Abstract

We report here on the theoretical performance of blazed binary diffractive elements composed of pillars carefully arranged on a two-dimensional grid whose period is smaller than the structural cutoff. These diffractive elements operate under unpolarized light. For a given grating geometry, the structural cutoff is a period value above which the grating no longer behaves like a homogeneous thin film. Because the grid period is smaller than this value, effective-medium theories can be fully exploited for the design, and straightforward procedures are obtained. The theoretical performance of the blazed binary elements is investigated through electromagnetic theories. It is found that these elements substantially outperform standard blazed échelette diffractive elements in the resonance domain. The increase in efficiency is explained by a decrease of the shadowing effect and by an unexpected sampling effect. The theoretical analysis is confirmed by experimental evidence obtained for a 3λ-period prismlike grating operating at 633 nm and for a 20°-off-axis diffractive lens operating at 860 nm.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]
  6. C. G. Blough, M. Rossi, S. K. Mack, R. L. Michaels, “Single-point diamond turning and replication of visible and near-infrared diffractive optical elements,” Appl. Opt. 36, 4648–4654 (1997).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. Z. Zhou, T. J. Drabik, “Optimized binary, phase-only, diffractive optical element with subwavelength features for 1.55 µm,” J. Opt. Soc. Am. A 12, 1104–1112 (1995).
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  21. P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
    [CrossRef]
  22. M. E. Warren, R. E. Smith, G. A. Vawter, J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm,” Opt. Lett. 20, 1441–1443 (1995).
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    [CrossRef] [PubMed]
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  26. S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, H. Launois, “High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm,” Opt. Lett. 23, 552–554 (1998).
    [CrossRef]
  27. Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
    [CrossRef]
  28. We believe that, with current technologies, the fabrication in glass of blazed binary diffractive elements is extremely difficult and probably impossible. Moreover, note that the use of a high-index material also has a beneficial effect on the theoretical performance (see Ref. 26).
  29. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  30. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. Ph. Lalanne, “Effective properties and band structures of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev. B 58, 9801–9807 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  36. W. Singer, H. Tiziani, “Born approximation for the nonparaxial scalar treatment of thick phase gratings,” Appl. Opt. 37, 1249–1255 (1997).
    [CrossRef]
  37. G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).
  38. See, for example, A. Bensoussan, J. L. Lions, G. Papanicolaou, “Asymptotic analysis for periodic structures,” in Study in Mathematics and Its Applications, J. L. Lions, G. Papanicolaou, eds. (North-Holland, Amsterdam, 1978), Chap. 4.
  39. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  40. One can qualitatively understand this by considering that the pillar surrounded by the low-index material (in our case, air) may be seen as the core of a 2D waveguide. For a given grating period and from well-known results on 1D waveguides, multimode operations are obtained for large core widths, i.e., for large fill factors, by analogy. Conversely, for a given fill factor and a given grating period, it is intuitively clear that the structure supports an increasing number of modes for increasing values of the core refractive index.
  41. H. Kikuta, Y. Ohira, H. Kubo, K. Iwata, “Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit,” J. Opt. Soc. Am. A 15, 1577–1585 (1998).
    [CrossRef]
  42. The computation was performed by the modal theory of Ref. 32, with square truncation. Nineteen orders along each periodicity axis were retained for the computation. No convergence problems were encountered, and the numerical results can be considered as exact. These numerical results strongly differ from those obtained by Chen and Craighead (see Fig. 1 of Ref. 24). For pillar sizes of approximately 400 nm, the zeroth-order diffraction efficiency is 65%, a value 15% smaller than that reported in Ref. 24. This difference is due to the fact that the numerical results of Ref. 24 are obtained for 16 retained orders by a slowly converging numerical method. We believe that our lower-efficiency prediction may explain the 12% discrepancy, observed by the authors of Ref. 24, between their experimental results and their numerical predictions. In our opinion, the existence of the higher-order modes is responsible for the drop in efficiency denoted by the multiplication signs; because of their oscillatory form, these modes appreciably excite the nonzero orders diffracted by the grating.
  43. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
  44. In our opinion, the pillar structure offers the advantage of open ridges that are suitable for removing material during the RIE process.
  45. The depth h is chosen such that a 2π-phase change occurs at normal incidence between two homogeneous thin films coated on a glass substrate and whose refractive indices are 1-(nmax-1)/(2N-2) and nmax+(nmax-1)/(2N-2). This phase change is straightforwardly obtained by use of the Airy formula [see M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), p. 62], for homogeneous thin films. The resulting formula for h is rather complex, and it turns out that, for the values of nmax considered hereafter, the simple expression given in Eq. (4) is valid.
  46. The value of 66.5% was computed by electromagnetic theory and holds for a grating etched into glass (refractive index, ng=1.52), with an optimized grating depth slightly larger than λ/(ng-1), and for unpolarized light at normal incidence from air. It is 1% larger than that obtained in Fig. 1 for a grating depth equal to λ/(ng-1).
  47. Also note that the maximum pillar aspect ratio is increased from 4.6, reported in Ref. 27, to 8.8 in this study because of the use of a smaller sampling period.
  48. Strictly speaking, the Airy formula for thin films has to be used, as mentioned above.
  49. The diffractive components designed along the lines of procedures 1 and 2 are weakly polarization dependent. In general, we found that the first-order diffraction efficiency is a few percent larger for TE than for TM (see, e.g., Table 1).
  50. B. Layet, M. R. Taghizadeh, “Electromagnetic analysis of fan-out gratings and diffractive lens arrays by field stitching,” J. Opt. Soc. Am. A 14, 1554–1561 (1997).
    [CrossRef]
  51. 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]
  52. Since the width of the shadowing zone of the blazed échelette gratings decreases as the value of n increases, the performance is expected to improve with the refractive index.

