Owing to the intrinsic breadth, depth, and informational load, the main text cannotbe fully decoupled from the Supplementary Notes, but we’ve maximized clarity tohelp readers — particularly in NCO — quickly grasp our LCO methods, results, andconclusions, which greatly benefit NCO.

Using this model, we create the material-matrix tetrahedral compass to conduct a detailed analysis of how the four fundamental characteristics of materials (linear/circular birefringence/dichroism) influence the eigensystems of the vector electric field in two-dimensional spatial frequency \bar{k}_{\mathrm{\rho}} domain and its distribution in three-dimensional \bar{r} space with a crystal-2f configuration.

We also present our model's early applications in focal engineering and NCO.

At the methodological level, three widely used plane wave bases are compared in terms of eigenanalysis in reciprocal space and light field propagation in real space.