Kramers–Kronig Relations: Can Materials Violate Them?




Do Any Materials Violate the Kramers–Kronig (KK) Relations?

Short answer: No passive, linear, time-invariant material violates the KK relations.
They follow from causality (response cannot precede the stimulus), together with linearity and time invariance.
If any of those assumptions is broken, the textbook KK formulas need modification and may not apply in their simple form.

1) Why KK holds

  • The KK relations link the real and imaginary parts of a linear response function (e.g., refractive index \(n(\omega)\),
    dielectric function \(\epsilon(\omega)\), magnetic permeability \(\mu(\omega)\)).
  • They are derived from the Cauchy integral formula applied to a complex susceptibility \(\chi(\omega)\) that is
    (i) analytic in the upper half of the complex \(\omega\)-plane and
    (ii) vanishes sufficiently fast as \(\lvert \omega \rvert \to \infty\).
  • Those properties follow from causality + stability for any linear, time-invariant, passive medium.

2) Situations where the standard form may appear to fail

Scenario What happens Does KK really fail?
Active / gain media Laser amplifiers or negative-resistance systems can have poles in the upper half-plane; dispersion–absorption pairing needs extra pole (residue) terms. No fundamental violation; use generalized KK including pole contributions.
Nonlinear response KK assumes linear response to the probe. Strong-field or intensity-dependent effects do not obey the linear KK pair. KK applies to the linear susceptibility (small-signal limit) only.
Time-variant / modulated systems Material parameters evolve in time; the system is not time-invariant, so stationary KK relations are inapplicable directly. Not a violation—assumptions changed.
Spatial dispersion Response depends on both frequency and wavevector, e.g., \(\epsilon(\omega,k)\); the simple 1D KK in \(\omega\) needs generalization. Generalized relations exist; the textbook form is insufficient.
Experimental “violations” Limited-band / noisy data can appear inconsistent because KK integrals require information over all frequencies. Artefacts, not real violations.

3) Reported “KK-violating” claims

Occasional claims involve, for example, negative-index metamaterials near resonances, “superluminal” pulse propagation,
or gain-assisted transparency windows. Closer analysis typically shows that the medium has gain (pumped/active),
is nonlinear, or the dataset is incomplete. When these are accounted for, the appropriate generalized KK relations hold.

4) Bottom line

As long as a system is linear, causal, and stable, its response functions obey Kramers–Kronig.

  • Apparent violations usually indicate gain/active elements, nonlinearity, time variance, spatial dispersion, or incomplete data.
  • No reproducible, peer-reviewed experiment has shown a passive, linear medium whose susceptibility truly violates KK.

References (classics & reviews)

  • L. D. Landau & E. M. Lifshitz, Electrodynamics of Continuous Media, §85.
  • R. W. Boyd, Nonlinear Optics, ch. 2 (KK in the context of nonlinear susceptibilities).
  • N. Nussenzveig, Causality and Dispersion Relations.
  • V. Lucarini, J. J. Saarinen, K.-E. Peiponen, E. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).