  In this paper, we numerically optimize broadband pulse shapes that maximize
Hahn echo amplitudes. Pulses are parameterized as neural networks (NN),
nonlinear amplitude limited Fourier series (FS), and discrete time series (DT).
These are compared to an optimized choice of the conventional hyperbolic secant
(HS) pulse shape. A power constraint is included, as are realistic shape
distortions due to power amplifier nonlinearity and the transfer function of
the microwave resonator. We find that the NN, FS, and DT parameterizations
perform equivalently, offer improvements over the best HS pulses, and contain a
large number of equivalent optimal solutions, implying the flexibility to
include further constraints or optimization goals in future designs.
