## A robust tool for photon source geometry measurements using the fractional Talbot effect |

Optics Express, Vol. 22, Issue 3, pp. 2745-2760 (2014)

http://dx.doi.org/10.1364/OE.22.002745

Acrobat PDF (1785 KB)

### Abstract

A reliable measurement of beam coherence is important for optimal performance of a number of coherence methods being utilized at third-generation synchrotrons and free-electron lasers. Various approaches have been proposed in the past for determining the source size, and hence the degree of coherence; however they often require complex setups with perfect optics and suffer from undefined uncertainties. We present a robust tool for X-ray source characterization with a full quantitative uncertainty analysis for fast on-the-fly coherence measurements. The influence of three multilayer monochromator crystals on the apparent source size is evaluated using the proposed method.

© 2014 Optical Society of America

## 1. Introduction

1. I. A. Vartanyants and A. Singer, “Coherence properties of hard X-ray synchrotron sources and X-ray free-electron lasers,” New J. Phys. **12**, 035004 (2010). [CrossRef]

2. S. Dierker, R. Pindak, R. Fleming, I. Robinson, and L. Berman, “X-ray photon correlation spectroscopy study of brownian motion of gold colloids in Glycerol,” Phys. Rev. Lett. **75**, 449–452 (1995). [CrossRef] [PubMed]

4. S. Roy, D. Parks, K. A. Seu, R. Su, J. J. Turner, W. Chao, E. H. Anderson, S. Cabrini, and S. D. Kevan, “Lensless X-ray imaging in reflection geometry,” Nat. Photonics **5**, 243–245 (2011). [CrossRef]

5. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. **2**, 473–475 (1996). [CrossRef] [PubMed]

8. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. **81**, 3287 (2002). [CrossRef]

9. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature **467**, 436–439 (2010). [CrossRef] [PubMed]

10. B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics **5**, 420–424 (2011). [CrossRef]

11. P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle X-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. **108**, 2–5 (2012). [CrossRef]

*et al.*[14

14. D. Paterson, B. Allman, P. McMahon, J. Lin, N. Moldovan, K. Nugent, I. McNulty, C. Chantler, C. Retsch, and T. Irving, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. **195**, 79–84 (2001). [CrossRef]

12. M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light* (Cambridge University, 1999). [CrossRef]

15. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. **47**, 895 (1957). [CrossRef]

16. C. Chang, P. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. **182**, 25–34 (2000). [CrossRef]

*et al.*[17

17. W. Leitenberger, S. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. **191**, 91–96 (2001). [CrossRef]

18. T. Panzner, W. Leitenberger, J. Grenzer, Y. Bodenthin, T. Geue, U. Pietsch, and H. Möhwald, “Coherence experiments at the energy-dispersive reflectometry beamline at BESSY II,” J. Phys. D: Appl. Phys. **36**, A93–A97 (2003). [CrossRef]

19. W. Leitenberger, H. Wendrock, L. Bischoff, and T. Weitkamp, “Pinhole interferometry with coherent hard X-rays,” J. Synchrotron Radiat. **11**, 190–197 (2004). [CrossRef] [PubMed]

13. V. Kohn, I. Snigireva, and A. Snigirev, “Direct measurement of transverse coherence length of hard X-rays from interference fringes,” Phys. Rev. Lett. **85**, 2745–2748 (2000). [CrossRef] [PubMed]

20. M. Yabashi, K. Tamasaku, and T. Ishikawa, “Characterization of the transverse coherence of hard synchrotron radiation by intensity interferometry,” Phys. Rev. Lett. **87**, 140801 (2001). [CrossRef] [PubMed]

21. F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. Robinson, “Shearing interferometer for quantifying the coherence of hard X-ray beams,” Phys. Rev. Lett. **94**, 1–4 (2005). [CrossRef]

22. K. S. Morgan, S. C. Irvine, Y. Suzuki, K. Uesugi, A. Takeuchi, D. M. Paganin, and K. K. Siu, “Measurement of hard X-ray coherence in the presence of a rotating random-phase-screen diffuser,” Opt. Commun. **283**, 216–225 (2010). [CrossRef]

23. J. Lin, D. Paterson, A. Peele, P. McMahon, C. Chantler, K. Nugent, B. Lai, N. Moldovan, Z. Cai, D. Mancini, and I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. **90**, 1–4 (2003). [CrossRef]

24. I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. **107**, 1–5 (2011). [CrossRef]

*et al.*[25

25. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard X-rays,” Opt. Lett. **22**, 1059–1061 (1997). [CrossRef] [PubMed]

