[indicated with a dashed arrow in Fig. 1 (a)] is split into two eigenstates that are similar to the TEM$_{01}$ and TEM$_{10}$ modes of transverse laser patterns. The interferogram of one of these eigenstates is shown in Fig. 1 (c). In order to specifically excite this eigenstate, one need to slightly red-shift the excitation laser with respect to the doublet energy. We have extracted the amplitude [Fig. 1 (d)] and the phase [Fig. 1 (e)] of the polariton wavefunction. In Fig. 1 (e), the π -phase shift between the two lobes is clearly visible. It indicates that the left lobe of the wavefunction is of opposite sign than the right lobe. Around the trap the phase is not defined, as the wavefunction is exponentially decaying outside the trap. The ellipticity being very small, the corresponding splitting is smaller than the linewidth. Therefore, the ellipticity can be overcome by pumping between the energies of the two split states, and excite a combination of them. In this case, the azimuthal dependence of the wavefunction reads as: where A and B are complex coefficients. The value of coefficients A and B is given by the overlap (in real space, reciprocal space and energy) between the ± m states and the pump. When the laser is focused on the mesa, both + m and − m components are generally excited, creating standing wave patterns, similar to the one observed in Fig. 1 (c). The excitation laser can be used to lock, and subsequently control, the pattern in any arbitrary direction $^{16}$. We are now going to show that by carefully selecting certain pumping conditions one can create “pure” + m or − m states, carrying an integer angular orbital momentum. Focusing the excitation laser on the side of the trap, with a finite excitation angle allows to inject polaritons with a well defined in-plane momentum mainly on one side of the trap. This can be used to select which one of the + m or − m state is injected. Such a state is experimentally produced and shown in Fig. 2 for ( n = 1 ,m = +1). In the interferogram displayed in Fig. 2 (a), one can observe a very clear fork-like dislocation, indicating the presence of a phase singularity. We extract from this interferogram the amplitude [Fig. 2 (b)] and phase [Figure 2 (c)] of the polariton field. The two characteristics of quantized vortices are observed: the intensity minimum at the center of the trap (b) and the 2 π -phase shift around the core (c) are straightforwardly visible. The core of this vortex is situated at the center of the trap, and the size of the vortex is delimited by the mesa diameter. This state is analogous to a TEM$_{01}$ ∗ “donut” transverse laser mode. 4 ψ ( φ ) = Aψ$_{+}$$_{m}$ ( φ ) + Bψ$_{−}$$_{m}$ ( φ ) = Ae imφ + Be − imφ