Abstract

The latest generation of synchrotron sources, so-called third generation sources, are able to produce copious amounts of coherent radiation. However it has become evident that the experimental systems that have been developed are unable to fully utilize the coherent flux. This has led to a perception that coherence is lost while the radiation is transported down the beamline. However it is well established that the degree of coherence must be preserved, or increased, by an experimental system, and so this apparent “decoherence” must have its origin in the nature of the measurement process. In this paper we use phase space methods to present an argument that the loss of useful coherent flux can be attributed to unresolved speckle in the x-ray beam.

© 2003 Optical Society of America

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References

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    [CrossRef]
  2. K.A. Nugent, T.E.Gureyev, D.F. Cookson, D.Paganin and Z.Barnea, �??Quantitative phase imaging using hard x-rays,�?? Phy. Rev. Letts. 77, 2961-2964 (1996)
    [CrossRef]
  3. P. Cloetens, M. Pateyron-Salome, J.Y.Buffiere, G.Peix, J.Baruchel, F.Peyrin and M.Sclenker, �??Observation of microstructure and damage in materials by phase sensitive radiography and tomography,�?? J. Appl. Phys. 81, 5878-5886 (1997)
    [CrossRef]
  4. S.W.Wilkins, T.E.Gureyev, D.Gao, A.Pogany and A.W.Stevenson, �??Phase-contrast imaging using polychromatic hard x-rays,�?? Nature 384, 335-338 (1996)
    [CrossRef]
  5. V.E. Coslett and W.C. Nixon, X-ray microscopy, (Cambridge University Press, Cambridge, 1960)
  6. M. Sutton, S.G.J. Mochrie, T. Greytak, S.E. Nagler, L.E. Berman, G.A. Held and G.B. Stephenson �??Observation Of Speckle By Diffraction With Coherent X-Rays,�?? Nature 352, 608-610 (1991)
    [CrossRef]
  7. I.A. Vartanyants and I.K. Robinson, �??Origins of decoherence in coherent X-ray diffraction experiments,�?? Opt. Commun. 222, 29-50 (2003).
    [CrossRef]
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J. Appl. Phys.

P. Cloetens, M. Pateyron-Salome, J.Y.Buffiere, G.Peix, J.Baruchel, F.Peyrin and M.Sclenker, �??Observation of microstructure and damage in materials by phase sensitive radiography and tomography,�?? J. Appl. Phys. 81, 5878-5886 (1997)
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

S.W.Wilkins, T.E.Gureyev, D.Gao, A.Pogany and A.W.Stevenson, �??Phase-contrast imaging using polychromatic hard x-rays,�?? Nature 384, 335-338 (1996)
[CrossRef]

M. Sutton, S.G.J. Mochrie, T. Greytak, S.E. Nagler, L.E. Berman, G.A. Held and G.B. Stephenson �??Observation Of Speckle By Diffraction With Coherent X-Rays,�?? Nature 352, 608-610 (1991)
[CrossRef]

Opt. Commun.

I.A. Vartanyants and I.K. Robinson, �??Origins of decoherence in coherent X-ray diffraction experiments,�?? Opt. Commun. 222, 29-50 (2003).
[CrossRef]

D. Paterson, B.E. Allman , P.J. McMahon, J.J.A. Lin, N. Moldovan, K.A. Nugent, I. McNulty, C.T. Chantler, C.C. Retsch, T.H.K. Irving and D.C. Mancini, �??Spatial coherence measurement of X-ray undulator radiation,�?? Opt. Commun. 195 (1-4): 79-84 (2001)
[CrossRef]

Phy. Rev. Letts.

K.A. Nugent, T.E.Gureyev, D.F. Cookson, D.Paganin and Z.Barnea, �??Quantitative phase imaging using hard x-rays,�?? Phy. Rev. Letts. 77, 2961-2964 (1996)
[CrossRef]

Rev. Sci. Instrum.

A. Snigirev, I Snigereva, V.Kohn, S.Kuznetsov, I Schelokov, �??On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,�?? Rev. Sci. Instrum. 66, 5846-5492 (1995).
[CrossRef]

Other

V.E. Coslett and W.C. Nixon, X-ray microscopy, (Cambridge University Press, Cambridge, 1960)

M. Born and E. Wolf, Principles of Optics, 6th edition, (Pergamon, Oxford, 1980).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995)

Supplementary Material (2)

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Figures (2)

Fig. 1.
Fig. 1.

