The detection of weak features in spectra remains a challenge. Differentiation remains a standard method, but maximum-entropy filters provide an alternative. Using dielectric-function data for MoSe2 as a test case, we compare the capabilities of two maximum-entropy filters, the original Burg version and its corrected offshoot, to detect the biexciton and determine its energy. In contrast to differentiation, both approaches detect the biexciton. However, neither provides accurate values of the excitonic transition energies in this spectral range, although the Burg version is somewhat better.

Considerable progress has been made in the last few years in removing white noise from visible–near-ultraviolet (UV/VIS) spectra while leaving information intact. For x-ray diffraction, the challenges are different: detecting and locating peaks rather than line shape analysis. Here, we investigate possibilities of state-of-the-art UV/VIS methods for noise reduction, peak detection, and peak location applied to x-ray diffraction data, in this case, data for a ZrO2 −33 mol. % TaO4 ceramic. The same advantages seen in UV/VIS spectroscopy are found here as well.

For nearly 50 years, the Burg/Andersen (BA) maximum‐entropy (ME) deconvolution procedure has been used to sharpen, and therefore accentuate, weak features in spectra, optical and otherwise. It is shown that the BA procedure can be further enhanced to yield even sharper features. Using this approach, the authors establish the existence of the biexciton in ellipsometric data of monolayer WS2 at 50 K, whereas differentiation and standard BA analysis cannot.

The molecule-plasmon interaction is the key to the mechanisms of surface enhanced infrared absorption (SEIRA) and surface enhanced Raman scattering (SERS). Since plasmons are well described by Maxwell's equations, one fundamental treatment involves the classical interpretation of infrared absorption and resonance Raman spectroscopies. We can understand the molecule-plasmon interaction using electromagnetic theory if the classical field effect on a transition dipole moment or transition polarizability is properly described. In previous work, we derived the Raman excitation profile of a model molecule using a classical driven spring attached to a charged mass with a perturbative force constant due to vibrational oscillations. In this study we generalize the interactions of plasmons with molecules by considering the N2O asymmetric stretch SEIRA signal on a Dy doped CdO (CdO:Dy) film. This semiconductor has tunable plasmon dispersion curves throughout the near-and mid-infrared that can interact directly with vibrational absorption transitions. We have demonstrated this using the Kretschmann configuration with a CaF2 prism and a MgO substrate. The model predicts the phase behavior of SEIRA. The calculated enhancement factor relative to an Au control is 6.2, in good agreement with the value of 6.8 ± 0.5 measured under the same conditions.

In linear filtering, high-frequency (white) noise is reduced by apodization, which is the attenuation or elimination of high-order Fourier coefficients followed by an inverse transformation. Unfortunately, apodization requires compromises to be made among noise leakage, information loss, and Gibbs oscillations. These shortcomings are avoided with the corrected maximum-entropy (CME) procedure, but this procedure applies only to Lorentzian or approximately Lorentzian features. We develop a generalized maximum-entropy method based on partial Hilbert transforms that allows CME to be applied to any spectrum, thereby eliminating white-noise completely with no deleterious side effects. As Hilbert transforms are exact Kramers–Kronig replicas of the original endpoint-discontinuity-corrected segment, new spectral processing opportunities are also realized.

Eliminating noise from spectra has been a goal in spectroscopy from its beginning. With recent advances, the goal of removing white noise completely from spectra with no deleterious side effects is now within sight. This review provides necessary background and summarizes the current state of the art.

In the last several years, considerable progress has been made in reducing or eliminating noise in ellipsometric spectra, including the development of algorithms for eliminating endpoint-discontinuity artifacts, a reciprocal-space expression for quantifying the effectiveness of linear filters, and the corrected maximum-entropy (CME) approach for eliminating apodization and its associated errors. These lead to new capabilities, together with additional opportunities for extracting information from spectra. These results enhance the utility of ellipsometry for analyzing surfaces, interfaces, materials, and structures. Examples are provided.

For over five decades, the mathematical procedure termed “maximum entropy” (M-E) has been used to deconvolve structure in spectra, optical and otherwise, although quantitative measures of performance remain unknown. Here, we examine this procedure analytically for the lowest two orders for a Lorentzian feature, obtaining expressions for the amount of sharpening and identifying how spurious structures appear. Illustrative examples are provided. These results enhance the utility of this widely used deconvolution approach to spectral analysis.

