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Zone-Plate X-Ray Microscopy

  • Chris JacobsenEmail author
  • Malcolm Howells
  • Tony Warwick
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

Fresnel zone plates are the most commonly used optic in x-ray microscopes. Following a short discussion of historical developments, the properties of zone plates are outlined, along with the microscope systems that employ them. A number of applications of x-ray microscopes are then surveyed, including in biology, environmental science, and materials science.

x-ray microscopy x-ray tomography Fresnel zone plates x-ray fluorescence trace element mapping cryo microscopy 

23.1 Background

In the 1949 issue of Scientific American, an article by Stanford physicist Paul Kirkpatrick, The x-ray microscope  [23.1], was described by the editors as follows:

It would be a big improvement on microscopes using light or electrons, for x-rays combine short wavelengths, giving fine resolution, and penetration. The main problems standing in the way have now been solved.

Now, with the perspective of a half century, we might change ‘‘improvement on'' to ‘‘complement to,'' and say that further problems were solved after 1949, but here in essence is the character of x-ray microscopes.

In this chapter, we outline some of the properties of x-ray microscope systems in operation today, and highlight some of their present applications. We will not discuss the history of x-ray microscopes prior to about 1975, but instead refer the reader to a series of conference proceedings known as X-Ray Optics and X-Ray Microanalysis, which began in 1956. Originally, these had valuable material on x-ray microscopy, but this diminished after about 1970. The first five had proceedings published: [23.2], Stockholm (1959) [23.3], Stanford (1962) [23.4], Orsay (1965) [23.5], and Tübingen (1968) [23.6]. We also recommend the historical perspectives by A. Baez [23.7, 23.8] and the book by Cosslett and Nixon [23.9]. There is a recognizable thread of continuity between the status of the field today and efforts that began slowly in the 1970s [23.10, 23.11, 23.12] and blossomed with the availability of synchrotron light sources and nanofabrication technologies; this thread can be traced in part via the proceedings of another conference series that began in 1984 [23.13] and has continued until today [23.14, 23.15, 23.16, 23.17, 23.18, 23.19, 23.20, 23.21, 23.22, 23.23, 23.24].

Zone-plate x-ray microscopes now exist at roughly two dozen international synchrotron radiation research centers; a listing of all such centers can be found at lightsources.org. In addition, commercial lab-based instruments are becoming increasingly available. Three types of x-ray microscopes are in especially widespread use:
  1. 1.

    Transmission x-ray microscopes ( s) specialize in the rapid acquisition of 2-D images using high-flux sources, and in the collection of tilt sequences of projection images for 3-D imaging by tomography (spectromicroscopy applications are also emerging [23.25]).

     
  2. 2.

    Scanning transmission x-ray microscopes ( s) specialize in the acquisition of reduced-dose images and point spectra with high energy resolution for elemental and chemical-state mapping, and require high source brightness.

     
  3. 3.

    Scanning fluorescence x-ray microprobes ( s) are similar to STXMs except that fluorescence x-rays are collected by energy-resolving detectors for trace element mapping.

     
All three approaches are now working below \({\mathrm{100}}\,{\mathrm{nm}}\) resolution, to the point of reaching \(10{-}12\,{\mathrm{nm}}\) resolution in some demonstrations [23.26, 23.27] and \({\mathrm{5}}\,{\mathrm{nm}}\) in the case of ptychography [23.28], which will be discussed in Sect. 23.3.1, Ptychography Layout. While many of the new technical developments continue to be pursued by specialists in x-ray optics and microscopy, much of present-day activity comes from scientists in other fields of research who are using x-ray microscopes to address their particular questions. This chapter is mainly aimed at scientists from the latter group, as well as those from the other communities represented in the content of this series of books.

23.1.1 X-Ray Interactions

A microscope requires illumination, magnification, and contrast. The characteristics of x-ray interactions with matter affect all three. In Fig. 23.1, we show the cross section [23.29] for photoelectric absorption, coherent (elastic or Thomson) scattering, and incoherent (inelastic or Compton) scattering for carbon. Below \({\mathrm{10}}\,{\mathrm{keV}}\), absorption dominates, so multiple x-ray scattering is usually not of concern (an x-ray is much more likely to be absorbed following any scattering event than to be scattered again), nor is inelastic scattering. However, what is not evident in Fig. 23.1 is the fact that the propagation of x-rays in materials can also include refractive effects, and in fact it was Einstein [23.30] who first pointed out that the refractive index is slightly less than unity. The x-ray refractive index for a wave forward-propagated as \(\exp[-\mathrm{i}\left(k\tilde{n}x-\omega\tau\right)]\) is often written as
$$\tilde{n}=1-\delta-\mathrm{i}\beta=1-\alpha\lambda^{2}(f_{1}+\mathrm{i}f_{2})\;,$$
(23.1)
where \(\delta\) represents the phase-shifting part of the refractive index and \(\beta\) represents absorption according to a linear coefficient \(\mu=4\uppi\beta/\lambda\) in the Lambert–Beer law
$$I=I_{0}\exp(-\mu t)$$
(23.2)
with \(t\) representing the thickness of the absorbing material along the beam direction. The latter form of (23.1) uses \(\alpha=n_{\text{a}}r_{\text{e}}/(2\uppi)\), where \(n_{\text{a}}=\rho N_{\text{A}}/A\) gives the number density of atoms, \(r_{e}={\mathrm{2.82\times 10^{-15}}}\,{\mathrm{m}}\) is the classical radius of the electron, and \(\left(f_{1}+\mathrm{i}f_{2}\right)\) represents the frequency-dependent oscillator strength of an atom. This oscillator strength \(\left(f_{1}+if_{2}\right)\) has been tabulated with very good absolute accuracy [23.31] for all of the naturally occurring elements over an energy range of \(10{-}30000\,{\mathrm{eV}}\) (Fig. 23.2). In examining Fig. 23.2, two features immediately jump out: \(f_{1}\) is somewhat constant except near absorption edges, so the thickness \(t_{\uppi}=\lambda/2\delta=1/\left(2\alpha\lambda f_{1}\right)\) needed to provide a phase advance \(\exp(\mathrm{i}k\delta t_{\pi})\) equal to \(\uppi\) increases as \(\lambda^{-1}\). Because \(f_{2}\) scales as \(E^{-2}\) or \(\lambda^{2}\), the thickness \(1/\mu=1/\left(4\uppi\alpha\lambda f_{2}\right)\) that produces an attenuation of \(1/e\) increases as \(E^{3}\) or \(\lambda^{-3}\). As a result, phase contrast becomes the dominant contrast mechanism as one goes to shorter wavelengths [23.32].
Fig. 23.1

X-ray interaction cross sections in carbon (\({\mathrm{1}}\,{\mathrm{barn}}={\mathrm{10^{-24}}}\,{\mathrm{cm^{2}}}\)). At energies below about \({\mathrm{10}}\,{\mathrm{keV}}\), absorption dominates, so that images are free from the complications of multiple scattering. Data from [23.29] and [23.31]

Much of modern x-ray microscopy centers on the exploitation of x-ray absorption edges. These absorption edges arise when the x-ray photon reaches the threshold energy needed to completely remove an electron from an inner-shell orbital. The energy at which this occurs is approximately given by the Bohr model as \(E_{n}=({\mathrm{13.6}}\,{\mathrm{eV}})(Z-z_{\text{shield}})^{2}/n^{2}\), where \(Z\) is the atomic number, \(z_{\text{shield}}\) approximates the partial screening of the nucleus' charge by other inner-shell electrons (\(z_{\text{shield}}\simeq 1\) for K edges), and \(n\) is the principal quantum number (\(n=1\) for K edges, \(n=2\) for L edges, and so on). This produces the step-like rise in the cross section for photoelectric absorption that can be seen in the plots of \(f_{2}\) in Fig. 23.2. If one takes one image \(I_{1}\) at an energy \(E_{1}\) just below an element's absorption edge where the incident flux is \(I_{01}\), and a similar image \(I_{2}\) at an energy just above an absorption edge, one can recover the mass per area \(m_{x}/A\) of the element \(x\) from [23.35]
$$\frac{m_{x}}{A}=\rho\frac{\left(E_{1}/E_{2}\right)^{3}\ln\left(I_{1}/I_{01}\right)-\ln\left(I_{2}/I_{02}\right)}{\mu_{2}-\mu_{1}\left(E_{1}/E_{2}\right)^{3}}\;.$$
(23.3)
This approach works well for mass concentrations greater than about \({\mathrm{1}}\%\). Another way in which x-ray absorption edges are exploited is by means of the water window . At x-ray energies between the carbon and oxygen absorption edges of 290 and \({\mathrm{540}}\,{\mathrm{eV}}\), respectively, organic materials show strong absorption contrast, while water layers up to several \(\mathrm{{\upmu}m}\) thick are reasonably transmissive [23.36]; this is particularly valuable for imaging hydrated biological and environmental science specimens.
Fig. 23.2

The frequency-dependent oscillator strength \((f_{1}+\mathrm{i}f_{2})\) for carbon and gold. At x-ray absorption edges (such as \({\mathrm{290}}\,{\mathrm{eV}}\) for carbon), \(f_{2}\) (dashed lines) has step increments, while \(f_{1}\) (solid lines) undergoes anomalous dispersion resonances. Data from [23.31] and [23.33]

Fig. 23.3a,b

Energies (a) and fluorescence yields (b) for K and L edge emission. Data from [23.34]

For those elements which have absorption edges below the energy of incident x-rays so that inner-shell ionization occurs, the aftermath of absorption involves the emission of either a fluorescent photon or an Auger electron of characteristic energy. The energy of these fluorescent photons, and the fluorescence yield [23.34] (the fraction of events which result in fluorescence rather than Auger electron emission), are both shown in Fig. 23.3a,b. At x-ray energies below \({\mathrm{1}}\,{\mathrm{keV}}\), Auger emission dominates, and scanning photoemission microscopes ( ) use electron spectrometers to exploit these electrons for surface studies [23.37, 23.38, 23.39, 23.40], although there have also been some demonstrations of \(<{\mathrm{1}}\,{\mathrm{keV}}\) fluorescence detection in soft x-ray scanning microscopes [23.41, 23.42]. At higher energies, the fluorescence signal dominates, and detection of these characteristic x-rays provides information on the concentration of various elements in the specimen. Most scanning fluorescence x-ray microprobes () [23.43, 23.44] use energy-dispersive detectors, where the number of electron–hole pairs created by each fluorescent photon is used to measure its energy, though crystal-based wavelength-dispersive spectrometers can also be used [23.45]. Exact quantitation of the elemental concentration requires accurate knowledge of a number of factors, including the solid-angle acceptance of the detector and its quantum efficiency, and the degree to which fluorescent photons are reabsorbed in the specimen, so that in most cases, comparison is made using standards with known elemental concentration and matrix concentration similar to that of the specimen under study. When compared with electron microprobes, x-ray microprobes do not suffer from expansion of the probe beam due to electron scattering, or a large continuum background, so that the sensitivity to trace elements is often in the 100-parts-per-billion range.

Because x-ray interactions are well understood and do not involve significant complications due to multiple scattering at energies below about \({\mathrm{10}}\,{\mathrm{keV}}\), reliable predictions of image contrast can be made. If we have a normalized signal \(I_{\text{f}}\) from a feature-containing pixel and \(I_{\text{b}}\) from a background region, the signal-to-noise ratio obtained with \(N\) illuminating photons is [23.46, 23.47]
$$\mathrm{SNR}=\frac{\mathrm{Signal}}{\mathrm{Noise}}=\sqrt{\bar{n}}\frac{I_{\text{f}}-I_{\text{b}}}{\sqrt{I_{\text{f}}+I_{\text{b}}}}=\sqrt{N}\Theta\;.$$
(23.4)
where we have used the Gaussian approximation to Poisson statistics (which is quite good for \(\bar{n}\) greater than about 10) and the assumption that there are no other noise sources with significant fluctuations. The contrast parameter \(\Theta\) is different from the usual definition of contrast, due to the square root in the denominator. With this definition, the number of average incident photons \(\bar{n}\) required to see a feature with a desired signal-to-noise ratio (SNR) is given by \(\bar{n}=\left(\mathrm{SNR}\right)^{2}/\Theta^{2}\), and a common choice for the minimum detectable signal-to-noise ratio is the Rose criterion of \(\mathrm{SNR}=5\) [23.48]. Using this approach, Sayre et al showed that water-window x-ray microscopes are able to image organic specimens in micrometer-thick water layers with greatly reduced radiation dose compared to electron microscopy [23.47, 23.49]. This conclusion remains true even when modern energy-filtered electron microscopes are considered [23.50, 23.51] (Fig. 23.4). Other investigators have extended the same approach to include the effects of phase contrast [23.52, 23.53] (Fig. 23.5) and the reduction in modulation transfer at high spatial frequencies [23.54], while Kirz et al have used this approach to compare elemental mapping using both differential absorption and x-ray fluorescence [23.55, 23.56, 23.57].
Fig. 23.4

Estimated radiation dose required for imaging \({\mathrm{30}}\,{\mathrm{nm}}\) protein features as a function of ice thickness for \({\mathrm{200}}\,{\mathrm{keV}}\) electron microscopes and \({\mathrm{520}}\,{\mathrm{eV}}\) soft x-ray microscopes

23.1.2 Focusing Optics

Microscopes require focusing optics, or some other means to provide a magnified view of the object. X-rays reflect well from single refractive interfaces only at grazing angles of incidence less than a  critical angle of \(\theta_{\text{c}}=\sqrt{2\delta}\), which is typically in the range of \(10^{-5}\) for soft x-rays. (Once a particular angle has been selected, this same relationship gives a critical energy above which the reflectivity becomes low; this can be used to low-pass-filter the energy spectrum from a radiation source). While a number of labs have explored the use of axially symmetric paraboloid or hyperboloid optics [23.36, 23.58], most present efforts center on the use of two orthogonal cylindrical grazing mirrors in the Kirkpatrick–Baez geometry  [23.59]. Advantageous characteristics of these optics include their relatively long focal length (several centimeters is typical) and their low chromaticity, so that the incident beam energy can be tuned for spectroscopy without any need to adjust the focus on the specimen. Optics of this sort have achieved sub-\({\mathrm{10}}\,{\mathrm{nm}}\) focus spots [23.60, 23.61], although the profile of the focus always has some degree of tail outside the geometrical image of the source due to small figure errors and surface roughness on even the best available mirrors. Synthetic multilayer x-ray mirrors [23.62] can increase the incidence angle well beyond \(\theta_{\text{c}}\) for narrow-bandwidth radiation, and can achieve good reflection efficiencies for normal incidence reflection at photon energies below about \({\mathrm{200}}\,{\mathrm{eV}}\). This approach has seen rapid improvements due to the development of extreme ultraviolet (EUV) projection lithography at \({\mathrm{95}}\,{\mathrm{eV}}\). However, notwithstanding recent progress in mirror manufacturing, it is important to recognize that even a perfectly made Kirkpatrick–Baez (KB) mirror system still suffers from aberrations such as obliquity of field which severely restrict its use in full-field microscopes. When imaging a small source to a nanofocus spot, this is not a limitation, and KB mirrors are used with much success in x-ray microprobe applications.

Fig. 23.5

Estimated radiation dose required for imaging \({\mathrm{20}}\,{\mathrm{nm}}\)-thick protein features in various ice thicknesses as a function of x-ray energy; \({\mathrm{100}}\%\) efficient optics and detectors are assumed

When Röntgen discovered x-rays, he immediately tried to focus them using refractive lenses , but without success. The reason for this is now well known [23.63]: the focal length for a planoconvex lens with radius of curvature \(R_{\text{c}}\) is given by \(f_{\text{R}}=-R_{\text{c}}/\delta\), so that at \({\mathrm{10}}\,{\mathrm{keV}}\), a glass lens with \(R_{\text{c}}={\mathrm{1}}\,{\mathrm{cm}}\) would have a focal length of about \({\mathrm{2}}\,{\mathrm{km}}\). This does not preclude the usefulness of refractive optics, however; a series of lenses with small \(R_{\text{c}}\) can be placed together to produce a significant net focusing effect. One simple way to achieve this result in one dimension is to drill a series of holes in a solid block [23.64], and more recent work using parabolic optics has demonstrated a resolution of about \({\mathrm{100}}\,{\mathrm{nm}}\) for hard x-ray imaging [23.65], with theoretical promise for sub-\({\mathrm{10}}\,{\mathrm{nm}}\)-resolution imaging [23.66]. Because the ratio of phase shift to absorption increases with increasing x-ray energy, these optics work primarily at energies above about \({\mathrm{5}}\,{\mathrm{keV}}\), though at higher energies one will ultimately need to consider the contributions of inelastic scattering to the image due to the overall thickness of the optic. Still, this approach is of interest especially since these optics can be easily water-cooled for high-power applications.

The third way to focus x-rays is to use diffraction. While bent crystals can provide focused beams of Bragg or Laue diffracted x-rays, most work in x-ray microscopy centers on the use of microfabricated diffractive optics in the form of Fresnel zone plates. Efforts in x-ray microscopy using zone-plate optics have a long history [23.67], and x-ray Fresnel zone plates are now benefiting from a high degree of development. Apart from detailed literature that we will cite in Sect. 23.2, general reviews are available [23.68, 23.69]. Because of their popularity as high-resolution optics for x-ray microscopy, the properties of Fresnel zone plates are described in some detail in Sect. 23.2.

23.1.3 Overview and Recent Trends

We now review some of the general trends in x-ray microscopy technology and how they might affect the planning of a new x-ray microscopy program today. In this section, the number of relevant references is essentially unlimited, so we cite instead the subsections of this article where many of the references can be found.

Modern x-ray microscopy was based on the twin ideas of the water window (Sect. 23.1) and microfabricated zone-plate lenses (Sect. 23.2). These led initially to life-science experiments consisting of 2-D imaging of wet samples in room-temperature air, with naturalness as the unique capability of the technique. Early versions of both the TXM (Sect. 23.3.1, Transmission X-Ray Microscope (TXM) Layout ) and STXM (Sect. 23.3.1, Scanning Transmission X-Ray Microscope (STXM) Layout ) were capable of such imaging, and this style of x-ray microscopy has proved to be good fit to many of the needs of the polymer (Sect. 23.4.3) and environmental research communities (Sect. 23.4.2) among others. However, the room-temperature approach could not be easily adapted to 3-D imaging of radiation-sensitive samples, and this was something for which there was, and still is, considerable demand. On the other hand, the STXM, even in its earliest realizations, was ready to begin the development of spectromicroscopy (Sects. 23.3.5 and 23.4). This development has continued over the past two decades or so, and spectromicroscopy is now quite a mature field that spans a wide range of application areas.

The step that has brought 3-D imaging of biological samples to routine use is the move towards imaging of hydrated biological samples at cryogenic temperatures. For this type of sample, x-ray microscopy fills an important gap in the range of resolution values and sample thicknesses that cannot be covered by other methods. The use of cryogenic temperatures is the key step in providing enough radiation damage protection to enable tomographic (three-dimensional) imaging (see Sect. 23.3.4), since Fig. 23.5 makes it clear that high radiation doses are involved, and the sample must remain unchanged while images are acquired over a range of rotations. However, 3-D cryomicroscopy requires significant technical additions to the x-ray microscope, including the use of either a vacuum sample chamber or a gas-stream approach for keeping the sample cold, plus the mechanical devices required for recording a tilt series.

The interest in 3-D imaging is also high in materials science and engineering, where there is a similar gap in coverage by other methods and where the samples generally have a higher atomic number and have nonaqueous background materials (microcircuits for example). For these samples, the water window offers no advantage, and x-ray microscopy at higher x-ray energies allows for the examination of larger samples and has, in fact, been going on for some time. Even in biology, there are factors pushing in the same direction. Ice is equally transparent at \({\mathrm{1.5}}\,{\mathrm{keV}}\) as in the water window. Moreover, the \(1.5{-}3.0\,{\mathrm{keV}}\) region gives less absorption and more phase contrast, so there is only a moderate increase in the required radiation dose compared to the water window (Fig. 23.5). In this energy range, zone plates also have longer focal lengths (important for sample tilting) and greater depth of focus (important for 3-D reconstruction; Sect. 23.3.4, The Depth-Of-Focus Limit). Overall, multi-keV 3-D x-ray microscopy holds great promise but poses somewhat different challenges with respect to the resolution/efficiency capabilities of the zone plates and the provision of a suitable condenser.

These advances in applications are enabled by ongoing advances in the underlying technical capabilities. While there have been examples of soft x-ray zone plates with sub-\({\mathrm{20}}\,{\mathrm{nm}}\) resolution for some time, a growing number of soft x-ray microscopes are using sub-\({\mathrm{30}}\,{\mathrm{nm}}\)-resolution optics with increasing efficiency for routine investigations. The changes have been even more dramatic in the hard x-ray range (Sect. 23.2.3, Hard X-Ray Zone Plates), where a variety of fabrication advances have enabled zone plates to reach \({\mathrm{10}}\,{\mathrm{nm}}\) resolution in some demonstrations [23.27], and sub-\({\mathrm{50}}\,{\mathrm{nm}}\) resolution in routine operation. These advances have been coupled with improved condenser optics based on single-bounce capillaries (Sect. 23.2.3, Alternatives to Condenser Zone Plates) in the case of full-field imaging systems, and the use of pixelated area detectors and ptychography in the case of scanning microscopes (Sect. 23.3.1, Ptychography Layout). In addition, improved interferometric position control systems are allowing tomography to be performed at \({\mathrm{16}}\,{\mathrm{nm}}\) three-dimensional resolution [23.70].

The TXM and STXM have always been seen as complementary devices; roughly equally popular and with important advantages on both sides. We do not believe that this general perception is likely to change in the near future. The STXM will always have the advantage in trace element mapping, the ability to instantly switch from high to low magnification, to hold constant magnification even when the x-ray energy is changing, to image at close to the theoretical minimum dose, and so forth. On the other hand, the TXM is undergoing a period of technical enhancement. It has been moving into the multi-keV region and into cryomicroscopy, which together with its traditional main asset, multiplexed data collection, is strengthening its capability in tomography, which appears to us to be its natural home.

When possible, we have cited publications in widely available journals rather than conference proceedings. However, good snapshots of the field are provided at two- or three-year intervals by the series of conference proceedings dating back to 1984 (Sect. 23.1).

23.2 Fresnel Zone Plates

A Fresnel zone plate is a circular diffraction grating that can be made to focus light waves in the manner of a lens. It consists of a series of concentric, usually metal, rings alternating with circular slots. Typically the rings are about equal in width to the slots and are fabricated on a thin membrane. The design is based on the idea that, by blocking, say, the even-numbered Fresnel half-period zones, the wavelets from the remaining (odd-numbered) zones will add constructively. To see this quantitatively, consider plane-wave illumination of a zone plate that has zones indexed by \(n\) up to a value of \(N\) for the outermost zone. Individual zones will have radii \(r_{n}\), and the zone plate as a whole will have a radius \(r_{N}\), outermost zone width \(\Updelta r_{N}\), and a focal length \(f\) at wavelength \(\lambda\) (Fig. 23.6). To obtain a first-order diffracted beam in which the signals from all the open zones reinforce at the focus, we need a path difference \(\lambda\) between neighboring open zones. As shown in Fig. 23.6, the optical path APB of
$$\mathrm{APB}=\sqrt{r_{n}^{2}+f^{2}}-\frac{n\lambda}{2}$$
(23.5)
must equal \(f\). Expanding the square root and neglecting terms above fourth order, we obtain
$$\frac{n\lambda}{2}=\frac{r_{n}^{2}}{2f}-\frac{r_{n}^{4}}{8f^{3}}+\cdots$$
(23.6)
Evidently, the focusing condition is
$$r_{n}^{\text{2}}=n\lambda f$$
(23.7)
to second order and
$$r_{n}^{2}=n\lambda f+\frac{n^{2}\lambda}{4}$$
(23.8)
to fourth order. Equations (23.7) and (23.8) are true for all zones \(n\); one can substitute the outermost zone \(N\) for \(n\) when considering the entire zone plate. In view of (23.10) below, we can neglect the fourth-order (spherical aberration) term of (23.6) if the numerical aperture ( )\({}\ll 1\), which is often the case for x-ray zone plates. If the fourth-order term is significant, then the zone plate can be made according to (23.8) and will be corrected for spherical aberration, but the correction will only apply near the chosen wavelength and conjugate distances (\(\infty\) and \(f\)). If the spherical aberration correction term \(n^{2}\lambda/4\) can be neglected, we have \(r_{n}=\sqrt{n\lambda f}\), and this defines a zone plate that will focus well for a range of wavelengths, although the focal length will vary inversely with wavelength. Thus the chromatic aberration of a zone-plate lens is much larger than that of a refractive lens for visible light, and to obtain a good focus, the zone plate needs to be illuminated by monochromatic light. The required degree of monochromaticity for achieving the diffraction-limited resolution is roughly given by \(\Updelta\lambda/\lambda\leq 1/N\) [23.71].
Fig. 23.6

Geometry to calculate the radius of the \(n\)-th half-period zone of a zone plate illuminated by parallel light. The path from the \(n\)-th zone to the focus must be equal to \(f\) plus \(n/2\) wavelengths \(\lambda\)

Some useful quantities follow from the fundamental zone plate equation of (23.7). First we can take the difference between the equations for \(n\) and \(\left(n-1\right)\) to get the width \(\Updelta r_{N}\) of the outermost zone \(N\) as
$$\Updelta r_{N}=\frac{\lambda f}{2r_{N}}\;.$$
(23.9)
This allows us to conclude that all of the zones have equal area and also gives us the numerical aperture
$$\mathrm{NA}\equiv\frac{r_{N}}{f}=\frac{\lambda}{2\Updelta r_{N}}\;.$$
(23.10)
and thence the Rayleigh resolution
$$\delta_{\mathrm{Rayleigh}}=\frac{0.61\lambda}{\mathrm{NA}}=1.22\Updelta r_{N}\;.$$
(23.11)
Thus we see that a given zone plate can be specified by its radius \(r_{N}\) and outermost zone width \(\Updelta r_{N}\) from which the resolution (which is independent of wavelength), focal length, and numerical aperture at a given wavelength follow from (23.11), (23.9), and (23.10), respectively. So far we have been discussing the first-order focus, but in general, beams of all integer orders may be produced. Thus there is a zero-order (unfocused) beam, a series of positive-order converging beams with focal distance \(f/m\), and a series of negative-order diverging beams with focal distance \(-f/m\) (Fig. 23.7). In \(m\)-th order, the numerical aperture is \(m\) times larger and hence the resolution is \(m\) times smaller (better) than in first order. As we will see, if the open and opaque zones are of equal width, then the even orders are missing.
Fig. 23.7

Diagram showing the shape of diffraction order numbers \(m=-5\), \(-3\), \(-1\), 1, 3, and 5 of a zone plate (ZP). Compared to the incident beam, the negative orders diverge, the positive orders converge, and the zero order (omitted for clarity) has the same shape as the incident beam. For plane-wave illumination, the virtual foci of the negative-order beams and the real foci of the positive-order beams are at distances \(|f|/m\) from the zone plate, where \(|f|\) is the focal distance of first order using an order-sorting aperture (OSA). After [23.69]

23.2.1 Zone-Plate Image Quality

Optical Path Function Analysis

Following the approach in an analysis by Kamiya [23.72], we consider the imaging of a general point \(\mathrm{A}\left(x,y,z\right)\) by a planar zone plate lying in the \(y-z\) plane (Fig. 23.8). We will use the method of the optical path function, so we start by calculating the optical path from A to a general point \(\mathrm{B}\left(x^{\prime},y^{\prime},z^{\prime}\right)\) that we will later identify as the Gaussian image point. Without loss of generality, we set \(y=y^{\prime}=0\). We calculate the path APB where \(\mathrm{P}\left(0,w,l\right)\) is a general point in the plane of the zone plate. The expression for the optical path will be a power series in the aperture coordinates, \(w\) and \(l\), and the field angle, \(z/x\), and each term in the series will represent a specific aberration. Evidently,
$$\mathrm{AP}=\sqrt{x^{2}+w^{2}+\left(z-l\right)^{2}}$$
(23.12)
so expanding the square root and keeping terms up to fourth order, we have
$$\begin{aligned}\displaystyle\mathrm{AP}&\displaystyle=x\left(1+\frac{1}{2}\frac{w^{2}+l^{2}}{x^{2}}+\frac{1}{2}\frac{z^{2}}{x^{2}}-\frac{2zl}{2x^{2}}\right.\\ \displaystyle&\displaystyle\qquad-\frac{1}{8}\frac{1}{x^{4}}\left[(w^{2}+l^{2})^{2}+z^{4}+4z^{2}l^{2}-4z^{3}l\right.\\ \displaystyle&\displaystyle\left.\left.\qquad+2(w^{2}+l^{2})z^{2}-2(w^{2}+l^{2})2zl\right]+\vphantom{\frac{1}{2}}\cdots\right).\end{aligned}$$
(23.13)
There is an identical series for PB, except that for PB, the \(x,y\) and \(z\) are replaced by \(x^{\prime}\), \(y^{\prime}\), and \(z^{\prime}\) (Fig. 23.8). We are now in a position to write the optical path function \(F\). Before doing so, we drop terms that do not depend on \(w\), \(l\), or \(z/x\), because they do not represent aberrations, and introduce the term \(-nm\lambda/2\) as we did in (23.5), except we now include the diffractive focal order \(m\). We choose to analyze the case of a parabolic zone plate, so we initially use (23.7) to write the optical path function up to third order only
$$\begin{aligned}\displaystyle F&\displaystyle=\mathrm{AP}+\mathrm{PB}-\frac{nm\lambda}{2}=\mathrm{AP}+\mathrm{PB}-\left|\frac{w^{2}+l^{2}}{2}\frac{1}{f/m}\right|\\ \displaystyle&\displaystyle=\frac{w^{2}+l^{2}}{2}\left(\frac{1}{x}+\frac{1}{x^{\prime}}-\frac{1}{f/m}\right)-l\left(\frac{z}{x}+\frac{z^{\prime}}{x^{\prime}}\right)+\cdots\end{aligned}$$
(23.14)
Specializing to the case when B is the Gaussian image point, the first (defocus) term vanishes, and by considering the ray AOB we obtain
$$\frac{z}{x}=\frac{z^{\prime}}{x^{\prime}}$$
(23.15)
so that the second term also vanishes. This leaves only the five fourth-order terms
$$\begin{aligned}\displaystyle F=&\displaystyle-\frac{(w^{2}+l^{2})^{2}}{8}\left(\frac{1}{x^{3}}+\frac{1}{x^{\prime 3}}\right)-\frac{l^{2}z^{2}}{2x^{2}}\frac{1}{f/m}\\ \displaystyle&\displaystyle+\frac{l}{2}\left(\frac{z^{3}}{x^{3}}+\frac{z^{\prime 3}}{x^{\prime 3}}\right)-\frac{(w^{2}+l^{2})}{4}\frac{z^{2}}{x^{2}}\frac{1}{f/m}\\ \displaystyle&\displaystyle+\frac{(w^{2}+l^{2})}{2}\frac{lz}{x}\left(\frac{1}{x^{2}}+\frac{1}{x^{\prime 2}}\right)+\cdots\end{aligned}$$
(23.16)
These are the five Seidel aberrations : spherical aberration , astigmatism , distortion , field curvature , and coma . Because of (23.15), the distortion term vanishes identically, which is a useful property of zone-plate lenses, and we can therefore turn our attention to the remaining aberrations.
Fig. 23.8

Notation for the optical path function analysis of a zone plate

Ray Aberrations

We need to know the ray pattern delivered by the zone plate for a given point object. That is, we want the ray aberrations \(\Updelta y^{\prime}\) and \(\Updelta z^{\prime}\) relative to the Gaussian image point (coordinates identified by subscript zero). For a normal-incidence optic these are given [23.73] by
$$\Updelta y^{\prime}=x_{0}^{\prime}\frac{\partial F}{\partial w},\quad\Updelta z^{\prime}=x_{0}^{\prime}\frac{\partial F}{\partial l}\;.$$
(23.17)
We now apply this to the Seidel aberrations individually.

