A large number of x-ray optical components are described in literature. One reason for this is the difficulty in manipulating x-rays due to their weak interaction with matter. There have been many complementary approaches to this problem, but none has proven significantly superior to the others. Indeed, as technology advances, approaching the physical limits, the performances of different optics become very similar. In this chapter, the most common x-ray optics are described.

Rotationally Parabolic Refractive X-ray Lenses While refractive optics are most common for visible light, the interplay between weak refraction and relatively strong absorption inside the lens material makes the design of refractive lenses for hard x-rays difficult (See in article of Introduction to X-Ray Optics). For instance, at 45°, the angular deviation of the refracted beam is δ, which is typically of the order of 10^-6 rad, compared to some 17° for visible light at an air-glass interface. From the discovery of x-rays to the 1990s it was therefore believed that refractive lenses for hard x-rays could not be made. In 1996 it was first demonstrated experimentally that refractive lenses can be fabricated and work despite this weak refraction and relatively strong absorption. Since then, a variety of refractive x-ray lenses have been designed.

As the refractive index of hard x-rays is smaller than one (n<1) in any lens material, a focusing lens has to have a concave shape (Fig. 1a), and since the refraction of hard x-rays in matter is about six orders of magnitude weaker than that of visible light in glass, the effective lens curvature must be accordingly stronger. For example, a biconvex lens for visible light with a radius of curvature of R=1m for both surfaces has a focal length of about f=R/2(n-1) ≈ 1m. A single x-ray lens with the same focal distance would need to have a radius of curvature R in the micrometer range. It is difficult to fabricate such a lens, in particular with a sufficiently large aperture. This difficulty can be overcome by making the radius of curvature R of an individual lens larger and compensating the lack of re-fraction by stacking many of these lenses behind each other, as shown in Fig. 1a.

Fig. 1 (a) Parabolic refractive x-ray lenses. A large number of individual lenses is stacked behind each other to form a refractive x-ray lens. (b) Partially assembled parabolic refractive x-ray lens made of beryllium

As the radius of curvature R has to be as small as possible to limit the number of single lenses, the spherical-lens approximation that is successfully applied in visible light optics does not apply to most x-ray lens designs. It would limit the aperture 2R₀ of the lens to be much smaller than the radius of curvature R (Fig. 1a). To avoid spherical aberration and to allow for apertures that are as large as possible, the lens surfaces have to have a parabolic shape. In the paraxial approximation this is the optimal aspherical lens shape. For parabolic lenses the aperture 2R₀ can be chosen independently of the radius of curvature R (Fig. 1a).

Attenuation inside the lens is significant for all lens materials (See in article of Introduction to X-Ray Optics), as there is no material that is as transparent in the hard x-ray range as glass is for visible light. Besides a reduction of the transmitted radiation, the attenuation inside the lens material reduces the aperture of the lens, since the thicker material in the outer parts of the lens (Fig. 1a) absorbs the x-rays more strongly than the thin material on the optical axis. In order to optimize the transmission and aperture it is thus important to choose the lens material carefully. As μ/ρ increases strongly with increasing atomic number Z (μ/ρ～Z³), atomic species with low Z, such as Li, Be, B, C, and compounds thereof, are good lens materials. In addition, the lenses should be made as thin as possible, reducing the distance d (Fig. 1a) between the apices of the parabolas to a minimum.

Ideal for imaging applications, in particular in X-ray microscopy, are rotationally parabolic lenses. At Aachen University, they are made by embossing the lens shape from both sides into the lens material, such as beryllium, aluminum, or nickel. These lenses are avail-able with different radii of curvature R, ranging from 50 to 2500 um and apertures 2R₀ between 0.4 and 3.1 mm. Typical focal distances lie in the range 0.3-10m, depending on the application. Figure. 1b shows a partly assembled beryllium lens. A single lens is held and centered by a hard metal ring of well-defined diameter. These coin-like single lenses are stacked along two parallel polished shafts or in a V groove to align their optical axes. In this design, the number of lenses can be adjusted starting from a single lens up to several hundred lenses in one stack, allowing one to control the optical properties, e.g., focal length, within a wide range in a large energy interval (2 to >100 keV). The lens is kept in an inert gas atmosphere or in vacuum to avoid corrosion of the beryllium in the intensive x-ray beam.

