The binary phase filters have been used to achieve an optical needle with small lateral size. Designing a binary phase filter is still a scientific challenge in such fields. In this paper, a hybrid genetic particle swarm optimization (HGPSO) algorithm is proposed to design the binary phase filter. The HGPSO algorithm includes self-adaptive parameters, recombination and mutation operations that originated from the genetic algorithm. Based on the benchmark test, the HGPSO algorithm has achieved global optimization and fast convergence. In an easy-to-perform optimizing procedure, the iteration number of HGPSO is decreased to about a quarter of the original particle swarm optimization process. A multi-zone binary phase filter is designed by using the HGPSO. The long depth of focus and high resolution are achieved simultaneously, where the depth of focus and focal spot transverse size are 6.05λ and 0.41λ, respectively. Therefore, the proposed HGPSO can be applied to the optimization of filter with multiple parameters.
© 2016 Optical Society of America
Recently, optical needle with a small lateral size and ultra-long focal depth has attracted much attention because of their widely potential applications in laser direct writing, particle acceleration and optical trapping [1–4]. In high numerical aperture (NA) system, polarized beam such as Bessel-Gaussian beam or Laguerre-Gaussian beam has been proposed to achieve an optical needle [5–7]. In such high NA systems, diffractive optical element (DOE) functions as a complex amplitude filter or focusing element. The parameters of a DOE should be elaborately determined. There are several methods employed to design a DOE. Zhan et al reversed the radiation field of a uniform line source to realize an optical needle . Huang et al designed a DOE by combining a global-search-optimization algorithm and the tight focusing properties of radially polarized light . The constrained simulation annealing algorithm has also been applied to obtain DOEs with super-resolution . However, the amplitude of the incident light is complex and the computational cost is large in aforementioned methods. The binary phase filter is one kind of DOE to realize the optical needle. Because of the difficulty in designing a multi-zone binary filter, most researchers only concentrated on the binary filter with small zone number [11–13]. For example, a five zone binary phase filter has been optimized in Wang’s work . A seventeen-belt binary phase pupil filter is designed to achieve tight focusing . By increasing the zone number of a binary phase filter, the focusing performance can be improved. However, there is not an easily extendable optimization algorithm to design a binary filter with arbitrary zone number. Among these optimization algorithms, particle swarm optimization (PSO) algorithm has been widely used to decide the parameter of a multi-zone binary optical element [16–19]. The PSO algorithm, which was developed by Eberhart and Kennedy, imitates the behavior of a swarm of bee . In PSO, particles move in the search space to find a best solution to minimize the objective function. Each particle adjusts its position according to its experience and the behavior of its neighboring particles. Each candidate solution therefore moves toward a global best optimum . A high NA binary lens with 107 zones is designed by using PSO with 5000 iterations . Improved particle swarm optimization method is successfully applied to design broadband and flat gain spectrum Raman fiber amplifiers [20–22]. Examples indicate that PSO is effective in solving non-linear multi-dimensional problems. However, PSO is slow to find the optimal solution, especially in the latter part of the search. PSO often falls into a local best solution . Obviously, an improved PSO algorithm, which is fast converging and global optimal, is needed. For example, a multi-objective PSO is proposed to optimize the amplitude annular element in a semiconductor solid immersion lens system . In order to accelerate the PSO algorithm and find the global best solution, a globally fast convergence PSO algorithm is proposed in this work to solve the multiple parameter problems. The proposed hybrid genetic PSO (HGPSO) combines the genetic algorithm, self-adaptive parameters, recombination, and mutation operation. For example, an arbitrary zone numbers binary phase filter is designed by using the proposed improved PSO. The HGPSO is verified by benchmark tests. By using the HGPSO, the binary phase filters are designed, and the corresponding focusing performance is calculated by vector integral formulae.
