OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 10 — May. 14, 2007
  • pp: 5991–5996
« Show journal navigation

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

Yi Yang and Guo Ping Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 10, pp. 5991-5996 (2007)
http://dx.doi.org/10.1364/OE.15.005991


View Full Text Article

Acrobat PDF (469 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Fig. 1. (color online) Scheme of the proposed 2D PC structures constructed with a portion of 12-fold symmetric PQ (orange circles). The primitive cell (dark diamonds) included seven cylinders is defined by two primitive translation vectors a1 and a2 .Six cylinders surrounded the central one are at positions u1, -u1,u2, -u2 ,u3 and -u3 , respectively.

2. Model of the complex 2D PC structures

Figure 1 schematically shows the proposed 2D PCs constructed with a local part of 12-fold symmetric PQs (orange circles). The primitive cell (dark diamonds) of the structure is formed through two primitive translation vectors a1= (1 + √3)a(1,0) and a2= (1 + √3)a(1/2,√3/2), of which, the central cylinder is surrounded by six ones at the positions u1 and-u1, u2 and -u2, and u3 and -u3 with u1 = a(√3/2,1/2), u2 = a(-√3/2,1/2), and u3=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 a1+ l 2 a2 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

In this paper, we will use the plane wave method (PWM) [13

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]

] to calculate PBSs of the proposed PCs for getting a systemic and exact insight into their interesting optical properties. In the framework of PWM, band structure problem can be reduced to the eigenvalue problem

det(κω2c2)=0
(1)

Where ω is the frequency, c is the velocity of light in vacuum. In 2D case, we can get κ(G,G′)=∣k + G′∣∣ k + Gη(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]

]. The Fourier transform is written as: ε(G)= (∫aa2∣ ε(RI)exp(-i∙(GRI))dRI)/ ∣ aa2 ∣, where the integration is taken over the primitive cell with area ∣ aa2 ∣. We see that ε(G) plays a central role on the formation of PBSs for both polarizations.

In the cases of dielectrics constructed PCs as shown in Fig. 1, we perform

ε(G)=fεa+(1f)εb,G=0
(2a)
ε(G)=2(εaεb)(1+2i=13cos(Gui))f0J1(GR)(GR),G0
(2b)

Where G= ∣ G ∣, εa and εb are the dielectric constants of cylinders and background, respectively. R is the radii of the cylinders, and f=7f0 is the filling fraction defined as the fraction of the primary cell area occupied by the cylinders (f0 =π∙R2/ ∣ aa2 ∣). J1(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

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]

].

Figure 2 displays the PBSs of a proposed PC with dielectric cylinders (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 ωg =0.365 can produce a complete PBG with width Δ ω = 0.016 and a gap-width ratio Δω/ωg =4.33%. Other complete gaps such as formed by gaps H-3 and E-2 around ωg =0.429 with Δω = 0.007 and Δω/ωg) =1.58%, H-4 and E-3 at ωg =0.563 with Δω = 0.021 and Δω/ωg =3.73%, H-5 and E-3 at ωg =0.588 with Δω= 0.021 and Δωg =3.49%, and H-6 and E-3 at ωg =0.613 with Δω = 0.009 and Δω ωg =1.53%, respectively, are also observed in the PBSs of the PC.

Fig. 2. (color online) Calculated PBSs of the proposed 2D PCs constructed with a portion of 12-fold symmetric PQ with dielectric cylinders (ε a =13 and R/a =0.35) in air (ε b =1) for (a) H- and (b) E- polarized modes, respectively. The PBGs are indicated with pink shadow.

By searching for the dependence of PBGs of the proposed PCs on dielectric contrast, we find that, for H-polarization, to open up a PBG, only 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 ωg =0.890 as 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

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

, 18

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]

]. These are attributed to the fact that our proposed PCs are with a short-range feature consistent with local 12-fold rotational symmetry.

Fig. 3. (color online) Calculated PBSs of the proposed 2D PCs constructed with a portion of 12-fold symmetric PQ with air-holes (ε a =1 and R/a =0.47) in a dielectric background (ε b =13) for E- polarization. The PBGs are indicated with pink shadow.

In the following, we will consider the case that air-holes are embedded in a dielectric background. Figure 3 shows the calculated PBSs as the proposed PC is with 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 Δω and the corresponding gap-width ratios Δω/ωg 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 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

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]

]. Even with a contrast as ε b =13, only one gap similar to E-3 of the present PCs is opened up for the simple triangular structures [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]

].

Different from the dielectrics, metal materials show frequency dispersion and their dielectric functions can be expressed by the Drude formula: ε(ω) = 1 - (ω 2 p/ω 2), where ωp 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

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

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]

]. In the following, we will investigate into the PBSs of the proposed PCs constructed by embedding metal cylinders into air. The Fourier transforms of dielectric functions of the metallic PCs can be read as [15

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(G)=1fωp2ω2,G=0
(3a)
ε(G)=2(ωp2ω2)(1+2i=13cos(Gui))f0J1(GRGR),G0
(3b)

By combining Eqs. (1) and (3), we can calculate out the PBSs of the PCs for E-polarization (Fig. 4). The filling fraction is assumed as R/a =0.45 and the frequency range is from ωa/2πc= 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]

]). We see that two large PBGs between ωa/2πc= 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 Δω = 0.101 and Δω/ωg=13.17% and ω = 0.031 and Δω/ωg=3.64%, respectively. In metals constructed simple triangular lattices, however, there exists no E-polarized PBG in the PBSs [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]

] even if the same parameters are used.

Fig. 4. (color online) Calculated PBSs of the proposed 2D PCs constructed with a portion of 12-fold symmetric PQ with metal cylinders (R/a =0.45) in air for E- polarization. Two large PBGs are indicated with pink shadow.

4. Conclusion

In conclusion, we have proposed a kind of complex 2D PCs constructed with a portion of different PQs. As an example, we have demonstrated theoretically that the proposed PCs with a local feature consistent with 12-fold rotational symmetry can indeed maintain the similar photonic properties of the original PQs. Additionally, such PCs can be formed with the local parts of any high symmetric PQs. Therefore, we believe that the complex PCs proposed in this paper may offer a big potential for creating novel photonic functional materials.

Acknowledgments

We are greatly grateful to Profs. C. T. Chan and Z. Y. Liu for fruitful discussions. This work is financially supported by the program of New Century Excellent Talents in University (NCET-04-0678), 973 Program (Grant 2006CB302900), and the National Natural Science Foundation of China (Grant 60278018, 10574101).

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. 80, 956–959 (1998). [CrossRef]

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 59, 4091–4099 (1999). [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]

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]

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]

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]

8.

K. Wang, “Light wave states in two-dimensional quasiperiodic media,” Phys. Rev. B 73, 235122 (2006). [CrossRef]

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. 53, 407–416 (2006). [CrossRef]

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]

11.

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]

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. 89, 111104 (2006). [CrossRef]

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]

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]

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]

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. 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]
  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]
  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]
  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]
  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]
  8. K. Wang, "Light wave states in two-dimensional quasiperiodic media," Phys. Rev. B 73, 235122 (2006). [CrossRef]
  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. 53, 407-416 (2006). [CrossRef]
  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]
  11. 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]
  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. 89, 111104 (2006). [CrossRef]
  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]
  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]
  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]
  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]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited