New amplitude-modulated radio-frequency pulses are proposed that allow one to achieve selective excitation, de-excitation, inversion, and refocusing of spin-1/2 systems in NMR. The rotation angles and axes are well defined over a chosen bandwidth, regardless of the initial conditions. These universal pulses are derived from the idea of Gaussian pulse cascades, but their design is guided by new principles. The amplitude profiles are optimized by calcg. the quaternions representing the propagators of the overall rotations as a function of offset. This procedure provides a more general approach to pulse design than the usual optimization of the response to some initial condition. [on SciFinder (R)]