Abstract

Volume phase holograms are formed by a standard process in commercially available photographic emulsion. The material is characterized before recording, and initial experimental results are presented for reconstruction under index-matched conditions. An initial comparison is made using two-wave coupled-wave theory and a technique of curve fitting with experimental measurements of transmission and diffraction efficiency. The model works well close to the Bragg condition, but several differences are noted between theory and experiment away from the Bragg condition. An anomalous absorptive effect is noted in transmission. An improved model is then formulated, again using coupled-wave theory, and capable of analytic solution, taking into account phase and absorption modulation, a second harmonic in the grating profile and the appearance of some higher diffraction orders. Using this model, all the initial experimental results are satisfactorily explained, and the effect of spurious gratings in the hologram response is noted. The model is then used with an extensive set of experimental results to deduce the major characteristics of the material, including saturation of the modulation with exposure. The formulation of a mixed grating and possible dispersion of the modulation are also investigated. Suggestions are made for the design of more complicated components using this material and for material improvement.

© 1983 Optical Society of America

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  1. D. H. Close, Opt. Eng. 14, 408 (1975).
    [CrossRef]
  2. W. C. Sweatt, Appl. Opt. 17, 1220 (1978).
    [CrossRef] [PubMed]
  3. M. R. Latta, R. V. Pole, Appl. Opt. 18, 2418 (1979).
    [CrossRef] [PubMed]
  4. O. D. D. Soares, Opt. Eng. 20, 740 (1981).
    [CrossRef]
  5. H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
    [CrossRef]
  6. J. L. Horner, J. E. Ludman, Appl. Opt. 20, 1845 (1981).
    [CrossRef] [PubMed]
  7. W-H. Lee, Appl. Opt. 16, 1392 (1977).
    [CrossRef] [PubMed]
  8. W. T. Welford, Opt. Commun. 14, 322 (1975).
    [CrossRef]
  9. J. N. Latta, Appl. Opt. 10, 2698 (1971).
    [CrossRef] [PubMed]
  10. H. W. Holloway, R. A. Ferrante, Appl. Opt. 20, 2081 (1981).
    [CrossRef] [PubMed]
  11. H. Kogelnik, Bell. Syst. Tech. J. 48, 2909 (1969).
  12. B. J. Chang, Proc. Soc. Photo. Opt. Instrum. Eng. 177, 71 (1979).
  13. A. Graube, Appl. Opt. 13, 2942 (1974).
    [CrossRef] [PubMed]
  14. N. Uchida, J. Opt. Soc. Am. 63, 280 (1973).
    [CrossRef]
  15. R. Kowarschik, Opt. Act. 23, 1039 (1976).
    [CrossRef]
  16. T. Kubota, Opt. Commun. 16, 347 (1976).
    [CrossRef]
  17. T. Kubota, Opt. Act. 25, 1035 (1976).
    [CrossRef]
  18. U. Killat, Opt. Commun. 21, 110 (1977).
    [CrossRef]
  19. D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
    [CrossRef]
  20. T. Kubota, Opt. Act. 26, 731 (1979).
    [CrossRef]
  21. M. P. Jordan, L. Solymar, Electron Lett. 14, 271 (1978).
    [CrossRef]
  22. J. A. Kong, J. Opt. Soc. Am. 67, 825 (1977).
    [CrossRef]
  23. H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
    [CrossRef]
  24. J. Ctyroki, Opt. Commun. 16, 259 (1976).
    [CrossRef]
  25. C. V. Raman, N. S. N. Nath, Proc. Ind. Acad. Sci. 2, 406, 413 (1935); 3, 75, 119, 459 (1936).
  26. R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 67, 1165 (1977).
    [CrossRef]
  27. B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
    [CrossRef]
  28. B. Benlarbi, L. Solymar, Int. J. Electron. 52, 95 (1982).
    [CrossRef]
  29. M. Chang, N. George, Appl. Opt. 9, 713 (1970).
    [CrossRef] [PubMed]
  30. B. Benlarbi, L. Solymar, Int. J. Electron. 48, 361 (1980).
    [CrossRef]
  31. B. Benlarbi, L. Solymar, Int. J. Electron. 48, 351 (1980).
    [CrossRef]
  32. H. M. Smith, Holographic Recording Materials (Springer, New York, 1977), pp. 22–74.
  33. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 2.
  34. D. Croucher, Holography Newletter 1, Agfa-Gevaert Ltd. (1979).
  35. R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 68, 809 (1978).
    [CrossRef]
  36. M. G. Moharam, L. Young, Appl. Opt. 17, 1757 (1978).
    [CrossRef] [PubMed]
  37. R. R. A. Syms, L. Solymar, Electron. Lett. 18, 249 (1982).
    [CrossRef]
  38. H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
    [CrossRef]
  39. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]
  40. P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
    [CrossRef]
  41. T. K. Gaylord, M. G. Moharam, Appl. Phys. B28, 1 (1982).
  42. I. S. Sokolnikoff, E. S. Sokolnikoff, Higher Mathematics for Physicists and Engineers (McGraw-Hill, New York, 1941).
  43. R. R. A. Syms, L. Solymar, Opt. Commun. 43, 107 (1982).
    [CrossRef]
  44. D. H. R. Vilkomerson, D. Bostwick, Appl. Opt. 6, 1270 (1967).
    [CrossRef] [PubMed]
  45. R. L. Lamberts, Appl. Opt. 11, 33 (1972).
    [CrossRef] [PubMed]
  46. O. Bryngdahl, Appl. Opt. 11, 195 (1972).
    [CrossRef] [PubMed]
  47. C. T. Chang, J. L. Bjorkstam, J. Opt. Soc. Am. 66, 558 (1976).
    [CrossRef]
  48. R. Alferness, J. Opt. Soc. Am. 66, 353 (1976).
    [CrossRef]
  49. B. J. Chang, S. K. Case, Appl. Opt. 15, 1800 (1976).
    [CrossRef] [PubMed]
  50. S. Sjolinder, Photogr. Sci. Eng. 25, 112 (1981).

