Abstract

Antireflection structured (ARS) surfaces on GaAs substrates for application with normally incident, randomly polarized, 10.6-μm-wavelength radiation are designed and analyzed. Both one-dimensional (1-D) and two-dimensional (2-D) multilevel profiles are examined with special attention given to multilevel approximations of 1-D triangular and 2-D pyramidal profiles. The 1-D profiles are designed by using second-order effective medium theory (EMT), as we have found zeroth-order EMT to be insufficient when ARS surfaces are designed for use with optically dense materials, e.g., most materials used in the infrared spectral region. We analyze both 1-D and 2-D profiles by using rigorous coupled-wave analysis and find that the more levels the profile contains, the better the ARS surface’s response to bias angles, wavelength detunings, and errors in etch depth. Although both 1-D and 2-D profiles can efficiently suppress reflections for unpolarized light, 2-D gratings are advantageous when randomly polarized light is of interest.

© 1993 Optical Society of America

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References

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  1. P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
    [CrossRef]
  2. M. C. Hutley, “Coherent photofabrication,” Opt. Eng. 15, 190–196 (1976).
  3. S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  5. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  6. 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]
  7. Y. Ono, Y. Kimura, Y. Ohta, N. Nishada, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
    [CrossRef] [PubMed]
  8. M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
  9. B. L. Sopori, “Broadband very low reflectance surface,” Appl. Opt. 27, 25–27 (1988).
    [CrossRef] [PubMed]
  10. N. F. Hartman, T. K. Gaylord, “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
    [CrossRef] [PubMed]
  11. D. H. Raguin, G. M. Morris, “Diffraction analysis of antireflection surface-relief gratings on lossless dielectric surfaces,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 122–123.
  12. 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]
  13. 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 (16April1991).
  14. D. H. Raguin, G. M. Morris, “Analysis of 1-D antireflection structured surfaces,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washing-ton, D.C., 1991), p. 153.
  15. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
    [CrossRef]
  16. C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).
  17. Lord Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
    [CrossRef]
  18. J. C. Maxwell Garnett, “On colours in metal glasses, in metallic films, and in metallic solutions,” Philos. Trans. R. Soc. London 205, 237–287 (1906).
    [CrossRef]
  19. J. D. Kraus, Antennas (McGraw-Hill, New York, 1950), pp. 390–394.
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.
  21. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 205–208.
  22. W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
    [CrossRef]
  23. J. M. Corless, M. W. Kaplan, “Structural interpretation of the birefringence gradient in retinal rod outer segments,” Biophys. J. 26, 543–556 (1979).
    [CrossRef] [PubMed]
  24. R. Oldenbourg, T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. 56, 195–205 (1989).
    [CrossRef] [PubMed]
  25. R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
    [CrossRef] [PubMed]
  26. S. M. Rytov, “The electromagnetic properties of finely layered medium,” Sov. Phys. JETP 2, 466–475 (1956).
  27. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc IEEE 73, 894–937 (1985).
    [CrossRef]
  28. W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).
  29. E. Hu, “Dry etching,” in Gallium Arsenide Technology, D. Ferry, ed. (Sams, Carmel, Ind., 1990), Chap. 10.
  30. H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), Chaps. 1 and 2.
  31. E. Palik, “Gallium arsenide (GaAs),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985), pp. 429–443.
  32. W. H. Emerson, “Electromagnetic wave absorbers and anechoic chambers through the years,” IEEE Trans. Antennas Propag. AP-21, 484–490 (1973).
    [CrossRef]

1991 (2)

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]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

1989 (2)

R. Oldenbourg, T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. 56, 195–205 (1989).
[CrossRef] [PubMed]

R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

1986 (1)

1985 (1)

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

1983 (1)

1982 (2)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

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

1979 (1)

J. M. Corless, M. W. Kaplan, “Structural interpretation of the birefringence gradient in retinal rod outer segments,” Biophys. J. 26, 543–556 (1979).
[CrossRef] [PubMed]

1976 (1)

M. C. Hutley, “Coherent photofabrication,” Opt. Eng. 15, 190–196 (1976).

1973 (2)

P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

W. H. Emerson, “Electromagnetic wave absorbers and anechoic chambers through the years,” IEEE Trans. Antennas Propag. AP-21, 484–490 (1973).
[CrossRef]

1967 (1)

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

1956 (1)

S. M. Rytov, “The electromagnetic properties of finely layered medium,” Sov. Phys. JETP 2, 466–475 (1956).

1953 (1)

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

1906 (1)

J. C. Maxwell Garnett, “On colours in metal glasses, in metallic films, and in metallic solutions,” Philos. Trans. R. Soc. London 205, 237–287 (1906).
[CrossRef]

1892 (1)

Lord Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
[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 (16April1991).

