Abstract

The validity of various homogeneous layer models for high-spatial-frequency rectangular-groove (binary) dielectric surface-relief gratings is examined for both nonconical and conical diffraction. In each model the grating is described by a slab of uniaxial material with its optic axis parallel to the grating vector. The ordinary and principal extraordinary indices of the slab depend on the grating filling factor, the substrate and cover refractive indices, and the ratio of the wavelength to the grating period. These indices can be determined by solving two transcendental equations. Higher-order indices are defined as the exact solution to these equations. Second-order indices (second-order dependence on the wavelength-to-period ratio) and first-order indices (no dependence on the wavelength-to-period ratio) are defined by approximate solutions to these equations. Layer models using higher-order and second-order indices are shown to be accurate for high-spatial-frequency gratings, even at wavelength-to-period ratios near the onset of higher-order propagating diffracted waves. These models are used to design example antireflecting gratings on silicon substrates, including designs for conical incidence. All designs are evaluated and optimized by exact rigorous coupled-wave analysis.

© 1994 Optical Society of America

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References

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  1. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  2. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [CrossRef] [PubMed]
  3. Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
    [CrossRef] [PubMed]
  4. T. K. Gaylord, E. N. Glytsis, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3123–3134 (1987).
    [CrossRef] [PubMed]
  5. E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
    [CrossRef] [PubMed]
  6. T. K. Gaylord, E. N. Glytsis, M. G. Moharam, W. E. Baird, “Technique for producing antireflection grating surfaces on dielectrics, semiconductors, and metals,” U.S. Patent5,007,708 (16-April1991).
  7. E. N. Glytsis, T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4469 (1992).
    [CrossRef] [PubMed]
  8. M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992).
    [CrossRef] [PubMed]
  9. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  10. D. H. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
  11. K. Knop, “Diffraction gratings for color filtering in the zero order,” Appl. Opt. 17, 3598–3603 (1978).
    [CrossRef] [PubMed]
  12. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [CrossRef]
  13. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
    [CrossRef] [PubMed]
  14. L. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
    [CrossRef] [PubMed]
  15. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  16. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [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. O. Wiener, “Die theorie des mischkorpers fur das feld der stationaren stromung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509–604 (1912).
  19. W. Thornburg, “The form birefringence of lamellar systems containing three or more components,” J. Biophys. Biochem. Cytol. 3, 413–419 (1957).
    [CrossRef] [PubMed]
  20. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Soviet Phys. JETP 2, 466–475 (1956).
  21. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.
  22. R. C. McPhedran, L. C. Botten, M. S. Craig, M. Neviere, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
    [CrossRef]
  23. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
    [CrossRef]
  24. G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromag. 5, 17–36 (1985).
    [CrossRef]
  25. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  26. D. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  27. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  28. R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 55–57.
  29. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]

1993 (2)

1992 (2)

1991 (1)

1990 (1)

1988 (1)

1987 (2)

1986 (1)

1985 (3)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

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

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1983 (3)

1982 (2)

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

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

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1979 (1)

1978 (1)

1972 (1)

1957 (1)

W. Thornburg, “The form birefringence of lamellar systems containing three or more components,” J. Biophys. Biochem. Cytol. 3, 413–419 (1957).
[CrossRef] [PubMed]

1956 (1)

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

1912 (1)

O. Wiener, “Die theorie des mischkorpers fur das feld der stationaren stromung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Baird, W. E.

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, W. E. Baird, “Technique for producing antireflection grating surfaces on dielectrics, semiconductors, and metals,” U.S. Patent5,007,708 (16-April1991).

Berreman, D.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

Botten, L. C.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Bouchitte, G.

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

Case, S. K.

Cescato, L.

Craig, M. S.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[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.

E. N. Glytsis, T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4469 (1992).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
[CrossRef] [PubMed]

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3123–3134 (1987).
[CrossRef] [PubMed]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

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

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, W. E. Baird, “Technique for producing antireflection grating surfaces on dielectrics, semiconductors, and metals,” U.S. Patent5,007,708 (16-April1991).

Gluch, E.

Glytsis, E. N.

Gunning, W. J.

Haidner, H.

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 55–57.

Kimura, Y.

Kipfer, P.

Knoesen, A.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Knop, K.

Maystre, D.

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

McPhedran, R. C.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3123–3134 (1987).
[CrossRef] [PubMed]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

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

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, W. E. Baird, “Technique for producing antireflection grating surfaces on dielectrics, semiconductors, and metals,” U.S. Patent5,007,708 (16-April1991).

Morris, G. M.

Motamedi, M. E.

Neviere, M.

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

Nishida, N.

Ohta, Y.

Ono, Y.

Petit, R.

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

Raguin, D. H.

Rytov, S. M.

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

Southwell, W. H.

Stork, W.

Streibl, N.

Thornburg, W.