Spherical Aberration

We can rearrange the spherical aberration term using the magnification \(M=x_{0}^{\prime}/x\) to obtain
$$F_{\text{sp\ ab}}=-\frac{\left(w^{2}+l^{2}\right)^{2}}{8f^{3}}\frac{M^{3}+1}{\left(M+1\right)^{3}}\;.$$
(23.18)
The last term of this expression, which we will denote by \(\mathcal{M}=(M^{3}+1)/(M+1)^{3}\), approaches unity in the cases of interest to us, namely \(M\) large (a microscope) or \(M\) small (a microprobe). If we consider the case that \(\mathcal{M}\) does approach unity, then (23.18) reduces to the fourth-order term of (23.6). This is expected because we are reverting to the conjugates used to derive (23.6) (\(\infty\) and \(f\)), and the parabolic zone plate we are analyzing is not corrected for spherical aberration. Treating the case of a microscope (\(x\approx f\)) and using the notation
$$r=\sqrt{w^{2}+l^{2}}\;,$$
we now substitute from (23.18) into (23.17) to obtain the ray aberrations as
$$\Updelta y^{\prime} =x_{0}^{\prime}\frac{1}{8f^{3}}2(w^{2}+l^{2})2w=\frac{M}{2}\left(\frac{r}{f}\right)^{2}w\;,$$
(23.19)
$$\Updelta z^{\prime} =y_{0}^{\prime}\frac{1}{8f^{3}}2(w^{2}+l^{2})2l=\frac{M}{2}\left(\frac{r}{f}\right)^{2}l\;.$$
(23.20)
Thus we see that the image-plane figure produced by a point object via the rays passing through the rim of the lens (\(r=r_{n}\)) is a circle of diameter \(D_{\mathrm{SA}}\), where
$$D_{\text{SA}}=M\left(\mathrm{NA}\right)^{2}r_{n}\;.$$
(23.21)
This is produced irrespective of the position of the object point. The presence of uncorrected spherical aberration means there will be an optimal value of the NA, that is, the largest NA for which the resolution is still diffraction-limited. This can be estimated [23.68] by requiring that the path error be less than the Rayleigh quarter-wave limit, giving
$$\mathrm{NA}_{\mathrm{opt}}=\sqrt[4]{\frac{2\lambda}{\Theta f}}\;.$$
(23.22)
In practice, the spherical aberration of a parabolic zone plate is often not negligible in the soft x-ray region (see Fig. 23.9a,b for example), and thus soft x-ray zone plates are usually made according to (23.8), which means that the spherical aberration is corrected for the chosen wavelength. In these cases we may ask what happens when such a zone plate is used with a wavelength other than the correction wavelength. We therefore consider a zone plate corrected for a given wavelength in first positive order with \(M\) very small, as shown in Fig. 23.6, and we assume that the correction is also small (\(\mathrm{NA}\ll 1\)). Now using a subscript 0 to represent the properties of the corrected zone plate and subscript \(1\) to represent the properties of the same zone plate operating at another wavelength, we can apply (23.8) for the \(n\)-th and \(\left(n-1\right)\)-th zones to show that \(\Updelta r_{N}\simeq\left(\lambda_{0}f_{0}+n\lambda_{0}^{2}/2\right)/2r_{N}\).
Fig. 23.9a,b

Size of the aberrations of a soft x-ray (\({\mathrm{500}}\,{\mathrm{eV}}\); (a)) and a hard x-ray (\({\mathrm{5000}}\,{\mathrm{eV}}\); (b)) zone plate as a function of object size. The soft x-ray zone plate has an outer zone width of \({\mathrm{30}}\,{\mathrm{nm}}\), diameter of \({\mathrm{62}}\,{\mathrm{{\upmu}m}}\), and a focal length of \({\mathrm{0.75}}\,{\mathrm{mm}}\) at \({\mathrm{500}}\,{\mathrm{eV}}\). The hard x-ray zone plate has an outer zone width of \({\mathrm{60}}\,{\mathrm{nm}}\), diameter \({\mathrm{124}}\,{\mathrm{{\upmu}m}}\), and focal length \({\mathrm{30}}\,{\mathrm{mm}}\) at \({\mathrm{5000}}\,{\mathrm{eV}}\)

The aberrations shown in Fig. 23.9a,b illustrate the general trend that hard x-ray zone plates have lower aberrations because of their lower NA. In that figure, the hard x-ray zone plate has only negligible aberrations up to a sample diameter of about ten radii, which is far beyond the sample size allowed by penetration requirements. On the other hand, the resolution of the soft x-ray zone plate is degraded by the field-angle-dependent aberrations for objects with a diameter more than about half a radius. This is still good for many experiments.

From that, we can use the grating equation to obtain the ray deviation angle \(\epsilon_{0}\) at the zone plate and thus the ray displacement \(\epsilon_{0}f_{0}\) at the detection plane when the zone plate images a distant axial point (\(M\) small)
$$\epsilon_{0}f_{0}=r_{n}\left\{1-\frac{n\lambda_{0}}{2f_{0}}\right\}=r_{n}-\frac{r_{n}}{2}\left(\mathrm{NA}_{0}\right)^{2},$$
(23.23)
where (23.7) and (23.10) have been used. For imaging an axial point with \(M\) large, the ray displacement would be approximately \(M\epsilon_{0}f_{0}\). The first term of these expressions, \(r_{n}\) or \(Mr_{n}\), is the ray displacement needed for correct imaging of an axial point, and the second term is an aberration equal to minus the radius of the spherical aberration disk of a parabolic zone plate (compare (23.23) with (23.21)). This results in perfect cancellation of the spherical aberration as intended.
When the same zone plate is used at another wavelength \(\lambda_{1}\) with allowance for the change in focal length, \(\Updelta r_{n}\) remains the same, so (23.23) becomes
$$\epsilon_{1}f_{1}=\frac{f_{1}\lambda_{1}}{f_{0}\lambda_{0}}r_{n}\left(1-\frac{n\lambda_{0}}{2f_{0}}\right)=r_{n}-\frac{r_{n}}{2}\left(\text{NA}_{0}\right)^{2}\;.$$
(23.24)
In other words, the amount of correction has not changed, whereas for exact correction it should now be \(r_{n}\left(\mathrm{NA}_{1}\right)^{2}/2\). Thus the residual error is an aberration disk of diameter
$$\left|r_{n}\left[\left(\mathrm{NA}_{1}\right)^{2}-\left(\mathrm{NA}_{0}\right)^{2}\right]\right|$$
or a new disk of diameter \(\left|\left(\lambda_{1}^{2}-\lambda_{0}^{2}\right)/\lambda_{0}^{2}\right|\) times the diameter of the original uncorrected disk. As an example, if the new wavelength differs by \({\mathrm{10}}\%\) from the wavelength of correction, then the aberration disk will be reduced to about \({\mathrm{20}}\%\) of its uncorrected size. This would be enough to make the aberration negligible in the soft x-ray example shown in Fig. 23.9a,b. Evidently this general conclusion applies equally to \(M\) very small or very large.

Astigmatism and Field Curvature

Taking the astigmatism and field curvature terms together and again calculating the ray aberrations by substituting the path function terms into (23.17), we obtain
$$\Updelta y^{\prime} =-\frac{1}{2}x_{0}^{\prime}w\frac{z^{2}}{x^{2}}\frac{1}{f}\;,$$
(23.25)
$$\Updelta z^{\prime} =-\frac{1}{2}x_{0}^{\prime}l\frac{z^{2}}{x^{2}}\frac{1}{f}\;.$$
(23.26)
Squaring and adding, we find that the marginal rays trace out an ellipse of major axis \(3k\) and minor axis \(k\) according to
$$\frac{\Updelta y^{\prime 2}}{\left(k/2\right)^{2}}+\frac{\Updelta z^{\prime 2}}{\left(3k/2\right)^{2}}=1\;,\quad\text{where }k=M\left(\mathrm{NA}\right)r_{n}^{2}\bar{z}^{2}\;.$$
(23.27)
The parameter \(z=\bar{z}/r_{n}\) expresses the position of the object point in units of the zone-plate radius. Therefore, unlike spherical aberration, the size of this aberration does depend on the position of the object point, and therefore will limit the field of view of the microscope.

Coma

In this case, the above procedure leads to
$$\begin{aligned}\displaystyle&\displaystyle\Updelta y^{\prime}=-2wlQ,\quad\Updelta z^{\prime}=-\left(w^{2}+3l^{2}\right)Q\;,\\ \displaystyle&\displaystyle\quad\text{where }Q=\frac{1}{2}\frac{z}{x}\frac{M-1}{f}\;.\end{aligned}$$
(23.28)
By expressing \(w\) and \(l\) in polar coordinates, one can show that each circle of radius \(r\) in the lens produces an aberration figure
$$\left(\Updelta z^{\prime}-2K^{2}\right)^{2}+\left(\Updelta y^{\prime}\right)^{2}=K^{2}\;,\quad\text{where }K=r^{2}Q\;.$$
(23.29)
This is a series of circles of radius \(K\), each shifted by \(2K\) from the origin, which is the usual comet-shaped figure associated with coma. The outer boundary of the figure has a length \(3K_{\mathrm{max}}\) and the largest circle has a diameter \(2K_{\mathrm{max}}\), where (assuming \(M\) is large so that \(M-1\approx M\)) we have
$$2K_{\mathrm{max}}=M\left(\mathrm{NA}\right)r_{n}\bar{z}\;.$$
(23.30)

Relative Size of the Aberrations

The relative size of these aberration figures can be obtained from (23.21), (23.27), and (23.30), respectively. The diameter of the spherical aberration circle, the diameter of the largest coma circle, and the major axis of the astigmatism ellipse are in the ratio \(1:\bar{z}:3\bar{z}^{2}\) [23.68]. For points close to the axis (\(\bar{z}\ll 1\)), the field-angle-dependent aberrations coma and astigmatism/field curvature will be negligible, and the resolution will be determined by diffraction if the numerical aperture is below \(\mathrm{NA}_{\mathrm{opt}}\) or by spherical aberration if it is above. On the other hand, for points further from the axis, coma and astigmatism/field curvature become more significant, and for points more than about a zone-plate-radius away from the axis (\(\bar{z}\geq 1\)), astigmatism/field curvature dominates. The general behavior of these aberrations is shown in Table 23.1, while Fig. 23.9a,b shows examples of parabolic zone plates working at 0.5 and \({\mathrm{5}}\,{\mathrm{keV}}\). However, note that the plots in the figure show the size of the outer boundary of the aberration figures which is a conservative estimate of their contribution to the resolution, because the light is somewhat concentrated near the center of the figure. For example, we can deduce from (23.21) that half of the light is concentrated in a circle with a diameter about one third of \(D_{\mathrm{SA}}\).

Table 23.1

Seidel aberrations of a planar zone plate

Name

Spherical aberration

Astigmatism/field curvature

Distortion

Coma

\(w\), \(l\), \(z/x\) term

\((w^{2}+l^{2})^{2}\)

\((w^{2}+3l^{2})(z/x)^{2}\)

\(l[(z/x)^{2}+(z^{\prime}/x^{\prime})^{2}]\)

\(l(w^{2}+l^{2})(z/x)\)

Aberration figure boundary

Circle

Ellipse

Vanishes identically

Two lines through O at \(\pm 30^{\circ}\) to O\(z\) touching a family of circles

\(y\) parameter

Circle diameter

Major axis

na

Diameter of the largest circle

Value \(y\) parameter

\(M(\mathrm{NA})^{2}r_{n}\)

\(3M(\mathrm{NA})^{2}r_{n}\bar{z}^{2}\)

na

\(M(\mathrm{NA})^{2}r_{n}\bar{z}\)

\(z\) parameter

Circle diameter

Minor axis

na

Distance from the origin to the far side of the largest circle

Value \(z\) parameter

Same

\(M(\mathrm{NA})^{2}r_{n}\bar{z}^{2}\)

na

\((3/2)M(\mathrm{NA})^{2}r_{n}\bar{z}\)

23.2.2 Zone Plate Efficiency

Idealized Structures

One can see from (23.7) that an ideal Fresnel zone plate is periodic in \(r^{2}\) space with period \(\lambda f\), which means that its amplitude transparency function \(T_{\mathrm{zp}}\) (shown in Fig. 23.10) can be written as the Fourier series
$$\begin{aligned}\displaystyle&\displaystyle T_{\text{zp}}=\sum_{-\infty}^{+\infty}a_{m}\exp\left(\frac{\mathrm{i}m\uppi r^{2}}{\lambda f}\right),\\ \displaystyle&\displaystyle\quad\text{where }a_{m}=q(-1)^{m}\operatorname{sinc}(mq)\;.\end{aligned}$$
(23.31)
The sinc function is defined by \(\operatorname{sinc}(x)=\sin(\uppi x)/(\uppi x)\), and \(q\) is the fraction of each period that is opaque. For a classical zone plate, \(q=0.5\), so in that case, (23.31) gives the power in the \(\pm m\)-th harmonic as \(\left|a_{m}\right|^{2}=1/\left(m\uppi\right)^{2}\) for \(m\) odd, zero for \(m\) even, and 0.25 for \(m=0\). These are therefore the intensity efficiencies of the classical zone plate in those orders. The main point here is that this type of zone plate has a maximum efficiency in the first-order focus of about \({\mathrm{10}}\%\). By taking the derivative of \(\left|a_{m}\right|^{2}\), we can show that the classical zone plate (\(q=0.5\)) is, in fact, the optimal choice of \(q\) for first-order efficiency. Similarly, the optimal choice for second-order efficiency is \(q=0.25\), or \(0.75\), and second-order light can appear with manufacturing errors [23.74]. By applying Parseval's theorem to the series in (23.31) with \(q=0.5\), one can determine the disposition of the energy for the classical zone plate as shown in Table 23.2. Another idealized type of zone plate proposed by Rayleigh [23.75] and implemented by Wood [23.76] over a century ago is the phase plate. Here, the opaque rings are replaced by transparent rings that impart a phase change of \(\phi=\uppi\). The phase plate transparency function is \(T_{\mathrm{pp}}=1+\left(\mathrm{e}^{\mathrm{i}\phi}-1\right)T_{\mathrm{zp}}\), so the Fourier series becomes
$$ T_{\text{pp}}=\sum_{-\infty}^{\infty}b_{m}\exp\left(\frac{\mathrm{i}m\uppi r^{2}}{\lambda f}\right),$$
(23.32)
$$ \quad\text{where }b_{m}=\delta_{m,0}+\left(\mathrm{e}^{\mathrm{i}\phi}-1\right)q\left(-1\right)^{m}\operatorname{sinc}\left(mq\right).$$
(23.33)
This shows that when \(\phi=\uppi\) and \(q=0.5\), which are the optimal values, the power in the \(\pm m\)-th harmonic is \(\left|b_{m}\right|^{2}-4/\left(m\uppi\right)^{2}\) for \(m\) odd, zero for \(m\) even, and \(\left|b_{0}\right|^{2}=\left|1+q\left(\mathrm{e}^{\mathrm{i}\phi}-1\right)\right|^{2}=0\) for \(m=0\). The efficiency in the first-order focus is now about \({\mathrm{40}}\%\), and the disposition of energy is again shown in Table 23.2.
Fig. 23.10

The transparency function of one period of a zone plate plotted in \(r^{2}\) space. The fraction that has a transparency of unity is \(q=c/L\)

Table 23.2

Efficiency of various types of zone plates

Type of zone plate

Fresnel

Rayleigh–Wood

Gabor amplitude

Gabor phase

Type of zone

Opaque, transparent

Phase-change \(\uppi\), transparent

Sine-wave transparency

Sine-wave phase change max \(={\mathrm{1.84}}\,{\mathrm{rad}}\)

Efficiency \(\pm 1\) order

\(1/4\)

\(4/\pi^{2}\)

\(1/16\)

0.34

Efficiency in general

\(1/(m^{2}\uppi^{2})\) \(m\) odd,

0 \(m\) even

\(4/(m^{2}\uppi^{2})\) \(m\) odd,

0 \(m\) even

0

\(\neq 0\)

Total positive orders

(\(m\neq 0\))

\(1/8\)

\(1/2\)

\(1/16\)

0.45

Total negative orders (\(m\neq 0\))

\(1/8\)

\(1/2\)

\(1/16\)

0.45

Efficiency zero order

\(1/4\)

0

\(1/4\)

0.10

Absorbed

\(1/2\)

0

\(5/8\)

0

Overall total of last four rows

1

1

1

1

A third idealized type of zone plate is the Gabor plate (the hologram of a point), in which the rectangular profile of the last two devices is replaced by a sinusoid in \(r^{2}\) space or a chirp function in \(r\) space. An absorption Gabor plate has only three orders, \(m=\pm 1\) and 0 of which the \({+}1\) order has efficiency \(1/16\). A phase Gabor plate, with a maximum phase change of \({\mathrm{1.84}}\,{\mathrm{rad}}\), has all the odd orders, and the \({+}1\) order receives \({\mathrm{34}}\%\) of the light (Table 23.2).

Real Structures

To make practically useful x-ray lenses, we must extend the treatment given so far to include the optical properties of real materials suitable for the manufacture of zone plates. Such an extension was first provided in an important paper by Kirz [23.77], in which the following was demonstrated:
  1. 1.

    Phase zone plates with primary efficiencies of \(20{-}40\%\) can be made from realistic materials.

     
  2. 2.

    Such zone plates can be designed to reduce or eliminate the zero-order beam and to reduce the absorbed fraction compared to a classical Fresnel zone plate.

     
  3. 3.

    These improvements can be effected essentially throughout the wavelength range \(0.1{-}80\,{\mathrm{nm}}\).

     
  4. 4.

    Realistic fabrication errors lead to only moderate deterioration in optical performance.

     
For a phase-reversal zone plate made of a material with a complex refractive index [23.31] of \(\tilde{n}=1-\delta-\mathrm{i}\beta\), the efficiency \(\left|b_{m}\right|^{2}\) can be found by making the replacement \(\phi=\phi_{1}+\mathrm{i}\phi_{2}\) in (23.33) where \(\phi_{1}=kt\delta\), \(\phi_{2}=kt\beta\), \(k=2\uppi/\lambda\), \(t\) is the thickness, and we use the shorthand \(r_{\text{a}}=\mathrm{e}^{-\phi_{2}}\) for the amplitude attenuation factor. For \(m\neq 0\), this gives
$$\left|b_{m}\right|^{2}=\frac{1}{\left(m\uppi\right)^{2}}\left(1+r_{\text{a}}^{2}-2r_{\text{a}}\cos\phi_{1}\right)$$
(23.34)
and for \(m=0\)
$$\left|b_{0}\right|^{2}=\frac{1}{4}\left(1+r_{\text{a}}^{2}+2r_{\text{a}}\cos\phi_{1}\right).$$
(23.35)
These equations give the efficiency of a planar zone plate of known thickness and refractive index. As has been noted [23.68, 23.77], the optimal phase change is no longer \(\uppi\), but is about \(10{-}20\%\) less. However, one can certainly choose a thickness to optimize the efficiency of a planar zone plate based on the use of (23.34). Plots of the theoretical efficiency, calculated using (23.34), for zone plates made of nickel and gold are shown in Fig. 23.11a,b.

23.2.3 Zone Plates: Fabricationand Examples

Fabrication Technique

The fabrication of x-ray zone plates involves several challenges. For high-resolution imaging, one wishes to obtain the smallest possible value for the outermost zone width \(\Updelta r_{N}\), or about \(15{-}80\,{\mathrm{nm}}\) in modern high-resolution examples. At the same time, the thickness \(t\) of the zone plate should ideally be that required to deliver a phase shift near \(\uppi\) so as to maximize efficiency; this generally implies a thickness of greater than \({\mathrm{100}}\,{\mathrm{nm}}\) for soft x-rays of \(100{-}1000\,{\mathrm{eV}}\) energy, and a thickness near \({\mathrm{1}}\,{\mathrm{{\upmu}m}}\) for zone plates designed to operate at multi-keV energies (Fig. 23.11a,b). These two requirements are difficult to meet at the same time, for they lead to a demand for the fabrication of nanostructures with very high aspect ratios (structure height over width). In addition, to obtain usable focal lengths on the order of \({\mathrm{1}}\,{\mathrm{mm}}\) or larger, most zone plates used as high-resolution objectives have diameters of \(50{-}200\,{\mathrm{{\upmu}m}}\), but to minimize loss of efficiency, all zones must be placed accurately to roughly one third of their width [23.74]. This implies absolute accuracy of zone placement of about \({\mathrm{0.01}}\%\), which is quite challenging but which can be achieved in modern \({\mathrm{100}}\,{\mathrm{keV}}\) electron-beam lithography systems that incorporate laser interferometer positioning control [23.78, 23.79]. Such zone plates typically have \(300{-}1000\) zones, and require corresponding quasimonochromatic illumination with \(E/\Updelta E> 300{-}1000\) or better [23.71].

Fig. 23.11a,b

Contour plot showing the efficiency of planar zone plates made of Ni (a) and Au (b) according to (23.32) and (23.34) as a function of thickness and x-ray energy

Early demonstrations of x-ray zone-plate fabrication used optical lithography to create freestanding zone plates with \(\Updelta r_{N}={\mathrm{20}}\,{\mathrm{{\upmu}m}}\) [23.67]. An important early advance was the use of holographic methods to create zone plates with submicron zone widths [23.80], eventually leading to outermost zone widths of \(\Updelta r_{N}={\mathrm{56}}\,{\mathrm{nm}}\) [23.81]. The method used by nearly all laboratories today involves electron-beam lithography, which after an early suggestion [23.82] was later demonstrated in several laboratories [23.83, 23.84, 23.85]. In order to obtain high-aspect-ratio nanostructures, most laboratories now use some variation of trilayer resist schemes [23.86] with electroplating [23.87], as illustrated in Fig. 23.12. This approach allows for the writing of fine, dense features in an electron-beam resist which is sufficiently thin that electron side scattering is minimized. A series of reactive ion etches are used to first transfer the e-beam pattern into a hard mask, and to then use that hard mask to transfer the pattern into a second polymer either to define an etch mask for the underlying zone material [23.88, 23.89] or, more commonly, as a mold for electroplating of the zone structures [23.87]. Using these approaches, many groups are now fabricating zone plates with outermost zone widths of \({\mathrm{20}}\,{\mathrm{nm}}\) or below. One promising approach [23.90] has been to write every other zone in one pass, and then to write the other half of the zones in a subsequent processing step (of course, extremely high overlay accuracy is required). By increasing the distance from the next zone written in one pass, the proximity effect of electron-beam lithography is reduced, leading to higher contrast and thus a higher aspect ratio in the developed resist. In addition, the width of the resist used as a plating mold is tripled, thus reducing collapse during processing. This approach has been used to fabricate zone plates with an outermost zone width of \({\mathrm{15}}\,{\mathrm{nm}}\) [23.90] and a theoretical zone efficiency of \({\mathrm{6}}\%\) (see next section). A key challenge lies in maintaining high efficiency as the outermost zone width is decreased. For applications requiring higher efficiency for greater flux in STXM or reduced radiation dose in TXM, efficiencies of \(10{-}18\%\) have been obtained at \({\mathrm{30}}\,{\mathrm{nm}}\) outermost zone width [23.91, 23.92], while even higher efficiencies have been obtained using multiple overwrite layers [23.93]. These efficiencies are for zone plates operating at \(400{-}550\,{\mathrm{eV}}\); zone plates for use at higher x-ray energies are discussed in Sect. 23.2.3, Zone Plates with Shaped Grooves.

Fig. 23.12

Schematic of one zone plate fabrication technique (trilevel resist and electroplating) as discussed in the text

Resolution-Determining Zone Plates

While a variety of groups are fabricating high-resolution zone plates as described above, we consider here one example which is the effort of the Center for X-Ray Optics at Lawrence Berkeley National Laboratory. By using a \({\mathrm{100}}\,{\mathrm{keV}}\) electron-beam lithography system with interferometric positioning control and customized circular pattern generation [23.94], this group created a series of zone plates with steadily improving resolution [23.78, 23.90, 23.95] and have developed a sophisticated technique for determining the resolution [23.96]. The tests were performed using the XM-1 microscope at beamline 6.1 at the Advanced Light Source in Berkeley (Sect. 23.3.1). One example result [23.90] showed a resolution of \(<{\mathrm{15}}\,{\mathrm{nm}}\) at a photon energy of \({\mathrm{815}}\,{\mathrm{eV}}\) (Fig. 23.13). This measured value depends on the degree of partial coherence of the illumination, and we discuss it in that context in Sect. 23.3.3. The zone plate had an outer zone width of \({\mathrm{15}}\,{\mathrm{nm}}\), which, with the XM-1 illumination system, gave a theoretically expected resolution of \({\mathrm{12}}\,{\mathrm{nm}}\). The zone plate was made by the above-mentioned process in which the \(n=2,6,10,\dots\) and the \(n=4,8,12,\dots\) opaque zones are made in two separate groups. In another, more recent development, a team at the Paul Scherrer Institute has written aligned zone-plate structures on either side of a thin membrane to achieve about \({\mathrm{10}}\,{\mathrm{nm}}\) resolution [23.27].

Fig. 23.13

The soft x-ray images of square-wave test objects used to demonstrate a microscope resolution of \(<{\mathrm{15}}\,{\mathrm{nm}}\) using a \({\mathrm{15}}\,{\mathrm{nm}}\)- and a \({\mathrm{25}}\,{\mathrm{nm}}\)-outer-zone-width zone plate as explained in the text. The half-period of the test object and the outer zone width of the zone plate for the four images are shown. From [23.90]

Condenser Zone Plates

Condenser zone plates serve the dual function of imaging the source onto the sample (in critical illumination) and, in combination with a pinhole close to the sample, of acting as a moderate-resolution monochromator. Ideally they should deliver a beam which (1) has a higher NA than the objective zone plate, (2) exactly fills the sample with light, and (3) has a spectral bandwidth equal to the reciprocal of the number of zones of the objective. In practice, given the fact that bending magnet synchrotron radiation sources usually have smaller phase-space area than the microscope, and because of the fabrication difficulties described below, conditions (1) and (2) cannot be fully met. Moreover, condition (3) implies that the condenser zone plate must be about \(5{-}10\,{\mathrm{mm}}\) in diameter. We discuss the trade-offs involved here in more detail in Sect. 23.3.1, Transmission X-Ray Microscope (TXM) Layout.

The first condenser zone plates were fabricated [23.81, 23.97] using roughly the same holographic process as was then used to make objective zone plates. As a result, the finest line widths (and thus the NAs) of the condenser and objective tended to match, which is broadly what is required for the best resolution (see for example Fig. 10.13 of [23.73]). With electron-beam lithography, new challenges have arisen. To achieve the same finest zone width and efficiency in the condenser as in the objective, one would require the use of the same high-resolution but necessarily slow-electron-beam resists. Since the area of a \({\mathrm{5}}\,{\mathrm{mm}}\)-diameter condenser is \(10^{4}\) times larger than a \({\mathrm{50}}\,{\mathrm{{\upmu}m}}\)-diameter objective zone plate, it would take \(10^{4}\) times as long to fabricate the condenser with the same process. (Because aberrations on condensers do not degrade image quality, the requirements for zone placement accuracy do not scale up in the same fashion.) For these reasons, as the resolution of the objectives has been pushed down below \({\mathrm{20}}\,{\mathrm{nm}}\), the condenser zone plates have not kept up. Instead, typical condenser zone plates fabricated by electron-beam lithography have had outer zone widths of \(50{-}60\,{\mathrm{nm}}\). As one example, the fabrication of a TXM condenser zone plate with a diameter of \({\mathrm{9}}\,{\mathrm{mm}}\) and outer-zone width of \({\mathrm{55}}\,{\mathrm{nm}}\) required a 48-hour writing time [23.78]. Although process improvements have since reduced that time, such large zone plates are still not widely available. The mismatch of numerical aperture between condenser and objective zone plates limits the modulation transfer at high spatial frequency [23.73] and thus limits the ability to detect small structures such as immunogold labels in bright-field or dark-field modes [23.98].

In addition to the challenges of matching the NA of the best objectives and their limited availability, condenser zone plates are usually required to operate in an unfriendly environment, relatively near the source in a synchrotron beamline. Even with the protection of an energy-filtering mirror, they still often take a significant heat load, and the heat removal pathways are poor. This is an undesirable circumstance for optics that take so much effort to build. The problem of power deposition in condenser zone plates can be understood by solving the boundary value problem of a uniformly heated membrane [23.99], the results of which were later applied to zone-plate monochromators [23.100]. Assuming a square membrane of side \(a\), thickness \(t\), and conductivity \(k\), the solution for the temperature is
$$\begin{aligned}\displaystyle&\displaystyle T\left(x,y\right)=\frac{16Qa^{2}}{\uppi^{4}kt}\\ \displaystyle&\displaystyle\times\sum_{m=1,3,5,\ldots}\sum_{n=1,3,5,\ldots}\frac{1}{mn}\frac{\sin\left(\frac{m\uppi x}{a}\right)\sin\left(\frac{n\uppi y}{a}\right)}{m^{2}+n^{2}}\end{aligned}$$
(23.36)
where \(Q\) is the absorbed power density. The double sum delivers a simple 2-D monolithic function that has a peak of height 0.448 in the center and is zero at the edges. The center temperature is then given by the useful relation
$$T_{\text{max}}=7.17\frac{Qa^{2}}{\uppi^{4}kt}\;.$$
(23.37)
Since the maximum temperature depends on \(Q/t\) and, for a simple thin membrane, \(Q\) is proportional to \(t\), the temperature is not reduced by making the membrane thicker. However, if the absorption is principally in the zone-plate rings, then a thicker membrane may help. A smaller or better-conducting membrane always helps.

The list of issues posed by condenser zone plates does not end here. Another serious limitation is that, in order to change the x-ray energy, one has to change the condenser zone plate, or at least change its object and image distances. This is so difficult that systems fed by a condenser zone plate are effectively not energy-tunable. Clearly, there are plenty of reasons to seek alternatives to condenser zone plates, and to this we now turn.

Alternatives to Condenser Zone Plates

Zone plates deflect the outermost ray by an angle equal to their NA, as shown by (23.10). As explained in the previous section, manufacturable NAs of condenser zone plates are too small to match those of the best objectives. Now we know that a mirror can deflect an x-ray roughly by \(2\sqrt{2\delta}\), or twice its critical angle [23.101]. Thus we can characterize a mirror by an effective outer zone width
$$\Updelta r_{N}=\lambda/\left(4\sqrt{2\delta}\right),$$
which a zone plate would need to have to produce a deflection equal to twice the critical angle of the mirror. Since \(\delta\) is proportional to \(\lambda^{2}\) (Sect. 23.1.1), one can see that the effective outer zone width will vary rather slowly with both energy and electron density. For example, for a platinum mirror it varies from \({\mathrm{6}}\,{\mathrm{nm}}\) at \({\mathrm{0.5}}\,{\mathrm{keV}}\) to \({\mathrm{4}}\,{\mathrm{nm}}\) at \({\mathrm{5}}\,{\mathrm{keV}}\), while for a silicon dioxide mirror it varies from 12 to \({\mathrm{10}}\,{\mathrm{nm}}\) over the same energy range. This shows that if a suitable geometry can be arranged, a single-reflection mirror system could be a very effective condenser. However, it is important to note that a reflective condenser will not provide a monochromatic beam, and that either a spectral line source or a monochromator will be required. As we shall see, a separate monochromator coupled with a reflective condenser will have major advantages. To produce the desired hollow-cone illumination, a grazing incidence ellipsoid of revolution in the form of a hollow tube would be suitable. The input angle should match the angle of the beam from the source or monochromator, and the output angle should match or exceed the objective NA. For a synchrotron beam, this design will normally lead to underfilling of the sample, which can be overcome by wobbling. The fabrication accuracy should be such that the point spread function of the mirror is small relative to the size of the object field. Mirrors of this general type have been made for some time, often starting from glass capillaries [23.102, 23.103]. Single-reflection monocapillary x-ray mirrors have now been in use for microscopes at both laboratory and synchrotron sources [23.104, 23.105, 23.106] and have enjoyed considerable success. These condenser mirrors have many advantages over condenser zone plates. Specifically, capillary-mirror condensers:
  1. 1.

    Are more readily available

     
  2. 2.

    When fed by a separate monochromator allow the TXM to be truly energy-tunable

     
  3. 3.

    Are able to match the NA of any currently available objective zone plate

     
  4. 4.

    Are a factor of \(3{-}15\) times more efficient, with no unwanted orders

     
  5. 5.

    Are more robust, longer-lived, and easier to clean

     
  6. 6.

    Are less fragile with respect to thermal or mechanical damage

     
  7. 7.

    Do not require a bandwidth-selecting pinhole close to the focus, and thus do not limit the size of the sample holder or its ability to rotate.

     
These capillary condensers have helped TXMs to be competitive for spectromicroscopy [23.25].

Zone Plates with Shaped Grooves

Until now, we have talked about square-wave zone plates with a gap-to-period ratio of 0.5 that behaved according to the theory of a thin zone plate, even if the thickness was greater than \(\Updelta r_{N}\). Just as a blazed reflection grating with a sawtooth profile has much better efficiency than a square-wave grating (even if the latter is a perfect phase grating with \(\uppi\) phase shifts), so can one obtain higher efficiency from zone plates with shaped-groove profiles. Considering that a zone plate is intended to synthesize a smooth spherical wave front from a succession of ring-shaped parts, we might expect that the optimal groove shape will be a parabola that increases the phase shift smoothly across the zone-plate period. In fact, the mathematical treatment [23.107] shows that the thickness function is
$$ t_{1}(r)=\begin{cases}\frac{\left(\sqrt{f^{2}+r_{i}^{2}}-\sqrt{f^{2}+r_{i-1}^{2}}\right)}{\delta}\;,&r_{i-1}\leq r<(r_{i}-d_{i})\;,\\ 0\;,&(r_{i}-d_{i})\leq r<r_{i}\;,\end{cases}$$
(23.38)
where \(d_{i}/r_{i}\) can be calculated and is the outer fraction of the \(i\)th period which is to be left open. The first-order efficiency of a nickel zone plate made according to this specification would be about \({\mathrm{80}}\%\) at \({\mathrm{7}}\,{\mathrm{keV}}\). Microfabrication of a smooth curve is difficult, but much of the advantage of this scheme can be achieved by approximating the parabolic profile by a stepped structure. In one example, a nickel zone plate was made with four equal width steps of optical delay 0, 0.25, 0.5, and 0.75 wavelengths [23.108]. The measured first-order efficiency of this zone plate was \({\mathrm{55}}\%\) at \({\mathrm{7}}\,{\mathrm{keV}}\), which represents a substantial improvement in efficiency and suppression of unwanted orders compared to traditional soft x-ray performance, and shows the benefits of both groove-shaping and phase plates. The use of several thickness steps evidently implies that the outermost zone must be several times wider than the finest line width that can be achieved with the particular fabrication process, so that this approach involves a trade-off between spatial resolution and efficiency.

