Development and application of variable-magnification x-rayBragg magnifiers

K. Hirano n, Y. Takahashi, H. SugiyamaInstitute of Materials Structure Science, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

a r t i c l e i n f o

Article history:Received 25 October 2013Received in revised form7 December 2013Accepted 13 December 2013Available online 24 December 2013

Keywords:Synchrotron radiationX-rayOpticsMagnifierImagingComputed tomography

a b s t r a c t

A novel x-ray Bragg magnifier of variable magnification is proposed. We describe the basic properties ofthe magnifier based on the dynamical theory of x-ray diffraction, and experimentally investigated itsperformance at the vertical wiggler beamline BL-14B of the Photon Factory. Using a Si(2 2 0) asymmetriccrystal as the Bragg magnifier, we successfully controlled the magnification factor, M, between 0.1 and10.0 at a wavelength of 0.112 nm. The observed spatial resolution with an x-ray CCD camera, the pixelsize of which was 23 μm�23 μm, was 10 μm at M¼10, 50 μm at M¼1 and 250 μm at M¼0.17. As ademonstration, we successfully used the x-ray magnifier for computed tomography for the non-destructive observation of a glass tube at several magnification factors.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Bragg magnifiers [1,2] have been widely used for hard x-rayfocusing [3,4], topography [5,6], radiography [7–15] and full-fieldmicroscopy [16–25] at x-ray beamlines of synchrotron facilities.In the conventional Bragg magnifier based on asymmetric x-raydiffraction by a nearly perfect crystal, an incident beam is eitherexpanded or compressed in one direction (in the plane of diffrac-tion). The magnification factor, M, is given by

M¼ sin ðθBþαÞsin ðθB�αÞ; ð1Þ

where θB is the Bragg angle and α is the angle between the crystalsurface and the diffracting lattice planes. To date, two methodshave been used for changing the magnification factor, M: one is tochange the asymmetric angle, α, by replacing the magnifier crystal,and the other is to change the Bragg angle, θB, by tuning the x-raywavelength. These methods, however, usually require rearrange-ment of the optics and are not practical in most cases. Accordingly,most experiments have been performed using a fixed magnifica-tion factor, hindering the use of the x-ray magnifier. To overcomethis problem, we have developed a novel x-ray Bragg magnifier ofvariable magnification.

In this paper, we first describe the basic properties of ourmagnifier. Next, we show the measured properties of the magni-fier at the vertical wiggler beamline BL-14B of the Photon Factory.Then, as a demonstration, we show tomograms of a glass tubeobtained at several magnification ratios.

2. Properties of variable-magnification x-ray Bragg magnifier

Fig. 1 schematically shows our variable-magnification x-rayBragg magnifier. The magnifier crystal can be rotated separatelyaround two different axes, θ̂ and ϕ̂. The angles θ and ϕ correspondto rotations around these axes. The θ̂-axis is perpendicular tothe direction of the incident beam. The ϕ̂-axis is mounted on theθ stage so that it is always perpendicular to the θ̂-axis, andperpendicular to the incident beam when θ equals zero. Anasymmetrically-cut magnifier crystal is mounted on the ϕ stagewith its diffracting lattice planes perpendicular to the ϕ̂-axis. Forϕ¼01, the intersection between one of the diffracting latticeplanes and the crystal surface is perpendicular to the θ̂-axis. Inthe operation of the magnifier, the magnification factor is con-trolled by the azimuthal angle ϕ with θ fixed at the Bragg angle, θB.The diffraction geometry shown in Fig. 1 is known as the rotated-inclined geometry and is widely used for high heat-load x-raymonochromators [26–29] and focusing [3]. The shape of thefootprint on the crystal surface was investigated by Smither andFernandez [26]. The special case of this geometry (ϕ¼01) is knownas the inclined geometry [30–34], where the magnification factor

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0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nima.2013.12.038

n Corresponding author. Tel.: +81 29 864 5596; fax: +81 29 864 2801.E-mail address: keiichi.hirano@kek.jp (K. Hirano).

Nuclear Instruments and Methods in Physics Research A 741 (2014) 78–83

is fixed at unity as described by Lee and Macrander [35]. Forϕ¼þ901 (ϕ¼�901), we have the conventional asymmetric situa-tion at low (high) incident angle with fixed magnification factor.In a sense, our variable-magnification x-ray Bragg magnifier is anatural expansion of the conventional fixed-magnification Braggmagnifiers.

