  We study spatial multimode quantum memories based on light storage in
extended ensembles of Lambda-type atoms. We show that such quantum light-matter
interfaces allow for highly efficient storage of many spatial modes. In
particular, forward operating memories possess excellent scaling with the
important physical parameters: quadratic scaling with the Fresnel number and
even cubic with the optical depth of the atomic ensemble. Thus, the
simultaneous use of both the longitudinal and transverse shape of the stored
spin wave modes constitutes a valuable and so far overlooked resource for
multimode quantum memories.