Bernhard, C. G.

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

Blank, P. S.

R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
[CrossRef] [PubMed]

Born, M.

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

Bragg, W. L.

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Carlson, F. D.

R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
[CrossRef] [PubMed]

Case, S. K.

Chen, C.-L.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Clapham, P. B.

P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Corless, J. M.

J. M. Corless, M. W. Kaplan, “Structural interpretation of the birefringence gradient in retinal rod outer segments,” Biophys. J. 26, 543–556 (1979).
[CrossRef] [PubMed]

Emerson, W. H.

W. H. Emerson, “Electromagnetic wave absorbers and anechoic chambers through the years,” IEEE Trans. Antennas Propag. AP-21, 484–490 (1973).
[CrossRef]

Enger, R. C.

Gaither, S. A.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Gaylord, T. K.

Glytsis, E. N.

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 (16April1991).

Haidner, H.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

Hartman, N. F.

Haskell, R. C.

R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
[CrossRef] [PubMed]

Hu, E.

E. Hu, “Dry etching,” in Gallium Arsenide Technology, D. Ferry, ed. (Sams, Carmel, Ind., 1990), Chap. 10.

Hutley, M. C.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

M. C. Hutley, “Coherent photofabrication,” Opt. Eng. 15, 190–196 (1976).

P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Kaplan, M. W.

J. M. Corless, M. W. Kaplan, “Structural interpretation of the birefringence gradient in retinal rod outer segments,” Biophys. J. 26, 543–556 (1979).
[CrossRef] [PubMed]

Kimura, Y.

Kipfer, P.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

Kraus, J. D.

J. D. Kraus, Antennas (McGraw-Hill, New York, 1950), pp. 390–394.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), Chaps. 1 and 2.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “On colours in metal glasses, in metallic films, and in metallic solutions,” Philos. Trans. R. Soc. London 205, 237–287 (1906).
[CrossRef]

Moharam, M. G.

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, 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]

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).

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 (16April1991).

Morris, G. M.

D. H. Raguin, G. M. Morris, “Diffraction analysis of antireflection surface-relief gratings on lossless dielectric surfaces,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 122–123.

D. H. Raguin, G. M. Morris, “Analysis of 1-D antireflection structured surfaces,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washing-ton, D.C., 1991), p. 153.

Nishada, N.

Ohta, Y.

Oldenbourg, R.

R. Oldenbourg, T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. 56, 195–205 (1989).
[CrossRef] [PubMed]

Ono, Y.

Osborne, T. R.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Palik, E.

E. Palik, “Gallium arsenide (GaAs),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985), pp. 429–443.

Pippard, A. B.

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Raguin, D. H.

D. H. Raguin, G. M. Morris, “Diffraction analysis of antireflection surface-relief gratings on lossless dielectric surfaces,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 122–123.

D. H. Raguin, G. M. Morris, “Analysis of 1-D antireflection structured surfaces,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washing-ton, D.C., 1991), p. 153.

Rayleigh, Lord

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

Ruiz, T.

R. Oldenbourg, T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. 56, 195–205 (1989).
[CrossRef] [PubMed]

Rytov, S. M.

S. M. Rytov, “The electromagnetic properties of finely layered medium,” Sov. Phys. JETP 2, 466–475 (1956).

Sopori, B. L.

Southwell, W. H.

Stork, W.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

Streibl, N.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

Swanson, G. J.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Veldkamp, W. B.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Wilson, S. J.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Wolf, E.

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

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 205–208.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 205–208.