W. Thornburg, “The form birefringence of lamellar systems containing three or more components,” J. Biophys. Biochem. Cytol. 3, 413–419 (1957).
[CrossRef] [PubMed]

Wiener, O.

O. Wiener, “Die theorie des mischkorpers fur das feld der stationaren stromung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

Yeh, P.

Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig (1)

O. Wiener, “Die theorie des mischkorpers fur das feld der stationaren stromung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Appl. Opt. (11)

L. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
[CrossRef] [PubMed]

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

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

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

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3123–3134 (1987).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4469 (1992).
[CrossRef] [PubMed]

M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992).
[CrossRef] [PubMed]

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
[CrossRef] [PubMed]

D. H. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional profiles,” Appl. Opt. 32, 2582–2598 (1993).
[CrossRef] [PubMed]

K. Knop, “Diffraction gratings for color filtering in the zero order,” Appl. Opt. 17, 3598–3603 (1978).
[CrossRef] [PubMed]

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Appl. Phys. Lett. (1)

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

Electromag. (1)

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

J. Biophys. Biochem. Cytol. (1)

W. Thornburg, “The form birefringence of lamellar systems containing three or more components,” J. Biophys. Biochem. Cytol. 3, 413–419 (1957).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (4)

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

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

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

Soviet Phys. JETP (1)

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

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, W. E. Baird, “Technique for producing antireflection grating surfaces on dielectrics, semiconductors, and metals,” U.S. Patent5,007,708 (16-April1991).

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 55–57.

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Figures (11)

Fig. 1
Fig. 1

Rectangular-groove (binary) gratings at general incidence: (a) low spatial frequency, (b) high spatial frequency. In both cases the plane of incidence is rotated through an azimuthal angle ϕ from the xz plane. In the high-spatial-frequency case all nonzero-diffracted orders are cut off.

Fig. 2
Fig. 2

Dielectric surface-relief grating in cross section. The grating is characterized by the region 1 and region 3 refractive indices n1 and n3, as well as its thickness d, its period Λ, and its filling factor F. The incident wave vector makes a polar angle θ1 from the surface normal, and the wave-vector direction in region 3 is determined by Snell's law.

Fig. 3
Fig. 3

Equivalent indices of the homogeneous layer model for a silicon grating (n3 = 3.5) in air (n1 = 1). (a) Indices are plotted as a function of F at a constant wavelength-to-period ratio of λ0/Λ = 5.0. For F = 0 all indices equal n1, whereas for F = 1 all indices equal n3. The first-order indices can deviate significantly from {nO(2), nE(2)} and {nO(H), nE(H)}. (b) Indices are plotted as a function of λ0/Λ at a constant filling factor of 0.5. Note that {nO(H), nE(H)} and {nO(2), nE(2)} both fall toward {nO(1), nE(1)} as λ0/Λ gets large.

Fig. 4
Fig. 4

Antireflection filling factors FAR for (a) TE polarization and (b) TM polarization, (c) antireflection groove depths dAR plotted as a function of angle of incidence in the decoupled planes of incidence. The Brewster angle is evident in (b) and (c). The antireflection filling factors and groove depths both vary slowly over the range 0° < θ1 < 50°.

Fig. 5
Fig. 5

Response of a grating designed to be antireflecting using {nO(1), nE(1)} with ϕ = 0°, θ1 = 0°, and TE polarization. The homogeneous layer reflectances using {nO(1), nE(1)}, {nO(2), nE(2)}, and {nO(H), nE(H)}, together with the RCWA zero-order backward diffraction efficiency, are plotted as a function of wavelength-to-period ratio. For λ0/Λ > 3.5 only the zeroth orders propagate, and the homogeneous layer models using {nO(2), nE(2)} and {nO(II), nE(II)} very closely approximate the RCWA result.

Fig. 6
Fig. 6

Relative performance of antireflection designs accomplished by the use of {nO(1), nE(1)}, {nO(2), nE(2)}, and {nO(H), nE(H)}, as calculated by RCWA: (a) DE0 as a function λ0/Λ, holding F,d, and θ1 at their design values. The bump in each curve at λ0/Λ = 4.0 is the cutoff point for the +1 diffracted order. (b) DE0 as a function of d0 holding λ0/Λ and F at their design values. Each of the TE designs has the same design groove thickness, whereas the TM designs have only slightly differing design thicknesses. (c) DE0 as a function of F while holding λ0/Λ, d, and θ1 at their design values. All TE designs have the same response because of their identical d values; the TM design responses are virtually identical for the same reason. (d) DE0 as a function of θ1, holding λ0/Λ, F, and d at their design values. Each design shows a large range of incidence angles for which the reflectance remains below 1%. Input parameters, design values, and design-point performances are given in Table 1. The vertical lines in each plot indicate the design values of the abscissa quantities (line types as in the previous plots; solid vertical lines indicate values common to the three designs).

