## Influence of partial coherence on analyzer-based imaging with asymmetric Bragg reflection

Optics Express, Vol. 17, Issue 14, pp. 11618-11637 (2009)

http://dx.doi.org/10.1364/OE.17.011618

Acrobat PDF (485 KB)

### Abstract

Image magnification via twofold asymmetric Bragg reflection (a setup called the ”Bragg Magnifier”) is a recently established technique allowing to achieve both sub-micrometer spatial resolution and phase contrast in X-ray imaging. The present article extends a previously developed theoretical formalism to account for partially coherent illumination. At a typical synchrotron setup polychromatic illumination is identified as the main influence of partial coherence and the implications on imaging characteristics are analyzed by numerical simulations. We show that contrast decreases by about 50% when compared to the monochromatic case, while sub-micrometer spatial resolution is preserved. The theoretical formalism is experimentally verified by correctly describing the dispersive interaction of the two orthogonal magnifier crystals, an effect that has to be taken into account for precise data evaluation.

© 2009 Optical Society of America

## 1. Introduction

1. E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. **15**, 937–945 (1980).
[CrossRef]

3. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature **373**, 595–598 (1995).
[CrossRef]

4. V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento **19**, 553–560 (1997).
[CrossRef]

5. K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. **78**, 132–134 (2001).
[CrossRef]

6. R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. **37**, 734–746 (2002).
[CrossRef]

7. D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. **36**, A65–A68 (2003).
[CrossRef]

8. M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A **551**, 119–124 (2005).
[CrossRef]

9. M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. **78**, 113708 (2007).
[CrossRef] [PubMed]

10. R. Spal, “Submicrometer resolution hard X-Ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. **86**, 3044–3047 (2001).
[CrossRef] [PubMed]

11. J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A **488**, 419–427 (2002).
[CrossRef]

12. Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. **37**1262–1274 (2004).
[CrossRef]

13. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B **74**054107 (2006).
[CrossRef]

14. J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. **270**, 180–188 (2007).
[CrossRef]

15. A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express **15**, 5641–5648 (2007).
[CrossRef] [PubMed]

16. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. **38**, 4259–4271 (2005).
[CrossRef]

13. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B **74**054107 (2006).
[CrossRef]

13. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B **74**054107 (2006).
[CrossRef]

## 2. Experimental background

**74**054107 (2006).
[CrossRef]

18. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) **204**, 2746–2752 (2007).
[CrossRef]

**74**054107 (2006).
[CrossRef]

*as seen from a single object point*(i.e. incident divergence) are the suitable quantities for the following discussion. Fig. 2 illustrates this argument and the corresponding quantitative values for the two example beamlines are shown in Tab. 2.

*σ*is transformed to the exit divergence

_{inc}*σ*by the equation

_{out}*ζ*incident on the sample is essentially given by the spectral width of the monochromator, if the spectral width of the source is much larger than the spectral acceptance of the monochromator crystals. The ID19 utilizes a silicon 111 double monochromator in a non-dispersive (n,-n) setup, resulting in an effective bandwidth of

*ζ*=1.3×10

^{-4}under the assumption of zero beam divergence. Comparing this to the spectral acceptance of the reflections of the analyzer crystals given in Tab. 1 clarifies that this effect is not negligible. Thus, polychromaticity is identified as the main influence of partially coherent illumination in the present context.

21. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. **29**, 133–146 (1996).
[CrossRef]

22. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6296–6304 (2005).
[CrossRef] [PubMed]

*µ*rad (i.e. smaller than about half of the Darwin widths of the reflections), which is essentially due to the small sample-to-detector distance as well as due to the utilization of asymmetric Bragg reflection. Therefore, the Bragg Magnifier opens the possibility of sub-micron resolution even for short beamlines like the TOPO-TOMO beamline.

## 3. Influence of polychromaticity at the analyzer crystal

*b*of the asymmetric reflection is defined by

*θ*is the Bragg angle,

_{B}*ρ*is the asymmetry angle between crystal surface and diffracting lattice planes, and the convention

*b*>1 for smaller incidence angles was used (see also Fig. 3).