26. J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. **11**, 476–482 (2004). [CrossRef] [PubMed]

21. F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. Robinson, “Shearing interferometer for quantifying the coherence of hard X-ray beams,” Phys. Rev. Lett. **94**, 1–4 (2005). [CrossRef]

24. I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. **107**, 1–5 (2011). [CrossRef]

18. T. Panzner, W. Leitenberger, J. Grenzer, Y. Bodenthin, T. Geue, U. Pietsch, and H. Möhwald, “Coherence experiments at the energy-dispersive reflectometry beamline at BESSY II,” J. Phys. D: Appl. Phys. **36**, A93–A97 (2003). [CrossRef]

19. W. Leitenberger, H. Wendrock, L. Bischoff, and T. Weitkamp, “Pinhole interferometry with coherent hard X-rays,” J. Synchrotron Radiat. **11**, 190–197 (2004). [CrossRef] [PubMed]

23. J. Lin, D. Paterson, A. Peele, P. McMahon, C. Chantler, K. Nugent, B. Lai, N. Moldovan, Z. Cai, D. Mancini, and I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. **90**, 1–4 (2003). [CrossRef]

26. J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. **11**, 476–482 (2004). [CrossRef] [PubMed]

26. J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. **11**, 476–482 (2004). [CrossRef] [PubMed]

27. T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. **86**, 054101 (2005). [CrossRef]

29. A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. **17**, 496–510 (2010). [CrossRef] [PubMed]

30. A. Yaroshenko, M. Bech, G. Potdevin, A. Malecki, T. Biernath, J. Wolf, A. Tapfer, M. Schüttler, J. Meiser, D. Kunka, M. Amberger, J. Mohr, and F. Pfeiffer, “Non-binary phase gratings for X-ray imaging with a compact Talbot interferometer,” Opt. Express **22**, 547 (2014). [CrossRef]

## 2. Methods

### 2.1. Experimental setup

*I*= 400mA, top-up mode), was monochromatized with a Si(111) double-crystal (DCM) and two different double-multilayer monochromators (DMM), respectively. The latter two consist of different compositions, coated on a Si(111) substrate, for use at different energy ranges: [Ru/C]

_{100}(100 layers, 4nm periodicity) for energies below 22keV and [W/Si]

_{100}(100 layers, 3nm periodicity) for energies above [31

31. M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE **6318**, 63180M (2006). [CrossRef]

_{4}C multilayer crystal (3.02nm periodicity) the X-ray energy was set to 18keV using the Si(111) monochromator and the crystal was placed on a goniometer 1480mm upstream from the grating. The detailed setup for the measurement of the V/B

_{4}C multilayer crystal has been reported elsewhere [29

29. A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. **17**, 496–510 (2010). [CrossRef] [PubMed]

*R*= 26.3m and

*R*= 26.5m distances from the source, respectively. A CCD detector coupled with visible light optics (20× magnification) and a 20

*μ*m thick scintillator was used for acquiring images, yielding an effective pixel size of 0.38

*μ*m. The grating was aligned with a goniometer (about the

*z*-axis), whereby smalls

*z*-tilts can also be corrected by rotating the detector, which introduces a negligible effect of projecting the respective source sizes onto one another. Tilts about the

*x*- and

*y*-axes were treated computationally and are discussed later on. Thus, it is noteworthy that, in the most simplified setup, the grating can be aligned purely by eye, making the use of high precision motors obsolete. For the characterization of the V/B

_{4}C multilayer, the pixels of the acquired images were binned by a factor of 2, yielding an effective pixel size of 0.76

*μ*m. The detector was moved to various propagation distances

*z*ranging from 0.002 – 2.102m. At each distance, an image was taken and corrected with the respective dark and flat-field images, prior to further analysis. Due to different X-ray fluxes under each experimental condition, the exposure times

*t*

_{exp}were adjusted as follows:

*t*

_{exp}= 0.2s for the [Ru/C]

_{100},

*t*

_{exp}= 1.5s for the [W/Si]

_{100}and

*t*

_{exp}= 36.0s for the combined Si(111) and V/B

_{4}C multilayer measurement. The time required for acquiring a full set of 106 Talbot images was between 30 and 120min.

