The frequency spectrum of the finite temperature correction to the Casimir force can be determined by use of the Lifshitz formalism for metallic plates of finite conductivity. We show that the correction for the $TE$ electromagnetic modes is dominated by frequencies so low that the plates cannot be modelled as ideal dielectrics. We also address issues relating to the behavior of electromagnetic fields at the surfaces and within metallic conductors, and calculate the surface modes using appropriate low-frequency metallic boundary conditions. Our result brings the thermal correction into agreement with experimental results that were previously obtained. We suggest a series of measurements that will test the veracity of our analysis.
We have performed comparative measurements of the Casimir force between a metallic plate and a transparent sphere coated with metallic films of different thicknesses. We have observed that, if the thickness of the coating is less than the skin-depth of the electromagnetic modes that mostly contribute to the interaction, the force is significantly smaller than that measured with a thick bulk-like film. Our results provide the first direct evidence of the skin-depth effect on the Casimir force between metallic surfaces.
We reply to the "Comment on 'The Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic mirrors'" [arXiv:1604.06432]. We believe the comment misrepresents our papers, and fails to provide a plausible resolution to the conflict between theory and experiment.
We derive an analytic behavior of the Casimir free energy, entropy and pressure of metallic films in vacuum at low temperature. It is shown that this behavior differs significantly depending on whether the plasma or the Drude model is used to describe the dielectric properties of film metal. For metallic films described by the lossless plasma model the thermal corrections to the Casimir energy and pressure drop to zero exponentially fast with increasing film thickness. There is no classical limit in this case. The Casimir entropy satisfies the Nernst heat theorem. For metallic films with perfe…
The Casimir force between two metallic plates is affected by their roughness state. This effect is usually calculated through the so-called `proximity force approximation' which is only valid for small enough wavevectors in the spectrum of the roughness profile. We introduce here a more general description with a wavevector-dependent roughness sensitivity of the Casimir effect. Since the proximity force approximation underestimates the effect, a measurement of the roughness spectrum is needed to achieve the desired level of accuracy in the theory-experiment comparison.
We calculate the Casimir force and free energy for plane metallic mirrors at non-zero temperature. Numerical evaluations are given with temperature and conductivity effects treated simultaneously. The results are compared with the approximation where both effects are treated independently and the corrections simply multiplied. The deviation between the exact and approximated results takes the form of a temperature dependent function for which an analytical expression is given. The knowledge of this function allows simple and accurate estimations at the % level.
We examine the conditions of validity for the Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic plane mirrors. As in the previously studied case of non-magnetic materials (Guerout et al, Phys. Rev. E 90 042125), we recover the usual expression for the lossy model of optical response, but not for the lossless plasma model. We also show that the modes associated with the Foucault currents play a crucial role in the limit of vanishing losses, in contrast to expectations.
We carefully re-examine the conditions of validity for the consistent derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors. We recover the usual expression for the lossy Drude model, but not for the lossless plasma model. We give an interpretation of this new result in terms of the modes associated with the Foucault currents which play a role in the limit of vanishing losses, in contrast to common expectations.
