Contracted nonlinear susceptibility notation ($jk\mapsto J\in \{xx,yy,zz,yz,zx,xy\}$)

For identical input frequencies, intrinsic permutation symmetry of nonlinear susceptibility makes the last two Cartesian indices symmetric. Store their six independent pairs as one index, so the quadratic response uses a three-by-six array $d_{iJ}$. In the convention $d_{ijk}={\chi }_{ijk}^{(2)}/2$, the contracted input vector is $(A_x^2,A_y^2,A_z^2,2A_y{A}_z,2A_z{A}_x,2A_x{A}_y{)}^T$. The factors of two preserve the sum over both off-diagonal orderings. This bookkeeping does not assert the stronger Kleinman symmetry and does not change the tensor's physical order.