We investigate the steady-state optical response of a Laguerre-Gaussian vortex cavity system integrated with cold atoms featuring a double-$\mathrm{\Lambda }$ energy level structure. Within this hybrid system, the atoms are driven by a cavity mode and three coherent vortex beams, each carrying independent orbital angular momentum. We first check the steady-state output spectrum of the hybrid system in the passive (active) case [without (with) external cavity driving]. Our findings reveal that the optomechanically induced transparency spectrum is modulated by the orbital angular momentum difference $\left(\mathrm{\Delta }\ell \hslash \right)$ from the atomic component throughout the four-wave-mixing process. The resulting loop phase (${\ell }_{\mathrm{\Theta }}\theta$) can achieve a switching effect on the absorption and gain behavior of the hybrid system for the probe beam. Additionally, the group delay, indicative of fast or slow light phenomena, is also tuned by $\mathrm{\Delta }\ell$. We further display how the atomic orbital angular momentum modulates the periodicity of the output spot pattern in the hybrid system. This research provides valuable insights into the modulation of optical responses in Laguerre-Gaussian vortex cavity systems.

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• Angular momentum of light
• Atomic ensemble
• Optical microcavities
• Four-wave mixing
• Rotating wave approximation