Abstract

We present a study on relation between the refraction and rightness effects in photonic crystals applied on a 2D square lattice photonic crystal. The plane wave (the band and equifrequency contour analyses) and FDTD calculations for both TM and TE modes revealed all possible refraction and rightness cases in photonic crystal structures in the first three bands. In particular, we show for the first time, a possibility of the left-handed positive refraction. This means that left-handedness does not necessarily imply negative refraction in photonic crystals.

© 2005 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,�?? Uspekhi Fiz. Nauk 92, 517-526 (1967); �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Soviet Physics Uspekhi 10, 509-514 (1968); �??Electrodynamics of materials with negative index of refraction,�?? Uspekhi Fiz. Nauk 173, 790-794 (2003).
    [CrossRef]
  2. J. B. Pendry and D. R. Smith, �??Reversing light with negative refraction,�?? Phys. Today, June, 37-43 (2004).
    [CrossRef]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. on Microwave Theory and Techniques 47, 2075-2084 (1999).
    [CrossRef]
  4. R. A. Shelby, D. R. Smith and S.Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, �??Refraction in media with a negative refraction index,�?? Physical Review Letters 90, 107402-1-107402-4 (2003); S. Foteinopoulou and C. M. Soukoulis, �??EM wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects,�?? Cond-Mat/0212434v1 (2004).
    [CrossRef]
  8. C. Luo, S. G. Johnson, J. D. Joannopoulos and J. B. Pendry, �??All-angle negative refraction without negative effective index,�?? Phys. Rev. B 65, 201104-1-4 (2002).
    [CrossRef]
  9. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, �??Negative refraction by photonic crystals,�?? Nature 423, 604 (2003); E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, �??Subwavelength resolution in a two-dimensional photonic-crystal-based superlense,�?? Physical Review Letters 91, 207401-1-4 (2003).
    [CrossRef] [PubMed]
  10. R. Gaji�? , 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,�?? Zeitschrift fuer Metallkunde 95, 618-623 (2004).
  11. 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,�?? Physical Review B 70, 205125-1-5 (2004).
    [CrossRef]
  12. R. Meisels, R. Gaji�? , F. Kuchar, K. Hingerl, J. Zarbakhsh, �??Negative refraction and left-handedness in a 2D square lattice photonic crystal at microwaves and millimetre waves,�?? submitted to Physical Review B (2005).
  13. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, �??Negative refraction and left-handed electromagnetism in microwave photonic crystals,�?? Phys. Rev. Lett. 92, 127401-1-4 (2004); P. Vodo, P. V. Parimi, W. T. Lu, S. Sridhar and R. Wing, �??Microwave photonic crystal with tailor-made negative refractive index,�?? Applied Physics Letters 85, 1858-1860 (2004).
    [CrossRef] [PubMed]
  14. BandSOLVE, RSoft Design Group Inc., URL: <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.
  15. FullWAVE, RSoft Design Group Inc., URL: <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.
  16. A.Martinez, H. Miguez, A. Griol and J. Marti, �??Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,�?? Phys. Rev. B 69, 165119-1-6 (2004).
    [CrossRef]

IEEE Trans. on Microwave Theory and Tech

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

Nature

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, �??Negative refraction by photonic crystals,�?? Nature 423, 604 (2003); E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, �??Subwavelength resolution in a two-dimensional photonic-crystal-based superlense,�?? Physical Review Letters 91, 207401-1-4 (2003).
[CrossRef] [PubMed]

Phys. Rev. B

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]

C. Luo, S. G. Johnson, J. D. Joannopoulos and J. B. Pendry, �??All-angle negative refraction without negative effective index,�?? Phys. Rev. B 65, 201104-1-4 (2002).
[CrossRef]

A.Martinez, H. Miguez, A. Griol and J. Marti, �??Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,�?? Phys. Rev. B 69, 165119-1-6 (2004).
[CrossRef]

Phys. Rev. Lett.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, �??Negative refraction and left-handed electromagnetism in microwave photonic crystals,�?? Phys. Rev. Lett. 92, 127401-1-4 (2004); P. Vodo, P. V. Parimi, W. T. Lu, S. Sridhar and R. Wing, �??Microwave photonic crystal with tailor-made negative refractive index,�?? Applied Physics Letters 85, 1858-1860 (2004).
[CrossRef] [PubMed]

Phys. Today

J. B. Pendry and D. R. Smith, �??Reversing light with negative refraction,�?? Phys. Today, June, 37-43 (2004).
[CrossRef]

Physical Review B

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]

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,�?? Physical Review B 70, 205125-1-5 (2004).
[CrossRef]

R. Meisels, R. Gaji�? , F. Kuchar, K. Hingerl, J. Zarbakhsh, �??Negative refraction and left-handedness in a 2D square lattice photonic crystal at microwaves and millimetre waves,�?? submitted to Physical Review B (2005).

