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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 17340–17351
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X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer

J. Rizzi, P. Mercère, M. Idir, P. Da Silva, G. Vincent, and Jérôme Primot  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 17340-17351 (2013)
http://dx.doi.org/10.1364/OE.21.017340


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Abstract

In this paper we present some quantitative measurements of X-ray phase contrast images and noise evaluation obtained with a recent grating based X-ray phase contrast interferometer. This device is built using a single phase grating and a large broadband X-ray source. It was calibrated using a reference sample and finally used to perform measurements of a biological fossil: a mosquito trapped in amber. As phase images, noise was evaluated from the measured interferograms.

© 2013 OSA

1. Introduction

Since Röntgen discovered X-rays in 1895, X-ray radiography has become a very common imaging technique. Nowadays, X-ray imaging systems based on absorption contrast are widely used in medical imaging, non-destructive control and many other industrial applications. However this method becomes inefficient when weakly absorbing tissues are of concern or when the probed object and its environment have similar properties of absorption. This aspect limits the impact of X-ray imaging on medical diagnosis. But observing the phase contrast instead of absorption can enhance the contrast of images. Since the end of the 1990s, relying on former works of opticians (in microscopy of biological tissues for example [1

1. G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).

]), the X-ray community has undertaken active studies on phase contrast imaging (XPCI).

The first phase sensitive X-ray interferometers, based on diffractive optics [2

2. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett 6(8), 155–156 (1965) [CrossRef] .

4

4. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003) [CrossRef] [PubMed] .

], enabled phase measurements of objects placed under monochromatic almost plane wave illumination. However the poor efficiency of X-ray optics required the use of high-brilliance light sources, restricting their implementation to synchrotron radiation facilities [5

5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995) [CrossRef] .

]. Some propagation based methods enabled the use of polychromatic divergent X-ray beams [6

6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996) [CrossRef] .

], but the resulting images were mainly governed by absorption contrast with an edge enhancement. Finally, important studies were devoted to grating-based interferometers (GBI) because of their strong potential for XPCI. The first GBIs created [7

7. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003) [CrossRef] .

9

9. 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(16), 6296–6304 (2005) [CrossRef] [PubMed] .

], which relied on the Talbot effect [10

10. H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 39, 401–407 (1836).

], enabled high spatial resolution imaging and easy wavefront reconstruction thanks to Fourier Transform algorithms (GBIs are gradient-sensitive devices) [11

11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982) [CrossRef] .

]. These interferometers provided interesting results in many laboratories applications with synchrotron light sources such as differential phase X-ray imaging microscopy [12

12. Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008) [CrossRef] .

] or at-wavelength characterization of refractive X-ray lenses [13

13. S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011) [CrossRef] .

]. However, they are challenging to be adapted to a product designed for medical diagnosis. In fact, Talbot GBIs require the use of multiple gratings leading to important constraints: the distance between the gratings is limited to discrete values, the alignment between the gratings has to be precise and is sensitive to vibrations and the use of the phase-stepping technique [14

14. M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.

] to perform imaging leads to significant total exposure times and hence dose on the sample. The first Talbot imaging systems were restricted to monochromatic beam and used one-dimensional periodic gratings. Those structures allowed gradient detection in only one direction. Thus two series of images in two orthogonal directions was often required to make the phase reconstruction robust. Moreover, one of these multiple gratings had to be a thick absorbing structure. In addition to fabrication challenges due to the high aspect ratio required in X-rays, this grating presents limitations when used on divergent X-ray beams. Nevertheless, the performances of Talbot GBIs increased significantly during the last years. In fact, some studies showed that they are compatible with white beam radiations from synchrotron sources [15

15. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009) [CrossRef] [PubMed] .

, 16

16. J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express 18(24), 24975–24982 (2010) [CrossRef] [PubMed] .

] and with standard low-brillance medical X-ray tubes [17

17. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006) [CrossRef] .

, 18

18. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007) [CrossRef] [PubMed] .

]. Following the evolution of classical lateral shearing interferometry in the 90ℙs[19

19. J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993) [CrossRef] [PubMed] .

, 20

20. J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995) [CrossRef] .

], X-ray GBIs using two-dimensional structures [21

21. A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006) [CrossRef] .

24

24. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010) [CrossRef] .