We consider two parallel corrugated plates and show that a Casimir torque arises when the corrugation directions are not aligned. We follow the scattering approach and calculate the Casimir energy up to second order in the corrugation amplitudes, taking into account nonspecular reflections, polarization mixing and the finite conductivity of the metals. We compare our results with the proximity force approximation, which overestimates the torque by a factor 2 when taking the conditions that optimize the effect. We argue that the Casimir torque could be measured for separation distances as large…
The frequency spectrum of the finite temperature correction to the Casimir force can be determined by the use of the Lifshitz formalism for metallic plates of finite conductivity. We show that the correction for the TE electromagnetic modes is dominated by frequencies so low that the plates cannot be modelled as ideal dielectrics. We also address the issues relating to the behavior of electromagnetic fields at the surfaces an within metallic conductors, and claculate the surface modes using appropriate low-frequency metallic boundary conditions. Our result brings the tehrmal correction into ag…
We study the influence of finite conductivity of metals on the Casimir effect. We put the emphasis on explicit theoretical evaluations which can help comparing experimental results with theory. The reduction of the Casimir force is evaluated for plane metallic plates. The reduction of the Casimir energy in the same configuration is also calculated. It can be used to infer the reduction of the force in the plane-sphere geometry through the `proximity theorem'. Frequency dependent dielectric response functions of the metals are represented either by the simple plasma model or, more accurately, b…
We propose a new theory of thermal Casimir effect, holding for the experimentally important case of metallic surfaces with a roughness having a spatial scale smaller than the skin depth. The theory is based on a simple phenomenological model for a rough conductor, that explicitly takes account of the fact that ohmic conduction in the immediate vicinity of the surface of a conductor is much impeded by surface roughness, if the amplitude of roughness is smaller than the skin depth. As a result of the new model, we find that surface roughness strongly influences the magnitude of the thermal corre…
We study the lateral Casimir force between corrugated dielectric plates. We use the dielectric contrast perturbation theory [R. Golestanian, {\it Phys. Rev. Lett.} {\bf 95}, 230601, (2005)], which allows us to consider arbitrary deformations with large amplitudes. We consider sinusoidal, rectangular, and sawtooth corrugations, for samples made of silicon and gold. We use the plasma and Drude-Lorentz models for the permittivity of gold and silicon, respectively. For these geometries and materials, the lateral Casimir force is {\it not} a sinusoidal function of the relative lateral displacement …
We calculate the second order roughness correction to the Casimir energy for two parallel metallic mirrors. Our results may also be applied to the plane-sphere geometry used in most experiments. The metallic mirrors are described by the plasma model, with arbitrary values for the plasma wavelength, the mirror separation and the roughness correlation length, with the roughness amplitude remaining the smallest length scale for perturbation theory to hold. From the analysis of the intracavity field fluctuations, we obtain the Casimir energy correction in terms of generalized reflection operators,…
The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within 1% of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically the non-vanishing Casimir…
Recently Guerout et al. [Phys. Rev. E, v.93, 022108 (2016)] advocated that the lossless plasma model has to be redefined as the limit of the Drude model when the relaxation parameter goes to zero. It was claimed that the previously used plasma model cannot correctly describe the Casimir pressure between two plates made of both nonmagnetic and magnetic metals and has to be replaced with the redefined one. We show that the suggested redefinition does not satisfy necessary physical requirements imposed on the dielectric permittivity. We also present a plausible explanation to the fact that the lo…
The frequencies of transitions between hyperfine levels of ground-state atoms can be measured with exquisite precision using magnetic-resonance techniques. This makes hyperfine transitions ideal probes of QED effects originating from the interaction of atoms with the quantized electromagnetic field. One of the most remarkable effects predicted by QED is the Casimir-Polder shift experienced by the energy levels of atoms placed near one or more dielectric objects. Here we compute the Casimir-Polder shift and the width of hyperfine transitions between ground-state Zeeman sub-levels of an hydrogen…
he Casimir force is calculated between plates with thin metallic coating. Thin films are described with spatially dispersive (nonlocal) dielectric functions. For thin films the nonlocal effects are more relevant than for half-spaces. However, it is shown that even for film thickness smaller than the mean free path for electrons, the difference between local and nonlocal calculations of the Casimir force is of the order of a few tenths of a percent. Thus the local description of thin metallic films is adequate within the current experimental precision and range of separations.
For the first time, we have studied the impact of magnetic polarizability of a neutral metallic particle on the dynamical Casimir-Polder force, when the particle is uniformly moving with relativistic velocity parallel to the surface of an ideally conducting plate. The material properties of particle are taken into account within the Drude model approach.
We propose a test on the role of relaxation properties of conduction electrons in the Casimir pressure between two parallel metal-coated plates kept at different temperatures. It is shown that for sufficiently thick metallic coatings the Casimir pressure and pressure gradient are determined by the mean of the equilibrium contributions calculated at temperatures of the two plates and by the term independent on separation. Numerical computations of the nonequilibrium pressures are performed for two parallel Au plates of finite thickness as a function of separation and temperature of one of the p…
Optical properties of the deposited gold films are discussed in connection with the Casimir force prediction. Voids in the films and electron scattering on the grain boundaries reduce the force at small separations on the 2% level in comparison with the bulk material prediction. The contribution of the patch potential due to polycrystalline structure of the films is shown to be small for the existing Casimir force experiments.
