  We demonstrate a novel algorithmic approach for the
second-harmonic-generation (SHG) frequency-resolved optical gating (FROG)
ultrashort-pulse-measurement technique that always converges and, for complex
pulses, is also much faster. It takes advantage of the Paley-Wiener Theorem to
retrieve the precise pulse spectrum (half the desired information) directly
from the measured trace. It also uses a multi-grid approach, permitting the
algorithm to operate on smaller arrays for early iterations and on the complete
array for only the final few iterations. We tested this approach on more than
25,000 randomly generated complex pulses, yielding SHG FROG traces to which
noise was added, and have achieved convergence to the correct pulse in all
cases. Moreover, convergence occurs in less than half the time for extremely
large traces corresponding to extremely complex pulses with time-bandwidth
products up to 100.
