Abstract

Magnification by the wavefront-reconstruction imaging method is discussed. An analysis is given of the aberrations which arise in this type of imagery. Conditions are derived which lead to aberration-free reconstructions.

© 1965 Optical Society of America

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References

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  1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949);Proc. Phys. Soc. (London) B64, 449 (1951).
  2. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  3. Paper presented at Washington Conference on Electron Microscopy, National Bureau of Standards, November 1951.
  4. H. M. A. El-Sum, “Reconstructed Wavefront Microscopy,” Ph.D. thesis, Stanford University (November1952).
  5. D. Gabor, “Light and Information”, in Progress in Optics. Vol. 1, E. Wolf, ed. (North Holland Publishing Co., Amsterdam, 1961).
    [Crossref]
  6. M. E. Haine and T. Mulvey, J. Opt. Soc. Am. 42, 763 (1952).
    [Crossref]
  7. A. V. Baez, J. Opt. Soc. Am. 42, 756 (1952).
    [Crossref]
  8. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 55, 569 (1965).
    [Crossref]
  9. As one of the r’s becomes infinite, the phase delay associated with it likewise becomes infinite. However, we ignore a constant phase delay, and consider only relative phase delays of different parts of the incident wavefront.
  10. K. Kamiya, Sci. Light 12, 35 (1963).
  11. The hologram was first analyzed as a Fresnel zone plate by G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).
  12. R. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [Crossref]
  13. This problem is touched upon briefly by Gabor, Ref. 1, p. 483.
  14. For a discussion of this type of optical system, see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 144–159.
  15. The use of the term pseudoscopic to describe these properties was pointed out to the authors by F. W. Brock and R. Innes.
  16. H. M. A. El-Sum and A. V. Baez, Phys. Rev. 99, 624 (1955).
  17. Encyclopedia of Microscopy, G. L. Clark, ed. (Reinhold Publishing Corp., New York, 1961).
  18. These problems were first called to our attention by C. R. Worthington of the University of Michigan, Physics Department, in private conversation in 1963.J. T. Winthrop and C. R. Worthington [Phys. Letters 15, 124 (1965)] have proposed Fourier-transform holograms as a method of solving these problems.
    [Crossref]

1965 (2)

1964 (1)

1963 (1)

K. Kamiya, Sci. Light 12, 35 (1963).

1955 (1)

H. M. A. El-Sum and A. V. Baez, Phys. Rev. 99, 624 (1955).

1952 (3)

The hologram was first analyzed as a Fresnel zone plate by G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

M. E. Haine and T. Mulvey, J. Opt. Soc. Am. 42, 763 (1952).
[Crossref]

A. V. Baez, J. Opt. Soc. Am. 42, 756 (1952).
[Crossref]

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949);Proc. Phys. Soc. (London) B64, 449 (1951).

Baez, A. V.

H. M. A. El-Sum and A. V. Baez, Phys. Rev. 99, 624 (1955).

A. V. Baez, J. Opt. Soc. Am. 42, 756 (1952).
[Crossref]

Born, M.

For a discussion of this type of optical system, see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 144–159.

Brock, F. W.

The use of the term pseudoscopic to describe these properties was pointed out to the authors by F. W. Brock and R. Innes.

El-Sum, H. M. A.

H. M. A. El-Sum and A. V. Baez, Phys. Rev. 99, 624 (1955).

H. M. A. El-Sum, “Reconstructed Wavefront Microscopy,” Ph.D. thesis, Stanford University (November1952).

Gabor,

This problem is touched upon briefly by Gabor, Ref. 1, p. 483.

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949);Proc. Phys. Soc. (London) B64, 449 (1951).

D. Gabor, “Light and Information”, in Progress in Optics. Vol. 1, E. Wolf, ed. (North Holland Publishing Co., Amsterdam, 1961).
[Crossref]

Haine, M. E.

Innes, R.

The use of the term pseudoscopic to describe these properties was pointed out to the authors by F. W. Brock and R. Innes.

Kamiya, K.

K. Kamiya, Sci. Light 12, 35 (1963).

Leith, E.

Meier, R.

Mulvey, T.

Rogers, G. L.

The hologram was first analyzed as a Fresnel zone plate by G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

Upatnieks, J.

Wolf, E.

For a discussion of this type of optical system, see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 144–159.

Worthington, C. R.

These problems were first called to our attention by C. R. Worthington of the University of Michigan, Physics Department, in private conversation in 1963.J. T. Winthrop and C. R. Worthington [Phys. Letters 15, 124 (1965)] have proposed Fourier-transform holograms as a method of solving these problems.
[Crossref]

J. Opt. Soc. Am. (5)

Phys. Rev. (1)

H. M. A. El-Sum and A. V. Baez, Phys. Rev. 99, 624 (1955).

Proc. Roy. Soc. (Edinburgh) (1)

The hologram was first analyzed as a Fresnel zone plate by G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949);Proc. Phys. Soc. (London) B64, 449 (1951).

Sci. Light (1)

K. Kamiya, Sci. Light 12, 35 (1963).

Other (9)

Encyclopedia of Microscopy, G. L. Clark, ed. (Reinhold Publishing Corp., New York, 1961).

These problems were first called to our attention by C. R. Worthington of the University of Michigan, Physics Department, in private conversation in 1963.J. T. Winthrop and C. R. Worthington [Phys. Letters 15, 124 (1965)] have proposed Fourier-transform holograms as a method of solving these problems.
[Crossref]

This problem is touched upon briefly by Gabor, Ref. 1, p. 483.