### 2.2. Theoretical background

*ψ*at a distant point

**r**can be described as with wave vector

**k**and amplitude

*A*. For simplicity, we disregard the time dependency by observing the wave only at

*t*= 0. The wave field disturbance after passing a diffraction object such as a 2D grating is described by Huygens-Fresnel principle stating that the result is a superposition of secondary wavelets emerging from every point of the initial wavefront [12

12. M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light* (Cambridge University, 1999). [CrossRef]

*G*is the area along the diffraction object (grating),

**n**the normal vector to it, and

**z′**and

**R′**represent position vectors indicated in Fig. 1. It is then possible to simplify the above expression by regarding a big source-to-grating distance (i. e. Fresnel approximation) to give: In the following formulation, we restrict ourselves to the case of a 2D grating with grating coordinates

*x*

_{0}and

*y*

_{0}that are normal to the axis

*z*. Since the grating can be seen as an operator changing the amplitude and/or phase of the impinging wavefront at defined values

*x*

_{0}and

*y*

_{0}, it can be expressed as a function multiplied with the wavefront at the grating position. Further on, we consider only small angles and utilize the fact that the aperture diameters (i. e. grating pitch) of the object are orders of magnitudes greater then the wavelength, which makes the Rayleigh-Sommerfeld and Fresnel-Kirchhoff formulas identical and omits the above cosine term [32]. Equation (3) then transforms to: with source-to-grating distance

*R*and

*k*= |

**k**|. After applying the so-called paraxial approximation and combining the grating function

*G*and the phasor of the impinging wavefront in function

*f*(

*x*

_{0},

*y*

_{0}) we finally get: where

*x*and

*y*are now the coordinates of the imaging plane. This equation can now be evaluated by two different approaches: either by applying the paraxial approximation to the phasor of the impinging wavefront [33], which leads to the definition of the defocussing distance

*D*and magnification

*M*: or by identifying above equation as a convolution of

*f*(

*x*,

*y*) with the Fresnel kernel. We choose the latter for simplicity reasons and get: where

*C*is a constant, proportional to the amplitude of the impinging wave.

1. I. A. Vartanyants and A. Singer, “Coherence properties of hard X-ray synchrotron sources and X-ray free-electron lasers,” New J. Phys. **12**, 035004 (2010). [CrossRef]

*σ*

_{proj,H}and

*σ*

_{proj,V}with where

*σ*is the FWHM of the horzontal and vertical source size, respectively [34

_{i}34. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE **6318**, 63180S (2006). [CrossRef]

*I*at

**r**= (

*x*,

*y*,

*z*) is thus given by: The function

*f*(

*x*,

*y*) describes the wavefront just after passing the grating and reads as: where

*h*(

*x*,

*y*) is the grating function with coordinates

*x*and

*y*,

*δ*and

*β*are the refractive and absorption indices, respectively. The function

*h*(

*x*,

*y*) can be expressed e. g. as a binary function with the grating’s gold pillar height

*h*, integer

*n*, grating period

*a*and grating’s gold pillar width

*w*. According to the fabrication process of the grating, function

*h*(

*x*,

*y*) was slightly modified in the numerical implementation to include a slight trapezoidal shape of the grating with angle

*α*.

### 2.3. Source size calculation

*Ĩ*(

*k*

_{x},

*k*

_{y},

*z*) = 𝔉[

*I*(

*x*,

*y*,

*z*)]. By calculating these coefficients at different propagation distances

*z*, it is possible to obtain the degree of coherence [26

_{i}**11**, 476–482 (2004). [CrossRef] [PubMed]

*n*

_{y}and

*n*

_{x}are the number of pixels in the vertical and horizontal directions, respectively. We then extract principle Fourier components of the line profiles, analog to [26

**11**, 476–482 (2004). [CrossRef] [PubMed]

*k*=

_{x}*p*

_{size}·

*n*

_{x}/(

*Ma*),

*k*=

_{y}*p*

_{size}·

*n*

_{y}/(

*Ma*) depending on the pixel size

*p*

_{size}, the grating period

*a*, magnification

*M*and the total number of pixels. From now on, we omit the explicit notation for horizontal and vertical Fourier coefficients, as they are treated equally further on, but distinguish those obtained from simulated and experimental Talbot images, denoted as

*F*

_{sim}(

*z*) and

*F*

_{exp}(

*z*). The further approach is as follows: based on the fact that Talbot images can be correctly modeled given that all experimental parameters are known [see Eq. (10)], the source size can be determined implicitly by using an appropriate set of simulation variables. More precisely, Fourier coefficients

*F*

_{sim}(

*z*) can be varied depending on the source size (and other parameters) and fitted by means of weighted least-squares to

*F*

_{exp}(

*z*).

