- Email: [email protected]
Genetic optimization of magneto-optic Kerr effect in lossy cavity-type magnetophotonic crystals
Journal of Magnetism and Magnetic Materials 323 (2011) 1823–1826
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Genetic optimization of magneto-optic Kerr effect in lossy cavity-type magnetophotonic crystals M. Ghanaatshoar , H. Alisafaee Laser and Plasma Research Institute, Shahid Beheshti University, G.C., Evin 1983963113, Tehran, Iran
a r t i c l e i n f o
abstract
Article history: Received 5 August 2010 Received in revised form 30 January 2011 Available online 17 February 2011
We have demonstrated an optimization approach in order to obtain desired magnetophotonic crystals (MPCs) composed of a lossy magnetic layer (TbFeCo) placed within a multilayer structure. The approach is an amalgamation between a 4 4 transfer matrix method and a genetic algorithm. Our objective is to enhance the magneto-optic Kerr effect of TbFeCo at short visible wavelength of 405 nm. Through the optimization approach, MPC structures are found meeting definite criteria on the amount of reflectivity and Kerr rotation. The resulting structures are fitted more than 99.9% to optimization criteria. Computation of the internal electric field distribution shows energy localization in the vicinity of the magnetic layer, which is responsible for increased light-matter interaction and consequent enhanced magneto-optic Kerr effect. Versatility of our approach is also exhibited by examining and optimizing several MPC structures. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magneto-optic Kerr effect Magnetophotonic crystal Genetic algorithm TbFeCo
1. Introduction Magnetophotonic crystals (MPCs) are the magnetic counterpart of photonic crystals (PCs), which exploit the properties of gyrotropic magnetic media [1,2]. Their structural designs can vary from highly symmetric to arbitrarily arranged, nonsymmetric quasi-periodic layer subsets. In a cavity-type MPC structure with single magnetic layer, the magnetic layer is sandwiched between stacks of two types of dielectric materials with higher and lower dielectric constants. Dielectric stacks form Bragg mirrors, and magnetic layer introduces a defect in the periodicity of the structure. As a result, the whole multilayer system exhibits a photonic band gap (PBG) leading to strong reflection of a certain wavelength range of light, but with the exception of a narrow resonance due to the presence of the defect. The incident light with a wavelength equal to that of resonance can propagate inside such MPC. It would be then localized around the magnetic layer in an optimal structure and hence, the MO (Kerr or Faraday) response of the structure increases remarkably. There are various theoretical and experimental studies aiming enhancement of the MO Kerr and Faraday effects using unique characteristics of MPCs [3–8]. In most of them, attempts have been made to achieve simultaneous large optical and MO responses in MPCs comprising non- or very low-absorptive
Corresponding author. Tel.: + 98 21 29902598; fax: + 98 21 22431775.
E-mail addresses: [email protected], [email protected] (M. Ghanaatshoar). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.02.019
magnetic materials for which conventional quarter-wavelength layer design can be easily applied. Furthermore, the investigations have been done mainly in the long visible or near infrared regions of the electromagnetic spectrum that MO response of materials is relatively large. Nevertheless, at short visible wavelengths which are desirable for important applications such as high density data storage, most magnetic materials have small MO constants and large extinction coefficients. Consequently, the material inevitably offers a low MO response accompanied with a high absorption amount of incident light which leads to a serious decrease of the structure’s reflectance or transmittance. As a result, although incorporating such an absorptive material into a 1-D PC can improve its performance, but a regular design based on quarterwave optical thickness is not adequately efficient and a nonquarter-wave design is required [9]. Here, to optimize MPC structures, we utilize the genetic algorithm (GA) as a versatile design solution that can involve global optimization of both regular and irregular structures. The GA is based on Darwinian theory of evolution, natural selection or survival of the fittest. It is a very powerful optimization tool and has been applied to a variety of problems such as prediction of hysteresis loop in magnetic cores [10], obtaining the configuration of moments with minimum energy in spin ice systems [11], design of the magnetic microactuators [12], modeling magnetostrictive transducers [13], and also design and optimization of PCs and PC fibers [14–17]. However, to our knowledge, its application to MPC optimization is not yet examined. Briefly, in optimizing with GA, the design variables are stored in a vector called chromosome. The GA begins with generating initial
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M. Ghanaatshoar, H. Alisafaee / Journal of Magnetism and Magnetic Materials 323 (2011) 1823–1826
population of such chromosomes (called parent population) and then repeatedly evaluates and evolves them until the algorithm terminates by reaching a generation of solutions that meets a convergence criterion; typically after a given number of generations, or when a particular ‘fit’ solution is found. Finally, the fittest chromosome would be reported as the optimized solution. The MPC, we are to optimize its MO Kerr response, consists of SiO2 and TiO2 as dielectric layers and TbFeCo as a magnetic defect layer, constructing (SiO2/TiO2)f TbFeCo (TiO2/SiO2)r structures on Al substrate (f and r are repetition numbers). TbFeCo is a rareearth transition-metal alloy that is desirable for data storage applications. This material is highly absorptive at short visible wavelengths. It should be noted that the optimization approach introduced in this paper can also be employed to optimize MPCs having lossless dielectric or magnetic materials. However, our purpose is to optimize a more general problem including absorption effects.