1998 (6)

1997 (6)

1996 (8)

1995 (5)

1994 (4)

1993 (2)

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

H. Haidner, J. T. Sheridan, N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993).
[CrossRef] [PubMed]

1992 (2)

W. M. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458 (1992).
[CrossRef] [PubMed]

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

1991 (2)

1982 (1)

1972 (1)

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

1948 (1)

W. E. Kock, “Metallic delay lenses,” Bell Syst. Tech. J. 27, 58–82 (1948).
[CrossRef]

Astilean, S.

Baraldi, L.

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

Bensoussan, A.

See, for example, A. Bensoussan, J. L. Lions, G. Papanicolaou, “Asymptotic analysis for periodic structures,” in Study in Mathematics and Its Applications, J. L. Lions, G. Papanicolaou, eds. (North-Holland, Amsterdam, 1978), Chap. 4.

Blough, C. G.

Bojko, R. J.

J. M. Finlan, K. M. Flood, R. J. Bojko, “Efficient f/1 binary-optics microlenses in fused silica designed using vector diffraction theory,” Opt. Eng. 34, 3560–3564 (1995).
[CrossRef]

Born, M.

The depth h is chosen such that a 2π-phase change occurs at normal incidence between two homogeneous thin films coated on a glass substrate and whose refractive indices are 1-(nmax-1)/(2N-2) and nmax+(nmax-1)/(2N-2). This phase change is straightforwardly obtained by use of the Airy formula [see M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), p. 62], for homogeneous thin films. The resulting formula for h is rather complex, and it turns out that, for the values of nmax considered hereafter, the simple expression given in Eq. (4) is valid.

Bryngdahl, O.

Cambril, E.

Chavel, P.

Chen, F. T.

Collischon, M.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

Craighead, H. G.

Craighhead, H. G.

d’Auria, L.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

de Beaucoudrey, N.

Drabik, T. J.

Farn, W. M.

Feng, D.

Finlan, J. M.

J. M. Finlan, K. M. Flood, R. J. Bojko, “Efficient f/1 binary-optics microlenses in fused silica designed using vector diffraction theory,” Opt. Eng. 34, 3560–3564 (1995).
[CrossRef]

Flood, K. M.

J. M. Finlan, K. M. Flood, R. J. Bojko, “Efficient f/1 binary-optics microlenses in fused silica designed using vector diffraction theory,” Opt. Eng. 34, 3560–3564 (1995).
[CrossRef]

Fujita, F.

Gale, M. T.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

Gaylord, T. K.

Goebel, B.

Granet, G.

Grann, E. B.

Guizal, B.

Haidner, H.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

H. Haidner, J. T. Sheridan, N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993).
[CrossRef] [PubMed]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
[CrossRef] [PubMed]

Harvey, A. F.

A. F. Harvey, Microwave Engineering (Academic, London, 1963), Sect. 13.3.

Huignard, J. P.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Hutfless, J.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

Iwata, K.

Kikuta, H.

