Analytic 3D vector non-uniform Fourier crystal optics in arbitrary $\bar{\bar{\varepsilon}}$ dielectric

Abstract: To find a suitable framework for nonlinear crystal optics(NCO), we have revisited linear crystal optics(LCO). 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. Inspired by complex ray tracing, we expand M.V. Berry & M.R. Dennis's 2003 uniform plane wave model to non-uniform Fourier crystal optics and ultimately derive the explicit form of its 3$\times$2 transition matrix, bridging the two major branches of crystal optics in reciprocal space, where either ray direction $\hat{k}$ or spatial frequency $\bar{k}{\mathrm{\rho}}$ serves as the input variable. 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. Along this journey, we have uncovered new territories in LCO in both real and reciprocal space, such as infinite singularities arranged in disk-, ring-, and crescent-like shapes, ``L shorelines'' resembling hearts, generalized haunting theorem, double conical refraction, and optical knots it induces. We also present our model's early applications in focal engineering and NCO. As the opening chapter in a trilogy, this work connects crystal optics, Fourier optics, and nonlinear optics, while integrating theoretical, computational, and experimental physics, advancing all six domains.