2. Approach By use of a universal matrix method [18], we wrote a computational code (MOCODE) for calculation of optical and MO responses of MPC structures and then connected this code to the GA code (GACODE) for data exchange. The optimization procedure is described in what follows. After creating initial population of first generation in GACODE, they are sent to MOCODE for calculation of MPC responses. Every population has chromosomes containing thickness values. Therefore, each chromosome represents a specific MPC structure. The chromosomes are evaluated using an objective function which is defined in MOCODE as F ¼ ð1jyKerr =ytarget jÞ2 þ ð1RKerr =Rtarget Þ2 , where yKerr and RKerr represent the Kerr rotation and reflectivity of each structure, respectively. This objective function can be minimized at target values of Kerr rotation ðyKerr ¼ ytarget Þ and reflectivity (RKerr ¼ Rtarget). After evaluating the objective function for each chromosome in the current generation, if the fitness values were not satisfactory, another population would be generated. This is done by evolving the current generation using combination of three different methods: selection (S), crossover (C) and mutation (M). The selection method chooses parents from the current generation for the next generation based on their fitness to objective function. There exist several schemes in the literature for accomplishing this. We use one of the most common; roulette-wheel selection [19] which simulates a roulette wheel with the area of each segment proportional to its expectation. Then, a random number is used to select one of the sections with a probability equal to its area. Once selection produces a new subpopulation, the crossover method creates another new subpopulation by combining (or mating) chromosomes. It randomly picks chromosome pairs from the population and crosses them over with some predetermined probability to produce child chromosomes. We use a crossover method called intermediate [20], which creates children by a weighted average of the parents. Chromosome pairs that are not chosen for crossover would be copied to the next generation without any change. Finally, a mutation method is applied to the population in order to prevent premature convergence of the GA, by making small random changes in some chromosomes, which provides genetic diversity and enables the GA to search a broader space. We use adaptive-feasible mutation method [21] that randomly generates directions and initial step sizes that are adaptive with respect to the last successful or unsuccessful generation, and then a mutated individual is generated and moved along a randomly chosen mutation direction vector. As a whole, if we denote the current generation as Gn, then the next generation Gn + 1 can be populated as Gn þ 1 ¼ M fC fS Gn gg.
Fig. 1. Schematic of optimization procedure and data flow between GACODE and MOCODE.
Therefore, by application of these methods a new generation would be created inheriting some features of the previous one. The procedure repeats until the fittest chromosome would be found. A schematic view of the optimization steps is illustrated in Fig. 1.
