## Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps

Optics Express, Vol. 13, Issue 11, pp. 4325-4330 (2005)

http://dx.doi.org/10.1364/OPEX.13.004325

Acrobat PDF (240 KB)

### Abstract

In holographic fabrication of photonic crystals the shape and size of the dielectric columns or particles (“atoms”) are determined by the isointensity surfaces of the interference field. Therefore their photonic band gap (PBG) properties are closely related to their fabrication design. As an example, we have investigated the PBGs of a kind of holographically formed two-dimensional (2-D) square lattice with pincushion columns rotated by 45°, and shown that this structure has complete PBGs in a wide range of dielectric contrast comparable to or even larger than those of the same lattice with square columns reported before. The optical design for making this structure is also given. This work may demonstrate the unique feature and advantages of photonic crystals made by holographic method and provide a guideline for their design and experimental fabrication.

© 2005 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

3. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. **61**, 495–497 (1992) [CrossRef]

10. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. **77**, 2574–2977 (1998). [CrossRef]

*p*) and TM (or

*s*) polarization modes and design methods to produce them.

3. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. **61**, 495–497 (1992) [CrossRef]

4. P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B **46**, 4969–4972 (1992). [CrossRef]

4. P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B **46**, 4969–4972 (1992). [CrossRef]

5. P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square lattices: Square and circular lattices,” Phys. Rev. B **46**, 4973–4975 (1992). [CrossRef]

6. D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B **10**, 399–403 (1993). [CrossRef]

10. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. **77**, 2574–2977 (1998). [CrossRef]

*p*polarization is favored if the dielectric regions are connected; and that the PBG property of a lattice is also dependent on the shape and size of the columns. Therefore it is not easy to give simple arguments predicting the existence of a complete PBG for a given lattice. Usually the trial and error method has to be used to find “good” structures [11

11. M. Qiu and S. He, “Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,” J. Opt. Soc. Am. A **17**, 1027–1030 (2000). [CrossRef]

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B **61**, 15519–15522 (2000). [CrossRef]

*ω/ω*=8.5% for

*ε*=8.9 (alumina) [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B **61**, 15519–15522 (2000). [CrossRef]

*ε*>12.3 for a complete PBG to exist [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B **61**, 15519–15522 (2000). [CrossRef]

## 2. Band gap analysis: relation between PBG and concrete structure

*c*=0 and

*I*=0, we can get a chessboard lattice with filling ratio (FR)

_{t}*f*=0.5. This structure can produce a complete band gap for a range of dielectric constant

*ε*>7 [12

**61**, 15519–15522 (2000). [CrossRef]

*c*in Eq. (1) and

*I*in fabrication process. Both of them have critical influences on PBG and therefore give us more freedom to improve band gap property. In the following we will make a systematic investigation on their effects. In the calculations we use plane wave method [18

_{t}18. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B **41**, 10188–10190 (1990). [CrossRef]

*s*and

*p*polarizations, and the accuracy within 1% is expected.

*I*

_{t}on the filling ratio, the column shape, and the PBG for a given

*c*, here

*c*is assumed to be 0.31 for calculations (the reason will be explained below). In Fig. 1 we draw the curves of FR versus

*I*

_{t}for both the normal and the inverse lattices. Fig. 2 gives the concrete shapes of several inverse lattices formed at different

*I*

_{t}. Evidently the shape and FR of the dielectric columns (black in Fig. 2) vary gradually with

*I*

_{t}. Generally a smaller It yields smaller dielectric columns with smaller FR, while a greater It gives rise to connected columns with a greater FR. The apparent difference here in holographic fabrication from the ideal chessboard design [12

**61**, 15519–15522 (2000). [CrossRef]

*I*

_{t}will lead to different PBG for a given intensity distribution. In Fig. 3 we give the curve of relative PBG versus It for the inverse structure in the case of

*c*=0.31 and

*ε*=8.9. The corresponding gap map is plotted in Fig. 4. From these two figures we can clearly see the general trend that the s gap is easier to occur for the case of non-overlapping dielectric columns (smaller

