Kramers-Kronig Dispersion Relations Research Articles

On the Applicability of Kramers–Kronig Dispersion Relations to Guided and Surface Waves

In unbounded media, the acoustic attenuation as function of frequency is related 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 control 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 physical acoustics measurements. In the present work, the applicability of Kramers–Kronig relations to guided and surface waves is investigated 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, e.g., those formed by rigid walls, Kramers–Kronig relations remain valid for both ideal and viscous liquids. Examples of numerical calculations of wave dispersion and attenuation using Kramers–Kronig relations, where applicable, are presented for unbounded media and for waveguides formed by two rigid walls.

Be10 -nucleus optical potentials developed from chiral effective field theory NN interactions

We present a determination of optical potentials for $^{10}\mathrm{Be}$-nucleus collisions using the double-folding method to compute the real part and Kramers-Kronig dispersion relations to derive the imaginary part. As microscopic inputs we use chiral effective field theory nucleon-nucleon interactions at next-to-next-to-leading order combined with state-of-the-art nucleonic densities. With these potentials, we compute elastic-scattering cross sections for the exotic nucleus $^{10}\mathrm{Be}$ off various targets, and compare them to experiment. Without any fitting parameter, we obtain good agreement with data. For collisions on light targets, we observe significant uncertainty related to the short-range physics, whereas for heavy targets that uncertainty remains small.

Statistical Inference of Sound Speed and Attenuation Dispersion of a Fine-Grained Marine Sediment

Acoustic recordings of signals in the 1.5–4.0-kHz band were analyzed for information about the sound speed and attenuation frequency dispersion of a fine-grained sediment found in the New England Mudpatch. Analysis of piston cores established prior bounds for a geophysical parameterization of a seabed model that predicts Kramers–Kronig dispersion relations. Sediment layers are described by the Buckingham viscous grain shearing (VGS) model that accounts for the effects of overburden pressure of compressional and shear speeds and attenuations. A statistical inverse problem was solved by using multiple samples of received levels recorded on two vertical line arrays as a function time and hydrophone depth for six frequencies in the 1.5–4.0-kHz band. A statistical inference model that assumed both model parameters and data samples are random variables quantified information content from marginalization of a conditional posterior probability distribution for the geophysical parameters that characterize the mud layer. From the inferred geophysical parameter point estimates the sediment sound speed and attenuation frequency dispersion are predicted and compared to previously reported direct measurements. Also, the predicted sound-speed gradient in the mud sediment from the VGS model is compared to a previous inference that utilized explosive sources.

Numerical Analysis of A Silicon-Based Electro-Optic Modulator

A phase shift electro-optic semiconductor structure that features a Mach-Zehnder Interferometer (MZI) is numerically analyzed. The interferometer structure considers a highly-doped Si-based waveguide that can control the phase shift by varying the refractive index through the carrier concentration while applying a control voltage. To analyze the structure, Kramers-Kronig dispersion relations, the effective index method and the Drude model are implemented within a finite element software. As a result, for an optical waveguide structure with a 9 mm length and 10 V, it is possible to obtain a phase shift by that allows, at the output of the MZI, total interference that is required to modulate an optical communication link.

α -nucleus optical potentials from chiral effective field theory NN interactions

We present a determination of optical potentials for $^4$He-target collisions using the double-folding method. We use chiral effective field theory nucleon-nucleon interactions at next-to-next-to-leading order combined with state-of-the-art nucleonic densities. The imaginary part of the optical potential is obtained from the real double-folding interaction either through a proportionality constant or applying Kramers-Kronig dispersion relations. With these potentials, we compute the elastic scattering of $^4$He off various targets, from $^4$He to $^{120}$Sn. We study the sensitivity of our predicted cross sections to the choice of nucleon-nucleon interactions and nuclear densities. Very good agreement is obtained with existing elastic-scattering data for $^4$He energies between 100 and 400 MeV.