Hard X-Ray Zone Plates

While much of the initial development of x-ray microscopes went into developing zone-plate microscopy in the \(290{-}540\,{\mathrm{eV}}\) water window region [23.36] for studies of \(0.1{-}10\,{\mathrm{{\upmu}m}}\)-thick specimens (an activity that of course continues with much vigor), there is increasing activity in hard-x-ray zone-plate imaging at energies of roughly \(2{-}15\,{\mathrm{keV}}\). Scanning fluorescence x-ray microprobes (SFXM) using zone-plate optics are providing new capabilities for trace element mapping, and hard x-ray transmission x-ray microscopes (TXMs) using absorption or especially phase contrast are able to image much thicker objects than their soft-x-ray counterparts. Zone plates for these energies must be much thicker to achieve good efficiency (Fig. 23.11a,b), which places increasing demands on the zone aspect ratio in lithographically patterned zone plates and means that the minimum zone widths (and thus first-order spatial resolution) were initially in the \(50{-}100\,{\mathrm{nm}}\) range. However, as noted in Sect. 23.1.3, higher-resolution hard x-ray zone plates are now in use at several synchrotrons. At the same time, because the ratio of phase shifting to absorption \(\delta/\beta\) (23.1) improves as the energy is increased, the achievable efficiency becomes much higher and the depth of focus increases considerably [23.109], which is helpful for applications such as tomography (Sect. 23.3.4, The Depth-Of-Focus Limit). Quantitatively, the transverse resolution of a zone plate is given by \(0.61\lambda/\mathrm{NA}=1.22\Updelta r_{N}\) (23.11) and the depth of focus by
$$\text{DOF}\simeq\frac{2\lambda}{\left(\text{NA}\right)^{2}}=\frac{8\Updelta r_{N}^{2}}{\lambda}$$
(23.39)
where the numerical prefactor should be considered as an inexact value. Therefore, a zone plate with \(\Updelta r_{N}={\mathrm{50}}\,{\mathrm{nm}}\) has a depth of focus of about \({\mathrm{160}}\,{\mathrm{{\upmu}m}}\) at \({\mathrm{10}}\,{\mathrm{keV}}\) as opposed to about \({\mathrm{8}}\,{\mathrm{{\upmu}m}}\) at \({\mathrm{500}}\,{\mathrm{eV}}\). In addition, the focal length \(f=2r_{n}\left(\Updelta r_{n}\right)/\lambda\) (23.9) for such a zone plate with \({\mathrm{100}}\,{\mathrm{{\upmu}m}}\) diameter increases from 2 to \({\mathrm{40}}\,{\mathrm{mm}}\), which significantly alleviates some of the challenges of mechanical design for specimen temperature control, insertion of fluorescence detectors, and so on.
Fig. 23.14

A hard x-ray zone plate with \({\mathrm{100}}\,{\mathrm{nm}}\) outermost zone width and \({\mathrm{1.6}}\,{\mathrm{{\upmu}m}}\) thickness of gold for use at \({\mathrm{5.4}}\,{\mathrm{keV}}\) in an older commercial x-ray microscope system (Xradia, Inc.; now Carl Zeiss x-ray Microscopy). The simultaneous achievement of narrow zone widths for high spatial resolution, and significant zone thickness so as to achieve a \(\uppi\) phase shift, means that the achievement of high-aspect-ratio nanostructures is important. This zone plate had an aspect ratio of \(16:1\) and a theoretical focusing efficiency of \({\mathrm{31}}\%\). Higher aspect ratios are now being produced, as noted in the text. Figure courtesy of Wenbing Yun

In spite of the challenges in fabricating thicker zone plates using lithographic techniques, much success has been achieved. In some cases the initial electron-beam lithography write has been transferred into a thicker plating mold using reactive ion etching as described above; this has led to the commercial availability [23.110] of a variety of high-aspect-ratio zone plates, including one with an outermost zone width of \({\mathrm{50}}\,{\mathrm{nm}}\) and thickness of \({\mathrm{700}}\,{\mathrm{nm}}\), or an aspect ratio of \(14:1\) (an example of an earlier zone plate of that type is shown in Fig. 23.14). Other approaches have involved using an electron-beam-written zone plate as a mask for the subsequent processing of a thicker zone plate using x-ray lithography [23.111, 23.83], including the fabrication of \({\mathrm{2.5}}\,{\mathrm{{\upmu}m}}\)-thick zone plates with a finest zone width of \({\mathrm{0.25}}\,{\mathrm{{\upmu}m}}\) [23.112]. Yet another approach is to stack two or more zone plates together [23.113, 23.114].

Sputter-sliced or jelly roll zone plates [23.115] represent a completely different approach to zone-plate fabrication. The goal of this approach is to start with a rotating wire and then build up alternating layers of weakly and strongly refractive material by sputtering or evaporation. The resulting structure is then sliced to yield zone plates of the appropriate thickness. In this case, the achievement of high aspect ratios is not at all challenging; instead, the challenges include avoiding error and roughness accumulation in realizing the proper zone radii, the difficulties of maintaining perfect cylindrical symmetry, and the challenges involved in slicing the structure. This approach has been the subject of ongoing development [23.116, 23.117], and in a notable demonstration, a central focal spot of better than \({\mathrm{5}}\,{\mathrm{nm}}\) was claimed from a sputter-sliced zone plate of only about \({\mathrm{2}}\,{\mathrm{{\upmu}m}}\) diameter onto which a beam was focused using a KB mirror pair [23.118]. These are intriguing results, though the focal length was impractically small. Thus, for the moment, the sputter-sliced approach has not yet produced practical optics with an optical performance consistent with the intended geometrical parameters.

Thick Zone Plates

Up until now we have used kinematic diffraction theory to understand the properties of zone plates. In this theory, the incident wave and the diffracted signals from each volume element are all treated as independent. However, in reality, the incident wave and the diffracted waves in the solid structure are coherently coupled, and if the zone plate is thick enough, the effect of this coupling will become evident at the output. Such coupling is known in perfect-crystal diffraction, where it leads to anomalous transmission in Laue geometry experiments (the Borrmann effect ), while on a larger size scale, Bragg effect holograms show similar behavior. Zone plates can be designed to exploit coupled-wave effects, and these devices offer the possibility of very high efficiency and resolution in high diffraction orders, thus exceeding the resolution limit of the outermost zone width that applies when operating in first order. Theoretical treatments of diffraction by thick periodic structures have been developed in the hard-x-ray community (dynamical diffraction by crystals) [23.119] and the optical holography community [23.120, 23.121], where it is termed coupled-wave theory . The coupled-wave method has been applied to x-ray gratings and zone plates [23.122, 23.123], with solutions that include the case of high orders and gap-to-period ratios other than \(0.5\). An alternative approach is to use the multislice method [23.124], which is able to handle arbitrary structures and was recently shown to reproduce the calculational results of coupled-wave theory when applied to x-ray optics [23.125]. Based on coupled-wave theory, it has been suggested [23.123, 23.126] that in the soft x-ray region, absolute efficiencies of \(30{-}50\%\) in a single high order are indeed possible, with line-to-space ratios of \(0.1{-}0.5\) and aspect ratios greater than 30 : 1. This has been tested in experiments using copolymer gratings with an aspect ratio of 10 : 1, where the predictions of the theory were broadly confirmed and maximum efficiency of \({\mathrm{15.3}}\%\) was achieved at \({\mathrm{13}}\,{\mathrm{nm}}\) wavelength [23.127]. This value was \({\mathrm{75}}\%\) of the prediction of the coupled-wave theory and 25 times the prediction of thin-grating (kinematic) theory. The authors suggested that zone plates based on this principle may find application as condensers for tabletop microscopes using isotropically emitting sources. A similar verification of dynamical diffraction theory for the case of sectioned multilayers illuminated with \({\mathrm{19.5}}\,{\mathrm{keV}}\) x-rays in Laue geometry has been obtained [23.128], and reflecting efficiency of \({\mathrm{70}}\%\) was observed. Both of these experiments used gratings as being representative of the diffraction-efficiency behavior of a conventional zone plate [23.127] and a sputter-sliced linear zone plate [23.128], respectively. However, as noted by Maser et al [23.129], thick zone plates, such as volume holograms, have a directional selectivity based on Bragg's law. That is, the zones must be oriented so that the incoming wave is locally Bragg-reflected by the zones, and there will be a rocking curve outside which the high efficiency is lost or moved to another order. Thus, (23.9) can be regarded as showing that Bragg's law is obeyed at every zone of a conventional zone plate operating at magnification unity. On the other hand, for high-magnification or demagnification applications, the zones must be tilted at an angle that varies with radius. The difficulty of doing this in practice is currently delaying the application of these ideas to practical zone plates.

A promising recent alternative to tilt angles that vary continuously with angle has been to apply the concepts of the sputter-sliced zone plate (see previous section) to produce linear half-zone plates called multilayer Laue lenses [23.129]. One starts with an atomically smooth flat and deposits alternating zone materials, starting with the highest-order zones, so that error accumulation mainly affects the coarser, low-order zones. Two multilayer Laue lenses together can achieve two-dimensional focusing in the manner of a Kirkpatrick–Baez mirror pair. Although tilting by a continuously variable angle is not exactly achieved with this approach, it can be approximated by tilting the whole lens to a compromise Bragg angle. One-dimensional line foci as narrow as \({\mathrm{11}}\,{\mathrm{nm}}\) have already been reported with this method [23.130], and further advances can be anticipated.

23.3 X-Ray Microscopes

In the previous sections we described some of the characteristics of x-ray interactions and focusing optics. We now turn our attention to a discussion of x-ray microscopes currently in operation. They fall into two classes, full-field imaging and scanning, which are both illustrated in Fig. 23.15a,b. We also describe three specific microscopes as examples: a transmission x-ray microscope (TXM) operated at Lawrence Berkeley National Laboratory, a scanning transmission x-ray microscope (STXM) formerly operated at Brookhaven National Laboratory, and a scanning fluorescence x-ray microprobe (SFXM) operated at Argonne National Laboratory.

Fig. 23.15a,b

Schematic of the main components of (a) a transmission x-ray microscope, or TXM (courtesy of D. Attwood, University of California Berkeley), (b) a  scanning transmission x-ray microscope, or STXM

Fig. 23.16

The aim in operating a scanning microscope or microprobe is to have a diffraction-limited focus. Therefore, the source must be sufficiently demagnified so that it contributes negligibly to the focal width. This contour plot shows how the modulation transfer function ( ) of an optic with a half-diameter central stop is affected by increasing the phase–space parameter \(p\) of the source. This parameter \(p=w\theta\) (source full width \(w\) times the full angle \(\theta\) accepted by the optic) should be less than the wavelength \(\lambda\) in both the \(x\) and \(y\) directions to achieve maximum spatial resolution. The normalized spatial frequency is defined to be unity at the MTF cutoff of \(1/\Updelta r_{N}\). From [23.131]. Reproduced with permission of the International Union of Crystallography

A key difference between TXM, STXM, and SFXM involves the illumination phase space that can be accepted. In STXM and SFXM, the size of the spot delivered by the zone-plate objective is a convolution of the geometric image of the source and the point spread function of the optic. As Fig. 23.16 shows, for an objective with diffraction-limited (as opposed to aberration-limited) resolution, the effect of the geometric source size becomes negligible if the product \(p=w\theta\) of source width \(w\) times the full angle \(\theta\) accepted by the optic is less than the wavelength \(\lambda\) in each dimension. This is commonly summarized by saying that scanning microscopes require single-mode illumination, although it is understood that a spatially filtered, incoherent source is not the exact equivalent of a single-mode optical cavity. The situation in TXM is much different; for incoherent bright-field imaging, each pixel in the object can be imaged independently of its neighbors (within good approximation), so one can illuminate all object pixels simultaneously and with nominally incoherent light. If object resolution elements are imaged in \(1:1\) correspondence to detector pixels in a TXM, the number of modes of phase space \(p/\lambda\) that can be accepted in the \(x\) direction is approximately equal to the number of detector pixels in that direction, and the same holds for \(y\). As a result, TXMs are often operated with bending magnet synchrotron radiation sources or laboratory sources that deliver high flux (photons per solid angle), while STXMs and SFXMs are often operated with undulator sources that deliver high brightness (photons per solid angle per source area). The issue of microscope illumination and its effects on image formation will be discussed in greater detail in Sect. 23.3.1.

23.3.1 Microscope Layoutsand Illumination Schemes

Transmission X-Ray Microscope (TXM) Layout

Full-field transmission x-ray microscopes ( s) typically use a zone plate to produce a magnified image of the specimen on a 2-D detector. This approach was pioneered by the group of G. Schmahl at the Universität Göttingen, who, after initial experiments including reflection/grating monochromators [23.132], switched to using a condenser zone plate as the sole monochromator [23.133]. This latter approach is now used by a number of TXMs, including the XM-1 at Lawrence Berkeley Lab [23.134], for which we provide some example numbers. As shown in Fig. 23.15a,ba, the beam from the synchrotron bending magnet source is deflected by a grazing-incidence mirror which filters out the power due to high-energy x-rays, passes through a thin metal filter to remove visible and ultraviolet radiation, and is then imaged by the condenser zone plate onto a pinhole located just upstream of the specimen. As noted in Sect. 23.2.3, on condenser zone plates, the condenser zone plate (of diameter \(D={\mathrm{9}}\,{\mathrm{mm}}\)) and the pinhole (of diameter \(d\approx 10{-}20\,{\mathrm{{\upmu}m}}\)) together are equivalent to a monochromator of resolving power equal to \(D/\left(2d\right)\) [23.132]. Because the light transmitted by the objective zone plate includes a significant undiffracted (zero-order) component which must not reach the detector, the illumination of the sample needs to be hollow-cone, and this is achieved by means of a stop built into the condenser, blocking a central circle with a radius of about one third to one half of the condenser radius. The objective zone plate used by XM-1 in the resolution test described above [23.90] had the following characteristics: outer zone width \(\Updelta r_{N}={\mathrm{15}}\,{\mathrm{nm}}\), diameter \(d={\mathrm{30}}\,{\mathrm{{\upmu}m}}\), 500 zones of \({\mathrm{80}}\,{\mathrm{nm}}\)-thick gold (giving a maximum aspect ratio of \(5:1\)), and focal length \(f={\mathrm{0.3}}\,{\mathrm{mm}}\) at \({\mathrm{815}}\,{\mathrm{eV}}\); however, zone plates with slightly larger zone widths and higher efficiency are used for routine user operations. The vertical phase-space area of the synchrotron source is generally smaller than its horizontal phase-space area and smaller than that of the microscope (which equals object full-width \(d\) times twice the objective NA). Since the condenser zone plate cannot expand the phase space, both the object width and the numerical aperture of the objective of a TXM will generally be underfilled. To counter the underfilling of the object field, the condenser is usually wobbled up and down during the course of an exposure. This type of microscope layout, in which the source is imaged onto the sample, is known as critical illumination [23.73] and is widely used for amplitude contrast. Traditionally, the specimen has been placed in an atmospheric pressure environment. Because the focal length of the objective zone plate is quite small (for example, in the case of a 25-nm-outermost-zone-width, 60-\(\mathrm{{\upmu}m}\)-diameter zone plate operating at \({\mathrm{530}}\,{\mathrm{eV}}\), it would be \({\mathrm{1.3}}\,{\mathrm{mm}}\)), the specimen region lying between these two windows is quite constrained.

The beam then re-enters a vacuum environment where the objective zone plate and the image detector are located. At energies below a few keV, the most common detector is a backside-thinned charge-coupled device (CCD), which is directly illuminated by the x-ray beam; at higher energies, phosphor screens imaged by a visible light lens onto a CCD detector are typically used. Because of the desire to deliver \(10{-}50\,{\mathrm{nm}}\) resolution using detectors with \(1{-}30\,{\mathrm{{\upmu}m}}\) pixel size, the distance from the zone-plate objective to the detector is often in the range of \(1{-}3\,{\mathrm{m}}\) to give acceptably high optical magnification. The approach described above is commonly used with bending magnet and laboratory sources. Particular challenges arise when the source phase-space area is dramatically smaller than desired, which is the case for undulator sources on low-emittance storage rings. A wide variety of solutions have been studied for the BESSY II undulator-based TXM [23.135]. The solution originally adopted [23.136] involved rotating flat mirrors, but capillary condensers (Sect. 23.2.3, Alternatives to Condenser Zone Plates) are now being used instead [23.105].

TXM Phase Contrast Layout

As noted in Sect. 23.1.1, phase contrast plays an important role in x-ray microscopy, particularly at higher photon energies. In order for phase variations at the specimen plane to produce intensity variations at the detector, some method of mixing the wave diffracted by the specimen with an undiffracted phase-reference wave must be employed. The most common approach in x-ray microscopes is that of Zernike [23.137]. In light microscopes, Köhler illumination is provided by using a relay lens to image the source onto the front focal plane of the condenser. Points at this front focal plane deliver parallel beams to the object plane, which (if undeviated by the object) are focused onto a phase ring at the back focal plane of the objective, where they are phase-shifted usually by \(\pm\uppi/2\). Thus the ring aperture at the front focal plane of the condenser provides a narrow, hollow cone of illumination of the specimen, and is conjugate to the phase ring. At the same time, light originating from a point scatterer in the object is focused to the detector, where it interferes with the phase-shifted unscattered light (Fig. 23.17).

Fig. 23.17

Two illustrations of a Zernike phase-contrast optical system . The upper diagram shows the classical scheme used in light microscopes based on Köhler illumination [23.73]. The lower diagram shows a practical synchrotron radiation implementation of a Zernike phase-contrast TXM that was realized at a \({\mathrm{4.1}}\,{\mathrm{keV}}\) branch beamline at ID21 at the European Synchrotron Radiation Facility in Grenoble [23.138]. The trade-offs involved in choosing the illuminated area of the condenser pupil are discussed in Sect. 23.3.3, Coherence in Zernike Phase Contrast. Figure courtesy J. Susini, ESRF

In x-ray microscopes, it is more difficult to use a relay optic to work in the Köhler illumination condition; instead, the front focal plane ring aperture is illuminated by nearly parallel light from the source (i. e., critical illumination) so that a much smaller area of the condenser is illuminated. Because the light from the front aperture is much more collimated than would have been the case with Köhler illumination, the longitudinal location of the aperture and its corresponding phase ring is much less critical than it is with visible light microscopes, so the primary alignment requirement is transverse to the x-ray beam direction. For computations, it is convenient to consider the two rings as built into the lens pupil functions. When the source is well-collimated, as in the case of a synchrotron, there is clearly no need for the two rings to be exactly conjugate. The use of the ring aperture obviously makes the illumination more coherent. The question of the best choice of width and radius for the two rings or, equivalently, how much coherence to have, we defer until Sect. 23.3.3, Coherence in Zernike Phase Contrast. The phase ring itself is constructed out of a material with a large ratio of phase shift to absorption or \(\delta/\beta\) (23.1), such as any material used at energies just below an absorption edge. Following initial discussion [23.32] and demonstrations [23.139] of phase contrast in x-ray microscopy, the Zernike approach was implemented with immediate success [23.140] in a configuration where the ring aperture and phase ring could be rapidly inserted or retracted. Phase contrast is arguably even more important at higher x-ray energies, where it is the dominant contrast mechanism. For example, a hard x-ray (\({\mathrm{4}}\,{\mathrm{keV}}\)) phase-contrast TXM, illuminated by a system using a crystal monochromator followed by a condenser zone plate, was first demonstrated at the ID21 beamline at the European Synchrotron Radiation Facility in Grenoble, France (Fig. 23.17). Imaging of functioning integrated circuits at \({\mathrm{60}}\,{\mathrm{nm}}\) resolution was demonstrated [23.138]; see Sect. 23.4 for more information on this. In another example of this configuration, the National Synchrotron Radiation Research Center (NSRRC) in Hsinchu, Taiwan, installed a commercial TXM from Xradia (now Carl Zeiss X-Ray Microscopy). This instrument (alluded to in Sects. 23.1.3 and 23.2.3, Alternatives to Condenser Zone Plates) has been used to demonstrate the long-standing idea of using zone-plate higher focal orders for imaging. In water-window instruments, which typically have focal lengths on the order of \(1{-}2\,{\mathrm{mm}}\) in first order (and therefore \(1/3\)\(2/3\) mm in third order), the idea has not been readily adopted in light of practical considerations of working distance. However, in the Taiwan experiment [23.104], a phase-contrast image of a fabricated test object was made at \({\mathrm{8}}\,{\mathrm{keV}}\) using a \({\mathrm{50}}\,{\mathrm{nm}}\)-outer-zone-width zone plate in third order. Lines of minimum width \({\mathrm{30}}\,{\mathrm{nm}}\) were imaged clearly, and the authors estimated a resolution below \({\mathrm{25}}\,{\mathrm{nm}}\).

Scanning Transmission X-Ray Microscope (STXM) Layout

Scanning transmission x-ray microscopes (STXMs) typically use a zone plate to demagnify a pinhole source to a small focus spot through which the specimen is scanned. While initial demonstrations using synchrotron radiation used pinhole optics [23.141, 23.43], the use of zone-plate optics in scanning microscopes was pioneered at Stony Brook University [23.142] and later by Göttingen University [23.143]. Since scanning microscopes require coherent illumination to reach their maximum resolution, they have often used undulators as high-brightness sources [23.144, 23.145, 23.146], though excellent performance has also been obtained using bending magnet sources on low-emittance storage rings [23.147]. While a large number of STXMs are now in operation, we describe here the characteristics of one of a series [23.145, 23.148, 23.149, 23.150] of undulator-based scanning microscopes that were built at Stony Brook University for operation at the National Synchrotron Light Source at Brookhaven National Laboratory in New York. A soft x-ray undulator plus spherical grating monochromator with an energy resolution that can be as good as \({\mathrm{0.06}}\,{\mathrm{eV}}\) at \({\mathrm{290}}\,{\mathrm{eV}}\) [23.131] was used to deliver soft x-rays to a 2-D exit slit which could limit the beam size in the range of \(25{-}120\,{\mathrm{{\upmu}m}}\) in both \(x\) and \(y\). This slit then served as a secondary radiation source for zone plates of either 80 or \({\mathrm{160}}\,{\mathrm{{\upmu}m}}\) diameter and zone widths of \(30{-}45\,{\mathrm{nm}}\) [23.151, 23.152, 23.91], producing a focal spot of \(36{-}54\,{\mathrm{nm}}\) Rayleigh resolution. The beam emerged from the ultrahigh-vacuum synchrotron beamline into an atmospheric pressure environment by passing through a \({\mathrm{100}}\,{\mathrm{nm}}\)-thick \(\mathrm{Si_{3}N_{4}}\) window. The zone plate included a central stop of about half the zone plate diameter; this stop must be made quite thick (\({\mathrm{0.3}}\,{\mathrm{{\upmu}m}}\) gold is common for soft x-ray applications) so that the undiffracted light transmitted through the large central stop is kept to a very low level compared to the flux in the focused x-ray beam. The zone plate was then followed by an order-sorting or order-selecting aperture (OSA) so that a pure first-order focal spot was obtained.

While steering mirrors are used to scan the beam in visible light scanning microscopes, it is easier to maintain signal uniformity by keeping the beam and zone plate fixed and scanning the specimen through the focal spot. This is accomplished using an \(x\)\(y\)\(z\) stack of stepping motor stages for large motion with \({\mathrm{1}}\,{\mathrm{{\upmu}m}}\) precision, and a piezo scanning stage for \(50{-}100\,{\mathrm{{\upmu}m}}\)-range and nanometer precision. Because piezos have nonlinearities and hysteresis in their response to scan voltages, some form of closed-loop feedback is generally used, based on position signals such as those provided by linear voltage differential transformers [23.153], capacitance micrometers [23.148], or laser interferometers [23.147, 23.154]. The specimen is then followed by a high efficiency x-ray detector; common choices include the use of gas-based proportional counters, which offer extremely high efficiency of detection for those x-rays that make it through a thin entrance window [23.141, 23.153, 23.155, 23.156] but which suffer from a count-rate limit of about \({\mathrm{1}}\,{\mathrm{MHz}}\). Alternatives are phosphor-coated screens followed by photomultipliers to detect the resulting visible light [23.157], and solid-state detectors which are capable of significantly higher signal rates [23.155, 23.158, 23.159, 23.160, 23.161]. In the Stony Brook STXM, the user was able to choose between proportional counter and segmented silicon detectors, and a visible light microscope was also placed on the detector stage with \(x\)\(y\)\(z\) motorized motion so as to prelocate desired regions of the specimen. Another approach, which works for either a TXM or an STXM, is co-indexing of offline light microscopes with the x-ray microscope [23.147, 23.163].

Scanning microscopes offer characteristics that differ from full-field imaging systems. These include the ability to quickly change from scanning very large areas at low resolution to taking high-resolution, small-field scans, and reduced radiation dose because the \(5{-}20\%\) efficient zone plate is located upstream of the specimen rather than downstream. Because of the need to mechanically scan the specimen in most present-day microscopes, and the need for coherent illumination, imaging times are generally longer (in the range of one or a few minutes, rather than seconds or shorter in the case of many TXMs). At the same time, the requirement for coherent illumination means that the étendue or phase space that the monochromator must accept is greatly reduced, so that aberrations are reduced and it is relatively easy to obtain very high spectral resolution. These characteristics make scanning transmission x-ray microscopes especially well suited to low-dose spectromicroscopy applications, as will be described below.

Phase contrast has historically seen less use in scanning transmission x-ray microscopes. However, refractive and diffractive effects by the specimen lead to a redistribution of signal on the detector, which can be interpreted to give phase-contrast images. One approach is to use quadrant detectors and their variants (Fig. 23.18a-c) to record differential phase-contrast images [23.164], wherein one analysis method involves the measure of the first moment [23.165] or the use of separate transfer function response filters for each segment [23.166, 23.167]. These detectors have the advantage that by using a limited number of specifically shaped pixels in the detector, one can reduce the volume of data that must be recorded. Other approaches include the use of zone-plate doublets [23.168] or phase modulators [23.169] to produce differential phase contrast, or implementation of the Zernike method in scanning microscopes [23.170]. The ultimate approach is to use a 2-D detector (such as a CCD camera) to detect the entire intensity distribution at each pixel of a scanned image; this was used in an impressive demonstration of Wigner deconvolution microscopy to recover the phase and magnitude distribution of the specimen as well as the zone-plate objective [23.171, 23.172], and this leads quite naturally into the following discussion of ptychography.

Fig. 23.18a-c

Amplitude (a) and phase (b) contrast images of a germanium test pattern imaged using a scanning transmission x-ray microscope with a segmented detector [23.162]; (c) schematic view of the detector. The undeflected beam from the first-order focus is directed into bright-field segments 1, 2, and 3 (these segments also allow differential interference contrast). The deflected beam is detected in the angular segments 4, 5, 6, and 7 for dark-field imaging and differential phase contrast

Ptychography Layout

By placing a pixelated area detector such as a soft x-ray CCD camera [23.173] or a hybrid pixel array detector [23.174] at some distance downstream from the focus, one can capture a far-field diffraction pattern from each beam position as a coherent beam is scanned across the specimen. It was first noted in the context of scanning transmission electron microscopy [23.177] that if one recorded successive diffraction patterns as the beam was moved by a fraction of its width, one would obtain common information between the two diffraction patterns that would aid in the recovery of the object from the far-field intensity data; this approach was coined ptychography based on the Greek word for folding. Following the introduction of iterative phase retrieval algorithms [23.178, 23.179] for single illumination positions, an adaptation of these algorithms was demonstrated for ptychographic object reconstruction [23.180, 23.181], with x-ray demonstrations following soon after [23.182, 23.183]. An important advantage of this approach over simpler methods such as differential phase contrast is that one can record coherent diffraction data at larger angles than the numerical aperture of the zone plate, and thus obtain a reconstructed image which is able to show detail much smaller than the size of the focused x-ray spot. Since STXMs require illumination with a high degree of coherence in order to obtain small focal spots, it becomes natural to equip them with pixelated area detectors and to make ptychography a routine imaging modality while also re-creating other transmission imaging modalities using the same data [23.184]. One can also combine a pixelated transmission detector with an energy-dispersive fluorescence detector (Sect. 23.3.1, Scanning Fluorescence X-Ray Microprobe (SFXM) Layout) to record simultaneous ptychography and fluorescence images [23.185, 23.186], as shown in Fig. 23.19a-c. In these experiments, phase contrast provides information on the lighter materials that have poor x-ray fluorescence yield, while fluorescence provides high sensitivity for trace elements. Since ptychography can be used to recover the probe function [23.187] (the magnitude and phase distribution of the coherent illumination beam), one can then use this probe function to improve the visibility of detail in the fluorescence image via probe deconvolution [23.176, 23.188]. Ptychographic reconstruction algorithms have been adapted to deal with illumination with multiple self-coherent but mutually incoherent probe modes [23.189], and this has enabled ptychography to be extended to apply to fast, continuous-motion scanning, first in simulations [23.190] and then in experiments [23.191, 23.192].

Fig. 23.19a-c

Combined x-ray fluorescence images (a), x-ray ptychography image (b), and an overlay of the two (c) obtained on the frozen hydrated algae Chlamydomonas reinhardtii. This image was obtained using \({\mathrm{5}}\,{\mathrm{keV}}\) x-rays focused by a Fresnel zone plate in an instrument designed for x-ray fluorescence imaging of cryogenic specimens [23.175]. Because the pixel array detector used to record the transmission signal extends out to a larger angle than that of the numerical aperture of the zone plate, the ptychography image (b) shows considerably more detail: in this case, sub-\({\mathrm{20}}\,{\mathrm{nm}}\)-resolution features from an \({\mathrm{80}}\,{\mathrm{nm}}\) zone-plate focus. From [23.176], published under CC-BY 4.0 license

Scanning Fluorescence X-Ray Microprobe (SFXM) Layout

Scanning fluorescence x-ray microprobes (SFXM) use a focused x-ray beam to stimulate the emission of characteristic fluorescence x-rays from specific elements in the specimen. When linearly polarized radiation (such as is usually obtained from synchrotron sources) is used, a fluorescence detector placed \(90^{\circ}\) to the beam in the polarization plane will detect a minimum of coherent scattering signal; this detector must then have some means of discriminating between different x-ray emission energies. Energy-dispersive detectors accomplish this by measuring the number of electron–hole pairs created by each x-ray in a semiconductor material, while wavelength-dispersive detectors use a crystal optic or a grating to separate the x-ray energies. Energy-dispersive detectors generally have large solid-angle collection, and multi-element detectors can be used to overcome the \(\approx{\mathrm{1}}\,{\mathrm{MHz}}\) count rate limit determined by charge readout time, while wavelength-dispersive detectors offer better separation between nearby spectral lines and larger dynamic range for detecting low-concentration elements among other fluorescing elements of higher concentration. There is a long and rich history of synchrotron-based microprobes [23.43, 23.44], and a variety of optical approaches including the use of compound refractive lenses and Kirkpatrick–Baez mirror optics are now achieving submicron resolution. We outline here some of the characteristics of microprobes using zone-plate optics [23.158, 23.193, 23.194] by considering the example of the 2-ID-E microprobe at the Advanced Photon Source at Argonne National Laboratory near Chicago.

This microprobe operates using a side-deflecting crystal monochromator to transfer an off-axis part of the central cone produced by a hard x-ray undulator. In the vertical direction, the objective zone plate images the source directly onto the specimen, while in the horizontal direction the variable-width monochromator exit slit is imaged. Astigmatism effects are avoided in the resulting focused beam by the fact that the depth of focus is much larger than the difference between the positions of the horizontal and vertical foci of the zone plate (in addition, the zone plate can be tilted to compensate for more severe source astigmatism). Zone plates of \(160{-}320\,{\mathrm{{\upmu}m}}\) diameter and outermost zone width of \({\mathrm{100}}\,{\mathrm{nm}}\) are typically used, giving focal lengths of \(12{-}25\,{\mathrm{cm}}\) at \({\mathrm{10}}\,{\mathrm{keV}}\). While the probe size can be as small as \({\mathrm{150}}\,{\mathrm{nm}}\), a larger horizontal source size is often chosen to give more flux, at the cost of resolution. The specimen is mounted at \(15^{\circ}\) to the incident beam to provide access to both the incident x-ray beam and the fluorescence detector, and it is scanned by motor-driven stages with \({\mathrm{0.1}}\,{\mathrm{{\upmu}m}}\) step size. A multi-element silicon drift diode fluorescence detector is used to collect the fluorescent signal; one can either record the signal in a limited number of predefined energy windows for rapid analysis with modest data file size, or record the full fluorescence spectrum per pixel for improved quantitation of elements with closely spaced fluorescence energies. The region consisting of the specimen and detector is located inside a glove box that can be purged with helium to eliminate fluorescence from argon in air which would otherwise obscure a number of low-\(Z\) elements, and to reduce the absorption of low-\(Z\) fluorescence signals by air. Per-pixel dwell times are on the order of \({\mathrm{100}}\,{\mathrm{ms}}\), so that the experimenter must be judicious in the choice of scan area (the use of common position indexing between a visible light microscope and the microprobe aids in rapid specimen location). Trace element mapping by fluorescence detection with sensitivities down to about \({\mathrm{100}}\,{\mathrm{ppb}}\), or about \({\mathrm{10^{-17}}}\,{\mathrm{g}}\) of iron within a (\({\mathrm{200}}\,{\mathrm{nm}}\))\({}^{2}\) spot, represents the majority of microprobe applications. However, x-ray microprobes can be used in a number of other ways as well, including measurement of crystal strain in small regions [23.195, 23.196, 23.197] and differential-aperture measurements of microstructure and strain [23.198]. The phase-contrast methods described above for STXM are equally applicable in SFXM, and offer a much-needed means of imaging the overall mass and ultrastructure of specimens while simultaneously forming trace element or strain maps.

