Abstract

We present a study on negative refractions in the four lowest bands of two-dimensional (2D) square lattices formed by holographic lithography (HL) and compare these features with those of a lattice of the same kind but with regular dielectric columns. The plane wave calculations and FDTD simulations have shown that in some bands or for some interfaces the negative refraction can only happen in holographic structures, and generally the rightness of holographic structures and regular structures of the same kind may be different.

© 2007 Optical Society of America

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References

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  1. V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Usp. Fiz. Nauk 92, 517-526 (1967).
  2. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans Microwave Theory Tech 47, 2075-2084 (1999).
    [CrossRef]
  4. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901-083904 (2003).
    [CrossRef] [PubMed]
  5. L. Chen, S. L. He, L. F. Shen, "Finite-size effects of a left-handed material Slab on the image quality," Phys. Rev. Lett. 92, 107404-1-4 (2004).
    [CrossRef] [PubMed]
  6. M. Notomi, "Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B. 62, 10696-10705 (2000).
    [CrossRef]
  7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-10099 (1998).
    [CrossRef]
  8. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
    [CrossRef] [PubMed]
  9. R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, "Physical and materials aspects of photonic crystals for microwaves and millimetre waves," Z. Metallkd. 95, 618-623 (2004).
  10. K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B. 70, 205125-1-5 (2004).
    [CrossRef]
  11. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and lefthanded electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401-1-4 (2004).
    [CrossRef] [PubMed]
  12. S. Foteinopoulou1 and C. M. Soukoulis "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B. 72, 165112-1-20 (2005).
    [CrossRef]
  13. R. Gajiæ, R. Meisels, F. Kuchar, and K. Hingerl, "Refraction and rightness in photonic crystals," Opt. Express 13, 8596-8605 (2005).
    [CrossRef] [PubMed]
  14. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295-2298 (1991).
    [CrossRef] [PubMed]
  15. X. Y. Ao and S. L. He, "Three-dimensional photonic crystal of negative refraction achieved by interference lithography," Opt. Lett. 29, 2542-2544 (2004).
    [CrossRef] [PubMed]
  16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, "Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap" J. Opt. Soc. Am. B 23, 1708-1711 (2006).
    [CrossRef]
  17. L. Z. Cai, C. S. Feng, M. Z. He, and X. L. Yang, "Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps" Opt. Express 13, 4325-4330 (2005).
    [CrossRef] [PubMed]
  18. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  19. 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]

2006 (1)

2005 (2)

2004 (2)

R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, "Physical and materials aspects of photonic crystals for microwaves and millimetre waves," Z. Metallkd. 95, 618-623 (2004).

X. Y. Ao and S. L. He, "Three-dimensional photonic crystal of negative refraction achieved by interference lithography," Opt. Lett. 29, 2542-2544 (2004).
[CrossRef] [PubMed]

2003 (2)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901-083904 (2003).
[CrossRef] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (2)

M. Notomi, "Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B. 62, 10696-10705 (2000).
[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 404, 53-56 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans Microwave Theory Tech 47, 2075-2084 (1999).
[CrossRef]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-10099 (1998).
[CrossRef]

1991 (1)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295-2298 (1991).
[CrossRef] [PubMed]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

1967 (1)

V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Usp. Fiz. Nauk 92, 517-526 (1967).

IEEE Trans Microwave Theory Tech (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans Microwave Theory Tech 47, 2075-2084 (1999).
[CrossRef]

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

Nature (2)

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]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-10099 (1998).
[CrossRef]

Phys. Rev. B. (1)

M. Notomi, "Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B. 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295-2298 (1991).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901-083904 (2003).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Usp. Fiz. Nauk (1)

V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Usp. Fiz. Nauk 92, 517-526 (1967).

Z. Metallkd. (1)

R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, "Physical and materials aspects of photonic crystals for microwaves and millimetre waves," Z. Metallkd. 95, 618-623 (2004).

Other (4)

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B. 70, 205125-1-5 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and lefthanded electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401-1-4 (2004).
[CrossRef] [PubMed]

S. Foteinopoulou1 and C. M. Soukoulis "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B. 72, 165112-1-20 (2005).
[CrossRef]

L. Chen, S. L. He, L. F. Shen, "Finite-size effects of a left-handed material Slab on the image quality," Phys. Rev. Lett. 92, 107404-1-4 (2004).
[CrossRef] [PubMed]

Supplementary Material (2)

» Media 1: GIF (202 KB)     
» Media 2: GIF (260 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Variation of the shape and size of the cross section of dielectric columns with the same I t = 2.72 and c= 0.16. (a)Holographic inverse structure, FR = 0.416; (b) holographic normal structure, FR = 0.584; and (c) regular structure, r = 0.364a, FR = 0.416.

Fig. 2.
Fig. 2.

EFCs plot and wave vector diagram for TE band 1 at the incident angle of 25° across the ΓM interface for the holographic inverse (a) and normal (b) structures and the regular structure (c). (a)f= 0.28 in air and frequency range of 0.15 to 0.30 with 0.01 step; (b)f= 0.24 in air, frequency range of 0.08 to 0.20 with 0.02 step and the range of 0.20 to 0.26 with 0.01 step; and (c)f= 0.28 in air and frequency range of 0.21 to 0.30 with 0.01 step.

Fig. 3.
Fig. 3.

Contour maps of H z simulated by FDTD showing the propagation wave patterns of TE1 wave for the cases in Figs. 2 (a), 2(b) and 2(c), respectively. The blue, red and yellow arrows denote K i, V g, and the wave vector of transmitted wave, respectively.

Fig. 4.
Fig. 4.

EFCs plot and wave vector diagrams for TE band 2 and an incident angle of 20°. (a), (b) and (c) are for the HL inverse structure, HL normal structure and the structure of regular columns shown in Fig.1 (a), (b) and (c), respectively. (a) f= 0.35 in air and frequency range of 0.29 to 0.36 with 0.01 step; (b) f = 0.32 in air and frequency range of 0.26 to 0.34 with 0.01step; and (c) f= 0.35 in air and frequency range of 0.29 to 0.41 with 0.01step.

Fig. 5.
Fig. 5.

GIF format images showing the propagation wave patterns of TE 2 wave on ΓX interface when incident angle changes from 5° to 60° with 5° step. (a)For inverse HL structure with incident frequency f = 0.35; [Media 1] (b) for normal HL structure and f= 0.32. [Media 2]

Fig. 6.
Fig. 6.

Propagation wave patterns of TE 2 wave at an incidence of 20° on the ΓM interface. (a), (b) and (c) are for the inverse holographic structure, normal holographic structure and regular structure, respectively.

Fig. 7.
Fig. 7.

EFCs plot and wave vector diagrams for TE band 4 and incident angle of 45°. (a) For HL inverse structure, f = 0.50 in air and frequency range of 0.44 to 0.53 with 0.01 step; (b) for HL normal structure, f = 0.43 in air and frequency range of 0.38 to 0.46 with 0.01step; and (c) for regular structures, f = 0.50 in air and frequency range of 0.48 to 0.53 with 0.01 step.

Fig. 8.
Fig. 8.

Propagation wave patterns of TE 4 wave at an incidence of 45° on the ΓX interface. (a), (b) and (c) are the contour maps of H z corresponding to Fig. 7 (a), (b) and (c), respectively.

Tables (1)

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Table 1: The distributions of relative frequencies that can intricate negative refractions

Equations (1)

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I ( x , y ) = 3 + cos ( 2 π x a ) + cos ( 2 π x a ) c { cos [ 4 π ( x + y ) a ] + cos [ 4 π ( x y ) a ] } .

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