The study focuses on reflection, transmission, and the related giant Goos-Hänchen shift through the Legendre-Gaussian polynomial for left and right circularly polarized RCP/LCP beam in a chiral medium. The phenomenon is studied with variation of the Legendre-Gaussian polynomial from \(P_0(x)\) to \(P_3(x)\) . The Legendre-Gaussian polynomial has a vital function in craters and peaks of the Goos-Hänchen shift. The crater- and peak-type reflection, transmission, and related shifts are recorded for both LCP and RCP beams. The number of craters and peaks enhances with increasing Legendre polynomial from \(P_0(x)\) to \(P_3(x)\) . The transmission, and reflection coefficients satisfy normalization, expressed as (T+R=1), ensuring energy conservation in the system. The highest value of reflection for RCP and LCP is observed to be \(R^{(\pm )}=\pm 0.4\) , and that for transmission is \(T^{(\pm )}=\pm 1\) . The non-negative transmission and reflection shifts are recorded for both LCP and RCP beams in chiral atomic medium. The optimum shift for the RCP beam in reflection is observed to be \(+6.5 \lambda \) , and for LCP it is \(+15 \lambda \) at incident angle \(\theta =\pi /6\) . The optimum shift for the RCP beam in transmission is observed to be \(+6.5 \lambda \) , and for LCP it is \(+9\lambda \) at incident angle \(\theta =\pi /6\) . When the incident angle decreases to \(\theta =\pi /10\) , the shift in transmission and reflection declines to \(S_{r,t}^{(+)}=3.2\lambda \) , \(S_{r}^{(-)}=8\lambda \) , and \(S_{t}^{(-)}=5\lambda \) . The observed results are useful for optical sensors, photonic circuits, optical resonators, optical imaging systems, surface metrology, plasmonic waveguides, data transmission, and quantum dot technologies.
