Eigenmode analysis per layer¶

Scope of this page¶

This page treats stage 2 of the pipeline. Given the 4x4 Berreman matrix \(\Delta_i\) produced in Foundations, we solve the algebraic eigenvalue problem in each homogeneous layer \(i\), partition the four resulting modes into forward and backward propagation, and impose the continuity ordering of Li, Sullivan, and Parsons [1] so that mode labels remain consistent when the dielectric tensor is swept smoothly across parameter space. The treatment follows Passler and Paarmann [2, Sec. 2.A.1 and 2.A.2, Eqs. (11)-(18)], who adapted the Li sorting rule into the 4x4 machinery.

The eigenvalue problem¶

Because \(\Delta_i\) is \(z\)-independent inside layer \(i\), the spatial wave equation \(\partial_z \Psi = i(\omega/c)\,\Delta_i\,\Psi\) decouples into four plane-wave solutions that share the same in-plane wave vector \(\xi\) but carry distinct out-of-plane components \(q_{ij}\) [2, Eq. (11)],

The four \(q_{ij}\) are the eigenvalues of \(\Delta_i\), which in numerical practice are obtained via a standard dense eigensolver. The four \(\Psi_{ij}\) are the right eigenvectors and carry the polarization content of each mode in the ordered component basis \((E_x, H_y, E_y, -H_x)^\top\) fixed by [2, Eq. (7)].

Forward and backward partition¶

The first sorting step separates the four modes into the two pairs that propagate or decay toward \(+\hat{z}\) (transmitted) and toward \(-\hat{z}\) (reflected). The rule given by [2, Eq. (12)] is

The two rules are consistent, because a real \(q_{ij}\) points a Poynting vector in its own direction whereas a complex \(q_{ij}\) with \(\operatorname{Im}(q_{ij}) > 0\) describes an exponentially damped wave, which in the assumed sign convention \(\exp(-i(\omega/c)q_{ij}z)\) corresponds to decay in the \(+\hat{z}\) direction.

The two transmitted modes are labeled \(q_{i1}\) and \(q_{i2}\), and the two reflected modes are labeled \(q_{i3}\) and \(q_{i4}\) [2, text following Eq. (12)]. Which of the two transmitted modes is assigned index \(1\) rather than \(2\) is the question answered by the next step.

Continuity sorting within each pair¶

A naive eigensolver returns eigenvectors in arbitrary permutation, so a dielectric tensor that is swept continuously through parameter space produces output spectra that flip labels discontinuously. The physical observables \(r_{kl}\) and \(t_{kl}\) then appear to jump between \(kl = pp\) and \(kl = ss\) branches across the sweep. This failure mode was diagnosed and corrected by Li, Sullivan, and Parsons in the context of magneto-optic recording layers [1], and we adopt their projection rule as the within-pair sorting criterion.

The rule uses a scalar projection functional of the eigenvector itself. For moderately anisotropic tensors,

is the ratio of the squared \(E_x\) amplitude to the squared in-plane electric amplitude of the mode [2, Eq. (13)]. For a canonically \(p\)-polarized mode the electric field lies in the \(x\)- \(z\) plane, so \(\Psi_{ij,3} = E_y = 0\) and \(C(q_{ij}) = 1\). For a canonically \(s\)-polarized mode the electric field is along \(\hat{y}\), so \(\Psi_{ij,1} = E_x = 0\) and \(C(q_{ij}) = 0\). The sorting rule is then [2, Eq. (14)]

so \(q_{i1}\) and \(q_{i3}\) are the \(p\)-like transmitted and reflected modes, and \(q_{i2}\) and \(q_{i4}\) are the \(s\)-like ones.