The Correlation between Structural and Optical Properties of Zinc Hydroxide Nanoparticle in Supports Photocatalytic Performance

- Green synthesized Zn(OH)2 mediated by Coleus blumei Benth leaf for pH: 4, 7, 10 and for calcination temperature: 400 °C, 450 °C, 500 °C, 550 °C, and 600 °C have been characterized by using Fourier transform infra-red (FTIR), ultraviolet–visible (Uv–Vis), and X-ray diffraction (XRD). The bonding formation present Zn– O –H, H– O –H, O –H, and Zn–O indicated Zn(OH)2 nanocrystal successfully synthesized. The optical, dielectric properties, and energy loss function were quantitatively determined from the FTIR spectra by using Kramers-Kronig (KK) dispersion relation. The distance between longitudinal (LO) and transverse (TO) optical phonon vibration mode Δ (LO-TO) and the bandgap increases with increasing the calcination temperature for all pH due to the amount of covalent bond is increase. The best calcination temperature is 600 °C and pH 7 indicated by the rate of the degradation performance is faster than the other temperature and pH. In this study shows FTIR spectra effective ways for determining the optical and dielectric properties, optical phonon vibration mode, and energy loss function of Zn(OH)2 material. • The Zn-O-H, H-O-H, O-H, and Zn-O bonding indicated formed of Zn(OH) 2 nanocrystal. • The n and k , ε 1 and ε 2 (ω), Im ( - 1 / ε 1 ( ω ) ) determined from FTIR spectra by applying K-K relation. • Distance optical phonon vibration (D(LO-TO)) is increases with increasing the calcination temperature for all pH. • The band gap increase with increasing the calcination temperature. • The best calcination temperature for photocatalyst is 600 o C.

A conformal dispersion relation: correlations from absorption

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block.

Benchmarking coherent radiation spectroscopy as a tool for high-resolution bunch shape reconstruction at free-electron lasers

We present a systematic comparison of two complementary methods for determining the longitudinal charge density profile of the compressed electron bunches in the soft x-ray free-electron laser FLASH: a frequency-domain technique---coherent transition radiation (CTR) spectroscopy---and a time-domain technique---streaking of the electron beam with a transversely deflecting microwave structure (TDS). While the direct time-profile measurement with a TDS is a well-established method invented at SLAC, our group has pioneered high-resolution bunch shape analysis based on coherent radiation spectroscopy. We have developed a broadband spectrometer covering the wavelength range from $5\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ to $433\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ with two sets of remotely interchangeable staged reflection gratings. The measured spectral intensity allows to compute the absolute magnitude of the bunch form factor but not its phase which, however, is needed to retrieve the bunch profile. Two phase retrieval methods are investigated in detail: analytic phase computation by means of the Kramers-Kronig dispersion relation, and iterative phase retrieval. Several computational techniques are compared and evaluated in view of their applicability and efficiency. For a large variety of bunch shapes, the time profiles derived from the spectroscopic data are compared with the TDS profiles, and generally excellent agreement is observed down to the 10 fs level. For part of the measurements, two independent CTR spectrometer systems have been available, yielding almost identical bunch shapes. In summary, we demonstrate that using well calibrated and broadband spectroscopy data, a fast and reliable phase reconstruction algorithm leads to bunch profiles competitive to high resolution TDS measurements.

Optical constants at complex energies: local deconvolution.

Causality implies that the optical constants of any material continue in the upper complex plane of photon energies or wavelengths as an analytic function. This is the basis for Kramers-Kronig dispersion relations to obtain ɛ1 from ɛ2, or n from k. However, there have not been attempts to explore this continuation. This research focuses on such continuation and on applications thereof. An interesting property has been found: optical constants progressively smoothen when entering the upper complex plane. The continuation to complex energies is found to result in an average of the optical constants with a Lorentzian weight function. This optical-constant smoothening originated in a shift to the upper complex plane is naturally produced in optical constants that have been obtained by means of an optical instrument with a Lorentzian slit function. This smoothening results in reduced resolution through convolution with the slit function. A procedure that takes advantage of optical constants at complex energies is developed for optical-constant deconvolution. Deconvolution is performed locally, i.e., with no integration, and it consists of shifting the energy of the optical constants by an imaginary amount by means of a Taylor series expansion. The first correction term involves the derivative of the other optical constant. Even though deconvolution of optical constants measured with a Gaussian slit cannot be directly performed with the present method, an approach based on powers of the Lorentz function is also proposed. This procedure could be implemented as an analysis tool of a spectrophotometer or an ellipsometer; this tool would enable one to measure optical constants with a modest resolution and to improve it by post-processing them with the present scheme.

