## Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three noncoplanar beams

Optics Express, Vol. 11, Issue 9, pp. 1050-1055 (2003)

http://dx.doi.org/10.1364/OE.11.001050

Acrobat PDF (651 KB)

### Abstract

Using the block-iterative frequency domain method and the non-orthogonal FDTD method, the photonic band gap (PBG) and spectral properties are investigated for a new class of two-dimensional (2-D) trigonal structures with an approximately circular or hexagonal “atom” shape formed by holographic lithography. Calculations of band structures as a function of the intensity threshold show that the PBG of 2-D titania arrays opens only for TM polarization, and directional PBG can open for TE and TM polarization simultaneously. In addition, up to four sizeable full PBGs can open for an inverted GaAs triangular structure.

© 2003 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]

2. S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

3. 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- (1993). [CrossRef]

4. 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. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

5. B. D’Urso, O. Painter, J.D. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal Microcavities,” J. Opt. Soc. Am. B. **15**, 1155–1159 (1998). [CrossRef]

6. 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 **404**, 53–56 (2000). [CrossRef] [PubMed]

10. X. L. Yang, L. Z. Cai, Y. R. Wang, and Q. Liu, “Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices,” Opt. Lett. **28**, 453–455 (2003). [CrossRef] [PubMed]

8. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. **26**, 1858–1860 (2001). [CrossRef]

8. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. **26**, 1858–1860 (2001). [CrossRef]

6. 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 **404**, 53–56 (2000). [CrossRef] [PubMed]

## 2. The PBG of two-dimensional triangular titania arrays

8. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. **26**, 1858–1860 (2001). [CrossRef]

*a*is the lattice constant. If the region of high intensity in the interference field is defined as bright lattices, we can obtain bright lattices shown in Fig. 1(a) with

**a**

_{1}=

*a*(1/2,3

^{1/2}/2) and

**a**

_{2}=

*a*(-1,0) as their translation basis vectors. The corresponding primitive vectors in reciprocal space are

**b**

_{1}=(2π/

*a*)(0,2/3

^{1/2}),

**b**

_{2}=(2π/

*a*)(-1,1/3

^{1/2}). The first Brillouin zone (BZ) turns out to be a hexagon as shown in Fig. 1(a). If the dark lattice whose intensity is lower than the given intensity threshold can be washed out, the shape of the remaining bright “atoms” changes from approximately hexagon to approximately circle when the intensity threshold is given from large to small. Obviously, the symmetry of the scatters and that of the lattices are the same. To calculate the photonic dispersion curves, the

**k**-points are traced along the boundary of the irreducible BZ determined by three symmetry points Γ=(0,0), M=(0,1/3

^{1/2}) and K=(1/3,1/3

^{1/2}), in units of 2π/

*a*.

11. A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. **79**, 3332–3334 (2001). [CrossRef]

12. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

^{-7}. By controlling the exposure intensity, introducing an intensity threshold

*I*(here

_{t}*I*ε (1.5, 6) for Eq. (1)), and washing out the underexposed regions, the filling ratio of titania can be selected. Consequently, the dielectric constant distribution ε(

_{t}*x, y*) of result lattice should be 1 in the region

*I*

_{0}(

*x, y*)<

*I*and 4 in the region

_{t}*I*

_{0}(

*x, y*)≥

*I*. The relation between

_{t}*I*and FR of the titania is shown in Fig. 1(b)-I. The derivative of curve-I is also given in Fig. 1(b)-II. It is clearly that the FR changes sharply when the intensity threshold is near 2. To optimize the structures so as to maximize the width of the PBG, the effect of different threshold and corresponding FR has been examined, and we present here some useful results.

_{t}### 2.1 The PBG for TE or TM polarization

*I*is.

_{t}*I*. Fig. 2(a) shows the dependence of the PBG on the

_{t}*I*, extracted from band structure calculations when

_{t}*I*changes from 1.5 to 6.0. All band gaps appearing for the ten lower-energy bands have been plotted for all possible values of the

_{t}*I*. The boundaries of shadows indicate the maxima and minima observed in the bands of the calculated band structure for various

_{t}*I*. Each shadow indicates a PBG region. Frequency axes in all plots of this paper are calculated using normalized frequency units (

_{t}*f*

_{n}=ω

*a*/2πc, where ω is the frequency and c is the speed of light). Clearly three gaps are observed for the ten lower-energy bands when

*I*is or above approximately 2.0 (the corresponding FR is about 74.73%). They locate between M

_{t}_{1}and M

_{2}, M

_{3}and M

_{4}, and M

_{6}and M

_{7}(where M

_{i}represents the

*i*th band for TM polarization, and E

_{i}below for TE polarization). The variation of the TM gap width with

*I*(measured by gap to midgap ratio Δω/ω

_{t}_{0}) is indicated in the inset of Fig. 2(a). Using Brent’s algorithm [13], we can find the favorable FR that maximizes the TM gaps. The maximum relative gaps for M

_{1–2}, M

_{3–4}, and M

_{6–7}(where M

_{i-j}denotes the gap appearing between the

*i*th and

*j*th TM bands) are 22.19%, 12.74% and 4.285%; the central normalized frequencies of these gaps are 0.4614, 0.7120 and 0.9993; the intensity thresholds are 4.100, 3.140 and 3.050; and the corresponding FRs are 21.23%, 37.06% and 38.94%, respectively. Figure 2(b) displays the photonic band structure for

*I*=3.0, which is the configuration yielding larger size of M

_{t}_{3–4}and M

_{6–7}gaps.

