Multiply subtractive kramers-kronig analysis of optical data

A unified framework for the numerical evaluation of the Q-subtractive Kramers-Kronig relations and application to the reconstruction of optical constants of quartz

The direct usage of the Kramers-Kronig (KK) relations is complicated by two factors: limited frequency range of the available spectra and experimental errors. Here, we reconsider the application of the KK relations to experimental data for the construction of a self-consistent set of optical constants over a wide spectral range: the real part of the complex optical constant, F₁, is reconstructed using the imaginary part F₂, obtained from an experiment. The focus is on multiply (Q-)subtractive KK relations, which in contrast to the standard KK transformation, exploit information about F₁ at a certain number Q of anchor frequencies. We develop a general mathematical framework of the Q-subtractive KK relations and analyze all sources of errors contributing to the inaccuracy of the reconstructed F₁. We show that for the reconstruction of F₁ only a single evaluation of the standard KK relation is needed together with a correction term given by an approximate evaluation of the error in the standard KK. It is demonstrated that in the classical form of the Q-subtractive KK relations, this correction term coincides with the Lagrange interpolation polynomial of the error with nodes at the anchor frequencies. Another correction term can also be constructed as a lower degree polynomial through a least squares fit, a particular realization of which is taking the average of Q singly subtractive KK relations. As a result, recommendations for the application of Q-subtractive KK relations are given. The accuracy of the considered approaches is illustrated on synthetic examples and experimental data of fused SiO₂.

A New Simple Analytical Method for a Highly Accurate Determination of the Optical Parameters of a Slab from Transmittance Data

To date, determining with high accuracy the optical parameters (extinction coefficient k and refractive index n) of a slab from the sole transmittance data requires an inverse method based on numerical iteration procedures. In this paper, we propose a new inverse analytical method of extracting (k, n) without numerical iterative processes. The high accuracy of this new inverse method is assessed, and as an application example, the optical parameters of CaF₂ and Si substrates are determined in the IR spectral range of 4-8 µm.

Ordinary and Extraordinary Complex Refractive Indices Extraction of a Mylar Film by Transmission Spectrophotometry

A new spectrophotometric technique for the determination of both ordinary and extraordinary complex refractive indices (CRIs) of a stretched polyethylene terephthalate (Mylar) film is proposed. The sample was placed between two identical polarizers, and the transmission spectra of two different configurations (incident polarization parallel and perpendicular to the Mylar film optical axis) were recorded. Ordinary and extraordinary complex refractive indices are then extracted by fitting the experimental spectra with a theoretical model that we had elaborated in advance. A new formula for transmittance dispersion, based on the Fresnel’s coefficients formalism and using the Cauchy model, was derived to describe n and κ wavelength dependence. The suggested theoretical model succeeded in reproducing the Mylar transmission spectra across the entire visible spectral range (400, 750 nm) for both configurations, and the retrieved dispersion curves of the refractive indices, extinction coefficients, and the birefringence are comparable to results found in the literature. The proposed method is fast, straightforward, easy to set up, and cost-effective. It proved to be an excellent alternative to more conventional methods such as spectroscopic ellipsometry.

Infrared refraction spectroscopy - Kramers-Kronig analysis revisited

Since about 60 years, it is possible to determine the set of optical constants from reflectance measurements by the Kramers-Kronig relations. Unfortunately, the potential of the method seems to be limited in practice by the need to extend measured data into unknown regions by extrapolation, which is prone to error and leads to deviations from the true values in the known region depending on the method of extension. With the advent of infrared refraction spectroscopy, which uses reflectance measurements at normal or near normal incidence, the development of a fast method to determine the complex index of refraction function reliably for these cases has become an even more interesting goal. In this work, we introduce a way to perform a Kramers-Kronig analysis of such reflectance spectra without the need to extrapolate the measured data for higher wavenumbers beyond the measurement range. Furthermore, we introduce and compare very simple and reliable material-independent ways to extrapolate the measured data for lower wavenumbers. As a result, we devise fast and sufficiently accurate methods, which are easily implementable into existing spectrometer software, to extract quantitative spectral information from reflectance measurements at near normal incidence.