3. Results and discussion At the first stage of the optimization procedure, the thickness values are assumed to be the same for identical layers. Therefore, the chromosomes should have three genes corresponding to three thicknesses: dSiO2 , dTiO2 , and dTbFeCo. In the computations, thickness values were limited to the range of 10–300 nm. In addition, several trial {f, r} sets are considered for investigation of their effects on the performance of the MPC. It has also been assumed that the light is incident on the samples perpendicularly with a linear polarization and a wavelength of 405 nm which is the operating wavelength of blue-violet laser diodes based on GaN. This short visible wavelength is utilized in MO data storage systems for which a Kerr rotation of about 11 and a reflectivity level more than 10% is well suited. Thus, we set ytarget and Rtarget in the objective function to be 11 and 20%, respectively, which are ample values for the mentioned application. In our study, a population of 400 chromosomes is generated randomly at the beginning and evolved during 200 generations. Fig. 2 shows the overall progress of evolution. As can be seen, at the end of a typical process with single-precision floating point variable types, the best fitness value reaches 4.9 10 4. This means that our criteria are met more than 99.9%. However, since we prefer the thicknesses to be integer values in nanometers unit, integerthickness chromosomes would be less fitted (Table 1). In Table 1, we can see 10 different MPC structures found by the GA exhibiting desirable Kerr rotation and sufficient reflection for the described application. For each structure, we have also calculated the Kerr ellipticity. A vanishing ellipticity avoids any additional compensating optical device, thereby reducing the size of detection system. It is worthy to note that the maximum Kerr rotation of a single TbFeCo layer on Al substrate is approximately 0.31 (which occurs for a thickness of 30 nm) and is also accompanied with an ellipticity of 0.11. Thus, regarding our results, we have enhanced the ratio of the Kerr rotation to the ellipticity from 3 (in the single layer case) to 92 (in the fourth structure of Table 1). By removing another design restriction and increasing the design flexibility, the MPC structures can be further developed. In
M. Ghanaatshoar, H. Alisafaee / Journal of Magnetism and Magnetic Materials 323 (2011) 1823–1826
used again for this new extra-variable problem. Details of two typical MPCs resulted from the developed procedure are listed in Table 2. It can be seen that the optical and MO responses of these MPCs are very close to what was expected. In other words, both structures are now fitted more than 99.9% to the objective function. More surprisingly is that despite having entirely different thicknesses (with the exception of magnetic layer thickness which is probably accidental) the structures are exhibiting very similar responses. This suitability is arisen from the definition of the objective function that assigns equal weights to each design
101 Best fitness Mean fitness
Fitness value
100
10−1
10−2
10−3
102
Fig. 2. The population evolution for 200 generations. Table 1 Some optimized MPC structures resulting from the GA. The thickness of identical layers was kept equal. dSiO2 ,
3, 3, 3, 3, 4, 4, 4, 4, 5, 6,
14, 29, 11, 10, 36, 16, 11, 85, 37, 26,
TiO2 , TbFeCo
21, 23, 25, 12, 24, 13, 23, 54, 28, 26,
(nm)
10 22 13 10 15 11 17 32 21 10
yKerr (deg)
RKerr (%)
cKerr (deg)
0.79 0.98 0.91 0.92 1.13 1.09 0.99 1.01 0.82 1.36
21.2 18.2 19.3 21.3 13.0 16.5 17.1 15.1 21.1 11.0
0.44 0.18 0.30 0.01 0.61 0.16 0.39 0.35 0.27 0.49
Table 2 Two completely random {3, 6} PBG structures optimized by the GA. For data representation of final chromosomes, we have presented the layers’ thicknesses according to their locations in the structure. The bold face numbers indicate the thickness of the magnetic layer. f, r
dChromosome (nm)
3, 6
33, 21, 20, 38, 12, 35, 12, 24, 12, 38, 26, 28, 14, 31, 41, 19, 14, 30, 41
3, 6
20, 41, 21, 23, 34, 26, 12, 12, 47, 39, 30, 38, 23, 35, 14, 22, 34, 16, 13
yKerr (deg)
RKerr (%)
cKerr (deg)
0.98
20.1
0.05
L H L H L HMH L H L H L H L H L H L 15 0.5 10 0.25 0 200
5
300
400
0 0
200
400 600 Structure depth (nm)
800
2.5 L H L H L H M H L H L H L H L H L H L 2 1.5 1 0.5 0 0
50
100 150 Structure depth (nm)
200
2.5
0.97
20.0
0.03
this order, we extend the size of chromosomes and allocate a separate gene to each layer; i.e. instead of identical thickness for layers with the same material, we assign different thicknesses to them. For applying this approach, we consider the fourth structure from Table 1 which possesses the least cKerr . Currently, this structure is fitted 98.9% to our criteria. There are 19 genes in each chromosome representing this structure: 18 genes for dielectric layers and one gene for magnetic layer. Since there are more genes in this developed approach, the procedure would take more time to find the optimal structures. However, as the procedure is the same as before and only minimal changes should be made, hence a lot of time can be conserved. The optimization cycle is
Electric field distribution (arb.)
f, r
Electric field distribution (arb.)
101 Generation
Electric field distribution (arb.)
