Abstract

We show that the refracted wave at the exit surface of a Photonic Crystal (PhC) slab is periodically modulated, in positive or in negative direction, changing the slab thickness. In spite of an always increasing literature, the effect of the thickness in negative refraction on PhC’s does not seem to be appropriately considered. However such an effect is not surprising if interpreted with the help of Dynamical Diffraction Theory (DDT), which is generally applied in the x-ray diffraction. The thickness dependence is a direct result of the so-called Pendellösung phenomenon. That explains the periodic exchange, inside the crystal, of the energy among direct beam (or positively refracted) and diffracted beam (or negatively refracted). The Pendellösung phenomenon is an outstanding example of the application of the DDT as a powerful and simple tool for the analysis of s electromagnetic interaction in PhC’s.

© 2005 Optical Society of America

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References

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    [Crossref]
  2. D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
    [Crossref] [PubMed]
  3. R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001)
    [Crossref] [PubMed]
  4. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
    [Crossref] [PubMed]
  5. P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
    [Crossref] [PubMed]
  6. D. R. Smith and N. Kroll “Negative Refractive Index in Left-Handed Material,” Phys. Rev. Lett. 85, 2933–2966 (2000).
    [Crossref] [PubMed]
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    [Crossref]
  8. J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
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  9. P. V. Parimi and W.T. Lu et al. “Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
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  10. P. Kolinko and D. R. Smith. “Numerical study of electromagnetic waves interacting with negative index materials,” Opt Express11, 640–648 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640.
    [Crossref] [PubMed]
  11. A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades. “Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial,” Opt. Express11, 696–708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
    [Crossref] [PubMed]
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  13. D. Maystre and S. Enoch. “Perfect lenses made with left-handed materials: Alice’s mirror?,” J. Opt. Soc. Am. A 21, 122–131 (2004).
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  14. A. Martinez and H. Miguez, et al. “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
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  15. C. G. Parazzoli and R. B. Greegor, et al. “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232–3234 (2004).
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  16. A. L. Pokrovsky and A. L. Efros. “Lens based on the use of left-handed materials,” Appl. Opt. 42, 5701–5705 (2003).
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  23. S. Foteinopoulou and C. M. Soukoulis. “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
    [Crossref]
  24. R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
    [Crossref]
  25. P. V. Parimi and W. T. Lu, et al. “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
    [Crossref] [PubMed]
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    [Crossref]
  27. A. Berrier and M. Mulot, et al. “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,”Phys. Rev. Lett. 93, 73902 (2004).
    [Crossref]
  28. B. W. Battermann and H. Cole, “Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
    [Crossref]
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  31. P.P. Ewald, “Zur Begründung der Kristalloptik-Teil III,” Ann. Physik 54, 519–597 (1917).
    [Crossref]
  32. P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
    [Crossref]
  33. J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).
  34. K. Sakoda, Optical Properties of Photonic Crystals, Springer Verlag (New York, 2001).
  35. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref] [PubMed]
  36. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref] [PubMed]
  37. O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
    [Crossref] [PubMed]
  38. Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
    [Crossref] [PubMed]
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    [Crossref]
  41. C. -H. Kuo and Z. Ye, “Negative-refraction behavior revealed by arrays of dielectric cylinders”, Phys. Rev. E 70, 026608 (2004).
    [Crossref]

2004 (10)

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

D. Maystre and S. Enoch. “Perfect lenses made with left-handed materials: Alice’s mirror?,” J. Opt. Soc. Am. A 21, 122–131 (2004).
[Crossref]

A. Martinez and H. Miguez, et al. “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

C. G. Parazzoli and R. B. Greegor, et al. “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232–3234 (2004).
[Crossref]

R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
[Crossref]

P. V. Parimi and W. T. Lu, et al. “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[Crossref] [PubMed]

A. Berrier and M. Mulot, et al. “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,”Phys. Rev. Lett. 93, 73902 (2004).
[Crossref]

C. Poulton, S. Guenneau, and A.B. Movchan, “Noncommuting limits and effective properties for oblique propagation of electromagnetic waves through an array of aligned fiber,” Phy. Rev. B 69, 195112 (2004).
[Crossref]

C. -H. Kuo and Z. Ye, “Negative-refraction behavior revealed by arrays of dielectric cylinders”, Phys. Rev. E 70, 026608 (2004).
[Crossref]

2003 (8)