23.3.2 Fundamentals of Contrast in the TXM

It is useful to have an analytical treatment that provides insight into the way a microscope produces contrast and at the same time enables practical calculations to assess experimental plans. This has been provided by Rudolph et al [23.52] in a form that allows amplitude-contrast, Zernike phase-contrast, and dark-field imaging to be included in a unified description that is largely independent of the microscope design. Assuming only that we have an imaging microscope, we consider first the Zernike phase-contrast TXM based on their treatment.

We are interested in the contrast \(C\) or the contrast parameter \(\Theta\) (Sect. 23.1) between an interesting feature \(F\) and a background feature \(B\) generated via the phase shifter \(S\). Here, \(F\) and \(B\) are defined as having the same thickness, but in reality the background material (water, for example) may be thicker than the feature, so we allow for that by adding a layer \(L\). If we define the complex transmission factors \(ap=a\exp(\mathrm{i}\varphi)\) for the feature \(F\), we have
$$a_{F}p_{F}\equiv\exp\left(-\displaystyle\frac{2\uppi\beta_{F}t_{F}}{\lambda}\right)\exp\left(\displaystyle\frac{2\uppi\mathrm{i}\delta_{F}t_{F}}{\lambda}\right)$$
(23.40)
with equivalent expressions for the background \(B\), phase shifter \(S\), and overlayer \(L\). In the above, \(1-\delta-\mathrm{i}\beta\) is the refractive index and \(t\) is the thickness. From these amplitudes we can obtain the image intensities (\(I_{F}\) and \(I_{B}\)) as
$$ \begin{aligned}\displaystyle I_{F}&\displaystyle=\left[a_{B}^{2}a_{S}^{2}+2a_{F}a_{B}a_{S}\mathfrak{Re}(p_{F}p_{B}^{*}p_{S}^{*})-2a_{B}^{2}a_{S}\mathfrak{Re}(p_{S}^{*})\right.\\ \displaystyle&\displaystyle\left.\quad\;+a_{F}^{2}-2a_{F}a_{B}\mathfrak{Re}(p_{F}p_{B}^{*})+a_{B}^{2}\right]a_{L}^{2}\end{aligned}$$
(23.41)
$$ I_{B}=a_{B}^{2}a_{S}^{2}a_{L}^{2}$$
(23.42)
and thence the contrast \(C\) and contrast parameter \(\Theta\) as
$$C =\frac{I_{F}-I_{B}}{I_{F}+I_{B}}\;,$$
(23.43)
$$\Theta =\frac{|I_{F}-I_{B}|}{\sqrt{I_{F}+I_{B}}}\;.$$
(23.44)
The dose \(D\) (the energy deposited per unit mass of sample) needed to detect a feature of area \(d^{2}\), thickness \(t_{F}=d\), and density \(\rho\) with signal-to-noise ratio SNR can now be calculated as
$$D=(\mathrm{SNR})^{2}\frac{hc}{\lambda\rho d^{3}}\frac{1-a_{F}^{2}a_{\text{L}}^{2}}{\Theta^{2}}\;,$$
(23.45)
where \(hc={\mathrm{1240}}\,{\mathrm{eV{\,}nm}}\) represents the product of Planck's constant and the velocity of light. The above relations are convenient because, in addition to phase contrast, they also describe amplitude-contrast (\(t_{\text{s}}=0\)) and dark-field (\(t_{\text{s}}=\) large) experiments. The formula for \(D\) yields dose plots such as Fig. 23.5 and also tells us that the number of x-rays (of energy \(E\)) per unit area required to make the measurement with the given resolution and signal-to-noise ratio is \(D\rho/(\mu E)\), where \(\mu\) is the x-ray absorption coefficient.

In the multi-keV x-ray energy range, phase contrast is substantially larger than the absorption contrast for suitable choices of the thickness of the phase shifter. The best result is typically achieved by attenuating the direct beam by the phase plate so that its amplitude is comparable to that of the scattered signal, resulting in an interference of two beams of similar amplitude. The available choices of phase plate thickness to optimize this are positive phase contrast (phase shift \(=\uppi/2\), \(5\uppi/2\), \(\ldots\)) or negative phase contrast (phase shift \(=3\uppi/2\), \(7\uppi/2\), \(\ldots\)). Figure 23.20a,b shows contrast plots of some of these possibilities. Although these plots are useful for providing comparative information, they represent a considerable idealization; the phase shift is assumed to be applied to \({\mathrm{100}}\%\) of the undiffracted light and \({\mathrm{0}}\%\) of the diffracted light, and the thickness of the phase shifter is chosen, at each energy, to give the stated phase shift, and the optical system is assumed to be \({\mathrm{100}}\%\) efficient. Under these assumptions, the dark-field contrast is identically equal to \(1\).

Fig. 23.20a,b

Intrinsic amplitude and Zernike phase contrast for two types of samples of thickness \({\mathrm{30}}\,{\mathrm{nm}}\) relative to a background material of the same thickness: protein in water (a) and vacuum in glass, representing a crack (b). The protein is modeled assuming a density of \({\mathrm{1.35}}\,{\mathrm{g/cm^{3}}}\), composition of \(\mathrm{H_{50}C_{30}N_{9}O_{10}S}\), and a phase ring made of copper. Glass is modeled assuming a density of \({\mathrm{2.5}}\,{\mathrm{g/cm^{3}}}\), composition \(\mathrm{Si_{16}Na_{12}K_{1}Ca_{7}Mg_{6}P_{1}O_{57}}\), and a phase ring made of gold. It is worth noting that the phase contrast can be much greater than the amplitude contrast even in the \(290{-}540\,{\mathrm{eV}}\) water window

23.3.3 Partial Coherence

History

The resolution of microscopes, including x-ray microscopes, depends on the angular widths of the light beams delivered to, and collected from, the sample. The analysis of this effect was pioneered in the 1950s by Hopkins, Wolf, and others, and was part of a movement to apply the linear-systems ideas, widely used by the engineering community, in the optical arena, so several review articles [23.199, 23.200] and books [23.201, 23.202, 23.73] cover the topic. The main point is that the finest features (highest spatial frequencies) in the sample diffract the illuminating beam by the largest angles \(\theta\). The best geometry to include such large deflection angles is therefore one that has a wide-angle beam both inward to, and outward from, the sample. This broadly implies that in a TXM, a large-area source providing spatially incoherent illumination gives better resolution than a point source giving coherent illumination (Fig. 23.21a,b), although such a comparison is not as simple as it sounds [23.203]. It would be wrong to conclude from this that STXMs which use coherent illumination have intrinsically worse resolution than TXMs. In fact, as we will see, all STXMs produce incoherent images when large detectors are used; their resolution depends not on the angle delivered by their source, but rather on the angle collected by their detector.

Fig. 23.21a,b

Schematic showing why the transfer function for incoherent imaging extends to twice the spatial frequency of coherent imaging for a given optical numerical aperture \(\theta\). For coherent imaging (a), the marginal ray is deviated by an angle \(\theta\) due to diffraction by the sample periodicity \(d\). For incoherent imaging (b) some light is deviated by \(2\theta\) due to the sample periodicity \(d/2\). Both TXM and STXM with large-area detectors deliver incoherent bright-field images. Figure adapted after [23.204]

The first application of linear-systems concepts in x-ray microscopy was in the analysis of STXM images [23.148] in which the intensity point spread function and its Fourier transform, the optical transfer function ( ), were calculated. In fact, the magnitude of the OTF, known as the modulation transfer function (MTF), was both calculated and measured for the Stony Brook STXM, and good agreement was obtained (Fig. 23.22). Similar analysis has been provided for TXMs [23.205, 23.206], and experimental measurements have recently been obtained [23.207].

Fig. 23.22

Measured and calculated modulation transfer function ( ) for STXM imaging with a nickel zone plate of outer zone width \({\mathrm{45}}\,{\mathrm{nm}}\) and diameter \({\mathrm{100}}\,{\mathrm{nm}}\). The calculated curve was derived from the known zone-plate aperture function and the size and distance of the source pinhole. After [23.148]

Fourier Optics Treatment

Partially coherent imaging by a microscope can generally be described by the methods of Fourier optics [23.201, 23.202, 23.73]. This method uses a real-space and a frequency-space description of waves in which frequencies (\(u\)) are closely related to directions (\(\theta\)) according to the general ( ) relation \(u=\sin\theta/\lambda\).

We follow here a treatment by Chapman et al [23.208], where we consider first an STXM with amplitude point spread function \(h(\boldsymbol{x})\) imaging a sample of amplitude transparency \(t(\boldsymbol{x})\). The bold characters \(\boldsymbol{x}\) and \(\boldsymbol{u}\) represent two-dimensional coordinates. Using capital letters to represent coordinates in the Fourier transform plane, and \(\mathcal{F}\) to represent the Fourier transform, the pupil function of the lens is \(H(\boldsymbol{u})\), where \(\boldsymbol{u}\) is the general frequency coordinate that is conjugate to the object-plane spatial coordinate \(\boldsymbol{x}\) and has a maximum value of \(\mathrm{NA}/\lambda\), where NA refers to the beam-limiting lens. Any point in the lens pupil or the detection plane can be represented by a \(\boldsymbol{u}\) value. Since the detector in an STXM is placed in the far field of the x-ray focal spot, \(H^{*}(-\boldsymbol{u})\) will represent the diffraction pattern formed in the detection plane in the absence of a sample. When the sample is present and the spot is at \(\boldsymbol{x}_{\text{s}}\), the wave field immediately behind the sample is \(h(\boldsymbol{x})t(\boldsymbol{x}-\boldsymbol{x}_{\text{s}})\), and the field in the far-field detection plane is given by the Fourier transform of that. The detected intensity is therefore
$$F(\boldsymbol{u},\boldsymbol{x}_{\text{s}})=\left|H(\boldsymbol{u})\otimes_{u}T(\boldsymbol{u})\,\mathrm{e}^{2\uppi\mathrm{i}\boldsymbol{x}_{\text{s}}\cdot\boldsymbol{u}}\right|^{2}$$
(23.46)
where \(\otimes\) represents convolution, and the convolution and shift theorems have been used. The same quantity \(F(\boldsymbol{u},\boldsymbol{x})\) can also be represented in another useful way. By inserting the representations of \(H(\boldsymbol{u})\) and \(T(\boldsymbol{u})\) as Fourier integrals into the convolution integral (23.46), and using the Fourier-integral definition of the delta function [23.73] and then its sifting property, one obtains [23.208]
$$F(\boldsymbol{u},\boldsymbol{x})=\left|h(\boldsymbol{x})\otimes_{x}t(\boldsymbol{x})\,\mathrm{e}^{-2\uppi\mathrm{i}\boldsymbol{x}\cdot\boldsymbol{u}}\right|^{2}\;.$$
(23.47)
Equation (23.46) represents the diffraction pattern formed in the detection plane by an STXM at each scan position as \(F(\boldsymbol{u},\boldsymbol{x}_{\text{s}})\), regarded as a function of \(\boldsymbol{u}\) for a given \(\boldsymbol{x}_{\text{s}}\). Equation (23.47) represents a coherent image in a TXM, for illumination direction \(\boldsymbol{u}\), as \(F(\boldsymbol{u},\boldsymbol{x})\), regarded as a function of \(\boldsymbol{x}\) for a given \(\boldsymbol{u}\). In (23.46) the exponential represents the scan shift \(\boldsymbol{x}_{\text{s}}\) and in (23.47) it represents the incoming plane wave at direction \(\boldsymbol{u}\). This optical equivalence of the STXM and TXM is known as reciprocity and is further discussed in Sect. 23.3.3, Reciprocity. To obtain the delivered intensity image \(I(\boldsymbol{x})\) in either case, one has to integrate the signal in the detection plane over the particular distribution of \(\boldsymbol{u}\) values that are used. That is, in STXM we integrate over the intensity response function of the detector \(|D(\boldsymbol{u})|^{2}\), while in TXM we similarly integrate over the intensity distribution in \(\boldsymbol{u}\) delivered by the source \(|S(\boldsymbol{x})|^{2}\), or
$$I_{\text{STXM}}(\boldsymbol{x}) =\int_{\text{DET}}F(\boldsymbol{u},\boldsymbol{x})\,|D(\boldsymbol{u})|^{2}\,\mathrm{d}\boldsymbol{u}\;,$$
(23.48)
$$I_{\text{TXM}}(\boldsymbol{x}) =\int_{\text{SOURCE}}F(\boldsymbol{u},\boldsymbol{x})\,|S(\boldsymbol{u})|^{2}\,\mathrm{d}\boldsymbol{u}\;.$$
(23.49)
If the condenser is approximately incoherently illuminated (as specified in (13) of Sect. 10.5.1 of [23.73]), which is often the case for TXMs [23.54, 23.98], then the effective source [23.199] will be the condenser lens aperture function \(|S(\boldsymbol{u})|^{2}\). The expression for the fully incoherent bright-field image (\(|D(\boldsymbol{u})|^{2}=1\) or \(|S(\boldsymbol{u})|^{2}=1\)) is the same in both TXM and STXM and is obtained by inserting (23.46) into (23.48) and applying Parseval's theorem  [23.208] to arrive at
$$I_{\text{BF}}(\boldsymbol{x})=|h(\boldsymbol{x})|^{2}\otimes_{x}|t(\boldsymbol{x})|^{2}\;.$$
(23.50)
The coherent bright-field image is available from an STXM by using an axial point detector and from a TXM by using an axial point source. Both are given by \(F(0,\boldsymbol{x})\), although neither is widely used in x-ray microscopy. The process of integrating over \(S\) or \(D\), which is carried out automatically by the hardware of the TXM or STXM, is generally convenient but it destroys potentially useful information about the sample. A procedure for capturing this information, by storing the full detection-plane pattern at every pixel position of the STXM image, was demonstrated some time ago [23.209], but at that time the limitations of soft x-ray CCD readout times (30 seconds per image) limited its utility. With modern high-frame-rate pixel detectors such as the PILATUS hybrid pixel array detector [23.174], this limitation has been removed, and detectors of this type are being implemented on a growing number of scanning x-ray microscopes (both STXM and SFXM), allowing one to obtain a variety of imaging modes based simply on the way the area detector data is processed [23.184], including ptychography as discussed in Sect. 23.3.1, Ptychography Layout. Equations (23.47) and (23.50) show that coherent imaging is linear in the amplitude, and incoherent imaging is linear in the intensity, respectively. On the other hand, as we see below, partially coherent imaging is not linear in either.

Contrast Transfer

In the case that we do not have \(|D(\boldsymbol{u})|^{2}=1\) or \(|S(\boldsymbol{u})|^{2}=1\), the above procedure used to obtain (23.50) does not lead to such a simple result, but rather to the following expression representing partially coherent imaging in an STXM [23.201, 23.210, 23.73] of
$$\begin{aligned}\displaystyle I(\boldsymbol{x})&\displaystyle=\iint_{-\infty}^{+\infty}\iint C(\boldsymbol{u};\boldsymbol{p})T(\boldsymbol{m})T^{*}(\boldsymbol{p})\\ \displaystyle&\displaystyle\quad\times\mathrm{e}^{-2\uppi\mathrm{i}[(\boldsymbol{m}-\boldsymbol{p})\cdot\boldsymbol{x}]}\,\mathrm{d}\boldsymbol{m}\,\mathrm{d}\boldsymbol{p}\;,\end{aligned}$$
(23.51)
with
$$C(\boldsymbol{m};\boldsymbol{p})=\iint_{-\infty}^{+\infty}|D(\boldsymbol{u})|^{2}H(\boldsymbol{u}-\boldsymbol{m})H^{*}(\boldsymbol{u}-\boldsymbol{p})\,\mathrm{d}\boldsymbol{u}\;.$$
(23.52)
For a TXM, \(S\) replaces \(D\) in the last equation. The integration variables \(\boldsymbol{m}\) and \(\boldsymbol{p}\) in (23.51) are frequencies similar to \(\boldsymbol{u}\), but \(\boldsymbol{m}\) represents a ray incident on the sample, while \(\boldsymbol{p}\) represents a ray emerging from it. The ranges of frequencies included in these beams by the form of \(S\) or \(D\) determine the range of periodicities \((\boldsymbol{m}-\boldsymbol{p})\) in the sample that contribute to the image and thus determine the extent of the modulation transfer function (MTF) in frequency space. The function \(C(\boldsymbol{m};\boldsymbol{p})\) is known in optics as the transmission cross coefficient [23.73] or the partially coherent transfer function [23.201], and provides a sample-independent description of the effect of both the illumination and the optical system on the transfer of information from object to image. It is not a true transfer function, since the transfer is not linear, but is a member of a wider class of bilinear transfer functions. These functions  [23.211] have been applied to partially coherent x-ray imaging [23.98].

The function \(C(\boldsymbol{m};\boldsymbol{p})\) is widely used in the optical and electron microscopy communities, and its properties have been worked out in detail [23.201, 23.212]. It is normally a  function, but in the case of a 1-D object it becomes the 2-D function \(C(m;p)\). The value of \(C(m;p)\) is then equal to the overlap integral of the three appropriately shifted aperture functions in the integrand of (23.52) [23.201, 23.210, 23.73]. For many cases of interest in both TXM and STXM, all three are circular disks or annuli (Figs. 23.23a-c and 23.24a,b). For the ideally incoherent bright-field image, the value of \(D\) or \(S\) is taken to be unity for all frequencies, and the overlap then depends only on the difference \(m-p\) of the shifts of \(H\) and \(H^{*}\). That is, there is only one response to the sample frequency \(s=m-p\) irrespective of \(m\), which indicates a linear system with MTF equal to \(C(m-p;0)/C(0;0)\). For forms of \(D\) or \(S\) corresponding to partial coherence, the system is not linear. On the other hand, a dark-field configuration must have detector (or source) and lens aperture functions that have zero overlap at \(m=p=0\). For example, \(D\) or \(S\) could be the Babinet inverse of \(H\). For circular functions of the latter type, \(C(m;p)=0\) if \(\mathrm{sign}(m)\neq\mathrm{sign}(p)\). Examples of both bright- and dark-field transfer functions for aperture geometries that are representative of an STXM and that show the above characteristics are shown in Fig. 23.24a,b. The response of the same systems to a grating-like object are also shown in Fig. 23.23a-c. Dark-field STXM is particularly well suited to imaging samples with small features, such as gold labels that scatter by large angles [23.208]. The procedures outlined above enable the calculation of the MTF and the resolution behavior of both types of x-ray microscopes based on a knowledge of the resolution-determining lens and the geometry of the source or detector. It is worth noting that, as in other types of microscopes, the resolution does not depend on aberrations of the condenser if there is one. As noted in Sect. 23.3.1, Transmission X-Ray Microscope (TXM) Layout, and illustrated in Fig. 23.17, the placement of a ring aperture and phase ring to obtain Zernike phase contrast in a TXM may be modeled as modifications of the source and lens aperture functions. By this means, the above method of analysis may be applied to this case as well [23.212, 23.213, 23.214].

Fig. 23.23a-c

Imaging of a one-dimensional cosine amplitude grating object using an STXM. The top row shows the signal in the detection plane. The mittle and bottom rows show gray-level plots of \(C(m,p)\) in \(m\)\(p\) space overlaid with the support boundary of \(C(m,p)\) (solid lines) and the spectra \(T(m)T^{*}(p)\) as in (23.51) (spots). The middle row shows bright field and the bottom row dark field. The three columns correspond to grating frequencies that are (a\(> 2\), (b) between 1 and 2, and (c\(<1\), expressed in units of the maximum values of \(m\) and \(p\), which are both equal to NA/\(\lambda\). Reprinted from [23.208], with permission from Elsevier

Fig. 23.24a,b

Bright-field (a) and dark-field (b) partially coherent transfer functions for a 1-D object and an annular lens with inner radius equal to 0.44 times the outer. The function \(C(m,p)\) is plotted against \(m\) and \(p\), expressed as multiples of their maximum value NA/\(\lambda\). Reprinted from [23.208], with permission from Elsevier

Reciprocity

The general conclusion of the above analysis is that the optical systems of the TXM and STXM are the same with the position of the lens, before or after the sample, interchanged and the role of the source and detector interchanged. This is the reciprocity relationship that has long been recognized in the electron [23.215, 23.216] and visible light [23.217] imaging communities, and which has since been discussed by Morrison et al [23.165, 23.214] in the context of x-ray imaging. Thus we might expect that, given identical resolution-determining lenses, a TXM and an STXM (both operating in incoherent bright-field mode) could utilize wide-angle beams and achieve good resolution equally well. For TXM, the requirement would be that the condenser should deliver a wide angle to the sample, and for STXM that the detector should collect a wide angle from the sample. However, in the past, the practical realization of a wide-angle condenser for a TXM has been much more difficult than a wide-angle detector for an STXM, as we discussed in Sect. 23.2.3, Condenser Zone Plates; in general it is easier to purchase a large detector than it is to obtain a high-quality condenser.

In practice, the TXM/STXM relationship is not quite as symmetrical as the above account suggests, because of the general use of zone plates with a central stop for STXM (Sect. 23.3.1, Scanning Transmission X-Ray Microscope (STXM) Layout) but not for TXM. The stop produces a point-spread function that has a narrower central peak but larger side lobes. As a consequence, the frequency response (the MTF, Fig 23.22) is increased in the high-frequency and decreased in the low-frequency regions.

The Influence of Coherence on Resolution

Calculations of the transfer function as an overlap area of three aperture functions in the integrand of (23.52) were discussed in Sect. 23.3.3, Contrast Transfer. For standard TXM, these functions are circular and two of them are the same. One can therefore follow [23.199] and characterize the illumination by a coherence parameter \(\sigma\) defined by the ratio of the condenser and objective numerical apertures or \(\sigma=\mathrm{NA}_{\text{c}}/\mathrm{NA}_{\text{o}}\). Full coherence is represented by \(\sigma=0\) and full incoherence by \(\sigma\rightarrow\infty\), although \(\sigma=1\) is usually sufficient to get close to fully incoherent behavior. We start by considering the modulation (the percentage dip in the valley between the peaks) for a two-point object with separation \(0.61\lambda/\mathrm{NA}_{\text{o}}\) . The modulation is \({\mathrm{26.5}}\%\) for incoherent illumination [23.73], and according to the Rayleigh criterion, the two points are just resolved. It is common practice [23.73] to extend the Rayleigh criterion to other pairs of objects and define them as resolved if the modulation is at least \({\mathrm{26.5}}\%\). An example is the two-point object with in-phase coherent illumination for which the just-resolvable separation is \(0.82\lambda/\mathrm{NA}_{\text{o}}\). Further detail of this [23.205] can be seen from the plots in Fig. 23.25. An important example for x-ray imaging is the modulation due to a periodic object, in particular a square-wave transmission object. Such an object be prepared can either by standard lithography methods [23.148] or, for finer line widths, by preparing thin cross sections of synthetic multilayers [23.96] to yield resolution test patterns for an x-ray microscope. If the resolution is defined as the half-period of the finest square wave that can be imaged with \({\mathrm{26.5}}\%\) modulation and is expressed as \(k_{1}\lambda/\mathrm{NA}_{\text{o}}\), then [23.90] the diffraction-limited value of \(k_{1}\) is 0.5 for a coherent system (\(\sigma=0\)) and 0.4 for \(\sigma=0.38\) (the actual value for XM-1 illuminating a \({\mathrm{15}}\,{\mathrm{nm}}\) zone plate). Thus the diffraction-limited resolution of their experiment was \(0.8\Updelta r_{N}\), while the value achieved was about \(1.0\Updelta r_{N}\) (\(<{\mathrm{15}}\,{\mathrm{nm}}\)). The data demonstrating the achieved resolution are shown in Fig. 23.13. By reciprocity, the same arguments about the value of the diffraction-limited resolution would apply for an STXM in incoherent bright-field mode using the same zone plate. With a large enough detector, one would achieve \(\sigma=1\) and the same diffraction-limited resolution as a TXM with \(\sigma=1\).

Fig. 23.25

Image contrast as a function of \(1/d\), where \(d\) is the point separation of a two-point object imaged by a lens with a circular pupil and coherence parameter \(\sigma\) equal to 0 (coherent case), 0.5, 1, and \(\infty\) (incoherent case). \(d\) is expressed in units of \(\lambda\)/NA. Note that the Rayleigh resolution corresponds to \({\mathrm{15.3}}\%\) intensity contrast (defined as \((I_{\text{max}}-I_{\text{min}})/(I_{\text{max}}+I_{\text{min}})\)), which is the same as \({\mathrm{26.5}}\%\) amplitude modulation. It occurs at \(d=0.61\lambda\)/NA for both \(\sigma=1\) and \(\sigma\rightarrow\infty\). After [23.205]

Coherence in Zernike Phase Contrast

We return now to the question of the choice of width and radius for the two rings in a Zernike phase-contrast configuration of the TXM, or, equivalently, how much coherence is desirable in this case. This choice has been discussed [23.213] and is essentially a trade-off between light collection and the distorting effects arising from the fact that the phase ring must have finite area, which applies an unintended phase change to a certain portion of the diffracted light, causing the so-called halo effect [23.218]. It is generally thought that one needs very little coherence; that is, the ring aperture can leave a large fraction of the condenser area open. This is true if the requirement is merely to make otherwise invisible phase features, especially phase jumps, become visible. However, with low coherence, a phase step is rendered as a double-peaked zero-crossing function, and a phase rect function is rendered as two such double peaks. How much coherence do we need to get anything resembling a faithful rendition of the object? We have not found much attention to this point in the literature, but one treatment [23.219] provides an answer, which is confirmed by our own computer modeling. To obtain a rendition of a rect function that looks like the original function, one needs to have the coherence width \(w_{\text{c}}=\lambda f_{\text{condenser}}/\Updelta r_{\text{ring}}\) of light arriving at the sample at least equal to the width of the rect function.

Propagation-Based Phase Contrast

Another way to achieve phase contrast is to exploit the \(\exp[\mathrm{i}\uppi(x^{2}+y^{2})/(\lambda z)]\) phase shifts that occur in the \((x,y)\) plane as a result of the propagation of a coherent wave field through a distance \(z\). This is exploited in x-ray holographic microscopy, which has had many successful demonstrations [23.220, 23.221, 23.222]. One major use of x-ray holography today is for the study of magnetic materials [23.223], where the relative simplicity of the experiment enables sophisticated control of magnetic fields synchronized with short-pulse illumination. Another is the use of holography for phase-contrast tomography at higher x-ray energies, in a point projection scheme [23.224] using a phosphor/lens/CCD detector system. While it lies beyond the scope of the present chapter's emphasis on zone-plate x-ray microscopy, this unique approach is providing impressive 3-D reconstructions of difficult specimens, including foams.

23.3.4 Tomography in X-Ray Microscopes

Principle of Operation

As was noted in Sect. 23.2.3 on hard x-ray zone plates, the transverse resolution of a zone plate operated in first diffraction order is given by \(0.61\lambda/\mathrm{NA}=1.22\Updelta r_{N}\), and the depth of focus is \(2\lambda/(\mathrm{NA})^{2}=8(\Updelta r_{N})^{2}/\lambda\). Since current zone plates have outermost zone widths \(\Updelta r_{N}\) that are much larger than the wavelength \(\lambda\), this means that the depth of focus is necessarily large compared to the resolution, as can be seen in the illustration of the 3-D modulation transfer function in Fig. 23.26a,b. This provides an opportunity for 3-D imaging if the object is smaller than the depth of focus, because a 2-D image can then be interpreted as providing a simple projection through the specimen, which is precisely what conventional tomography requires at each viewing angle. Tomography with electron microscopes is long established, and following earlier demonstrations using the Stony Brook STXM [23.225] and the Göttingen TXM at BESSY I [23.226], a number of groups are now using x-ray microscopes for tomographic studies of frozen hydrated biological specimens and integrated circuits (among other applications that will be described below in Sect. 23.4). Although the technique used in these studies is improving, they have still not reached the resolution achieved by the same microscopes in 2-D, in part because of the challenges in aligning projection images onto the true rotation axis.

Fig. 23.26a,b

Properties of soft x-ray tomography using zone-plate optics. (a) The 3-D modulation transfer function for monochromatic, spatially incoherent bright-field imaging with a 45-nm-outer-zone-width zone plate with a half-diameter central stop, as a function of depth. One can see that the good in-focus frequency response is preserved over a total depth of about \({\mathrm{8}}\,{\mathrm{{\upmu}m}}\). This is useful for many tomography experiments that rely on the fact that the delivered image is a projection of the object. (b) The same information for a \({\mathrm{20}}\,{\mathrm{nm}}\) zone plate; as can be seen, the figure scales quite well from the figure at the left according to the square of the ratio of the finest zone widths. With a \({\mathrm{20}}\,{\mathrm{nm}}\) zone plate and monochromatic illumination, good frequency response is preserved only over a depth of about \(0.5{-}1\,{\mathrm{{\upmu}m}}\), which is much more restrictive, illustrating the challenges of improved resolution in TXM zone-plate tomography

The Depth-Of-Focus Limit

Given the successes achieved and the amount of current interest, what are the issues to be faced in improving the resolution of tomography in x-ray microscopes? One of them concerns the same depth of focus that makes such tomography straightforward. To our knowledge, all demonstrations of soft x-ray tomography reconstructions have used zone plates with outermost zone width \(\Updelta r_{N}\) no finer than \({\mathrm{35}}\,{\mathrm{nm}}\), so that the depth of focus in the water-window region is at least \({\mathrm{4}}\,{\mathrm{{\upmu}m}}\), which has been comparable to the specimen size. As higher-resolution zone plates are employed, the depth of focus will decrease as the square of improvements in transverse resolution, so that a zone plate with an outermost zone width of \({\mathrm{15}}\,{\mathrm{nm}}\) would have a depth of focus of about \({\mathrm{0.8}}\,{\mathrm{{\upmu}m}}\). This approaches the \(\approx{\mathrm{0.5}}\,{\mathrm{{\upmu}m}}\) thickness accessible to cryo-electron tomography of frozen hydrated cell regions at \(6{-}8\,{\mathrm{nm}}\) resolution [23.50]. The depth-of-focus limit clearly poses a considerable challenge to those who would exploit zone-plate x-ray microscopes for tomography. There are several strategies for overcoming this challenge:
  • Local tomography: This method [23.227] reconstructs a smaller region out of a larger object, where the smaller region is fully within the field of view in all projection angles. In this case, the depth-of-focus limit applies to the partial volume rather than the whole volume, and the signals generated by the out-of focus parts of the sample reduce, after combining all members of the tilt series, to a fairly smooth background. The size of the region that is always in focus will be roughly a sphere of diameter equal to the depth of focus.

  • Increased bandwidth: The depth-of-focus calculation given above is for monochromatic illumination, which is applicable to demonstrations of tomography of frozen hydrated cells in an STXM with a high-energy-resolution monochromator [23.228]. If one uses an objective zone plate with \(N=375\) zones, the recommended bandwidth for such an objective is \(E/\Updelta E> N\) [23.71]. However, it has been shown that the use of a condenser zone plate and pinhole size leading to \(E/\Updelta E\simeq 200\) leads to significant changes in the modulation transfer function (MTF) as a function of defocus [23.229] (similar results for different cases have also been reported [23.230]). In calculations for a zone plate with \({\mathrm{40}}\,{\mathrm{nm}}\) outermost zone width, the MTF at a spatial frequency of \({\mathrm{15}}\,{\mathrm{{\upmu}m}}\)\({}^{-1}\) (corresponding to a spatial half-period of \({\mathrm{33}}\,{\mathrm{nm}}\)) declines from about 0.28 in focus, to about 0.015 at a defocus of \({\mathrm{4}}\,{\mathrm{{\upmu}m}}\) in the monochromatic case. When the bandwidth is increased to \(E/\Updelta E\simeq 200\), the MTF at \({\mathrm{15}}\,{\mathrm{{\upmu}m^{-1}}}\) is reduced in focus to only about 0.11, but at the same \({\mathrm{4}}\,{\mathrm{{\upmu}m}}\) defocus it degrades much less to 0.08. Although the transverse resolution and efficiency for the collection of structural information are both made worse by the use of high-bandwidth radiation, the consistency of imaging conditions is maintained over a larger depth of field. (To our knowledge, no calculations have yet addressed the possibly interesting question of how this trade-off with nonmonochromatic illumination compares with a trade-off of simply using a lower-resolution zone plate to improve the depth of focus).

