Abstract

It is known that the zero-order two-phase-level gratings with a period much smaller and a thickness much larger than the wavelength may have antireflection properties the same as appropriate dielectric layers under normal incidence. On the basis of the rigorous coupled-wave analysis method formulation, it is shown that multilevel unidimensional phase gratings, for both TE and TM polarization, are functionally equivalent to antireflection structures of multilevel dielectric layers, even if the period is close to the wavelength.

© 1997 Optical Society of America

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References

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  1. Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection effect in ultra-high spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1983).
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  3. 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).
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  4. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
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  5. E. B. Grann, M. G. Moharam, “Comparison between continuous and discrete subwavelength grating structures for antireflection surfaces,” J. Opt. Soc. Am. A 13, 988–992 (1996).
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  6. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
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  7. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
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  9. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
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    [CrossRef]

1996 (1)

1995 (2)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (1)

1982 (1)

1981 (1)

Gaylord, T. K.

Grann, E. B.

Gunning, W. J.

Haidner, H.

Kimura, Y.

Kipfer, P.

Moharam, M. G.

Morris, G. M.

Motamedi, M. E.

Nishida, N.

Ohta, Y.

Ono, Y.

Pommet, D. A.

Raguin, D. H.

Southwell, W. H.

Stork, W.

Streibl, N.

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

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

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Multilayer dielectric coating and (b) equivalent multilevel phase grating. Because the duty cycle must increase as we approach the substrate, the condition of Eq. (2) must be satisfied.

Fig. 2
Fig. 2

Reflected zeroth-order diffraction efficiency of a multilayer antireflection dielectric coating with two, three, and four layers as a function of d0. The external index is n0 = 1.0 and the substrate index is ns = 1.5125. The values of intermediate indices are given by Eq. (4).

Fig. 3
Fig. 3

Reflected zeroth-order diffraction efficiency of a single-layer phase grating, as a function of Λ/λ0, for n0 = 1.0, ns = 1.5, θ = 0°, f = 0.5, TE polarization, and normal incidence. When Λ/λ0 is close to 2/3, the cutoff limit, the effective index is not independent of d0, as occurs in simple dielectric layers. If Λ/λ0 is far from cutoff, the effective index becomes constant with d0. The curve for Λ/λ0 = 0.1 resembles the curve for the dielectric layer with n = 1.282.

Fig. 4
Fig. 4

Effective index of refraction of a grating as a function of d0, for different values of Λ/λ0. For sufficiently high values of d0, the index becomes independent of d0. We can notice also that this variation with d0 does not occur for the values of Λ/λ0 far from the cutoff limit (Λ/λ0 = 2/3).

Fig. 5
Fig. 5

Single-layered grating surrounded by media with indices (n0 and n3) different from the grating indices (n1 and n2).

Fig. 6
Fig. 6

Reflected diffraction efficiency of the zeroth-order of the grating depicted in Fig. 5, far from cutoff, with Λ/λ0 = 0.1. The indices n0 = 1.0, n1 = 1.0, and n2 = 1.5 are fixed. As n3 increases from 1.3 to 1.6, the amplitude of the diffraction efficiency changes; the period of the curves and, consequently, the effective index of the grating remain constant.

Fig. 7
Fig. 7

Reflected diffraction efficiency of the zeroth order of the grating depicted in Fig. 5, in the cutoff limit, with Λ/λ0 = 2/3. The other parameters are the same as those in Fig. 6. The effective index now varies with both d0 and n3.

Fig. 8
Fig. 8

Reflected diffraction efficiency of the zeroth order for the designed three-layer grating. The reflectance of the equivalent three-layer dielectric coating is also shown for comparison.

Fig. 9
Fig. 9

Sawtooth multilevel phase grating profile. The ith layer duty cycle is the same as in Fig. 1(b).

Equations (6)

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Λc=λ0ns-n0sin θ,
n0<n1<n2<<nN<ns.
ni2=ni-1ni+1,  i=1, , N,  nN+1=ns.
n1N+1=n0Lns,  ni=n1i/n0i-1i=2, ,  N.
d1n1=d2n2==dNnN=λ0/4,  di=d,
di=dj=0,jiNninj+1.

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