Z. Y. Li and L. L. Lin. “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Phys. Rev. B 68, 245110 (2003).
[Crossref]

A. L. Pokrovsky and A. L. Efros. “Lens based on the use of left-handed materials,” Appl. Opt. 42, 5701–5705 (2003).
[Crossref] [PubMed]

E. Cubukcu and K. Aydin, et al. “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[Crossref] [PubMed]

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

S. Foteinopoulou and C. M. Soukoulis. “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
[Crossref]

P. V. Parimi and W.T. Lu et al. “Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
[Crossref] [PubMed]

P. F. Loschialpo and D. L. Smith, et al. “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602 (2003).
[Crossref]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

2002 (3)

P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
[Crossref] [PubMed]

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

C. Luo and S. G. Johnson, et al. “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104, (2002).
[Crossref]

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 (4)

D. R. Smith and N. Kroll “Negative Refractive Index in Left-Handed Material,” Phys. Rev. Lett. 85, 2933–2966 (2000).
[Crossref] [PubMed]

B. Gralak, S. Enoch, and G. Tayeb. “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. of Am. A 17, 1012–1020 (2000).
[Crossref]

M. Notomi. “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

1990 (1)

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[Crossref] [PubMed]

1989 (1)

O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
[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]

1968 (1)

V. G. Veselago “The electrodynamics of substances with simultaneously negative value of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1965 (1)

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
[Crossref]

1964 (1)

B. W. Battermann and H. Cole, “Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

1917 (1)

P.P. Ewald, “Zur Begründung der Kristalloptik-Teil III,” Ann. Physik 54, 519–597 (1917).
[Crossref]

1913 (1)

P.P. Ewald, “Zur Theorie der Interferenzen der Röntgenstrahlen,” Physik Z. 14, 465–472 (1913).

Authier, A.

A. Authier, Dynamical Theory of X-ray Diffraction, Oxford University Press (Oxford, 2001).

Aydin, K.

E. Cubukcu and K. Aydin, et al. “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[Crossref] [PubMed]

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

Battermann, B. W.

B. W. Battermann and H. Cole, “Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

Berrier, A.

A. Berrier and M. Mulot, et al. “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,”Phys. Rev. Lett. 93, 73902 (2004).
[Crossref]

Cole, H.

B. W. Battermann and H. Cole, “Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

Cubukcu, E.

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

E. Cubukcu and K. Aydin, et al. “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[Crossref] [PubMed]

De Leo, R.

O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
[Crossref] [PubMed]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Efros, A. L.

Eleftheriades, G. V.

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades. “Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial,” Opt. Express11, 696–708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[Crossref] [PubMed]

Enoch, S.

D. Maystre and S. Enoch. “Perfect lenses made with left-handed materials: Alice’s mirror?,” J. Opt. Soc. Am. A 21, 122–131 (2004).
[Crossref]

B. Gralak, S. Enoch, and G. Tayeb. “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. of Am. A 17, 1012–1020 (2000).
[Crossref]

Ewald, P.P.

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
[Crossref]

P.P. Ewald, “Zur Begründung der Kristalloptik-Teil III,” Ann. Physik 54, 519–597 (1917).
[Crossref]

P.P. Ewald, “Zur Theorie der Interferenzen der Röntgenstrahlen,” Physik Z. 14, 465–472 (1913).

Foteinopoulou, S.

R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis. “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
[Crossref]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Francescangeli, O.

O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
[Crossref] [PubMed]

Gralak, B.

B. Gralak, S. Enoch, and G. Tayeb. “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. of Am. A 17, 1012–1020 (2000).
[Crossref]

Greegor, R. B.

C. G. Parazzoli and R. B. Greegor, et al. “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232–3234 (2004).
[Crossref]

Guenneau, S.

C. Poulton, S. Guenneau, and A.B. Movchan, “Noncommuting limits and effective properties for oblique propagation of electromagnetic waves through an array of aligned fiber,” Phy. Rev. B 69, 195112 (2004).
[Crossref]

Iyer, A. K.

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades. “Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial,” Opt. Express11, 696–708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[Crossref] [PubMed]

Joannopoulos, J. D.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Joannopulos, J.D.

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Johnson, S. G.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

C. Luo and S. G. Johnson, et al. “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104, (2002).
[Crossref]

Kolinko, P.

