TY - JOUR
AU1 - Takeuchi,, Akihisa
AU2 - Suzuki,, Yoshio
AB - Abstract The advent of high-flux, high-brilliance synchrotron radiation (SR) has prompted the development of high-resolution X-ray imaging techniques such as full-field microscopy, holography, coherent diffraction imaging and ptychography. These techniques have strong potential to establish non-destructive three- and four-dimensional nano-imaging when combined with computed tomography (CT), called nano-tomography (nano-CT). X-ray nano-CTs based on full-field microscopy are now routinely available and widely used. Here we discuss the current status and some applications of nano-CT using a Fresnel zone plate as an objective. Optical properties of full-field microscopy, such as spatial resolution and off-axis aberration, which determine the effective field of view, are also discussed, especially in relation to 3D tomographic imaging. X-ray microscopy, X-ray nanotomography, Fresnel zone plate Introduction The highly penetrating power of X-rays allows non-destructive investigation of three-dimensional (3D) internal structures of objects. Computed tomography (CT) is a well-known technique that takes advantage of this feature. These techniques dedicated to microscopic applications are called X-ray microtomography (micro-CT) and nano-tomography (nano-CT). In terms of spatial resolution, X-ray CT surpasses optical microscopy, but 3D visualization techniques such as transmission electron microscopic (TEM) tomography and serial sectioning with focused ion beam–scanning electron microscopy (FIB–SEM) are superior to X-ray CT [1–4]. Nevertheless, high-resolution X-ray imaging is an indispensable technique for imaging in the mesoscopic region, connecting the observable regions of optical and electron microscopy. The high penetration power of X-rays allows a very large D/∆ ratio, where D is the diameter (or thickness) of the object and ∆ is the spatial resolution. Measurements can be performed in the ambient atmosphere with no sample pre-treatment, allowing non-destructive 3D observation of bulky objects. Four-dimensional (4D) measurements such as in situ, in vivo and operando CT can also be easily undertaken. X-ray micro- and nano-imaging are one of the few methods capable of non-destructive 3D–4D high-resolution study of internal structures. Much information is contained in 3D and 4D data, enabling the understanding of complex and local phenomena in various fields, including material engineering, earth and planetary sciences, astronomy, biomedical engineering and industrial applications. While the penetrating power of X-rays is an advantage in non-destructive observations, it has a disadvantage in high-sensitivity microscopic imaging because there is a trade-off between the penetrating power and interaction with the object, i.e. the ‘detection sensitivity’. X-ray phase contrast imaging methods have been developed with high sensitivity to overcome this disadvantage. The complex refractive index, n, of X-rays is expressed in terms of the refractive index decrement, δ, and absorption factor, β, as: $$\begin{equation} n=1\hbox{--} \delta \hbox{--} i\beta \end{equation}$$(1) Although both δ and β are much smaller than unity, δ is 1–3 orders of magnitude greater than β. X-ray phase contrast imaging uses δ as the image contrast, allowing up to three orders of magnitude greater sensitivity than that of conventional X-ray absorption contrast imaging based on β alone. It is impossible to measure δ directly from X-ray intensity (unlike β), so its measurement requires interferometric or holographic techniques. Although many studies have been attempted since the 1960s [5–7], the practical utilization of this technique stagnated for a long time. There has been rapid progress prompted by practical phase contrast methods developed in the 1990s [8, 9]. The advent of synchrotron radiation (SR), which allows use of highly coherent X-rays, has promoted development of various phase contrast imaging methods [10–22], particularly for applications with light materials and in biology and medicine. Furthermore, as both high-penetrating power and high sensitivity can be achieved with high-energy X-ray phase contrast imaging, the technique can now be applied to heavy element bulk samples, such as metals and ceramics [23–25]. In recent decades, various types of SR X-ray CT methods have been developed and are now widely used. They are classified broadly into two types according to a boundary spatial resolution of ~1 μm: one based on simple X-ray projection optics, known as micro-imaging or micro-CT, and used mainly for observing structures with dimensions >1 μm, and the other, nano-imaging or nano-CT, for finer observations. Various types of nano-CT have been developed so far. Among them, full-field X-ray microscope is one of the most promising for 3D/4D nano-CT because of its high throughput and D/∆ ratio. Here, micro-CT is described in Section X-ray micro-imaging (micro-CT), and nano-CTs are introduced in Section X-ray nano-imaging (nano-CT). Nano-CT based on full-field X-ray microscopy, developed at the large synchrotron radiation facilicy SPring-8 (Hyogo, Japan), is also described. The basic concept and aberration theory of the Fresnel zone plate (FZP), which is a key technology in full-field microscopy, is discussed in Section Fresnel zone plate for X-ray nano-imaging and nano-CT. Section Full-field X-ray microscopy for nano-CT considers the optical properties of full-field nano-CT microscopes, and examples of their application at SPring-8 are described. X-ray micro-imaging (micro-CT) The high penetration power of X-rays means that while other imaging methods often aim at observing an object only in 2D plane view, X-ray imaging allows 3D observations including the longitudinal direction. Another unique feature of X-rays is their strong ‘straightness’. For example, in the case of medical radiography, X-rays are considered to travel in straight lines with no deflection between source and detector. Projection optics is based on this consideration. The reconstruction principle of conventional CT is based on the Radon transform, which is a back-projection algorithm that assumes this geometric approximation. This is useful for medical CT scanners because of the short X-ray wavelength, high energy (several tens of keV) and low spatial resolution of ~1 mm. However, the assumption of straightness becomes inappropriate at longer wavelengths and higher spatial resolution because blurring of the image due to Fresnel diffraction is more noticeable. The spatial resolution of projection optics, ∆, ultimately restricted by Fresnel diffraction, is expressed as: $$\begin{equation} \varDelta ={\left(\lambda L\right)}^{1/2}, \end{equation}$$(2) where λ is the wavelength and L is the distance between the sample and detector. When λ = 1 Å, L = 1 cm, and the blurring of the image, ∆, is 1 μm. Although ∆ can be reduced by shortening the distance L, that increases the influence of scattering from the sample and reduces image contrast. Therefore, structures resolvable by projection CT are practically restricted to dimensions of ~1 μm, with projection CT for such applications thus being termed ‘micro-CT’. A typical setup for SR X-ray micro-CT at SPring-8 is shown in Fig. 1. The sample is irradiated with a quasi-parallel beam from the SR source, and transmitted images of the rotating sample are captured with an image detector installed immediately behind the sample. Such simple systems can routinely perform 3D observations at ~1 μm spatial resolution [26]. In most laboratory-based micro-CT systems, projection magnification using a spherical wave from a point-like source is employed, whereas a quasi-parallel beam is used mainly in SR micro-CT, with the beam being monochromatized by passing through a silicon double-crystal monochromator. Monochromaticity given by ∆λ/λ is ~10−4, where ∆λ is the bandwidth. For high-speed measurements, a ‘white’ beam from a bending magnet or quasi-monochromatic ‘pink’ beam from an undulator source is used [27, 28]. A ball-bearing-type rotation stage is typically used for sample rotation, with a wobbling inaccuracy of <1 μm. Transmission images of the sample are acquired at every fixed angle during 180° of rotation. Fig. 1 Open in new tabDownload slide Schematic diagram of a SR X-ray micro-CT system. Fig. 1 Open in new tabDownload slide Schematic diagram of a SR X-ray micro-CT system. A visible light conversion-type image detector which comprises a scintillator screen, visible light microscope objective and scientific complementary metal oxide semiconductor (sCMOS) detector (Fig. 2) is generally arranged behind the sample. Spatial resolution is restricted by the wavelength of light emitted by the scintillator, although the practically achievable spatial resolution is determined mainly by the scattering and thickness of the scintillator [29, 30]. A 200 