Abstract

Using the effective medium theory, I interpret the band-gap opening in photonic crystals with simple geometries as an interference effect between alternating layers of high and low optical indices and introduce the interesting concept of multidimensional quarter-wave stacks. The interpretation provides a simple insight into band-gap opening processes. For several simple crystal geometries, I analyze the variations of the gap width and depth with respect to the light polarization, the incident angle, and contrast inversion. For two- and three-dimensional structures composed of cubic and square cylinders, I show that the effective medium theory can be used to predict accurately the gap width, the central wavelength, and the attenuation at the central wavelength. The validity domain of the effective medium theory predictions is checked with results from rigorous computations.

© 1996 Optical Society of America

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References

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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. See, for example, recent special issues on photonic band structures: J. Mod. Opt. 41(2) (1994)and J. Opt. Soc. Am. B 10(2) (1993).
  4. J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
    [CrossRef]
  5. W. H. Southwell, “Pyramid-array surface-relief structures producing antireflection index matching on optical surfaces,” J. Opt. Soc. Am. A 8, 549–553 (1991).
    [CrossRef]
  6. R. Brauer, A. Bryngdahl, “Design of antireflection gratings with approximate and rigorous methods,” Appl. Opt. 33, 7875–7882 (1994).
    [CrossRef] [PubMed]
  7. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
  8. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  9. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [CrossRef]
  10. F. T. Chen, H. G. Craighhead, “Diffractive phase elements on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995).
    [CrossRef] [PubMed]
  11. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  12. R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
    [CrossRef]
  13. G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
    [CrossRef]
  14. J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
    [CrossRef]
  15. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt., to be published.
  16. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  17. Ph. Lalanne, G. M. Morris, “Highly improved convergence rate of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  18. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 1, p. 55.
  20. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 5, p. 102.
  21. Ref. 20, Chap. 7, p. 144.
  22. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Nature of photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B 10, 328–332 (1993).
    [CrossRef]
  23. M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  24. R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
    [CrossRef]
  25. D. Maystre, “Electromagnetic study of photonic bandgaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  26. Y. Ono, Y. Kimura, Y. Otha, N. Nishida, “Antireflection effects in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
    [CrossRef] [PubMed]
  27. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  28. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).
  29. J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
    [CrossRef]
  30. In fact, as was confirmed by RCWA, the structure of Fig. 17, similar to its corresponding 3-D version, does not present any gap in the x and y directions. In that particular case, the EMT does not fail, but, in general, the conclusion derived here holds. For a discussion regarding the 3-D chessboard crystal see E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
    [CrossRef]

1996

1995

1994

In fact, as was confirmed by RCWA, the structure of Fig. 17, similar to its corresponding 3-D version, does not present any gap in the x and y directions. In that particular case, the EMT does not fail, but, in general, the conclusion derived here holds. For a discussion regarding the 3-D chessboard crystal see E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
[CrossRef]

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

See, for example, recent special issues on photonic band structures: J. Mod. Opt. 41(2) (1994)and J. Opt. Soc. Am. B 10(2) (1993).

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

R. Brauer, A. Bryngdahl, “Design of antireflection gratings with approximate and rigorous methods,” Appl. Opt. 33, 7875–7882 (1994).
[CrossRef] [PubMed]

1993

1992

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

1991

M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

W. H. Southwell, “Pyramid-array surface-relief structures producing antireflection index matching on optical surfaces,” J. Opt. Soc. Am. A 8, 549–553 (1991).
[CrossRef]

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]

Y. Ono, Y. Kimura, Y. Otha, N. Nishida, “Antireflection effects in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
[CrossRef] [PubMed]

1985

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

1983

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
[CrossRef] [PubMed]

1982

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

1977

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).

1956

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Bell, J. M.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 1, p. 55.

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Bouchitté, G.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Brauer, R.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Nature of photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B 10, 328–332 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

Bryngdahl, A.

Case, S. K.

Chen, F. T.

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Craighhead, H. G.

Derrick, G. H.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Enger, R. C.

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Haus, J. W.

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Nature of photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B 10, 328–332 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

John, S.

