Maximizing spectral flux from self-seeding hard x-ray free electron lasers

Fully coherent x rays can be generated by self-seeding x-ray free electron lasers (XFELs). Self-seeding by a forward Bragg diffraction (FBD) monochromator has been recently proposed [G. Geloni, V. Kocharyan, and E. Saldin, J. Mod. Opt. 58, 1391 (2011)] and demonstrated [J. Amann et al., Nat. Photonics 6, 693 (2012)]. Characteristic time ${\mathcal{T}}_0$ of FBD determines the power, spectral, and time characteristics of the FBD seed [Yu. Shvyd’ko and R. Lindberg, Phys. Rev. ST Accel. Beams 15, 100702 (2012)]. Here we show that for a given electron bunch with duration ${\sigma }_{\mathrm{e}}$ the spectral flux of the self-seeding XFEL can be maximized, and the spectral bandwidth can be respectively minimized by choosing ${\mathcal{T}}_0\sim {\sigma }_{\mathrm{e}}/\pi$ and by optimizing the electron bunch delay ${\tau }_e$. The choices of ${\mathcal{T}}_0$ and ${\tau }_e$ are not unique. In all cases, the maximum value of the spectral flux and the minimum bandwidth are primarily determined by ${\sigma }_{\mathrm{e}}$. Two-color seeding takes place if ${\mathcal{T}}_0\ll {\sigma }_{\mathrm{e}}/\pi$. The studies are performed, for a Gaussian electron bunch distribution with the parameters, close to those used in the short-bunch (${\sigma }_{\mathrm{e}}\simeq 5\mathrm{fs}$) and long-bunch (${\sigma }_{\mathrm{e}}\simeq 20\mathrm{fs}$) operation modes of the Linac Coherent Light Source XFEL.

Figure 1

Hard x-ray FEL self-seeding scheme uses x rays from the first half of the magnetic undulator system U1 to seed the electron bunch in the second half U2 via a single crystal x-ray monochromator. The monochromator produces the delayed monochromatic seed under the forward Bragg diffraction (FBD) conditions [21].

Figure 2

(left) Schematic of Bragg diffraction (BD) and forward Bragg diffraction (FBD) of x rays from a crystal in Bragg-case geometry. (a) Example of energy dependence of x-ray reflectivity $|R_{0H}(E){|}^2$ in Bragg diffraction and (b) of energy dependence of forward Bragg diffraction $|R_{00}(E)-R_{00}(\infty ){|}^2$ from a diamond crystal in the 004 Bragg reflection, $E_{\mathrm{c}}=8.33\mathrm{keV}$, $\theta =56.6°$, $\eta =0°$, $d=0.1\mathrm{mm}$. (c) Intensities of corresponding time dependencies of crystal response to an excitation with ultrashort x-ray pulse in BD $|G_{0H}(t){|}^2$ (blue) and in FBD $|G_{00}(t){|}^2$ (red), respectively.

Figure 3

(left) Schematic of Bragg diffraction (BD) and forward Bragg diffraction (FBD) of x rays from a crystal in Laue-case scattering geometry. All notations are as in Fig. 2. Example of energy and time dependencies from a diamond crystal in the 004 Bragg reflection, $E_{\mathrm{c}}=8.33\mathrm{keV}$, $\theta =56.6°$, $\eta =90°$ $d=0.1\mathrm{mm}$.

Figure 4

2D plot of the universal FBD response function intensity $|{\tilde{G}}_{00}(t){|}^2$ (in relative units) plotted for different values of the time delay $t$ to the instantaneous excitation of the crystal, and for different values of the FBD time constant ${\mathcal{T}}_0$, using Eq. (9).

Figure 5

Similar to Fig. 4, however, Eq. (4) for the Laue-case FBD response function is used for a particular case of ${\mathcal{T}}_d\simeq 104{\mathcal{T}}_0$, which permits the duration of the FBD peaks to be stretched.