We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to…
We consider the Casimir-Polder interaction energy between a metallic nanoparticle and a metallic plate, as well as the Casimir interaction energy between two macroscopic metal plates, in terms of the many-body dispersion interactions between their constituents. Expressions for two- and three-body dispersion interactions between the microscopic parts of a real metal are first obtained, both in the retarded and non-retarded limits. These expressions are then used to evaluate, a compare each other, the overall two- and three-body contributions to the macroscopic Casimir-Polder and Casimir force, …
We give an example of a geometry in which two metallic objects in vacuum experience a repulsive Casimir force. The geometry consists of an elongated metal particle centered above a metal plate with a hole. We prove that this geometry has a repulsive regime using a symmetry argument and confirm it with numerical calculations for both perfect and realistic metals. The system does not support stable levitation, as the particle is unstable to displacements away from the symmetry axis.
The Casimir interaction between one-dimensional metallic objects (cylinders, wires) displays unconventional features. Here we study the orientation dependence of this interaction by computing the Casimir energy between two inclined cylinders over a wide range of separations. We consider Dirichlet, Neumann and perfect metal boundary conditions, both at zero temperature and in the classical high temperature limit. For all types of boundary conditions, we find that at large distances the interaction decays slowly with distance, similarly to the case of parallel cylinders, and at small distances s…
We study the Casimir interaction between a dielectric nanosphere and a metallic plane, using the multiple scattering theory. Exact results are obtained with the dielectric described by a Sellmeier model and the metal by a Drude model. Asymptotic forms are discussed for small spheres, large or small distances. The well-known Casimir-Polder formula is recovered at the limit of vanishingly small spheres, while an expression better behaved at small distances is found for any finite value of the radius. The exact results are of particular interest for the study of quantum states of nanospheres in t…
We give an exact series expansion of the Casimir force between plane and spherical metallic surfaces in the non trivial situation where the sphere radius $R$, the plane-sphere distance $L$ and the plasma wavelength $λ_¶$ have arbitrary relative values. We then present numerical evaluation of this expansion for not too small values of $L/R$. For metallic nanospheres where $R, L$ and $λ_¶$ have comparable values, we interpret our results in terms of a correlation between the effects of geometry beyond the proximity force approximation (PFA) and of finite reflectivity due to material properties. …
We report on the measurement of the Casimir force between conducting surfaces in a parallel configuration. The force is exerted between a silicon cantilever coated with chromium and a similar rigid surface and is detected looking at the shifts induced in the cantilever frequency when the latter is approached. The scaling of the force with the distance between the surfaces was tested in the 0.5 - 3.0 $μ$m range, and the related force coefficient was determined at the 15% precision level.
We study the thermal interaction between two nearby thin metallic wires, at finite temperature. It is shown that the Johnson currents in the wires give rise, via inductive coupling, to a repulsive force between them. This thermal interaction exhibits all the puzzling features found recently in the thermal Casimir effect for lossy metallic plates, suggesting that the physical origin of the difficulties encountered in the Casimir problem resides in the inductive coupling between the Johnson currents inside the plates. We show that in our simple model all puzzles are resolved if account is taken …
The Casimir force between bodies in vacuum can be understood as arising from their interaction with an infinite number of fluctuating electromagnetic quantum vacuum modes, resulting in a complex dependence on the shape and material of the interacting objects. Becoming dominant at small separations, the force plays a significant role in nanomechanics and object manipulation at the nanoscale, leading to a considerable interest in identifying structures where the Casimir interaction behaves significantly different from the well-known attractive force between parallel plates. Here we experimentall…
We investigate the magnetic dipole coupling between a metallic surface and an atom in a thermal state, ground state and excited hyperine state. This interaction results in a repulsive correction and - unlike the electrical dipole contribution - depends sensitively on the Ohmic dissipation in the material.
