Closed Form KramersKronig Relations With Shape Preserving Piecewise Cubic Interpolation

The derivation of the Kramers-Kronig relations is revisited based on system theory considerations to show their general validity and to emphasize how these relations are connected to the causality of a linear system. Then, the closed form Kramers-Kronig and subtractive Kramers-Kronig relations are introduced based on shape-preserving third-order piecewise cubic interpolation. The advantage of this formulation is that a small number of properly chosen data points are sufficient to produce accurate results over a broad frequency range. The developed formulas are applied to validate the transfer function of a bandpass filter, to determine the phase of the transmission coefficient of a metasurface from the measured amplitude, and to uniquely extract the refractive index of metamaterials.

Automated summary

By revisiting the derivation of the Kramers-Kronig relations based on system theory considerations, their general validity and connection to the causality of a linear system are demonstrated. The closed form Kramers-Kronig and subtractive Kramers-Kronig relations are introduced using shape-preserving third-order piecewise cubic interpolation, allowing accurate results to be obtained over a broad frequency range with only a small number of properly chosen data points. These developed formulas are applied to validate transfer functions, determine phase shifts, and extract refractive indices in various applications.