Kramers—Kronig Relations and Sum Rules in Nonlinear Optical Spectroscopy

Peiponen, Kai–Erik; Lucarini, Valerio; Saarinen, Jarkko J.; Vartiainen, Erik M.

doi:10.1366/000370204774103309

Cited by 23 publications

(11 citation statements)

References 73 publications

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“…n, is sufficiently fast, the nth order complex moduli are guaranteed to satisfy a set of Kramers-Kronig relations [28][29][30]. For multi-dimensional analytic functions, a number of different relations of this type are known [31,32]. These results are extensive, and we do not review them here as it is unclear yet whether they serve any useful purpose in the context of rheological measurements.…”

Section: A Strain Controlled Frequency Responsementioning

confidence: 99%

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

Lennon

McKinley

Swan

2020

Journal of Rheology

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A new mathematical representation for nonlinear viscoelasticity is presented based on application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call Medium Amplitude Parallel Superposition (MAPS) Rheology, reveals a new material property, the third order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain or stress controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a super-set of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the data sets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol.

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Section: A Strain Controlled Frequency Responsementioning

confidence: 99%

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

Lennon

McKinley

Swan

2020

Journal of Rheology

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“…KK analysis can be appropriate if the response is causal and holomorphic (analytic), as CARS is [3]; however, this type of analysis is strictly valid only if experimental data cover an infinite frequency range. Extrapolation approaches have been devised to deal with a finite data range by approximating the missing data, but these are often difficult to apply [4,5], because ln|χ(ω)|→–∞, as |χ(ω)|→0, which generally occurs as ω→∞.…”

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confidence: 99%

Broadband CARS spectral phase retrieval using a time-domain Kramers–Kronig transform

2009

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We describe a closed-form approach for performing a Kramers–Kronig (KK) transform that can be used to rapidly and reliably retrieve the phase, and thus the resonant imaginary component, from a broadband coherent anti-Stokes Raman scattering (CARS) spectrum with a nonflat background. In this approach we transform the frequency-domain data to the time domain, perform an operation that ensures a causality criterion is met, then transform back to the frequency domain. The fact that this method handles causality in the time domain allows us to conveniently account for spectrally varying nonresonant background from CARS as a response function with a finite rise time. A phase error accompanies KK transform of data with finite frequency range. In examples shown here, that phase error leads to small (<1%) errors in the retrieved resonant spectra.

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“…This analysis is usually performed on linear systems, however it can be extended to non-linear systems. [22] Each harmonic of the response should respect Kramer Kronig relations. In our case we have verified these relations for the first and the second-order terms of the impedance of the system.…”

Section: Kramers-kronig Relationsmentioning

confidence: 99%

Non-linear impedance spectroscopy for complete thermoelectric characterization: Beyond the zT estimation

Thiebaut

Pesty

Goupil

et al. 2018

Journal of Applied Physics

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Thermoelectric measurement of the dimensionless zT parameter requires multiple physical quantities to be measured, therefore there is great interest to find an experimental setup capable of measuring all these properties at once. Previous works on impedance spectroscopy have shown promising results in this direction, however, this technique does not lead to a complete characterization of the thermoelectric system without additional measurement. In order to extend impedance spectroscopy, we have investigated the measurement of the non-linear harmonic response of a Peltier device. The experiments are analyzed using an analytic model obtained by solving the heat equation in the frequency regime. Our work shows that fitting the experimental response of the system in the harmonic regime can leads to a complete characterization of the thermoelectric properties without the need of additional measurement.

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Kramers—Kronig Relations and Sum Rules in Nonlinear Optical Spectroscopy

Cited by 23 publications

References 73 publications

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

Broadband CARS spectral phase retrieval using a time-domain Kramers–Kronig transform

Non-linear impedance spectroscopy for complete thermoelectric characterization: Beyond the zT estimation

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