Abstract

Diffraction of plane waves by dielectric surface-relief gratings with conic (elliptic, hyperbolic, parabolic) cross-sectional grating shapes is analyzed, using a simple transmittance theory. Integral expressions for the complex amplitudes of the diffracted waves are given for these structures. The calculated results indicate new types of diffraction behavior compared with corresponding results for the commonly studied grating shapes (sinusoidal, rectangular, triangular). In particular, these structures are shown to exhibit a large number of diffracted waves with similar intensities. It is also shown that a bandpasslike behavior (a variable number of fairly even-intensity orders with reasonably sharp cutoff) is obtainable.

© 1989 Optical Society of America

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References

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  1. R. Magnusson, T. K. Gaylord, “Diffraction regimes of transmission gratings,”J. Opt. Soc. Am. 68, 809–814 (1978).
    [CrossRef]
  2. R. Magnusson, T. K. Gaylord, “Diffraction efficiencies of thin phase gratings with arbitrary grating shape,”J. Opt. Soc. Am. 68, 806–809 (1978).
    [CrossRef]
  3. L. P. Boivin, “Multiple imaging using various types of simple phase gratings,” Appl. Opt. 11, 1782–1792 (1972).
    [CrossRef] [PubMed]
  4. W. H. Lee, “High efficiency multiple beam gratings,” Appl. Opt. 18, 2152–2158 (1979).
    [CrossRef] [PubMed]
  5. H. Machida, J. Nitta, A. Seko, H. Kobayashi, “High-efficiency fiber grating for producing multiple beams of uniform intensity,” Appl. Opt. 23, 330–332 (1984).
    [CrossRef] [PubMed]
  6. R. Magnusson, D. Shin, “Diffraction by periodic arrays of dielectric cylinders,” J. Opt. Soc. Am. A 6, 412–414 (1989).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman–Nath diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
    [CrossRef]
  8. T. K. Gaylord, M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271–3273 (1981).
    [CrossRef] [PubMed]
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    [CrossRef]

1989 (1)

1984 (1)

1982 (1)

1981 (1)

1980 (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman–Nath diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

1979 (1)

1978 (2)

1972 (1)

Boivin, L. P.

Gaylord, T. K.

Kobayashi, H.

Lee, W. H.

Machida, H.

Magnusson, R.

Moharam, M. G.

Nitta, J.

Seko, A.

Shin, D.

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Figures (7)

Fig. 1
Fig. 1

Geometry of a dielectric surface-relief grating. The normally incident plane wave is diffracted into multiple output waves.

Fig. 2
Fig. 2

Example of a single-period structure used in this analysis.

Fig. 3
Fig. 3

Diffraction efficiency of a surface-relief grating with elliptic grating shape for c = 1 as a function of the order of diffraction i and the modulation parameter g.

Fig. 4
Fig. 4

Diffraction efficiency of a surface-relief grating with elliptic grating shape as a function of the order index i with varying c for g = 25.

Fig. 5
Fig. 5

Diffraction efficiency of a surface-relief grating with elliptic grating shape as a function of the modulation parameter g for several of the lowest diffracted orders for c = 2.

Fig. 6
Fig. 6

Diffraction efficiency of a surface-relief grating with hyperbolic grating shape as a function of the order number i and the modulation parameter g for C = 1.

Fig. 7
Fig. 7

Diffraction efficiency of a surface-relief grating with parabolic grating shape as a function of the order number i and the modulation parameter g.

Tables (2)

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Table 1 Diffraction by Surface-Relief Gratings with Conic Cross-Sectional Grating Shapes

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Table 2 Diffraction Efficiencies of Gratings with Classical Grating Shapes

Equations (7)

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ϕ ( x ) = k ( n c - n 0 ) Δ ( x ) + k n 0 d ,
T ( x ) = t ( x ) * l = - δ ( x - l Λ ) .
T ( x ) = i = - S i exp ( j i K x ) ,
S i = 1 Λ - Λ / 2 Λ / 2 t ( x ) exp ( - j i K x ) d x .
η i = S i S i * .
i = - η i = 1 ,
g = π Δ n d / λ ,

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