1982 (4)

B. Benlarbi, L. Solymar, Int. J. Electron. 52, 95 (1982).
[CrossRef]

R. R. A. Syms, L. Solymar, Electron. Lett. 18, 249 (1982).
[CrossRef]

T. K. Gaylord, M. G. Moharam, Appl. Phys. B28, 1 (1982).

R. R. A. Syms, L. Solymar, Opt. Commun. 43, 107 (1982).
[CrossRef]

1981 (4)

1980 (4)

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 361 (1980).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 351 (1980).
[CrossRef]

B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
[CrossRef]

P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
[CrossRef]

1979 (3)

M. R. Latta, R. V. Pole, Appl. Opt. 18, 2418 (1979).
[CrossRef] [PubMed]

T. Kubota, Opt. Act. 26, 731 (1979).
[CrossRef]

B. J. Chang, Proc. Soc. Photo. Opt. Instrum. Eng. 177, 71 (1979).

1978 (4)

1977 (4)

1976 (7)

J. Ctyroki, Opt. Commun. 16, 259 (1976).
[CrossRef]

R. Kowarschik, Opt. Act. 23, 1039 (1976).
[CrossRef]

T. Kubota, Opt. Commun. 16, 347 (1976).
[CrossRef]

T. Kubota, Opt. Act. 25, 1035 (1976).
[CrossRef]

B. J. Chang, S. K. Case, Appl. Opt. 15, 1800 (1976).
[CrossRef] [PubMed]

R. Alferness, J. Opt. Soc. Am. 66, 353 (1976).
[CrossRef]

C. T. Chang, J. L. Bjorkstam, J. Opt. Soc. Am. 66, 558 (1976).
[CrossRef]

1975 (3)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

W. T. Welford, Opt. Commun. 14, 322 (1975).
[CrossRef]

D. H. Close, Opt. Eng. 14, 408 (1975).
[CrossRef]

1974 (1)

1973 (1)

1972 (2)

1971 (1)

1970 (1)

1969 (2)

D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
[CrossRef]

H. Kogelnik, Bell. Syst. Tech. J. 48, 2909 (1969).

1967 (2)

1966 (1)

1962 (1)

H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
[CrossRef]

1935 (1)

C. V. Raman, N. S. N. Nath, Proc. Ind. Acad. Sci. 2, 406, 413 (1935); 3, 75, 119, 459 (1936).

Alferness, R.

Benlarbi, B.

B. Benlarbi, L. Solymar, Int. J. Electron. 52, 95 (1982).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 351 (1980).
[CrossRef]

B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 361 (1980).
[CrossRef]

Bjorkstam, J. L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 2.

Bostwick, D.

Bryngdahl, O.

Case, S. K.

Chang, B. J.

