Abstract

A qualitative analysis of various diffraction phenomena, visible in scattering volume gratings, like scattering rings, Kossel lines, dark scattering rings, is made for gratings including higher order harmonics and for gratings with strong coupling. An interference phenomenon, explained by the dynamical theory of diffraction, is demonstrated. In order to simplify the analysis of these phenomena and diffraction in volume gratings in general, a modification of the traditional Ewald construction is introduced. This modification is geometrically equivalent but physically more consistent and, for instance, defines how the observer is related to the wave vector sphere and also permits the location and extent of the intermodulation spectrum to be easily constructed. The construction is applied to the recording and readout of multiple recording by superposition in volume holograms as an illustration.

© 1978 Optical Society of America

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References

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  1. R. W. James, The Optical Principles of Diffraction of X-Rays, Vol. 2: The Crystalline State (G. Bell & Sons, London, 1950).
  2. E. Bauer, Elektronenbeugung, Verlag Moderne Industrie, München2 (1958).
  3. J. M. Moran, I. P. Kaminow, Appl. Opt. 12, 1964 (1973).
    [CrossRef] [PubMed]
  4. K. Biedermann, Optik 31, 367 (1970).
  5. W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).
  6. F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
    [CrossRef]
  7. F. P. Laming, Polym. Eng. Sci. 11, 421 (1971).
    [CrossRef]
  8. W. S. Colburn, K. A. Haines, Appl. Opt. 10, 1636 (1971).
    [CrossRef] [PubMed]
  9. V. Files, Appl. Phys. Lett. 19, 451 (1971).
    [CrossRef]
  10. K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
    [CrossRef]
  11. M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
    [CrossRef]
  12. S.-I. Ragnarsson, Opt. Commun. 14, 39 (1975).
    [CrossRef]
  13. P. P. Ewald, Rev. Mod. Phys. 37, 46 (1965).
    [CrossRef]
  14. E. J. Saccocio, Appl. Phys. 38, 3995 (1967).
  15. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  16. V. V. Aristov, V. Sh. Shekhtman, Sov. Phys. Usp. 14, 263 (1971).
    [CrossRef]
  17. M. R. B. Forshaw, Opt. Commun. 15, 218 (1975).
    [CrossRef]
  18. R. Kötz, Diploma work 1974 at Fritz-Haber-Institut der Max-Planck-Gesellschaft, 1 Berlin 33/Dahlem.
  19. K. Biedermann, Opt. Acta 22, 103 (1975).
    [CrossRef]
  20. M. R. B. Forshaw, Appl. Opt. 13, 2 (1974).
    [CrossRef] [PubMed]
  21. R. Magnusson, T. K. Gaylord, Appl. Opt. 13, 1545 (1974).
    [CrossRef]
  22. S. F. Su, T. K. Gaylord, J. Opt. Soc. Am. 65, 61 (1975).
    [CrossRef]
  23. M. Chang, N. George, Appl. Opt. 9, 713 (1970).
    [CrossRef] [PubMed]
  24. B. W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
    [CrossRef]
  25. E. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]
  26. M.R. B. Forshaw, Opt. Commun. 12, 279 (1974).
    [CrossRef]
  27. S.-I. Ragnarsson (to be published).

1975 (4)

S.-I. Ragnarsson, Opt. Commun. 14, 39 (1975).
[CrossRef]

M. R. B. Forshaw, Opt. Commun. 15, 218 (1975).
[CrossRef]

K. Biedermann, Opt. Acta 22, 103 (1975).
[CrossRef]

S. F. Su, T. K. Gaylord, J. Opt. Soc. Am. 65, 61 (1975).
[CrossRef]

1974 (3)

1973 (2)

1972 (2)

K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
[CrossRef]

W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).

