## Two-dimensional photonic crystals constructed with a portion of photonic quasicrystals

Optics Express, Vol. 15, Issue 10, pp. 5991-5996 (2007)

http://dx.doi.org/10.1364/OE.15.005991

Acrobat PDF (469 KB)

### Abstract

Photonic quasicrystals (PQs) can produce interesting photonic properties. However, the exact prediction on photonic band structures (PBSs) of the PQs is currently a fundamental challenge due to the lack of periodicity of the structures. Here, we propose a kind of complex periodic two-dimensional photonic crystal (PC) structures constructed with a small portion of different PQs for the purposes of overcoming the difficulty of numerical calculations on the PBSs but maintaining the photonic properties of the original PQs owned. Theoretically calculated results on PBSs of the complex PCs with a local feature consistent with 12-fold rotational symmetry show that, in the cases of dielectric cylinders in air, air-holes in a dielectric, and metal cylinders in air, respectively, the complex PCs can indeed produce similar photonic properties of the original 12-fold PQs such as the uniform or isotropic PBGs under much lower dielectric contrast etc. Because the complex PCs can be constructed with the local parts of any high symmetric PQs, we believe that the PCs presented in this article may provide a way for creating novel photonic functional materials.

© 2007 Optical Society of America

## 1. Introduction

1. Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. **80**, 956–959 (1998). [CrossRef]

4. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature **436**, 993–996 (2005). [CrossRef] [PubMed]

5. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature **404**, 740–743 (2000). [CrossRef] [PubMed]

10. X. Wang, C. Y. Ng, W. T. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. **15**, 1526–1528 (2003). [CrossRef]

7. M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals: from diffraction pattern to band structure,” Nanotechnology **11**, 274–280 (2000). [CrossRef]

3. C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. **75**, 1848–1850 (1999). [CrossRef]

1. Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. **80**, 956–959 (1998). [CrossRef]

## 2. Model of the complex 2D PC structures

**a**1= (1 + √3)a(1,0) and

**a**2= (1 + √3)a(1/2,√3/2), of which, the central cylinder is surrounded by six ones at the positions

**u**1 and

**-u**1,

**u**2 and

**-u**2, and

**u**3 and

**-u**3 with

**u**1 = a(√3/2,1/2),

**u**2 = a(-√3/2,1/2), and

**u**3=a(0,1), respectively (a is the lattice constant). The whole structure is a perfect periodic triangular lattice. Hence dielectric constants of the PC structures can be written as

*ε*(

**r**+

**R**

_{I})=

*ε*(

**r**), where

**r**is the position vector. For any integers

*l*

_{12},

**R**

_{I}=

*l*

_{1}

**a**1+

*l*

_{2}

**a**2 defines the Bravais lattice. Thus, PBSs of the PCs can exactly be calculated.

## 3. PBSs of the complex 2D PCs under the situations of dielectric cylinders in air, air-hole in dielectric, and metal cylinders in air, respectively

13. 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**, 2646–2649 (1990). [CrossRef] [PubMed]

*ω*is the frequency, c is the velocity of light in vacuum. In 2D case, we can get κ(

**G**,

**G**′)=∣

**k**+

**G**′∣∣

**k**+

**G**∣

*η*(

**G**-

**G**′) for E- polarized modes (electric field parallel to the cylinders) and κ(

**G**,

**G**′) = (

**k**+

**G**)(

**k**+

**G**′)

*η*(

**G**-

**G**′) for H- polarized modes (magnetic field parallel to the cylinders), where

**k**is the reciprocal vector lying inside the Brillouin zone,

**G**is the translation vector of reciprocal lattice, and

*η*(

**G**) is a matrix found by inverting the Fourier transform of the dielectric constant [14–16

14. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B **44**, 8565–8571 (1991). [CrossRef]

