  We present a combined analytical and numerical investigation to show how an
optimal control field can be designed to generate maximum field-free
orientation of molecules for three populated rotational states. Based on a
model involving pure rotational ladder-climbing excitation between rotational
states, a set of optimal amplitude and phase conditions are analytically
derived for the applied control fields. The maximum degree of orientation can
be achieved when the field satisfies amplitude and phase conditions at the two
transition frequencies. Multiple optimal solutions exist and to examine these
conditions, we devise a quantum coherent control scheme using two terahertz
pulses and successfully apply it to the linear polar molecule HCN at ultracold
temperature. The sensitivity of both populations and phases of rotational
states to control field parameters, i.e., the detuning, bandwidth, and time
delay, is analyzed for understanding the optimal orientation mechanism. This
work thus examines the frequency domain landscape belonging to optimal pulses.
