Microbunching instability in storage rings: Link between phase-space structure and terahertz coherent synchrotron radiation radio-frequency spectra

In storage rings, when the number of electrons in a bunch exceeds a threshold value, microstructures appear spontaneously in the bunch. Experimental observations of these microstructures and their evolution are indirect to obtain. In practice, most information is obtained from the terahertz coherent synchrotron radiation (THz CSR), since microstructures in the millimeter range radiate THz CSR. Here we show, thanks to a simple analytical formula associated to numerical simulations, that a frequency component in the THz CSR signal permits one to obtain information about the structure of the electron bunch longitudinal phase space during the microbunching instability. The proposed formula permits one also to link experimental information together, in particular the bunch length and the THz CSR spectrum.

Figure 1

From microstructures in the electron-bunch longitudinal phase space $f(z,p,t)$ to modulation of the THz CSR power $P_{\mathrm{CSR}}(t)$. Numerical results with (a), (c), (e), (g), and (i) SOLEIL parameters and (b), (d), (f), (h), and (j) UVSOR II parameters. (a) and (b) Electron bunch phase space $f(z,p,0)$ at the time $t=0$. (c) and (d) Associated longitudinal charge density $\rho (z,0)$. (e) and (f) Time evolution of $\rho (z,t)$. (g) and (h) Associated THz power $P_{\mathrm{CSR}}(t)$. (i) and (j) Norm of the Fourier transform of $P_{\mathrm{CSR}}(t)$: (black line) calculated from all the 2D phase-space distribution, (red line) calculated from the 2D distribution, where $-0.5<p<0$ (distribution between the red dashed lines in (b). In (h), the red line is shifted vertically for figure clarity.

Figure 2

Angle $\phi$ calculated from the Eq. (6) (in red) superposed to the numerical phase space (in grey scale). (a) and (b) SOLEIL case and (c) and (d) UVSOR II case. (a) and (c) Cartesian $(q,p)$ coordinates and (b) and (d) polar $(r,\theta )$ coordinates (same parameters as Fig. 1).

Figure 3

Schematic representation of an electron-bunch phase space with a modulation located at a distance ${\sigma }_q$ from the bunch center. The distance $\lambda$ is associated to the projection on the horizontal axis (in unit of meters) of the arclength corresponding to one modulation period of the phase space. The distance $\lambda$ is generally associated to a coherent emission, and thus can be measured experimentally.

Figure 4

Average THz CSR spectrum $⟨P_{\mathrm{CSR}}(\nu ,\tau )⟩$ in the case of (a) SOLEIL and (b) UVSOR-II. Vertical lines indicate ${\lambda }_{\mathrm{CSR}\text{peak}}$ (same parameters as in Fig. 1).