David E. Rourke

The inverse scattering transform and its use in the exact inversion of the Bloch equation for noninteracting spins

Abstract A general analytic inversion of the undamped Bloch equation to produce frequency-selective pulses is described. It is shown how the results of inverse scattering theory can be applied in order to determine the possibly complex pulse that will produce any desired magnetization response. The theory is directed principally at the problem of determining phase-compensated selective pulses. The easier problem of determining selective pulses that require subsequent refocusing is also discussed. Symmetry properties of the system are examined and a powerful result—that a symmetric magnetization response can only be achieved with a real pulse—is proved. The inversion is used to determine phase-compensated selective 90° and doubly selective 90° pulses. It is shown that the pulses can be designed to make the responses as close to perfect as is required. An arbitrarily good selective 90° pulse that requires subsequent refocusing is also obtained, and its relationship with the equivalent pulse obtained via Fourier analysis is discussed. The sensitivity to B 1 inhomogeneity is considered—the pulses that need subsequent refocusing are found to be less sensitive than the self-refocused equivalents.