Abstract

Dynamic theory of volume holography is used to calculate the variations in the thickness direction of the hologram-constituting refractive-index modulation and the externally observable effects of these variations in electrooptic materials. It is shown that thick holographic gratings may exhibit significant amplitude and grating phase variations with thickness including amplitude sign reversal. These nonuniformities strongly affect holographic grating recording and readout characteristics such as maximum possible diffraction efficiency and angular selectivity. Thus a variety of grating applications will be affected by these nonuniformities.

© 1976 Optical Society of America

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References

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  1. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [CrossRef]
  2. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  3. Y. Ninomiya, J. Opt. Soc. Am. 63, 1124 (1973).
    [CrossRef]
  4. R. Magnusson, T. K. Gaylord, J. Appl. Phys. 47, 190 (1976).
    [CrossRef]
  5. D. Kermisch, J. Opt. Soc. Am. 61, 1202 (1971).
    [CrossRef]
  6. W. J. Tomlinson, Appl. Opt. 14, 2456 (1975).
    [CrossRef] [PubMed]
  7. G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).
  8. S. F. Su, T. K. Gaylord, J. Appl. Phys. 46, 5208 (1975).
    [CrossRef]
  9. S. F. Su, T. K. Gaylord, J. Appl. Phys. 47, 2757 (1976).
    [CrossRef]
  10. D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
    [CrossRef]
  11. J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).
  12. D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
    [CrossRef]
  13. L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
    [CrossRef]
  14. R. Magnusson, Ph.D. Thesis, Georgia Institute of Technology (1976).
  15. The finding reported here that the intensity inequality of the incident writing beams indeed has significant effects on thick hologram behavior seems to be in contradiction with a statement by Ninomiya3 that the intensity ratio of the incident beams does not influence unslanted transmission holograms.
  16. D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
    [CrossRef]
  17. N. Uchida, J. Opt. Soc. Am. 63, 280 (1973).
    [CrossRef]
  18. From a theoretical point of view, it is interesting to note that H. Kogelnik, Bell Syst. Tech. J. 55, 109 (1976), in analyzing Bragg filtering of structures with nonuniform coupling coefficients and period in the direction of propagation, obtains similar sidelobe obliteration (in his reflectivity function) for (linear) nonuniformity in the coupling coefficient and asymmetric behavior relative to the Bragg frequency for (quadratic) nonuniformity in the period.

1976 (3)

S. F. Su, T. K. Gaylord, J. Appl. Phys. 47, 2757 (1976).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Appl. Phys. 47, 190 (1976).
[CrossRef]

From a theoretical point of view, it is interesting to note that H. Kogelnik, Bell Syst. Tech. J. 55, 109 (1976), in analyzing Bragg filtering of structures with nonuniform coupling coefficients and period in the direction of propagation, obtains similar sidelobe obliteration (in his reflectivity function) for (linear) nonuniformity in the coupling coefficient and asymmetric behavior relative to the Bragg frequency for (quadratic) nonuniformity in the period.

1975 (4)

W. J. Tomlinson, Appl. Opt. 14, 2456 (1975).
[CrossRef] [PubMed]

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

S. F. Su, T. K. Gaylord, J. Appl. Phys. 46, 5208 (1975).
[CrossRef]

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

1974 (1)

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

1973 (2)

1972 (2)

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

1971 (1)

1969 (2)

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

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

1966 (1)

Alig, R. C.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

Alphonse, G. A.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

Amodei, J. J.

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Burckhardt, C. B.

Cornish, W. D.

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

Gaylord, T. K.

S. F. Su, T. K. Gaylord, J. Appl. Phys. 47, 2757 (1976).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Appl. Phys. 47, 190 (1976).
[CrossRef]

S. F. Su, T. K. Gaylord, J. Appl. Phys. 46, 5208 (1975).
[CrossRef]

Kermisch, D.

Kogelnik, H.

From a theoretical point of view, it is interesting to note that H. Kogelnik, Bell Syst. Tech. J. 55, 109 (1976), in analyzing Bragg filtering of structures with nonuniform coupling coefficients and period in the direction of propagation, obtains similar sidelobe obliteration (in his reflectivity function) for (linear) nonuniformity in the coupling coefficient and asymmetric behavior relative to the Bragg frequency for (quadratic) nonuniformity in the period.