Acta Crystallogr. (1)

W. L. Bragg, A. B. Pippard, “The form birefringence of macromolecules,” Acta Crystallogr. 6, 865–867 (1953).
[CrossRef]

Appl. Opt. (5)

Biophys. J. (3)

J. M. Corless, M. W. Kaplan, “Structural interpretation of the birefringence gradient in retinal rod outer segments,” Biophys. J. 26, 543–556 (1979).
[CrossRef] [PubMed]

R. Oldenbourg, T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. 56, 195–205 (1989).
[CrossRef] [PubMed]

R. C. Haskell, F. D. Carlson, P. S. Blank, “Form birefringence of muscle,” Biophys. J. 56, 401–413 (1989).
[CrossRef] [PubMed]

Endeavour (1)

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

IEEE Trans. Antennas Propag. (1)

W. H. Emerson, “Electromagnetic wave absorbers and anechoic chambers through the years,” IEEE Trans. Antennas Propag. AP-21, 484–490 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Opt. Acta (1)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Opt. Eng. (1)

M. C. Hutley, “Coherent photofabrication,” Opt. Eng. 15, 190–196 (1976).

Opt. Lett. (1)

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
[CrossRef]

Philos. Mag. (1)

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

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, “On colours in metal glasses, in metallic films, and in metallic solutions,” Philos. Trans. R. Soc. London 205, 237–287 (1906).
[CrossRef]

Proc IEEE (1)

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

Sov. Phys. JETP (1)

S. M. Rytov, “The electromagnetic properties of finely layered medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (11)

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” MIT Lincoln Laboratory Project Rep. ODT 20, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

E. Hu, “Dry etching,” in Gallium Arsenide Technology, D. Ferry, ed. (Sams, Carmel, Ind., 1990), Chap. 10.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), Chaps. 1 and 2.

E. Palik, “Gallium arsenide (GaAs),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985), pp. 429–443.

J. D. Kraus, Antennas (McGraw-Hill, New York, 1950), pp. 390–394.

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

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 205–208.

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 (16April1991).

D. H. Raguin, G. M. Morris, “Analysis of 1-D antireflection structured surfaces,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washing-ton, D.C., 1991), p. 153.

D. H. Raguin, G. M. Morris, “Diffraction analysis of antireflection surface-relief gratings on lossless dielectric surfaces,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 122–123.

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).

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Figures (19)

Fig. 1
Fig. 1

ARS surface with grating vector K. For a given angle of incidence θi, all the orders are evanescent except the zeroth orders, R0 and T0. To achieve this, the grating period Λ must be smaller than the incident free-space wavelength λ; see Eq. (2).

Fig. 2
Fig. 2

Effective mediums: (a) for a multilevel surface profile the effective medium is a film stack, (b) for a continuous surface profile the effective medium is a gradient-index film.

Fig. 3
Fig. 3

Stratified medium composed of two distinct materials: a substrate medium having a permittivity ∊s and an incident medium having a permittivity ∊i. The filling factor f of this stratified medium is given by f = b/Λ and represents the fraction of substrate material within a period Λ. K is the stratification or grating vector.

Fig. 4
Fig. 4

Effective film stack of an N-level ARS surface profile. Note that an N-level surface profile’s effective film stack has N − 1 layers. The effective indices of refraction, nj, are calculated from Eq. (14). The propagation angles, θj, are calculated from Snell’s law, and rj are the polarization-dependent Fresnel field-reflection coefficients for the boundary between the jth and j + 1th layer. The thickness of each layer, dj, is equal to the thickness of the equivalent step of the N-level ARS surface profile.

Fig. 5
Fig. 5

(a) One-dimensional binary ARS surface with grating vector K. (b) Effective medium corresponding to the binary ARS surface. Angles θ1 and θs are determined from Snell’s law. For a binary profile, the duty cycle is identical to the filling factor: both are defined by b/Λ.

Fig. 6
Fig. 6

Power reflected as a function of duty cycle for ZnSe (ns = 2.42) 1-D binary ARS surfaces; see Fig. 5. Zeroth-order EMT, second-order EMT, and RCWA data are displayed for normally incident radiation polarized such that (a) EK and (b) EK. The incident medium is air (ni = 1), the profile period Λ is specified by Eq. (8) with β = 1, and the profile depth satisfies Eq. (20). For generality, dimensions are in units of λ0, the design wavelength. Note that there is good correlation between the second-order EMT and RCWA data. For both polarizations, a ZnSe 1-D binary ARS surface is able to reduce the power reflected to below 8 × 10−5, although two different duty cycles are required (a reflection of a 1-D ARS profile’s effective birefringence).

