Abstract

A systematically theoretical study on how to form two-dimensional photonic lattices with various plane groups by three elliptically polarized beams is presented. It is shown that nine plane groups can be formed in the photonic lattices by use of an intuitionistic intensity pattern–superposition method; however, we demonstrate that the other eight plane groups cannot be constructed. A phase shift associated with the interference intensity and the elliptic polarization is derived, and a relevant formula for interference intensity is deduced. The phase shift can be used to obtain the lower symmetries in some wallpaper groups such as p1, pm, cm, and p3m1 without introducing additional undesired symmetries. This analysis may lay the foundation for the study of space groups in holographic three-dimensional photonic crystals and multidimensional photonic quasicrystals.

© 2005 Optical Society of America

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2004 (2)

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]

A. Rung and C. G. Ribbin, "Polaritonic and photonic gap interactions in a two-dimensional photonic crystal," Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

2003 (7)

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]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, E. L. Thomas, C. A. White, and S. Yang, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[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, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

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

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

J. Bravo-Abad, T. Ochiai, and J. Sánchez-Dehesa, "Anomalous refractive properties of a two-dimensional photonic band-gap prism," Phys. Rev. B 67, 115116 (2003).
[CrossRef]

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

2002 (1)

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

2001 (1)

2000 (3)

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]

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]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

1999 (1)

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

1998 (1)

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]

1997 (2)

A. van Blaaderen, R. Ruel, and P. Wiltzius, "Template-directed colloidal crystallization," Nature (London) 385, 321-324 (1997).
[CrossRef]

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

1996 (1)

C. M. Anderson and K. P. Giapis, "Two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2952 (1996).
[CrossRef] [PubMed]

1994 (1)

K. I. Petsas, A. B. Coates, and G. Grynberg, "Crystallography of optical lattices," Phys. Rev. A 50, 5173-5197 (1994).
[CrossRef] [PubMed]

1987 (2)

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]

Aizenberg, J.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

Ananthavel, S. P.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Anderson, C. M.

C. M. Anderson and K. P. Giapis, "Two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2952 (1996).
[CrossRef] [PubMed]

Arsh, A.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Bagnall, D. M.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Barlow, S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Biswas, R.

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]

Blur, J.

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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Bravo-Abad, J.

J. Bravo-Abad, T. Ochiai, and J. Sánchez-Dehesa, "Anomalous refractive properties of a two-dimensional photonic band-gap prism," Phys. Rev. B 67, 115116 (2003).
[CrossRef]

Cai, L. Z.

Campbell, M.

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]

Chaikin, P. M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

Chan, C. T.

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

Chen, G.

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]

Chen, Y. L.

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

Chutinan, A.

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]

Coates, A. B.

K. I. Petsas, A. B. Coates, and G. Grynberg, "Crystallography of optical lattices," Phys. Rev. A 50, 5173-5197 (1994).
[CrossRef] [PubMed]

Coles, H. J.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Cumpston, B. H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Denning, R. G.

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]

Dyer, D. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Ehrlich, J. E.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Erskine, L. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Feigel, A.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Fleming, J. G.

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]

Giapis, K. P.

C. M. Anderson and K. P. Giapis, "Two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2952 (1996).
[CrossRef] [PubMed]

Grynberg, G.

K. I. Petsas, A. B. Coates, and G. Grynberg, "Crystallography of optical lattices," Phys. Rev. A 50, 5173-5197 (1994).
[CrossRef] [PubMed]

Han, Y. J.

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]

Harrison, M. T.

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]

Heikal, A. A.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Hetherington, D. L.

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]

Ho, K. M.

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]

John, S.

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Klebanov, M.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Kotler, Z.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Kuebler, S. M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Kurtz, S. R.

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]

Lee, I. Y. S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Lehmann, O.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

Lin, S. Y.

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]

Liu, Q.

Lyubin, V.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Maldovan, M.

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]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, E. L. Thomas, C. A. White, and S. Yang, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

Marder, S. R.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

McCord-Maughon, D.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Megens, M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

Müller, K.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

Ng, C. Y.

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

Noda, S.

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]

Ochiai, T.

J. Bravo-Abad, T. Ochiai, and J. Sánchez-Dehesa, "Anomalous refractive properties of a two-dimensional photonic band-gap prism," Phys. Rev. B 67, 115116 (2003).
[CrossRef]

Pang, Y. K.

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

Papakostas, A.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Perry, J. W.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Petsas, K. I.

K. I. Petsas, A. B. Coates, and G. Grynberg, "Crystallography of optical lattices," Phys. Rev. A 50, 5173-5197 (1994).
[CrossRef] [PubMed]

Potts, A.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Prosvirnin, S. L.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Qin, J.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Ribbin, C. G.

A. Rung and C. G. Ribbin, "Polaritonic and photonic gap interactions in a two-dimensional photonic crystal," Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Röckel, H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Ruel, R.

