**Multiple azimuths and incidence angles make tensor recovery identifiable.** Rotating the sample around its normal changes the projection of material axes into the p/s basis. Cross-polarized coefficients often exhibit characteristic angular symmetries, while diagonal coefficients constrain average response and thickness. A joint fit should use all azimuths with one consistent tensor and orientation rather than fit independent optical constants at each angle. Symmetry-related azimuths provide strong diagnostics. For some reciprocal sample classes, measurements at positive and negative azimuth or after 180-degree rotation obey defined sign and interchange relations. Violations can expose azimuth offset, sample tilt, wrong handedness, nonreciprocity, patterned asymmetry, or calibration error. The expected relation depends on crystal class and reference convention and must be derived for the actual geometry. Changing incidence angle alters sensitivity to in-plane and out-of-plane dielectric response, propagation distance, and interface phase. It also changes footprint size and position. On laterally heterogeneous samples, multiple angles may interrogate different material, breaking the assumption of one stack. Registration and footprint overlap must be verified before joint fitting. Different surface cuts add independent tensor projections. Bulk anisotropic crystals can be measured on several known faces to reduce orientation and dielectric-function ambiguity. For thin films, sample azimuth, incidence angle, wavelength, and sometimes transmission data play an analogous role. X-ray diffraction, polarized microscopy, or known growth axes can anchor the coordinate transformation. |Sample class|Minimum useful model|Why conventional ellipsometry can fail|Helpful measurement diversity|Critical validity check| |---|---|---|---|---| |Tilted uniaxial film|Ordinary and extraordinary dielectric functions, axis angles, thickness|Optic-axis projection generates p–s coupling|Several azimuths and incidence angles|Axis-angle covariance and stage-zero calibration| |Biaxial or low-symmetry crystal|Full symmetry-allowed complex dielectric tensor|Three axes or off-diagonal response mix in p/s basis|Multiple cuts, azimuths, and broad spectrum|Causal tensor model and crystallographic registration| |Oriented molecular or columnar film|Anisotropic effective-medium tensor plus orientation distribution|Form and intrinsic anisotropy create cross-polarization|Azimuth series with structural texture measurement|Depolarization and nonuniqueness of effective medium| |Periodic grating or device pattern|RCWA or another rigorous electromagnetic geometry model|Pattern converts polarization and diffracts light|Azimuth, angle, wavelength, and design constraints|Pitch regime, diffraction orders, and footprint registration| |Magneto-optic or chiral structure|Symmetric and antisymmetric tensor components|Circular and nonreciprocal coupling are outside scalar model|Field reversal, direction reversal, and azimuth|Instrument handedness and linear-anisotropy artifacts| **Forward propagation through anisotropic layers requires coupled-wave electromagnetics.** Isotropic 2×2 characteristic matrices can propagate s and p separately. In an anisotropic layer they are coupled, so Berreman-type 4×4 formalisms or equivalent eigenmode solvers propagate tangential electric and magnetic field components through the stack. Boundary conditions then yield the Jones reflection and transmission matrices. A schematic first-order propagation equation is $$ \frac{d\mathbf F}{dz}=ik_0\mathbf G(\boldsymbol\epsilon,\boldsymbol\mu,\mathbf k_{\parallel})\mathbf F $$ where $\mathbf F$ contains tangential field components, $k_0$ is vacuum wavenumber, and $\mathbf G$ depends on material tensors and conserved in-plane wavevector. Numerical stability matters for thick, absorbing, evanescent, or highly anisotropic layers; scattering-matrix or stabilized algorithms may be preferable to naive transfer multiplication. Eigenmode ordering and branch selection need consistent treatment across wavelength. Abruptly swapping modes can create discontinuities in predicted spectra or gradients used for regression. Passive materials should follow causal sign conventions for complex wavevectors and decay. Solver validation against isotropic limits, analytic uniaxial cases, energy balance, and independent implementations reduces subtle convention errors. Surface roughness or mixed composition is often represented by anisotropic effective-medium theory. The chosen inclusion shape, volume fractions, host, and axis distribution strongly affect the effective tensor. A fitted void fraction is model-dependent and not automatically porosity; a fitted optical axis is not automatically the crystallographic axis. Microscopy, density, diffraction, or porosimetry should constrain the microstructure. Interfaces may have their own anisotropy through bonding, reconstruction, strain, or graded orientation. Adding an anisotropic interface layer can improve fit while introducing severe covariance with bulk tensor and thickness. Use residual signatures, multiple specimens, or thickness series to establish whether the interface is identifiable. **Dielectric-tensor dispersion must obey symmetry and causality.** Each independent tensor component is complex and spectral. Transparent regions can use suitable dispersion forms, while absorbing regions require causal oscillators or another Kramers–Kronig-consistent representation. Fitting every wavelength independently can reveal trends but may produce nonphysical discontinuities and mix changing principal axes with oscillator parameters. For orthorhombic symmetry with frequency-independent axes, each principal component can be modeled causally. In monoclinic or triclinic material, electronic transitions can have different dipole directions, and the apparent principal axes may rotate with energy. Forcing one diagonal tensor basis across all energies can bias optical constants. A dyadic oscillator model can assign each transition an amplitude, line shape, and polarization direction while maintaining a shared crystallographic frame. Kramers–Kronig relations apply to causal response components expressed in an appropriate fixed basis. Diagonalizing the complex tensor independently at every energy can generate axes that lack a simple causal interpretation. Report the basis and oscillator construction used to claim principal optical functions. Thickness and tensor amplitude remain correlated, especially for ultrathin films. A thickness series with shared dielectric functions is powerful: different optical path lengths constrain the common tensor while allowing specimen-specific thickness. Independent thickness, mass density, or composition data can reduce degeneracy. A single perfect-looking spectrum rarely proves all tensor elements. Model comparison should test whether anisotropy is required. Fit an isotropic baseline, then a symmetry-constrained anisotropic model, and examine residual structure, parameter uncertainty, and predictive improvement at withheld azimuths. Extra tensor elements that only absorb noise or calibration error should not be promoted to material physics. **Depolarization marks the boundary between generalized Jones and Mueller descriptions.