Abstract

A property of duality is used that describes the interaction between image-point astigmatism and apparent image-surface deformation when a holographic image is modified by reconstruction geometry.

© 1981 Optical Society of America

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References

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  1. J. A. Armstrong, “Fresnel holograms: their imaging properties and aberrations,” IBM J. Res. Dev. 9, 171–178 (1965).
    [CrossRef]
  2. E. N. Leith, J. Upatnieks, and K. A. Haines, “Microscopy by wavefront reconstruction,” J. Opt. Soc. Am. 55, 981–986 (1965).
    [CrossRef]
  3. R. W. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  4. E. B. Champagne, “Non paraxial imaging, magnification and aberration properties in holography,” J. Opt. Soc. Am. 57, 51–55 (1967).
    [CrossRef]
  5. I. A. Abramowitz and J. M. Ballantyne, “Evaluation of hologram aberrations by ray tracing,” J. Opt. Soc. Am. 57, 1522–1526 (1967).
    [CrossRef]
  6. J. F. Miles, “Imaging and magnification properties in holography,” Opt. Acta 19, 165–186 (1972).
    [CrossRef]
  7. I. Prikryl, “Studying hologram imaginery by a ray tracing method,” Opt. Acta 19, 623–631 (1972).
    [CrossRef]
  8. R. Dändliker, K. Hess, and Th. Sidler, “Astigmatic pencils of rays reconstructed from holograms,” Isr. J. Technol. (in press).
  9. J. Nowak and M. Zajac, Geometric analysis of holographic imaging,” Optik 55, 93–103 (1980).
  10. W. Schumann and M. Dubas, Holographic Interferometry from the Scope of Deformation Analysis of Opaque Bodies, Springer Series in Optical Science, No. 16 (Springer-Verlag, Berlin, 1979).
  11. K. A. Stetson, “The argument of the fringe function in hologram interferometry of general deformations,” Optik 31, 576–591 (1970).

1980 (1)

J. Nowak and M. Zajac, Geometric analysis of holographic imaging,” Optik 55, 93–103 (1980).

1972 (2)

J. F. Miles, “Imaging and magnification properties in holography,” Opt. Acta 19, 165–186 (1972).
[CrossRef]

I. Prikryl, “Studying hologram imaginery by a ray tracing method,” Opt. Acta 19, 623–631 (1972).
[CrossRef]

1970 (1)

K. A. Stetson, “The argument of the fringe function in hologram interferometry of general deformations,” Optik 31, 576–591 (1970).

1967 (2)

1965 (3)

Abramowitz, I. A.

Armstrong, J. A.

J. A. Armstrong, “Fresnel holograms: their imaging properties and aberrations,” IBM J. Res. Dev. 9, 171–178 (1965).
[CrossRef]

Ballantyne, J. M.

Champagne, E. B.

Dändliker, R.

R. Dändliker, K. Hess, and Th. Sidler, “Astigmatic pencils of rays reconstructed from holograms,” Isr. J. Technol. (in press).

Dubas, M.

W. Schumann and M. Dubas, Holographic Interferometry from the Scope of Deformation Analysis of Opaque Bodies, Springer Series in Optical Science, No. 16 (Springer-Verlag, Berlin, 1979).

Haines, K. A.

Hess, K.

R. Dändliker, K. Hess, and Th. Sidler, “Astigmatic pencils of rays reconstructed from holograms,” Isr. J. Technol. (in press).

Leith, E. N.

Meier, R. W.

Miles, J. F.

J. F. Miles, “Imaging and magnification properties in holography,” Opt. Acta 19, 165–186 (1972).
[CrossRef]

Nowak, J.

J. Nowak and M. Zajac, Geometric analysis of holographic imaging,” Optik 55, 93–103 (1980).

Prikryl, I.

I. Prikryl, “Studying hologram imaginery by a ray tracing method,” Opt. Acta 19, 623–631 (1972).
[CrossRef]

Schumann, W.

W. Schumann and M. Dubas, Holographic Interferometry from the Scope of Deformation Analysis of Opaque Bodies, Springer Series in Optical Science, No. 16 (Springer-Verlag, Berlin, 1979).

Sidler, Th.

R. Dändliker, K. Hess, and Th. Sidler, “Astigmatic pencils of rays reconstructed from holograms,” Isr. J. Technol. (in press).

Stetson, K. A.

K. A. Stetson, “The argument of the fringe function in hologram interferometry of general deformations,” Optik 31, 576–591 (1970).

Upatnieks, J.

Zajac, M.

J. Nowak and M. Zajac, Geometric analysis of holographic imaging,” Optik 55, 93–103 (1980).

IBM J. Res. Dev. (1)

J. A. Armstrong, “Fresnel holograms: their imaging properties and aberrations,” IBM J. Res. Dev. 9, 171–178 (1965).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Acta (2)

J. F. Miles, “Imaging and magnification properties in holography,” Opt. Acta 19, 165–186 (1972).
[CrossRef]

I. Prikryl, “Studying hologram imaginery by a ray tracing method,” Opt. Acta 19, 623–631 (1972).
[CrossRef]

Optik (2)

J. Nowak and M. Zajac, Geometric analysis of holographic imaging,” Optik 55, 93–103 (1980).

K. A. Stetson, “The argument of the fringe function in hologram interferometry of general deformations,” Optik 31, 576–591 (1970).

Other (2)

W. Schumann and M. Dubas, Holographic Interferometry from the Scope of Deformation Analysis of Opaque Bodies, Springer Series in Optical Science, No. 16 (Springer-Verlag, Berlin, 1979).

R. Dändliker, K. Hess, and Th. Sidler, “Astigmatic pencils of rays reconstructed from holograms,” Isr. J. Technol. (in press).

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Figures (2)

Fig. 1
Fig. 1

Holographic imaging in the case of a modification. P, object point at the recording; R ˜, position of the observer at the reconstruction; Ĥ, H , hologram points in both positions; n ˆ , n , unit normals; k, k ˜, c, c ˜, unit direction vectors. The hologram is usually plane, but it could also be curved. (a) Formation of the astigmatic image { P ˜} of the point P of the object. (b) Apparent deformation of the neighborhood { P ¯} of P on the surface of the object.

Fig. 2
Fig. 2

The analog of the ray tracing. 〈R1,R2〉, astigmatic intervals; r1, r2, analog of the focal lines.

Equations (31)

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θ P = 2 π λ ˜ ( p ˜ - q ˜ ) - 2 π λ ( p - q )
θ R ˜ = - 2 π λ ˜ ( f ˜ + q ˜ ) + 2 π λ ( f + q ) ,
θ = 2 π λ ˜ ( l ˜ - q ˜ ) - 2 π λ ( l - q ) ,
d θ = 2 π λ ˜ d r · n ( l ˜ - q ˜ ) - 2 π λ d r ˆ · n ˆ ( l - q ) .
n = N ,             n ˆ = N ˆ
d r = F ˆ d r ˆ             or             d r ˆ = F d r ,
N F ˆ N ˆ = N F ˆ = F ˆ N ˆ = å α â α .
1 λ ˜ N ˆ F ˆ T ( k ˜ - c ˜ ) - 1 λ N ˆ ( k - c ) = 0
1 λ ˜ N ( k ˜ - c ˜ ) - 1 λ N F T ( k - c ) = 0 ,
d 2 θ = 2 π λ ˜ d r · [ n n ( l ˜ - q ˜ ) ] d r - 2 π λ d r ˆ · [ n ˆ n ˆ ( l - q ) ] d r ˆ + 2 π λ ˜ b · n ( l ˜ - q ˜ ) d r 2 - 2 π λ b ˆ · n ˆ ( l - q ) d r ˆ 2 .
( n n l ˜ ) N = N [ ( N ) l ˜ ] N + N ( l ˜ ) N = B ( n · k ˜ ) + 1 l ˜ N K ˜ N .
d 2 θ = 2 π λ ˜ d r · N [ 1 l ˜ K ˜ - 1 q ˜ C ˜ + B n · ( k ˜ - c ˜ ) ] N d r - 2 π λ d r ˆ · N ˆ [ 1 l K - 1 q C + B ˆ n ˆ · ( k - c ) ] N ˆ d r ˆ .
d k ˜ = m ˜ d α ˜ ,             d k = m d α
d r = l ˜ M T d k ˜ ,             d r ˆ = l M ˆ T d k ,
M = I - n k ˜ n · k ˜ ,             M ˆ = I - n ˆ k n ˆ · k ,
d 2 θ P = p ˜ 2 2 π λ ˜ m ˜ · ( 1 p ˜ K ˜ - T ˜ ) m ˜ d a ˜ 2 ,
T ˜ = λ ˜ M { 1 λ ˜ [ 1 q ˜ C ˜ - B n · ( k ˜ - c ˜ ) ] + 1 λ F T [ 1 p K - 1 q C + B ˆ n ˆ · ( k - c ) ] F } M T ,
d 2 θ R ˜ = f 2 2 π λ m · ( 1 f K - T ) m d α 2 ,
T = - λ M ˆ { 1 λ [ 1 q C - B ˆ n ˆ · ( k - c ) ] + 1 λ ˜ F ˆ T [ - 1 f ˜ K ˜ - 1 q ˜ C ˜ + B n · ( k ˜ - c ˜ ) ] F ˆ } M ˆ T .
1 p ˜ = m ˜ · T ˜ m ˜ ,
1 f = m · Tm ,
K d r = - ( f + p ) d k + χ E k d k ,
d k · d r = - f p ( 1 f + 1 p ) d α 2 = - f p m · ( T + 1 p K ) m d α 2 ,
d k · d r = - f ˜ p d k · ( T + 1 p K ) F M T d k ˜ ,
d k ¯ = f T d k ,
K d r = - ( p d k ¯ + f d k )
d r = - f ˜ p M T ( T + 1 p K ) F M T d k ˜ .
M ˆ F ˆ T = M ˆ N ˆ F ˆ T = M ˆ F ˆ T N = M ˆ F ˆ T M , , F M T F ˆ M ˆ T = F ˆ - 1 F ˆ M ˆ T = M ˆ T , .
T + 1 p K = λ λ ˜ M ˆ F ˆ T ( T ˜ + 1 f ˜ K ˜ ) F ˆ M ˆ T
λ ˜ λ M F T ( T + 1 p K ) F M T = T ˜ + 1 f ˜ K ˜ .
d r = - λ λ ˜ f ˜ p M T M ˆ F ˆ T ( T ˜ + 1 f ˜ K ˜ ) d k ˜ .