Abstract

Fringe sharpening occurs in the higher diffraction orders of nonlinear two-exposure holographic interferograms. The fringe functions in these various orders are modified by diffraction. At the Talbot distance the fringes replicate those near the plate, and a variety of fringe patterns can be obtained at different propagation distances.

© 1976 Optical Society of America

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References

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  1. K. Matsumoto and M. Takashima, J. Opt. Soc. Am. 60, 30 (1970).
    [Crossref]
  2. O. Bryngdahl, J. Opt. Soc. Am. 59, 142 (1969).
    [Crossref]
  3. C. H. F. Velzel, Opt. Commun. 2, 289 (1970).
    [Crossref]
  4. S. Toyooka, App. Opt. 13, 2014 (1974).
    [Crossref]
  5. K. Matsumoto, J. Opt. Soc. Am. 59, 777 (1969).
    [Crossref]
  6. O. Bryngdahl, J. Opt. Soc. Am. 59, 1171 (1969).
    [Crossref]
  7. K. Matsumoto, J. Opt. Soc. Am. 61, 176 (1971).
    [Crossref]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

1974 (1)

S. Toyooka, App. Opt. 13, 2014 (1974).
[Crossref]

1971 (1)

1970 (2)

1969 (3)

App. Opt. (1)

S. Toyooka, App. Opt. 13, 2014 (1974).
[Crossref]

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

C. H. F. Velzel, Opt. Commun. 2, 289 (1970).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

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Figures (4)

FIG. 1
FIG. 1

Fringe functions for diffraction orders 1, 3, and 5 of a nonlinear two-exposure hologram. These functions are given by Eq. (4) and normalized with respect to I0 = 2n.

FIG. 2
FIG. 2

Fringe patterns in diffraction orders 1, 3, and 5 of a nonlinear two-exposure hologram. Each photograph was recorded on Polaroid PN-55 film with an exposure of 0.04 μJ/cm−2. (a) Order 1; (b) Order 3; (c) Order 5.

FIG. 3
FIG. 3

Fringe functions in diffraction orders 1, 2, and 3 of a nonlinear two-exposure hologram, evaluated at the distance 1 2 Z T = 1 2 L 2 / λ from the hologram. These functions are obtained by evaluation of the intensity corresponding to the amplitude given by Eq. (11), normalized with respect to the maximum intensity I0.

FIG. 4
FIG. 4

Fringe patterns in diffraction order 3 of a nonlinear two-exposure hologram. (a) Photographed at the distance ZT = L2/λ from the hologram. (b) Photographed at the distance 1 2 Z T from the hologram.

Equations (8)

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t ( x , y ) = n = 0 b n E n .
t ( x , y ) = n = 0 b n ( U O + U R 2 + U O + U R 2 ) n ,
U O ( x , y ) = exp [ - i ϕ 1 ( x , y ) ] ,             U O ( x , y ) = exp [ - i ϕ ( x , y ) ] .
I n ( x , y ) = [ 1 + cos Δ ϕ ( x , y ) ] n ,             n = 1 , 2 , ,
U n ( x , y ; 0 ) = [ 1 + exp ( i 2 π X / L ) ] n ,
H ( f x , f y ) = exp [ - i π λ z ( f x 2 + f y 2 ) ] .
U n ( x , y ; z ) = k = 0 n ( n k ) exp ( - i π λ z k 2 / L 2 ) exp ( i 2 π k X / L ) .
I n [ x , y ; ( 2 m + 1 ) z T ] = [ 1 - cos ( 2 π X / L ) ] n ,