We investigate the Casimir free energy of a metallic film either sandwiched between two dielectric plates or in vacuum. It is shown that even for a thin film of several tens of nanometer thickness the Casimir free energy and pressure calculated with the Lifshitz theory using the Drude model and the plasma model approaches take significantly different values and can be easily discriminated. According to our results, the classical limit is already achieved for films of about 100\,nm thickness if the Drude model approach is used in calculations. In this case the classical expressions for the Casi…
We analyze some consequences of the Casimir-type zero-point radiation pressure. These include macroscopic "vacuum" forces on a metallic layer in-between a dielectric medium and an inert ($ε(ω) = 1$) one. Ways to control the sign of these forces, based on dielectric properties of the media, are thus suggested. Finally, the large positive Casimir pressure, due to surface plasmons on thin metallic layers, is evaluated and discussed.
We show that exact results are obtained for the calculation of Casimir forces between arbitrary materials using the concept of surface impedances, obtaining in a trivial way the force in the limit of perfect conductors and also Lifshitz formula in the limit of semi-infinite media. As an example we present a full and rigorous calculation of the Casimir force between two metallic half-spaces described by a hydrodynamic nonlocal dielectric response.
The Casimir effect is investigated in cylindrical and toroidal carbon nanotubes within the framework of the Dirac-like model for the electronic states. The topological Casimir energy is positive for metallic cylindrical nanotubes and is negative for semiconducting ones. The toroidal compactification of a cylindrical nanotube along its axis increases the Casimir energy for metallic-type (periodic) boundary conditions along its axis and decreases the Casimir energy for the semiconducting-type compactifications. For finite length metallic nanotubes the Casimir forces acting on the tube edges are …
By using silicon oxide based aerogels we show numerically that the Casimir force can be reduced several orders of magnitude, making its effect negligible in nanodevices. This decrease in the Casimir force is also present even when the aerogels are deposited on metallic substrates. To calculate the Casimir force we model the dielectric function of silicon oxide aerogels using an effective medium dielectric function such as the Clausius-Mossotti approximation. The results show that both the porosity of the aerogel and its thickness can be use as control parameters to reduce the magnitude of the …
We consider the dispersion energy of a pair of dipoles embedded in a metallic waveguide with transverse dimension $a$ smaller than the characteristic dipolar wavelength. We find that $a$ sets the scale that separates retarded, Casimir-Polder-like, from quasistatic, van der Waals-like, interactions. Whereas in the retarded regime, the energy decays exponentially with inter-dipolar distance, typical of evanescent waves, in the van der Waals regime, the known free-space result is obtained. This short-range scaling implies that the additivity of the dispersion interactions inside a waveguide exten…
The Casimir free energy and pressure of thin metal films deposited on metallic plates are considered using the Lifshitz theory and the Drude and plasma model approaches to the role of conduction electrons. The bound electrons are taken into account by using the complete optical data of film and plate metals. It is shown that for films of several tens of nanometers thickness the Casimir free energy and pressure calculated using these approaches differ by hundreds and thousands percent and can be easily discriminated experimentally. According to our results, the free energy of a metal film does …
In Phys. Rev. {\bf E 70}, 047102 (2004), J.R. Torgerson and S.K. Lamoreaux investigated for the first time the real-frequency spectrum of finite temperature correction to the Casimir force, for metallic plates of finite conductivity. The very interesting result of this study is that the correction from the TE mode is dominated by low frequencies, for which the dielectric description of the metal is invalid. However, their analysis of the problem, based on more appropriate low-frequency metallic boundary conditions, uses an incorrect form of boundary conditions for TE modes. We repeat their ana…
We find that the Casimir pressure in peptide films deposited on metallic substrates is always repulsive which makes these films less stable. It is shown that by adding a graphene sheet on top of peptide film one can change the sign of the Casimir pressure by making it attractive. For this purpose, the formalism of the Lifshitz theory is extended to the case when the film and substrate materials are described by the frequency-dependent dielectric permittivities, whereas the response of graphene to the electromagnetic field is governed by the polarization tensor in (2+1)-dimensional space-time f…
We extend our previous work [Phys. Rev. Lett. {\bf 100}, 183602 (2008)] on the generalization of the Casimir-Lifshitz theory to treat anisotropic magnetodielectric media, focusing on the forces between metals and magnetodielectric metamaterials and on the possibility of inferring magnetic effects by measurements of these forces. We present results for metamaterials including structures with uniaxial and biaxial magnetodielectric anisotropies, as well as for structures with isolated metallic or dielectric properties that we describe in terms of filling factors and a Maxwell Garnett approximatio…
Based on a generalization of the Lifshiftz theory we calculate Casimir forces involving magnetodielectric and possibly anisotropic metamaterials, focusing on the possibility of repulsive forces. It is found that Casimir repulsion decreases with magnetic dissipation, and even a small Drude background in metallic-based metamaterials acts to make attractive a Casimir force that would otherwise be predicted to be repulsive. The sign of the force also depends sensitively on the degree of optical anisotropy of the metamaterial, and on the form of the frequency-dependency of the magnetic response.
