Abstract

An effective grating model, which generalizes effective-medium theory to the case of resonance domain surface-relief gratings, is presented. In addition to the zero order, it takes into account the first diffraction order, which obeys the Bragg condition. Modeling the surface-relief grating as an effective grating with two diffraction orders provides closed-form analytical relationships between efficiency and grating parameters. The aspect ratio, the grating period, and the required incidence angle that would lead to high diffraction efficiencies are predicted for TE and TM polarization and verified by rigorous numerical calculations.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
    [CrossRef]
  2. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  3. N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, E. Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt. 40, 2076–2080 (2001).
    [CrossRef]
  4. J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 2000), pp. 343–388.
    [CrossRef]
  5. T. Shiono, T. Hamamoto, K. Takahara, “High-efficiency blazed diffractive optical elements for the violet wavelength fabricated by electron-beam lithography,” Appl. Opt. 41, 2390–2393 (2002).
    [CrossRef] [PubMed]
  6. 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]
  7. Y. Sheng, D. Feng, S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled wave theory,” J. Opt. Soc. Am. A 14, 1562–1568 (1997).
    [CrossRef]
  8. D. Meshulach, D. Yelin, Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615–1619 (1998).
    [CrossRef]
  9. R. Shechter, Y. Amitai, A. A. Friesem, “Compact beam expander with linear gratings,” Appl. Opt. 41, 1236–1240 (2002).
    [CrossRef] [PubMed]
  10. S. J. Walker, U. 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]
  11. A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
    [CrossRef] [PubMed]
  12. I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
    [CrossRef]
  13. G. W. Stroke, “Ruling, testing and use of optical gratings for high resolution spectroscopy,” in Progress in Optics, Vol. II, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1963), pp. 343–388.
  14. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  15. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  16. K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiency,” Appl. Opt. 23, 2303–2310 (1984).
    [CrossRef] [PubMed]
  17. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  18. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  19. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  20. L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
    [CrossRef]
  21. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  22. E. Popov, E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4, Sect. 4.2.3.
  23. M. Nevière, E. Popov, Light Propagation in Periodic Media (Marcel Dekker, New York, 2003).
  24. H. J. Gerritsen, D. K. Thornton, S. R. Bolton, “Application of Kogelnik’s two-wave theory to deep, slanted, highly efficient, relief transmission gratings,” Appl. Opt. 30, 807–814 (1991).
    [CrossRef] [PubMed]
  25. M. A. Golub, A. A. Friesem, “Analytical theory for efficient surface relief gratings in the resonance domain,” in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2004), Chap. 19, pp. 307–328.
  26. M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
    [CrossRef]
  27. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  28. R. Magnusson, T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  29. N. Chateau, J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  30. J.-P. Plumey, B. Guizal, J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]
  31. J. M. Miller, N. Beaucoudrey, P. Chavel, J. Turunen, E. Cambril, “Design and fabrication of binary slanted surface-relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
    [CrossRef] [PubMed]
  32. L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. A 16, 2521–2531 (1999).
    [CrossRef]
  33. P. Laakkonen, M. Kuittinen, J. Simonen, J. Turunen, “Electron-beam-fabricated asymmetric transmission gratings for microspectroscopy,” Appl. Opt. 39, 3187–3191 (2000).
    [CrossRef]
  34. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
    [CrossRef]
  35. M. Breidne, D. Maystre, “Equivalence of ruled, holographic, and lamellar gratings in constant deviation mountings,” Appl. Opt. 19, 1812–1821 (1980).
    [CrossRef] [PubMed]
  36. M. A. Golub, “Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1194–1201 (1999).
    [CrossRef]

2004 (1)

M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
[CrossRef]

2003 (1)

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

1999 (3)

1998 (1)

1997 (3)

1996 (1)

1995 (2)

1994 (1)

1993 (3)

1992 (1)

1991 (2)

1984 (1)

1982 (1)

1980 (2)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. Breidne, D. Maystre, “Equivalence of ruled, holographic, and lamellar gratings in constant deviation mountings,” Appl. Opt. 19, 1812–1821 (1980).
[CrossRef] [PubMed]

1977 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Ailawadi, N. K.

Amitai, Y.

Awada, K. A.

Beaucoudrey, N.

Beyer, O.

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

Bokor, N.

Bolton, S. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Breidne, M.

Buse, K.

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

Cambril, E.

Chandezon, J.

Chateau, N.

Chavel, P.

Davidson, N.

Eisen, L.

M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
[CrossRef]

Engel, H.

Feng, D.

Friesem, A. A.

