Crystal X-ray phase plates (Skalicky & Malgrange, 1972; Annaka, 1982; Golovchenko et al., 1986; Mills, 1987; Belyakov & Dmitrienko, 1989; Hirano et al., 1991) are the most popular optics components for polarization control in the hard X-ray region. This particular crystal optics setup has several advantages over helical undulators: (i) faster switching of photon helicities, (ii) a variety of polarization states (horizontal/vertical linear polarization is available in addition to right/left circular polarization), (iii) no depolarization effects from other optics components such as crystal monochromators and (iv) no influence on the electron orbit and X-ray beam path at other beamlines.

The polarization state conversion is based on X-ray bi­refringence due to dynamical multiple diffraction (Batterman & Cole, 1964; Ishikawa & Kohra, 1991), which occurs under conditions close to the diffraction conditions of a so-called perfect crystal, such as silicon, germanium or diamond. Extensive studies employing several different materials and geometries (reflection, transmission, Bragg, Laue and so on) have concluded that the use of a light element in transmission Laue or transmission Bragg geometry is the most efficient method of controlling the X-ray polarization states (Hirano et al., 1993). As synthetic diamond crystals are the most suitable material for this technique, they are typically employed (Sumiya & Tamasaku, 2012).

A crystal phase plate functions as a quarter-wave plate that converts horizontally polarized X-rays from a bending magnet or planar undulator to circularly polarized rays. Fig. 1 shows a typical scheme of a crystal X-ray phase plate in transmission geometry using a symmetric Laue reflection. A diamond crystal in (111) orientation is mounted on a rotation stage with the axis tilted by 45° such that the (220) diffraction plane is also tilted. An incident electromagnetic wave with horizontal polarization can be decomposed into σ and π polarization components with respect to the diffraction plane, which have the same amplitude and are oscillating in phase. The diffracted X-ray beam is monitored in order to determine the Bragg condition of the (220) reflection. The transmitted beam (forward diffraction) is used as a variably polarized X-ray.

Figure 1 Schematic of a crystal X-ray phase plate in transmission Laue geometry.

According to the dynamical theory of diffraction, the phase retardation between the σ and π components generated by a crystal of thickness t is given by (Giles et al., 1994) where r_e is the classical electron radius, F_h and are the structure factors of the hkl and hkl reflections, respectively, V is the unit-cell volume, λ is the X-ray wavelength, θ_B is the Bragg angle and . Further, note that δ is a function of the angular offset Δθ = θ − θ_B. Fig. 2 plots δ versus Δθ for values calculated for a diamond crystal with t = 0.5 mm at a photon energy of 7.0 keV, where At = 80.4 arcsec. The incoming horizontal polarization can be converted to circular polarization with positive helicity at Δθ = At, at which the crystal generates a retardation of δ = −π/2. The helicity of the circular polarization can be reversed by changing the sign of the offset angle, i.e. Δθ = −At. The crystal angular motion required to reverse the helicity is very small, typically being a few tens of arcseconds (∼0.01°). Note that fast switching of photon helicities has previously been achieved at 100 Hz through the use of a piezo actuator (Hirano et al., 1992) and at frequencies of up to 2 kHz using a galvano scanner (Suzuki et al., 2003).

Figure 2 Phase shift δ and degree of circular polarization PC as functions of the offset angle Δθ calculated for a diamond crystal with 0.5 mm thickness at 7.0 keV. For these plots, the energy band width ΔE/E = 1.3 × 10−4 of a Si(111) monochromator is assumed. δ and PC will be impacted by the energy and angular spread of the radiation incoming on the crystal, both of which will reduce PC from ideal values.

The crystal can also function as a half-wave plate, rotating the linear polarization direction from horizontal to vertical. This rotation is achieved by tuning the offset angle to Δθ = ±At/2 so as to yield a phase shift of δ = ∓π. This feature is particularly useful for examining the polarization dependence of the X-ray absorption fine structure (XAFS) at a liquid surface and for polarized hard X-ray photoemission spectroscopy (HAXPES). It should be noted that with the use of a crystal half-wave plate, the degree of vertical polarization, P_L = (I_H − I_V)/(I_H + I_V), where I_H and I_V are the intensities of the horizontal and vertical polarization components, respectively, is typically not very high () due to the beam divergence and energy spread.

