Abstract

The ideal photographic material in holography would have a linear relationship between amplitude transmittance and exposure. Here we study the case where this relationship can be described instead by a polynomial. This nonlinearity in reconstruction gives rise to some extra images, autocorrelations, autoconvolutions, and ambiguity functions of the object, which may be found superposed on normal images, or spatially separated both laterally and in depth.

© 1968 Optical Society of America

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References

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  1. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]
  2. A. Kozma, in Introduction to Optical Data Processing (McGraw-Hill Book Co., New York, 1967), Vol. 1, Ch. 9.
  3. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [Crossref]
  4. A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
    [Crossref] [PubMed]
  5. J. W. Goodman, J. Opt. Soc. Am. 57, 560A (1967).
    [Crossref]
  6. G. R. Knight, Appl. Opt. 7, 205 (1968).
    [Crossref] [PubMed]
  7. A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).
  8. H. J. Gerritsen, Appl. Phys. Letters 10, 239 (1967).
    [Crossref]
  9. H. J. Gerritsen, E. G. Ramberg, and S. Freeman in Modern Optics, Symposium 1967, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1968; John Wiley & Sons., Inc., New York).
  10. O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968).
    [Crossref]
  11. In the following, an asterisk between two functions indicates a convolution.

1968 (3)

1967 (4)

A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).

H. J. Gerritsen, Appl. Phys. Letters 10, 239 (1967).
[Crossref]

A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
[Crossref] [PubMed]

J. W. Goodman, J. Opt. Soc. Am. 57, 560A (1967).
[Crossref]

1966 (1)

Bryngdahl, O.

Freeman, S.

H. J. Gerritsen, E. G. Ramberg, and S. Freeman in Modern Optics, Symposium 1967, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1968; John Wiley & Sons., Inc., New York).

Friesem, A. A.

Gerritsen, H. J.

H. J. Gerritsen, Appl. Phys. Letters 10, 239 (1967).
[Crossref]

H. J. Gerritsen, E. G. Ramberg, and S. Freeman in Modern Optics, Symposium 1967, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1968; John Wiley & Sons., Inc., New York).

Goodman, J. W.

Knight, G. R.

Komar, A. P.

A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).

Kozma, A.

A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
[Crossref]

A. Kozma, in Introduction to Optical Data Processing (McGraw-Hill Book Co., New York, 1967), Vol. 1, Ch. 9.

Lohmann, A. W.

Ramberg, E. G.

H. J. Gerritsen, E. G. Ramberg, and S. Freeman in Modern Optics, Symposium 1967, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1968; John Wiley & Sons., Inc., New York).

Stabnikov, M. V.

A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).

Turukhano, B. G.

A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).

Zelenka, J. S.

Appl. Opt. (2)

Appl. Phys. Letters (1)

H. J. Gerritsen, Appl. Phys. Letters 10, 239 (1967).
[Crossref]

J. Opt. Soc. Am. (4)

Sov. Phys.—Dokl. (1)

A. P. Komar, M. V. Stabnikov, and B. G. Turukhano, Sov. Phys.—Dokl. 11, 712 (1967).

Other (3)

H. J. Gerritsen, E. G. Ramberg, and S. Freeman in Modern Optics, Symposium 1967, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1968; John Wiley & Sons., Inc., New York).

A. Kozma, in Introduction to Optical Data Processing (McGraw-Hill Book Co., New York, 1967), Vol. 1, Ch. 9.

In the following, an asterisk between two functions indicates a convolution.

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Figures (10)

Fig. 1
Fig. 1

Reconstructions of real images from nonlinear holograms of (a) a transparent letter on an opaque background and (b) an opaque letter on a transparent background. The upper records of (a) and (b) are taken at the object–hologram distance from the hologram, the middle at one half of that distance, and the lower at one third of that distance.

Fig. 2
Fig. 2

Reconstructions of real images from nonlinear holograms of (a) a transparent diffuse letter on an opaque background and (b) an opaque letter on a transparent diffuse background. The upper records of (a) and (b) are taken at the object–hologram distance from the hologram, the middle at one half of that distance, and the lower at one third of that distance.

