Abstract

We extend the rigorous eigenmode theory of binary surface-relief gratings to accommodate three-dimensional-modulation profiles, to formulate a synthesis problem for doubly periodic resonance-domain diffractive elements, and to demonstrate some of the problem’s symmetry properties. Several solutions for multiple beam splitters with ~90% transmission-mode diffraction efficiency are obtained by nonlinear parametric optimization. The polarization sensitivity and the required fabrication accuracy are analyzed for some solutions.

© 1994 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).
    [CrossRef]
  2. A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
    [CrossRef]
  3. E. V. Jull, J. W. Heath, G. R. Ebbeson, “Gratings that diffract all incident energy,”J. Opt. Soc. Am. 67, 557–560 (1977).
    [CrossRef]
  4. L. S. Cheo, J. Schmoys, A. Hessel, “On simultaneous blazing of triangular groove diffraction gratings,”J. Opt. Soc. Am. 67, 1686–1688 (1977).
    [CrossRef]
  5. E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
    [CrossRef] [PubMed]
  6. E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 32, 5010–5012 (1992).
  7. 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]
  8. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  9. D. H. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
  10. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  11. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]
  12. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
    [CrossRef]
  13. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  14. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  15. M. C. Gupta, S. T. Peng, “Diffraction characteristics of surface-relief gratings,” Appl. Opt. 32, 2911–2917 (1993).
    [CrossRef] [PubMed]
  16. J. Turunen, P. Blair, J. M. Miller, M. R. Taghizadeh, E. Noponen, “Bragg holograms with binary synthetic surface-relief profile,” Opt. Lett. 18, 1022–1024 (1993).
    [CrossRef] [PubMed]
  17. H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
    [CrossRef]
  18. E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
    [CrossRef]
  19. R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
    [CrossRef]
  20. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  21. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  22. 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).
    [CrossRef]
  23. S. T. Han, Y.-L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  24. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  25. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  26. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,”J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  27. The dimensions of the matrix eigenvalue problem are halved, which implies an approximately eightfold reduction in the computational effort of its numerical solution.
  28. P. B. Fischer, S. Y. Chou, “Sub-50 nm high aspect-ratio silicon pillars, ridges, and trenches fabricated using ultra-high resolution electron beam lithography and reactive ion etching,” Appl. Phys. Lett. 62, 1414–1416 (1993).
    [CrossRef]
  29. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  30. S. J. Walker, J. Jahns, L. Li, W. M. Mansfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
    [CrossRef] [PubMed]

1994 (1)

1993 (10)

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

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

M. C. Gupta, S. T. Peng, “Diffraction characteristics of surface-relief gratings,” Appl. Opt. 32, 2911–2917 (1993).
[CrossRef] [PubMed]

J. Turunen, P. Blair, J. M. Miller, M. R. Taghizadeh, E. Noponen, “Bragg holograms with binary synthetic surface-relief profile,” Opt. Lett. 18, 1022–1024 (1993).
[CrossRef] [PubMed]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

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 surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
[CrossRef] [PubMed]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

P. B. Fischer, S. Y. Chou, “Sub-50 nm high aspect-ratio silicon pillars, ridges, and trenches fabricated using ultra-high resolution electron beam lithography and reactive ion etching,” Appl. Phys. Lett. 62, 1414–1416 (1993).
[CrossRef]

S. J. Walker, J. Jahns, L. Li, W. M. Mansfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
[CrossRef] [PubMed]

1992 (5)

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]

1979 (2)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
[CrossRef] [PubMed]

1978 (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,”J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

1977 (3)

1975 (2)

A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Ailawadi, N. K.

Baird, W. E.

Becker, M. F.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Blair, P.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Cheo, L. S.

Chou, S. Y.

P. B. Fischer, S. Y. Chou, “Sub-50 nm high aspect-ratio silicon pillars, ridges, and trenches fabricated using ultra-high resolution electron beam lithography and reactive ion etching,” Appl. Phys. Lett. 62, 1414–1416 (1993).
[CrossRef]

Collischon, M.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Ebbeson, G. R.

Fischer, P. B.

P. B. Fischer, S. Y. Chou, “Sub-50 nm high aspect-ratio silicon pillars, ridges, and trenches fabricated using ultra-high resolution electron beam lithography and reactive ion etching,” Appl. Phys. Lett. 62, 1414–1416 (1993).
[CrossRef]

Gaylord, T. K.

Gupta, M. C.

Haidner, H.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Han, S. T.

Heath, J. W.

Hessel, A.

Hutfless, J.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Ichikawa, H.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
[CrossRef]

Jaakkola, T.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
[CrossRef]

Jahns, J.

