  Gradient echo memory (GEM) stores and retrieves photon wave packet in forward
direction with high efficiency and fidelity using photon-echo mechanism. It is
an important technique for quantum memory applications. By breaking the
continuity of the gradient absorption structure, the scheme becomes gradient
frequency comb (GFC), which is a hybrid of GEM and atomic frequency comb (AFC).
To elucidate the non-trivial field-atom evolution of gradient echo, we derive
its exact analytical solution in a medium of arbitrary optical thickness
subjecting to any linear gradient of transition frequency, and discuss its
physical processes. Based on this solution, we further suggest and analysis two
types of GFC, one with stepwise gradient and the other with discontinuous
gradient.
