Abstract

Light described by a certain class of coherence function is investigated, and simple results are found to describe the diffraction patterns. A physical basis for these results is presented. The results are discussed in the context of coherence measurement techniques based on diffraction measurements.

© 1991 Optical Society of America

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References

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  1. J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
    [Crossref] [PubMed]
  2. M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
    [Crossref] [PubMed]
  3. For example, N. Iskander, N. Wang, “Partially coherent radiation from lasers, undulators and laser produced plasmas,” in Short Wavelength Coherent Radiation: Generation and Applications, D. T. Attwood, J. Baker, eds., AIP. Conf Proc.147, 346–353 (1986).
  4. R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
    [Crossref] [PubMed]
  5. M. D. Feit, J. A. Fleck, “Wave optics description of laboratory soft x-ray lasers,” J. Opt. Soc. Am. B 7, 2048–2060 (1990).
    [Crossref]
  6. M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).
  7. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 513–516.
  8. J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.
  9. A. C. Schell, “Multiple plate antenna,” Ph.D. dissertation (MIT, Cambridge, Mass., 1961);J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 222–228.
  10. K. A. Nugent, “A generalization of Schell’s theorem,” Opt. Commun. 79, 267–269 (1990).
    [Crossref]
  11. For example, A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 90–95.
  12. W. H. Carter, E. Wolf, “Coherence and radiometry withquasihomogenous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [Crossref]
  13. K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A 246, 71–76 (1986).
    [Crossref]
  14. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  15. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [Crossref]
  16. For example, M. Kalal, K. A. Nugent, B. Luther-Davies, “Phase-amplitude imaging: its application to fully automated analysis of magnetic field measurements in laser-produced plasmas,” Appl. Opt. 26, 1674–1679 (1987).
    [Crossref] [PubMed]
  17. T. Afshar-rad, O. Willi, “A novel technique for x-ray laser beam characterization,” Appl. Phys. B 50, 287–290 (1990).
    [Crossref]
  18. Ref. 11, pp. 160–167.

1990 (4)

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

K. A. Nugent, “A generalization of Schell’s theorem,” Opt. Commun. 79, 267–269 (1990).
[Crossref]

T. Afshar-rad, O. Willi, “A novel technique for x-ray laser beam characterization,” Appl. Phys. B 50, 287–290 (1990).
[Crossref]

M. D. Feit, J. A. Fleck, “Wave optics description of laboratory soft x-ray lasers,” J. Opt. Soc. Am. B 7, 2048–2060 (1990).
[Crossref]

1987 (4)

For example, M. Kalal, K. A. Nugent, B. Luther-Davies, “Phase-amplitude imaging: its application to fully automated analysis of magnetic field measurements in laser-produced plasmas,” Appl. Opt. 26, 1674–1679 (1987).
[Crossref] [PubMed]

M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

1986 (2)

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[Crossref]

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A 246, 71–76 (1986).
[Crossref]

1982 (1)

1977 (1)

Afshar-rad, T.

T. Afshar-rad, O. Willi, “A novel technique for x-ray laser beam characterization,” Appl. Phys. B 50, 287–290 (1990).
[Crossref]

Barbee, T.

J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 513–516.

Brown, S. B.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Campbell, E. M.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Carter, W. H.

Feder, R.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Feit, M. D.

Fleck, J. A.

Howells, M. R.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Iskander, N.

For example, N. Iskander, N. Wang, “Partially coherent radiation from lasers, undulators and laser produced plasmas,” in Short Wavelength Coherent Radiation: Generation and Applications, D. T. Attwood, J. Baker, eds., AIP. Conf Proc.147, 346–353 (1986).

Jacobson, C.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Kalal, M.

Kim, K. J.

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A 246, 71–76 (1986).
[Crossref]

Kirz, J.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

London, R. A.

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Luther-Davies, B.

Marathay, A. S.

For example, A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 90–95.

Matthews, D. L.

M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Mcquaid, K.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Nathal, H.

J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.

Nilson, D. G.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Nugent, K. A.

Rosen, M. D.

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).

Rothman, S.

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Schell, A. C.

A. C. Schell, “Multiple plate antenna,” Ph.D. dissertation (MIT, Cambridge, Mass., 1961);J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 222–228.

Stone, G. F.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Strauss, M.

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Szoke, A.

J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.

Trebes, J. E.

