Roughness model, Nevot-Croce¶

Scope of this page¶

This page documents the Nevot-Croce factor [1, 2] as the multiplicative roughness correction applied to the eight Passler-Paarmann amplitudes \(r_{kl}, t_{kl}\) at each interface in refloxide. The treatment follows the derivation that Esashi and coworkers [3, Sec. 3] reproduce from the distorted-wave Born approximation [4], adapted to the tensor 4x4 setting. The correction has the lowest computational cost of the three roughness models in the kernel and is the default choice in the small-roughness regime defined by \(\sigma k_{i,z} < 1\) in either of the two adjacent media.

The companion pages Debye-Waller and Graded interface treat the long-correlation limit and the structural discretization respectively. The Selection guide gives the decision criterion that maps a given physical regime onto one of the three.

Closed-form correction factors¶

The Nevot-Croce factor models a Gaussian distribution of interface heights with rms amplitude \(\sigma_{i,i+1}\) across the boundary between layers \(i\) and \(i+1\). For the reflection and transmission amplitudes at a single interface, the corrections read [3, Eq. (1)]

\[ r_{i,i+1}^{\text{rough}} = r_{i,i+1}^{\text{sharp}}\, \exp\!\big(-2\,k_{i,z}\,k_{i+1,z}\,\sigma_{i,i+1}^{2}\big), \]

\[ t_{i,i+1}^{\text{rough}} = t_{i,i+1}^{\text{sharp}}\, \exp\!\big((k_{i,z} - k_{i+1,z})^{2}\,\sigma_{i,i+1}^{2}/2\big), \]

where \(k_{i,z} = (\omega/c)\,q_{ij}\) for the relevant mode \(j\) of layer \(i\), and the factor on the transmission coefficient is the symmetric counterpart of the reflection factor in the limit of vanishing index contrast. Two structural features distinguish the Nevot-Croce form from the Debye-Waller form of the next page. First, the reflection exponent contains the product \(k_{i,z}k_{i+1,z}\) rather than \(k_{i,z}^{2}\), which embeds the index contrast across the boundary. Second, the transmission exponent carries the squared difference \((k_{i,z} - k_{i+1,z})^{2}\) with a positive sign in the exponent, which encodes the amplitude enhancement of forward-propagating modes when the roughness scatters energy out of the specular reflection channel.

For absorbing media the wave-vector projections \(k_{i,z}\) are complex, and the exponential factors carry both modulus and phase. The phase contribution modifies the field reconstruction inside the layer through stage 6 of the pipeline and is the mechanism by which the Nevot-Croce factor influences observables beyond the specular reflectance, including ellipsometric phase and standing-wave field profiles. We surmise that any implementation that drops the phase by extracting only the magnitude of the exponential will silently introduce errors in the X-ray standing wave intensity that scale linearly with \(\sigma\).

Application to the eight Passler-Paarmann amplitudes¶

The Esashi treatment is scalar and addresses only the two co-polarized channels \(r_{pp}, r_{ss}\) and \(t_{pp}, t_{ss}\). The 4x4 formalism carries four additional cross-polarization channels \(r_{ps}, r_{sp}, t_{ps}, t_{sp}\) that arise in birefringent layers. The kernel applies the Nevot-Croce factor to each of the eight channels individually using the mode-resolved \(k_{i,z}\) on each side of the interface.

For the co-polarized reflection amplitudes the assignment is unambiguous,

\[ r_{pp}^{\text{rough}} = r_{pp}^{\text{sharp}}\, \exp\!\big(-2\,k_{i,z}^{(p)}\,k_{i+1,z}^{(p)}\,\sigma^{2}\big), \quad r_{ss}^{\text{rough}} = r_{ss}^{\text{sharp}}\, \exp\!\big(-2\,k_{i,z}^{(s)}\,k_{i+1,z}^{(s)}\,\sigma^{2}\big), \]

with \(k_{i,z}^{(p)} = (\omega/c)\,q_{i1}\) and \(k_{i,z}^{(s)} = (\omega/c)\,q_{i2}\) in the Passler sorted-mode labels of Eigenmode analysis.

For the cross-polarization channels the natural choice is the geometric mean of the two polarization-specific projections,

\[ r_{ps}^{\text{rough}} = r_{ps}^{\text{sharp}}\, \exp\!\Big(-2\sqrt{k_{i,z}^{(p)}k_{i,z}^{(s)}}\, \sqrt{k_{i+1,z}^{(p)}k_{i+1,z}^{(s)}}\,\sigma^{2}\Big), \]

and analogously for \(r_{sp}^{\text{rough}}\). The geometric-mean choice reduces to the co-polarized form in the limit of weak birefringence, where \(q_{i1} \approx q_{i2}\), and makes physical sense because \(r_{ps}\) couples a \(p\) incident mode to an \(s\) reflected mode, so its decoherence integral involves both projections in symmetric combination. We hypothesize that this prescription is correct to leading order in the roughness but note that no published derivation extends Nevot-Croce to the birefringent cross-polarization regime, and the regression scaffold should compare it against a graded-interface calculation in a uniaxial substrate before adopting it as the production default. If the comparison fails, the kernel should fall back to the graded approach in the cross-polarized channels and emit a KernelError::UnsupportedConversion diagnostic flagging the discrepancy.

The transmission corrections follow the same logic with the substitution

\[ t_{kl}^{\text{rough}} = t_{kl}^{\text{sharp}}\, \exp\!\Big((k_{i,z}^{(\text{in})} - k_{i+1,z}^{(\text{out})})^{2}\, \sigma^{2}/2\Big), \]

where the in and out projections are determined by the input and output polarization labels of the channel.