20
10−4 100
3 4 5 6 4 5 6 7 5 6
1825
L H L H L H MH L H
L H L H
L
HL H
L
2 1.5 1 0.5 0 0
100
200 300 Structure depth (nm)
400
Fig. 3. Electric field distribution in the interior of three different {3,6} structures: (a) a quarter-wave structure with 10 nm magnetic layer, (b) the fourth GA-structure of Table 1 and (c) the first structure of Table 2.
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parameter, and consequently the optimization procedure tries to find the best value of them without giving any priority. In addition to the large Kerr effect enhancement, the total length of structures optimized here using GA, comparing to conventional designs based on quarter-wave layers, is reduced remarkably. The total thickness of our optimal structures in the 3-gen and 19-gen cases are 208 and 489 nm, respectively. By comparison with an usual quarter-wave structure, that typically would be thicker than 900 nm, it is obvious that genetic structures are more desirable for practical purposes. To investigate the effectiveness of GA-generated structures, we have further computed the electric field distribution in the interior of them. In this way, three structures with {f,r} ¼{3,6} are considered: (a) a quarter-wave structure with 12 nm magnetic layer, (b) the fourth GA-structure of Table 1 and (c) the first structure of Table 2. The results are plotted in Fig. 3. As can be seen in Fig. 3a, there is no light localization in the vicinity of the magnetic layer for this ordinary quarter-wave design. Contrasting to this, the localization effect is obvious in the other two structures that have been generated using GA. This effect is the main reason for considerable enhancement of the Kerr rotation by increasing interaction of light and matter.
4. Conclusion In this paper, we have demonstrated the application of genetic algorithm for optimization of lossy cavity-type MPCs through fully non-quarter-wave design. To do this, we have introduced a tunable two-value fitness function. It has been confirmed that a natural evolution eventually leads to the most fitted solutions which in our case are the optimum structures exhibiting simultaneous sufficient reflectivity and enhanced Kerr rotation due to effective light localization. Further, comparing to conventional designs, the total length of final structures can be reduced remarkably, which is desirable in practical purposes. It should be noted that, although we have considered the Kerr effect enhancement in absorptive MPCs, non-absorptive structures can also be included in GA optimization procedure to examine the results of non-quarter-wave designs in enhancement of MO effects. References [1] M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P.B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, A. Granovsky, Magnetophotonic crystals, Journal of Physics D: Applied Physics 39 (8) (2006) R151–R161.
[2] M. Levy, A.A. Jalali, X. Huang, Magnetophotonic crystals: nonreciprocity, birefringence and confinement, Journal of Materials Science: Materials in Electronics 20 (Suppl. 1) (2009) S43–S47. [3] M.J. Steel, M. Levy, R.M. Osgood Jr., Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects, Journal of Lightwave Technology 18 (9) (2000) 1289–1296. [4] M.J. Steel, M. Levy, R.M. Osgood Jr., Photonic bandgaps with defects and the enhancement of Faraday rotation, Journal of Lightwave Technology 18 (9) (2000) 1297–1308. [5] H. Kato, M. Inoue, Reflection-mode operation of one-dimensional magnetophotonic crystals for use in film-based magneto-optical isolator devices, Journal of Applied Physics 91 (10 II) (2002) 7017. [6] M. Vasiliev, K.E. Alameh, V.I. Belotelov, V.A. Kotov, A.K. Zvezdin, Magnetic photonic crystals: 1-D optimization and applications for the integrated optics devices, Journal of Lightwave Technology 24 (5) (2006) 2156–2162. [7] S. Erokhin, Yu. Boriskina, A. Vinogradov, M. Inoue, D. Kobayashi, A. Fedyanin, E. Ganshina, M. Kochneva, A. Granovsky, Transverse Kerr effect in onedimensional magnetophotonic crystals: experiment and theory, Journal of Magnetism and Magnetic Materials 300 (2006) e257–e259. [8] A.G. Zhdanov, A.A. Fedyanin, O.A. Aktsipetrov, D. Kobayashi, H. Uchida, M. Inoue, Enhancement of Faraday rotation at photonic-band-gap edge in garnet-based magnetophotonic crystals, Journal of Magnetism and Magnetic Materials 300 (2006) e253–e256. [9] M. Moradi, H. Alisafaee, M. Ghanaatshoar, The Kerr effect enhancement in non-quarter-wave lossy magnetophotonic crystals, Physica B: Condensed Matter 405 (2010) 4488–4491. [10] I. Kucuk, Prediction of hysteresis loop in magnetic cores using neural network and genetic algorithm, Journal of Magnetism and Magnetic