Abstract

A novel thoretical treatment of antireflection-structured surfaces possessing general one-dimensional continuous profiles is presented. Closed-form solutions for the field reflection coefficients of these antireflection-structured surfaces are obtained through the use of effective medium theory and tapered transmission-line theory. Two specific surface profiles (sinusoidal and triangular) are analyzed in detail. Both the sinusoidal and triangular profiles are found to exhibit low reflectances over a broad range of angles and wavelengths. Results obtained with effective medium theory and transmission-line theory are compared with results obtained through the application of rigorous coupled-wave analysis.

© 1993 Optical Society of America

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    [CrossRef]
  3. N. S. Gluck, J. P. Heuer, “Properties of mixed composition IR optical thin films,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 176.
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  5. P. B. Clapham, M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature (London) 244, 281–282 (1973).
    [CrossRef]
  6. M. C. Hutley, “Coherent photofabrication,” Opt. Eng. 15, 190–196 (1976).
  7. S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
    [CrossRef]
  8. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  9. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  10. 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]
  11. Y. Ono, Y. Kimura, Y. Ohta, N. Nishada, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
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    [CrossRef]
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  18. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 24, 1921–1923 (1991).
    [CrossRef]
  19. D. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  20. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  21. W. Thornburg, “The form birefringence of lamellar systems containing three or more components,” J. Biophys. Biochem. Cytol. 3, 413–419 (1957).
    [CrossRef] [PubMed]
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    [CrossRef]
  25. 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 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).
  26. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 205–208.
  27. J. D. Kraus, Electromagnetics (McGraw-Hill, New York, 1984), pp. 378–494.
  28. S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding, and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).
  29. L. R. Walker, N. Wax, “Nonuniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
    [CrossRef]
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    [CrossRef]
  33. E. Hu, “Dry etching,” in Gallium Arsenide Technology, D. Ferry, ed. (Sams, Carmel, Ind., 1990), Ch. 10.
  34. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 589–590.
  35. M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
    [CrossRef]
  36. S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
    [CrossRef]
  37. D. Raguin, “Subwavelength structured surfaces: theory and experiments,” Ph.D. dissertation (University of Rochester, Rochester, N.Y.1993).

1993 (1)

1991 (3)

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

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (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]

1989 (1)

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

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]

1984 (1)

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]

1980 (1)

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

1979 (1)

1976 (2)

1975 (1)

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

1973 (1)

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

1964 (1)

G. Franceschetti, “Scattering from plane layered media,” IEEE. Trans. Antennas Propat. 12, 754–763 (1964).
[CrossRef]

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

1946 (1)

L. R. Walker, N. Wax, “Nonuniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

1938 (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding, and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

Aiki, K.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[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).

Bethea, C. G.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Born, M.

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

Bour, D. P.

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

Carlson, N. W.

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

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 (Massachusetts Institute of Technology 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]

Enger, R. C.

Evans, G. A.

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

Franceschetti, G.

G. Franceschetti, “Scattering from plane layered media,” IEEE. Trans. Antennas Propat. 12, 754–763 (1964).
[CrossRef]

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 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

Gaylord, T. K.

Gieson, E. V.

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

Gluck, N. S.

N. S. Gluck, J. P. Heuer, “Properties of mixed composition IR optical thin films,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 176.

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.

Hasnain, G.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Heuer, J. P.

N. S. Gluck, J. P. Heuer, “Properties of mixed composition IR optical thin films,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 176.

Hu, E.

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

Johnson, L. F.

Katzir, A.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

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, Electromagnetics (McGraw-Hill, New York, 1984), pp. 378–494.

Levine, B. F.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Liew, S. K.

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

Logan, R. A.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Lowdermilk, W. H.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Macleod, H. A.

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

Malik, R. J.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Milam, D.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Minot, M. J.

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

Nakamura, M.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

Nishada, N.

Ohta, Y.

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 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

Raguin, D.

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

D. Raguin, “Subwavelength structured surfaces: theory and experiments,” Ph.D. dissertation (University of Rochester, Rochester, N.Y.1993).