*ω*(i.e. the angular acceptance) to the corresponding spectral acceptance (or wavelength band)

_{D}*ζ*

23. M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. **12**, 23–50 (1982).
[CrossRef]

*θ*=0). It is important to carefully distinguish between the angular offset on the reflection curve Dqd and the angular offset from the reference beam after reflection Δ

_{D}*θ*. From geometrical considerations as indicated by Fig. 3 we conclude that Dqad is given by

_{ad}*x*is obviously proportional to the incident spectral width

*ζ*and the distance between analyzer crystal and detector

*d*

*ζ*(see Tab. 1) and the analyzer-to-detector distance of about

*d*=

*d*

_{2}+

*d*

_{3}=195 mm for the first reflection lead to a total blurring of Δ

*x*≈25

*µ*m. Although this is comparable to the physical pixel size (15

*µ*m) of the used camera it is still smaller than the width of the point spread function (about two pixels) of the camera. Thus, the effect of dispersion induced divergence can be neglected in this case.

## 4. Theoretical formalism

### 4.1. Case of fully coherent illumination

**74**054107 (2006).
[CrossRef]

*D*̂

*(*

_{in}*q*) to the output wave field

*D*(

_{out}*x*) on the detector

*K*=2

*π/λ*is the modulus of the wave vector,

*R*̂

_{1}is the reflection curve of the analyzer crystal,

*ω*

_{1}is the angular position of the main beam direction on the reflection curve (”the working point”) and the definition of the remaining quantities can be found in Fig. 4.

*z*=

_{eff}*a*

^{2}/

*λ*, where a denotes the size of typical sample features. After asymmetric Bragg reflection the contrast is expanded by the magnification factor

*b*, so that

*z*scales with

_{eff}*b*

^{2}. This means that the propagation distance after asymmetric reflection is effectively decreased by

*b*

^{2}. For a magnification factor of 40, this implies an effective reduction of the propagation distance by the factor 1600. Thus, free-space propagation after asymmetric reflection can be neglected.

*z*; interpolation is necessary to retrieve the output wave field on the image position

*s*. However, in a second step [25

25. P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express **16**, 5141–5149 (2008).
[CrossRef] [PubMed]

*D̂*(

_{in}*q*), which is usually defined on a regular grid in

*q*, it is necessary to use interpolation for retrieving

*D̂*(

_{in}*q*(

*f*)), because the numerical Fourier transform requires a regular grid in

*f*. But if sufficient sampling points are used, linear interpolation seems to be adequate and the entire calculation time is only slightly increased by the interpolation.

### 4.2. Case of partially coherent illumination

*λ*and a certain angular deviation Δ

*α*from the main beam direction. Each plane wave delivers an output wave field that is generally dependent on Δ

*α*and

*λ*:

*D*(

_{out}*x*)=

*D*

_{Δα,λ}(

*x*). Then, the observable intensity is given by the incoherent sum over all incident directions and wavelengths [26]

*I*(Δ

_{b}*α,λ*). We would like to remind the reader that

*I*(Δ

_{b}*α,λ*) is defined in the object plane (see Fig. 2) and in Appendix A we briefly lay out a ray-tracing approach to express

*I*(Δ

_{b}*α,λ*) at the position of the sample in terms of monochromator and source parameters.

*µ*m at a wavelength of 1 Å, which seems to imply that the used plane wave approximation is only valid within the same range. However, if the

*detectable*interference phenomena due to a single object point extends less than the first Fresnel zone the plane wave approach is applicable over the entire field of view. Our experimental and theoretical results in the case of monochromatic illumination [13

**74**054107 (2006).
[CrossRef]

27. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. **90**, 193501 (2007).
[CrossRef]

*D*

_{Δα,λ}(

*x*) in dependence of the wavelength and the angular deviation Δ

*α*. Since the angular deviation from the main beam direction is proportional to the derivative of the phase

*ϕ*(

*x*) of the wave field

*D*(

_{in}*x*) with the corresponding phase factor, yielding

*q*0=-Δ

*αK*

*q*→

*q*-

*q*

_{0}in integral (9) and following a discussion analogous to section 4.1, an adapted relation between

*q*and

*f*

*α*illuminating the sample is three-fold: First, an additional angular offset occurs in the argument of the reflection curve. Secondly, the image position appears shifted on the detector, which is reflected in the additional

*q*

_{0}in Eq. (18). Lastly, the mean propagation distance is changed according to Eq. (19).

