  The measurement of optical ultrafast laser pulses is done indirectly because
the required bandwidth to measure these pulses exceeds the bandwidth of current
electronics. As a result, this measurement problem is often posed as a 1-D
phase retrieval problem, which is fraught with ambiguities. The phase retrieval
method known as ptychography solves this problem by making it possible to
measure ultrafast pulses in either the time domain or the frequency domain. One
well known algorithm is the principal components generalized projections
algorithm (PCGPA) for extracting pulses from Frequency-Resolved Optical Gating
(FROG) measurements. Here, we discuss the development of the PCPGA and
introduce new developments including an operator formalism that allows for the
convenient addition of external constraints and the development of more robust
algorithms. A close cousin, the ptychographic iterative engine will also be
covered and compared to the PCGPA. Additional developments using other
algorithmic strategies will also be discussed along with new developments
combining optics and high-speed electronics to achieve megahertz measurement
rates.
