Abstract

It has previously been shown that the phase-related refractive index is positive in photonic crystals that display negative refraction at higher bands. We hypothesize that the phase velocity is governed by a wave that can be related to the dominant Bloch mode. This dominant wave can be identified from an approximate solution of Maxwell Equations using a homogeneously averaged dielectric constant and the dominant wavevector is related to the fundamental wavevector and the reciprocal lattice vectors. We validate this hypothesis by numerical Fourier decomposition of the field in the entire simulation domain. It confirms that for negative refraction at higher bands, the phase-related refractive index is indeed positive and differs significantly from the negative value of effective refractive index calculated from the band structure.

© 2007 Optical Society of America

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  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, 10096-10099, (1998).
    [CrossRef]
  2. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
    [CrossRef]
  3. J.P. Dowling and C.M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt.,  41, 345-351, (1994).
    [CrossRef]
  4. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals, " J. Opt. Soc. Am, A,  17, 1012-1020, (2000).
    [CrossRef]
  5. 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]
  6. 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, (2002).
    [CrossRef]
  7. V. Veselago, "The electrodynamics of substances with simultaneously negative values of and ," Soviet Phys.Uspekhi, 10, 509-514, (1968). (in Russian,1964)
    [CrossRef]
  8. J.B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett.,  85, 3966-3969, (2000).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005)
    [CrossRef]
  13. M.  Born, E.  Wolf, Principles of Optics, Pergamon, Oxford, 1989.
  14. P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A,  33, 3232-3242, (1986)
    [CrossRef]
  15. K. Sakoda, "Optical properties of photonic crystals,"Springer-Verlag, New York,2nd ed., 2005
  16. 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]
  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.
  18. 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]
  19. G. Sun, A. Bakhtazad, A. Jugessur, and A. Kirk, "Open cavities using photonic crystals with negative refraction," Proc. SPIE, 6343, ed. P. Mathieu, (2006), doi:10.1117/12.708026
  20. A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation, Wiley, New York, 2003.
  21. S. Foteinopoulou and C.M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
    [CrossRef]
  22. G. Sun and A. G. Kirk, "Lattice resonance inside photonic crystal slab with negative refraction," OSA Frontiers in Optics 2006/Laser Science XXII conferences, October 8-12, 2006, Rochester, New York.

2006 (2)

2005 (3)

2003 (1)

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]

2002 (1)

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, (2002).
[CrossRef]

2000 (4)

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]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (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, 10096-10099, (1998).
[CrossRef]

1994 (1)

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

1986 (1)

P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A,  33, 3232-3242, (1986)
[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]

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]

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]

Dunbar, L. A.

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.

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

Ferrini, R.

Foteinopoulou, S.

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

Gralak, B.

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

Houdré, R.

Joannopoulos, J.D.

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, (2002).
[CrossRef]

Johnson, S.G.

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, (2002).
[CrossRef]

Jugessur, A.S.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

Kirk, A.G.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

Lombardet, B.

Luo, C.

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, (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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

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]

Pendry, J.B.

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, (2002).
[CrossRef]

Pentry, J.B.

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

Russel, P. St.J.

P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A,  33, 3232-3242, (1986)
[CrossRef]

Sánchez-Dehesa, J.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

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]

Smith, D.R.

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.

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

Sun, G.

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

Tayeb, G.

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

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[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]

Appl. Phys. Lett. (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett.,  74, 1212-1214, (1999)
[CrossRef]

IEEE Antennas & Wireless Propag. Lett. (1)

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. (1)

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 (1)

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

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

Opt. Express (3)

Phys. Rev. A (1)

P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A,  33, 3232-3242, (1986)
[CrossRef]

Phys. Rev. B (1)

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

Phys. Rev. Lett. (1)

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

Phys. Rev., B (3)

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, (2002).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev., B, 58, 10096-10099, (1998).
[CrossRef]

Phys.Rev.Lett. (1)

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]

Other (7)

M.  Born, E.  Wolf, Principles of Optics, Pergamon, Oxford, 1989.

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

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

G. Sun, A. Bakhtazad, A. Jugessur, and A. Kirk, "Open cavities using photonic crystals with negative refraction," Proc. SPIE, 6343, ed. P. Mathieu, (2006), doi:10.1117/12.708026

A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation, Wiley, New York, 2003.

G. Sun and A. G. Kirk, "Lattice resonance inside photonic crystal slab with negative refraction," OSA Frontiers in Optics 2006/Laser Science XXII conferences, October 8-12, 2006, Rochester, New York.

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.

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Figures (4)

Fig. 1.
Fig. 1.

The reciprocal lattice of the hexagonal photonic crystals. First Brillouin zone is inside the blue hexagon.

Fig. 2.
Fig. 2.

Fourier decomposition of Bloch modes at the first band of the PhC with r/a=0.4 and host material n=3.6 for TE polarization

Fig. 3.
Fig. 3.

Fourier decomposition of Bloch modes of the same photonic crystal as in Fig. 2 but at the second band at normalized frequency 0.30.

Fig. 4.
Fig. 4.

Fourier decomposition of Bloch modes of the PhC (r/a=0.35) at the 3rd band for TM-polarization at normalized frequency 0.428.

Equations (3)

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u ( r ) = u m , n ( r ) e j ( k ' + m G 1 + n G 2 ) r
q = k ' + ( m G 1 + n G 2 ) 2 ω 2 μ ε 0 ε ¯
n p = k M k 0

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