nm line-and-space pattern was resolved using a transparent ceramic LuAG:Ce layer with 5 μm thickness bonded by solid-state diffusion as a scintillator [31]. In addition to these visible light conversion-type detectors, direct sensing X-ray detectors may be used for acquiring X-ray images. When X-rays are directly incident upon a Si-type charge-coupled device (CCD) or CMOS device, ~700 electron–hole pairs are created for each X-ray photon with 10 keV. Considering the charge saturation of a typical device (typically 105 to 106 electrons), the dynamic range should be about tens to hundreds of X-ray photons. With visible light conversion detectors, several electron–hole pairs are created by each X-ray photon, sufficiently higher than the achieved noise level, with a wide dynamic range of up to 105 being achievable without loss of X-ray signal. Therefore, visible light conversion-type detectors are generally more suitable for wide dynamic range imaging. The direct sensing-type detector may also incur more radiation damage, and thus visible light conversion-type detectors are used more widely in SR X-ray imaging. Fig. 2 Open in new tabDownload slide Photograph (left) and schematic drawing (right) of a visible light conversion X-ray camera. Fig. 2 Open in new tabDownload slide Photograph (left) and schematic drawing (right) of a visible light conversion X-ray camera. CCD cameras have been widely used for image collection, but CMOS cameras are now mainstream because they are versatile and have no extra development costs. Furthermore, CCDs transfer data as charge signals that are likely to generate read-out noise during transfer, restricting transfer speed (with a trade-off between noise and transfer speed). On the other hand, CMOS devices transfer the signal after amplification in each pixel, so read-out noise is not an issue, allowing both high-speed and high-dynamic-range imaging. Typical CT measurement parameters at SPring-8 are as follows: voxel size of 0.5 μm; number of pixels of 2048 × 2048; effective field of view (FOV) ~ 1 mm; exposure time of several tens of ms; and scan time for 1800 projections per 180° of 3–5 min, with higher speed scans being possible. For example, 4D CT applications with a scan speed of several tens of Hz have been performed by decreasing the spatial resolution to several μm [32, 33]. Millisecond-order CT measurements have been demonstrated by correcting the signal reduction due to ultra-high-speed measurement using a phase contrast method [34]. An example of 4D CT measurement at SPring-8 is provided in Fig. 3, which shows the solidification process of ductile cast iron during repeated CT measurements as the iron melt cooled from ~1200°C at a cooling rate of 0.5°C s−1. The two figures represent a difference of 4 s (2°C), showing that austenite (face-centered cubic) dendrite and graphite are formed simultaneously at different positions. Fig. 3 Open in new tabDownload slide In situ CT image of the solidification process of cooled ductile cast iron solution. The sample was cooled from ~1200°C at ~0.5° s−1. Two images show the effect of a 2° temperature difference. Voxel size is 2 μm, CT rotation speed is 0.25 rpm, and frame camera image rate is 200 frames s−1. Fig. 3 Open in new tabDownload slide In situ CT image of the solidification process of cooled ductile cast iron solution. The sample was cooled from ~1200°C at ~0.5° s−1. Two images show the effect of a 2° temperature difference. Voxel size is 2 μm, CT rotation speed is 0.25 rpm, and frame camera image rate is 200 frames s−1. X-ray nano-imaging (nano-CT) Resolution of X-ray nano-imaging Conventional tomographic reconstruction based on the Radon transform is effective as long as the assumption of the X-ray beam going straight through the object without diffraction, scattering or reflection holds. In the case of high-resolution tomography, with 1 μm resolution or better, where beam deflection cannot be ignored, this condition restricts the measurable sample size (depth of the optical axis) as defined by the Fresnel diffraction formula. When the object center is located on the object plane, the image blur due to beam deflection inside the object is represented by: $$\begin{equation} \varDelta ={\left(\lambda D\right)}^{1/2}, \end{equation}$$(3) where D is the diameter (or the thickness) of the sample. Equations (2) and (3) are similar, but note that L in Eq. (2) is the sample-to-camera distance, while D in Eq. (3) represents the sample size. Tomographic reconstruction based on the conventional back-projection algorithm is effective within the range where the amount of blur expressed in Eq. (3) does not exceed spatial resolution. This limitation is related to that of the depth of field of an optical system, as described in Section Principles and properties of FZP for X-rays. X-ray deflection thus restricts the spatial resolution of micro-CT while ultimately determining the observable FOV in the depth direction of nano-CT. X-ray nano-CT techniques In recent years, highly coherent SR X-ray beams and rapid progress in computer science have led to remarkable development of holography, coherent diffraction imaging (CDI) [35] and ptychography [36]. High-resolution and high-sensitivity 3D imaging methods combining holographic image reconstruction with tomographic measurements have been implemented at the European Synchrotron Radiation Facility (ESRF) [20, 21]. CDI is a ‘lensless’ technique for reconstruction of images of nanoscale structures. An object is irradiated with completely coherent X-rays, its diffraction pattern collected by a detector, and the data is used in computer reconstruction. However, it is impossible to obtain phase information directly from the diffraction data that correspond to the absolute square of the electric field of the X-ray (a phase problem). CDI measurements are constrained by the size of an isolated object being less than half the illuminating beam size (termed ‘oversampling’ conditions). Based on this constraint and diffraction pattern data, the phase distribution of the object is gradually recovered by repeatedly reciprocating between the detector surface plane and sample plane, which have Fourier and inverse transform relationships [37]. Since the spatial resolution of CDI is determined simply by the diffraction angle that can be recorded by the detector, aberration-free and high spatial resolution measurements are possible. A spatial resolution of 2 nm has been achieved with a sample of gold particles [38]. However, this method requires an isolated object smaller than the irradiating beam size, limiting samples to very small scales, with CDI thus having a narrow FOV. Ptychography has been developed to overcome this shortcoming. Beam scans overlap spatially on an object, and a coherent diffraction intensity pattern is obtained at each scan position, as with CDI. By using data of the overlapping region as a constraint condition, it is possible to retrieve the phase of objects larger than the irradiated region. Many phase retrieval algorithms have been proposed. In particular, the extended ptychographical iterative engine (ePIE) enables reconstruction of complex amplitude distributions of illuminating light and complex object transmission functions, and this has become a popular method of the accurate evaluation of X-ray optical devices [39]. Ptychography has a large numerical aperture (NA) as an X-ray optical system, with a correspondingly limited depth of field. Taking advantage of this, a multi-slice ptychographic method has been proposed for high-resolution 3D imaging, in which the object is regarded as a stack of thin layers in the focal direction [40, 41]. X-ray full-field microscopy has basically the same optical system as that of optical microscopy, and, unlike other nano-imaging methods, high-spatial coherence is not required. The transmitted X-ray image of an object is magnified with an objective, and the magnified image is captured by an image detector. However, since the refractive index of X-rays expressed by Eq. (1) is almost in unity for all materials, refractive/reflective devices such as lens and mirrors used in the visible light region do not work effectively in the X-ray region. Therefore, optical devices for the X-ray region have been uniquely developed, taking into account optical characteristics peculiar to X-rays. Of these, there are three general types using (1) total reflection such as with a Wolter mirror [42] and advanced Kirkpatrick–Baez (AKB) mirror [43, 44]; (2) diffraction (e.g. FZP [45]); and (3) an arrangement of many refractive lenses to amplify the effect of refraction (i.e. compound refractive lenses (CRL) [46]). All of these satisfy Abbe’s sine condition and are now used as high-precision objectives for the X-ray region, achieving a spatial resolution of the order of sub-ten to several tens of nm [47–51]. Each of these techniques has its