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

Magnusson, R.

R. Magnusson, T. K. Gaylord, J. Appl. Phys. 47, 190 (1976).
[CrossRef]

R. Magnusson, Ph.D. Thesis, Georgia Institute of Technology (1976).

Ninomiya, Y.

Phillips, W.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

Staebler, D. L.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Su, S. F.

S. F. Su, T. K. Gaylord, J. Appl. Phys. 47, 2757 (1976).
[CrossRef]

S. F. Su, T. K. Gaylord, J. Appl. Phys. 46, 5208 (1975).
[CrossRef]

Thewalt, M. L. W.

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

Tomlinson, W. J.

Uchida, N.

Vahey, D. W.

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

Wong, W. K. Y.

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

Young, L.

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Young, W. K. Y. Wong, M. L. W. Thewalt, W. D. Cornish, Appl. Phys. Lett. 24, 264 (1974).
[CrossRef]

Bell Syst. Tech. J. (2)

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

From a theoretical point of view, it is interesting to note that H. Kogelnik, Bell Syst. Tech. J. 55, 109 (1976), in analyzing Bragg filtering of structures with nonuniform coupling coefficients and period in the direction of propagation, obtains similar sidelobe obliteration (in his reflectivity function) for (linear) nonuniformity in the coupling coefficient and asymmetric behavior relative to the Bragg frequency for (quadratic) nonuniformity in the period.

J. Appl. Phys. (5)

R. Magnusson, T. K. Gaylord, J. Appl. Phys. 47, 190 (1976).
[CrossRef]

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

S. F. Su, T. K. Gaylord, J. Appl. Phys. 46, 5208 (1975).
[CrossRef]

S. F. Su, T. K. Gaylord, J. Appl. Phys. 47, 2757 (1976).
[CrossRef]

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

J. Opt. Soc. Am. (5)

RCA Rev. (2)

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, RCA Rev. 36, 213 (1975).

Other (2)

R. Magnusson, Ph.D. Thesis, Georgia Institute of Technology (1976).

The finding reported here that the intensity inequality of the incident writing beams indeed has significant effects on thick hologram behavior seems to be in contradiction with a statement by Ninomiya3 that the intensity ratio of the incident beams does not influence unslanted transmission holograms.

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Figures (4)

Fig. 1
Fig. 1

Calculated refractive-index amplitude variations with normalized thickness along an amplitude peak in the holographic grating pattern. The hologram thickness is 2.00 mm, a = 10−11 (V/m)−2 sec−1, αo′ = αo = 102 m−1, and θ = 2.23° (5.00° angle of incidence external to a crystal of no = 2.243). The curves are calculated at the times corresponding to the first maxima of the recording characteristics shown in the inset.

Fig. 2
Fig. 2

Calculated angular selectivity characteristics for the gratings of Fig. 1. Vertical dashed line indicates position of Bragg angle.

Fig. 3
Fig. 3

Peak amplitudes of the refractive-index gratings (solid curves) as functions of distance through the grating after an exposure time T = 25 sec. Also shown are the positions of the grating peaks (dashed curves) as functions of thickness indicating the bending of the grating. All parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Calculated angular selectivity characteristics for the gratings of Fig. 3. Vertical dashed line shows position of Bragg angle.

Equations (5)

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d n / d E = a = constant ,
E = 0 T E ¯ · E ¯ * d t ,
n 1 ( x , z , T ) = 2 a cos 2 θ ( M 2 + N 2 ) 1 / 2 × cos { ( σ ¯ - ρ ¯ ) · r ¯ + ϕ n + cos - 1 [ M / ( M 2 + N 2 ) 1 / 2 ] } ,
M M ( z , T ) 0 T Re { R S * } d t , N N ( z , T ) 0 T Im { R S * } d t ,
n 1 p ( z , T ) = 2 a cos 2 θ ( M 2 + N 2 ) 1 / 2 .

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