*ε*(

**G**)= (∫

_{∣a1×a2∣}

*ε*(

**R**)exp(-i∙(

_{I}**G**∙

**R**))d

_{I}**R**)/ ∣

_{I}**a**1×

**a**2 ∣, where the integration is taken over the primitive cell with area ∣

**a**1×

**a**2 ∣. We see that

*ε*(

**G**) plays a central role on the formation of PBSs for both polarizations.

**G**∣, ε

_{a}and ε

_{b}are the dielectric constants of cylinders and background, respectively. R is the radii of the cylinders, and f=7f

_{0}is the filling fraction defined as the fraction of the primary cell area occupied by the cylinders (f

_{0}=

*π*∙R

^{2}/ ∣

**a**1×

**a**2 ∣). J

_{1}(GR) is the first-order Bessel function. Substituting Eq. (2) into Eq. (1), we can calculate out the PBSs of the proposed PCs [14

14. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B **44**, 8565–8571 (1991). [CrossRef]

15. P. R. Villeneuve and M. Piche, “Photonic bandgaps: What is best numerical representation of periodic structures,” J. Mod. Opt. **41**, 241–256 (1994). [CrossRef]

*ε*

_{a}=13 and R/a =0.35) embedded in air (

*ε*

_{b}=1) for both polarizations. One can see that, for H-polarization, six band gaps appear in the PBSs in the normalized frequency ranges (

*ω*a/2π c) between 0.291 and 0.318 (H-1), 0.355 and 0.373 (H-2), 0.426 and 0.433 (H-3), 0.553 and 0.574 (H-4), 0.578 and 0.598 (H-5), and 0.608 and 0.622 (H-6), respectively [Fig. 2(a)]. For E-polarization, there exist three PBGs between 0.201 and 0.262 (E-1), 0.357 and 0.437 (E-2), and 0.536 and 0.617 (E-3), respectively [Fig. 2(b)]. Furthermore, some PBGs for both polarizations can overlap with each other to form a series of complete PBGs. For instance, gaps H-2 and E-2 overlapped with each other around the central frequency

*ω*=0.365 can produce a complete PBG with width Δ

_{g}*ω*= 0.016 and a gap-width ratio Δ

*ω*/

*ω*=4.33%. Other complete gaps such as formed by gaps H-3 and E-2 around

_{g}*ω*=0.429 with Δ

_{g}*ω*= 0.007 and Δ

*ω*/

*ω*) =1.58%, H-4 and E-3 at

_{g}*ω*=0.563 with Δ

_{g}*ω*= 0.021 and Δ

*ω*/

*ω*=3.73%, H-5 and E-3 at

_{g}*ω*=0.588 with Δ

_{g}*ω*= 0.021 and Δ

*ω*=3.49%, and H-6 and E-3 at

_{g}*ω*=0.613 with Δ

_{g}*ω*= 0.009 and Δ

*ω*

*ω*=1.53%, respectively, are also observed in the PBSs of the PC.

_{g}*ε*

_{a}>4.5 is needed. However, for the simple triangular structures (one cylinder in a primitive cell), to open up a H-polarized PBG, at least

*ε*

_{a}>9 is needed. Furthermore, the present PC can even produce a complete gap around

*ω*=0.890 as

_{g}*ε*

_{a}is equal to 4.5 (not shown here). While for the simple triangular structures, no any complete gap will appear even if the structure is with any high dielectric contrast [14

14. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B **44**, 8565–8571 (1991). [CrossRef]

17. C. M. Anderson and K. P. Giapis, “Large two-dimensional photonic band gaps,” Phys. Rev. Lett. **77**, 2949–2951 (1996). [CrossRef] [PubMed]

18. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. **81**, 2574–2576 (1998). [CrossRef]

*ε*

_{a}=1,

*ε*

_{b}=13, and R/a =0.47 for E-polarization. One sees that there exist five PBGs in the frequency regions between 0.173 and 0.179 (E-1), 0.203 and 0.220 (E-2), 0.407 and 0.458 (E-3), 0.476 and 0.492 (E-4), and 0.518 and 0.543 (E-5), respectively. The gap widths Δ