  • Through-focus deconvolution: One possible approach to beat the depth-of-focus limit is to use through-focus deconvolution as is done in light and electron microscopy. In electron microscopy, the recording of defocus image sequences is routine; each defocus provides positive and negative phase contrast at various bands of spatial frequencies along with zeros in the transfer function, and the combination of several images can provide a complete image of the specimen [23.231]. In fluorescence light microscopy, through-focus image sequences can yield a high-quality 3-D image through the use of deconvolution of the 3-D point spread function [23.232, 23.233]. However, there are important differences between these examples and the situation present in x-ray microscopy. In electron microscopy, this approach is usually applied to thin samples for which phase contrast dominates (indeed the specimen focus can be quickly estimated by looking for a minimum in image contrast). In light microscopy, the use of fluorescence means that the object is a sparse, pure-real function (incoherent emission from independent fluorescence emitters with no sensitivity to the relative phase of the illumination), so that the deconvolution can be done based on the intensity point spread function. In x-ray microscopy, one must account for the fact that biological specimens imaged at water-window energies produce both absorption and phase contrast, thus requiring exact knowledge of the complex bilinear transfer function of the zone-plate optic and illumination system. In other words, the problem of 3-D deconvolution of a strongly absorbing, optically thick, complex object with partial coherence is much more difficult than the cases in which optical sectioning is typically used at present. However, one can take an approximate approach by collecting tomography data at a set of focus positions for each rotation angle, and then combining the data in a way that emphasizes the high-contrast, in-focus features from each of these planes [23.207, 23.234].

  • Use of higher photon energies: The use of shorter wavelengths (higher x-ray energies) to increase the monochromatic depth of field \(\simeq 8\Updelta r_{N}^{2}/\lambda\) of (23.39) is a guaranteed way to extend the depth of focus. The associated questions as to what the cost in resolution, contrast, and efficiency will be are also now beginning to be answered. This approach has been used to obtain sub-\({\mathrm{100}}\,{\mathrm{nm}}\)-resolution tomographic reconstructions of metallic layers within thinned integrated circuits using a laboratory x-ray microscope operating at \({\mathrm{5.4}}\,{\mathrm{keV}}\) [23.235] and \({\mathrm{8}}\,{\mathrm{keV}}\) [23.106], as will be described in Sect. 23.4.3. For lower-density specimens, the use of hard x-rays naturally leads to the use of phase contrast, which is far more dose-efficient than absorption contrast in this energy region. As shown in Fig 23.5, this enables multi-keV imaging at dose levels similar to the water window. As noted in Sect. 23.3.3, Propagation-Based Phase Contrast, there are numerous successful demonstrations of phase-contrast tomography using hard x-rays, including at the NSRRC synchrotron light source in Taiwan, where they obtained 3-D images of a microcircuit at \({\mathrm{60}}\,{\mathrm{nm}}\) resolution [23.236].

23.3.5 X-Ray Spectromicroscopy

As noted in Sect. 23.1.1, x-ray absorption edges arise when the x-ray photon reaches the threshold energy needed to completely remove an electron from an inner-shell orbital. At photon energies within about \({\mathrm{10}}\,{\mathrm{eV}}\) of the edge, electrons can also be promoted to unoccupied or partially occupied molecular orbitals (Fig. 23.27); photons over a narrow energy range are sometimes able to excite inner-shell electrons into such orbitals, giving rise to absorption resonances. This so-called x-ray absorption near-edge structure ( ) or near-edge x-ray absorption fine structure ( ) is highly sensitive to the local chemical bonding state of the atom in question [23.237].

Fig. 23.27

Schematic of a K-shell x-ray absorption edge, which involves the removal of a \(n=1\) state inner-shell electron, and a near-edge absorption resonance in which the electron is promoted to a partially occupied or vacant molecular orbital. These resonances are referred to as the x-ray absorption near-edge structure (XANES) or near-edge x-ray absorption fine structure (NEXAFS)

One can exploit these resonances as an additional contrast mechanism in soft x-ray imaging. In electron energy-loss spectroscopy ( ), the equivalent contrast mechanism is known as the energy-loss near-edge structure ( ), and its use in energy-loss spectrum imaging [23.238, 23.239] is described elsewhere in this volume. Early efforts in x-ray imaging included the use of XANES resonances to enhance the sensitivity of differential absorption measurements of calcium in bone [23.153], spectrum imaging [23.240] and microspectroscopy in photoelectron microscopes [23.241], and photoelectron and transmission imaging at selected photon energies [23.242, 23.37]. It is now common to take image sequences across x-ray absorption edges [23.243], yielding data sets with a full near-edge spectrum per pixel. When comparing spectrum imaging in electron versus x-ray microscopes, a few comments are in order:
  • ELNES is typically performed using a fixed electron energy in the range of \(80{-}200\,{\mathrm{keV}}\). The ideal specimen thickness is under \({\mathrm{100}}\,{\mathrm{nm}}\) in most cases.

  • In ELNES, one obtains spectroscopic information over a wide range of energies, including plasmon energies of \(\approx{\mathrm{10}}\,{\mathrm{eV}}\), in a single measurement. However, plural inelastic scattering dominates the signal at higher energies (for example, electrons can lose \({\mathrm{300}}\,{\mathrm{eV}}\) once or \({\mathrm{50}}\,{\mathrm{eV}}\) six times), resulting in poorer signal-to-background ratios.

  • In x-ray absorption spectroscopy, one must tune the incident x-ray energy across each absorption edge of interest. The optimal specimen thickness of about \(1/\mu(E)\) changes accordingly, so that in the ideal case, one would require samples of several different thicknesses to study chemical speciation of several elements. However, x-rays suffer almost no plural inelastic scattering , which leads to an improved signal-to-background ratio.

  • It is common to find scanning x-ray microscopes operating with monochromators with an energy resolution of \(0.03{-}0.1\,{\mathrm{eV}}\) or better. Most electron microscopes have an energy resolution of \(0.5{-}0.7\,{\mathrm{eV}}\), which leads to blurring of near-edge spectral features, although some electron microscopes now include electron-optical monochromators to narrow the energy bandwidth from the electron gun, achieving \({\mathrm{0.014}}\,{\mathrm{eV}}\) energy resolution [23.244].

  • Using XANES, one can exploit the favorable characteristics of x-ray microscopes, including the ability to study hydrated specimens and/or specimens in an ambient atmosphere environment. This is more difficult to do in electron microscopes because of the shorter penetration power in windows and liquids, and restricted space between electron pole pieces, although there is increasing activity in this area.

Other comparisons between EEELS and NEXAFS have led to the conclusion that the radiation dose imparted in x-ray spectromicroscopy is often substantially reduced relative to EELS [23.245, 23.246].

We now introduce multivariate analysis methods for treating spectromicroscopy data [23.247], using here a notation developed for x-ray applications [23.248]. In x-ray microscopes, we obtain images (maps of transmitted flux \(I(E)\) at each of the energies \(E\)) according to the Lambert–Beer law for absorption (23.2) of \(I(E)=I_{0}(E)\exp[-\mu(E)t]\). The value of \(\mu(E)\) for near-edge absorption resonances can be calculated based on the electronic structure of specific molecules, and this has been employed in detailed studies via microscopy of the absorption spectra of polymers [23.249] and amino acids [23.250] (Fig. 23.28), to name two examples.

Fig. 23.28

Near-carbon-edge absorption spectra of several amino acids, showing the effects of various molecular bonds in the absorption spectrum. These resonances can be used for chemical contrast in x-ray microscopy. Adapted with permission from [23.250]. Copyright 2002 American Chemical Society

For a thickness \(t\) of a single material, a measurement of the transmitted flux \(I(E)\) relative to the incident flux \(I_{0}(E)\) provides a means of calculating the energy-dependent optical density
$$D(E)=-\ln\left[\frac{I(E)}{I_{0}(E)}\right]=\mu(E)t\;.$$
(23.53)
If, however, we measure the optical density not over a continuous energy range \(E\), but at some set of \(n=1,\dots,N\) discrete energies \(E_{n}\), we then measure
$$D_{n}=\mu_{n}t$$
(23.54)
for each of the \(n=1,\dots,N\) photon energies. Let us next consider a mixture of \(s=1,\dots,S\) different materials with partial thicknesses \(t_{s}\); our total measurement of optical density \(D_{n}\) at one photon energy is given by the combined absorption of all the materials, or
$$D_{n}=\mu_{n1}t_{1}+\mu_{n2}t_{2}+\dots+\mu_{ns}t_{S}\;.$$
(23.55)
Finally, if we carry out this measurement not from a single homogeneous uniform film, but from heterogeneous pixels \(p=1\ldots P\) indexed by \(p=i_{\text{column}}+(i_{\text{row}}-1)\cdot(\#\ \mathrm{columns})\) in an image, the optical density measured at one pixel \(p\) is given by
$$D_{np}=\mu_{n1}t_{1p}+\mu_{n2}t_{2p}+\dots+\mu_{nS}t_{Sp}\;.$$
(23.56)
When all \(N\) photon energies are considered, we see that we have a data matrix \(\mathbf{D}_{NP}\) of
$$\begin{aligned}\displaystyle&\displaystyle\begin{bmatrix}D_{11}&\cdots&D_{1P}\\ \vdots&\ddots&\vdots\\ D_{N1}&\cdots&D_{NP}\end{bmatrix}=\\ \displaystyle&\displaystyle\begin{bmatrix}\mu_{11}&\cdots&\mu_{1S}\\ \vdots&\ddots&\vdots\\ \mu_{N1}&\cdots&\mu_{NS}\end{bmatrix}\begin{bmatrix}t_{11}&\cdots&t_{1P}\\ \vdots&\ddots&\vdots\\ t_{S1}&\cdots&t_{SP}\end{bmatrix}\end{aligned}$$
(23.57)
or \(\mathbf{D}_{N\times P}=\boldsymbol{\upmu}_{N\times S}\times\mathbf{t}_{S\times P}\). In other words, the data represent a series of spectral signatures \(\boldsymbol{\upmu}_{N\times S}\) and thickness maps \(\mathbf{t}_{S\times P}\) for the \(S\) chemical components.
When we acquire a series of images at different photon energies \(N\), we are in fact measuring the data matrix \(\mathbf{D}_{N\times P}\). If we know the exact absorption spectrum \(\upmu_{Ns}\) for each of the \(s=1,\dots,S\) components in the sample, then we can find the spatially resolved thicknesses \(\mathbf{t}_{S\times P}\) of the components by matrix inversion
$$\mathbf{t}_{S\times P}=\boldsymbol{\upmu}^{-1}_{S\times N}\times\mathbf{D}_{N\times P}\;.$$
(23.58)
The inversion of the matrix of spectra from all known components can be accomplished in a robust fashion using singular-value decomposition [23.251, 23.252]. This approach, as well as approaches that involve pixel-by-pixel least-squares fits of all reference spectra, work well with specimens that involve mixtures of components that can all be measured separately. Examples using this approach are shown in the chemical imaging section of this chapter.

In many areas of research, such as biology or environmental science, the complexity of the specimen and the possibility of reactions between components means that one cannot know in advance the set of all absorption spectra \(\boldsymbol{\upmu}_{N\times S}\) present in the specimen. In this case, one approach that has yielded success is to first use principal component analysis [23.240, 23.253, 23.254] to orthogonalize and noise-reduce the data matrix \(\mathbf{D}_{N\times P}\), and then use cluster analysis (a method of unsupervised pattern recognition [23.255]) to group pixels together based on similarity of spectral signatures [23.248, 23.256] (Fig. 23.35). This method yields a set of absorption spectra \(\boldsymbol{\upmu}_{N\times S}\), where \(S\) now indexes the set of characteristic spectra found from the data. The power of this approach lies in its ability to improve the signal-to-noise ratio of spectra of heterogeneous specimens by averaging noncontiguous pixels, to find even quite small regions with distinct spectroscopic signatures, and to deliver continuous thickness maps \(\mathbf{t}_{S\times P}\) based on the distribution of the discovered signature spectra. A further improvement on this approach is provided by using nonnegative matrix analysis in an optimization approach [23.257].

For studies at the carbon edge, one can characterize the observed set of near-edge resonances in terms of a limited number of functional group types [23.258, 23.259]. While there are a number of open questions regarding this approach (for example, how many resonances should be used, with what range of allowed center photon energies, and what range of energy widths), confidence in it can be enhanced by correlation with other spectroscopies such as solid-state nuclear magnetic resonance [23.258, 23.259] and Fourier transform infrared spectroscopy [23.260].

23.4 Applications

Two decades ago, nearly all research using x-ray microscopes was performed by the groups that had developed the instruments. Today, most x-ray microscopes are operated as user facilities at synchrotron radiation research centers, and are used both by their developers and by a wider community of scientists. As a result, while it was originally possible to see the major applications of x-ray microscopes in conference proceedings [23.13, 23.14, 23.15], papers in which x-ray microscopes were used to address the problem of interest now appear across a very wide array of scientific journals. In what follows, we do not presume to be exhaustive in covering all research using x-ray microscopes; instead, we will briefly highlight a few examples from some of the areas of activity using the same examples as in an earlier edition of this volume.

23.4.1 Biology

X-ray microscopes using zone plates have been employed from the start for studies of biological specimens [23.132, 23.141], as discussed in several older reviews [23.261, 23.262, 23.263]. One emphasis has been on high-resolution imaging of whole cells at water-window wavelengths between the K edges of carbon (about \({\mathrm{290}}\,{\mathrm{eV}}\)) and oxygen (about \({\mathrm{540}}\,{\mathrm{eV}}\)), where organic materials show strong contrast against water. While most of these studies have been carried out at synchrotron radiation light sources, laboratory sources have advanced to the point that they can deliver high-resolution TXM images [23.264, 23.265, 23.266]. As soft x-ray microscopes are pushed to higher spatial resolution, views through whole cells will involve a great deal of overlap of structure, but several developments offer information beyond two-dimensional images with natural contrast. One of these is the use of molecular labeling methods to tag specific proteins (such as is done with great success in visible light microscopy). Several groups have demonstrated the use of gold labeling in x-ray microscopes, including detection by dark-field [23.267] (Fig. 23.29) and bright-field [23.268, 23.269] approaches. One of the challenges faced thus far is that for efficient detection, the label must be comparable in size to the resolution of the microscope [23.208, 23.98], which means that in all studies carried out thus far, the cell membrane has been permeabilized by agents such as methanol to allow relatively large labels to reach the cell's interior, and this step must be preceded by chemical fixation. As a result, improvements in x-ray microscope resolution will not only lead to improved visualization of unlabeled ultrastructure, but will also make it possible to use smaller immunolabels with more natural preparation protocols.

Fig. 23.29

Human fibroblast with immunogold labeling for tubulin. This is a composite of two images: a bright-field image (gray tones) to image overall mass, and a dark-field image (red tones) to selectively image the silver-enhanced gold labels. This whole-mount cell was fixed and then permeabilized to allow for introduction of the immunogold labels, after which it was air-dried. From [23.267]. Reprinted with permission

X-ray microscopy can be combined with spectroscopy for the study of biological specimens. One can use (23.3) to measure the concentration of particular elements by imaging on either side of an absorption edge, especially for elements at high concentrations such as calcium in bone [23.153, 23.270]. XANES spectromicroscopy (Sect. 23.3.5) adds information on chemical speciation, as has been used for transmission-mode studies in biology [23.251, 23.271] and of biomaterials [23.272, 23.273], or in biological studies using fluorescence analysis [23.274, 23.275, 23.276]. X-ray fluorescence can also be used to study trace elements with very high sensitivity, as described in several recent review papers [23.277, 23.278], including in 3-D [23.279], and in fact there are now large communities making use of these capabilities.

Because x-rays offer advantages over electron microscopes mainly for imaging of biological specimens that are thicker than about \({\mathrm{1}}\,{\mathrm{{\upmu}m}}\), it is important that tomography is performed in x-ray microscopes, as was discussed in Sect. 23.3.4, to deal with the complexity of overlapping features in thick specimens. In biology, this was first used to study algae in a thin capillary [23.229] (Fig. 23.30a,b) and to study whole-mount eukaryotic cells [23.228], followed by studies of yeast in capillaries [23.280] (Fig. 23.31a,b). In all of these cases, cells were studied in the frozen hydrated state for reasons that will be discussed in the following paragraph. Soft x-ray microscopes optimized for x-ray tomography of frozen hydrated biological specimens are now in operation at the Advanced Light Source in Berkeley, BESSY II in Berlin, ALBA in Spain, and Diamond in the UK, while laboratory-based soft x-ray tomography instruments are also in use [23.265].

Fig. 23.30a,b

3-D rendering (a) and reconstruction slices (b) of the algae Chlamydomonas reinhardtii viewed by soft x-ray tomography at the former BESSY I synchrotron. This alga was plunge-frozen in liquid ethane and imaged over a \(180^{\circ}\) rotation sequence. The reconstruction is given in terms of the quantitative linear absorption coefficient for \({\mathrm{517}}\,{\mathrm{eV}}\) x-rays. Reprinted from [23.229], with permission from Elsevier

Fig. 23.31a,b

Single projection image (a) and slice from a tomographic reconstruction (b) of a frozen hydrated yeast Saccharomyces cerevisiae. A number of cells were loaded into a thin-walled, \({\mathrm{10}}\,{\mathrm{{\upmu}m}}\)-diameter glass capillary and rapidly frozen using a jet of helium gas cooled by liquid nitrogen. A series of 45 images through a \(180^{\circ}\) tilt range was then acquired using the XM-1 TXM at the Advanced Light Source. This illustrates the ability of soft x-ray tomography to image the interior detail of cells rapidly frozen from a living state. This work is now carried out on a dedicated TXM as part of the National Center for X-ray Tomography. Reprinted with permission from [23.280]

When studying biological specimens, attention must be paid to the limitations imposed by radiation damage. Basic considerations of signal to noise and absorption indicate that the radiation dose that is necessarily imparted for x-ray imaging at a resolution of \({\mathrm{50}}\,{\mathrm{nm}}\) or better is in excess of \({\mathrm{10^{6}}}\,{\mathrm{Gy}}\) [23.47, 23.54]. This is well in excess of the \(<{\mathrm{10}}\,{\mathrm{Gy}}\) (\({\mathrm{1}}\,{\mathrm{Gy}}={\mathrm{100}}\,{\mathrm{rd}}\)) dose that is lethal to humans when received over a short period of time. Studies of initially living cells have shown that doses of \({\mathrm{10^{6}}}\,{\mathrm{Gy}}\) are at the approximate threshold for producing immediate changes in bacteria [23.261] and are well above the dose that will affect more complex cells in x-ray microscopy investigations [23.261, 23.281, 23.282, 23.283, 23.284]. One of the main damage mechanisms is the creation of radiolytic free radicals in water. Chemical fixation can help increase radiation robustness, but some fixed hydrated specimens still show considerable mass loss and shrinkage upon irradiation [23.286, 23.287] (of course, chemical fixation also produces its own changes in many specimens [23.288, 23.289, 23.290, 23.291, 23.292]). Fortunately, a ready solution was developed some years ago by electron microscopists: the use of rapidly frozen specimens which are then imaged under cryogenic conditions [23.293, 23.294, 23.295, 23.296]. In x-ray microscopes, frozen hydrated biological specimens have been shown to be well preserved and free of easily visible structural changes and mass loss at radiation doses (Fig. 23.32) up to about \({\mathrm{10^{10}}}\,{\mathrm{Gy}}\), thus providing the required conditions for a variety of biological studies [23.157, 23.54]. The situation for spectroscopy is not yet so clear; cryo methods have been shown to be less effective in preserving XANES resonances in dry polymers [23.297], but they may be more advantageous in studies of frozen hydrated organic specimens due to the inactivation of the diffusion of free radicals [23.54].

Fig. 23.32

Whole fibroblast imaged in the frozen hydrated state. The cell was cultured on a formvar-coated gold electron microscope grid, and rapidly frozen by plunging into liquid ethane. It was then imaged using a cryo-STXM operated at \({\mathrm{516}}\,{\mathrm{eV}}\). In addition to this 2-D image, 3-D reconstructions were obtained using tomography. Reprinted with permission from [23.157]. John Wiley and Sons

For additional examples of x-ray microscopy applications in biology, see for example a recent special journal issue [23.263]. Other important developments have included correlative microscopy for studying the location of molecule-specific visible light fluorophores [23.298, 23.299], and correlative microscopy with both visible light and electron microscopy [23.300].

23.4.2 Environmental Science

Environmental science using synchrotron radiation is a broad topic, as discussed in various reviews [23.301]; we note here just a few examples using x-ray microscopes.

By placing microliter drops between two silicon nitride windows, which are then drawn together by surface tension and some sort of seal, it is straightforward to make a specimen chamber with micrometer-thick water layers and to study samples wet and at room temperature [23.302] (Fig. 23.33). Using this approach, one can use soft x-ray spectromicroscopy to study the role of bacteria and their biofilms in changing the reduction/oxidation state or sequestration of various elements in the environment [23.273, 23.303, 23.304] (Fig. 23.34), the growth of crystalline materials [23.305], and other geochemical reactions [23.306, 23.307, 23.308]. Spectromicroscopy at the carbon edge can be used to study a variety of organic processes, from the diagenetic breakdown of organic material over geological timescales and its presence and preservation in fossilized plants and wood [23.309, 23.310] and coals [23.311, 23.312], to the role of natural organic matter in the properties of soils [23.258, 23.259, 23.260, 23.313, 23.314, 23.315, 23.316], including its role in the groundwater transport of radionuclides [23.317] (Fig. 23.35). Tomography has also been used to study bacterial microhabitats [23.318]. Other studies have considered the functional groups present in the soot produced by combustion in diesel engines [23.319].

Fig. 23.33

Images of a colloidal chemistry sample consisting of oil in water with clays and calcium-rich layered double hydroxides used to cage the oil droplet where present (left and bottom edges of the droplet). This illustrates the ability to highlight various elemental components in a room-temperature wet specimen. From [23.285]

Fig. 23.34

(a) Quantitative chemical maps of protein, K\({}^{+}\), lipids, and polysaccharides from a wet microbial colony from the South Saskatchewan River, derived from STXM images (880\(\times\)880 pixels) and image sequences (52 energies, 230\(\times\)230 pixels). The spatial distributions of the various chemical species are determined by fitting the spectra from each pixel with a linear combination of the absorption spectra of the constituents. X-ray absorption spectra in the C \({\mathrm{1}}\,{\mathrm{s}}\) region are shown for \(\mathrm{CaCO_{3}}\), \(\mathrm{K^{+}}\), silicate, lipids, polysaccharides, and protein (b). The spectrum of \(\mathrm{CaCO_{3}}\) is from pure material. Those of the other five species are derived from the C \({\mathrm{1}}\,{\mathrm{s}}\) image sequence recorded from this biofilm using pixel identification and (for lipid, polysaccharides) spectral subtractions based on fits of the image sequence to the spectra of pure reference materials. (Courtesy A.P. Hitchcock, McMaster University)

Fig. 23.35

Cluster analysis in a spectromicroscopy study of lutetium in hematite. Lutetium is serving as a homologue to americium in an investigation of the uptake and transport of nuclear waste products in groundwater colloids [23.320]. By using a pattern recognition algorithm to search for pixels with spectroscopic similarities, a set of signature spectra is automatically recovered from the data (shown here in a color-coded classification map), and thickness maps can be formed based on these signature spectra. Analysis at the oxygen edge reveals two different phases of reactivity for lutetium with hematite. Reprinted from [23.248], with permission from Elsevier

The trace-element mapping capabilities of x-ray microprobes are also very useful for studies in environmental science. Low concentration of iron sets a biotic limit to carbon uptake in the southern Pacific; Twining et al have used microprobe studies to investigate this on a cell-by-cell basis [23.321] (Fig. 23.36), since bulk chemistry measurements do not allow one to differentiate between protist types and particulate matter at the same size scale. Other studies using zone-plate microprobes have concentrated on the speciation of metals near the roots of healthy and diseased plants [23.193]; the presence of metals in soil bacteria [23.275]; sulfur speciation in bacteria [23.322], natural silicate glasses [23.323], and microbial filaments [23.324]; and elemental concentrations in atmospheric particles [23.325]. These represent only early examples, as the number of projects carried out using zone-plate microprobes is increasing rapidly.

Fig. 23.36

Visible light and epifluorescence micrographs, and false-color x-ray fluorescence element maps of a centric diatom collected from the southern Pacific. In this region of the ocean, iron availability is a biolimiter with an impact on oceanic uptake of carbon dioxide from the atmosphere. X-ray microprobes allow one to study iron content on a protist-specific basis. Adapted with permission from [23.321]. Copyright 2003 American Chemical Society

23.4.3 Materials Science

Applications of x-ray microscopes to materials science include five broad categories of study: chemical state mapping in polymer systems using spectromicroscopy, studies of catalytic and energy storage materials, imaging of the structure and electromigration failure of integrated circuits, measurements of strain in crystalline materials using microdiffraction, and studies of surface properties using photoelectrons.

Polymer systems represent one of the first uses of zone-plate spectromicroscopy [23.242] (Fig. 23.37), and subsequent work has ranged from exploring fundamental questions such as confinement-induced miscibility [23.328] to studies of specific industrially useful materials using both absorption contrast [23.329, 23.330, 23.331, 23.332] and linear dichroism [23.333]. Other studies have measured the degree to which polymers can seep into wood at the cellular level in particleboard [23.334]. These represent only a few examples; a much wider survey is given in reviews [23.335].

Fig. 23.37

One of the first applications of x-ray transmission spectromicroscopy was to the study of polymers, where the chemical selectivity of near-edge absorption resonances allows one to create maps based on XANES spectral signatures. In this example, polymethyl methacrylate (PMMA) was spun-cast with polystyrene (PS) before annealing, giving rise to phase segregation. Images acquired at specific absorption resonances show very different contrast and can be used to form compositional maps of the polymers. Provided by D. Winnesett and H. Ade based on data acquired for a published study [23.326]

The elemental and chemical imaging capabilities of x-ray microscopes are well suited to the study of catalytic and energy storage materials. A TXM at SSRL/Stanford has been used for tomography of the chemical composition and pore structure of a fluid catalytic cracking particle nearly \({\mathrm{40}}\,{\mathrm{{\upmu}m}}\) across, shedding light on the aging of these particles [23.336]. Smaller catalyst particles have been studied at higher resolution using soft x-ray ptychography [23.337]. The SSRL TXM has also been used to image the change in polysulfides in Li-S battery particles during electrochemical cycling [23.338]. These and other nanomaterial x-ray microscopy applications are the subject of a recent review [23.339].

Fig. 23.38a-c

Interferometric TXM imaging of polystyrenes at \({\mathrm{9}}\,{\mathrm{keV}}\). In these experiments, a hard x-ray micro-interferometer has been constructed using two overlying objective zone plates with a slight transverse offset to produce an interferometric fringe pattern as shown in (b). Compared to the single-objective image (a), interference fringes with visibility of as high as \({\mathrm{60}}\%\) can clearly be seen. Analysis of the interferometric image (b) is then used to obtain the quantitative contrast image of the polystyrene spheres shown in (c). This example shows how low-absorption-contrast objects can be imaged in hard x-ray microscopes. From [23.327]. Copyright 2004 The Japan Society of Applied Physics. From T. Koyama et al. [23.327]

Modern integrated circuits are incredibly intricate, with oxidation layers sometimes only a few molecular layers thick, and metallization planes and vias which connect them having dimensions in the \({\mathrm{100}}\,{\mathrm{nm}}\) range. The ability of x-ray microscopes to image thick specimens (especially using phase contrast at higher energies; Fig. 23.39a,b) is well suited to studies of the properties and failure modes of such circuits. As one example, electromigration failures have been studied as they take place, leading to observations of the propagation of voids from the point of their original formation [23.340, 23.341, 23.342] (Fig. 23.39a,b). For industrial applications of chip inspection, a very significant development has been the commercial availability (Xradia, Inc.; now Carl Zeiss X-ray Microscopy) of laboratory-based tomography systems using zone-plate optics and operating at sufficiently high energy (\({\mathrm{5.4}}\,{\mathrm{keV}}\)) to allow tomographic data sets to be acquired and reconstructed, thus allowing one to study various metallization layers in intact, working chips [23.343] (Fig. 23.40). At the same time, synchrotron-based x-ray ptychography has been used to image circuit detail in \({\mathrm{300}}\,{\mathrm{{\upmu}m}}\)-thick Si wafers at sub-\({\mathrm{15}}\,{\mathrm{nm}}\) resolution in 2-D [23.344], and sub-\({\mathrm{15}}\,{\mathrm{nm}}\) detail in 3-D within a \({\mathrm{15}}\,{\mathrm{{\upmu}m}}\) pillar carved out from an integrated circuit using focused ion beam (FIB) methods [23.345].

Fig. 23.39a,b

Zernike phase contrast provides one means of imaging the metallic layers of integrated circuits in regions where the underlying silicon wafer has been thinned. A common failure mode in integrated circuits is electromigration, in which voids in a conducting layer or via are formed. These images obtained using a TXM formerly operated at \({\mathrm{4}}\,{\mathrm{keV}}\) at the European Synchrotron Radiation Research Facility (ESRF) show what appear to be such voids (circles) within test structures for advanced microprocessors. From [23.138]. © IOP Publishing. Reproduced with permission. All rights reserved

Fig. 23.40

Tomographic imaging of an integrated circuit performed with a commercial laboratory x-ray microscope. An integrated circuit had the silicon wafer underneath a region of interest thinned to about \({\mathrm{15}}\,{\mathrm{{\upmu}m}}\), after which a tilt series of TXM images was acquired over 8 h using a rotating anode source operating at \({\mathrm{5.4}}\,{\mathrm{keV}}\). The figure shows slices extracted at depths corresponding to the center of three Cu interconnect layers in the tomographic reconstruction with an estimated resolution of \({\mathrm{60}}\,{\mathrm{nm}}\) in the transverse dimension and \({\mathrm{90}}\,{\mathrm{nm}}\) in depth. This system can be used for chip inspection at a chip fabrication plant, among other applications. Provided by Wenbing Yun, based on data acquired for a published study [23.343]

Another way in which zone-plate x-ray microscopes are used to study material properties is through microdiffraction, where one examines not the undeviated transmission image through the specimen, but the signal that is Bragg-diffracted (usually in the Laue geometry) by specific crystalline regions within the specimen [23.346]. Measurement of the position of the Bragg peaks can give values of the local lattice constants so that repetition of the measurement over a grid of points provides a strain map of the sample. This has been applied to optoelectronic devices [23.196], in magnetic-domain evolution [23.347], and for examination of the strain at the midpoint and edges of mesoscopic structures [23.348] (Fig. 23.41).

Fig. 23.41

In x-ray microdiffraction, a detector is set to collect diffraction from small crystalline features of the specimen that can be selectively illuminated by the microfocus beam. Local variations from perfect crystal order are seen as changes in the width or angle of the diffraction peaks. In this example, a \({\mathrm{20}}\,{\mathrm{{\upmu}m}}\)-wide, \({\mathrm{0.24}}\,{\mathrm{{\upmu}m}}\)-thick \(\mathrm{Si_{0.86}Ge_{0.14}}\) pseudomorphically strained film is located on a Si\(\langle 001\rangle\) surface. A determination of the angle of the SiGe\(\langle 008\rangle\) diffraction peak as a function of position on the sample reveals elastic relaxation at the free edges of the SiGe feature, and demonstrates the ability of a zone-plate STXM to study the strain distribution of patterned microstructures. Reprinted from [23.348], with the permission of AIP Publishing

Since photoelectrons emerge only from within the top \({\mathrm{100}}\,{\mathrm{nm}}\) or so of a bulk specimen, methods that use photoelectron detection are ideal for studies of surface phenomena. Photoelectron emission microscopes using x-ray illumination of a broad area, and sub-\({\mathrm{30}}\,{\mathrm{nm}}\)-resolution electron optics are beyond the scope of our concentration on zone-plate microscopes, though we note that they are used with great success and at very high spatial resolution (Fig. 23.42). Another type of photoelectron microscope is a scanning photoemission microscope (SPEM), using a zone plate to produce a fine focus and an electron spectrometer for signal detection [23.349, 23.37, 23.40] (Fig. 23.43); earlier activities in this area are part of a comprehensive review [23.350].

