KramersKronigBeziehungen

The Kramers-Kronig relations are a pair of mathematical formulas that relate the real and imaginary parts of a complex response function. In physics and engineering, these relations are fundamental for understanding how materials respond to external stimuli, particularly in the context of electromagnetic fields, acoustics, and mechanical vibrations. Essentially, they state that if you know the full spectrum of either the real part or the imaginary part of a system's response, you can calculate the other part.

These relations are derived from causality, the principle that an effect cannot precede its cause. For a system to be causal, its response to a stimulus at a given time can only depend on the stimulus at present or in the past, not the future. This causality imposes constraints on the analytic properties of the complex response function in the complex frequency plane, leading directly to the Kramers-Kronig relations.

The real part of the response function typically describes the dispersive properties of a medium (how the speed of a wave changes with frequency), while the imaginary part describes the absorptive or dissipative properties (how much energy is lost or absorbed). Therefore, the Kramers-Kronig relations imply a deep connection between dispersion and absorption. For example, if a material strongly absorbs light at a particular frequency, it will also exhibit significant dispersion (a rapid change in refractive index) around that same frequency. These relations are widely used in spectroscopy, optical engineering, and material science for data analysis and prediction.