Abstract

A modified genetic algorithm is proposed for the optimization of fiber birefringent filters. The orientation angles and the element lengths are determined by the genetic algorithm to minimize the sidelobe levels of the filters. Being different from the normal genetic algorithm, the algorithm proposed reduces the problem space of the birefringent filter design to achieve faster speed and better performance. The design of 4-, 8-, and 14-section birefringent filters with an improved sidelobe suppression ratio is realized. A 4-section birefringent filter designed with the algorithm is experimentally realized.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. J. M. Becker, "High-resolution measurements of photosphere and sun-spot velocity and magnetic fields using a narrow-band birefringent filter," Sol. Phys. 3, 258-263 (1968).
    [CrossRef]
  10. J. M. Beckers, L. Dickson, and R. S. Joyce, "Observing the sun with a fully tunable Lyot-Ohman filter," Appl. Opt. 14, 2061-2066 (1975).
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    [CrossRef]
  15. P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
    [CrossRef]
  16. L. J. November and F. Stauffer, "Derivation of the universal wavelength tuning formula for a Lyot birefringent filter," Appl. Opt. 23, 2333-2341 (1984).
    [CrossRef] [PubMed]
  17. Y. Zhou, L. R. Liu, J. Zhang, D. Liu, and Z. Luan, "Nearly-off-axis transmissivity of Solc birefringent filters," J. Opt. Soc. Am. A 20, 733-740 (2003).
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  18. A. Lien, "A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays," Liq. Cryst. 22, 171-175 (1997).
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  20. Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 2nd ed. (Springer-Verlag, 1994).
  21. A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
    [CrossRef]
  22. J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
    [CrossRef]
  23. R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms (Wiley, 1998).
  24. J. R. Perez and J. Basterrechea, "Near to far-field transformation using GA based optimization: real versus binary encoding schemes," in Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 1122-1125.
  25. S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.
  26. V. S. Gordon and T. J. Slocum, "The knight's tour--Evolutionary vs. depth-first search," in Congress on Evolutionary Computation (IEEE, 2004), pp. 1435-1440.
    [CrossRef]
  27. K. Sastry and D. E. Goldberg, "Modeling tournament selection with replacement using apparent added noise," in Intelligent Engineering Systems through Artificial Neural Networks (American Society of Mechanical Engineering, 2001), pp.129-134.
  28. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
    [CrossRef] [PubMed]
  29. S. Kirkpatrick, "Optimization by simulated annealing: quantitative studies," J. Stat. Phys. 34, 975-986 (1984).
    [CrossRef]
  30. D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
    [CrossRef]
  31. H. Sakanashi and Y. Kakazu, "Co-evolving genetic algorithm with filtered evaluation function," in IEEE Symposium on Emerging Technologies and Factory Automation (IEEE, 1994), pp. 454-457.
  32. J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
    [CrossRef]

2004 (2)

V. S. Gordon and T. J. Slocum, "The knight's tour--Evolutionary vs. depth-first search," in Congress on Evolutionary Computation (IEEE, 2004), pp. 1435-1440.
[CrossRef]

J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
[CrossRef]

2003 (1)

2002 (2)

2001 (2)

K. Sastry and D. E. Goldberg, "Modeling tournament selection with replacement using apparent added noise," in Intelligent Engineering Systems through Artificial Neural Networks (American Society of Mechanical Engineering, 2001), pp.129-134.

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

2000 (1)

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

1999 (2)

A. Carballar, M. A. Muriel, and J. Azafi, "Fiber grating filter for WDM systems: an improved design," IEEE Photon. Technol. Lett. 11, 694-696 (1999).
[CrossRef]

A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
[CrossRef]

1997 (1)

A. Lien, "A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays," Liq. Cryst. 22, 171-175 (1997).
[CrossRef]

1996 (1)

S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.

1994 (1)

1993 (1)

D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
[CrossRef]

1990 (1)

D. Bonaccini and F. Stauffer, "High resolution solar bidimensional spectroscopy with a universal birefringent filter in tandem with a Fabry-Perot interferometer--Tests and experimental results," Astron. Astrophys. 229, 272-278 (1990).

1989 (1)

1984 (2)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

1981 (1)

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

1975 (1)

1973 (1)

1968 (2)

B. M. Schiffman and L. Young, "Birefringent filter for millimeter waves," IEEE Trans. Microwave Theory Tech. 16, 351-360 (1968).
[CrossRef]

J. M. Becker, "High-resolution measurements of photosphere and sun-spot velocity and magnetic fields using a narrow-band birefringent filter," Sol. Phys. 3, 258-263 (1968).
[CrossRef]

1966 (1)

1964 (1)

1958 (1)

1949 (1)

Ammann, E. O.

Azafi, J.

A. Carballar, M. A. Muriel, and J. Azafi, "Fiber grating filter for WDM systems: an improved design," IEEE Photon. Technol. Lett. 11, 694-696 (1999).
[CrossRef]

Basterrechea, J.

J. R. Perez and J. Basterrechea, "Near to far-field transformation using GA based optimization: real versus binary encoding schemes," in Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 1122-1125.

Beasley, D.

D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
[CrossRef]

Becker, J. M.

J. M. Becker, "High-resolution measurements of photosphere and sun-spot velocity and magnetic fields using a narrow-band birefringent filter," Sol. Phys. 3, 258-263 (1968).
[CrossRef]

Beckers, J. M.

Bonaccini, D.

D. Bonaccini and F. Stauffer, "High resolution solar bidimensional spectroscopy with a universal birefringent filter in tandem with a Fabry-Perot interferometer--Tests and experimental results," Astron. Astrophys. 229, 272-278 (1990).

Bull, D. R.

D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
[CrossRef]

Carballar, A.

A. Carballar, M. A. Muriel, and J. Azafi, "Fiber grating filter for WDM systems: an improved design," IEEE Photon. Technol. Lett. 11, 694-696 (1999).
[CrossRef]

Chang, I. C.

Chipman, R. A.

Chu, R. H.

Conroy, R. S.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

Dickson, L.

Eiben, A. E.

A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
[CrossRef]

Evans, J. W.

Friel, G. J.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

Fujii, Y.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Goldberg, D. E.

K. Sastry and D. E. Goldberg, "Modeling tournament selection with replacement using apparent added noise," in Intelligent Engineering Systems through Artificial Neural Networks (American Society of Mechanical Engineering, 2001), pp.129-134.

Gordon, V. S.

V. S. Gordon and T. J. Slocum, "The knight's tour--Evolutionary vs. depth-first search," in Congress on Evolutionary Computation (IEEE, 2004), pp. 1435-1440.
[CrossRef]

Gunning, W.

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

Hall, R.

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

Harris, S. E.

Haupt, R. L.

R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms (Wiley, 1998).

Haupt, S. E.

R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms (Wiley, 1998).

He, Y.

J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
[CrossRef]

Hinterding, R.

A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
[CrossRef]

Joyce, R. S.

Kakazu, Y.

H. Sakanashi and Y. Kakazu, "Co-evolving genetic algorithm with filtered evaluation function," in IEEE Symposium on Emerging Technologies and Factory Automation (IEEE, 1994), pp. 454-457.

Kemp, A. J.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

Kimura, T.

Kirkpatrick, S.

S. Kirkpatrick, "Optimization by simulated annealing: quantitative studies," J. Stat. Phys. 34, 975-986 (1984).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Lake, T. K.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

Lien, A.

A. Lien, "A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays," Liq. Cryst. 22, 171-175 (1997).
[CrossRef]

Liu, D.

Liu, L. R.

Lu, S. Y.

Luan, Z.

Marom, E.

Martin, R. R.

D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
[CrossRef]

Mendlovic, D.

Michalewicz, Z.

A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
[CrossRef]

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 2nd ed. (Springer-Verlag, 1994).

Minowa, J.

Muriel, M. A.

A. Carballar, M. A. Muriel, and J. Azafi, "Fiber grating filter for WDM systems: an improved design," IEEE Photon. Technol. Lett. 11, 694-696 (1999).
[CrossRef]

November, L. J.

Otsuka, K.

Perez, J. R.

J. R. Perez and J. Basterrechea, "Near to far-field transformation using GA based optimization: real versus binary encoding schemes," in Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 1122-1125.

Ramirez, J. A.

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

Rochet, S.

S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.

Sakanashi, H.

H. Sakanashi and Y. Kakazu, "Co-evolving genetic algorithm with filtered evaluation function," in IEEE Symposium on Emerging Technologies and Factory Automation (IEEE, 1994), pp. 454-457.

Saldanha, R. R.

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

Saruwatari, M.

Sastry, K.

K. Sastry and D. E. Goldberg, "Modeling tournament selection with replacement using apparent added noise," in Intelligent Engineering Systems through Artificial Neural Networks (American Society of Mechanical Engineering, 2001), pp.129-134.

Schiffman, B. M.

B. M. Schiffman and L. Young, "Birefringent filter for millimeter waves," IEEE Trans. Microwave Theory Tech. 16, 351-360 (1968).
[CrossRef]

Sinclair, B. D.

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

Slimane, M.

S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.

Slocum, T. J.

V. S. Gordon and T. J. Slocum, "The knight's tour--Evolutionary vs. depth-first search," in Congress on Evolutionary Computation (IEEE, 2004), pp. 1435-1440.
[CrossRef]

Stauffer, F.