Birefringent fallback to the Poynting criterion¶

When a principal axis of the dielectric tensor lies outside the \(x\)-\(z\) plane and outside \(\hat{y}\), the eigenmodes are no longer purely \(p\) or \(s\) and the \(E\)-field projection becomes ambiguous. Both \(\Psi_{ij,1}\) and \(\Psi_{ij,3}\) are non-zero for all four modes, and the \(C\)-functional no longer cleanly separates the pair. In this regime Passler and Paarmann replace the electric projection with the analogous Poynting-vector projection [2, Eq. (15)],

where \(\vec{S}_{ij} = \vec{E}_{ij} \times \vec{H}_{ij}\) is evaluated component-wise. The tangential Poynting components are recovered from the eigenvector using the identifications \(E_{ij,x} = \Psi_{ij,1}\), \(E_{ij,y} = \Psi_{ij,3}\), \(H_{ij,x} = -\Psi_{ij,4}\), \(H_{ij,y} = \Psi_{ij,2}\), with the longitudinal components \(E_{ij,z}\) and \(H_{ij,z}\) recovered from the auxiliary linear combinations of the \(a_{3n}\) and \(a_{6n}\) coefficients,

as enumerated in [2, Eqs. (17)-(18)]. This is the same \(a_{3n}\) and \(a_{6n}\) tabulation [2, Eq. (9)] that populates \(\Delta\) itself, now reused for the Poynting reconstruction.

The Poynting criterion is more expensive than the electric criterion because it requires assembling \(\vec{E}\) and \(\vec{H}\) fully before the projection, whereas the electric criterion reads directly off the eigenvector. We surmise that a practical implementation should dispatch on the tensor geometry, defaulting to the cheaper electric projection and switching to the Poynting projection whenever the principal-axis alignment fails.

What the sorting buys the rest of the pipeline¶

The ordered mode quadruplet \((q_{i1}, q_{i2}, q_{i3}, q_{i4})\) becomes the convention relied on in stages 3 through 6. Stage 3 (see Interface matrices) assembles the interface matrix \(A_i\) with column layout \((\text{p-trans}, \text{s-trans}, \text{p-refl}, \text{s-refl})\), the Xu \(\gamma_{ij}\) eigenvectors [2, Eqs. (19)-(20); 3] are written in the same order, and the Yeh-layout permutation \(\Lambda_{1324}\) introduced in Propagation and assembly relies on exactly this ordering. A violation of the Li sorting would therefore propagate as a mis-permutation of \(\Gamma_N\) and would surface as discontinuous \(r_{kl}(\omega)\) or \(r_{kl}(\theta)\) spectra in whatever observable is requested.

Where the code lives¶

In refloxide, the eigensolve and the sorting live in the core::modes module. The module consumes \(\Delta_i\) from core::delta, returns the quadruplet \((q_{ij}, \Psi_{ij})\) in the ordered layout above, and exposes the sorting criterion as a user-visible switch so that the Poynting fallback can be forced on when the caller knows a principal axis is tilted. Targeted unit tests against the Fresnel limit, against Yeh's birefringent multilayer benchmark, and against a smoothly swept dielectric tensor near a degenerate point are the appropriate validation battery.

Edge cases and limitations¶

The Li criterion is well defined whenever exactly two of the four modes are forward-propagating, which is the generic case. Four-fold degeneracies, which occur in exactly isotropic media at exactly normal incidence, leave the within-pair order formally ambiguous. In that limit the \(p\) and \(s\) labels carry no physical content and the pipeline returns the correct answer regardless of the label assignment. We note that this degeneracy is the one Xu, Wood, and Golding treat separately at the eigenvector level [3; 2, Eq. (19) branch \(q_{i1} = q_{i2}\)], and the Xu piecewise construction (see Interface matrices) is the mechanism that keeps the rest of the pipeline finite through the degenerate limit.

The forward and backward classification on \(\operatorname{Im}(q)\) alone presumes the loss sign convention built into the dielectric tensor. A caller who feeds a gain medium with \(\operatorname{Im}(\varepsilon) < 0\) inverts the classification and produces physically backward-decaying modes labeled forward. The library does not guard against this, because a consistent gain convention is the caller's responsibility.

References¶

1. Z.-M. Li, B. T. Sullivan, and R. R. Parsons, "Use of the 4x4 matrix method in the optics of multilayer magneto-optic recording media," Appl. Opt. 27, 1334 (1988). DOI.
2. N. C. Passler and A. Paarmann, "Generalized 4x4 matrix formalism for light propagation in anisotropic stratified media," J. Opt. Soc. Am. B 34, 2128 (2017). DOI.
3. W. Xu, L. T. Wood, and T. D. Golding, "Optical degeneracies in anisotropic layered media," Phys. Rev. B 61, 1740 (2000). DOI.