  Several Beyond the Standard Model theories are proposed that fermions might
have composite substructure. The existence of excited quarks is going to be the
noticeable proof for the compositeness of Standard Model fermions. For this
reason, excited quarks have been investigated by phenomenological and
experimental high energy physicists at various collider options for the last
few decades. The Future Circular Collider (FCC) has been recently planned as
particle accelerator to be established at CERN. Beside the $\sqrt{s}$ = 100 TeV
proton-proton collisions, the FCC includes electron-positron and
electron-proton collision options. Construction of linear $e^- e^+$ colliders
(or dedicated e-linac) tangential to the FCC will afford an opportunity to
handle multi-TeV $ep$ and $\gamma p$ collisions. In this respect, we executed a
simulation of the resonance production of the excited $u$ quark at the FCC
based $\gamma p$ colliders with choosing both the polarized and unpolarized
photon beams. The findings revealed that the chirality structure of the
$q^*$-$q$-$\gamma$ vertex can be determined by the photon beam polarization.
The attainable mass limits of the excited $u$ quark reached the highest values
when the polarized photon beam was chosen. In addition, the ultimate
compositeness scale values can be handled by appropriate choice of the photon
beam polarization.
