Polarization-dependent effects in vibrational absorption spectra of 2D finite-size adsorbate islands on dielectric substrates.

In the last few years, infrared reflection-absorption spectroscopy (IRRAS) has become a standard technique to study vibrational excitations of molecules. These investigations are strongly motivated by potential applications in monitoring chemical processes. For a better understanding of the adsorption mechanism of molecules on dielectrics, the polarization-dependence of an interaction of infrared light with adsorbates on dielectric surfaces is commonly used. Thus, the peak positions in absorption spectra could be different for s- and p-polarized light. This shift between the peak positions depends on both the molecule itself and the dielectric substrate. While the origin of this shift is well understood for infinite two-dimensional adsorbate layers, finite-size samples, which consist of 2D islands of a small number of molecules, have never been considered. Here, we present a study on polarization-dependent finite-size effects in the optical response of such islands on dielectric substrates. The study uses a multi-scale modeling approach that connects quantum chemistry calculations with Maxwell scattering simulations. We distinguish the optical response of a single molecule, a finite number of molecules, and a two-dimensional adsorbate layer. We analyze CO and CO2 molecules deposited on CeO2 and Al2O3 substrates. The evolution of the shift between the polarization-dependent absorbance peaks is first studied for a single molecule, which does not exhibit any shifting at all, and for finite molecular islands, where it increases with increasing island size, as well as for an infinite two-dimensional adsorbate layer. In the latter case, the agreement between the obtained results and the experimental IRRAS data and more traditional three/four-layer model theoretical studies supports the predictive power of the multi-scale approach.

Understanding Refractive Index Changes in Homologous Series of Unbranched Organic Compounds Based on Beer's Law.

Changes of the refractive index for homologous series of hydrocarbons are usually plotted versus the density. While there is a clear linear dependence for alkanes and alkenes, the linearity deteriorates for homologous series with functional groups involving heteroatoms. The slope can even become negative, e. g., for carboxylic acids. For gaining a deeper understanding and to establish a more general correlation, we reinvestigate the corresponding theories starting with the Newton-Laplace, Gladstone-Dale and the Lorentz-Lorenz rules. We revisit the concept of molar refractivity pioneered by Landolt and Brühl and show that it is closely connected with a twin of Beer's law. We conclude that the refractive index of homologues series should better be plotted versus the molar concentration of the main UV-chromophore, the C-H bond, which actually causes the refractive index changes. This new approach is not limited to alkanes and alkenes but holds for homologous series with functional groups including heteroatoms.

Hybrid 2D Correlation-Based Loss Function for the Correction of Systematic Errors.

We present the derivation of a new kind of loss function from the symmetry rules of synchronous and asynchronous two-dimensional correlation maps. This loss function, which takes into account correlations that are based on causal relations among the members of a series of spectra, can be employed to solve non-linear inverse problems that are plagued by systematic multiplicative errors. This possibility results from the correlation-based loss function being practically insensitive to such systematic errors, which often arise in spectroscopy because sample spectra are usually ratioed against reference spectra. Using dispersion analysis, a sophisticated method of band fitting, of the spectra of poly(methyl methacrylate) films deposited on gold, we demonstrate the applicability and validity of the new loss function. If gold is used as a substrate, experimental spectra are often unphysical, that is, they display reflectance values larger than unity. In such cases, our correlation-based loss function not only helps to achieve accurate fits but also provides corrections to obtain physically meaningful spectra, which leads to results that are superior to conventional correction methods. The validity of the results is checked and proved with help of the results of dispersion analysis of spectra of films of poly(methyl methacrylate) on calcium fluoride (CaF2) and silicon (Si), which do not suffer from the systematic errors.

To generate a photonic nanojet outside a high refractive index microsphere illuminated by a Gaussian beam.

A non-resonant, concentrated, narrow beam of light emerging from an illuminated microlens is called a photonic nanojet (PNJ). According to currently prevailing opinion, microspheres and microcylinders are only able to generate a PNJ in their exterior when their refractive index ns (or refractive index contrast) is less than 2. In this Letter we demonstrate that a PNJ can emerge from a microsphere even when ns > 2: first by employing the laws of geometrical optics for a divergent light source; then, by using ray transfer matrix analysis, a mathematical condition for the Gaussian beam (GB) outside the high ns microsphere is derived. The PNJ outside the microsphere with ns = 2.5 is simulated using Generalized Lorenz-Mie theory (GLMT), by using a front focused GB source. The simulated difference between front and back focusing on the dependence of ns is confirmed experimentally by Raman imaging. By opening the PNJ field for high refractive index materials, we believe this work will be a nucleus for new ideas in the field and enable new PNJ applications.

