"An effect does not precede its cause" — that alone binds the real and imaginary parts of the index by an integral
When you turned up the absorption slider on the phasor diagram in Episode 1, the tip of the arrow curled inward. That is this whole episode. The refractive index is really a complex number: the real part says how much the phase is delayed, the imaginary part says how much is absorbed. So far that is just notation. The substance begins here — the real and imaginary parts cannot be chosen independently. Impose one single requirement on physics — "an effect does not precede its cause" — and the two are bound together by an integral. These are the Kramers–Kronig relations. And the consequence is merciless. A material that is perfectly transparent at every frequency and still bends light cannot exist, in principle. Glass is transparent in the visible and has \(n=1.5\) — because glass absorbs ultraviolet.
Write \(\tilde n=n+i\kappa\) and put it in a plane wave:
$$E=E_0\,e^{i(k_0\tilde n z-\omega t)} =E_0\underbrace{e^{ik_0nz}}_{\text{phase delay}}\ \underbrace{e^{-k_0\kappa z}}_{\text{decay}}\ e^{-i\omega t}$$Intensity is amplitude squared, so the absorption coefficient is
$$\alpha=2k_0\kappa=\frac{4\pi\kappa}{\lambda_0}$$In Episode 1's phasor, whether the tip stayed on the circle or spiralled inward was exactly \(\kappa=0\) versus \(\kappa>0\). Refraction and absorption were two components of one complex number from the start.
Write down the material response. The polarization at time \(t\) depends on every field that has acted so far:
The boxed line is causality — a field that has not been applied yet does not affect the present polarization. Nothing more.
But "vanishes for \(\tau<0\)" acquires enormous force under Fourier transform. If \(\chi(\tau)\) is zero for \(\tau<0\), then its transform \(\chi(\omega)\) is analytic in the upper half of the complex \(\omega\) plane — no poles, no cuts. The \(\mathrm{Im}\,\omega>0\) part of \(e^{i\omega\tau}\) suppresses only the \(\tau>0\) side, and that is what makes the integral converge.
And an analytic function is subject to Cauchy's integral theorem. The rest is automatic.
(P is the principal value; Kronig and Kramers, independently, 1926–27.)
Know either one at all frequencies and the other is a calculation. Real and imaginary parts were never independent information.
Put \(\kappa(\omega')\equiv0\) — absorbing nothing at any frequency — into the first relation.
$$\kappa\equiv0\ \Longrightarrow\ n(\omega)-1=0\ \text{for all }\omega\ \Longrightarrow\ n\equiv1$$A material that absorbs nothing bends nothing. It is vacuum.
Turned around: every material that bends light is absorbing light at some frequency. A "perfectly transparent lens" cannot exist, in principle.
Now look at a real material. Fused silica is transparent in the visible, with enormous absorption on either side.
| Band | What happens | Glass |
|---|---|---|
| Ultraviolet (\(\gtrsim4\) eV) | electrons are excited (band gap) | absorbs strongly |
| Visible (1.6–3.2 eV) | too little for electrons, too fast for the lattice | transparent (the window) |
| Infrared (beyond \(\sim\)4 μm) | Si–O bond vibrations (phonons) | absorbs strongly |
The visible transparency is a window between two absorptions. And what Kramers–Kronig says is that the value \(n=1.5\) inside the window is created by the absorptions on either side, with the ultraviolet side dominant.
It is not that "glass is transparent, so there is nothing there." Glass's \(n=1.5\) in the visible is the long reach of an absorption sitting out in the invisible ultraviolet.
Press "remove the ultraviolet absorption" on the right of the figure below — the refractive index inside the window drops straight to 1.
On the left is a Lorentz oscillator, the simplest model of a bound electron, with \(n\) and \(\kappa\) drawn. The green dots are \(n\) reconstructed by numerically evaluating the Kramers–Kronig integral using \(\kappa\) alone — they should land right on the curve. One part really does give the other.
