Kramers-Kronig Dispersion Relations Research Articles

On the transient solutions of three acoustic wave equations: van Wijngaarden’s equation, Stokes’ equation and the time-dependent diffusion equation

Acoustic wave propagation in a dispersive medium may be described by a wave equation containing one or more dissipation terms. Three such equations are examined in this article: van Wijngaarden's equation (VWE) for sound propagating through a bubbly liquid; Stokes' equation for acoustic waves in a viscous fluid; and the time-dependent diffusion equation (TDDE) for waves in the interstitial gas in a porous solid. The impulse-response solution for each of the three equations is developed and all are shown to be strictly causal, with no arrivals prior to the activation of the source. However, the VWE is nonphysical in that it predicts instantaneous arrivals, which are associated with infinitely fast, propagating Fourier components in the Green's function. Stokes' equation and the TDDE are well behaved in that they do not predict instantaneous arrivals. Two of the equations, the VWE and Stokes' equation, satisfy the Kramers-Kronig dispersion relations, while the third, the TDDE, does not satisfy Kramers-Kronig, even though its impulse-response solution is causal and physically realizable. The Kramers-Kronig relations are predicated upon the (mathematical) existence of the complex compressibility, a condition which is not satisfied by the TDDE because the Fourier transform of the complex compressibility is not square-integrable.

The refractive index of curved spacetime: The fate of causality in QED

It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime. Assuming the validity of the Kramers–Kronig dispersion relation, this would imply a superluminal wavefront velocity and the violation of causality. Here, we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic. We find that the high-frequency limit of the phase velocity (i.e. the wavefront velocity) is always equal to c and causality is assured. However, the Kramers–Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane, whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points. This makes the issue of micro-causality, i.e. the vanishing of commutators of field operators at spacelike separated points, a subtle one in local quantum field theory in curved spacetime.

Causality and micro-causality in curved spacetime

We consider how causality and micro-causality are realised in QED in curved spacetime. The photon propagator is found to exhibit novel non-analytic behaviour due to vacuum polarization, which invalidates the Kramers–Kronig dispersion relation and calls into question the validity of micro-causality in curved spacetime. This non-analyticity is ultimately related to the generic focusing nature of congruences of geodesics in curved spacetime, as implied by the null energy condition, and the existence of conjugate points. These results arise from a calculation of the complete non-perturbative frequency dependence of the vacuum polarization tensor in QED, using novel world-line path integral methods together with the Penrose plane-wave limit of spacetime in the neighbourhood of a null geodesic. The refractive index of curved spacetime is shown to exhibit superluminal phase velocities, dispersion, absorption (due to γ→e+e−) and bi-refringence, but we demonstrate that the wavefront velocity (the high-frequency limit of the phase velocity) is indeed c, thereby guaranteeing that causality itself is respected.

Model-independent sum rule analysis based on limited-range spectral data

Partial sum rules are widely used in physics to separate low- and high-energy degrees of freedom of complex dynamical systems. Their application, though, is challenged in practice by the always finite spectrometer bandwidth and is often performed using risky model-dependent extrapolations. We show that, given spectra of the real and imaginary parts of any causal frequency-dependent response function (for example, optical conductivity, magnetic susceptibility, acoustical impedance etc) in a limited range, the sum-rule integral from zero to a certain cutoff frequency inside this range can be safely derived using only the Kramers–Kronig dispersion relations without any extra model assumptions. This implies that experimental techniques providing both active and reactive response components independently, such as spectroscopic ellipsometry in optics, allow an extrapolation-independent determination of spectral weight ‘hidden’ below the lowest accessible frequency.

Superluminality and UV completion

The idea that the existence of a consistent UV completion satisfying the fundamental axioms of local quantum field theory or string theory may impose positivity constraints on the couplings of the leading irrelevant operators in a low-energy effective field theory is critically discussed. Violation of these constraints implies superluminal propagation, in the sense that the low-frequency limit of the phase velocity v ph ( 0 ) exceeds c. It is explained why causality is related not to v ph ( 0 ) but to the high-frequency limit v ph ( ∞ ) and how these are related by the Kramers–Kronig dispersion relation, depending on the sign of the imaginary part of the refractive index Im n ( ω ) which is normally assumed positive. Superluminal propagation and its relation to UV completion is investigated in detail in three theories: QED in a background electromagnetic field, where the full dispersion relation for n ( ω ) is evaluated numerically and the role of the null energy condition T μ ν k μ k ν ⩾ 0 is highlighted; QED in a background gravitational field, where examples of superluminal low-frequency phase velocities arise in violation of the positivity constraints; and light propagation in coupled laser–atom Λ-systems exhibiting Raman gain lines with Im n ( ω ) < 0 . The possibility that a negative Im n ( ω ) must occur in quantum field theories involving gravity to avoid causality violation, and the implications for the relation of IR effective field theories to their UV completion, are carefully analysed.