Materials 305 (2006) 423–427. [11] A. Leo´n, J. Pozo, Using a genetic algorithm to study properties of minimum energy states and geometrical frustration in artificial ‘spin ice’ systems, Journal of Magnetism and Magnetic Materials 320 (2008) 210–216. [12] C.H. Ko, J.C. Chiou, Optimal design of the magnetic microactuator using the genetic algorithm, Journal of Magnetism and Magnetic Materials 263 (2003) 38–46. [13] L.A.L. Almeida, G.S. Deep, A.M.N. Lima, H. Neff, Modeling a magnetostrictive transducer using genetic algorithm, Journal of Magnetism and Magnetic Materials 266–230 (2001) 1262–1264. [14] W. Paszkowicz, Genetic algorithms, a nature-inspired tool: survey of applications in materials science and related fields, Materials and Manufacturing Processes 24 (2) (2009) 174–197. [15] J. Goh, I. Fushman, D. Englund, J. Vukovi, Genetic optimization of photonic bandgap structures, Optics Express 15 (13) (2007) 8218–8230. [16] L. Shen, Z. Ye, S. He, Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm, Physical Review B—Condensed Matter and Materials Physics 68 (3) (2003) 351091–351095. [17] E. Kerrinckx, L. Bigot, M. Douay, Y. Quiquempois, Photonic crystal fiber design by means of a genetic algorithm, Optics Express 12 (9) (2004) 1990–1995. [18] Z.Q. Qiu, S.D. Bader, Surface magneto-optic Kerr effect, Review of Scientific Instruments 71 (3) (2000) 1243–1255. [19] T. Baeck, Evolutionary Algorithms in Theory and Practice, Oxford University Press, Oxford, 1996. [20] T.D. Gwiazda, Genetic Algorithms Reference. Volume 1. Crossover for SingleObjective Numerical Optimization Problems, Tomasz Gwiazda, 2006. [21] R. Kumar, System and method for the use of an adaptive mutation operator in genetic algorithms, U.S. Patent # 7,660,773 issued 2/9/2010.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Genetic optimization of magneto-optic Kerr effect in lossy cavity-type magnetophotonic crystals M. Ghanaatshoar , H. Alisafaee Laser and Plasma Research Institute, Shahid Beheshti University, G.C., Evin 1983963113, Tehran, Iran
a r t i c l e i n f o
abstract
Article history: Received 5 August 2010 Received in revised form 30 January 2011 Available online 17 February 2011
We have demonstrated an optimization approach in order to obtain desired magnetophotonic crystals (MPCs) composed of a lossy magnetic layer (TbFeCo) placed within a multilayer structure. The approach is an amalgamation between a 4 4 transfer matrix method and a genetic algorithm. Our objective is to enhance the magneto-optic Kerr effect of TbFeCo at short visible wavelength of 405 nm. Through the optimization approach, MPC structures are found meeting definite criteria on the amount of reflectivity and Kerr rotation. The resulting structures are fitted more than 99.9% to optimization criteria. Computation of the internal electric field distribution shows energy localization in the vicinity of the magnetic layer, which is responsible for increased light-matter interaction and consequent enhanced magneto-optic Kerr effect. Versatility of our approach is also exhibited by examining and optimizing several MPC structures. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magneto-optic Kerr effect Magnetophotonic crystal Genetic algorithm TbFeCo
1. Introduction Magnetophotonic crystals (MPCs) are the magnetic counterpart of photonic crystals (PCs), which exploit the properties of gyrotropic magnetic media [1,2]. Their structural designs can vary from highly symmetric to arbitrarily arranged, nonsymmetric quasi-periodic layer subsets. In a cavity-type MPC structure with single magnetic layer, the magnetic layer is sandwiched between stacks of two types of dielectric materials with higher and lower dielectric constants. Dielectric stacks form Bragg mirrors, and magnetic layer introduces a defect in the periodicity of the structure. As a result, the whole multilayer system exhibits a photonic band gap (PBG) leading to strong reflection of a certain wavelength range of light, but with the exception of a narrow resonance due to the presence of the defect. The incident light with a wavelength equal to that of resonance can propagate inside such MPC. It would be then localized around the magnetic layer in an optimal structure and hence, the MO (Kerr or Faraday) response of the structure increases remarkably. There are various theoretical and experimental studies aiming enhancement of the MO Kerr and Faraday effects using unique characteristics of MPCs [3–8]. In most of them, attempts have been made to achieve simultaneous large optical and MO responses in MPCs comprising non- or very low-absorptive
Corresponding author. Tel.: + 98 21 29902598; fax: + 98 21 22431775.