** A Jones matrix maps fully polarized coherent input to fully polarized output. If a measured beam is partially polarized because the instrument averages domains, thickness variation, roughness scattering, angular spread, backside paths, or temporal fluctuations, no single Jones matrix captures the ensemble. The degree of depolarization should be measured with a capable Mueller instrument or bounded using repeatable polarization-state tests. A low residual in a generalized Jones fit does not prove nondepolarization if the instrument observes only a subset of states. Conversely, small apparent depolarization may be the instrument floor from retardance calibration, beam walk, bandwidth, or detector drift. Anisotropy does not imply depolarization, and roughness does not always imply it. A homogeneous birefringent crystal is deterministic. Subwavelength roughness may be represented coherently by an effective interface under suitable conditions. Large or heterogeneous roughness can scatter and mix states incoherently. Choose the formalism from measured polarization behavior and spatial scales, not from a material label. When depolarization is modest, some workflows fit a nondepolarizing model to a dominant component and treat the remainder statistically. Such approximations need a stated mixture model and uncertainty. Forcing all data into a Jones matrix can map heterogeneity into false birefringence, axis tilt, or thickness. Mueller matrix ellipsometry can also measure deterministic anisotropy, so the categories overlap. The practical distinction is the observable and model: generalized ellipsometry emphasizes complex co- and cross-polarization amplitudes for nondepolarizing response; Mueller analysis uses Stokes transfer and can represent partial depolarization. Report which was measured. **Periodic structures require symmetry-aware scatterometry rather than a blanket-film tensor alone.** Gratings, fin arrays, line-space patterns, metasurfaces, and overlay structures couple p and s depending on azimuth and geometry. Rigorous coupled-wave analysis, finite-element, finite-difference, or another validated Maxwell solver predicts the reflected Jones matrix and any propagating diffraction orders. If pitch is deeply subwavelength, an anisotropic effective-medium approximation may capture the zeroth order over a bounded range. Near diffraction onset or when critical dimensions are comparable to wavelength, homogenization fails. Sidewall angle, height, linewidth, corner rounding, pitch walk, overlay, material optical constants, and line roughness can have correlated signatures. Generalized polarization data add constraints but do not guarantee unique optical critical dimension extraction. Use design priors, multiple azimuths and angles, sensitivity analysis, and orthogonal CD-SEM, AFM, or x-ray measurements. Synthetic recovery and profile likelihood reveal which geometric combinations the dataset actually identifies. The illuminated region must contain a consistent periodic structure. Finite arrays, scribe boundaries, multiple device orientations, focus variation, and spot placement can mix Jones responses or depolarize. Record beam footprint and pattern azimuth, and verify repeatability after translating within the target. For reciprocal symmetric gratings, Jones elements can obey useful azimuth and mirror relations. These relations are excellent alignment and model checks. Fabrication asymmetry may break them, but so can stage offset or an inconsistent p/s convention; controls decide which interpretation is justified. ```flowchart Identify the anisotropic, patterned, magneto-optic, or chiral decision variable -> Define p/s order, phase sign, handedness, crystal frame, and sample azimuth zero -> Test whether one nondepolarizing Jones matrix describes the footprint -> Choose wavelengths, angles, azimuths, sample cuts, and reference measurements -> Calibrate polarization states, cross-talk, retardance, angle, and registration -> Build a symmetry-constrained tensor or rigorous patterned-structure forward model -> Fit all configurations jointly with covariance and alternate initializations -> Inspect cross-polarization, symmetry relations, residuals, and parameter profiles -> Validate axes, thickness, tensor elements, or geometry with orthogonal metrology ``` **A traceable generalized ellipsometry result preserves conventions and identifiability evidence.** Freeze wavelength range, incidence angles, beam footprint, sample azimuths, input and analyzer states, p/s ordering, phase and handedness convention, stage zero, calibration artifacts, detector settings, and environmental conditions. Changing a sign convention can change cross-polarization phase and fitted axis direction without any new physics. Store raw intensity modulation, reconstructed Jones parameters, covariance, normalization, absolute reflectance when available, model graph, dielectric-tensor basis, Euler convention, parameter bounds, solver version, residuals, and fit restarts. Report parameter correlations and symmetry-equivalent orientation solutions rather than select one axis angle without qualification. Validation should include an isotropic sample that drives off-diagonal terms to the calibrated floor, a known anisotropic crystal or retarder, symmetry-related azimuths, and a reference outside the calibration set. For patterned structures, use a design-known or independently measured geometry. Repeat after any polarizer, compensator, objective, source, detector, alignment, or software change. The strongest conclusion uses the simplest tensor or geometry model that predicts all angles and azimuths within uncertainty and remains valid under a depolarization test. Extra Jones terms reveal missing scalar physics; they do not license unconstrained complexity. The durable way to interpret generalized ellipsometry is through a Jones-coupling-dielectric-tensor-coordinate-rotation-multi-azimuth-forward-model-depolarization-boundary-and-identifiability lens.