Abstract

We present a numerical study of bidimensional photonic crystals with an emphasis on the behavior of the gaps versus the polarization and the conicity of the incident plane wave. We use a rigorous modal theory of diffraction at oblique incidence by a set of arbitrarily shaped parallel fibers. This theory allows the study of the refractive properties of bidimensional photonic crystals. We develop a heuristic method of homogenization that allows us to predict the position of the gaps and their behavior with respect to the polarization and the conicity angle. With this homogenization scheme, we also present some important elements for obtaining full gaps.

© 2000 Optical Society of America

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References

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  1. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  2. D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
    [CrossRef]
  3. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  4. S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
    [CrossRef]
  5. L. M. Li, Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic bandgap materials,” Phys. Rev. B 58, 9587–9590 (1998).
    [CrossRef]
  6. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).
  7. P. Vincent, R. Petit, “Sur la diffraction d’une onde plane par un cylindre diélectrique,” Opt. Commun. 5, 261–266 (1972).
    [CrossRef]
  8. D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique par un objet cylindrique non infiniment conducteur et de section quelconque,” Opt. Commun. 5, 327–330 (1972).
    [CrossRef]
  9. M. Nevière, E. Popov, “New theoretical method for electromagnetic wave diffraction by metallic or dielectric cylinder, bare or coated with thin dielectric layer,” J. Electromagn. Waves Appl. 12, 1265–1296 (1998).
    [CrossRef]
  10. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  11. J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  12. J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
    [CrossRef]
  13. T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
    [CrossRef]
  14. D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–255 (1997).
    [CrossRef]
  15. H. Hasimoto, “On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres,” J. Fluid Dynamics 5, 317–328 (1959).
  16. M. Zukovski, H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” Z. Angew. Math. Phys. 28, 979–992 (1977).
    [CrossRef]
  17. J. W. S. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of the medium,” Philos. Mag. 34, 481–502 (1892).
    [CrossRef]
  18. G. Tayeb, D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
    [CrossRef]
  19. E. Centeno, D. Felbacq, “Characterization of defect modes in finite bidimensional photonic crystals,” J. Opt. Soc. Am. A 16, 2705–2712 (1999).
    [CrossRef]
  20. “Photonic Band Structure,” special issue, J. Mod. Opt. 41, 171–404 (1994).
    [CrossRef]
  21. “Development and Applications of Materials Exhibiting Photonic Band Gaps,” special issue, J. Opt. Soc. Am. B 10, 279–413 (1993).
  22. E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
    [CrossRef]

1999 (2)

1998 (4)

M. Nevière, E. Popov, “New theoretical method for electromagnetic wave diffraction by metallic or dielectric cylinder, bare or coated with thin dielectric layer,” J. Electromagn. Waves Appl. 12, 1265–1296 (1998).
[CrossRef]

L. M. Li, Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic bandgap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[CrossRef]

T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
[CrossRef]

1997 (2)

1995 (1)

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

1994 (3)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

“Photonic Band Structure,” special issue, J. Mod. Opt. 41, 171–404 (1994).
[CrossRef]

1993 (1)

“Development and Applications of Materials Exhibiting Photonic Band Gaps,” special issue, J. Opt. Soc. Am. B 10, 279–413 (1993).

1992 (1)

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

1977 (1)

M. Zukovski, H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” Z. Angew. Math. Phys. 28, 979–992 (1977).
[CrossRef]

1972 (2)

P. Vincent, R. Petit, “Sur la diffraction d’une onde plane par un cylindre diélectrique,” Opt. Commun. 5, 261–266 (1972).
[CrossRef]

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique par un objet cylindrique non infiniment conducteur et de section quelconque,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

1959 (1)

H. Hasimoto, “On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres,” J. Fluid Dynamics 5, 317–328 (1959).

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

1892 (1)

J. W. S. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of the medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Barkou, S. E.

T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
[CrossRef]

J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[CrossRef]

Bouchitté, G.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–255 (1997).
[CrossRef]

Brenner, H.

M. Zukovski, H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” Z. Angew. Math. Phys. 28, 979–992 (1977).
[CrossRef]

Broeng, J.

J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[CrossRef]

T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
[CrossRef]

Centeno, E.

Felbacq, D.

E. Centeno, D. Felbacq, “Characterization of defect modes in finite bidimensional photonic crystals,” J. Opt. Soc. Am. A 16, 2705–2712 (1999).
[CrossRef]

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–255 (1997).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

Hasimoto, H.

