Abstract

Blazed-binary optical elements with only binary ridges or pillars are diffractive components that mimic standard blazed-échelette diffractive elements. We report on the behavior of one-dimensional blazed-binary optical elements with local periods much larger than the wavelength. For this purpose, an approximate model based on both scalar and electromagnetic theory is proposed. The model is tested against electromagnetic-theory computational results obtained for one-dimensional blazed-binary gratings with large periods. An excellent agreement is obtained, showing that the model is able to predict quantitatively the wavelength and the incidence-angle dependences of the diffraction efficiency of blazed-binary structures.

© 2000 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16, 1143–1156 (1999).
    [CrossRef]
  4. Ph. Lalanne, “Waveguiding in blazed-binary diffractive elements,” J. Opt. Soc. Am. A 16, 2517–2520 (1999).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 8, p. 387.
  6. 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]
  7. 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]
  8. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.
  9. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

1999 (2)

1998 (1)

1996 (1)

1995 (1)

1991 (1)

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Astilean, S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 8, p. 387.

Cambril, E.

Chavel, P.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.

Gaylord, T. K.

Grann, E. B.

Haidner, H.

Kipfer, P.

Lalanne, Ph.

Launois, H.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Stork, W.

Streibl, N.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 8, p. 387.

J. Opt. Soc. Am. A (4)

Opt. Lett. (2)

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (2)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1989), Chap. 3.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Macmillan, New York, 1964), Chap. 8, p. 387.

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Figures (8)

Fig. 1
Fig. 1

Encoding a phase transfer function with blazed-binary structures: (a) unwrapped phase transfer function, (b) phase transfer function of the corresponding diffractive optical element, and (c) corresponding blazed-binary structure.

Fig. 2
Fig. 2

Modulus of the normalized amplitude of the zero-transmitted order for three sampling periods [Λs=λ0/2 (solid curve), Λs=λ0/5 (dashed curve), and Λs=λ0/10 (dotted curve)] and for TM polarization. Because normal incidence from the glass substrate at the nominal wavelength is assumed for the computation, the argument of T0(ϕ) is simply ϕ in this specific case.

Fig. 3
Fig. 3

One-dimensional blazed-binary gratings for TM polarization. First-transmitted-order efficiency as a function of the period-to-wavelength ratio for a normal incidence from the glass substrate. The horizontal lines correspond to the asymptotic predictions for Λ/λ0.

Fig. 4
Fig. 4

Same as Fig. 3 but for TE polarization.

Fig. 5
Fig. 5

First-transmitted-order diffraction efficiency versus angle of incidence θ1 in glass for TM polarization. Solid (Λs=λ0/2) and dashed (Λs=λ0/5) curves, asymptotic predictions of the model for Λ/λ0. Pluses and crosses, RCWA computational results obtained for 50λ0-period blazed-binary gratings with Λs=λ0/2 and λ0/5, respectively.

Fig. 6
Fig. 6

Same as Fig. 5 but for TE polarization.

Fig. 7
Fig. 7

First-transmitted-order diffraction efficiency versus normalized wavelength λ/λ0 for TM polarization. Solid (Λs=λ0/2) and dashed (Λs=λ0/5) curves, asymptotic predictions of the model for Λ/λ0. Pluses and crosses, RCWA computational results obtained for 50λ0-period blazed-binary gratings with Λs=λ0/2 and λ0/5, respectively.

Fig. 8
Fig. 8

Same as Fig. 7 but for TE polarization.

Tables (2)

Tables Icon

Table 1 TE Polarization. Comparison of the Asymptotic Values η1 of the First-Transmitted-Order Efficiency Predicted by the Approximate Model and the Extrapolated Values a0 Obtained from the RCWA Results of Fig. 3 a

Tables Icon

Table 2 Same As Table 1 Except for TE Polarization and RCWA Values Obtained from Fig. 4

Equations (8)

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

t(ϕ)=-+cn exp(jnϕ).
cn=12π 02πt(ϕ)exp(-jnϕ)dϕ,
ηn=|cn|2.
Ai=exp[-jk0n1(sin θ1x+cos θ1z)],
At=T0(ϕ)exp[-jk0(n1 sin θ1x+n3 cos θ3z)].
t(ϕ)=n3 cos θ3n1 cos θ11/2T0(ϕ).
t(ϕ)=n1 cos θ3n3 cos θ11/2T0(ϕ).
η1λ0Λ=i=0N-1aiλ0Λi,

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