*p*as a measure for the quality of the fit between

*F*

_{sim}(

*z*) and

*F*

_{exp}(

*z*): with propagation distances

**z**= (

*z*

_{1},...,

*z*,...), energy

_{i}*E*, source size

*σ*, grating’s duty cycle

*d*

_{c}, trapezoidal angle

*α*, total number of propagation distances

*n*

_{z}and being the normalized experimental and simulated principal Fourier coefficients. In the following, we denote

*p*as the “weighted LSE”. Obviously, if a particular set of parameters minimizes

*p*so that it approaches 0, the solution is found. Considering that, we only need to define minimum and maximum margins for each parameter that are consistent with the experimental setup and conduct an efficient search in the parameter space for minimizing

*p*.

*E*,

*σ*,

*d*

_{c}and

*α*,

*n*

_{max}intervals are “nested” between given minimum and maximum margins. Fourier coefficients for the simulated and experimental Talbot images are then calculated for each direction (horizontal and vertical) independently. Within this step, the averaged line profiles from Eq. (14) are additionally multiplied with a Hanning window function to reduce leakage and aliasing effects [35

35. R. Kluender, F. Masiello, P. van Vaerenbergh, and J. Härtwig, “Measurement of the spatial coherence of synchrotron beams using the Talbot effect,” Phys. Status Solidi A **206**, 1842–1845 (2009). [CrossRef]

*p*is calculated and compared to the value of

*p*

_{start}, which upon initialization can be any big number (e. g.

*p*

_{start}= ∞) and for any further iteration contains the lowest occurrence of

*p*heretofore. If

*p*is lower than

*p*

_{start}at this point then the whole set of parameters is written to temporary memory and

*p*

_{start}is overwritten by the current value of

*p*. These steps are repeated until all intervals for all parameters have been run through. Thereafter, the parameters corresponding to the lowest value of

*p*are loaded and used to nest intervals for the next iteration. The nesting is implemented similarly to a bisection method (also known as binary search algorithm), where

*n*

_{max}sections are used rather than only two. Finally, after repeating the nesting

*k*

_{max}times, the calculated source size along with other parameters can be loaded from temporary memory.

*E*

_{min}= 21.5keV,

*E*

_{max}= 22.5keV,

*σ*

_{min}= 0

*μ*m,

*σ*

_{max}= 200

*μ*m,

*d*

_{c,min}= 0.50,

*d*

_{c,max}= 0.54,

*α*

_{min}= 0° and

*α*

_{max}= 4.2°.

*z*when fitting to the experimental data. The weighting in Eq. (16) follows another simple principle: small values of Fourier coefficients contain less information (more noise), and thus, deviations arising from these coefficients are penalized by the additional weighting when minimizing

_{i}*p*. The only remaining ambiguity in this respect arises from the fitting procedure itself, namely the treatment of local minima which represents a common challenge for most iterative processes. In the present work we address this issue by multiplying each interval, that is parted upon interval nesting, with an additional factor

*s*. To be exact, each parted interval is additionally “stretched” by an arbitrary factor

*s*(as indicated in Fig. 3), thus enabling each parameter to “climb out” of a local minimum in a subsequent iteration step. Naturally, there exist more sophisticated methods for treating local extrema which are not discussed further, as they were not required in the present study. Finally, all parameter margins are set in consistence with the experimental setup: the X-ray energy margin originates from the fact that the aligned energy might not be correct due to impreciseness in the monochromator calibration; the source size margin can be set arbitrarily in a region where the source size is approximately expected; and the margins for the grating’s duty cycle and angle on the one hand account for grating’s fabrication impreciseness and on the other hand for imprecise grating alignment in the beam. In particular, the latter two account for tilts about

*x*- and

*y*-axes causing more pronounced trapezoidal pillar shapes [30

30. A. Yaroshenko, M. Bech, G. Potdevin, A. Malecki, T. Biernath, J. Wolf, A. Tapfer, M. Schüttler, J. Meiser, D. Kunka, M. Amberger, J. Mohr, and F. Pfeiffer, “Non-binary phase gratings for X-ray imaging with a compact Talbot interferometer,” Opt. Express **22**, 547 (2014). [CrossRef]

*N*

_{max}= 120, 204 for the present example of

*n*

_{max}= 3 intervals,

*k*

_{max}= 7 iterations with

*s*= 1.5 and

*n*= 106 propagation distances. The complete calculation took about 10min on a single PC (Pentium i5, Mathworks Matlab), whereas a single iteration step lasted around 6ms. In order to boost the calculation time, the simulation from Eq. (10) was conducted in 1D independently for the horizontal and vertical directions.