B. J. Chang, Proc. Soc. Photo. Opt. Instrum. Eng. 177, 71 (1979).

B. J. Chang, S. K. Case, Appl. Opt. 15, 1800 (1976).
[CrossRef] [PubMed]

Chang, C. T.

Chang, M.

Close, D. H.

D. H. Close, Opt. Eng. 14, 408 (1975).
[CrossRef]

Cooke, D. J.

B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
[CrossRef]

Croucher, D.

D. Croucher, Holography Newletter 1, Agfa-Gevaert Ltd. (1979).

Ctyroki, J.

J. Ctyroki, Opt. Commun. 16, 259 (1976).
[CrossRef]

Ferrante, R. A.

Gaylord, T. K.

George, N.

Graube, A.

Hashimoto, H.

H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
[CrossRef]

Holloway, H. W.

Horner, J. L.

Howie, A.

H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
[CrossRef]

Inohara, S.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

Jordan, M. P.

M. P. Jordan, L. Solymar, Electron Lett. 14, 271 (1978).
[CrossRef]

Kermisch, D.

Killat, U.

U. Killat, Opt. Commun. 21, 110 (1977).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Bell. Syst. Tech. J. 48, 2909 (1969).

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
[CrossRef]

Kong, J. A.

Kowarschik, R.

R. Kowarschik, Opt. Act. 23, 1039 (1976).
[CrossRef]

Koyama, J.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

Kozma, A.

Kubota, T.

T. Kubota, Opt. Act. 26, 731 (1979).
[CrossRef]

T. Kubota, Opt. Commun. 16, 347 (1976).
[CrossRef]

T. Kubota, Opt. Act. 25, 1035 (1976).
[CrossRef]

Lamberts, R. L.

Latta, J. N.

Latta, M. R.

Lee, W-H.

Leith, E. N.

Ludman, J. E.

Magnusson, R.

Marks, J.

Massey, N.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, Appl. Phys. B28, 1 (1982).

M. G. Moharam, L. Young, Appl. Opt. 17, 1757 (1978).
[CrossRef] [PubMed]

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, Proc. Ind. Acad. Sci. 2, 406, 413 (1935); 3, 75, 119, 459 (1936).

Nishihara, H.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

Pole, R. V.

Raman, C. V.

C. V. Raman, N. S. N. Nath, Proc. Ind. Acad. Sci. 2, 406, 413 (1935); 3, 75, 119, 459 (1936).

Russell, J.

P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
[CrossRef]

Sjolinder, S.

S. Sjolinder, Photogr. Sci. Eng. 25, 112 (1981).

Smith, H. M.

H. M. Smith, Holographic Recording Materials (Springer, New York, 1977), pp. 22–74.

Soares, O. D. D.

O. D. D. Soares, Opt. Eng. 20, 740 (1981).
[CrossRef]

Sokolnikoff, E. S.

I. S. Sokolnikoff, E. S. Sokolnikoff, Higher Mathematics for Physicists and Engineers (McGraw-Hill, New York, 1941).

Sokolnikoff, I. S.

I. S. Sokolnikoff, E. S. Sokolnikoff, Higher Mathematics for Physicists and Engineers (McGraw-Hill, New York, 1941).

Solymar, L.

B. Benlarbi, L. Solymar, Int. J. Electron. 52, 95 (1982).
[CrossRef]

R. R. A. Syms, L. Solymar, Opt. Commun. 43, 107 (1982).
[CrossRef]

R. R. A. Syms, L. Solymar, Electron. Lett. 18, 249 (1982).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 361 (1980).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 351 (1980).
[CrossRef]

B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
[CrossRef]

P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
[CrossRef]

M. P. Jordan, L. Solymar, Electron Lett. 14, 271 (1978).
[CrossRef]

St, P.

P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
[CrossRef]

Suhara, T.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

Sweatt, W. C.

Syms, R. R. A.

R. R. A. Syms, L. Solymar, Opt. Commun. 43, 107 (1982).
[CrossRef]

R. R. A. Syms, L. Solymar, Electron. Lett. 18, 249 (1982).
[CrossRef]

Uchida, N.

Upatnieks, J.

Vilkomerson, D. H. R.

Welford, W. T.

W. T. Welford, Opt. Commun. 14, 322 (1975).
[CrossRef]

Whelan, M. T.

H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 2.

Young, L.

