PHACT - Phase-Contrast Imaging

Grating-based Interferometer

Phase-contrast imaging is an approach to gain information about a sample by measuring the local phase of X-rays passing through it. Nowadays experimental techniques have become applicable not only for optical wavelengths but also for wavelengths in the X-ray range [2]. One of them is the grating based interferometry [3], [1], which resolves both the phase and the absorption of the investigated sample.

Setup

Talbot-grating interferometer [1] (Figure 1) consists of three gratings the source grating G0, phase grating G1 and analyzer grating G3. Each grating has a specific period p and distances l and d. As the name implies this approach is based on the Talbot effect.

Figure 1: Basic setup of a Talbot-grating interferometer

Talbot effect

Talbot effect was first reported by Talbot in 1836 [4]. It states that a coherent illumination of a periodic structure leads to repetitive intensity patterns in certain distances downstream the wave propagation, frequently called self images. For a phase-grating (with a period p₁) impressing a phase shift of π onto an impinging plane wave-front leads to an intensity distribution with a periodicity of p₁ / 2 in the following distances

Differential Phase-Contrast Imaging

Objects placed right in front of the phase-grating G1 distort the interference pattern in the detection plane, which is chosen to be in one of the Talbot- distances. The distortion of a phase-object is given by a local angular deviation  of the pattern. For X-rays, with λ ≤ 10^-10m, the angle is typically in the order of a few microradians. The principle of differential phase contrast imaging is to detect these deviations by placing an absorption grating G2 with the periodicity of the interference fringes p₂ right in front of the detection plane [1]. In case of a X-ray tube with a large focal spot p₀ >> p₂ · l / d an absorption grating G0 has to be placed directly in front of the latter to achieve a sufficient degree of spatial coherence [1]. To gain the differential phase of an object one now acquires a sequence of images with an appropriate pixelated detector behind the analyzer grating [5]. After each image the grating is moved a fraction of its own periodicity until at least one period of p₂ is stepped. Doing this for each single pixel an intensity-modulation with respect to the grating position can be seen. In an ideal setup this modulation is described by a triangle-function as both the interfering intensity pattern and the absorption grating G2 are equal rectangle functions. If an object in the way of the wave-field shifts the intensity pattern laterally the detected modulation in each pixel will as well be shifted.

Figure 2: The triangle function as the ideal intensity modulation.

Figure 2 depicts the ideal case of the described procedure for one pixel. The shifted phase between the images without and with an object in the wave-field is the desired measure of the angular deviation. Knowing this local deviation it is straight forward to calculate the gradient of the total phase shift Φ throughout the object by [6]

Herein x denotes the axis perpendicular to the grating bars. The phase shift can be subsequently obtained after one dimensional integration.

Examples

Following images were taken in our Lab at the ECAP Institute. Figure 3 shows a small fish and Figure 4 a rasin. Both data were acquired with a 80 keV tungsten spectrum and the Medipix 2 detector. The images show from left to right the differential phase-contrast image, absorption image and a so called dark-field image.

Figure 3: From left to right: differential phase-contrast image, absorption image and dark-field image of a small fish

Figure 4: From left to right: differential phase-contrast image, absorption image and dark-field image of a raisin

References

1. F. Pfeiffer, T.Weitkamp, O. Bunk, C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources," Nature Phys. 2, 258-261, 2006
2. S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany, A.W. Stevenson, "Phase-contrast imaging using polychromatic hard x-rays," Nature 384, 335-337, 1996
3. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys., 45 (6A), 5254-5262, 2006
4. Talbot, H.F., Philos. Mag., 9, 403, 1836
5. Weitkamp, T., et al., "X-ray phase imaging with a grating interferometer," Optics Express Vol. 13, No. 16, 6296-6304, 2005
6. Born, M., Wolf E., "Principles of Optics," Cambridge University Press, 1997