Abstract

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

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References

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    [Crossref]

2015 (2)

Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23(6), 7527–7534 (2015).
[Crossref] [PubMed]

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]

2014 (2)

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]

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]

2013 (2)

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]

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]

2012 (2)

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]

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]

2011 (5)

2010 (6)

2009 (1)

Z. H. Zhan, J. Zhang, Y. Li, and H. S. Chung, “Adaptive particle swarm optimization,” IEEE Trans. Syst. Man Cybern. B Cybern. 39(6), 1362–1381 (2009).
[Crossref] [PubMed]

2008 (2)

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]

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]

2007 (2)

V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[Crossref] [PubMed]

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).

2006 (2)

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]

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006).
[Crossref] [PubMed]

2004 (2)

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43(22), 4322–4327 (2004).
[Crossref] [PubMed]

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]

1994 (1)

Banaee, M. G.

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]

Baskar, S.

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]

Cao, Y.

Chong, C. T.

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]

Choudhury, A.

Chung, H. S.

Z. H. Zhan, J. Zhang, Y. Li, and H. S. Chung, “Adaptive particle swarm optimization,” IEEE Trans. Syst. Man Cybern. B Cybern. 39(6), 1362–1381 (2009).
[Crossref] [PubMed]

Eberhart, R.

J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proc. IEEE Int. Conf. Neural Networks, 1942–1948 (1995).
[Crossref]

Eberhart, R. C.

Y. Shi and R. C. Eberhart, “Comparison between genetic algorithms and particle swarm optimization,” Proc. 7th Int. Conf. Evol. Program.1447, 611–616 (1998).

Gan, F.

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]

Goldberg, B. B.

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]

Golub, I.

Gu, M.

Guo, H.

Halgamuge, S. K.

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]

Harada, Y.

Harley, R. G.

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]

Hashimoto, N.

Hernandez, J. C.

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]

Hibi, T.

Ho, S. T.

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]

Hong, M.

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]

Horanai, H.

Hu, Q.

Huang, K.

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]

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[Crossref] [PubMed]

Jia, B.

Jia, W.

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]

Jiang, H.

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]

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]

Jiang, H. M.

Jiang, M.

Jiao, J.

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]

Jin, N.

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).

Kalosha, V. P.

Kang, X. L.

Kennedy, J.

J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proc. IEEE Int. Conf. Neural Networks, 1942–1948 (1995).
[Crossref]

Khonina, S. N.

Kozawa, Y.

Kurihara, M.

Li, B.

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]

Li, X.

Li, Y.

J. Lin, K. Yin, Y. Li, and J. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36(7), 1185–1187 (2011).
[Crossref] [PubMed]

Z. H. Zhan, J. Zhang, Y. Li, and H. S. Chung, “Adaptive particle swarm optimization,” IEEE Trans. Syst. Man Cybern. B Cybern. 39(6), 1362–1381 (2009).
[Crossref] [PubMed]

Li, Y. P.

Liang, J. J.

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]

Lin, H.

Lin, J.

Liu, J.

Liu, T.

Lukyanchuk, B.

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]

Luo, X.

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]

Mohagheghi, S.

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]

Nemoto, T.

Nie, Z.

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]

Qin, A. K.

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]

Qin, F.

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]

Qiu, C.

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]

Ratnaweera, A.

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]

Ravi, K.

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]

Samii, Y. R.

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).

Sato, A.

Sato, S.

Sheppard, C.

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]

Sheppard, C. J. R.

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]

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43(22), 4322–4327 (2004).
[Crossref] [PubMed]

Shi, G.

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]

Shi, L.

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]

Shi, P.

Shi, Y.

Y. Shi and R. C. Eberhart, “Comparison between genetic algorithms and particle swarm optimization,” Proc. 7th Int. Conf. Evol. Program.1447, 611–616 (1998).

Song, Y.

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]

Suganthan, P. N.

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]

Sui, G.

Tan, J.

Tian, Q.

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]

Ünlü, M. S.

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]

Valle, Y.

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]

Venayagamoorthy, G. K.

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]

Vienne, G.

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]

Volotovsky, S. G.

Wang, H.

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]

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]

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]

Wang, J.

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]

Wang, Y.

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]

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]

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]

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]

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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).
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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).
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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).
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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).
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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).
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Appl. Opt. (1)

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IEEE Trans. Syst. Man Cybern. B Cybern. (1)

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Inf. Sci. (1)

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).
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J. Opt. (1)

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).
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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).
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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).
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Opt. Express (5)

Opt. Lasers Eng. (1)

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).
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Opt. Lett. (7)

Sci. Rep. (1)

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).
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Figures (5)

Fig. 1
Fig. 1 Front view of multi-zone binary phase filter.
Fig. 2
Fig. 2 Landscape maps of two-dimension benchmarks. (a) Sphere function and (b) Rastrigin’s function.
Fig. 3
Fig. 3 Iteration number in HGPSO and PSO for filters with different belt. The detail of the curves is shown in the inset table.
Fig. 4
Fig. 4 Intensity distribution along the z-axis and x-axis for the results generated by PSO and HGPSO.
Fig. 5
Fig. 5 (a) and (b) Field intensity distributions for five- and ten-belt filters, respectively. (a1) and (b1) Radially polarized, longitudinally polarized components and total intensity on the focal plane. (a2) and (b2) Total intensity distribution along z-axis. (a3) and (b3) Intensity distribution on the x-z plane.

Tables (3)

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Table 1 Pseudocode for HGPSO

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Table 2 Comparison of PSO and HGPSO Strategies for Two Benchmark Functions after 1000 Iterations (50 Trials)

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Table 3 Optimizing Normalized R and the Focus Parameters for Different Belt Binary Phase Filters

Equations (11)

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E r ( r,z )=A 0 α cos 1/2 θsin(2θ) l 0 (θ)T(θ) J 1 (krsinθ)exp(ikzcosθ)dθ ,
E z ( r,z )=2iA 0 α cos 1/2 θ sin 2 θ l 0 (θ)T(θ) J 0 (krsinθ)exp(ikzcosθ)dθ ,
l 0 (θ)=exp[ β 2 ( sinθ sinα ) 2 ] J 1 ( 2β sinθ sinα ),
T(θ)={ 1, θ 2m <θ θ 2m+1 1, θ 2m+1 <θ θ 2m+2 .
Minimize δ(R)=|sFWHMDOF|; Subject to: 0< r i < r i+1 <1.
V i t+1 =w V i t + c 1 Ran d 1 ()( Pbes t i t X i t )+ c 2 Ran d 2 ()( Gbes t t X i t ), X i t+1 = X i t + V i t+1 .
w j = w min +( w max w min ) (Nj) (N1) ,
c 1j = c 2j = c min +( c max c min ) (Nj) (N1) .
V j t+1 =w j V j t + c 1j Ran d 1 ()( Pbes t j t X j t )+ c 2j Ran d 2 ()( Gbes t t X j t ); X j t+1 = X j t + V j t+1 .
V N t+1 =Rand() V 1 t +(1Rand()) V 2 t , X N t+1 =Rand() X 1 t +(1Rand()) X 2 t ,
X N t+1 = X N t+1 + X random ,

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