Fig. 3
Fig. 3

Reconstructions of real images from nonlinear holograms of (a) a transparent letter and an adjacent strong point source on an opaque background and (b) an opaque letter on a transparent background and an adjacent strong point source. The upper records of (a) and (b) are taken at the object–hologram distance from the hologram, the middle at one half of that distance, and the lower at one third of that distance.

Fig. 4
Fig. 4

Reconstruction of real images from nonlinear hologram of a transparent diffuse letter and an adjacent strong point source on an opaque background. The upper record was taken at the object–hologram distance from the hologram, the middle at one half of that distance, and the lower at one third of that distance.

Fig. 5
Fig. 5

Reconstruction of real images from nonlinear hologram of an opaque letter on a transparent diffuse background and an adjacent strong point source. The upper record was taken at the object–hologram distance from the hologram, the middle at one half of that distance, and the lower at one third of that distance.

Fig. 6
Fig. 6

Reconstructions of real images from a hologram recorded with severe nonlinearity. The object is a transparent diffuse letter on an opaque background. (a) reconstruction in the first diffraction order recorded at the object to hologram distance behind the hologram; (b) reconstruction in the second diffraction order at half that distance and enlarged two times; (c) reconstruction in the third diffraction order at one third of the object to hologram distance and enlarged three times.

Fig. 7
Fig. 7

Reconstructions of real images from a hologram recorded with severe nonlinearity. The object is a transparent diffuse letter and an adjacent strong point source on an opaque background. (a) reconstruction in the first diffraction order recorded at the object to hologram distance behind the hologram; (b) reconstruction in the second diffraction order at half that distance and enlarged two times; (c) reconstruction in the third diffraction order at one third of the object to hologram distance and enlarged three times.

Fig. 8
Fig. 8

Schematic representation of the letter L and the configuration of its autocorrelation and autoconvolution functions in the same scale.

Fig. 9
Fig. 9

Reconstructions corresponding to those in Fig. 6 but with the letter L as object.

Fig. 10
Fig. 10

Reconstructions corresponding to those in Fig. 7 but with the letter L and a strong point source as object.

Equations (24)

