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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 4 — Feb. 24, 2003
  • pp: 317–323
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Synthesis of 2-dimensional photonic crystals

Caihua Chen, Ahmed Sharkway, Shouyuan Shi, and Dennis W. Prather  »View Author Affiliations


Optics Express, Vol. 11, Issue 4, pp. 317-323 (2003)
http://dx.doi.org/10.1364/OE.11.000317


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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

2. Band gap optimization

The main idea behind this method is the sampling of the unit cell by a factor of 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]

]. Subsequent to analysis, the material properties within the sampled unit cell are systematically changed until either performance criteria, e.g., wider band gap, is obtained or convergence is observed.

Because a photonic structure is typically composed of two materials: a background material, A, and a constituent material, B, the discretized the unit cell of a structure consists of many grids points, where each point is filled either by the material A or material B, as shown in Fig. 1 for the case of a honeycomb lattice with hexagonal high dielectric cylinders in air with a filling factor of 0.143.

It should be noted that in Fig. 1, we use black grids to represent high dielectric materials and white grids to represent air. From this point of view, any arbitrary PBG structure can be parameterized as a binary structure. Consequently, the optimization of this type of structure is very similar to the optimization of Computer Generated Holograms (CGH) [33

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]

35

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 12, 69 (1998).

]. Thus, if the total number of grid points for one discretized unit cell is N, the solution space will be 2N. By assuming a diagonal symmetry in one unit cell we can significantly reduce the solution space to the order of 2N/4, where 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.

Fig. 1. The discretized unit cell of a honeycomb lattice structure of dielectric cylinders in air with a resolution a/17
Fig. 2. The unit cells and the numbered grids used for synthesis (a) the triangular/honeycomb lattice case (b) the rectangular lattice case

Fig. 3. The brillouin zones and the irreducible brillouin zones for unit cell with diagonal symmetry (a) the rectangular lattice case (b) the triangular lattice case

Where the regions inside the thick black lines represent the Brillouin zones. The shadow regions are the irreducible Brillouin zones, which are defined by 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

We first implemented this procedure on a honeycomb lattice of hexagonal GaAs cylinders in air. The initial structure is the honeycomb lattice of hexagonal GaAs (εr = 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]

]. To optimize this structure, we choose the resolution Δ=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.

Fig. 4. The convergence of the maximum absolute bandgap-to-midgap ratio for optimization of a honeycomb lattice of hexagonal GaAs cylinders in air Figure 4 indicates that some

300 iterations are needed in this case. For a structure with diagonal symmetry which we applied in the optimization, each iteration takes about 10 seconds, and in total, 300 iterations need 3000 seconds, which is less than one hour.

The unit cells for both initial and optimized structures and the dispersion diagrams for these two structures are shown in Fig. 5.

Fig. 5. Optimization of a honeycomb lattice of hexagonal GaAs cylinders in air (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram

Where the blue solid lines represent TM dispersion curves, the red dotted lines represent TE dispersion curves and the yellow shadow regions represent absolute band gap for both TE and TM modes. The rest paper uses the same representations, unless indicated specifically. For initial structure, the MAGTMR is 5.76%, while for the optimized structure the MAGTMR was numerically measured to be 13.29%. This represents an improvement of more than twice that for the initial structure. As additional examples, we optimized two of the best found hexagonal and square lattice structures, as reported by Wang et al. [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]

]. One is a hexagonal lattice consisting of hexagonal air holes with the orientation θ = 24° in GaAs and the filling factor ƒ = 0.805. Figure 6 shows the unit cells and the dispersion diagrams for the structures before and after optimization.

Fig. 6. Optimization of a hexagonal lattice of hexagonal air holes with orientation θ = 24≊ in GaAs and the filling factor ƒ = 0.805 (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram

In this case, the MAGTMR is 20.71% for the initial structure, and for the optimized structure the MAGTMR is 21.42%.

The other structure is the square lattice of square air holes with the orientation θ = 30° in GaAs with the filling factor ƒ = 0.68. Similarly, applying the procedure presented above on this structure with a resolution of Δ = a/17 increase its MAGTMR from 13.6% to 21.2%. The corresponding results are shown in Fig. 7.

Fig. 7. Optimization of a square lattice of square air holes with orientation θ = 30° in GaAs and the filling factor ƒ = 0.68 (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram

4. Conclusion

In conclusion, we have presented an efficient synthesis procedure for improving MAGTMR of 2-D photonic crystal structures. The basic steps of this procedure include: (1) discretizing the unit cell of a PBG structure, (2) choosing the number of grid points for optimization according to the required symmetry in the unit cell and creating a binary sequence for parameterizing the unit cell, (3) applying the DBS method to optimize this binary sequence, by retrieving the PBG structure from the binary sequence and calculating the MAGTMR of the corresponding structure with PWM, and (4) repeating this sequence until optimal performance is achieved. Three examples have been shown to illustrate the utility of the procedure, with one of them showing an improvement of more than 200% in the MAGTMR.

References and links

1.

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

2.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals:Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

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]

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. E85C, 1025 (2002).

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.

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]

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. 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]

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]

19.

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

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. 71, 1780 (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]

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]

26.

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]

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]

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]

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]

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]

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]

34.

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

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 12, 69 (1998).

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

  1. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987). [CrossRef] [PubMed]
  2. J. D.Joannopoulos, R. D.Meade, and J. N.Winn, Photonic Crystals:Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).
  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]
  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. E85C, 1025 (2002).
  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," <a href="http://www.phys.ksu.edu/~namkv/photonic.html">http://www.phys.ksu.edu/~namkv/photonic.html</a>.
  11. 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]
  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," <a href="http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/">http://www.ee.ucla.edu/~pbmuri/1999-review/shirk/</a>.
  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]
  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]
  19. D. Cassagne, C. Jouanin, and D. Bertho, "Hexagonal photonic-band-gap structures," Phys. Rev. B 53, 7134 (1996). [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]
  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. 71, 1780 (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]
  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]
  26. 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]
  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]
  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]
  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]
  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]
  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]
  34. V. Boutenko and R. Chevallier, "Second order direct binary search algorithm for the synthesis of computergenerated holograms," Opt. Commun. 125, 43 (1996). [CrossRef]
  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 12, 69 (1998).

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