Abstract

Tomographic studies of submicrometer samples in materials science using electron microscopy have been inhibited by diffraction effects. In the present work, we describe a practical method for ameliorating these effects. First, we find an analytic expression for the mutual coherence function for hollow-cone illumination. Then, we use this analytic expression to propose a Gaussian weighting of hollow-cone illumination, which we name tapered solid-cone illumination, and present an analytic expression for its mutual coherence function. Finally, we investigate numerically an n-ring approximation to tapered solid-cone illumination. The results suggest a method for removing diffraction effects and hence enabling tomography.

© 2007 Optical Society of America

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  1. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).
  2. V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
    [CrossRef] [PubMed]
  3. A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
    [CrossRef] [PubMed]
  4. P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).
  5. P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
    [CrossRef]
  6. Z. H. Levine, "Synthetic incoherence via scanned Gaussian beams," J. Res. Natl. Inst. Stand. Technol. 111, 429-433 (2006).
    [CrossRef]
  7. W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).
  8. W. Krakow and L. A. Howland, "A method for producing hollow cone illumination electronically in the conventional transmission microscope," Ultramicroscopy 2, 53-67 (1976).
    [CrossRef] [PubMed]
  9. G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
    [CrossRef]
  10. J. H. C. Spence, High-Resolution Electron Microscopy (Oxford U. Press, 2003).
  11. P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic, 1994), Vol. 3, p. 1724.
  12. P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Asken, 1991), pp. 90-91.
  13. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1944), p. 48, Eq. .
  14. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978), p. 572, Eq. .
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic Press, 1980), p. 717, Eq. (6.631.4).
  16. P. J. Davis and I. Polonsky, "Numerical interpolation, differentiation, and integration," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), p. 890, Eq. (25.4.45) and p. 923, Table 25.9; also, for n=1, x1=1, and w1=1.

2006

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

Z. H. Levine, "Synthetic incoherence via scanned Gaussian beams," J. Res. Natl. Inst. Stand. Technol. 111, 429-433 (2006).
[CrossRef]

2005

V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
[CrossRef] [PubMed]

2001

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

1999

G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
[CrossRef]

1978

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

1976

W. Krakow and L. A. Howland, "A method for producing hollow cone illumination electronically in the conventional transmission microscope," Ultramicroscopy 2, 53-67 (1976).
[CrossRef] [PubMed]

1970

A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
[CrossRef] [PubMed]

Banerjee, P. P.

P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Asken, 1991), pp. 90-91.

Baumeister, W.

V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
[CrossRef] [PubMed]

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978), p. 572, Eq. .

Crewe, A. V.

A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
[CrossRef] [PubMed]

Davis, P. J.

P. J. Davis and I. Polonsky, "Numerical interpolation, differentiation, and integration," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), p. 890, Eq. (25.4.45) and p. 923, Table 25.9; also, for n=1, x1=1, and w1=1.

Ercius, P.

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

Ewert, J. C.

G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
[CrossRef]

Foster, F.

V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
[CrossRef] [PubMed]

Freeman, L. A.

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

Gignac, L.

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic Press, 1980), p. 717, Eq. (6.631.4).

Hartung, F.

G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
[CrossRef]

Hawkes, P. W.

P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic, 1994), Vol. 3, p. 1724.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

Howland, L. A.

W. Krakow and L. A. Howland, "A method for producing hollow cone illumination electronically in the conventional transmission microscope," Ultramicroscopy 2, 53-67 (1976).
[CrossRef] [PubMed]

Jenkins, W. K.

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

Johnson, B. F. G.

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

Kasper, E.

P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic, 1994), Vol. 3, p. 1724.

Krakow, W.

W. Krakow and L. A. Howland, "A method for producing hollow cone illumination electronically in the conventional transmission microscope," Ultramicroscopy 2, 53-67 (1976).
[CrossRef] [PubMed]

Langmore, J.

A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
[CrossRef] [PubMed]

Levine, Z. H.

Z. H. Levine, "Synthetic incoherence via scanned Gaussian beams," J. Res. Natl. Inst. Stand. Technol. 111, 429-433 (2006).
[CrossRef]

Lucic, V.

V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
[CrossRef] [PubMed]

Midgley, P. A.

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

Muller, D.

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978), p. 572, Eq. .

Polonsky, I.

P. J. Davis and I. Polonsky, "Numerical interpolation, differentiation, and integration," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), p. 890, Eq. (25.4.45) and p. 923, Table 25.9; also, for n=1, x1=1, and w1=1.