] and single-shot interferometers [25

25. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010) [CrossRef] [PubMed] .

] have been recently developed enabling simultaneous measurement of both orthogonal gradients with an important reduction of exposure time of the sample during XPCI. Moreover, the use of bent gratings [26

26. V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011) [CrossRef] .

] provided increase of the field of view of Talbot based inteferometers.

However, despite all this recent breakthroughs, X-ray interferometers using multiple gratings are not well suited for industrial purposes. A new approach using a single absorption grating as X-ray Hartmann mask has already been tested [27

27. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011) [CrossRef] [PubMed] .

]. This study gave interesting results but the range of applications with such a grating is limited.

In this context, we proposed to realize an X-ray GBI as simple as possible. It only requires a single phase grating and an X-ray source with a large spectral bandwidth. The scope of this paper is to present some quantitative phase contrast and noise measurements obtained with this device. We first used a reference sample in order to calibrate our interferometer, then we performed measurements of a biological fossil: a mosquito trapped in amber. A method to evaluate the noise in the reconstructed phase images directly from the interferograms is also presented.

2. X-ray phase contrast imaging with a single phase grating interferometer (SPGI)

2.1. Methodology and experimental setup

The GBI presented here belongs to the class of multi-wave lateral shearing interferometers, a well-known technique for measuring the phase gradients in at least two orthogonal directions. The incoming wavefront is divided into identical but tilted replicas by a specific two-dimensional (2D) grating. After propagation, their mutual interference leads to a fringe pattern whose spatial distribution is directly linked to the incoming wavefront. For some specific 2D-structures, these interferences can produce an achromatic and propagation invariant fringe pattern. Such gratings are called Continuously Self-Imaging Gratings (CSIG) and have been studied in [28

28. J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).

, 29

29. N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999) [CrossRef] .

]. However, the transmittance of these ideal objects is hard to achieve and only approximated CSIGs can be realized. The simplest way to approximate a CSIG is to use binary (amplitude or phase) gratings. Martin Piponnier [30

30. M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011) [CrossRef] [PubMed] .

] showed that a binary grating coupled to a spectrally large source will have the properties of a CSIG after a certain distance of propagation, Zpanchro, called the panchromatic distance [31

31. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000) [CrossRef] .

]. The larger the spectral bandwidth of the incoming radiation, the shorter the panchromatic distance. But if the spectral bandwidth of the source is too small, or is reduced too much due to the passing through an object, the binary grating will not generate any interferogram, except at its fractionnal Talbot distances [32

32. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990) [CrossRef] [PubMed] .

, 33

33. P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997) [CrossRef] [PubMed] .

]. In that case, the contrast of the fringe pattern will become distance dependant. Nevertheless, if the changings in the spectral content of the beam due to the passing through an object are small, the interferograms will remain achromatic and propagation invariant. The only negative impact will be an increasing of the panchromatic distance.

In the visible light and infrared domains, an approximated CSIG was created using a binary phase grating coupled to a binary transmission grating: the Modified Hartmann Mask (MHM) [34

34. J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000) [CrossRef] .

].

The experimental setup is presented in Fig. 1. The GBI is composed of a single binary phase grating and a dedicated indirect detection system. The phase grating is a 2D periodic chess-board made of gold structures (3μm cubes) deposited on a 300μm thick silicon wafer. The lattice generates a π-phase shift for a wavelength of 0.088nm (14.1keV). Experiments were performed on the Metrology and Tests Beamline at the French national synchrotron radiation facility SOLEIL. This beamline uses the radiation emitted by a bending magnet. The bending magnetic field is 1.72 Tesla and the electron beam energy is 2.75GeV. The transverse size of the source is 140(H) × 60(V) μm2 FWHM. For the experiment, the bending magnet radiation was used without any spectral or spatial conditioning (no monochromator or focusing optics were involved in the beamline), and propagated in free space down to the beamline end station. The GBI was implemented, at atmospheric pressure, at a distance S = 32m from the source. As predicted, when illuminated under such conditions, the binary phase grating possesses a panchromatic regime. The beam spectrum was determined taking into account the source characteristics and the attenuation of the X-rays by the different materials in the beam path: the vacuum exit window of the beamline (150μm thick CVD diamond), the 300μm thick silicon substrate of the grating and the 1.5m air path between the vacuum window and the detector. The X-ray energy spectrum impinging on the detector is centered on 18keV (0.07nm) with a spectral bandwidth of 20keV (corresponding to approximately 0.09nm). Considering that this grating is an approximation of a 4 orders CSIG, according to reference [38