_{z}### 2.4. Uncertainty analysis

*R*,

*R′*,

*z*,

*z′*as well as all object’s structures (such as grating pitch etc.) are orders of magnitudes greater than the wavelength

*λ*[32]. This condition is in full agreement with the investigated energy range of 10 – 40keV. The paraxial approximation from Eq. (6) is justified by the fact that we are concerned with regions very close to the optical axis. On the other hand it has also been shown that combined paraxial-Fresnel approximation is more accurate than either one imposed separately [36

36. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. **71**, 7 (1981). [CrossRef]

*α*= 0°,

*d*

_{c}= 0.5), that all of its parameters are known (

*h*,

*w*) and it is perfectly aligned in the beam, the experimental results will still differ from the theoretical prediction due to the uncertainty in the adjusted propagation distance of the detector, which will affect the Fresnel kernel and for our experiment we estimate to be less than Δ

*z*= 2mm. The maximal error margin for the source size can be evaluated numerically from this value by including it in the weighted LSE from Eq. (16) and postulating that the precision of

*z*is directly determining the precision of the source size

*σ*: Thus, the source size uncertainty Δ

*σ*is calculated by two subsequent simulations for finding an appropriate value of Δ

*σ*that equally alters the weighted LSE

*p*as Δ

*z*. This relation is derived from the fact that it is impossible to define the origin of

*p*≠ 0 from the acquired experimental data and the subsequent calculation of

*p*. Likewise,

*z*is also directly connected with the energy uncertainty, which arises from the fact that the energy cannot be selected very precisely by the monochromator (see above). For this reason, the energy uncertainty is treated the same way by requiring that meaning that it is impossible to determine whether

*p*≠ 0 arises from Δ

*z*or Δ

*E*, which is why they have to be treated equally in order to cover maximal error margins.

*k*

_{eff}and require that it incorporates all uncertainties originating from the grating’s fabrication process: where Δ

*h*is the uncertainty of the fabricated grating’s pillar height and Δ

*δ*and Δ

*β*are the dispersion and absorption uncertainties, respectively. The latter two originate from imperfections in the grating’s gold structures, that have a slightly altered gold density and thus induce slightly different phase shifts and absorption levels for X-rays. By evaluating Eq. (21) we obtain: which can further be approximated by making use of

*c*and Planck’s constant

*h̄*. In the latter approximation we have used the fact that

*k*·

*𝒪*(10

^{−2}) ≈ Δ

*k*and 2Δ

*k·𝒪*(10

^{−2}) ≪ Δ

*k*. We can thus state that the most pronounced error from the grating’s fabrication process will result in a slightly modified phase shift and absorption, which can be consolidated into an effective energy uncertainty. The only remaining error sources in this respect are the grating’s duty cycle and angle that are determined very precisely in the fitting process as they significantly influence the shapes of the curves in Fig. 4. This means that, once the best fit has been found, the source size uncertainty no longer depends on the uncertainties of these parameters. As mentioned above, their values depend mainly on the grating’s fabrication process as well as the alignment of the grating in the beam.

*k*and the propagation distance Δ

*z*affecting the Fresnel kernel in Eq. (10). Following the same strategy as before to consolidate both values into an effective wave vector uncertainty, we write and obtain: To evaluate this equation we regard two marginal conditions. For Δ

*z*≪

*z*, it is trivial to show that Δ

*k*

_{eff}≈ Δ

*k*, meaning that for longer propagation distances the precision of the detector’s travel range will be negligible. For Δ

*z*≈

*z*we get the relatively big value of Δ

*k*

_{eff}≈

*k*/2, but it can be shown that the magnitude of Δ

*k*is proportional to the magnitude of Δ

*z*. This means, if we require Eq. (28) to have the same order of magnitude as Δ

*k*

_{eff}from before, it is in fact enough to measure the first Fourier coefficients several centimeters away from the grating. For our case, this constraint is obsolete since the fitting algorithm equally takes all Fourier coefficients into account and if a few coefficients are incorrect, the final results will not be altered. Moreover, uncertainties affecting the Fresnel kernel have already been taken into account in Eqs. (19) and (20).

## 3. Results and discussion

_{4}C multilayer crystal is shown in Fig. 7. The source size values obtained with the fitting algorithm as well as all calculated uncertainties are listed in Tab. 1 and graphically represented in Fig. 8.