Complex conductivity of monolayer graphene and Zitterbewegung

A recently derived formula for complex conductivity of monolayer graphene is analyzed. We show that the real and imaginary parts in this formula obey the Kramers-Kronig dispersion relations, which are a good test for validity of the formula for complex conductivity of monolayer graphene. We consider also an additional test for this formula, sensitive to the integral characteristic of the conductance such as the famous f-sum rule. We write it in the two dimensional form and show that it fulfils identically if we admit the cyclotron mass as an effective one and take the principal value of the integral. We find a deep relation between the graphene complex optical conductivity singularities and electrons Zitterbewegung (ZB) in graphene with frequency Namely, the value of ZB frequency is related with the recently found magnitudes of the inductance L and capacitance C by Thomson’s formula, that is,

Optical Analysis Using Effective Medium Theory and Finite Element Method to Study the Enhanced Light Absorption in Porous BaMgAl10O17:Eu2+ Phosphor

The main objective of this paper is to reveal the mechanism of enhanced excitation light absorption in nano-pores structure BaMgAl10O17:Eu2+ (BAM) phosphor by optical analysis. The optical refractive index of the BAM was calculated from the reflectance spectra by Kramers–Kronig dispersion relation. And based on the effective medium theory, the anisotropic optical properties of porous BAM layer and its relations of absorption enhancement with porosity and thickness were investigated. A finite element simulation model was used for study the influence of pores size on optical properties. All the numerically evaluated results were match the experimental data.

Study of vibrational modes in (Ga2S3)x−(Ga2Se3)1-x mixed crystals by Raman and infrared reflection measurements

Raman and infrared (IR) reflection characteristics were investigated in the frequency region of 100–450 cm−1 for (Ga2S3)x − (Ga2Se3)1-x mixed crystals for compositions of x increasing from 0.0 to 1.0 by intervals of 0.25 obtained by Bridgman crystal growth technique. In the Raman spectra of these crystals four dominant peak features were observed while two bands were detected in the IR spectra of interest samples. Kramers-Kronig dispersion relations applied to IR spectra presented the frequencies of transverse optical modes. The compositional dependencies of revealed Raman- and IR-active mode frequencies on (Ga2S3)x − (Ga2Se3)1-x crystals were established. One-mode behavior was displayed from indicated dependencies.

Two-model spectral study of volume holograms in materials with diffusion-based mechanisms

We devote this article to the spectral analysis of volume holograms recorded in materials with diffusion-based formation mechanisms. Two media of different natures were examined: polymer material with dye molecules photoattachable to macromolecules (PQ-PMMA) and photochromic additively colored fluorite crystal (CaF2). Differential spectra of holograms refractive index and absorbance were determined by two approaches: numerical approximation of both spectral and angular hologram response (selectivity hypercontour) by coupled-wave theory and processing hologram spectra by Kramers–Kronig dispersion relations. The principles of the two approaches, results of their application, and advantages and disadvantages are discussed. Kramers–Kronig relations can be usefully applied in cases where the nature of holograms formation is well studied, whereas the hypercontour approach is indispensable for obtaining the values of absorbance and refractive index modulation; therefore, a combination of both the techniques is a powerful tool for fundamental investigation of the hologram formation mechanisms.

Optical analysis using effective medium theory and finite element method to study the enhanced light absorption in porous BaMgAl-=SUB=-10-=/SUB=-O-=SUB=-17-=/SUB=- : Eu-=SUP=-2+-=/SUP=- phosphor

AbstractThe main objective of this paper is to reveal the mechanism of enhanced excitation light absorption in nano-pores structure BaMgAl_10O_17:Eu^2+ (BAM) phosphor by optical analysis. The optical refractive index of the BAM was calculated from the reflectance spectra by Kramers–Kronig dispersion relation. And based on the effective medium theory, the anisotropic optical properties of porous BAM layer and its relations of absorption enhancement with porosity and thickness were investigated. A finite element simulation model was used for study the influence of pores size on optical properties. All the numerically evaluated results were match the experimental data.

Determination of complex refractive index of SU-8 by Kramers–Kronig dispersion relation method at the wavelength range 2.5–22.0 µm

Accurate determination of the complex refractive index of SU-8 epoxy has significant for the wide variety of applications in optical sensor technology at IR range. The complex refractive index of SU-8 is determined by recording the transmission of light spectra for the wavelength range of 2.5–22.0 µm. The data analysis is based on the Kramers–Kronig dispersion relation method. The method has several merits, such as ease of operation, non-contact technique, measurement accuracy, and rapid measurement. The present method is not restricted to the case of SU-8 but it is also proposed to be applicable across a broad range of applications, such as assessment of the optical properties of paints and biomedical samples.

The Study on Dispersion Characteristic of Dusty Plasma

Dispersion is one of the important characteristics of dusty plasma. In this paper, the real and imaginary parts of the complex refractive index of dusty plasma are first deduced and verified by the Kramers-Kronig dispersion relation. And then, the dependence of dispersion ability on the temperature, pressure, angular frequency of incident electromagnetic wave, as well as the number density, radius, and charge of dust particles is given. Finally, the theoretical results indicate that: 1) the dispersion ability decreases with the increase of angular frequency; 2) the dispersion ability monotonically varies with the pressure approximately, but nonmonotonically varies with the temperature; and 3) the dispersion ability monotonically increases with the increase of dust particle's charge, but nonmonotonically changes with the number density and radius of dust particles. In particular, the involved physical explanations are proposed.

