Photoproduction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>B</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math> meson at future <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math> colliders

In the present paper, we study photoproduction of various $S$-wave $B_c$ states, i.e. $B_c(1{}^1S_0)$, $B_c^*(1{}^1S_0)$, $B_c(2{}^1S_0)$ and $B^*_c(2{}^3S_1)$, at future $e^+e^-$ colliders within framework non-relativistic QCD. Two sources initial photons can be realized in production, i.e., LBS photon WWA photon. addition to direct via sub-process, $\gamma + \gamma \rightarrow B_c b \bar{c}$, also calculate resolved sub-processes, g \bar{c}$ $q \bar{q} with $q=u,d,s,g$, respectively. Numerical results indicate that contributions from single are significant even dominant certain collision configuration. At high-energy high-luminosity colliders, meson generated mechanism is promisingly observable well studied.