Kramers-Kronig analysis of attenuation and dispersion in trabecular bone

A restricted-bandwidth form of the Kramers-Kronig dispersion relations is applied to in vitro measurements of ultrasonic attenuation and dispersion properties of trabecular bone specimens from bovine tibia. The Kramers-Kronig analysis utilizes only experimentally measured properties and avoids extrapolation of ultrasonic properties beyond the known bandwidth. Compensation for the portions of the Kramers-Kronig integrals over the unknown bandwidth is partially achieved by the method of subtractions, where a subtraction frequency acts as an adjustable parameter. Good agreement is found between experimentally measured and Kramers-Kronig reconstructed dispersions. The restricted-bandwidth approach improves upon other forms of the Kramers-Kronig relations and may provide further insight into how ultrasound interacts with trabecular bone.

Optical properties of plasmas based on an average-atom model

Optical properties of plasmas, including dielectric constants, indices of refraction, and absorption coefficients, are determined from an average-atom point of view. Linear response of an average atom to a harmonic electric field leads to an average-atom version of the Kubo–Greenwood formula, which is used to calculate the frequency-dependent electric conductivity of the plasma. The frequency-dependent dielectric function is determined from the conductivity using Kramers–Kronig dispersion relations. The index of refraction and absorption coefficient of the plasma are subsequently obtained from the dielectric function. Comparing the present results with the free-electron model helps one understand anomalies observed recently in space and time resolved interferograms of Al plasmas produced by 13.9- and 14.7-nm X-ray lasers.

Frequency power law of material damping

The experimental studies often show that the damping in various solid materials increases with frequency over a finite bandwidth, and the increase is weak if the damping is low. A frequency power law is suggested and discussed in this paper to describe this damping behaviour with special respect to the low loss resilient materials used for sound and vibration control. The dynamic modulus as a function of frequency is determined from the loss modulus through the Kramers–Kronig dispersion relations to satisfy the causality requirement. It is proved that the dynamic modulus obeys the frequency power law of the same type as the loss modulus. In addition, it is proved that the weak frequency dependences of damping properties are inevitably associated with the low loss factor. Examples of fitting the frequency power law to experimental data on some materials of acoustical purposes are presented.

Kramers-Kronig relations and sum rules in nonlinear optical spectroscopy.

The full potential of the Kramers-Kronig relations and sum rules for nonlinear susceptibilities has unfortunately drawn relatively little attention in nonlinear optical spectra analysis. In this feature article a simple treatment of an anharmonic oscillator model in description of the nonlinear susceptibility of media and holomorphic properties of the nonlinear susceptibility were utilized. Using such concepts, conventional Kramers-Kronig, multiply-subtractive Kramers-Kronig, and generalized Kramers-Kronig dispersion relations can be derived. We demonstrate how in practice the variety of different Kramers-Kronig relations mentioned above, as well as various sum rules, can be applied in nonlinear optical spectra analysis. As an example we treat the third-harmonic wave generation spectrum from a polymer.

The general failure of Kleinman symmetry in practical nonlinear optical applications

Evidence is presented to refute the general applicability of the overall permutation symmetry condition originally proposed by Kleinman in describing the non-resonant nonlinear optical properties of the overwhelming majority of practical materials and systems. Kramers–Kronig dispersion relations demonstrate that the range of validity of Kleinman symmetry is substantially more restrictive than is widely assumed. These findings are supported by numerous experimental observations of effects explicitly forbidden by Kleinman symmetry but allowed by the dispersion relations. Considerations of molecular symmetry coupled with the Kramers–Kronig dispersion relations yield alternative simple and general macroscopic symmetry relations that are consistent with experimental observations.

On Titanium Carbide Nanoparticles as the Origin of the 21 Micron Emission Feature in Post-asymptotic Giant Branch Stars

Titanium carbide (TiC) nanocrystals were recently proposed as the carrier of the mysterious 21$\mum$ emission feature observed in post-asymptotic giant branch stars, based on their close spectral match and the presolar nature of meteoritic TiC nanograins (which reveals their stellar ejecta origin). But we show in this {\it Letter} that the Kramers-Kronig dispersion relations, which relate the wavelength-integrated extinction cross section to the total dust mass, would impose a lower bound on the TiC mass. This Kramers-Kronig lower limit exceeds the maximum available TiC mass by a factor of at least $\simali$50, independent of the absolute value of the ultraviolet/visible absorptivity of nano TiC. The TiC model is therefore readily ruled out by the Kramers-Kronig physical principle.

Demonstration of superluminal effects in an absorptionless, nonreflective system.

We present an experimental and theoretical study of a simple, passive system consisting of a birefringent, two-dimensional photonic crystal and a polarizer in series, and show that superluminal dispersive effects can arise even though no incident radiation is absorbed or reflected. We demonstrate that a vector formulation of the Kramers-Kronig dispersion relations facilitates an understanding of these counterintuitive effects.