*ω*and the corresponding gap-width ratios Δ

*ω*/

*ω*are 0.007 and 3.81%, 0.017 and 7.86%, 0.052 and 11.93%, 0.016 and 3.29%, and 0.024 and 4.58%, respectively. The lowest dielectric contrast for opening up a PBG is just

_{g}*ε*

_{b}>3.5. For a simple triangular structure, however, to open up a PBG, around

*ε*

_{b}>6 should be satisfied for Ref. [14

**44**, 8565–8571 (1991). [CrossRef]

*ε*

_{b}=13, only one gap similar to E-3 of the present PCs is opened up for the simple triangular structures [14

**44**, 8565–8571 (1991). [CrossRef]

5. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature **404**, 740–743 (2000). [CrossRef] [PubMed]

6. X. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B **63**, 081105 (2001). [CrossRef]

6. X. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B **63**, 081105 (2001). [CrossRef]

*ε*(

*ω*) = 1 - (

*ω*

^{2}

_{p}/

*ω*

^{2}), where

*ω*is the plasma frequency of the conduction electrons. Metals constructed PCs have attracted tremendous interests because metals can not only effectively enhance the dielectric contrast of the PCs, but also make optical properties of the PCs tunable by external fields [19

_{p}19. J. Zhou, Y. Zhou, S. L. Ng, H. Zhang, W. X. Que, Y. L. Lam, Y. C. Chan, and C. H. Kam, “Three-dimensional photonic band gap structure of a polymer-metal composite,” Appl. Phys. Lett. **76**, 3337–3339 (2000). [CrossRef]

20. W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. **84**, 2853–2856 (2000). [CrossRef] [PubMed]

15. P. R. Villeneuve and M. Piche, “Photonic bandgaps: What is best numerical representation of periodic structures,” J. Mod. Opt. **41**, 241–256 (1994). [CrossRef]

*ω*a/2

*π*c= 0.6 to 1.1 (the lowest frequency band does not tend to zero [16

16. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two -dimensional systems containing metallic components,” Phys. Rev. B **50**, 16835–16844 (1994). [CrossRef]

*ω*a/2

*π*c= 0.710 and 0.816 (E-1), and 0.835 and 0.866 (E-2) appear in the band structures. The corresponding gap widths and gap-width ratios are Δ

*ω*= 0.101 and Δ

*ω*/

*ω*=13.17% and

_{g}*ω*= 0.031 and Δ

*ω*/

*ω*=3.64%, respectively. In metals constructed simple triangular lattices, however, there exists no E-polarized PBG in the PBSs [16

_{g}16. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two -dimensional systems containing metallic components,” Phys. Rev. B **50**, 16835–16844 (1994). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. |

2. | S. S. M. Cheng, L. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B |

3. | C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. |

4. | W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature |

5. | M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature |

6. | X. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B |

7. | M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals: from diffraction pattern to band structure,” Nanotechnology |

8. | K. Wang, “Light wave states in two-dimensional quasiperiodic media,” Phys. Rev. B |

9. | D. T. Roper, D. M. Beggs, M. A. Kaliteevski, S. Brand, and R. A. Abram, “Properties of two-dimensional photonic crystals with octagonal quasicrystalline unit cells,” J. Mod. Opt. |

10. | X. Wang, C. Y. Ng, W. T. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. |

11. | Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quaiscrystals by single-beam holography,” Appl. Phys. Lett. |

12. | Y. Yang and G. P. Wang, “Realization of periodic and quasiperiodic microstructures with sub-diffraction-limit feature sizes by far-field holographic lithography,” Appl. Phys. Lett. |

13. | K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. |

14. | M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B |

15. | P. R. Villeneuve and M. Piche, “Photonic bandgaps: What is best numerical representation of periodic structures,” J. Mod. Opt. |

