Abstract

The factors which affect resolution in diffraction microscopy are studied. The inequality pλ>4nd/N is a necessary condition for making good holograms, where p is the distance between source and film, d is the diameter of the source, and λ is its wavelength, n is the number of recorded central bands of the diffraction pattern, and N is the resolving power of the film. The minimum theoretical resolvable distance between two point objects is limited by d and N but not by the wavelengths used in the analysis and reconstruction. Possible applications to the focusing of x-rays by Gabor techniques and by a single Fresnel zone plate are considered.

© 1952 Optical Society of America

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References

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  1. P. Kirkpatrick and A. V. Baez, J. Opt. Soc. Am. 38, 766 (1948).
    [Crossref] [PubMed]
  2. P. Kirkpatrick, J. Opt. Soc. Am. 39, 796 (1949).
    [Crossref]
  3. Ora E. Myers, Am. J. Phys. 19, 359 (1951).
    [Crossref]
  4. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
    [Crossref]
  5. G. L. Rogers, Nature 166, 237 (1950).
    [Crossref]
  6. G. L. Rogers, Proc. Roy. Soc. (Edin.) A58, Part III, 193 (1950–51).

1951 (1)

Ora E. Myers, Am. J. Phys. 19, 359 (1951).
[Crossref]

1950 (1)

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

1949 (2)

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

P. Kirkpatrick, J. Opt. Soc. Am. 39, 796 (1949).
[Crossref]

1948 (1)

Baez, A. V.

Gabor, D.

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

Kirkpatrick, P.

Myers, Ora E.

Ora E. Myers, Am. J. Phys. 19, 359 (1951).
[Crossref]

Rogers, G. L.

G. L. Rogers, Proc. Roy. Soc. (Edin.) A58, Part III, 193 (1950–51).

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

Am. J. Phys. (1)

Ora E. Myers, Am. J. Phys. 19, 359 (1951).
[Crossref]

J. Opt. Soc. Am. (2)

Nature (1)

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

Proc. Roy. Soc. (Edin.) (1)

G. L. Rogers, Proc. Roy. Soc. (Edin.) A58, Part III, 193 (1950–51).

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

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

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Figures (9)

Fig. 1
Fig. 1

Top: A photographic enlargement of a microscope scale. Center: A hologram made from the same scale with filtered visible light. Bottom: A reconstruction made from the above hologram. A Cenco point light source (arc) was used at pa=35 cm.

Fig. 2
Fig. 2

The transmission, T, of a Fresnel zone plate is plotted against the distance, x, from the center, O. The radius of the first circle is x1, and the radii of successive circles are proportional to the square roots of the integers.

Fig. 3
Fig. 3

An extended source, SS′, illuminates a point object, O, to produce a diffraction pattern in the hologram plane, H. The center of the plate pattern due to S is P, and the center of the pattern due to S′ is P′.

Fig. 4
Fig. 4

This figure illustrates how the reconstruction deteriorates as p decreases. At the top is a photographic enlargement of the object, consisting of two pin points and one pin head. Beneath it on the left is a series of holograms made from the above object. Reading down, they were made at pa=90 cm, qa=30 cm; pa=50 cm, qa=16.7 cm; and pa=30 cm, qa=10 cm, respectively. On the right are the corresponding reconstructions made at pr=90 cm, pr=49.86 cm; pr=50 cm, qr=27.6 cm; and pr=30 cm, qr=20.0 cm, respectively. See Figs. 8 and 9 for meanings of symbols.

Fig. 5
Fig. 5

An idealized diagram of the central rings of the hologram of a point object. The rings get successively narrower proceeding outward until they cannot be adequately recorded on the photographic film.

Fig. 6
Fig. 6

An idealized diagram of the grain structure of a photographic film. The thin outer rings of the Fresnel zone plate cannot be resolved on this grainy film. Thus grain size limits the effective aperture of the zone plate.

Fig. 7
Fig. 7

If the bands shown were barely resolved on a film of resolving power N, the distance AB would be 1/N.

Fig. 8
Fig. 8

Sa is a point source of a wavelength λa illuminating two point objects separated by a distance y. On the hologram film, H, the centers of the diffraction patterns due to the point objects are P1 and P2, separated by a distance z.

Fig. 9
Fig. 9

Sr is a point source of wavelength λr illuminating the enlarged and processed hologram, H, on which the separation between zone plate centers is now Mz. Q1 and Q2 are the centers of the images produced by reconstruction. To the right of the reconstruction plane R are the diffraction patterns produced in the reconstruction plane indicating objects “barely resolved.” The angular separation between the first maximum and the first minimum is β.

Equations (27)

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P P / S S = q / ( p - q ) .
d = k n [ p λ ( m - 1 ) ] 1 2 ,
d < k n [ p λ ( m - 1 ) ] 1 2             or             ( m - 1 ) > d 2 / p k 2 n λ .
p > d 2 / [ ( m - 1 ) k n 2 λ ] - 1 ,
{ [ 2 ( n + 1 ) ] 1 2 - ( 2 n + 1 ) 1 2 } x 1 = k n x 1 .
k n x 1 > 1 / 2 N k n [ p λ / ( m - 1 ) ] 1 2 > 1 / 2 N m - 1 < 4 N 2 p λ k n 2 .
q r β / M z = ( p r + q r ) / p r             and             z / y = p a / ( p a - q a ) y = ( p r q r p r + q r ) ( p a - q a p a q a ) 1.22 λ r q a M a r y = ( f r / f a ) ( 1.22 λ r q a / M a r ) ,
f r / f a = x 1 r 2 λ a / x 1 a 2 λ r , f r / f a = M 2 λ a / λ r ,
y = 1.22 λ a q a M / a r .
a r = 2 x 1 r ( 2 n - 1 ) 1 2 2 x 1 r ( 2 n ) 1 2
a r 2 x 1 r ( 2 n ) 1 2 < 2 M x 1 a 2 N
y 1 = 1.22 λ a q a / 2 N x 1 a 2 ;
x 1 a 2 = p a λ a / ( m - 1 ) ,
y 1 = 0.61 ( m - 1 ) / m N .
k 2 n > d 2 / p λ ( m - 1 ) .
8 ( n + 1 ) < p a λ a ( m - 1 ) / d 2
n < p a λ a ( m - 1 ) / 8 d 2
a r 2 x 1 r ( 2 n ) 1 2 < x 1 r [ p a λ a ( m - 1 ) / d ] 1 2
y 2 = 1.22 d λ a q a M / x 1 r [ p a λ a ( m - 1 ) ] 1 2 .
x 1 r = M x 1 a = M ( f a λ a ) 1 2             and             f a = p a / ( m - 1 )
y 2 = 1.22 d / m .
m - 1 > d 2 / p λ k n 2
m - 1 < 4 N 2 p λ k n 2 .
p λ > d / 2 N k n 2 .
λ a = 10 - 8 cm N = 10 4 cm - 1 d = 10 - 3 cm p = 500 cm .
p λ = 5.0 × 10 - 6             and             d / 2 N k n 2 = 2.4 × 10 - 6
f 1 / f 2 = x 1 2 λ 2 / x 2 2 λ 1 = ( M ) 2 λ 2 / λ 1 .