In spectroscopy, the objective is to obtain information by analyzing spectra that ideally are undistorted and noise-free. In standard Fourier-space filtering, this goal cannot be achieved because of apodization, which forces a trade-off among errors arising from distortion, noise leakage, and Gibbs oscillations. We show that low-order coefficients can be preserved and apodization, and its associated errors eliminated with the corrected maximum-entropy (M-E) filter obtained here. Although the Burg derivation begins as M-E, by making certain assumptions the Burg approach yields a procedure that deconvolves (sharpens) structure in spectra, thereby violating the basic M-E principle of leaving the low-order coefficients intact. The corrected solution preserves these data and projects the trends established by them into the white-noise region in a model-independent way, thereby eliminating apodization and its associated errors. For a single Lorentzian line, the corrected M-E approach has an exact analytic solution, which reveals not only how M-E performs its extension but also why it works particularly well for line shapes resulting from first-order decay processes. The corrected M-E filter is quantitatively superior to any previous filtering method, including recently proposed high-performance linear filters, yet requires only minimal computational effort. Examples, including multiple differentiation, are provided.

For over five decades the procedure termed maximum-entropy (M-E) has been used to sharpen structure in spectra, optical and otherwise. However, this is a contradiction: by modifying data, this approach violates the fundamental M-E principle, which is to extend, in a model-independent way, trends established by low-index Fourier coefficients into the white-noise region. The Burg derivation, and indirectly the prediction-error equations on which sharpening is based, both lead to the correct solution, although this has been consistently overlooked. For a single Lorentzian line these equations can be solved analytically. The resultant lineshape is an exact autoregressive model-1 (AR(1)) replica of the original, demonstrating how the M-E reconstruction extends low-index Fourier coefficients to the digital limit and illustrating why this approach works so well for lineshapes resulting from first-order decay processes. By simultaneously retaining low-index coefficients exactly and eliminating Gibbs oscillations, M-E noise filtering is quantitatively superior to that achieved by any linear method, including the high-performance filters recently proposed. Examples are provided.

Linear noise-reduction filters used in spectroscopy must strike a balance between reducing noise and preserving lineshapes, the two conflicting requirements of interest. Here, we quantify this tradeoff by capitalizing on Parseval's Theorem to cast two measures of performance, mean-square error (MSE) and noise, into reciprocal- (Fourier-) space (RS). The resulting expressions are simpler and more informative than those based in direct- (spectral-) space (DS). These results provide quantitative insight not only into the effectiveness of different linear filters, but also information as to how they can be improved. Surprisingly, the rectangular ("ideal" or "brick wall") filter is found to be nearly optimal, a consequence of eliminating distortion in low-order Fourier coefficients where the major fraction of spectral information is contained. Using the information provided by the RS version of MSE, we develop a version that is demonstrably superior to the brick-wall and also the Gauss-Hermite filter, its former nearest competitor.

We present a method of reducing noise in spectra that is based on eliminating low-order derivatives of reciprocal-space (RS) filter functions, yet ensuring that the functions roll off smoothly to minimize Gibbs oscillations. The approach takes advantage of the fact that information and noise are separated in RS. The method preserves as much information as possible, while reducing or even eliminating unwanted contributions (noise). To demonstrate the method we apply it to a model spectrum, data including an XPS spectrum of S2p in hierarchical NiCo2S4 nanosheets, and the Raman spectrum of 10-layer film of FePS3 with polarization direction of 90° with respect to the a-axis.

A classical correlation model (CCM), based on forces instead of potentials, is developed and applied to resonance Raman scattering to provide a foundation for further advances in understanding the effects of fields and vibronic perturbations on the optical properties of materials by a simple, yet versatile, description. The model consists of a charge connected by a classical spring to a surface and driven by an external electric field. The spring represents the charge cloud of the electrons and the transition strength, and the surface represents the nucleus or molecule. Molecular vibrations are assumed to be many-body effects that change the configuration and hence modify the spring constant directly, as opposed to all previous classical models of Raman scattering, and opposed to the anisotropic bond model (ABM) of nonlinear optics, by adding anharmonic terms to the potential. The resulting expression agrees exactly with quantum mechanical models of resonance Raman scattering in the limit of weak electron-phonon coupling, and it agrees well when the coupling becomes strong. The result is a classical derivation of Kramers-Heisenberg-Dirac scattering theory. We show that the difference between classical and quantum approaches lies only in the interpretation of the prefactor. In particular, the Raman excitation profile shows excellent agreement with all other methods of calculation. By comparing complementary classical and quantum solutions of the same complex system, understanding of both is enhanced.