### 2.2 Directional PBG

*I*. Many gaps appear for TE or TM polarization. The essential result here is that numerous gaps existing in TM polarization (M

_{t}_{1–2}, M

_{3–4}, M

_{4–5}and M

_{10–11}) eventually overlap TE gaps (E

_{1–2}, E

_{2–3}, E

_{4–5}and E

_{9–10}), respectively, which leads to four directional PBGs (E

_{1}-M

_{2}, E

_{2}-E

_{3}or E

_{2}-M

_{4}, E

_{4}-M

_{5}and M

_{10–11}). The optimized relative gaps calculated by Brent’s algorithm for these directional PBGs are 8.923%, 7.396%, 4.276% and 3.154%; the central frequencies of these gaps are 0.5194, 0.6774, 1.042 and 1.477; The intensity thresholds are 4.5881, 2.982, 4.961 and 3.942; and the corresponding FRs are 14.75%, 40.29%, 10.49% and 23.44%, respectively. Figures 4(a) and (c) illustrate the directional photonic band structures of the PHCs in the case of

*I*=3.0 and 4.6, respectively. The former corresponds to a larger E

_{t}_{2–3}gap and the latter corresponds to larger E

_{1}-M

_{2}and E

_{4}-M

_{5}gaps. For comparisons, in Figs. 4(b) and (d) we also give the corresponding transmission spectra (in Γ-M direction) calculated with non-orthogonal FDTD method [14

14. A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B **58**, 7252–7259 (1998). [CrossRef]

15. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

^{1/2}

*a*, and the electromagnetic fields are integrated for 16 blocks of 1024 time steps each. Obviously, the gaps in the band structure are exactly corresponding to minima of the transmittance power.

## 3. The PBG of two-dimensional triangular inverted GaAs structure

*I*on the size and position of the band gaps. The gap map clearly indicates that the gap for TE or TM polarization drops off sharply if

_{t}*I*is less than about 1.6 or greater than about 2.5, this fact limits the overlap of TE and TM polarization gaps. On the other hand, up to four full PBGs appear during

_{t}*I*changes from 1.6 to 2.5, and their number varies with the

_{t}*I*. The PBG B

_{t}_{12–13}(where B

_{i-j}presents the band gap locating between the

*i*th and the

*j*th bands) corresponding to the largest

*I*range (from about 1.6 to 2.5) is due to the overlap of E

_{t}_{5–6}and M

_{7–8}gaps. In the following we will discuss the evolution of B

_{12–13}as an example. Figure 5(a) shows the result of band calculation for

*I*=1.6. Band structure analysis indicates that degeneration at the symmetry points of the BZ closes the TE PBG in the case of lower intensity threshold. The bands will move downward with the increase of intensity threshold, and the down-moving speed of E

_{t}_{5}at the three symmetry points is a bit faster than that of E

_{6}when

*I*changes from 1.6 to 1.7. Then the full PBG B

_{t}_{12–13}determined by E

_{5}(Γ) and E

_{6}(Γ) gradually appears (where E

_{5}(Γ) presents the symmetry point Γ of E

_{5}). During

*I*increases from 1.7 to 1.9, the down-moving speed of E

_{t}_{5}(Γ) is a bit faster at the beginning stage and a bit slower at the ending stage than that of E

_{6}(Γ), and a maximum 6.684% full PBG calculated by Brent’s algorithm appears from 0.9653 to 1.032 when

*I*=1.762 (corresponding FR is 10.78%). The down-moving speed of M

_{t}_{7}(M) (or M

_{7}(K)) is slower than E

_{5}(Γ) when

*I*changes from 1.9 to 2.0, and the B

_{t}_{12–13}translates into being bounded on the lower side by the TM polarization gap boundary M

_{7}(M) (or M

_{7}(K)) and on its upper side by the TE polarization gap boundary E

_{6}(Γ). Adopting a similar analysis for other

*I*and other three PBGs, B

_{t}_{3–4}, B

_{9–10}and B

_{22–23}(due to the overlap of E

_{1–2}and M

_{2–3}, E

_{3–4}and M

_{6–7}, and E

_{9–10}and M

_{13–14}), one can find their evolution at different intensity threshold. By using Brent’s algorithm in the calculation of band structure again, B

_{3–4}, B

_{9–10}and B

_{22–23}can also be optimized. The maximum relative gaps for B

_{3–4}, B

_{9–10}and B

_{22–23}are 18.71%, 8.235% and 2.552%; the central frequencies of these gaps are 0.4039, 0.7098 and 0.9985; the intensity thresholds are 2.000, 1.944 and 2.000; and the corresponding FRs are 25.27%, 20.65% and 25.27%, respectively. The most valuable result that has not been reported before [18

18. R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. **41**, 295–310 (1994). [CrossRef]

*I*is about 2.0 (see Fig. 5(b)). It has shown the important effect of “atom” shape introduced by holography on PBG properties.