D. Bonaccini and F. Stauffer, "High resolution solar bidimensional spectroscopy with a universal birefringent filter in tandem with a Fabry-Perot interferometer--Tests and experimental results," Astron. Astrophys. 229, 272-278 (1990).

L. J. November and F. Stauffer, "Derivation of the universal wavelength tuning formula for a Lyot birefringent filter," Appl. Opt. 23, 2333-2341 (1984).
[CrossRef] [PubMed]

Takahashi, R. H. C.

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

Town, G.

Tracy, J.

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

Vasconcelos, J. A.

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Venturini, G.

S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.

von Willisen, F. K.

Xiong, J.

J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
[CrossRef]

Yeh, P.

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

Young, L.

B. M. Schiffman and L. Young, "Birefringent filter for millimeter waves," IEEE Trans. Microwave Theory Tech. 16, 351-360 (1968).
[CrossRef]

Yu, F.

J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
[CrossRef]

Zalevsky, Z.

Zhang, J.

Zhou, Y.

Appl. Opt. (6)

Astron. Astrophys. (1)

D. Bonaccini and F. Stauffer, "High resolution solar bidimensional spectroscopy with a universal birefringent filter in tandem with a Fabry-Perot interferometer--Tests and experimental results," Astron. Astrophys. 229, 272-278 (1990).

Congress on Evolutionary Computation (1)

V. S. Gordon and T. J. Slocum, "The knight's tour--Evolutionary vs. depth-first search," in Congress on Evolutionary Computation (IEEE, 2004), pp. 1435-1440.
[CrossRef]

Evol. Comput. (1)

D. Beasley, D. R. Bull, and R. R. Martin, "A sequential niche technique for multimodal function optimization," Evol. Comput. 1, 101-125 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. J. Kemp, G. J. Friel, T. K. Lake, R. S. Conroy, and B. D. Sinclair, "Polarization effects, birefringent filtering, and single-frequency operation in lasers containing a birefringent gain crystal," IEEE J. Quantum Electron. 35, 228-235 (2000).
[CrossRef]

P. Yeh, W. Gunning, R. Hall, and J. Tracy, "Dispersive birefringent filters for laser communications," IEEE J. Quantum Electron. 17, 2424 (1981).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Carballar, M. A. Muriel, and J. Azafi, "Fiber grating filter for WDM systems: an improved design," IEEE Photon. Technol. Lett. 11, 694-696 (1999).
[CrossRef]

IEEE Trans. Evol. Comput. (1)

A. E. Eiben, R. Hinterding, and Z. Michalewicz, "Parameter control in evolutionary algorithms," IEEE Trans. Evol. Comput. 3, 124-141 (1999).
[CrossRef]

IEEE Trans. Magn. (1)

J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi, and R. R. Saldanha, "Improvements in genetic algorithms," IEEE Trans. Magn. 37, 3414-3417 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

B. M. Schiffman and L. Young, "Birefringent filter for millimeter waves," IEEE Trans. Microwave Theory Tech. 16, 351-360 (1968).
[CrossRef]

Intelligent Engineering Systems through Artificial Neural Networks (1)

K. Sastry and D. E. Goldberg, "Modeling tournament selection with replacement using apparent added noise," in Intelligent Engineering Systems through Artificial Neural Networks (American Society of Mechanical Engineering, 2001), pp.129-134.

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Stat. Phys. (1)

S. Kirkpatrick, "Optimization by simulated annealing: quantitative studies," J. Stat. Phys. 34, 975-986 (1984).
[CrossRef]

Liq. Cryst. (1)

A. Lien, "A detailed derivation of extended Jones matrix representation for twisted nematic liquid crystal displays," Liq. Cryst. 22, 171-175 (1997).
[CrossRef]

Opt. Eng. (1)

J. Xiong, F. Yu, and Y. He, "Direct design of a polarization interference filter by genetic algorithm," Opt. Eng. 43, 1200-1205 (2004).
[CrossRef]

Opt. Express (1)

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Sol. Phys. (1)

J. M. Becker, "High-resolution measurements of photosphere and sun-spot velocity and magnetic fields using a narrow-band birefringent filter," Sol. Phys. 3, 258-263 (1968).
[CrossRef]

Other (5)

R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms (Wiley, 1998).

J. R. Perez and J. Basterrechea, "Near to far-field transformation using GA based optimization: real versus binary encoding schemes," in Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 1122-1125.

S. Rochet, M. Slimane, and G. Venturini, "Epistasis for real encoding in genetic algorithms," in Proceedings of the Australian and New Zealand Conference on Intelligent Information Systems (IEEE, 1996), pp. 268-271.

