Abstract

The lateral coherence properties of two illumination modes in electron interferometry and holography are investigated within the framework of the mutual coherence function. It is shown that the results obtained can be considered particular realizations of the general, anisotropic, Gaussian Schell model, which plays an important role in the classical coherence theory in optics. Another property of this model is described: the ratio between the coherence and illumination areas is constant along every section of the beam. This invariant parameter is shown to be related to another geometric optical invariant of the beam, i.e., its etendue. Numerical calculations showing the relation between illumination and coherence areas in the image, Fresnel, and far-field (Fraunhofer) regions are presented and discussed with regard to the experimental implications.

© 1990 Optical Society of America

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  1. G. Fontaine, “Electron sources,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. P. Goedgebuer, C. Malgrange, J. Ch. Viénot, R. H. Wade, eds. (Springer-Verlag, Berlin, 1980), pp. 28–36.
    [CrossRef]
  2. D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–156.
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).
  4. P. W. Hawkes, “Coherence in electron optics,” Adv. Opt. Electron Microsc. 7, 101–184 (1978).
  5. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  6. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schellmodel sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  7. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  8. P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
    [CrossRef]
  9. G. Möllenstedt, H. Düker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
    [CrossRef]
  10. G. Möllenstedt, H. Düker, “Fresnelscher Interferenzversuch mit einem Biprisma für Elektronenwellen,” Naturwissenschaften 42, 41 (1955).
    [CrossRef]
  11. G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
    [CrossRef]
  12. G. Möllenstedt, H. Wahl, “Elektronenholographie und Rekonstruktion mit Laserlicht,” Naturwissenschaften 55, 340–341 (1968).
    [CrossRef]
  13. H. Wahl, “Bildebenenholographie mit Elektronen,” Ph.D. dissertation (University of Tübingen, Tübingen, Federal Republic of Germany, 1975).
  14. R. Lauer, “Electron microscopy and holography employing coherent quasi-cylindrical illumination waves,” in Proceedings of the 10th International Congress on Electron Microscopy, (Deutsche Gesellschaft für Elektronenmikroskopie, Hamburg, 1982), Vol. 1, pp. 427–428.
  15. K. J. Hanszen, “Methods of off-axis electron holography and investigations of the phase structure in crystals,” J. Phys.D 19, 373–395 (1986).
    [CrossRef]
  16. J. F. Hainfeld, “Understanding and using field emission sources,” in Scanning Electron Microscopy 1977, O. Johari, ed. (ITT Research Institute, Chicago, Ill., 1977), Vol. 1, pp. 591–604.
  17. G. Pozzi, “Theoretical considerations on the spatial coherence in field emission electron microscopes,” Optik 77, 69–73 (1987).
  18. A. Maréchal, “Optique géometrique générate,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 24, pp. 44–170.
    [CrossRef]
  19. W. Glaser, “Elektronen und Ionenoptik,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 33,pp.123–395.
  20. P. W. Hawkes, “Image processing based on the linear theory of image formation,” in Computer Processing of Electron Microscope Images, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 1–33.
    [CrossRef]
  21. A. T. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
    [CrossRef]
  22. G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

1987 (1)

G. Pozzi, “Theoretical considerations on the spatial coherence in field emission electron microscopes,” Optik 77, 69–73 (1987).

1986 (3)

K. J. Hanszen, “Methods of off-axis electron holography and investigations of the phase structure in crystals,” J. Phys.D 19, 373–395 (1986).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

1983 (3)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

A. T. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
[CrossRef]

1982 (1)

1981 (1)

G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
[CrossRef]

1978 (1)

P. W. Hawkes, “Coherence in electron optics,” Adv. Opt. Electron Microsc. 7, 101–184 (1978).

1968 (1)

G. Möllenstedt, H. Wahl, “Elektronenholographie und Rekonstruktion mit Laserlicht,” Naturwissenschaften 55, 340–341 (1968).
[CrossRef]

1956 (1)

G. Möllenstedt, H. Düker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[CrossRef]

1955 (1)

G. Möllenstedt, H. Düker, “Fresnelscher Interferenzversuch mit einem Biprisma für Elektronenwellen,” Naturwissenschaften 42, 41 (1955).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Carpenter, R. W.