Fig. 7
Fig. 7

Power reflected as a function of duty cycle for germanium (ns = 4.0) 1-D binary ARS surfaces; see Fig. 5. Zeroth-order EMT, second-order EMT, and RCWA data are displayed for normally incident radiation polarized such that (a) EK and (b) EK. The incident medium is air (ni = 1), the profile period Λ is specified by Eq. (8) with β = 1, and the profile depth d satisfies Eq. (20). For generality, dimensions are in units of λ0, the design wavelength. Note that, for EK, the second-order EMT and RCWA data are in good agreement, while for EK there is a large discrepancy. Two different duty cycles are required to suppress Fresnel reflections for EK and EK: an indication of a 1-D ARS profile’s effective birefringence.

Fig. 8
Fig. 8

ARS surface with a 1-D) triangular profile. The corresponding four-level approximation to this profile is illustrated along with its effective dielectric film stack equivalent. K is the grating vector of the profile.

Fig. 9
Fig. 9

Power reflected as a function of profile depth for fused silica (ns = 1.46) four-level approximations to a 1-D 100% duty cycle triangular profile; see Fig. 8. Zeroth-order EMT, second-order EMT, and RCWA data are displayed for normally incident radiation polarized such that (a) EK and (b) EK. The incident medium is air (ni = 1), the profile period Λ is specified by Eq. (8) with β = 1, and the profile depth d satisfies Eq. (20). For generality, dimensions are in units of λ0, the design wavelength. The second-order EMT is a good match with RCWA for EK, while for EK the match is not as exact. Note that the reflection versus profile depth curves are different for the two polarization states: an indication of the profiles’ effective birefringence.

Fig. 10
Fig. 10

Power reflected as a function of profile depth for silicon approximations to a 1-D 100% duty cycle (ns = 3.5) four-level triangular profile; see Fig. 8. Zeroth-order EMT, second-order EMT, and RCWA data are displayed for normally incident radiation polarized such that (a) EK and (b) EK. The incident medium is air (ni = 1), the profile period Λ is specified by Eq. (8) with β = 1, and the profile depth d satisfies Eq. (20). For generality, dimensions are in units of λ0, the design wavelength. The second-order EMT is a good match with RCWA for EK. Although for EK the second-order EMT does not match RCWA as well, it matches better than the zeroth-order EMT. Note that the curves are different for the two polarization states: an indication of the profiles’ effective birefringence.

Fig. 11
Fig. 11

Two different ARS surface profiles on the same substrate material that have the equivalent effective dielectric film stack. Since the dimensions b1, b2, and b3 and the period Λ are the same for both profiles, the filling factor at each level, fj = bj/Λ, is the same. From Eqs. (3)(7) the effective permittivity or index of refraction at each level is therefore equivalent for the two profiles.

Fig. 12
Fig. 12

RCWA data for power transmitted as a function of angle of incidence for binary, four-level, and eight-level approximations to a 1-D triangular profile; see Fig. 8. The profiles were designed by using second-order EMT and their parameters are given in Table 3. Incident radiation is randomly polarized. Because of the effective birefringence of 1-D ARS surfaces, the binary profile’s peak transmission is barely greater than 89%. Peak transmission increases the more levels the ARS structure contains, but the eight-level profile achieves less than 99.5% peak transmission. Response to bias angles improves as the number of levels increases.

Fig. 13
Fig. 13

RCWA data showing half-field of view in degrees as a function of wavelength for (a) four-level and (b) eight-level approximations to a 1-D triangular profile; see Fig. 8. The profiles were designed by using second-order EMT and their parameters are given in Table 3. Half-fields of view are indicated when 90%, 95%, and 99% of the randomly polarized, incident radiation must be transmitted. One notes that the more levels the ARS surface contains, the better the profile’s operating field of view.

Fig. 14
Fig. 14

RCWA data for power transmitted as a function of manufacturing depth tolerance for binary, four-, and eight-level approximations to a 1-D triangular profile; see Fig. 8. The profiles were designed by using second-order EMT and their parameters are given in Table 3. Note that the transmission for the design wavelength, λ0, at normal incidence, is relatively insensitive to the depth of the profile for changes as much as ±10% from the design depth d0.

Fig. 15
Fig. 15

Creation of a 2-D surface profile by the logical or operation between two orthogonal periodic patterns. Profiles of this type are generally termed crossed profiles or crossed gratings.

Fig. 16
Fig. 16

2-D ARS surface with a pyramidal profile. By making the duty cycles the same in both dimensions (bxx = byy), the profile’s effective medium is nearly isotropic.