In Section 2, vector representation of a high NA focusing system and the filter model are introduced. In Section 3, the HGPSO algorithm for designing multi-zone binary phase filter is proposed. The HGPSO is verified by benchmark tests. Optimized filters and their focusing performance are given in Section 4. Analysis and discussion are further provided in this section. In Section 5, conclusions are given.
2. Vector representation and the optimizing model
In a high NA aplanatic focusing system illuminated by a radially polarized beam, the vector electric field near the focus in a homogeneous dielectric medium can be calculated by Richards and Wolf’s theory. The radially polarized component Er(r, z) and the longitudinally polarized component Ez(r, z) are expressed as [15, 25]5]
The multi-zone binary filter used in this work is presented as Fig. 1. The binary optical element consists of n belts (n is a variable) with 0 phase and π phase alternately in the aperture. The corresponding amplitude transmission is + 1 and −1. ri is the normalized radius of i-th belt. The aperture half angle of the i-th belt is calculated by. Therefore, the transmission function of the multi-zone binary filter is expressed as
Theoretical researches have indicated that the portion of the longitudinal field component is increasing for large belt number . However, more belts require more variables to be determined. A fast algorithm with global optimization is necessary to design R. We firstly introduce a constraint equation in optimization algorithm. The non-linear constraint equation is expressed as26], which can control the DOF. s should be carefully considered in designing ring belt filter. When s is small, we will obtain a focal spot with a small DOF or a large FWHM. Especially, when sFWHM < DOF, a focal spot with larger FWHM and smaller DOF is achieved. Therefore, to achieve small FWHM and large DOF, a large s is necessary. Based on the existed results [10, 14, 15], we assume that s is 20 in the following calculation. The uniformity of the optical needle is also important and is considered in the optimization.
3. Procedure of HGPSO algorithm
The PSO algorithm originated from the social behavior of individuals in a swarm. The algorithm starts with a random initialization of particle (solution) in search space. The trajectory of each individual is adjusted by the velocity vector of particles, which depends on its flying experience and the flying experience of the other particles. In a d-dimensional search space, the position and velocity vector of i-th particle are and , respectively. For the i-th particle of PSO, the position of (t + 1)-th generation particle is calculated from the t-th generation by [16, 17],23]. The best position of the i-th particle at time t is. The best fitted particle at time t is . The fitness values of each particle is calculated by.
Obviously, the iteration of PSO is often time consuming and sometimes falls into a local best. In order to achieve a small iteration number and a global optimal resolution, a hybrid genetic PSO is proposed by changing the parameter sets and introducing the recombination and mutation operations.
3.1 Parameters in HGPSO
In HGPSO, their parameters are provided in a new way to effectively realize the global search and convergence to the globally optimal solution. Shi et al had proposed the linearly varying inertial weight w over the generations . A linearly time-varying method was applied to deal with the acceleration coefficientsand. Meanwhile, nonlinearly time-varying methods have been investigated. However, there is no significant improvement for nonlinear varying . These methods based on predefined number of generations, but one can’t find out a properly predefined iteration number in the beginning of optimizing. Additionally, PSO is fast converging in the initial stage of iteration, while the convergence is very slow in the final phase of iteration [28, 29]. Therefore, the linearly-varying, self-adaption inertial weight and acceleration coefficients are introduced into the PSO algorithm. The corresponding algorithm, which unites certain concept in genetic algorithm, is named as hybrid genetic PSO algorithm. For HGPSO, all particles at time t are sorted by their fitness values from small (better solution) to big (worse solution). The inertial weight and the acceleration coefficients of j-th particle (has been sorted) are defined asEq. (5) for the j-th particle (after sorted) can be rewritten as27, 28]. In HGPSO, different solution carries different searching task in one generation. A better solution assigned a larger w can explore in a wilder space to reach the global best solution. For a bad solution, a small w is applied, and it will converge to good solution fast . Therefore, the improved algorithm can quickly achieve global best solution, i.e., we can obtain the good solution in smaller iteration comparing with that of PSO.
