Abstract

We present a simplified model and experimental verification for multiplexed substrate-mode volume holograms. The analysis is based on coupled-wave theory for uncoupled S- and P-polarization states of the incident beam. The theory is used to investigate a practical situation with the reconstruction beam normally incident upon the substrate plane and the diffracted multimode beams propagating in orthogonal directions within the substrate. Model results are then used to optimize the performance of experimental holograms formed in dichromated gelatin. The resultant multiplexed grating splits the incident beam into four orthogonal beams with nearly equal intensities, for a total efficiency of 80%.

© 1990 Optical Society of America

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References

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  1. R. K. Kostuk, L. Wang, Y.-T. Huang, “Optical clock signal distribution with holographic optical elements,” in Real-Time Signal Processing XI, J. P. Letellier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.977, 24–36 (1988).
    [CrossRef]
  2. F. Sauer, “Fabrication of diffractive-refractive optical interconnects for infrared operation based on total internal reflection,” Appl. Opt. 28, 386–388 (1989).
    [CrossRef] [PubMed]
  3. H. Kogelnik, T. P. Sosnowski, “Holographic thin film couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
  4. S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
    [CrossRef]
  5. R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
    [CrossRef]
  6. L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
    [CrossRef]
  7. R. Kowarschik, “Diffraction of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
    [CrossRef]
  8. J. W. Lewis, L. Solymar, “Spurious wave in thick phase gratings,” Opt. Commun. 47, 23–26 (1983).
    [CrossRef]
  9. J. J. A. Couture, R. A. Lessard, “Diffraction efficiency changes induced by coupling effects between gratings of transmission holograms,” Optik 68, 69–80 (1984).
  10. J. J. A. Couture, R. A. Lessard, “Effective thickness determination for volume transmission multiplex holograms,” Can. J. Phys. 64, 553–557 (1986).
    [CrossRef]
  11. C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
    [CrossRef]
  12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  13. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  14. E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
    [CrossRef] [PubMed]
  15. S. K. Case, “Multiple exposure holography in volume materials,” Ph.D. dissertation, University Microfilm order 76-27,461 (University of Michigan, Ann Arbor, Mich., 1976).
  16. L. Solymar, C. J. R. Sheppard, “A two-dimensional theory of volume gratings with electric polarization in the plane of the grating,” J. Opt. Soc. Am. 69, 491–495 (1979).
    [CrossRef]
  17. T. G. Georgekutty, H.-K. Liu, “Simplified dichromated gelatin hologram recording process,” Appl. Opt. 26, 372–376 (1987).
    [CrossRef] [PubMed]

1989

1987

T. G. Georgekutty, H.-K. Liu, “Simplified dichromated gelatin hologram recording process,” Appl. Opt. 26, 372–376 (1987).
[CrossRef] [PubMed]

C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
[CrossRef]

1986

J. J. A. Couture, R. A. Lessard, “Effective thickness determination for volume transmission multiplex holograms,” Can. J. Phys. 64, 553–557 (1986).
[CrossRef]

1984

J. J. A. Couture, R. A. Lessard, “Diffraction efficiency changes induced by coupling effects between gratings of transmission holograms,” Optik 68, 69–80 (1984).

1983

1979

1978

R. Kowarschik, “Diffraction of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

1977

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[CrossRef]

1975

1970

H. Kogelnik, T. P. Sosnowski, “Holographic thin film couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

1969

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

Alferness, R.

Case, S. K.

Couture, J. J. A.

J. J. A. Couture, R. A. Lessard, “Effective thickness determination for volume transmission multiplex holograms,” Can. J. Phys. 64, 553–557 (1986).
[CrossRef]

J. J. A. Couture, R. A. Lessard, “Diffraction efficiency changes induced by coupling effects between gratings of transmission holograms,” Optik 68, 69–80 (1984).

Gaylord, T. K.

Georgekutty, T. G.

Glytsis, E. N.

Huang, Y.-T.

R. K. Kostuk, L. Wang, Y.-T. Huang, “Optical clock signal distribution with holographic optical elements,” in Real-Time Signal Processing XI, J. P. Letellier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.977, 24–36 (1988).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, “Holographic thin film couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

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

Kostuk, R. K.

R. K. Kostuk, L. Wang, Y.-T. Huang, “Optical clock signal distribution with holographic optical elements,” in Real-Time Signal Processing XI, J. P. Letellier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.977, 24–36 (1988).
[CrossRef]

Kowarschik, R.

