Layer-by-layer entangled spin-orbital texture of the topological surface state in Bi$_2$Se$_3$

With their spin-helical metallic surface state, topological insulators (TI) define a new class of materials with a strong application potential in quantum electronic devices. Technological exploitation depends on the degree of spin polarization of the topological surface state (TSS) - assumed to be 100$\%$ in phenomenological models. Yet in real materials, spin- and angle-resolved photoemission spectroscopy (ARPES) showed that the TSS spin polarization varies over a wide range: 20-85$\%$. This striking variation in TSS spin polarization has remained unexplored, leaving an undefined application prospect of TIs. Here we present a light-polarization study of ARPES momentum maps to unveil the entangled spin-orbital texture of the TSS in Bi$_2$Se$_3$. By determining the layer-by-layer evolution of this spin-orbital entanglement, we solve the puzzle of the observed TSS spin polarization and also provide means to manipulate the spin polarization of photoelectrons and photocurrents in TI devices.