own advantages and disadvantages, and they are selected according to purpose rather than simply to achieve high spatial resolution. The advantages of total reflection mirrors, such as high efficiency, large NA and achromaticity, have promoted the use of Wolter and AKB mirrors in the imaging of X-ray fluorescence and scattering [52–54]. CRLs are often used in the high-energy X-ray region where the δ/β ratio is relatively high [55]. FZPs are regarded as the only ‘thin lens’ in the X-ray region and are often used as the objective in analytical microscopy applications, such as nano-CT, due to its low off-axis aberration (Section Fresnel zone plate for X-ray nano-imaging and nano-CT). Full-field microscopy is capable of live imaging and 3D CT measurement in relatively short timeframes and is therefore suitable for intact 3D–4D measurement which is one of the greatest advantages of X-ray imaging. SR X-ray full-field microscopes are now available worldwide, as discussed further in Section Full-field X-ray microscopy for nano-CT. Fresnel zone plate for X-ray nano-imaging and nano-CT Principles and properties of FZP for X-rays The FZP is one of the most widely used optical devices for the X-ray region because of its low geometrical off-axis aberration and simple optical configuration. The spatial resolution of X-ray microscopes with FZP optics is now approaching 10 nm, which is near the theoretical limit of simple FZP optics [56]. Great progress has been achieved mainly by utilizing modern silicon nanotechnology, involving very large-scale integrated (VLSI) circuit fabrication processes. An FZP is a diffractive optical device and can be considered as a concentric transmission grating with radially decreasing grating period, as shown schematically in Fig. 4. The grating period of the zone pattern is adjusted according to d = λ/sinθ, as shown in Fig. 5 to focus diffracted X-rays at a point, where θ is determined by tanθ = r/f, r is the radius of the circular grating and f is a specific distance from the zone plate called the ‘focal length’. The FZP thus acts as a convex lens does for visible light and simultaneously as a concave lens for negative-order diffraction. The 0th and higher-order diffractions usually coexist, so some form of spatial filtering is needed for FZP optics to function as an optical lens. Fig. 4 Open in new tabDownload slide Fresnel zone plate (FZP): schematic drawing and photomicrograph. This FZP has a diameter of 100 μm and an outermost zone width of 250 nm. It was fabricated by electron beam lithography. Fig. 4 Open in new tabDownload slide Fresnel zone plate (FZP): schematic drawing and photomicrograph. This FZP has a diameter of 100 μm and an outermost zone width of 250 nm. It was fabricated by electron beam lithography. Fig. 5 Open in new tabDownload slide Schematic of diffraction at an FZP. Fig. 5 Open in new tabDownload slide Schematic of diffraction at an FZP. The precise FZP structure is derived from the optical path difference equation. For a zone plate comprising alternating transparent and opaque zones, fully constructive interference occurs (i.e. all waves through each zone are in phase at the focal point) when the optical path difference between neighboring zones is a half wavelength. The nth zone boundary between opaque and transparent zones of a FZP that focuses the spherical wave emitted from point A to point B is then expressed by the following equation: |$\Big({R}_a+{R}_b\Big)= n\lambda /2+\Big(a+b\Big).$| as shown in Fig. 6, and: $$\begin{equation} {\left({a}^2+{r_n}^2\right)}^{1/2}+{\left({b}^2+{r_n}^2\right)}^{1/2}\hbox{---} \left(a+b\right)=n\;\lambda /2, \end{equation}$$(4) Fig. 6 Open in new tabDownload slide Optical path description of an FZP. Fig. 6 Open in new tabDownload slide Optical path description of an FZP. where a is distance from point A to the FZP, b is distance between B and FZP, rn is the nth boundary of the zone structure and n is an integer. Expanding the square root, and omitting higher-order terms, Eq. (4) can be rewritten as: $$\begin{equation} a+{r_n}^2/2a+b+{r_n}^2/2b\hbox{---} \left(a+b\right)= n\lambda /2, \end{equation}$$(5) and then: $$\begin{equation} {r_n}^2/a+{r_n}^2/b= n\lambda . \end{equation}$$(6) By defining 1/a + 1/b = 1/f (the well-known lens equation), Eq. (6) can be rewritten as: $$\begin{equation} {r_n}^2= n\lambda f. \end{equation}$$(7) Thus, an FZP whose zone boundary is defined by the above equation works as a lens with a focal length of f, which varies inversely with X-ray wavelength (proportional to X-ray energy). Therefore, a monochromatic X-ray beam is required for FZP microscopy. An FZP can be fabricated exactly according to Eq. (4), although such FZPs are not useful for general purposes because they can be used only in a fixed configuration at fixed wavelength. Conventional FZPs defined by Eq. (7), known as ‘parabolic’ zone plates, are generally used in the X-ray region, with limitations as discussed later. The spatial resolution limit for a microscope with an objective of circular aperture (axis-symmetric optics) and incoherent illumination (as in micro-focus applications) is expressed by the well-known Rayleigh criterion as: $$\begin{equation} \varDelta =0.61\lambda / NA, \end{equation}$$(8) where ∆ is generally known as the ‘diffraction-limited’ resolution, representing the smallest observable distance between two points. NA is the numerical aperture of the objective and is defined as NA = sinθ, where θ is the angle between the marginal rays and optical axis. The resolution, ∆, is also apparent from the diffraction of a linear grating where sinθ = λ/d, where 2d is the outermost zone period of the FZP. Then, the following equation, $$\begin{equation} \varDelta =1.22{d}_N, \end{equation}$$(9) is frequently used to calculate the spatial resolution of the FZP, where dN is the width of the outermost zone. The depth of focus, ∆f, is expressed by the well-known formula: $$\begin{equation} \varDelta f=\pm \lambda /\left(2{NA}^2\right). \end{equation}$$(10) Using Eqs. (8) and (10), Eq. (3) showing the relationship between measurable sample thickness and spatial resolution can be derived. Chromatic aberration, which influences the finite bandwidth of the X-ray beam, can be discussed in relation to the depth of focus. When an FZP defined by the equation rn2 = nλf is used, the focal length, f, is proportional to 1/λ because rn and n are constant for a given FZP. Therefore, the depth of focus, ∆f, caused by the difference in wavelength, ∆λ, can be written as: $$\begin{equation} \varDelta f/f=\varDelta \lambda /\lambda . \end{equation}$$(11) When the defocusing is less than the depth of focus, defocusing is considered negligible. Therefore, based on Eqs. (10) and (11), the bandwidth tolerance, ∆λ, is defined by the relationship: $$\begin{equation} \varDelta f= f\varDelta \lambda /\lambda <0.5\lambda /{NA}^2. \end{equation}$$(12) By using the approximations NA ≈ rN/f and rN2 = Nλf, where N is the zone number of the outermost zone, the following useful formula is derived: $$\begin{equation} \varDelta \lambda /\lambda <0.5/N. \end{equation}$$(13) Taking both sides of defocusing into account, the total bandwidth tolerance is 2∆λ. Therefore, the requirement for monochromaticity of the X-ray beam in FZP microscopy, 2∆λ/λ, is approximately equal to the inverse of the total number of zones, 1/N. Diffraction efficiency is an important characteristic of diffraction-based optics. The mth-order diffraction efficiency of an FZP with an even zone width (i.e. a 1:1 zone ratio) is expressed as: $$\begin{eqnarray} {\varepsilon}_m=1/\left(2{\pi}^2{m}^2\right)\ \left[1\hbox{---} \mathit{\cos}\left(\pi m\right)\right]\ \left[1+\mathit{\exp}\left(-2 k\beta t\right)\hbox{---} \right.