Kipfer, P.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
[CrossRef] [PubMed]

Kock, W. E.

W. E. Kock, “Metallic delay lenses,” Bell Syst. Tech. J. 27, 58–82 (1948).
[CrossRef]

Koyama, J.

Kubo, H.

Kuittinen, M.

M. Kuittinen, J. Turunen, P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998).
[CrossRef]

Lalanne, Ph.

Larochelle, S.

Launois, H.

Layet, B.

Lemercier-Lalanne, D.

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Li, L.

Lions, J. L.

See, for example, A. Bensoussan, J. L. Lions, G. Papanicolaou, “Asymptotic analysis for periodic structures,” in Study in Mathematics and Its Applications, J. L. Lions, G. Papanicolaou, eds. (North-Holland, Amsterdam, 1978), Chap. 4.

Mack, S. K.

Mait, J. N.

Marz, M.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

Michaels, R. L.

Miller, J. M.

Mirtoznik, M. S.

Moharam, M. G.

Morris, G. M.

Nishihara, H.

Noponen, E.

Ogawa, H.

Ohira, Y.

Oppliger, Y.

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

Papanicolaou, G.

See, for example, A. Bensoussan, J. L. Lions, G. Papanicolaou, “Asymptotic analysis for periodic structures,” in Study in Mathematics and Its Applications, J. L. Lions, G. Papanicolaou, eds. (North-Holland, Amsterdam, 1978), Chap. 4.

Pedersen, J.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Pommet, D. A.

Prather, D. W.

Régnault, P.

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

Rossi, M.

C. G. Blough, M. Rossi, S. K. Mack, R. L. Michaels, “Single-point diamond turning and replication of visible and near-infrared diffractive optical elements,” Appl. Opt. 36, 4648–4654 (1997).
[CrossRef] [PubMed]

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Roy, A. M.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Schmitz, M.

Schütz, H.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Schwider, J.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

Sheng, Y.

Sheridan, J. T.

H. Haidner, J. T. Sheridan, N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993).
[CrossRef] [PubMed]

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

Shiono, T.

Singer, W.

Smith, R. E.

Spitz, E.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Stauffer, J. M.

J. M. Stauffer, Y. Oppliger, P. Régnault, L. Baraldi, M. T. Gale, “Electron beam writing of continuous resist profiles for optical applications,” J. Vac. Sci. Technol. B 10, 2526–2529 (1992).
[CrossRef]

Stork, W.

Streibl, N.

H. Haidner, P. Kipfer, J. T. Sheridan, J. Schwider, N. Streibl, M. Collischon, J. Hutfless, M. Marz, “Diffraction grating with rectangular grooves exceeding 80% diffraction efficiency,” Infrared Phys. 34, 467–475 (1993).
[CrossRef]

H. Haidner, J. T. Sheridan, N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993).
[CrossRef] [PubMed]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
[CrossRef] [PubMed]

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1989).

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

Taghizadeh, M. R.

Tiziani, H.

Tschudi, T.

Turunen, J.

M. Kuittinen, J. Turunen, P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998).
[CrossRef]

E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
[CrossRef]

Vahimaa, P.

M. Kuittinen, J. Turunen, P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998).
[CrossRef]

Vawter, G. A.

Wang, L. L.

Warren, M. E.

Wendt, J. R.

Wolf, E.

The depth h is chosen such that a 2π-phase change occurs at normal incidence between two homogeneous thin films coated on a glass substrate and whose refractive indices are 1-(nmax-1)/(2N-2) and nmax+(nmax-1)/(2N-2). This phase change is straightforwardly obtained by use of the Airy formula [see M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), p. 62], for homogeneous thin films. The resulting formula for h is rather complex, and it turns out that, for the values of nmax considered hereafter, the simple expression given in Eq. (4) is valid.

Zhou, Z.

Appl. Opt. (6)

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Other (14)

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

See, for example, A. Bensoussan, J. L. Lions, G. Papanicolaou, “Asymptotic analysis for periodic structures,” in Study in Mathematics and Its Applications, J. L. Lions, G. Papanicolaou, eds. (North-Holland, Amsterdam, 1978), Chap. 4.

We believe that, with current technologies, the fabrication in glass of blazed binary diffractive elements is extremely difficult and probably impossible. Moreover, note that the use of a high-index material also has a beneficial effect on the theoretical performance (see Ref. 26).

A. F. Harvey, Microwave Engineering (Academic, London, 1963), Sect. 13.3.

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1989).

