The formation of novel two-dimensional electron gas (2DEG) with high mobility in metal/amorphous interfaces has motivated an ongoing debate regarding the formation and novel characteristics of these 2DEGs. Here we report an optical study, based on infrared spectroscopic ellipsometry, of nonmagnetic metal and amorphous semiconducting oxide (Cu/Bi$_2$O$_3$) interfaces that confirms the formation of a 2DEG with spin orbit coupling (SOC). The 2DEG optical response was simulated with a uniaxial diagonal dielectric tensor within a sub-nanometer thin layer, where its $x$ and $z$ components lineshapes resolved in both free-electron and peak-like contributions, resulted very similar to theoretical predictions [M. Xie et al., Phys. Rev. B $bf{89}$, 245417 (2014)] of a two dimensional electron gas confined in the normal direction of a perovskite interface. In particular, the small but finite conducting character of the $z$ component provides a unambiguous signature of the presence of the 2DEG in the Cu/Bi$_2$O$_3$ system. Although the original constituent materials do not possess spin-orbit coupling (SOC), the resulting interfacial hybridization of such states induce electronic asymmetric wave functions. This work demonstrates the detection of 2DEG in amorphous crystals allowing to study its challenging interfacial phenomena such as SOC and interface-bulk coupling, overcoming an experimental impediment which has hold back for decades important advancements for the understanding of 2DEGs in amorphous materials.
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