We present systematic measurements of the Casimir force between a gold-coated plate and a sphere coated with a Hydrogen Switchable Mirror (HSM). HSMs are shiny metals that can become transparent by hydrogenation. In spite of such a dramatic change of the optical properties of the sphere, we did not observe any significant decrease of the Casimir force after filling the experimental apparatus with hydrogen. This counterintuitive result can be explained by the Lifshitz theory that describes the Casimir attraction between metallic and dielectric materials.
The Casimir force between uncharged metallic surfaces originates from quantum mechanical zero point fluctuations of the electromagnetic field. We demonstrate that this quantum electrodynamical effect has a profound influence on the oscillatory behavior of microstructures when surfaces are in close proximity (<= 100 nm). Frequency shifts, hysteretic behavior and bistability caused by the Casimir force are observed in the frequency response of a periodically driven micromachined torsional oscillator.
We study the Casimir pressure between two similar plates of finite thickness kept at different temperatures in the case when the dielectric permittivity of the plates depends on temperature. It is suggested to consider the dielectric permittivity at two different temperatures as the permittivities of two dissimilar bodies, thus allowing to apply the theory of Casimir forces out of thermal equilibrium developed earlier in the literature. Following this approach, we show that, in addition to the equilibrium contribution to the nonequilibrium Casimir pressure, a proper nonequilibrium contribution…
We discuss the temperature correction to the Casimir force between nonideal metallic bodies which caused disagreement in the literature. A general method to find the troubling term is proposed that does not require a direct reference to the Lifshitz formula. The linear in temperature correction is shown to survive for nonideal metals. It is important for small separations between bodies tested in the recent experiments.
The Casimir-Polder (CP) potential and transition rates of a Rydberg atom above a plane metal surface at finite temperature are discussed. As an example, the CP potential and transition rates of a rubidium atom above a copper surface at room temperature is computed. Close to the surface we show that the quadrupole correction to the force is significant and increases with increasing principal quantum number n. For both the CP potential and decay rates one finds that the dominant contribution comes from the longest wavelength transition and the potential is independent of temperature. We provide …
The interaction between arbitrary dielectric heterostructures is studied within the framework of a recently developed dielectric contrast perturbation theory. It is shown that periodically patterned dielectric or metallic structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as modulations in the normal force as they are displaced with respect to one another. The strength of these oscillatory contributions increases with decreasing gap size and increasing contrast in the dielectric properties of the materials used in the heterostructures.
Casimir-Polder potential is investigated for a polarizable microparticle in the geometry of a straight cosmic string with a metallic cylindrical shell. The electromagnetic field Green tensor is evaluated on the imaginary frequency axis. The expressions for the Casimir-Polder potential is derived in the general case of anisotropic polarizability for the both interior and exterior regions of the shell. The potential is decomposed into pure string and shell-induced parts. The latter dominates for points near the shell, whereas the pure string part is dominant near the string and at large distance…
We calculate the roughness correction to the Casimir effect in the parallel plates geometry for metallic plates described by the plasma model. The calculation is perturbative in the roughness amplitude with arbitrary values for the plasma wavelength, the plate separation and the roughness correlation length. The correction is found to be always larger than the result obtained in the Proximity Force Approximation.