M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
[CrossRef]

R. Shechter, Y. Amitai, A. A. Friesem, “Compact beam expander with linear gratings,” Appl. Opt. 41, 1236–1240 (2002).
[CrossRef] [PubMed]

N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, E. Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt. 40, 2076–2080 (2001).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

M. A. Golub, A. A. Friesem, “Analytical theory for efficient surface relief gratings in the resonance domain,” in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2004), Chap. 19, pp. 307–328.

Gaylord, T. K.

Gerritsen, H. J.

Golub, M. A.

M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
[CrossRef]

M. A. Golub, “Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1194–1201 (1999).
[CrossRef]

M. A. Golub, A. A. Friesem, “Analytical theory for efficient surface relief gratings in the resonance domain,” in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2004), Chap. 19, pp. 307–328.

Granet, G.

Grann, E. B.

Guizal, B.

Hamamoto, T.

Hasman, E.

Hugonin, J.-P.

Jahns, U.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kuittinen, M.

P. Laakkonen, M. Kuittinen, J. Simonen, J. Turunen, “Electron-beam-fabricated asymmetric transmission gratings for microspectroscopy,” Appl. Opt. 39, 3187–3191 (2000).
[CrossRef]

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 2000), pp. 343–388.
[CrossRef]

Laakkonen, P.

Larochelle, S.

Li, L.

Loewen, E. G.

E. Popov, E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4, Sect. 4.2.3.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

Mansfield, W. M.

Maystre, D.

Meshulach, D.

Miller, J. M.

Moharam, M. G.

Morris, G. M.

Mulgrew, P.

Muller, M.

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

Nee, I.

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

Nevière, M.

M. Nevière, E. Popov, Light Propagation in Periodic Media (Marcel Dekker, New York, 2003).

Noponen, E.

Pai, D. M.

Peng, S.

Plumey, J.-P.

Pommet, D. A.

Popov, E.

E. Popov, E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4, Sect. 4.2.3.

M. Nevière, E. Popov, Light Propagation in Periodic Media (Marcel Dekker, New York, 2003).

Raguin, D. H.

Roberts, C. W.

Rosenblatt, D.

Sharon, A.

Shechter, R.

Sheng, Y.

Shiono, T.

Silberberg, Y.

Simonen, J.

Steingrueber, R.

Stroke, G. W.

G. W. Stroke, “Ruling, testing and use of optical gratings for high resolution spectroscopy,” in Progress in Optics, Vol. II, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1963), pp. 343–388.

Taghizadeh, M. R.

Takahara, K.

Tennant, D. M.

Thornton, D. K.

Turunen, J.

Vasara, A.

Walker, S. J.

Weber, H. G.

West, L. C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Wyrowski, F.

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 2000), pp. 343–388.
[CrossRef]

Yelin, D.

Yokomori, K.

Appl. Opt. (11)

T. Shiono, T. Hamamoto, K. Takahara, “High-efficiency blazed diffractive optical elements for the violet wavelength fabricated by electron-beam lithography,” Appl. Opt. 41, 2390–2393 (2002).
[CrossRef] [PubMed]

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

N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, E. Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt. 40, 2076–2080 (2001).
[CrossRef]

R. Shechter, Y. Amitai, A. A. Friesem, “Compact beam expander with linear gratings,” Appl. Opt. 41, 1236–1240 (2002).
[CrossRef] [PubMed]

S. J. Walker, U. 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]

K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiency,” Appl. Opt. 23, 2303–2310 (1984).
[CrossRef] [PubMed]

L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

H. J. Gerritsen, D. K. Thornton, S. R. Bolton, “Application of Kogelnik’s two-wave theory to deep, slanted, highly efficient, relief transmission gratings,” Appl. Opt. 30, 807–814 (1991).
[CrossRef] [PubMed]

J. M. Miller, N. Beaucoudrey, P. Chavel, J. Turunen, E. Cambril, “Design and fabrication of binary slanted surface-relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
[CrossRef] [PubMed]

P. Laakkonen, M. Kuittinen, J. Simonen, J. Turunen, “Electron-beam-fabricated asymmetric transmission gratings for microspectroscopy,” Appl. Opt. 39, 3187–3191 (2000).
[CrossRef]

M. Breidne, D. Maystre, “Equivalence of ruled, holographic, and lamellar gratings in constant deviation mountings,” Appl. Opt. 19, 1812–1821 (1980).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (11)

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

I. Nee, O. Beyer, M. Muller, K. Buse, “Multichannel wavelength-division multiplexing with thermally fixed Bragg gratings in photorefractive lithium niobate crystals,” J. Opt. Soc. Am. A 20, 1593–1602 (2003).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

