Abstract

The transmittance of a hologram is described in terms of Fresnel transformation. This description facilitates the establishment of requirements for the hologram-recording material and it leads naturally to a classification of holograms made from plane transmitting objects illuminated by a point source. Four subtypes are singled out: the Fresnel-transform hologram, geometrical shadowgram, quasi-Fourier-transform hologram, and Fourier-transform hologram. The type of hologram produced depends on the spatial-frequency content and over-all dimension of the object. Carrier-frequency and film requirements vary with the type of hologram. Experimental arrangements with and without lenses are briefly described.

© 1966 Optical Society of America

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References

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  1. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 55, 569 (1965).
    [Crossref]
  2. E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [Crossref]
  3. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [Crossref]
  4. J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
    [Crossref]
  5. J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).
    [Crossref]
  6. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  7. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  8. G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A58, 193 (1950–51).
  9. H. M. A. El Sum, “Reconstructed Wavefront Microscopy,” Ph.D. thesis, Stanford University (November1952).
  10. J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
    [Crossref]
  11. We also note recent work along these lines by H. W. Rose, J. Opt. Soc. Am. 55, 1565A (1965); and R. F. van Ligten, J. Opt. Soc. Am. 55, 1570A (1965); J. Opt. Soc. Am. 56, 1009 (1966).

1966 (1)

1965 (6)

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

We also note recent work along these lines by H. W. Rose, J. Opt. Soc. Am. 55, 1565A (1965); and R. F. van Ligten, J. Opt. Soc. Am. 55, 1570A (1965); J. Opt. Soc. Am. 56, 1009 (1966).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 55, 569 (1965).
[Crossref]

E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
[Crossref]

R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
[Crossref]

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

1962 (1)

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Armstrong, J. A.

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

El Sum, H. M. A.

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

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Haines, K. A.

Leith, E. N.

Meier, R. W.

Rogers, G. L.

G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A58, 193 (1950–51).

Rose, H. W.

We also note recent work along these lines by H. W. Rose, J. Opt. Soc. Am. 55, 1565A (1965); and R. F. van Ligten, J. Opt. Soc. Am. 55, 1570A (1965); J. Opt. Soc. Am. 56, 1009 (1966).

Upatnieks, J.

Winthrop, J. T.

J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).
[Crossref]

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Worthington, C. R.

J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 588 (1966).
[Crossref]

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

IBM J. Res. Develop. (1)

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

J. Opt. Soc. Am. (6)

Phys. Letters (1)

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

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

G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A58, 193 (1950–51).

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

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

The first stage of a holographic microscope. O=object, H=hologram recording plane.

Fig. 2
Fig. 2

The reconstruction stage of a holographic microscope. H=hologram, R=image plane.

Fig. 3
Fig. 3

Production of a Fresnel-transform hologram. The primary and reference sources lie in a plane parallel to the plane of the object. O=object, H=hologram recording plane.

Fig. 4
Fig. 4

Production of a Fresnel-transform hologram by means of Lloyd’s mirror. O=object, M=mirror, H=hologram recording plane.

Fig. 5
Fig. 5

Production of a quasi-Fourier-transform hologram without use of lenses. The object is placed over an aperture in a screen that is otherwise opaque except for a small pinhole. Radiation from the pinhole provides reference illumination. O=object, P=pinhole, H=hologram recording plane.

Fig. 6
Fig. 6

Production of a quasi-Fourier-transform hologram, using lenses. O=object, H=hologram recording plane.

Fig. 7
Fig. 7

Production of a Fourier-transform hologram. Object illumination converges to a point on the recording plane. O=object, H=hologram recording plane.

Tables (2)

Tables Icon

Table I Four special forms of the Fresnel-interference transform expression t ˜(r) and its Fourier transform T ˜(R).

Tables Icon

Table II Minimum values of carrier frequency Ω and film resolving power N for reconstruction of an object of radius a and maximum spatial frequency R0.

Equations (65)