H. Hasimoto, “On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres,” J. Fluid Dynamics 5, 317–328 (1959).

Joannopoulos, J. D.

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Lee, S. C.

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

Li, L. M.

L. M. Li, Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic bandgap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Maystre, D.

G. Tayeb, D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique par un objet cylindrique non infiniment conducteur et de section quelconque,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

Meade, R.

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Nevière, M.

M. Nevière, E. Popov, “New theoretical method for electromagnetic wave diffraction by metallic or dielectric cylinder, bare or coated with thin dielectric layer,” J. Electromagn. Waves Appl. 12, 1265–1296 (1998).
[CrossRef]

Petit, R.

P. Vincent, R. Petit, “Sur la diffraction d’une onde plane par un cylindre diélectrique,” Opt. Commun. 5, 261–266 (1972).
[CrossRef]

Popov, E.

M. Nevière, E. Popov, “New theoretical method for electromagnetic wave diffraction by metallic or dielectric cylinder, bare or coated with thin dielectric layer,” J. Electromagn. Waves Appl. 12, 1265–1296 (1998).
[CrossRef]

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of the medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Sondergaard, T.

T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
[CrossRef]

St. J. Russell, P.

J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[CrossRef]

Tayeb, G.

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

Vincent, P.

P. Vincent, R. Petit, “Sur la diffraction d’une onde plane par un cylindre diélectrique,” Opt. Commun. 5, 261–266 (1972).
[CrossRef]

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique par un objet cylindrique non infiniment conducteur et de section quelconque,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

Winn, J.

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Zhang, Z. Q.

L. M. Li, Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic bandgap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Zukovski, M.

M. Zukovski, H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” Z. Angew. Math. Phys. 28, 979–992 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Sondergaard, J. Broeng, S. E. Barkou, “Suppression of spontaneous emission in two-dimensional honeycomb photonic band structure estimated using a new effective-index model,” IEEE J. Quantum Electron. 34, 2308–2313 (1998).
[CrossRef]

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. Electromagn. Waves Appl. (1)

M. Nevière, E. Popov, “New theoretical method for electromagnetic wave diffraction by metallic or dielectric cylinder, bare or coated with thin dielectric layer,” J. Electromagn. Waves Appl. 12, 1265–1296 (1998).
[CrossRef]

J. Fluid Dynamics (1)

H. Hasimoto, “On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres,” J. Fluid Dynamics 5, 317–328 (1959).

J. Mod. Opt. (2)

“Photonic Band Structure,” special issue, J. Mod. Opt. 41, 171–404 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

“Development and Applications of Materials Exhibiting Photonic Band Gaps,” special issue, J. Opt. Soc. Am. B 10, 279–413 (1993).

J. Quant. Spectrosc. Radiat. Transfer (1)

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

Opt. Commun. (4)

P. Vincent, R. Petit, “Sur la diffraction d’une onde plane par un cylindre diélectrique,” Opt. Commun. 5, 261–266 (1972).
[CrossRef]

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique par un objet cylindrique non infiniment conducteur et de section quelconque,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

J. Broeng, S. E. Barkou, P. St. J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[CrossRef]

Philos. Mag. (1)

J. W. S. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of the medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Phys. Rev. B (1)

L. M. Li, Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic bandgap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Waves Random Media (1)

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–255 (1997).
[CrossRef]

Z. Angew. Math. Phys. (1)

M. Zukovski, H. Brenner, “Effective conductivity of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix,” Z. Angew. Math. Phys. 28, 979–992 (1977).
[CrossRef]

Other (2)

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).

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Figures (15)

Fig. 1
Fig. 1

Definition of incident angle φ0, polarization angle δ0, and conicity angle θ0 in the Cartesian system (O, x, y, z).

Fig. 2
Fig. 2

Scattering by a set of parallel fibers of arbitrary shape, index, and position.

Fig. 3
Fig. 3

Finite crystal 9×9, with hexagonal symmetry. The line segment below the structure is used for computation of the transmission coefficient T.

Fig. 4
Fig. 4

Logarithm of transmission T versus wavelength λ and polarization angle δ0 in normal incidence θ0=90° for the crystal of Fig. 3. (δ0=90°  s polarization and δ0=0°  p polarization.) The gap vanishes in p polarization.