1971 (4)

F. P. Laming, Polym. Eng. Sci. 11, 421 (1971).
[CrossRef]

W. S. Colburn, K. A. Haines, Appl. Opt. 10, 1636 (1971).
[CrossRef] [PubMed]

V. Files, Appl. Phys. Lett. 19, 451 (1971).
[CrossRef]

V. V. Aristov, V. Sh. Shekhtman, Sov. Phys. Usp. 14, 263 (1971).
[CrossRef]

1970 (2)

1969 (1)

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

1968 (1)

F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

1967 (1)

E. J. Saccocio, Appl. Phys. 38, 3995 (1967).

1966 (1)

1965 (1)

P. P. Ewald, Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

1964 (1)

B. W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Amodei, J. J.

W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).

Aristov, V. V.

V. V. Aristov, V. Sh. Shekhtman, Sov. Phys. Usp. 14, 263 (1971).
[CrossRef]

Batterman, B. W.

B. W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Bauer, E.

E. Bauer, Elektronenbeugung, Verlag Moderne Industrie, München2 (1958).

Biedermann, K.

K. Biedermann, Opt. Acta 22, 103 (1975).
[CrossRef]

K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
[CrossRef]

K. Biedermann, Optik 31, 367 (1970).

Chang, M.

Chen, F. S.

F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Colburn, W. S.

Cole, H.

B. W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Ewald, P. P.

P. P. Ewald, Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

Files, V.

V. Files, Appl. Phys. Lett. 19, 451 (1971).
[CrossRef]

Forshaw, M. R. B.

M. R. B. Forshaw, Opt. Commun. 15, 218 (1975).
[CrossRef]

M. R. B. Forshaw, Appl. Opt. 13, 2 (1974).
[CrossRef] [PubMed]

M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
[CrossRef]

Forshaw, M.R. B.

M.R. B. Forshaw, Opt. Commun. 12, 279 (1974).
[CrossRef]

Fraser, D. B.

F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Gaylord, T. K.

S. F. Su, T. K. Gaylord, J. Opt. Soc. Am. 65, 61 (1975).
[CrossRef]

R. Magnusson, T. K. Gaylord, Appl. Opt. 13, 1545 (1974).
[CrossRef]

George, N.

Haines, K. A.

James, R. W.

R. W. James, The Optical Principles of Diffraction of X-Rays, Vol. 2: The Crystalline State (G. Bell & Sons, London, 1950).

Kaminow, I. P.

Kogelnik, H.

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

Komlos, P.

K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
[CrossRef]

Kötz, R.

R. Kötz, Diploma work 1974 at Fritz-Haber-Institut der Max-Planck-Gesellschaft, 1 Berlin 33/Dahlem.

Kozma, A.

LaMacchia, J. T.

F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Laming, F. P.

F. P. Laming, Polym. Eng. Sci. 11, 421 (1971).
[CrossRef]

Leith, E.

Magnusson, R.

Marks, J.

Massey, N.

Moran, J. M.

Phillips, W.

W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).

Ragnarsson, S.-I.

S.-I. Ragnarsson, Opt. Commun. 14, 39 (1975).
[CrossRef]

K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
[CrossRef]

S.-I. Ragnarsson (to be published).

Saccocio, E. J.

E. J. Saccocio, Appl. Phys. 38, 3995 (1967).

Shekhtman, V. Sh.

V. V. Aristov, V. Sh. Shekhtman, Sov. Phys. Usp. 14, 263 (1971).
[CrossRef]

Staebler, D. L.

W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).

Su, S. F.

S. F. Su, T. K. Gaylord, J. Opt. Soc. Am. 65, 61 (1975).
[CrossRef]

Upatnieks, J.

Appl. Opt. (6)

Appl. Phys. (1)

E. J. Saccocio, Appl. Phys. 38, 3995 (1967).

Appl. Phys. Lett. (2)

F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

V. Files, Appl. Phys. Lett. 19, 451 (1971).
[CrossRef]

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. (1)

S. F. Su, T. K. Gaylord, J. Opt. Soc. Am. 65, 61 (1975).
[CrossRef]

Opt. Acta (1)

K. Biedermann, Opt. Acta 22, 103 (1975).
[CrossRef]

Opt. Commun. (5)

M.R. B. Forshaw, Opt. Commun. 12, 279 (1974).
[CrossRef]

