  Quantum states and the modes of the optical field they occupy are
intrinsically connected. Here, we show that one can trade the knowledge of a
quantum state to gain information about the underlying mode structure and, vice
versa, the knowledge about the modal shape allows one to perform a complete
tomography of the quantum state. Our scheme can be executed experimentally
using the interference between the signal and probe states on an unbalanced
beam splitter with a single on/off-type detector. By changing the temporal
overlap between the signal and the probe, the imperfect interference is turned
into a powerful tool to extract the information about the signal mode
structure. A single on/off detector is already sufficient to collect the
necessary measurement data for the reconstruction of the diagonal part of the
density matrix of an arbitrary multi-mode signal. Moreover, we experimentally
demonstrate the feasibility of our scheme with just one control parameter --
the time-delay of a coherent probe field.