G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

Düker, H.

G. Möllenstedt, H. Düker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[CrossRef]

G. Möllenstedt, H. Düker, “Fresnelscher Interferenzversuch mit einem Biprisma für Elektronenwellen,” Naturwissenschaften 42, 41 (1955).
[CrossRef]

Fontaine, G.

G. Fontaine, “Electron sources,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. P. Goedgebuer, C. Malgrange, J. Ch. Viénot, R. H. Wade, eds. (Springer-Verlag, Berlin, 1980), pp. 28–36.
[CrossRef]

Friberg, A. T.

A. T. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
[CrossRef]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

Gabor, D.

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–156.
[CrossRef]

Glaser, W.

W. Glaser, “Elektronen und Ionenoptik,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 33,pp.123–395.

Gori, F.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Hainfeld, J. F.

J. F. Hainfeld, “Understanding and using field emission sources,” in Scanning Electron Microscopy 1977, O. Johari, ed. (ITT Research Institute, Chicago, Ill., 1977), Vol. 1, pp. 591–604.

Hanszen, K. J.

K. J. Hanszen, “Methods of off-axis electron holography and investigations of the phase structure in crystals,” J. Phys.D 19, 373–395 (1986).
[CrossRef]

Hawkes, P. W.

P. W. Hawkes, “Coherence in electron optics,” Adv. Opt. Electron Microsc. 7, 101–184 (1978).

P. W. Hawkes, “Image processing based on the linear theory of image formation,” in Computer Processing of Electron Microscope Images, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 1–33.
[CrossRef]

Lauer, R.

R. Lauer, “Electron microscopy and holography employing coherent quasi-cylindrical illumination waves,” in Proceedings of the 10th International Congress on Electron Microscopy, (Deutsche Gesellschaft für Elektronenmikroskopie, Hamburg, 1982), Vol. 1, pp. 427–428.

Li, Y.

Maréchal, A.

A. Maréchal, “Optique géometrique générate,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 24, pp. 44–170.
[CrossRef]

Matteucci, G.

G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

Missiroli, G. F.

G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
[CrossRef]

Möllenstedt, G.

G. Möllenstedt, H. Wahl, “Elektronenholographie und Rekonstruktion mit Laserlicht,” Naturwissenschaften 55, 340–341 (1968).
[CrossRef]

G. Möllenstedt, H. Düker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[CrossRef]

G. Möllenstedt, H. Düker, “Fresnelscher Interferenzversuch mit einem Biprisma für Elektronenwellen,” Naturwissenschaften 42, 41 (1955).
[CrossRef]

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

Pozzi, G.

G. Pozzi, “Theoretical considerations on the spatial coherence in field emission electron microscopes,” Optik 77, 69–73 (1987).

G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

Valdrè, U.

G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
[CrossRef]

Wahl, H.

G. Möllenstedt, H. Wahl, “Elektronenholographie und Rekonstruktion mit Laserlicht,” Naturwissenschaften 55, 340–341 (1968).
[CrossRef]

H. Wahl, “Bildebenenholographie mit Elektronen,” Ph.D. dissertation (University of Tübingen, Tübingen, Federal Republic of Germany, 1975).

Wolf, E.

A. T. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
[CrossRef]

Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schellmodel sources,” Opt. Lett. 7, 256–258 (1982).
[CrossRef] [PubMed]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Adv. Opt. Electron Microsc. (1)

P. W. Hawkes, “Coherence in electron optics,” Adv. Opt. Electron Microsc. 7, 101–184 (1978).

J. Electron. Microsc. Suppl. (1)

G. Pozzi, G. Matteucci, R. W. Carpenter, “Spatial coherence in field emission electron microscopes,” J. Electron. Microsc. Suppl. 35, 275–276 (1986).

