  We study the phenomenon of electromagnetically induced transparency and
absorption (EITA) using a control laser with a Laguerre-Gaussian (LG) profile
instead of the usual Gaussian profile, and observe resonances with width up to
20 times below the natural linewidth. Aligning the probe beam to the central
hole in the doughnut-shaped LG control beam allows simultaneously a strong
control intensity required for high signal-to-noise ratio and a low intensity
in the probe region required to get narrow resonances. Experiments with a
second-order LG beam show that transit time and orbital angular momentum do not
play a significant role. This explanation is borne out by a density-matrix
analysis with a radially varying control Rabi frequency. We observe these
resonances using degenerate two-level transitions in the $D_2$ line of Rb in a
room temperature vapor cell. We show that the linewidth is reduced by about a
factor of 4 compared to the use of a Gaussian control beam, which should prove
advantageous in all applications of EITA and other kinds of pump-probe
spectroscopy.
