In the last decade outstanding progress has been made in x-ray optics. This development has been triggered by the availability of high-brilliance synchrotron radiation sources. Well-known optical schemes have been improved and new ones have been invented. Important fields of application for these optics are collimation and focusing, both at laboratory and synchrotron radiation sources, and hard x-ray microscopy, which is a growing field mainly at synchrotron radiation sources.

Although there is a large variety of x-ray optics with very different designs, they are all based on the same physical principles, i.e., elastic scattering of x-rays in matter. Refractive x-ray lenses make use of refraction inside the lens material, while mirror optics, capillaries, and waveguides use total external reflection, closely related to the refraction inside the optic’s material. Diffractive optics, such as Fresnel zone plates, make use of attenuation and refraction to reduce and shift the amplitudes of the x-rays to generate a desired interference pattern, e.g., a small focal spot in the focal plane. Multilayer or crystal optics exploit Bragg reflection to focus x-rays. These physical mechanisms that underlie x-ray optics are reviewed in below Sect. 2 (Interaction of X-rays with Matter)

At laboratory x-ray sources, e.g., x-ray tubes, x-ray optics are mostly used to capture the radiation from a large solid angle and concentrate it either on the sample or on a detector. For this purpose very efficient optics are required that capture a large solid angle, such as for example polycapillary, multilayer or crystal optics. At modern synchrotron radiation sources, hard x-ray microscopy has developed quickly over the past decade. Both full-field and scanning techniques find a growing number of applications, requiring both imaging and focusing optics. The most important x-ray optics are reviewed in the article Type of X-Ray Optics.

For imaging, i.e., in full-field microscopy, Fresnel zone plates and refractive x-ray lenses are most commonly used. They are used as objective lens, generating a magnified image of the specimen on the detector. In this way, spatial resolutions down to 40nm in the hard and to below 20nm in the soft x-ray range can be achieved. The key strength of this type of microscopy is the large penetration depth of x-rays in matter, which allows one to image the interior of an object with-out destructive sample preparation. By combining this technique with tomography, the three-dimensional inner structure of an object can be reconstructed at high spatial resolution.

Scanning microscopy, on the other hand, allows one to perform with high spatial resolution hard x-ray analytical techniques, such as diffraction, fluorescence analysis, or absorption spectroscopy, that yield the local (nano-) structure, the elemental composition, or the chemical state of an element in the sample, respectively. When combined with tomography, spectroscopic information from inside a specimen can be obtained. The small beam for these scanning techniques is of-ten generated by means of an x-ray optic, such as zone plates, refractive lenses, or curved total reflection or multilayer mirrors. Currently, all these optical schemes are capable of generating intensive beams with a lateral extension well below 100nm at third-generation synchrotron radiation sources. Recently, hard x-rays have been focused down to 7nm in one dimension by curved mulilayer mirrors. In addition to scanning microscopy applications, the small beam can also be used as a small source for magnified projection microscopy. This scheme is recently used in full-field microscopy to obtain magnified high-resolution images of a specimen using Kirkpatrick-Baez mirrors or waveguides.

Currently, all x-ray optics are technology limited in their performance, but significant technological advances have been made, approaching physical limits. Recently, these limits have been addressed theoretically for several x-ray optics, such as waveguides, refractive lenses, and Fresnel zone plates. While beams in the range from 1-10nm are conceivable with ideal optics, atomic resolution, e.g., at I A seems to be out of reach.

As smaller and smaller x-ray beams are produced, their characterization becomes more and more difficult. Usually, well-characterized test patterns or sharp knife-edges are used to determine the beam size. Their fabrication and characterization become increasingly difficult, potentially introducing systematic errors. Recently, scanning coherent diffraction microscopy (SCDM), also known as ptychography, has been introduced for nanobeam characterization. As this method gives full access to the complex wave field in a reference plane and does not require any prior knowledge of a test object, it has revolutionized nanobeam characterization. A brief introduction to this method is given in Article X-ray Nanobeam Characteristics.