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T ( E ) = T 0 + T 1 E .
T ( E ) = n = 0 n = 3 T n E n .
E ( x , y ) = K exp ( 2 π i α x ) + u ( x , y ) 2 .
exp ( 2 π i α x ) = p ;             u * = v ;             u 2 = φ ;             K = 1.
E = 1 + p - 1 u + p v + φ ;
T ( x , y ) = t 0 + ( p - 1 u + p v ) t 1 + ( p - 2 u 2 + p 2 v 2 ) t 2 + ( p - 3 u 3 + p 3 v 3 ) t 3 ; t 0 = T 0 + ( 1 + φ ) T 1 + ( 1 + 4 φ + φ 2 ) T 2 + ( 1 + 9 φ + 9 φ 2 + φ 3 ) T 3 ; t 1 = T 1 + 2 ( 1 + φ ) T 2 + 3 ( 1 + 3 φ + φ 2 ) T 3 ; t 2 = T 2 + 3 ( 1 + φ ) T 3 ; t 3 = T 3 .
T ( x , y ) = 1 + cos ( 2 π α x - ψ ) = 1 + cos [ 2 π α ( x - ψ / 2 π α ) ] .
I = 1 + cos [ 2 π α x - 3 ψ ( x , y ) ]
u ( x , y ) = ũ 0 ( x λ f , y λ f ) ; ũ 0 ( ν , μ ) = u 0 ( x , y ) exp [ - 2 π i ( x ν + y μ ) ] d x d y .
T ( x , y ) exp [ 2 π i ( x x + y y ) / ( λ f ) ] d x d y ,
u u 0 ( x ) ; v u 0 * ( - x ) ; u 2 u 0 ( x ) u 0 ( x - x ) d x ; v 2 u 0 * ( x ) u 0 * ( - x - x ) d x ; φ u 0 ( x ) u 0 * ( x - x ) d x ; u 3 u 0 ( x ) u 0 ( x ) u 0 ( x - x - x ) d x d x ; u φ u 0 ( x ) u 0 * ( x ) u 0 ( x - x + x ) d x d x ; u 2 φ u 0 ( x ) u 0 ( x ) u 0 ( x ) × u 0 * ( - x + x + x + x ) d x d x d x ; φ 2 u 0 ( x ) u 0 * ( x ) u 0 ( x ) × u 0 * ( - x + x - x + x ) d x d x d x .
u 0 ( x , y ) = δ ( x , y + y 0 ) + Δ ( x , y ) ;
ũ 0 ( ν , μ ) = exp ( 2 π i μ y 0 ) + Δ ˜ ( ν , μ ) = q + Δ ˜ ( ν , μ ) .
u = q + Δ ˜ ;             v = q - 1 + Δ ˜ * ; u 2 = q 2 + 2 q Δ ˜ + ;             φ = 1 + q Δ ˜ * + q - 1 Δ ˜ + ; u 3 = q 3 + 3 q 2 Δ ˜ + ;             u φ = q + q 2 Δ ˜ * + 2 Δ ˜ + ; φ 2 = 1 + 2 q Δ ˜ * + 2 q - 1 Δ ˜ + ; u 2 φ = q 2 + 3 q Δ ˜ + q 3 Δ ˜ * + ; u φ 2 = q + 2 q 2 Δ ˜ * + 3 Δ ˜ + .
Δ ( x , y ) [ T 1 + 6 T 2 + 30 T 3 ] + Δ * ( - x , - y - 2 y 0 ) [ 2 T 2 + 15 T 3 ] .
Δ ( x - 3 λ f α , y + 2 y 0 ) ;             Δ * ( - x - 3 λ f α , - y - 2 y 0 ) .
u ( x , y ) = u 0 ( x , y ) S ( x - x , y - y ; z 1 ) d x d y ;             S ( x , y ; z 1 ) ( λ z 1 ) - 1 exp [ i π ( x 2 + y 2 ) / λ z 1 ] ,
T ( x , y ) S ( x - x , y - y ; z ) d x d y .
ϕ = ( j , k , l ) a j k l x j x k x l ; 0 j , k , l 2 ;             0 j + k + l 2.
S ( x , y ; z ) = ( λ z ) - 1 exp [ i π ( x 2 + y 2 ) / λ z ] ; S ( x , y ; z ) = S ( - x , y ; z ) = S ( x , - y ; z ) = S ( - x , - y ; z ) ; S ( x , y ; z ) = S * ( x , y ; - z ) ; a 2 S ( a x , a y ; z ) = S ( x , y ; z / a 2 ) ; lim ( z = 0 ) S ( x , y ; z ) = i δ ( x , y ) ; - S ( x , y ; z ) d x d y = λ z ; - S ( x , y ; z ) S ( x - x , y - y ; z ) d x d y = - i S ( x , y ; z + z ) ; S n ( x , y ; z ) = ( λ z ) 1 - n S ( x , y ; z / n ) ; u ( x , y ) = - u 0 ( x , y ) S ( x - x , y - y ; z 1 ) d x d y ; u 0 ( x , y ) = - u ( x , y ) S ( x - x , y - y ; - z 1 ) d x d y ; - u 0 ( x , y ) exp ( 2 π i α x ) S ( x - x , y - y ; z 1 ) d x d y = u ( x - α λ z 1 , y ) exp { 2 π i ( α x - α 2 λ z 1 / 2 ) } .
u 0 ( x + x ) u 0 ( x - x ) exp [ 2 π i ( 2 x 2 + x 2 ) / λ z 1 ] d x .
exp ( 4 π i x 2 / λ z 1 ) u 0 ( x + x ) u 0 ( x - x ) d x .
u * 2 ( x , y ) = u 0 * ( x , y ) u 0 * ( x , y ) S * ( x , y ; 2 z 1 )
S * ( x , y ; 2 z 1 ) = 1 4 S * ( x / 2 , y / 2 ; z 1 / 2 )