Fig. 7
Fig. 7

Optimization of TE and TM higher-order homogeneous layer model (HLM) designs by RCWA. Solid curves indicate contours of constant DE0 given in percent. The input parameters are the same as those of Fig. 6.

Fig. 8
Fig. 8

Grating of Fig. 5, now illuminated at ϕ = 45°, θ1 = 45°, with TE polarization. (a) TE component of the zero-order backward diffraction efficiency DE0TE, the TM component DE0TM, the total DE0, and (b) the phase difference from TE to TM components are each plotted as a function of the wavelength-to-period ratio.

Fig. 9
Fig. 9

Antireflection grating illuminated through a small f-number system. Linearly polarized light is deflected toward the grating, with electric-field vectors indicated by arrows.

Fig. 10
Fig. 10

Plane-wave impulse responses calculated by RCWA for the optimized antireflection grating design from Table 1 for ϕ = 0°, θ1 = 30°, and TM polarization illuminated as in Fig. 9. Responses are calculated in the range −46° < α < 46° and −45° < δ < 45°, thus covering the solid angle described by 0° < θ1 < 45°.

Fig. 11
Fig. 11

Higher-order antireflection grating design from Table 1 for ϕ = 0°, θ1 = 30°, and TM polarization optimized by RCWA for ϕ = 45°, θ1 = 30°, and the polarization of Fig. 9; HLM, homogeneous layer model.

Tables (1)

Tables Icon

Table 1 Example Antireflection Design Parametersa

Equations (17)

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

n 2 TE = [ ɛ 2 TE ] 1 / 2 = [ n 1 2 ( 1 F ) + n 3 2 F ] 1 / 2 .
n 2 TM = [ ɛ 2 TM ] 1 / 2 = [ ( 1 F ) n 1 2 + F n 3 2 ] 1 / 2 .
( n 1 2 n O 2 ) 1 / 2 tan [ π Λ λ 0 ( 1 F ) ( n 1 2 n O 2 ) 1 / 2 ] = ( n 3 2 n O 2 ) 1 / 2 tan [ π Λ λ 0 F ( n 3 2 n O 2 ) 1 / 2 ] , ( n 1 2 n E 2 ) 1 / 2 n 1 2 tan [ π Λ λ 0 ( 1 F ) ( n 1 2 n E 2 ) 1 / 2 ] = ( n 3 2 n E 2 ) 1 / 2 n 3 2 tan [ π Λ λ 0 F ( n 3 2 n E 2 ) 1 / 2 ] .
n O ( 2 ) = { [ n O ( 1 ) ] 2 + 1 3 [ π Λ λ 0 F ( 1 F ) ] 2 ( n 3 2 n 1 2 ) 2 } 1 / 2 n E ( 2 ) = { [ n E ( 1 ) ] 2 + 1 3 [ π Λ λ 0 F ( 1 F ) ] 2 ( 1 n 3 2 1 n 1 2 ) 2 × [ n E ( 1 ) ] 6 [ n O ( 1 ) ] 2 } 1 / 2 .
r = ( N 1 N in ) / ( N 1 + N in ) ,
N in = N 2 ( N 3 + j N 2 tan Δ ) ( N 2 + j N 3 tan Δ ) ,
d AR = λ 0 4 n 2 cos θ 2 p , p = 1 , 3 , 5 , .
N 2 = ( N 1 N 3 ) 1 / 2 .
n O 2 n 1 cos θ 1 n 3 cos θ 3 n 1 2 sin 2 θ 1 = 0 .
F AR = n 1 cos θ 1 n 1 cos θ 1 + n 3 cos θ 3 .
n O 2 n E 2 n O 2 ( cos θ 1 cos θ 3 n 1 n 3 ) n 1 2 sin 2 θ 1 = 0 .
F AR = B ± [ B 2 4 A C ] 1 / 2 2 A ,
A = ( n 3 2 n 1 2 ) 2 , B = ( n 3 2 n 1 2 ) ( n 1 cos θ 1 n 3 cos θ 3 + n 1 2 cos 2 θ 1 n 3 2 ) , C = n 1 2 ( n 1 cos θ 1 n 3 cos θ 3 n 3 2 cos 2 θ 1 ) .
n E 2 n 1 cos θ 1 n 3 cos θ 3 n 1 2 sin 2 θ 1 = 0 ,
F AR = n 3 cos θ 1 n 1 cos θ 3 + n 3 cos θ 1 .
n O 4 ( n 1 n 3 cos θ 1 cos θ 3 ) ( n O 2 n 1 2 sin 2 θ 1 ) = 0 ,
A = ( n 3 2 n 1 2 ) 2 , B = ( n 3 2 n 1 2 ) [ 2 n 1 n 3 / ( cos θ 1 cos θ 3 ) ] , C = n 1 2 ( n 1 2 n 1 n 3 cos θ 1 / cos θ 3 ) .

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