# K-K from Reflectance — The log-r Trick

*How to recover the missing phase of a complex reflectance from intensity-only measurements, by applying Kramers–Kronig to log r(ω) — the standard workhorse for extracting optical constants of opaque materials.*

## The measurement problem

At normal incidence, the Fresnel reflection coefficient of a sample
(in vacuum) is

```
r(ω) = (1 − ñ(ω)) / (1 + ñ(ω)),   ñ = n + iκ
```

a complex quantity with magnitude and phase

```
r(ω) = √R(ω) · exp[i θ(ω)]
```

where R(ω) = |r(ω)|² is the intensity reflectance actually delivered
by a spectrometer. A photon detector integrates intensity; it responds
to R(ω), but not to the phase θ(ω). Yet, to invert Fresnel and recover
ñ(ω) = n(ω) + iκ(ω) at every frequency, both R(ω) and θ(ω) are
needed. Interferometric techniques that measure θ directly exist, but
over a broadband spectrum they are awkward; for opaque samples in the
visible, UV and IR, only R is routinely measured well.

The **log-r trick**, worked out in the 1950s (notably by Philipp and
Taft 1959, Roessler 1965, F. Wooten's 1972 monograph), uses Kramers–
Kronig to reconstruct θ(ω) from R(ω) alone.

## r(ω) is causal ⇒ log r(ω) obeys K–K

The reflected electric-field envelope at the surface responds causally
to the incident envelope. In a linear approximation, the reflection
operator is an LTI system with impulse response r(t), vanishing for
t < 0. Its Fourier transform r(ω) is therefore the boundary value of a
function analytic in the upper half of the complex ω-plane, by the
same Titchmarsh theorem used throughout this series.

For the logarithm to also be analytic in the UHP we need r(ω) to have
no **zeros** in the UHP. A zero of r(ω) at some ω_z would correspond
to a Brewster-like phenomenon in the complex plane and would introduce
a branch point in log r. For most real materials at normal incidence
this condition is satisfied (bulk opaque samples, no total transmission
resonances in the UHP); when it fails, Blaschke-product corrections are
needed (Chapter 18).

Assuming no UHP zeros, write

```
log r(ω) = (1/2) ln R(ω) + i θ(ω)
```

and apply the K–K pair to the real and imaginary parts on the real axis.
After exploiting the crossing symmetry r(−ω) = r*(ω) (real impulse
response), the result is the celebrated single-integral relation

```
θ(ω₀) = −(ω₀/π) P ∫₀^∞  ln R(ω) / (ω² − ω₀²)   dω
```

Factor of ω₀ inside the integral can be rewritten using the identity

```
1/(ω² − ω₀²) = (1/2ω₀) [1/(ω − ω₀) − 1/(ω + ω₀)]
```

to show explicitly that only the *variation* of ln R with ω matters;
a constant ln R contributes nothing to θ by the principal-value
prescription. The phase at ω₀ is therefore determined by the
log-magnitude of the reflectance at every other frequency, weighted by
a kernel peaked at ω = ω₀.

## From θ(ω) to n(ω) and κ(ω)

Once θ(ω) is known, invert Fresnel:

```
ñ(ω) = (1 − r(ω)) / (1 + r(ω))
     = (1 − √R e^{iθ}) / (1 + √R e^{iθ})
```

Rationalising gives explicit closed-form expressions for n and κ in
terms of R and θ. The standard textbook formulae (see Wooten) are

```
n(ω) = (1 − R) / (1 + R − 2√R cos θ)
κ(ω) = −2√R sin θ / (1 + R − 2√R cos θ)
```

with signs convention-dependent. The conductivity, dielectric function,
absorption coefficient, and loss function all follow. For a metal, the
low-frequency tail of κ determines the DC conductivity; for a
semiconductor, sharp structures in κ map optical interband transitions.
Entire volumes of optical-constants data for solids (the Palik
handbooks) rest on this workflow.

## Practical caveats

The theorem is exact, but the integral is evaluated numerically on a
finite range of measured R values. Three practical problems dominate.

- **Finite integration range.** The integral runs to ∞, but data
  exist only over a measured window [ω_min, ω_max]. Both tails must be
  extrapolated. Low-frequency extrapolation uses a Hagen–Rubens form
  for metals (R(ω) ≈ 1 − A√ω) or a fit to known phonon structure for
  insulators; high-frequency extrapolation uses a power-law
  R(ω) ∝ ω^{−p} motivated by the f-sum rule (Chapter 16). Tail choices
  shift θ(ω) near the band edges, and tail sensitivity is the main
  source of systematic error in K–K-derived optical constants.
- **Zeros of r in the UHP.** If present, they add a Blaschke factor
  ∏ (ω − ω_k*)/(ω − ω_k) to r that has |B(ω)| = 1 on the real axis
  and therefore contributes nothing to R but *does* contribute to θ.
  The K–K integral alone cannot detect them. Anchoring the reconstructed
  θ to an independent measurement (e.g. an ellipsometric data point at
  one frequency) is standard practice to pin this ambiguity.
- **Oblique incidence, anisotropy, thin films.** The scalar Fresnel
  formula above assumes normal incidence on a bulk isotropic sample.
  Oblique incidence splits into s and p polarisations with different
  r(ω); thin films add interference fringes; anisotropic materials need
  a tensor treatment. Each generalisation has its own K–K flavour, but
  the principle — phase from log-magnitude under UHP analyticity — is
  the same.

## Where it's used

K–K reflectance analysis is standard in:

- **Condensed-matter optical spectroscopy.** Opaque metals,
  superconductors, oxides, correlated electron systems. Basov and
  Timusk's reviews of infrared spectroscopy on high-T_c
  superconductors rely pervasively on K–K-inverted reflectance.
- **Semiconductor characterisation.** Fundamental absorption edge,
  excitonic features, interband transitions — mapped by reflecting off
  a polished wafer and inverting.
- **IR reflectance of minerals and polymers.** Phonon spectra and
  absorption bands extracted without need for transmission cells.
- **Hard-X-ray reflectivity.** Near-edge structure (XANES) and
  anomalous-scattering factors are analysed with a K–K pair between
  the real and imaginary parts of the atomic form factor; the log-r
  trick is its reflectance counterpart.

The method is powerful because it requires only a single-channel,
intensity-measuring spectrometer — a photodiode behind a
monochromator — to extract the full complex dielectric response of a
material that may be completely opaque to transmission. It is a
textbook payoff of causality: with one integral you trade away a
phase-sensitive experiment for a single-detector one.

## See also

- [20_bode_gain_phase.md](20_bode_gain_phase.md)
- [22_optical_constants.md](22_optical_constants.md)
- [24_fdtd_causal_materials.md](24_fdtd_causal_materials.md)

## References

- F. Wooten, *Optical Properties of Solids*, Academic Press, 1972 —
  Chapter 6.
- M. Dressel and G. Grüner, *Electrodynamics of Solids*, Cambridge,
  2002.
- E. D. Palik (ed.), *Handbook of Optical Constants of Solids*,
  Academic Press, 1985–1998 (three volumes).
- H. R. Philipp and E. A. Taft, "Kramers–Kronig analysis of reflectance
  data for diamond", *Phys. Rev.* 136, A1445 (1964).
- [Wikipedia: Kramers–Kronig relations §Applications](https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations)