## Synthesis of 2-dimensional photonic crystals

Optics Express, Vol. 11, Issue 4, pp. 317-323 (2003)

http://dx.doi.org/10.1364/OE.11.000317

Acrobat PDF (144 KB)

### Abstract

We present a procedure for optimizing two-dimensional (2D) Photonic Band Gap (PBG) structures. The procedure discretizes the unit cell of a PBG structure into a binary cell and uses Direct Binary Search to search through a terrain of possible solutions in order to find a more optimal one. This process is designed either for improving the absolute band gap or opening a new one, for a predefined PBG structure. By applying the procedure on a honeycomb array of high dielectric objects in an air background, we increased its Maximum Absolute Gap-to-Midgap Ratio (MAGTMR) to more than twice that of the initial structure. To further prove the utility of this procedure, we also present other examples.

© 2003 Optical Society of America

## 1. Introduction

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987). [CrossRef] [PubMed]

3. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

4. A. R. McGurn, “Photonic crystal circuits: A theory for two- and three-dimensional networks,” Phys. Rev. B **61**, 13235 (2000). [CrossRef]

6. S. John and M. Florescu, “Photonic bandgap materials:towards an all-optical micro-transistor,” J. Opt. A:Pure Appl. Opt. **3**, S103 (2001). [CrossRef]

7. O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science **284**, 1819 (1999). [CrossRef] [PubMed]

8. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic Crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. **87**, 603 (2000). [CrossRef]

9. A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” Appl. Phy. Lett. **78**, 3184 (2001). [CrossRef]

10. K. Nam, “Photonic Crystals,” http://www.phys.ksu.edu/~namkv/photonic.html.

10. K. Nam, “Photonic Crystals,” http://www.phys.ksu.edu/~namkv/photonic.html.

11. J. Moosburger, M. Kamp, F. Klopf, M. Fischer, and A. Forchel, “Fabrication of semiconductor lasers with 2D-photonic crystal mirrors using a wet oxidized Al2O3-mask,” Microelectron. Eng. **57**, 1017 (2001). [CrossRef]

13. J. S. Shirk, R. G. S. Pong, S. R. Flom, and E. A. Bolden, “Nonlinear 2-d Photonic Crystals for Optical Limiting,” http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/.

14. M. Imada, S. Noda, A. Chutinan, M. Mochizuk, and T. Tanaka, “Channel Drop Filter Using a Single Defect in a 2-D Photonic Crystal Slab Waveguide,” J. Lightwave Technol. **20**, 873 (2002). [CrossRef]

15. M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A **64**, 033801 (2001). [CrossRef]

16. Z. Li, J. Wang, and B. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys.Rev. B. **58**, 3721 (1998). [CrossRef]

17. Z. Li, B. Gu, and G. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. **81**, 2574 (1998). [CrossRef]

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987). [CrossRef] [PubMed]

18. C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys. Rev. E. **56**, 6291 (1997). [CrossRef]

22. P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B **46**, 4969 (1992). [CrossRef]

23. W. H. R. Hillebrand and W. Harms, “Theoretical Band Gap Studies of Two-Dimensional Photonic Crystals with Varying Column Roundness,” Phys. Stat. Sol. **217**, 981 (2000). [CrossRef]

18. C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys. Rev. E. **56**, 6291 (1997). [CrossRef]

23. W. H. R. Hillebrand and W. Harms, “Theoretical Band Gap Studies of Two-Dimensional Photonic Crystals with Varying Column Roundness,” Phys. Stat. Sol. **217**, 981 (2000). [CrossRef]

29. M. Qiu and S. He, “Large Complete band gap in two-dimensional photonic crystals with elliptic air holes,” Phys. Rev. B. **60**, 10610 (1999). [CrossRef]

20. E. Yablonovitch, T. J. Gmitter, and K.M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. **67**, 2295 (1991). [CrossRef] [PubMed]

*et al.*[30

30. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. **61**, 495 (1992). [CrossRef]

*et al.*[31

31. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys.Rev.Lett. **65**, 3152 (1990). [CrossRef] [PubMed]

*et al.*[19

19. D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B. **53**, 7134 (1996). [CrossRef]

*et al.*[16

16. Z. Li, J. Wang, and B. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys.Rev. B. **58**, 3721 (1998). [CrossRef]

17. Z. Li, B. Gu, and G. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. **81**, 2574 (1998). [CrossRef]

27. C. M. Anderson and K. P. Giapis, “Symmetry reduction in grounp 4mm Photonic crystals,” Phys. Rev. B. **56**, 7313 (1997). [CrossRef]

