Abstract

It is shown that for one-dimensional dielectric photonic crystals, the Bloch modes, a vital tool in the analysis of these structures, cannot provide a complete representation of the electromagnetic field at the edges of bandgaps. On these points, the couple of Bloch modes representing the propagation on both sides of the crystal reduces to a single one, with a stationary field, and a complete representation of the field inside the crystal illuminated by a plane wave must include a linearly damped mode (LDM), the amplitude of which behaves linearly in space. The theory of transfer matrices and the use of basic properties of the field allow a precise description of the LDM from a few parameters. An extension to two-dimensional photonic crystals is proposed.

© 2010 Optical Society of America

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References

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  1. E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
    [CrossRef]
  2. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).
  3. C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).
  4. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).
  5. J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
    [CrossRef]
  6. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
    [CrossRef]
  7. G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
    [CrossRef]
  8. S. Y. Lin, V. M. Hietala, L. Wang, and E. D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).
    [CrossRef] [PubMed]
  9. 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–R10099 (1998).
    [CrossRef]
  10. D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).
  11. J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
    [CrossRef]
  12. S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
    [CrossRef]
  13. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
    [CrossRef]
  14. V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  15. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  16. 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]
  17. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
    [CrossRef] [PubMed]
  18. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
    [CrossRef] [PubMed]
  19. M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
    [CrossRef] [PubMed]
  20. M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
    [CrossRef] [PubMed]
  21. T. Grzegorczyk, B. Kemp, and J. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A 23, 2324–2330 (2006).
    [CrossRef]
  22. T. Grzegorczyk, B. Kemp, and J. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
    [CrossRef] [PubMed]
  23. M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
    [CrossRef]
  24. D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A, Pure Appl. Opt. 8, 1059–1066 (2006).
    [CrossRef]
  25. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
    [CrossRef] [PubMed]
  26. D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” J. Opt. Soc. Am. A 24, 2383–2393 (2007).
    [CrossRef]
  27. L. Ferrier, X. Letartre, P. Rojo Romeo, E. Drouard, P. Viktorovitch, and J. M. Fedeli, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express 16, 3136–3145 (2008).
    [CrossRef] [PubMed]
  28. S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, “Microlasers based on effective index confined slow light modes in photonic crystal waveguides,” Opt. Express 16, 6331–6339 (2008).
    [CrossRef] [PubMed]
  29. A. A. Cottey, “Solutions of Schrödinger equation at a band edge in a one-dimensional crystal,” J. Phys. C 5, 2583–2590 (1972).
    [CrossRef]

2008 (3)

2007 (1)

2006 (4)

D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A, Pure Appl. Opt. 8, 1059–1066 (2006).
[CrossRef]

T. Grzegorczyk, B. Kemp, and J. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A 23, 2324–2330 (2006).
[CrossRef]

T. Grzegorczyk, B. Kemp, and J. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

2005 (1)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).

2000 (2)

1999 (2)

S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (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, R10096–R10099 (1998).
[CrossRef]

1997 (1)

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (1)

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

1994 (4)

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

1990 (1)

M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

1989 (1)

M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

1987 (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

1972 (1)

A. A. Cottey, “Solutions of Schrödinger equation at a band edge in a one-dimensional crystal,” J. Phys. C 5, 2583–2590 (1972).
[CrossRef]

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]

Antonoyiannakis, M.

M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
[CrossRef] [PubMed]

Benisty, H.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

Berger, V.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

Bordas, F.

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Bozio, R.

Burns, M.

M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Cottey, A. A.

A. A. Cottey, “Solutions of Schrödinger equation at a band edge in a one-dimensional crystal,” J. Phys. C 5, 2583–2590 (1972).
[CrossRef]

Decoopman, T.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Drouard, E.

Enoch, S.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[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]

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

S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Fedeli, J. M.

Ferrier, L.

Fournier, J. -M.

M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Gardin, S.

Gérard, J. -M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

Golovshenko, J.

M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Gralak, B.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[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]

Grzegorczyk, T.

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Hietala, V. M.

Joannopoulos, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Jones, E. D.

Kawakami, S.

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–R10099 (1998).
[CrossRef]

Kawashima, T.

