# Optical Constants — Refractive Index and Extinction *The complex refractive index ñ(ω) = n(ω) + iκ(ω) is the K–K pair that optics students meet first: anomalous dispersion near an absorption line is a direct, quantitative consequence of causality, and Sommerfeld–Brillouin showed that signal fronts still never outrun c.* ## Complex refractive index A plane wave in an isotropic absorbing medium is written ``` E(z, t) = E₀ exp[i(k z − ωt)], k = ñ(ω) ω / c ``` with the complex refractive index ñ(ω) = n(ω) + iκ(ω). The real part n(ω) is the ordinary refractive index (phase velocity v_p = c/n), and κ(ω) is the **extinction coefficient**: the amplitude decays as exp(−κ ω z /c), so the intensity absorption coefficient is α(ω) = 2κω/c. ñ(ω) is related to the relative permittivity of Chapter 21 by ε(ω) = ñ(ω)², i.e. ``` ε′ = n² − κ², ε″ = 2 n κ ``` for non-magnetic materials (μ_r ≈ 1 across optical frequencies). Since ε obeys K–K and analyticity in the UHP is preserved by taking a principal square root (with the branch cut chosen so that ñ → 1 at infinity), ñ(ω) − 1 is itself a causal response function satisfying dispersion relations. ## Subtracted K–K for n and κ At high frequency the refractive index asymptotes to 1 (the electrons cannot follow the field). This forbids the "unsubtracted" K–K of Chapter 1 from converging directly on ñ; instead one works with ñ − 1, which *does* decay. Exploiting the crossing symmetry ñ(−ω) = ñ*(ω) — n even, κ odd in ω — and folding the integrals onto the positive axis gives the **once-subtracted** optical K–K relations: ``` n(ω) − 1 = (2/π) P ∫₀^∞ ω′ κ(ω′) / (ω′² − ω²) dω′ κ(ω) = −(2ω/π) P ∫₀^∞ [n(ω′) − 1] / (ω′² − ω²) dω′ ``` (The second relation is the physically useful one only if you have reliable refractive-index data across the entire spectrum, which is rare; in practice one runs the first direction — extract n from a measured absorption spectrum — much more often.) Because of the ω′/(ω′² − ω²) weighting, the main contribution to n(ω) − 1 comes from absorption features at comparable frequencies. Strong UV absorption therefore dominates the visible refractive index of most transparent solids; transparent glasses have n > 1 in the visible precisely because they absorb strongly in the ultraviolet. ## Worked check — Lorentzian oscillator as a K–K pair Take a single oscillator contribution to χ(ω) = ñ²(ω) − 1: ``` χ(ω) = ω_p² / (ω₀² − ω² − iγω) ``` with plasma frequency ω_p, resonance ω₀, damping γ. Split into real and imaginary parts: ``` χ′(ω) = ω_p² (ω₀² − ω²) / [ (ω₀² − ω²)² + γ²ω² ] χ″(ω) = ω_p² γ ω / [ (ω₀² − ω²)² + γ²ω² ] ``` χ″(ω) is a Lorentzian peak centred at ω₀ with FWHM γ (at small damping). Substitute χ″ into the one-sided K–K relation of Chapter 1: ``` χ′(ω) = (2/π) P ∫₀^∞ ω′ · [ω_p² γ ω′ / ((ω₀²−ω′²)² + γ²ω′²)] / (ω′² − ω²) dω′. ``` The integrand has poles where `(ω₀² − ω′²)² + γ²ω′² = 0`, i.e. at ω′ = ±ω₀ ± iγ/2 + O(γ²). Two of these four roots sit in the upper half- plane of ω′. Close the contour in the UHP and apply the residue theorem: summing the two enclosed residues, plus the usual `−iπ·residue` for the simple pole on the real axis at ω′ = ω, and collecting the pieces yields *exactly* the expression for χ′(ω) above. The Lorentzian susceptibility is therefore an explicit, closed-form instance of a K–K pair: you can verify the integrals line by line, and nothing is adjustable — the dispersive S-curve in n(ω) is *completely determined* by the absorption peak in κ(ω) and the dispersion relation, no extra information required. This is the canonical worked example that every K–K tutorial in condensed-matter optics carries at some point. Extending to a weighted sum of oscillators (the Lorentz–Drude model) is immediate: each term contributes its own closed-form pair and the total still satisfies K–K by linearity. ## Anomalous dispersion Around an isolated absorption line at ω₀, κ(ω) is a peak of width γ and n(ω) − 1 shows a characteristic **S-shape**: it rises toward ω₀ from below, swings through a local maximum just below ω₀, drops through a local minimum just above ω₀, and returns to its background value above the line. In the narrow range between those extrema, n(ω) **decreases** with increasing ω — the sign of