Abstract

An analysis of the function of a spatial domain filter employed in recent optical syntheses of molecular images from electron holograms is presented. The device is effective in ridding the primary image region of the naturally arising background intensity which, unless filtered, can severely obscure the image. The filter also introduces artifacts, however, which may confuse the interpretation of images, and excessive filtering reduces resolving power. It is shown how to calculate the effect of the filter variables in order to secure a reasonable compromise between resolution, background noise, and spurious rings.

© 1979 Optical Society of America

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References

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  1. L. S. Bartell and R. D. Johnson, “Molecular images by electron-wave holography,” Nature 268, 707–708 (1977).In prior work, G. Saxon [“Division of wavefront side-band Fresnel holography with electrons,” Optik 35, 195–210 (1972)] achieved a resolving power of 103Å by a different variant of electron holography.
    [Crossref]
  2. L. S. Bartell and W. J. Gignac, “Images of gas molecules by electron holography. II. Experiment and comparison with theory,” J. Chem. Phys. 70, 3958–3964 (1979).
    [Crossref]
  3. G. S. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966), p. 119.
  4. L. S. Bartell, “Images of gas molecules by electron holography. I. Theory.” J. Chem. Phys. 70, 3952–3957 (1979).
    [Crossref]
  5. L. S. Bartell, “Images of gas atoms by electron holography. I. Theory; II. Experiment and comparison with theory,” Optik 43, 373–393, 403–418 (1975).
  6. E. N. Leith, “Photographic film as an element of a coherent optical system,” Photogr. Sci. Eng. 6, 75–80 (1962).
  7. M. Born and E. Wolf, Principles of Optics, 2nd Ed. (Pergamon, New York, 1964), Chap. 8.
  8. F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), pp. 93, 102.
  9. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, 4th Ed. (Academic, New York, 1965), p. 667.
  10. And, indeed, did in the case of images of CF3OOCF3 molecules in Ref. 2.
  11. Data from F. B. Clippard and L. S. Bartell, “Molecular structures of arsenic trifluoride and arsenic pentafluoride as determined by electron diffraction,” Inorg. Chem. 9, 805–811 (1970).
    [Crossref]
  12. Note that the F… F image is not a primary holographic image arising from the electron interference fringes produced when the subject wave mixes with the reference wave. It arises from interference fringes produced by the mixing of waves from one part of the subject with waves from another part. See Refs. 2 and 4 for details.

1979 (2)

L. S. Bartell and W. J. Gignac, “Images of gas molecules by electron holography. II. Experiment and comparison with theory,” J. Chem. Phys. 70, 3958–3964 (1979).
[Crossref]

L. S. Bartell, “Images of gas molecules by electron holography. I. Theory.” J. Chem. Phys. 70, 3952–3957 (1979).
[Crossref]

1977 (1)

L. S. Bartell and R. D. Johnson, “Molecular images by electron-wave holography,” Nature 268, 707–708 (1977).In prior work, G. Saxon [“Division of wavefront side-band Fresnel holography with electrons,” Optik 35, 195–210 (1972)] achieved a resolving power of 103Å by a different variant of electron holography.
[Crossref]

1975 (1)

L. S. Bartell, “Images of gas atoms by electron holography. I. Theory; II. Experiment and comparison with theory,” Optik 43, 373–393, 403–418 (1975).

1970 (1)

Data from F. B. Clippard and L. S. Bartell, “Molecular structures of arsenic trifluoride and arsenic pentafluoride as determined by electron diffraction,” Inorg. Chem. 9, 805–811 (1970).
[Crossref]

1962 (1)

E. N. Leith, “Photographic film as an element of a coherent optical system,” Photogr. Sci. Eng. 6, 75–80 (1962).

Bartell, L. S.

L. S. Bartell and W. J. Gignac, “Images of gas molecules by electron holography. II. Experiment and comparison with theory,” J. Chem. Phys. 70, 3958–3964 (1979).
[Crossref]

L. S. Bartell, “Images of gas molecules by electron holography. I. Theory.” J. Chem. Phys. 70, 3952–3957 (1979).
[Crossref]

L. S. Bartell and R. D. Johnson, “Molecular images by electron-wave holography,” Nature 268, 707–708 (1977).In prior work, G. Saxon [“Division of wavefront side-band Fresnel holography with electrons,” Optik 35, 195–210 (1972)] achieved a resolving power of 103Å by a different variant of electron holography.
[Crossref]

L. S. Bartell, “Images of gas atoms by electron holography. I. Theory; II. Experiment and comparison with theory,” Optik 43, 373–393, 403–418 (1975).

Data from F. B. Clippard and L. S. Bartell, “Molecular structures of arsenic trifluoride and arsenic pentafluoride as determined by electron diffraction,” Inorg. Chem. 9, 805–811 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd Ed. (Pergamon, New York, 1964), Chap. 8.