*R*̂. In the present context, the angular offset Δ

*θ*has to be written in terms of deviation from the reference wavelength

_{λ}*λ*and thus reads

_{ref}*i*=1,2 applies for the first and second reflection, respectively (the second reflection will be used in section 6). This is the only influence of polychromatic illumination and therefore, the generalized propagator in the case of partially coherent illumination reads

### 4.3. Case of polychromatic illumination

*q*

_{0}equal zero and the angular and spectral distribution of the beam

*I*can be approximated by

_{b}*δ*denotes the Dirac-

_{D}*δ*function and

*W*(

*λ*) is given by the spectral distribution of the double crystal reflection curve of the monochromator. With the approximations mentioned, the generalized propagator reduces to

## 5. Numerical investigations

*σ*

*b*: 40fold

*ω*

_{1}: left slope of RC

*z*

_{0}: 5 mm

*λ*): Gaussian

*ζ*: variable

### 5.1. Polychromatic response function

*D*(

_{in}*x*)=

*δ*(

_{D}*x*), or in Fourier space

*D̂*(

_{in}*q*)=1) in Eq. (23) and Eq. (25) yields the polychromatic response function for the Bragg Magnifier

*ζ*, where

*ζ*=0 corresponds to the case of monochromatic illumination. As expected, with increasing spectral width

*ζ*the interference fringes cancel out and the width of the maximum is broadened. Generally speaking, this will tend to decrease visible contrast and degrade spatial resolution in the images. However, we emphasize the fact that this is only a qualitative and not a quantitative argument since this interpretation of the response function is implicitly based on the assumption of two incoherent object points. But for every given wavelength λ the imaging process is, in fact, coherent and only the contributions of different wavelengths are incoherent to each other.

### 5.2. Influence of polychromatic illumination on visible contrast

*h*(

*x*) of the sample be given by

*h*

_{1}=5

*µ*m is the maximum thickness and

*σ*=1

*µ*m corresponds to the width of the sample feature (see also Fig. 6(a)). Thus, the input wave field equals

*n*is the complex refractive index of the material at the given x-ray photon energy. The material was chosen to be amorphous carbon, modeling a biological sample with a maximum attenuation of 0.2% and a maximum phase shift of about

*π*/2. Fourier transform and linear interpolation according to Eq. (11) was used to retrieve the input wave field in Fourier space

*D*(

_{in}*q*(

*f*)).

*ζ*. Although contrast decreases with increasing

*ζ*, the general shape of the intensity distribution is preserved. Regarding the post-detection analysis of experimental images, this implies that the influence of polychromatic illumination leads to no (additional) qualitative artifacts.

25. P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express **16**, 5141–5149 (2008).
[CrossRef] [PubMed]

*The visible contrast decreases to 50%, when the spectral width of the incident beam becomes comparable to the spectral acceptance of the reflection curves*.

*ζ*=1.3×10

^{-4}. The dashed lines in Fig. 6(b) indicate that the visible contrast degrades to about 30% (mean of both reflections) due to polychromatic instead of monochromatic illumination.

### 5.3. Influence of polychromatic illumination on spatial resolution

**74**054107 (2006).
[CrossRef]

29. C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. **44**, 76–86 (1916).
[CrossRef]

*δ*-functions:

*D*(

_{in}*x*)=

*δ*(

_{D}*x*)+

*δ*(

_{D}*x*+

*x*

_{0}). The corresponding intensity distribution, which is calculated by Eqs. (23)–(25), can easily be checked for a minimum between the two image points. The distance between the two object points x0 can then be varied until the minimum has a certain contrast compared to the smaller of both maxima. We have chosen the contrast to be 5%, which is experimentally well observable.

*ζ*. Although this reminds of a focussing effect, we cannot explain it completely. But we point out that extensive investigations have shown that this is not a result of numerical artifacts.

## 6. Comparison with experiment

*θλ*

_{2}and

*ω*

_{2}for the second reflection are defined analogously to the first. According to Eq. 25 the result is then given by

*ω*

_{1}and

*ω*

_{2}of the first and second reflection. Equation 32 can now be used to compare the theoretical and experimental result in the case of an absent sample.

*ω*

_{1}and

*ω*

_{2}as experimental parameters was performed and by calculating the mean intensity of each experimental image, the CCD-detector was used as point detector. The experimental intensity map of the scan is shown in Fig. 8(a).

*W*(

*λ*), that is offered to the Bragg Magnifier, was taken as the Si-111 double-crystal reflection curve of the monochromator. The corresponding full width at half maximum (FWHM) of 47.4

*µ*rad implies a wavelength band of

*ζ*=1.3×10·

^{4}at the given photon energy. The reflection curves were calculated according to standard dynamic theory [17]. The theoretical intensity map is shown in Fig. 8(b). The excellent agreement between experiment and theory validates the theoretical formalism developed in this article.