3.2 Recombination and mutation in HGPSO
The genetic algorithm can avoid local optimization through recombination of genetic material between individuals and individual’s mutation during the search. PSO will falls into local best when it is fast converging. Therefore, the recombination and mutation, which are concepts in genetic algorithm, are introduced into PSO to enhance the global search capability by providing additional diversity. The HGPSO combines the advantage of genetic algorithm and PSO algorithm. The convergence of HGPSO is faster than that of PSO. Meanwhile, HGPSO can easily escape from a local best. Then a global optimal resolution can be obtained. The worst solution is replaced by a recombination solution. The recombination operation can be written as
The position vector mutates with certain probability. A big mutation probability provides additional diversity. However, a big mutation probability will destroy the original proceeding of PSO. The mutation probability is chosen as 0.3. The mutation operation, which is the fluctuation of solution, is given by
The new solution obtained by recombination and mutation operation is reserved when the solution is better than the original. While it is worse, the new solution is reserved with small probability of 0.05. The reserved probability depends the amount of bad solutions achieved by recombination and mutation. The bad solution achieved by recombination and mutation can provide additional diversity. However, much bad solutions will reduce the quality of the particle population. The recombination and mutation ensure HGPSO to avoid local best. The convergence at the global best solution is also accelerating. The position vector is normalized between the neighboring two belts with the radius and. Therefore, allshould be larger than zero. R will be normalized as 0<ri<ri+1<1.
The procedure and pseudocode of HGPSO is given in Table 1. The inertial weight w and acceleration coefficients are calculated in the initialization. These steps require little calculation time. The recombination and mutation operation in the loop is also a single formula and take few time, as well as mutation operation occurs in a small probability. Therefore, the calculated amount in a loop for HGPSO is same as that of PSO. According to [27–31], the iteration number is used as the computation time of the algorithm.
3.4 Benchmark test of HGPSO
In order to validate the proposed HGPSO algorithm, two well-known benchmark functions are used to evaluate the performance of HGPSO. These benchmarks are widely used in evaluating optimize methods [30, 31]. The landscape maps of the two-dimension benchmark functions are given in Fig. 2. The positions of the minimum of the two benchmark functions are both at the original point. The Sphere function is a unimodal function, while the Rastrigin’s function is multimodal function designed with a considerable amount of local optima. When one solves Rastrigin’s function, algorithms may fall into local best. Hence, an algorithm capable of maintaining a larger diversity is likely to yield better results.
Simulations were carried out to find the global minimum of each benchmark function. The optimum solution after 1000 iterations and the best solution, the worst solution, the mean solution and the standard deviation (St. d) in 50 trials are calculated in the empirical simulation. These calculated results are shown in Table 2. A significant improvement of HGPSO is observed. The faster converging velocity and the better optimum results are obtained by using HGPSO. The HGPSO takes 0.72s and 0.78s to achieve the best results of the two benchmark functions in Table 2, respectively. In contrast, the performance of the optimum solution is significantly poor for the same benchmark by using PSO. The comparison indicates that the effect of parameters set in HGPSO is significant for the benchmark of sphere function. The mutation is highly effective to the benchmark of multimodal function. Meanwhile, HGPSO is fast to obtain the global optimum. One can achieve good solution with small iteration number in HGPSO.