Line A–A in Fig. 1 is the intersection between the crystal surfaceand the plane of diffraction defined by the incident beam and thediffracted beam. Line B–B is the intersection between the plane ofdiffraction and one of the diffracting lattice planes. The effectiveasymmetric angle, α0, is defined as the angle between line A–A andline B–B and is given by

tan α0 ¼ tan α sin ϕ: ð2ÞThe magnification factor is given by

M¼ sin ðθBþα0Þsin ðθB�α0Þ: ð3Þ

Substituting Eq. (2) into Eq. (3), we obtain

M¼ cos α sin θBþ sin α cos θB sin ϕ

cos α sin θB� sin α cos θB sin ϕ: ð4Þ

Under the conditions of the conventional magnifier (ϕ¼ 790 3 ),Eq. (4) reduces to Eq. (1). Fig. 2 shows the calculated magnificationfactors of a Si(2 2 0) asymmetric magnifier crystal at the wave-length of 0.112 nm for α¼121 (solid line), 141 (dashed line) and 161(dashed–dotted line). This result shows that the magnificationfactor, M, is tunable through the azimuthal angle ϕ. For example,we can control the magnification factor, M, between 0.03 and 32for α¼161.

Fig. 3 shows the cross-section of (a) the incident beam and(b) the diffracted beam. Due to the rotated-inclined geometry ofthe magnifier, the rectangular incident beam is modified to theparallelogram-shaped exit beam. The symbols in Fig. 3 are definedas follows: p (p0) is the length of the incident (exit) beam in thedirection perpendicular to the plane of diffraction and q (q0) is thelength of the incident (exit) beam in the plane of diffraction. Whilethe former is kept unchanged (p0 ¼ p), the latter is either expandedor compressed (q0 ¼Mq). The extra length added by the slantingsides of the parallelogram-shaped footprint, r0, is given by

r0 ¼ tan α cos α0 sin ð2θÞ cos ϕ

sin ðθ�α0Þ p: ð5Þ

In the inclined geometry (ϕ¼01), this equation is reduced to thesimpler form given by Lee and Macrander [35]

r0 ¼ 2 cos θ tan α: ð5Þ0

The extra length is zero for the conventional Bragg magnifiers(ϕ¼ 790 3 ). In the general case (ϕa790 3 ), the observed imageafter the magnifier is deformed due to this extra length. However,in modern x-ray imaging with the widespread use of x-ray digitalarea sensors such as x-ray CCD cameras, flat panel sensors andpixel-array detectors, we can easily correct this image deformationusing Eq. (5).

According to the dynamical theory of x-ray diffraction [36–39],the angular acceptance of the magnifier, ωO, and the angulardivergence of the diffracted beam, ωH , are given by

ωO ¼ffiffiffiffiffiM

pωS;

ωH ¼ωS=ffiffiffiffiffiM

p;

ð6Þ

where ωS is the acceptance angle for symmetric diffraction for thesame diffraction plane and is given by

ωS �2

sin ð2θBÞreλ2

πVC Fh ;j������ ð7Þ

Fig. 1. Variable-magnification x-ray Bragg magnifier.

Fig. 2. Calculated magnification factors of a Si(2 2 0) asymmetric magnifier crystalat the wavelength of 0.112 nm for α¼121 (solid line), 141 (dashed line) and 161(dashed-dotted line).

Fig. 3. Cross-section of (a) the incident beam and (b) the diffracted beam.

K. Hirano et al. / Nuclear Instruments and Methods in Physics Research A 741 (2014) 78–83 79

where re is the classical electron radius, λ the wavelength, V thevolume of the unit cell, C the polarization factor and Fh the crystalstructure factor.