*ω*

_{1}as the scan parameter and

*ω*

_{2}as a fixed parameter. The half widths of each rocking curve can be determined and the result is the half width in dependence of

*ω*

_{2}: FWHM

*ω*

_{1}(

*ω*

_{2}). The same can be done with the vertical lines resulting in FWHM

*ω*

_{2}(

*ω*

_{1}). Both half widths are shown in Fig. 9.

*ω*

_{2}is the scan parameter; bottom curve in Fig. 9) for our explanation. Assuming for the moment that all reflection curves can be approximated by Gaussian functions, then the integral in Eq. 32 in fact describes a convolution of the function |

*R*̂

_{2}|

^{2}with the function |

*W*×

*R*̂

_{1}|

^{2}. With this assumption, the observable half width after convolution (FWHM

*) is given by*

_{out}*is the width of the function*

_{WR1}*W*×

*R*̂

_{1}and FWHM

*is the width of the second reflection curve. But as Fig. 10 shows, FWHMWR1 depends on the chosen working point of the first reflection. This means that one reflection limits the available wavelength band for the other.*

_{R2}27. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. **90**, 193501 (2007).
[CrossRef]

## 7. Conclusion

### A. Back-tracing

*I*(Δ

_{b}*α,λ*) of the x-ray beam at the position of the sample. In this appendix, we want to present a ray-tracing approach to connect

*I*(Δα,

_{b}*λ*) to the parameters of the monochromator and the source. The direction and intensity of the rays correspond to the propagation direction and intensity of the plane waves used in section 4.2. We also want to investigate an implicit assumption about

*I*used in section 4.2. In general, the spectral and angular distribution of the x-ray beam at the sample will depend on the location x in the object plane, i.e.

_{b}*I*(Δ

_{b}*α,λ*)=

*I*(Δ

_{b}*α,λ,x*) and the aim of this section will be to justify this assumption. The definition of quantities not defined here, can be found in section 3. Please note that we denote Δ

*α*as

*ϕ*in the following.

*m*=1,2) to the position on the source. Each ray is characterized by three quantities (see also Fig. 12) expressed in the appropriate deviations from the reference beam: x the starting position in the object plane, Δ

*λ/λ*the relative deviation from the reference wavelength and

*ϕ*the angular deviation from the reference beam. Thus, the reference beam (i.e.

*x*=0, Δ

*λ/λ*=0,

*ϕ*=0) defines the optical axis of the system.

#### A.1. Dispersion at the monochromator crystal

30. B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. **36**, 681–716 (1964).
[CrossRef]

*θ*and the angular deviation from the reference beam

_{m}*ϕ*for the exit side and

*ϕ*′ for the incident side, respectively. In consistency with the theoretical formalism developed in section 4 we use a formulation of dynamic theory that uses angular deviations at the incidence side of the reflection Δ

*θ*as the input argument of the reflection curve

_{m}*R̂*. From Fig. 11 it can be concluded that Δ

_{cm}*θ*is given by

_{m}#### A.2. Ray-tracing

*ϕ*′. Therefore, we have connected the ray characterized by (

*x*,

*ϕ*,Δ

*λ/λ*) before reflection to the ray characterized by (

*x*′,

*ϕ*′,Δ

*λ/λ*) after reflection. Naturally, this can be done a second time in order to connect the intermediate plane with the source plane using the same equations. The result for the position on the source plane

*x*″ is

*θ*

_{1}=

*θ*

_{2}=

*θ*

_{B}).

*θm*(Eq. 34) on the corresponding rocking curve: |

*R*̂

*(Δ*

_{cm}*θ*)|

_{m}^{2}(with

*m*=1,2). On the presumption that the source is completely incoherent, the angular and spectral intensity distribution at the position of the sample is finally given by

*I*(

_{s}*θ*″,

*λ,x*″) denotes the intensity distribution of the source in dependence of the emission angle

*ϕ*″, the wavelength

*λ*and the position

*x*″. With the help of the ray-tracing approach developed here, it is possible to calculate the characteristics of the beam

*I*for nearly all practical relevant setups.