Poon, T. C.

P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Asken, 1991), pp. 90-91.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic Press, 1980), p. 717, Eq. (6.631.4).

Saxton, W. O.

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

Schmitz, G.

G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
[CrossRef]

Smith, D. J.

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

Spence, J. H. C.

J. H. C. Spence, High-Resolution Electron Microscopy (Oxford U. Press, 2003).

Thomas, J. M.

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

Wall, J.

A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
[CrossRef] [PubMed]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1944), p. 48, Eq. .

Weyland, M.

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

Annu. Rev. Biochem.

V. Lucic, F. Foster, and W. Baumeister, "Structural studies by electron tomography: from cells to molecules," Annu. Rev. Biochem. 74, 833-865 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett.

P. Ercius, M. Weyland, D. Muller, and L. Gignac, "Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography," Appl. Phys. Lett. 84, 243116 (2006).
[CrossRef]

Comput. Graph.

P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G. Johnson, "Z-contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering," Comput. Graph. 10, 907-908 (2001).

J. Res. Natl. Inst. Stand. Technol.

Z. H. Levine, "Synthetic incoherence via scanned Gaussian beams," J. Res. Natl. Inst. Stand. Technol. 111, 429-433 (2006).
[CrossRef]

Optik

W. O. Saxton, W. K. Jenkins, L. A. Freeman, and D. J. Smith, "TEM observations using bright field hollow cone illumination," Optik 49, 505-510 (1978).

Science

A. V. Crewe, J. Wall, and J. Langmore, "Visibility of single atoms," Science 168, 1338-1340 (1970).
[CrossRef] [PubMed]

Ultramicroscopy

W. Krakow and L. A. Howland, "A method for producing hollow cone illumination electronically in the conventional transmission microscope," Ultramicroscopy 2, 53-67 (1976).
[CrossRef] [PubMed]

G. Schmitz, J. C. Ewert, and F. Hartung, "Chemical analysis by high-angle hollow cone illumination," Ultramicroscopy 77, 49-63 (1999).
[CrossRef]

Other

J. H. C. Spence, High-Resolution Electron Microscopy (Oxford U. Press, 2003).

P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic, 1994), Vol. 3, p. 1724.

P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Asken, 1991), pp. 90-91.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1944), p. 48, Eq. .

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978), p. 572, Eq. .

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic Press, 1980), p. 717, Eq. (6.631.4).

P. J. Davis and I. Polonsky, "Numerical interpolation, differentiation, and integration," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), p. 890, Eq. (25.4.45) and p. 923, Table 25.9; also, for n=1, x1=1, and w1=1.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

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Figures (2)

Fig. 1
Fig. 1

Absolute value of the normalized mutual coherence function for hollow-cone illumination calculated numerically (solid curve), absolute value of the difference between that and the analytic solution of Eq. (13) (dashed bottom curve), and the absolute value of the difference between the asymptotic expression of Eq. (17) and the analytic solution (dotted middle curve). The parameters are θ = 3 mrad , θ 1 = 3 mrad , λ = 1.969 pm (corresponding to a 300 keV electron energy), z 1 = z 2 = 1 mm , and x 1 = y 1 = y 2 = 0 .

Fig. 2
Fig. 2

Real part of normalized mutual coherence function for the average of one, two, four, and eight hollow cones using Laguerre integration points and weights [16] and the analytic limit (labeled ), given in Eq. (25). The imaginary part is very small. Parameters are given in the text. For small x 2 , not all of the curves are distinguishable.

Equations (31)