E-mail addresses: [email protected], [email protected] (M. Ghanaatshoar). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.02.019
magnetic materials for which conventional quarter-wavelength layer design can be easily applied. Furthermore, the investigations have been done mainly in the long visible or near infrared regions of the electromagnetic spectrum that MO response of materials is relatively large. Nevertheless, at short visible wavelengths which are desirable for important applications such as high density data storage, most magnetic materials have small MO constants and large extinction coefficients. Consequently, the material inevitably offers a low MO response accompanied with a high absorption amount of incident light which leads to a serious decrease of the structure’s reflectance or transmittance. As a result, although incorporating such an absorptive material into a 1-D PC can improve its performance, but a regular design based on quarterwave optical thickness is not adequately efficient and a nonquarter-wave design is required [9]. Here, to optimize MPC structures, we utilize the genetic algorithm (GA) as a versatile design solution that can involve global optimization of both regular and irregular structures. The GA is based on Darwinian theory of evolution, natural selection or survival of the fittest. It is a very powerful optimization tool and has been applied to a variety of problems such as prediction of hysteresis loop in magnetic cores [10], obtaining the configuration of moments with minimum energy in spin ice systems [11], design of the magnetic microactuators [12], modeling magnetostrictive transducers [13], and also design and optimization of PCs and PC fibers [14–17]. However, to our knowledge, its application to MPC optimization is not yet examined. Briefly, in optimizing with GA, the design variables are stored in a vector called chromosome. The GA begins with generating initial
1824
M. Ghanaatshoar, H. Alisafaee / Journal of Magnetism and Magnetic Materials 323 (2011) 1823–1826
population of such chromosomes (called parent population) and then repeatedly evaluates and evolves them until the algorithm terminates by reaching a generation of solutions that meets a convergence criterion; typically after a given number of generations, or when a particular ‘fit’ solution is found. Finally, the fittest chromosome would be reported as the optimized solution. The MPC, we are to optimize its MO Kerr response, consists of SiO2 and TiO2 as dielectric layers and TbFeCo as a magnetic defect layer, constructing (SiO2/TiO2)f TbFeCo (TiO2/SiO2)r structures on Al substrate (f and r are repetition numbers). TbFeCo is a rareearth transition-metal alloy that is desirable for data storage applications. This material is highly absorptive at short visible wavelengths. It should be noted that the optimization approach introduced in this paper can also be employed to optimize MPCs having lossless dielectric or magnetic materials. However, our purpose is to optimize a more general problem including absorption effects.
2. Approach By use of a universal matrix method [18], we wrote a computational code (MOCODE) for calculation of optical and MO responses of MPC structures and then connected this code to the GA code (GACODE) for data exchange. The optimization procedure is described in what follows. After creating initial population of first generation in GACODE, they are sent to MOCODE for calculation of MPC responses. Every population has chromosomes containing thickness values. Therefore, each chromosome represents a specific MPC structure. The chromosomes are evaluated using an objective function which is defined in MOCODE as F ¼ ð1jyKerr =ytarget jÞ2 þ ð1RKerr =Rtarget Þ2 , where yKerr and RKerr represent the Kerr rotation and reflectivity of each structure, respectively. This objective function can be minimized at target values of Kerr rotation ðyKerr ¼ ytarget Þ and reflectivity (RKerr ¼ Rtarget). After evaluating the objective function for each chromosome in the current generation, if the fitness values were not satisfactory, another population would be generated. This is done by evolving the current generation using combination of three different methods: selection (S), crossover (C) and mutation (M). The selection method chooses parents from the current generation for the next generation based on their fitness to objective function. There exist several schemes in the literature for accomplishing this. We use one of the most common; roulette-wheel selection [19] which simulates a roulette wheel with the area of each segment proportional to its expectation. Then, a random number is used to select one of the sections with a probability equal to its area. Once selection produces a new subpopulation, the crossover method creates another new subpopulation by combining (or mating) chromosomes. It randomly picks chromosome pairs from the population and crosses them over with some predetermined probability to produce child chromosomes. We use a crossover method called intermediate [20], which creates children by a weighted average of the parents. Chromosome pairs that are not chosen for crossover would be copied to the next generation without any change. Finally, a mutation method is applied to the population in order to prevent premature convergence of the GA, by making small random changes in some chromosomes, which provides genetic diversity and enables the GA to search a broader space. We use adaptive-feasible mutation method [21] that randomly generates directions and initial step sizes that are adaptive with respect to the last successful or unsuccessful generation, and then a mutated individual is generated and moved along a randomly chosen mutation direction vector. As a whole, if we denote the current generation as Gn, then the next generation Gn + 1 can be populated as Gn þ 1 ¼ M fC fS Gn gg.