Appl. Opt. (14)

Appl. Phys. (2)

P. St, J. Russell, L. Solymar, Appl. Phys. 22, 335 (1980).
[CrossRef]

T. K. Gaylord, M. G. Moharam, Appl. Phys. B28, 1 (1982).

Bell. Syst. Tech. J. (1)

H. Kogelnik, Bell. Syst. Tech. J. 48, 2909 (1969).

Electron Lett. (1)

M. P. Jordan, L. Solymar, Electron Lett. 14, 271 (1978).
[CrossRef]

Electron. Lett. (1)

R. R. A. Syms, L. Solymar, Electron. Lett. 18, 249 (1982).
[CrossRef]

IEEE J. Quantum Electron (1)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, IEEE J. Quantum Electron QE-11, 794 (1975).
[CrossRef]

Int. J. Electron. (3)

B. Benlarbi, L. Solymar, Int. J. Electron. 52, 95 (1982).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 361 (1980).
[CrossRef]

B. Benlarbi, L. Solymar, Int. J. Electron. 48, 351 (1980).
[CrossRef]

J. Opt. Soc. Am. (8)

Opt. Act. (4)

T. Kubota, Opt. Act. 26, 731 (1979).
[CrossRef]

B. Benlarbi, D. J. Cooke, L. Solymar, Opt. Act. 27, 885 (1980).
[CrossRef]

T. Kubota, Opt. Act. 25, 1035 (1976).
[CrossRef]

R. Kowarschik, Opt. Act. 23, 1039 (1976).
[CrossRef]

Opt. Commun. (5)

T. Kubota, Opt. Commun. 16, 347 (1976).
[CrossRef]

W. T. Welford, Opt. Commun. 14, 322 (1975).
[CrossRef]

U. Killat, Opt. Commun. 21, 110 (1977).
[CrossRef]

J. Ctyroki, Opt. Commun. 16, 259 (1976).
[CrossRef]

R. R. A. Syms, L. Solymar, Opt. Commun. 43, 107 (1982).
[CrossRef]

Opt. Eng. (2)

D. H. Close, Opt. Eng. 14, 408 (1975).
[CrossRef]

O. D. D. Soares, Opt. Eng. 20, 740 (1981).
[CrossRef]

Photogr. Sci. Eng. (1)

S. Sjolinder, Photogr. Sci. Eng. 25, 112 (1981).

Proc. Ind. Acad. Sci. (1)

C. V. Raman, N. S. N. Nath, Proc. Ind. Acad. Sci. 2, 406, 413 (1935); 3, 75, 119, 459 (1936).

Proc. R. Soc. London Ser. A (1)

H. Hashimoto, A. Howie, M. T. Whelan, Proc. R. Soc. London Ser. A 269, 80 (1962).
[CrossRef]

Proc. Soc. Photo. Opt. Instrum. Eng. (1)

B. J. Chang, Proc. Soc. Photo. Opt. Instrum. Eng. 177, 71 (1979).

Other (4)

H. M. Smith, Holographic Recording Materials (Springer, New York, 1977), pp. 22–74.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 2.

D. Croucher, Holography Newletter 1, Agfa-Gevaert Ltd. (1979).

I. S. Sokolnikoff, E. S. Sokolnikoff, Higher Mathematics for Physicists and Engineers (McGraw-Hill, New York, 1941).

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Figures (28)

Fig. 1
Fig. 1

8E56 holographic plate dimensions and technical data.

Fig. 2
Fig. 2

Apparatus for determination of emulsion layer thickness and refractive index.

Fig. 3
Fig. 3

Transmission characteristic of 8E56 holographic plate in air over portions of the visible spectrum at different angles of incidence θ.

Fig. 4
Fig. 4

Apparatus for determination of absorption level in emulsion layer.

Fig. 5
Fig. 5

Transmission characteristic of 8E56 holographic plate and of a plate containing pure gelatin only under index-matched conditions over portions of the visible spectrum at normal incidence.

Fig. 6
Fig. 6

Recording conditions for A series holograms using an index-matching tank.

Fig. 7
Fig. 7

Recording conditions for B series holograms using an index-matched neutral density filter.

Fig. 8
Fig. 8

Apparatus for testing holograms under index-matched conditions using a laser probe beam at a 5145-Å wavelength.

Fig. 9
Fig. 9

Apparatus for testing holograms under index-matched conditions using a white light source and measuring transmission over a narrow wavelength band.

Fig. 10
Fig. 10

Variation of peak diffraction efficiency and minimum transmission with exposure for A series holograms measured at 5145 Å under index-matched conditions.

