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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 9 — May. 5, 2003
  • pp: 1050–1055
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Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three noncoplanar beams

X. L. Yang, L. Z. Cai, and Q. Liu  »View Author Affiliations


Optics Express, Vol. 11, Issue 9, pp. 1050-1055 (2003)
http://dx.doi.org/10.1364/OE.11.001050


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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

In our last letter [8

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]

], we demonstrated an interference technique of three noncoplanar beams (ITNB) to fabricate 2-D triangular lattices. The remaining question is what effect the intensity threshold has on the filling ratio (FR), the shape of “atoms” and then the PBG of the final structure, and how we should control the intensity threshold experimentally to optimize the PBG. In this letter, we will further provide a theoretical modeling of the PBG and spectral properties of the 2-D trigonal lattices proposed in Ref. [8

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]

]. Because it is easy to produce a PHC with a thickness larger than several dozen micrometers while the corresponding lattice constant is usually less than one micrometer [6

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]

], we can reasonably treat the PHCs as ideal 3-D structures with 2-D periodicity in PBG analysis.

2. The PBG of two-dimensional triangular titania arrays

As shown in Ref. [8

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]

], the intensity of interference field for this lattice can be expressed as

I0=3+cos[2π3a(2y)]+cos[2π3a(33x+y)]+cos[2π3a(33xy)],
(1)

where 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(1/2,31/2/2) and a 2=a(-1,0) as their translation basis vectors. The corresponding primitive vectors in reciprocal space are b 1=(2π/a)(0,2/31/2), b 2=(2π/a)(-1,1/31/2). 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=(0,1/31/2) and K=(1/3,1/31/2), in units of 2π/a.

Fig. 1. (a) 2-D triangular photonic lattice fabricated by the interference technique of three noncoplanar beam and the first Brillouin zone with the symmetry points indicated; (b) and (c), dotted lines (I) present the relation between the intensity threshold and the FR of dielectric, and solid lines (II) present the derivative of curves (I), where (b) is for titania and (c) is for GaAs.

Using the photolithography technique reported by Shishido etc. [11

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]

], 2-D titania triangular arrays can be directly fabricated by polymerization of a photosensitive titanium-containing monomer film. The effective dielectric constant of the titania array is found to be about 4. In the following, the block-iterative frequency-domain method for Maxwell’s equations in a plane wave basis [12

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]

] is employed to investigate the gap properties for the 2-D triangular titania arrays. In the calculations, grid size is 32×32×1, mesh size is 7 and tolerance is 1×10-7. By controlling the exposure intensity, introducing an intensity threshold It (here Itε (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(x, y) of result lattice should be 1 in the region I 0(x, y)<It and 4 in the region I 0(x, y)≥It. The relation between It 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.

2.1 The PBG for TE or TM polarization

In this paper, TM polarization corresponds to E field in the z direction (see Fig. 1(a)) and vice versa for TE. For TE polarization, numerous symmetry-induced degeneracies are observed at the symmetry points of the BZ. As a result, no gap is observed whatever the It is.

For TM polarization, we have studied the evolution of PBG’s as function of It. Fig. 2(a) shows the dependence of the PBG on the It, extracted from band structure calculations when It 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 It. The boundaries of shadows indicate the maxima and minima observed in the bands of the calculated band structure for various It. Each shadow indicates a PBG region. Frequency axes in all plots of this paper are calculated using normalized frequency units (f na/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 It is or above approximately 2.0 (the corresponding FR is about 74.73%). They locate between M1 and M2, M3 and M4, and M6 and M7 (where Mi represents the ith band for TM polarization, and Ei below for TE polarization). The variation of the TM gap width with It (measured by gap to midgap ratio Δω/ω0) is indicated in the inset of Fig. 2(a). Using Brent’s algorithm [13

13. Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).

], we can find the favorable FR that maximizes the TM gaps. The maximum relative gaps for M1–2, M3–4, and M6–7 (where Mi-j denotes the gap appearing between the ith and jth 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 It=3.0, which is the configuration yielding larger size of M3–4 and M6–7 gaps.

Fig. 2. (a) TM gap map for the 2-D triangular titania arrays; (b) TM photonic band structure for It=3.0.
Fig. 3. (a) Gap map of directional PBG for 2-D titania arrays; (b) Gap map of full PBG for the inverted GaAs structure, where blue area is for TM polarization, red is for TE polarization and yellow area is for both.

2.2 Directional PBG

Although no full PBG is expected for PHCs discussed here, several directional PBGs can be obtained. For example, we show in Fig. 3(a) the evolutions of PBG in Γ-M direction with respect to the intensity threshold It. Many gaps appear for TE or TM polarization. The essential result here is that numerous gaps existing in TM polarization (M1–2, M3–4, M4–5 and M10–11) eventually overlap TE gaps (E1–2, E2–3, E4–5 and E9–10), respectively, which leads to four directional PBGs (E1-M2, E2-E3 or E2-M4, E4-M5 and M10–11). 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 It=3.0 and 4.6, respectively. The former corresponds to a larger E2–3 gap and the latter corresponds to larger E1-M2 and E4-M5 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]

]. In these calculations, a Berenger type of perfectly matched layer (PML) [15

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

] is implemented as absorbing layer, the total length of the scattering region is 10×31/2 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.

Fig. 4. The directional photonic band diagrams (a and c) and calculated transmission spectra (b and d) for TE (red line) and TM (blue line) polarizations respectively, where (a) and (b) are for It=3.0, and (c) and (d) are for It=4.6.

