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

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. B. Pendry and D. R. Smith. "Reversing light with negative refraction,�?? Physics Today 57, 37-43 (2004).
    [CrossRef]
  2. D. R. Smith, J. B. Pendry, 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]
  7. V. G. Veselago �?? The electrodynamics of substances with simultaneously negative value of ε and µ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  8. J. B. Pendry. "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  9. P. V. Parimi , W.T. Lu et al. �??Imaging by flat lens using negative refraction,�?? Nature 426, 404 (2003).
    [CrossRef] [PubMed]
  10. P. Kolinko and D. R. Smith. "Numerical study of electromagnetic waves interacting with negative index materials," Opt Express 11, 640-648 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640</a>
    [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. Express 11, 696-708 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696. </a>
    [CrossRef] [PubMed]
  12. P. F. Loschialpo, D. L. Smith, et al. "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
    [CrossRef]
  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).
    [CrossRef]
  14. A. Martinez, 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]
  15. C. G. Parazzoli, R. B. Greegor, et al. "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
    [CrossRef]
  16. A. L. Pokrovsky and A. L. Efros. "Lens based on the use of left-handed materials," Appl. Opt. 42, 5701-5705 (2003).
    [CrossRef] [PubMed]
  17. B. Gralak, S. Enoch and G. Tayeb. "Anomalous refractive properties of photonic crystals," J. Opt. Soc. of Am. A 17, 1012-1020 (2000).
    [CrossRef]
  18. 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]
  19. C. Y. Luo, S. G. Johnson and J. D. Joannopoulos. "All-angle negative refraction in a three-imensionally periodic photonic crystal," Appl. Phys. Lett. 81, 2352-2354 (2002).
    [CrossRef]
  20. C. Luo, S. G. Johnson, et al. "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104, (2002).
    [CrossRef]
  21. E. Cubukcu, K. Aydin, et al. "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
    [CrossRef] [PubMed]
  22. E. Cubukcu, K. Aydin, et al. "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
    [CrossRef] [PubMed]
  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, W. T. Lu, et al. "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  26. Z. Y. Li and L. L. Lin. "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Phys. Rev. B 68, 245110 (2003).
    [CrossRef]
  27. A. Berrier, 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, H. Cole, �??Dynamical diffraction theory of X rays by perfect crystals,�?? Rev. Mod. Phys. 36, 681-717 (1964).
    [CrossRef]
  29. A. Authier, Dynamical Theory of X-ray Diffraction, Oxford University Press (Oxford, 2001).
  30. P.P. Ewald, �??Zur Theorie der Interferenzen der Röntgenstrahlen,�?? Physik Z. 14, 465-472 (1913).
  31. P.P. Ewald, �??Crystal optics for visible light and X rays,�?? Rev. Mod. Physics 37, 46-56 (1965).
    [CrossRef]
  32. J.D. Joannopulos, R.D. Mead, J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).
  33. K. Sakoda, Optical Properties of Photonic Crystals, Springer Verlag (New York, 2001).
  34. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  35. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  36. O. Francescangeli, S. Melone, R. De Leo, "Dynamical diffraction of microwaves by periodic dielectric media,�?? Phys. Rev. A 40, 4988-4996 (1989).
    [CrossRef] [PubMed]
  37. Z. Zhang, S. Satpathy, �??Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell's equations,�?? Phys. Rev. Lett. 65, 2650-2653 (1990).
    [CrossRef] [PubMed]
  38. P. St. J. Russel, Designing photonic crystals in Electron and Photon Confinement in Semiconductor Nanostructures , IOP Press (Amsterdam, 2003), p. 79-103.
  39. C. Poulton, S. Guenneau, 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]
  40. C. �??H. Kuo and Z. Ye, �??Negative-refraction behavior revealed by arrays of dielectric cylinders�??, Phys. Rev. E 70, 026608 (2004).
    [CrossRef]
  41. P.P. Ewald, �??Crystal optics for visible light and X rays,�?? Rev. Mod. Physics 37, 46-56 (1965).

Appl. Opt. (1)

Appl. Phys. Lett. (3)

C. G. Parazzoli, R. B. Greegor, et al. "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

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

E. Cubukcu, K. Aydin, et al. "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

P. V. Parimi , W.T. Lu et al. �??Imaging by flat lens using negative refraction,�?? Nature 426, 404 (2003).
[CrossRef] [PubMed]

Opt Express (1)

P. Kolinko and D. R. Smith. "Numerical study of electromagnetic waves interacting with negative index materials," Opt Express 11, 640-648 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640</a>
[CrossRef] [PubMed]

Opt. Express (1)

Phy. Rev. B (1)

C. Poulton, S. Guenneau, 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, R. De Leo, "Dynamical diffraction of microwaves by periodic dielectric media,�?? Phys. Rev. A 40, 4988-4996 (1989).
[CrossRef] [PubMed]

Phys. Rev. B (4)

A. Martinez, 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]

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

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

Phys. Rev. B. (1)

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]

Phys. Rev. E (2)

P. F. Loschialpo, 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 (1)

Z. Zhang, S. Satpathy, �??Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell's equations,�?? Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (9)

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]

J. B. Pendry. "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[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]

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

P. V. Parimi, 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, M. Mulot, et al. "Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,�??Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

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, H. Cole, �??Dynamical diffraction theory of X rays by perfect crystals,�?? Rev. Mod. Phys. 36, 681-717 (1964).
[CrossRef]

Rev. Mod. Physics (2)

P.P. Ewald, �??Crystal optics for visible light and X rays,�?? Rev. Mod. Physics 37, 46-56 (1965).
[CrossRef]

P.P. Ewald, �??Crystal optics for visible light and X rays,�?? Rev. Mod. Physics 37, 46-56 (1965).

Science (2)

D. R. Smith, J. B. Pendry, 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 (4)

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

J.D. Joannopulos, R.D. Mead, 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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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

Metrics