Abstract

The spatial coherence properties of a monochromatic component of synchrotron radiation from an insertion device in the Fraunhofer limit are analyzed in the general case when the coherence distance is comparable with the beam width, expressing them by simple products and convolutions of Fourier transforms and autocorrelations on the single-electron field amplitude and the electron-beam position and angular distributions. In particular, the Gaussian approximation is discussed, in which case the far-field amplitude satisfies the Schell condition 1its statistical properties can be described by a coherence factor depending only on the difference of the reciprocal space coordinates2, and this discussion leads to simple estimates of the coherence widths. The coherence widths deviate from the Van Cittert–Zernike values when one or more of the phase space widths of the electron beam are close to (or smaller than) the diffraction limit.

© 1995 Optical Society of America

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References

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  1. See, for example, Handbook on Synchrotron Radiation, E. E. Koch, ed. (North-Holland, Amsterdam, 1983), Vols. 1–4.
  2. K.-J. Kim, “A new formulation of synchrotron radiation optics using the Wigner distribution,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 2–9 (1986).
  3. S. Bartalucci, “An overview of programs for calculation of undulator radiation spectra,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 32–37 (1986).
  4. M. Cornacchia, H. Winick, presented at the Fifteenth International Conference on High Energy Acceleration, Hamburg, 1992.
  5. D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
    [CrossRef] [PubMed]
  6. D. Attwood, “New opportunities at soft X-ray wavelengths,” Phys. Today 45 (8), 24–31 (1992).
    [CrossRef]
  7. A. M. Kondratenko, A. N. Skrinsky “Use of radiation of electron storage rings in X-ray holography of objects,” Opt. Spectrosc. (USSR) 42, 189–192 (1977);Opt. Spektrosk. 42, 338–344 (1975).
  8. D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).
  9. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977);J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.
  11. A. Friberg, E. Wolf, “Reciprocity relations with partially coherent sources,” Opt. Acta 30, 1417–1435 (1983).
    [CrossRef]
  12. R. Coïsson, “Source and far field coherence functions,” Note SPS/ABM/RC 81-11 (CERN, Geneva, 1981).
  13. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  14. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  15. R. Coïsson, R. P. Walker, “Phase space distribution of brilliance of undulator sources,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 24–29 (1986).
  16. R. Coïsson, “Effective phase space widths of undulator radiation,” Opt. Eng. 27, 250–252 (1988).
  17. R. Coïsson, B. Diviacco, “Practical estimates of peak flux and brilliance of undulator radiation on even harmonics,” Appl. Opt. 27, 1376–1377 (1988).
    [CrossRef] [PubMed]
  18. G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.
  19. L. Mandel, “Concept of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [CrossRef]
  20. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1968), Chap. 12.

1992 (1)

D. Attwood, “New opportunities at soft X-ray wavelengths,” Phys. Today 45 (8), 24–31 (1992).
[CrossRef]

1988 (2)

1985 (1)

D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
[CrossRef] [PubMed]

1983 (1)

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

1978 (1)

D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).

1977 (2)

A. M. Kondratenko, A. N. Skrinsky “Use of radiation of electron storage rings in X-ray holography of objects,” Opt. Spectrosc. (USSR) 42, 189–192 (1977);Opt. Spektrosk. 42, 338–344 (1975).

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977);J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

1974 (1)

1968 (1)

1961 (1)

Abernathy, D.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Alferov, D. F.

D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).

Als-Nielsen, J.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Attwood, D.

D. Attwood, “New opportunities at soft X-ray wavelengths,” Phys. Today 45 (8), 24–31 (1992).
[CrossRef]

D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
[CrossRef] [PubMed]

Bartalucci, S.

S. Bartalucci, “An overview of programs for calculation of undulator radiation spectra,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 32–37 (1986).

Bashmakov, Yu. A.

D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).

Bessonov, E. G.

D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).

Brauer, S.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Carter, W. H.

Coïsson, R.

R. Coïsson, “Effective phase space widths of undulator radiation,” Opt. Eng. 27, 250–252 (1988).

R. Coïsson, B. Diviacco, “Practical estimates of peak flux and brilliance of undulator radiation on even harmonics,” Appl. Opt. 27, 1376–1377 (1988).
[CrossRef] [PubMed]

R. Coïsson, R. P. Walker, “Phase space distribution of brilliance of undulator sources,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 24–29 (1986).

R. Coïsson, “Source and far field coherence functions,” Note SPS/ABM/RC 81-11 (CERN, Geneva, 1981).

Cornacchia, M.

M. Cornacchia, H. Winick, presented at the Fifteenth International Conference on High Energy Acceleration, Hamburg, 1992.

Dierker, S.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Diviacco, B.

Fleming, R.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Friberg, A.

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

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

Grübel, G.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Halbach, K.

D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
[CrossRef] [PubMed]

Kim, K.-J.

D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
[CrossRef] [PubMed]

K.-J. Kim, “A new formulation of synchrotron radiation optics using the Wigner distribution,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 2–9 (1986).

Kondratenko, A. M.

A. M. Kondratenko, A. N. Skrinsky “Use of radiation of electron storage rings in X-ray holography of objects,” Opt. Spectrosc. (USSR) 42, 189–192 (1977);Opt. Spektrosk. 42, 338–344 (1975).

Leg-rand, J. F.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Mandel, L.

Mochrie, S. G. J.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Papoulis, A.

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1968), Chap. 12.

Pindak, R.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Robinson, I. K.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Skrinsky, A. N.

A. M. Kondratenko, A. N. Skrinsky “Use of radiation of electron storage rings in X-ray holography of objects,” Opt. Spectrosc. (USSR) 42, 189–192 (1977);Opt. Spektrosk. 42, 338–344 (1975).