P_C is determined using δ and the transmitted amplitudes of the σ and π polarizations, E_σ and E_π, respectively, according to For a perfect horizontally polarized beam incident on the phase-plate crystal with the 45° tilted diffraction plane, E_σ ≃ E_π and thus P_C ≃ −sinδ. The transmittance of the crystal in a symmetric Laue geometry is given by where μ is the linear absorption coefficient of diamond. The figure of merit is then given by .

The polarization conversion efficiency is maximized for a parallel and monochromatic incident X-ray beam. The combination of a modern undulator source and a Si(111) double-crystal monochromator yields a desirable X-ray beam for a crystal phase plate that allows excellent performance. A high P_C (>0.95) and a reasonable T (0.25–0.5) can practically be obtained using a diamond crystal with t equal to a few hundred micrometres at photon energies of 5–10 keV. Fig. 3 shows the calculated performance for diamond phase plates with different t values and geometries. In practical use, the available X-ray energy range for a diamond phase plate is between 3.3 and 20 keV. The lowest energy limit is determined by the small lattice constant of diamond and the absorption losses, although a high-quality thin diamond crystal of 50 µm thickness is currently available. Ultrathin silicon crystals can be used as phase plates to cover X-ray energies between 2.4 and 3.2 keV (Bouchenoire et al., 2012). The use of phase plates in this energy range is of great advantage over the use of helical undulators since the degree of circular polarization produced by a helical undulator is degraded by the Si(111) monochromator working as a linear polarizer around 2.8 keV, at which the Bragg angle is close to 45°. The higher energy side is primarily determined by the size of the available crystals. The use of a diamond phase plate at higher X-ray energies requires a thicker crystal. The effects of the degradation induced by the energy spread and angular divergence are stronger at higher energies, and a thicker crystal is required in order to eliminate this influence. To improve the polarization conversion efficiency and the controllability of the polarization states, a method involving the combination of multiple diamond crystals has been proposed (Okitsu et al., 2002).

Figure 3 Calculated performance of diamond phase plates for different crystal thicknesses t and geometries: t = 0.1 mm for (111) Laue diffraction (thin solid line) and t = 0.5 and 1.5 mm for (220) Laue diffraction (thick solid and dashed lines, respectively). (a) Degree of circular polarization PC, (b) transmittance T and (c) figure of merit as functions of X-ray energy. The calculation assumes a relative band width ΔE/E = 1.3 × 10−4, corresponding to a Si(111) monochromator.

Giles and coworkers performed the first XMCD measurements using a diamond phase plate with a polychromator in the energy-dispersive mode (Giles et al., 1994). The angle of the crystal phase plate was essentially fixed, but was optimized so as to yield appropriate phase retardation for divergent and polychromatic X-ray beams to cover the X-ray absorption edge simultaneously. This skilfully established setup enabled energy-dispersive XMCD measurements, which are particularly useful for both time-resolved XMCD analyses and XMCD spectroscopy conducted under high pressure using a diamond anvil cell (Torchio et al., 2014).

In another energy-scanning mode, a crystal phase plate is combined with a silicon double-crystal monochromator (Hirano & Maruyama, 1997). The angle of the phase plate crystal is tuned to follow the X-ray energy scanning at every energy point of the spectrum and therefore magnetic extended X-ray absorption fine-structure (EXAFS) measurements over a wide energy range have become feasible (Suzuki et al., 2001). In the energy-scanning mode, the helicity-modulation (lock-in detection) technique can be employed to significantly improve the signal-to-noise ratio of the XMCD measurements (Suzuki et al., 1998). XMCD signals as small as ∼10⁻⁵, which correspond to a paramagnetic moment of ∼10⁻⁴μ_B per atom in bulk gold, have been detected using this technique (Suzuki et al., 2012). In addition to the XMCD applications, diamond X-ray phase plates have facilitated the development of polarization-dependent HAXPES (Ueda et al., 2008; Ouardi et al., 2011).