Abstract

We make use of a dual beam multiple-exposure (DBME) holographic technique for the formation of all 14 Bravais lattices of three-dimensional photonic crystal microstructures. For simplicity of experimental implementation, the DBME method has been modified such that, prior to each exposure, once the proper angle between the wave vectors of the interfering beams is chosen, a single axis rotation of the recording medium gives the desired results. The parameters required for the generation of the lattice structures have been derived by appropriate modification of interference of four noncoplanar beams (IFNB) analysis for corresponding implementation in the DBME technique, and the results have been verified by computer simulations. After giving a comparative study of the results with the IFNB method, recording geometries for the DBME approach are also proposed in order to realize all 14 Bravais lattices experimentally.

© 2008 Optical Society of America

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References

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2006

2005

2004

R. C. Gauthier and K. W. Mnaymneh, “Design of photonic band gap structures through a dual-beam multiple exposure technique,” Opt. Laser Technol. 36, 625-633 (2004).
[CrossRef]

M. J. Escuti and G. P. Crawford, “Holographic photonic crystals,” Opt. Eng. 43, 1973-1987 (2004).
[CrossRef]

L. Carretero, M. Ulibarrena, P. Acebal, S. Blaya, R. Madrigal, and A. Fimia, “Multiplexed holographic gratings for fabricating 3-D photonic crystals in BB640 photographic emulsions,” Opt. Express 12, 2903-2908 (2004).
[CrossRef] [PubMed]

2003

L. Z. Cai, X. L. Yang, and Y. R. Wang, “What kind of Bravais lattices can be made by the interference of four umbrella beams?,” Opt. Commun. 224, 243-246 (2003).
[CrossRef]

L. Yuan, G. P. Wang, and X. Huang, “Arrangement of four beams for any Bravais lattice,” Opt. Lett. 28, 1769-1771(2003).
[CrossRef] [PubMed]

H. M. Su, Y. C. Zhong, X. Wang, X. G. Zheng, J. F. Xu, and H. Z. Wang, “Effects polarization on laser holography for microstructure fabrication,” Phys. Rev. E 67, 056619-1-056619-6(2003).
[CrossRef]

V. Y. Miklyaev, C. D. Meisel, and A. Blanco, “Three dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations,” Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

2002

2001

A. J. Turberfield, “Photonic crystals made by holographic lithography,” Mater. Res. Soc. Bull. 26, 632-636 (2001).
[CrossRef]

2000

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]

1997

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

1993

Appl. Opt.

Appl. Phys. Lett.

V. Y. Miklyaev, C. D. Meisel, and A. Blanco, “Three dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations,” Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Mater. Res. Soc. Bull.

A. J. Turberfield, “Photonic crystals made by holographic lithography,” Mater. Res. Soc. Bull. 26, 632-636 (2001).
[CrossRef]

Nature

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]

Opt. Commun.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “What kind of Bravais lattices can be made by the interference of four umbrella beams?,” Opt. Commun. 224, 243-246 (2003).
[CrossRef]

Opt. Eng.

M. J. Escuti and G. P. Crawford, “Holographic photonic crystals,” Opt. Eng. 43, 1973-1987 (2004).
[CrossRef]

Opt. Express

Opt. Laser Technol.

R. C. Gauthier and K. W. Mnaymneh, “Design of photonic band gap structures through a dual-beam multiple exposure technique,” Opt. Laser Technol. 36, 625-633 (2004).
[CrossRef]

Opt. Lett.

Phys. Rev. E

H. M. Su, Y. C. Zhong, X. Wang, X. G. Zheng, J. F. Xu, and H. Z. Wang, “Effects polarization on laser holography for microstructure fabrication,” Phys. Rev. E 67, 056619-1-056619-6(2003).
[CrossRef]

Other

J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).

P. N. Prasad, Nanophotonics (Wiley-Interscience, 2004).
[CrossRef]

J. -M. Lourtioz, H. Benisty, V. Berger, J .-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Longman Group, 1984), pp. 268-269.

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Figures (5)

Fig. 1
Fig. 1

Orientation of beam and the recording medium for recording of holographic photonic crystal structures by the DBME technique; θ s are the mutual angles between the wave vectors of the beams, φ s are the azimuthal angles (Fig. 2), and α s are angle of rotation before the exposures: (i) first exposure [ α 1 = φ 2 = 0 ], (ii) second exposure [ α 2 = φ 3 ], and (iii) third exposure [ α 2 + α 3 = φ 3 + ( φ 4 φ 3 ) = φ 4 ].

Fig. 2
Fig. 2

Beam representation, showing mutual angle between the wave vectors of the beams ( θ ij ) and the azimuthal angle ( φ j ).

Fig. 3
Fig. 3

Simulated 3D photonic crystal lattice structures by the DBME technique (scale in μm ). The orthogonal axes are given with respect to the direction of orthogonal planes [-1,0,0], [0,-1,0], [0,0,1].

Fig. 4
Fig. 4

Comparison between the 3D crystal lattice structures (fcc, bcc, and hexagonal), generated by the IFNB and DBME techniques (scale in μm ). The orthogonal axes are given with respect to the direction of orthogonal planes [- 1,0,0], [0,-1,0], [0,0,1].

Fig. 5
Fig. 5

Experimental setup (symmetric beam arrangement) for recording of the 3D periodic structures by the DBME method: S, shutter; WP, wave plate; PBS, polarizing beam splitter; SF, spatial filter; CL, collimating lens; M, mirror.

Tables (1)

Tables Icon

Table 1 List of Different Parameters Calculated by MATLAB Simulation at ~ 0.532 μm Wavelength

Equations (10)

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I = E 1 2 + E 2 2 + 2 E 1 E 2 e 12 cos [ ( k 1 k 2 ) · r ] ,
I 12 = 2 [ 1 + cos ( ( k 1 k 2 ) · r ) ] I 13 = 2 [ 1 + cos ( ( k 1 k 3 ) · r ) ] I 14 = 2 [ 1 + cos ( ( k 1 k 4 ) · r ) ] .
I = 6 + 2 { cos [ ( k 1 k 2 ) · r ] + cos [ ( k 1 k 3 ) · r ] + cos [ ( k 1 k 4 ) · r ] } .
I = j = 1 4 E j 2 + 2 i < j E i E j e i j cos [ ( k i k j ) · r + i j ] ,
k 1 = π ( 1 a + 3 a b 2 , 2 b , 1 c ) k 2 = π ( 1 a + 3 a b 2 , 4 b , 1 c ) k 3 = π ( 1 a + 3 a b 2 , 2 b , 1 c ) k 4 = π ( 1 a + 3 a b 2 , 2 b , 1 c ) ,
λ = 2 ( 1 a 2 + 10 b 2 + 1 c 2 + 9 a 2 b 4 ) 1 / 2 .
k i · k j = | k i | | k j | cos ( θ i j ) .
cos θ 23 = cos θ 12 cos θ 13 + sin θ 12 sin θ 13 cos ( φ 2 φ 3 ) ,
cos θ 24 = cos θ 12 cos θ 14 + sin θ 12 sin θ 14 cos ( φ 2 φ 4 ) ,
cos θ 34 = cos θ 13 cos θ 14 + sin θ 13 sin θ 14 cos ( φ 3 φ 4 ) .

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