Abstract

From the view of crystallography, a systematic theoretical study on one-step formation of two-dimensional compound photonic lattices by four noncoplanar elliptical waves is presented. A general formula for the interference intensity of N elliptically polarized waves, and relevant phase shifts that compensate for the initial phases and control the relative position and size of the motifs, have been deduced. Using appropriate polarization configurations, four kinds of beam geometries can be used to form various compound lattices. This provides an ideal new experimental platform for fabricating large-area compound photonic lattices.

© 2005 Optical Society of America

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References

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  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. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Blur, "A three-dimensional photonic crystal operating at infrared wavelengths," Nature (London) 394, 251-253 (1998).
    [CrossRef]
  4. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).
    [CrossRef] [PubMed]
  5. Y.A. Vlasov, X.Z. Bo, J.C. Sturm, and D.J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (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 (London) 404, 53-56 (2000).
    [CrossRef]
  7. X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang, and W. Y. Tam, "Threedimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
    [CrossRef]
  8. X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, "Large-area two-dimensional mesoscale quasi-crystals," Adv. Mater. 15,1526-1528 (2003).
    [CrossRef]
  9. C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. J. Han, and S. Yang, "Photonic crystals through holographic lithography: simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434- 5436 (2004).
    [CrossRef]
  10. A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, and J. Y. Courtois, "An optical drill for the fabrication of photonic crystals," J. Opt. A Pure Appl. Opt. 1, L3�?? L6 (1999).
    [CrossRef]
  11. A. Feigel, Z. Kotler, and B. Sfez, "Scalable interference lithography alignment for fabrication of three-dimensional photonic crystals," Opt. Lett. 27, 746-748 (2002).
    [CrossRef]
  12. Max Born, and Emil Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
  13. W.D. Mao, Y.C. Zhong, J.W. Dong, and H.Z. Wang, "Crystallography of two-dimensional photonic lattices formed by holography of three noncoplanar beams," J. Opt. Soc. Am. B (to be published).

Adv. Mater.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, "Large-area two-dimensional mesoscale quasi-crystals," Adv. Mater. 15,1526-1528 (2003).
[CrossRef]

Appl. Phys. Lett.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. J. Han, and S. Yang, "Photonic crystals through holographic lithography: simple cubic, diamond-like, and gyroid-like structures," Appl. Phys. Lett. 84, 5434- 5436 (2004).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang, and W. Y. Tam, "Threedimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

J. Opt. A Pure Appl. Opt.

A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, and J. Y. Courtois, "An optical drill for the fabrication of photonic crystals," J. Opt. A Pure Appl. Opt. 1, L3�?? L6 (1999).
[CrossRef]

J. Opt. Soc. Am. B

W.D. Mao, Y.C. Zhong, J.W. Dong, and H.Z. Wang, "Crystallography of two-dimensional photonic lattices formed by holography of three noncoplanar beams," J. Opt. Soc. Am. B (to be published).

Nature

Y.A. Vlasov, X.Z. Bo, J.C. Sturm, and D.J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[CrossRef]

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 (London) 404, 53-56 (2000).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Blur, "A three-dimensional photonic crystal operating at infrared wavelengths," Nature (London) 394, 251-253 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

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]

Science

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).
[CrossRef] [PubMed]

Other

Max Born, and Emil Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Beam geometry for 2D compound lattices. Four laser beams are placed around Z axis and make the same angle with OZ. (b) Primitive rectangular compound lattice. (c) 3D structure of the square compound lattice. (d) Change of the relative size (left) and position (right) of the motifs in two unit cells due to the phase shifts.

Fig. 2.
Fig. 2.

(a) Hexagonal compound lattice. (b) Schematic illustration of the choices of the G ij vectors for the hexagonal compound lattice. Inset, the change of the relative size and shape of the motifs in a unit cell. (c) Compound rectangular p-lattice. (d) G ij vector choices for the lattice in (c).

Fig. 3.
Fig. 3.

(a) New hexagonal compound lattice. (b) Schematic illustration of the G ij vector choice for the hexagonal compound lattice. Inset is the superposition of two different simple lattices corresponding to the top left corner of (a). (c) Compound rectangular c-lattice. (d) G ij vector choices for the lattice in (c).

Fig. 4.
Fig. 4.

(a) Square compound lattice that contains four simple lattices. (b) G ij vector choices and the determination of corresponding k j vectors projected onto the xy-plane. Inset, the change of relative size of the motifs in a unit cell.

Equations (6)

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I ( r ) = j E aj exp [ i ( k j · r + δ j ) ] e aj + j E bj exp [ i ( k j · r + δ j π 2 ) ] e bj 2
= j E aj 2 + j E bj 2 + i < j 2 E ij 2 cos [ ( k j k i ) · r + δ j δ i t shift _ ij ]
i , j = 1 , 2 , 3 , , N ,
E ij 2 = ( E ai E aj cos θ aiaj + E bi E bj cos θ bibj ) 2 + ( E ai E bj cos θ aibj E bi E aj cos θ biaj ) 2 ,
cos [ ( δ 2 δ 1 + δ 4 δ 3 ) 2 ] · cos [ ( k 2 k 1 ) · r + ( δ 2 δ 1 + δ 3 δ 4 ) 2 ]
cos [ ( δ 3 δ 2 + δ 1 δ 4 ) 2 ] · cos [ ( k 3 k 2 ) · r + ( δ 3 δ 2 + δ 4 δ 1 ) 2 ]

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