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–R10099 (1998).
[CrossRef]

Kemp, B.

Kong, J.

Kosaka, H.

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–R10099 (1998).
[CrossRef]

Letartre, X.

Lin, S. Y.

Lourtioz, J. -M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

Maystre, D.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” J. Opt. Soc. Am. A 24, 2383–2393 (2007).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A, Pure Appl. Opt. 8, 1059–1066 (2006).
[CrossRef]

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]

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).

S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

McPhedran, R. C.

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Meade, R.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Milton, G. W.

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Nicorovici, N. -A. P.

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Notomi, M.

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–R10099 (1998).
[CrossRef]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Pendry, J.

M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[CrossRef]

Pendry, J. B.

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

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

Rahmani, A.

Rojo Romeo, P.

Sato, T.

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–R10099 (1998).
[CrossRef]

Seassal, C.

Soukoulis, C.

C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

Tamamura, T.

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–R10099 (1998).
[CrossRef]

Tayeb, G.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[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]

S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Tchelnokov, A.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

Tomita, A.

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–R10099 (1998).
[CrossRef]

Veselago, V. G.

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

Viktorovitch, P.

Vincent, P.

D. Maystre and P. Vincent, “Phenomenological study of binding in optically trapped photonic crystals,” J. Opt. Soc. Am. A 24, 2383–2393 (2007).
[CrossRef]

D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A, Pure Appl. Opt. 8, 1059–1066 (2006).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Wang, L.

Winn, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Yablonovitch, E.

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

Zengerle, R.

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

J. Mod. Opt. (5)

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. Opt. A, Pure Appl. Opt. 8, 1059–1066 (2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. C (1)

A. A. Cottey, “Solutions of Schrödinger equation at a band edge in a one-dimensional crystal,” J. Phys. C 5, 2583–2590 (1972).
[CrossRef]

Nature (1)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (3)

M. Antonoyiannakis and J. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[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, R10096–R10099 (1998).
[CrossRef]

Phys. Rev. Lett. (4)

T. Grzegorczyk, B. Kemp, and J. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

M. Burns, J.-M. Fournier, and J. Golovshenko, “Optical binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

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

Proc. Natl. Acad. Sci. U.S.A. (1)

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
[CrossRef] [PubMed]

Science (1)

M. Burns, J.-M. Fournier, and J. Golovshenko, “Lateral binding effect, due to particle’s optical interaction,” Science 249, 749–754 (1990).
[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 (3)

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, 2nd ed. (Springer, 2008).

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Figures (10)

Fig. 1
Fig. 1

Notations for a stack with N = 5 .

Fig. 2
Fig. 2

Dispersion curve of a 1D photonic crystal. The width w of the dielectric layers is equal to 0.5 and the period is equal to 1. The index of the dielectric layers is equal to 2. The wavenumber corresponding to the top of the conduction band is equal to 1.6822, which corresponds to a wavelength equal to 3.7352, normalized with respect to the period d. The part corresponding to negative values can be deduced by symmetry with respect to the k axis.

Fig. 3
Fig. 3

Representation of the field proportional to the Bloch mode in a finite photonic crystal illuminated on both sides by two plane waves with wavenumber at the top of the first conduction band. The parameters are the same as in Fig. 2. The hatched regions represent the dielectric films. The wavenumber, which corresponds to the top of the first conduction band, is equal to 1.6822.

Fig. 4
Fig. 4

Representation of the field inside a finite size crystal ( N = 5 ) illuminated from the left-hand side by a single plane wave with k = 1.6822 (top of the conduction band). The width w of the dielectric layers is equal to 0.5 and the period is equal to 1. The index of the dielectric layers is equal to 2.

Fig. 5
Fig. 5

The same as Fig. 4, but with N = 25 .

Fig. 6
Fig. 6

Property of space symmetry: schematic representations of the directions of propagation (arrows) and amplitudes (legends of arrows) of the plane waves in vacuum on both sides of a dielectric layer and consequences on the transfer matrix T vac . (a) Waves involved in the definition of the matrix; (b) new waves after a symmetry of the field with respect to the Y Z plane.