38. N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001) [CrossRef] .

], the panchromatic distance is given by:
Zpanchro=2a02η2Δλ
(1)
where a0 is the pitch of the grating, a0=2p2 (because the lattice of the chessboard is inclined at 45° with respect to the axes of the camera) with p the size of the gold cubes and η2 = 4 for a 4 orders CSIG. This leads to a panchromatic distance of about 40cm. Because of the limited transverse coherence of the source, the phase grating to detector distance D was adjusted to this minimum propagation distance given by Zpanchro. Observation of the fringe pattern was performed by use of a 20μm thick YAG:Ce crystal relayed through a (×5.5) magnification optical system onto a highly sensitive cooled 14 bits visible CCD camera (PCO 2000s, 2048×2048 pixels, 7.4μm/pixel). The resulting effective pixel size was 1.34μm in the detection plane. The recorded and presented interferograms follow from the integration of 155 CCD images (5ms exposure time per CCD image), leading to an overall exposure time of 775ms per interfogram.

Fig. 1 Experimental set-up as implemented on the Metrology and Tests Beamline at SOLEIL (S = 32 m, D = 40 cm and L = 20 cm).

2.2. Experimental results

2.2.1. Calibration of the interferometer

In order to calibrate the SPGI, we performed phase-contrast imaging of a reference sample. This one was placed, between the phase grating and the detector, at a distance L = 20cm from the detection plane. As presented in Figs. 2, our reference sample is a silicon wafer chemically etched in the two orthogonal directions X and Y providing with two slopes perfectly controlled in the two directions where the interferometer is gradient sensitive. Fig. 2(a) gives a sketch of the sample and Fig. 2(b) is a Scanning Electron Microscope (SEM) image of the area of interest. The width of the chemically etched bevel has been measured with the SEM to 151μm, which considering the orientation of the crystal (111) reticular planes leads to an etching depth of 211μm. The equivalent Optical Path Difference (OPD) expected from XPCI of the sample can be estimated to 0.21nm for the average working photon energy of 18keV. This is a benchmark value: all the measurements performed in the same configuration of use of the interferometer will be scaled according to this reference. In this case, absolute precision of the SPGI relies on the precision of SEM measurements, the precision of the silicon index of refraction and the accuracy of the slope measurements achieved by the interferometer. The advantage of this calibration technique is that we do not need to know precisely the value of specific parameters (such as the distance between the grating and the detector plane, the magnification of the lens or the tilt of the grating respect to the optical axis). Such parameters are difficult to determine in absolute at the high level of precision required for X-ray differential phase contrast imaging.

Fig. 2 Bevels along the (111) reticular planes of a chemically etched silicon wafer. Fig. 2(a): Sketch of the sample. Fig. 2(b): SEM measurement of the sample.

The interferograms obtained successively with and without sample in the beam path can be shown in Figs. 3. The area of interest is limited to a square area of 512 × 512 pixels, corresponding to 1/16 of the detector usable field of view. The contrast of the fringe pattern [39

39. A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).

] is about 0.32, which is high enough to enable relevant phase contrast imaging. We can observe that the intensity is not homogeneous on the reference interferogram. This is due to the optical lens we used to collect the light from the scintillator. In fact, this optical lens has been used several times in a lot of other experiments that took place at the beam line, and has been a little bit affected by those multiple exposures to the X-ray beam.

Fig. 3 Interferograms. Fig. 3(a): raw interferogram with sample. Fig. 3(b): reference interferogram without sample. Fig. 3(c): enlarged part of the fringe pattern (without sample)

The sample phase derivatives along the X and Y directions, reconstructed from the interferograms of Fig. 3(a), are presented in Figs. 4. Since the SPGI is a gradient sensitive device, interferograms contain the local slope information which can be easily retrieved by Fourier analysis. These two local slopes are equal. The signal-to-noise ratio is better in Fig. 4(b) because the synchrotron source has an higher degree of spatial coherence along the horizontal direction.

Fig. 4 Sample Derivatives. Fig. 4(a): Dx, derivative along the X direction. Fig. 4(b): Dy, derivative along the Y direction.