On the right is a model of glass, with one oscillator in the infrared and one in the ultraviolet. The horizontal axis is log frequency, and the pale band in the middle is the visible window.
| Field | The same structure |
|---|---|
| Optics (this episode) | causality → analyticity of \(\tilde n(\omega)\) → refraction and absorption linked |
| Statistical mechanics (Temperature That Clicks) | the fluctuation–dissipation theorem: the imaginary part of the response is dissipation, the real part is reaction. Same causality |
| Particle physics | dispersion relations and the optical theorem: the imaginary part of the forward amplitude is the total cross-section. S-matrix analyticity reflects causality |
| Circuits | the real part (resistance) and imaginary part (reactance) of a passive impedance \(Z(\omega)\) obey the same relation |
The imaginary part is "losing," the real part is "shifting," and causality is the glue between them.
This structure is a cousin of a shape that recurs throughout the sister series — in Tunneling That Clicks, making time imaginary turns a barrier into a valley; here, making frequency complex links absorption to refraction. Both are cases of going out into the complex plane and finding that two apparently separate things are one.
Established: the complex index \(\tilde n=n+i\kappa\) and \(\alpha=4\pi\kappa/\lambda_0\); that a linear, time-translation-invariant, causal response function is analytic in the upper half plane (Titchmarsh's theorem) and the Kramers–Kronig relations that follow (1926–27); \(\kappa\equiv0\Rightarrow n\equiv1\); the f-sum rule \(\int_0^\infty\omega\,\mathrm{Im}\,\varepsilon_r\,d\omega=(\pi/2)\omega_p^2\); the Lorentz oscillator model and normal/anomalous dispersion; the ultraviolet electronic absorption edge and infrared phonon absorption of fused silica with the visible window between them. All standard physics.
Caveats: (1) Kramers–Kronig assumes a linear, time-translation-invariant, causal, passive response. It does not apply in this form to nonlinear optics at high intensity or to time-varying media. (2) The derivation assumes \(\chi\to0\) as \(\omega\to\infty\). In a conductor \(\varepsilon\) diverges as \(\omega\to0\), so one must work with \(\sigma\) instead of \(\varepsilon\), or otherwise handle the pole. (3) "Transparent means it does not refract" is an idealised statement. What actually follows is "if there is no absorption at any frequency then \(n\equiv1\)"; a band in which absorption is below the measurement floor can perfectly well have \(n\neq1\) — and the point of the text is that a different band's absorption is producing that \(n\). (4) The "glass" on the right is a two-oscillator model, not a reproduction of silica's optical constants. The resonance positions and strengths were chosen to give \(n\approx1.5\) in the visible, for teaching. (5) The KK reconstruction on the left integrates numerically over a finite range (\(\omega'\le40\,\omega_0\)), so there is a small truncation error in principle. (6) The \(N\) in the sum rule is the total electron density including core electrons, and is not a quantity you can verify by looking at the visible region alone.
The refractive index is complex, \(\tilde n=n+i\kappa\): real part phase delay (refraction), imaginary part decay (absorption, \(\alpha=4\pi\kappa/\lambda_0\)). And the two cannot be chosen independently. From one line — "an effect does not precede its cause," \(\chi(\tau)=0\) for \(\tau<0\) — \(\tilde n(\omega)\) becomes analytic in the upper half plane, and Cauchy's theorem forces the Kramers–Kronig relations.
The consequences are merciless. Put \(\kappa\equiv0\) in and you get \(n\equiv1\): a material that absorbs nothing bends nothing. And the f-sum rule \(\int_0^\infty\omega\,\mathrm{Im}\,\varepsilon_r\,d\omega=(\pi/2)\omega_p^2\) says that the total absorption is fixed by the number of electrons. You can move it in frequency; you cannot reduce it.
So — glass has \(n=1.5\) in the visible because glass absorbs ultraviolet. The visible transparency is a window between the ultraviolet and infrared absorptions, and the index inside the window is the shadow of an absorption you cannot see. Remove the ultraviolet oscillator in the figure and \(n\) drops straight to 1. The same structure appears in the fluctuation–dissipation theorem, in the optical theorem of particle physics, and in passive circuit impedance — the imaginary part loses, the real part shifts, and causality is the glue.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, move γ on the left and watch the green dots (n reconstructed from κ) stay on the blue curve. Press "remove the UV absorption" on the right and the refractive index in the visible window falls to 1. "Show answer" reveals the solutions.