Abstract

The properties of guided-mode resonance reflection filters constructed with multiple thin-film layers are addressed. Greatly improved filter characteristics are shown to follow by the incorporation of multiple homogeneous layers with the spatially modulated layer. Calculated results for single-layer, double-layer, and triple-layer filter structures are presented. Whereas good filter characteristics are obtainable with single layers that are half-resonance-wavelength thick, there remains a residual reflection in the sidebands unless the cover and the substrate permittivities are equal. With double-layer and triple-layer designs, extensive wavelength ranges with low sideband-reflectance values are shown to be possible without requiring equal cover and substrate permittivities. The antireflection properties of the layer stack can be understood if the modulated layer is modeled as a homogeneous layer characterized by its average relative permittivity. However, as the grating-modulation index increases, this approximation deteriorates. In particular it is found that, for a given high modulation index, the double-layer antireflection thin-film approximation fails, whereas for the same modulation in a triple-layer system it holds firmly. Multilayer designs can thus have significantly large filter passbands, as they may contain heavily modulated resonant gratings without corruption of the ideal filter characteristics.

© 1995 Optical Society of America

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References

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  1. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
    [CrossRef]
  2. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.
  3. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
    [CrossRef]
  4. M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  5. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  6. L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
    [CrossRef]
  7. L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
    [CrossRef]
  8. I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
    [CrossRef]
  9. I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
    [CrossRef]
  10. H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
    [CrossRef]
  11. M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1210, 83–89 (1990).
  12. R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
    [CrossRef]
  13. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  14. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]
  15. S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
    [CrossRef] [PubMed]
  16. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  17. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  19. J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
    [CrossRef]

1994 (2)

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1990 (1)

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

1989 (3)

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1988 (1)

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (2)

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1984 (1)

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

1982 (1)

1973 (1)

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

1965 (1)

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[CrossRef]

Avrutsky, I. A.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

Bagby, J. S.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.

Bertoni, H.

H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Black, T. D.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Cadilhac, M.

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Cheo, L.

H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

DeSandre, L. F.

Elson, J. M.

Gale, M. T.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1210, 83–89 (1990).

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Hessel, A.

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[CrossRef]

Knop, K.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1210, 83–89 (1990).

Magnusson, R.

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

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

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Moharam, M. G.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.

Morf, R. H.

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1210, 83–89 (1990).

Neviere, M.

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Oliner, A. A.

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[CrossRef]

Petit, R.

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

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

Sohn, A.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

Stanford, J. L.

Svakhin, A. S.

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

Sychugov, V. A.

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

Tamir, T.

H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Vincent, P.

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Wang, S. S.

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (2)

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[CrossRef]

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

H. Bertoni, L. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

J. Mod. Opt. (2)

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (3)

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

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

M. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonance of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

M. T. Gale, K. Knop, R. H. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1210, 83–89 (1990).

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Waveguide mode-induced resonances in planar diffraction gratings,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 117.

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Figures (11)

Fig. 1
Fig. 1

Multiple-layer, square-wave-profile waveguide-grating model.

Fig. 2
Fig. 2

Resonance relation of wavelength versus thickness parameter (t) for a single-layer waveguide-grating filter. The lines labeled by the integer m are given by t = mλ, where the grating thickness is d 1 = t / 4 1 , eff . The parameters are ∊ c = ∊ s = 2.31, ∊1,eff = 4.0, Λ = 300 nm, and θ′ = 0° (normal incidence).

Fig. 3
Fig. 3

TE spectral response of a single-layer waveguide-grating filter, where the resonance wavelength λres ≅ 526 nm and the grating thickness d 1 = 131 nm (half-wavelength). The parameters are ∊ c = ∊ s = 2.31 (AR design), ∊1 H = 4.4, ∊1 L = 3.6, Λ = 300 nm, and θ′ = 0°.

Fig. 4
Fig. 4

TE spectral response of a single-layer waveguide-grating filter, where the resonance wavelength λres ≅ 512 nm and the grating thickness d 1 = 125 nm (near half-wavelength). The other parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1 H = 4.4, ∊1 L = 3.6, Λ = 300 nm, and θ′ = 0°.