Fig. 7
Fig. 7

Property of time reversal: the same as Fig. 6 but with time reversal instead of space symmetry.

Fig. 8
Fig. 8

Variations of the parameter τ and of Im { T 12 vac } with the wavenumber. The dashed regions represent the bandgaps.

Fig. 9
Fig. 9

The same as Fig. 5 ( N = 25 ) but with wavenumbers k = 1.68 (left) and 1.667 (right) below the gap edge ( k = 1.6822 ) .

Fig. 10
Fig. 10

A 2D photonic crystal with square symmetry.

Equations (75)

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2 E X 2 + k 2 ε ( X ) E = 0 ,
ε ( X ) = { 1 in   vacuum ν 2 in   the   dielectric   films . }
E = a j   exp ( i k ν j x j ) + b j   exp ( + i k ν j x j ) ,
ν j = { 1     if   j   is   odd ν     if   j   is   even , }
E ( α , X ) = exp ( i α X ) P ( α , X ) ,     K / 2 α + K / 2 ,
P ( α , X ) = m = , + p m ( α ) exp ( i m K X ) ,
E ( α , X ) 2 = P ( α , X ) 2 = n = , + | p n ( α ) | 2 = 1.
E ( α , X ) = E ( α , X ) ,     thus   P ( α , X ) = P ( α , X ) .
E ( α , X ) = exp ( i α X ) P ( α , X ) .
E ( α , X ) = exp ( i ψ ( α ) ) E ( α , X ) ,     thus   P ( α , X ) = exp ( i ψ ( α ) ) P ( α , X ) ,
P ( α , X ) = exp ( i ψ ( α ) ) P ( α , X ) .
P ( α , X ) = P ( α , X ) = P ( α , X ) ,
E ( α , X ) = E ( α , X ) = E ( α , X ) .
n ,     Im { p n ( α ) } = 0.
E ( + K / 2 , X ) = M E ( K / 2 , X ) ,     M = ± 1.
E ( + K / 2 , X ) = M E ( K / 2 , X ) ,     M = ± 1.
E ( K / 2 , X + d ) = E ( K / 2 , X ) ,
E ( + K / 2 , X ) = M E ( K / 2 , X ) ,     with   X = X + d / 2.
m p m ( K / 2 ) exp ( i K ( m + 1 / 2 ) X ) = M m p m ( K / 2 ) exp ( i K ( m + 1 / 2 ) X ) ,
m p m ( K / 2 ) exp ( i K ( m + 1 ) X ) = M m p m ( K / 2 ) exp ( i K m X ) ,
p m ( K / 2 ) = M p m 1 ( K / 2 ) .
ε h = 0 d ε ( X ) | E ( K / 2 , X ) | 2 d X ,
E ( α , X ) = M E ( α + K , X ) .
Re { E ( α , X ) } = Re { E ( + α , X ) } = M   Re { E ( α + K , X ) } .
Re { E ( K / 2 α , X ) } M   Re { E ( K / 2 + α , X ) } = 0.
Re { | E ( α , X ) α | α = K / 2 } = 0.
Im { E ( K / 2 , X ) } = 0.
Re { E ( K / 2 , X ) } = 0 ,
Im { | E ( α , X ) α | α = K / 2 } = 0.
E = 2   sin ( k x 2 N + 1 )     in   region   2 N + 1.
E = 2 sin ( k w / 2 ) cos ( k ν w / 2 ) cos ( k ν x 2 N )     in   region   2 N .
P 2 n + 1 = T vac P 2 n 1 ,     n ( 1 , N ) ,
P 2 n + 2 = T diel P 2 n ,     n ( 1 , N 1 ) .
ρ 2 ( T 11 vac + T 22 vac ) ρ + T 11 vac T 22 vac T 12 vac T 21 vac = 0 ,
