Birefringence of lateral shifts in reflection and transmission based on milnor polynomial and quantum tunnelling in induced chiral medium

We theoretically investigate birefringence of lateral Goos-Hänchen (GH) shift in a tunnelling chiral atomic medium. The Milnor polynomial of controlled fields and tunnelling modulate the light-mater interactions. We analyze the GH shift for right- and left-circularly polarized (RCP and LCP) beams for varying incident angles. We found a lossless condition R ⁽⁺⁾ + T ⁽⁺⁾ = 1 for RCP beams at the incident angle θ _i = π/3, while LCP beams shows loss R ⁽⁻⁾ + T ⁽⁻⁾ + A ⁽⁻⁾ = 1 and A ⁽⁻⁾ = 0.2. The lateral shift in reflection of RCP beam varies in range of 0 λ ⩽ S_r⁽⁺⁾ ⩽ 60 λ and transmission of RCP beam varies in the range of 0 λ ≤ S_t⁽⁺⁾ ⩽ 100 λ at θ _i = π/3, while the shift in LCP beam decreases 0 λ ⩽ S_r⁽⁻⁾ ⩽ 40 λ and transmission increases 0 λ ⩽ S_t⁽⁻⁾ ≤ 150 λ for the same parameters. Further, for incident angle θ _i = π/4, we found an increase in lateral shifts S_r,t⁽⁻⁾ ⩽ 150 λ for RCP and 0 λ ⩽ S_r,t⁽⁻⁾ ⩽ 200, 1500 λ for LCP for incident angle θ _i = π/4. These findings enhances our understanding of GH shifts in chiral medium for potential application in high speed communication, imaging and sensing.