Abstract

The parameters of several stratified volume holographic gratings are optimized in order to obtain high diffraction efficiency beam spliters. Two-, three-, five-, seven-, and nine-beam fan-outs are theoretically studied and compared with what has been achieved by other means. With a simple optical setup, a fan-out element that generates seven beams of equal intensity is experimentally realized and a comparison with theory is made.

© 1993 Optical Society of America

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References

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  1. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  2. S. J. Walker, J. Jahns, “Array generation with multilevel phase gratings,” J. Opt. Soc. Am. A 7, 1509–1513 (1990).
    [CrossRef]
  3. D. Daly, S. M. Hodson, M. C. Hutley, “Fan-out gratings with a continuous profile,” Opt. Commun. 82, 4 (1990).
  4. P. Langlois, R. Beaulieu, “Phase relief gratings with conic section profile used in the production of multiple beams,” Appl. Opt. 29, 3434–3439 (1990).
    [CrossRef]
  5. D. Shin, R. Magnusson, “Diffraction by surface-relilef gratings with conic cross-sectional grating shapes,” J. Opt. Soc. Am. 6, 1249–1253 (1989).
    [CrossRef]
  6. A. W. Lohmann, J. A. Thomas, “Making an array illumination based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  7. H. Machida, J. Nitta, A. Seko, H. Kobayashi, “High-efficiency fiber grating for producing multiple beams of uniform intensity,” Appl. Opt. 23, 330–332 (1984).
    [CrossRef] [PubMed]
  8. H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
    [CrossRef]
  9. R. V. Johnson, A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189–191 (1989).
    [CrossRef]
  10. A. P. Yakimovich, “Multilayer three-dimensional holographic optical elements,” Opt. Spectrosc. (USSR) 49, 85–88 (1981).
  11. B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 69–70.
  13. L. Song, R. A. Lessard, P. Galarneau, “Diffraction efficiency of a thin amplitude-phase holographic grating: a convolution approach,” J. Mod. Opt. 37, 1319–1328 (1990).
    [CrossRef]
  14. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
    [CrossRef] [PubMed]
  15. L. Solymar, D. J. Cooke, “Volume holography and volume gratings,” (Academic, New York, 1981), pp. 127–128.
  16. J. R. Léger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase grating,” Appl. Opt. 26, 4337–4399 (1987).
    [CrossRef]
  17. A. Granger, “Diviseur de faisceau par hologramme de volume à réseaux stratifiés,” M.Sc. thesis (Université Laval, Québec, Canada, 1992).

1992 (1)

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

1990 (5)

1989 (2)

D. Shin, R. Magnusson, “Diffraction by surface-relilef gratings with conic cross-sectional grating shapes,” J. Opt. Soc. Am. 6, 1249–1253 (1989).
[CrossRef]

R. V. Johnson, A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189–191 (1989).
[CrossRef]

1987 (1)

J. R. Léger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase grating,” Appl. Opt. 26, 4337–4399 (1987).
[CrossRef]

1984 (2)

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

H. Machida, J. Nitta, A. Seko, H. Kobayashi, “High-efficiency fiber grating for producing multiple beams of uniform intensity,” Appl. Opt. 23, 330–332 (1984).
[CrossRef] [PubMed]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
[CrossRef] [PubMed]

1981 (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic optical elements,” Opt. Spectrosc. (USSR) 49, 85–88 (1981).

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Beaulieu, R.

Cooke, D. J.

L. Solymar, D. J. Cooke, “Volume holography and volume gratings,” (Academic, New York, 1981), pp. 127–128.

Daly, D.

D. Daly, S. M. Hodson, M. C. Hutley, “Fan-out gratings with a continuous profile,” Opt. Commun. 82, 4 (1990).

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Galarneau, P.

L. Song, R. A. Lessard, P. Galarneau, “Diffraction efficiency of a thin amplitude-phase holographic grating: a convolution approach,” J. Mod. Opt. 37, 1319–1328 (1990).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 69–70.

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Granger, A.

A. Granger, “Diviseur de faisceau par hologramme de volume à réseaux stratifiés,” M.Sc. thesis (Université Laval, Québec, Canada, 1992).

Hodson, S. M.

D. Daly, S. M. Hodson, M. C. Hutley, “Fan-out gratings with a continuous profile,” Opt. Commun. 82, 4 (1990).

Hutley, M. C.

D. Daly, S. M. Hodson, M. C. Hutley, “Fan-out gratings with a continuous profile,” Opt. Commun. 82, 4 (1990).

Jahns, J.

Johnson, R. V.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
[CrossRef] [PubMed]

Kobayashi, H.

Kobolia, H.

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Langlois, P.

Léger, J. R.

J. R. Léger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase grating,” Appl. Opt. 26, 4337–4399 (1987).
[CrossRef]

Lessard, R. A.

L. Song, R. A. Lessard, P. Galarneau, “Diffraction efficiency of a thin amplitude-phase holographic grating: a convolution approach,” J. Mod. Opt. 37, 1319–1328 (1990).
[CrossRef]

Lohmann, A. W.

Machida, H.