P. Kolinko and D. R. Smith. “Numerical study of electromagnetic waves interacting with negative index materials,” Opt Express11, 640–648 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640.
[Crossref] [PubMed]

Kremer, P. C.

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades. “Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial,” Opt. Express11, 696–708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[Crossref] [PubMed]

Kroll, N.

D. R. Smith and N. Kroll “Negative Refractive Index in Left-Handed Material,” Phys. Rev. Lett. 85, 2933–2966 (2000).
[Crossref] [PubMed]

Kuo, C. -H.

C. -H. Kuo and Z. Ye, “Negative-refraction behavior revealed by arrays of dielectric cylinders”, Phys. Rev. E 70, 026608 (2004).
[Crossref]

Li, Z. Y.

Z. Y. Li and L. L. Lin. “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Phys. Rev. B 68, 245110 (2003).
[Crossref]

Lin, L. L.

Z. Y. Li and L. L. Lin. “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Phys. Rev. B 68, 245110 (2003).
[Crossref]

Loschialpo, P. F.

P. F. Loschialpo and D. L. Smith, et al. “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602 (2003).
[Crossref]

Lu, W. T.

P. V. Parimi and W. T. Lu, et al. “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[Crossref] [PubMed]

Lu, W.T.

P. V. Parimi and W.T. Lu et al. “Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
[Crossref] [PubMed]

Luo, C.

C. Luo and S. G. Johnson, et al. “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104, (2002).
[Crossref]

Luo, C. Y.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Martinez, A.

A. Martinez and H. Miguez, et al. “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Maystre, D.

Mead, R.D.

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

Melone, S.

O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
[Crossref] [PubMed]

Miguez, H.

A. Martinez and H. Miguez, et al. “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Moussa, R.

R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
[Crossref]

Movchan, A.B.

C. Poulton, S. Guenneau, and A.B. Movchan, “Noncommuting limits and effective properties for oblique propagation of electromagnetic waves through an array of aligned fiber,” Phy. Rev. B 69, 195112 (2004).
[Crossref]

Mulot, M.

A. Berrier and M. Mulot, et al. “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,”Phys. Rev. Lett. 93, 73902 (2004).
[Crossref]

Notomi, M.

M. Notomi. “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

Parazzoli, C. G.

C. G. Parazzoli and R. B. Greegor, et al. “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232–3234 (2004).
[Crossref]

Parimi, P. V.

P. V. Parimi and W. T. Lu, et al. “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[Crossref] [PubMed]

P. V. Parimi and W.T. Lu et al. “Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
[Crossref] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

Pokrovsky, A. L.

Poulton, C.

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[Crossref]

Russel, P.St.J.

P.St.J. Russel, Designing photonic crystals in Electron and Photon Confinement in Semiconductor Nanostructures, IOP Press (Amsterdam, 2003), p. 79–103.

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R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001)
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Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001)
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P. F. Loschialpo and D. L. Smith, et al. “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602 (2003).
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Smith, D. R.

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
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R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001)
[Crossref] [PubMed]

D. R. Smith and N. Kroll “Negative Refractive Index in Left-Handed Material,” Phys. Rev. Lett. 85, 2933–2966 (2000).
[Crossref] [PubMed]

P. Kolinko and D. R. Smith. “Numerical study of electromagnetic waves interacting with negative index materials,” Opt Express11, 640–648 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640.
[Crossref] [PubMed]

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R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis. “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
[Crossref]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
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P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
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P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
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P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
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D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
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Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
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C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
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C. G. Parazzoli and R. B. Greegor, et al. “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232–3234 (2004).
[Crossref]

R. Moussa, S. Foteinopoulou, and C. M. Soukoulis. “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85, 1125–1127 (2004).
[Crossref]

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

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

B. Gralak, S. Enoch, and G. Tayeb. “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. of Am. A 17, 1012–1020 (2000).
[Crossref]

Nature (2)

P. V. Parimi and W.T. Lu et al. “Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
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E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
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Phy. Rev. B (1)

C. Poulton, S. Guenneau, and A.B. Movchan, “Noncommuting limits and effective properties for oblique propagation of electromagnetic waves through an array of aligned fiber,” Phy. Rev. B 69, 195112 (2004).
[Crossref]

Phys. Rev. A (1)

O. Francescangeli, S. Melone, and R. De Leo, “Dynamical diffraction of microwaves by periodic dielectric media,” Phys. Rev. A 40, 4988–4996 (1989).
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Phys. Rev. B (5)