The spatial resolution depends on the penetration depth ofx-rays into the magnifier crystal. The penetration depth, zpd, isdefined as the perpendicular distance from the crystal surface forwhich the attenuation factor is 1=e. Rigorous calculations of thepenetration depth can be performed by the dynamical theory ofx-ray diffraction. For the non-absorption case, the penetrationdepth at the diffraction condition is given by

zpd �V

ffiffiffiffiffiffiffiffiffiffiffiffiffiγH�� ��γO

q2reλjCjjFhj

; ð8Þ

where γO and γH are the directional cosines of the incident beamand the exit beam, respectively, with respect to the inward surfacenormal. γO and γH are given by

γO ¼ cos α sin θB� sin α cos θB sin ϕ; ð9Þ

γH ¼ � cos α sin θB� sin α cos θB sin ϕ: ð10ÞThe geometrical blur, Δxpd, due to the penetration depth isestimated as

Δxpd � zpd cos ðθB�α0Þþ 1M

cos ðθBþα0Þ� �

: ð11Þ

Fig. 4(a) shows the calculated spatial resolution due to thegeometrical blur for s-polarization (C¼1). The calculation condi-tions are the same as in Fig. 2.

In experiments such as x-ray holography where the passageof refracted/scattered radiation needs to be preserved, one has toconsider the wave-optical spatial resolution as discussed by Spal [2].The wave-optical spatial resolution can be estimated from theangular acceptance of the magnifier, ωO, which provides the max-imum spatial frequency which is unattenuated. Because the angularacceptance, ωO, depends on the magnification factor, M, as shownin Eq. (6), the wave-optical spatial resolution also depends on themagnification factor, M. This spatial resolution is estimated as

Δxwo �2λωO

¼ 2λffiffiffiffiffiM

pωS

: ð12Þ

Fig. 4(b) shows the calculated wave-optical spatial resolution fors-polarization (C¼1). The calculated value of ωS is 3.58″.

Deterioration of the spatial resolution will be increased by theblurring due to the finite source size. This so-called penumbralblurring, Δxp, is estimated as

Δxp ¼ sDsd=Dss; ð13Þwhere s is the source size, Dss is the distance between the sourceand the sample and Dsd is that between the sample and thedetector. In the case of the bending-magnet light source, thepenumbral blurring in the horizontal direction is larger than thatin the vertical direction.

Fig. 4. Calculated (a) geometrical spatial resolution, Δxpd , and (b) wave-opticalspatial resolution, Δxwo , for s-polarization (C¼1). The solid lines are for α¼121, thedotted lines for α¼141 and the dashed-dotted lines for α¼161.

Fig. 5. Experimental setup.

K. Hirano et al. / Nuclear Instruments and Methods in Physics Research A 741 (2014) 78–8380

Further deterioration of the spatial resolution will be caused byan area detector. This effect is estimated by

Δxd ¼Δxdet=M; ð14Þwhere Δxdet is the spatial resolution of the area detector.

3. Experiment and results

3.1. Estimation of the performance of the x-ray magnifier

In order to estimate the performance of our variable-magnification Bragg magnifier, we carried out experiments atthe vertical wiggler beamline BL-14B of the Photon Factory.The experimental setup is schematically shown in Fig. 5. Thewhite beam from the vertical wiggler was monochromatized at awavelength of 0.112 nm by a pair of Si(1 1 1) crystals. The size ofthe incident beamwas limited to 4 mm (H)�4 mm (V) in size by aslit. The beam transmitted through a sample was either expandedor magnified by a Si(2 2 0) asymmetric magnifier crystal (α¼141,θB¼16.961) in the horizontal plane. The θ̂-axis is in the verticaldirection and the ϕ̂-axis is in the horizontal plane. The surface ofthe magnifier crystal was mechano-chemically polished to removedefects and strain fields. The beam diffracted by the magnifier wasobserved by a fiber-coupled x-ray CCD camera (Photonic ScienceLtd., X-ray Coolview FDI 40 mm) consisting of a GdO2S:Tb scintil-lator, a 3.43: 1.00 glass fiber plate and a CCD. At the scintillator,x-rays were converted to visible light, which was transmittedthrough the glass fiber plate and detected by the CCD. Theeffective pixel size was 23 μm (H)�23 μm (V) and the number ofpixels was 1384 (H)�1032 (V).

We observed x-ray test charts at ϕ¼�901, �601, �301, 01, 301,601 and 901. Fig. 6 shows the estimated (a) magnification factorand (b) ratio between r0 and p. The theoretical values (solid lines)and the experimental values (filled circles) agree well. In Fig. 6(a),the magnification factor is 0.1 at ϕ¼�901, 1.0 at ϕ¼01 and 10 atϕ¼901. In Fig. 6(b), the ratio between r0 and p is zero at theconventional asymmetric geometry (ϕ¼ 790 3 ) and reaches itsmaximum value (0.83) at around ϕ¼541.

