole and silica, respectively. Because a structural factor adds to the cost value, the dark grids are all connected together. In other words, there are no floating structure in our simulated result. Comparing Figs. 4(a) and 4(b), the most field is constrained in the central silica region. Comparing Figs. 4(b) and 3(b), the results of the second GAs case are distributed smoothly and like a circle. It is the same as the results of the first GAs case, only one confined mode is found in the fiber. V parameter, half diffraction angle and field confinement ratio are 4.38, 24.73° and 77.97%, respectively. Figure 4(c) shows the normalized far-field pattern of the initial structure (blue solid line) and the second GAs case (black dash line) in Y axis. The width of the central lobe of the far-field pattern is reduced in the second GAs case indicating the reduction of the half diffraction angle. We can observe that the side lobe intensity is significant reduced in the Y-direction far-field pattern. (The side lobe is also not observed in the X-direction far-field pattern.) The half diffraction angle is decreased 8.75% comparing with the initial case [Fig. 1] and the field confinement ratio is improved to 77.97%. Comparing these two different GAs calculation results: 1. For decreasing the half diffraction angle, the first GAs case reduces the mean FWHM of the central lobe of the far field pattern and the second case decreases the side lobe intensity. 2. The half diffraction angle in the second GAs case is better than the first one, because the chosen structure can more easily modify the distribution of electromagnetic field. However, the larger numbers of variable require more population for finding good solution. This also means large requirement about computation resource. Thus, this kind of optimization cannot optimize extreme fine structure for retrenching computation resource.
Fig. 4 (a) The optimized structure after the 2nd GAs case, (b) the fundamental mode of Hx field, and (c) normalized field intensity between the initial and the 2nd GAs cases in Y axis.
Recently, 3D printing technique has been commercialized to make a three-dimensional solid object of arbitrary shape. In 2012, Karl D.D. Willis et al. have used this technique to manufacture an optical element [14]. However, this equipment is expansive for fabricating a fine structure such as the PCF. In the future, when the manufacturing technique of 3D printing is mature, the fabricating cost can be lower. Then, it is possible to use this technique for PCF with arbitrary structure.
3. Conclusion
In this work, we demonstrate two optimization cases by using the GAs to optimize the PCFs. Furthermore, this article shows the potentials of this study, such as optimizing three purposes at the same time and the arbitrary structure design is achieved. Comparing with the initial structure, the half diffraction angle is decreased 5.7% in the first GAs case and 8.75% in the second GAs case. In the first GAs case, only the radius of the air hole is varied, and in the second GAs case, the arbitrary shape are obtained. In our simulation and analysis, the degree of freedom of the structure design in the second GAs case is larger than the first one. Thus, in this study, the results in second GAs case are better than the results in the first one.
References and links
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