Stephenson, G. B.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Sutton, M.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Vignaud, G.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

Walker, R. P.

R. Coïsson, R. P. Walker, “Phase space distribution of brilliance of undulator sources,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 24–29 (1986).

Walther, A.

Winick, H.

M. Cornacchia, H. Winick, presented at the Fifteenth International Conference on High Energy Acceleration, Hamburg, 1992.

Wolf, E.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

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

Opt. Eng. (1)

R. Coïsson, “Effective phase space widths of undulator radiation,” Opt. Eng. 27, 250–252 (1988).

Opt. Spectrosc. (USSR) (1)

A. M. Kondratenko, A. N. Skrinsky “Use of radiation of electron storage rings in X-ray holography of objects,” Opt. Spectrosc. (USSR) 42, 189–192 (1977);Opt. Spektrosk. 42, 338–344 (1975).

Phys. Today (1)

D. Attwood, “New opportunities at soft X-ray wavelengths,” Phys. Today 45 (8), 24–31 (1992).
[CrossRef]

Science (1)

D. Attwood, K. Halbach, K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
[CrossRef] [PubMed]

Zh. Tekh. Fiz. (1)

D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of undulator radiation,” Zh. Tekh. Fiz. 48, 1592–1597, 1598–1606 (1978);Phys. Tech. Phys. 23, 902–904, 905–909 (1978);E. G. Bessonov, “On the space–time coherence of undulator radiation,” Zh. Tekh. Fiz. 58, 498–505 (1988).

Other (9)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

See, for example, Handbook on Synchrotron Radiation, E. E. Koch, ed. (North-Holland, Amsterdam, 1983), Vols. 1–4.

K.-J. Kim, “A new formulation of synchrotron radiation optics using the Wigner distribution,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 2–9 (1986).

S. Bartalucci, “An overview of programs for calculation of undulator radiation spectra,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 32–37 (1986).

M. Cornacchia, H. Winick, presented at the Fifteenth International Conference on High Energy Acceleration, Hamburg, 1992.

G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, J. F. Leg-rand, “Scattering with coherent X-rays,” ESRF Newsletter (European Synchrotron Radiation Facility, Grenoble, France, 1994), pp. 14–15.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1968), Chap. 12.

R. Coïsson, “Source and far field coherence functions,” Note SPS/ABM/RC 81-11 (CERN, Geneva, 1981).

R. Coïsson, R. P. Walker, “Phase space distribution of brilliance of undulator sources,” in Insertion Devices for Synchrotron Sources, I. Lindau, R. Tatchyn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.582, 24–29 (1986).

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Equations (26)

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k = 2 π λ θ ,
f ( k ) = F f ( k ) = 1 ( 2 π ) 1 / 2 f ( x ) exp ( ikx ) d x ,
f * ( k η / 2 ) f ( k + η / 2 ) = 1 2 π f * ( x ξ / 2 ) f ( x + ξ / 2 ) × exp [ i ( k ξ + η x ) ] d x d ξ .
M f ( x , ξ ) = f ( x ξ / 2 ) f ( x + ξ / 2 ) .
I f ( x ) = M f ( x , 0 )
C f ( ξ ) = f ( x ξ / 2 ) f ( x + ξ / 2 ) d x = M f ( x , ξ ) d x .
FCf ( k ) = IFf ( k ) ,
FIf ( η ) = CFf ( η ) .
W f ( x , k ) = f ( x ξ / 2 ) f * ( x + ξ / 2 ) exp ( i ξ k ) d ξ = f ( k η / 2 ) f * ( k + η / 2 ) exp ( i η x ) d η .
A f ( ξ , η ) = f ( x ξ / 2 ) f * ( x + ξ / 2 ) exp ( i x η ) d x .
g ( x ) = 1 2 π σ 1 σ 2 exp [ 1 2 ( x 1 2 σ 1 2 + x 2 2 σ 2 2 ) ] ,
γ ( k ) = 1 2 π s 1 s 2 exp [ 1 2 ( k 1 2 s 1 2 + k 2 2 s 2 2 ) ] ,
s = 2 π λ σ ,
I f ( x ) = g ( x ) I a ( x ) ,
I f ( k ) = γ ( k ) I a ( k ) .
f m n ( k ) = a ( k k m ) exp ( i k x n ) exp ( i θ m n ) .
M f ( k , η ) = g ( η ) [ γ ( k ) M a ( k , η ) ]
M f ( x , ξ ) = γ ( ξ ) [ g ( x ) M a ( x , ξ ) ] .
M f ( x , ξ ) = I f ( x ) F I f ( ξ ) ,
M f ( k , η ) = I f ( k ) FIf ( η ) .
R s , 1 / R σ ,
s c = ( σ 2 + 1 2 R 2 ) 1 / 2 1 σ .
a ( k ) = 1 2 π R exp ( 1 2 k 2 R 2 )
M f ( k , η ) = exp ( 1 2 σ 2 η 2 ) { exp ( 1 2 k 2 s 2 ) exp [ 1 2 ( k η / 2 ) 2 R 2 ] × exp [ 1 2 ( k + η / 2 ) 2 R 2 ] } = exp [ 1 2 ( σ 2 + 1 2 R 2 ) η 2 ] × exp ( 1 2 k 2 s 2 + R 2 / 2 ) .
M f ( k , η ) = exp [ 1 2 ( σ 2 + 1 2 R 2 1 2 R 2 + 4 s 2 ) η 2 ] × exp [ 1 2 ( k η / 2 ) 2 2 ( s 2 + R 2 / 2 ) ] × exp [ 1 2 ( k + η / 2 ) 2 2 ( s 2 + R 2 / 2 ) ] .
s c = 1 ( σ 2 + 1 2 R 2 1 2 R 2 + 4 s 2 ) 1 / 2 .

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