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]

Y. Sheng, D. Feng, S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled wave theory,” J. Opt. Soc. Am. A 14, 1562–1568 (1997).
[CrossRef]

N. Chateau, J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

J.-P. Plumey, B. Guizal, J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. A 16, 2521–2531 (1999).
[CrossRef]

M. A. Golub, “Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1194–1201 (1999).
[CrossRef]

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

Opt. Commun. (2)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. A. Golub, A. A. Friesem, L. Eisen, “Bragg properties of efficient surface relief gratings in the resonance domain,” Opt. Commun. 235, 261–267 (2004).
[CrossRef]

Opt. Lett. (1)

Other (7)

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, Vol. XL, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 2000), pp. 343–388.
[CrossRef]

G. W. Stroke, “Ruling, testing and use of optical gratings for high resolution spectroscopy,” in Progress in Optics, Vol. II, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1963), pp. 343–388.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

M. A. Golub, A. A. Friesem, “Analytical theory for efficient surface relief gratings in the resonance domain,” in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2004), Chap. 19, pp. 307–328.

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

E. Popov, E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), Chap. 4, Sect. 4.2.3.

M. Nevière, E. Popov, Light Propagation in Periodic Media (Marcel Dekker, New York, 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Relevant parameters and geometry of two grooves from a surface-relief grating: (a) actual geometry and parameters, (b) normalized geometry and parameters.

Fig. 2
Fig. 2

Geometry and parameters for a surface-relief grating after coordinate rotation: (a) rotated coordinate system, (b) normalized rotated coordinate system and sublayers.

Fig. 3
Fig. 3

Characterization of a surface-relief grating, both overall and in each layer, by sinusoidal graded-index gratings with a three-dimensional grating vector.

Fig. 4
Fig. 4

Relative refractive-index modulation term Δ n 1 Δ n s as a function of groove deviation Δ p from the slant of the effective grating. For the effective grating Δ n 1 = Δ n s , δ low is the threshold value, 2 Δ p low the bandwidth, and p low and p low + the boundaries.

Fig. 5
Fig. 5

Bragg diffraction efficiency as a function of aspect ratio h Λ , for a surface relief grating with symmetrical ( q c = 0.5 ) triangular groove profile. The Bragg incidence angle was θ inc , 1 = 30 ° , λ Λ = 1.0 , refractive index of grooves n M = 2.5 , and TE polarization. Bold curves, effective grating model; thin curves, numerical RCWA.[14]

Fig. 6
Fig. 6

Bragg diffraction efficiency of the + 1 st order as a function of aspect ratio h Λ , for a surface relief grating with symmetrical ( q c = 0.5 ) groove profile: (a) sinusoidal, (b) triangular. The Bragg incidence angle was θ inc , 1 = 45 ° , λ Λ = 1.414 , refractive-index of grooves n M = 1.66 , and TE polarization. Bold curves, effective grating model; thin curves, numerical RCWA.[16]

Fig. 7
Fig. 7

Bragg diffraction efficiency of the + 1 st order as a function of ratio λ Λ for a surface-relief grating with symmetrical ( q c = 0.5 ) sinusoidal groove profile. The Bragg incidence angle θ inc , 1 was set at each λ Λ , h Λ = 2.0 ; refractive index of grooves n M = 1.66 . Bold curves, effective grating model; thin curves, numerical RCWA.[16]

Fig. 8
Fig. 8

Optimal aspect ratio h Λ as a function of Λ λ and their boundaries for obtaining high Bragg diffraction efficiency η Bragg for a surface-relief grating with triangular sawtooth groove profile ( q c = 1 ) . Refractive index of grooves n M = 1.46 . ρ 2 is a two-wave criterion of the Bragg diffraction regime with η Bragg = 100 % in TE polarization. From bold to thin curves: η Bragg =100%, 98%, 96%, 94%; dashed curves, 92%, dotted curves, 90%.

Fig. 9
Fig. 9

Optimal aspect ratio h Λ as a function of Λ λ and their limits for different Bragg diffraction efficiencies η Bragg for surface-relief gratings with sinusoidal slanted groove profile. Slant angle ϕ s = 17 ° , refractive index of grooves n M = 1.66 . ρ 2 is a two-wave criterion of Bragg diffraction regime with η Bragg = 100 % . Curves as in Fig. 8.

