Abstract

Specific configurations of three linearly polarized, monochromatic plane waves have previously been shown to be capable of producing interference patterns exhibiting symmetry inherent in 5 of the 17 plane groups. Starting with the general expression for N linearly polarized waves, three-beam interference is examined in detail. The totality of all possible sets of constraints for producing the five plane groups is presented. In addition, two uniform contrast conditions are identified and discussed. Further, it is shown that when either of the uniform contrast conditions is applied and the absolute contrast is maximized, unity absolute contrast is achievable.

© 2008 Optical Society of America

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References

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  1. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 75, 894-937 (1985).
    [CrossRef]
  2. J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
    [CrossRef]
  3. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).
  4. J. L. Stay and T. K. Gaylord, “Photo-mask for wafer-scale fabrication of two- and three-dimensional photonic crystal structures,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FThC5.
  5. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858-60 (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 404, 53-56 (2000).
  7. M. Weidong, Z. Yongchun, D. Jianwen, and W. Hezhou, “Crystallography of two-dimensional photonic lattices formed by holography of three noncoplanar beams,” J. Opt. Soc. Am. B 22, 1085-1091 (2005).
    [CrossRef]
  8. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
    [CrossRef]
  9. T. Hahn, ed., International Tables for Crystallography, Volume A: Space Group Symmetry (Springer, 2002).

2007 (1)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

2006 (2)

J. L. Stay and T. K. Gaylord, “Photo-mask for wafer-scale fabrication of two- and three-dimensional photonic crystal structures,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FThC5.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

2005 (1)

2002 (2)

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
[CrossRef]

T. Hahn, ed., International Tables for Crystallography, Volume A: Space Group Symmetry (Springer, 2002).

2001 (1)

2000 (1)

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).

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 75, 894-937 (1985).
[CrossRef]

Cai, L. Z.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858-60 (2001).
[CrossRef]

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 404, 53-56 (2000).

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 404, 53-56 (2000).

Dolling, G.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

Gaylord, T. K.

J. L. Stay and T. K. Gaylord, “Photo-mask for wafer-scale fabrication of two- and three-dimensional photonic crystal structures,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FThC5.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 75, 894-937 (1985).
[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 404, 53-56 (2000).

Hezhou, W.

Jianwen, D.

Linden, S.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 75, 894-937 (1985).
[CrossRef]

Moon, J. H.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[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 404, 53-56 (2000).

Soukoulis, C. M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

Stay, J. L.

J. L. Stay and T. K. Gaylord, “Photo-mask for wafer-scale fabrication of two- and three-dimensional photonic crystal structures,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FThC5.

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 404, 53-56 (2000).

Wang, Y. R.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858-60 (2001).
[CrossRef]

Wegener, M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

Weidong, M.

Yang, S.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

Yang, S.-M.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

Yang, X. L.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858-60 (2001).
[CrossRef]

Yongchun, Z.

Appl. Phys. Lett. (1)

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. 19, 2238-2244 (2002).
[CrossRef]

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

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 75, 894-937 (1985).
[CrossRef]

Other (4)

T. Hahn, ed., International Tables for Crystallography, Volume A: Space Group Symmetry (Springer, 2002).

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, pp. 53-55 (2007).

J. L. Stay and T. K. Gaylord, “Photo-mask for wafer-scale fabrication of two- and three-dimensional photonic crystal structures,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FThC5.

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).

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Figures (9)

Fig. 1
Fig. 1

Locations of symmetry elements for the 17 plane groups [9]. Unit cell outlines correspond to conventional primitive unit cells, illustrated in subsequent figures as dashed lines.

Fig. 2
Fig. 2

Design parameters and associated interference pattern exhibiting pmm plane group symmetry. The UCC-2 has been applied and absolute contrast maximized, resulting in unity absolute contrast ( V abs = 1 ) , with zero intensity at intensity nulls. The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 3
Fig. 3

Design parameters and associated interference pattern exhibiting pmm plane group symmetry. No uniform contrast condition has been applied. The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 4
Fig. 4

Design parameters and associated interference pattern exhibiting cmm plane group symmetry. This design results in one of two fundamentally different interference patterns when UCC-1 is applied, possessing intensity peaks at lattice points. The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 5
Fig. 5

Design parameters and associated interference pattern exhibiting pmm plane group symmetry. The UCC-2 has been applied and absolute contrast maximized, resulting in unity absolute contrast ( V abs = 1 ) , with zero intensity at intensity nulls. The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 6
Fig. 6

Design parameters and associated interference pattern exhibiting p4m plane group symmetry. The UCC-2 has been applied and absolute contrast maximized, resulting in unity absolute contrast ( V abs = 1 ) , with zero intensity at intensity nulls. The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 7
Fig. 7

Design parameters and associated interference pattern exhibiting p6m plane group symmetry. This design results in one of two fundamentally different interference patterns when UCC-1 is applied, possessing intensity nulls at lattice points. Polarization unit vectors are coplanar ( x y plane) and 120 ° apart from one another . The conventional primitive unit cell (dashed lines) and the Wigner–Seitz proximity unit cell (dotted lines) are shown.