R. Kowarschik, “Diffraction of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

Lessard, R. A.

J. J. A. Couture, R. A. Lessard, “Effective thickness determination for volume transmission multiplex holograms,” Can. J. Phys. 64, 553–557 (1986).
[CrossRef]

J. J. A. Couture, R. A. Lessard, “Diffraction efficiency changes induced by coupling effects between gratings of transmission holograms,” Optik 68, 69–80 (1984).

Lewis, J. W.

J. W. Lewis, L. Solymar, “Spurious wave in thick phase gratings,” Opt. Commun. 47, 23–26 (1983).
[CrossRef]

Liu, H.-K.

Moharam, M. G.

Sauer, F.

Sheppard, C. J. R.

Slinger, C. W.

C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
[CrossRef]

Solymar, L.

C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
[CrossRef]

J. W. Lewis, L. Solymar, “Spurious wave in thick phase gratings,” Opt. Commun. 47, 23–26 (1983).
[CrossRef]

L. Solymar, C. J. R. Sheppard, “A two-dimensional theory of volume gratings with electric polarization in the plane of the grating,” J. Opt. Soc. Am. 69, 491–495 (1979).
[CrossRef]

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[CrossRef]

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, “Holographic thin film couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

Syms, R. R. A.

C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
[CrossRef]

Wang, L.

R. K. Kostuk, L. Wang, Y.-T. Huang, “Optical clock signal distribution with holographic optical elements,” in Real-Time Signal Processing XI, J. P. Letellier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.977, 24–36 (1988).
[CrossRef]

Appl. Opt.

Appl. Phys. B

C. W. Slinger, R. R. A. Syms, L. Solymar, “Multiple holographic transmission gratings in silver halide emulsion,” Appl. Phys. B 42, 121–128 (1987).
[CrossRef]

Bell Syst. Tech. J.

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

H. Kogelnik, T. P. Sosnowski, “Holographic thin film couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).

Can. J. Phys.

J. J. A. Couture, R. A. Lessard, “Effective thickness determination for volume transmission multiplex holograms,” Can. J. Phys. 64, 553–557 (1986).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

R. Kowarschik, “Diffraction of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

Opt. Commun.

J. W. Lewis, L. Solymar, “Spurious wave in thick phase gratings,” Opt. Commun. 47, 23–26 (1983).
[CrossRef]

L. Solymar, “Two-dimensional N-coupled wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[CrossRef]

Optik

J. J. A. Couture, R. A. Lessard, “Diffraction efficiency changes induced by coupling effects between gratings of transmission holograms,” Optik 68, 69–80 (1984).

Other

S. K. Case, “Multiple exposure holography in volume materials,” Ph.D. dissertation, University Microfilm order 76-27,461 (University of Michigan, Ann Arbor, Mich., 1976).

R. K. Kostuk, L. Wang, Y.-T. Huang, “Optical clock signal distribution with holographic optical elements,” in Real-Time Signal Processing XI, J. P. Letellier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.977, 24–36 (1988).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Multiplexed SMH with (a) two parallel gratings, (b) two orthogonal gratings, (c) two parallel and two orthogonal gratings. S is light polarized perpendicular to the plane of incidence.

Fig. 2
Fig. 2

Diffraction efficiency versus refractive-index modulation calculated for two multiplexed SMH’s recorded with equal index modulation (n1 = n2). The diffraction angle in the medium θ is 45 deg. η 1 ( S ) is the efficiency of grating K1 with an S-polarized reconstruction beam, and η 2 ( P ) is the efficiency of grating K2 with a P-polarized reconstruction beam. ηR is the zero-order transmittance.

Fig. 3
Fig. 3

Diffraction efficiency versus refractive-index modulation calculated for the four-grating SMH with equal index modulation (n1 = n2 = n3 = n4). The diffraction angles in the medium θi, equal 45 deg. η 1 ( S ) and η 3 ( S ) are the efficiencies of gratings K1 and K3 illuminated with an S-polarized beam. η 2 ( P ) and η 4 ( P ) are efficiencies of gratings K2 and K4 illuminated with a P-polarized beam, ηR is the zero-order transmittance.

Fig. 4
Fig. 4

Diffraction efficiency versus refractive-index modulation calculated for the two-grating SMH with the index-modulation ratio of n2/n1 = 1/cos θ.

Fig. 5
Fig. 5

Diffraction efficiency versus refractive-index modulation calculated for the four-grating SMH with the index-modulation ratio of n1:n2:n3:n4 = cos θ:l:cos θ:1.