_{100}monochromator at lower energies was smaller during the second experiment. Table 1 also highlights the actual value of the X-ray energy that resulted from the fitting procedure. The values were found to be systematically lower than the aligned energy based on the pre-calibrated monochromators, being indicative of how precise the X-ray energies can actually be set during beamline operation. The respective energy uncertainties Δ

*E*

_{eff}correspond to effective uncertainties in the aligned mean energy. Thus they must not be confused with the energy bandwidth of the monochromated beam, which is in the range of a few percent [31

31. M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE **6318**, 63180M (2006). [CrossRef]

_{4}C multilayer crystal on the X-ray beam, the previously introduced fitting algorithm had to be slightly modified. First, since the energy

*E*is determined precisely during an initial reference measurement with the Si(111) monochromator and the grating,

*E*can be regarded as a constant parameter (from that point on). Secondly, the source-to-sample distance

*R*in Eq. (11) was added as a fitting variable, while in Eq. (9) it was left unchanged with

*R*= 26.5m. By doing so, the wavefront curvature of the X-ray beam after being reflected from the V/B

_{4}C multilayer was studied independently from the source size. Thus, we found that the horizontal beam characteristics were not altered significantly by the multilayer, yielding a horizontal source size of 154

*μ*m and a curvature radius of 25.0m. In the vertical direction, the curvature radius of the beam was decreased to 4.5m, while the source size was found to be 43

*μ*m. This effect is obvious from the shifted value of the Talbot plane to

*z*= 0.7m [in Fig. 7(b)] in the vertical direction as compared to

*z*= 0.6m in the horizontal direction. From Fig. 7 it is also clearly visible that in the range of

*z*= 0–0.4m, the fitting results for the vertical coefficients are not as good as for the horizontal ones. As discussed above, this effect can be affiliated to a local minimum in the fitting procedure regarding the grating’s duty cycle and/or trapezoidal angle.

**11**, 476–482 (2004). [CrossRef] [PubMed]

29. A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. **17**, 496–510 (2010). [CrossRef] [PubMed]

*z*, but also on the quality of the data. For instance, in the presence of noisy data, many Talbot images are required for obtaining a correct fit and for reducing statistical errors (e. g. from beam fluctuations). Decreasing the travel range of the detector, on the other hand, simplifies the experimental setup, but may increase the calculated uncertainty which we showed to be directly connected to the uncertainty of the adjusted detector’s propagation distance (Δ

*z*). Applied to the [Ru/C]

_{100}multilayer at 21.3keV, a reduction to 56 Talbot images and

*z*= 1.1m yields

*σ*

_{H}= (131 ± 3)

*μ*m and

*σ*

_{V}= (44 ± 2)

*μ*m, which represents only a marginally greater uncertainty. For other experimental conditions, however, these results may vary as the values depend both on the curve shapes and the measured range of

*z*. Concerning the data from Tab. 1, the same parameters were used in the fitting procedure, while the source size and energy uncertainties were calculated for each dataset separately.

## 4. Conclusion

37. G. Lovric, “Source size calculator,” http://www.psi.ch/sls/tomcat/ (2013).

## 5. Appendix

*et al.*[38

38. S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard X-rays by 100 keV e-beam lithography and electroplating,” J. Synchrotron Radiat. **18**, 442–446 (2011). [CrossRef] [PubMed]

39. S. Yasin, D. Hasko, and H. Ahmed, “Comparison of MIBK/IPA and water/IPA as PMMA developers for electron beam nanolithography,” Microelectron. Eng. **61–62**, 745–753 (2002). [CrossRef]

^{3}and a nominal cobalt content of 1–2%, corresponding to three components contributing to the optical properties in the X-ray range.

## Acknowledgments

## References and links

1. | I. A. Vartanyants and A. Singer, “Coherence properties of hard X-ray synchrotron sources and X-ray free-electron lasers,” New J. Phys. |

2. | S. Dierker, R. Pindak, R. Fleming, I. Robinson, and L. Berman, “X-ray photon correlation spectroscopy study of brownian motion of gold colloids in Glycerol,” Phys. Rev. Lett. |

3. | I. Robinson, J. Libbert, I. Vartanyants, J. Pitney, D. Smilgies, D. Abernathy, and G. Grübel, “Coherent X-ray diffraction imaging of silicon oxide growth,” Phys. Rev. A |

4. | S. Roy, D. Parks, K. A. Seu, R. Su, J. J. Turner, W. Chao, E. H. Anderson, S. Cabrini, and S. D. Kevan, “Lensless X-ray imaging in reflection geometry,” Nat. Photonics |

5. | A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. |

6. | K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. |

7. | P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. |

8. | C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. |

9. | M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature |

10. | B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics |

11. | P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle X-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. |

12. | M. Born and E. Wolf, |

13. | V. Kohn, I. Snigireva, and A. Snigirev, “Direct measurement of transverse coherence length of hard X-rays from interference fringes,” Phys. Rev. Lett. |