4. Numerical results and discussions
HGPSO algorithm is fast converging and high-dimensional effective. Therefore, HGPSO is employed to design binary phase filters. The focusing performance is calculated by Eq. (1). A series of optimized normalization belt radii of the binary phase filters is achieved. Their focusing performance in a high NA system is tabulated in Table 3. The real time to achieve the results in Table 3 by the HGPSO are 107.8s, 91.3s, 121.7s, 142.6s, 169.1s, and 180.2s, respectively. Obviously, the lateral super-resolution and long depth of focus (DOF) are realized, simultaneously. The FWHM is smaller than 0.43λ, while the depth of focus is increasing with the increase of belt number. Meanwhile, the focusing field is longitudinally polarized. The maximal electric density of the transversal polarization is only 8% of the longitudinally polarized component. The polarization conversion efficiency η is defined as Where is defined as the first zero point of the radial electric intensity on the focal plane, andis about. The polarization conversion efficiency is about 83% for all belt number, which is larger than that of a focusing BG beam with. Therefore, the focusing field is dominated by longitudinally polarized component. For all these optimized binary filters, the conversion efficiency and the depth of focus are better than that (,) given in . One can see that FWHM is decreasing with the increase of belt number, while the depth of focus and polarization conversion efficiency are increasing.
Importantly, the iterations of HGPSO are significantly reduced compared with the PSO. As shown in Fig. 3, there are only ~30 iterations in HGPSO to achieve the excellent focus as mentioned in Table 3. There are hundred iterations for PSO. The iteration is quarter of that in PSO. It is indicated that the self-adaptive inertial weight and acceleration coefficients are effective in speeding convergence. The recombination and mutation operation also enhanced the convergence at global best solution because the mutated solutions will speed up the convergence by increasing the diversity of solution group . The solution is not in a connected-domain. The PSO easily falls into a local-domain and it will take several iterations to jump out. It is the reason that more iteration number is needed in the PSO. Due to the recombination and mutation operations, the iteration of HGPSO easily leaps to global best from local-domain. Because the randomness is in PSO and HGPSO, there is no exactly same solution in two optimizing algorithms for the same belt filter optimizing. One can obtain similar solution with same FWHM and DOF. Obviously, the improved algorithm is effective for arbitrary belts filter optimizing. The decrease of iteration number will bring a remarkable reduction in the computational cost.
The iteration number for different belt number is given in Fig. 3. One can obviously know that more belt number there is, more iteration number there is. However, when HGPSO is applied, the iteration number is sharply decreasing even for large belt number. For example, the iteration number is 23 for binary filter with 5 belts for HGPSO, while it is as high as 117 for PSO. The blue curve in Fig. 2 shows the iteration number ratio of PSO to HGPSO. One can see that the iteration number of PSO is always about four times of HGPSO. In Fig. 3, data indicated that the iteration number of HGPSO is much smaller than that of PSO.
Furthermore, optimized results generated by PSO and HGPSO are shown in Fig. 4 for a ten-belt binary phase filter. The R-distribution generated by the HGPSO has been given in Table 1. The result of PSO is R = [0.0728 0.1298 0.2396 0.3780 0.5081 0.5918 0.6468 0.7312 0.8393 1]. The FWHM and DOF in the two cases are 0.41λ and 6.07λ. As shown in Fig. 4, there is only small intensity fluctuation along the DOF. It indicates that the computational cost will be remarkably reduced and the quality of the focal spot didn’t worsen for HGPSO.
For filters with five and ten belts, the radially polarized component, longitudinally polarized component and the total field intensity on the focal plane and along the z-axis are displayed in Fig. 5. The corresponding FWHM and DOF are shown in Table 3. The intensity of side-lobe is smaller than 0.170. Although the side-lobe intensity 0.17 is little larger than that in , there is a significant reduction of belt number (seventeen belt in the reference). The fluctuation of the total intensity along the z-axis is no larger than 8% for all filters. Especially for the seven-, nine- and ten-belt filter, the fluctuation does not exceed 3%. There is no obviously intensity fluctuation for six-belt filtering.