Verification of Kramers–Kronig relationship in porous materials having a rigid frame

The propagation of acoustic waves in porous materials having a rigid frame is well described by several models. A doubt about the causality of these models has been raised recently in the literature. A verification of the causality of these models is studied in this paper using the Kramers–Kronig dispersion relations adapted to the frequency power law dependence of the attenuation. It is shown that these models are causal in the high- and low-frequency range. A time domain wave equation and time-causal theory have been treated.

Differential forms of the Kramers-Kronig dispersion relations

Differential forms of the Kramers-Krönig dispersion relations provide an alternative to the integral Kramers-Krönig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Krönig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (alpha(omega) = alpha0 + alpha1(absolute value of omega)y). Dispersion predictions using the differential dispersion relations are compared to the measured dispersion for a series of specimens (two polymers, an egg yolk, and two liquids) exhibiting attenuation obeying a frequency power law (1.00 < or = y < or = 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Krönig dispersion prediction is found to be identical to the (integral) Kramers-Krönig dispersion prediction.

First-principle calculations of the dielectric function of zinc-blende andwurtzite InN

The imaginary part of the dielectric function of zinc-blende and wurtziteInN has been calculated using a full-potential linearized augmented planewave method. We show that the exchange potential of Engel and Vosko givesan insulating ground state for both structures. The real part of thedielectric function has been obtained from the Kramers-Kronig dispersionrelations, assuming a quasi-particle band-gap correction according toBechstedt and Del Sole. We have found that it is necessary to have a goodaccount of the band gap in order to derive the low-frequency opticalproperties. We present the longitudinal as well as the transversecomponents in wurtzite InN, showing that the anisotropy is small.

First-principle calculations of optical properties of wurtzite AlN and GaN

The imaginary part of the dielectric function of wurtzite AlN and GaN has been calculated in the long wavelength limit, using two different first-principle electronic structure methods. The first method is a full-potential linearized augmented plane wave method and the second is a full-potential linear muffin-tin orbital method. From the Kramers–Kronig dispersion relations the real part of the dielectric function has been obtained, taking into account a quasi-particle band-gap correction according to Bechstedt and Del Sole. Absorption due to optical phonons is treated as a delta function in the imaginary part of the dielectric function. Both the longitudinal as well as the transverse components of the dielectric function are presented, showing that the anisotropy is small in these materials. Although we use different correlation potentials in the two methods, the results are similar. We compare our calculated dielectric functions with spectroscopic ellipsometry and reflectance spectra measurements.

Conductance and density of states from the Kramers-Kronig dispersion relation

By applying the Kramers-Kronig dispersion relation to the transmission amplitude, a direct connection of the conductance with the density of states is given in quantum scattering systems connected to two one-channel leads. Using this method we show that in the Fano resonance the peak position of the density of states is generally different from the position of the corresponding conductance peak, whereas in the Breit-Wigner resonance those peak positions coincide. The line shapes of the density of states are well described by a Lorentz type in both the resonances. These results are verified by another approach using a specific form of the scattering matrix to describe the scattering resonances.

On a time-domain representation of the Kramers-Kronig dispersion relations

The development of Kramers-Kronig dispersion relations is typically carried out in the frequency domain. An alternative approach known as the time-causal theory develops dispersion relations for media with attenuation obeying a frequency power law through analysis in the time domain [T. L. Szabo, J. Acoust. Soc. Am. 96, 491-500 (1994)]. Although both approaches predict identical dispersion relations, it is perceived that these two approaches are distinct from each other. It is shown, however, that the time-causal theory is in essence a time-domain formulation of the Kramers-Kronig dispersion relations for the special case of media with attenuation obeying a frequency power law. Additionally, it is shown that time-domain representations of the Kramers-Kronig dispersion relations are available for a broader class of media than simply those with power law attenuation. The time-causal theory and the Kramers-Kronig dispersion relations can be viewed as two complementary, yet equivalent, approaches to the study of dispersion.

On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law

In the recent literature concern has been raised regarding the validity of Kramers-Kronig relations for media with ultrasonic attenuation obeying a frequency power law. It is demonstrated, however, that the Kramers-Kronig dispersion relations for application to these types of media are available. The developed dispersion relations are compared with measurements on several liquids, and agreement is found to better than 1 m/s over the experimentally available bandwidth. A discussion regarding the validity of these dispersion relations, in particular how the dispersion relations relate to the so-called Paley-Wiener conditions, forms the conclusion.

Radiation of a point charge uniformly moving in a dielectric medium

We analyse how the frequency dependence of the dielectric permittivity affects the electromagnetic field radiated by a point charge uniformly moving in a medium. It turns out that a moving charge radiates at every velocity. We study the space distribution of the electromagnetic field and show that its oscillations are due to the time-dependent polarization of the medium induced by the moving charge. Spectral distributions of the radiated energy and the photon number are given. Consequences arising from a choice of polarization different from the usual one are discussed. An analysis of the Kramers-Kronig dispersion relations for the treated problem is given. This consideration may be useful for the experiments recently discussed in the literature.