(344KB) Left hand panel shows the Generalized Radiance as a function of position x and u x . The panel on the right shows the absolute value of the Mutual Optical Intensity as a function of x1 and x2. The movie shows the relationship between these two representations as the coherence changes.

Fig. 2.
Fig. 2.

(550KB) Gaussian-Schell model coherence functions for 1nm x-rays after having passed through a random Gaussian phase screen with a characteristic spatial frequency of 1 µm-1 and a standard deviation of 0.5 radians. The beam width wx is 39.9 µm and σx is 10.0 µm. The image sizes are 120 µm×120 µm. (a) The result obtained for a system with perfect resolution. (b) The apparent coherence function when the experimental system has finite spatial resolution of 18.6 µm. The bright streak diagonally across the screen corresponds to an apparent low coherence component.

Equations (26)

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Γ ( r 1 , r 2 , τ ) = E ( r 1 , t ) E * ( r 2 , t + τ ) ,
r ( r 1 + r 2 ) 2 ; q r 1 r 2 ,
B ( r , u ) = J ( r + q 2 , r q 2 ) e 2 π i u q λ d q ,
B z = z 0 ( r , u ) = B z = 0 ( r z 0 u , u ) .
G ( r , u ) T ( r + q 2 ) T * ( r q 2 ) e 2 π i u q λ d q ,
B diff ( r , u ) = B ( r , u ) G ( r , u u ) d u .
G ( r , u ) = G 1 ( r , u ) G 2 ( r , u u ) d u .
J ( r 1 , r 2 ) = I 0 exp ( x 2 w x 2 + y 2 w y 2 ) exp ( q x 2 4 σ x 2 + q y 2 4 σ y 2 ) ,
B ( r , u ) = 4 π σ x σ y I 0 exp { x 2 w x 2 + y 2 w y 2 } exp ( π 2 λ 2 ) { σ x 2 u x 2 + σ y 2 u y 2 } .
B exit ( x , u x ) = B source ( x z 1 u x , u x ) G exp ( x , u x u x ) d u x ,
G exp ( x , u x ) = G ideal ( x , u x ) G error ( x , u x u x ) d u x ;
G ideal ( x , u x ) = P ( z + q x 2 ) P * ( z q x 2 ) e 2 π i u x q x λ d q x ,
G error ( x , u x ) e i [ Φ ( x + q x 2 ) Φ ( x q x 2 ) ] e 2 π i u x q x λ d q x ,
B det ( x , u x ) = B exit ( x z 2 u x , u x ) .
B meas ( x 0 , u x ) = B det ( x , u x ) G det ( x x 0 , u x u x ) d u x dx .
G det ( x , u x ) D ( x ) δ ( u x ) ,
B meas ( x 0 , u x ) B det ( x , u x ) D ( x x 0 ) dx .
B meas ( x 0 , u x ) B source ( x ( z 1 + z z ) u x , u x ) G exp ( x z 2 u x , u x u x ) d u x ×
D ( x z 2 u x x 0 ) dx .
B meas ( x 0 , u x ) B source ( x 0 ( z 1 + z 2 ) u x , u x ) G ideal ( x 0 z 2 u x , u x u x ) d u x ×
G error ( x z 2 u x , u x ) D ( x z 2 u x x 0 ) dx d u x .
G error ( x , u x ) δ ( u x ) + i Δ Φ ( x , q x ) e 2 π i u x q x λ d q x ,
Δ Φ ( q x ) = Δ Φ ( x , q x ) D ( x x 0 ) dx ;
S ( u x ) i Δ Φ ( q x ) e 2 π i u x q x λ d q x .
B meas ( x , u x ) = B ideal ( x , u x ) + B ideal ( x , u x u x ) S ( u x ) d u x ,
J meas ( x 1 , x 2 ) = J ideal ( x 1 , x 2 ) ( 1 + i Δ Φ ( x 1 x 2 ) ) .

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