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 2nd ed. (Springer-Verlag, 1994).

H. Sakanashi and Y. Kakazu, "Co-evolving genetic algorithm with filtered evaluation function," in IEEE Symposium on Emerging Technologies and Factory Automation (IEEE, 1994), pp. 454-457.

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Figures (12)

Fig. 1
Fig. 1

Lyot filter.

Fig. 2
Fig. 2

(Color online) Solc filter with arbitrary angles.

Fig. 3
Fig. 3

(Color online) Spectral responses with different weights.

Fig. 4
Fig. 4

(Color online) Spectral response of an 8-section birefringent filter.

Fig. 5
Fig. 5

(Color online) Flow chart of the modified GA.

Fig. 6
Fig. 6

(Color online) Family of an 8-section birefringent filter designed by the modified GA.

Fig. 7
Fig. 7

(Color online) 14-section birefringent filter designed by the modified GA.

Fig. 8
Fig. 8

(Color online) 4-section birefringent filter with nonidentical section lengths.

Fig. 9
Fig. 9

Experimental setup. SM, single-mode; PM, polarization-maintaining fiber.

Fig. 10
Fig. 10

(Color online) Theoretical (solid curve) and experimental (dashed curve) spectral responses.

Fig. 11
Fig. 11

(Color online) Precise angles (solid curve) versus angles with up to 1° error (dashed curve).

Fig. 12
Fig. 12

(Color online) Precise lengths (solid curve) versus imprecise lengths up to 1 mm error (dashed curve).

Tables (1)

Tables Icon

Table 1 Comparison of Cost Functions between the Normal Genetic Algorithm and the Modified Genetic Algorithm

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

( E A 1 E A 2 ) = R ( φ p ) M ( φ i , η ) P x ( E x 0 E y 0 ) ,
R ( φ p ) = [ cos φ p sin φ p sin φ p cos φ p ] ,
P x = [ 1 0 0 0 ] ,
R ( φ p ) M ( φ i , η ) = R ( φ p φ n ) R η R ( φ n φ n 1 ) × R η R ( φ 2 φ 1 ) R η R ( φ 1 ) = R ( θ n ) R η R ( θ n 1 ) R η R ( θ 1 ) R η R ( θ 0 ) .
R ( θ i ) = [ cos θ i sin θ i sin θ i cos θ i ] .
R η = [ exp ( j η / 2 ) 0 0 exp ( j η / 2 ) ] ,
η = 2 π f ( Δ n L / c ) ,
I = { 1 + δ pass ,               0 f f p n ot  of  c oncern ,           f p < f < f s 0 + δ stop , f s f } ,
cos t ( φ 1 , φ 2 , , φ n ) = passband | I pass 1 | + W stop  band | I stop | ,
φ i = π ( r 0.5 ) ,
φ k ν + = ( 1 a ) φ k ν + a φ k w ,
φ k w + = ( 1 a ) φ k w + a φ k ν ,
φ k t + 1 = φ k t ± Δ ,
Δ = π r [ 1 ( t T ) x ] ,
η = 2 π f ( Δ n L / c ) = 2 π ( Δ n L / λ ) = 2 k π ,
k λ = Δ n L .
FSR  =   c Δ n L .
F ( x ) = F ( x ) G ( x , s 1 , s 2 , , s n ) ,
G ( x , s 1 , s 2 , , s n ) = { [ d ( x , s 1 , s 2 , , s n ) / r ] α , d ( x , s 1 , s 2 , …  , s n )  <  r 1 , otherwise ,
R ( θ 2 ) R η R ( θ 1 ) R η R ( θ 0 )
= [ cos θ 2 sin θ 2 sin θ 2 cos θ 2 ] [ exp ( j η / 2 )                 0                   0 exp ( - j η / 2 ) ] × [ cos θ 1 sin θ 1 sin θ 1 cos θ 1 ] [ exp ( j η / 2 )                   0                   0 exp ( - j η / 2 ) ] × [ cos θ 0 sin θ 0 sin θ 0 cos θ 0 ] .
[ exp ( j η / 2 )               0                   0 exp ( - j η / 2 ) ] [ cos θ 1 sin θ 1 sin θ 1 cos θ 1 ] × [ exp ( j η / 2 )             0                 0 exp ( - j η / 2 ) ] = [ A B * B A * ] ,
| A cos θ 2 cos θ 0 A * sin θ 2 sin θ 0 + B sin θ 2 cos θ 0 + B * cos θ 2 sin θ 0 | 2 ;
[ A B * B A * ] ,
| A cos θ n + 2 cos θ 0 A * sin θ n + 2 sin θ 0 + B sin θ n + 2 cos θ 0 + B * cos θ n + 2 sin θ 0 | 2 ,

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