Abstract

Zero-order gratings are grating structures with a period that is small compared with the wavelength of light. Only the directly transmitted or reflected light, the zero diffraction order, is nonevanescent and propagates in a distance from the grating. Thus the grating behaves like a slab of ordinary homogeneous material with an effective refractive index. By varying the material composition, i.e., by variation of the duty cycle of the grating, the effective refractive index can be changed. A grating with variable duty cycle therefore behaves like a material with distributed index. Based on such artificial materials, distributed-index elements are proposed. The physical principle is demonstrated with water waves.

© 1991 Optical Society of America

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References

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  1. C. G. Bernhardt, Endeavour 26, 79 (1967).
  2. S. J. Wilson, M. C. Hutley, Opt. Acta 29, 993 (1982).
    [CrossRef]
  3. M. T. Gale, Opt. Commun. 18, 292 (1976).
    [CrossRef]
  4. K. Knop, Opt. Commun. 18, 298 (1976).
    [CrossRef]
  5. D. C. Flanders, Appl. Phys. Lett. 42, 492 (1983).
    [CrossRef]
  6. L. C. Cescato, E. Gluch, N. Streibl, Appl. Opt. 29, 3286 (1990).
    [CrossRef] [PubMed]
  7. W. Southwell, J. Opt. Soc. Am. A 8, 549 (1991).
    [CrossRef]
  8. T. Ishikawa, K. Tada, Jpn. J. Appl. Phys. 28, 1982 (1989).
    [CrossRef]
  9. E. Keilmann, German patent OffenlegungsschriftDE 3707984 Al (September22, 1988).
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 14.5.2.
  11. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 165.
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 1.6.4.

1991 (1)

1990 (1)

1989 (1)

T. Ishikawa, K. Tada, Jpn. J. Appl. Phys. 28, 1982 (1989).
[CrossRef]

1983 (1)

D. C. Flanders, Appl. Phys. Lett. 42, 492 (1983).
[CrossRef]

1982 (1)

S. J. Wilson, M. C. Hutley, Opt. Acta 29, 993 (1982).
[CrossRef]

1976 (2)

M. T. Gale, Opt. Commun. 18, 292 (1976).
[CrossRef]

K. Knop, Opt. Commun. 18, 298 (1976).
[CrossRef]

1967 (1)

C. G. Bernhardt, Endeavour 26, 79 (1967).

Bernhardt, C. G.

C. G. Bernhardt, Endeavour 26, 79 (1967).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 1.6.4.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 14.5.2.

Cescato, L. C.

Flanders, D. C.

D. C. Flanders, Appl. Phys. Lett. 42, 492 (1983).
[CrossRef]

Gale, M. T.

M. T. Gale, Opt. Commun. 18, 292 (1976).
[CrossRef]

Gluch, E.

Hutley, M. C.

S. J. Wilson, M. C. Hutley, Opt. Acta 29, 993 (1982).
[CrossRef]

Ishikawa, T.

T. Ishikawa, K. Tada, Jpn. J. Appl. Phys. 28, 1982 (1989).
[CrossRef]

Keilmann, E.

E. Keilmann, German patent OffenlegungsschriftDE 3707984 Al (September22, 1988).

Knop, K.

K. Knop, Opt. Commun. 18, 298 (1976).
[CrossRef]

Southwell, W.

Streibl, N.

Tada, K.

T. Ishikawa, K. Tada, Jpn. J. Appl. Phys. 28, 1982 (1989).
[CrossRef]

Wilson, S. J.

S. J. Wilson, M. C. Hutley, Opt. Acta 29, 993 (1982).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 1.6.4.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 14.5.2.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 165.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 165.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. C. Flanders, Appl. Phys. Lett. 42, 492 (1983).
[CrossRef]

Endeavour (1)

C. G. Bernhardt, Endeavour 26, 79 (1967).

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

Jpn. J. Appl. Phys. (1)

T. Ishikawa, K. Tada, Jpn. J. Appl. Phys. 28, 1982 (1989).
[CrossRef]

Opt. Acta (1)

S. J. Wilson, M. C. Hutley, Opt. Acta 29, 993 (1982).
[CrossRef]

Opt. Commun. (2)

M. T. Gale, Opt. Commun. 18, 292 (1976).
[CrossRef]

K. Knop, Opt. Commun. 18, 298 (1976).
[CrossRef]

Other (4)

E. Keilmann, German patent OffenlegungsschriftDE 3707984 Al (September22, 1988).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 14.5.2.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 165.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1985), Sec. 1.6.4.

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Figures (4)

Fig. 1
Fig. 1

Schematic of a binary surface-relief grating with period p and duty cycle γ at the interface of two materials with refractive indices n1 and n2.

Fig. 2
Fig. 2

Dependence of the effective refractive index for ordinary (TE) and extraordinary (TM) waves within a binary grating on the duty cycle. Dashed curves, small-period approximation of Eqs. (2). Solid curves, calculation from Bloch waves [Eq. (5)]. The grating was assumed to have a period p = 1.5 μm, the wavelength is λ = 10.6 μm, and the materials are air (n1 = 1) and silicon (n2 = 3.4). Normal incidence on the grating (φ = 0° is assumed.

Fig. 3
Fig. 3

Side view of the zero-order grating for water waves. The duty cycle increases in such a way that a linear increase in phase is expected along the component. Thus an index gradient or a microprism is emulated.

Fig. 4
Fig. 4

Experimental result showing the deflection of water waves by the component shown in Fig. 3.

Equations (7)

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p λ / ( 2 n 1 ) ,             p λ / ( 2 n 2 ) .
n exo 2 = n 1 2 n 2 γ n 1 2 + ( 1 - γ ) n 2 2 , n ord 2 = γ n 2 2 + ( 1 - γ ) n 1 2 .
u ( x , z , t ) = τ ( x ) exp ( i K x ) exp ( i β z - i ω t ) ,
cos ( K p ) = cos ( k 1 γ p ) cos [ k 2 ( 1 - γ ) p ] - 1 / 2 ( j 1 / j 2 + j 2 / j 1 ) sin ( k 1 γ p ) sin [ k 2 ( 1 - γ ) p ] ,
n eff 2 = ( λ / 2 π K ) 2 + cos 2 φ .
h wave = λ / ( n 2 - n 1 ) .
n eff 2 = d o [ γ / d 1 + ( 1 - γ ) / d 2 ] .

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