A. van Blaaderen, R. Ruel, and P. Wiltzius, "Template-directed colloidal crystallization," Nature (London) 385, 321-324 (1997).
[CrossRef]

Rumi, M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Rung, A.

A. Rung and C. G. Ribbin, "Polaritonic and photonic gap interactions in a two-dimensional photonic crystal," Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Russel, W. B.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
[CrossRef]

Sánchez-Dehesa, J.

J. Bravo-Abad, T. Ochiai, and J. Sánchez-Dehesa, "Anomalous refractive properties of a two-dimensional photonic band-gap prism," Phys. Rev. B 67, 115116 (2003).
[CrossRef]

Sfez, B.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[CrossRef]

Sharp, D. N.

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]

Sheng, P.

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

Sigalas, M. M.

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]

Smith, B. K.

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]

Stuke, M.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

Su, H. M.

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

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

Tam, W. Y.

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

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

Thomas, E. L.

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]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, E. L. Thomas, C. A. White, and S. Yang, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

Tomoda, K.

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]

Turberfield, A. J.

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]

Ullal, C. K.

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]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, E. L. Thomas, C. A. White, and S. Yang, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

van Blaaderen, A.

A. van Blaaderen, R. Ruel, and P. Wiltzius, "Template-directed colloidal crystallization," Nature (London) 385, 321-324 (1997).
[CrossRef]

Wang, H. Z.

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

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

Wang, X.

H. M. Su, Y. C. Zhong, X. Wang, X. G. Zheng, J. F. Xu, and H. Z. Wang, "Effects of polarization on laser holography for microstructure fabrication," Phys. Rev. E 67, 056619 (2003).
[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, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

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

Wang, Y. R.

Wanke, M. C.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

Wen, Q.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

White, C. A.

Wiltzius, P.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

A. van Blaaderen, R. Ruel, and P. Wiltzius, "Template-directed colloidal crystallization," Nature (London) 385, 321-324 (1997).
[CrossRef]

Wohlgemuth, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Wu, X. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[CrossRef]

Xu, J. F.

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

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

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yamamoto, N.

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]

Yang, S.

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]

C. K. Ullal, M. Maldovan, M. Wohlgemuth, E. L. Thomas, C. A. White, and S. Yang, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003).
[CrossRef]

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

Yang, X. L.

Zeng, Z. H.

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

Zheludev, N. I.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

Zheng, X. G.

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

Zhong, Y. C.

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

Zubrzycki, W.

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]

Adv. Mater. (Weinheim, Ger.) (1)

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

Appl. Phys. Lett. (3)

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]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Chalcogenide glass-based three-dimensional photonic crystals," Appl. Phys. Lett. 77, 3221-3223 (2000).
[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, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Chem. Phys. (1)

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Phys. 14, 2831-2833 (2002).

J. Opt. Soc. Am. A (1)

Nature (London) (4)

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]

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three dimensional optical data storage and microfabrication," Nature (London) 398, 51-54 (1999).
[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]

A. van Blaaderen, R. Ruel, and P. Wiltzius, "Template-directed colloidal crystallization," Nature (London) 385, 321-324 (1997).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

K. I. Petsas, A. B. Coates, and G. Grynberg, "Crystallography of optical lattices," Phys. Rev. A 50, 5173-5197 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

J. Bravo-Abad, T. Ochiai, and J. Sánchez-Dehesa, "Anomalous refractive properties of a two-dimensional photonic band-gap prism," Phys. Rev. B 67, 115116 (2003).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (5)

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]

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003).
[CrossRef] [PubMed]

C. M. Anderson and K. P. Giapis, "Two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2952 (1996).
[CrossRef] [PubMed]

A. Rung and C. G. Ribbin, "Polaritonic and photonic gap interactions in a two-dimensional photonic crystal," Phys. Rev. Lett. 92, 123901 (2004).
[CrossRef] [PubMed]

Science (2)

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, and M. Stuke, "Laser rapid prototyping of photonic band-gap microstructures," Science 275, 1284-1286 (1997).
[CrossRef] [PubMed]

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 (2)

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Beam geometry for 2D lattices. Three laser beams are placed around the Z axis and make the same angle (38.9°) with O , Z .

Fig. 2
Fig. 2

(a) Single point splits into three points after elliptic polarization is introduced; the dashed and solid lines represent the directions of the intensity stripes. (b) The three longer dotted-line segments indicate the directions of the blocky intensity patterns. The G vectors marked by the shorter dotted lines are perpendicular to the intensity stripes. (c) Two kinds of blocky patterns have been created when two G vectors have the same conditions, except for those perpendicular to each other (the left pattern) and oblique crossing (the right pattern).

Fig. 3
Fig. 3

Plane group p2 formed by three-beam holography.

Fig. 4
Fig. 4

Plane group p1 formed by three-beam holography where elliptic polarization was introduced.

Fig. 5
Fig. 5

Plane group p2mm formed by three-beam holography.