M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.

Wang, N.

For example, N. Iskander, N. Wang, “Partially coherent radiation from lasers, undulators and laser produced plasmas,” in Short Wavelength Coherent Radiation: Generation and Applications, D. T. Attwood, J. Baker, eds., AIP. Conf Proc.147, 346–353 (1986).

Whelan, D. A.

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

Willi, O.

T. Afshar-rad, O. Willi, “A novel technique for x-ray laser beam characterization,” Appl. Phys. B 50, 287–290 (1990).
[Crossref]

Wolf, E.

Appl. Opt. (1)

Appl. Phys. B (1)

T. Afshar-rad, O. Willi, “A novel technique for x-ray laser beam characterization,” Appl. Phys. B 50, 287–290 (1990).
[Crossref]

Comments Plasma Phys. Controlled Fusion (1)

M. D. Rosen, J. E. Trebes, D. L. Matthews, “A strategy for achieving spatially coherent output from laboratory x-ray lasers,” Comments Plasma Phys. Controlled Fusion 10(5), 245–252 (1987).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nucl. Instrum. Methods A (1)

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A 246, 71–76 (1986).
[Crossref]

Opt. Commun. (1)

K. A. Nugent, “A generalization of Schell’s theorem,” Opt. Commun. 79, 267–269 (1990).
[Crossref]

Phys. Rev. Lett. (1)

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Science (2)

J. E. Trebes, S. B. Brown, E. M. Campbell, D. L. Matthews, D. G. Nilson, G. F. Stone, D. A. Whelan, “Demonstration of x-ray holography with an x-ray laser,” Science 238, 517–519 (1987).
[Crossref] [PubMed]

M. R. Howells, C. Jacobson, J. Kirz, R. Feder, K. Mcquaid, S. Rothman, “X-ray holograms at improved resolution: a study of Zymogen granules,” Science 238, 514–517 (1987).
[Crossref] [PubMed]

Other (6)

For example, N. Iskander, N. Wang, “Partially coherent radiation from lasers, undulators and laser produced plasmas,” in Short Wavelength Coherent Radiation: Generation and Applications, D. T. Attwood, J. Baker, eds., AIP. Conf Proc.147, 346–353 (1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 513–516.

J. E. Trebes, T. Barbee, H. Nathal, A. Szoke, “Proposed method for the measurement of the spatial coherence of x-ray lasers,” in Short Wavelength Radiation: Generation and Applications, R. Falcone, J. Kirz, eds., Vol. 2 of the OSA Proceedings Series (Optical Society of America, Washington, D.C., 1988), pp. 350–354.

A. C. Schell, “Multiple plate antenna,” Ph.D. dissertation (MIT, Cambridge, Mass., 1961);J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 222–228.

For example, A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 90–95.

Ref. 11, pp. 160–167.

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Figures (1)

Fig. 1
Fig. 1

The wave field leaving this (infinite) lens is statistically stationary and is specified completely by the distribution of k vectors. In this case the coherence is fully determined by the angular distribution of the incoherent source as viewed from the center of the lens. The solid arrow illustrates the k vector of the plane wave resulting from one point in the source.

Equations (41)

