Abstract

A new and useful method for obtaining diffraction efficiencies from holograms manufactured practically is presented. Applying the rigorous coupled-wave analysis, we express each difference between the practical and the ideal as a mathematical component that can be easily integrated. In Part 1 the effects due to thickness change in the hologram layer (observed frequently after the development process) are treated. Although uniform swelling or shrinking causes a simple reconstruction wavelength or incidence-angle shift, nonuniform thickness extends the capacity of the Bragg condition matching, creating a diffraction efficiency curve in the asymmetric profile. Other characteristics of diffraction are also maintained. A refractive-index change has an effect that is similar to the thickness change. Higher-order terms in permittivity modulation create negligible effects in general holograms when used at or near the simple first-order Bragg condition.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  5. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
    [CrossRef]
  6. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
    [CrossRef]
  30. L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
    [CrossRef]
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    [CrossRef]

1995

1994

1992

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

A. Beléndez, I. Pascual, A. Fimia, “Model for analyzing the effects of processing on recording material in thick holograms,” J. Opt. Soc. Am. A 9, 1214–1223 (1992).
[CrossRef]

1991

1987

L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

1985

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

1981

1980

P. Hariharan, “Pseudocolor images with volume reflection holograms,” Opt. Commun. 35, 42–44 (1980).
[CrossRef]

1979

1978

1977

1976

R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
[CrossRef]

P. Hariharan, “Longitudinal distortion in images reconstructed by reflection holograms,” Opt. Commun. 17, 52–54 (1976).
[CrossRef]

1974

1973

1971

1970

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

1969

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

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[CrossRef]

1967

1966

Alferness, R.

Au, L. B.

L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

Awada, K. A.

Beléndez, A.

Benton, S. A.

J. L. Walker, S. A. Benton, “In-situ swelling for holographic color control,” in Practical Holography III, S. A. Benton, ed., Proc. SPIE1051, 192–199 (1989).

Burckhardt, C. B.

Campbell, G.

Chateau, N.

Chu, R. S.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

Coleman, D. J.

Cooke, D. J.

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

Fillmore, G. L.

Fimia, A.

Gaylord, T. K.

Hariharan, P.

P. Hariharan, “Pseudocolor images with volume reflection holograms,” Opt. Commun. 35, 42–44 (1980).
[CrossRef]

P. Hariharan, “Longitudinal distortion in images reconstructed by reflection holograms,” Opt. Commun. 17, 52–54 (1976).
[CrossRef]

Hugonin, J.-P.

Kaspar, F. G.

Kermisch, D.

Kim, T. J.

Kogelnik, H.

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

Kong, J. A.

Kostuk, R. K.

Kozma, A.

Kubota, T.

Leith, E. N.

Lin, L. H.

LoBianco, C. V.

Magariños, J.

Magnusson, R.

Marks, J.

Massey, N.

McGrew, S. P.

S. P. McGrew, “Color control in dichromated gelatin reflection holograms,” in Recent Advances in Holography, P. N. Tamura, ed., Proc. SPIE215, 24–31 (1980).
[CrossRef]

Moharam, M. G.

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

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

Newell, J. C. W.

L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

Ose, T.

Pai, D. M.

Pascual, I.

Rigrod, W. W.

Sheridan, J. T.

J. T. Sheridan, “Generalization of the boundary diffraction method for volume gratings,” J. Opt. Soc. Am. A 11, 649–656 (1994).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

Solymar, L.

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

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).

Tamir, T.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

Tynan, R. F.

Uchida, N.

Upatnieks, J.

Walker, J. L.

J. L. Walker, S. A. Benton, “In-situ swelling for holographic color control,” in Practical Holography III, S. A. Benton, ed., Proc. SPIE1051, 192–199 (1989).

Appl. Opt.

Bell Syst. Tech. J.

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

IEEE Trans. Microwave Theory Tech.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

J. Mod. Opt.

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

L. B. Au, J. C. W. Newell, L. Solymar, “Nonuniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

J. Opt. Soc. Am.

C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
[CrossRef]

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[CrossRef]

G. L. Fillmore, R. F. Tynan, “Sensitometric characteristics of hardened dichromated-gelatin films,” J. Opt. Soc. Am. 61, 199–203 (1971).
[CrossRef]

F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with a complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
[CrossRef]

N. Uchida, “Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector,” J. Opt. Soc. Am. 63, 280–287 (1973).
[CrossRef]

W. W. Rigrod, “Diffraction efficiency of nonsinusoidal Bragg reflection gratings,” J. Opt. Soc. Am. 64, 97–99 (1974).
[CrossRef]

R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
[CrossRef]

J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Equivalence of multiwave coupled-wave theory and modal theory for periodic-media diffraction,” J. Opt. Soc. Am. 68, 1777–1779 (1978).
[CrossRef]

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

J. Opt. Soc. Am. A

Opt. Acta

T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
[CrossRef]

Opt. Commun.