M. R. B. Forshaw, Opt. Commun. 15, 218 (1975).
[CrossRef]

K. Biedermann, S.-I. Ragnarsson, P. Komlos, Opt. Commun. 6, 205 (1972).
[CrossRef]

M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
[CrossRef]

S.-I. Ragnarsson, Opt. Commun. 14, 39 (1975).
[CrossRef]

Optik (1)

K. Biedermann, Optik 31, 367 (1970).

Polym. Eng. Sci. (1)

F. P. Laming, Polym. Eng. Sci. 11, 421 (1971).
[CrossRef]

RCA Rev. (1)

W. Phillips, J. J. Amodei, D. L. Staebler, RCA Rev. 33, 94 (1972).

Rev. Mod. Phys. (2)

P. P. Ewald, Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

B. W. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Sov. Phys. Usp. (1)

V. V. Aristov, V. Sh. Shekhtman, Sov. Phys. Usp. 14, 263 (1971).
[CrossRef]

Other (4)

R. Kötz, Diploma work 1974 at Fritz-Haber-Institut der Max-Planck-Gesellschaft, 1 Berlin 33/Dahlem.

R. W. James, The Optical Principles of Diffraction of X-Rays, Vol. 2: The Crystalline State (G. Bell & Sons, London, 1950).

E. Bauer, Elektronenbeugung, Verlag Moderne Industrie, München2 (1958).

S.-I. Ragnarsson (to be published).

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Figures (20)

Fig. 1
Fig. 1

Ewald construction for a 1-D sinusoidal grating. The vectors Kr and Ko represent the reconstructing and the reconstructed wave vectors, respectively, and H represents the grating vector. O is the origin, and β is the Bragg angle.

Fig. 2
Fig. 2

Modified Ewald construction illustrating the recording of a two-beam interference hologram. The vectors Koi and Koi represent the two wavefields incident on each point in the medium, while wave vectors Krt and Kot represent the coherently forwardscattered wavefields. The vectors H and H* represent the time independent wave vectors, obtained as the difference between the vectors Koi and Krt and Kri and Kot, respectively, while the wave vectors representing the sum indicate the directions of the energy flow. The points A and A* and the origin O represent the spectral points of the sinusoidal grating, convolved with a weight function.

Fig. 3
Fig. 3

Recording of an extended object. The spatial spectrum is obtained as the difference between the reference wave vector Krt and all the vectors within the sector B01OB02. The spectrum is composed of the (spherical) segment A1A2, the origin O, and the conjugate part, segment A1*A2* each convolved with a weight function. The extent of the intermodulation noise is directly obtained as the area common to spheres centered at object extremum points B01, B02, C01, and C02 (the hatched area).

Fig. 4
Fig. 4

Readout of hologram. The observer located at Cr is reached by wavefields corresponding to the intersection between the spatial spectrum and a wave vector sphere centered at Cr. The effect of rotating the readout beam an angle γ is illustrated by the dashed vectors. The observer sees a reconstruction corresponding to the intersection between the sphere Sr and the weight function of the spectral point, i.e., A′.

Fig. 5
Fig. 5

Spatial spectrum of a hologram consisting of five incoherently superimposed extended objects, where the reference beam has been rotated an angle λ between each exposure. Readout with reference beam from direction Br3 is illustrated. Readout of conjugate image from direction Brk is illustrated with dashed vectors.

Fig. 6
Fig. 6

Spatial spectrum of a hologram consisting of four incoherently superimposed extended objects, where the hologram has been rotated an angle γ between each exposure.

Fig. 7
Fig. 7

Construction for explanation of scattering rings. The observer at Cr is reached by wavefields from directions corresponding to the intersection between the observation sphere Sr and the spectral surfaces resulting from the interference between scattered light and the two beams from Br1 and Br2. A beam incident from the direction Br creates two rings with angular diameters OCrP and OCrQ.

Fig. 8
Fig. 8

Normalized grating vectors H as a function of the different grating spacings Λ(φ) experienced by a wave propagating at an angle φ relative to the plane of incidence. The plane of incidence is perpendicular to the plane of the paper through Ao, O, and Ao*.

