Reflection and transmission coefficients¶

Scope of this page¶

This page treats stage 5 of the pipeline. Given the Yeh-layout transfer matrix \(\tilde{\Gamma}_N\) from Propagation and assembly, we extract the eight complex amplitude coefficients \(r_{pp}, r_{ss}, r_{ps}, r_{sp}, t_{pp}, t_{ss}, t_{ps}, t_{sp}\) from rational combinations of its matrix elements, and we document the sign and normalization corrections imposed by the Passler-Paarmann erratum [1]. We also record the relabeling \(p/s \to o/e\) required for birefringent substrates, and the open status of the transmittance identity in anisotropic substrates.

The closed-form expressions follow Yeh [2] and are tabulated by Passler and Paarmann [3, Eqs. (33)-(36)], with the sign corrections of [1, Eqs. (34)-(36)]. The \(t_{pp}\) formula itself is unchanged and carries the unstarred Eq. (33) label in the erratum; only \(t_{ss}\), \(t_{ps}\), and \(t_{sp}\) required sign flips [1, Sec. 2.B]. The \(r_{kl}\) expressions are untouched by the erratum and remain those of [3, Eqs. (33)-(36)].

From \(\tilde{\Gamma}_N\) to the amplitude coefficients¶

Let \(\tilde{\Gamma}_N\) have matrix elements \(\Gamma_{\alpha\beta}\), where the \((\alpha, \beta)\) indexing uses the Yeh row order \((E^p_{\text{trans}}, E^p_{\text{refl}}, E^s_{\text{trans}}, E^s_{\text{refl}})\) [3, Eq. (30)]. For an incident medium that is isotropic and non-absorbing, the reflection and transmission coefficients read, with the transmission signs following the erratum-corrected convention [1, Eqs. (34)-(36)] and the reflection expressions unchanged from [3, Eqs. (33)-(36)],

The subscript convention is the usual one in ellipsometry. The first subscript \(k\) labels the outgoing polarization and the second \(l\) labels the incident polarization, so \(r_{ps}\) is the amplitude of the outgoing \(p\) wave per unit incident \(s\) wave [3, text preceding Eq. (33)]. The common denominator \(\Gamma_{11}\Gamma_{33} - \Gamma_{13}\Gamma_{31}\) is the same in all eight expressions and is the determinant of the forward-transmitted sub-block of \(\tilde{\Gamma}_N\).

The sign correction of the 2019 erratum¶

The 2017 paper [3, Eqs. (33)-(36)] gave \(t_{ss}\), \(t_{ps}\), and \(t_{sp}\) with an opposite sign convention from Yeh's original derivation [2], carried through deliberately for consistency with the field reconstruction of the 2017 paper. The 2019 erratum restores the Yeh convention, because the opposite convention produced incorrect electric-field components in birefringent media [1, Sec. 2.B]. The transmittance \(T_{kl} = |t_{kl}|^2\) is unchanged, but the phase of the transmitted field is not, and the phase matters for the field reconstruction of stage 6 (see Electric field distribution). refloxide implements the erratum-corrected signs. A library that implements the 2017 signs verbatim and then uses the 2017 field reconstruction recipe produces self-consistent but incorrect fields in birefringent stacks.

Reflectance¶

Because the incident medium is isotropic and non-absorbing (see the assumptions in Overview), the reflectance is the modulus-squared of the amplitude coefficient,

This identity follows from the Poynting flux projection in the incident half-space and does not require any anisotropy adjustment on the incident side [3, Sec. 2.B]. The full angular distribution of reflected intensity is the \(2 \times 2\) Jones reflectance matrix with components \(R_{pp}\), \(R_{ss}\), \(R_{ps}\), \(R_{sp}\).

Transmittance, and the open question¶

The symmetric identity \(T_{kl} = |t_{kl}|^2\) is true only in the degenerate case where the substrate is vacuum. For a general anisotropic substrate the transmittance depends on the Poynting projection on the substrate side, which in turn depends on the eigenvalue \(q_{i,N+1}\) of the transmitted mode, on the dielectric tensor of the substrate, and on the refractive index of the incident medium [1, Sec. 2.B, discussion preceding Eqs. (34)-(36)]. The correct expression was deferred by Passler and Paarmann to a later publication [1, Ref. 6 of the erratum]. As of the sources we cite here, no general closed-form transmittance identity has been published.