As the attenuation increases towards the outer parts of the lens, the rays passing the lens in these parts do not contribute as much to the image as those traveling through the lens close to the optical axis. Therefore, the effective aperture D_eff describing diffraction at the lens is smaller than the geometric aperture 2R₀. It typically lies in the range of a few hundred micrometers to more than a millimeter. Attenuation inside the lens material also reduces the transmission T_p of the optic. Details about these quantities can be found in.

These optics are ideal for imaging applications with hard x-rays above 5 keV, such as hard x-ray microscopy. They are free of spherical aberration. 100nm spatial resolution has been achieved with beryllium lenses at 12 keV in an x-ray microscope. Potentially, the diffraction limit of these optics can be pushed to below 50nm. The lenses can be used to generate a small and intensive microbeam, by imaging the x-ray source onto the sample in a strongly reducing geometry. Besides for x-ray microscopy, these optics are commonly used for beam conditioning at third-generation synchrotron radiation sources, since they can withstand the high heat load of modern undulator sources and match their beam size very well. In addition, they find wide application in scanning microscopy, where they are used to generate intensive hard x-ray microbeams.

According to the refractive index depends on the x-ray energy, leading to chromatic aberration. For the many experiments carried out with monochromatic synchrotron radiation, chromatic effects are irrelevant.

In order to keep the focal length approximately constant at different energies, the number of lenses can be varied. This has been automated at several synchrotron radiation beamlines at the ESRF and PETRA III (positron-electron tandem-ring accelerator). There are, however, also many experiments which require a broad x-ray spectrum or scanning the energy while keeping the focus unchanged. If the energy range is sufficiently small, a slight defocusing of the lenses can be tolerated in some cases. In others, one should resort to genuinely achromatic optics, such as total reflection mirrors.

To generate beams with a lateral extension well below one micrometer, the geometric image of the source must be made sufficiently small. This is possible by reducing the focal length f of a refractive lens, thus increasing the demagnification ratio. In addition, the diffraction limit decreases with decreasing focal length for refractive lenses.

Fig. 2(a) Scanning electron micrograph of nanofocusing lenses. A single lens and a nanofocusing lens are outlined by dark shaded areas. (b) Nanoprobe setup: the x-ray beam is focused onto the sample by two crossed nanofocusing lenses

As it is difficult to reduce the focal length of the rotationally parabolic lenses to well below 0.3m, a new type of parabolic lenses with extremely short focal distance was developed.

In order to achieve focal lengths in the centimeter range at hard-x-ray energies, the radii of curvature of the individual single lenses (Fig. 2a) need to be in the range of a few micrometers. This can for example be achieved with nanofabrication techniques for planar lenses, such as those shown in Fig. 2a. They are made of silicon by electron-beam lithography and subsequent deep reactive-ion etching. These planar structures only focus in one dimension. Thus, in order to obtain a point focus, two such lenses with appropriate focal lengths need to be crossed. Figure 2b shows a nanoprobe setup based on two crossed nanofocusing lenses (NFLs). With this setup, hard-x-ray beams down to about 50×50nm² have been generated at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France.

With optimized parameters, these optics are expected to generate beams with a lateral size between 10 and 20nm. This is a physical limit for the given lens design. There are more-complicated lens designs that will in principle allow one to overcome this limit and focus hard x-rays down to below 5 nm.

The reflectivity of a surface or interface is negligible for large angles. Only in grazing incidence below the critical angle of total external reflection θc are high reflectivities possible (Sect.22.1). This makes x-ray mirrors and capillaries very slender, i.e., much longer than wide. According to equation 2 and equation 3 θ_c depends mainly on the x-ray wavelength λ and on the density ρ of the mirror material. The higher the x-ray energy（E=hc/λ）the smaller the critical angle θ_c. In order to have sufficiently large reflection angles, materials with high density, such as palladium or platinum, are often used.

Two effects reduce the ability of a mirror to reflect x-rays without losses: absorption and surface rough-ness. X-rays penetrate into the mirror surface to a depth of a few nanometers. Here, they are partially absorbed, so that just below the critical angle the reflectivity is often far below 50%. For heavy elements and just above absorption edges this is particularly strong.

The second detrimental effect to the reflectivity is surface roughness. Due to the short wavelength of hard x-rays roughness brings the reflection amplitudes of a photon in the different areas inside each Fresnel zone out of phase, so that the total amplitude and hence the reflectivity are reduced. For high-quality mirror optics, the surface roughness must be in the 0.1 nm range. This effect and the high requirements on form fidelity make X-ray mirrors very expensive optical components.