J. Phys. E (1)

G. F. Missiroli, G. Pozzi, U. Valdrè, “Electron interferometry and interference electron microscopy,” J. Phys. E 14, 649–671 (1981).
[CrossRef]

J. Phys.D (1)

K. J. Hanszen, “Methods of off-axis electron holography and investigations of the phase structure in crystals,” J. Phys.D 19, 373–395 (1986).
[CrossRef]

Naturwissenschaften (2)

G. Möllenstedt, H. Wahl, “Elektronenholographie und Rekonstruktion mit Laserlicht,” Naturwissenschaften 55, 340–341 (1968).
[CrossRef]

G. Möllenstedt, H. Düker, “Fresnelscher Interferenzversuch mit einem Biprisma für Elektronenwellen,” Naturwissenschaften 42, 41 (1955).
[CrossRef]

Opt. Acta (3)

P. De Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

A. T. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
[CrossRef]

Opt. Commun. (1)

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Opt. Lett. (1)

Optik (1)

G. Pozzi, “Theoretical considerations on the spatial coherence in field emission electron microscopes,” Optik 77, 69–73 (1987).

Z. Phys. (1)

G. Möllenstedt, H. Düker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[CrossRef]

Other (9)

H. Wahl, “Bildebenenholographie mit Elektronen,” Ph.D. dissertation (University of Tübingen, Tübingen, Federal Republic of Germany, 1975).

R. Lauer, “Electron microscopy and holography employing coherent quasi-cylindrical illumination waves,” in Proceedings of the 10th International Congress on Electron Microscopy, (Deutsche Gesellschaft für Elektronenmikroskopie, Hamburg, 1982), Vol. 1, pp. 427–428.

A. Maréchal, “Optique géometrique générate,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 24, pp. 44–170.
[CrossRef]

W. Glaser, “Elektronen und Ionenoptik,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1956), Vol. 33,pp.123–395.

P. W. Hawkes, “Image processing based on the linear theory of image formation,” in Computer Processing of Electron Microscope Images, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 1–33.
[CrossRef]

G. Fontaine, “Electron sources,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. P. Goedgebuer, C. Malgrange, J. Ch. Viénot, R. H. Wade, eds. (Springer-Verlag, Berlin, 1980), pp. 28–36.
[CrossRef]

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–156.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

J. F. Hainfeld, “Understanding and using field emission sources,” in Scanning Electron Microscopy 1977, O. Johari, ed. (ITT Research Institute, Chicago, Ill., 1977), Vol. 1, pp. 591–604.

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Figures (7)

Fig. 1
Fig. 1

Scheme of the illumination system: S′, electron source; L, condenser lens; A, aperture; S, image of the source in the object space; OP observation plane.

Fig. 2
Fig. 2

Fig. 2, Relation between illumination (solid curves) and coherence (dashed curves) areas for the slit-source case: Ry/Rx = 5/2. Coherence parameter, E = 0.1; a, image region p = 0; b, Fresnel region p = 1; c, Fraunhofer region p = 10.

Fig. 3
Fig. 3

Same as Fig. 2. The coherence parameter is E = 1.

Fig. 4
Fig. 4

Same as Fig. 2. The coherence parameter is E = 10.

Fig. 5
Fig. 5

Relation between illumination (solid curves) and coherence (dashed curves) areas for the astigmatic case. The value of the astigmatic parameter C = 0.1. Coherence parameter, E = 0.1; a, image region p = 0; b, Fresnel region (near the stigmatic line) p = 2.7; c, Fraunhofer region p = 10.

Fig. 6
Fig. 6

Same as Fig. 5. The coherence parameter is E = 1.

Fig. 7
Fig. 7

Same as Fig. 5. The coherence parameter is E = 10.

