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
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 ), the X-ray community has undertaken active studies on phase contrast imaging (XPCI).
The first phase sensitive X-ray interferometers, based on diffractive optics [2–4], 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 . Some propagation based methods enabled the use of polychromatic divergent X-ray beams , 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 [7–9], which relied on the Talbot effect , enabled high spatial resolution imaging and easy wavefront reconstruction thanks to Fourier Transform algorithms (GBIs are gradient-sensitive devices) . These interferometers provided interesting results in many laboratories applications with synchrotron light sources such as differential phase X-ray imaging microscopy  or at-wavelength characterization of refractive X-ray lenses . 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  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, 16] and with standard low-brillance medical X-ray tubes [17, 18]. Following the evolution of classical lateral shearing interferometry in the 90ℙs[19, 20], X-ray GBIs using two-dimensional structures [21–24] and single-shot interferometers  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  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 . 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, 29]. 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  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 . 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, 33]. 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) .
In the hard X-ray domain, binary transmission gratings are challenging to make, because of the high aspect ratio they require. So we decided to create an interferometer that uses a single binary phase grating. When illuminated with a sufficiently large spectrum of radiations, this device is an approximated CSIG that produces an interference pattern that is achromatic and propagation invariant after its panchromatic distance. The contrast of the fringes in the pattern is limited because of the binary approximation, but remains sufficiently high to envision phase contrast imaging or wavefront metrology. This result was demonstrated in a previous paper . Yet, fringe contrast can also be affected by the limited spatial coherence of the source. In fact, the tolerance of the device to partial coherence is inversely proportional to the propagation distance of the fringe pattern . A higher limit to the propagation distance can also be established. Nevertheless, Xin Ge et al. showed  that a chessboard-type phase grating minimizes the limitation due to partial coherence and thus is the most suitable candidate for implementation on a GBI design for XPCI.
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 , the panchromatic distance is given by:
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.
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  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.
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.
We used a recent non-iterative boundary artifact free algorithm in order to reconstruct the wavefront from its derivatives developed by  based on the Ghiglia reconstruction method . In this method, we can obtain a phase reconstruction using the following expression:Figs. 4.
Figure 5(a) represents the final reconstructed OPD. A qualitative phase reconstruction was already obtained for this particular sample in . Whereas the interferograms are (512 × 512) pixels matrices, the reconstructed images are only (128 × 128) pixels maps. This reduction in size of the support, which is due to the Fourier based reconstruction algorithm, leads to a reconstructed effective pixel size of 5.36μm (instead of 1.34μm for the raw interferograms). From the sample SEM measurements, the OPD was estimated to be 0.21nm from one edge to the other of the bevel. Since the refractive index is wevelength dependant, the value is calculated at 0.07 nm (18 keV), which is the central wavelength of the spectrum resulting after passing through this phantom. It corresponds to the height of 0.6 Arbitrary Units (A.U) given by the OPD horizontal cross-section in Fig. 5(b). We can also notice in Fig. 5(b) that the area corresponding to the slope is perfectly flat (even when using a thiner scale to analyse this area) while areas corresponding to the wafer are incurved. This curvature represents the non-uniformity of the thickness of the wafer. This has no influence on the calibration because only the thickness of the areas of the slopes are used to perform this calibration. In order to estimate the signal to noise ratio of the reconstruction, a 20 × 20 pixels area was extracted from the bevel and corrected from its tilts. Fig. 5(c) shows the resulting image. The standard deviation σb in this image gives an upper limit of the noise level: 2 × 10−3A.U. which corresponds to 0.7pm according the previous calibration. The signal-to-noise ratio (SNR) could be estimated to 300. It is a conservative value, as it is assumed that the observed slope of the sample is perfect. This high SNR shows that the SPGI offers interferogram patterns whose contrast is sufficiently high to perform quantitative wavefront reconstructions. This result is not linked to any error in the reconstruction process since the bevels are seen perfectly flat. However, this method to evaluate the noise in the reconstructed image is not valid for a more complex sample, as it can contain phase dislocations. Another method to calculate the noise in the reconstructed image of unkown samples is given in the next section.
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.
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 , 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:
So we can define the phase derivatives closure map (PDCM) as:
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 . 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):
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  then Freischlad and Koliopoulos  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:
To obtain an estimate Φ̃LS of Φ̃, we calculate the two-dimensionnal quadratic cost function by the use of:
The Fourier Transform of the quantity ΦLS(x, y) is computed as the minimizer of this cost function  as:
We remind the expression Φ̃G(u, v) of the Fourier Transform of ΦG(x, y) phase reconstructed with the Ghiglia method:
One can show that:
However, Φ̃G(u, v), Φ̃LS(u, v) and H̃(u, v) are just the integrated linear combination of D̃x(u, v) and D̃y(u, v). D̃x(u, v) and D̃y(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, H̃(u, v) is a real quantity. So,
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.
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 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.
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.
Using a single phase grating and an easy experimental set up (white beam coming directly from the X-ray source), this fast single-shot technique is well suited for measuring OPDs created by a biological sample with weakly absorbing tissues. The high signal-to-noise ratio achievable makes the SPGI an interesting set-up for industrial implementation of XPCI. Our next step is to perform the same measurements using a low-brilliance and divergent X-ray source. Giving the fact that this technique can give quantitative noise evaluation makes it interesting for other studies on synchrotron light sources, such as X-ray mirror metrology or quantitative X-ray phase contrast tomography.
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.
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