Magnusson, R.

D. Shin, R. Magnusson, “Diffraction by surface-relilef gratings with conic cross-sectional grating shapes,” J. Opt. Soc. Am. 6, 1249–1253 (1989).
[CrossRef]

Mirovitskii, D. I.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

Nitta, J.

Rostovseva, N. V.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

Schmidt, J.

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Seko, A.

Serov, O. B.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

Sheridan, J. T.

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Shin, D.

D. Shin, R. Magnusson, “Diffraction by surface-relilef gratings with conic cross-sectional grating shapes,” J. Opt. Soc. Am. 6, 1249–1253 (1989).
[CrossRef]

Solymar, L.

L. Solymar, D. J. Cooke, “Volume holography and volume gratings,” (Academic, New York, 1981), pp. 127–128.

Song, L.

L. Song, R. A. Lessard, P. Galarneau, “Diffraction efficiency of a thin amplitude-phase holographic grating: a convolution approach,” J. Mod. Opt. 37, 1319–1328 (1990).
[CrossRef]

Streibl, N.

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Swanson, G. J.

J. R. Léger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase grating,” Appl. Opt. 26, 4337–4399 (1987).
[CrossRef]

Tanguay, A. R.

Thomas, J. A.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
[CrossRef] [PubMed]

Veldkamp, W. B.

J. R. Léger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase grating,” Appl. Opt. 26, 4337–4399 (1987).
[CrossRef]

Volkel, R.

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Walker, S. J.

Yakimovich, A. P.

A. P. Yakimovich, “Multilayer three-dimensional holographic optical elements,” Opt. Spectrosc. (USSR) 49, 85–88 (1981).

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (2)

L. Song, R. A. Lessard, P. Galarneau, “Diffraction efficiency of a thin amplitude-phase holographic grating: a convolution approach,” J. Mod. Opt. 37, 1319–1328 (1990).
[CrossRef]

H. Kobolia, J. Schmidt, J. T. Sheridan, N. Streibl, R. Volkel, “Holographic optical beam splitters in dichromated gelatin,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

D. Shin, R. Magnusson, “Diffraction by surface-relilef gratings with conic cross-sectional grating shapes,” J. Opt. Soc. Am. 6, 1249–1253 (1989).
[CrossRef]

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

Opt. Commun. (2)

D. Daly, S. M. Hodson, M. C. Hutley, “Fan-out gratings with a continuous profile,” Opt. Commun. 82, 4 (1990).

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic optical elements,” Opt. Spectrosc. (USSR) 49, 85–88 (1981).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–679 (1983).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

B. Ya. Zel’dovich, D. I. Mirovitskii, N. V. Rostovseva, O. B. Serov, “Characteristics of two-layer phase holograms,” Sov. J. Quantum Electron. 14, 364–369 (1984).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 69–70.

L. Solymar, D. J. Cooke, “Volume holography and volume gratings,” (Academic, New York, 1981), pp. 127–128.

A. Granger, “Diviseur de faisceau par hologramme de volume à réseaux stratifiés,” M.Sc. thesis (Université Laval, Québec, Canada, 1992).

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Figures (3)

Fig. 1
Fig. 1

SVHG as a beam splitter: n, n′, refractive indices of modulated layers and buffer, respectively.

Fig. 2
Fig. 2

Billet system for recording the SVHG.

Fig. 3
Fig. 3

Theoretical and experimental efficiencies of a seven-beam beam splitter.

Tables (2)

Tables Icon

Table 1 Theoretical Diffraction Efficiencies of Stratified Volume Holographic Gratings as Beam Splitters

Tables Icon

Table 2 Comparison of Theoretical Diffraction Efficiencies η of Beam Splitters

Equations (13)

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t ( x ) = m = A m exp ( i m 2 π x Δ ) ,
A m = i m J m ( φ ) ,
{ A m } j = { i m J m ( φ j ) } ,
P m exp { i 2 π λ 0 n l [ 1 ( m λ 0 n Δ ) 2 ] 1 / 2 }
F ( ξ ) = F [ t 1 ( x ) P 1 2 t 2 ( x ) ] = m i m J m ( φ 1 ) P m , 1 2 δ ( ζ m Δ ) n i n J n ( φ 2 ) δ ( ζ n Δ ) .
F ( ξ ) = m l A l , 1 P l , 1 2 A m 1 , 2 δ ( ξ m Δ )
{ A m } exit = { { A m } 1 · { P m } 1 2 } { A m } 2 .
{ A m } exit = { { A m } 1 · { P m } 1 2 } { A m exp ( i 2 π m Δ ) } 2 .
{ A m } exit = { { { { A m } 1 · { P m } 1 2 } { A m } 2 } · { P m } 2 3 } { A m } 3 .
η m = | A m | exit 2
max E p orders T = m = p orders η m .
max σ p orders T = m = p orders ( η m η mean ) 2 .
f ( α 1 , , α N ) = ( σ p orders 2 E p orders T ) 2 = [ m = p orders ( | A m | 2 1 p k = p orders | A k | 2 ) 2 m = p orders | A m | 2 ] 2 ,

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