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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. From the reconstructed phase space, we derive macroscopic beam observables including the longitudinal bunch length, energy-spread related horizontal bunch size, and coherent synchrotron radiation (CSR) power. The reconstructed observables are validated against synchronized measurements from three independent diagnostics. 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 density dynamics from EOSD measurements.
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