Fig. 8
Fig. 8

Plot of maximum contrast obtainable for an interference pattern exhibiting p6m plane group symmetry as a function of wave vector zenith angle when UCC-1 is applied and optimized for V ( 1 ) > 0 . The minimum occurs for V abs = 0.6 at θ = tan 1 2 54.7 ° or when all three recording wave vectors are orthogonal. The minimum shown, in actuality, is the global minimum for all configurations of three wave vectors.

Fig. 9
Fig. 9

Flow chart illustrating relationships between conditions required for each of the five plane symmetry groups to exist in three-beam interference. In general, p2 plane group symmetry occurs for general three-beam interference. As relationships between the basis vectors (a and b) and interference coefficients ( V i j ) emerge, the other four plane symmetry groups can exist.

Equations (28)

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I i ( r ) = i ( r , t ) · i ( r , t ) = 1 2 Re [ E i ( r ) · E i * ( r ) ] = 1 2 E i 2 .
I T ( r ) = T ( r , t ) · T ( r , t ) = I 1 ( r ) + I 2 ( r ) + 2 J 12 ( r ) ,
J 12 ( r ) = 1 ( r , t ) · 2 ( r , t ) = 1 2 Re [ E 1 ( r ) · E 2 * ( r ) ] = 1 2 E 1 E 2 e 12 cos ( ( k 2 k 1 ) · r + ϕ 1 ϕ 2 ) ,
I T ( r ) = T ( r , t ) · T ( r , t ) = i = 1 N ( I i ( r ) + j > i N 2 J i j ( r ) ) = i = i N ( 1 2 E i 2 + j > i N E i E j e i j cos ( ( k j k i ) · r + ϕ i ϕ j ) ) .
A = 2 π b × z ^ a · b × z ^ , B = 2 π z ^ × a a · b × z ^ .
λ n 2 sin 2 γ ( 1 | a | 2 + 1 | b | 2 + 2 cos γ | a | | b | ) 1 / 2 ,
I T = I 0 [ 1 + V 12 cos ( G 21 · r ) + V 13 cos ( G 31 · r ) + V 23 cos ( G 32 · r ) ] ,
I 0 = 1 2 k = 1 3 E k 2 ,
V i j = E i E j e i j I 0 ,
V abs = I max - I min I max + I min ,
V ( 1 ) = V 12 = V 13 = V 23 .
E 2 = e 13 e 23 E 1 , E 3 = e 12 e 23 E 1 .
V ( 1 ) = 2 e 12 e 13 e 23 e 12 2 + e 13 2 + e 23 2 .
V abs = | 9 4 / V ( 1 ) + 3 | .
V ( 2 ) = V 12 = V 13 , V 23 = 0.
e 23 = 0 , E 3 = e 12 e 13 E 2 .
V ( 2 ) = 2 E 1 E 2 e 12 e 13 2 E 1 2 e 13 2 + E 2 2 e 13 2 + E 2 2 e 12 2 .
V abs = | 2 V ( 2 ) | .
| e 13 + e 23 | 2 cos ( cos - 1 ( e 12 ) 2 ) ,
| e 12 + e 23 | 2 cos ( cos - 1 ( e 13 ) 2 ) ,
| e 12 + e 13 | 2 cos ( cos - 1 ( e 23 ) 2 ) ,
e 12 = ± 1 , e 13 = ± 1 , e 23 = ± 1.
E 2 = ± E 1 , E 3 = ± E 1 .
e 12 = ± 1 2 , e 13 = ± 1 2 , e 23 = ± 1 2 .
E 2 = ± E 1 , E 3 = ± E 1 .
| e 12 + e 13 | 2 / 2 ,
e 12 = ± 2 2 , e 13 = ± 2 2 , e 23 = 0.
E 2 = ± 2 2 E 1 , E 3 = ± 2 2 E 1 .

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