We determine the infrared absorption spectra of a gas due to evanescent plasmonic electromagnetic fields in a system where surface interactions (physisorption and chemisorption) are demonstrably negligible. The plasmonic host material, the degenerate semiconductor CdO:Dy, has high mobility (366-450 cm2/(V s)) and carrier density ((0.6-3.5) × 1020 cm-3), and therefore supports low-loss surface plasmon resonances in the mid-IR. This high-mobility layer gives the highest resolution observed in a plasmonic conducting material in the infrared, higher than that of gold and rivaling that of silver. The high resolution permits a new understanding of the nature of the interaction of emerging fields with molecular transitions. Using different carrier concentrations, the resonance condition of the surface plasmon polariton (SPP) frequency (ωSPP) and N2O vibrational absorption spectral frequency (ωN2O) can be controlled, thereby allowing a critical test of field-molecule interactions. Experiment and theory both indicate a dispersive N2O line shape for ωSPP < ωN2O, an absorptive line shape for ωSPP < ωN2O, and an abrupt change between the two when the resonance condition ωSPP < ωN2O is reached. A first-order expansion of the Airy equation describes this behavior analytically. The SPP surface enhancement is 6.8 ± 0.5 on-resonance, lower than enhancements observed in other systems, but in agreement with recent quantitative reports of surface enhanced infrared reflection absorption spectroscopy (SEIRA). Our results show that interactions of infrared SPPs with molecular vibrations are in the weak coupling limit, and that enhancements comparable those reported for noble metals can be achieved.

We present a systematic method of removing endpoint-discontinuity artifacts in the Fourier analysis of spectral segments, enabling the more accurate extraction of information. This principal-component-removal approach differs from a previous version by using extrapolated (or extended) data outside rather than inside the spectral range. This not only allows coefficients to be accessed to the white-noise limit with no distortion of the segment, but also generates interpolated coefficients for improved analytic insight. Examples are provided.

We report the pseudodielectric functions and the critical points of GaAs x Sb 1− x ternary alloy films. Data were obtained by performing spectroscopic ellipsometry on 1-μm-thick films grown on (001) GaAs by using molecular beam epitaxy. Artifacts from surface contaminants, including oxide overlayers, were minimized by using in-situ chemical cleaning, leading to accurate representations of the bulk dielectric responses of these materials. We determined the energies of the E 1 , E 1 +Δ 1 , E ′ 0 , E ′ 0 +Δ′ 0 , E 2 , E ′ 2 , and E ′ 1 critical points from numerically calculated second energy derivatives, as well as their compositional dependences by using lineshape fitting.

The authors present a simple, convenient, and accurate noise-reduction approach for interpolating spectra, in particular, for converting those available as discrete points equally spaced in wavelength, acquired, for example, by a photodiode-array detector, to equivalent spectra equally spaced in energy, as needed for analysis. Based on continuum mathematics, the algorithm uses Gaussian kernels and capitalizes on the fact that trapezoidal-rule integration is accurate to fourth order in the ratio of point separation to width when applied to Gaussian functions. The approach can be expanded to perform differentiation and other operations. Examples include false-data verification, wavelength-to-energy conversion of near-bandgap interference oscillations of a GaN film, and calculation of the second energy derivative of monolayer MoS2 in the exciton region.

The authors investigate linear and nonlinear methods of reducing noise while preserving information in spectra, optical and otherwise. The optimum linear and nonlinear approaches are Gauss–Hermite and maximum-entropy, respectively. However, intelligent processing still requires an initial assessment of the data in reciprocal space.

The electron-core-hole interaction is studied via energy derivative reflectance spectra of 20-eV transitions from Ga 3dcore levels to lower conduction-band final states in GaAs1−xPx alloys. A two-level anticrossing behavior of line shapes and threshold energies as the relative positions of the L and X minima invert yields a previously unanticipated L−Xmixing energy |VLX|∼50 meV.