We used a recent non-iterative boundary artifact free algorithm in order to reconstruct the wavefront from its derivatives developed by [40

40. P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012) [CrossRef] [PubMed] .

] based on the Ghiglia reconstruction method [41

41. D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.

]. In this method, we can obtain a phase reconstruction using the following expression:
Φ(x,y)=FT1[D˜x(u,v)+i*D˜y(u,v)u+i*v](x,y)
(2)
where FT−1[g] is the inverse Fourier Transform of g, (u, v) represents the Fourier Transform of X(x, y), u and v are the Fourier coordinates dual to x and y, Dx(x, y) and Dy(x, y) are the derivatives of the interferogram along the directions x and y as presented in Figs. 4.

Fig. 5 Final reconstruction. Fig. 5(a): optical path difference (OPD) of the reference sample. Size of the reconstructed images size is (128 × 128) pixels, instead of (512 × 512) pixels for the interferograms. The resulting effective pixel size in the reconstructed images is 5.36μm. Fig. 5(b): OPD horizontal cross-section [row 80 out of the (128, 128) OPD map given in Fig. 5(a)]. The bevel height is 0.6 A.U., which corresponds to an OPD of 0.21nm as estimated from the SEM measurements. The thin green lines represent a Cartesian mapping useful to read the axes more easily. The two other thick lines are here to mark out the slope. Fig. 5(c): 20 × 20 pixels area out of the bevel, after tilt subtraction. The standard deviation in this area is equal to 2 × 10−3A.U., which corresponds to 0.7pm. It leads to a SNR of 300. We can also notice in Fig. 5(b) that the areas corresponding to the bevels are flat whereas those corresponding to the rest of the wafer are slightly non-uniform.

In summary, the evaluation of the deviation introduced by the Si-prism is measured from the difference between the interferograms obtained with and without the sample in the beam path. The noise is then estimated from the plane part of the wafer (i.e. the chemically etched bevel) of known orientation, angle and refraction index. Indeed the etched bevel is given by the Silicon (111) reticular planes, ensuring perfect flatness in this part of the wafer. It should be noticed that the outer un-etched parts of the wafer, which present important fluctuations from a plane structure, are neither used as a reference not to quantify the noise in the reconstructed OPD.

2.2.2. Application: X-ray phase imaging of a biological sample

Using our calibrated interferometer, we can measure the phase shift introduced by an unknown and more complex sample. We chose to perform X-ray phase contrast imaging of a mosquito trapped in amber. The sample was placed at the same distance L = 20cm in front of the detection plane. Fig. 6(a) shows the reconstructed OPD of the mosquito, while Fig. 6(b) a visible light image of this sample. Figures 6 show that XPCI enables observation of some details of the mosquito anatomy with a good spatial resolution. Thanks to the calibration step, the OPD introduced by this mosquito can be estimated to 34pm. For this measurement, we used a reference interferogram of an area of the sample containing only the amber. Of course, this value of the OPD in the mosquito is calculated for a wavelength of 0.07 nm (18keV) because the calibration with the phantom was performed at this wavelength. It should be noticed that the contrast of the fringes in the interferogram of the mosquito was the same than the contrast of the fringes in the interferogram of the phantom. It means that the changings in the spectral content of the beam due to the introduction of the amber are not important enough to cancel the panchromatic effect generated by the phase grating and the X-ray white beam. The value of the panchromatic distance remains unaffected by the introduction of the amber in the beam too. As it is explained in section 2.1, it means that the interferogram of the mosquito is still achromatic and propagation invariant. As a result, we were able to measure the optical path difference at 18 keV introduced by the mosquito without the need to know the thickness of the amber or its index at this wavelength. This demonstrates that our SPGI can measure OPDs in the picometer range, which is one order of magnitude lower than the average working wavelength.

Fig. 6 Application of the SPGI on a biological fossil. Fig. 6(a): reconstructed OPD of a mosquito trapped in amber. Fig. 6(b): mosquito viewed under a microscope. The red square marks out in Fig. 6(b) the area of the mosquito observed in Fig. 6(a). We can clearly observe some details of the anatomy of the mosquito in the OPD image.