The spatial resolution in the horizontal direction was estimatedfrom the modular transfer functions (MTFs) [9] as shown in Fig. 6(c),where the spatial resolution was specified at 25% modulation transfer.The estimated spatial resolution was 250 μm at ϕ¼�601, 112 μm atϕ¼�301, 50 μm at ϕ¼01, 23 μm at ϕ¼301, 15 μm at ϕ¼601 and10 μm at ϕ¼901. The solid line shows the theoretical values, where weassumed that Δxdet¼45 μm and Δxp¼9 μm. The wave-optical spatialresolution, Δxwo, was neglected because the experiment was a simpleabsorption imaging setup. We found that the most dominant term isΔxdet for ϕo541 and Δxp for ϕ4541.

Since the Photon Factory is a second generation synchrotronradiation facility, the best attainable resolution is around a fewmicrometers in the horizontal direction. We expect, however, thatthe spatial resolution will be improved at third generation syn-chrotron radiation facilities where sub-μm resolutions are alreadyachieved with the conventional fixed-magnification Bragg magni-fiers [20,21,25].

3.2. Application of the x-ray magnifier to computed tomography

The one-dimensional (1D) magnification system shown inFig. 5 has a much higher throughput than the two-dimensional(2D) magnification system. This kind of 1D optical system will beuseful especially for observing fibrous objects that have ratheruniform structures in one direction. As a demonstration, weapplied our variable-magnification Bragg magnifier to x-ray com-puted tomography (CT). A glass tube was observed at ϕ¼�301

(M¼0.42), 301 (M¼2.38) and 901 (M¼10.0). The sample wasrotated around the vertical axis from 01 to 1801 in steps of 0.721.At each angle, the projection image of the sample was observed

Fig. 6. Estimated (a) magnification factor, M, (b) ratio between r0 and p and(c) spatial resolution.

K. Hirano et al. / Nuclear Instruments and Methods in Physics Research A 741 (2014) 78–83 81

with the x-ray CCD camera. The exposure time was 60 ms forϕ¼�301, 200 ms for ϕ¼301 and 300 ms for ϕ¼901. Since therotation axis of the sample was perpendicular to the direction of

the image (de)magnification, we could carry out the imagereconstructions by the routine procedures. Fig. 7 shows the CTimages reconstructed by the filtered back-projection method for (a)ϕ¼�301, (b) ϕ¼301 and (c) ϕ¼901. The filter was a Shepp–Loganfilter. The diameter of the reconstructed image of the glass tube is0.6 mm for ϕ¼�301, 3.45 mm for ϕ¼301 and 14.3 mm for ϕ¼901.From these results the diameter of the glass tube is estimated to be1.45 mm.

4. Conclusions

We have proposed a variable-magnification x-ray Bragg mag-nifier in rotated-inclined diffraction geometry. We obtained thebasic properties of the magnifier, such as the magnification factor,M, and the spatial resolution based on the dynamical theory of x-ray diffraction. In experiments performed at the vertical wigglerbeamline BL-14B of the Photon Factory, we successfully controlledthe magnification factor, M, between 0.1 and 10 at the wavelengthof 0.112 nm using an asymmetric Si(2 2 0) magnifier crystal(α¼141, θB¼16.961). The spatial resolution estimated from theMTFs was 10 μm at M¼10, 50 μm at M¼1 and 250 μm at M¼0.17.Since the Photon Factory is a second generation synchrotronradiation facility, the best attainable resolution is around a fewmicrometers in the horizontal direction. We expect, however, thatthe spatial resolution will be improved at third generation syn-chrotron radiation facilities. As a demonstration, the magnifier wassuccessfully applied to x-ray CT for the non-destructive observa-tion of a glass tube at several magnification factors. Although wehave considered only one-dimensional (1D) magnification in thispaper, two-dimensional (2D) magnification could be attained bytwo crossed asymmetric crystals, where the diffraction planes ofthe first and second crystals are set perpendicular to one another.

Acknowledgments

This work was performed with the approval of the ProgramAdvisory Committee of the Photon Factory (2011G032 and 2013G054).

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