Using angular dispersion and anomalous transmission to shape ultramonochromatic x rays

Optical spectrometers, instruments that work with highly monochromatic light, are commonly rated by the spectral bandwidth, which defines the ability to resolve closely spaced spectral components. Another equally important feature is the spectral contrast, the ability to detect faint objects among these components. Here we demonstrate that a combined effect of angular dispersion (AD) and anomalous transmission (AT) of x rays in Bragg reflection from asymmetrically cut crystals can shape spectral distributions of x rays to profiles with high contrast and small bandwidths. The AD&AT x-ray optics is implemented as a five-reflection, three-crystal arrangement featuring a combination of the above-mentioned attributes so desirable for x-ray monochromators and analyzers: a spectral contrast of $\simeq 500$, a bandwidth of $\simeq 0.46$ meV, and a remarkably large angular acceptance of $\simeq 107\mu$rad with 9.1 keV x rays. The new optics can become a foundation for the next-generation inelastic x-ray scattering spectrometers for studies of atomic dynamics.

Figure 1

Basic phenomena underlying the AD&AT x-ray optics. In x-ray Bragg diffraction from atomic planes composing nonzero angle $\eta$ to the crystal entrance face, the crystal acts (a) like an optical prism dispersing the photons into a divergent x-ray fan with photons of different energies $E$ propagating at different reflection angles ${\theta }^{\prime }\left(E\right)$—the effect of angular dispersion (AD) [1, 10, 11, 12] and (b) as a filter with anomalously high transparency for x rays with incidence angles $\theta -\Psi$ in the immediate vicinity ($\Psi \approx 5\mu$rad) but smaller than the Bragg angle $\theta$—the effect of anomalous transmission (AT) [13, 14, 15].

Figure 2

Optical scheme of the in-line three-crystal five-reflection $CD$$F$$DW$ monochromators (r) red-winged and (b) blue-winged, respectively. The $CFW$ crystal executes three key functions: a collimator $C$, an anomalous transmission filter $F$, and a wavelength selector $W$ in successive reflections. The crystals $D$${}_1$ and $D$${}_2$ are dispersing elements. All crystals are asymmetrically cut with the reflecting atomic planes shown by the white lines perpendicular to the diffraction vectors $\mathbf{H}$ ($\mathbf{H}=\mathbf{C}$, $\mathbf{F}$, ${\mathbf{D}}_1$, ${\mathbf{D}}_2$, or $\mathbf{W}$), composing nonzero asymmetry angle ${\eta }_H$ to the entrance surface. ${\theta }_H$ and ${\theta }_H^{\prime }$ are the glancing angles of incidence and reflection, respectively.

Figure 3

Dynamical theory calculations of the spectral distribution of x rays after each successive reflection, indicated by number and color (gray scale), from the crystals of the $CD$$F$$DW$ optics in the red-winged [Fig. 2r], and blue-winged [Fig. 2b] configurations, respectively. Black dashed lines show Gaussian distribution of the same full width at half maximum. Insets show the distributions on the linear scale.

Figure 4

DuMond diagrams for a sequence of four asymmetric Bragg reflections from crystals functioning as $C$, $D$${}_1$, $D$${}_2$, or $W$ elements. Anomalous transmission in the $F$ element is not taken into account. Green (dark gray) and yellow (light gray) stripes are the regions of Bragg reflections in the space of x-ray wavelengths $\lambda$ and angles of incidence ${\theta }_H$ or reflection ${\theta }_H^{\prime }$ from an $H$ element ($H=C$, $D$${}_1$, $D$${}_2$, or $W$). Blue (black) stripes display the overlapping reflection regions of the $C$ and $D$ elements. Orange (medium gray) tetragons display the reflection region common for all elements. The $D$ element is set into backscattering (${\theta }_D\to \pi /2$) with the center of the reflection spectral range at ${\lambda }_R=hc/E_R$ [Eq. (3)].

Figure 5

Angular distributions of x-ray intensity upon a sequence of interactions $C\to D_1\to F$, schematically shown in the insets, representing (s) effect of anomalous transmission, and (b),(r) combined effect of angular dispersion and anomalous transmission of x rays in Bragg diffraction from asymmetrically cut crystals. The narrow line ($\simeq 4\mu$rad) in (s) is due to backreflection and subsequent anomalous transmission taking place for all photon energies at the same incidence angle to crystal $D$${}_1$ (same rotation angle ${\tilde{\Theta }}_{D_1}=0$). In contrast, in (b) and (r) backreflection takes place at different angles for different photon energies, indicated by color (gray scale). Solid and dashed lines are dynamical theory calculations, and solid circles are experimental data. (s${}^{\prime }$) shows the same dependences as (s) but on the linear scale.

Figure 6

Experimental scheme and angular dependence of transmission through $CD$$F$$DW$ monochromators. Solid blue (black) circles show results for the blue-winged monochromator with incident beam collimated to $\simeq 15\mu$rad, as measured by the Det${}_1$ x-ray detector. Solid line, theory. Open purple (dark gray) circles show results for the red-winged analyzer with the incident beam emanating from the monochromator, set to the transmission maximum, as measured by the Det${}_2$ x-ray detector.

Figure 7

Experimental scheme and a spectral tuning curve of the $CD$$F$$DW$ monochromator measured by rotation of the $D$ crystals. The blue (black) circles show experimental data, while the solid line represents calculations using multicrystal dynamical theory of x-ray diffraction.

Figure 8

Experimental scheme and combined spectral resolution function of the blue-winged $CD$$F$$DW$ monochromator measured against the red-winged analyzer. The purple (dark gray) circles show experimental spectral resolution function; the black solid line shows spectral functions calculated using multicrystal dynamical theory of x-ray diffraction. Other functions with the same FWHM are shown for comparison: green (dark gray) dashed line, Gaussian; dark red (gray) dash-dotted line, Lorentzian; orange (light gray) open circles, experimental dependence for a four-crystal monochromator [23].