A hybrid genetic particle swarm optimization algorithm is proposed in this paper. In HGPSO algorithm, self-adaptive parameters, recombination and mutation operations are introduced. Based on the benchmarks tests, the HGPSO achieves improved global optimization and is efficient for multiple parameter models. In an easy-to-perform optimizing procedure, a series of optimal structures of multiple zones binary phase filters are obtained. The FWHM is super-resolution and a long DOF is achieved. The iteration number of HGPSO is a quarter of that in the PSO algorithm. It indicates that the self-adaptive inertial weight and the acceleration coefficients are effective for accelerating convergence. The recombination and mutation operations guarante the HGPSO algorithm can easily escape from a locally optimal. HGPSO can be extended to an arbitrary zone binary filters optimization and will greatly reduce the computational cost.
This work is financially supported by National Natural Science Foundation of China (NSFC) (Grant No. 51275111).
References and links
3. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011). [CrossRef] [PubMed]
4. H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6(3), 354–392 (2012). [CrossRef]
7. Z. Nie, G. Shi, X. Zhan, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014). [CrossRef]
10. J. Yu, C. Zhou, and W. Jia, “Transverse superresolution with extended depth of focus using binary phase filters for optical storage system,” Opt. Commun. 283(21), 4171–4177 (2010). [CrossRef]
12. H. Lin, B. Jia, and M. Gu, “Generation of an axially super-resolved quasi-spherical focal spot using an amplitude-modulated radially polarized beam,” Opt. Lett. 36(13), 2471–2473 (2011). [CrossRef] [PubMed]
14. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
15. H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21(5), 5363–5372 (2013). [CrossRef] [PubMed]
16. J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proc. IEEE Int. Conf. Neural Networks, 1942–1948 (1995). [CrossRef]
17. Y. Valle, G. K. Venayagamoorthy, S. Mohagheghi, J. C. Hernandez, and R. G. Harley, “Particle swarm optimization: basic concepts, variants and applications in power systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008). [CrossRef]
18. Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of ultra-long depth of focus super-resolution longitudinally polarized beam with ternary optical element,” J. Opt. 15(7), 075703 (2013). [CrossRef]
19. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015). [CrossRef] [PubMed]
20. H. M. Jiang, K. Xie, and Y. F. Wang, “Pump scheme for gain-flattened Raman fiber amplifiers using improved particle swarm optimization and modified shooting algorithm,” Opt. Express 18(11), 11033–11045 (2010). [CrossRef] [PubMed]
21. H. Jiang, K. Xie, and Y. Wang, “Shooting algorithm and particle swarm optimization based Raman fiber amplifiers gain spectra design,” Opt. Commun. 283(17), 3348–3352 (2010). [CrossRef]
22. H. Jiang, K. Xie, and Y. Wang, “Flat gain spectrum design of Raman fiber amplifiers based on particle swarm optimization and average power analysis technique,” Opt. Lasers Eng. 50(2), 226–230 (2012). [CrossRef]
23. N. Jin and Y. R. Samii, “Advances in particle swarm optimization for antenna designs: real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Ante. Prop. 55, 556–567 (2007).
24. M. G. Banaee, M. S. Ünlü, and B. B. Goldberg, “Sub-λ/10 spot size in semiconductor solid immersion lens microscopy,” Opt. Commun. 315, 108–111 (2014). [CrossRef]
25. S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010). [CrossRef] [PubMed]
27. Y. Shi and R. C. Eberhart, “Comparison between genetic algorithms and particle swarm optimization,” Proc. 7th Int. Conf. Evol. Program.1447, 611–616 (1998).
28. A. Ratnaweera, S. K. Halgamuge, and H. C. Watson, “Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients,” IEEE Trans. Evol. Comput. 8(3), 240–255 (2004). [CrossRef]
30. Y. Wang, B. Li, T. Weise, J. Wang, B. Yuan, and Q. Tian, “Self-adaptive learning based particle swarm optimization,” Inf. Sci. 181(20), 4515–4538 (2011). [CrossRef]
31. J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar, “Comprehensive learning particle swarm optimizer for global optimization of multimodal functions,” IEEE Trans. Evol. Comput. 10(3), 281–295 (2006). [CrossRef]