P. Hariharan, “Pseudocolor images with volume reflection holograms,” Opt. Commun. 35, 42–44 (1980).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in dielectric gratings of the reflection type: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

P. Hariharan, “Longitudinal distortion in images reconstructed by reflection holograms,” Opt. Commun. 17, 52–54 (1976).
[CrossRef]

Proc. IEEE

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

Other

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

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

J. L. Walker, S. A. Benton, “In-situ swelling for holographic color control,” in Practical Holography III, S. A. Benton, ed., Proc. SPIE1051, 192–199 (1989).

S. P. McGrew, “Color control in dichromated gelatin reflection holograms,” in Recent Advances in Holography, P. N. Tamura, ed., Proc. SPIE215, 24–31 (1980).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Definition of the holographic grating used in this paper: (a) before and (b) after thickness change and shear effect during processing, (c) relationship between the grating parameters before and after processing.

Fig. 2
Fig. 2

Vector diagram of incident light and diffracted light.

Fig. 3
Fig. 3

Exposure model geometry used in the calculations.

Fig. 4
Fig. 4

Diffraction efficiencies of each reflection order from an unaffected standard hologram: (a) with cover plate, (b) without cover plate.

Fig. 5
Fig. 5

Diffraction efficiencies of each transmission order from an unaffected standard hologram: (a) with cover plate, (b) without cover plate.

Fig. 6
Fig. 6

Relations between the first reflection diffraction and swelling rate m: (a), (b) Bragg condition matching parameter B 3,1,1 and diffraction efficiency with constant m; (c), (d) m and diffraction efficiency while m is expressed as a linear function; (e)–(h) m and diffraction efficiency while m is expressed as a parabolic function with (e), (f) G 3 = 3 and (g), (h) G 3 = 10.

Fig. 7
Fig. 7

Relationships between the first reflection diffraction and mean index 3: (a) Bragg condition matching parameter B 3,1,1 and (b) diffraction efficiency with constant 3.

Fig. 8
Fig. 8

Influence of higher-order permittivity modulation on reflection diffractions: (a) normal grating period with Δε c,3,1 = 0.05 (standard); (b) normal grating period with Δε c,3,1 = 0.05 and Δε c,3,2 = 0.025; (c) half-grating period with Δε c,3,1 = 0.05; (d) half-grating period with Δε c,3,1 = 0.05 and Δε c,3,2 = 0.025.

Fig. 9
Fig. 9

Absolute error from unity in the case of a swollen hologram, shown as a dotted curve in Fig. 6(d).

Equations (22)