A hard x-ray photon interacts with an atom mainly through scattering at its electrons or by absorption. Scattering can be both elastic or inelastic. In the cases of inelastic scattering, also called Compton scattering, and photoabsorption the photon is lost for image formation, while elastic scattering is responsible for refractive and diffractive effects. The refraction of hard x-rays in matter is typically expressed by the index of refraction given in the equation 1 below

where δ describes the deviation of the real part of the refractive index from unity and is referred to as the index

Fig. 1 (a)For visible light, the refractive index n in matter is larger than one. Therefore, light rays are refracted toward the surface normal when entering matter from the vacuum. (b) For hard x-rays, the refractive index n of matter is smaller than one. There-fore, x-rays impinging from vacuum onto a surface are refracted away from the surface normal. (c) If the angle of incidence θ1 falls below the critical angle of total reflection, the x-rays do not propagate deeply into the material but are totally reflected at the surface

of refraction decrement. As (equation 1) suggests (with positive δ), the refractive index of hard x-rays in matter is smaller than unity, i.e., the vacuum is x-ray optically denser than matter. Figure 1b illustrates this effect as compared to the refraction of visible light in glass (Fig. 1a). For a given atomic species, δ in below equation 2 is given by

where Na is Avogadro’s constant, r0 is the classical electron radius, λ and E are the wavelength and energy of the x-rays, respectively, ρ is the mass density of the material, Z＋f'(E)is the real part of its atomic form factor in the forward direction, and A is the material’s atomic mass. Figure 2a shows δ/ρ for different materials as a function of x-ray energy. Away from absorption edges, f'(E) is small and δ is proportional to λ²～E^-2 and ρ. Since Z/A does not vary much between most elements, δ/ρ varies very little as a function of the atomic species away from absorption edges. As compared to that of visible light in glass, the refraction of hard x-rays is several orders of magnitude weaker. While δ/ρ is of the order of 10^-6 cm³/g for hard x-rays between 10 and 20 keV (Fig. 2a),that of visible light in glass is about – δ/ρ ≈ 0.2cm³/g.

Due to this extremely small refraction, the reflectivity at a surface or interface is extremely low. Therefore, there are no mirrors for hard x-rays reflecting at large angles. However, when an x-ray beam impinges on

Fig. 2 (a) δ/ρ (the index of refraction decrement divided by the density of the material) for Be, Al, and Ni as a function of x-ray energy. (b)Mass attenuation coefficient μ/p for the same elements.

a plane interface between vacuum (or air) and a medium of index n under a sufficiently small angle θ₁, it is totally reflected (Fig. 1c). This total external reflection occurs for θ₁ smaller than the critical angle for total reflection θ_c in below equation 3, i.e.

For hard x-rays, this angle lies below about 0.5° for all materials. In other words, total reflection occurs only at grazing incidence. The imaginary part β of the refractive index in (equation 1) describes the attenuation of x-rays in matter and is related to the linear attenuation coefficient μ in below equation 4 by

Lambert-Beer law (equation 5)

describes the transmitted intensity I(z) through a homogeneous piece of material of thickness z, given an incident intensity I₀. The linear attenuation coefficient μ is the inverse of the characteristic length of this exponential decay.

β and μ include photoabsorption as well as the attenuation of the incident beam by elastic (Rayleigh) and inelastic (Compton) scattering. The dependence of μ/ρ on the x-ray energy is shown in Fig. 2b. At low energies, photoabsorption 𝜏 dominates μ/ρ. As opposed to refraction, photoabsorption is strongly dependent on the atomic number. Between absorption edges it scales approximately like 𝜏 ~ Z³/E³. With increasing x-ray energy, Compton scattering contributes increasingly to μ/ρ. Compton scattering is only weakly dependent on the atomic species and limits the mass attenuation coefficient from below to about μ/ρ > 0.1 cm²/g in the energy range from a few keV to several hundred keV. As a consequence, there is no material that is as transparent for hard x-rays as is glass for visible light. For comparison, the attenuation of visible light in glass can be as small as μsilica ≈ 10^-7 cm^-1 for pure silica used for fiber optics. In the general case, elastic scattering contributes little to the attenuation of the transmitted beam. In some cases, however, when f0r example a Bragg reflection is excited in a crystalline material, elastic scattering can significantly attenuate the transmitted beam. In that case, the x-ray beam can be efficiently diffracted away from the forward and into another direction. Figure 3 illustrates this situation, in which the scattered amplitudes from a large number of atoms interfere constructively.

Fig. 3 Bragg scattering of hard x-rays from a crystalline material. The lattice planes have a spacing d and the x-rays impinge on the lattice planes under the angle θ

This is the case when the path length difference 2d sinθ between amplitudes reflected from neighboring lattice planes is an integer multiple m of the wavelength λ of the x-rays in below equation 6, i.e.，

While elastic scattering preserves the wave number (k=2π/λ), the wave vector changes its direction, see in below equation 7

where G=mG0 is the reciprocal space vector associated with the lattice planes and the reflection order m (Fig. 3). For arbitrary alignment of a monocrystalline sample in a monochromatic beam, this virtually never occurs. However, some x-ray optics, such as crystal or multilayer optics, exploit this effect to monochromatize and focus the x-rays.