For a given grazing incidence geometry with reflection angle θ₁, total-reflection mirrors reflect x-rays up to the energy where θ₁=θ_e. They act like a low-pass filter for x-rays and are therefore often used to remove the high-energy content from the beam. In conjunction with a crystal monochromator, this allows one to strongly reduce higher harmonic radiation (m≥2 in equation 6 )and generate a clean monochromatic beam.

To focus or collimate x-rays, mirrors can be curved. To focus the x-rays, i.e., to image the source from point to point, the mirror surface needs have ellipsoidal shape (Fig. 3a). For collimation, i.e., generating a parallel x-ray beam, the mirror surface needs to be parabolic (Fig. 3b). To achieve focusing or parallelization in two dimensions with a single reflection, the mirror must have the shape of an ellipsoid or paraboloid of rotation, respectively. Due to the very small reflection angle, this rotational body is very elongated (needle-like) having extremely different curvatures in the sagittal and meridional direction. Since such a mirror is extremely difficult to fabricate with high accuracy, there are only a few realizations. These optics are also referred to as elliptical or parabolic monocapillaries.

To avoid strong sagittal curvature, two reflections from flat mirrors are often used to obtain two-dimensional focusing. Figure 4 illustrates this scheme, which was first described by Kirkpatrick and Baez in 1948. Today, this technique is widespread as a focusing optic for scanning microscopy at synchrotron radiation sources. While it is relatively difficult to align, it has the great advantage of being achromatic for x-rays below a critical energy given by the maximal angle of incidence. This makes these optics particularly useful for white-beam microdiffraction techniques and for absorption spectroscopy studies.

Today, these mirrors can be fabricated with extremely small surface roughness and excellent form fidelity, generating hard x-ray beams with a lateral size as small as 25 nm. Since the reflection angles are limited by θc, the numerical aperture and thus the diffraction-limited focus size is limited, too. It is expected that beam sizes down to below 20nm can be reached with total reflection mirrors. One way to overcome this limit is to use multilayer mirrors (see in article X-ray Nanobeam Characteristics) or use more-complicated multiple-reflection schemes.

Another optical component which is based on total re-flection and which has been developed recently are x-ray waveguides. They consist of a substrate made of a material with high density (e.g., nickel), a light-element (e.g., carbon) layer on top of it and a cap layer made of a dense material again (Fig. 5). The middle layer is the guiding layer, which is typically a few tens of nm thick. The x-rays are confined inside the waveguide by total external reflection at the side walls defined by the denser materials. The device works in analogy to a waveguide for microwaves. There are two ways to couple x-rays into a wave-guide. A well-collimated plane wave can be coupled into the waveguide through the thin cap layer or a focused beam can be directly coupled into the waveguide from the side, as shown in Fig. 5a, b, respectively.

Fig. 3 (a)Elliptical mirror for focusing form point to point. (b)Parabolic mirror to collimate the beam from a point source

Fig. 4 Kirkpatrick-Baez focusing geometry: focusing from point to point using two crossed elliptical mirrors

Fig. 5 X-ray waveguide.(a) The x-rays are coupled into the waveguide through the cap layer.(b)Focused x-rays are directly coupled into the waveguide

Only those modes that survive by constructive interference can propagate in the waveguide. Depending on the illumination, different modes can be systematically excited, corresponding to different standing waves in the transverse direction. The first waveguides were planar structures, confining the beam only in one direction. Today, using nanofabrication techniques, two-dimensionally confining waveguides can be made.

Waveguides are mostly used for two types of applications: to illuminate a sample confined inside the waveguide or to generate a very fine beam that is emerging from its exit. At the exit, the beam size is comparable to the dimensions of the waveguide.Lateral dimensions of 25 nm × 45 nm can be generated. Recently, more efficient planar waveguides making use of a more sophisticated stack of materials were introduced, generating confined beams below 20 nm.

This beam can be used for projection microscopy and has a high degree of lateral coherence.If a single mode is excited inside the waveguide, the exiting beam is perfectly coherent. This can be exploited for coherent imaging experiments and holography. The smallest size that x-rays can be confined to by waveguides is also limited by total-reflection effects and is expected to lie slightly below 10nm. Until now, these components are in the stage of development and have not yet found widespread application.