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

Kimura, Y.

Lalanne, Ph.

Ph. Lalanne, G. M. Morris, “Highly improved convergence rate of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt., to be published.

Lemercier-Lalanne, D.

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt., to be published.

Maradudin, A. A.

M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Maystre, D.

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

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Nature of photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B 10, 328–332 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Nishida, N.

Noponen, E.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Ono, Y.

Otha, Y.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).

Petit, R.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Plihal, M.

M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Pommet, D. A.

Raguin, D. H.

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Nature of photonic band gap: some insights from a field analysis,” J. Opt. Soc. Am. B 10, 328–332 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Saarinen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Southwell, W. H.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).

Turunen, J.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 1, p. 55.

Yablonovitch, E.

In fact, as was confirmed by RCWA, the structure of Fig. 17, similar to its corresponding 3-D version, does not present any gap in the x and y directions. In that particular case, the EMT does not fail, but, in general, the conclusion derived here holds. For a discussion regarding the 3-D chessboard crystal see E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

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

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 5, p. 102.

Appl. Opt.

Appl. Phys. Lett.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

Electromagnetics

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1977).

J. Mod. Opt.

See, for example, recent special issues on photonic band structures: J. Mod. Opt. 41(2) (1994)and J. Opt. Soc. Am. B 10(2) (1993).

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

In fact, as was confirmed by RCWA, the structure of Fig. 17, similar to its corresponding 3-D version, does not present any gap in the x and y directions. In that particular case, the EMT does not fail, but, in general, the conclusion derived here holds. For a discussion regarding the 3-D chessboard crystal see E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Opt. Eng.

J. Saarinen, E. Noponen, J. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. B

M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Phys. Rev. Lett.

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]

Pure Appl. Opt.

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

Sov. Phys. JETP

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt., to be published.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 1, p. 55.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 5, p. 102.

Ref. 20, Chap. 7, p. 144.

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Figures (18)

Fig. 1
Fig. 1

(a) 2-D square lattice composed of square cylinders with a relative permittivity ɛ2 immersed in a homogeneous medium with a relative permittivity ɛ1. (b) Equivalent stack of homogeneous thin films for a wave propagating in the x direction. ɛ is the effective relative permittivity and depends on the fill factor f, on ɛ1, ɛ2, and α, according to Eqs. (1) and (2).

Fig. 2
Fig. 2

Diffraction problem used to check the validity of the quarter-wave photonic crystal design. The periodic structure is the same as in Fig. 1(a) and has an infinite spatial extent in the y direction. L denotes the number of grids, each composed of one grating and one thin film. The fill factor and the period-to-wavelength ratio are given by Eqs. (4) and (5). ɛ2 = 4 and ɛ1 = 1.

Fig. 3
Fig. 3

(a) Logarithm of transmittance T as a function of the number L of grids for the diffraction problem of Fig. 22 = 4 and ɛ1 = 1). The solid curves correspond to the EMT approximation of Eq. (6). Circles are numerical results obtained with the RCWA. In this example, θ = 0, λ = 1, and the fill factors f and the period-to-wavelength ratios α are given in Eqs. (4) and (5), respectively. (b) The dashed curves and the pluses were obtained with Eq. (6) and the RCWA computation for the contrast-inversion problem (ɛ2 = 1 and ɛ1 = 4). In that case, fTE = 0.44, αTE = 0.284, and nTE = 1.57 for TE polarization; fTM = 0.37, αTM = 0.33, and nTM = 1.29 for TM polarization.

Fig. 4
Fig. 4

Wavelength dependence of the transmittance of the six-grid stack shown in Fig. 2 for TE polarization and normal incidence. The solid and dashed curves correspond to the RCWA and EMT results. The EMT model assumes an effective index of n = 1.566 independent of the wavelength.

Fig. 5
Fig. 5

Enlarged view of Fig. 4 centered around the nominal wavelength. The solid and dashed curves are those of Fig. 4, and the dash–dot curve corresponds to a more accurate EMT model of the six-grid stack with an effective index n varying with the wavelength according to Eq. (1).

