$\pi/3$ is also an eigenstate. If we consider a dipole-like mode, which returns to itself only under a $2\pi$ rotation, this gives six possible eigenstates, which are in general not linearly independent. It can be shown[@kim2003] that only two modes are linearly independent and that if $\psi_1 (r)$ is a dipole-like eigenstate, then, under the approximation of scalar fields, the orthogonal linearly-independent dipole-mode is: $$\psi_2 (r) = \frac{1}{\sqrt{3}} \left[ R_{2\pi/3} \psi_1 (r) + R_{\pi/3} \psi_1 
(r) \right]$$ where $R_x$ indicates a rotation of $x$ around the cavity center. These two degenerate orthogonally-polarized dipole-modes are perfectly suited for polarization-degenerate light emission. These considerations are valid in the ideal case of a perfect triangular lattice; in practice, however, fabrication imperfections lift the ideal structure symmetry and remove the degeneracy of the two dipole modes. This mode splitting is illustrated in the characteristic photoluminescence measurement in Fig. \[fignfff\](c). These two modes, labeled H and V, each have both $x$ and $y$ E-field components, as shown in Fig. \[fignfff\](d). For the H mode, $E_y$ has an antinode at the center of the cavity region, whereas the less intense $E_x$ has a node at the center, so a perfectly centered QD would primarily couple to 