Equations (25)

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Γ ( r s , r s M ) = I ( r s ) δ ( r s r s ) .
Γ ( r 0 , r 0 ) = Γ ( r s , r s ) Ψ ( r 0 , r s ) Ψ * ( r 0 , r s ) d r s d r s ,
T ( k ) = λ 2 exp ( k 2 / K A 2 ) ,
I ( r s ) = I s π R 2 exp ( r s 2 / R 2 ) ,
I ( r s ) = I s π R x R y exp [ ( x s 2 R x 2 + y s 2 R y 2 ) ] .
Ψ ( r 0 , r s ) = 2 π λ K A 2 1 + 2 π i λ Z K A 2 exp [ π 2 ( r 0 r s ) 2 ( 1 / 2 K A 2 ) + i π λ Z ]
Γ ( r 0 , r 0 ) = I s λ 2 K A 2 A x A y exp { [ x 0 2 + x 0 2 2 A x 2 + y 0 2 + y 0 2 2 A y 2 ] [ ( x 0 x 0 ) 2 B x 2 + ( y 0 y 0 ) 2 B y 2 ] } × exp [ i k ( x 0 2 x 0 2 2 L x + y 0 2 y 0 2 2 L y ) ] ,
A μ 2 = 1 4 π 2 K A 2 + R μ 2 + λ 2 K A 2 Z 2 , B μ 2 = 1 π 2 K A 2 + 1 4 π 4 K A 4 R μ 2 + λ 2 Z 2 π 2 R μ 2 , μ = x , y , L μ = Z + R μ 2 + 1 4 π 2 K A 2 λ 2 K A 2 Z .
I ( r 0 ) = Γ ( r 0 , r 0 ) = I s λ 2 K A 2 A x A y exp [ ( x 0 2 A x 2 + y 0 2 A y 2 ) ]
γ 12 ( r 0 r 0 ) = | Γ ( r 0 , r 0 ) | [ I ( r 0 ) I ( r 0 ) ] 1 / 2 = exp { [ ( x 0 x 0 ) 2 B x 2 + ( y 0 y 0 ) 2 B y 2 ] } .
exp [ i π λ a ( k x 2 k y 2 ) ] ,
Ψ ( r 0 , r s ) = π λ H exp { π 2 [ ( x 0 x s ) 2 ( 1 / 2 K A 2 ) + i π ( λ Z a / λ ) + ( y 0 y s ) 2 ( 1 / 2 K A 2 ) + i π ( λ Z a / λ ) ] } ,
H = ( 1 2 K A 2 + i π λ Z ) 2 + π 2 a 2 λ 2 ,
Γ ( r 0 , r 0 ) = I s λ 2 K A 2 A x A y exp { ( x 0 2 + x 0 2 2 A x 2 + y 0 2 + y 0 2 2 A y 2 ) [ ( x 0 x 0 ) 2 B x 2 + ( y 0 y 0 ) 2 B y 2 ] } × exp { i k 2 R 2 [ ( λ 2 Z a ) ( x 0 2 x 0 2 ) D x + ( λ 2 Z + a ) ( y 0 2 y 0 2 ) D y ] } ,
{ A μ 2 = K A 2 R 2 D μ B μ 2 = D μ π 2 μ = x , y ,
D y = 1 4 π 2 K A 4 R 2 + 1 K A 2 + ( λ Z + a λ ) 2 R 2
D x = D y 4 a Z R 2 .
{ A μ 2 ( Z ) = 2 σ μ 2 ( Z ) B μ 2 ( Z ) = 2 δ μ 2 ( Z ) μ = x , y .
E μ = B μ / A μ = 1 / ( π R μ K A ) , μ = x , y .
E = E x E y = π B x B y π A x A y = S C / S I .
E = 1 / π 2 K A 2 R x R y ,
φ A = λ K A ,
E = λ 2 / U ,
U = π 2 R x R y φ 2 = d S d Ω
p = λ Z / R x R y ,

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