Fig. 6
Fig. 6

Wavelength dependence of the transmittance of the six-grid stack shown in Fig. 2 for TM polarization and normal incidence. The solid and dashed curves correspond to RCWA and EMT results. The EMT model assumes an effective index of n = 1.287 independent of the wavelength.

Fig. 7
Fig. 7

Transmittance of the six-grid stack shown in Fig. 2 for TE polarization and for two angles of incidence, θ = π/12 and θ = π/6. The solid, dashed and dash–dot curves are the same as in the captions for Figs. 4 and 5.

Fig. 8
Fig. 8

Transmittance of the six-grid stack shown in Fig. 2 for TE polarization and for θ = π/4. The solid and dashed curves are the same as in the caption for Fig. 4.

Fig. 9
Fig. 9

2-D circular rod profiles on a 2-D triangular lattice with an hexagonal symmetry. Six grids are represented.

Fig. 10
Fig. 10

Homogenized medium associated with the 2-D hexagonal structure of Fig. 9. The figure shows a rough approximation composed by a stack of four thin films.

Fig. 11
Fig. 11

Decimal logarithm of the transmittance as a function of the number L of grids in the periodic structure of Fig. 9. Circles represent results from rigorous computation (see Table 1 in Ref. 23). The solid curve represents results obtained by EMT modeling, when 1000 films are used to approximate the cylindrical rod shape.

Fig. 12
Fig. 12

Elementary cell of a simple cubic lattice with cubic filling geometries. The corresponding 3-D crystal is composed of cubes of size fΛ and relative permittivity ɛ2, immersed in a medium of relative permittivity ɛ1.

Fig. 13
Fig. 13

u(f) and υ(f) as functions of the fill factor f, ɛ1 = 13 and ɛ2 = 1.

Fig. 14
Fig. 14

Comparison between RCWA (solid curve) and EMT (dashed curve) results of the transmittance of a five-grid photonic crystal with the elementary cell of Fig. 12. 81 orders are retained for RCWA computations; ɛ1 = 13 and ɛ2 = 1.

Fig. 15
Fig. 15

Same as Fig. 14 except that ɛ1 = 13 and ɛ2 = 1.

Fig. 16
Fig. 16

Wavelength dependence of the transmittance of the photonic crystal shown in Fig. 2 for TE polarization and normal incidence. All the results were obtained with RCWA: solid curve, L = 1; dotted curve, L = 2; dash–dot curve, L = 3; dashed curve, L = 6.

Fig. 17
Fig. 17

2-D version of the 3-D chessboard structure originally proposed by Yablonovitch.1 Gray areas represent high-index media.

Fig. 18
Fig. 18

Two different structures that cannot be distinguished from the chessboard striker of Fig. 17 by homogenization.

Equations (15)

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ɛ = ɛ 0 + π 2 3 [ f ( 1 f ) ( ɛ 2 ɛ 1 ) ] 2 α 2 + O ( α 4 ) ,
ɛ = 1 a 0 + π 2 3 [ f ( 1 f ) ( ɛ 2 ɛ 1 ) ɛ 2 ɛ 1 ] 2 ɛ 0 a 0 3 α 2 + O ( α 4 ) ,
ɛ f Λ = λ / 4 ,
ɛ 1 ( 1 f ) Λ = λ / 4 .
f TE = 0.390 , f TM = 0.437 ,
α TE = 0.410 , α TM = 0.444 .
T = 1 | n 1 n e n 1 + n e | 2 with n e = ( n n 1 ) 2 L n 1 ,
log ( T ) = c LA ,
λ edge = λ / ( m ± Δ ) ,
Δ = 2 π arcsin ( n H n L n H + n L ) ,
Δ λ = 4 π λ arcsin ( n n 1 n + n 1 ) ,
ϕ = 4 π n i d i ( cos θ i / λ )
ɛ = η 0 + η 2 ( Λ / λ ) 2 ,
16 ɛ 1 2 ( 1 f ) 4 = f 2 [ 16 ɛ 2 ( 1 f ) 2 η 0 + η 2 ] .
τ = exp k z ( 1 f ) Λ ,

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