Jull, E. V.

Kipfer, P.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Knop, K.

Kuisma, S.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
[CrossRef]

Lang, A.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Li, L.

Lindolf, J.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Loewen, E. G.

Magnusson, R.

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]

Mansfield, W. M.

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
[CrossRef] [PubMed]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Miller, J. M.

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, “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).
[CrossRef]

Morris, G. M.

Mulgrew, P.

Nevière, M.

E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
[CrossRef] [PubMed]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Noponen, E.

Peng, S. T.

M. C. Gupta, S. T. Peng, “Diffraction characteristics of surface-relief gratings,” Appl. Opt. 32, 2911–2917 (1993).
[CrossRef] [PubMed]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Raguin, D. H.

Roberts, C. W.

Schmoys, J.

Schwider, J.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Sheridan, J. T.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Streibl, N.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[CrossRef]

Taghizadeh, M. R.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tennant, D. M.

Tsao, Y.-L.

Tseng, D. Y.

Turunen, J.

Vasara, A.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Walker, S. J.

Walser, R. M.

Wang, S. S.

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]

West, L. C.

Westerholm, J.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
[CrossRef]

Appl. Opt. (10)

E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
[CrossRef] [PubMed]

E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 32, 5010–5012 (1992).

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]

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 surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
[CrossRef] [PubMed]

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

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3220–3236 (1992).
[CrossRef]

M. C. Gupta, S. T. Peng, “Diffraction characteristics of surface-relief gratings,” Appl. Opt. 32, 2911–2917 (1993).
[CrossRef] [PubMed]

S. T. Han, Y.-L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
[CrossRef] [PubMed]

S. J. Walker, J. Jahns, L. Li, W. M. Mansfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
[CrossRef] [PubMed]

Appl. Phys. (2)

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Appl. Phys. Lett. (2)

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

P. B. Fischer, S. Y. Chou, “Sub-50 nm high aspect-ratio silicon pillars, ridges, and trenches fabricated using ultra-high resolution electron beam lithography and reactive ion etching,” Appl. Phys. Lett. 62, 1414–1416 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Eng. (1)

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, M. Collischon, A. Lang, J. Hutfless, “Polarizing reflection grating beamsplitter for the 10.6 μm wavelength,” Opt. Eng. 32, 1861–1865 (1993).
[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)

The dimensions of the matrix eigenvalue problem are halved, which implies an approximately eightfold reduction in the computational effort of its numerical solution.

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

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Figures (8)

Fig. 1
Fig. 1

Structure of a doubly periodic binary surface-relief grating.

Fig. 2
Fig. 2

Definition of the incident plane wave.

Fig. 3
Fig. 3

Convergence of the efficiencies of the central diffraction orders produced by a checkerboard grating when the number of Rayleigh orders L × L is increased; the periods are (a) d = 2.5λ and (b) d = 5.0λ.

Fig. 4
Fig. 4

(a) Goal signal and (b)–(d) the grating structures (top and side views) for 1 → 4 beam-splitter elements. The shaded regions are occupied by the dielectric substrate material.

Fig. 5
Fig. 5

Same as Fig. 4 but for 1 → 6 beam splitters. The parts of circular features that fall outside the period are sketched only to illustrate the hexagonal symmetry.

Fig. 6
Fig. 6

Same as Fig. 4 but for 1 → 7 beam splitters.

Fig. 7
Fig. 7

Same as Fig. 4 but for 1 → 9 beam splitters.

Fig. 8
Fig. 8

Fabrication tolerances for the 1 → 4 beam splitter of Fig. 4(c) (solid curves) and the 1 → 9 beam splitter of Fig. 7(b) (dashed curves). In (a) and (b) the diffraction efficiencies are evaluated as a function of deviations from the optimum relief depth and from the feature size, respectively; the corresponding uniformity errors are plotted in (c) and (d).

Tables (2)

Tables Icon

Table 1 Efficiencies (%) of the Transmitted Diffraction Orders for a Checkerboard Grating with Period d = 2.5λ and Relief Depth h = λ

Tables Icon

Table 2 Grating Parameters of the Elements Illustrated in Figs. 4 and 7, with Theoretical Diffraction Efficiencies and Uniformity Errors

Equations (73)

