Longitudinal Bunch Diagnostics using Coherent Transition Radiation Spectroscopy

The generation and properties of transition radiation (TR) are thoroughly treated. The spectral energy density, as described by the Ginzburg-Frank formula, is computed analytically, and the modifications caused by the finite size of the TR screen and by near-field diffraction effects are carefully analyzed. The principles of electron bunch shape reconstruction using coherent transition radiation are outlined. Spectroscopic measure- ments yield only the magnitude of the longitudinal form factor but not its phase. Two phase retrieval methods are investigated and illustrated with model calculations: analytic phase computation by means of the Kramers- Kronig dispersion relation, and iterative phase retrieval. Particular attention is paid to the ambiguities which are unavoidable in the reconstruction of longitudinal charge density profiles from spectroscopic data. The origin of these ambiguities has been identified and a thorough mathematical analysis is presented. The experimental part of the paper comprises a description of our multichannel infrared and THz spectrometer and a selection of measurements at FLASH, comparing the bunch profiles derived from spectroscopic data with those determined with a transversely deflecting microwave structure. A rigorous derivation of the Kramers-Kronig phase formula is presented in Appendix A. Numerous analytic model calculations can be found in Appendix B. The differences between normal and truncated Gaussians are discussed in Appendix C. Finally, Appendix D contains a short description of the propagation of an electromagnetic wave front by two-dimensional fast Fourier transformation. This is the basis of a powerful numerical Mathematica code THzTransport, which permits the propagation of electromagnetic wave fronts through a beam line consisting of drift spaces, lenses, mirrors and apertures.