Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane

We present multiply subtractive Kramers-Kronig (MSKK) relations for the moments of arbitrary order harmonic generation susceptibility. Using experimental data on third-harmonic wave from polysilane, we show that singly subtractive Kramers-Kronig (SSKK) relations provide better accuracy of data inversion than the conventional Kramers-Kronig (K-K) relations. The fundamental reason is that SSKK and MSKK relations have strictly faster asymptotic decreasing integrands than the conventional K-K relations. Therefore SSKK and MSKK relations can provide a reliable optical data inversion procedure based on the use of measured data only.Comment: 14 pages, 2 figure

Kramers-Kronig, Bode, and the meaning of zero

The implications of causality, as captured by the Kramers-Kronig relations between the real and imaginary parts of a linear response function, are familiar parts of the physics curriculum. In 1937, Bode derived a similar relation between the magnitude (response gain) and phase. Although the Kramers-Kronig relations are an equality, Bode's relation is effectively an inequality. This perhaps-surprising difference is explained using elementary examples and ultimately traces back to delays in the flow of information within the system formed by the physical object and measurement apparatus.Comment: 8 pages; American Journal of Physics, to appea

Modified Hilbert transform pair and Kramers-Kronig relations for complex permittivities

Modified versions of the Hilbert transform pair and the Kramers-Kronig relations are derived for the complex permittivity of a plasma/dielectric medium which is singular at the frequency of the applied electric field equal to 0. Such a complex permittivity exists when the plasma/dielectric model allows a loss term but no restoring term. Permittivity, in which both loss and restoring terms are included, is shown to satisfy the standard Hilbert transform pair and, thus, the Kramers-Kronig relations

Phase retrieval of reflection and transmission coefficients from Kramers-Kronig relations

Analytic and passivity properties of reflection and transmission coefficients of thin-film multilayered stacks are investigated. Using a rigorous formalism based on the inverse Helmholtz operator, properties associated to causality principle and passivity are established when both temporal frequency and spatial wavevector are continued in the complex plane. This result extends the range of situations where the Kramers-Kronig relations can be used to deduce the phase from the intensity. In particular, it is rigorously shown that Kramers-Kronig relations for reflection and transmission coefficients remain valid at a fixed angle of incidence. Possibilities to exploit the new relationships are discussed.Comment: submitted for publicatio

Magneto-optical Kramers-Kronig analysis

We describe a simple magneto-optical experiment and introduce a magneto-optical Kramers-Kronig analysis (MOKKA) that together allow extracting the complex dielectric function for left- and right-handed circular polarizations in a broad range of frequencies without actually generating circularly polarized light. The experiment consists of measuring reflectivity and Kerr rotation, or alternatively transmission and Faraday rotation, at normal incidence using only standard broadband polarizers without retarders or quarter-wave plates. In a common case, where the magneto-optical rotation is small (below $\sim$ 0.2 rad), a fast measurement protocol can be realized, where the polarizers are fixed at 45$^\circ$ with respect to each other. Apart from the time-effectiveness, the advantage of this protocol is that it can be implemented at ultra-high magnetic fields and in other situations, where an \emph{in-situ} polarizer rotation is difficult. Overall, the proposed technique can be regarded as a magneto-optical generalization of the conventional Kramers-Kronig analysis of reflectivity on bulk samples and the Kramers-Kronig constrained variational analysis of more complex types of spectral data. We demonstrate the application of this method to the textbook semimetals bismuth and graphite and also use it to obtain handedness-resolved magneto-absorption spectra of graphene on SiC.Comment: 11 pages, 4 figur

On Kramers-Kronig relations for guided and surface waves

It is well known that in unbounded media the acoustic attenuation as function of frequency is linked to the frequency-dependent sound velocity (dispersion) via Kramers-Kronig dispersion relations. These relations are fundamentally important for better understanding of the nature of attenuation and dispersion and as a tool in physical acoustics measurements, where they can be used for verification purposes. However, physical acoustic measurements are frequently carried out not in unbounded media, but in acoustic waveguides, e.g. inside liquid-filled pipes. Surface acoustic waves are also often used for measurements. In the present work, the applicability of Kramers-Kronig relations to guided and surface waves is discussed using the approach based on the theory of functions of complex variables. It is demonstrated that Kramers-Kronig relations have limited applicability to guided and surface waves. In particular, they are not applicable to waves propagating in waveguides characterised by the possibility of wave energy leakage from the waveguides into the surrounding medium. For waveguides without leakages, Kramers-Kronig relations may remain valid for both ideal and viscous liquids. In the former case, Kramers-Kronig relations express the exponential decay of non-propagating (evanescent) higher-order acoustic modes below the cut-off frequencies via the dispersion of the same modes above the cut-off frequencies. Examples of numerical calculations of wave dispersion and attenuation using Kramers-Kronig relations, where applicable, are presented for different types of guided and surface waves