Benchmarking coherent radiation spectroscopy as a tool for high-resolution bunch shape reconstruction at free-electron lasers

We present a systematic comparison of two complementary methods for determining the longitudinal charge density profile of the compressed electron bunches in the soft x-ray free-electron laser FLASH: a frequency-domain technique---coherent transition radiation (CTR) spectroscopy---and a time-domain technique---streaking of the electron beam with a transversely deflecting microwave structure (TDS). While the direct time-profile measurement with a TDS is a well-established method invented at SLAC, our group has pioneered high-resolution bunch shape analysis based on coherent radiation spectroscopy. We have developed a broadband spectrometer covering the wavelength range from $5\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ to $433\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ with two sets of remotely interchangeable staged reflection gratings. The measured spectral intensity allows to compute the absolute magnitude of the bunch form factor but not its phase which, however, is needed to retrieve the bunch profile. Two phase retrieval methods are investigated in detail: analytic phase computation by means of the Kramers-Kronig dispersion relation, and iterative phase retrieval. Several computational techniques are compared and evaluated in view of their applicability and efficiency. For a large variety of bunch shapes, the time profiles derived from the spectroscopic data are compared with the TDS profiles, and generally excellent agreement is observed down to the 10 fs level. For part of the measurements, two independent CTR spectrometer systems have been available, yielding almost identical bunch shapes. In summary, we demonstrate that using well calibrated and broadband spectroscopy data, a fast and reliable phase reconstruction algorithm leads to bunch profiles competitive to high resolution TDS measurements.