Fig. 10
Fig. 10

The diffraction efficiency of the + 1 st order as a function of incidence angle for surface-relief gratings of sinusoidal slanted groove profile with angle ϕ s = 17 ° . Optimal aspect ratio h Λ = 1.555 in TE polarization and h Λ = 1.897 in TM polarization lead to 98% diffraction efficiency at Bragg incident angle θ inc , 1 = 8.8 ° . Refractive index of grooves n M = 1.66 , n sub = 1.52 , Λ λ = 0.96 . Bold curves, effective-grating model; thin curves, numerical RCWA. Analytical: TE, --- TM. Numerical: — TE, ----- TM.

Fig. 11
Fig. 11

Diffraction efficiency of the + 1 st order as a function of incidence angle for sinusoidal surface-relief gratings with aspect ratio h Λ = 1.615 optimized so as to achieve the same high diffraction efficiency of 93% for both TE and TM polarization at Bragg incident angle θ inc , 1 = 7.5 ° . Slant angle is ϕ s = 17 ° , Λ λ = 1.0 , refractive index of grooves n M = 1.66 , n sub = 1.52 . Bold curves effective grating model; thin curves numerical RCWA. Curves as in Fig. 10.

Fig. 12
Fig. 12

The diffraction efficiency of the + 1 st order as a function of incidence angle, with aspect ratio h Λ optimized for 100% diffraction efficiency at appropriate Bragg incident angles θ inc , 1 for surface-relief gratings with triangular groove profiles at three different groove peak positions q c : (a) overhanging of q c = 1.4 , h Λ = 2.091 , θ inc , 1 = 0.0 ° , (b) sawtooth of q c = 1.0 , h Λ = 2.246 , θ inc , 1 = 11.7 ° , and (c) symmetrical of q c = 0.5 , h Λ = 2.309 , θ inc , 1 = 27.0 ° . Refractive index of grooves n M = 1.46 , n sub = 1.46 , Λ λ = 1.1 , and TE polarization. Bold curves, effective grating model; thin curves, numerical RCWA.

Equations (57)

