  The $B_c$ meson pair, including pairs of both pseudoscalar states and vector
states, productions in high energy photon-photon interaction are investigated
at the next-to-leading order (NLO) accuracy in the nonrelativistic quantum
chromodynamics (NRQCD) factorization formalism. The corresponding cross
sections at the future $e^+e^-$ colliders with $\sqrt{s}=250$ GeV and $500$ GeV
are evaluated. Numerical result indicates that the inclusion of the NLO
corrections shall greatly suppress the scale dependence and enhance the
prediction reliability. In addition to the phenomenological meaning, the NLO
QCD calculation of this process subjects to certain technical issues, which are
elucidated in details and might be applicable to other relevant investigations.