Fig. 6
Fig. 6

SMH geometries: (a) recording, (b) reconstruction.

Fig. 7
Fig. 7

Experimental diffraction efficiency versus exposure for two-grating SMH with equal exposure for the gratings (E1 = E2).

Fig. 8
Fig. 8

Experimental efficiency versus exposure ratio for a two-grating SMH. Experimental exposure ratios are indicated next to the curve for η 1 ( S ).

Fig. 9
Fig. 9

Diffraction efficiency versus exposure ratio for a four-grating SMH. Experimental exposure ratios (E2/E1 = E4/E3) are indicated next to the curve for η 1 ( S ) , η 3 ( S ).

Fig. 10
Fig. 10

Experimental diffraction efficiency versus angle of incidence for a four-grating SMH with (a) an S-polarized reconstruction beam, (b) a P-polarized reconstruction beam.

Fig. 11
Fig. 11

Experimental diffraction efficiency versus angle of incidence for a two-grating SMH recorded with the exposure ratio of E2/E1 = 1.7 (E1 = 45 mJ/cm2, E2 = 75 mJ/cm2). The polarization state of the incident beam is S for K1 and P for K2.

Fig. 12
Fig. 12

Experimental diffraction efficiency versus angle of incidence for a four-grating SMH recorded with the exposure ratio of E2/E1 = E4/E3 = 1.7 (E1 = E3 = 30 mJ/cm2, E2 = E4 = 50 mJ/cm2). The polarization state of the incident beam is S for K1 and K3 and P for K2 and K4.

Fig. 13
Fig. 13

Photographs of multiplexed SMH’s: (a) two orthogonal gratings and (b) two parallel and two orthogonal gratings reconstructed with normally incident light at 632.8 nm showing diffracted beams propagating by total internal reflection. As indicated, there is little evidence of strong secondary diffraction orders.

Equations (14)

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ε = ε 0 + m = 1 4 ε m cos ( K m · x ) ,
E = R ( z ) exp [ i ( ρ · x ) ] + m = 1 4 S m ( z ) exp [ i ( σ m · x ) ] ,
σ m = ρ K m ( m = 1 , 2 , 3 , 4 ) .
R ( z ) = R y ( z ) j ˆ , S m ( z ) = S m y ( z ) j ˆ ( m = 1 , 3 ) , S m ( z ) = S m y ( z ) j ˆ + S m z ( z ) k ˆ ( m = 2 , 4 ) ,
E x = 0 , E y = R y ( z ) exp [ i ( ρ · x ) ] + m = 1 4 S m y ( z ) exp [ i ( σ m · x ) ] , E z = S 2 z ( z ) exp [ i ( σ 2 · x ) ] + S 4 z ( z ) exp [ i ( σ 4 · x ) ] .
2 E ( · E ) + ( ε ω 2 / c 2 ) E = 0 ,
0 = c S m S m ( z ) + i κ m R ( z ) ( m = 1 , 3 ) , 0 = c S m S m ( z ) + i κ m R ( z ) ( s ˆ m · r ˆ ) ( m = 2 , 4 ) , 0 = c R R ( z ) + i κ 1 S 1 ( z ) + i κ 2 S 2 ( z ) ( s ˆ 2 · r ˆ ) + i κ 3 S 3 ( z ) + i κ 4 S 4 ( z ) ( s ˆ 4 · r ˆ ) ,
c S m = σ m z / β , c R = ρ z / β , β = 2 π n 0 / λ ,
κ m = π n m / λ , ( m = 1 , 2 , 3 , 4 ) .
η S i = ν i 2 sin 2 ( m = 1 4 ν m 2 ) 1 / 2 m = 1 4 ν m 2 ( i = 1 , 2 , 3 , 4 ) , η R = cos 2 ( m = 1 4 ν m 2 ) 1 / 2 ,
ν m = π n m d λ ( c S m c R ) ( m = 1 , 3 ) , ν m = π n m d λ ( c S m c R ) 1 / 2 ( s ˆ m · r ˆ ) ( m = 2 , 4 ) ,
η 2 ( P ) η 1 ( S ) = ( ν 2 ν 1 ) 2 = ( κ 2 κ 1 ) 2 = ( n 2 n 1 ) 2 cos 2 θ = cos 2 θ = 0.5 ,
η 2 ( P ) η 1 ( S ) = ( n 2 n 1 ) 2 cos 2 θ = 1 ,
n 1 = n 2 cos θ ( n 3 = n 1 , n 4 = n 2 ) .

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