dn/dω is opposite to the "normal" trend of n rising toward blue — and this is the original 1870s meaning of "anomalous dispersion" as seen by Christiansen and Kundt in fuchsin dye. This behaviour is what the Lorentz-oscillator susceptibility of Chapter 1 predicts, and it is exactly what the K–K integral produces from a Lorentzian κ(ω). The qualitative message: you cannot have a sharp absorption band without a matching S-curve in n nearby. Choosing a narrow-band absorber to remove a wavelength automatically produces a region of anomalous dispersion that distorts pulses passing through it. Prisms *disperse normally* in the visible (blue refracted more than red) because the resonances driving the K–K tail are in the UV and we sit on the low-frequency side of every one, where n rises with ω. A prism built of a material with a visible-range absorption line would show anomalous dispersion across that line, reversing the colour ordering across it. Wood's 1904 observations of sodium vapour in the yellow D lines are the classical demonstration. ## Group velocity and the speed-of-light puzzle In the anomalous-dispersion region the group velocity v_g = dω/dk can exceed c — and even become negative — without violating relativity. This worried physicists immediately after special relativity in 1905: if the group velocity is the signal velocity, how can it be superluminal? The resolution, worked out by Arnold Sommerfeld (1907) and Léon Brillouin (1914) and documented in Brillouin's *Wave Propagation and Group Velocity* (1960), is that "group velocity" is not in general the signal velocity. They analysed a sharp-fronted pulse propagating through a Lorentz medium by evaluating the inverse Fourier integral by steepest descent. Three regimes appear. - **Sommerfeld forerunner**: the very high-frequency Fourier components of the pulse front propagate essentially unattenuated at phase velocity c/n(∞) = c. A small, high-frequency, low-amplitude precursor arrives first, at exactly t = z/c. - **Brillouin forerunner**: a low-frequency, oscillatory precursor follows, propagating at roughly c/n(0). - **Main signal**: the bulk of the pulse arrives later, travelling near the group velocity where that is well-defined. The **front** of any signal — the first moment at which the field is non-zero — travels at exactly c, independently of n(ω) and κ(ω). Causality, formulated as analyticity of ñ(ω) in the UHP, is exactly what guarantees this: the same Titchmarsh theorem that gives K–K also gives the Paley–Wiener result that a causal field profile cannot be band-limited, so its Fourier reconstruction always contains the c-propagating high-frequency tail. Superluminal group velocity is real and measurable, but it is a property of the peak of a pre-existing pulse, not of any genuine signal information, which is always locked to the front. ## Practical use in optical physics - **Ellipsometry** measures the complex reflectance ratio ρ(ω) of a sample; inverting Fresnel relations yields ñ(ω) directly without needing K–K. But the K–K-consistency of the extracted ñ(ω) is a standard cross-check. - **Absorption spectroscopy + K–K** is the historical workflow for opaque or strongly scattering samples for which transmission measurements fail. See Chapter 23 on the closely related reflectance method. - **Ultrafast optics**: group-delay dispersion (GDD) in pulse compressors is governed by d²n/dω², itself determined by the material's absorption spectrum via K–K. Chirped mirrors are designed by engineering a complex reflectance whose phase (via K–K) gives the desired GDD. ## See also - [01_introduction.md](01_introduction.md) - [21_dielectric_spectroscopy.md](21_dielectric_spectroscopy.md) - [23_reflectance_spectroscopy.md](23_reflectance_spectroscopy.md) ## References - M. Born and E. Wolf, *Principles of Optics*, 7th ed., Cambridge, 1999. - L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, *Electrodynamics of Continuous Media*, 2nd ed., Pergamon, 1984 — §82, §83. - L. Brillouin, *Wave Propagation and Group Velocity*, Academic Press, 1960. - J. D. Jackson, *Classical Electrodynamics*, 3rd ed., Wiley, 1999 — §7.10, §7.11. - [Wikipedia: Refractive index](https://en.wikipedia.org/wiki/Refractive_index)