_{t}## 4. Conclusions

**26**, 1858–1860 (2001). [CrossRef]

*I*indicate that PBGs of 2-D triangular titania arrays open only for TM polarization, and four broad directional PBGs along Γ-M can be obtained. Furthermore, for a higher dielectric contrast of 13.6:1, up to four sizeable full PBGs can be produced with an 18.15% relative gap-width of the maximum full PBG. These results have shown the potential of the holographic method in controlling the “atom” shape of the PHC and then improving their PBG properties, and are useful in the fabrication of 2-D PHCs with the technique of ITNB.

_{t}## Acknowledgement

## References and links

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

2. | S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. |

3. | 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 |

4. | 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. |

5. | B. D’Urso, O. Painter, J.D. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal Microcavities,” J. Opt. Soc. Am. B. |

6. | 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 |

7. | 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. |

8. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. |

9. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. |

10. | X. L. Yang, L. Z. Cai, Y. R. Wang, and Q. Liu, “Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices,” Opt. Lett. |

11. | A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. |

12. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

13. | Richard Brent, |

14. | A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B |

15. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

16. | X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B |

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

18. | R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. |

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

20. | C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B |

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

22. | M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. |

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

24. | V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. |

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

**OCIS Codes**

(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: April 7, 2003

Revised Manuscript: April 24, 2003

Published: May 5, 2003

**Citation**

X. L. Yang, L. Cai, and 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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1050

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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]
- S. John, �??Strong Localization of Photons in Certain Disordered Dielectric Superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- 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- (1993). [CrossRef]
- 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]
- B. D'Urso, O. Painter, J.D. O'Brien, T. Tombrello, A. Yariv, A. Scherer, �??Modal reflectivity in finite-depth twodimensional photonic-crystal Microcavities,�?? J. Opt. Soc. Am. B. 15, 1155-1159 (1998). [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]
- 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]
- L. Z. Cai, X. L. Yang, Y. R. Wang, �??Formation of a microfiber bundle by interference of three noncoplanar beams,�?? Opt. Lett. 26, 1858-1860 (2001). [CrossRef]
- L. Z. Cai, X. L. Yang, Y. R. Wang, �??All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,�?? Opt. Lett. 27, 900-902 (2002). [CrossRef]
- X. L. Yang, L. Z. Cai, Y. R. Wang, Q. Liu, �??Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices,�?? Opt. Lett. 28, 453-455 (2003). [CrossRef] [PubMed]
- A. Shishido, Ivan B. Diviliansky, I. C. Khoo, T. S. Mayer, �??Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,�?? Appl. Phys. Lett. 79, 3332-3334 (2001). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,�?? Opt. Express 8, 173-190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
- Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).
- A. J. Ward and J. B. Pendry, �??Calculating photonic Green's functions using a nonorthogonal finite-difference time-domain method,�?? Phys. Rev. B 58, 7252-7259 (1998). [CrossRef]
- J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, B. Cheng, �??Enlarging a photonic band gap by using insertion,�?? Phys. Rev. B 61, 1892-1897 (2000). [CrossRef]
- C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996). [CrossRef] [PubMed]
- R. Padjen, J. M. Gerard, J. Y. Marzin, �??Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,�?? J. Mod. Opt. 41, 295-310 (1994). [CrossRef]
- X. H. Wang, B. Y. Gu, Z. Y. Li, G. Z. Yang, �??Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,�?? Phys. Rev. B 60, 11417�??11421 (1999). [CrossRef]
- C. M. Anderson and K. P. Giapis, �??Symmetry reduction in group 4 mm photonic crystals�??, Phys. Rev. B 56, 7313-7320 (1997). [CrossRef]
- Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Large absolute band gap in 2D anisotropic photonic crystals,�?? Phys. Rev. Lett. 81, 2574-2577 (1998). [CrossRef]
- M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, �??Two-dimensional photonic band structures,�?? Opt. Commun. 80, 199-204 (1991) [CrossRef]
- M. Plihal and A. A. Maradudin, �??Photonic band structure of two-dimensional systems: The triangular lattice,�?? Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
- V. Berger, O. Gauthier-Lafaye, E. Costard, �??Photonic band gaps and holography,�?? J. Appl. Phys. 82, 60-64 (1997). [CrossRef]
- D. Cassagne, C. Jouanin, D. Bertho, �??Hexagonal photonic-band-gap structures,�?? Phys. Rev. B 53, 7134-7142 (1996). [CrossRef]

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