Raguin, D. H.

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

Rytov, S. M.

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

Sankur, H.

Schelkunoff, S. A.

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding, and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

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]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 589–590.

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 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

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]

Umeda, J.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

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 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

Walker, J.

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

Walker, L. R.

L. R. Walker, N. Wax, “Nonuniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

Wax, N.

L. R. Walker, N. Wax, “Nonuniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

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.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

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.

Yen, H. W.

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (3)

G. Hasnain, B. F. Levine, C. G. Bethea, R. A. Logan, J. Walker, R. J. Malik, “GaAs/AlGaAs multiquantum well infrared detector arrays using etched gratings,” Appl. Phys. Lett. 54, 2515–2517 (1989).
[CrossRef]

S. K. Liew, N. W. Carlson, D. P. Bour, G. A. Evans, E. V. Gieson, “Demonstration of InGaAs/AlGaAs strained-layer distributed-feedback grating-surface-emitting lasers with a buried second-order grating structure,” Appl. Phys. Lett. 58, 228–230 (1991).
[CrossRef]

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Bell Syst. Tech. J. (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding, and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

IEEE J. Quantum Electron. (1)

M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, H. W. Yen, “GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback,” IEEE J. Quantum Electron. QE-11, 436–439 (1975).
[CrossRef]

IEEE. Trans. Antennas Propat. (1)

G. Franceschetti, “Scattering from plane layered media,” IEEE. Trans. Antennas Propat. 12, 754–763 (1964).
[CrossRef]

J. Appl. Phys. (1)

L. R. Walker, N. Wax, “Nonuniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[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. (2)

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)

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Figures (22)

Fig. 1
Fig. 1

Arbitrary ARS surface profile. For a given angle of incidence θi, all diffraction orders are evanescent except the zeroth orders, R0 and T0. For this to occur, the grating period Λ must be smaller than the incident wavelength λ [see inequality (2)].

Fig. 2
Fig. 2

Geometry of a 1-D ARS surface with a continuous surface profile: (a) The filling factor f varies as a function of z, the depth into the profile. Since a 1-D ARS surface behaves effectively as a uniaxial crystal whose crystal axis is along the same direction as the grating vector K, it is important to define the polarization of the incident electric field E in relation to K. (b) The incident field is polarized such that EK; (c) the electric field is polarized such that EK. ∊i and ∊s represent the permittivities of the incident and substrate materials, respectively.

Fig. 3
Fig. 3

Maximum values of second-order correction factors as a function of αn(ns/ni): (a) Δ p max; (b) functional dependence of Δ n p max, where the subscript p represents the polarization state of the incident light, either EK or EK.

Fig. 4
Fig. 4

Minimum β (βmin) as a function of αn(ns/ni) such that the maximum value for the second-order correction term for the effective index of refraction Δ n p max remains below (a) 5% and (b) 2%. In (a) βmin may equal 1 until αn reaches ~2, while in (b) βmin equals unity only until αn reaches ~1.5.

Fig. 5
Fig. 5

General gradient medium sandwiched between two homogeneous isotropic mediums.

Fig. 6
Fig. 6

Example of an effective index of refraction gradient encountered by light incident on an ARS surface. The index of refraction need not be continuous at z = 0 or z = d. Similar discontinuities will exist, in general, for the permeability and conductivity gradient profiles. ns, substrate medium refractive index; ni, incident medium refractive index; nf, refractive index of gradient medium at z = 0; ns, refractive index of gradient medium at z = d.

Fig. 7
Fig. 7

General 1-D triangular-profile ARS surface. The grating vector K lies in the x direction, and the incident and substrate media are taken to be dielectrics (μi = μs = μ0 and σi = σs = 0).

Fig. 8
Fig. 8

Three examples of ARS surface profiles that may be described by the same effective gradient profile.