3. Noise measurements in the phase reconstruction

It is impossible to evaluate directly the noise in the reconstructed OPD image of a sample like the mosquito, like we did with the reference sample. In fact, for a more complex sample the phase can present multiple dislocations randomly dispersed in the reconstruction and it would be meaningless to isolate very small areas and compute the standard deviation in these areas. To overcome this limitation, we present in this chapter a method that enables determination of the noise in the reconstructed OPD image directly from the interferogram analysis, thanks to the phase gradients. A quantity called the phase derivatives closure map (PDCM), defined for the first time by R. C. Jennison in 1958 [43

43. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).

], will be used to perform a rigorous noise analysis. The noise in the PDCM can be calculated easily, and there is a simple relation between this noise and the noise in the reconstructed image. We will use the phantom to establish experimentally this relation. Then we will be able to deduce the noise in the reconstructed OPD image of any sample from the noise in its PDCM, assuming that XPCI is performed in the same configuration of the SPGI for the phantom and the studied sample.

3.1. A priori noise evaluation

Let’s consider a wavefront Φ(x, y). As it is assumed that Φ is a surface, we can write:
(Φ(x,y)/y)x=(Φ(x,y)/x)y
(3)

So we can define the phase derivatives closure map (PDCM) as:
C(x,y)=DyxDxy
(4)

With
Dx=Φ(x,y)xandDy=Φ(x,y)y
(5)

So C(x, y) = 0, except in the three following cases: 1- the image of the wavefront Φ(x, y) is noisy. 2- the pitch sampling of the image is too large. 3- the phase signal is dislocated [44

44. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977) [CrossRef] .

]. So if we are sure that a sample is not dislocated, and if we use a correct sampling in the phase reconstruction, we can compute the noise in PDCM and deduce from this measurement the noise in the OPD image. However, the interferogram of a sample gives access to the Fourier Transform of C(x, y):
C˜(u,v)=2iπ[u*D˜y(u,v)v*D˜x(u,v)]
(6)

But we just have to perform the inverse Fourier Transform in order to obtain the PDCM.

3.2. Noise evaluation in the reconstructed image

The aim of this section is to find experimentally the relation between the standard deviation of the noise in the PDCM and the standard deviation of the noise in the reconstructed OPD image.

Southwell [45

45. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980) [CrossRef] .

] then Freischlad and Koliopoulos [46

46. K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 3(11), 1852–1861 (1986) [CrossRef] .

] showed that in a Least Square reconstruction, the standard deviation of the noise in the PDCM σPDCM is proportional to the standard deviation of the noise in the reconstructed OPD image σOPD, regardless of the sampling used to perform the reconstruction.

We will now demonstrate that Ghiglia’s reconstruction is equivalent to least square reconstruction. First, let’s consider the expression of the reconstructed phase ΦLS in the case of the Least Square reconstruction. For each direction x and y, the Fourier Transform of the derivatives is given by:
D˜x(u,v)=i2πuΦ˜(u,v)andD˜y(u,v)=i2πvΦ˜(u,v)
(7)

To obtain an estimate Φ̃LS of Φ̃, we calculate the two-dimensionnal quadratic cost function by the use of:
E(Φ˜)=Dx(u,v)i2πuΦ˜2Dy(u,v)i2πvΦ˜
(8)

The Fourier Transform of the quantity ΦLS(x, y) is computed as the minimizer of this cost function [47

47. S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005) [CrossRef] [PubMed] .

] as:
Φ˜LS(u,v)=i2π(u*D˜x(u,v)+v*D˜y(u,v))u2+v2
(9)

We remind the expression Φ̃G(u, v) of the Fourier Transform of ΦG(x, y) phase reconstructed with the Ghiglia method:
Φ˜G(u,v)=D˜x(u,v)+i*D˜y(u,v)u+i*v
(10)

One can show that:
Φ˜G(u,v)=2π(H˜(u,v)+iΦ˜LS(u,v))
(11)

With
H˜(u,v)=[C˜(u,v)4π2(u2+v2)]
(12)

However, Φ̃G(u, v), Φ̃LS(u, v) and (u, v) are just the integrated linear combination of x(u, v) and y(u, v). x(u, v) and y(u, v) are the derivatives of the initial image, and are obviously a real quantity. The derivative or the integer of a real quantity remains a real quantity. Thus, (u, v) is a real quantity. So,
Im[Φ˜G(u,v)]=2πΦ˜LS(u,v)
(13)

So, in our particular configuration, Ghiglia’s reconstruction is equivalent to a Least Square reconstruction and the result obtained by Freischlad is valid: σOPD and σPDCM are proportionnal. Thus, it is possible to evaluate the noise in a reconstructed OPD image of a sample from the noise in its PDCM. Figures 7 are an image of the PDCMs obtained for both the calibration phantom and the mosquito. It should be noticed, that the PDCM is a linear combination of the gradients and is computed only using the interferogram.

