Localization in inhomogeneously broadened systems using the Gibbs phenomenon

Abstract

Spectra and images derived from the Fourier transformation of time-domain signals can often exhibit overshoots and/or “ringing” near sharp features. Such artifacts are due to the slow convergence of the Fourier series near such features, an effect referred to as the Gibbs phenomenon. While usually viewed as being purely mathematical in origin, the Gibbs phenomenon can often be found in a variety of physical situations, such as in imaging and spectroscopy. In this work, a physical description of the Gibbs phenomenon is presented where it is interpreted as an interference effect whereby slower destructive interference or “Fourier dephasing” occurs near sharp spectral features compared with the Fourier dephasing observed away from such features. Differences in Fourier dephasing can be exploited to localize magnetization near physical boundaries on timescales about an order of magnitude faster than can be achieved using conventional frequency or spatially selective pulses. This localization, which is reversible, also occurs on much faster timescales than can be attributed to irreversible sources, such as restricted diffusion or spatial variations of the intrinsic spin relaxation within the sample.