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

The important remaining question is how to turn these aforementioned directional PBGs into the full PBGs. Several methods have been suggested for obtaining full PBGs in 2-D situations. For example, it has shown that large band gaps can be obtained by such as varying dielectric contrast ratio and filling factors, inserting a third component into the existing photonic crystals [16

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 61, 1892–1897 (2000). [CrossRef]

], reducing the structure symmetry [17

17. C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996). [CrossRef] [PubMed]

], using non-circular rods [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]

], and subsequently by rotating the non-circular rods [19

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 60, 11417–11421 (1999). [CrossRef]

], rotating the lattice structure [20

20. C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997). [CrossRef]

], using anisotropic dielectric materials [21

21. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998). [CrossRef]

], placing rods of various shapes on different lattice configurations such as square [22

22. M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991). [CrossRef]

], triangular [21

21. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998). [CrossRef]

, 23

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

], honeycomb [17

17. C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996). [CrossRef] [PubMed]

] and so on. Each approach may have its advantages and shortcomings. For holographic technique, one can obtain an inverted crystal with higher refractive-index contrast by using some photoresist to fabricate template for in-filling other high refractive index materials or by using a photoresist that has been exposed to the interference field as mask for the GaAs etching [24

24. V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997). [CrossRef]

]. In the following, we examine an inverted structure by supposing the dielectric contrast ratio reaches 13.6:1 (13.6 is the GaAs’s dielectric contrast) as other authors used [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]

, 25

25. D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996). [CrossRef]

]. Then the dielectric constant ε(x, y) equals 13.6 in the area of I 0(x, y)<It and 1 otherwise. Both the relation between It and FR of GaAs and its derivative are given in Fig. 1(c).

Fig. 5. Photonic band structures when the intensity threshold is 1.6 (a) and 2.0 (b) for TE (red line) and TM (blue line) polarizations, respectively.

Our aim here is to modify the intensity threshold to obtain a full PBG. Figure 3(b) shows the effect of the parameter It 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 It 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 It changes from 1.6 to 2.5, and their number varies with the It. The PBG B12–13 (where Bi-j presents the band gap locating between the ith and the jth bands) corresponding to the largest It range (from about 1.6 to 2.5) is due to the overlap of E5–6 and M7–8 gaps. In the following we will discuss the evolution of B12–13 as an example. Figure 5(a) shows the result of band calculation for It=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 E5 at the three symmetry points is a bit faster than that of E6 when It changes from 1.6 to 1.7. Then the full PBG B12–13 determined by E5(Γ) and E6(Γ) gradually appears (where E5(Γ) presents the symmetry point Γ of E5). During It increases from 1.7 to 1.9, the down-moving speed of E5(Γ) is a bit faster at the beginning stage and a bit slower at the ending stage than that of E6(Γ), and a maximum 6.684% full PBG calculated by Brent’s algorithm appears from 0.9653 to 1.032 when It=1.762 (corresponding FR is 10.78%). The down-moving speed of M7(M) (or M7(K)) is slower than E5(Γ) when It changes from 1.9 to 2.0, and the B12–13 translates into being bounded on the lower side by the TM polarization gap boundary M7(M) (or M7(K)) and on its upper side by the TE polarization gap boundary E6(Γ). Adopting a similar analysis for other It and other three PBGs, B3–4, B9–10 and B22–23 (due to the overlap of E1–2 and M2–3, E3–4 and M6–7, and E9–10 and M13–14), one can find their evolution at different intensity threshold. By using Brent’s algorithm in the calculation of band structure again, B3–4, B9–10 and B22–23 can also be optimized. The maximum relative gaps for B3–4, B9–10 and B22–23 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]

] is that the four sizeable full PBGs simultaneously appear when It is about 2.0 (see Fig. 5(b)). It has shown the important effect of “atom” shape introduced by holography on PBG properties.

4. Conclusions

In conclusion, using the block-iterative frequency domain method and the non-orthogonal FDTD method, we have shown the existence of PBGs for a new class of 2-D triangular lattice, which can be fabricated by an interference technique of three noncoplanar beams proposed by us [8

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 specific problem in holographic lithography, the effect of threshold selection on the FR, the “atom” shape and then the PBG properties, is comprehensively investigated. Our calculations of band structures as a function of It 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.

Acknowledgement

We sincerely thank Dr. Steven G. Johnson of MIT for his help in using the software MPB as well as his valuable suggestions for the calculation of PBG. This research is supported by the National Natural Science Foundation of China (grant 60177002).

References and links

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]

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. 79, 725–727 (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]

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. 27, 900–902 (2002). [CrossRef]

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]

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]

13.

Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).

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]

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 61, 1892–1897 (2000). [CrossRef]

17.

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996). [CrossRef] [PubMed]

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]

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 60, 11417–11421 (1999). [CrossRef]

20.

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997). [CrossRef]

21.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998). [CrossRef]

22.

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991). [CrossRef]

23.

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

24.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997). [CrossRef]

25.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996). [CrossRef]

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

  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, 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, 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, A. Scherer, �??Modal reflectivity in finite-depth twodimensional 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, A. J. Turberfield, �??Fabrication of photonic crystals for the visible spectrum by holographic lithography, �?? Nature 404, 53-56 (2000). [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  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), <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]
  13. Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).
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