Dispersion and causal response

In a linear, time-translation-invariant, spatially local medium, polarization is a convolution of the field with a response kernel. Causality requires the kernel to vanish for negative delay: an output cannot depend on a future input. Linearity is an approximation valid for sufficiently small fields, and spatial locality can fail when microscopic structure matters. These assumptions are separate from causality and should be stated rather than hidden inside one symbol.

Choose fields proportional to $e^{-i\omega t}$. Fourier transformation turns the convolution into multiplication by complex $\chi(\omega)$. Its real part contributes to in-phase polarization, while its imaginary part describes the quadrature response and dissipation. For an ordinary passive medium at positive frequency, this convention gives nonnegative imaginary susceptibility. Reversing the time-exponential convention reverses the corresponding imaginary sign without changing the physical absorption.

Model a bound charge as a driven oscillator with positive damping and a restoring frequency. Its displacement produces a dipole moment, and a density of such oscillators produces polarization. The steady-state response has a resonance rather than an instantaneous constant permittivity. Damping keeps the real-frequency response finite and transfers energy irreversibly to degrees of freedom omitted from the oscillator model.

Near resonance the imaginary part can be substantial even when the real susceptibility crosses through zero. A vanishing real part therefore does not mean the material is optically indistinguishable from vacuum. A material can contain multiple resonances, free-carrier terms, anisotropy, or nonlinearities; the one-resonance model is a useful interpretable fit only over a range where those omissions are acceptable.

A causal kernel makes susceptibility analytic in the appropriate complex-frequency half-plane. With sufficient high-frequency decay, its real and imaginary parts satisfy Kramers–Kronig relations. Changing absorption across a frequency band therefore constrains dispersion elsewhere; the two curves cannot be specified independently while retaining this causal linear model. Applying the relation to finite measurement data needs extrapolations, and those assumptions can dominate an inferred refractive index.

For a weakly absorbing narrow-band pulse, group velocity follows the derivative of its dispersion relation. Strong absorption or rapid dispersion can reshape a pulse so severely that the location of its maximum is a poor marker for transmitted information. A superluminal or negative formal group velocity in such a region does not establish faster-than-light signaling. The earliest causal front depends on the full broadband response, not one derivative at a carrier frequency.