Fig. 6 (a)High-resolution transmission electron micrograph of a multilayer optic consisting of N=40 double layers of Mo and Si. The lattice spacing is d =7nm. The inset shows a detail of the multilayer including the interface with the monocrystalline substrate (courtesy of S. Braun, Fraunhofer IWS, Dresden). (b) Schematic drawing of a multilayer optic. The x-rays are reflected from the multilayer if the Bragg condition is fulfilled, i.e., the dark brown segment of the lower optical path is an integer multiple of the x-ray wavelength λ

Multilayer optics consist of a series of alternating layers of high- and low-density materials. Figure 6a shows a transmission electron micrograph of a multilayer made of 40 double layers of Mo and Si. Figure 6b shows the operating principle of these optics: the x-rays impinging at the angle e are reflected from the multi-layer if the Bragg condition (equation 6) for the multilayer is fulfilled, i.e., the path-length difference marked in dark brown in Fig. 6b is an integer multiple of the wavelength λ. Here, d is the period of the multilayer, which can be chosen freely and typically lies in the range from about 2 to 20nm, and the resulting Bragg angles lie in the range of degrees for hard x-rays. The reflectivity can be close to one (70-90%) for hard x-rays. It depends on the number of layers and the difference in electron density between the two multilayer materials. As for total-reflection mirrors, the reflectivity is also deteriorated by attenuation inside the optic and by roughness of the interfaces.

Similar to crystal optics, multilayers can be used to monochromatize the x-ray beam. The monochromaticity and the angular acceptance depend on the number of periods that contribute to the reflection. This number is either limited by the total number of periods N in the stack or by the extinction length inside the multilayer. A detailed description of the reflectivity of multilayer systems can be found in.

To use these optics to focus or collimate the x-ray beam in the geometries shown in Fig. 7, the spacing d of the multilayer needs to be adapted to the reflection angle that varies along the optic. Such laterally graded multilayers were first introduced by Schuster and Gobel to capture a large solid angle from an x-ray tube and collimate it for diffraction experiments. A variety of optics exist today that either focus or collimate the beam from an x-ray tube. Since the reflection angles are larger than in the total-reflection case (see in the Sect. 2 Reflective Optics) there are sagittally curved elliptical and parabolic mirrors in addition to double-bounce systems of Kirkpatrick-Baez (KB) type.

For scanning microscopy, curved multilayer systems in the KB geometry are used with great success, generating very intensive nanobeams down to well below 100nm. These optics have the advantage that they moderately monochromatize the beam at the same time. In this way, an additional crystal monochromator can be avoided.

Fig. 7(a)Elliptical graded multilayer for focusing.(b)Parabolic graded multilayer for focusing and collimation. The period d changes as a function of position on the optic, locally adjusting the Bragg angle to the angle of reflection

This can be advantageous for experiments that do not require a high degree of monochromatization, since the larger energy band pass leads to a higher flux. Due to the relatively large reflection angles, the numerical aperture of these optics can be relatively large. Recently, one-dimensional focusing of hard x-rays down to 7nm was demonstrated with these optics.

The flexibility in the choice of the lattice parameters and its variation over the mirror open possibilities that cannot be realized with crystal optics. Besides lateral grading of the multilayers, depth grading can also be useful, e.g., to tailor the angular acceptance and energy bandwidth of the device. As there are some applications of these optics in conjunction with laboratory x-ray tubes, there is a larger market for these optics, and they are available from a number of laboratories and commercial companies. Multilayers are also of great importance in the extreme ultraviolet (EUV) range, where they are used in lithographic steppers.

The main field of application of crystal optics is monochromatization of x-rays using Bragg reflection (see in the Article Introduction to X-Ray Optics).Depending on the crystal, crystal cut, reflection, and the arrangement of different crystals, relative energy bandwidths △E/E from about 10^-3 to 10^-7 can be reached, finding wide application at nearly every beam line at synchrotron radiation sources. High energy resolutions are needed in particular in inelastic x-ray scattering and nuclear resonance scattering.

Besides pure monochromatization, crystal optics can also be used to focus x-rays from point to point. As one cannot adjust the d-spacing for varying reflection angles as for multilayer optics (Fig. 7) a geometry in which the reflection angle is constant and equal to the Bragg angle θ_B for every point on the optic has to be used. Figure 8 shows a bent crystal in Johann geometry. In Johann geometry, the Bragg planes are parallel to the crystal’s surface. The meridional bending radius Rh is twice the radius of the so-called Rowland circle (Fig. 8). To focus also out of plane, the crystal has to be sagittally bent as well. In that direction the bending radius is R_v = R_hsin² θ_B (Fig. 8). Both the source and the focus lie on the Rowland circle, forming an isosceles triangle with the crystal.

For larger and larger crystals, the surface deviates increasingly from the Rowland circle. To fulfill the Bragg condition strictly in that case, the crystal needs to be cut to follow the Rowland circle when bent. This is called the Johansson geometry, which is very difficult to realize and is therefore not used very often.

The bending of the crystal optic has to be very ac-curate in order to stay within the width of the Bragg reflection over the whole surface. In addition, the crystal must not be damaged. As this bending procedure is difficult, these optics are not very widespread.

Fig. 8 Focusing with a bent crystal optic in the Johann geometry. The source and focus position lie on the Rowland circle

Fig. 9 Fresnel zone plate.(a) Schematic diagram, (b) Scanning electron microscope(SEM) image

The most important diffractive optical components are Fresnel zone plates. They have found very widespread applications, particularly at energies below a few keV, like for example in the water window. As a genuine imaging optic, they can be used as objective lens in full-field microscopy and to generate a small focus for scanning microscopy.

Fresnel zone plates consist in their simplest form of a set of alternating transparent and opaque rings (Fig. 9), so-called zones, with an optical path length difference of λ/2 between neighboring zones for x-rays propagating from infinity through the zone plate to the focus at distance f behind it. The amplitudes of all transparent zones superpose constructively when the radii of all zones are given by

where the even zones are transparent and odd zones are opaque or vice versa. Here, f is the focal length of the optic for a given wavelength λ.first diffraction order, the zone plate is most efficient, and about 10% of the radiation falling onto its aperture can be focused into this diffraction order. The spatial resolution depends on the numerical aperture NA =r_n/f and is in this case given by

which is approximately equal to the width of the outermost zone △r_n. The factor 1.22 comes from the Rayleigh criterion for the spatial resolution of a circular optic. This means that the spatial resolution corresponds to that of the smallest features in the zone plate. For increasing x-ray energy, it becomes increasingly difficult to make a zone plate with opaque zones. Therefore, this type of optic is mostly used in the soft-x-ray regime, in particular in the so-called water window between the absorption edges of carbon and oxygen. As the opaque zones become more transparent, the zone plate loses its efficiency.

For harder x-rays one therefore pursues another scheme.Rather than blanking out the amplitudes of every other zone, one can shift them by π, adding to the constructive interference in the focus. Such phase zone plates are much more efficient than zone plates with opaque zones and have an optimal efficiency of about 40% in the first diffraction order. Despite this new scheme, the zones still need to have a thickness in the micrometer range for hard x-rays around 10 keV. To reach high spatial resolution efficiently the fabrication of very thin but high zone plate structures is required. While there have been tremendous improvements over the last view years, it is still a challenge to make these structures. However, there are several institutions and commercial sources for these optics. Another phase-shifting optic is a Bragg-Fresnel optic, in which the Fresnel zones are etched into a single crystal surface. Tuned to Bragg condition, the x-rays are diffracted from different heights inside the zone plate structure. Refraction introduces the desired phase shift between the even and odd zones.

In the soft-x-ray range, zone plates with spatial resolution down to 15 nm have been made. Even higher spatial resolutions have been demonstrated with zone plates made with a special thin film technique. In the hard x-ray range, the high aspect ratio of the outermost zones currently limits the spatial resolution to about 50nm. For such a high-resolution zone plate, the aperture is typically in the range of a couple of hundred microns and is made of several hundred zones.

As the aspect ratio increases with decreasing outer-most zone width, the thin object approximation for the Fresnel zone plates becomes less appropriate. Propagation effects inside the zone plate structure need to be taken into account. It was shown theoretically that the zones of a zone plate need to be tilted to fulfill a local Bragg condition in order not to lose their focusing efficiency for smaller outermost zone widths. While these structures are even more difficult to fabricate, they are predicted to be able to generate focal sizes down to I nm and below. These structures are currently approximated by so-called multilayer Laue lenses. Focusing down to below 20nm has been demonstrated with these devices.