Fig. 7 Phase Derivatives Closure Map (PDCM) of the calibration phantom [Fig. 7(a)] and the mosquito [Fig. 7(b)].

When analysing Fig. 7(a), which represents the PDCM of the phantom, we notice that the level of noise is homogeneous on almost all the map except on the edges of the chemical etching. This is because edges are undersampled points in the image. This is confirmed when analysing the PDCM of the mosquito in Fig. 7(b), where some edges of the mosquito, which represents strong gradients, appear more noisy than large flat areas. Moreover, the noise generated by the edges of the mosquito is more important than the noise generated by the edges of the reference sample. Due to normalisation, the noise in the background of Fig. 7(b) is less visible than the one of Fig. 7(a).

In Fig. 7(a), the standard deviation was measured in four square areas of 20*20 pixels (named area 1, 2, 3 and 4). Their values are named σPDCMi=1,2,3,4. The standard deviation of these areas was computed in the OPD images too (named σOPDi), as we are sure they contain no phase dislocations for this sample. The ratio Rnoise=σPDCMiσOPDi was computed for each value of i. As predicted, this ratio is a constant value which is estimated here to 4.0 × 103. We can use this reference value of Rnoise, obtained with the phantom, to deduce the standard deviation of the noise in a reconstructed OPD image from the standard deviation of the noise the PDCM of another sample. We assume that the configuration of the SPGI is the same during the imaging of both the phantom and the sample. In Fig. 7(b), which represent the PDCM of the mosquito, the measured standard deviation of the noise in area A is equal to 9.4 × 10−2A.U. Thus noise level in the corresponding area of the reconstructed OPD can be estimated to 2.35×10−5A.U.. It leads to a noise with standard deviation of 0.08pm. Finally, with an OPD mean value of 28pm in this area, the SNR reaches 340.

4. Conclusion

An X-ray phase contrast imaging method was demonstrated, using a simple set-up consisting of a phase chessboard grating and an X-ray source with a very large spectral distribution. After calibration of the interferometer with a reference sample, both optical path difference and noise measurements have been performed on a mosquito trapped in amber. A recently developed boundary-artifacts-free algorithm was used and provided high spatial resolution on the reconstructed images. A method to deduce the noise in the reconstructed OPD map directly from the interferograms was also presented. The constrast of the interferograms was estimated to 0.32. On the reference sample, we measured a noise of 0.7pm for an OPD of about 210pm, leading to a signal-to-noise ratio of 300. In the case of the mosquito, noise was evaluated to 0.08pm for a measured OPD of 28pm.

Acknowledgments

The research presented here is supported by Triangle de la Physique and by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract No. DE-AC-02-98CH10886.

References and links

1.

G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium 16, S88–S88 (1955).

2.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett 6(8), 155–156 (1965) [CrossRef] .

3.

R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000) [CrossRef] .

4.

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003) [CrossRef] [PubMed] .

5.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum 66(12), 5486–5492 (1995) [CrossRef] .

6.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996) [CrossRef] .

7.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys 42(7B), 866–868 (2003) [CrossRef] .

8.

A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys 44(9A), 6355–6367 (2005) [CrossRef] .

9.

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(16), 6296–6304 (2005) [CrossRef] [PubMed] .

10.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 39, 401–407 (1836).

11.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982) [CrossRef] .

12.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express 1, 117002 (2008) [CrossRef] .

13.

S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett 99, 221104 (2011) [CrossRef] .

14.

M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.

15.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009) [CrossRef] [PubMed] .

16.

J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express 18(24), 24975–24982 (2010) [CrossRef] [PubMed] .

17.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys. 2, 258–261 (2006) [CrossRef] .

18.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett 98, 108105 (2007) [CrossRef] [PubMed] .

19.

J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt 32(31), 6242–6249 (1993) [CrossRef] [PubMed] .

