Kramers-Kronig relations

The Kramers-Kronig relations are mathematical formulas that connect how a material responds to a wave of energy, like light or sound, across different frequencies. They show that information about how a material absorbs energy at one frequency determines how it disperses or shifts that energy at other frequencies. This means if you know how a material attenuates a wave, you can predict how it will change the wave’s speed or shape, and vice versa. These relations ensure the responses are consistent with fundamental physical principles like causality.