Abstract

We consider the conditions for negative refraction in the partial bandgap of photonic crystals and show that, in contrast to previously published studies, anisotropy is not a necessary condition for negative refraction, and that unrestricted imaging is possible. This analysis is made possible by the introduction of a negative local wavevector. In addition, we analyze the origins of the previously reported restricted and adjacent imaging properties and extend negative refraction to higher bands.

© 2008 Optical Society of America

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  1. V. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514, (1968, in Russian,1964).
    [CrossRef]
  2. J. B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef]
  3. D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett. 2, 103-106 (2003).
    [CrossRef]
  5. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  6. D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 1074011 (2003).
    [CrossRef]
  7. J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
    [CrossRef]
  8. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals, " J. Opt. Soc. Am. A 17, 1012-1020 (2000).
    [CrossRef]
  9. 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]
  10. 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(R) (2002).
    [CrossRef]
  11. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
    [CrossRef]
  12. Z.-Y. Li and L.-L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
    [CrossRef]
  13. G. Sun, A. S. Jugessur, and A. G. Kirk, "Imaging properties of dielectric photonic crystal slabs for large object distances, " Opt. Express 14, 6755-6765 (2006).
    [CrossRef] [PubMed]
  14. K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, New York, 2nd ed., 2005).
  15. A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, "Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes," Opt. Express 13, 4160-4174 (2005).
    [CrossRef] [PubMed]
  16. A. Martínez and J. Martí, "Positive phase evolution of waves propagating along a photonic crystal with negative index of refraction," Opt. Express 14, 9805-9814 (2006).
    [CrossRef] [PubMed]
  17. G. Sun and A. G. Kirk, "Pseudo-interference and its application in determining averaged phase refractive index of photonic crystals," IEEE LEOS 2006 Annual Meeting, 29 October - 2 November 2006, Montreal, Canada.
  18. G. Sun and A. G. Kirk, "On the relationship between Bloch modes and phase-related refractive index of photonic crystals," Opt. Express 15, 13149-13154 (2007).
    [CrossRef] [PubMed]
  19. Z. Tang, R. Peng, Y. Ye, C. Zhao, D. Fan, H. Zhang, and S. Wen, "Optical properties of a square-lattice photonic crystal within the partial bandgap," J. Opt. Soc. Am. A 24, 379-384 (2007).
    [CrossRef]
  20. Z. Lu, S. Shi, C. Schuetz, and D. Prather, "Experimental demonstration of negative refraction imaging in both amplitude and phase," Opt. Express 13, 2007-2012 (2005).
    [CrossRef] [PubMed]
  21. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
    [CrossRef] [PubMed]
  22. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
    [CrossRef]
  23. S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
    [CrossRef]

2007

2006

2005

2004

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

2003

D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 1074011 (2003).
[CrossRef]

M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett. 2, 103-106 (2003).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (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]

2002

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(R) (2002).
[CrossRef]

2000

J. B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef]

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals, " J. Opt. Soc. 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]

1994

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

1968

V. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514, (1968, in Russian,1964).
[CrossRef]

Anioniades, M.

M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett. 2, 103-106 (2003).
[CrossRef]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

Chen, L.

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

Eleftheriades, G.V.

M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett. 2, 103-106 (2003).
[CrossRef]

Enoch, S.

Fan, D.

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Gralak, B.

He, S.

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[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(R) (2002).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[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(R) (2002).
[CrossRef]

Jugessur, A. S.

Kirk, A. G.

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

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]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

Lu, Z.

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[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(R) (2002).
[CrossRef]

Martí, J.

Martínez, A.

Míguez, H.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Padina, W. J.

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[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(R) (2002).
[CrossRef]

Peng, R.

Pentry, J. B.

J. B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef]

Prather, D.

Ruan, Z.

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

Sánchez-Dehesa, J.

Schuetz, C.

Schultz, S.

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 1074011 (2003).
[CrossRef]

Shen, J.

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

Shi, S.

Smith, D. R.

D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 1074011 (2003).
[CrossRef]

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Sun, G.

Tang, Z.

Tayeb, G.

Veselago, V.

V. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514, (1968, in Russian,1964).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

Wen, S.

Ye, Y.

Zhang, H.

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Zhao, C.

IEEE Antennas & Wireless Propag. Lett.

M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett. 2, 103-106 (2003).
[CrossRef]

J. Mod. Opt.

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Nature

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 27 (2003).
[CrossRef]

Opt. Express

Phys. Rev. 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]

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(R) (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, " Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (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]

S. He, Z. Ruan, L. Chen, and J. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phys. Rev. B 70, 115113 (2004).
[CrossRef]

Phys. Rev. Lett.

J. B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef]

D. R. Smith, W. J. Padina, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 1074011 (2003).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514, (1968, in Russian,1964).
[CrossRef]

Other

K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, New York, 2nd ed., 2005).

G. Sun and A. G. Kirk, "Pseudo-interference and its application in determining averaged phase refractive index of photonic crystals," IEEE LEOS 2006 Annual Meeting, 29 October - 2 November 2006, Montreal, Canada.

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Figures (8)

Fig. 1.
Fig. 1.

The equifrequency contours calculated from band structure and the conservation of the momentum. The interface is along NN.

Fig. 2
Fig. 2

The effective refractive index (solid red line) and the local effective refractive index (black line) calculated from band structure. The circles and diamonds are numerical results extracted from simulations in Sec.III.

Fig. 3.
Fig. 3.

The reciprocal lattice space and the first few Bloch modes (red spots) from simulations.

Fig. 4(a) and (b)
Fig. 4(a) and (b)

(a) Ω=0.168 (no image) (b) Ω=0.192 (no image) Fig. 4. (a) (b) Imaging of a 7- row PhC slab at four different normalized frequencies.

Fig. 4(c) and (d)
Fig. 4(c) and (d)

(c) Ω=0.200(image formed) (d) Ω=0.225 (non-adjacent image) Fig. 4. (c) (d) Imaging of a 7-row PhC slab at four different normalized frequencies.

Fig. 5.
Fig. 5.

2nd band of a photonic crystal with rod in air (rod-radius-period ratio 0.38 and dielectric constant 8.9)

Fig. 6.
Fig. 6.

Negative imaging at Ω=0.41 in the partial bandgap at the 2nd band (rod in air with permittivity 8.9 and radius-period ratio 0.38).

Fig.7.
Fig.7.

Equifrequency contours of the PhC in Fig. 6 (Notice the contours surrounding M satisfy the two conditions of negative refraction).

Equations (4)

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

u ( r ) = u m , n ( r ) e j ( k + m G 1 + n G 2 ) r
G 1 = G 2 = 2 π 2 a
u ( r ) = e j G 1 r u m , n ( r ) e j { k + ( m G 1 + n G 2 ) } r = u m , n ( r ) e j { k + ( m G 1 + n G 2 ) } r
k = k G 1

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