Bowman, F.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), pp. 93, 102.

Clippard, F. B.

Data from F. B. Clippard and L. S. Bartell, “Molecular structures of arsenic trifluoride and arsenic pentafluoride as determined by electron diffraction,” Inorg. Chem. 9, 805–811 (1970).
[Crossref]

Gignac, W. J.

L. S. Bartell and W. J. Gignac, “Images of gas molecules by electron holography. II. Experiment and comparison with theory,” J. Chem. Phys. 70, 3958–3964 (1979).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, 4th Ed. (Academic, New York, 1965), p. 667.

Johnson, R. D.

L. S. Bartell and R. D. Johnson, “Molecular images by electron-wave holography,” Nature 268, 707–708 (1977).In prior work, G. Saxon [“Division of wavefront side-band Fresnel holography with electrons,” Optik 35, 195–210 (1972)] achieved a resolving power of 103Å by a different variant of electron holography.
[Crossref]

Leith, E. N.

E. N. Leith, “Photographic film as an element of a coherent optical system,” Photogr. Sci. Eng. 6, 75–80 (1962).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, 4th Ed. (Academic, New York, 1965), p. 667.

Stroke, G. S.

G. S. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966), p. 119.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd Ed. (Pergamon, New York, 1964), Chap. 8.

Inorg. Chem. (1)

Data from F. B. Clippard and L. S. Bartell, “Molecular structures of arsenic trifluoride and arsenic pentafluoride as determined by electron diffraction,” Inorg. Chem. 9, 805–811 (1970).
[Crossref]

J. Chem. Phys. (2)

L. S. Bartell and W. J. Gignac, “Images of gas molecules by electron holography. II. Experiment and comparison with theory,” J. Chem. Phys. 70, 3958–3964 (1979).
[Crossref]

L. S. Bartell, “Images of gas molecules by electron holography. I. Theory.” J. Chem. Phys. 70, 3952–3957 (1979).
[Crossref]

Nature (1)

L. S. Bartell and R. D. Johnson, “Molecular images by electron-wave holography,” Nature 268, 707–708 (1977).In prior work, G. Saxon [“Division of wavefront side-band Fresnel holography with electrons,” Optik 35, 195–210 (1972)] achieved a resolving power of 103Å by a different variant of electron holography.
[Crossref]

Optik (1)

L. S. Bartell, “Images of gas atoms by electron holography. I. Theory; II. Experiment and comparison with theory,” Optik 43, 373–393, 403–418 (1975).

Photogr. Sci. Eng. (1)

E. N. Leith, “Photographic film as an element of a coherent optical system,” Photogr. Sci. Eng. 6, 75–80 (1962).

Other (6)

M. Born and E. Wolf, Principles of Optics, 2nd Ed. (Pergamon, New York, 1964), Chap. 8.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), pp. 93, 102.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, 4th Ed. (Academic, New York, 1965), p. 667.

And, indeed, did in the case of images of CF3OOCF3 molecules in Ref. 2.

G. S. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York, 1966), p. 119.

Note that the F… F image is not a primary holographic image arising from the electron interference fringes produced when the subject wave mixes with the reference wave. It arises from interference fringes produced by the mixing of waves from one part of the subject with waves from another part. See Refs. 2 and 4 for details.

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Figures (8)

FIG. 1
FIG. 1

Schematic drawings of optical reconstruction stage of holographic microscope. (a) Filter configuration adopted in experiments. (b) Equivalent configuration treated in text.

FIG. 2
FIG. 2

Calculated characteristic intensities F 0 2 received at position I2, Fig. 1, if H1 is of uniform transmission. Solid curve, pattern obtained with filter parameters of α = 14 and m = 0.9. Dashed curve, Airy pattern if filter elements S and D2 are removed. Abscissa given both in terms of natural dimensionless variable ξ (see text) and impact parameter in Å at the holographic subject [for se(max) = 21.6 Å−1].

FIG. 3
FIG. 3

Points corresponding to intensities F 0 2 surviving optical filter as a function of ξ or b (see caption, Fig. 2) plotted for various diaphragm openings m at stop radius α = 11. For clarity of comparison, the oscillating functions F 0 2 are plotted only at their maxima, and smooth curves are drawn through these points. Because image intensities tend to be proportional to m2, intensities passed by the filter are here divided by m2 to give a truer idea of relative signal to noise ratios. The Airy intensity plotted corresponds to m = 1 and α = 0. The vertical scale for all curves is such that the intensity of the Airy pattern goes to unity at ξ = 0.

FIG. 4
FIG. 4

Points corresponding to ratios (I/IAiry) between intensities F 0 2 surviving optical filter and the Airy intensities obtained if stop S is removed but diaphragm D2 is retained. Ratios are shown for a diaphragm opening of m = 0.9 at various stop radii α. As in Fig. 3, only the curve maxima are shown, and these are connected by smooth curves. Moreover, again for simplicity, the Airy intensities IAiry in the denominator are taken as the smooth curve through the successive maxima instead of the actual local values. In viewing this figure it should be noted that IAiry, at large ξ, falls off as ξ−3.