28. C. M. Anderson and K. P. Giapis, “Larger Two-Dimensional Photonic Band Gaps,” Phys. Rev. Lett. **77**, 2949 (1996). [CrossRef] [PubMed]

18. C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys. Rev. E. **56**, 6291 (1997). [CrossRef]

24. X. Wang, B. Gu, Z. Li, and G. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B. **60**, 11417 (1999). [CrossRef]

25. R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. **90**, 4307 (2001). [CrossRef]

29. M. Qiu and S. He, “Large Complete band gap in two-dimensional photonic crystals with elliptic air holes,” Phys. Rev. B. **60**, 10610 (1999). [CrossRef]

## 2. Band gap optimization

*a*/17, where

*a*is the lattice constant. The resolution, i.e., the minimum feature size, is actually the spatial step Δ used to discretize the unit cell of a PBG structure before optimization. The dispersion properties of the sampled grid, which contains the initial unit cell, are then determined using the Plane Wave Method (PWM) [32

32. K.M. Leung and Y.F. Liu, “Full Vector Wave Calculation of Photonic Band Structures in Face-Centered-Cubic Dielectric Media,” Phys. Rev. Lett. **65**, 2646 (1990). [CrossRef] [PubMed]

33. C. Lemmi, S. Ledesma, J. Campos, and M. Villarreal, “Gray-level computer-generated hologram filters for multiple-object correlation,” Appl. Opt. **39**, 1233 (2000). [CrossRef]

*N*, the solution space will be 2

*. By assuming a diagonal symmetry in one unit cell we can significantly reduce the solution space to the order of 2*

^{N}*, where*

^{N/4}*N*can be selected according to computational cost and possible application-specific constraints, such as fabrication constraints. In our examples, we choose the spatial step size, i.e., resolution, to be Δ =

*a*/17, which thereby represents the discretization of the unit cell that we will optimize. At this point, we then draw two diagonals of this unit cell to partition it into four parts. Since we already assumed diagonal symmetry for the PBG structures, these four parts are identical. As such, we can take only one part for optimization and obtain the other parts through symmetry relations within the unit cell. Next, we number the grid points, including those crossed by the diagonals. For example, if we take Δ =

*a*/3, the grid points that need to be optimized can be numbered as shown in Fig. 2, for both the hexagonal lattice and the square case.

34. V. Boutenko and R. Chevallier, “Second order direct binary search algorithm for the synthesis of computergenerated holograms,” Opt. Commun. **125**, 43 (1996). [CrossRef]

*GXM*for the rectangular lattice case and

*GKM*for the triangular lattice case, the arrows on the thick lines indicate the directions of the sampled wave vector,

*k⇀*. Symmetry constraints can be easily applied in the optimization by changing the grids to inverse and defining the irreducible Brillouin zones for bandgap calculation.

## 3. Example designs

_{r}= 12.96) cylinders in air, with a filling factor of 0.143. This has been shown to be one of the best honeycomb lattice structures in terms of MAGTMR [25

25. R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. **90**, 4307 (2001). [CrossRef]

*a*/17 to discretize the unit cell of the initial structure. Figure 4 shows the MAGTMR convergence verse iteration times for the optimization of this structure.

25. R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. **90**, 4307 (2001). [CrossRef]

*a*/17 increase its MAGTMR from 13.6% to 21.2%. The corresponding results are shown in Fig. 7.

## 4. Conclusion

## References and links

1. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

2. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

3. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

4. | A. R. McGurn, “Photonic crystal circuits: A theory for two- and three-dimensional networks,” Phys. Rev. B |

5. | M. Notomi, A. Shinya, E. Kuramochi, I. Yokohama, C. Takahashi, K. Yamada, J. Takahashi, T. Kawashima, and S. Kawakami, “Si-based photonic crystals and photonic-bandgap waveguides,” IEICE Trans. Electro. |

6. | S. John and M. Florescu, “Photonic bandgap materials:towards an all-optical micro-transistor,” J. Opt. A:Pure Appl. Opt. |

7. | O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science |

8. | B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic Crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. |

9. | A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” Appl. Phy. Lett. |

10. | K. Nam, “Photonic Crystals,” http://www.phys.ksu.edu/~namkv/photonic.html. |

11. | J. Moosburger, M. Kamp, F. Klopf, M. Fischer, and A. Forchel, “Fabrication of semiconductor lasers with 2D-photonic crystal mirrors using a wet oxidized Al2O3-mask,” Microelectron. Eng. |