Fig. 9
Fig. 9

Effective index of refraction gradient for an ARS surface with a triangular profile [see Eqs. (42) and (44)]. For this plot the incident medium was taken to be air (ni = 1), the substrate medium to be ZnSe (ns = 2.42), and the dimensions a and b (see Fig. 7) to be, respectively, zero and Λ. Owing to the more gradual slope of nEK at z = 0, the reflection coefficient for EK is less than that for EK.

Fig. 10
Fig. 10

Graphical analysis of the EK field reflection-coefficient integral [see Eq. (50)]. The triangular-profile parameters used are a = 0, b = Λ, ni = 1, and ns = 2.42 (ZnSe substrate) (see Fig. 7). The plots in (a) and (b) represent the rotation of the integrand of Eq. (50) in the complex plane for two different profile depths as integration progresses from w0 to wd: (a) the profile depth d1 = 0.496λ corresponds to a local maximum of the power reflection coefficient |ρ|2 with respect to profile depth, (b) the depth d2 = 0.707λ (OPL = λ.) corresponds to a local minimum of ρ [see Eq. (54)]. For local minima, the initial and final expressions of the integrand, exp(iw)/w, differ in phase by exactly an even multiple of π.

Fig. 11
Fig. 11

Reflection characteristics of a fused-silica (ns = 1.46) triangular ARS surface profile as a function of profile depth for EK. The incident medium is air (ni = 1), and θi = 0°. The behavior of (a) the power reflection coefficient and (b) the reflected phase are illustrated. For RDE results, power reflection minima occur at depths predicted by using Eq. (53). Period Λ used for the RCWA results corresponds to β = 1 in Eq. (7).

Fig. 12
Fig. 12

Same as Fig. 11 except that EK. For RDE results power reflection minima occur at depths predicted by using Eq. (54).

Fig. 13
Fig. 13

Reflection characteristics of a ZnSe (ns = 2.42) triangular ARS surface profile as a function of profile depth for EK. The incident medium is air (ni = 1), and θi = 0°. The behavior of (a) the power reflection coefficient and (b) the reflected phase are illustrated. For RDE results, power reflection minima occur at depths predicted by using Eq. (53). Periods Λ used for the RCWA, from largest to smallest, correspond to β = 1, β = βmin, where βmin is defined in Fig. 4(a), and β = βmin, where βmin is defined in Fig. 4(b). The design constant β is defined in Eq. (7).

Fig. 14
Fig. 14

Same as Fig. 13 except that EK. For RDE results power reflection minima occur at depths predicted by Eq. (54).

Fig. 15
Fig. 15

Same as Fig. 13 except that GaAs (ns = 3.27) is used.

Fig. 16
Fig. 16

Same as Fig. 13 except that GaAs (ns = 3.27) is used for EK. For RDE results, power reflection minima occur at depths predicted by Eq. (54).

Fig. 17
Fig. 17

RCWA and RDE data representing the power reflection characteristics of triangular ARS surface profiles as a function of wavelength. For both plots [(a) and (b)], ni = 1, θi = 0°, a = 0, and b = Λ (see Fig. 7). The depths of the profiles were chosen such that for the design wavelength λ0 the reflection coefficient for EK is a minimum [see Table 1 and Eq. (53)]. Periods Λ for the RCWA analysis correspond to β = 1 [see Eq. (7)].

Fig. 18
Fig. 18

Power reflection characteristics as a function of incident angle for a fused-silica triangular profile. Profile parameters for both plots [(a) and (b)] are a = 0, b = Λ, d = 0.804λ, ni = 1, and ns = 1.46 (see Fig. 7). For both EK and EK [(a) and (b), respectively] the power reflection coefficient remains below 1% for a 50° half field of view.

Fig. 19
Fig. 19

RDE data representing the power reflection characteristics as a function of incident angle for a ZnSe triangular profile. The profile is designed such that the power reflection coefficient for EK is a minimum at θi = 32°. Profile parameters are a = 0, b = Λ, d = 0.75λ, ni = 1, and ns = 2.42 (see Fig. 7).

