Misalignment effects on the performance and stability of x-ray free electron laser oscillator

We report on the effects of transverse spatial misalignment on the performance and stability of x-ray free electron laser oscillators (XFELOs). For this study, we adopt the FEL driven paraxial resonator model in which the transverse profile of the radiation field is represented by Gauss-Hermite mode expansion. Then, we apply Gaussian optics for transporting the radiation field along with the misalignment in the optical cavity. We divide the misalignment into relevant categories of static and periodic types to understand their effects on FEL gain, lasing, and saturation. After that, we associate the first order moments of the radiation field to that of driven simple harmonic oscillator models to identify regions of instability. Throughout this report, we validate our models and analyses via theoretical calculations, analytical approximations, and simulations.

Figure 1

Schematic of an x-ray free electron laser oscillator. The figure is not drawn to scale.

Figure 2

Unfolded representation of XFELO cavity of Fig. 1. The optical axis is defined along the unperturbed ray path (red) which overlaps with that of the electron beam (blue) in the undulator.

Figure 3

(a) Density plot of (a) normalized 3D gain calculated for mismatch between radiation and electron beams at the waist location in one dimension, with error values normalized to rms widths of radiation beam, and (b) difference in normalized gain calculated from theory [Eq. (23)] and the Gaussian approximation [Eq. (25)] for a range of error values used in Fig. 3.

Figure 4

XFELO cavity configurations based on (a) two focusing mirrors (green) and two flat crystals (purple) for XFELO-A, and (b) two compound refractive lenses (green) and four crystals (purple) forming a bow-tie pattern for XFELO-B (with $\theta =30.56°$ for C444 crystal) and XFELO-C (with $\theta =23.275°$ for C333 crystal). Figures are not drawn to scale. For (a), the geometrical shape is only approximate and determined by the choice of crystals and focusing mirrors.

Figure 5

Evolution of (a) transverse centroid phase space of the radiation beam at the waist along the misalignment dimension, (b) total radiation power, and (c) radiation power gain per pass for constant crystal tilts from 0 to 125 nrad in the first crystal of XFELO-A defined by Table 1 and Fig. 4.

Figure 6

Comparison of normalized gain values obtained from simulations, calculated gain formulas of Eq. (23), and approximate gain formulas of Eq. (25) for steady-state coordinates of the radiation beam obtained from simulations.

Figure 7

Evolution of total radiation power per pass for periodic misalignment with the crystal tilt modeled as a sinusoid given by $\delta \theta (t)=\delta \theta \cos (\omega t)$ for six different frequencies in XFELO-A. The label T indicates the period in turns associated with each frequency.

Figure 8

Evolution of total radiation power per pass for scans conducted (a) for two frequencies with the second frequency component kept fixed with its period at 80 turns, and (b) for three frequencies with two frequencies kept fixed close to each other with their periods corresponding to 20 and 25 turns.

Figure 9

Plot of (a) mean position and (b) mean divergence of the radiation beam minus their steady-state values versus pass for static misalignment cases of Fig. 5 obtained from the simulations (solid lines) and harmonic oscillator model of Eq. (32) (dashed lines). The $x$-axis scale is ${\log }_{10}$.

Figure 10

Plots of normalized amplitude $|A({\omega }_{\beta },\omega ,\beta )|$ with rms size versus misalignment periods in XFELO-A with $\beta =0.1,\widehat{q}=4$, and crystal tilt amplitude of $-45\mathrm{nrad}$ obtained from simulations (diamonds for position and circles for angle) and harmonic oscillator model (solid lines, blue for position, and red for angle) of Eq.(35).

Figure 11

Density plot of threshold value $\sqrt{2\ln [G_0\widehat{q}]}$ for static misalignment for various $\widehat{q}$ and $G_0$ values. Both $x$ axis and $y$ axis are on a ${\log }_{10}$ scale.

Figure 12

Power evolution in XFELO-A with $R=0.8$ $(\widehat{q}=4)$ subject to periodic misalignments with frequencies between 0.67 and 1.46 times the Betatron frequency with the crystal tilt amplitude of (a) 45 nrad, (b) 67.5 nrad, and (c) 78.75 nrad. The $y$-axis scale is ${\log }_{10}$.

Figure 13

Power evolution in XFELO-A with (a) $R=0.8333$ $(\widehat{q}=5)$, and (b) $R=0.75$ $(\widehat{q}=3)$ subject to periodic misalignments with single frequency between 0.77 and 1.46 times the Betatron frequency with an amplitude of crystal tilt of $-45\mathrm{nrad}$. The $y$ axis is on a ${\log }_{10}$ scale.

Figure 14

Normalized excitation amplitude plotted against normalized frequency for the periodic misalignment of case studies of (a) Fig. 13 for cavities with $\widehat{q}=3$, $\widehat{q}=4$, and $\widehat{q}=5$ and (b) Fig. 12 in which crystal tilt of amplitude 45 nrad is amplified by a factor of 1.5 and 1.75. The resonance frequency widths are obtained when normalized excitation equals 1 based on Eq. (40), beyond which instability kicks in.