12. | T. D. Happ, A. Markard, M. Kamp, J. L. Gentner, and A. Forchel, “Short cavity InP-lasers with 2D photonic crystal mirrors,” presented at Optoelectronics, 2001. |

13. | J. S. Shirk, R. G. S. Pong, S. R. Flom, and E. A. Bolden, “Nonlinear 2-d Photonic Crystals for Optical Limiting,” http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/. |

14. | M. Imada, S. Noda, A. Chutinan, M. Mochizuk, and T. Tanaka, “Channel Drop Filter Using a Single Defect in a 2-D Photonic Crystal Slab Waveguide,” J. Lightwave Technol. |

15. | M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A |

16. | Z. Li, J. Wang, and B. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys.Rev. B. |

17. | Z. Li, B. Gu, and G. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. |

18. | C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys. Rev. E. |

19. | D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B. |

20. | E. Yablonovitch, T. J. Gmitter, and K.M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. |

21. | F. Gadot, A. Chelnokov, A. D. Lustrac, P. Crozat, J.-M. Lourtioz, D. Cassagne, and C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. |

22. | P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B |

23. | W. H. R. Hillebrand and W. Harms, “Theoretical Band Gap Studies of Two-Dimensional Photonic Crystals with Varying Column Roundness,” Phys. Stat. Sol. |

24. | X. Wang, B. Gu, Z. Li, and G. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B. |

25. | R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. |

26. | P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square lattices: Square and circular rods,” Phys. Rev. B. |

27. | C. M. Anderson and K. P. Giapis, “Symmetry reduction in grounp 4mm Photonic crystals,” Phys. Rev. B. |

28. | C. M. Anderson and K. P. Giapis, “Larger Two-Dimensional Photonic Band Gaps,” Phys. Rev. Lett. |

29. | M. Qiu and S. He, “Large Complete band gap in two-dimensional photonic crystals with elliptic air holes,” Phys. Rev. B. |

30. | R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. |

31. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys.Rev.Lett. |

32. | K.M. Leung and Y.F. Liu, “Full Vector Wave Calculation of Photonic Band Structures in Face-Centered-Cubic Dielectric Media,” Phys. Rev. Lett. |

33. | C. Lemmi, S. Ledesma, J. Campos, and M. Villarreal, “Gray-level computer-generated hologram filters for multiple-object correlation,” Appl. Opt. |

34. | V. Boutenko and R. Chevallier, “Second order direct binary search algorithm for the synthesis of computergenerated holograms,” Opt. Commun. |

35. | N. Wang, Y. Chen, Z. Nakao, S. Tamura, and H. Aritome, “Sythesis of Binary Computer-generated holograms based on a coding and frequency domain optimization algorithm,” International J. Optoelectronics |

**OCIS Codes**

(000.0000) General : General

(350.4600) Other areas of optics : Optical engineering

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 14, 2003

Revised Manuscript: February 11, 2003

Published: February 24, 2003

**Citation**

Caihua Chen, Ahmed Sharkway, Shouyuan Shi, and Dennis Prather, "Synthesis of 2-dimensional photonic crystals," Opt. Express **11**, 317-323 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-4-317