Fig. 20
Fig. 20

ARS surface with 1-D sinusoidal profile. The grating vector K lies in the x direction, and the incident and substrate media are taken to be dielectrics (μi = μs = μ0 and σi = σs = 0).

Fig. 21
Fig. 21

Effective index of refraction gradient for an ARS surface with a sinusoidal profile [see Eqs. (63) and (65)]. Because of the sharp slope of nEK at z = 0 and nEK at z = d, a sinusoidal ARS surface profile, in general, exhibits higher reflectivities than a triangular profile.

Fig. 22
Fig. 22

RDE method data representing the power reflection coefficients for sinusoidal ARS surface profiles as a function of profile depth for EK and EK. For all profiles the incident medium is air (ni = 1), and θi = 0°. The substrate materials analyzed are (a) fused silica, (b) ZnSe, and (c) GaAs.

Tables (1)

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Table 1 Physical Parameters of Selected Optical Materials

Equations (79)

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n 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 ) ( z ) = E K ( 0 ) ( z ) [ 1 + π 2 3 ( Λ λ ) 2 f 2 ( z ) [ 1 f ( z ) ] 2 ( s i ) 2 0 E K ( 0 ) ( z ) ]
E K ( 2 ) ( z ) = E K ( 0 ) ( z ) [ 1 + π 2 3 ( Λ λ ) 2 f 2 ( z ) [ 1 f ( z ) ] 2 ( s i ) 2 × E K ( 0 ) 0 ( E K ( 0 ) ( z ) i s ) 2 ]
E K ( 0 ) ( z ) = f ( z ) s + [ 1 f ( z ) ] i ,
1 E K ( 0 ) ( z ) = f ( z ) s + 1 f ( z ) i ,
β = 1 ( n i + n s ) ( Λ λ ) .
p ( 2 ) ( z ) = p ( 0 ) ( z ) [ 1 + Δ p ( z ) β 2 ] ,
Δ E K ( z ) = π 2 3 f 2 ( z ) [ 1 f ( z ) ] 2 ( α n 1 ) 2 1 + f ( z ) ( α n 2 1 )
Δ E K ( z ) = π 2 3 f 2 ( z ) [ 1 f ( z ) ] 2 ( α n 1 ) 2 × 1 + f ( z ) ( α n 2 1 ) [ α n 2 f ( z ) ( α n 2 1 ) ] 2
α n = n s n i .
n p ( 2 ) ( z ) = n p ( 0 ) ( z ) [ 1 + Δ n p ( z ) ] ,
Δ n p ( z ) = [ 1 + Δ p ( z ) / β 2 ] 1 / 2 1
n p ( 0 ) ( z ) = [ p ( 0 ) ( z ) / 0 ] 1 / 2 .
× E = B t ,
× H = D t + J ,
E = y ˆ E y , H = x ˆ H x + z ˆ H z ,
E y z = + i μ ( z ) ω H x ,
H x z = + i ω [ ˜ ( z ) k x 2 μ ( z ) ω 2 ] E y ,
˜ ( z ) = ( z ) i σ ( z ) ω
d V d z = Z ( z ) I ,
d I d z = Y ( z ) V ,
Z TE ( z ) = i μ ( z ) ω ,