*I*

_{t}and

*f*), while the

*p*gap is favored in the case of air columns (greater

*I*

_{t}and

*f*); and the largest PBG occurs as a compromise of the two factors. However, different from the case of exact square columns, now the columns begin to be overlapped at a smaller

*f*, about 0.355 instead of 0.5, due to the pincushion shape of the columns. Figs. 3 and 4 indicate that the complete PBG can be obtained in the range of

*I*

_{t}=1.31 to 1.45 or equally

*f*=0.307 to 0.405; and that the largest relative band gap width appears in

*I*=1.37 when the corresponding FR is

_{t}*f*=0.347. In this optimal condition we have Δ

*ω/ω*=8.8 %, a little larger than the result of best design for square columns [12

**61**, 15519–15522 (2000). [CrossRef]

*c*in Fig. (1). Because the shape and size of dielectric columns and consequently the PBG of the resultant structure are dependent on both

*c*and

*I*

_{t}, we must scan

*I*

_{t}for each given

*c*to find the maximum PBG. In the scanning process the step size we used is 0.01 for both

*c*and

*I*

_{t}. In Fig. 6 we give some lattice pictures to show the column shape and size for several values of

*c*and its corresponding optimized

*I*

_{t}yielding maximum PBG. From these pictures a general rule may be concluded: The maximum PBG always appears for the structure in which the dielectric columns are just close to be but have not yet connected; and when the factor

*c*increases the required optimized

*I*

_{t}usually decreases leading to a smaller optimized FR

*f*.

*c*and the corresponding maximum relative band width Δ

*ω/ω*for

*ε*=8.9 is depicted by the curve in Fig. 7. It is clear from this figure that the complete PBG can be obtained in a quite wide range of

*c*, the maximum relative PBG (8.8 %) is realized for

*c*=0.31 (that is the reason we use it in above calculations), and the maximum Δ

*ω/ω*keeps a large value in a considerable range of

*c*, specifically, Δ

*ω/ω*>7% for

*c*=0.05~0.36. But we must indicate that the maximum PBG in Fig. 7 is reached for the optimal intensity threshold

*I*

_{t}, and generally different

*c*has different optimal It. As mentioned above, the optimal

*I*

_{t}decreases when

*c*increases, and the maximum PBG always occurs for the columns of pincushion shape rather than exact square shape. For example, the case of

*c*=0 and

*I*

_{t}=2 yields an exact chessboard lattice with square columns of

*f*=0.5, its PBG is 4.3 %, coincident with previous result [12

**61**, 15519–15522 (2000). [CrossRef]

*I*

_{t}=1.98 for the same light intensity distribution will generate non-square columns of

*f*=0.485 and the corresponding Δ

*ω/ω*=6.6%, considerably larger than the former. This fact has again convincingly verified the unique feature and usefulness of holographic method in band gap improvement.

**61**, 15519–15522 (2000). [CrossRef]

*c*=0.27 and

*I*

_{t}=1.51), lower than 7 in the case of square columns [12

**61**, 15519–15522 (2000). [CrossRef]

**61**, 15519–15522 (2000). [CrossRef]

## 3. Optical design of holographic fabrication

*θ*with z axis shown in Fig. 9 to fabricate this structure. In this geometry the four wave vectors are

*K*=2

*π/λ, λ*is the wavelength. This symmetric arrangement, which can be realized with the use of a diffraction beam splitter (DBS) [19

19. T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. **79**, 725–727 (2001). [CrossRef]

*a*=

*λ*/(2

^{1/2}sin

*θ*),

*e*

_{ij}=

*e*

_{i}·

*e*

_{j},

*e*

_{j}is the unit polarization vector of the

*j*th plane wave. Comparing the cosine terms in Eqs. (3) and (1), we can find that the varying parts in these two equations have the same relation on the condition

*l, m, n, p, q*and

*r*are all real numbers. Using normalization condition of each polarization vector, orthogonal condition of

*e*

_{j}and

**K**_{j}, and Eq. (4), we have

*c*and

*θ*we can obtain all the six parameters. A possible solution for

*c*=0.31 and

*θ*=30° is

*I*

_{t}’ corresponding to the

*I*

_{t}for Eq. (1) to generate the same structure should be modulated as

*I*

_{t}=1.37 for

*ε*=8.9 should now become

*I*

_{t}’=2.197, readers can verify the same FR

*f*=0.347 as mentioned before.

