inverse problems · configuration space what the projection keeps · underdetermined experiments, collapsed configuration spaces, and the tools brought from outside · started October 2026

Dielectric functions from infrared spectra (FOCUS, difes)

Dielectric functions from infrared reflectance and emittance

The group's own case, and Iñigo's field: the infrared emissivity of materials at high temperature, interpreted through a dielectric-function model. The machinery is FOCUS (CEMHTI) and its reconstruction difes; the facts below come from the difes documentation and from practice, and this is the page most in need of Iñigo's correction.

The projection

The configuration is a parametrised dielectric function: \( \epsilon(\omega) = \epsilon_\infty\prod(\text{multiplicative terms}) + \sum(\text{additive terms}) \), in difes' convention — four-parameter semi-quantum modes as factors on \( \epsilon_\infty \); Drude, Lorentz and causal Gaussian terms as sums, each Lorentz term \( \omega_p^2/(\omega_0^2 - \omega^2 - i\gamma\omega) \). Three or four parameters per oscillator, and the number of oscillators is itself a choice. The forward map is exact and short: \( \tilde n = \sqrt{\epsilon} \), \( \tilde r = (\tilde n - 1)/(\tilde n + 1) = r e^{i\theta} \), the reflectance \( R = |\tilde r|^2 \), the opaque emittance \( E = 1 - R \); for a thin sample the transmittance with the thickness. The data are \( R \) or \( E \) against wavenumber on a finite range, often a series of them in temperature.

Where it folds

  • Exactly. The reflectance is the modulus of \( \tilde r \); the phase \( \theta \) is not measured. Two dielectric functions with the same \( |\tilde r| \) on the range are indistinguishable by that measurement. And a smooth \( \epsilon(\omega) \) can be written as a sum of overlapping Lorentzians in many ways: the decomposition of a continuous function into a finite set of oscillators is not unique, so the oscillator parameters are not identifiable even when \( \epsilon(\omega) \) itself is. Everything outside the measured range is in the null space, and the Kramers–Kronig integrals reach into it.
  • At the noise. The width of one mode against the strength of its neighbour when they overlap; \( \epsilon_\infty \) against the tail at the high-frequency end; a Drude term against the lowest wavenumbers measured; the four parameters of a semi-quantum mode when its longitudinal and transverse frequencies are close.
  • Off the manifold. Lorentzian line shapes for modes that are anharmonic at high temperature; a Kramers–Kronig transform on a finite range without its correction; a bulk model on a sample with an oxide layer or a rough surface; an emittance model that assumes opacity on a sample that is not.

The external tools

The exact constraint that plays the part symmetry plays in crystallography is causality. The real and imaginary parts of \( \epsilon \), and the modulus and phase of \( \tilde r \), are Kramers–Kronig pairs: the phase is fixed by the modulus over the whole axis, which recovers what the measurement did not record — at the price of the range, which is finite, hence the finite-range corrections difes carries and the anchoring of the transform (Kuzmenko 2005, treats the dielectric function as a free causal function anchored by a few oscillators, which is the cleanest statement that the data determine \( \epsilon(\omega) \) on the range and not a unique set of oscillators). The sum rules are further exact constraints. The second external tool is another projection: transmission on a thin sample, ellipsometry, which measures the phase. The third is a prior that is the analogue of sparsity in mode space: continuity along a series. The parameters of a mode measured at twenty temperatures must move smoothly; a set of fits that satisfies each spectrum and jumps between neighbours has found a valley in each and not a material in any.

What is already scripted

difes carries the expert's steps as recipes: bench verbs in a text file that run the same fit over every spectrum of a series, with the model, the range, the kind of spectrum, the anchoring. That is T10 before the fact, written for one purpose — reproducibility of the group's fits — and not yet for the other: keeping, for every spectrum, the tube of the fit, the singular spectrum of its Jacobian and the alternative minima, so that the identifiability of each oscillator parameter can be learned against the range measured and the number of oscillators chosen.

What it gives the programme

P5: the difes and FOCUS libraries of dielectric functions are a corpus of configurations; spectra simulated from them on realistic ranges with realistic noise are the training set; the question to learn is which oscillator parameters a given range and a given noise determine, and how the answer changes when a mode is added or removed. The result would be, for the first time in the group's practice, a statement of what an emittance spectrum on 400–4000 \(\mathrm{cm}^{-1}\) can and cannot say about a material — before the fit, and independent of the fitter's hand.