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

E I ( r ) = u ^ exp ( i k 0 · r ) ,
k 0 = α 0 x ^ + β 0 y ^ + r 00 z ^ ,
u ^ = ( cos ψ cos θ cos ϕ - sin ψ sin ϕ ) x ^ + ( cos ψ cos θ sin ϕ + sin ψ cos ϕ ) y ^ - cos ψ sin θ z ^ .
E R ( r ) = m , n R n m exp ( i k 1 m n · r ) ,
E T ( r ) = m , n T m n exp [ i k 2 m n · ( r - h z ^ ) ] ,
k 1 m n = α m x ^ + β n y ^ - r m n z ^ ,
k 2 m n = α m x ^ + β n y ^ + t m n z ^ ,
r m n = { [ ( n 1 k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ( n 1 k ) 2 i [ α m 2 + β n 2 - ( n 1 k ) 2 ] 1 / 2 α m 2 + β n 2 > ( n 1 k ) 2 ,
t m n = { [ ( n 2 k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ( n 2 k ) 2 i [ α m 2 + β n 2 - ( n 2 k ) 2 ] 1 / 2 α m 2 + β n 2 > ( n 2 k ) 2 .
H I ( r ) = ( ω μ 0 ) - 1 k 0 × u ^ exp ( i k 0 · r ) ,
H R ( r ) = ( ω μ 0 ) - 1 m , n k 1 m n × R m n exp ( i k 1 m n · r ) ,
H T ( r ) = ( ω μ 0 ) - 1 m , n k 2 m n × T m n exp [ i k 2 m n · ( r - h z ^ ) ] ,
× E ( r ) = i ω μ 0 H ( r ) ,
× H ( r ) = - i ω E ( r ) ,
z E x = i ω μ 0 H y - x [ ( i ω ) - 1 ( x H y - y H x ) ] ,
z E y = - i ω μ 0 H x - y [ ( i ω ) - 1 ( x H y - y H x ) ] ,
z H x = - i ω E y + ( i ω μ 0 ) - 1 x ( x E y - y E x ) ,
z H y = i ω E x + ( i ω μ 0 ) - 1 y ( x E y - y E x ) .
E x ( x , y , z ) = m , n E x m n exp [ i ( α m x + β n y + γ z ) ] ,
( x , y ) = 0 p , q p q exp [ i 2 π ( p x / d x + q y / d y ) ] ,
[ ( x , y ) ] - 1 = 0 - 1 p , q ζ p q exp [ i 2 π ( p x / d x + q y / d y ) ] ,
ω 0 γ E x m n = k 2 H y m n - α m p , q ζ m - p , n - q × ( α p H y p q - β q H x p q ) ,
ω 0 γ E y m n = - k 2 H x m n - β n p , q ζ m - p , n - q × ( α p H y p q - β q H x p q ) ,
ω μ 0 γ H x m n = - k 2 p , q m - p , n - q E y p q + α m ( α m E y m n - β n E x m n ) ,
ω μ 0 γ H y m n = k 2 p , q m - p , n - q E x p q + β n ( α m E y m n - β n E x m n ) ,
E x ( x , y , z ) = l = 1 2 L { A l exp ( i γ l z ) + B l exp [ - i γ l ( z - h ) ] } × m , n E x m n l exp [ i ( α m x + β n y ) ] ,
E y ( x , y , z ) = l = 1 2 L t { A l exp ( i γ l z ) + B l exp [ - i γ l ( z - h ) ] } × m , n E y m n l exp [ i ( α m x + β n y ) ] ,
H x ( x , y , z ) = k l = 1 2 L t { A l exp ( i γ l z ) - B l exp [ - i γ l ( z - h ) ] } × m , n H x m n l exp [ i ( α m x + β n y ) ] ,
H y ( x , y , z ) = k l = 1 2 L t { A l exp ( i γ l z ) - B l exp [ - i γ l ( z - h ) ] } × m , n H y m n l exp [ i ( α m x + β n y ) ] ,
u x δ m 0 δ n 0 + R x m n = l [ A l + B l exp ( i γ l h ) ] E x m n l ,
u y δ m 0 + R y m n = l [ A l + B l exp ( i γ l h ) ] E y m n l ,
( β 0 u z - r 00 u y ) δ m 0 δ n 0 + β n R z m n + r m n R y m n = k l [ A l - B l exp ( i γ l h ) ] H x m n l ,
( r 00 u x - α 0 u z ) δ m 0 δ n 0 - r m n R x m n - α m R z m n = k l [ A l - B l exp ( i γ l h ) ] H y m n l .
l [ A l exp ( i γ l h ) + B l ] E x m n l = T x m n ,
l [ A l exp ( i γ l h ) + B l ] E y m n l = T y m n ,
k l [ A l exp ( i γ l h ) - B l ] H x m n l = β n T z m n - t m n T y m n ,
k l [ A l exp ( i γ l h ) - B l ] H y m n l = t m n T x m n - α m T z m n .
η R m n = R ( r m n / r 00 ) R m n 2 ,
η T m n = R ( t m n / r 00 ) T m n 2 ,
T m n = 1 d x d y 0 d x 0 d y t ( x , y ) × exp [ - i 2 π ( m x / d x + n y / d y ) ] d y d x ,
t ( x , y ) = exp { i k h [ ( x , y ) / 0 ] 1 / 2 }
= { η ^ m n η ^ m n { 0 , 1 } , ( m , n ) W } .
N = ( m , n ) W η ^ m n ,
η = ( m , n ) W η ^ m n η m n ,
E = max ( m , n ) W { η ^ m n 1 - N η m n / η } .
E ˜ = ( E ˜ x E ˜ y ) ,
H ˜ = ( H ˜ x H ˜ y ) ,
E ˜ x n = E ˜ x , p + L x q = E x p q ,
E ˜ y n = E ˜ y , p + L x q = E y p q ,
H ˜ x n = H ˜ x , p + L x q = ( μ 0 / 0 ) 1 / 2 H x p q ,
H ˜ y n = H ˜ y , p + L x q = ( μ 0 / 0 ) 1 / 2 H y p q .
k γ E ˜ = F H ˜ ,
k γ H ˜ = G E ˜ ,
F = [ ( α ˜ n η ˜ n - p β ˜ p ) ( k 2 δ n p - α ˜ n η ˜ n = p α ˜ p ) ( - k 2 δ n p + β ˜ n η ˜ n - p β ˜ p ) ( - β ˜ n η ˜ n - p α ˜ p ) ] ,
G = [ ( - α ˜ n β ˜ n δ n p ) ( - k 2 ˜ n - p + α ˜ n 2 δ n p ) ( k 2 ˜ n - p - β ˜ n 2 δ n p ) ( α ˜ n β ˜ n δ n p ) ] .
FG E ˜ = k 2 γ 2 E ˜ = g E ˜
R z m n = ( α m R x m n + β n R y m n ) / r m n ,
T z m n = - ( α m T x m n + β n T y m n ) / t m n .
P = C 1 A + C 2 B ,
Q = C 3 A + C 4 B ,
P n = [ - α ˜ 0 β ˜ 0 r ˜ 0 u x - ( 2 r ˜ 0 + β ˜ 0 2 r ˜ 0 ) u y + β ˜ 0 u z ] δ n 0 ,
C 1 n l = [ k H ˜ x n l - α ˜ n β ˜ n r ˜ n E ˜ x n l - ( r ˜ n + β ˜ n 2 r ˜ n ) E ˜ y n l ] ,
C 2 n l = [ - k H ˜ x n l - α ˜ n β ˜ n r ˜ n E ˜ x n l - ( r ˜ n + β ˜ n 2 r ˜ n ) E ˜ y n l ] × exp ( i γ l h ) ,
Q n = [ ( 2 r ˜ 0 + α ˜ 0 2 r ˜ 0 ) u x + α ˜ 0 β ˜ 0 r ˜ 0 u y - α ˜ 0 u z ] δ n 0 ,
C 3 n l = [ k H ˜ y n l + ( r ˜ n + α ˜ n 2 r ˜ n ) E ˜ x n l + α ˜ n β ˜ n r ˜ n E ˜ y n l ] ,
C 4 n l = [ - k H ˜ y n l + ( r ˜ n + α ˜ n 2 r ˜ n ) E ˜ x n l + α ˜ n β ˜ n r ˜ n E ˜ y n l ] × exp ( i γ l h ) .
D 1 A + D 2 B = 0 ,
D 3 A + D 4 B = 0 ,
D 1 n l = [ k H ˜ x n l + α ˜ n β ˜ n t ˜ n E ˜ x n l + ( t ˜ n + β ˜ n 2 t ˜ n ) E ˜ y n l ] exp ( i γ l h ) ,
D 2 n l = [ - k H ˜ x n l + α ˜ n β ˜ n t ˜ n E ˜ x n l + ( t ˜ n + β ˜ n 2 t ˜ n ) E ˜ y n l ] ,
D 3 n l = [ k H ˜ y n l - ( t ˜ n + α ˜ n 2 t ˜ n ) E ˜ x n l - α ˜ n β ˜ n t ˜ n E ˜ y n l ] exp ( i γ l h ) ,
D 4 n l = [ - k H ˜ y n l - ( t ˜ n + α ˜ n 2 t ˜ n ) E ˜ x n l - α ˜ n β ˜ n t ˜ n E ˜ y n l ] .
[ C 1 C 2 C 3 C 4 D 1 D 2 D 3 D 4 ] ( A B ) = ( P Q 0 0 ) ,

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