Abstract

A single geometric model based on a new concept of a reciprocal primitive pyramid (RPP) in reciprocal space is proposed for investigation of relationships between any three-dimensional (3D) lattice and arrangements of four beams (AFBs) that produce the lattice. A ternary linear equation set, described for the one-to-one correspondence between a RPP and AFB, can readily reveal all AFBs for the same lattice (AFBSLs). Quantitative AFBs for bcc and fcc real lattices are illustrated to show that various AFBSLs can modulate the properties of a photonic bandgap (PBG) both by tuning the lattice constant and by changing the lattice-point shape. This fact may yield the appropriate AFB for a complete 3D PBG with the desired center wavelength. The nonuniqueness of AFBSLs can provide abundant choices for persons who plan interference experiments, especially for holographic fabrication of 3D photonic crystals (PCs).

© 2003 Optical Society of America

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References

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I. Divliansky and T. S. Mayer, Appl. Phys. Lett. 82, 1667 (2003).
[CrossRef]

2002

X. L. Yang and L. Z. Cai, Opt. Commun. 208, 293 (2002).
[CrossRef]

N. Susa, J. Appl. Phys. 91, 3501 (2002).
[CrossRef]

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, Opt. Lett. 27, 900 (2002).
[CrossRef]

2001

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

2000

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

S. Shoji and S. Kawata, Appl. Phys. Lett. 76, 2668 (2000).
[CrossRef]

1995

1994

K. I. Petsas, A. B. Coates, and G. Grynberg, Phys. Rev. A 50, 5173 (1994).
[CrossRef] [PubMed]

1993

Bunning, T. J.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Cai, L. Z.

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Cheng, B. Y.

Coates, A. B.

K. I. Petsas, A. B. Coates, and G. Grynberg, Phys. Rev. A 50, 5173 (1994).
[CrossRef] [PubMed]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Divliansky, I.

I. Divliansky and T. S. Mayer, Appl. Phys. Lett. 82, 1667 (2003).
[CrossRef]

Grynberg, G.

K. I. Petsas, A. B. Coates, and G. Grynberg, Phys. Rev. A 50, 5173 (1994).
[CrossRef] [PubMed]

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
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Hu, W.

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

Kawata, S.

S. Shoji and S. Kawata, Appl. Phys. Lett. 76, 2668 (2000).
[CrossRef]

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

Li, Z. L.

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

Mayer, T. S.

I. Divliansky and T. S. Mayer, Appl. Phys. Lett. 82, 1667 (2003).
[CrossRef]

Mei, D. B.

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

Natarajan, L. V.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Petsas, K. I.

K. I. Petsas, A. B. Coates, and G. Grynberg, Phys. Rev. A 50, 5173 (1994).
[CrossRef] [PubMed]

Shan, H.

G. P. Wang, C. L. Tan, Y. X. Yi, and H. Shan, “Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals by using a single cw laser beam,” J. Mod. Opt. (to be published).

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Shoji, S.

S. Shoji and S. Kawata, Appl. Phys. Lett. 76, 2668 (2000).
[CrossRef]

Susa, N.

N. Susa, J. Appl. Phys. 91, 3501 (2002).
[CrossRef]

Sutherland, R. L.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Tan, C. L.

G. P. Wang, C. L. Tan, Y. X. Yi, and H. Shan, “Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals by using a single cw laser beam,” J. Mod. Opt. (to be published).

Tomlin, D.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Tondiglia, V. P.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Wang, G. P.

G. P. Wang, C. L. Tan, Y. X. Yi, and H. Shan, “Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals by using a single cw laser beam,” J. Mod. Opt. (to be published).

Wang, Y. R.

Yablonovitch, E.

Yang, X. L.

Yi, Y. X.

G. P. Wang, C. L. Tan, Y. X. Yi, and H. Shan, “Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals by using a single cw laser beam,” J. Mod. Opt. (to be published).

Zhang, D. Z.

Adv. Mater.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, Adv. Mater. 14, 187 (2002).
[CrossRef]

Appl. Phys. Lett.

S. Shoji and S. Kawata, Appl. Phys. Lett. 76, 2668 (2000).
[CrossRef]

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, Appl. Phys. Lett. 79, 725 (2001).
[CrossRef]

I. Divliansky and T. S. Mayer, Appl. Phys. Lett. 82, 1667 (2003).
[CrossRef]

J. Appl. Phys.

N. Susa, J. Appl. Phys. 91, 3501 (2002).
[CrossRef]

J. Mod. Opt.

G. P. Wang, C. L. Tan, Y. X. Yi, and H. Shan, “Holography for one-step fabrication of three-dimensional metallodielectric photonic crystals by using a single cw laser beam,” J. Mod. Opt. (to be published).

J. Opt. Soc. Am. B

Nature

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Opt. Commun.

X. L. Yang and L. Z. Cai, Opt. Commun. 208, 293 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. A

K. I. Petsas, A. B. Coates, and G. Grynberg, Phys. Rev. A 50, 5173 (1994).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Geometric correspondence between a RPP and an AFB. B0,B1,B2, and B3 are all lattice points in reciprocal space, and no other lattice point inside or on the boundary of triangular pyramid B0B1B2B3 or B0,B1,B2,B3 specifies a circumsphere of radius k=k=2π/λ with its center at C. ki=BiC i=0,1,2,3 represent relative directions of four incident beams.

Fig. 2
Fig. 2

Choosing possible RPPs in a reciprocal lattice. Lattice vectors b1,b2, and b3 can be selected in the reciprocal lattice at will. Pyramid B0B1B2B3 becomes a RPP if and only if the volume of pyramid B0B1B2B3 is equal to 1/6Ω (Ω is the volume of a primitive cell in the reciprocal lattice).

Fig. 3
Fig. 3

Choosing RPPs in a reciprocal lattice. a, fcc forms an interference pattern of lattice bcc; b, bcc forms an interference pattern of lattice fcc. Here all real and reciprocal lattice constants are of 2-unit length, but the real unit length is 1 and the reciprocal unit length is π or vice versa.

Fig. 4
Fig. 4

Simulation of patterns in the z=0 plane with changeable lattice-point shape, formed by interference of AFBs listed in Table 1. a–f, bcc; g–j, fcc lattices. The coordinate system is the same as that used in Fig. 3 and Table 1.

Fig. 5
Fig. 5

Normalized lattice constants a/λ of some tested AFBs for fcc and bcc lattices. One AFB is represented by one star. The more highly tested the AFB, the denser the dotted line.

Tables (1)

Tables Icon

Table 1 Tunable Lattice Constants and Changeable Lattice-Point Shapes of Various AFBsa

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

I=i=03Ei2+20i<j3×EiEj cos θijcoski-kj·r+φ0i-φ0j,
b11b12b13b21b22b23b31b32b33xcyczc=12b112+b122+b132b212+b222+b232b312+b322+b332.
xcyczc=12b11b12b13b21b22b23b31b32b33-1b112+b122+b132b212+b222+b232b312+b322+b332.
ki=xc-bi1,yc-bi2,zc-bi3,    i=0,1,2,3.

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