Figure 2
(a) Reconstruction principle of the PSD distribution with electro-optical sampling (illustrated by a simulated phase space configuration). Top: we assume a rigid rotation. Shown are the PSD configurations at different rotation angles and the final reconstruction. Bottom: the EO measurement (projection on the time axis) of the different PSD configurations can be interpreted as a Radon transform. Each PSD configuration is represented by a single column in the corresponding sinogram. The Radon transform is connected to the 2D Fourier transform of the original image via the Fourier slice theorem40. The original PSD can be reconstructed with a filtered back-projection. (b) Validation of the reconstruction algorithm with simulated data obtained from the Vlasov–Fokker–Planck solver Inovesa25. We used a standard ramp filter to avoid the blurring effect of the back-projection40 in combination with a high frequency cut-off at a relative frequency of 0.1 to reduce the exaggeration of edges. Left side: comparison between the original and reconstructed PSD. The “original” PSD is evaluated at the \(90^\circ \) frame, while for the reconstruction the complete sinogram has to be taken into the account. Right side: several comparisons during different times of a bursting cycle. Even in situations with prominent substructures, where bunch self-interaction is expected to be strong, the reconstruction shows a reasonable agreement with the original PSD, also in fine details.