Generalized framework for determining time origin in terahertz emission spectroscopy on the basis of causality

We develop a generalized causality-based framework for determining the time origin in terahertz emission spectroscopy. Our framework is formulated in terms of a multiply subtractive Kramers-Kronig relation and can treat all major mechanisms of terahertz emission, which include the occurrence of a delta-function-like instantaneous polarization observed typically in nonlinear optical processes. We show that a function derived within our framework properly determines the positions of t = 0 both for simulated terahertz waveforms and for a measured one obtained in biased conjugated polymers. This function will be useful for an in-depth understanding of ultrafast phenomena involving terahertz emission in various optoelectronic materials.

Wavelength-dependent excess permittivity as indicator of kerosene in diesel oil

Adulteration of diesel oil by kerosene is a serious problem because of air pollution resulting from car exhaust gases. The objective of this study was to develop a relatively simple optical measurement and data analysis method to screen low-adulterated diesel oils. For this purpose, we introduce the utilization of refractive index measurement with a refractometer, scanning of visible-near-infrared transmittance, transmittance data inversion using the singly subtractive Kramers-Kronig relation, and exploitation of so-called wavelength-dependent relative excess permittivity. It is shown for three different diesel oil grades, adulterated with kerosene, that the excess permittivity is a powerful measure for screening fake diesel oils. The excess relative permittivity of such binary mixtures also reveals hidden spectral fingerprints that are neither visible in dispersion data alone nor in spectral transmittance measurements alone. We believe that the excess permittivity data are useful in the case of screening adulteration of diesel oil by kerosene and can further be explored for practical sensing solutions, e.g., in quality inspection of diesel oils in refineries.

Optical and terahertz spectra analysis by the maximum entropy method

Phase retrieval is one of the classical problems in various fields of physics including x-ray crystallography, astronomy and spectroscopy. It arises when only an amplitude measurement on electric field can be made while both amplitude and phase of the field are needed for obtaining the desired material properties. In optical and terahertz spectroscopies, in particular, phase retrieval is a one-dimensional problem, which is considered as unsolvable in general. Nevertheless, an approach utilizing the maximum entropy principle has proven to be a feasible tool in various applications of optical, both linear and nonlinear, as well as in terahertz spectroscopies, where the one-dimensional phase retrieval problem arises. In this review, we focus on phase retrieval using the maximum entropy method in various spectroscopic applications. We review the theory behind the method and illustrate through examples why and how the method works, as well as discuss its limitations.

Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge

A practical and accurate method to obtain the index of refraction, especially the decrement δ, across the carbon 1s absorption edge is demonstrated. The combination of absorption spectra scaled to the Henke atomic scattering factor database, the use of the doubly subtractive Kramers-Kronig relations, and high precision specular reflectivity measurements from thin films allow the notoriously difficult-to-measure δ to be determined with high accuracy. No independent knowledge of the film thickness or density is required. High confidence interpolation between relatively sparse measurements of δ across an absorption edge is achieved. Accurate optical constants determined by this method are expected to greatly improve the simulation and interpretation of resonant soft x-ray scattering and reflectivity data. The method is demonstrated using poly(methyl methacrylate) and should be extendable to all organic materials.

Causality-based method for determining the time origin in terahertz emission spectroscopy

We propose a method for determining the time origin on the basis of causality in terahertz (THz) emission spectroscopy. The method is formulated in terms of the singly subtractive Kramers-Kronig relation, which is useful for the situation where not only the amplitude spectrum but also partial phase information is available within the measurement frequency range. Numerical analysis of several simulated and observed THz emission data shows that the misplacement of the time origin in THz waveforms can be detected by the method with an accuracy that is an order of magnitude higher than the given temporal resolutions.

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

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.

Retrieval of linear optical functions from finite range spectra

There are many experimental situations in which infrared reflectivity spectra can be acquired only over a limited spectral range. It is therefore necessary to find computing procedures that allow the efficient analysis of such data. In this paper, we propose a new procedure labeled constrained finite range correction (CFRC) that can be advantageously substituted to multiply subtractive Kramers-Kronig relations. The constrained finite range correction is able to produce realistic results even when very little supplementary information is available. For semitransparent crystals, the hypothesis of the phase spectrum positiveness alone is often sufficient to compute satisfactory approximations of the optical functions. The efficiency of the new method is shown through the analysis of several synthetic and experimental spectra.