The Effect of Uniaxial Pressures on the Infrared Spectra of LiNH4SO4 Crystals

The infrared reflection spectra of mechanically free and uniaxially compressed LiNH4SO4 crystals are studied for the first time in the spectral range of 800–1700 сm–1 along three crystallophysical directions. The Kramers–Kronig dispersion relations are used to determine the dispersion and baric dependences of refractive index n and the real e1 and imaginary e2 parts of the dielectric constant and to calculate the frequencies of longitudinal ωLO and transverse ωТO vibrations, decay constant γ, and oscillator strength f of mechanically free and compressed LiNH4SO4 crystals. The considerable changes observed in the main reflection bands are explained by the effect of uniaxial pressures on the NH4 and SO4 tetrahedra.

UV–Vis–IR spectral complex refractive indices and optical properties of brown carbon aerosol from biomass burning

Constraining the complex refractive indices, optical properties and size of brown carbon (BrC) aerosols is a vital endeavor for improving climate models and satellite retrieval algorithms. Smoldering wildfires are the largest source of primary BrC, and fuel parameters such as moisture content, source depth, geographic origin, and fuel packing density could influence the properties of the emitted aerosol. We measured in situ spectral (375–1047 nm) optical properties of BrC aerosols emitted from smoldering combustion of Boreal and Indonesian peatlands across a range of these fuel parameters. Inverse Lorenz–Mie algorithms used these optical measurements along with simultaneously measured particle size distributions to retrieve the aerosol complex refractive indices (m = n + iκ). Our results show that the real part n is constrained between 1.5 and 1.7 with no obvious functionality in wavelength (λ), moisture content, source depth, or geographic origin. With increasing λ from 375 to 532 nm, κ decreased from 0.014 to 0.003, with corresponding increase in single scattering albedo (SSA) from 0.93 to 0.99. The spectral variability of κ follows the Kramers-Kronig dispersion relation for a damped harmonic oscillator. For λ ≥ 532 nm, both κ and SSA showed no spectral dependency. We discuss differences between this study and previous work. The imaginary part κ was sensitive to changes in FPD, and we hypothesize mechanisms that might help explain this observation.

Synthesis and characterization of nanosized pure α-cordierite glass-ceramic powders

This work reports the synthesis and characterization of 2MgO·2Al2O3·5SiO2 glass-ceramic powder produced through the Pechini method. The synthesized glass powder was pressed to form disk pellets, which were sintered at 1000, 1100, 1150, 1200, 1250, and 1300 °C. Moreover, the Raman and FTIR spectroscopies were used for better understanding of the phase transformation. Refractive index and extinction coefficient of the synthesized samples were determined by Kramers-Kronig dispersion relations using FTIR transmittance spectra. The SEM micrographs of the glass-ceramic showed a uniform distribution of the particles and growth of nanocrystallines with particle sizes between 42 and 48 nm. Besides, energy-dispersive spectroscopy proved stoichiometric composition of α-cordierite sintered at 1300 °C. The dielectric constant and the hardness of the glass-ceramic sintered at 1300 °C were measured. The value of the dielectric constant as low as 5.19 has been mainly ascribed to the high purity of the α-cordierite.

Subtleties in the acoustics of marine sediments

Based on the pioneering work of David Farmer, it is now is well recognized that the layer immediately beneath the sea surface is a highly dynamic, two-phase region, where bubbles created by wave breaking form an upward refracting sound speed profile that acts as an acoustic waveguide. The bottom boundary, although less dynamic than the sea surface, possesses its own unique complexities that are no less challenging to understand than those of the near-surface bubble layer. For instance, a growing body of experimental evidence indicates that the acoustic attenuation in a marine sediment obeys a frequency power law, extending over a wide bandwidth, in which the exponent takes a value close to unity. A long-standing problem has been to identify the frequency dispersion in the sound speed associated with such a power-law attenuation. Several solutions to this problem have been proposed over recent decades but are in fact unphysical in that they fail to obey the Kramers-Kronig dispersion relations. An alternative solution that has recently been developed, which overcomes the previous difficulties, contains a number of subtleties that, it is hoped, will appeal to David Farmer’s keen sense of scientific curiosity. [Research supported by ONR.]