2.1

Phase-stepping Technique

Stepping a phase grating using a step size of fractions of period of a grating we can create an intensity profile for each step of the phase grating, which can be written in mathematical terms as,

Here, I is the intensity pattern of a single pixel, a₀ and a₁ are the first and second coefficients of the Fourier transform or fitting function. x_g is the phase stepping period, p₂ the period of the interferogram at the detector position, and ϕ the phase of the sinusoidal curve representing the phase stepping. Measuring with sample and without sample in the beam we can obtain two phase stepping curves and deduce the Absorption (A), differential phase contrast (DPC) and dark-field (DF) images by comparing the two curves, as given below,

Here, superscripts, ‘s’ and ‘r’ represent images taken with and without the sample in the beam, respectively. Evaluating this for all of the pixels we can build three image contrast modes representing different physical characteristics of the sample. For a thorough reading please check^1,2 and the references there in.

2.2

Harmonic imaging technique

If we place a sample upstream of a phase grating, the interferogram at the fractional Talbot distance is proportional to the product of the transmittance of the object with the grating pattern^3,9, which is given by,

Here we have assigned w(x, y) and g(x,y) as the transmittance functions of the sample and the phase grating, respectively. Therefore, the Fourier transform(W_g(k)) of the measured interferogram is convolution of the spectrum of the sample with the spectrum of the self-image of the grating pattern. For a 1D grating with its grid lines aligned in y direction, parallel to lab vertical, we have

Here, W_g(k), F(k) and G(k) represent the Fourier transform of w_g(x, y), w(x, y) and g(x, y), respectively. G_m is the amplitude of the m^th order harmonic peak. The pattern around the harmonic peaks represents the spectrum of the entire sample, assuming that the harmonic spectra from adjacent peaks do not overlap, or the high-frequency components of the sample are negligible. We can then obtain the m^th order spatial harmonic image h_m(x, y) by the inverse Fourier transform of the m^th order spectrum peak within the interval k_x ∈ [π(2m-1)/p, π(2m+1)/p], which leads to

It can be inferred from Eq. 6 that the harmonic images are identical containing the information of the entire sample and are only weighted by the amplitude of the harmonic peak of the grating grid. The phase of the high-order (m>0) harmonic images is influenced by the x-ray refraction of the sample and one can extract differential phase images of the sample by inverse Fourier transforming the harmonic spectrum. The details can be found in Ref. 3 and references there in.

4.1

Spatial Resolution

In the present experiment a 1D phase grating with lines in vertical direction was used. Therefore, the peaks in the Fourier-transformed image can be found in the horizontal direction. The distance between the 0^th and 1^st harmonic peak is 429 pixels. As described in Section 2.2, the maximum distance from a peak in the horizontal direction without overlapping of the harmonic spectra is half the distance between two peaks. Hence we selected harmonic images with a width of 427 pixels symmetrically around the center of the peaks. Even though the height of the harmonic images can be as large as the original image itself, here we choose it similar to the width. As can be recognized, for the PS technique, the whole height and width of the original image (i.e., 2560 × 2160 pixels) is used for the image formation. Therefore the resolution of the SHI technique is about 6 × 5 times inferior to the PS technique in the width and height direction, respectively.

Figure 1 shows the absorption image of the mouse embryo heart sample processed with SHI (see Fig. 1(a)) and the PS (see Fig. 1(c)) techniques. The insets Fig. 1(b) and Fig. 1(d) compare the spatial resolution of the image across the red lines in Fig. 1(a) and Fig. 1(c). The line profile across the feature is represented by 4 and 24 pixels in Fig. 1(b) and Fig. 1(d), respectively, indicating a 6 times inferior spatial resolution in horizontal (width) direction.

Figure 1.

The absorption image of mouse embryo heart reconstructed from SHI (a) and PS (c) techniques. The colour bar represents arbitrary units. The inset (b) and (d) are the line profiles across red horizontal line in (a) and (b), respectively, comparing the spatial resolution.