Fig. 1. Schematic of optimization procedure and data flow between GACODE and MOCODE.
Therefore, by application of these methods a new generation would be created inheriting some features of the previous one. The procedure repeats until the fittest chromosome would be found. A schematic view of the optimization steps is illustrated in Fig. 1.
3. Results and discussion At the first stage of the optimization procedure, the thickness values are assumed to be the same for identical layers. Therefore, the chromosomes should have three genes corresponding to three thicknesses: dSiO2 , dTiO2 , and dTbFeCo. In the computations, thickness values were limited to the range of 10–300 nm. In addition, several trial {f, r} sets are considered for investigation of their effects on the performance of the MPC. It has also been assumed that the light is incident on the samples perpendicularly with a linear polarization and a wavelength of 405 nm which is the operating wavelength of blue-violet laser diodes based on GaN. This short visible wavelength is utilized in MO data storage systems for which a Kerr rotation of about 11 and a reflectivity level more than 10% is well suited. Thus, we set ytarget and Rtarget in the objective function to be 11 and 20%, respectively, which are ample values for the mentioned application. In our study, a population of 400 chromosomes is generated randomly at the beginning and evolved during 200 generations. Fig. 2 shows the overall progress of evolution. As can be seen, at the end of a typical process with single-precision floating point variable types, the best fitness value reaches 4.9 10 4. This means that our criteria are met more than 99.9%. However, since we prefer the thicknesses to be integer values in nanometers unit, integerthickness chromosomes would be less fitted (Table 1). In Table 1, we can see 10 different MPC structures found by the GA exhibiting desirable Kerr rotation and sufficient reflection for the described application. For each structure, we have also calculated the Kerr ellipticity. A vanishing ellipticity avoids any additional compensating optical device, thereby reducing the size of detection system. It is worthy to note that the maximum Kerr rotation of a single TbFeCo layer on Al substrate is approximately 0.31 (which occurs for a thickness of 30 nm) and is also accompanied with an ellipticity of 0.11. Thus, regarding our results, we have enhanced the ratio of the Kerr rotation to the ellipticity from 3 (in the single layer case) to 92 (in the fourth structure of Table 1). By removing another design restriction and increasing the design flexibility, the MPC structures can be further developed. In
M. Ghanaatshoar, H. Alisafaee / Journal of Magnetism and Magnetic Materials 323 (2011) 1823–1826
used again for this new extra-variable problem. Details of two typical MPCs resulted from the developed procedure are listed in Table 2. It can be seen that the optical and MO responses of these MPCs are very close to what was expected. In other words, both structures are now fitted more than 99.9% to the objective function. More surprisingly is that despite having entirely different thicknesses (with the exception of magnetic layer thickness which is probably accidental) the structures are exhibiting very similar responses. This suitability is arisen from the definition of the objective function that assigns equal weights to each design
101 Best fitness Mean fitness
Fitness value
100
10−1
10−2
10−3
102
Fig. 2. The population evolution for 200 generations. Table 1 Some optimized MPC structures resulting from the GA. The thickness of identical layers was kept equal. dSiO2 ,
3, 3, 3, 3, 4, 4, 4, 4, 5, 6,
14, 29, 11, 10, 36, 16, 11, 85, 37, 26,
TiO2 , TbFeCo
21, 23, 25, 12, 24, 13, 23, 54, 28, 26,
(nm)
10 22 13 10 15 11 17 32 21 10
yKerr (deg)
RKerr (%)
cKerr (deg)
0.79 0.98 0.91 0.92 1.13 1.09 0.99 1.01 0.82 1.36
21.2 18.2 19.3 21.3 13.0 16.5 17.1 15.1 21.1 11.0
0.44 0.18 0.30 0.01 0.61 0.16 0.39 0.35 0.27 0.49
Table 2 Two completely random {3, 6} PBG structures optimized by the GA. For data representation of final chromosomes, we have presented the layers’ thicknesses according to their locations in the structure. The bold face numbers indicate the thickness of the magnetic layer. f, r