Fig. 17
Fig. 17

RCWA data for power transmitted as a function of angle of incidence for 2-D multilevel pyramidal ARS surface profiles; see Fig. 16. Profile parameters are given in Table 4 and the incident radiation is randomly polarized. In contrast with the 1-D profiles (see Fig. 12) the 2-D profiles all achieve essentially 100% transmission at normal incidence for the design wavelength λ0 = 10.6 μm. Note that the half-field of view is essentially the same for all three profiles.

Fig. 18
Fig. 18

RCWA data showing half-field of view in degrees as a function of wavelength for (a) binary, (b) four-level, and (c) eight-level approximations to a 2-D pyramidal profile; see Fig. 16. The profile parameters are given in Table 4, and the incident light is randomly polarized. Half-fields of view are indicated when 90%, 95%, 99%, and 99.9% of the incident radiation must be transmitted. One notes that the more levels the ARS structure contains, the better the profile’s operating field of view and that the 2-D profiles, unlike the 1-D profiles, can be used when more than 99.9% of the incident light must be transmitted.

Fig. 19
Fig. 19

RCWA data for power transmitted as a function of profile depth for 2-D ARS surfaces with multilevel pyramidal profiles; see Fig. 16. Profile parameters are listed in Table 4 and the incident light is randomly polarized. The effect, at the design wavelength, of depth error of as much as ±10% is displayed. Note that the more levels the profile contains, the better the tolerance to etch-depth error.

Tables (4)

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Table 1 Deviation of the Zeroth-Order EMT from the Second-Order EMT

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Table 2 Computation Time for 50 Data Points Using RCWA (diffract) and EMT

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Table 3 1-D ARS Surface Parameters for Multilevel Triangular Profilesa

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Table 4 2-D ARS Surface Parameters for Multilevel Pyramidal Profilesa

Equations (22)

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n t sin θ m - n i sin θ i = m λ Λ ,
Λ λ < 1 max [ n s , n i ] + n i sin θ max ,
E K ( 2 ) = E K ( 0 ) [ 1 + π 2 3 ( Λ λ ) 2 f 2 ( 1 - f ) 2 ( s - i ) 2 0 E K ( 0 ) ] ,
E K ( 2 ) = E K ( 0 ) [ 1 + π 2 3 ( Λ λ ) 2 f 2 ( 1 - f ) 2 × ( s - i ) 2 E K ( 0 ) 0 ( E K ( 0 ) i s ) 2 ] ,
f = b / Λ .
E K ( 0 ) = f s + ( 1 - f ) i ,
1 E K ( 0 ) = f s + 1 - f i ,
Λ λ = 1 β ( n i + n s ) .
E K ( 2 ) = E K ( 0 ) [ 1 + π 2 0 3 β 2 f 2 ( 1 - f ) 2 ( n s - n i ) 2 E K ( 0 ) ] ,
E K ( 2 ) = E K ( 0 ) { 1 + π 2 0 3 β 2 f 2 ( 1 - f ) 2 ( n s - n i ) 2 × E K ( 0 ) [ E K ( 0 ) i s ] 2 } .
p ( 2 ) = p ( 0 ) ( 1 + Δ p β 2 ) ,
Δ E K = π 2 3 f 2 ( 1 - f ) 2 ( α n - 1 ) 2 1 + f ( α n 2 - 1 )
Δ E K = π 2 3 f 2 ( 1 - f ) 2 ( α n - 1 ) 2 1 + f ( α n 2 - 1 ) [ α n 2 - f ( α n 2 - 1 ) ] 2
n p ( 2 ) = n p ( 0 ) ( 1 + Δ n p ) ,
n p ( 0 ) = [ p ( 0 ) / 0 ] 1 / 2 ,
Δ n p = ( 1 + Δ p / β 2 ) 1 / 2 - 1.
Δ E K < Δ E K for     f > 1 / 2 , Δ E K = Δ E K for     f = 1 / 2 , Δ E K > Δ E K for     f < 1 / 2.
ρ j = r j + ρ j + 1 exp ( 2 i δ j + 1 ) 1 + r j ρ j + 1 exp ( 2 i δ j + 1 ) ,
δ j = 2 π n j d j λ cos θ j .
d = λ 0 4 n i n s ,
n film = n i n s .
f ( z ) = γ ( z / d ) ,

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