14. | D. Paterson, B. Allman, P. McMahon, J. Lin, N. Moldovan, K. Nugent, I. McNulty, C. Chantler, C. Retsch, and T. Irving, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. |

15. | B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. |

16. | C. Chang, P. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. |

17. | W. Leitenberger, S. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. |

18. | T. Panzner, W. Leitenberger, J. Grenzer, Y. Bodenthin, T. Geue, U. Pietsch, and H. Möhwald, “Coherence experiments at the energy-dispersive reflectometry beamline at BESSY II,” J. Phys. D: Appl. Phys. |

19. | W. Leitenberger, H. Wendrock, L. Bischoff, and T. Weitkamp, “Pinhole interferometry with coherent hard X-rays,” J. Synchrotron Radiat. |

20. | M. Yabashi, K. Tamasaku, and T. Ishikawa, “Characterization of the transverse coherence of hard synchrotron radiation by intensity interferometry,” Phys. Rev. Lett. |

21. | F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. Robinson, “Shearing interferometer for quantifying the coherence of hard X-ray beams,” Phys. Rev. Lett. |

22. | K. S. Morgan, S. C. Irvine, Y. Suzuki, K. Uesugi, A. Takeuchi, D. M. Paganin, and K. K. Siu, “Measurement of hard X-ray coherence in the presence of a rotating random-phase-screen diffuser,” Opt. Commun. |

23. | J. Lin, D. Paterson, A. Peele, P. McMahon, C. Chantler, K. Nugent, B. Lai, N. Moldovan, Z. Cai, D. Mancini, and I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. |

24. | I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. |

25. | P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard X-rays,” Opt. Lett. |

26. | J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. |

27. | T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. |

28. | A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. |

29. | A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. |

30. | A. Yaroshenko, M. Bech, G. Potdevin, A. Malecki, T. Biernath, J. Wolf, A. Tapfer, M. Schüttler, J. Meiser, D. Kunka, M. Amberger, J. Mohr, and F. Pfeiffer, “Non-binary phase gratings for X-ray imaging with a compact Talbot interferometer,” Opt. Express |

31. | M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE |

32. | J. W. Goodman, |

33. | P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation: principles, implementation and applications,” Phd thesis, Vrije Universiteit Brussel (1999). |

34. | T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE |

35. | R. Kluender, F. Masiello, P. van Vaerenbergh, and J. Härtwig, “Measurement of the spatial coherence of synchrotron beams using the Talbot effect,” Phys. Status Solidi A |

36. | W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. |

37. | G. Lovric, “Source size calculator,” http://www.psi.ch/sls/tomcat/ (2013). |

38. | S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard X-rays by 100 keV e-beam lithography and electroplating,” J. Synchrotron Radiat. |

39. | S. Yasin, D. Hasko, and H. Ahmed, “Comparison of MIBK/IPA and water/IPA as PMMA developers for electron beam nanolithography,” Microelectron. Eng. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(340.0340) X-ray optics : X-ray optics

(340.6720) X-ray optics : Synchrotron radiation

(340.7440) X-ray optics : X-ray imaging

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 3, 2013

Revised Manuscript: January 16, 2014

Manuscript Accepted: January 17, 2014

Published: January 30, 2014

**Citation**

Goran Lovric, Peter Oberta, Istvan Mohacsi, Marco Stampanoni, and Rajmund Mokso, "A robust tool for photon source geometry measurements using the fractional Talbot effect," Opt. Express **22**, 2745-2760 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2745

