Goos-Hänchen shifts in a combined tripod and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math> atom-light coupling scheme

We investigate the controlled manipulation of the Goos-H\"anchen (GH) shift in probe light beams, encompassing both plane and Gaussian beams, when introduced into a cavity housing a highly resonant five-level atomic system exhibiting a combined tripod and $\mathrm{\ensuremath{\Lambda}}$ (CTL) configuration. This complex scheme arises from the interaction of three atomic ground states with two excited states through five distinct light fields. Such a system effectively reduces to a $\mathrm{\ensuremath{\Lambda}}$- or N-shaped configuration by manipulating the light fields, resulting in the alteration of the dispersion behavior of the probe beam, ultimately inducing either a positive or negative GH shift. This unique behavior is a direct consequence of the closed-loop structure inherent to the five-level atomic scheme. For both plane and Gaussian beams, we demonstrate the superiority of CTL over $\mathrm{\ensuremath{\Lambda}}$ and $N$ schemes in achieving substantial positive GH shifts. When considering Gaussian probe light, we observe a critical role played by the beam width in controlling the magnitude and sign of the GH shifts. Our proposed approach for studying the GH shift carries significant practical implications, particularly in its capacity to monitor media with either left-handed characteristics displaying negative permittivity and permeability or right-handed characteristics displaying positive permittivity and permeability through the manipulation of externally controlled parameters. This approach may find applications in various fields, including optical heterodyne sensors used to measure beam angle, displacement, temperature, and refractive index, thereby advancing our understanding and control of optical phenomena.