Fig. 5
Fig. 5

Logarithm of transmission T versus wavelength λ and conicity angle θ0 in s polarization δ0=90° for the crystal of Fig. 3.

Fig. 6
Fig. 6

Transmission T versus wavelength λ and polarization angle δ0 in normal incidence θ0=90° for the inverse-contrast crystal defined in Subsection 3.B (δ0=90° in s polarization and δ0=0° in p polarization). For the wavelengths belonging to (21; 24), the crystal behaves like a mirror for both polarization cases.

Fig. 7
Fig. 7

Transmission T versus wavelength λ and conicity angle θ0 in s polarization (δ0=90°) for the inverse-contrast crystal, defined in Subsection 3.B.

Fig. 8
Fig. 8

Transmission T versus wavelength λ and conicity angle θ0 in p polarization (δ0=0°) for the inverse-contrast crystal defined in Subsection 3.B.

Fig. 9
Fig. 9

Superposition of the diagram of transmission in Figs. 7 and 8 for both polarizations. The crystal defined in Subsection 3.B presents a full gap for [λ,θ0](21; 24)×(70°; 90°).

Fig. 10
Fig. 10

Homogenized bidimensional crystal. The rods of permittivity 1 embedded in a medium of permittivity 2 are replaced by a periodic stack of two layers. The first (resp. the second) layer is defined by a permittivity eq (resp. 2) and a width 2r (resp. h).

Fig. 11
Fig. 11

Logarithm of transmission T versus wavelength λ in normal incidence (θ0=90°), in s polarization (δ0=90°), for a bidimensional direct-contrast crystal (dotted–dashed curve) and for a unidimensional homogenized crystal (solid curve).

Fig. 12
Fig. 12

Transmission T versus wavelength λ in normal incidence (θ0=90°), in p polarization (δ0=0°), for a bidimensional inverse-contrast crystal (dotted–dashed curve) and for a unidimensional homogenized crystal (solid curve).

Fig. 13
Fig. 13

Transmission T versus wavelength λ and conicity angle θ0 in s polarization (δ0=90°) for the homogenized inverse-contrast crystal.

Fig. 14
Fig. 14

Transmission T versus wavelength λ and conicity angle θ0 in p polarization (δ0=0°) for the homogenized inverse-contrast crystal.

Fig. 15
Fig. 15

Index contrast Δn versus the filling ratio ρ for both polarizations and for direct- and inverse-contrast crystals defined in Subsections 3.A and 3.B, respectively.

Equations (25)

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

k0=k0 sin θ0 cos ϕ0k0 sin θ0 sin ϕ0k0 cos θ0.
Ed=E-Ei,
Hd=H-Hi.
F=EzHz.
χ02=k02-γ 2.
F(r, φ)=exp(-iγ z)m=-+[FmiJm(χ0r)+FmdHm(2)(χ0r)]exp(imφ),
Fmi=sin(δ0)sin(θ0)exp[-im(π/2+φ0)]0/μ0 cos(δ0)sin(θ0)exp[-im(π/2+φ0)].
Fmd...F-md=SFmi...F-mi.
Fj(P)=m=-{Fj,mlocalJm[χ0rj(P)]+Fj,mdHm(2)(χ0rj(P)]}exp[imφj(P)].
Fjlocal=QjFi+kjTj,kFkd.
Qj=exp[-iχ0rj cos(φj-φ0)],
Tj,k,m,q=exp[i(q-m)φjk]Hm-q(2)(χ0rjk).
Fjd=SjFjlocal.
j=1N(δl,j-SlTl,j)Fjd=SlQlFi,l{1,,N},
Fjd=Ej,-mdEj,mdHj,-mdHj,md,Fi=E-miEmiH-miHmi.
S-1Fd=Gi,
F(P)=Fi(P)+exp[-iγz(P)]×j=1Nm=-+Fj,mdHm(2)[χ0rj(P)]exp[imφj(P)].
spolarization
eq=β(1-2)+2,
ppolarization
1eq=111-3β2/1+1/21/1-1/2+β-α(1/1-1/2)4/31+1/2β10/3+O(β14/3),
nhs=2r×(neqs)2+h×(n2)22r+h1/2,
1nhp=2r/(neqp)2+h/(n2)22r+h1/2.
αls=arccos(nhs/n2)=24°,
αlp=arccos(nhp/n2)=33°.

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