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### References

- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987). [CrossRef] [PubMed]
- J. D.Joannopoulos, R. D.Meade, and J. N.Winn, Photonic Crystals:Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).
- E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- A. R.McGurn, "Photonic crystal circuits: A theory for two- and three-dimensional networks," Phys. Rev. B 61, 13235 (2000). [CrossRef]
- M. Notomi, A. Shinya, E. Kuramochi, I. Yokohama, C. Takahashi, K. Yamada, J. Takahashi, T. Kawashima, and S. Kawakami, "Si-based photonic crystals and photonic-bandgap waveguides," IEICE Trans. Electro. E85C, 1025 (2002).
- S. John and M. Florescu, "Photonic bandgap materials:towards an all-optical micro-transistor," J. Opt. A:Pure Appl. Opt. 3, S103 (2001). [CrossRef]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P.D. Dapkus, and I. Kim, "Two-Dimensional Photonic Band-Gap Defect Mode Laser," Science 284, 1819 (1999). [CrossRef] [PubMed]
- B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, "Photonic Crystal-based resonant antenna with a very high directivity," J. Appl. Phys. 87, 603 (2000). [CrossRef]
- A. Ferrando and J. J. Miret, "Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers," Appl. Phy. Lett. 78, 3184 (2001). [CrossRef]
- K. Nam, "Photonic Crystals," <a href="http://www.phys.ksu.edu/~namkv/photonic.html">http://www.phys.ksu.edu/~namkv/photonic.html</a>.
- J. Moosburger, M. Kamp, F. Klopf, M. Fischer, and A. Forchel, "Fabrication of semiconductor lasers with 2Dphotonic crystal mirrors using a wet oxidized Al2O3-mask," Microelectron. Eng. 57, 1017 (2001). [CrossRef]
- T. D. Happ, A. Markard, M. Kamp, J. L. Gentner, and A. Forchel, "Short cavity InP-lasers with 2D photonic crystal mirrors," presented at Optoelectronics, 2001.
- J. S. Shirk, R. G. S. Pong, S. R. Flom, and E. A. Bolden, "Nonlinear 2-d Photonic Crystals for Optical Limiting," <a href="http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/">http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/</a>.
- M. Imada, S. Noda, A. Chutinan, M. Mochizuk, and T. Tanaka, "Channel Drop Filter Using a Single Defect in a 2-D Photonic Crystal Slab Waveguide," J. Lightwave Technol. 20, 873 (2002). [CrossRef]
- M. Florescu and S. John, "Single-atom switching in photonic crystals," Phys. Rev. A 64, 033801 (2001). [CrossRef]
- Z. Li, J. Wang, and B. Gu, "Creation of partial band gaps in anisotropic photonic-band-gap structures," Phys. Rev. B 58, 3721 (1998). [CrossRef]
- Z. Li, B. Gu, and G. Yang, "Large Absolute Band Gap in 2D Anisotropic Photonic Crystals," Phys. Rev. Lett. 81, 2574 (1998). [CrossRef]
- C. S. Kee, J. E. Kim, and H. Y. Park, "Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air," Phys. Rev. E 56, 6291 (1997). [CrossRef]
- D. Cassagne, C. Jouanin, and D. Bertho, "Hexagonal photonic-band-gap structures," Phys. Rev. B 53, 7134 (1996). [CrossRef]
- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295 (1991). [CrossRef] [PubMed]
- F. Gadot, A. Chelnokov, A. D. Lustrac, P. Crozat, J.-M. Lourtioz, D. Cassagne, and C. Jouanin, "Experimental demonstration of complete photonic band gap in graphite structure," Appl. Phys. Lett. 71, 1780 (1997). [CrossRef]
- P. R. Villeneuve and M. Piche, "Photonic band gaps in two-dimensional square and hexagonal lattices," Phys. Rev. B 46, 4969 (1992). [CrossRef]
- W. H. R.Hillebrand, and W.Harms, "Theoretical Band Gap Studies of Two-Dimensional Photonic Crystals with Varying Column Roundness," Phys. Stat. Sol. 217, 981 (2000). [CrossRef]
- X. Wang, B. Gu, Z. Li, and G. Yang, "Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices," Phys. Rev. B 60, 11417 (1999). [CrossRef]
- R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, "Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals," J. Appl. Phys. 90, 4307 (2001). [CrossRef]
- P. R. Villeneuve and M. Piche, "Photonic band gaps in two-dimensional square lattices: Square and circular rods," Phys. Rev. B 46, 4973 (1992). [CrossRef]
- C. M. Anderson and K. P. Giapis, "Symmetry reduction in grounp 4mm Photonic crystals," Phys. Rev. B 56, 7313 (1997). [CrossRef]
- C. M. Anderson and K. P. Giapis, "Larger Two-Dimensional Photonic Band Gaps," Phys. Rev. Lett. 77, 2949 (1996). [CrossRef] [PubMed]
- M. Qiu and S. He, "Large Complete band gap in two-dimensional photonic crystals with elliptic air holes," Phys. Rev. B 60, 10610 (1999). [CrossRef]
- R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495 (1992). [CrossRef]
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152 (1990). [CrossRef] [PubMed]
- K. M. Leung and Y. F. Liu, "Full Vector Wave Calculation of Photonic Band Structures in Face-Centered-Cubic Dielectric Media," Phys. Rev. Lett. 65, 2646 (1990). [CrossRef] [PubMed]
- C. Lemmi, S. Ledesma, J. Campos, and M. Villarreal, "Gray-level computer-generated hologram filters for multiple-object correlation," Appl. Opt. 39, 1233 (2000). [CrossRef]
- V. Boutenko and R. Chevallier, "Second order direct binary search algorithm for the synthesis of computergenerated holograms," Opt. Commun. 125, 43 (1996). [CrossRef]
- N. Wang, Y. Chen, Z. Nakao, S.Tamura, and H. Aritome, "Sythesis of Binary Computer-generated holograms based on a coding and frequency domain optimization algorithm," International J. Optoelectronics 12, 69 (1998).

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