Fig. 23.42

In electron spectroscopy for chemical analysis (ESCA) microscopy, a monochromatic beam is used to illuminate a region several micrometers across; electron optics are then used to image a tunable electron ejection energy to reveal surface chemistry. Though this does not involve zone-plate imaging, we include it here because of its widespread use with tunable X rays. In this case, a \({\mathrm{90}}\,{\mathrm{nm}}\)-resolution ESCA microscope was used to locate aligned \(\mathrm{MoS_{2}}\) nanotube bundles and select certain areas along the axes of the tubes for detailed examination. The image at the left was acquired using Mo 3d electrons, while S 2p, Mo 3d, and valence band spectra taken at the tips and sidewalls and the growth base from the Si wafer appear strongly affected by the low dimensionality of the nanotubes and differ significantly from the corresponding spectra taken on a reference \(\mathrm{MoS_{2}}\) crystal

Fig. 23.43

Scanning photoemission microscope (SPEM) study of a plasma display cell. In this microscope, the specimen is scanned through the zone-plate focus while photoelectrons are collected by an electron spectrometer. This figure shows a SPEM image, a scanning electron micrograph, and photoelectron spectra from several regions of the sample. In a plasma display cell, light of the appropriate color emerges through a front glass window which is protected from plasma damage by a composite insulating layer including MgO. The photoelectron spectra show aging in the \(\mathrm{Mg(OH)_{2}}\) component of the layer over the life of the display cell. From [23.349]. Copyright 2005 The Japan Society of Applied Physics

23.4.4 Magnetic Materials

X-ray magnetic circular dichroism ( ) exploits changes in absorption due to the relative orientation of magnetic domains and incident circularly polarized radiation. It draws upon the fact that in magnetic materials, the density of certain electronic states is different for electrons with spin parallel to the magnetization compared to electrons with spin that is antiparallel. The absorption of circularly polarized photons selects between electron spins, depending on the component of spin parallel to the helicity of the photon (the direction of the photon beam). Images taken with a particular polarization of the illumination beam, at saturated magnetization states, or at L\({}_{2}\) versus L\({}_{3}\) absorption edges can by themselves show magnetic contrast effects, while difference images between two polarization states at an absorption edge can be used to obtain element-specific images of magnetic contrast only. Although much work has been done using photoemission microscopes [23.351] and x-ray holography [23.223], with zone-plate microscopy there are two primary approaches. One method involves the use of a TXM with a large-angle-collection condenser zone plate and exploits the fact that the radiation from synchrotron bending magnet sources is circularly polarized above and below the synchrotron plane; this was the first method demonstrated [23.352], and it has led to considerable success for the study of out-of-specimen-plane magnetism [23.353] (Fig. 23.44a-d) and has been extended to the study of in-specimen-plane magnetic structure as well [23.354]. Another approach involves the use of an STXM with a variable-polarization undulator source. In either case, the pulsed nature of synchrotron radiation from electron bunches means that one can cycle an applied magnetic field in synchrony with the arrival of short (\(\approx{\mathrm{100}}\,{\mathrm{ps}}\)) pulses of x-rays, and thereby accumulate images corresponding to controlled time delays before and after application of the pulsed field [23.355] (Fig. 23.45a,b). A more extended discussion of magnetic contrast x-ray microscopy is provided in a recent review by Fischer [23.356].

Fig. 23.44a-d

X-ray magnetic circular dichroism (XMCD) images of the magnetic domain structure of a \({\mathrm{50}}\,{\mathrm{nm}}\)-thick \(\mathrm{(Co_{83}Cr_{17})_{87}Pt_{13}}\) alloy film recorded at the Co L\({}_{3}\) absorption edge (\({\mathrm{777}}\,{\mathrm{eV}}\)) and in an external field of (a) \({\mathrm{0}}\,{\mathrm{Oe}}\), (b\({+}{\mathrm{400}}\,{\mathrm{Oe}}\), and (c\({-}{\mathrm{400}}\,{\mathrm{Oe}}\). (dM versus H hysteresis loop obtained via vibrating-sample magnetometer (VSM) measurement. The arrows indicate the point in the reversal cycle at which each image is recorded. The domain structure is apparent as the magnetization of the film is driven around the hysteresis loop and the net magnetization reversal can be seen to be the average of the reversal of individual domains, with the number of reversed domains increasing as the strength of the applied field is increased. Reprinted from [23.357], with the permission of AIP Publishing

Fig. 23.45a,b

Time-resolved XMCD imaging of a magnetized Ni-Fe film patch as the magnetization is reversed in an applied magnetic field. (a) The \(z\)-component of the dynamic magnetization at selected time delays obtained from the micro-magnetic simulation program Object Oriented MicroMagnetic Framework or OOMMF. (b) XMCD images from the XM-1 TXM taken with various time delays between the application of the pulsed magnetic field and the arrival of radiation from electron bunches in the storage ring. By integrating over many bunches with a particular time delay, one can study the temporal evolution of the \(z\)-component of the magnetization at delay times varying from probe pulse \({\mathrm{400}}\,{\mathrm{ps}}\) before the pump, up to \({\mathrm{2400}}\,{\mathrm{ps}}\) after the pump. Reprinted from [23.355], with the permission of AIP Publishing

23.5 Conclusion

In this chapter, we have outlined some of the principles and characteristics of x-ray microscopes using zone-plate optics, and have attempted to convey a representative, albeit incomplete, survey of their application in scientific studies. We have seen that the resolution and efficiency of zone plates has improved considerably over the lifetime of the field, although, despite constant efforts and the application of the best technology, the rate of improvement has been slow. For some time, the Moore's Law graph for zone-plate resolution has had a slope of about a factor of 2 per decade. However, as we have seen, this area of development has been especially active in recent times. The \({\mathrm{10}}\,{\mathrm{nm}}\) barrier has been broken (although with reduced focusing efficiency) and the present art is coming closer to fundamental limits. Resolution is not the whole story, however; many applications are combining imaging with tilt of the specimen for tomography, with energy tunability for spectromicroscopy, and with fluorescence detection for elemental identification. These represent the application of zone-plate optics to extend the boundaries of previously existing techniques with active communities, so these areas are likely to expand. Another general trend of the past few years has been the growth in hard x-ray applications of zone-plate imaging. This has been especially beneficial for tomography and microanalysis, and as recent experiments have shown, the use of hard x-ray zone plates in high order may soon approach the best resolution of soft x-ray zone plates in first order. At the time of this writing it seems that technical developments in x-ray microscopy and its marriage with promising application areas is happening at an ever-increasing pace, and we can now forecast with more confidence than ever before that these activities have a bright future.

Notes

Acknowledgements

Naturally, an enterprise like writing this review depends greatly on the willingness of our colleagues around the x-ray microscopy community to provide us with advice, information, and images, and we thank the many people who have done that. We especially thank Graeme Morrison for helpful comments and technical assistance in the preparation of this updated version, and Janos Kirz and Henry Chapman for additional comments. We also thank our immediate colleagues at Argonne, Northwestern, and Berkeley for many helpful discussions. Work by MH and TW was supported by the Advanced Light Source, which is a DOE Office of Science User Facility under Contract No. DE-AC02-05CH11231. Work by CJ was supported by the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

