It is shown that perpendicular gradients in shear Alfv¿n wave (SAW) dispersion regulate the localization of wave power on nightside geomagnetic L-shells where narrow Field Line Resonances (FLRs) form. We estimate the timescale for this process, ω0tc = 1/$sqrt{left vertbeta right vertalpha}$, and demonstrate that it is analogous to optical wave focusing. Here, ¿ is the gradient in the global wave dispersion parameter across L-shells in the equatorial plane, and α is the gradient in the SAW eigenfrequency. It is demonstrated that dispersive SAWs with wave numbers and frequencies in a certain range, are subject to magnetospheric focusing onto L-shells where they reach large amplitude and are expected to dissipate. Our theory addresses a class of arc scales that are comparable to the electron inertial length near the ionosphere, or the ion gyroradius near the equatorial plane. We further demonstrate that when the gradient in the SAW frequency reverses across the edges of auroral density cavities, it naturally traps dispersive SAWs and focuses them down to the inertial scale.