Hard-x-ray phase-imaging microscopy using the self-imaging phenomenon of a transmission grating

We report on a hard-x-ray imaging microscope consisting of a lens, a sample, and a transmission grating. After the theoretical framework of self-imaging phenomenon by the grating in the system is presented, equations for the electric field on the image plane are derived for ideal and real lenses and an equation for the intensity on the image plane for partially coherent illumination is derived. The equations are simple and similar to those applying to a projection microscope consisting of a transmission grating except that there is no defocusing effect, regardless of whether the grating is in front of or behind the lens. This means that x-ray phase-imaging microscopy can be done without the defocusing effect. It is also shown that, by resolving the self-image on the image plane, high-sensitive x-ray phase-imaging microscopy can be attained without degradation in the spatial resolution due to diffraction by the grating. Experimental results obtained using partially coherent illumination from a synchrotron x-ray source confirm that hard-x-ray phase-imaging microscopy can be quantitatively performed with high sensitivity and without the spatial resolution degradation.

Figure 1

X-ray imaging microscopy system consisting of a sample, a lens, and a grating. The sample is located on the object plane of the microscopy system. In (a) the grating is located between the object plane and the lens, while in (b) it is between the lens and the image plane.

Figure 2

X-ray projection microscopy system consisting of a sample, a lens, and a grating.

Figure 3

Setup of phase-difference microscopy with a single-phase grating.

Figure 4

Self-images without a sample. (a), (b), and (d) were obtained at $p=1/2$ by using $\pi /2$ phase gratings with pitches of $4.3$, $6.0$, and $8.0$ $\mathrm{\mu }\mathrm{m}$, while (c) was obtained at $p=3/2$ by using a $\pi /2$ phase grating a pitch of $6.0$ $\mathrm{\mu }\mathrm{m}$.

Figure 5

Self-images obtained with polystyrene (PS) spheres on a glass plate as the sample (left); corresponding phase-difference images obtained by the fringe scanning technique (right; gray scale, $-0.5\pi ~0.5\pi$). Images in (a)–(d) correspond to those in (a)–(d) of Fig. 4.

Figure 6

(a) Enlarged phase-difference image of the PS spheres in the square on the right-hand side of figure of Fig. 5a (gray scale, $-0.3\pi ~0.3\pi$). (b) Absorption image of the same area as Fig. 6a obtained by simply removing the grating. (c) Section profile along the dashed line in Fig. 6a.

Figure 7

Edge profiles of phase-difference (open circles) and absorption (solid circles) images of a 1-$\mathrm{\mu }\mathrm{m}$-thick tantalum pattern. Open circles are the edge profile of the phase-difference image and solid circles are that of the absorption image.

Figure 8

(a) The differential phase image of a 1-$\mu$m-thick tantalum Siemens star chart and (b) its phase image retrieved from (a).