Fig. 9
Fig. 9

Nonsinusoidal grating f(x) composed of a fundamental sinusoidal grating (—), and a second harmonic (- - -). The second harmonic is decomposed into two virtual gratings tilted by the Bragg angle β relative to the fundamental grating.

Fig. 10
Fig. 10

Construction representing diffraction in a grating with a second harmonic. (a) The reconstructing beam impinges along the hologram normal. (b) The reconstructing beam impinges at an angle to the hologram normal corresponding to the Bragg angle for the second harmonic.

Fig. 11
Fig. 11

Scattering rings from a nonsinusoidal grating, rotated around a vertical axis, for various angles of incidence. The Bragg angle is 7.25°. In (c) is a strange bending of a Kossel line, which has not been explained, and a large diffuse scattering circle due to the near coincidence between the spheres Sr and S1 in Fig. 10(b). Near fulfillment of 1st order diffraction is seen in (d), and the fulfillment of 2nd order diffraction is seen in (h). The scattering ring around the 1st order diffraction is related to the 2nd harmonic of the grating.

Fig. 12
Fig. 12

Scattering rings from the hologram when rotated around a horizontal axis. Reconstructing beam incident normal to the hologram.

Fig. 13
Fig. 13

Microdensitometer recording of scattering rings and Kossel lines.

Fig. 14
Fig. 14

Scattering rings from the hologram when reconstructed with a monochromatic extended source. In the 1st order reconstruction in (a) the (horizontal) sinc-function intensity profile can be seen. In (b) on fulfillment of the Bragg condition a central maximum and secondary maxima of sinc-function profile are visible while, in (c) for the 2nd order reconstruction only the central maximum is visible.

Fig. 15
Fig. 15

Construction describing reconstruction with an extended monochromatic source as shown in Fig. 14(a).

Fig. 16
Fig. 16

Construction describing reconstruction with a white light point source as in Fig. 17(a).

Fig. 17
Fig. 17

Scattering rings from the hologram when reconstructed with a white light (tungsten) point source. Note the vertical sinc-function profile of the reconstruction.

Fig. 18
Fig. 18

Detail of Fig. 11(h), showing circular and linear interference fringes due to interference between the two sets of wavefields predicted by the dynamical theory.

Fig. 19
Fig. 19

Scattering rings from a hologram recorded in PMMA with λ = 365 nm and reconstructed with λ = 633 nm at two angles of incidence. The additional 3rd ring through the origin (the largest ring) can be explained, using Fig. 10(b) as the intersection between the observation sphere Sr (now with smaller radius than the spectrum spheres) and the sphere S1 at a point to the right of A1. Very diffuse scattering circle in Fig. 11(c) is explained likewise, but in this case Sr and S1 never intersect since they have the same radii. The figures are taken from Ref. 18.

Fig. 20
Fig. 20

Geometric parameters for deriving the relation between the refractive index difference and the angular fringe spacing in Fig. 18. This is the modified Ewald construction in the case of strong coupling (the dynamical theory). The linewidths of the dispersion surfaces indicate the relative intensity of the propagating wavefields for various angles of incidence.

Equations (7)

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f ( x ) = a + b cos 2 π x H c cos 4 π x H ,
F ( ξ ) = F { f ( x ) } = a δ ( ξ ) + b δ ( ξ H ) + b δ ( ξ + H ) c δ ( ξ 2 H ) c δ ( ξ 2 H ) ,
F ( ξ ) = c δ ( ξ + 2 H ) c δ ( ξ ) + b δ ( ξ + H ) + ( a + 2 c ) δ ( ξ ) + b δ ( ξ H ) c δ ( ξ ) c δ ( ξ 2 H ) ,
K sin β = K sin ( β ψ ) , ( n 2 n 1 ) sin β n 2 sin ψ cos β .
( Δ n ) / n = ψ / ( tan β ) .
q = 2 ψ 2 β = 2 arctan h / K 2 arctan H / 2 K 2 h H 2 K ψ 2 K tan β = ψ tan β .
( Δ n ) / n q .

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