Y TE ( z ) = i ω [ ˜ ( z ) k x 2 μ ( z ) ω 2 ] ,
Z g = E y H x = V I
d Z g d z + Z ( z ) Y ( z ) Z g 2 = 0 .
E ( z ) = E ( z ) + E + ( z ) , H ( z ) = H ( z ) + H + ( z ) ,
ρ ( z ) = E ( z ) E + ( z )
ρ ( z ) = Z g ( z ) Z 0 ( z ) Z g ( z ) + Z 0 ( z ) ,
Z 0 ( z ) = E + H + = E H = [ Z ( z ) Y ( z ) ] 1 / 2 .
d ρ d z 2 γρ + 1 2 Z 0 d Z 0 d z ( 1 ρ 2 ) = 0 ,
γ ( z ) = [ Y ( z ) Z ( z ) ] 1 / 2 .
γ ( z ) = i ω [ μ ( z ) ˜ ( z ) k x 2 ω 2 ] 1 / 2 ,
1 Z 0 d Z 0 d z = 1 2 μ ( z ) [ ˜ ( z ) 2 k x 2 μ ( z ) ω 2 ] d μ ( z ) d z μ ( z ) d ˜ ( z ) d z ˜ ( z ) k x 2 μ ( z ) ω 2 .
d ρ d z 2 γρ + 1 2 Z 0 d Z 0 d z = 0 .
ρ ( z ) = z d 1 2 Z 0 d Z 0 d z exp [ 2 z z γ ( s ) d s ] d z + ρ ( d ) exp [ 2 z d γ ( s ) d s ] ,
ρ ( 0 ) = r i + ρ ( 0 + ) 1 + r i ρ ( 0 + ) .
ρ TE ( z ) = z d 1 2 n ˜ a d n ˜ a d z exp [ i 2 k z z n ˜ a ( s ) d s ] d z + ρ ( d ) exp [ i 2 k z d n ˜ a ( s ) d s ] ,
0 n ˜ a 2 ( z ) = ˜ ( z ) i sin 2 θ i .
ρ ( z ) = z d 1 2 n ˜ d n ˜ d z exp [ i 2 k z z n ˜ ( s ) d s ] d z + ρ ( d ) exp [ i 2 k z d n ˜ ( s ) d s ] ,
f ( z ) = a Λ + b a Λ ( z d ) .
E K ( 0 ) ( z ) = A E K + B E K z = b + ( f + b ) z / d ,
b = a Λ ( s i ) + i , f = b Λ ( s i ) + i .
E K ( 0 ) ( z ) = ( A E K B E K z ) 1 = [ 1 b ( 1 b 1 f ) z d ] 1 ,
b = 1 i a Λ ( 1 i 1 s ) , f = 1 i b Λ ( 1 i 1 s ) .
ρ E K ( 0 + ) = exp ( i u 0 ) [ ρ ( d ) exp ( i u d ) 1 6 u 0 u d exp ( i u ) u d u ] ,
u d = 8 π 3 ( d λ ) f cos 3 θ f f b n f , u 0 = 8 π 3 ( d λ ) b cos 3 θ b f b n b .
ρ E K ( 0 + ) = exp ( i ϕ 0 ) { ρ ( d ) exp ( i ϕ d ) + 1 2 2 θ b 2 θ f d ϕ sin ϕ exp [ i a ( sin ϕ + ϕ ) ] } ,
ϕ 0 = 4 π ( d λ ) f n b f b ( cos θ b + θ b sin θ b ) , ϕ d = 4 π ( d λ ) b n f f b [ cos θ f + θ f sin θ f ] , a = 2 π ( d λ ) f f b n b sin θ b .
ρ E K ( 0 + ) = exp ( i w 0 ) [ ρ ( d ) exp ( i w d ) + 1 2 w 0 w d exp ( + i w ) w d w ] ,
w 0 = 8 π ( d λ ) f f b n b , w d = 8 π ( d λ ) b f b n f .
R p d = d | ρ p | 2 = 0 .
d λ = 3 m 4 n s 2 n i 2 n s 3 cos 3 θ s n i 3 cos 3 θ i ,
d λ = m 4 n s + n i n s n i ,
( z ) = b ( 1 + Δ z ) m ,
Δ = B p A p ,
E ( z ) = ( W W 0 ) 1 q [ c 1 H q 1 ( 1 ) ( ω μ 0 W ) + c 2 H q 1 ( 2 ) ( ω μ 0 W ) ] ,
