Abstract

A periodic array of dielectric rods or holes, known as two-dimensional photonic crystal, is shown to have blazing properties similar to those of classical diffraction gratings. Several different optogeometric configurations are shown numerically to exhibit an almost perfect blazing in the -1st reflected order with a plateaulike spectral dependence in nonpolarized light.

© 2001 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  4. P. Villeneuve, S. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Academic, The Netherlands, 1996), pp. 133–151.
    [CrossRef]
  5. 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]
  6. R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
    [CrossRef]
  7. T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
    [CrossRef]
  8. D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
    [CrossRef]
  9. See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
  10. L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
    [CrossRef]
  11. M. Perry, D. Boyd, J. Britten, B. Shore, C. Shannon, E. Shults, “High-efficiency multilayer dielectric diffraction gratings,” Opt. Lett. 20, 940–942 (1995).
    [CrossRef] [PubMed]
  12. E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 8.
  13. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).
  14. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
    [CrossRef]
  15. E. Popov, B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926–4932 (2000).
    [CrossRef]
  16. E. Popov, M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
    [CrossRef]

2000

1999

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

1997

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

1996

T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

1995

1991

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

1990

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

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]

1987

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]

1984

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Bardinal, V.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

Benisty, H.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Boyd, D.

Bozhkov, B.

Brand, S.

T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Britten, J.

Brommer, K.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Cassagne, D.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

Chan, C.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

De La Rue, R.

T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Fan, S.

P. Villeneuve, S. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Academic, The Netherlands, 1996), pp. 133–151.
[CrossRef]

Ho, K.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Houdré, R.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Joannopoulos, J.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

P. Villeneuve, S. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Academic, The Netherlands, 1996), pp. 133–151.
[CrossRef]

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Jouanin, C.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

Krauss, T.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Labilloy, D.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Loewen, E.

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 8.

Mashev, L.

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Meade, R.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

Nevière, M.

Oesterle, U.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Perry, M.

Popov, E.

E. Popov, M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
[CrossRef]

E. Popov, B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926–4932 (2000).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 8.

Rappe, A.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Satpathy, S.

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]

Shannon, C.

Shore, B.

Shults, E.

Soukoulis, C.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Villeneuve, P.

P. Villeneuve, S. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Academic, The Netherlands, 1996), pp. 133–151.
[CrossRef]

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

Weisbuch, C.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Zhang, Z.

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]

Appl. Opt.

Appl. Phys. Lett.

D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

IEEE J. Quantum Electron.

See, for example, D. Labilloy, H. Benisty, C. Weisbuch, T. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).

Nature (London)

T. Krauss, R. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Opt. Commun.

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Opt. Lett.

Phys. Rev. B

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Phys. Rev. Lett.

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]

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]

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Other

P. Villeneuve, S. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Academic, The Netherlands, 1996), pp. 133–151.
[CrossRef]

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 8.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of the cross section of a periodic system of circular rods: dashed, dotted lines, two different slice cuts; n 1, n 2, refractive indices of the bulk material and the rods, respectively.

Fig. 2
Fig. 2

Spectral dependence of the imaginary part of the eigenvalue of the transmission matrix that has a minimum imaginary part modulus. The slice is cut along the dashed lines in Fig. 1. Solid curve, TM polarization; squares, TE polarization. The band gaps for which min|Im(γ)| ≠ 0 are well presented. The system parameters are d′ = 1 µm, n 1 = 1, n 2 = 2.9833, r = 200 nm, incidence normal to the dashed lines in Fig. 1.

Fig. 3
Fig. 3

Schematic representation of a photonic crystal obtained by a cut, as represented by the dotted lines in Fig. 1 and preserving all parameters. Owing to the change in the cut direction, the incidence is changed to 45° and the period is multiplied by 2. This permits propagation of the -1st reflected and transmitted order.

Fig. 4
Fig. 4

Forbidden bands for the system in Fig. 3 with d = h = 1.414 µm, c = h 1 = d/2, n 1 = 1, n 2 = 2.9833. The polarization is indicated: (a) r = 350 nm, incidence 45°; (b) r = 150 nm; bare lines, 45° incidence; squares, 60° incidence.

Fig. 5
Fig. 5

Total reflected energy (zero and minus first diffracted order) as a function of the wavelength for the system with parameters in Fig. 4(a) and 21 cylinder rows (M = 10).

Fig. 6
Fig. 6

(a) Spectral dependence of the absolute efficiency in the nonspecular (-1st) reflected order for the system with parameters in Fig. 4(a) (r = 350 nm) and 21 rows of rods (M = 10). (b) As in (a), but presenting the relative efficiency, defined as the energy diffracted in this order divided by the total reflected energy. The polarization is indicated as well as the number of vertical periods. M = 1 corresponds to three layers of rods, whereas M = 10 corresponds to 21 layers.

Fig. 7
Fig. 7

Spectral dependence of the absolute efficiency in the nonspecular (-1st) reflected order for the system with parameters in Fig. 4(b) (r = 150 nm) and 21 rows of rods (M = 10). Angle of incidence: (a) 45°; (b) 60°.

Fig. 8
Fig. 8

Same as Fig. 7(a), but with a set of square rods instead of the circular rods. The rods have cross sections of 300 × 300 nm.

Fig. 9
Fig. 9

Forbidden band for the system in Fig. 3 with the following parameters: d = h = 0.9617 µm, c = h 1 = d/2, n 1 = 1, n 2 = 2.3, r = 140 nm, incidence 45°.

Fig. 10
Fig. 10

Spectral dependence of the absolute efficiency in the nonspecular (-1st) reflected order for the system with parameters in Fig. 9. Angle of incidence: (a) 45°; (b) 60°.

Fig. 11
Fig. 11

Forbidden band for a system presented in Fig. 3 with the following parameters: d = h = 0.38 µm, c = h 1 = d/2, n 1 = 3.2, n 2 = 1, r = 80 nm, incidence 60°.

Fig. 12
Fig. 12

Diffraction efficiency bare lines, in the reflected -1st order and, squares, the sum of the two reflected orders. The system parameters are as in Fig. 11.

Equations (6)

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

Fx, y0+dy=expiγ Fx, y0.
Fx+d, y=expiα0dFx, y,
Fx, y=expiα0xm FmyexpimKx,
Fy+dy=T1Fy.
T1=VTΦVT-1,
F˜y+dy=ΦF˜y,

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