Abstract

The diffraction efficiencies of practical dielectric holograms are evaluated with rigorous coupled-wave analysis. The cases of the hologram surfaces eroded in several shapes are treated and compared with those in which the surfaces are not eroded and with those in which there are only surface gratings. Eroding the surface will increase the higher-order reflection diffraction efficiencies and the transmissions, thus reducing the first reflection and the zeroth transmission. However, sealing the hologram with a cover plate, as is done in manufacturing many holograms, extinguishes the erosion effect.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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1998 (1)

1995 (1)

1994 (2)

1993 (1)

1991 (1)

1985 (1)

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

1982 (1)

1981 (2)

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

S. Sjölinder, “Dichromated gelatin and the mechanism of hologram formation,” Photogr. Sci. Eng. 25, 112–118 (1981).

1977 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1966 (2)

Awada, K. A.

Burckhardt, C. B.

Campbell, G.

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Gaylord, T. K.

Kamiya, N.

Kim, T. J.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kostuk, R. K.

Kozma, A.

Leith, E. N.

Li, Lifeng

Loewen, E. G.

Marks, J.

Massey, N.

Maystre, D.

Moharam, M. G.

Nevière, M.

Pai, D. M.

Rallison, R. D.

R. D. Rallison, S. R. Schicker, “Polarization propaties of gelatin holograms,” in Practical Holography VI, S. A. Benton, ed., Proc. SPIE1667, 266–275 (1992).
[CrossRef]

Schicker, S. R.

R. D. Rallison, S. R. Schicker, “Polarization propaties of gelatin holograms,” in Practical Holography VI, S. A. Benton, ed., Proc. SPIE1667, 266–275 (1992).
[CrossRef]

Sheridan, J. T.

Sjölinder, S.

S. Sjölinder, “Dichromated gelatin and the mechanism of hologram formation,” Photogr. Sci. Eng. 25, 112–118 (1981).

Solymar, L.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Syms, R. R. A.

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

Upatnieks, J.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am. (3)

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

Photogr. Sci. Eng. (1)

S. Sjölinder, “Dichromated gelatin and the mechanism of hologram formation,” Photogr. Sci. Eng. 25, 112–118 (1981).

Proc. IEEE (1)

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

Other (3)

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

R. D. Rallison, S. R. Schicker, “Polarization propaties of gelatin holograms,” in Practical Holography VI, S. A. Benton, ed., Proc. SPIE1667, 266–275 (1992).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Example of atomic-force microscope photograph of a DCG hologram surface developed through a wet process. The viewed area is 20 μm × 25 μm, and the height scale is 0–140 nm. The grating slant angle is less than 5 deg. The observed surfacelooks nearly sinusoidal in shape, and the depth is estimated as 10–20 nm. Some peaks on the surface are probably dust.

Fig. 2
Fig. 2

Holographic grating, including the surface region.

Fig. 3
Fig. 3

(a) Division of surface region 2. (b) Permittivity distribution in divided subregion 2, g.

Fig. 4
Fig. 4

Vector diagram of incident light and diffracted light.

Fig. 5
Fig. 5

Exposure model geometry used in the calculations.

Fig. 6
Fig. 6

Diffraction efficiencies for eroded-region thicknesses of 20, 50, and 100 nm and sinusoidally shaped surfaces, without cover plates, compared with those for noneroded surfaces. In noneroded surfaces, region 2 does not exist, and the hologram thickness is 20 μm: (a) reflection, (b) transmission.

Fig. 7
Fig. 7

Values of the internal amplitude S l,(g),j (z) S l,(g),j *(z) for a sinusoidally shaped eroded hologram, 100-nm erosion, without a cover plate, compared with a noneroded hologram. In the noneroded hologram, region 2 does not exist, and the hologram thickness is 20 μm. The incident angle in air is 30 deg.: (a) along the entire thickness, (b) near the surface region only. Values may be greater than 1 because the coefficient ξ1,j 1,0 or ξ4,j 1,0 was not applied.

Fig. 8
Fig. 8

Diffraction efficiencies for eroded-surface shape variation, 100-nm erosion, without a cover plate: (a) reflection, (b) transmission.