H ( z ) = i ( b μ 0 ) 1 / 2 ( W W 0 ) q [ c 1 H q ( 1 ) ( ω μ 0 W ) + c 2 H q ( 2 ) ( ω μ 0 W ) ] ,
W ( z ) = W 0 ( 1 + Δ z ) ( m + 2 ) / 2 , W 0 = 2 b Δ ( m + 2 ) , q = m + 1 m + 2 .
R = | ρ | 2 = | M + M | 2 ,
M ± = [ ( f s ) 1 / 2 J q ( ξ d ) + i J q 1 ( ξ d ) ] × [ ± ( b i ) 1 / 2 Y q ( ξ 0 ) + i Y q 1 ( ξ 0 ) ] [ ( f s ) 1 / 2 Y q ( ξ d ) + i Y q 1 ( ξ d ) ] × [ ± ( b i ) 1 / 2 J q ( ξ 0 ) + i J q 1 ( ξ 0 ) ] , ξ 0 = 4 π n b m + 2 ( d λ ) [ ( f b ) 1 / m 1 ] 1 , ξ d = ξ 0 ( f b ) ( m + 2 ) / 2 m .
f ( z ) = 1 2 + 1 π arcsin ( 2 z d 1 ) ,
E K ( 0 ) ( z ) 0 = [ n E K ( 0 ) ( z ) ] 2 = A E K + B E K arcsin ( 2 z d 1 ) ,
A E K = 1 2 ( n s 2 + n i 2 ) , B E K = 1 π ( n s 2 n i 2 ) .
E K ( 0 ) ( z ) 0 = [ n E K ( 0 ) ( z ) ] 2 = [ A E K B E K arcsin ( 2 z d 1 ) ] 1 ,
A E K = 1 2 ( 1 n i 2 + 1 n s 2 ) , B E K = 1 π ( 1 n i 2 1 n s 2 ) .
ρ E K ( 0 ) = exp ( i ϕ ) 2 ν 1 ν 2 d ν ν × exp { i α d λ [ ν sin ( π ν 2 / 2 r ) cos ( r ) S ( ν ) + sin ( r ) C ( ν ) ] } ,
ϕ = α d λ [ ν 1 cos ( r ) C ( ν 1 ) + sin ( r ) S ( ν 1 ) ] , α = π [ 2 ( n s 2 n i 2 ) ] 1 / 2 , r = A E K n i 2 sin 2 θ i B E K , ν 1 = ( 2 i s i ) 1 / 2 cos ( θ i ) , ν 2 = ( 2 s s i ) 1 / 2 cos ( θ s ) .
S ( z ) = 0 z sin ( π t 2 2 ) d t , C ( z ) = 0 z cos ( π t 2 2 ) d t .
ρ E K ( 0 ) = exp ( i φ ) 2 u 1 u 2 d u u × exp { i γ d λ [ cos ( ψ ) C ( u ) sin ( ψ ) S ( u ) ] } ,
φ = γ d λ [ cos ( ψ ) C ( u 2 ) + sin ( ψ ) S ( u 2 ) ] , γ = 2 π 2 ( 2 s i i s ) 1 / 2 , ψ = A E K B E K , u 1 = ( 2 i s i ) 1 / 2 , u 2 = 2 s s i .
E = x ˆ E x + z ˆ E z , H = y ˆ H y ,
E x z = i ω [ μ ( z ) k x 2 ( z ) ω 2 ] H y ,
H y z = i ω ˜ ( z ) E x ,
Z TM ( z ) = + i ω [ μ ( z ) k x 2 ˜ ( z ) ω 2 ] ,
Y TM ( z ) = + i ω ˜ ( z ) ,
γ ( z ) = i ω [ μ ( z ) ˜ ( z ) k x 2 ω 2 ] 1 / 2 ,
1 Z 0 d Z 0 d z = 1 2 ˜ ( z ) ˜ ( z ) d μ ( z ) d z [ μ ( z ) 2 k x 2 ˜ ( z ) ω 2 ] d ˜ ( z ) d z μ ( z ) k x 2 ˜ ( z ) ω 2 .
ρ TM ( z ) = z d 1 2 n ˜ a d n ˜ a d z n ˜ a 2 n i 2 sin 2 θ i n ˜ 2 × exp [ i 2 k z z n ˜ a ( s ) d s ] d z + ρ ( d ) exp [ i 2 k z d n ˜ a ( s ) d s ] ,

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