Band-Selective Pulses without Phase Distortion. A Simulated Annealing Approach

The design of shaped pulses for selective excitation experiments has received much attention in the fields of both high-resolution NMR and magnetic resonance imaging. The simplest pulse shape is rectangular, but the excitation profile has an undesirable “sine function” shape with side lobes extending a considerable distance from resonance. Pulse shaping can improve the frequency-domain response by localizing the excitation to a selected region around resonance, the most popular time-domain shaping functions being the Gaussian ( 1), Hermite (2), and “sine” pulses (3, 4). Although these shaped pulses succeed in eliminating side-lobe responses, they have the disadvantage that the excited signals exhibit a large frequency-dependent phase shift. Several authors have addressed this important problem (5-7). Fourier theory (8) predicts that all symmetric, purely amplitude-modulated pulses give rise to a linear phase error. Gaussian, Hermite, and sine pulses fall into this category. In many simple experiments a linear phase error across the spectrum can be corrected by conventional software routines, but if the spectrum contains broad lines this introduces an undesirable “rolling baseline.” In magnetic resonance imaging, a linear phase error can be corrected by reversing the direction of the field gradient immediately after the selective pulse, and allowing a suitable delay during which the magnetization can refocus. Field gradient reversal is not an option in high-resolution NMR, but the same effect can be achieved by a hard 180” refocusing pulse followed by a period of free precession ( 9). These methods suffer from the disadvantage that the refocusing delay causes extra signal loss due to relaxation; in addition, the 180” pulse method aggravates phase errors due to homonuclear spin-spin coupling. We describe here the design of shaped pulses which minimize such phase distortions. The resulting magnetization trajectories can be said to possess a self-focusing property. All signals are then in the pure absorption mode, an important consideration for coherence transfer and other experiments. In magnetic resonance imaging, much attention has been given to the problem of designing a pulse with a rectangular frequency-domain excitation profile-the socalled “top-hat” response. Computer optimization methods have proved popular, leading to a variety of elaborate modulation schemes (10-12). For practical reasons