References

  1. P. Kirkpatrick: The x-ray microscope, Sci. Am. 180, 44–47 (1949)CrossRefGoogle Scholar
  2. V.E. Cosslett, A. Engström, H.H. Pattee Jr. (Eds.): International Symposium on X-Ray Optics and X-Ray Microanalysis (Academic Press, New York 1957)Google Scholar
  3. A. Engström, V.E. Cosslett, H.H. Pattee Jr. (Eds.): X-Ray Microscopy and X-Ray Microanalysis (Elsevier, Amsterdam 1960)Google Scholar
  4. H.H. Pattee Jr., V.E. Cosslett, A. Engström (Eds.): X-Ray Optics and X-Ray Microanalysis (Academic Press, New York 1963)Google Scholar
  5. R. Castaing, P. Deschamps, J. Philibert: X-Ray Optics and Microanalysis (Hermann, Paris 1966)Google Scholar
  6. G. Möllenstedt, K.H. Gaukler: X-Ray Optics and Microanalysis (Springer, Berlin 1969)Google Scholar
  7. A.V. Baez: The early days of x-ray optics: A personal memoir, J. Xray Sci. Technol. 1, 3–6 (1989)Google Scholar
  8. A.V. Baez: Anecdotes about the early days of x-ray optics, J. Xray Sci. Technol. 7(2), 90–97 (1997)Google Scholar
  9. V.E. Cosslett, W.C. Nixon: X-Ray Microscopy (Cambridge Univ. Press, London 1960)Google Scholar
  10. D.F. Parsons (Ed.): Short wavelength microscopy, Annals of the New York Academy of Sciences, Vol. 306 (NYAS, New York 1978)Google Scholar
  11. E.A. Ash (Ed.): Scanned Image Microscopy (Academic Press, London 1980)Google Scholar
  12. D.F. Parsons (Ed.): Ultrasoft X-Ray Microscopy: Its Application to Biological and Physical Sciences, Annals of the New York Academy of Sciences, Vol. 342 (NYAS, New York 1980)Google Scholar
  13. G. Schmahl, D. Rudolph (Eds.): X-Ray Microscopy, Springer Series in Optical Sciences, Vol. 43 (Springer, Berlin 1984)Google Scholar
  14. D. Sayre, M.R. Howells, J. Kirz, H. Rarback (Eds.): X-Ray Microscopy II, Springer Series in Optical Sciences, Vol. 56 (Springer, Berlin 1988)Google Scholar
  15. A.G. Michette, G.R. Morrison, C.J. Buckley (Eds.): X-Ray Microscopy III, Springer Series in Optical Sciences, Vol. 67 (Springer, Berlin 1992)Google Scholar
  16. V.V. Aristov, A.I. Erko (Eds.): X-Ray Microscopy IV (Bogorodskii Pechatnik, Chernogolovka 1994)Google Scholar
  17. J. Thieme, G. Schmahl, E. Umbach, D. Rudolph (Eds.): X-Ray Microscopy and Spectromicroscopy (Springer, Berlin 1998)Google Scholar
  18. W. Meyer-Ilse, T. Warwick, D. Attwood (Eds.): X-Ray Microscopy: Proceedings of the Sixth International Conference (AIP, College Park 2000)Google Scholar
  19. J. Susini, D. Joyeux, F. Polack: 7th international conference on x-ray microscopy: Preface, J. Phys. IV 104, V (2003)Google Scholar
  20. S. Aoki, Y. Kagoshima, Y. Suzuki (Eds.): X-Ray Microscopy: Proceedings of the 8th International Conference, IPAP Conference Series, Vol. 7 (IPAP, Tokyo 2006)Google Scholar
  21. F. Pfeiffer, C. David, C. Quitmann (Eds.): 9th International Conference on X-Ray Microscopy, Vol. 186 (IOP, Bristol 2009)Google Scholar
  22. I. McNulty, C. Eyeberger, B. Lai (Eds.): The 10th International Conference on X-Ray Microscopy, American Institute of Physics Conference Proceedings, Vol. 1365 (AIP, College Park 2011)Google Scholar
  23. H. Xu, Z. Wu, R. Tai: 11th International Conference on X-Ray Microscopy (XRM2012), J. Phys. Conf. Ser. 463, 011001 (2013)CrossRefGoogle Scholar
  24. M.D. de Jonge, D.J. Paterson, C.G. Ryan: Preface: The International Conference on X-Ray Microscopy, AIP Conf. Proc. 1696, 010001 (2016)CrossRefGoogle Scholar
  25. P. Guttmann, C. Bittencourt, S. Rehbein, P. Umek, X. Ke, G. Van Tendeloo, C.P. Ewels, G. Schneider: Nanoscale spectroscopy with polarized x-rays by NEXAFS-TXM, Nat. Photonics 6(1), 25–29 (2011)CrossRefGoogle Scholar
  26. W. Chao, J. Kim, S. Rekawa, P. Fischer, E.H. Anderson: Demonstration of 12 nm resolution Fresnel zone plate lens based soft x-ray microscopy, Opt. Express 17(20), 17669–17677 (2009)CrossRefGoogle Scholar
  27. I. Mohacsi, I. Vartiainen, B. Rösner, M. Guizar-Sicairos, V.A. Guzenko, I. McNulty, R. Winarski, M.V. Holt, C. David: Interlaced zone plate optics for hard x-ray imaging in the 10 nm range, Sci. Rep. 7, 43624 (2017)CrossRefGoogle Scholar
  28. D.A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A.L.D. Kilcoyne, F. Maia, S. Marchesini, Y.S. Meng, T. Warwick, L.L. Yang, H.A. Padmore: Chemical composition mapping with nanometre resolution by soft x-ray microscopy, Nat. Photonics 8(10), 765–769 (2014)CrossRefGoogle Scholar
  29. J.H. Hubbell, H.A. Gimm, I. Øverbø: Pair, triplet, and total atomic cross sections (and mass attenuation coefficients) for 1 MeV–100 GeV photons in elements Z=1–100, J. Phys. Chem. Ref. Data 9, 1023–1147 (1980)CrossRefGoogle Scholar
  30. A. Einstein: Lassen sich Brechungsexponenten der Körper für Röntgenstrahlen experimentell ermitteln?, Verh. Dtsch. Phys. Ges. 9(12), 86–87 (1918)Google Scholar
  31. B.L. Henke, E.M. Gullikson, J.C. Davis: X-ray interactions: Photoabsorption, scattering, transmission, and reflection at \(E\)=50–30,000 eV, \(Z\)=1–92, At. Data Nucl. Data Tables 54, 181–342 (1993)CrossRefGoogle Scholar
  32. G. Schmahl, D. Rudolph: Proposal for a phase contrast x-ray microscope. In: X-Ray Microscopy: Instrumentation and Biological Applications, ed. by P.C. Cheng, G.J. Jan (Springer, Berlin 1987) pp. 231–238CrossRefGoogle Scholar
  33. E.B. Saloman, J.H. Hubbell: Critical analysis of soft x-ray cross section data, Nucl. Instrum. Methods Phys. Res. A 255, 38–42 (1987)CrossRefGoogle Scholar
  34. M.O. Krause: Atomic radiative and radiationless yields for K and L shells, J. Phys. Chem. Ref. Data 8, 307–327 (1979)CrossRefGoogle Scholar
  35. A. Engström: Quantitative micro- and histochemical elementary analysis by Röntgen absorption spectrography, Acta Radiol. 27(S63), 1–106 (1946)Google Scholar
  36. H. Wolter: Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen, Ann. Phys. 10(286), 94–114 (1952)CrossRefGoogle Scholar
  37. H. Ade, J. Kirz, S.L. Hulbert, E. Johnson, E. Anderson, D. Kern: X-ray spectromicroscopy with a zone plate generated microprobe, Appl. Phys. Lett. 56, 1841–1843 (1990)CrossRefGoogle Scholar
  38. S. Günther, A. Kolmakov, J. Kovac, L. Casalis, L. Gregoratti, M. Marsi, M. Kiskinova: Scanning photoelectron microscopy of a bimetal/Si interface: Au coadsorbed on Ag/Si(111), Surf. Sci. 377, 145–149 (1997)CrossRefGoogle Scholar
  39. T. Warwick, H. Ade, A.P. Hitchcock, H. Padmore, E.G. Rightor, B.P. Tonner: Soft x-ray spectromicroscopy development for materials science at the Advanced Light Source, J. Electron Spectrosc. Relat. Phenom. 84, 85–98 (1997)CrossRefGoogle Scholar
  40. C.-H. Ko, R. Klauser, D.-H. Wei, H.-H. Chan, T.J. Chuang: The soft x-ray scanning photoemission microscopy project at SRRC, J. Synchrotron Radiat. 5(3), 299–304 (1998)CrossRefGoogle Scholar
  41. B. Kaulich, A. Gianoncelli, A. Beran, D. Eichert, I. Kreft, P. Pongrac, M. Regvar, K. Vogel-Mikuš, M. Kiskinova: Low-energy x-ray fluorescence microscopy opening new opportunities for bio-related research, J. R. Soc. Interface 6(5), S641–S647 (2009)Google Scholar
  42. A.P. Hitchcock, M. Obst, J. Wang, Y.S. Lu, T. Tyliszczak: Advances in the detection of As in environmental samples using low energy x-ray fluorescence in a scanning transmission x-ray microscope: Arsenic immobilization by an Fe(II)-oxidizing freshwater bacteria, Environ. Sci. Technol. 26, 2821–2829 (2012)CrossRefGoogle Scholar
  43. P. Horowitz, J.A. Howell: A scanning x-ray microscope using synchrotron radiation, Science 178, 608–611 (1972)CrossRefGoogle Scholar
  44. C.J. Sparks Jr.: X-ray fluorescence microprobe for chemical analysis. In: Synchrotron Radiation Research, ed. by H. Winick, S. Doniach (Plenum, New York 1980) pp. 459–512CrossRefGoogle Scholar
  45. J. Szlachetko, M. Cotte, J. Morse, M. Salomé, P. Jagodzinski, J.C. Dousse, J. Hoszowska, Y. Kayser, J. Susini: Wavelength-dispersive spectrometer for x-ray microfluorescence analysis at the x-ray microscopy beamline ID21 (ESRF), J. Synchrotron Radiat. 17(3), 400–408 (2010)CrossRefGoogle Scholar
  46. R.M. Glaeser: Limitations to significant information in biological electron microscopy as a result of radiation damage, J. Ultrastruct. Res. 36, 466–482 (1971)CrossRefGoogle Scholar
  47. D. Sayre, J. Kirz, R. Feder, D.M. Kim, E. Spiller: Transmission microscopy of unmodified biological materials: Comparative radiation dosages with electrons and ultrasoft x-ray photons, Ultramicroscopy 2, 337–341 (1977)CrossRefGoogle Scholar
  48. A. Rose: Unified approach to performance of photographic film, television pickup tubes, and human eye, J. Soc. Motion Pict. Eng. 47, 273–294 (1946)CrossRefGoogle Scholar
  49. D. Sayre, J. Kirz, R. Feder, D.M. Kim, E. Spiller: Potential operating region for ultrasoft x-ray microscopy of biological specimens, Science 196, 1339–1340 (1977)CrossRefGoogle Scholar
  50. R. Grimm, M. Bärmann, W. Häckl, D. Typke, E. Sackman, W. Baumeister: Energy filtered electron tomography of ice-embedded actin and vesicles, Biophys. J. 72, 482–489 (1997)CrossRefGoogle Scholar
  51. C. Jacobsen, R. Medenwaldt, S. Williams: A perspective on biological x-ray and electron microscopy. In: X-Ray Microscopy and Spectromicroscopy, ed. by J. Thieme, G. Schmahl, E. Umbach, D. Rudolph (Springer, Berlin 1998) pp. 93–102Google Scholar
  52. D. Rudolph, G. Schmahl, B. Niemann: Amplitude and phase contrast in x-ray microscopy. In: Modern Microscopies, ed. by P.J. Duke, A.G. Michette (Plenum, New York 1990) pp. 59–67CrossRefGoogle Scholar
  53. P. Gölz: Calculations on radiation dosages of biological materials in phase contrast and amplitude contrast x-ray microscopy. In: X-Ray Microscopy III, Springer Series in Optical Sciences, Vol. 67, ed. by A.G. Michette, G.R. Morrison, C.J. Buckley (Springer, Berlin 1992) pp. 313–315CrossRefGoogle Scholar
  54. G. Schneider: Cryo x-ray microscopy with high spatial resolution in amplitude and phase contrast, Ultramicroscopy 75, 85–104 (1998)CrossRefGoogle Scholar
  55. J. Kirz, D. Sayre, J. Dilger: Comparative analysis of x-ray emission microscopies for biological specimens. In: Short Wavelength Microscopy, Annals of the New York Academy of Sciences, Vol. 306, ed. by D.F. Parsons (NYAS, New York 1978) pp. 291–305Google Scholar
  56. J. Kirz: Mapping the distribution of particular atomic species. In: Ultrasoft X-Ray Microscopy: Its Application to Biological and Physical Sciences, Annals of the New York Academy of Sciences, Vol. 342, ed. by D.F. Parsons (NYAS, New York 1980) pp. 273–287Google Scholar
  57. J. Kirz: Specimen damage considerations in biological microprobe analysis. In: Scanning Electron Microscopy, Vol. 2 (SEM Inc, Chicago 1980) pp. 239–249Google Scholar
  58. S. Aoki: Recent developments in x-ray microscopy at the photon factory. In: X-Ray Microscopy IV, ed. by V.V. Aristov, A.I. Erko (Bogorodskii Pechatnik, Chernogolovka 1994) pp. 35–40Google Scholar
  59. P. Kirkpatrick: X-ray images by refractive focusing, J. Opt. Soc. Am. 39(9), 796 (1949)CrossRefGoogle Scholar
  60. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, K. Yamauchi: Breaking the 10 nm barrier in hard-x-ray focusing, Nat. Phys. 6(2), 122–125 (2010)CrossRefGoogle Scholar
  61. M. Yabashi, K. Tono, H. Mimura, S. Matsuyama, K. Yamauchi, T. Tanaka, H. Tanaka, K. Tamasaku, H. Ohashi, S. Goto, T. Ishikawa: Optics for coherent x-ray applications, J. Synchrotron Radiat. 21(5), 976–985 (2014)CrossRefGoogle Scholar
  62. E. Spiller: Low-loss reflection coatings using absorbing materials, Appl. Phys. Lett. 20, 365–367 (1972)CrossRefGoogle Scholar
  63. B.X. Yang: Fresnel and refractive lenses for x-rays, Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993)CrossRefGoogle Scholar
  64. A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler: A compound refractive lens for focusing high energy x-rays, Nature 384, 49–51 (1996)CrossRefGoogle Scholar
  65. B. Lengeler, C.G. Schroer, B. Benner, A. Gerhardus, T.F. Günzler, M. Kuhlmann, J. Meyer, C. Zimprich: Parabolic refractive x-ray lenses, J. Synchrotron Radiat. 9, 119–124 (2002)CrossRefGoogle Scholar
  66. C.G. Schroer, B. Lengeler: Focusing hard x rays to nanometer dimensions by adiabatically focusing lenses, Phys. Rev. Lett. 94, 54802 (2005)CrossRefGoogle Scholar
  67. A.V. Baez: A self-supporting metal Fresnel zone-plate to focus extreme ultra-violet and soft x-rays, Nature 186, 958 (1960)CrossRefGoogle Scholar
  68. A.G. Michette: Optical Systems for Soft X Rays (Plenum, New York 1986)CrossRefGoogle Scholar
  69. D. Attwood: Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge Univ. Press, Cambridge 1999)CrossRefGoogle Scholar
  70. M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, O. Bunk: X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution, Sci. Rep. 4, 3857 (2014)CrossRefGoogle Scholar
  71. J. Thieme: Theoretical investigations of imaging properties of zone plates and zone plate systems using diffraction theory. In: X-Ray Microscopy II, Springer Series in Optical Sciences, Vol. 56, ed. by D. Sayre, M.R. Howells, J. Kirz, H. Rarback (Springer, Berlin 1988) pp. 70–79CrossRefGoogle Scholar
  72. K. Kamiya: Theory of Fresnel zone plate, Sci. Light 12(3), 35–49 (1963)Google Scholar
  73. M. Born, E. Wolf: Principles of Optics, 7th edn. (Cambridge Univ. Press, Cambridge 1999)CrossRefGoogle Scholar
  74. M.J. Simpson, A.G. Michette: The effects of manufacturing inaccuracies on the imaging properties of Fresnel zone plates, Opt. Acta 30, 1455–1462 (1983)CrossRefGoogle Scholar
  75. J.W. Strutt: Wave theory of light. In: Scientific Papers by John William Strutt, Baron Rayleigh, Vol. 3 (Dover, New York 1888) pp. 47–187Google Scholar
  76. R.W. Wood: Phase-reversal zone-plates, and diffraction telescopes, Philos. Mag. 45(277), 511–522 (1898)CrossRefGoogle Scholar
  77. J. Kirz: Phase zone plates for X rays and the extreme UV, J. Opt. Soc. Am. 64, 301–309 (1974)CrossRefGoogle Scholar
  78. E.H. Anderson, D.L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W.L. Chao, A. Lucero, L. Johnson, D. Attwood: Nanofabrication and diffractive optics for high-resolution x-ray applications, J. Vac. Sci. Technol. B 18(6), 2970–2975 (2000)CrossRefGoogle Scholar
  79. D. Tennant, S. Spector, A. Stein, C. Jacobsen: Electron beam lithography of Fresnel zone plates using a rectilinear machine and trilayer resists. In: Xray Microsc. Proc. Sixth Int. Conf., ed. by W. Meyer-Ilse, T. Warwick, D. Attwood (AIP, College Park 2000) pp. 601–606Google Scholar
  80. G. Schmahl, D. Rudolph: Lichtstarke Zonenplatten als abbildende Systeme für weiche Röntgenstrahlung (High power zone plates as image forming systems for soft x-rays), Optik 29, 577–585 (1969)Google Scholar
  81. G. Schmahl, D. Rudolph, P. Guttmann, O. Christ: Zone plate lenses for x-ray microscopy. In: X-Ray Microscopy, Springer Series in Optical Sciences, Vol. 43, ed. by G. Schmahl, D. Rudolph (Springer, Berlin 1984) pp. 63–74CrossRefGoogle Scholar
  82. D. Sayre: Proposal for the utilization of electron beam technology in the fabrication of an image forming device for the soft x-ray region, Technical Report RC 3974 (#17965) (IBM T. J. Watson Research Laboratory, Yorktown Heights 1972)Google Scholar
  83. D.C. Shaver, D.C. Flanders, N.M. Ceglio, H.I. Smith: X-ray zone plates fabricated using electron-beam and x-ray lithography, J. Vac. Sci. Technol. 16, 1626–1630 (1980)CrossRefGoogle Scholar
  84. D. Kern, P. Coane, R. Acosta, T.H.P. Chang, R. Feder, P. Houzego, W. Molzen, J. Powers, A. Speth, R. Viswanathan, J. Kirz, H. Rarback, J. Kenney: Electron beam fabrication and characterization of Fresnel zone plates for soft x-ray microscopy, Proc. SPIE 447, 204–213 (1984)CrossRefGoogle Scholar
  85. C.J. Buckley, M.T. Browne, P.S. Charalambous: Contamination lithography for the fabrication of zone plate x-ray lenses, Proc. SPIE 537, 213–217 (1985)CrossRefGoogle Scholar
  86. D.M. Tennant, L.D. Jackel, R.E. Howard, E.L. Hu, P. Grabbe, R.J. Capik, B.S. Schneider: Twenty-five nm features patterned with trilevel e-beam resist, J. Vac. Sci. Technol. B 19(4), 1304–1307 (1981)CrossRefGoogle Scholar
  87. G. Schneider, T. Schliebe, H. Aschoff: Cross-linked polymers for nanofabrication of high-resolution zone plates in nickel and germanium, J. Vac. Sci. Technol. B 13(6), 2809–2812 (1995)CrossRefGoogle Scholar
  88. D.M. Tennant, J.E. Gregus, C. Jacobsen, E.L. Raab: Construction and test of phase zone plates for x-ray microscopy, Opt. Lett. 16, 621–623 (1991)CrossRefGoogle Scholar
  89. C. David, B. Kaulich, R. Medenwaldt, M. Hettwer, N. Fay, M. Diehl, J. Thieme, G. Schmahl: Low-distortion electron-beam lithography for fabrication of high-resolution germanium and tantalum phase zone plates, J. Vac. Sci. Technol. B 13(6), 2762–2766 (1995)CrossRefGoogle Scholar
  90. W. Chao, B.D. Harteneck, J.A. Liddle, E.H. Anderson, D.T. Attwood: Soft x-ray microscopy at a spatial resolution better than 15 nm, Nature 435, 1210–1213 (2005)CrossRefGoogle Scholar
  91. S. Spector, C. Jacobsen, D. Tennant: Process optimization for production of sub-20 nm soft x-ray zone plates, J. Vac. Sci. Technol. B 15(6), 2872–2876 (1997)CrossRefGoogle Scholar
  92. M. Peuker: High-efficiency nickel phase zone plates with 20 nm minimum outermost zone width, Appl. Phys. Lett. 78(15), 2208–2210 (2001)CrossRefGoogle Scholar
  93. S. Werner, S. Rehbein, P. Guttmann, G. Schneider: Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency, Nano Res. 7(4), 1–8 (2014)CrossRefGoogle Scholar
  94. E.H. Anderson, V. Boegli, L.P. Muray: Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures, J. Vac. Sci. Technol. B 13, 2529–2534 (1995)CrossRefGoogle Scholar
  95. W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa: Real space soft x-ray imaging at 10 nm spatial resolution, Opt. Express 20(9), 9777 (2012)CrossRefGoogle Scholar
  96. W. Chao, E.H. Anderson, G.P. Denbeaux, B. Harteneck, J.A. Liddle, D.L. Olynick, A.L. Pearson, F. Salmassi, C.Y. Song, D.T. Attwood: 20-nm resolution soft x-ray microscopy demonstrated by use of multilayer test structures, Opt. Lett. 28(21), 2019–2021 (2003)CrossRefGoogle Scholar
  97. M. Hettwer, D. Rudolph: Fabrication of the x-ray condenser zone plate KZP 7. In: X-Ray Microscopy and Spectromicroscopy, ed. by J. Thieme, G. Schmahl, E. Umbach, D. Rudolph (Springer, Berlin 1998) pp. 313–318CrossRefGoogle Scholar
  98. S. Vogt, H.N. Chapman, C. Jacobsen, R. Medenwaldt: Dark field x-ray microscopy: the effects of condenser/detector aperture, Ultramicroscopy 87, 25–44 (2001)CrossRefGoogle Scholar
  99. M. Howells: Design of X-Ray-Heated Beamline Windows, Technical Report LSBL-0168 (Lawrence Berkeley National Laboratory 1992) revised (2004)Google Scholar
  100. M.R. Howells, P. Charalambous, H. He, S. Marchesini, J.C.H. Spence: An off-axis zone-plate monochromator for high-power undulator radiation, Proc. SPIE 4783, 65–73 (2002)CrossRefGoogle Scholar
  101. A.H. Compton: The total reflexion of x-rays, Philos. Mag. 45(270), 1121–1131 (1923)CrossRefGoogle Scholar
  102. D.H. Bilderback: Review of capillary x-ray optics from the 2nd International Capillary Optics Meeting, Xray Spectrom. 32(3), 195–207 (2003)CrossRefGoogle Scholar
  103. X. Zeng, F. Duewer, M. Feser, C. Huang, A. Lyon, A. Tkachuk, W. Yun: Ellipsoidal and parabolic glass capillaries as condensers for x-ray microscopes, Appl. Opt. 47(13), 2376–2381 (2008)CrossRefGoogle Scholar
  104. G.-C. Yin, Y.-F. Song, M.-T. Tang, F.-R. Chen, K.S. Liang, F.W. Duewer, M. Feser, W. Yun, H.-P.D. Shieh: 30 nm resolution x-ray imaging at 8 keV using third order diffraction of a zone plate lens objective in a transmission microscope, Appl. Phys. Lett. 89(22), 221122 (2006)CrossRefGoogle Scholar
  105. G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J.B. Heymann, W.G. Müller, J.G. McNally: Three-dimensional cellular ultrastructure resolved by x-ray microscopy, Nat. Methods 7(12), 985–987 (2010)CrossRefGoogle Scholar
  106. A. Tkachuk, F. Duewer, H. Cui, M. Feser, S. Wang, W. Yun: X-ray computed tomography in Zernike phase contrast mode at 8 keV with 50-nm resolution using Cu rotating anode x-ray source, Z. Kristallogr. 222, 650–655 (2007)Google Scholar
  107. R. Tatchyn, P.L. Csonka, I. Lindau: The constant-thickness zone plate as a variational problem, Opt. Acta 31, 729–733 (1984)CrossRefGoogle Scholar
  108. E. Di Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, R. Barrett: High-efficiency multilevel zone plates for keV x-rays, Nature 401, 895–898 (1999)CrossRefGoogle Scholar
  109. C. Jacobsen: Making soft x-ray microscopy harder: Considerations for sub-0.1 $$\upmu$$m resolution imaging at $$\sim 4$$ Å wavelengths. In: X-Ray Microscopy III, Springer Series in Optical Sciences, Vol. 67, ed. by A.G. Michette, G.R. Morrison, C.J. Buckley (Springer, Berlin 1992) pp. 274–277CrossRefGoogle Scholar
  110. Y. Feng, M. Feser, A. Lyon, S. Rishton, X. Zeng, S. Chen, S. Sassolini, W. Yun: Nanofabrication of high aspect ratio 24 nm x-ray zone plates for x-ray imaging applications, J. Vac. Sci. Technol. B 25(6), 2004–2007 (2007)CrossRefGoogle Scholar
  111. B. Lai, W.B. Yun, D. Legnini, Y. Xiao, J. Chrzas, P.J. Viccaro, V. White, S. Bajikar, D. Denton, F. Cerrina, E. Di Fabrizio, M. Gentili, L. Grella, M. Baciocchi: Hard x-ray phase zone plate fabricated by lithographic techniques, Appl. Phys. Lett. 61, 1877–1879 (1992)CrossRefGoogle Scholar
  112. A.A. Krasnoperova, J. Xiao, F. Cerrina, E. Di Fabrizio, L. Luciani, M. Figliomeni, M. Gentili, W. Yun, B. Lai, E. Gluskin: Fabrication of hard x-ray phase zone plate by x-ray lithography, J. Vac. Sci. Technol. B 11, 2588–2591 (1993)CrossRefGoogle Scholar
  113. S.D. Shastri, J.M. Maser, B. Lai, J. Tys: Microfocusing of 50 keV undulator radiation with two stacked zoneplates, Opt. Commun. 197, 9–14 (2001)CrossRefGoogle Scholar
  114. S.-C. Gleber, M. Wojcik, J. Liu, C. Roehrig, M. Cummings, J. Vila-Comamala, K. Li, B. Lai, D. Shu, S. Vogt: Fresnel zone plate stacking in the intermediate field for high efficiency focusing in the hard x-ray regime, Opt. Express 22(23), 28142–28153 (2014)CrossRefGoogle Scholar
  115. G. Schmahl, D. Rudolph, B. Niemann: Imaging and scanning soft x-ray microscopy with zone plates. In: Scanned Image Microscopy, ed. by E.A. Ash (Academic Press, London 1980) pp. 393–412Google Scholar
  116. R.M. Bionta, K.M. Skulina, J. Weinberg: Hard x-ray sputtered-sliced phase zone plates, Appl. Phys. Lett. 64(8), 945–947 (1994)CrossRefGoogle Scholar
  117. S. Tamura, M. Yasumoto, N. Kamijo, Y. Suzuki, M. Awaji, A. Takeuchi, K. Uesugi, Y. Terada, H. Takano: New approaches to fabrication of multilayer Fresnel zone plate for high-energy synchrotron radiation x-rays, Vacuum 80, 823–827 (2006)CrossRefGoogle Scholar
  118. F. Döring, A.L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, H.U. Krebs: Sub-5 nm hard x-ray point focusing by a combined Kirkpatrick-Baez mirror and multilayer zone plate, Opt. Express 21(16), 19311 (2013)CrossRefGoogle Scholar
  119. B.W. Batterman, H. Cole: Dynamical diffraction of x rays by perfect crystals, Rev. Modern Phys. 36(3), 681 (1964)CrossRefGoogle Scholar
  120. H. Kogelnik: Coupled wave theory for thick hologram gratings, Bell Syst. Tech. J. 48(9), 2909–2947 (1969)CrossRefGoogle Scholar
  121. L. Solymar, D.J. Cooke: Volume Holography and Volume Gratings (Academic Press, Cambridge 1981)Google Scholar
  122. J. Maser, G. Schmahl: Coupled wave description of the diffraction by zone plates with high aspect ratios, Opt. Commun. 89, 355–362 (1992)CrossRefGoogle Scholar
  123. G. Schneider, S. Rehbein, S. Werner: Volume effects in zone plates. In: Modern Developments in X-Ray and Neutron Optics, ed. by A. Erko, M. Idir, T. Krist, A.G. Michette (Springer, Berlin, Heidelberg 2008) pp. 137–171CrossRefGoogle Scholar
  124. J.M. Cowley, A.F. Moodie: The scattering of electrons by atoms and crystals. I. A new theoretical approach, Acta Crystallogr. 10(10), 609–619 (1957)CrossRefGoogle Scholar
  125. K. Li, M. Wojcik, C. Jacobsen: Multislice does it all—Calculating the performance of nanofocusing x-ray optics, Opt. Express 25(3), 1831–1846 (2017)CrossRefGoogle Scholar
  126. G. Schneider, J. Maser: Zone plates as imaging optics in high diffraction orders described by coupled wave theory. In: X-Ray Microscopy and Spectromicroscopy, ed. by J. Thieme, G. Schmahl, E. Umbach, D. Rudolph (Springer, Berlin 1998) pp. IV-71–IV-76Google Scholar
  127. D. Hambach, G. Schneider, E. Gullikson: Efficient high-order diffraction of extreme-ultraviolet light and soft x-rays by nanostructured volume gratings, Opt. Lett. 26(15), 1200–1202 (2001)CrossRefGoogle Scholar
  128. H.C. Kang, G.B. Stephenson, C. Liu, R. Conley, A.T. Macrander, J. Maser, S. Bajt, H.N. Chapman: High-efficiency diffractive x-ray optics from sectioned multilayers, Appl. Phys. Lett. 86, 151109 (2005)CrossRefGoogle Scholar
  129. J. Maser, G.B. Stephenson, S. Vogt, W. Yun, A. Macrander, H.C. Kang, C. Liu, R. Conley: Multilayer Laue lenses as high-resolution x-ray optics, Proc. SPIE 5539, 185–194 (2004)CrossRefGoogle Scholar
  130. X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I.K. Robinson, Y.S. Chu: 11 nm hard x-ray focus from a large-aperture multilayer Laue lens, Sci. Rep. 3, 3562 (2013)CrossRefGoogle Scholar
  131. B. Winn, H. Ade, C. Buckley, M. Feser, M. Howells, S. Hulbert, C. Jacobsen, K. Kaznacheyev, J. Kirz, A. Osanna, J. Maser, I. McNulty, J. Miao, T. Oversluizen, S. Spector, B. Sullivan, S. Wang, S. Wirick, H. Zhang: Illumination for coherent soft x-ray applications: The new X1A beamline at the NSLS, J. Synchrotron Radiat. 7, 395–404 (2000)CrossRefGoogle Scholar
  132. B. Niemann, D. Rudolph, G. Schmahl: Soft x-ray imaging zone plates with large zone numbers for microscopic and spectroscopic applications, Opt. Commun. 12(2), 160–163 (1974)CrossRefGoogle Scholar
  133. D. Rudolph, B. Niemann, G. Schmahl, O. Christ: The Göttingen x-ray microscope and x-ray microscopy experiments at the BESSY storage ring. In: X-Ray Microscopy, Springer Series in Optical Sciences, Vol. 43, ed. by G. Schmahl, D. Rudolph (Springer, Berlin 1984) pp. 192–202CrossRefGoogle Scholar
  134. W. Meyer-Ilse, G.P. Denbeaux, L.E. Johnson, W. Bates, A. Lucero, E.H. Anderson: The high resolution x-ray microscope XM-1. In: Xray Microsc. Proc. Sixth Int. Conf., ed. by W. Meyer-Ilse, T. Warwick, D. Attwood (American Institute of Physics, College Park 2000) pp. 129–134Google Scholar
  135. B. Niemann: High numerical-aperture x-ray condensers for transmission x-ray microscopes. In: X-ray Microscopy and Spectromicroscopy, ed. by J. Thieme, G. Schmahl, E. Umbach, D. Rudolph (Springer, Berlin 1998) pp. 337–347CrossRefGoogle Scholar
  136. S. Oestreich, G. Rostaing, B. Niemann, B. Kaulich, J. Susini, R. Barret: Dynamical coherent illumination for x-ray microscopy at 3rd generation synchrotron radiation sources: First results with x-rays at the Ca-K edge (4 keV). In: Xray Microsc. Proc. Sixth Int. Conf., ed. by W. Meyer-Ilse, T. Warwick, D. Attwood (American Institute of Physics, College Park 2000) pp. 464–467Google Scholar
  137. F. Zernike: Phase contrast, a new method for microscopic observation of transparent objects. Part I, Physica 9, 686–698 (1942)CrossRefGoogle Scholar
  138. U. Neuhäusler, G. Schneider, W. Ludwig, M.A. Meyer, E. Zschech, D. Hambach: X-ray microscopy in Zernike phase contrast mode at 4 keV photon energy with 60 nm resolution, J. Phys. D 36, A79–A82 (2003)CrossRefGoogle Scholar
  139. G. Schmahl, D. Rudolph, P. Guttmann: Phase contrast x-ray microscopy—Experiments at the BESSY storage ring. In: X-Ray Microscopy II, Springer Series in Optical Sciences, Vol. 56, ed. by D. Sayre, M.R. Howells, J. Kirz, H. Rarback (Springer, Berlin 1988) pp. 228–232CrossRefGoogle Scholar
  140. G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, B. Niemann: Phase contrast x-ray microscopy studies, Optik 97, 181–182 (1994)Google Scholar
  141. H. Rarback, J. Kenney, J. Kirz, X.S. Xie: Scanning x-ray microscopy—First tests with synchrotron radiation. In: Scanned Image Microscopy, ed. by E.A. Ash (Academic Press, London 1980) pp. 449–456Google Scholar
  142. H. Rarback, J.M. Kenney, J. Kirz, M.R. Howells, P. Chang, P.J. Coane, R. Feder, P.J. Houzego, D.P. Kern, D. Sayre: Recent results from the Stony Brook scanning microscope. In: X-Ray Microscopy, Springer Series in Optical Sciences, Vol. 43, ed. by G. Schmahl, D. Rudolph (Springer, Berlin 1984) pp. 203–215CrossRefGoogle Scholar
  143. B. Niemann: The Göttingen scanning x-ray microscope, Proc. SPIE 733, 422–427 (1987)CrossRefGoogle Scholar
  144. G.R. Morrison, M.T. Borwne, C.J. Buckley, R.E. Burge, R.C. Cave, P. Charalambous, P.J. Duke, A.R. Hare, C.P.B. Hills, J.M. Kenney, A.G. Michette, K. Ogawa, A.M. Rogoyski, T. Taguchi: Early experience with the King's College--Daresbury x-ray microscope. In: X-Ray Microscopy II, Springer Series in Optical Sciences, Vol. 56, ed. by D. Sayre, M.R. Howells, J. Kirz, H. Rarback (Springer, Berlin 1988) pp. 201–208CrossRefGoogle Scholar
  145. H. Rarback, D. Shu, S.C. Feng, H. Ade, J. Kirz, I. McNulty, D.P. Kern, T.H.P. Chang, Y. Vladimirsky, N. Iskander, D. Attwood, K. McQuaid, S. Rothman: Scanning x-ray microscope with 75-nm resolution, Rev. Sci. Instrum. 59, 52–59 (1988)CrossRefGoogle Scholar
  146. J.M. Kenney, G.R. Morrison, M.T. Browne, C.J. Buckley, R.E. Burge, R.C. Cave, P.S. Charalambous, P.D. Duke, A.R. Hare, C.P.B. Hills, A.G. Michette, K. Ogawa, A.M. Rogoyski: Evaluation of a scanning transmission x-ray microscope using undulator radiation at the SERC Daresbury Laboratory, J. Phys. E 22, 234–238 (1989)CrossRefGoogle Scholar
  147. A.L.D. Kilcoyne, T. Tyliszczak, W.F. Steele, S. Fakra, P. Hitchcock, K. Franck, E. Anderson, B. Harteneck, E.G. Rightor, G.E. Mitchell, A.P. Hitchcock, L. Yang, T. Warwick, H. Ade: Interferometer-controlled scanning transmission x-ray microscopes at the Advanced Light Source, J. Synchrotron Radiat. 10(2), 125–136 (2003)CrossRefGoogle Scholar
  148. C. Jacobsen, S. Williams, E. Anderson, M.T. Browne, C.J. Buckley, D. Kern, J. Kirz, M. Rivers, X. Zhang: Diffraction-limited imaging in a scanning transmission x-ray microscope, Opt. Commun. 86, 351–364 (1991)CrossRefGoogle Scholar
  149. M. Feser, T. Beetz, C. Jacobsen, J. Kirz, S. Wirick, A. Stein, T. Schäfer: Scanning transmission soft x-ray microscopy at beamline X-1A at the NSLS—Advances in instrumentation and selected applications, Proc. SPIE 4506, 146–153 (2001)CrossRefGoogle Scholar
  150. T. Beetz, M. Feser, H. Fleckenstein, B. Hornberger, C. Jacobsen, J. Kirz, M. Lerotic, E. Lima, M. Lu, D. Sayre, D. Shapiro, A. Stein, D. Tennant, S. Wirick: Soft x-ray microscopy at the NSLS, Synchrotron Radiat. News 16(3), 11–15 (2003)CrossRefGoogle Scholar
  151. A. Stein, C. Jacobsen, K. Evans-Lutterodt, D.M. Tennant, G. Bogart, F. Klemens, L.E. Ocola, B.J. Choi, S.V. Sreenivasan: Diffractive x-ray optics using production fabrication methods, J. Vac. Sci. Technol. B 21(1), 214–219 (2003)CrossRefGoogle Scholar
  152. M. Lu, D.M. Tennant, C.J. Jacobsen: Orientation dependence of linewidth variation in sub-50-nm Gaussian e-beam lithography and its correction, J. Vac. Sci. Technol. B 24(6), 2881–2885 (2006)CrossRefGoogle Scholar
  153. J.M. Kenney, C. Jacobsen, J. Kirz, H. Rarback, F. Cinotti, W. Thomlinson, R. Rosser, G. Schidlovsky: Absorption microanalysis with a scanning soft x-ray microscope: Mapping the distribution of calcium in bone, J. Microsc. 138(3), 321–328 (1985)CrossRefGoogle Scholar
  154. D. Shu, D.P. Siddons, H. Rarback, J. Kirz: Two-dimensional laser interferometric encoder for the soft x-ray scanning microscope at the NSLS, Nucl. Instrum. Methods Phys. Res. A 266, 313–317 (1988)CrossRefGoogle Scholar
  155. P. Guttmann, B. Niemann: A detector system for high photon rates for a scanning x-ray microscope. In: X-Ray Microscopy II, Springer Series in Optical Sciences, Vol. 56, ed. by D. Sayre, M.R. Howells, J. Kirz, H. Rarback (Springer, Berlin 1988) pp. 154–159CrossRefGoogle Scholar
  156. M. Feser, M. Carlucci-Dayton, C.J. Jacobsen, J. Kirz, U. Neuhäusler, G. Smith, B. Yu: Applications and instrumentation advances with the Stony Brook scanning transmission x-ray microscope, Proc. SPIE 3449, 19–29 (1998)CrossRefGoogle Scholar
  157. J. Maser, A. Osanna, Y. Wang, C. Jacobsen, J. Kirz, S. Spector, B. Winn, D. Tennant: Soft x-ray microscopy with a cryo STXM: I. Instrumentation, imaging, and spectroscopy, J. Microsc. 197(1), 68–79 (2000)CrossRefGoogle Scholar
  158. R. Barrett, B. Kaulich, S. Oestreich, J. Susini: Scanning microscopy end station at the ESRF x-ray microscopy beamline, Proc. SPIE 3449, 80–90 (1998)CrossRefGoogle Scholar
  159. U. Wiesemann, J. Thieme, R. Früke, P. Guttmann, B. Niemann, D. Rudolph, G. Schmahl: Construction of a scanning transmission X-ray microscope at the undulator U-41 at BESSY II, Nucl. Instrum. Methods Phys. Res. A 467–468, 861–863 (2001)CrossRefGoogle Scholar
  160. M. Feser, C. Jacobsen, P. Rehak, G. DeGeronimo, P. Holl, L. Strüder: Novel integrating solid state detector with segmentation for scanning transmission soft x-ray microscopy, Proc. SPIE 4499, 117–125 (2001)CrossRefGoogle Scholar
  161. P. Guttmann, B. Niemann, J. Thieme, D. Hambach, G. Schneider, U. Wiesemann, D. Rudolph, G. Schmahl: Instrumentation advances with the new x-ray microscopes at BESSY II, Nucl. Instrum. Methods Phys. Res. A 467, 849–852 (2001)CrossRefGoogle Scholar
  162. M. Feser, C. Jacobsen, P. Rehak, G. DeGeronimo: Scanning transmission x-ray microscopy with a segmented detector, J. Phys. IV 104, 529–534 (2003)Google Scholar
  163. W. Meyer-Ilse, D. Attwood, M. Koike: The x-ray microscopy resource center at the Advanced Light Source. In: Synchrotron Radiation in the Biociences, ed. by B. Chance, D. Deisenhober, S. Ebashi, D.T. Goodhead, J.R. Helliwell, H.E. Huxley, T. Iizuka, J. Kirz, T. Mitsui, E. Rubenstein, N. Sakabe, T. Sasaki, G. Schmahl, H. Sturhmann, K. Wüthrich, G. Zaccai (Clarendon, Oxford 1994) pp. 624–636Google Scholar
  164. G.R. Morrison: X-ray imaging with a configured detector. In: X-Ray Microscopy IV, ed. by V.V. Aristov, A.I. Erko (Bogorodskii Pechatnik, Chernogolovka 1994) pp. 478–486Google Scholar
  165. G. Morrison, W.J. Eaton, R. Barnett, P. Charalambous: STXM imaging with a configured detector, J. Phys. IV 104, 547–550 (2003)Google Scholar
  166. M. Feser, B. Hornberger, C. Jacobsen, G. De Geronimo, P. Rehak, P. Holl, L. Strüder: Integrating silicon detector with segmentation for scanning transmission x-ray microscopy, Nucl. Instrum. Methods Phys. Res. A 565, 841–854 (2006)CrossRefGoogle Scholar
  167. B. Hornberger, M.D. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, S. Vogt: Differential phase contrast with a segmented detector in a scanning x-ray microprobe, J. Synchrotron Radiat. 15(4), 355–362 (2008)CrossRefGoogle Scholar
  168. B. Kaulich, T. Wilhein, E. Di Fabrizio, F. Romanato, M. Altissimo, S. Cabrini, B. Fayard, J. Susini: Differential interference contrast x-ray microscopy with twin zone plates, J. Opt. Soc. Am. A 19(4), 797–806 (2002)CrossRefGoogle Scholar
  169. F. Polack, D. Joyeux, M. Feser, D. Phalippou, M. Carlucci-Dayton, K. Kaznacheyev, C. Jacobsen: Demonstration of phase contrast in scanning transmission x-ray microscopy: Comparison of images obtained at NSLS X1-A with numerical simulations. In: Xray Microsc. Proc. Sixth Int. Conf., ed. by W. Meyer-Ilse, T. Warwick, D. Attwood (American Institute of Physics, College Park 2000) pp. 573–580Google Scholar
  170. C. Holzner, M. Feser, S. Vogt, B. Hornberger, S.B. Baines, C. Jacobsen: Zernike phase contrast in scanning microscopy with x-rays, Nat. Phys. 6, 883–887 (2010)CrossRefGoogle Scholar
  171. H.N. Chapman: Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution, Ultramicroscopy 66, 153–172 (1996)CrossRefGoogle Scholar
  172. H.N. Chapman: Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution: Signal processing, Scanning Microsc. 11, 67–80 (1997)Google Scholar
  173. W. Meyer-Ilse, T. Wilhein, P. Guttmann: Thinned back-illuminated CCD for x-ray microscopy, Proc. SPIE 1900, 241–245 (1993)CrossRefGoogle Scholar
  174. C. Broennimann, E.F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, A. Wagner: The PILATUS 1M detector, J. Synchrotron Radiat. 13, 120–130 (2006)CrossRefGoogle Scholar
  175. S. Chen, J. Deng, Y. Yuan, C. Flachenecker, R. Mak, B. Hornberger, Q. Jin, D. Shu, B. Lai, J. Maser, C. Roehrig, T. Paunesku, S.-C. Gleber, D.J. Vine, L. Finney, J. Von Osinski, M. Bolbat, I. Spink, Z. Chen, J. Steele, D. Trapp, J. Irwin, M. Feser, E. Snyder, K. Brister, C. Jacobsen, G. Woloschak, S. Vogt: The Bionanoprobe: Hard x-ray fluorescence nanoprobe with cryogenic capabilities, J. Synchrotron Radiat. 21(1), 66–75 (2014)CrossRefGoogle Scholar
  176. J. Deng, D.J. Vine, S. Chen, Q. Jin, Y.S.G. Nashed, T. Peterka, S. Vogt, C. Jacobsen: X-ray ptychographic and fluorescence microscopy of frozen-hydrated cells using continuous scanning, Sci. Rep. 7(1), 445 (2017)CrossRefGoogle Scholar
  177. W. Hoppe: Beugung im Inhomogenen Primärstrahlwellenfeld. I. Prinzip einer Phasenmessung, Acta Crystallogr. A 25, 495–501 (1969)CrossRefGoogle Scholar
  178. R.W. Gerchberg, W.O. Saxton: Phase determination from image and diffraction plane pictures in the electron microscope, Optik 34(3), 275–284 (1971)Google Scholar
  179. J.R. Fienup: Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett. 3(1), 27–29 (1978)CrossRefGoogle Scholar