## 4. Conclusions

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

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

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

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

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

6. | D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B |

7. | C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. |

8. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode fiber with photonic crystal cladding,” Opt. Lett. |

9. | S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. |

10. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. |

11. | M. Qiu and S. He, “Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,” J. Opt. Soc. Am. A |

12. | M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B |

13. | M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature |

14. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. A |

15. | Y.A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bangap crystals,” Nature |

16. | X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express |

17. | X. L. Yang, L. Z. Cai, Q. Liu, and H. K. Liu, “Theoretical bandgap modeling of two-dimensional square photonic crystals fabricated by interference technique of three-noncoplanar beams,” J. Opt. Soc. Am. B |

18. | M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B |

19. | T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(220.4000) Optical design and fabrication : Microstructure fabrication

(260.2110) Physical optics : Electromagnetic optics

(260.3160) Physical optics : Interference

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 25, 2005

Revised Manuscript: May 18, 2005

Published: May 30, 2005

**Citation**

L. Cai, C. S. Feng, M.Z. He, X. L. Yang, X. Meng, G. Y. Dong, and X. Yu, "Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps," Opt. Express **13**, 4325-4330 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4325

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### References

- E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
- R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannapoulos, �??Existence of a photonic band gap in two- dimensions ,�?? Appl. Phys. Lett. 61, 495-497 (1992) [CrossRef]
- P. R. Villeneuve, M. Piche, �??Phonic band gaps in two-dimensional square and hexagonal lattices,�?? Phys. Rev. B 46, 4969-4972 (1992). [CrossRef]
- P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square lattices: Square and circular lattices,�?? Phys. Rev. B 46, 4973-4975 (1992). [CrossRef]
- D. L. Bullock, C. Shih, R. S. Margulies, �??Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,�?? J. Opt. Soc. Am. B 10, 399-403 (1993). [CrossRef]
- C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996). [CrossRef] [PubMed]
- J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, �??All-silica single-mode fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- S. Y. Lin, G. Arjavalingam, W. M. Robertson, �??Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,�?? J. Mod. Opt. 41, 385-393 (1994) [CrossRef]
- Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Large absolute band gaps in two-dimensional anisotropic photonic crystals,�?? Phys. Rev. Lett. 77, 2574-2977 (1998). [CrossRef]
- M. Qiu, S. He, �??Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,�?? J. Opt. Soc. Am. A 17, 1027-1030 (2000). [CrossRef]
- M. Agio, L. C. Andreanm, �??Complete photonic band gap in a two-dimensional chessboard lattice,�?? Phys. Rev. B 61, 15519-15522 (2000). [CrossRef]
- M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, �??Fabrication of photonic crystals for the visible spectrum by holographic lithography,�?? Nature 404, 53-56 (2000). [CrossRef] [PubMed]
- L. Z. Cai, X. L. Yang, Y .R. Wang, �??Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,�?? J. Opt. Soc. Am. A 19, 2238-2244 (2002). [CrossRef]
- Y.A. Vlasov, X. Z. Bo, J. C. Sturm, D. J. Norris, �??On-chip natural assembly of silicon photonic bangap crystals,�?? Nature 414, 289-293 (2001). [CrossRef] [PubMed]
- X. L. Yang, L. Z. Cai, Q. Liu, �??Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,�?? Opt. Express 11, 1050-1055 (2003). [CrossRef] [PubMed]
- X. L. Yang, L. Z. Cai, Q. Liu, H. K. Liu, �??Theoretical bandgap modeling of two-dimensional square photonic crystals fabricated by interference technique of three-noncoplanar beams,�?? J. Opt. Soc. Am. B 21, 1050-1055 (2004). [CrossRef]
- M. Leung, Y. F. Liu, �??Photon band structures: The plane-wave method,�?? Phys. Rev. B 41, 10188-10190 (1990). [CrossRef]
- T. Kondo, S. Matsuo, S. Juodkazis, H. Misawa, �??Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,�?? Appl. Phys. Lett. 79, 725-727 (2001). [CrossRef]

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