Validity regime¶

The Nevot-Croce factor descends from a perturbative expansion in \(\sigma k_z\) and is valid only when the expansion parameter remains small. Esashi [3, text following Eq. (1)] cites \(\sigma k_{i,z} < 1\) as the canonical bound. Two secondary conditions also apply. First, the index contrast across the interface must be small enough that the unperturbed Fresnel amplitude is not itself sensitive to the roughness profile shape, which is the regime in which X-ray and EUV optics naturally sit but is violated for visible-light multilayers with metal-dielectric contrast. Second, the correlation length of the roughness must be short compared to the lateral coherence length of the probe, which selects the short-correlation branch of the Sinha-Sirota-Garoff-Stanley classification [4] and excludes the long-correlation Debye-Waller regime.

For X-ray reflectivity at \(\lambda = 0.154\) nm and \(\theta = 5 \,\text{deg}\) from grazing in vacuum, the bound translates to \(\sigma < 0.28\) nm. For EUV reflectivity at \(\lambda = 13.5\) nm and \(\theta = 30\,\text{deg}\) from grazing, it relaxes to \(\sigma < 4.3\) nm [3, text after Eq. (1)]. For soft X-ray and visible-wavelength reflectometry the bound is correspondingly relaxed, but the index-contrast condition becomes the binding constraint instead. The kernel evaluates both conditions in RoughnessModel::validate and returns KernelError::InvalidGeometry with both numerical bounds in the diagnostic message when either fails.

Where the code lives¶

In refloxide, the Nevot-Croce model is implemented in core::roughness::nevot_croce. The module exposes a single NevotCroce { sigma_nm: f64 } struct that implements RoughnessModel. The correct_interface method reads the mode-resolved \(q_{ij}\) from the upstream stack output, computes the eight exponential corrections, and applies them in place to a four-by-four channel-resolved amplitude matrix that the amplitude solver carries forward. The validate method checks the two regime conditions above and returns the diagnostic error.

The Python wrapper exposes the model through the convenience constructor Roughness.nevot_croce(sigma_nm=...). No additional parameters are required because the correlation length does not enter the closed-form factor. Users who need to specify a correlation length for the validity-region test pass it as Roughness.nevot_croce(sigma_nm=..., correlation_length_nm=...), which the validator uses but the algorithm ignores.

Validation¶

Three regression tests gate the Nevot-Croce implementation.

The first is sharp-interface recovery, \(\sigma \to 0 \Rightarrow r_{kl}^{\text{rough}} \to r_{kl}^{\text{sharp}}\), which holds analytically and is asserted at \(10^{-12}\) absolute tolerance in tests/regression/test_roughness_sharp_limit.py.

The second is co-polarized agreement with the graded model in the small-roughness regime, where the Nevot-Croce co-polarized amplitudes must agree with a fully discretized graded-interface calculation to within \(10^{-6}\) when \(\sigma k_z < 0.3\). This test exercises the regime overlap and catches sign or factor-of-two errors in the exponent. It lives in tests/regression/test_roughness_nevot_vs_graded.py.

The third is cross-polarization sanity in a uniaxial substrate, where the geometric-mean prescription for \(r_{ps}, r_{sp}\) is compared against a graded-interface calculation with the same roughness. The test passes when the two agree to \(10^{-4}\) in the small-roughness regime, which is looser than the co-polarized tolerance because the geometric-mean ansatz is itself approximate. The test lives in tests/regression/test_roughness_cross_pol_uniaxial.py and is the only regression test in the roughness battery that may xfail permanently if the geometric-mean ansatz is found to be inadequate.

Limitations and known failure modes¶

The Nevot-Croce factor fails in three regimes that are catalogued here so that downstream users do not rediscover them by debugging mismatched fits.

The first failure is the large-roughness regime \(\sigma k_z \gtrsim 1\), where the perturbative expansion breaks down and the correction overcorrects the amplitudes. The graded-interface model is the correct fallback.

The second failure is the high-index-contrast regime characteristic of metal-dielectric multilayers in the visible. The unperturbed Fresnel amplitude depends sensitively on the roughness profile shape, and the closed-form Gaussian-derived factor produces results that disagree with both the graded model and with experiment. There is no clean fallback in this regime, and the kernel emits a warning rather than an error because the user may still want a fast estimate.

The third failure is non-Gaussian roughness profiles. The closed-form factor descends specifically from a Gaussian height distribution. For exponential or Lorentzian distributions, a distinct factor is required and is documented in the roughness-distribution literature [5]. The kernel does not currently implement the non-Gaussian variants, and the Roughness.nevot_croce constructor rejects any ProfileShape other than Gaussian at validation time.

References¶

L. Nevot and P. Croce, "Caracterisation des surfaces par reflexion rasante de rayons X. Application a l'etude du polissage de quelques verres silicates," Rev. Phys. Appl. 15, 761 (1980). DOI.
P. Croce and L. Nevot, "Etude des couches minces et des surfaces par reflexion rasante, speculaire ou diffuse, de rayons X," Rev. Phys. Appl. 11, 113 (1976). DOI.
Y. Esashi, M. Tanksalvala, Z. Zhang, N. W. Jenkins, H. C. Kapteyn, and M. M. Murnane, "Influence of surface and interface roughness on X-ray and extreme ultraviolet reflectance: A comparative numerical study," OSA Continuum 4, 1497 (2021). DOI.
S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38, 2297 (1988). DOI.
D. K. G. de Boer, "X-ray reflection and transmission by rough surfaces," Phys. Rev. B 51, 5297 (1995). DOI.