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M 1 = 1 + q 1 / p 1 .
M 2 = 1 + q 2 / p 2 .
c = A c exp [ i π λ 1 - 1 ( p c + q 1 ) - 1 ( r - r c ) 2 ] ,
0 = A 0 exp [ i π λ 1 - 1 ( p 1 + q 1 ) - 1 r 2 ] .
τ ( ϱ ) = t ( ϱ ) * z ( ϱ ) .
z ( ϱ ) = - i B exp ( i π B ϱ 2 ) ,
B = M 1 / λ 1 q 1 ,
ϱ = r / M 1 .
t H = c 2 + 0 τ 2 + t R + t V ,
t V = t R * = C t ˜ ( r ) exp ( i 2 π Ω · r ) ,
t ˜ ( r ) = τ ( ϱ ) exp ( - i π λ 2 - 1 f I - 1 r 2 ) ,
1 / f I = ( μ / L 2 ) [ 1 / ( p c + q 1 ) - 1 / ( p 1 + q 1 ) ] ,
μ = λ 2 / λ 1 ,
Ω = ( 1 / λ 1 L ) [ r c / ( p c + q 1 ) ] ,
C = A c * A 0 exp [ - i π λ 1 - 1 ( p c + q 1 ) - 1 r c 2 ] .
M = L M 1 M 2
= ± ( μ / L ) ( q 2 / q 1 ) ,
± 1 / f G = 1 / p 2 + 1 / q 2 = ± μ / L 2 M 1 q 1 .
M = [ 1 / L M 1 ( L / μ ) ( q 1 / p 2 ) ] - 1 .
± 1 / f L = 1 / p 2 + 1 / q 2 = ± ( 1 / f G - 1 / f I ) .
1 / f L = μ / L 2 M c q 1 ,
M c = 1 + q 1 / p c .
M = [ 1 / L M c ( L / μ ) ( q 1 / p 2 ) ] - 1 .
M = L M c M 2 .
T ( R ) = t ( r ) exp ( - i 2 π r · R ) d σ ,
t ( r ) = T ( R ) exp ( i 2 π r · R ) d Σ .
Z ( R ) = exp ( - i π B - 1 R 2 ) .
τ ( ϱ ) = Z * ( B ϱ ) [ T ( B ϱ ) * Z ( B ϱ ) ] .
t ˜ ( r ) = exp ( - i π λ 2 - 1 f I - 1 r 2 ) [ t ( ϱ ) * z ( ϱ ) ] ,
t ˜ ( r ) = exp ( - i π λ 2 - 1 f I - 1 r 2 ) T ( R ) Z ( R ) × exp ( i 2 π ϱ · R ) d Σ .
t ˜ ( r ) = exp ( i π λ 2 - 1 f L - 1 r 2 ) [ T ( B ϱ ) * Z ( B ϱ ) ] ,
t ˜ ( r ) = exp ( i π λ 2 - 1 f L - 1 r 2 ) t ( r ) z ( r ) × exp ( - i 2 π B ϱ · r ) d σ .
l = B - 1 2 = λ 1 q 1 / M 1 - 1 2 .
l < R 0 - 1             geometrical-shadow region
l > a             far-field diffraction region
R 0 - 1 < l < a             Fresnel-diffraction region
t ˜ ( r ) = t ( ϱ ) exp ( - i π λ 2 - 1 f I - 1 r 2 ) .
t ˜ ( r ) = - i B T ( B ϱ ) exp ( i π λ 2 - 1 f L - 1 r 2 ) .
T ˜ ( R ) = exp ( i π λ 2 f I R 2 ) T ( M 1 U ) × exp { i π λ 2 f I [ ( 1 - f G f I - 1 ) U 2 - 2 U · R ] } d U .
T ˜ ( R ) = exp ( - i π λ 2 f L R 2 ) t ( - λ 1 q 1 U ) × exp { - i π λ 2 f L [ ( 1 - f G f L - 1 ) U 2 - 2 U · R ] } d U .
T ˜ ( R ) = exp ( i π λ 2 f I R 2 ) T ( M 1 U ) × exp [ i π λ 2 f I ( U 2 - 2 U · R ) ] d U .
T ˜ ( R ) = exp ( - i π λ 2 f L R 2 ) t ( - λ 1 q 1 U ) × exp [ - i π λ 2 f L ( U 2 - 2 U · R ) ] d U .
T ˜ ( R ) t ( - λ 2 f I M 1 - 1 R ) exp ( i π λ 2 f I R 2 ) ,
T ˜ ( R ) T ( M c R ) exp ( - i π λ 2 f L R 2 ) .
1 2 λ 2 f I - f G M 1 - 2 R 0 2 1 ,
1 2 λ 2 f L - f G ( λ 1 q 1 ) - 2 a 2 1.
λ 2 f I - f G M 1 - 2 = λ 2 - 1 f L - f G - 1 ( λ 1 q 1 ) 2 = l F 2 ,
l F = λ 1 q 1 / ( M c - M 1 ) 1 2 .
1 2 l F 2 R 0 2 1 ,
1 2 l F - 2 a 2 1 ,
l F < R 0 - 1             quasi-Fourier transform hologram
l F > a             Fresnel-transform hologram
R 0 - 1 < l F < a             general hologram .
1 2 l G S 2 R 0 2 1 ,
l G S = l M c / ( M c - M 1 ) 1 2 ,
1 2 l G S - 2 a 2 1.
T ˜ ( R ) T ( M 1 R ) ,
l G S < R 0 - 1             quasi-Fourier transform hologram
l G S > a             geometrical shadowgram
R 0 - 1 < l G S < a             general hologram .
T ˜ t ( - λ 1 q 1 R ) .
l F F = λ 1 q 1 / M c 1 2 .
l F F < R 0 - 1             Fourier-transform hologram
l F F > a             Fresnel-transform hologram
R 0 - 1 < l F F < a             general hologram .