Fig. 11
Fig. 11

Diagrammatic representation of output beams observed from an A series hologram replayed at 5145 Å under index-matched conditions for various angles of incidence θ.

Fig. 12
Fig. 12

Experimental measurement of the variation of A series hologram 2 0th- and +1st-order beam efficiencies at a replay wavelength of 5145 Å under index-matched condition with incident angle θ.

Fig. 13
Fig. 13

Simplified boundary matching for the Kogelnik model.

Fig. 14
Fig. 14

Prediction of the Kogelnik model corresponding to a portion of the experimental results of Fig. 12 with parameter values d = 4.7 μm, ɛ 0 = 1.59 , ɛ 0 = 8 × 10 - 3 , ɛ 1 / ɛ 0 = 0.057, and (i) ɛ 1 / ɛ 0 = 0, (ii) ɛ 1 / ɛ 0 = + 0.486, (iii) ɛ 1 / ɛ 0 = - 0.486

Fig. 15
Fig. 15

Comparison of the prediction of the Kogelnik model with parameter values d = 4.7 μm, ɛ 0 = 1.59 , ɛ 0 = 8 × 10 - 3 , ɛ 1 / ɛ 0 = 0.057 , ɛ 1 / ɛ 0 = 0.486 with the experimental results of Fig. 12.

Fig. 16
Fig. 16

Slab hologram geometry for the multiwave model.

Fig. 17
Fig. 17

Vector construction for the wave vector of a diffraction order (i) under normal conditions, (ii) close to cutoff, (iii) beyond cutoff.

Fig. 18
Fig. 18

Comparison of the prediction of three-wave theory with parameter values d = 4.6 μm, ɛ 0 = 1.59 , ɛ 0 = 8.15 × 10 - 13 , ɛ 1 / ɛ 0 = 0.059 , ɛ 1 / ɛ 0 = 0.486 , ɛ 2 / ɛ 1 = ɛ 2 / ɛ 1 = 0 with the experimental results of Fig. 12.

Fig. 19
Fig. 19

Comparison of the prediction of three-wave theory with parameter values d = 4.6 μm, ɛ 0 = 1.59 , ɛ 0 = 8.15 × 10 - 13 , ɛ 1 / ɛ 0 = 0.059 , ɛ 1 / ɛ 0 = 0.486 , ɛ 2 / ɛ 1 = ɛ 2 / ɛ 1 = - 0.145 with the experimental results of Fig. 12.

Fig. 20
Fig. 20

Comparison of the prediction of three-wave theory with experimental measurement of transmission under index-matched conditions of A series hologram 2 at replay wavelengths of 4067, 5145, 6330, and 7525 Å.

Fig. 21
Fig. 21

Proposed geometry for off-axis lens fabricated in 8E56.

Fig. 22
Fig. 22

Comparison of the prediction of three-wave theory with experimental measurement of transmission under index-matched conditions of A series hologram 1 at replay wavelengths of 4067, 5145, and 7525 Å.

Fig. 23
Fig. 23

Comparison of the prediction of three-wave theory with experimental measurement of transmission under index-matched conditions of A series hologram 3 at replay wavelengths of 4067, 5145, and 7525 Å.

Fig. 24
Fig. 24

Variation of peak diffraction efficiency and minimum transmission with exposure for B series holograms measured at 5145 Å under index-matched conditions.

Fig. 25
Fig. 25

Variation of κ 1 d and κ 2 d with exposure for B series holograms; κ 2 d is plotted positively for simplicity.

Fig. 26
Fig. 26

Variation of αd with exposure for B series holograms.

Fig. 27
Fig. 27

Comparison of the prediction of three-wave theory with experimental measurement of transmission at 4067 Å under conditions for B series holograms (i) 4, (ii) 5 (iii) 6, (iv) 7. Also illustrated are representations of the grating modulation profile constructed by Fourier synthesis.