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Γ ( r 1 , r 2 , T ) = lim τ 1 2 τ τ τ d t ψ * ( r 1 , t ) ψ ( r 2 , t + T ) ,
γ ( r 1 , r 2 ) = Γ ( r 1 , r 2 ) [ Γ ( r 1 , r 1 ) Γ ( r 2 , r 2 ) ] 1 2 .
ψ ( x , y , z ) = 1 1 + i Z exp ( X 2 + Y 2 1 + i Z ) ,
X = x r 0 ,
Y = y r 0 ,
Z = λ z π r 0 2 = z l R ,
r 0 = λ π θ 1 .
( x y z ) = [ cos θ cos φ cos θ sin φ sin θ sin φ cos φ 0 sin θ cos φ sin θ sin φ cos θ ] ( x y z ) .
( X Y Z ) [ cos φ r 0 sin φ r 0 θ r 0 sin φ r 0 cos φ r 0 0 0 0 1 l R ] ( x y z ) .
Γ HC ( r 1 , r 2 ) = 1 2 π 0 2 π d φ j = 1 2 ( 1 i z j l R ) 1 exp ( x j 2 + y j 2 2 θ z j ( x j cos φ + y j sin φ ) + θ 2 z j 2 r 0 2 ( 1 i z j l R ) ) .
α = 2 θ r 0 2 j = 1 2 z j x j i ± z j l R ,
β = 2 θ r 0 2 j = 1 2 z j y j i ± z j l R .
α = Ω cos ψ ,
β = Ω sin ψ .
α 2 + β 2 = Ω 2 ,
I = 0 2 π exp [ i Ω ( cos ψ cos φ + sin ψ sin φ ) ] = 0 2 π exp [ i Ω cos ( φ ψ ) ] = 2 π J 0 ( Ω ) .
Γ HC ( r 1 , r 2 ) = [ j = 1 2 ( 1 i z j l R ) 1 exp ( x j 2 + y j 2 + θ 2 z j 2 r 0 2 ( 1 i z j l R ) ) ] J 0 ( Ω ) ,
Ω = 2 θ r 0 2 [ ( j = 1 2 z j x j i ± z j l R ) 2 + ( j = 1 2 z j y j i ± z j l R ) 2 ] 1 2 .
Γ HC ( r j , r j ) = ( 1 + z j 2 l R 2 ) 1 exp ( 2 x j 2 + y j 2 + θ 2 z j 2 r 0 2 [ 1 + ( z j l R ) 2 ] ) I 0 ( 4 θ r 0 2 z j ( x j 2 + y j 2 ) 1 2 1 + ( z j l R ) 2 ) ,
γ HC ( r 1 , r 2 ) = { j = 1 2 exp [ ± i tan 1 ( z j l R ) i x j 2 + y j 2 + θ 2 z j 2 r 0 2 z j l R 1 + ( z j l R ) 2 ] } J 0 ( Ω ) [ j = 1 2 I 0 ( 4 θ r 0 2 z j ( x j 2 + y j 2 ) 1 2 1 + ( z j l R ) 2 ) ] 1 2 .
J 0 ( z ) ( 2 π z ) 1 2 cos ( z π 4 ) ,
W ( θ ) = θ θ 0 2 exp ( θ 2 2 θ 0 2 ) ,
I = 0 d θ θ exp ( θ 2 2 θ 2 2 ) J 0 ( ω θ ) ,
θ 2 2 = θ 0 2 + 2 r 0 2 j = 1 2 z j 2 1 i z j l R .
0 d θ θ exp ( a θ 2 ) J 0 ( b θ ) = 1 2 a exp ( b 2 4 a ) ,
Γ TC ( r 1 , r 2 ) = [ j = 1 2 ( 1 i z j l R ) 1 exp ( x j 2 + y j 2 r 0 2 ( 1 i z j l R ) ) ] θ 2 2 θ 0 2 exp ( θ 2 2 ω 2 2 ) .
Γ TC ( r j , r j ) = ( 1 + z j 2 l R 2 ) 1 exp ( 2 r 0 2 x j 2 + y j 2 1 + ( z j l R ) 2 ) θ 2 j 2 θ 0 2 exp ( 8 θ 2 j 2 r 0 4 z j 2 ( x j 2 + y j 2 ) [ 1 + ( z j l R ) 2 ] 2 ) ,
θ 2 j 2 = θ 0 2 + 4 r 0 2 z j 2 1 + ( z j l R ) 2 .
γ TC ( r 1 , r 2 ) = { j = 1 2 θ 2 θ 2 j exp [ ± i tan 1 ( z j l R ) i x j 2 + y j 2 r 0 2 z j l R 1 + ( z j l R ) 2 ] } exp ( θ 2 2 ω 2 2 j = 1 2 4 θ 2 j 2 r 0 4 z j 2 ( x j 2 + y j 2 ) [ 1 + ( z j l R ) 2 ] 2 ) .
γ TC exp ( 2 θ 2 2 l R 2 r 0 4 ( x 1 x 2 ) 2 ) = exp [ ( 2 1 2 π θ 2 λ ) 2 ( x 1 x 2 ) 2 2 ] .
I = θ 2 2 0 d u J 0 ( 2 θ 2 ω u ) exp ( u ) .

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