Theory of angular-dispersive, imaging hard-x-ray spectrographs

A spectrograph is an optical instrument that disperses photons of different energies into distinct directions and space locations and that images photon spectra on a position-sensitive detector. Spectrographs consist of collimating, angular dispersive, and focusing optical elements. Bragg reflecting crystals arranged in an asymmetric scattering geometry can be used as the dispersing elements in the hard-x-ray regime. A ray-transfer matrix technique is applied to propagate x-rays through the optical elements. Several optical designs of hard-x-ray spectrographs are proposed and their performance is analyzed. Spectrographs with an energy resolution of 0.1 meV and a spectral window of imaging up to a few tens of meVs are shown to be feasible for inelastic x-ray scattering (IXS) spectroscopy applications. In another example, a spectrograph with a 1-meV spectral resolution and 85-meV spectral window of imaging is considered for Cu $K$-edge resonant IXS.

Figure 1

Time-length $\left(t\text{−}\lambda \right)$ and relevant energy-momentum $\left(\varepsilon \text{−}Q\right)$ space of excitations in condensed matter and how it is accessed by different inelastic scattering probes: neutron (INS), x-ray (IXS), ultraviolet (IUVS), and Brillouin (BLS). The ultra-high-resolution IXS spectrometer presented in Ref. [1] entered the previously inaccessible region marked in green (gray). The capabilities discussed in the present paper will enable IXS experiments with even higher resolution, 0.1 meV and $0.01 {\mathrm{nm}}^{-1}$, in the region marked in light green (light gray), and will close completely the existing gap between the high-frequency and low-frequency probes. The energy $\varepsilon =E_{{}_{\mathrm{f}}}-E_{{}_{\mathrm{i}}}$ and the momentum $\mathit{Q}={\mathit{k}}_{\mathrm{f}}-{\mathit{k}}_{\mathrm{i}}$ transfers from initial to final photon and neutron states are measured in inelastic scattering experiments, schematically shown in the oval inset.

Figure 2

Scheme of the Czerny-Turner-type spectrograph [12] with (a) a diffraction grating or (b) a crystal in asymmetric x-ray Bragg diffraction as dispersing element (DE). Other components include radiation source S, collimating and focusing mirrors ${\mathrm{M}}_{\mathrm{C}}$ and ${\mathrm{M}}_{\mathrm{F}}$, and position sensitive detector Det. (c) Multicrystal multireflection CDDW optic, an example of a hard-x-ray diffraction grating (DE element) with enhanced dispersion rate, suitable for hard-x-ray spectrographs [10].

Figure 3

Schematic elucidating the definition of an absolute right-handed coordinate system $\{x,y,z\}$ with the $\widehat{\mathit{z}}$ axis always looking in the direction of the optical axis (dash-dotted line) both before and after each optical element, a Bragg reflecting crystal in this particular case. By definition, positive is the counterclockwise sense of angular variations $\xi$ of ray slopes in the $(x,z)$ plane. Shown are examples of an optical element “Bragg reflection from a crystal” with (a) counterclockwise deflection, for which deflection sign $s=+1$, and (b) clockwise deflection of the reflected beam with $s=-1$. In the input reference system, the incident x-ray beam (wavy vector line) impinges onto the crystal at a glancing angle of incidence $\theta -s{\xi }_{{}_1}$ to the Bragg reflecting planes and with coordinate $x_{{}_1}$. It is reflected at a glancing angle of reflection ${\theta }^{\prime }+s{\xi }_{{}_2}$ with coordinate $x_{{}_2}$, as seen in the output reference system. Both $x$ and $\xi$ change signs upon reflection.

Figure 4

Schematics of the four-crystal CDDW and three-crystal CDW optic as dispersing elements (“diffraction gratings”) of the Czerny-Turner-type hard-x-ray imaging spectrographs (top row) and their spectral transmittance functions (solid, dark blue [dark gray] lines in the bottom row) calculated by the dynamical theory of Bragg diffraction using crystal parameters from Table 2. Angular spread of incident x rays is $20\mu \mathrm{rad}$ in panels (a) and (b) and $50\mu \mathrm{rad}$ in panels (c) and (d). Black spectral lines with a 0.1-meV width indicate the target spectral resolution of the spectrographs.

Figure 5

Schematics of two-crystal optic in the (+–) arrangement designed as dispersing elements DE (“diffraction gratings”) of the Czerny-Turner-type hard-x-ray imaging spectrographs (Fig. 2). Identical Bragg reflections with asymmetry parameter $\left|b\right|<1$ are used, with crystal parameters presented in Tables 3(a) and 3(b), and with spectral transmittance functions presented in Figs. 6 and 6, respectively.

Figure 6

Spectral transmittance functions [solid dark blue (dark gray) lines in panels (a) and (b)] of the (+–)-type two-crystal dispersing elements DE, schematically shown in Figs. 5 and 5, respectively. Transmittance is calculated using the dynamical theory of Bragg diffraction with crystal parameters from Tables 3(a) and 3(b). Angular spread of incident x rays is $50\mu \mathrm{rad}$ in both cases. Black spectral lines with a 0.1-meV width in panel (a) and with a 1-meV width in panel (b) represent the spectral resolution of the spectrographs in the particular $(\theta ,\eta )$ configurations highlighted by magenta (gray) dots in Figs. 7 and 8, respectively.

Figure 7

Properties of the spectrographs with the two-crystal diffraction grating (Fig. 5): (a) spectral resolution $\Delta E$; (b) image to source size ratio $\Delta x^{\prime }/\Delta x$; and (c) the lateral beam size enlargement $\Delta x_{{}_{\mathrm{b}}}^{\prime }/\Delta x_{{}_{\mathrm{b}}}$ by crystal optics, shown as a function of Bragg's angle $\theta$ and the asymmetry angle $\eta$. A particular case is presented for $\Delta x=5\mu \mathrm{m},f_{{}_1}=1$ m, and $f_{{}_2}=5$ m. Magenta (gray) dots highlight the case with the spectrograph resolution $\Delta E=0.1$ meV, $\Delta x^{\prime }/\Delta x\simeq 0.55$, and $\Delta x_{{}_{\mathrm{b}}}^{\prime }/\Delta x_{{}_{\mathrm{b}}}\simeq 9$, attainable with crystal parameters presented in Table 3(a). The (008) Bragg reflections from Si crystals enable a diffraction grating with a $\Delta {\theta }_{{}_{\cup }}=284\mu \mathrm{rad}$ angular acceptance and $\Delta E_{{}_{\cup }}=47$ meV imaging window for $9.13294$-keV x rays.

Figure 8

Properties of a spectrograph for Cu $K$-edge RIXS applications. Notations are the same as in Fig. 7 for a particular case of $\Delta x=5\mu \mathrm{m},f_{{}_1}=0.2$ m, and $f_{{}_2}=2$ m. The (337) Bragg reflection of the $8.99$-keV x rays from Ge crystals with crystal parameters presented in Table 3(b) provide a diffraction grating featuring a 133-$\mu \mathrm{rad}$ angular acceptance and a 85-meV bandwidth. In this case, the spectrograph resolution should be $\Delta E=1$ meV, $\Delta x^{\prime }/\Delta x\simeq 0.3$, and $\Delta x_{{}_{\mathrm{b}}}^{\prime }/\Delta x_{{}_{\mathrm{b}}}\simeq 16$.