dChromosome (nm)
3, 6
33, 21, 20, 38, 12, 35, 12, 24, 12, 38, 26, 28, 14, 31, 41, 19, 14, 30, 41
3, 6
20, 41, 21, 23, 34, 26, 12, 12, 47, 39, 30, 38, 23, 35, 14, 22, 34, 16, 13
yKerr (deg)
RKerr (%)
cKerr (deg)
0.98
20.1
0.05
L H L H L HMH L H L H L H L H L H L 15 0.5 10 0.25 0 200
5
300
400
0 0
200
400 600 Structure depth (nm)
800
2.5 L H L H L H M H L H L H L H L H L H L 2 1.5 1 0.5 0 0
50
100 150 Structure depth (nm)
200
2.5
0.97
20.0
0.03
this order, we extend the size of chromosomes and allocate a separate gene to each layer; i.e. instead of identical thickness for layers with the same material, we assign different thicknesses to them. For applying this approach, we consider the fourth structure from Table 1 which possesses the least cKerr . Currently, this structure is fitted 98.9% to our criteria. There are 19 genes in each chromosome representing this structure: 18 genes for dielectric layers and one gene for magnetic layer. Since there are more genes in this developed approach, the procedure would take more time to find the optimal structures. However, as the procedure is the same as before and only minimal changes should be made, hence a lot of time can be conserved. The optimization cycle is
Electric field distribution (arb.)
f, r
Electric field distribution (arb.)
101 Generation
Electric field distribution (arb.)
20
10−4 100
3 4 5 6 4 5 6 7 5 6
1825
L H L H L H MH L H
L H L H
L
HL H
L
2 1.5 1 0.5 0 0
100
200 300 Structure depth (nm)
400
Fig. 3. Electric field distribution in the interior of three different {3,6} structures: (a) a quarter-wave structure with 10 nm magnetic layer, (b) the fourth GA-structure of Table 1 and (c) the first structure of Table 2.
1826
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parameter, and consequently the optimization procedure tries to find the best value of them without giving any priority. In addition to the large Kerr effect enhancement, the total length of structures optimized here using GA, comparing to conventional designs based on quarter-wave layers, is reduced remarkably. The total thickness of our optimal structures in the 3-gen and 19-gen cases are 208 and 489 nm, respectively. By comparison with an usual quarter-wave structure, that typically would be thicker than 900 nm, it is obvious that genetic structures are more desirable for practical purposes. To investigate the effectiveness of GA-generated structures, we have further computed the electric field distribution in the interior of them. In this way, three structures with {f,r} ¼{3,6} are considered: (a) a quarter-wave structure with 12 nm magnetic layer, (b) the fourth GA-structure of Table 1 and (c) the first structure of Table 2. The results are plotted in Fig. 3. As can be seen in Fig. 3a, there is no light localization in the vicinity of the magnetic layer for this ordinary quarter-wave design. Contrasting to this, the localization effect is obvious in the other two structures that have been generated using GA. This effect is the main reason for considerable enhancement of the Kerr rotation by increasing interaction of light and matter.
4. Conclusion In this paper, we have demonstrated the application of genetic algorithm for optimization of lossy cavity-type MPCs through fully non-quarter-wave design. To do this, we have introduced a tunable two-value fitness function. It has been confirmed that a natural evolution eventually leads to the most fitted solutions which in our case are the optimum structures exhibiting simultaneous sufficient reflectivity and enhanced Kerr rotation due to effective light localization. Further, comparing to conventional designs, the total length of final structures can be reduced remarkably, which is desirable in practical purposes. It should be noted that, although we have considered the Kerr effect enhancement in absorptive MPCs, non-absorptive structures can also be included in GA optimization procedure to examine the results of non-quarter-wave designs in enhancement of MO effects. References [1] M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P.B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, A. Granovsky, Magnetophotonic crystals, Journal of Physics D: Applied Physics 39 (8) (2006) R151–R161.
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