  The time-of-flight method is a fundamental approach for characterizing the
transport properties of semiconductors. Recently, the transient photocurrent
and optical absorption kinetics have been simultaneously measured for thin
films; pulsed-light excitation of thin films should give rise to non-negligible
in-depth carrier injection. Yet, the effects of in-depth carrier injection on
the transient currents and optical absorption have not yet been elucidated
theoretically. Here, by considering the in-depth carrier injection in
simulations, we found a 1/t^{1-alpha/2} initial time (t) dependence rather than
the conventional $1/t^{1-alpha}$ dependence under a weak external electric
field, where alpha<1 is the index of dispersive diffusion.The asymptotic
transient currents are not influenced by the initial in-depth carrier injection
and follow the conventional 1/t^{1+alpha} time dependence. We also present the
relation between the field-dependent mobility coefficient and the diffusion
coefficient when the transport is dispersive. The field dependence of the
transport coefficients influences the transit time in the photocurrent kinetics
dividing two power-law decay regimes. The classical Scher--Montroll theory
predicts a_1+a_2=2 when the initial photocurrent decay is given by 1/t^{a_1}
and the asymptotic photocurrent decay is given by 1/t^{a_2}. The results shed
light on the interpretation of the power-law exponent of 1/t^{a_1} when
a_1+a_2neq 2.