For a discussion of this type of optical system, see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 144–159.

The use of the term pseudoscopic to describe these properties was pointed out to the authors by F. W. Brock and R. Innes.

As one of the r’s becomes infinite, the phase delay associated with it likewise becomes infinite. However, we ignore a constant phase delay, and consider only relative phase delays of different parts of the incident wavefront.

Paper presented at Washington Conference on Electron Microscopy, National Bureau of Standards, November 1951.

H. M. A. El-Sum, “Reconstructed Wavefront Microscopy,” Ph.D. thesis, Stanford University (November1952).

D. Gabor, “Light and Information”, in Progress in Optics. Vol. 1, E. Wolf, ed. (North Holland Publishing Co., Amsterdam, 1961).
[Crossref]

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Figures (1)

F. 1
F. 1

Two-beam Gabor microscope.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

U ( x 2 , y 2 ) = i / ( λ 1 z 1 ) s ( x 1 , y 1 ) exp [ i k 1 r 0 ( x 1 , y 1 ) i k 1 r 1 ( x 2 x 1 , y 2 y 1 ) ] d x 1 d y 1 ,
U 0 = a 0 e k 1 r 2 ( x 2 , y 2 ) ,
T = | U 0 + U | 2 = | U 0 | 2 + | U | 2 + 2 Re ( U 0 U * ) .
{ U 0 U * U 0 * U } = i / ( λ 1 z 1 ) a e i k 1 r 2 s e ± i k 1 ( r 0 + r 1 ) d x 1 d y 1 ,
U r = C e i k 1 r 2 [ s e ± i k 1 ( r 0 + r 1 ) d x 1 d x 1 ] × e i k 2 ( r a + r b ) d x 2 d y 2 ,
U r = C s ( x 1 , y 1 ) f 1 f 2 d x 1 d y 1 d x 2 d y 2 ,
f 1 = exp [ i ( k 1 r 2 k 1 r 0 k 1 r 1 ± k 2 r a ) ] ,
f 2 = e i k 2 r b .
f 1 = exp [ i k 2 r b ( x 2 x 1 M , y 2 y 1 M ) + i h ( x 1 , y 1 ) ] ,
s ( x / M , y / M ) f 1 d x d y
f 1 = exp [ i k 2 r 1 ( x 2 x 1 , y 2 y 1 ) ] exp [ i h ( x 1 , y 1 ) ] ,
r 0 = [ Z 0 2 + ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 ] 1 2 = Z 0 + 1 2 ( x 1 x 0 ) 2 / Z 0 + 1 2 + ( y 1 y 0 ) 2 / Z 0
f 2 = exp i 2 [ k 2 ( x 3 x 2 ) 2 / Z b ] ,
f 1 = exp i 2 [ k 1 x 2 2 Z 2 ± k 1 ( x 1 x 0 ) 2 Z 0 ± k 1 ( x 2 x 1 ) 2 Z 1 k 2 x 2 2 Z a ] ,
f 1 = exp i 2 [ k 2 ( x 2 x 1 M ) 2 Z b + i h ( x 1 , y 1 ) ] .
Z b = ± Z 1 Z a Z 2 Z a Z 2 ( λ 2 / λ 1 ) Z 1 Z 2 Z 1 Z a ( λ 2 / λ 1 )
M = Z b = ( 1 Z 1 Z a λ 1 λ 2 Z 1 Z 2 ) 1 .
U k = b 0 + b 1 cos b 2 x 2 ,
U m = c 0 + c 1 cos k 1 ( r 1 r 2 ) = c 0 + c 1 cos k 1 { [ Z 1 2 + ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] 1 2 [ Z 2 2 + x 2 2 + y 2 2 ] 1 2 } .
Z a ± ( λ 1 / λ 2 ) 1 3 Z 1 ,
Δ ρ = 1 2 r b σ 3 [ Z b 3 ( λ 2 / λ 1 ) Z 2 3 ] .
M x = p Z 2 Z a ( λ 2 / λ 1 ) Z 2 Z a ( λ 2 / λ 1 ) Z 1 Z a ( λ 2 / λ 1 ) p 2 Z 1 Z 2 ,
M Z = d Z b / d Z 1 = ( λ 1 / λ 2 ) M x 2 .
Z 2 / Z a = ( λ 2 / λ 1 ) p 2
M x = p ,
Δ ρ = 1 2 r b σ 3 [ Z a 3 + Z b 3 ( λ 2 / p 4 λ 1 ) ( Z 1 3 Z 2 3 ) ] ,
[ Δ y 3 r b σ 2 ( y a Z a 3 + y 3 Z b 3 λ 2 p 3 λ 1 y 1 Z 1 3 ) ] 2 + [ Δ x 3 r b ] 2 = [ σ 2 2 ( y 3 Z a 3 + y a Z b 3 λ 2 p 3 λ 1 y 1 Z 1 3 ) ] 2 .
( Δ y 3 / { 3 2 σ r b ( y a 2 Z a 3 + y 3 2 Z b 3 λ 2 p 2 λ 1 y 1 2 Z 1 3 ) } ) 2 + ( Δ x 3 / { 1 2 σ r b [ y a 2 Z a 3 + y 3 2 Z b 3 λ 2 p 2 λ 1 y 1 2 Z 1 3 ] } ) 2 = 1 ,