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This study presents the first experimental application of longitudinal phase space tomography constrained by the Vlasov–Fokker–Planck equation (VFPE) using electro-optical spectral decoding (EOSD) diagnostics at the Karlsruhe Research Accelerator (KARA). The EOSD measurements are modeled as the convolution of the system’s impulse response with the charge density profile at the time of acquisition. Combining this model with the VFPE we formulate a partial differential equation (PDE)-constrained optimization framework for the inverse tomography.
Using this framework, we successfully reconstruct the longitudinal phase space of the electron bunch across different dynamical regimes, ranging from the stable state to the onset of micro-bunching. The reconstructed phase space density reproduce expected features of the bunch evolution and allow us to compute the corresponding coherent synchrotron radiation (CSR), which exhibits clear changes correlated with the reconstructed micro-structures. Overall, the results demonstrate that the VFPE-constrained approach, combined with a detailed EOSD forward model, provides a physically consistent reconstruction of the phase space dynamics from EOSD measurements.
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