16. | V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two -dimensional systems containing metallic components,” Phys. Rev. B |

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

18. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. |

19. | J. Zhou, Y. Zhou, S. L. Ng, H. Zhang, W. X. Que, Y. L. Lam, Y. C. Chan, and C. H. Kam, “Three-dimensional photonic band gap structure of a polymer-metal composite,” Appl. Phys. Lett. |

20. | W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 2, 2007

Revised Manuscript: April 16, 2007

Manuscript Accepted: April 17, 2007

Published: May 1, 2007

**Citation**

Yi Yang and Guo Ping Wang, "Two-dimensional photonic crystals constructed with a portion of photonic quasicrystals," Opt. Express **15**, 5991-5996 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-5991

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### References

- Y. S. Chan, C. T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998). [CrossRef]
- S. S. M. Cheng, L. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091-4099 (1999). [CrossRef]
- C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and B. Sun, "Band gap and wave guiding effect in a quasiperiodic photonic crystal," Appl. Phys. Lett. 75, 1848-1850 (1999). [CrossRef]
- W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, "Experimental measurement of the photonic properties of icosahedral quasicrystals," Nature 436, 993-996 (2005). [CrossRef] [PubMed]
- M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000). [CrossRef] [PubMed]
- X. Zhang, Z. Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, 081105 (2001). [CrossRef]
- M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. DeLa Rue, and P. Millar, "Two-dimensional Penrose-tiled photonic quasicrystals: from diffraction pattern to band structure," Nanotechnology 11, 274-280 (2000). [CrossRef]
- K. Wang, "Light wave states in two-dimensional quasiperiodic media," Phys. Rev. B 73, 235122 (2006). [CrossRef]
- D. T. Roper, D. M. Beggs, M. A. Kaliteevski, S. Brand and R. A. Abram, "Properties of two-dimensional photonic crystals with octagonal quasicrystalline unit cells," J. Mod. Opt. 53, 407-416 (2006). [CrossRef]
- X. Wang, C. Y. Ng, W. T. Tam, C. T. Chan and P. Sheng, "Large-area two-dimensional mesoscale quasi-crystals," Adv. Mater. 15, 1526-1528 (2003). [CrossRef]
- Y. Yang, S. Zhang, and G. P. Wang, "Fabrication of two-dimensional metallodielectric quaiscrystals by single-beam holography," Appl. Phys. Lett. 88, 251104 (2006). [CrossRef]
- Y. Yang and G. P. Wang, "Realization of periodic and quasiperiodic microstructures with sub-diffraction-limit feature sizes by far-field holographic lithography," Appl. Phys. Lett. 89, 111104 (2006). [CrossRef]
- 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, 2646-2649 (1990). [CrossRef] [PubMed]
- M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
- P. R. Villeneuve and M. Piche, "Photonic bandgaps: What is best numerical representation of periodic structures," J. Mod. Opt. 41, 241-256 (1994). [CrossRef]
- V. Kuzmiak, A. A. Maradudin and F. Pincemin, "Photonic band structures of two -dimensional systems containing metallic components," Phys. Rev. B 50, 16835-16844 (1994). [CrossRef]
- C. M. Anderson and K. P. Giapis, "Large two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2951 (1996). [CrossRef] [PubMed]
- Z. Y. Li, B. Y. Gu, and G. Z. Yang, "Large absolute band gap in 2D anisotropic photonic crystals," Phys. Rev. Lett. 81, 2574-2576 (1998). [CrossRef]
- J. Zhou, Y. Zhou, S. L. Ng, H. Zhang, W. X. Que, Y. L. Lam, Y. C. Chan, and C. H. Kam, "Three-dimensional photonic band gap structure of a polymer-metal composite," Appl. Phys. Lett. 76, 3337-3339 (2000). [CrossRef]
- W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, " Robust photonic band gap from tunable scatterers," Phys. Rev. Lett. 84, 2853-2856 (2000). [CrossRef] [PubMed]

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