Shearing x-ray interferometer with an x-ray prism and its improvement
Yoshiki Kohmura, Hidekazu Takano, Kentaro Uesugi, Yoshio Suzuki, Tetsuya Ishikawa
SPring-8, 1-1-1, Kouto, Mikazuki, Sayo-gun, Hyogo, 679-5198, JapanAbstract
A wave-front dividing interferometer was devised in which half of a highly spatially coherent beam deflected by an x-rayprism is overlapped with another half. By placing the samples in the two beams and the x-ray imaging detector closely
downstream of the sample, the differential phase of the sample is recorded on the imaging detector. The separation of the twopaths at the sample plane, the amount of shear, was approximately 2 µm in the present experiment. This is easily controllable bychanging the distance between the sample and the imaging detector. Phase retrieval using the fringe scanning method wassuccessfully demonstrated for various kinds of weakly absorbing samples.

1. INTRODUCTION
The various methods for phase retrieval are reported for the scattered / diffracted x-rays [1,2] and thetrasmitted x-rays [3,4]. The interferometers, using the coherent superposition of transmitted and reference
beams, offer the most direct method for unique retrieval of the phase. The crystal Bonse-Hart interferometers(usually as amplitude dividing interferometers) are widely used for the phase retrieval in the hard x-rays[5,6]. Although a shearing geometry has also been attempted [7], the narrow angular acceptance of thecrystal optics limits the spatial resolution. In this paper, we will show that the wave-front division using an
x-ray prism [8,9] can be used for an x-ray shearing interferometer without significant angular windows andwith a better spatial resolution.
2. EXPERIMENTAL SETUP
To obtain a high spatial coherence demanded for the wave-front dividing interferometer, a 250 m longbeamline with an undulator source, BL20XU, of SPring-8 was utilized [9]. Monochromatic x-rays at 12.4
keV (λ= 1Å) through a silicon monochromator and cross-slits, 18×19 µm2 at 195 m upstream of the end
station, make the spatial coherence length of x-rays to be approximately 1 mm at the end station. An x-ray
prism [9] was set so as to make its edge line vertical.
A cubic prism made of aclyric resin with 20 mm sides was inserted in the beam to deflect half of it [9,10]. The x-ray glancing angle onto the prism and the deflection angle were 4 deg and 23 µrad (∆θ). Thedirect and the refracted beams overlap with the width of 150 µm (L1 ∆θ ) at 6.5 m (L1) downstream of theprism where the photo cathode of x-ray zooming tube detector (modified C5333, Hamamatsu Photonics
Co.) [11] was placed (Fig.1). The interference fringe spacing is 4 µm (d=λ/(∆θ)). The deflection angle ischosen so as to compromise of the large interference region and the detectable fringe spacing. The detector
has the magnification factor, the pixel size at the sample plane and the field of view of 55, 0.22 µm and 220µmφ. The measured spatial resolution of the detector was 0.7 µm at 8 keV [12]. The observed visibility ofthe fringes was 0.6.
X-ray prism

Phase-shift (Sample)
X

Φ(A)Path APath B

Φ(B)L1L2

Figure 1: The schematic diagram of the two-beam shearing x-ray interferometer.

A sample was placed in front of the image detector by 96 mm (L2). Interference signal between twoslightly tilted beams are recorded with the detector. The phases of the two beams suffered local shifts intransmitting through the sample, causing a modulated interference pattern. The distance L2 was close to theminimum possible due to the structure of the detector. The amount of shear was 2.2 µm (L2 ∆θ) at thesample position. When Fresnel number, NF (= a2/( λ L2)), is much greater than unity, the phase is easily
retrieved without significant effect of Fresnel diffraction, where a is the sample structure size.
Samples observed are, (a) a Kapton foil (125 µmt with rough tapered edges, (b) a Nylon mesh (pitch 93µm) and ascidian larvae (a biological specimen). The optical microscope images and the observed x-ray
interference images (1 minute exposure) of the samples are shown in Fig.2. The upward / downward bentsof the individual fringes were observed where the sample thickness has a positive / negative gradient in the
downward direction (Fig.2(d)-(f)). Where the phase gradient of the samples are large (≧ 2π over the shear
length), interference fringes appear discontinuous (e.g. along the lower Kapton foil edge in Fig.2(d)).