Fig. 6
Fig. 6

(a) Approximate treatment. Let G h , G v , and G o represent the G vectors in horizontal, vertical, and oblique angles, respectively. The solid rhomboid pattern formed by G v and G h has a larger intensity area so that those patterns (whose directions are marked by two dotted-line segments 1 and 2) formed by G v and G o , G o and G h have no effect on the shape of the rhomboid pattern. Translating G o from the oblique dashed line to the solid line, the intensity pattern formed by G v and G o will have an effect on the solid rhombus. The approximate intensity pattern marked by a dashed rhombus implies the group pm in the lattice. (b) When we introduced elliptic polarization, the plane group pm was formed.

Fig. 7
Fig. 7

Plane group c2mm formed by three-beam holography.

Fig. 8
Fig. 8

(a) When elliptic polarization is introduced, the plane group cm was formed. (b) The motif of the group cm was formed by a phase shift.

Fig. 9
Fig. 9

Plane group p4mm formed by three-beam holography.

Fig. 10
Fig. 10

Obviously, whether adjusting the amplitudes of one or two electric vectors, or introducing the elliptic polarization, there should be face-to-face direction deviations of two blocky intensity patterns. Therefore rotation of the intensity pattern with rotational symmetries of order 3, 4, and 6 cannot be realized by only three beams.

Fig. 11
Fig. 11

Plane group p6mm formed by three-beam holography.

Fig. 12
Fig. 12

(a) After the elliptical waves are introduced, the threefold rotational symmetry emerges; thus the plane group p3m1 can be obtained in a holographic photonic lattice. (b) The plane group p3m1 formed by three-beam holography.

Equations (15)

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

I ( r ) = j E j 2 + i < j 2 E i E j cos θ i j cos ( G j r + Δ δ j ) , i , j = 1 , 2 , 3 ,
I 12 = 2 E 1 E 2 cos θ 12 cos ( G 1 r + Δ δ 1 ) ,
I 23 = 2 E 2 E 3 cos θ 23 cos ( G 2 r + Δ δ 2 ) ,
I 31 = 2 E 3 E 1 cos θ 31 cos ( G 3 r + Δ δ 3 ) ,
I ( r ) = j E j 2 + ( I 12 + I 23 + I 31 ) ,
I ( r ) = j E j 2 + 1 2 [ ( I 12 + I 23 ) + ( I 23 + I 31 ) + ( I 12 + I 31 ) ] ,
j = 1 , 2 , 3 .
{ E a j = E a j exp [ i ( k j r + δ j ) ] e a j E b j = E b j exp [ i ( k j r + δ j π 2 ) ] e b j } j = 1 , 2 , 3 ,
I ( r ) = j E a j exp [ i ( k j r + δ j ) ] e a j + j E b j exp [ i ( k j r + δ j π 2 ) ] e b j 2 = j E a j 2 + j E b j 2 + i < j 2 E a i E a j cos θ a i a j cos [ ( k j k i ) r + δ j δ i ] + i < j 2 E b i E b j cos θ b i b j cos [ ( k j k i ) r + δ j δ i ] + i < j 2 E a i E b j cos θ a i b j cos [ ( k j k i ) r + δ j δ i π 2 ] + i < j 2 E b i E a j cos θ b i a j cos [ ( k j k i ) r + δ j δ i + π 2 ] ,
i , j = 1 , 2 , 3 .
I ( r ) = j E a j 2 + E b 3 2 + 2 E a 1 E a 2 cos θ a 1 a 2 cos ( G 1 r + Δ δ 1 ) + 2 E a 2 E a 3 cos θ a 2 a 3 cos ( G 2 r + Δ δ 2 ) + 2 E a 1 E a 3 cos θ a 1 a 3 cos ( G 3 r + Δ δ 3 ) + 2 E a 1 E b 3 cos θ a 1 b 3 cos ( G 3 r + Δ δ 3 + π 2 ) + 2 E a 2 E b 3 cos θ a 2 b 3 cos ( G 2 r + Δ δ 2 π 2 ) ,
i , j = 1 , 2 , 3 ,
cos t 23 = E a 3 cos θ a 2 a 3 [ ( E a 3 cos θ a 2 a 3 ) 2 + ( E b 3 cos θ a 2 b 3 ) 2 ] 1 2 ,
cos t 13 = E a 3 cos θ a 1 a 3 [ ( E a 3 cos θ a 1 a 3 ) 2 + ( E b 3 cos θ a 1 b 3 ) 2 ] 1 2 ,
I ( r ) = j E a j 2 + E b 3 2 + 2 E a 1 E a 2 cos θ a 1 a 2 cos τ 1 + 2 E a 2 [ ( E a 3 cos θ a 2 a 3 ) 2 + ( E b 3 cos θ a 2 b 3 ) 2 ] 1 2 cos ( τ 2 t 23 ) + 2 E a 1 [ ( E a 3 cos θ a 1 a 3 ) 2 + ( E b 3 cos θ a 1 b 3 ) 2 ] 1 2 cos ( τ 3 + t 13 ) .

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