\nonumber \\ \left.2\mathit{\exp}\left(- k\beta t\right)\ \mathit{\cos}\left( k\delta t\right)\right], \end{eqnarray}$$(14) where δ and β are the real and imaginary parts of the complex refraction index of the FZP pattern material, respectively, t is the pattern thickness and k is the wave number representing k = 2π/λ. For first-order diffraction (m = 1): $$\begin{equation} {\varepsilon}_1=1/{\pi}^2\left[1+\mathit{\exp}\left(-2 k\beta t\right)\hbox{---} 2\mathit{\exp}\left(- k\beta t\right)\mathit{\cos}\left( k\delta t\right)\right]. \end{equation}$$(14′) The efficiency of mth-order diffraction for transparent and opaque zones (i.e. black and white zones) is 1/(mπ)2 (~10% for first-order diffraction), where m = ±1, ±3, ±5, etc. and even-order diffraction does not exist. More efficient gratings can be achieved by phase modulation structures that utilize phase shift rather than stopping the beam. Maximum diffraction efficiency is attained at a phase shift of a half wavelength (tδ = λ/2), with the efficiency of mth-order diffraction for an ideal phase grating being 4/(mπ) 2 (m = ±1, ±3, ±5,….). First-order diffraction thus has an efficiency of ~40% in the case of an ideal phase modulated (without absorption loss) zone plate. However, in the X-ray region, such pure phase material does not exist, and all optical media involve a complex mix of phase and absorption effects. Most FZPs for X-ray microscopy are fabricated using state-of-the-art technology developed in the semiconductor industry, involving electron beam lithographic techniques. Current integrated circuit technology makes it possible to fabricate 10-nm-level nanostructures on silicon surfaces, and this is key technology for X-ray microscopy. The FZP structure is illustrated in Fig. 7. The zone plate pattern is drawn by an electron beam on a photoresist supported on a thin membrane on a silicon wafer; the membrane comprises silicon nitride or silicon carbide and is usually a few μm thick. The pattern transfer from photoresist to zone material is done by dry etching or wet electroplating processes. Finally, the silicon wafer containing the patterned area is removed by chemical etching (Fig. 7). Fig. 7 Open in new tabDownload slide Structure of an FZP: cross-sectional view (upper) and scanning electron microscope images (lower). Fig. 7 Open in new tabDownload slide Structure of an FZP: cross-sectional view (upper) and scanning electron microscope images (lower). A difficulty of fabrication of X-ray zone plates is that the high aspect ratio (zone height/zone width) of the zone structure is required in marginal areas. In the X-ray region, the phase shift is much greater than the absorption contrast, so most X-ray FZPs are designed as phase modulation zone plates. Assuming a free electron approximation, the real part of the refraction index, δ, for X-rays is given by: $$ \delta =1.35\times{10}^{-6}\times{\rho \lambda}^2=2.08\times{10}^{-4}\times \rho /{E}^2, $$ where ρ is density of the material (g cm−3), λ is in Å units, E is the X-ray energy (keV) and a Z/A ratio of 1/2 is assumed (Z = atomic number; A = atomic weight). As described above, the optimum thickness of the zone plate, t, is equivalent to an optical path difference of λ/2 expressed by tδ = λ/2, which gives maximum efficiency for first-order diffraction, and can be calculated as: $$\begin{equation} t=\lambda /\left(2\delta \right)={\left(2.7\times{10}^{-2}\rho \lambda \right)}^{-1}=3.0\times E/\rho \mu m. \end{equation}$$(15) This means that thicker zones and higher density zone material are preferable at higher X-ray energies. Gold or tantalum are usually chosen for zone material for FZPs for hard X-rays. The optimized thickness and δ value for a tantalum (ρ = 16.654 g cm−3) zone plate, for example, are 2.9 μm and 1.71 × 10−5 at an X-ray wavelength of 1.0 Å (12.4 keV), respectively. The aspect ratio should then be 29 for an FZP with a 100 nm zone width. Achievement of this value is challenging even with current nano-fabrication technology. The practical X-ray energy region of FZPs therefore remains at ~10 keV or lower. Various attempts to realize a higher aspect ratio have been made [57–67]. The total number of zones, N, is usually 100–10 000; with >100 zones, FZP imaging quality is similar to that of a conventional lens [68]. Although larger N values are preferable in terms of numerical aperture and FOV, aberration of optical systems limits the value, as discussed below. Aberrations of X-ray FZP optics Here, the imaging properties of FZP microscopes are discussed on the basis of the optical path method and wavefront aberration analysis. The optical path equation is as follows: $$\begin{eqnarray} &&{\left\{{\left({r}_a\hbox{---} {r}_n cos\phi \right)}^2+{\left({r}_n sin\phi \right)}^2+{a}^2\right\}}^{1/2} \nonumber \\ &&+{\left\{{\left({r}_b+{r}_n cos\phi \right)}^2+{\left({r}_n sin\phi \right)}^2+{b}^2\right\}}^{1/2} \nonumber \\ &&= n\lambda /2+{\left({a}^2+{r_a}^2\right)}^{1/2}+{\left({b}^2+{r_b}^2\right)}^{1/2} \end{eqnarray}$$(16) where ra/a = rb/b (Fig. 8) and rb/ra (= b/a) represent the geometrical magnification of the microscope [69] and ϕ is the angle at a point on the FZP objective (Fig. 8). By expanding the square root and omitting higher-order terms, Eq. (16) can be approximated as: $$\begin{eqnarray} \nonumber && \frac{1}{2}\left({r_a}^2\hbox{---} 2{r}_a{r}_n cos\phi +{r_n}^2\right)/a\hbox{---} \frac{1}{8} \\ \nonumber &&{\left({r_a}^2\hbox{---} 2{r}_a{r}_n cos\phi +{r_n}^2\right)}^2/{a}^3 \\ \nonumber{}&&+\frac{1}{2}\left({r_b}^2+2{r}_b{r}_n cos\phi +{r_n}^2\right)/b\hbox{---} \frac{1}{8}\\&&{\left({r_b}^2+2{r}_b{r}_n cos\phi +{r_n}^2\right)}^2/{b}^3 \nonumber \\ &&= n\lambda /2+\frac{1}{2}\left({r_a}^2/a\right)\hbox{---} \frac{1}{8}\left({r_a}^4/{a}^3\right)+ \nonumber \\ &&\frac{1}{2}\left({r_b}^2/b\right)\hbox{---} \frac{1}{8}\left({r_b}^4/{b}^3\right). \end{eqnarray}$$(17) Fig. 8 Open in new tabDownload slide Schematic diagram of the off-axis optical path for FZP imaging optics. Fig. 8 Open in new tabDownload slide Schematic diagram of the off-axis optical path for FZP imaging optics. Wavefront aberrations, Ф, can be determined using the optical path equation (Eq. (17)) and can be evaluated using the Rayleigh’s quarter-wavelength rule, which states that when the optical path discrepancy from the primary path is within a quarter of the wavelength, the effect of wavefront aberration on spatial resolution may be ignored, with diffraction-limited spatial resolution thus being achieved. The Rayleigh’s quarter-wavelength rule is expressed as: $$\begin{eqnarray} \mid \varPhi \mid\!\!\!\!\!\! &=&\!\!\!\!\!\!\mid \left\{-{r}_a{r}_n cos\phi /a+{r}_b{r}_n cos\phi /b\right\}+ \nonumber \\ \nonumber&&\!\!\!\!\!\!\!\! \frac{1}{2}\left\{{r_n}^2/a+{r_n}^2/b\hbox{---} n\lambda \right\}\\ \nonumber &&\!\!\!\!\!\!\!\!{}\hbox{--} \frac{1}{8}\left\{{\left({r_a}^2\hbox{---} 2{r}_a{r}_n cos\phi +{r_n}^2\right)}^2/{a}^3\hbox{---} {r_a}^4/{a}^3 \right.\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\left.+{\left({r_b}^2+2{r}_b{r}_n cos\phi +{r_n}^2\right)}^2/{b}^3\hbox{---} {r_b}^4/{b}^3\right\}\mid{}<\lambda /4. \end{eqnarray}$$(18) The first term represents the geometrical magnification equation of ra/a = rb/b, and the second corresponds to the basic parabolic zone plate equation of rn2 = nλf. The residual aberration can then be rewritten as: $$\begin{eqnarray} &&\mid \varPhi \mid = \mid \frac{1}{8}\left\{\left(-4{r_a}^3{r}_n cos\phi /{a}^3+2{r_a}^2{r_n}^2/{a}^3\nonumber\right.\right.\\&& \left.\left.+{\left(2{r}_a{r}_n cos\phi \right)}^2/{a}^3\hbox{---} 4{r}_a{r_n}^3 cos\phi \right)/{a}^3+{r_n}^4/{a}^3\right)\nonumber\\ && {}+\left(4{r_b}^3{r}_n cos\phi /{b}^3+2{r_b}^2{r_n}^2/{b}^3 \right.\\&& \left.\left.\left.+{\left(2{r}_b{r}_n cos\phi \right)}^2/{b}^3+4{r}_b{r_n}^3 cos\phi \right)/{b}^3+{r_n}^4/{b}^3\right)\right\}\mid{}<\lambda /4. \nonumber \end{eqnarray}$$(19) The wavefront aberration Ф consists of the first, second, third and fourth orders of rn, corresponding to image distortion, astigmatism and field curvature, coma and spherical aberration, respectively, of the Seidel aberrations [70]. The first order of rN, which corresponds to image distortion [70], is always eliminated, with the residual aberration being written as: $$\begin{equation} \mid \hbox{--} \frac{1}{8}\left\{\hbox{--} 4{r_a}^3{r}_N cos\phi /{a}^3+4{r_b}^3{r}_N cos\phi /{b}^3\right\}\mid, \end{equation}$$(20) which is always zero because ra/a = rb/b. Therefore, there is no image distortion in FZP microscopy, with this being specific to thin lens optics such as FZP, unlike other X-ray optical devices. FZPs are thus the most suitable objectives for analytical microscopes used for quantitative imaging, as in tomography. The residual off-axis aberration consisting of the second order (rN2) is extracted from Eq. (19) as: $$\begin{eqnarray} &&\mid \frac{1}{8}\left\{{\left(2{r}_a{r}_N cos\phi \right)}^2/{a}^3+2{r_a}^2{r_N}^2/{a}^3\nonumber\right. \\&& \left.