( ρ exp ( i α d ) ) ( ρ exp ( i α d ) ) = ρ 2 ( 2   cos ( α d ) ) ρ + 1 = 0.
T 11 vac = cos ( α d ) + γ ,     T 22 vac = cos ( α d ) γ ,
[ a 2 n + 1 b 2 n + 1 ] = T vac [ a 2 n 1 b 2 n 1 ] ,
[ b 2 n 1 a 2 n 1 ] = T vac [ b 2 n + 1 a 2 n + 1 ] ,
[ a 2 n 1 b 2 n 1 ] = sym ( T vac ) [ a 2 n + 1 b 2 n + 1 ] .
sym ( T vac ) × T vac = 1 .
T 12 vac = T 21 vac ,    
( T 12 vac ) 2 = ( T 21 vac ) 2 = 1 T 11 vac T 22 vac = ( sin ( α d ) 2 + γ 2 ) .
T 21 vac = T 12 vac = ± ( sin ( α d ) 2 + γ 2 ) .
E ̃ ( X , t ) = Re { E ( X ) exp ( i ω t ) } ,
E ̃ ( X , t ) = Re { E ( X ) exp ( + i ω t ) } .
E ̃ ( X , t ) = Re { E ( X ) exp ( i ω t ) } .
[ b 2 n + 1 a 2 n + 1 ] = T vac [ b 2 n 1 a 2 n 1 ] .
P 2 n + 1 = sym ( T vac ) P 2 n 1 .
T vac = sym ( T vac ) .
( T vac ) T vac = 1 ,     T 22 = T 11 ,     T 21 = T 12 .
T vac = [ cos ( α d ) + i τ ± i τ 2 sin ( α d ) 2 i τ 2 sin ( α d ) 2 cos ( α d ) i τ ] ,     τ   real .
T vac = [ 1 + i τ ± i τ i τ 1 i τ ] ,     τ   real .
P 2 n + 1 = ( T vac ) n P 1 = b ( 1 ) ( T vac ) n [ 0 1 ] ,     n ( 0 , N ) .
A = [ 1 ± 1 1 1 ] ,
P 2 n + 1 = ( 1 ) n b ( 1 ) ( [ 0 1 ] + i n τ [ 1 + 1 ] ) .
a ( 2 n + 1 ) = ( 1 ) n i n τ ,     b ( 2 n + 1 ) = ( 1 ) n ( 1 + i n τ ) .
C ± ( α , X ) = E ( α , X ) ± E ( α , X ) .
C ( α , X ) = E ( α , X ) E ( α , X ) = E ( α , X ) E ( α , X ) = 2 i   Im { E ( α , X ) } ,
E ( α , X ) ( α K / 2 ) | E ( α , X ) α | α = K / 2 .
C ( α , X ) 2 i ( α K / 2 ) Im { | E ( α , X ) α | α = K / 2 }     as   α K / 2.
E LDM ( X ) = κ   Im { | E ( α , X ) α | α = K / 2 } .
E LDM ( X ) = κ ( X E ( K / 2 , X ) + Im { exp ( i α X ) | P ( α , X ) α | α = K / 2 } ) .
2 f x 2 + κ 2 f = 0
T vac = [ cos ( α d ) + i τ ± i τ 2 sin ( α d ) 2 i τ 2 sin ( α d ) 2 cos ( α d ) i τ ] .
T ( u ) = exp ( u Γ ) ,
Γ = i α sin ( α d ) [ τ i τ 2 sin ( α d ) 2 ± i τ 2 sin ( α d ) 2 + τ ] ,
u = X N d / 2 ,
T ( 2 n d ) = ( T vac ) 2 n ,
2 P ( u ) u 2 = Γ 2 T P 1 = ( α ) 2 P ( u ) ,
α ( 2 E ( α , X ) X 2 ) + k 2 α ε ( X ) E ( α , X ) + k 2 ε ( X ) E ( α , X ) α = 0.
E ( α , X ) = exp ( i α X ) P ( α , X ) ,
P ( α , X ) = n = , + m = , + p n , m ( α ) exp ( i n K X + i m K Y ) .
E LDM = lim α K / 2 [ E ( α , X ) E ( α , X ) 2 ( K / 2 α ) ] .
E LDM = lim α K / 2 [ 2 i   Im ( E ( α , X ) ) 2 ( K / 2 α ) ] .
E LDM ( X ) = κ ( X E ( K / 2 , X ) + Im { exp ( i α X ) | P ( α , X ) α | α = K / 2 } ) .

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