  180. H.M.L. Faulkner, J. Rodenburg: Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm, Phys. Rev. Lett. 93(2), 023903 (2004)CrossRefGoogle Scholar
  181. J.M. Rodenburg, H.M.L. Faulkner: A phase retrieval algorithm for shifting illumination, Appl. Phys. Lett. 85(20), 4795–4797 (2004)CrossRefGoogle Scholar
  182. J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson: Hard-x-ray lensless imaging of extended objects, Phys. Rev. Lett. 98(3), 034801 (2007)CrossRefGoogle Scholar
  183. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pfeiffer: High-resolution scanning x-ray diffraction microscopy, Science 321, 379–382 (2008)CrossRefGoogle Scholar
  184. P. Thibault, M. Dierolf, C.M. Kewish, A. Menzel, O. Bunk, F. Pfeiffer: Contrast mechanisms in scanning transmission x-ray microscopy, Phys. Rev. A 80(4), 043813 (2009)CrossRefGoogle Scholar
  185. A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, C.G. Schroer: Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes, Appl. Phys. Lett. 100(25), 253112 (2012)CrossRefGoogle Scholar
  186. J. Deng, D.J. Vine, S. Chen, Y.S.G. Nashed, Q. Jin, N.W. Phillips, T. Peterka, R. Ross, S. Vogt, C.J. Jacobsen: Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae, Proc. Natl. Acad. Sci. U.S.A. 112(8), 2314–2319 (2015)CrossRefGoogle Scholar
  187. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, F. Pfeiffer: Probe retrieval in ptychographic coherent diffractive imaging, Ultramicroscopy 109(4), 338–343 (2009)CrossRefGoogle Scholar
  188. D.J. Vine, D. Pelliccia, C. Holzner, S.B. Baines, A. Berry, I. McNulty, S. Vogt, A.G. Peele, K.A. Nugent: Simultaneous x-ray fluorescence and ptychographic microscopy of Cyclotella meneghiniana, Opt. Express 20(16), 18287–18296 (2012)CrossRefGoogle Scholar
  189. P. Thibault, A. Menzel: Reconstructing state mixtures from diffraction measurements, Nature 494(7435), 68–71 (2013)CrossRefGoogle Scholar
  190. J.N. Clark, X. Huang, R.J. Harder, I.K. Robinson: A continuous scanning mode for ptychography, Opt. Lett. 39(20), 6066–6069 (2014)CrossRefGoogle Scholar
  191. P.M. Pelz, M. Guizar-Sicairos, P. Thibault, I. Johnson, M. Holler, A. Menzel: On-the-fly scans for x-ray ptychography, Appl. Phys. Lett. 105, 251101 (2014)CrossRefGoogle Scholar
  192. J. Deng, Y.S.G. Nashed, S. Chen, N.W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, D.J. Vine: Continuous motion scan ptychography: Characterization for increased speed in coherent x-ray imaging, Opt. Express 23(5), 5438–5451 (2015)CrossRefGoogle Scholar
  193. W. Yun, S.T. Pratt, R.M. Miller, Z. Cai, D.B. Hunter, A.G. Jarstfer, K.M. Kemner, B.P. Lai, H.-R. Lee, D.G. Legnini, W. Rodrigues, C.I. Smith: X-ray imaging and microspectroscopy of plants and fungi, J. Synchrotron Radiat. 5(6), 1390–1395 (1998)CrossRefGoogle Scholar
  194. Y. Suzuki, A. Takeuchi, H. Takano: Diffraction-limited microbeam with Fresnel zone plate optics in hard x-ray regions, Jpn. J. Appl. Phys. 40, 1508–1510 (2001)CrossRefGoogle Scholar
  195. R. Rebonato, G.E. Ice, A. Habenschuss, J.C. Bilello: High-resolution microdiffraction study of notch-tip deformation in Mo single crystals using x-ray synchrotron radiation, Philos. Mag. A 60(5), 571–583 (1989)CrossRefGoogle Scholar
  196. Z.H. Cai, W. Rodrigues, P. Ilinski, D. Legnini, B.P. Lai, W. Yun, E.D. Isaacs, K.E. Lutterodt, J. Grenko, R. Glew, S. Sputz, J. Vandenberg, R. People, M.A. Alam, M. Hybertsen, L.J.P. Ketelsen: Synchrotron x-ray microdiffraction diagnostics of multilayer optoelectronic devices, Appl. Phys. Lett. 75(1), 100–102 (1999)CrossRefGoogle Scholar
  197. Y.A. Soh, P.G. Evans, Z. Cai, B.P. Lai, C.Y. Kim, G. Aeppli, N.D. Mathur, M.G. Blamire, E.D. Isaacs: Local mapping of strain at grain boundaries in colossal magnetoresistive films using x-ray microdiffraction, J. Appl. Phys. 91(10), 7742–7743 (2002)CrossRefGoogle Scholar
  198. B.C. Larson, W. Yang, G.E. Ice, J.D. Budai, J.Z. Tischler: Three-dimensional x-ray structural microscopy with submicrometre resolution, Nature 415(6874), 887–890 (2002)CrossRefGoogle Scholar
  199. H.H. Hopkins: Applications of coherence theory in microscopy and interferometry, J. Opt. Soc. Am. 47(6), 508–526 (1957)CrossRefGoogle Scholar
  200. B.J. Thompson: Image formation with partially coherent light, Prog. Opt. 7, 169–230 (1969)CrossRefGoogle Scholar
  201. T. Wilson, C. Sheppard: Theory and Practice of Scanning Optical Microscopy (Academic Press, Cambridge 1984)Google Scholar
  202. J.W. Goodman: Statistical Optics (Wiley, New York 1985)Google Scholar
  203. J.W. Goodman: Introduction to Fourier Optics, 3rd edn. (Roberts & Company, Greenwood Village 2005)Google Scholar
  204. C. Jacobsen, J. Kirz, S. Williams: Resolution in soft x-ray microscopes, Ultramicroscopy 47, 55–79 (1992)CrossRefGoogle Scholar
  205. L. Jochum, W. Meyer-Ilse: Partially coherent image formation with x-ray microscopes, Appl. Opt. 34(22), 4944–4950 (1995)CrossRefGoogle Scholar
  206. J.M. Heck, D.T. Attwood, W. Meyer-Ilse: Resolution determination in x-ray microscopy: An analysis of the effects of partial coherence and illumination spectrum, J. Xray Sci. Technol. 8(2), 95–104 (1998)Google Scholar
  207. J. Otón, E. Pereiro, A.J. Pérez-Berná, L. Millach, C.O.S. Sorzano, R. Marabini, J.M. Carazo: Characterization of transfer function, resolution and depth of field of a soft x-ray microscope applied to tomography enhancement by Wiener deconvolution, Biomed. Opt. Express 7(12), 5092–5103 (2016)CrossRefGoogle Scholar
  208. H.N. Chapman, C. Jacobsen, S. Williams: A characterisation of dark-field imaging of colloidal gold labels in a scanning transmission x-ray microscope, Ultramicroscopy 62, 191–213 (1996)CrossRefGoogle Scholar
  209. H.N. Chapman, C. Jacobsen, S. Williams: Applications of a CCD detector in scanning transmission x-ray microscopy, Rev. Sci. Instrum. 66(2), 1332–1334 (1995)CrossRefGoogle Scholar
  210. E.C. Kintner: Method for the calculation of partially coherent imagery, Appl. Opt. 17(17), 2747–2747 (1978)CrossRefGoogle Scholar
  211. B.E.A. Saleh: Optical bilinear transformations: General properties, Opt. Acta 26(6), 777–799 (1979)CrossRefGoogle Scholar
  212. C.J.R. Sheppard, T. Wilson: Fourier imaging of phase information in scanning and conventional optical microscopes, Philos. Trans. R. Soc. A 295(1415), 513–536 (1980)CrossRefGoogle Scholar
  213. P.K. Mondal, S. Slansky: Influence de la position de l'anneau de phase sur le contraste de l'image dans un objectif à contraste dephase, Appl. Opt. 9(8), 1879–1882 (1970)CrossRefGoogle Scholar
  214. G.R. Morrison: Some aspects of quantitative x-ray microscopy, Proc. SPIE 1140, 41–49 (1989)CrossRefGoogle Scholar
  215. E. Zeitler, M.G.R. Thomson: Scanning transmission electron microscopy. I, Optik 31(3), 258–280 (1970)Google Scholar
  216. E. Zeitler, M.G.R. Thomson: Scanning transmission electron microscopy. II, Optik 31(4), 359–366 (1970)Google Scholar
  217. C.J.R. Sheppard, T. Wilson: Reciprocity and equivalence in scanning microscopes, J. Opt. Soc. Am. A 3(5), 755–756 (1986)CrossRefGoogle Scholar
  218. T. Wilson, C.J.R. Sheppard: The halo effect of image processing by spatial frequency filtering, Optik 59(1), 19–23 (2003)Google Scholar
  219. L.C. Martin: The Theory of the Microscope (Elsevier, New York 1966)Google Scholar
  220. S. Aoki, S. Kikuta: X-ray holographic microscopy, Jpn. J. Appl. Phys. 13, 1385–1392 (1974)CrossRefGoogle Scholar
  221. C. Jacobsen, M. Howells, J. Kirz, S. Rothman: X-ray holographic microscopy using photoresists, J. Opt. Soc. Am. A 7, 1847–1861 (1990)CrossRefGoogle Scholar
  222. I. McNulty, J. Kirz, C. Jacobsen, E. Anderson, D. Kern, M. Howells: High-resolution imaging by Fourier transform x-ray holography, Science 256, 1009–1012 (1992)CrossRefGoogle Scholar
  223. S. Eisebitt, J. Lüning, W.F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, J. Stöhr: Lensless imaging of magnetic nanostructures by x-ray spectro-holography, Nature 432, 885–888 (2004)CrossRefGoogle Scholar
  224. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.P. Guigay, M. Schlenker: Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays, Appl. Phys. Lett. 75(19), 2912–2914 (1999)CrossRefGoogle Scholar
  225. W.S. Haddad, I. McNulty, J.E. Trebes, E.H. Anderson, R.A. Levesque, L. Yang: Ultra high resolution x-ray tomography, Science 266, 1213–1215 (1994)CrossRefGoogle Scholar
  226. J. Lehr: 3D x-ray microscopy: Tomographic imaging of mineral sheaths of bacteria Leptothrix ochracea with the Göttingen x-ray microscope at BESSY, Optik 104(4), 166–170 (1997)Google Scholar
  227. A. Faridani, E.L. Ritman, K.T. Smith: Local tomography, SIAM J. Appl. Math. 52(2), 459–484 (1992)CrossRefGoogle Scholar
  228. Y. Wang, C. Jacobsen, J. Maser, A. Osanna: Soft x-ray microscopy with a cryo STXM: II. Tomography, J. Microsc. 197(1), 80–93 (2000)CrossRefGoogle Scholar
  229. D. Weiß, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, G. Schmahl: Computed tomography of cryogenic biological specimens based on x-ray microscopic images, Ultramicroscopy 84(3–4), 185–197 (2000)CrossRefGoogle Scholar
  230. G. Schneider, E. Anderson, S. Vogt, C. Knöchel, D. Weiss, M. Legros, C. Larabell: Computed tomography of cryogenic cells, Surf. Rev. Lett. 9(1), 177–183 (2002)CrossRefGoogle Scholar
  231. L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, Springer Series in Optical Sciences, Vol. 36, 3rd edn. (Springer, Berlin 1993)CrossRefGoogle Scholar
  232. D.A. Agard, J.W. Sedat: Three-dimensional architecture of a polytene nucleus, Nature 302(5910), 676–681 (1983)CrossRefGoogle Scholar
  233. W.A. Carrington, R.M. Lynch, E.D. Moore, G. Isenberg, K.E. Fogarty, F.S. Fay: Superresolution three-dimensional images of fluorescence in cells with minimal light exposure, Science 268(5216), 1483–1487 (1995)CrossRefGoogle Scholar
  234. M. Selin, E. Fogelqvist, S. Werner, H.M. Hertz: Tomographic reconstruction in soft x-ray microscopy using focus-stack back-projection, Opt. Lett. 40(10), 2201–2204 (2015)CrossRefGoogle Scholar
  235. S. Wang, F. Duewer, S. Kamath, C. Kelly, A. Lyon, K. Nill, P. Pombo, D. Scott, D. Trapp, W. Yun, S. Neogi, M. Kuhn, C. Bennet, P. Coon, S. Yan: A transmission x-ray microscope (TXM) for non-destructive 3D imaging of ICs at sub-100 nm resolution. In: Proc. 28th Int. Symp. Test. Fail. Anal., ed. by C. Boit (ASM International, Materials Park 2002) pp. 227–233Google Scholar
  236. G.-C. Yin, M.-T. Tang, Y.-F. Song, F.-R. Chen, K.S. Liang, F.W. Duewer, W. Yun, C.-H. Ko, H.-P.D. Shieh: Energy-tunable transmission x-ray microscope for differential contrast imaging with near 60 nm resolution tomography, Appl. Phys. Lett. 88(24), 241115 (2006)CrossRefGoogle Scholar
  237. J. Stöhr: NEXAFS Spectroscopy, Springer Series in Surface Sciences, Vol. 25, 1st edn. (Springer, Berlin 1992)CrossRefGoogle Scholar
  238. C. Jeanguillaume, C. Colliex: Spectrum-image: The next step in EELS digital acquisition and processing, Ultramicroscopy 28, 252–257 (1989)CrossRefGoogle Scholar
  239. J.A. Hunt, D.B. Williams: Electron energy-loss spectrum-imaging, Ultramicroscopy 38, 47–73 (1991)CrossRefGoogle Scholar
  240. P.L. King, R. Browning, P. Pianetta, I. Lindau, M. Keenlyside, G. Knapp: Image-processing of multispectral x-ray photoelectron-spectroscopy images, J. Vac. Sci. Technol. A 7(6), 3301–3304 (1989)CrossRefGoogle Scholar
  241. G.R. Harp, Z.-L. Han, B.P. Tonner: Spatially-resolved x-ray absorption near-edge spectroscopy of silicon in thin silicon-oxide films, Phys. Scr. 31, 23–27 (2007)Google Scholar
  242. H. Ade, X. Zhang, S. Cameron, C. Costello, J. Kirz, S. Williams: Chemical contrast in x-ray microscopy and spatially resolved XANES spectroscopy of organic specimens, Science 258, 972–975 (1992)CrossRefGoogle Scholar
  243. C. Jacobsen, G. Flynn, S. Wirick, C. Zimba: Soft x-ray spectroscopy from image sequences with sub-100 nm spatial resolution, J. Microsc. 197(2), 173–184 (2000)CrossRefGoogle Scholar
  244. O.L. Krivanek, T.C. Lovejoy, N. Dellby, T. Aoki, R.W. Carpenter, P. Rez, E. Soignard, J. Zhu, P.E. Batson, M.J. Lagos, R.F. Egerton, P.A. Crozier: Vibrational spectroscopy in the electron microscope, Nature 514(7521), 209–212 (2014)CrossRefGoogle Scholar
  245. M. Isaacson, M. Utlaut: A comparison of electron and photon beams for determining micro-chemical environment, Optik 50(3), 213–234 (1978)Google Scholar
  246. E.G. Rightor, A.P. Hitchcock, H. Ade, R.D. Leapman, S.G. Urquhart, A.P. Smith, G. Mitchell, D. Fischer, H.J. Shin, T. Warwick: Spectromicroscopy of poly(ethylene terephthalate): Comparison of spectra and radiation damage rates in x-ray absorption and electron energy loss, J. Phys. Chem. B 101(11), 1950–1960 (1997)CrossRefGoogle Scholar
  247. N. Bonnet, E. Simova, S. Lebonvallet, H. Kaplan: New applications of multivariate statistical analysis in spectroscopy and microscopy, Ultramicroscopy 40, 1–11 (1992)CrossRefGoogle Scholar
  248. M. Lerotic, C. Jacobsen, T. Schäfer, S. Vogt: Cluster analysis of soft x-ray spectromicroscopy data, Ultramicroscopy 100(1/2), 35–57 (2004)CrossRefGoogle Scholar
  249. O. Dhez, H. Ade, S.G. Urquhart: Calibrated NEXAFS spectra of some common polymers, J. Electron Spectrosc. Relat. Phenom. 128(1), 85–96 (2003)CrossRefGoogle Scholar
  250. K. Kaznacheyev, A. Osanna, C. Jacobsen, O. Plashkevych, O. Vahtras, H. Ågren, V. Carravetta, A.P. Hitchcock: Innershell absorption spectroscopy of amino acids, J. Phys. Chem. A 106(13), 3153–3168 (2002)CrossRefGoogle Scholar
  251. X. Zhang, R. Balhorn, J. Mazrimas, J. Kirz: Mapping and measuring DNA to protein ratios in mammalian sperm head by XANES imaging, J. Struct. Biol. 116, 335–344 (1996)CrossRefGoogle Scholar
  252. I.N. Koprinarov, A.P. Hitchcock, C.T. McCrory, R.F. Childs: Quantitative mapping of structured polymeric systems using singular value decomposition analysis of soft x-ray images, J. Phys. Chem. B 106, 5358–5364 (2002)CrossRefGoogle Scholar
  253. E.R. Malinowski: Factor Analysis in Chemistry, 2nd edn. (Wiley, New York 1991)Google Scholar
  254. A. Osanna, C. Jacobsen: Principle component analysis for soft x-ray spectromicroscopy. In: Xray Microsc. Proc. Sixth Int. Conf., ed. by W. Meyer-Ilse, T. Warwick, D. Attwood (American Institute of Physics, College Park 2000) pp. 350–357Google Scholar
  255. B.S. Everitt, S. Landau, M. Leese: Cluster Analysis, 4th edn. (Arnold, London 2001)Google Scholar
  256. M. Lerotic, C. Jacobsen, J.B. Gillow, A.J. Francis, S. Wirick, S. Vogt, J. Maser: Cluster analysis in soft x-ray spectromicroscopy: Finding the patterns in complex specimens, J. Electron Spectrosc. Relat. Phenom. 144–147, 1137–1143 (2005)CrossRefGoogle Scholar
  257. R. Mak, M. Lerotic, H. Fleckenstein, S. Vogt, S.M. Wild, S. Leyffer, Y. Sheynkin, C. Jacobsen: Non-negative matrix analysis for effective feature extraction in x-ray spectromicroscopy, Faraday Discuss. 171(1), 357–371 (2014)CrossRefGoogle Scholar
  258. A. Scheinost, R. Kretzschmar, I. Christl, C. Jacobsen: Carbon group chemistry of humic and fulvic acid: A comparison of C-1s NEXAFS and \({}^{13}\)C-NMR spectroscopies. In: Humic Substances: Structures, Models and Functions, ed. by E.A. Ghabbour, G. Davies (Royal Society of Chemistry, Cambridge 2001) pp. 37–45Google Scholar
  259. M. Schumacher, I. Christl, A. Scheinost, C. Jacobsen, R. Kretzschmar: Chemical heterogeneity of organic soil colloids investigated by scanning transmission x-ray microscopy and C-1s NEXAFS microscopy, Environ. Sci. Technol. 39, 9094–9100 (2005)CrossRefGoogle Scholar
  260. D. Solomon, J. Lehmann, J. Kinyangi, B. Liang, T. Schäfer: Carbon K-edge NEXAFS and FTIR-ATR spectroscopic investigation of organic carbon speciation in soils, Soil Sci. Soc. Am. J. 69, 107–119 (2005)CrossRefGoogle Scholar
  261. J. Kirz, C. Jacobsen, M. Howells: Soft x-ray microscopes and their biological applications, Q. Rev. Biophys. 28(1), 33–130 (1995), also available as Lawrence Berkeley Laboratory report LBL-36371CrossRefGoogle Scholar
  262. J.V. Abraham-Peskir: X-ray microscopy with synchrotron radiation: Applications to cellular biology, Cell. Mol. Biol. 46, 1045–1052 (2000)Google Scholar
  263. J.L. Carrascosa, R.M. Glaeser: Focused issue on x-ray microscopy of biological materials, J. Struct. Biol. 177(2), 177–178 (2012)CrossRefGoogle Scholar
  264. M. Berglund, L. Rymell, M. Peuker, T. Wilhein, H.M. Hertz: Compact water-window transmission x-ray microscopy, J. Microsc. 197(3), 268–273 (2000)CrossRefGoogle Scholar
  265. M. Bertilson, O. von Hofsten, U. Vogt, A. Holmberg, A.E. Christakou, H.M. Hertz: Laboratory soft-x-ray microscope for cryotomography of biological specimens, Opt. Lett. 36(14), 2728–2730 (2011)CrossRefGoogle Scholar
  266. E. Duke, K. Dent, M. Razi, L.M. Collinson: Biological applications of cryo-soft x-ray tomography, J. Microsc. 255(2), 65–70 (2014)Google Scholar
  267. H.N. Chapman, J. Fu, C. Jacobsen, S. Williams: Dark-field x-ray microscopy of immunogold-labeled cells, Microsc. Microanal. 2(2), 53–62 (1996)CrossRefGoogle Scholar
  268. W. Meyer-Ilse, D. Hamamoto, A. Nair, S.A. Lelievre, G. Denbeaux, L. Johnson, A.L. Pearson, D. Yager, M.A. Legros, C.A. Larabell: High resolution protein localization using soft x-ray microscopy, J. Microsc. 201(3), 395–403 (2001)CrossRefGoogle Scholar
  269. S. Vogt, M. Jäger, G. Schneider, E. Schulze, H. Saumweber, D. Rudolph, G. Schmahl: Visualizing specific nuclear proteins in eukaryotic cells using soft x-ray microscopy, Nucl. Instrum. Methods Phys. Res. A 467–468, 1312–1314 (2001)CrossRefGoogle Scholar
  270. C.J. Buckley, N. Khaleque, S.J. Bellamy, M. Robins, X. Zhang: Mapping the organic and inorganic components of tissue using NEXAFS, J. Phys. IV 7(C2), 83–90 (1997)Google Scholar
  271. A. Ito, K. Shinohara, H. Nakano, T. Matsumura, K. Kinoshita: Measurement of soft x-ray absorption spectra and elemental analysis in local regions of mammalian cells using an electronic zooming tube, J. Microsc. 181, 54–60 (1996)CrossRefGoogle Scholar
  272. A.P. Hitchcock, C. Morin, Y.M. Heng: Towards practical soft x-ray spectromicroscopy of biomaterials, J. Biomater. Sci. Polym. Ed. 13(8), 919–937 (2002)CrossRefGoogle Scholar
  273. A.P. Hitchcock, J.J. Dynes, J.R. Lawrence, M. Obst, G.D.W. Swerhone, D.R. Korber, G.G. Leppard: Soft x-ray spectromicroscopy of nickel sorption in a natural river biofilm, Geobiology 7(4), 432–453 (2009)CrossRefGoogle Scholar
  274. J. Kawai, K. Takagawa, S. Fujisawa, A. Ektessabi, S. Hayakawa: Microbeam XANES and x-ray fluorescence analysis of cadmium in kidney, J. Trace Microprobe Tech. 19(4), 541–546 (2007)CrossRefGoogle Scholar
  275. K.M. Kemner, S.D. Kelly, J. Maser, E.J. O'Loughlin, D. Sholto-Douglas, Z. Cai, M.A. Schneegurt, C.F. Kulpa, K.H. Nealson: Elemental and redox analysis of single bacterial cells by x-ray microbeam analysis, Science 306(5696), 686–687 (2004)CrossRefGoogle Scholar
  276. R. Ortega, G. Devès, A. Carmona: Bio-metals imaging and speciation in cells using proton and synchrotron radiation x-ray microspectroscopy, J. R. Soc. Interface 6, S649–S658 (2009)CrossRefGoogle Scholar
  277. T. Paunesku, S. Vogt, J. Maser, B. Lai, G. Woloschak: X-ray fluorescence microprobe imaging in biology and medicine, J. Cell. Biochem. 99, 1489–1502 (2006)CrossRefGoogle Scholar
  278. C.J. Fahrni: Biological applications of x-ray fluorescence microscopy: Exploring the subcellular topography and speciation of transition metals, Curr. Opin. Chem. Biol. 11, 121–127 (2007)CrossRefGoogle Scholar
  279. M.D. de Jonge, S. Vogt: Hard x-ray fluorescence tomography – An emerging tool for structural visualization, Curr. Opin. Struct. Biol. 20(5), 606–614 (2010)CrossRefGoogle Scholar
  280. C.A. Larabell, M.A. Le Gros: X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution, Mol. Biol. Cell 15, 957–962 (2004)CrossRefGoogle Scholar
  281. J. Pine, J. Gilbert: Live cell specimens for x-ray microscopy. In: X-Ray Microscopy III, Springer Series in Optical Sciences, Vol. 67, ed. by A.G. Michette, G.R. Morrison, C.J. Buckley (Springer, Berlin 1992) pp. 384–387CrossRefGoogle Scholar
  282. J.R. Gilbert, J. Pine: Imaging and etching: Soft x-ray microscopy on whole wet cells, Proc. SPIE 1741, 402–408 (1992)CrossRefGoogle Scholar
  283. G.F. Foster, C.J. Buckley, P.M. Bennett, R.E. Burge: Investigation of radiation damage to biological specimens at water window wavelengths, Rev. Sci. Instrum. 63, 599–600 (1992)CrossRefGoogle Scholar
  284. P.M. Bennett, G.F. Foster, C.J. Buckley, R.E. Burge: The effect of soft x-radiation on myofibrils, J. Microsc. 172, 109–119 (1993)CrossRefGoogle Scholar
  285. U. Neuhäusler, S. Abend, G. Lagaly, C. Jacobsen: Soft x-ray spectromicroscopy on solid stabilized emulsions, Colloid Polym. Sci. 277, 719–726 (1999)CrossRefGoogle Scholar
  286. T.W. Ford, A.M. Page, G.F. Foster, A.D. Stead: Effects of soft x-ray irradiation on cell ultrastructure, Proc. SPIE 1741, 325–332 (1992)CrossRefGoogle Scholar
  287. S. Williams, X. Zhang, C. Jacobsen, J. Kirz, S. Lindaas, J. van't Hof, S.S. Lamm: Measurements of wet metaphase chromosomes in the scanning transmission x-ray microscope, J. Microsc. 170, 155–165 (1993)CrossRefGoogle Scholar
  288. J. Coetzee, C.F. van der Merwe: Extraction of substances during glutaraldehyde fixation of plant cells, J. Microsc. 135(2), 147–158 (1984)CrossRefGoogle Scholar
  289. J. Coetzee, C.F. van der Merwe: Extraction of carbon 14-labeled compounds from plant tissue during processing for electron microscopy, J. Electron Microsc. Tech. 11(2), 155–160 (1989)CrossRefGoogle Scholar
  290. A.D. Stead, R.A. Cotton, A.M. Page, M.D. Dooley, T.W. Ford: Visualization of the effects of electron microscopy fixatives on the structure of hydrated epidermal hairs of tomato (Lycopersicum peruvianum) as revealed by soft x-ray contact microscopy, Proc. SPIE 1741, 351–362 (1992)CrossRefGoogle Scholar
  291. E. O'Toole, G. Wray, J. Kremer, J.R. McIntosh: High voltage cryomicroscopy of human blood platelets, J. Struct. Biol. 110, 55–66 (1993)CrossRefGoogle Scholar
  292. S. Jearanaikoon: An x-ray microscopy perspective on the effect of glutaraldehyde fixation on cells, J. Microsc. 218(2), 185–192 (2005)CrossRefGoogle Scholar
  293. K. Taylor, R. Glaeser: Electron diffraction of frozen, hydrated protein crystals, Science 106, 1036–1037 (1974)CrossRefGoogle Scholar
  294. R.A. Steinbrecht, K. Zierold (Eds.): Cryotechniques in Biological Electron Microscopy (Springer, Berlin 1987)Google Scholar
  295. J. Dubochet, M. Adrian, J.J. Chang, J.-C. Homo, J. Lepault, A.W. McDowell, P. Schultz: Cryo-electron microscopy of vitrified specimens, Q. Rev. Biophys. 21, 129–228 (1988)CrossRefGoogle Scholar
  296. P. Echlin: Low-Temperature Microscopy and Analysis (Plenum, New York 1992)CrossRefGoogle Scholar
  297. T. Beetz, C. Jacobsen: Soft x-ray radiation-damage studies in PMMA using a cryo-STXM, J. Synchrotron Radiat. 10(3), 280–283 (2003)CrossRefGoogle Scholar
  298. M.A. Le Gros, G. McDermott, M. Uchida, C.G. Knoechel, C.A. Larabell: High-aperture cryogenic light microscopy, J. Microsc. 235(1), 1–8 (2009)CrossRefGoogle Scholar
  299. E.A. Smith, B.P. Cinquin, M. Do, G. Mcdermott, M.A. Le Gros, C.A. Larabell: Correlative cryogenic tomography of cells using light and soft x-rays, Ultramicroscopy 143(C), 33–40 (2014)CrossRefGoogle Scholar
  300. C. Hagen, P. Guttmann, B. Klupp, S. Werner, S. Rehbein, T.C. Mettenleiter, G. Schneider, K. Grünewald: Correlative VIS-fluorescence and soft x-ray cryo-microscopy/tomography of adherent cells, J. Struct. Biol. 177(2), 193–201 (2012)CrossRefGoogle Scholar
  301. G.E. Brown Jr., N.C. Sturchio: An overview of synchrotron radiation applications to low temperature geochemistry and environment science, Rev. Mineral. Geochem. 49(1), 1–115 (2002)CrossRefGoogle Scholar
  302. U. Neuhäusler, C. Jacobsen, D. Schulze, D. Stott, S. Abend: A specimen chamber for soft x-ray spectromicroscopy on aqueous and liquid samples, J. Synchrotron Radiat. 7, 110–112 (2000)CrossRefGoogle Scholar
  303. J.R. Lawrence, G.D.W. Swerhone, G.G. Leppard, T. Araki, X. Zhang, M.M. West, A.P. Hitchcock: Scanning transmission x-ray, laser scanning, and transmission electron microscopy mapping of the exopolymeric matrix of microbial biofilms, Appl. Environ. Microbiol. 69(9), 5543–5554 (2003)CrossRefGoogle Scholar
  304. T.H. Yoon, S.B. Johnson, K. Benzerara, C.S. Doyle, T. Tyliszczak, D.K. Shuh, G.E. Brown: In situ characterization of aluminum-containing mineral-microorganism aqueous suspensions using scanning transmission x-ray microscopy, Langmuir 20(24), 10361–10366 (2004)CrossRefGoogle Scholar
  305. C.S. Chan, G. de Stasio, S.A. Welch, M. Girasole, B.H. Frazer, M.V. Nesterova, S. Fakra, J.F. Banfield: Microbial polysaccharides template assembly of nanocrystal fibers, Science 303(5664), 1656–1658 (2004)CrossRefGoogle Scholar
  306. S.C.B. Myneni, T.K. Tokunaga, G.E. Brown Jr.: Abiotic selenium redox transformations in the presence of Fe(II,III) oxides, Science 278, 1106–1109 (1997)CrossRefGoogle Scholar
  307. B.P. Tonner, T. Droubay, J. Denlinger, W. Meyer-Ilse, T. Warwick, J. Rothe, E. Kneedler, K. Pecher, K. Nealson, T. Grundl: Soft x-ray spectroscopy and imaging of interfacial chemistry in environmental specimens, Surf. Interface Anal. 27(4), 247–258 (1999)CrossRefGoogle Scholar
  308. K. Pecher, D. McCubbery, E. Kneedler, J. Rothe, J. Bargar, G. Meigs, L. Cox, K. Nealson, B.P. Tonner: Quantitative charge state analysis of manganese biominerals in aqueous suspension using scanning transmission x-ray microscopy (STXM), Geochim. Cosmochim. Acta 67(6), 1089–1098 (2003)CrossRefGoogle Scholar
  309. C.K. Boyce, G.D. Cody, M. Feser, C. Jacobsen, A.H. Knoll, S. Wirick: Organic chemical differentiation within fossil plant cell walls detected with x-ray spectromicroscopy, Geology 30, 1039–1042 (2002)CrossRefGoogle Scholar
  310. C.K. Boyce, M.A. Zwieniecki, G.D. Cody, C. Jacobsen, S. Wirick, A.H. Knoll, N.M. Holbrook: Evolution of xylem lignification and hydrogel transport regulation, Proc. Natl. Acad. Sci. U.S.A. 101(50), 17555–17558 (2004)CrossRefGoogle Scholar
  311. R.E. Botto, G.D. Cody, J. Kirz, H. Ade, S. Behal, M. Disko: Selective chemical mapping of coal microheterogeneity by scanning transmission x-ray microscopy, Energy Fuels 8, 151–154 (1994)CrossRefGoogle Scholar
  312. G.D. Cody, R.E. Botto, H. Ade, S. Behal, M. Disko, S. Wirick: Inner-shell spectroscopy and imaging of a subbituminous coal: In-situ analysis of organic and inorganic microstructure using C(1s)-, Ca(2p)-, and Cl(2s)-NEXAFS, Energy Fuels 9, 525–533 (1995)CrossRefGoogle Scholar
  313. J. Thieme, J. Niemeyer, P. Guttmann, T. Wilhein, D. Rudolph, G. Schmahl: X-ray microscopy studies of aqueous colloid systems, Prog. Colloid Polym. Sci. 95, 135–138 (1994)CrossRefGoogle Scholar
  314. J. Thieme, J. Niemeyer: Interaction of colloidal soil particles, humic substances and cationic detergents studied by x-ray microscopy, Prog. Colloid Polym. Sci. 111, 193–201 (1998)CrossRefGoogle Scholar
  315. T. Schäfer, N. Hertkorn, R. Artinger, F. Claret, A. Bauer: Functional group analysis of natural organic colloids and clay association kinetics using C(1s) spectromicroscopy, J. Phys. IV 104, 409–412 (2003)Google Scholar
  316. J. Lehmann, D. Solomon, J. Kinyangi, L. Dathe, S. Wirick, C. Jacobsen: Spatial complexity of soil organic matter forms at nanometre scales, Nat. Geosci. 1, 238–242 (2008)CrossRefGoogle Scholar
  317. T. Schäfer, G. Buckau, R. Artinger, J.I. Kim, S. Geyer, M. Wolf, W.F. Bleam, S. Wirick, C. Jacobsen: Origin and mobility of fulvic acids in the Gorleben aquifer system: Implications from isotopic data and carbon/sulfur XANES, Org. Geochem. 36, 567–582 (2005)CrossRefGoogle Scholar
  318. J. Thieme, G. Schneider, C. Knöchel: X-ray tomography of a microhabitat of bacteria and other soil colloids with sub-100 nm resolution, Micron 34(6/7), 339–344 (2003)CrossRefGoogle Scholar
  319. A. Braun, N. Shah, F.E. Huggins, G.P. Huffman, S. Wirick, C. Jacobsen, K. Kelly, A.F. Sarofim: A study of diesel PM with x-ray microspectroscopy, Fuel 83(7/8), 997–1000 (2004)CrossRefGoogle Scholar
  320. K. Dardenne, T. Schäfer, M.A. Denecke, J. Rothe, J.I. Kim: Identification and characterization of sorbed lutetium species on 2-line ferrihydrite by sorption data modeling, TRLFS and EXAFS, Radiochim. Acta 89(7), 469–479 (2001)CrossRefGoogle Scholar
  321. B.S. Twining, S.B. Baines, N.S. Fisher, J. Maser, S. Vogt, C. Jacobsen, A. Tovar-Sanchez, S.A. Sañudo-Wilhelmy: Quantifying trace elements in individual aquatic protist cells with a synchrotron x-ray fluorescence microprobe, Anal. Chem. 75, 3806–3816 (2003)CrossRefGoogle Scholar
  322. M. Labrenz, G.K. Druschel, T. Thomsen-Ebert, B. Gilbert, S.A. Welch, K.M. Kemner, G.A. Logan, R.E. Summons, G. De Stasio, P.L. Bond, B. Lai, S.D. Kelly, J.F. Banfield: Formation of sphalerite (ZnS) deposits in natural biofilms of sulfate-reducing bacteria, Science 290(5497), 1744–1747 (2000)CrossRefGoogle Scholar
  323. M. Bonnin-Mosbah, N. Metrich, J. Susini, M. Salomé, D. Massare, B. Menez: Micro x-ray absorption near edge structure at the sulfur and iron \(K\)-edges in natural silicate glasses, Spectrochim. Acta B 57(4), 711–725 (2002)CrossRefGoogle Scholar
  324. J. Foriel, P. Philippot, J. Susini, P. Dumas: High-resolution imaging of sulfur oxidation states, trace elements, and organic molecules distribution in individual microfossils and contemporary microbial filaments, Geochim. Cosmochim. Acta 68(7), 1561–1569 (2004)CrossRefGoogle Scholar
  325. C.J. Ma, S. Tohno, M. Kasahara, S. Hayakawa: Determination of the chemical properties of residues retained in individual cloud droplets by XRF microprobe at SPring-8, Nucl. Instrum. Methods Phys. Res. B 217(4), 657–665 (2004)CrossRefGoogle Scholar
  326. D.A. Winesett, S. Story, J. Luning, H. Ade: Tuning substrate surface energies for blends of polystyrene and poly(methylmethacrylate), Langmuir 19, 8526–8535 (2003)CrossRefGoogle Scholar
  327. T. Koyama, Y. Kagoshima, I. Wada, A. Saikubo, K. Shimose, K. Hayashi, Y. Tsusaka, J. Matsui: High-spatial-resolution phase measurement by micro-interferometry using a hard x-ray imaging microscope, Jpn. J. Appl. Phys. 43, 421 (2004)CrossRefGoogle Scholar
  328. S. Zhu, Y. Liu, M.H. Rafailovich, J. Sokolov, D. Gersappe, A. Winesett, H. Ade: Confinement-induced miscibility in polymer blends, Nature 400, 49–51 (1999)CrossRefGoogle Scholar
  329. A.P. Smith, H. Ade, C.C. Koch, R.J. Spontak: Cryogenic mechanical alloying as an alternative stragetgy for there cycling of tires, Polymer 42, 4453–4457 (2001)CrossRefGoogle Scholar
  330. E.G. Rightor, S.G. Urquhart, A.P. Hitchcock, H. Ade, A.P. Smith, G.E. Mitchell, R.D. Priester, A. Aneja, G. Appel, G. Wilkes, W.E. Lidy: Identification and quantitation of urea participates in flexible polyurethane foam formulations by x-ray spectromicroscopy, Macromolecules 35, 5873–5882 (2002)CrossRefGoogle Scholar
  331. A.P. Hitchcock, T. Araki, H. Ikeura-Sekiguchi, N. Iwata, K. Tani: 3d chemical mapping of toners by serial section scanning transmission x-ray microscopy, J. Phys. IV 104, 509–512 (2003)Google Scholar
  332. L.M. Croll, H.D.H. Stöver, A.P. Hitchcock: Composite tectocapsules containing porous polymer microspheres as release gates, Macromolecules 38(7), 2903–2910 (2005)CrossRefGoogle Scholar
  333. H. Ade, B. Hsiao: X-ray linear dichroism microscopy, Science 262, 1427–1429 (1993)CrossRefGoogle Scholar
  334. C.J. Buckley, C. Phanopoulous, N. Khaleque, A. Engelen, M.E.J. Holwill, A.G. Michette: Examination of the penetration of polymeric methylene di-phenyl-di-isocyanate (pMDI) into wood structure using chemical-state x-ray microscopy, Holzforschung 56(2), 215–222 (2002)CrossRefGoogle Scholar
  335. H. Ade, S.G. Urquhart: NEXAFS spectroscopy and microscopy of natural and synthetic polymers. In: Chemical Applications of Synchrotron Radiation, ed. by T.K. Sham (World Scientific, Singapore 2002) pp. 285–355CrossRefGoogle Scholar
  336. Y. Liu, F. Meirer, C.M. Krest, S. Webb: Relating structure and composition with accessibility of a single catalyst particle using correlative 3-dimensional micro-spectroscopy, Nat. Commun. 7, 12634 (2016)CrossRefGoogle Scholar
  337. A.M. Wise, J.N. Weker, S. Kalirai, M. Farmand, D.A. Shapiro, F. Meirer, B.M. Weckhuysen: Nanoscale chemical imaging of an individual catalyst particle with soft x-ray ptychography, ACS Catalysis 6, 2178–2181 (2016)CrossRefGoogle Scholar
  338. J. Nelson, S. Misra, Y. Yang, A. Jackson, Y. Liu, H. Wang, H. Dai, J.C. Andrews, Y. Cui, M.F. Toney: In operando x-ray diffraction and transmission x-ray microscopy of lithium sulfur batteries, J. Am. Chem. Soc. 134(14), 6337–6343 (2012)CrossRefGoogle Scholar
  339. J.N. Weker, X. Huang, M.F. Toney: In situ x-ray-based imaging of nano materials, Curr. Opin. Chem. Eng. 12, 14–21 (2016)CrossRefGoogle Scholar
  340. Z.H. Levine, A.R. Kalukin, M. Kuhn, S.P. Frigo, I. McNulty, C.C. Retsch, Y. Wang, U. Arp, T.B. Lucatorto, B.D. Ravel, C. Tarrio: Tomography of an integrated circuit interconnect with an electromigration void, J. Appl. Phys. 87(9), 4483–4488 (2000)CrossRefGoogle Scholar
  341. G. Schneider, D. Hambach, B. Niemann, B. Kaulich, J. Susini, N. Hoffmann, W. Hasse: In situ x-ray microscopic observation of the electromigration in passivated Cu interconnects, Appl. Phys. Lett. 78(13), 1936–1938 (2001)CrossRefGoogle Scholar
  342. E. Zschech, R. Huebner, D. Chumakov, O. Aubel, D. Friedrich, P. Guttmann, S. Heim, G. Schneider: Stress-induced phenomena in nanosized copper interconnect structures studied by x-ray and electron microscopy, J. Appl. Phys. 106(9), 093711–093715 (2009)CrossRefGoogle Scholar
  343. A. Tkachuk, M. Feser, H. Cui, F. Duewer, H. Chang, W. Yun: High-resolution x-ray tomography using laboratory sources, Proc. SPIE 63181, 63181D (2006)CrossRefGoogle Scholar
  344. J. Deng, Y.P. Hong, S. Chen, Y.S.G. Nashed, T. Peterka, A.J.F. Levi, J. Damoulakis, S. Saha, T. Eiles, C. Jacobsen: Nanoscale x-ray imaging of circuit features without wafer etching, Phys. Rev. B 95(10), 104111 (2017)CrossRefGoogle Scholar
  345. M. Holler, M. Guizar-Sicairos, E.H.R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, G. Aeppli: High-resolution non-destructive three-dimensional imaging of integrated circuits, Nature 543(7645), 402–406 (2017)CrossRefGoogle Scholar
  346. P. Engström, S. Fiedler, C. Riekel: Microdiffraction instrumentation and experiments on the microfocus beamline at the ESRF, Rev. Sci. Instrum. 66(2), 1348–1350 (1995)CrossRefGoogle Scholar
  347. P.G. Evans, E.D. Isaacs, G. Aeppli, Z. Cai, B. Lai: X-ray microdiffraction images of antiferromagnetic domain evolution in chromium, Science 295(5557), 1042–1045 (2002)CrossRefGoogle Scholar
  348. C.E. Murray, H.F. Yan, I.C. Noyan, Z. Cai, B.P. Lai: High-resolution strain mapping in heteroepitaxial thin-film features, J. Appl. Phys. 98(1), 013504–013509 (2005)CrossRefGoogle Scholar
  349. Y. Yi, S. Cho, M. Noh, C.-N. Whang, K. Jeong, H.-J. Shin: Characterization of surface chemical states of a thick insulator: Chemical state imaging on MgO surface, Jpn. J. Appl. Phys. 44(2), 861–864 (2005)CrossRefGoogle Scholar
  350. S. Günther, B. Kaulich, L. Gregoratti, M. Kiskinova: Photoelectron microscopy and applications in surface and materials science, Prog. Surf. Sci. 70(4-8), 187–260 (2002)CrossRefGoogle Scholar
  351. J. Stöhr, Y. Wu, B.D. Hermsmeier, M.G. Samant, G.R. Harp, S. Koranda, D. Dunham, B.P. Tonner: Element-specific magnetic microscopy with circularly polarized x-rays, Science 259, 658–661 (1993)CrossRefGoogle Scholar
  352. P. Fischer, T. Eimüller, G. Schütz, P. Guttmann, G. Schmahl, K. Pruegl, G. Bayreuther: Imaging of magnetic domains by transmission x-ray microscopy, J. Phys. D 31, 649–655 (1998)CrossRefGoogle Scholar
  353. P. Fischer, T. Eimuller, G. Schütz, G.P. Denbeaux, A. Pearson, L. Johnson, D.T. Attwood, S. Tsunashima, M. Kumazawa, N. Takagi, M. Kohler, G. Bayreuther: Element-specific imaging of magnetic domains at 25 nm spatial resolution using soft x-ray microscopy, Rev. Sci. Instrum. 72(5), 2322–2324 (2001)CrossRefGoogle Scholar
  354. P. Fischer, T. Eimuller, G. Schütz, M. Kohler, G. Bayreuther, G.P. Denbeaux, D.T. Attwood: Study of in-plane magnetic domains with magnetic transmission x-ray microscopy, J. Appl. Phys. 89(11), 7159–7161 (2001)CrossRefGoogle Scholar
  355. H. Stoll, A. Puzic, B. van Waeyenberge, P. Fischer, J. Raabe, M. Buess, T. Haug, R. Höllinger, C. Back, D. Weiss, G. Denbeaux: High-resolution imaging of fast magnetization dynamics in magnetic nanostructures, Appl. Phys. Lett. 84(17), 3328–3330 (2004)CrossRefGoogle Scholar
  356. P. Fischer: Viewing spin structures with soft x-ray microscopy, Mater. Today 13(9), 14–22 (2010)CrossRefGoogle Scholar
  357. M.-Y. Im, P. Fischer, T. Eimüller, G.P. Denbeaux, S.-C. Shin: Magnetization reversal study of CoCrPt alloy thin films on a nanogranular-length scale using magnetic transmission soft x-ray microscopy, Appl. Phys. Lett. 83(22), 4589–4591 (2003)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Advanced Photon SourceArgonne National Laboratory and Northwestern UniversityArgonne, ILUSA
  2. 2.Advanced Light SourceLawrence Berkeley National LaboratoryBerkeley, CAUSA

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