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Linear response theory approximates a small change in an observable by a linear function of an applied perturbation around a specified reference state. A weak force can produce a proportional displacement, and a weak applied magnetic field can produce proportional magnetization. The response coefficient depends on the material and state; linearity is an approximation to test by changing the drive size.

For a causal time-dependent response, writeThe kernel $\chi$ describes the delayed effect of the perturbation $f$. A sinusoidal input probes the frequency-dependent response: its in-phase part and out-of-phase part determine the response magnitude and phase lag. Causality links the real and imaginary parts through Kramers-Kronig relations under the appropriate convergence assumptions. Comparing the response at several small drive sizes tests linearity; a linear-looking response at one size alone cannot establish the range of validity.

For independent electronic transitions, add the electric dipole moment responses:The positive constants $S_j$ include electron density and transition atomic oscillator strength. Each term adds an absorption peak to ${ϵ}_r^{\prime \prime }$ and a dispersive positive-to-negative contribution to ${ϵ}_r^{\prime }-1$, as in the middle row of the figure.

Between well-separated absorption lines, where the medium is transparent, the refractive index generally increases with frequency: $d{n}_r/d\omega >0$, or $d{n}_r/d\lambda <0$. This is normal dispersion. Near a resonance the real response turns down with increasing frequency, producing an interval of anomalous dispersion, $d{n}_r/d\omega <0$, accompanied by appreciable absorption. The absorptive and dispersive parts are linked by the Kramers-Kronig relations; the steep variation is not independent of the loss spectrum.