_{b}*b*

_{1}=

*b*

_{2}=1), which implies

*ϕ*=

*ϕ*′=

*ϕ*″. Furthermore, the spectral acceptance of the monochromator crystals is much smaller than the spectrum provided by the source and as it was shown in section 2 the influence of divergence is negligible. Thus, the intensity distribution of the source can be written as

*I*(

_{s}*ϕ*″,

*λ,x*″)=

*I*(

_{s}*x*″)=

*I*

_{0}

*δ*(

_{D}*x*″) with

*I*

_{0}a factor of proportionality. According to Eq. (36) and Eq. (37) with

*x*″=0 and

*z*=

_{g}*z*

_{1}+

*z*

_{2}+

*z*

_{3}+

*z*

_{4}the angular offset

*ϕ*is now given by

*ϕ*=

*x/z*. This directional change corresponds to a shift of the center of the wavelength band (Eq. 34) of

_{g}*x*. The field of view of the Bragg Magnifier is typically 1 mm in width, which results into a shift of the center of the wavelength band of about ≈2×10

^{-5}. This is one order of magnitude smaller than the wavelength band itself. Therefore, this effect is not visible in the experiment and thus the assumption of the independency of

*I*on

_{b}*x*is justified.

## References and links

1. | E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. |

2. | M. Kuriyama, R. C. Dobbyn, R. D. Spal, H. E. Burdette, and D.R. Black, “Hard x-ray microscope with submicrometer spatial resolution,” J. Res. Natl. Inst. Stand. Technol. |

3. | T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S.W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature |

4. | V. N. Ingal and E. A. Beliaevskaya, “Imaging of biological objects in the plane-wave diffraction scheme,” Nuovo Cimento |

5. | K. Kobayashi, K. Izumi, H. Kimura, S. Kimura, T. Ibuki, Y. Yokoyama, Y. Tsusaka, Y. Kagoshima, and J. Matsui, “X-ray phase-contrast imaging with submicron resolution by using extremely asymmetric Bragg diffractions,” Appl. Phys. Lett. |

6. | R. Köhler and P. Schäfer, “Asymmetric Bragg reflection as magnifying optics,” Cryst. Res. Technol. |

7. | D. Korytár, P. Mikulík, C. Ferrari, J. Hrdý, T. Baumbach, A. Freund, and A. Kubena, “Two-dimensional x-ray magnification based on a monolithic beam conditioner,” J. Phys. D: Appl. Phys. |

8. | M. Stampanoni, G. Borchert, and R. Abela, “Towards nanotomography with asymmetrically cut crystals,” Nucl. Instrum. Meth. A |

9. | M. G. Hönnicke and C. Cusatis, “Analyzer-based x-ray phase-contrast microscopy combining channel-cut and asymmetrically cut crystals,” Rev. Sci. Instrum. |

10. | R. Spal, “Submicrometer resolution hard X-Ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. |

11. | J. Keyriläinen, M. Fernandez, and P. Suortti, “Refraction contrast in x-ray imaging,” Nucl. Instrum. Meth. A |

12. | Ya.I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S.W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. D: Appl. Phys. |

13. | P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B |

14. | J.P. Guigay, E. Pagot, and P. Cloetens, “Fourier optics approach to X-ray analyser-based imaging,” Opt. Commun. |

15. | A. Bravin, V. Mocella, P. Coan, A. Astolfo, and C. Ferrero, “A numerical wave-optical approach for the simulation of analyzer-based x-ray imaging,” Opt. Express |

16. | Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. D: Appl. Phys. |

17. | A. Authier |

18. | P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Spatial resolution in Bragg-magnified X-ray images as determined by Fourier analysis,” Phys. Status Solidi (a) |

19. | A. Rack, H. Riesemeier, S. Zabler, T. Weitkamp, B. Müller, G. Weidemann, P. Modregger, J. Banhart, L. Helfen, A. Danilewsky, H. Gräber, R. Heldele, B. Mayzel, J. Goebbels, and T. Baumbach, “The high resolution synchrotron-based imaging stations at the BAMline (BESSY) and TopoTomo (ANKA),” Proc. SPIE |

20. | P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synch. Rad. |

21. | P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. |

22. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

23. | M. Kuriyama, W. J. Boettinger, and G. G. Cohen, “Synchrotron radiation topography,” Annu. Rev. Mater. Sci. |

24. | J. Als-Niehlsen and D. McMorrow |

25. | P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express |

26. | J. W. Goodman, |

27. | P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. |

28. | E. Wilson |

29. | C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. |

30. | B. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(110.2990) Imaging systems : Image formation theory

(340.7460) X-ray optics : X-ray microscopy

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 12, 2009

Revised Manuscript: June 12, 2009

Manuscript Accepted: June 19, 2009

Published: June 25, 2009

**Citation**

Peter Modregger, Daniel Lübbert, Peter Schäfer, Jane Richter, Rolf Köhler, and Tilo Baumbach, "Influence of partial coherence on analyzer-based imaging with asymmetric Bragg reflection," Opt. Express **17**, 11618-11637 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11618

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### References

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