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

Λ r = Λ cos ϕ r , h r = h cos ϕ r .
χ r = x r cos ϕ r Λ , ζ r = z r h r .
g r ( χ r ) = g ( χ ) , χ r = χ p g ( χ ) ,
p = h tan ϕ r Λ .
n 2 = n ¯ 2 + Δ n M 2 Δ j G ¯ Δ j exp ( i Δ j K r r ) ,
n ¯ 2 = 0 1 0 1 n 2 d χ r d ζ r , G ¯ Δ j = 0 1 G Δ j ( ζ r ) d ζ r ,
G Δ j ( ζ r ) = 0 1 n 2 n ¯ 2 Δ n M 2 exp ( i 2 π Δ j χ r ) d χ r
ϵ = 0 1 0 1 n 4 d χ r d ζ r Δ n M 4 Δ j G ¯ Δ j 2 .
n ¯ 2 = n i 2 + Δ n M 2 g ¯ , g ¯ = 0 1 g ( χ ) d χ ,
G ¯ Δ j = 0 1 g r ( χ r ) exp ( i 2 π Δ j χ r ) d χ r , Δ j 0 ,
sin θ r j = j λ 2 n ¯ Λ r , θ r j = θ j + ϕ r .
n ¯ sin θ j = j λ 2 Λ Λ p h [ n ¯ 2 1 + ( Λ p h ) 2 ( j λ 2 Λ ) 2 ] 1 2 .
σ j = σ 0 j K r , σ j = k n ¯ ( s j r , 0 , c 0 r ) ,
s j r = sin θ r j 2 sin θ r j , c 0 r = cos θ r = ( 1 s 0 r 2 ) 1 2 .
Λ λ > Λ subw λ = ( n ¯ + n ¯ sin θ 1 ) 1 ,
ρ 2 ε high ,
ρ = ( λ Λ r ) 2 n ¯ Δ n 1 , ρ 2 = ( Λ λ ) 4 ( n ¯ Δ n 1 ) 2 cos 4 ϕ r .
T ( ζ r ; Δ z r ) = exp [ i k n ¯ M ( ζ r ) Δ z r ] .
Q 12 = Q 12 slant + ϑ 1 Q 12 Bragg ,
Q 12 slant = c 0 r 2 [ κ 01 ( ζ r 1 ) κ 01 * ( ζ r 2 ) κ 01 * ( ζ r 1 ) κ 01 ( ζ r 2 ) ] [ 1 0 0 1 ] ,
Q 12 Bragg = c 0 r 2 [ 0 Δ κ 01 Δ κ 01 * 0 ] , Δ κ 01 = κ 01 ( ζ r 1 ) κ 01 ( ζ r 2 ) ,
ϑ 1 = 1 2 ( 1 s 1 r 2 c 0 r 2 ) ,
T ¯ = exp ( i k n ¯ M ¯ h r ) , M ¯ = 0 1 M ( ζ r ) d ζ r ,
M ¯ j , j = c 0 r 1 ( κ ¯ j , j + ϑ j δ j j ) ,
κ ¯ j , j = δ n M 2 e j e j G ¯ j j , δ n M 2 = Δ n M 2 2 n ¯ 2 ,
p s = h tan φ s Λ ,
S 0 ( out ) = exp ( i ξ ) ( cos μ i ξ sin μ μ ) ,
S 1 ( out ) = i exp ( i ξ ) ν sin μ μ ,
μ = ( ν 2 + ξ 2 ) 1 2 ,
ν = h s k n ¯ κ ¯ 01 c 0 r , ξ = h s k n ¯ ϑ 1 2 c 0 r ,
η Bragg = sin 2 ν Bragg ,
ν Bragg = 2 π h λ n ¯ κ ¯ 01 cos ϕ s c 0 s Bragg
c 0 s Bragg = ( 1 sin 2 θ s 1 ) 1 2 , sin θ s 1 = λ 2 n ¯ Λ cos ϕ s .
κ ¯ 01 TE = δ n M 2 G 1 s , κ ¯ 01 TM = κ ¯ 01 TE ( 1 2 sin 2 θ s 1 ) .
h λ = c 0 s Bragg cos ϕ s 2 π n ¯ κ ¯ 01 [ π 2 ± ( π 2 arcsin η Bragg ) ] .
tan ϕ s = n ¯ Λ λ { cos θ req [ 1 ( sin θ req λ n ¯ Λ ) 2 ] 1 2 } .
q c = 0.5 + p s .
η mis = sin 2 [ ( ν 2 + ξ 2 ) 1 2 ] sin 2 ν ( 1 + ξ 2 ν 2 ) 1 ,
sin θ 1 ± = F ± ( tan ϕ s ) [ cos 2 ϕ s ( F ± ) 2 ] 1 2 ,
F ± = λ 2 n ¯ Λ ± ξ mis cos 3 ϕ s π Λ h c 0 s Bragg .
θ 1 , low = θ 1 .
g ( χ ) = { χ q c , 0 χ q c ( 1 χ ) ( 1 q c ) , q c χ 1 } .
G ¯ Δ j = 2 exp ( i Δ α ) ( π Δ j ) 2 ( 4 Δ p 2 1 ) { i sin ( Δ α ) when Δ j = 2 , 4 , cos ( Δ α ) when Δ j = 1 , 3 , } ,
G Δ j ( ζ r ) = ζ r sinc ( Δ j ζ r ) exp ( i 2 Δ α ζ r ) + 0.5 δ Δ j ,
g ( χ ) = g sym ( χ sym ) , χ = χ sym + ( q c 0.5 ) g sym ( χ sym ) ,
G ¯ Δ j = J Δ j ( Δ α ) 2 Δ α i exp ( i Δ α i π Δ j 2 ) ,
G Δ j ( ζ r ) = α r sinc ( Δ j α r ) exp ( i 2 Δ α ζ r ) + 0.5 δ Δ j ,
n 2 = Δ j [ Δ n M 2 G Δ j ( ζ r ) + n ¯ 2 δ Δ j ] exp ( i Δ j K r r ) ,
G Δ j ( ζ r ) = Δ χ r sinc ( Δ j Δ χ r ) exp ( i 2 π Δ j χ r c ) g ¯ δ Δ j ,
χ r c = 0.5 or χ r 1 + χ r 2 = 1 ,
E = j = e j S j ( z r ; ζ r ) exp ( i σ j r ) ,
2 E ( E ) + k 2 n ¯ 2 E + k 2 [ n 2 ( x r , z r ) n ¯ 2 ] E = 0 .
i ( 1 e j z r 2 ) 2 k n ¯ c 0 r S j ( z r ; ζ r ) + S j ( z r ; ζ r ) = i k n ¯ j M j j ( ζ r ) S j ( z r ; ζ r ) ,
M j , j ( ζ r ) = c 0 r 1 [ κ j , j ( ζ r ) + ϑ j δ j j ] ,
κ j , j ( ζ r ) = δ n M 2 e j e j G j j ( ζ r ) ,
ϑ j = 1 2 ( 1 s j r 2 c 0 r 2 ) ,
S ( z r ; ζ r ) = i k n ¯ M ( ζ r ) S ( z r ; ζ r )

Metrics