S. Foteinopoulou and C. M. Soukoulis. “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
[Crossref]

Z. Y. Li and L. L. Lin. “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Phys. Rev. B 68, 245110 (2003).
[Crossref]

M. Notomi. “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
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C. Luo and S. G. Johnson, et al. “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104, (2002).
[Crossref]

A. Martinez and H. Miguez, et al. “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004).
[Crossref]

Phys. Rev. E (2)

P. F. Loschialpo and D. L. Smith, et al. “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602 (2003).
[Crossref]

C. -H. Kuo and Z. Ye, “Negative-refraction behavior revealed by arrays of dielectric cylinders”, Phys. Rev. E 70, 026608 (2004).
[Crossref]

Phys. Rev. Lett. (10)

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[Crossref] [PubMed]

J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

E. Cubukcu and K. Aydin, et al. “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[Crossref] [PubMed]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis. “Refraction in media with a negative refractive index,”, Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
[Crossref] [PubMed]

D. R. Smith and N. Kroll “Negative Refractive Index in Left-Handed Material,” Phys. Rev. Lett. 85, 2933–2966 (2000).
[Crossref] [PubMed]

A. Berrier and M. Mulot, et al. “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,”Phys. Rev. Lett. 93, 73902 (2004).
[Crossref]

P. V. Parimi and W. T. Lu, et al. “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[Crossref] [PubMed]

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

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Physics Today (1)

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

Physik Z. (1)

P.P. Ewald, “Zur Theorie der Interferenzen der Röntgenstrahlen,” Physik Z. 14, 465–472 (1913).

Rev. Mod. Phys. (1)

B. W. Battermann and H. Cole, “Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
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Rev. Mod. Physics (1)

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
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Science (2)

D. R. Smith, J. B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001)
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago “The electrodynamics of substances with simultaneously negative value of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (6)

P. Kolinko and D. R. Smith. “Numerical study of electromagnetic waves interacting with negative index materials,” Opt Express11, 640–648 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640.
[Crossref] [PubMed]

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades. “Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial,” Opt. Express11, 696–708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[Crossref] [PubMed]

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals, Springer Verlag (New York, 2001).

A. Authier, Dynamical Theory of X-ray Diffraction, Oxford University Press (Oxford, 2001).

P.St.J. Russel, Designing photonic crystals in Electron and Photon Confinement in Semiconductor Nanostructures, IOP Press (Amsterdam, 2003), p. 79–103.

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Figures (4)

Fig. 1.
Fig. 1.

Dispersion surfaces in crystal in the long wavelength limit: the medium can be considered as homogeneous (a). Decreasing the wavelength, the spheres approach and a Bragg gap appears (b).

Fig. 2.
Fig. 2.

The crystal dispersion surface, which determines the permitted wavevectors in the structure for a given frequency is a hyperbola (thick line) close to the Bragg gap. In the figure are also shown dispersion surfaces in the air, which are spheres. The intersections of the hyperbola asymptotes in vacuum, the Lorentz point (Lo) and the Laue point (La) respectively, are also indicated.

Fig. 3.
Fig. 3.

Electric field obtained via a FEM simulation in a 2D square lattice PhC with filling factor r/a=0.1. The cylinders have a real dielectric constant ε=3 embedded in vacuum. The polarization is chosen with the electric field (shown in the figure) parallel to the cylinder axis. The wavelength λ satisfies the Bragg law in vacuum with an incidence angle θi =60°, i.e. I0La in Fig. 2. (a) Slab thickness t=6a0/2: the maximum intensity is in the diffracted direction and exhibits negative refraction behavior. (b) t=12a0 ; (c) t=18a=3/2 Λ0 ; (d) t=24a=2Λ0. The Pendellösung exchange of energy between positive and negative refracted beam corresponds to the thickness period Λ0 as calculated from (2).

Fig. 4.
Fig. 4.

FEM simulation of a PhC with the same characteristics as in Fig. 3 and a thickness t=6a. Square modulus of the electric field parallel to the cylinder axis for an incident angle θi =55° (a), and θi =65° (b). The detail (c) shows the modulus of the electric field inside the PhC.

Equations (2)

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X 0 X h = k 2 χ h χ h 4 ( 1 + χ 0 )
Λ 0 = λ cos θ B 1 + χ 0 χ h χ h

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