10µm(a)10µm(b)

(c)

10µm

10µm

10µm(d)100µm(e)(f)

Figure 2: The visible microscope image for the samples (a) an edge of a Kapton foil (125 µmt), (b) a Nylon mesh (pitch93 µm) and (c) an ascidian larvae (biological specimen). The observed x-ray interference images (d, e, f) for the samples(a, b, c).

2. EXPERIMENTAL RESULT
For the precise phase retrieval using the interferometers, the fringe scanning method [13] and theFourier transform method [14] are widely used. We applied the fringe scanning method, since the higher
spatial resolution close to that of the detector is easily achieved. For the phase shift in one of the opticalpaths the translational motion of the prism was introduced.
For the 4-step fringe scanning method, 0, 1/4π, 2/4π and 3/4π phase-shifts were given to path A bymoving the prism by 0, 1/4d, 2/4d and 3/4d (where d is the fringe spacing). The intensity distributions at thedetector plane, I0, I1, I2 and I3 were measured with and without the sample. The phase at the detector plane iscalculated with the following equation, where ∆φ is the differential phase due to the sample (similarequation is derived with ∆φ=0 for the dataset taken without the sample).
2π(kA-kB)(x)-∆φ=tan-1((I3- I1)/( I0- I2))(1)
and
∆φ=φ(A)−φ(B) ∼ δφ/δx (∆x)(2),

where kA-kB are the X-ray wave-vectors along the path A and path B.
The phase distributions at the detector plane are calculated using eq.(1). Since the phase values calculated
are the principal values from -π/2 to π/2, the phase unwrapping is carried out for the calculated phase
distributions with and without the sample, independently. Subtraction between these two unwrapped phase
gives the differential phase distribution of the sample without the complete knowledge of the carrier fringe.
The differential phase of samples with various thickness gradient can be retrieved by adjusting the amount ofshear.
The box areas in Fig.2(d)-(f) are analyzed. The calculated differential phase image for Fig.2(d) is shownin Fig.3(a). The areas where the thickness gradient is positive (negative) is shown in dark (bright) color. For
the upper Kapton foil edge, the differential phase image shows almost monotonious thickness increase
downward (Fig.3(a)). The measured differential phase for the upper Kapton edge is integrated along the line
in Fig.3(a) and is shown in Fig.3(b). The derived phase due to the foil thickness was 15 rad (2.4 λ) close to
the calculated value of 16 rad (2.5 λ), assuming the thickness of 125 µm and the index of refraction n=1- 2.0×10-6 at λ= 1Å. The several horizontal lines in the differential phase images, in and out of the samples, arepartly caused by the insufficient surface finish of the prism.
One dimensional phase distribution
161412phi(rad)1086420010µm

10203040506070y(micrometer)(b)(a)
Figure 3: The calculated differential phase (a) for the upper Kapton foil edge inside the region of interest in Fig.2(d)using the 4-step method and (b) The phase distribution of the upper Kapton foil edge by integrating the measured
differential phase distribution.
Similar phase determination was done for one of the end of the nylon and an ascidian larva (almosttransparent for the 1Å X-ray) as shown in Fig.4(a) and (b), using 12-step method. In Fig.4(b), othoris, thegravity sensor with the diameter of about 10 µm is clearly observed.

(a)

10µm

(b)

10µm

Figure 4: The calculated phase (a) for the Nylon mesh and (b) for the ascidian larvae using the 12-step method.

3. IMPROVEMENT
To improve the precision of the phase retrieval, the surface of the X-ray prism needs to be improved.
For this purpose, ultra precision cutting has been done on the surface of the acrylic resin using a single
crystalline diamond tool which was developed at RIKEN. The surface roughness was evaluated toapproximately 30 nm (RMS) with a Zygo NewView 200.

Figure 5:The surface roughness of improved acrylic X-ray prism evaluated with Zygo Newview200.

5. FUTURE PROSPECTThe present method is easily combined with the tomographic technique. The three dimensional
refractive index distribution will be measured for visible-opaque and x-ray-weakly-absorbing samples withburied structures. Application of this new technique to material science, biological and medical sciences
will be promising and fruitful.
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