We study the quantum and thermal fluctuations of eddy (Foucault) currents in thick metallic plates. A Casimir interaction between two plates arises from the coupling via quasi-static magnetic fields. As a function of distance, the relevant eddy current modes cross over from a quantum to a thermal regime. These modes alone reproduce previously discussed thermal anomalies of the electromagnetic Casimir interaction between good conductors. In particular, they provide a physical picture for the Casimir entropy whose nonzero value at zero temperature arises from a correlated, glassy state.
Kelvin probe force microscopy at normal pressure was performed by two different groups on the same Au-coated planar sample used to measure the Casimir interaction in a sphere-plane geometry. The obtained voltage distribution was used to calculate the separation dependence of the electrostatic pressure $P_{\rm res}(D)$ in the configuration of the Casimir experiments. In the calculation it was assumed that the potential distribution in the sphere has the same statistical properties as the measured one, and that there are no correlation effects on the potential distributions due to the presence o…
We describe a numerical method to compute Casimir forces in arbitrary geometries, for arbitrary dielectric and metallic materials, with arbitrary accuracy (given sufficient computational resources). Our approach, based on well-established integration of the mean stress tensor evaluated via the fluctuation-dissipation theorem, is designed to directly exploit fast methods developed for classical computational electromagnetism, since it only involves repeated evaluation of the Green's function for imaginary frequencies (equivalently, real frequencies in imaginary time). We develop the approach by…
We show the influence of surface plasmons on the Casimir effect between two plane parallel metallic mirrors at arbitrary distances. Using the plasma model to describe the optical response of the metal, we express the Casimir energy as a sum of contributions associated with evanescent surface plasmon modes and propagative cavity modes. In contrast to naive expectations, the plasmonic modes contribution is essential at all distances in order to ensure the correct result for the Casimir energy. One of the two plasmonic modes gives rise to a repulsive contribution, balancing out the attractive con…
The Casimir-Lifshitz force is calculated between two inhomogeneous composite slabs, each made of a homogeneous matrix with spherical metallic inclusions. The effective dielectric function of the slabs is calculated using several effective medium approximations and we compare the resulting forces as a function of slab separation and filling fraction. We show that the choice of effective medium approximation is critical in making precise comparisons between theory and experiment. The role that the spectral representation of the effective medium plays in making a Wick rotation to the complex freq…
We compute the Casimir interaction between a plane and a sphere, the configuration employed in the most precise experiments. The scattering formula is developed by taking a suitably chosen plane-wave and multipole basis and is valid for arbitrary values of the sphere radius, inter-plate distance, temperature and arbitrary dielectric functions for both sphere and plate. Our analytical and numerical results for metallic surfaces show a non-trivial interplay between the effects of curvature, temperature, finite conductivity and dissipation.
We obtain constraints on parameters of the Yukawa-type corrections to Newton's gravitational law from measurements of the gradient of the Casimir force between surfaces coated with ferromagnetic metal Ni and from measurements of the Casimir force between Au-coated sinusoidally corrugated surfaces at various angles between corrugations. It is shown that constraints following from the experiment with magnetic surfaces are slightly weaker than currently available strongest constraints, but benefit from increased reliability and independence of systematic effects.The constraints derived from the e…
We study the spatial fluctuations of the Casimir-Polder force experienced by an atom or a small sphere moved above a metallic plate at fixed separation distance. We demonstrate that unlike the mean force, the magnitude of these fluctuations crucially relies on the relaxation of conduction electron in the metallic bulk, and even achieves values that differ by orders of magnitude depending on the amount of dissipation. We also discover that fluctuations suffer a spectacular decrease at large distances in the case of nonzero temperature.
The possibility of making precise predictions for the Casimir force is essential for the theoretical interpretation of current precision experiments on the thermal Casimir effect with metallic plates, especially for sub-micron separations. For this purpose it is necessary to estimate very accurately the dielectric function of a conductor along the imaginary frequency axis. This task is complicated in the case of ohmic conductors, because optical data do not usually extend to sufficiently low frequencies to permit an accurate evaluation of the standard Kramers-Kronig integral used to compute $ε…