Fig. 9
Fig. 9

Normalized coefficients of coupling factors ε̅2,g /ε̅3andΔε c,2,g,h /ε̅3 for h = 1–3 in the eroded region. The rectangular shape has the largest Δε c,2,g,1/ε̅3 through the entire region 2.

Fig. 10
Fig. 10

Diffraction efficiencies for the eroded-surface shape variation, 100-nm erosion, surface region only, without a cover plate: (a) reflection, (b) transmission.

Fig. 11
Fig. 11

Diffraction efficiencies for eroded-surface shape variation, 100-nm erosion, with a cover plate: (a) reflection, (b) transmission.

Equations (8)

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ε 2 ,   g x = n 2 , g x 2 = ε ¯ 2 , g + h = 1 Δ ε c , 2 , g , h cos 2 π hx / p + Δ ε s , 2 , g , h sin 2 π hx / p , ε ¯ 2 ,   g = 1 p 0 p   ε 2 , g x d x , Δ ε c , 2 , g , h = 2 p 0 p   ε 2 , g x cos 2 π hx / p d x , Δ ε s , 2 , g , h = 2 p 0 p   ε 2 , g x sin 2 π hx / p d x   h = 1 ,   2 , ,
ε 2 , g x = ε ¯ 2 , g + 1 2 h = 1 Δ ε c , 2 , g , h + i Δ ε s , 2 , g , h × exp - ihK 3 , g sin   ϕ 3 , g x + Δ ε c , 2 , g , h - i Δ ε s , 2 , g , h × exp ihK 3 , g sin   ϕ 3 , g x .
ε 3 K 3 · r = ε ¯ 3 + 1 2 h = 1 Δ ε c , 3 , h + i Δ ε s , 3 , h × exp - ih K 3 · r + Δ ε c , 3 , h - i Δ ε s , 3 , h × exp ih K 3 · r .
E 1 = j = - Ref j exp - i - ξ 1 , j z + β j x + exp - i ξ 1,0 z + β 0 x , E 2 ,   g = j = -   S 2 , g , j z exp - i ξ 2 , g , j z - g = 1 g - 1   t 2 , g + β j x + g = 1 g - 1   ξ 2 , g , j t 2 , g ¯ , E 3 = j = -   S 3 , j z exp - i ξ 3 , j z - T 2 + β j x + g = 1 G 2   ξ 2 ,   g ,   j t 2 ,   g ¯ , E 4 = j = - Tra j exp - i ξ 4 ,   j z - T 2 - T 3 + β j x + g = 1 G 2   ξ 2 , g , j t 2 , g + ξ 3 , j T 3 ¯ ,
β j = k 1 sin   θ 1 - jp , = k 1 sin   θ 1 - jK 3 , g sin   ϕ 3 , g independent   of   g , ξ l , j = k l 2 - β j 2   l = 1 ,   4 ;   region   number , ξ 2 , g , j = k 2 , g cos   θ 2 , g   l = 2 , ξ 3 , j = k 3 cos   θ 3 - jK 3 cos   ϕ 3     l = 3 , n 1 sin   θ 1 = n ¯ l ,   g sin   θ l ,   g , n ¯ l , g = ε ¯ l , g   l = 2 ,   3 , k l = 2 π n l λ   l = 1 ,   4 , k l , g = 2 π n ¯ l , g λ   l = 2 ,   3 ,
2 E + k 2 ε E = 0 ,
d 2 d z 2   S l , g , j z - 2 i ξ l , g , j d d z   S l , g , j z + k l , g 2 B l , g , j S l , g , j z + k l , g 2 2 h = 1 Δ ε c , l , g , h + i Δ ε s , l , g , h S l , g , j - h z + Δ ε c , l , g , h - i Δ ε s , l , g , h S l , g , j + h z = 0 ,
η 1 ,   j = ξ 1 ,   j ξ 1,0 Ref j   Ref j * = ξ 1 ,   j ξ 1,0   S 2,1 ,   j 0 S 2,1 ,   j * 0 , η 4 ,   j = ξ 4 ,   j ξ 1,0 Tra j   Tra j * = ξ 4 ,   j ξ 1,0   S 3 ,   j T 2 + T 3 S 3 ,   j * T 2 + T 3 ,

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