Finite-bandwidth Kramers-Kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions

Kramers-Kronig (KK) analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of KK integrals for a complete infinite spectrum of input information. For data exhibiting localized extrema, KK relations can provide accurate transforms over finite bandwidths due to the local-weighting properties of the KK kernel. Recently, acoustic KK relations have been derived for the determination of the group velocity (cg) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number). These relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters. In contrast to twice-subtracted finite-bandwidth KK predictions for phase velocity and attenuation coefficient (components of the undifferentiated wave number), these more recently derived relations for cg and alpha' provide stricter tests of causal consistency because the resulting shapes are invariant with respect to subtraction constants. The integrals in these relations can be formulated so that they only require the phase velocity and attenuation coefficient data without differentiation. Using experimental data from suspensions of encapsulated microbubbles, the finite-bandwidth KK predictions for cg and alpha' are found to provide an accurate mapping of the primary wave number quantities onto their derivatives.

Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory

A comparison of two theoretical methods based on time-dependent density functional theory for the calculation of the linear dispersive and absorptive properties of chiral molecules has been made. For this purpose, a recently proposed computational method for the calculation of circular dichroism (CD) spectra from the imaginary part of the optical rotation parameter has been applied to six rigid organic molecules. The results have been compared to the CD spectra obtained from the rotatory strengths and from the Kramers-Kronig transformation of optical rotatory dispersion (ORD) curves. We have also investigated a criterion based on the Kramers-Kronig integration formula to determine a number of excitations in truncated CD spectra which may yield a reasonable low frequency resonant ORD. It has been tested by calculating the ORD from the sum-over-states formula both in the nonresonant and resonant regions. Finally, we have applied these methods to model the resonant optical activity of proline at low pH.

Comparison of subtractive Kramers-Kronig analysis and maximum entropy model in resolving phase from finite spectral range reflectance data

The maximum entropy model (MEM) and Kramers-Kronig (K-K) analysis were compared with the aim of phase retrieval from reflectance. The object was to test two different phase-retrieval methods when reflectance is known at a finite frequency range and data fitting is not performed beyond the finite frequency band. In addition, it was assumed that the phase is known only at one or two anchor points. As an example, we study the terahertz reflection spectrum related to a semiconductor and an optical spectrum of potassium chloride. It is shown that the MEM resolves the complex refractive index of a medium, in the vicinity of initial and final points of the spectra, better than singly and doubly subtractive K-K relations. Both methods give only satisfactory results in the event of one anchor point, but in the case of two anchor points, the MEM is better than doubly subtractive K-K. It is proposed that the MEM should be used instead of K-K analysis, for a priori information of phase at two anchor points, for the purpose of resolving the complex refractive index of a medium from reflectance with high accuracy.

Causal determination of acoustic group velocity and frequency derivative of attenuation with finite-bandwidth Kramers-Kronig relations

Kramers-Kronig (KK) analyses of experimental data are complicated by the extrapolation problem, that is, how the unexamined spectral bands impact KK calculations. This work demonstrates the causal linkages in resonant-type data provided by acoustic KK relations for the group velocity (c(g)) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number) without extrapolation or unmeasured parameters. These relations provide stricter tests of causal consistency relative to previously established KK relations for the phase velocity (c(p)) and attenuation coefficient (alpha) (components of the undifferentiated acoustic wave number) due to their shape invariance with respect to subtraction constants. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the entire infinite spectrum, but differ when restricted to bandlimited spectra. Using experimental data from suspensions of elastic spheres in saline, the accuracy of finite-bandwidth KK predictions for c(g) and alpha' is demonstrated. Of the multiple methods, the most accurate were found to be those whose integrals were expressed only in terms of the phase velocity and attenuation coefficient themselves, requiring no differentiated quantities.

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.

Estimation of the wavelength-dependent effective refractive index of spherical plastic pigments in a liquid matrix

Refractive-index data are important for the prediction of light scattering from spherical pigments. Reflectance from a slurry that contains plastic pigments was studied with the aid of a reflectometer. The effective refractive index of spherical plastic pigments in a slurry was determined by use of reflectance data and a phase-retrieval procedure based on the maximum-entropy method. This method provides a simple way to estimate the effective refractive index of pigments in a liquid matrix.