20.

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12(12), 2679–2685 (1995) [CrossRef] .

21.

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys 45(1A), 314–316 (2006) [CrossRef] .

22.

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007) [CrossRef] [PubMed] .

23.

H. Itoh, K. Nagai, G. sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase contrast imaging using Fourier transform phase retrieval,” Opt. Express 19(4), 3339–3346 (2011) [CrossRef] [PubMed] .

24.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett 105, 248102 (2010) [CrossRef] .

25.

H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett 35(12), 1932–1934 (2010) [CrossRef] [PubMed] .

26.

V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A 648, 302–305 (2011) [CrossRef] .

27.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011) [CrossRef] [PubMed] .

28.

J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A 2, 110 (1985).

29.

N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999) [CrossRef] .

30.

M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19(23), 23054–23066 (2011) [CrossRef] [PubMed] .

31.

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com 180, 199–203 (2000) [CrossRef] .

32.

J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett 15(5), 288–290 (1990) [CrossRef] [PubMed] .

33.

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett 22(14), 1059–1061 (1997) [CrossRef] [PubMed] .

34.

J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt 39(31), 5715–5720 (2000) [CrossRef] .

35.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett 36(8), 1398–1400 (2011) [CrossRef] [PubMed] .

36.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE 631863180S (2006) [CrossRef] .

37.

X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem 401, 865–870 (2011) [CrossRef] [PubMed] .

38.

N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett 26(7), 411–413 (2001) [CrossRef] .

39.

A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).

40.

P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt 51(23), 5698–5704 (2012) [CrossRef] [PubMed] .

41.

D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.

42.

J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser. 425,192002 (2013) [CrossRef] .

43.

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc. 118(3), 276–284(1958).

44.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67(3), 370–375 (1977) [CrossRef] .

45.

W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 70(8), 998–1006 (1980) [CrossRef] .

46.

K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A 3(11), 1852–1861 (1986) [CrossRef] .

47.

S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett 30(3), 245–247 (2005) [CrossRef] [PubMed] .

OCIS Codes
(340.6720) X-ray optics : Synchrotron radiation
(340.7440) X-ray optics : X-ray imaging
(340.7450) X-ray optics : X-ray interferometry

ToC Category:
X-ray Optics

History
Original Manuscript: April 24, 2013
Revised Manuscript: May 27, 2013
Manuscript Accepted: May 31, 2013
Published: July 12, 2013

Citation
J. Rizzi, P. Mercère, M. Idir, P. Da Silva, G. Vincent, and Jérôme Primot, "X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer," Opt. Express 21, 17340-17351 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17340