Sort: Year | Journal | Reset

### References

- I. A. Vartanyants and A. Singer, “Coherence properties of hard X-ray synchrotron sources and X-ray free-electron lasers,” New J. Phys.12, 035004 (2010). [CrossRef]
- S. Dierker, R. Pindak, R. Fleming, I. Robinson, and L. Berman, “X-ray photon correlation spectroscopy study of brownian motion of gold colloids in Glycerol,” Phys. Rev. Lett.75, 449–452 (1995). [CrossRef] [PubMed]
- I. Robinson, J. Libbert, I. Vartanyants, J. Pitney, D. Smilgies, D. Abernathy, and G. Grübel, “Coherent X-ray diffraction imaging of silicon oxide growth,” Phys. Rev. A60, 9965–9972 (1999).
- S. Roy, D. Parks, K. A. Seu, R. Su, J. J. Turner, W. Chao, E. H. Anderson, S. Cabrini, and S. D. Kevan, “Lensless X-ray imaging in reflection geometry,” Nat. Photonics5, 243–245 (2011). [CrossRef]
- A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast X-ray computed tomography for observing biological soft tissues,” Nat. Med.2, 473–475 (1996). [CrossRef] [PubMed]
- K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett.75, 2912–2914 (1999). [CrossRef]
- C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett.81, 3287 (2002). [CrossRef]
- M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010). [CrossRef] [PubMed]
- B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics5, 420–424 (2011). [CrossRef]
- P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle X-ray scattering distribution with grating interferometry,” Phys. Rev. Lett.108, 2–5 (2012). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999). [CrossRef]
- V. Kohn, I. Snigireva, and A. Snigirev, “Direct measurement of transverse coherence length of hard X-rays from interference fringes,” Phys. Rev. Lett.85, 2745–2748 (2000). [CrossRef] [PubMed]
- D. Paterson, B. Allman, P. McMahon, J. Lin, N. Moldovan, K. Nugent, I. McNulty, C. Chantler, C. Retsch, and T. Irving, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun.195, 79–84 (2001). [CrossRef]
- B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am.47, 895 (1957). [CrossRef]
- C. Chang, P. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun.182, 25–34 (2000). [CrossRef]
- W. Leitenberger, S. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun.191, 91–96 (2001). [CrossRef]
- T. Panzner, W. Leitenberger, J. Grenzer, Y. Bodenthin, T. Geue, U. Pietsch, and H. Möhwald, “Coherence experiments at the energy-dispersive reflectometry beamline at BESSY II,” J. Phys. D: Appl. Phys.36, A93–A97 (2003). [CrossRef]
- W. Leitenberger, H. Wendrock, L. Bischoff, and T. Weitkamp, “Pinhole interferometry with coherent hard X-rays,” J. Synchrotron Radiat.11, 190–197 (2004). [CrossRef] [PubMed]
- M. Yabashi, K. Tamasaku, and T. Ishikawa, “Characterization of the transverse coherence of hard synchrotron radiation by intensity interferometry,” Phys. Rev. Lett.87, 140801 (2001). [CrossRef] [PubMed]
- F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. Robinson, “Shearing interferometer for quantifying the coherence of hard X-ray beams,” Phys. Rev. Lett.94, 1–4 (2005). [CrossRef]
- K. S. Morgan, S. C. Irvine, Y. Suzuki, K. Uesugi, A. Takeuchi, D. M. Paganin, and K. K. Siu, “Measurement of hard X-ray coherence in the presence of a rotating random-phase-screen diffuser,” Opt. Commun.283, 216–225 (2010). [CrossRef]
- J. Lin, D. Paterson, A. Peele, P. McMahon, C. Chantler, K. Nugent, B. Lai, N. Moldovan, Z. Cai, D. Mancini, and I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett.90, 1–4 (2003). [CrossRef]
- I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett.107, 1–5 (2011). [CrossRef]
- P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard X-rays,” Opt. Lett.22, 1059–1061 (1997). [CrossRef] [PubMed]
- J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat.11, 476–482 (2004). [CrossRef] [PubMed]
- T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett.86, 054101 (2005). [CrossRef]
- A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat.17, 299–307 (2010). [CrossRef] [PubMed]
- A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat.17, 496–510 (2010). [CrossRef] [PubMed]
- A. Yaroshenko, M. Bech, G. Potdevin, A. Malecki, T. Biernath, J. Wolf, A. Tapfer, M. Schüttler, J. Meiser, D. Kunka, M. Amberger, J. Mohr, and F. Pfeiffer, “Non-binary phase gratings for X-ray imaging with a compact Talbot interferometer,” Opt. Express22, 547 (2014). [CrossRef]
- M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE6318, 63180M (2006). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Companies, 1996).
- P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation: principles, implementation and applications,” Phd thesis, Vrije Universiteit Brussel (1999).
- T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S (2006). [CrossRef]
- R. Kluender, F. Masiello, P. van Vaerenbergh, and J. Härtwig, “Measurement of the spatial coherence of synchrotron beams using the Talbot effect,” Phys. Status Solidi A206, 1842–1845 (2009). [CrossRef]
- W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am.71, 7 (1981). [CrossRef]
- G. Lovric, “Source size calculator,” http://www.psi.ch/sls/tomcat/ (2013).
- S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, and C. David, “High-efficiency Fresnel zone plates for hard X-rays by 100 keV e-beam lithography and electroplating,” J. Synchrotron Radiat.18, 442–446 (2011). [CrossRef] [PubMed]
- S. Yasin, D. Hasko, and H. Ahmed, “Comparison of MIBK/IPA and water/IPA as PMMA developers for electron beam nanolithography,” Microelectron. Eng.61–62, 745–753 (2002). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.