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d 3 , g cos   ϕ 3 , g = M e 3 , g d o cos   ϕ o ,     d 3 , g sin   ϕ 3 , g = d o sin   ϕ o ,
M e 3 , g M 3 , g 1 - M 3 , g tan   ϕ o   tan   δ 3 , g .
M p 3 , g d 3 , g d o = M e 3 , g cos 2   ϕ o + M e 3 , g 2 sin 2   ϕ o 1 / 2 ,
cos   ϕ 3 , g = M p 3 , g M e 3 , g cos   ϕ o , sin   ϕ 3 , g = M p 3 , g sin   ϕ o , K 3 , g = K o M p 3 , g ,
K 3 , g = K 3 , g cos   ϕ 3 , g ,   sin   ϕ 3 , g = K o 1 M e 3 , g cos   ϕ o ,   sin   ϕ o .
T 3 = g = 1 G 3   t 3 , g ,     where   t 3 , g M 3 , g T o G 3 ,
ε 3 , g K 3 , g · r = n 3 , g K 3 , g · r 2 = ε ¯ 3 , g + h = 1 Δ ε c , 3 , g , h cos h K 3 , g · r + Δ ε s , 3 , g , h sin h K 3 , g · r , ε ¯ 3 , g = 1 d 3 , g 0 d 3 , g   ε 3 , g K 3 , g · r d K 3 , g · r , Δ ε c , 3 , g , h = 2 d 3 , g 0 d 3 , g   ε 3 , g K 3 , g · r cos h K 3 , g · r × d K 3 , g · r , Δ ε s , 3 , g , h = 2 d 3 , g 0 d 3 , g   ε 3 , g K 3 , g · r sin h K 3 , g · r × d K 3 , g · r h = 1 ,   2 , .
ε 3 , g K 3 , g · r = ε ¯ 3 , g + 1 2 h = 1 Δ ε c , 3 , g , h + i Δ ε s , 3 , g , h × exp - ih K 3 , g · r + Δ ε c , 3 , g , h - i Δ ε s , 3 , g , h × exp ih K 3 , g · 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 3 , g = j = -   S 3 , g , j z exp - i ξ 3 , g , j z - g = 1 g - 1   t 3 , g + β j x + g = 1 g - 1   ξ 3 , g , j t 3 , g ¯ , E 4 = j = - Tra j exp - i ξ 4 , j z - T 3 + β j x + g = 1 G 3   ξ 3 , g , j t 3 , g ¯ ,
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 1 / 2 l = 1 ,   4 ;   region   number , ξ 3 , g , j = k 3 , g cos   θ 3 , g - jK 3 , g cos   ϕ 3 , g , n 1 sin   θ 1 = n ¯ 3 , g sin   θ 3 , g ,   n ¯ 3 , g = ε ¯ 3 , g , k l = 2 π n l λ     l = 1 ,   4 , k 3 , g = 2 π n ¯ 3 , g λ ,
2 E + k 2 ε E = 0 ,
d 2 d z 2   S 3 , g , j z - 2 i ξ 3 , g , j d d z   S 3 , g , j z + k 3 , g 2 B 3 , g , j S 3 , g , j z + k 3 , g 2 2 h = 1 Δ ε c , 3 , g , h + i Δ ε s , 3 , g , h S 3 , g , j - h z + Δ ε c , 3 , g , h - i Δ ε s , 3 , g , h S 3 , g , j + h z = 0 ,
a 3 , g , h = - k 2 2 Δ ε c , 3 , g , h + i Δ ε s , 3 , g , h , a 3 , g , h * = - k 2 2 Δ ε c , 3 , g , h - i Δ ε s , 3 , g , h , b 3 , g , j = - k 3 , g 2 B 3 , g , j ,     c 3 , g , j = 2 i ξ 3 , g , j , S 3 , g , j d d z   S 3 , g , j z ,     S 3 , g , j d 2 d z 2   S 3 , g , j z .
S 3 , g , u j z = m C 3 , g , m w 3 , g , u j , m × exp q 3 , g , m z - g = 1 g - 1   t 3 , g ,
w 3 , g v 1 , g , 1 v 1 , g , 2     v 1 , g , m .
m ξ 1 , j + ξ 3,1 , j + iq 3,1 , m C 3,1 , m w 3,1 , u j , m = 0 j 0 2 ξ 1,0 j = 0 ,
m C 3 , g - 1 , m w 3 , g - 1 , u j , m exp q 3 , g - 1 , m t 3 , g - 1 = m C 3 , g , m w 3 , g , u j , m , m q 3 , g - 1 , m - i ξ 3 , g - 1 , j C 3 , g - 1 , m w 3 , g - 1 , u j , m × exp q 3 , g - 1 , m t 3 , g - 1 = m q 3 , g , m - i ξ 3 , g , j C 3 , g , m w 3 , g , u j , m ,
m ξ 4 , j - ξ 3 , G 3 , j - iq 3 , G 3 , m C 3 , G 3 , m w 3 , G 3 , u j , m = 0 .
η 1 , j = ξ 1 , j ξ 1,0 Ref j   Ref j * = ξ 1 , j ξ 1,0   S 3,1 , j 0 S 3,1 , j * 0 , η 4 , j = ξ 4 , j ξ 1,0 Tra j   Tra j * = ξ 4 , j ξ 1,0   S 3 , G 3 , j T 3 S 3 , G 3 , j * T 3 .
k 3 , g h = 1 κ c , 3 , g , h + i κ s , 3 , g , h S 3 , g , j + h z + κ c , 3 , g , h - i κ s , 3 , g , h S 3 , g , j - h z ,
κ c , 3 , g , h k 3 , g 4 Δ ε c , 3 , g , h n ¯ 3 , g ,   κ s , 3 , g , h k 3 , g 4 Δ ε s , 3 , g , h n ¯ 3 , g ,
κ c , 3 , g , h = π Δ n c , 3 , g , h λ ,   κ s , 3 , g , h = π Δ n s , 3 , g , h λ .

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