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References

  1. G. Nomarski, “Nouveau dispositif pour l’observation en contraste de phase differentiel,” J.Phys.Radium16, S88–S88 (1955).
  2. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett6(8), 155–156 (1965). [CrossRef]
  3. R. Fitzgerald, “Phase sensitive X-ray imaging,” Phys. Today53, 23–26 (2000). [CrossRef]
  4. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express11(19), 2303–2314 (2003). [CrossRef] [PubMed]
  5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum66(12), 5486–5492 (1995). [CrossRef]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996). [CrossRef]
  7. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jap. Jour. Appl. Phys42(7B), 866–868 (2003). [CrossRef]
  8. A. Momose, “Recent advances in X-ray phase imaging,” Jap. Jour. Appl. Phys44(9A), 6355–6367 (2005). [CrossRef]
  9. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express13(16), 6296–6304 (2005). [CrossRef] [PubMed]
  10. H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series39, 401–407 (1836).
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72(1), 156–160 (1982). [CrossRef]
  12. Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, “Differential phase X-ray imaging microscopy with Talbot interferometer,” Appl. Phys. Express1, 117002 (2008). [CrossRef]
  13. S. Ruthishauer, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength charcterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett99, 221104 (2011). [CrossRef]
  14. M. Creath, “Phase-Measurement Interferometry Techniques” (Elsevier Science, 1988) pp. 349–393.
  15. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express17(15), 12540–12545 (2009). [CrossRef] [PubMed]
  16. J. M. Kim, I. H. Cho, S. Y. Lee, H. C. Kang, R. Conley, C. Liu, A. T. Macrander, and D. Y. Noh, “Observation of the Talbot effect using broadband hard x-ray beam,” Opt. Express18(24), 24975–24982 (2010). [CrossRef] [PubMed]
  17. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brillance X-ray sources,” Nature Phys.2, 258–261 (2006). [CrossRef]
  18. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brillance sources,” Phys. Rev. Lett98, 108105 (2007). [CrossRef] [PubMed]
  19. J. Primot, “Three-wave lateral shearing interferometry,” Appl. Opt32(31), 6242–6249 (1993). [CrossRef] [PubMed]
  20. J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A12(12), 2679–2685 (1995). [CrossRef]
  21. A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jap. Jour. Appl. Phys45(1A), 314–316 (2006). [CrossRef]
  22. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express15(3), 1175–1181 (2007). [CrossRef] [PubMed]
  23. H. Itoh, K. Nagai, G. sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase contrast imaging using Fourier transform phase retrieval,” Opt. Express19(4), 3339–3346 (2011). [CrossRef] [PubMed]
  24. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional X-ray grating interferometer,” Phys. Rev. Lett105, 248102 (2010). [CrossRef]
  25. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission grating,” Opt. Lett35(12), 1932–1934 (2010). [CrossRef] [PubMed]
  26. V. Revol, C. Kottler, R. Kaufmann, I. Jerjen, T. lüthi, F. Cardot, P. Niedermann, U. Straumann, U. Sennhauser, and C. Urban, “X-ray interferometer with bent grating: toward larger fields of view,” Nucl. Instr. Meth. Phys. Res. A648, 302–305 (2011). [CrossRef]
  27. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express19(20), 19781–19789 (2011). [CrossRef] [PubMed]
  28. J. Durnin, “Continuously self-imaging fields of finite aperture,” J. Opt. Soc. Am. A2, 110 (1985).
  29. N. Guérineau and J. Primot, “Non diffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A16(2), 293–298 (1999). [CrossRef]
  30. M. Piponnier, G. Druart, N. Guérineau, J. L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express19(23), 23054–23066 (2011). [CrossRef] [PubMed]
  31. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Com180, 199–203 (2000). [CrossRef]
  32. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett15(5), 288–290 (1990). [CrossRef] [PubMed]
  33. P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “fractionnal Talbot imaging of phase gratings with hard x rays,” Opt. Lett22(14), 1059–1061 (1997). [CrossRef] [PubMed]
  34. J. Primot and N. Guérineau, “Extended Hartmann test based on the pseudo-giuding property of a Hartmann mask completed by a phase chessboard,” Appl. Opt39(31), 5715–5720 (2000). [CrossRef]
  35. J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. Da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett36(8), 1398–1400 (2011). [CrossRef] [PubMed]
  36. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brillance sources,” Proc. of SPIE631863180S (2006). [CrossRef]
  37. X. Ge, Z. Wang, K. Gao, K. Zhang, Y. Hong, D. Wang, Z. Zhu, P. Zhu, and Z. Wu, “Inverstigation of the partially coherent effects in a 2D Talbot interferometer,” Anal. Bioanal. Chem401, 865–870 (2011). [CrossRef] [PubMed]
  38. N. Guérineau, B. Harchaoui, K. Heggarty, and J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. lett26(7), 411–413 (2001). [CrossRef]
  39. A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1927).
  40. P. Bon, S. Monneret, and B. Wattelier, “Noninterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt51(23), 5698–5704 (2012). [CrossRef] [PubMed]
  41. D. Ghilia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998) pp. 34.
  42. J. Rizzi, P. Mercre, M. Idir, N. Gurineau, E. Sakat, R. Hadar, G. Vincent, P. Da Silva, and J. Primot, “X-ray phase contrast imaging using a broadband X-ray beam and a single phase grating used in its achromatic and propagation-invariant regime,” J. Phys.: Conf. Ser.425,192002 (2013). [CrossRef]
  43. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier Transfoms of spatial brightness distributions of small angular extent,” Mon. Not. Roy. Astron. Soc.118(3), 276–284(1958).
  44. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A67(3), 370–375 (1977). [CrossRef]
  45. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A70(8), 998–1006 (1980). [CrossRef]
  46. K. R. Freischlad and C. L. Koliopoulos, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986). [CrossRef]
  47. S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett30(3), 245–247 (2005). [CrossRef] [PubMed]

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