Signatures of misalignment in x-ray cavities of cavity-based x-ray free-electron lasers

Cavity-based x-ray free-electron lasers (CBXFEL) will allow the use of optical cavity feedback to support generation of fully coherent x-rays of high brilliance and stability by electrons in undulators. CBXFEL optical cavities comprise Bragg-reflecting flat crystal mirrors, which ensure x-rays circulation on a closed orbit, and x-ray refractive lenses, which stabilize the orbit and refocus the x-rays back on the electrons in the undulator. Depending on the cavity design, there are tens of degrees of freedom of the optical elements, which can never be perfectly aligned. Here, we study signatures of misalignment of the optical components and of the undulator source with the purposes of understanding the effects of misalignment on x-ray beam dynamics, understanding misalignment tolerances, and developing cavity alignment procedures. Betatron oscillations of the x-ray beam trajectory (both symmetric and asymmetric) are one of the characteristic signatures of cavity misalignment. The oscillation period is in the general case a noninteger number of round-trip passes of x-rays in the cavity. This period (unlike the amplitude and offset of the oscillations) is independent of the type of misalignment and is defined by cavity parameters. The studies are performed on an example of a four-crystal rectangular cavity using analytical and numerical wave optics as well as ray-tracing techniques. Both confocal and generic stable cavity types are studied.

Figure 1

Schematic of a cavity-based XFEL with an undulator x-ray source together with a four-crystal (${\mathrm{C}}_1–{\mathrm{C}}_4$), two-lens (${\mathrm{L}}_1–{\mathrm{L}}_2$) rectangular x-ray cavity and x-ray diagnostics [23]. Cavity round-trip length $2(L+W)\simeq 65\mathrm{m}$. X-ray diagnostics tools include x-ray beam intensity monitors (XBIMs) and an x-ray beam position, profile, and intensity monitor (XBPM).

Figure 2

Equivalent optical scheme of the unfolded symmetric cavity with two lenses.

Figure 3

Gaussian beam profiles for self-consistent solutions in the symmetric confocal cavity, shown for different initial Rayleigh range values $Z_o=f^{(1)}/M$. The second Gaussian beam waist is in the midpoint of the round-trip path, with the size of the beam waist (relative to the original waist size) scaling with the magnification factor $M$.

Figure 4

Schematic of incident and adjusted rays clarifying calculation of the x-ray trajectory distorted by a misaligned optical element.

Figure 5

A diagram of the simulation work flow. (OE stands for optical elements).

Figure 6

Numerical simulations of x-ray beam dynamics in the perfectly aligned generic four-crystal rectangular cavity shown in Fig. 1. Values are presented simulated at cavity midpoint (top, left column), source center (top, right column), and beam intensity monitor positions (bottom row). The cavity and source parameters are provided in Table 1. Two resolutions are shown: $2\times {10}^5$ rays (red) and $2\times {10}^6$ rays (blue). The positional and angular oscillations are due to numerical simulation artifacts, as discussed in the text. See text for details.

Figure 7

Schematics of crystal misalignment scenarios. One-crystal misalignment: (a) yaw angle $\delta$, (b) roll angle $\rho$, and (c) position $\xi$ for crystal ${\mathrm{C}}_4$. Systematic multi-crystal misalignment: (d) yaw angles $\delta$, $3\delta$, $5\delta$, and $7\delta$ for crystals ${\mathrm{C}}_1–{\mathrm{C}}_4$, respectively.

Figure 8

Additional misalignment scenarios. (a) Lens ${\mathrm{L}}_1$ with yaw angle error $\delta$ or (b) spatial error $\xi$; (c) undulator source with yaw angle error $\delta$ or (d) position misalignment $\xi$.

Figure 9

Simulation results of the beam dynamics in the rectangular four-crystal cavity without lenses with systematically misaligned crystal yaw angles.

Figure 10

Similar to Fig. 9, but with yaw angle error $\delta$ in crystal ${\mathrm{C}}_4$.

Figure 11

Similar to Fig. 9, but with roll angle error $\rho$ in crystal ${\mathrm{C}}_4$. The dashed lines show out-of-cavity-plane deflections.

Figure 12

Similar to Fig. 9, but with spatial error $\xi$ in crystal ${\mathrm{C}}_4$.

Figure 13

Numerical simulation results of x-ray beam dynamics in the rectangular four-crystal generic cavity with systematically misaligned crystal yaw angles.

Figure 14

Similar to Fig. 13, but with yaw angle error $\delta$ in crystal ${\mathrm{C}}_4$.

Figure 15

Similar to Fig. 14, but with roll angle error $\rho$ in ${\mathrm{C}}_4$. The dashed lines represent out-of-cavity-plane deviations.

Figure 16

Similar to Fig. 14, but with position error $\xi$ in crystal ${\mathrm{C}}_4$.

Figure 17

Simulation results of x-ray beam dynamics in the four-crystal generic cavity with yaw angle error $\delta$ at lens ${\mathrm{L}}_1$.

Figure 18

Similar to Fig. 17, but with position error $\xi$ at lens ${\mathrm{L}}_1$.

Figure 19

Simulation results of x-ray beam dynamics in the rectangular four-crystal generic cavity with angle error $\delta$ at source.

Figure 20

Similar to Fig. 19, but with position error $\xi$ at source.

Figure 21

Numerical simulation results of x-ray beam dynamics in the confocal rectangular four-crystal cavity with systematically misaligned crystal yaw angles.

Figure 22

Similar to Fig. 21, but with yaw angle error $\delta$ at crystal ${\mathrm{C}}_4$.

Figure 23

Similar to Fig. 21, but with position error $\xi$ at lens ${\mathrm{L}}_1$.

Figure 24

Similar to Fig. 21, but with position error $\xi$ at crystal ${\mathrm{C}}_4$.