One can qualitatively understand this by considering that the pillar surrounded by the low-index material (in our case, air) may be seen as the core of a 2D waveguide. For a given grating period and from well-known results on 1D waveguides, multimode operations are obtained for large core widths, i.e., for large fill factors, by analogy. Conversely, for a given fill factor and a given grating period, it is intuitively clear that the structure supports an increasing number of modes for increasing values of the core refractive index.

In our opinion, the pillar structure offers the advantage of open ridges that are suitable for removing material during the RIE process.

The depth h is chosen such that a 2π-phase change occurs at normal incidence between two homogeneous thin films coated on a glass substrate and whose refractive indices are 1-(nmax-1)/(2N-2) and nmax+(nmax-1)/(2N-2). This phase change is straightforwardly obtained by use of the Airy formula [see M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), p. 62], for homogeneous thin films. The resulting formula for h is rather complex, and it turns out that, for the values of nmax considered hereafter, the simple expression given in Eq. (4) is valid.

The value of 66.5% was computed by electromagnetic theory and holds for a grating etched into glass (refractive index, ng=1.52), with an optimized grating depth slightly larger than λ/(ng-1), and for unpolarized light at normal incidence from air. It is 1% larger than that obtained in Fig. 1 for a grating depth equal to λ/(ng-1).

Also note that the maximum pillar aspect ratio is increased from 4.6, reported in Ref. 27, to 8.8 in this study because of the use of a smaller sampling period.

Strictly speaking, the Airy formula for thin films has to be used, as mentioned above.

The diffractive components designed along the lines of procedures 1 and 2 are weakly polarization dependent. In general, we found that the first-order diffraction efficiency is a few percent larger for TE than for TM (see, e.g., Table 1).

The computation was performed by the modal theory of Ref. 32, with square truncation. Nineteen orders along each periodicity axis were retained for the computation. No convergence problems were encountered, and the numerical results can be considered as exact. These numerical results strongly differ from those obtained by Chen and Craighead (see Fig. 1 of Ref. 24). For pillar sizes of approximately 400 nm, the zeroth-order diffraction efficiency is 65%, a value 15% smaller than that reported in Ref. 24. This difference is due to the fact that the numerical results of Ref. 24 are obtained for 16 retained orders by a slowly converging numerical method. We believe that our lower-efficiency prediction may explain the 12% discrepancy, observed by the authors of Ref. 24, between their experimental results and their numerical predictions. In our opinion, the existence of the higher-order modes is responsible for the drop in efficiency denoted by the multiplication signs; because of their oscillatory form, these modes appreciably excite the nonzero orders diffracted by the grating.

Since the width of the shadowing zone of the blazed échelette gratings decreases as the value of n increases, the performance is expected to improve with the refractive index.

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Figures (15)

Fig. 1
Fig. 1

(a) Blazed grating with a sawtooth échelette profile. (b) Blazed-index grating with a real graded index; along the period, the refractive index is linearly varying from 1 to n. The incident medium is air, the substrate is glass (refractive index, 1.52), and normal incidence from air is assumed. The concept of geometrically tracing rays through the finite depth of the gratings is used to sketch the light-shadowing effect.

Fig. 2
Fig. 2

First-order diffraction efficiency of the gratings considered in Fig. 1 as a function of the period-to-wavelength ratio for n=1.52 and for unpolarized light. Solid and dotted curves correspond to blazed-index and blazed gratings, respectively.

Fig. 3
Fig. 3

First-order diffraction efficiency of blazed-index gratings as a function of the period-to-wavelength ratio for several values of n (n=1.52, 2, 2.5) and for unpolarized light.

Fig. 4
Fig. 4

Same as in Fig. 3, except that the relative efficiency, defined as the percentage of the total transmitted light diffracted into the first order, is plotted versus Λ/λ.

Fig. 5
Fig. 5

Calibration curve. Effective index of a 2D grating composed of a 272-nm-period array of square pillars engraved in a 2.3-refractive-index material versus the fill factor.

Fig. 6
Fig. 6

Solid curves: Transmitted (0, 0)th-order diffraction efficiency of a 2D grating composed of square pillars placed on a square grid of period Λ as a function of the fill factor. The pillar height is 1.032 µm, the wavelength used is 0.6328 µm, and the pillars are assumed to be etched in a glass substrate of refractive index 1.46. ×’s, Λ=700 nm; circles, Λ=440 nm. Dotted curves: n values of all the propagating modes supported by the biperiodic structure for Λ=700 nm. The upper dotted curve (n varying between 1 and 1.46) corresponds to the grating effective index.