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Γ ( s 1 , s 2 , τ ) = E * ( s 1 , t ) E ( s 2 , t + τ ) ,
J ( s 1 , s 2 ) = Γ ( s 1 , s 2 , 0 ) .
J ( r 1 , r 2 , z ) = 1 λ 2 z 2 J ( r a 1 , r a 2 , 0 ) × exp [ i π ( r 1 r a 1 ) 2 ( r 2 r a 2 ) 2 λ z ] d r a 1 d r a 2 .
r = ( r 1 + r 2 ) / 2 , r a = ( r a 1 + r a 2 ) / 2 , x = r 1 r 2 , x a = r a 1 r a 2 .
J ( r , x , z ) = 1 λ 2 z 2 ( 2 π i r · x λ z ) J ( r a , x a , 0 ) × exp ( 2 π i r a · x a λ z ) exp [ 2 π i ( r a · x + r · x a ) λ z ] d r a d x a .
J o ( r , x , 0 ) = J ( r , x , 0 ) A ( r + x / 2 ) A * ( r x / 2 ) ,
I ( r , z ) = 1 λ 2 z 2 J ( r a , x a , 0 ) { A ( r a + x a 2 ) × exp [ i π ( J ( r a + x a / 2 ) 2 λ z ] } { A ( r a x a 2 ) × exp [ i π ( r a x a / 2 ) 2 λ z ] } * exp ( 2 π i r · x a λ z ) d r a d x a .
J ( r , x , 0 ) = Ψ ( r + x / 2 ) Ψ * ( r x / 2 ) g ( x ) ,
I ( r , 0 ) = Ψ ( r ) Ψ * ( r ) ,
J ( r , x , 0 ) = [ I ( r + x / 2 ) ] 1 / 2 [ I ( r x / 2 ) ] 1 / 2 g ( x ) .
I ( r , z ) = 1 λ 2 z 2 g ( x a ) ( { Ψ ( r a + x a / 2 ) A ( r a + x a / 2 ) × exp [ i π ( r a + x a / 2 ) 2 λ z ] } { Ψ ( r a x a / 2 ) A ( r a x a / 2 ) × exp [ i π ( r a x a / 2 ) 2 λ z ] } * d r a ) exp ( 2 π i r · x a λ z ) d x a .
T ( r ) Ψ ( r ) A ( r ) exp ( i π r 2 / λ z )
K ( x a ) T ( r ) T * ( r x a ) d r .
I ( r , z ) = 1 λ 2 z 2 g ( x a ) K ( x a ) exp ( 2 π i r · x a λ z ) d x a .
P ( r , z ) = 1 λ 2 z 2 K ( x a ) exp ( 2 π i r · x a λ z ) d x a ,
G ( u ) g ( x a ) exp ( 2 π i u · x a λ ) d x a ,
I ( r / z ) = G ( r / z ) * * P ( r , z ) ,
T ( r ) = A ( r ) exp ( i π r 2 / λ z ) .
T ( r ) = A ( r ) ,
J ( r , x , 0 ) = [ I ( r + x / 2 ) ] 1 / 2 [ I ( r x / 2 ) ] 1 / 2 g ( x ) ,
J ( r , x , 0 ) = exp { i [ ϕ ( r + x / 2 ) ϕ ( r x / 2 ) ] } g ( x ) .
J ( r , x , 0 ) = 1 λ 2 z 2 exp ( 2 π i r · x λ l ) S ( r ) exp ( 2 π i r · x λ l ) d r .
g ( x ) = S ( r ) exp ( 2 π i r · x λ l ) d r ,
ϕ ( r ) = r 2 / λ l ,
I ( r , z ) = S ( l r / z ) * * P l ( r / z ) ,
P l ( r , z ) = | exp [ i π ( l + z ) r 2 λ l z ] A * ( r ) exp ( 2 π i r · r λ z ) d r | 2 ,
Î ( k , z ) = I ( r , z ) exp ( 2 π i r · k λ z ) d r ,
Î ( k , z ) = g ( k ) K ( k ) .
K ( x ) constant ,
B ( r , u ) = J ( r , x ) exp ( 2 π i u · x λ ) d x ,
K ( x ) = exp { i [ ϕ ( r a + x / 2 ) ϕ ( r a x / 2 ) ] } × A ( r a + x / 2 ) A * ( r a x 2 ) exp ( 2 π i r a · x λ z ) d r a .
ϕ ( r a + x / 2 ) ϕ ( r a x / 2 ) ϕ ( r a ) · x ,
K ( x ) A ( r a + x / 2 ) A * ( r a x / 2 ) × exp { 2 π i [ r a + λ z ϕ ( r a ) ] · x a λ z } d r a ,
( r a + x / 2 ) 2 ( r a x / 2 ) = 2 r a · x ,
K ( x ) { A ( r a + x / 2 ) [ i π ( r + x / 2 ) 2 λ z ] } × { A ( r a x / 2 ) [ i π ( r x / 2 ) 2 λ z ] } d r ,
r = r a + λ z ϕ ( r a )
Ψ ( r ) = | Ψ ( r ) | exp [ i ϕ ( r ) ] ,
r · x / λ z 1
I ( u , z ) = T ( u , z ) * * G ( u ) / λ 2 z 2 ,
T ( u , z ) Ψ ( r + 1 2 x a ) Ψ * ( r 1 2 x a ) d r × exp ( 2 π i u · x a λ ) d x a ,
G ( u , z ) g ( x a ) exp ( 2 π i u · x a λ ) d x a .

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