+{\left(2{r}_b{r}_N cos\phi \right)}^2/{b}^3+2{r_b}^2{r_N}^2/{b}^3\right)\mid <\lambda /4. \end{eqnarray}$$(21) This term is considered to be related to astigmatism and field curvature [70]. By using the conditions −1 ≤ cosϕ ≤ +1, a < < b and a ≈ f, the following expression is derived: $$\begin{equation} \frac{3}{4}{r_a}^2{r_N}^2/{f}^3<\lambda /4. \end{equation}$$(22) Concerning the third order (rN3) that corresponds to coma, the following relationship can be extracted from Eq. (19): $$\begin{equation} \mid \frac{1}{8}\left\{\left(\hbox{--} 4{r}_a{r_N}^3 cos\phi \right)/{a}^3+\left(4{r}_b{r_N}^3 cos\phi \right)/{b}^3\right\}\mid <\lambda /4. \end{equation}$$(23) This expression may also be reduced to: $$\begin{equation} \frac{1}{2}{r}_a{r_N}^3/{f}^3<\lambda /4. \end{equation}$$(24) For the fourth order of rN, the aberration is expressed as: $$\begin{equation} \mid \frac{1}{8}\left({r_N}^4/{a}^3+{r_N}^4/{b}^3\right)\mid <\lambda /4. \end{equation}$$(25) This term represents spherical aberrations [69, 70], which are not zero, even on the optical axis. An approximation for Eq. (25) can be made for high magnifications (or low magnifications with micro-focusing optics) as follows: $$\begin{equation} \frac{1}{8}{r_N}^4/{f}^3<\lambda /4. \end{equation}$$(26) In FZP optics, spatial filters are needed for separating undesired orders of diffraction. This means that for bright-field imaging, the FOV cannot exceed the radius of the FZP. Aberrations can be evaluated assuming three typical conditions: ra ≈ 0 (a very small FOV); ra = rN/2; and ra = rN (the largest FOV in bright field imaging). The condition ra ≈ 0 corresponds to micro-focus applications. Using an approximation of NA ≈ rN/f and ∆ = 0.61 λ/NA, the requirement for diffraction-limited resolution can be expressed as: $$\begin{equation} \frac{1}{8}{r}_N{\left(0.61\lambda /\varDelta \right)}^3<\lambda /4, \end{equation}$$(27) for spherical aberrations (on-axis aberration). The total aberration can be obtained by summing Eqs. (22), (24) and (26) to give: $$\begin{equation} \frac{3}{4}{r_a}^2/{r}_N{\left(0.61\lambda /\varDelta \right)}^3+\frac{1}{2}{r}_a{\left(0.61\lambda /\varDelta \right)}^3+\frac{1}{8}{r}_N{\left(0.61\lambda /\varDelta \right)}^3<\lambda /4. \end{equation}$$(28) Therefore, for the largest (ra = rN) and medium (ra = rN/2) FOVs, the total aberration is: $$\begin{equation} 0.31{r}_N{\left(\lambda /\varDelta \right)}^3<\lambda /4,{forr}_a={r}_N. \end{equation}$$(28′) $$\begin{equation} 0.12{r}_N{\left(\lambda /\varDelta \right)}^3<\lambda /4,{forr}_a={r}_N/2. \end{equation}$$(28″) The radius of FZP and FOV are restricted by Eqs. (27) and (28). Calculation results for spatial resolutions of 10 and 30 nm, in the wavelength range 1–10 Å, are shown in Fig. 9. An FOV larger than 600 μm is achievable at 30 nm spatial resolution with X-ray wavelengths of <3 Å. An FOV of >50 μm is achievable with 10 nm resolution. The total number of zones, N, is also restricted by Eq. (7) as: $$\begin{equation} N\approx{r}_N/\left(2\varDelta \right). \end{equation}$$(29) Fig. 9 Open in new tabDownload slide Limiting FZP radius as related to diffraction-limited spatial resolution and X-ray wavelength. Plot range beyond zone number N = 10 000 is omitted. Radius of field of view is assumed to be half of the FZP radius. Fig. 9 Open in new tabDownload slide Limiting FZP radius as related to diffraction-limited spatial resolution and X-ray wavelength. Plot range beyond zone number N = 10 000 is omitted. Radius of field of view is assumed to be half of the FZP radius. because of the limited FZP radius for geometrical aberration Eq. (26). Equations (22), (24) and (26) give restrictions on the maximum FOV (ra) for a given FZP to attain a diffraction-limited spatial resolution. On the other hand, for tomographic microscopy applications where rotated samples are observed, the diameter of observable sample is restricted intrinsically by the depth of focus, ∆f. Therefore, off-axis aberration-free CT measurements are possible whenever the effective FOV is larger than the depth of focus (ra > ∆f). In a comparison of Eqs. (10), (22), (24) and (26), it is apparent that the above conditions are generally well satisfied in all X-ray regions, because of the small NA of X-ray optics. Therefore, full-field microscopy with an FZP objective can be regarded as an aberration-free system for tomographic applications in which the 3D FOV is restricted only by the depth of focus. Off-axis imaging properties with coherent beams With the FZP effectively providing off-axis aberration-free optics for X-ray nano-CT, the whole effective FOV can be regarded as an isoplanatic region where the relationships between an object at object plane, t(x), and intensity at the conjugate plane, I(x), are described by a convolution with a probe function, p(x) [69]: $$\begin{equation} I(x)={\left|t(x)\right|}^2\bigotimes{\left|p(x)\right|}^2\ \mathrm{for}\ \mathrm{incoherent}\ \mathrm{illumination} \end{equation}$$(30a) $$\begin{equation} I(x)={\left|t(x)\bigotimes p(x)\right|}^2\ \mathrm{for}\ \mathrm{coherent}\ \mathrm{illumination}. \end{equation}$$(30b) In practice, however, this is not the case, especially with coherent illumination. The objective generally has a finite aperture that may deteriorate the intensity distribution at the image plane due to the interference. Geometrical off-axis aberration (Section Aberrations of X-ray FZP optics) does not take this into account, and it may affect the image especially in off-axis conditions, with image properties being much different in the FOV. As shown in Fig. 10, edge-enhanced fringe noise (‘ringing’) and image contrast modulation are remarkable, especially in the peripheral FOV (top in the figure). Below we consider why such noise occurs and how it can be reduced. Fig. 10 Open in new tabDownload slide Full-field X-ray microscope image obtained with a conventional FZP objective. The sample is a tantalum test chart that is 0.5 μm thick. Flat-field correction was applied. X-ray energy was 8 keV. Image contrast is obviously modulated by edge enhancement and fringe noise. Fig. 10 Open in new tabDownload slide Full-field X-ray microscope image obtained with a conventional FZP objective. The sample is a tantalum test chart that is 0.5 μm thick. Flat-field correction was applied. X-ray energy was 8 keV. Image contrast is obviously modulated by edge enhancement and fringe noise. If the off-axis distance of an objective with a radius of r0 is r1 (r1 ≤ r0), then the difference in optical path length from the optical axis is xr1/f. Therefore, Eq. (30a) and Eq. (30b) can be rewritten as: $$\begin{eqnarray} I\left(x;{r}_1\right)={\left|t(x)\right|}^2\bigotimes{\left|p(x)\exp \left\{-\frac{2\pi i{r}_1x}{f\lambda}\right\}\right|}^2\nonumber \\=I\left(x;0\right)\ \mathrm{for}\ \mathrm{incoherent}\ \mathrm{illumination} \end{eqnarray}$$(30a’) $$\begin{eqnarray} I\left(x;{r}_1\right)={\left|t(x)\bigotimes p(x)\exp \left\{-2\pi i{r}_1x/\left(f\lambda \right)\right\}\right|}^2 \nonumber \\ \ne I\left(x;0\right).\mathrm{for}\ \mathrm{coherent}\ \mathrm{illumination} \end{eqnarray}$$(30b’) In the case of incoherent illumination, where the off-axis components cancel each other, the relation I(x;r1) = I(x;0) is always valid. On the other hand, under coherent illumination, the off-axis component remains and I(x;r) ≠ I(x;0) if r ≠ 0. The isoplanatic condition is thus satisfied only when the illumination is completely incoherent. In the case of partially coherent illumination, the off-axis component remains, depending on the degree of coherence. With FZP optics, which require diffraction-order sorting, incoherent illumination is unavailable, so the off-axis condition with coherent illumination (Eq. (30b')) must be considered. Here we estimate how image contrast modulation is caused by off-axis conditions. Equation 30b can be rewritten by convolution theory as: $$\begin{equation} V(r)=T(r)P(r), \end{equation}$$(31) where V(r), T(r) and P(r) are the Fourier transforms of the amplitude at the image plane v(x), object t(x) and probe function p(x), respectively. P(r) corresponds to the pupil function of the objective, which can be redefined as P(r) = P0(r)rect(r/r0) where P(r) = P0(r) for |r| < r0 and P(r) = 0 for |r| > r0. Equation (31) is rewritten as: $$\begin{equation} V(r)=T(r){P}_0 rect\ \left(r/{r}_0\right). \end{equation}$$(32) Here, due to the nature of the pupil function, P0 has a more gradual change than T(r) and is regarded as constant when the objective has a uniform efficiency distribution. The inverse Fourier transform of Eq. (32) is given by: $$\begin{eqnarray} v(x)={\int}_{-\infty}^{\infty }T(r){P}_0(r)\ rect\left(r/{r}_0\right)\mathit{\exp} \nonumber \\ \left\{-2\pi irx/ f\lambda \right\} dr={\int}_{-{r}_0}^{r_0}T(r){P}_0(r)\mathit{\exp}\left\{-2\pi irx/ f\lambda \right\} dr. \end{eqnarray}$$(33) Equation (33) describes an isoplanatic system with an objective with a pupil function of radius of r0. In the case of an off-axis optical system, when the offset distance of the objective from the axis is r1 (0 < r1 < r0), Eq. (31) can be rewritten as |$(30{\mathrm{b}}^{\prime})$|⁠: $$\begin{equation} V(r)=T(r)P\left(r\hbox{---} {r}_1\right)=T(r){P}_0 rect\left(\left(r\hbox{---} {r}_1\right)/{r}_0\right). \end{equation}$$(34) The inverse Fourier transform of Eq. (34) is given by: $$ v\left(x;{r}_1\right)={\int}_{-\infty}^{\infty }T(r){P}_0(r)\ rect\left\{\left(\frac{r-{r}_1}{r_0}\right)\right\}\ \mathit{\exp}\left\{-2\pi irx/ f\lambda \right\} dr. $$ $$\begin{equation} ={\int}_{-r0-r1}^{r0-r1}T(r){P}_0(r)\mathit{\exp}\left\{-2\pi irx/ f\lambda \right\} dr \end{equation}$$(35) By separating the isoplanatic components from the above equation, the remaining is extracted as the error component due to the off-axis condition, shown as: $$\begin{eqnarray} &&v\left(x;{r}_1\right)={\int}_{-\left(r0+r1\right)}^{r0+r1}T(r){P}_0(r) \\ \nonumber &&\mathit{\exp}\left\{-\frac{2\pi irx}{f\lambda}\right\} dr\hbox{--} {\int}_{r0-r1}^{r0+r1}T(r){P}_0(r)\mathit{\exp}\left\{-\frac{2\pi irx}{f\lambda}\right\} dr. \end{eqnarray}$$(35′) The first term on the right side of Eq. (35) can be regarded as representing axially symmetric isoplanatic optics with a pupil radius of r0 + r1; the second term is the off-axis error component. Computer simulation results are shown in Fig. 11, with the intensity profile at the image plane of a tantalum sphere (0.5 μm diameter) located in the object plane, with an X-ray energy of 8 keV, and in both coherent and incoherent illumination. Figure 11a represents the on-axis condition in which both the object and objective are aligned on the optical axis. Correct intensity distributions are recognized whether the illumination is coherent (solid line) or incoherent (broken line). Figure 11b shows the off-axis condition in which the object is located 0.8r0 from the optical axis with respect to the objective radius r0. The incoherent illumination condition exhibits almost the same intensity distribution as in Fig. 11a, confirming that the system satisfies the isoplanatic condition. On the other hand, in the case of coherent illumination, the intensity distribution obviously deteriorates with the fringes and the edge-enhanced contrast. This simulation indicates that incoherent illumination is necessary to satisfy the isoplanatic condition in a wide FOV. However, since incoherent illumination is not available for FZP optics requiring diffraction-order sorting, a certain degree of image contrast disturbance is unavoidable. Such image disturbance can be a ‘fatal’ error in quantitative CT. As a method of reducing this effect, the installation of a beam diffuser to reduce coherence is well known [71, 72]. However, this causes an overlap of unwanted diffraction orders, resulting in a decrease in image contrast and sensitivity. Fig. 11 Open in new tabDownload slide Computer simulation considering the relationship between sample position at the object plane and image contrast at the image plane. A tantalum sphere with 0.5 μm in diameter was assumed to be the object. Photon energy is 8 keV, with an object transmittance of 0.87. Solid and broken lines in intensity profiles represent illumination by coherent and incoherent beams, respectively. (a) and (c) Object located on axis; (b) and (d) object located off-axis at a distance of 0.8 times the radius, r, of the objective. A conventional lens was used as the objective in (a) and (b), and an apodization lens in (c) and (d). Fig. 11 Open in new tabDownload slide Computer simulation considering the relationship between sample position at the object plane and image contrast at the image plane. A tantalum sphere with 0.5 μm in diameter was assumed to be the object. Photon energy is 8 keV, with an object transmittance of 0.87. Solid and broken lines in intensity profiles represent illumination by coherent and incoherent beams, respectively. (a) and (c) Object located on axis; (b) and (d) object located off-axis at a distance of 0.8 times the radius, r, of the objective. A conventional lens was used as the objective in (a) and (b), and an apodization lens in (c) and (d). Apodization filtering is known to provide partial suppression of secondary maxima of the point-spread function by modification of the aperture function [69, 73], which reduces ringing and edge-enhanced contrast effectively even with coherent illumination [74]. Apodization lenses for the visible light region are designed such that transmittance decreases around the marginal region of the aperture as shown in Fig. 12 [75]. In the case of a normal lens, pupil function of the objective, P0 (Eq. (35)), is constant, while an apodization lens has a shape such that P0 approaches zero at both ends, as in a Gaussian shape. The value of the second term on the right side of Eq. (35), which was considered as the off-axis error component, then becomes small, reducing the effect of ringing and edge-enhanced contrast. Isoplanatic conditions are thus relaxed by employing apodization. The profile of the pupil function is similar to a Gaussian curve, and the probe function corresponding to the Fourier transform of the pupil function is also a Gaussian-like shape. This optical system is thus termed ‘Gaussian beam optics’. Fig. 12 Open in new tabDownload slide Comparison between a conventional lens (FZP) and an apodization lens (A-FZP). FT and iFT represent Fourier and inverse Fourier transforms, respectively. The top, middle, and bottom columns display the overviews, the two-dimensional aperture functions and their cross-sectional profiles, and the point-spread functions corresponding to the Fourier transform of the aperture functions, respectively. Fig. 12 Open in new tabDownload slide Comparison between a conventional lens (FZP) and an apodization lens (A-FZP). FT and iFT represent Fourier and inverse Fourier transforms, respectively. The top, middle, and bottom columns display the overviews, the two-dimensional aperture functions and their cross-sectional profiles, and the point-spread functions corresponding to the Fourier transform of the aperture functions, respectively. Figure 11c and d illustrates the use of an apodization lens as the objective, compared with a normal lens in Fig. 11a and b. While strong ringing and edge-enhanced contrast are evident under off-axis conditions with the normal lens in Fig. 11b, they are barely visibly with the apodization lens in Fig. 11d, which represents almost the same image contrast between coherent and incoherent illumination. The apodization lens thus satisfies isoplanatic conditions regardless of coherence, and it is of interest to determine whether such a lens can be realized with FZP optics. In the hard X-ray region, where the effect of absorption is small, Eq. (14) indicates that FZP diffraction efficiency increases with respect to the zone thickness, when it is smaller than the optimized thickness which satisfies the condition kδt = π. Therefore, it is possible to create an apodized aperture by gradually reducing zone thickness from the center of the FZP pattern to its margins [73, 74]. A comparison between conventional and apodization FZPs is given in Fig. 12. As the conventional FZP has a uniform zone depth, its pupil function is defined as a definite circle with a rectangle profile (left side of Fig. 12). In contrast, an apodization FZP (A-FZP) has a zone structure in which the zone thickness decreases toward the outer region (right side of Fig. 12). Diffraction efficiency then gradually decreases in the outer region, realizing an apodized aperture. Such a zone structure can be realized by using the ‘micro-loading’ effect, which decreases etching depth with decreasing pattern width [76]. Although the diffraction efficiency of the marginal region is less than that of the central region, it has been experimentally confirmed that almost diffraction-limited spatial resolution can be obtained [73]. A comparison of nano-CT images obtained with a conventional FZP and an A-FZP are shown in Fig. 13a and b, for a sample of Kilabo meteorite. In CT images obtained with a conventional FZP (Fig. 13a), image contrast modulation such as edge enhancement, streak noise and contrast unevenness is obvious, while in images obtained with the A-FZP (Fig. 13b), such noises are remarkably suppressed. The linear absorption coefficient (LAC) histograms of these CT images are shown in Fig. 13c and d. Comparing the histograms, Fig. 13c shows a single broad peak. Originally, this peak includes two different mineral peaks, but the sensitivity is not enough (histogram peak width is too broad) to separate them. On the contrary, Fig. 13d clearly shows two separated sharp peaks representing two different minerals. These results show that the application of Gaussian beam optics thus provides remarkable noise suppression and improves the sensitivity of X-ray nano-CT. Fig. 13 Open in new tabDownload slide X-ray nano-CT images of the Kilabo meteorite and their linear absorption coefficient (LAC) histograms. (a) and (c) Obtained with a conventional FZP objective; (b) and (d) obtained with an A-FZP objective. X-ray energy of 8 keV; exposure time of 500 ms; voxel size of 84 nm; 1800 projections over a 180° rotation. Fig. 13 Open in new tabDownload slide X-ray nano-CT images of the Kilabo meteorite and their linear absorption coefficient (LAC) histograms. (a) and (c) Obtained with a conventional FZP objective; (b) and (d) obtained with an A-FZP objective. X-ray energy of 8 keV; exposure time of 500 ms; voxel size of 84 nm; 1800 projections over a 180° rotation. A-FZP also has another advantage. Whereas conventional FZP has a uniform zone depth, A-FZP can be designed to have thinner and finer zone structures in the outer region or a deeper zone in the central region (Fig. 14). The former enables higher spatial resolution, while the latter provides higher efficiency in the high-energy X-ray region than the conventional