Fig. 7
Fig. 7

Modified calibration curve. Thin curve: same as in Fig. 5. Vertical dotted lines: limits imposed by fabrication constraints for Δ1=Δ2=80 nm. On the left-hand side (f<Δ1/Λ1) the pillar width is too small for stable fabrication. On the right-hand side (f>1-Δ2/Λ1), the spacing between two adjacent pillars is too narrow for a reliable RIE process. The central part (Δ1/Λ1<f<1-Δ2/Λ1) corresponds to fill factors effectively manufacturable. Thick curve: modified calibration curve used effectively for the design of prismlike blazed binary gratings. Fill factors larger than 1-Δ2/Λ1 are not considered, and, for 0<f<Δ1/Λ1, the continuous calibration (thin) curve is replaced by a steplike function for which only two values (1 and nmin) of the effective index are encoded.

Fig. 8
Fig. 8

Design procedure 1 for prismlike blazed binary gratings. One period of length Λ is shown. This period is divided into N intervals. The sampling period is equal to Λ1=Λ/N. The x’s at the interval centers indicate the location of the sampling points. The effective indices associated with every sampling point are denoted by n(i), i=1, 2, N and are linearly varying between 1 and nmax according to Eq. (3).

Fig. 9
Fig. 9

Theoretical performance for design procedure 1. The diffraction efficiency of blazed binary gratings with a 272-nm sampling period is considered for different period-to-wavelength ratios. Both unpolarized light and normal incidence from air are assumed for the computation. Solid curve: first-order diffraction efficiencies; circles: percentage of the total transmitted light diffracted into the first order.

Fig. 10
Fig. 10

Scanning-electron micrograph of a blazed binary subwavelength grating etched in TiO2. The period along the vertical axis is 1.9 µm, and the period along the horizontal axis is equal to the sampling period (272 nm). The grating depth is ≈816 nm, and the maximum pillar aspect ratio is ≈8.8.

Fig. 11
Fig. 11

Calibration curve used for the design of blazed binary diffractive components as a function of the fill factors. Thin curves: n values of all the propagating modes supported by a biperiodic structure composed of a 405-nm-period array of square pillars engraved in a 2.23-refractive-index material. The upper curve (n varying from 1 to 2.23) corresponds to the fundamental mode or effective index. Vertical dotted lines: limits imposed by fabrication constraints for Δ1=100 nm and Δ2=75 nm. Thick curve: modified calibration curve.

Fig. 12
Fig. 12

Theoretical performance for design procedure 2. First-order diffraction efficiency of blazed binary subwavelength gratings as a function of the normalized depth for different grating periods Λ1.4λ, 2.3λ, 3.3λ, 4.2λ, 8.0λ. The numerical values are obtained for λ=860 nm, for gratings etched in a TiO2 layer (refractive index, 2.23), for normal incidence from air, and for a 405-nm sampling period.

Fig. 13
Fig. 13

First-order diffraction efficiency of blazed binary subwavelength gratings as a function of Λ1/λ for Λ=1.41λ, 4.23λ. The numerical values are obtained for λ=860 nm and for gratings etched in a TiO2 material (refractive index, 2.23). The gratings are designed according to procedure 2 with Δ1=0. For each sampling period, Δ2 is chosen so that nmax is equal to 1.87. The two horizontal dotted lines correspond to the first-order diffraction efficiencies of two blazed-index gratings for n=1.87 and for Λ=1.41λ, 4.23λ.

Fig. 14
Fig. 14

Scanning-electron micrograph (located not far from a corner) of the off-axis diffractive lens.

Fig. 15
Fig. 15

Spot profiles measured at a rear distance of 400 mm with a 400-µm-diameter photodiode. Solid curves are fits by Gaussian functions.

Tables (1)

Tables Icon

Table 1 Diffraction Efficiencies (in Percent) of the Different Transmitted Orders for the Grating Shown in Fig. 10a

Equations (6)

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

wa/Λ=1n(n-1)λΛ2.
wb/Λ=12n(n-1)λΛ2,
n(i)=nmax-1N-1(i-1)+1.
h=N-1Nλnmax-1.
h=ϕM2πλnmax-1.
nij=(nmax-1) ϕijϕM+1.

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