4.2

Fast Imaging

As described in Section 2 the PS technique uses 5 images per projection to reconstruct a single projection of the tomogram. However, the SHI technique uses only one image per projection. Considering the tomography data acquisition this implies the SHI technique is also suitable for fly-scans where the sample rotation stage is continuously rotated in synchronization with camera acquisition, reducing the dead time for rotation stage movements after every projection. The exposures needed for SHI is almost same as normal absorption imaging since the phase grating absorption is low. Using a fly-scan mode and 0.5 s exposure time, the SHI technique is typically 8 times quicker than the PS. When comparing PS and SHI, the PS requires an exposure time of 0.1 s/image (total 0.5 s/projection), while 0.5 s/image for the SHI. This way we have same exposure per projection. However, it is important to note that the images from the SHI technique shown here were reconstructed from single image of 0.1 s exposure time (i.e., the 1^st images from a phase-stepping set of images for each projection). This implies we can further improve the signal-to-noise ratio for SHI.

Also we can use the SHI technique to do a quick scan of the sample if the sample structure is unknown or if we are looking at a particular region within a sample where we want to do high spatial resolution imaging using PS. This may be important for dose-sensitive samples.

4.3

Reconstruction of probe with SHI

One of the unique features of the SHI technique is that we can obtain the phase profile of the illuminating beam. What we mean by this is that when we take the reference images without the sample in the beam, we can process this in a similar way as we do to the sample images to obtain the amplitude profile of 0^th and 1^st harmonic image as well as the differential phase contrast (DPC) image of the 1^st harmonic image. For the PS technique, to obtain a phase image, you need both the reference as well as the sample image. One can argue that the reference image from the PS measurement can also be used to process like the SHI technique to obtain the phase images, but what we are saying is that it is not integral to the PS system. However, for the SHI technique it is one of the processes in the chain.

Getting the phase information of the illuminating beam can be very useful for reconstruction of the wavefront. It may also be useful to correct for the wavefront aberrations by having an adaptive optics element in a feed-back loop with the single-grating harmonic imaging interferometer [9]. The known wavefront information from the reconstruction can also be fed to the algorithms for phase reconstruction. Figure 2 illustrates the DPC of the reference image obtained using SHI technique. Fig. 2(a) shows the wrapped DPC image which has a size of 427 × 427 pixels and Fig. 2(b) the line profile across (a) at y=200^th pixel. Fig. 2(c) displays the unwrapped DPC and Fig. 2(d) the line profile across (c) at y=200^th pixel.

Figure 2.

The DPC of the reference image obtained using SHI technique. (a) Wrapped DPC (427 × 427 pixels), (b) line profile across (a) at y=200^th pixel. (c) Unwrapped DPC, (d) line profile across (c) at y= 200^th pixel.

4.4

Differential phase contrast and phase wrapping

The pco.edge5.5 camera has a chip size of 2560 x 2160 pixels. The SHI technique makes use of only 427 x 427 pixels to plot the same image. In the case of the SHI technique it has an effect of binning since the information spread over 6 x 5 pixels in the original image is merged into single pixel in SHI. Figures 3(a) and 3(c) and Figs. 3(b) and 3(d) show the DPC images obtained from the SHI and PS techniques, respectively, for two different orientations of the sample. Figures 3 (a) and 3(c) were produced by phase unwrapping both the reference and the sample DPC images. Figures 3(a) and 3(c) have similar contrast compared to Fig. 3(b) and Fig. 3(d), however the spatial resolution is better for the latter. In Fig. 3(b) and 3(d) the white arrow indicates the position of a horizontal band of very faint vertical lines, which is due to some instabilities in the setup. The beam from the MLM has a horizontal stripe structure, typical for this type of monochromators. A small drift of the beam during the measurement caused this horizontal band.

Figure 3.

DPC images of a mouse embryo heart reconstructed with the SHI (a,c) and the PS (b,d) techniques at 0° (a,b) and 12.42°(c,d) rotation of the sample. The effect of phase wrapping is visible in (c), indicated by the red arrow. A horizontal band of vertical lines in (b) and (d) is caused by beam drift. The colour bar is in radians.

In the case of SHI technique we are measuring the differential phase image (DPC) of the reference image and subtracting this image from the sample DPC to produce the final DPC image. The change in DPC image with beam drift is small; hence the SHI technique is more robust to beam instabilities. Also the inferior spatial resolution of the SHI technique helps to reduce this effect. However, for the PS technique we compare the intensity profiles of reference and sample images which are much more prone to this beam drift.

The black region (indicated by the red arrow) in the top right hand side of the DPC image of Fig. 3(c) is the incomplete phase-unwrapping of the image. Figure 3(d) shows images processed with the PS technique at the same orientation of the sample. The phase-wrapping issue seems less pronounced than for the SHI technique. We attribute this to the lower spatial resolution of the SHI technique; however more quantitative analysis is needed.