Fig. 28
Fig. 28

Variation of the value of ɛ 1 / ɛ 0 obtained by curve-fitting for A series holograms 1–3 and B series holograms 4–7 with wavelength.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

Ω = K 2 2 β κ 1 , κ 1 = β ɛ 1 4 ɛ 0 , K = 2 π Λ , β = 2 π ɛ 0 λ ,
κ 1 d cos θ 0 = π 2 ,
sin 2 θ 0 cos θ 0 = Ω λ 8 0 d .
A 0 2 = exp ( - 2 α d cos θ 0 ) cos 2 ( κ 1 d cos θ 0 ) , A - 1 2 = exp ( - 2 α d cos θ 0 ) sin 2 ( κ 1 d cos θ 0 ) ,
κ 1 d cos θ 0 π 2
ɛ 1 av = 1 d 0 d ɛ 1 ( x ) d x .
ɛ ( r ) = ɛ 0 - j ɛ 0 + i = 1 ( ɛ 1 - j ɛ 0 ) cos ( i K · r ) ,
ρ 0 = β , β = 2 π ɛ 0 λ , ρ 0 y β = ɛ e ɛ 0 sin θ .
2 E + β 2 [ 1 - j ɛ 0 ɛ 0 + i = 1 ( ɛ i ɛ 0 - j ɛ i ɛ 0 ) cos ( iK · r ) ] E = 0.
ρ l = ρ 0 + l K .
ρ l y = ρ 0 y + l K y .
E = l = - A l ( x ) exp ( - j ρ l · r ) .
d 2 A l d x 2 - 2 j ρ l x d A l d x + [ ( β 2 - ρ l 2 ) - j β 2 ɛ 0 ɛ 0 ] A l + β 2 2 i = 1 ( ɛ i ɛ 0 - j ɛ i ɛ 0 ) { A l + i exp [ - j ( ρ l + i , x - ρ l x - i K x ) ] + A l - i exp [ - j ( ρ l - i , x - ρ l x + i K x ) ] } = 0.
ρ l x β d A l d x + [ α + j ( β 2 - ρ l 2 2 β ) ] A l + j i = 1 ( κ i - j κ i ) × { A l + i exp [ - j ( ρ l + i , x - ρ l x - i K x ) ] + A l - i exp [ - j ( ρ l - i , x - ρ l x + i K x ) ] } = 0 ,
α = β ɛ 0 2 ɛ 0 , κ i = β ɛ i 4 ɛ 0 , κ i = β ɛ i 4 ɛ 0 .
B l = A l exp [ - j ( ρ l - ρ l ) · r ] ,
ρ l x β d B l d x + { α + j [ ( ρ l x 2 - ρ l x ρ l x β ) + ( β 2 - ρ l 2 2 β ) ] } B l + j i = 1 ( κ i - j κ i ) ( B l + i + B l - i ) = 0.
ρ 2 x β d B 2 d x + [ α + j ( β 2 - ρ 2 · ρ 2 β ) ] × B 2 + j ( κ 1 - j κ 1 ) B 1 + j ( κ 2 - j κ 2 ) B 0 = 0 , ρ 1 x β d B 1 d x + [ α + j ( β 2 - ρ 1 · ρ 1 β ) ] × B 1 + j ( κ 1 - j κ 1 ) ( B 2 + B 0 ) = 0 , ρ 0 x β d B 0 d x + α B 0 + j ( κ 1 - j κ 1 ) B 1 + j ( κ 2 - j κ 2 ) B 2 = 0.
ρ 1 x β d B 1 d x + [ α + j ( β 2 - ρ 1 · ρ 1 β ) ] × B 1 + j ( κ 1 - j κ 1 ) B 0 + j ( κ 2 - j κ 2 ) B - 1 = 0 , ρ 0 x β d B 0 d x + α B 0 + j ( κ 1 - j κ 1 ) ( B 1 + B - 1 ) = 0 , ρ - 1 x β d B - 1 d x + [ α + j ( β 2 - ρ - 1 · ρ - 1 β ) ] × B - 1 + j ( κ 1 - j κ 1 ) B 0 + j ( κ 2 - j κ 2 ) B 1 = 0.
ρ 0 , ρ 1 , ρ 2 ,
ɛ = ɛ 0 - j ɛ 0 + ( ɛ 0 - j ɛ 1 ) cos ( K · r ) + ( ɛ 2 - j ɛ 0 ) cos ( 2 K · r ) .
η l = ρ l x ρ 0 x B l 2 .
exp ( - 2 α d cos θ 0 ) 0.76 ,
sin 2 ( κ 1 d cos θ 0 ) 0.5 ,
κ 1 d cos θ 0 < π 2 .
κ 1 d = ( cos θ 0 ) tan - 1 ( A - 1 2 / A 0 2 ) ,
α d = - ½ ( cos θ 0 ) log e [ A 0 2 / cos 2 ( κ 1 d / cos θ 0 ) ] .
κ 1 d cos θ 1 cos θ 2 = π / 2 ;

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