FIG. 5
FIG. 5

Absorbance of electron diffraction pattern of AsF5 transmitted through an r3 sector-filter; plotted as a function of plate radius for Le = 21 cm, λe = 0.06 Å.

FIG. 6
FIG. 6

Airy amplitude F 0 and AsF5 image amplitude F m passed by filter; calculated, assuming α = 15, m = 0.9, for a negative photographic copy of the plate illustrated in Fig. 5.

FIG. 7
FIG. 7

Calculated AsF5 (plus Airy) image intensities ( F 0 + F m ) 2. Solid curve for filter parameters α = 15 and m = 0.9. Dashed curve assumes no filtering and is reduced to half the vertical scale of the solid curve in order to fit in the same frame.

FIG. 8
FIG. 8

Experimental AsF5 (plus Airy) image intensities. Hologram a negative copy of the plate of Fig. 5 reduced 16-fold. (On left) Photographed without filter. (On right) Photographed with filter; α ≈ 15, m ≈ 0.9. Scale bar, 1 Å.

Equations (17)

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b j = λ L j s j / 2 π .
H 1 F I 1 ( s 2 ) = N 1 0 2 π a 0 a 1 exp ( i s 2 b 1 ) U ( b 1 ) b 1 d b 1 d ϕ 1 = 2 π N 1 a 0 a 1 J 0 ( s 2 b 1 ) U ( b 1 ) b 1 d b 1 ,
I 1 F H 2 ( s 3 ) = N 2 0 2 π a 2 exp ( i s 3 b 2 ) × H 1 F I 1 ( b 2 ) b 2 d b 2 d ϕ 2 = 2 π N 2 a 2 J 0 ( s 3 b 2 ) H 1 F I 1 ( b 2 ) b 2 d b 2 = ( 2 π ) 2 N 1 N 2 a 2 a 1 a 0 a 1 J 0 ( s 3 b 2 ) × J 0 ( s 2 b 1 ) U ( b 1 ) b 1 b 2 d b 1 d b 2 ,
H 2 F I 2 ( s 4 ) = N 3 0 2 π 0 a 3 exp ( i s 4 b 3 ) × I 1 F H 2 ( b 3 ) b 3 d b 3 d ϕ 3 = 2 π N 3 0 a 3 J 0 ( s 4 b 3 ) I 1 F H 2 ( b 3 ) b 3 d b 3 = ( 2 π ) 3 N 3 N 2 N 1 0 a 3 a 2 a 0 a 1 J 0 ( s 4 b 3 ) J 0 ( s 3 b 2 ) × J 0 ( s 2 b 1 ) U ( b 1 ) b 1 b 2 b 3 d b 1 d b 2 d b 3 ,
U ( b 1 ) = U 0 ( b 1 ) + U m ( b 1 ) ,
H 1 F I 1 ( s 2 ) = ( N 1 U 01 ° L 2 / b 2 ) [ a 1 J 1 ( β b 2 ) a 0 J 1 ( a 0 β b 2 / a 1 ) ] ,
[ H 2 F I 2 ( s 4 ) ] 0 = ( U 01 ° K / m ) × α I u [ a 1 J 1 ( υ ) a 0 J 1 ( a 0 υ / a 1 ) ] d υ ,
I u = 0 m J 0 ( ξ u ) J 0 ( υ u ) u d u
F 0 ( ξ ) = K U 0 1 ° [ ( h 1 h 0 ) + ( H 1 + H 0 ) ] ,
h k = a 1 m k J 1 ( m k ξ ) / ξ
H k = m a 1 k [ J 0 ( m k ξ k ) I 1 J 1 ( m k ξ k ) I 2 ]
I 1 ( m k , α k ) = 0 α k J 1 ( υ k ) J 1 ( m k υ k ) ( υ k 2 ξ k 2 ) 1 υ k d υ k
I 2 ( m k , α k ) = 0 α k J 1 ( υ k ) J 0 ( m k υ k ) ( υ k 2 ξ k 2 ) 1 ξ k d υ k ,
F i j ( ξ ) = K U i j ° [ ( h j h i ) + ( H j H i ) ] ,
α = z 1 a 2 / a 2 A ,
F 0 ( ξ ) = K U 0 1 ° a 1 [ J 1 ( ξ ) / ξ ] ,
ξ = 2 π L e L 3 b 4 R λ L 2 L 4 tan { 2 [ arcsin ( λ e s e max 4 π ) ] } S e max b 4 / ( R λ L 2 L 4 / λ e L e L 3 ) = ( s e max / M ) b 4 = s e max b ,