Physical Review Letters

S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, �??Refraction in media with a negative refraction index,�?? Physical Review Letters 90, 107402-1-107402-4 (2003); S. Foteinopoulou and C. M. Soukoulis, �??EM wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects,�?? Cond-Mat/0212434v1 (2004).
[CrossRef]

Science

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

Uspekhi Fiz. Nauk

V. G. Veselago, �??Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,�?? Uspekhi Fiz. Nauk 92, 517-526 (1967); �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Soviet Physics Uspekhi 10, 509-514 (1968); �??Electrodynamics of materials with negative index of refraction,�?? Uspekhi Fiz. Nauk 173, 790-794 (2003).
[CrossRef]

Zeitschrift fuer Metallkunde

R. Gaji�? , 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,�?? Zeitschrift fuer Metallkunde 95, 618-623 (2004).

Other

BandSOLVE, RSoft Design Group Inc., URL: <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.

FullWAVE, RSoft Design Group Inc., URL: <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.

Supplementary Material (2)

» Media 1: AVI (606 KB)     
» Media 2: AVI (650 KB)     

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Figures (13)

Fig. 1.
Fig. 1.

LH- (a) and RH- (b) refraction (the sign- denotes negative refraction). For LH-, v ph v gr < 0 whereas for RH- , v ph v gr > 0.

Fig. 2.
Fig. 2.

Band structure of a 2D PhC made of n = 3.1 rods. The boxes determine the regions where negative refraction occurs.

Fig. 3.
Fig. 3.

EFC plot of the TM1 for a 2D PhC made of rods with n = 3.1. EFCs are presented for 10, 15, …,30 and 33, 34, …39 GHz.

Fig. 4.
Fig. 4.

Wave pattern of the TM1 wave incident at 45° on the ΓM interface of the PhC (FDTD) atf= 36 GHz. The black lines represent the wave front in the PhC. The red and yellow arrows denote v gr and v ph, respectively.

Fig. 5.
Fig. 5.

EFC plot for the second TE band of a 2D square lattice PhC of n =3.1 rods in the case of an EMW incidence of 15 ° in respect to the normal to the ΓM interface. The red and yellow arrows denote the incident EMW of 67 GHz and the wave vectors ( k PhC and k PhC), respectively. The dashed arrows represent group velocities. EFCs are shown with 0.5 GHz steps between 65 and 67.5 GHz. The wave with k PhC at point A does not exist since the corresponding v gr would not give an energy flow away from the source.

Fig. 6.
Fig. 6.

Propagation wave pattern of the 67 GHz TE2 wave at an incidence of 15° across the ΓM interface of the PhC (FDTD simulation) exhibiting the Veselago LH- refraction. The red and yellow arrows denote v gr and v ph, respectively.

Fig. 7.
Fig. 7.

Propagation wave pattern of the 67 GHz TE wave at an incidence of 15° on the ΓX interface of the PhC (FDTD) exhibiting the Veselago LH- refraction. The dashed gray outgoing arrows denote the temporary positions of the beams. The red and yellow arrows denote v gr and v ph, respectively, as before.

Fig. 8.
Fig. 8.

Similarly as in Fig. 1, positive refraction could be realized as a LH+ (a) or a RH+ (b) beam.

Fig. 9.
Fig. 9.

EFC plot of the TM2 band of a 2D square lattice PhC of n = 3.1 rods. The red circle denotes the 64 GHz EFC in air. The PhC EFCs are shown at 50, 55, 60, 62, 64 and 66 GHz. The point B corresponds to an outgoing wave as before.

Fig. 10.
Fig. 10.

EFC plot for the TM3 band of a 2D square lattice PhC made of n = 3.1 rods in the case of the 64 GHz EMW incident at 10° across the ΓM interface. The black and white arrows denote the incident air wave vector and phase velocity in the crystal (k PhC), respectively. The dash blue arrow represents the group velocity. The EFCs are shown in the range 60 to 68 GHz with 2 GHz steps.

Fig. 11.
Fig. 11.

Propagation wave pattern of the 64 GHz TM wave at an incidence of 10° on the ΓM interface of the PhC. Both beams are LH whereas just the left one (from the TM2 band) exhibits the LH negative refraction. The right beam corresponds to the TM3 band and shows coexistence between LH and positive refraction as in Fig. 8. LH± refers to positive (negative) LH refraction. The black lines represent the wave fronts of the beams. The red and yellow arrows denote v gr and v ph as defined in Figs. 1 and 8.

Fig. 12.
Fig. 12.

Backward waves of the 67 GHz TE2 mode (FDTD) as in Fig. 6 exhibiting the Veselago LH- refraction [Media 1]

Fig. 13.
Fig. 13.

Backward waves in the LH- (left, TM2) and LH+ beams (right, TM3) as in Fig. 11. [Media 2]

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