Fig. 5
Fig. 5

TE spectral response of a single-layer waveguide-grating filter, where the resonance wavelength λres ≅ 535 nm and the grating thickness d 1 = 175 nm (near three-quarter wavelength). The other parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Resonance relation of wavelength versus thickness parameter (t) for a double-layer waveguide-grating filter. The lines labeled by the integer m are given by t = mλ, where the layer thicknesses are d 1 = t / 4 1 , eff and d 2 = t / 4 2 . The other parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1,eff = 1.77, ∊2 = 2.56, Λ = 300 nm, and θ′ = 0°.

Fig. 7
Fig. 7

TE spectral response of a double-layer waveguide-grating filter. The parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1,eff = 1.77, ∊2 = 2.56, Λ = 300 nm, θ′ = 0°, d 1 = 256 nm, and d 2 = 213 nm (AR design). For the dashed curve ∊1 H = 2.56 and ∊1 L = 1.0, whereas for the solid curve ∊1 H = 1.85 and ∊1 L = 1.71.

Fig. 8
Fig. 8

TE spectral response of a double-layer waveguide-grating filter. The parameters are ∊ c = 1.0, ∊ s = 2.31, ∊2,eff = 2.56, ∊2 H = 2.82, ∊2 L = 2.31, Λ = 300 nm, θ′ = 0°, d 1 = 256 nm, d 2 = 213 nm, and ∊1 = 1.21 (non-AR design, dashed curve) or ∊1 = 1.77 (AR design, solid curve).

Fig. 9
Fig. 9

TE spectral response of a triple-layer waveguide-grating filter. The parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1 H = 2.56, ∊1 L = 1.0, ∊2 = 4.0, ∊3 = 3.42, Λ = 300 nm, θ′ = 0°, d 1 = 94 nm, d 2 = 63 nm, and d 3 = 68 nm (AR design). The quarter-wave thicknesses are determined at a wavelength of 500 nm.

Fig. 10
Fig. 10

TE spectral response of a triple-layer waveguide-grating filter. The parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1 = 1.77, ∊2 H = 4.75, ∊2 L = 3.24, ∊3 = 3.42, θ′ = 0°, d 1 = 94 nm, d 2 = 63 nm, and d 3 = 65 nm (AR design).

Fig. 11
Fig. 11

TE spectral response of a triple-layer waveguide-grating filter. The parameters are ∊ c = 1.0, ∊ s = 2.31, ∊1 = 1.77, ∊2 = 4.0, ∊3 H = 4.2, ∊3 L = 2.66, Λ = 300 nm, θ′ = 0°, d 2 = 63 nm, and d 3 = 68 nm. The dashed curve d 1 = 188 nm (half-wavelength) corresponds to non-AR design, and the solid curve d 2 = 94 nm (quarter-wavelength) represents the AR design.

Equations (16)

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

β n = k n sin θ n ,
P n = [ n ( β n / k ) 2 ] 1 / 2 ,
P c = [ c ( β n / k ) 2 ] 1 / 2 ,
P s = [ s ( β n / k ) 2 ] 1 / 2 ,
γ n = k d n P n ,
m 11 , n = cos γ n ,
m 12 , n = ( j sin γ n ) / P n ,
m 21 , n = j P n sin γ n ,
m 22 , n = cos γ n ,
[ A B C D ] = [ m 11 , 1 m 12 , 1 m 21 , 1 m 22 , 1 ] [ m 11 , 2 m 12 , 2 m 21 , 2 m 22 , 2 ] [ m 11 , N m 12 , N m 21 , N m 22 , N ] .
[ 1 A + B P s P c C + D P s ] [ E r 0 E t 0 exp ( j k s cos θ t Z N ) ] = [ E i 0 P c E i 0 ] ,
P c A + P c P s B + C + P s D = 0 ,
β i / k = n sin θ n i λ / Λ ,
max { c , s } β i / k < max { n , eff | n = 1 , 2 , 3 } .
tan ( k P 1 i d 1 ) = j P 1 i ( P c i + P s i ) P 1 i 2 + P c i P s i ,
max { c , s } β i / k < 1 , eff .

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