We flag this as a known gap. The kernel computes and returns the eight \(t_{kl}\) amplitudes, which are physically meaningful and complete. We do not currently expose \(T_{kl}\) as a derived observable, and we prefer to leave the conversion to the caller (with the caveat that the isotropic-substrate identity \(T_{kl} = |t_{kl}|^2 \operatorname{Re}(n_{N+1}\cos\theta_{N+1}) / \operatorname{Re}(n_0\cos\theta_0)\) is the one classical result that we cover in a docstring example). A generalized transmittance module will be added once the Passler followup publication is available, or once we are willing to derive it independently.

Birefringent substrates and the \(o/e\) relabeling¶

When the substrate is birefringent, the two transmitted modes are no longer purely \(p\)-polarized and \(s\)-polarized but are the ordinary (o) and extraordinary (e) eigenmodes of the substrate's dielectric tensor [1, Sec. 2.B, text following Eqs. (34)-(36)]. Both eigenmodes carry non-zero \(E_x\) and non-zero \(E_y\) in general, so the Jones layout \((p, s)\) on the outgoing side is no longer natural.

The relabeling takes the 2017 coefficients and renames them [1, Sec. 2.B],

with the analogous renaming applied to \(r_{kl}\). The numerical values of the matrix elements \(\Gamma_{\alpha\beta}\) are unchanged, only the semantic labeling of which transmitted mode each coefficient addresses. The calling code is responsible for interpreting these in the ordinary and extraordinary eigenbasis of the substrate.

For lab-frame-diagonal substrates, which is the common case, the ordinary and extraordinary modes coincide with the canonical \(s\) and \(p\) labels, and the relabeling is the identity. A library that wants to support both regimes uniformly, as refloxide does, returns the eight coefficients under the neutral names \(t_{11}\), \(t_{22}\), \(t_{12}\), \(t_{21}\) (and analogously for \(r\)) and exposes label aliases in a separate semantic layer.

Ellipsometric observables¶

The amplitude coefficients feed directly into the standard ellipsometric observables. For a single reflected \(p\) and \(s\) ratio,

which defines the ellipsometric angles \((\Psi, \Delta)\) consumed by fitting frameworks. For anisotropic samples with non-negligible cross-polarization, the full \(2 \times 2\) Jones or \(4 \times 4\) Mueller matrix constructed from the \(r_{kl}\) set is the appropriate observable. We do not construct Mueller matrices in the kernel, because the incoherent averaging they encode is a depolarization operation that belongs in the fitting layer.

Where the code lives¶

Stage 5 is the core::coefficients module. It consumes \(\tilde{\Gamma}_N\) and returns the eight amplitudes as a named struct. The rational expressions are written out verbatim from [3, Eqs. (33)-(36)] for the reflection coefficients and from [1, Eqs. (33), (34)-(36)] for the transmission coefficients, rather than being rederived in code, so that a reader comparing the implementation to the erratum can trace the mapping line by line. The common denominator is computed once per call and shared across the eight returns.

Numerical notes¶

The denominator \(\Gamma_{11}\Gamma_{33} - \Gamma_{13}\Gamma_{31}\) vanishes in two regimes. The first is total reflection with a non-absorbing substrate, where the incident mode couples entirely to the reflected basis and the forward-transmitted sub-block is rank-deficient. The second is at a guided-mode resonance of the stack, where a lossless pole of \(r_{kl}\) coincides with a zero of the denominator. Both regimes produce physically finite observables when evaluated carefully, because the numerators vanish at the same rate as the denominator. The library does not branch on proximity to these zeros, because the complex arithmetic survives the \(0/0\) cancellation at standard double precision when the stack is built correctly.

References¶

1. N. C. Passler and A. Paarmann, "Generalized 4x4 matrix formalism for light propagation in anisotropic stratified media, erratum," J. Opt. Soc. Am. B 36, 3246 (2019). DOI.
2. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742 (1979). DOI.
3. 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.