Figure 6

Spectra of the seeding radiation as “seen” by the electron bunch (solid black lines). The spectra are calculated for different delays ${\tau }_e$ of the electron bunch (solid orange lines in the insets) with respect to the FBD intensity (solid green lines in the insets). The dotted lines show for reference the time-integrated FBD spectra. Calculations are presented for an electron bunch with an rms duration ${\sigma }_{\mathrm{e}}=5.3\mathrm{fs}$, Bragg-case FBD time constant ${\mathcal{T}}_0=0.75\mathrm{fs}$ (a), and ${\mathcal{T}}_0=0.07\mathrm{fs}$ (b), respectively. Case (a) with ${\mathcal{T}}_0\simeq {\sigma }_{\mathrm{e}}/\pi$ is optimal for the single-color seeding, while case (b) with ${\mathcal{T}}_0\ll {\sigma }_{\mathrm{e}}/\pi$ is better suited for the two-color seeding.

Figure 7

Time dependencies of power and phase of the entire radiation interacting with the electron bunch at the beginning of the U2 undulator system. The entire radiation comprises both the SASE radiation [the first term of Eq. (5)], and the monochromatic seed generated by FBD of the SASE radiation from a crystal [the second term of Eq. (5)]. Examples of different SASE pulses in the FEL long-pulse operation mode are shown, demonstrating jitter in the seed power, and positions of the seed maxima. The crystal monochromator with ${\mathcal{T}}_0=0.75\mathrm{fs}$ is used.

Figure 8

Time (left) and spectral (right) dependencies of the XFEL radiation in the self-seeding short-bunch mode. Graphs in the top row present the dependencies at the entrance of the U2 undulator system. Seeding is performed by the FBD monochromator with the FBD time constant ${\mathcal{T}}_0=0.21\mathrm{fs}$. The top right spectrum of the seeding radiation is time integrated. The lower graphs show in solid dark blue lines the time and spectral dependencies at the end of the U2 system for different electron bunch delays ${\tau }_e$. Orange lines show the electron current profiles. The dashed gray lines in the left graphs are the time profile of the seed. The dashed gray lines in the right graphs represent the seed spectrum as it is seen by the electron bunch at appropriate time delays. Red dashed lines show the same, however, for an idealistic case of instantaneous excitation of the crystal with a white spectrum. The nominal photon energy $E_x=8.3\mathrm{keV}$.

Figure 9

Similar to Fig. 8, however, showing results of calculations in the long-bunch mode, with seeding by the FBD monochromator with the FBD time constant ${\mathcal{T}}_0=0.75\mathrm{fs}$. The nominal photon energy $E_x=9.1\mathrm{keV}$.

Figure 10

Time dependencies of the monochromatic seed power (magenta lines in the $\alpha$-row graphs), seed plus SASE power (dashed blue line in same graphs), 10 times increased shot-noise level (dashed black line), and the time dependencies of the seed phase—graphs in the $\beta$ row. The seed is generated in the long-bunch mode by a SASE radiation from the U1 undulator system (cyan lines in the $\alpha$-row graphs) in forward Bragg diffraction (FBD) from crystals with FBD time constant ${\mathcal{T}}_0=0.75\mathrm{fs}$ (a), ${\mathcal{T}}_0=1.5\mathrm{fs}$ (b), ${\mathcal{T}}_0=3.0\mathrm{fs}$ (c), ${\mathcal{T}}_0=4.1\mathrm{fs}$ (d), and ${\mathcal{T}}_0=8\mathrm{fs}$ (e). Graphs in other rows represent properties of the radiation at the end of the undulator system U2, as a function of the electron bunch delay ${\tau }_e$ in the magnetic chicane.

Figure 11

Same as Fig. 10, but for the short-bunch mode calculations with (a) ${\mathcal{T}}_0=0.07\mathrm{fs}$ (Bragg case), (b) ${\mathcal{T}}_0=0.07\mathrm{fs}$ (Laue case), (c) ${\mathcal{T}}_0=0.21\mathrm{fs}$ (Bragg case), (d) ${\mathcal{T}}_0=0.21\mathrm{fs}$ (Laue case), and (e) ${\mathcal{T}}_0=0.73\mathrm{fs}$ (Bragg case).

Figure 12

Time (left) and spectral (right) dependencies of the XFEL radiation in the self-seeding long-bunch mode with the FBD monochromator having ${\mathcal{T}}_0=0.07\mathrm{fs}$. The graphs show in solid dark blue lines the time and spectral dependencies at the end of the U2 system for different electron bunch delays ${\tau }_e$. Orange lines show the electron current profiles. The dashed gray lines in the left graphs are the time profiles of the seed. The nominal photon energy $E_x=9.1\mathrm{keV}$.