ones [73]. It is technically difficult to fabricate conventional FZPs exceeding the limiting aspect ratio (Section Aberrations of X-ray FZP optics), but A-FZP can realize a performance beyond the technical limit without exceeding the limiting aspect ratio. Fig. 14 Open in new tabDownload slide Comparison of cross-sections between (a) conventional FZP and (b) apodization FZP. rN, ΔrN, and t represent the radius, the outermost zone width and the thickness of the FZPs. Subscripts ‘1’ and ‘2’ indicate conventional and apodization FZP, respectively. Fig. 14 Open in new tabDownload slide Comparison of cross-sections between (a) conventional FZP and (b) apodization FZP. rN, ΔrN, and t represent the radius, the outermost zone width and the thickness of the FZPs. Subscripts ‘1’ and ‘2’ indicate conventional and apodization FZP, respectively. Full-field X-ray microscopy for nano-CT As described in Section X-ray micro-imaging (micro-CT), the spatial resolution of 3D CT with projection optics is restricted to ≥1 μm. X-ray nano-CT based on full-field X-ray microscopy was developed to overcome this limitation, first in the soft X-ray region and then in the hard X-ray region [78, 79]. For high-resolution CT applications, the FOV is ultimately limited by the depth of focus (Section Resolution of X-ray nano-imaging). X-ray optical systems are required to be aberration-free in such applications, as satisfied by FZPs (Section Fresnel zone plate for X-ray nano-imaging and nano-CT). Therefore, FZPs are widely used as the objective for full-field X-ray microscopes for nano-CT applications [80–86]. It is technically difficult to fabricate deep- and fine-zoned structures with the high aspect ratios required for the high-energy X-ray region (Section Principles and properties of FZP for X-rays), so FZPs are generally used with energies of ≤10 keV. However, applications in the high-energy region (ca. 30 keV) have also been attempted [74, 87, 88]. Some examples of applications of nano-CT with full-field X-ray microscopy at SPring-8 are described in the following section. X-ray nano-CT at SPring-8 Full-field X-ray microscopes are currently open for users at the Beamlines (BL)20XU, 37XU and 47XU of SPring-8. An in-vacuum-type undulator is used as the X-ray source. The undulator radiation is monochromatized to ∆λ/λ ≈ 10−4 by a liquid nitrogen-cooled silicon 111 double-crystal monochromator. This monochromaticity is sufficient for the chromatic aberration conditions of the FZP used (N ≈ 103–104), as expressed in Eq. (13). Figure 15 illustrates a typical SPring-8 X-ray nano-CT setup. The system comprises a hollow-cone beam illumination system with a condenser zone plate (CZP), sample stages, an objective (FZP) and an image detector. The Zernike phase contrast method, which is widely used for high-sensitivity imaging in visible light and electron microscopy [69, 89], is applied by inserting a phase plate at the back-focal plane of the objective. The sample is placed on the object plane and the detector at the image plane. Magnification of the optical system, M, is derived from the Gaussian lens formula: $$\begin{equation} L=\frac{{\left(M+1\right)}^2}{M}f, \end{equation}$$(36) Fig. 15 Open in new tabDownload slide Schematic diagram of an X-ray nano-CT system based on a full-field X-ray microscope at SPring-8. Fig. 15 Open in new tabDownload slide Schematic diagram of an X-ray nano-CT system based on a full-field X-ray microscope at SPring-8. where L is the distance between the sample and detector and can be approximated as L ≈ Mf for large M values. As the focal length of an FZP is proportional to X-ray energy (Section Principles and properties of FZP for X-rays), a large distance is required for high magnification in the high-energy X-ray region, although in practice L is limited by physical constraints such as the size of the experimental hutch. In the case of BL47XU, where L is limited to being <7 m, M is typically 30–70 for X-ray energies of ~10 keV. Higher magnifications (M ≈ 270) are achievable at BL37XU, where L ≈ 27 m. High-energy X-ray nano-CT has been developed at BL20XU by using two experimental hutches separated by 165 m, with M values of 150–300 achievable at X-ray energies of 20–37.7 keV. By using a CZP (described below), a photon flux of 1012–1013 photons s−1 is now available at the object plane with an FOV diameter of 100 μm. Samples are illuminated with a hollow-cone beam. Tapered capillary and CZPs with concentric gratings of even pitch (Fig. 16a) may be used as condensers to provide a hollow-cone beam [82, 86]. However, they cannot produce a uniform intensity distribution in the FOV, as they give extremely large intensities only in the paraxial region [90]. We have employed a condenser arranged with multiple linear diffraction gratings, called a ‘beam-shaping CZP’ (BS-CZP), as shown in Fig. 16b and c [85, 91–93]. The first-order diffracted beam from each diffraction grating segment with an even pitch in the CZP is collected in an arbitrary region on the optical axis. Hollow-cone illumination is performed by placing the sample at the circle of minimum confusion. As the SR X-ray beam is coherent, speckle noise appears in the image field. To generate a uniform flat field with no such noise, the CZP is rotated on its optical axis in synchronization with exposure [90]. Fig. 16 Open in new tabDownload slide Condenser zone plate (CZP) with schematic drawings of zone pattern (left) and focusing properties (right) of: (a) CZP with concentric grating of even pitch; (b) beam-shaping CZP with single pitch; (c) beam-shaping CZP with multiple pitches; and (d) an FZP. Fig. 16 Open in new tabDownload slide Condenser zone plate (CZP) with schematic drawings of zone pattern (left) and focusing properties (right) of: (a) CZP with concentric grating of even pitch; (b) beam-shaping CZP with single pitch; (c) beam-shaping CZP with multiple pitches; and (d) an FZP. An A-FZP is used as the objective. The zone pattern of the FZP was drawn with tantalum on a silicon carbide membrane. Since the CZP and FZP are diffractive optics, diffraction-order sorting is needed to select only the first-order diffraction and to cut out unwanted diffraction orders (e.g. 0th, higher and negative orders). Beam stops and pinholes are installed in the system for this purpose as shown in Fig. 15. A visible light conversion-type sCMOS camera is used as the image detector. Powdered P43 (Gd2O2S:Tb) and GAGG (Gd3Al2Ga3O12:Ce) and LuAG (Lu3Al5O12:Ce) single crystals are used as scintillators as appropriate for conditions such as X-ray energy, magnification and measurement speed. Stage accuracy and drift during measurement may reduce spatial resolution. For example, tomographic measurements require the sample rotation stage to have a higher axial wobbling accuracy than the spatial resolution. We employ a slide-guide bearing-type rotation stage with which the wobbling accuracy is measured to be <±70 nm per 360° rotation. Stage drifts due to the temperature change during measurements make accurate CT imaging difficult or impossible. In the case of BL47XU, temperature change inside the experimental hutch is controlled to ~ ± 0.1° d−1 by the air conditioning of the facility, with stage drift during a typical CT measurement of ~10 min being <20 nm. Tomographic reconstruction from the transmitted X-ray image raw dataset involves the convolution back-projection method [94]. Three-dimensional CT measurements are routinely possible with an accuracy of <100 nm without any data correction. Measurement parameters vary with X-ray energy and purpose but, for BL47XU, typically include a pixel size of 30–70 nm with a 2048 × 2048 pixel format and a CT scan time of 3–10 min. Spatial resolution of full-field microscopes The spatial resolution limit for a periodic structure placed at the center of the FOV, ∆S, is expressed approximately as: $$\begin{equation} {\varDelta}_S=\lambda /\left({NA}_I+{NA}_O\right) \end{equation}$$(37) where NAI and NAO are the numerical apertures of the illumination system and objective, respectively [77]. In the case of annular hollow-cone illumination, the FOV of bright-field images is restricted by the acceptance of objective, expressed as: $$\begin{equation} FOV\le 2{r}_O\left(1\hbox{---} {NA}_I/{NA}_O\right), \end{equation}$$(38) where rO is the radius of the objective. The FOV is also limited if the FZP is used as the objective, when overlap of the direct beam at the image plane must be considered through the following inequality: $$\begin{equation} FOV\le 2{r}_O{NA}_I/{NA}_O. \end{equation}$$(39) If zone plates are used for both condenser and objective, the relationship between the outermost zone width and numerical aperture (d = λ/(2NA); Eqs. (37)–(39)) can be rewritten as: $$\begin{equation} 2/{\varDelta}_S=1/{d}_I+1/{d}_O \end{equation}$$(37′) $$\begin{equation} FOV\le 2{r}_O\left(1\hbox{---} {d}_O/{d}_I\right)\kern2.75em \left({d}_O/{d}_I>0.5\right) \end{equation}$$(38′) $$\begin{equation} FOV\le 2{r}_O{d}_O/{d}_I\kern2em \left({d}_O/{d}_I<0.5\right) \end{equation}$$(39′) where dI and dO represent the half-pattern pitch of the CZP and outermost zone width of the FZP objective, respectively. Equations (37)–(39) (or (37′)–(39′)) indicate that optimal spatial resolution is achieved when NAI = NAO (dI = dO), with reducing FOV to be zero. However, in some nano-imaging methods, the advantage of full-field X-ray microscopy with an FZP objective is a wide FOV and, more specifically, a wide 3D FOV (Section Fresnel zone plate for X-ray nano-imaging and nano-CT). It is therefore reasonable to consider the optimum condition is satisfied by maximizing the FOV/∆S ratio. This condition is satisfied when NAI = 0.5NAO (dI = 2dO). In the case of hollow-cone illumination, optimal optical conditions are attained when the NA value of the illumination is one half that of the objective, and, based on Eqs. (37′)–(39′), the spatial resolution is ∆S = 4/3dO and FOV = rO, when dI = 2dO. Measured spatial resolution X-ray images of the tantalum test chart (ATN/XRESO-20; NTT-AT) obtained at BL37XU are shown in Fig. 17. An FZP outermost zone width of 35 nm was used as an objective. A BS-CZP with a half-pitch of 100 nm was used to produce hollow-cone illumination. Although the innermost region of the test chart, with 20 nm line and 20 nm space (40 nm pitch), is barely resolved, lines wider than ~25 nm (~50 nm pitch) are clearly observed. This is reasonably consistent with the expected spatial resolution of 51.9 nm based on Eq. (37′). A Zernike phase contrast nano-CT image of a diatom fossil is shown in Fig. 18 as a resolution test for 3D imaging. Fine hole structures of diameter > 70 nm are clearly observed. Fig. 17 Open in new tabDownload slide X-ray image of a tantalum test chart (ATN/XRESO-20; NTT-AT). (a) Whole image; (b) magnified view of the central region of (a). X-ray energy of 6 keV; exposure time of 1000 s; CZP half pitch of dI = 100 nm; outermost FZP zone width of dO = 35 nm. Fig. 17 Open in new tabDownload slide X-ray image of a tantalum test chart (ATN/XRESO-20; NTT-AT). (a) Whole image; (b) magnified view of the central region of (a). X-ray energy of 6 keV; exposure time of 1000 s; CZP half pitch of dI = 100 nm; outermost FZP zone width of dO = 35 nm. Fig. 18 Open in new tabDownload slide X-ray Zernike phase contrast nano-CT images of a diatom fossil. X-ray energy of 8 keV; voxel size of 37 nm; exposure time of 0.5 s; 1800 images over a 180° CT scan; scan time of 15 min. Fig. 18 Open in new tabDownload slide X-ray Zernike phase contrast nano-CT images of a diatom fossil. X-ray energy of 8 keV; voxel size of 37 nm; exposure time of 0.5 s; 1800 images over a 180° CT scan; scan time of 15 min. Applications X-ray nano-CT is now widely used in the medical, biological, material science, mineralogical, astronomy and industrial fields, with some applications at SPring-8 being described here as examples. Spectroscopic nano-CT Nano-CT is frequently used in spectroscopic applications such as X-ray absorption fine-structure (XAFS) and near-edge structure (XANES) analyses to investigate nm-order distributions of chemical states [95–97]. Figure 19 shows nano-CT images of tiny particles from the Itokawa asteroid. Tomographic measurements were performed at two different monochromatic X-ray energies (7 and 8 keV; below and above the Fe K-absorption edge). Mineral compositions were non-destructively determined by quantitative analysis of LACs at the two X-ray energies [98, 99]. These 3D CT data revealed that Itokawa has a mineral composition very similar to that of the LL4–6 ordinary chondrites found on Earth’s surface. Fig. 19 Open in new tabDownload slide 3D CT images showing mineral phase distributions in a tiny particle from asteroid Itokawa collected by the Hayabusa spacecraft. Fig. 19 Open in new tabDownload slide 3D CT images showing mineral phase distributions in a tiny particle from asteroid Itokawa collected by the Hayabusa spacecraft. Multiscale CT Multiscale 2D imaging is an easily adopted technique to switch between wide-field whole and high-resolution ROI observations, by changing objectives with different magnifications on an optical microscope. In 3D imaging, however, it is not so easy to perform such measurement non-destructively because the FOV in the thickness as well as the width direction is also different between the wide-field and the high-resolution observations. This requires the probe to have two competing characteristics: high penetration power for a whole object and strong interaction with matter to detect tiny image contrasts of the ROI. High-energy X-ray phase contrast nano-CT is a promising technique for this. X-ray nano-CT at BL20XU is dedicated to the high-energy X-ray region of 20–37.7 keV and is now frequently used for multiscale 3D imaging. The system comprises a micro-CT (Fig. 1) for macro imaging and a Zernike phase contrast nano-CT (Fig. 15) for fine imaging of an ROI. Besides the advantage of nm-scale non-destructive 3D imaging inside bulky objects, this system also makes 4D nano-CT time-resolved, in situ, ex situ and operando observations much easier in combination with micro-CT measurements. As an example, Fig. 20 shows the non-destructive observation of interior fatigue cracks in a titanium alloy, Ti-6Al-4 V, which has a dual-phase microstructure with α (hexagonal close-packed lattice) and β (body centered cubic lattice) phases. In spite of their excellent mechanical properties, titanium alloys have a critical problem that may cause internal fatigue fracture in very high-cycle (>107 cycles) applications [100]. In this example, the entire body of a sample of 0.45 mm diameter was first studied in micro-CT mode (Fig. 20a and b). The position of the internal crack was determined from the micro-CT data (surrounded with yellow lines in Fig. 20a and b), and the identified region was imaged non-destructively in nano-CT mode (Fig. 20c and d). The 3D positional relationship between the (α + β) dual-phase microstructure and initial cracks is clearly visible. It is also possible to reveal their propagation process by employing an in situ measurement method. Fig. 20 Open in new tabDownload slide Multiscale-CT images of interior fatigue cracks in Ti-6Al-4 V alloy. (a) and (b) micro-CT images (voxel size of 0.5 μm; 250 ms exposure; 1800 images over 180°). (c) and (d) Zernike phase contrast nano-CT images of the region surrounded with yellow lines in (a) and (b) (voxel size of 65 nm; 1 s exposure, 1800 images over 180°; X-ray energy of 30 keV). Fig. 20 Open in new tabDownload slide Multiscale-CT images of interior fatigue cracks in Ti-6Al-4 V alloy. (a) and (b) micro-CT images (voxel size of 0.5 μm; 250 ms exposure; 1800 images over 180°). (c) and (d) Zernike phase contrast nano-CT images of the region surrounded with yellow lines in (a) and (b) (voxel size of 65 nm; 1 s exposure, 1800 images over 180°; X-ray energy of 30 keV). Summary Details and the current status of SR X-ray imaging, especially X-ray micro- and nano-CT, have been described. Spatial resolutions of <100 nm have been achieved with X-ray nano-CT. Although spatial resolutions achieved with measurement techniques other than those using X-rays may be superior to that of X-ray imaging, the characteristics of X-ray probes such as high penetrating power and high FOV/resolution ratios provide information-rich 3D data. Furthermore, by using high-intensity SR X-rays, the FOV/resolution/time ratio can be increased to enable 4D observations. This makes X-ray nano-CT a unique technique. It is not always advantageous to improve spatial resolution as much as possible, because the effective FOV is limited by the depth of focus, which is proportional to the square of the resolution and becomes extremely small at high resolutions as shown in Eq. (3). Therefore, a reasonable goal for the development of X-ray nano-CT is a stable and robust operation with a spatial resolution of 10–30 nm. Ultra-high-resolution ‘monumental’ measurements, which require sample processing and subdivision, can be left to other methods. The role of X-ray micro- and nano-CT should be to specialize in 3D–4D non-destructive analyses to reveal features of unaltered samples. A possible direction for further development would be to seek a dynamic approach to the complex and dynamic phenomena through the nano-macro-scale structural analyses of materials, both inanimate and living. 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TI - Recent progress in synchrotron radiation 3D–4D nano-imaging based on X-ray full-field microscopy
JF - Microscopy
DO - 10.1093/jmicro/dfaa022
DA - 2020-10-30
UR - https://www.deepdyve.com/lp/oxford-university-press/recent-progress-in-synchrotron-radiation-3d-4d-nano-imaging-based-on-x-T7uQfNICnV
SP - 259
EP - 279
VL - 69
IS - 5
DP - DeepDyve
ER -