The Negative Solvatochromism of Reichardt's Dye B30 - A Complementary Study.

The UV/Vis spectra of a hypothetical negative solvatochromic dye in a solvent are theoretically calculated assuming the classical damped harmonic oscillator model and the Lorentz-Lorenz relation. For the simulations, the oscillator strength of the solvent was varied, while for the solute all oscillator parameters were kept constant. As a result, a simple change of the oscillator strength of the solute can explain the redshift and intensity increase of the UV/Vis band of the solute. Simulated results are compared with measured UV/Vis spectroscopic data of 2,6-diphenyl-4-(2,4,6-triphenylpyridinium-1-yl) phenolate B30 (Reichardt's dye) Significant correlations of the absorption energy (1/λmax ) with the molar absorption coefficient ϵ as function of solvent polarity are demonstrated for several derivatives of B30. The approach presented is only applicable to negative solvatochromism. Therefore, while the approach is vital to fully understand solvatochromism, it needs to be complemented by other approaches, e. g., to describe the changes of the chemical interactions based on the nature of the solvent, to explain all its various aspects.

CaF2: An Ideal Substrate Material for Infrared Spectroscopy?

CaF2 seems to be the ideal substrate material for the infrared spectroscopy of organic and biological layers, since its refractive index is very similar to that of these materials. As a consequence of this similarity, the baseline, i.e., the signal strength in nonabsorbing regions, is nearly flat and does not show notable interference fringes. Nevertheless, as absorption is always accompanied by changes of the refractive index, the refractive indices of substrate and layer can substantially deviate around absorption bands. As a consequence, changes in band intensity, shape, and position result, which aggravate a correct interpretation of the spectra. For layers with thicknesses between 1 and 2 µm, we show experimentally, that deviations from the Beer-Lambert law of up to ±10% occur. Calculations reveal that for thinner layers these deviations are even higher. These results suggest the application of a wave-optics based formalism to correct the deviations. We introduce such a formalism and prove that it is able to remove the errors. In addition, it also corrects band shape and position changes.

Beyond Beer's Law: Revisiting the Lorentz-Lorenz Equation.

In this contribution we show how the Lorentz-Lorenz and the Clausius-Mosotti equations are related to Beer's law. Accordingly, the linear concentration dependence of absorbance is a consequence of neglecting the difference between the local and the applied electric field. Additionally, it is necessary to assume that the absorption index and the related refractive index change is small. By connecting the Lorentz-Lorenz equations with dispersion theory, it becomes obvious that the oscillators are coupled via the local field. We investigate this coupling with numerical examples and show that, as a consequence, the integrated absorbance of a single band is in general no longer linearly depending on the concentration. In practice, the deviations from Beer's law usually do not set in before the density reaches about one tenth of that of condensed matter. For solutions, the Lorentz-Lorenz equations predict a strong coupling also between the oscillators of solute and solvent. In particular, in the infrared spectral region, the absorption coefficients are prognosticated to be much higher due to this coupling compared to those in the gas phase.

The Bouguer-Beer-Lambert Law: Shining Light on the Obscure.

The Beer-Lambert law is unquestionably the most important law in optical spectroscopy and indispensable for the qualitative and quantitative interpretation of spectroscopic data. As such, every spectroscopist should know its limits and potential pitfalls, arising from its application, by heart. It is the goal of this work to review these limits and pitfalls, as well as to provide solutions and explanations to guide the reader. This guidance will allow a deeper understanding of spectral features, which cannot be explained by the Beer-Lambert law, because they arise from electromagnetic effects/the wave nature of light. Those features include band shifts and intensity changes based exclusively upon optical conditions, i. e. the method chosen to record the spectra, the substrate and the form of the sample. As such, the review will be an essential tool towards a full understanding of optical spectra and their quantitative interpretation based not only on oscillator positions, but also on their strengths and damping constants.

Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration.

Beer's empiric law states that absorbance is linearly proportional to the concentration. Based on electromagnetic theory, an approximately linear dependence can only be confirmed for comparably weak oscillators. For stronger oscillators the proportionality constant, the molar attenuation coefficient, is modulated by the inverse index of refraction, which is itself a function of concentration. For comparably weak oscillators, the index of refraction function depends, like absorbance, linearly on concentration. For stronger oscillators, this linearity is lost, except at wavenumbers considerably lower than the oscillator position. In these transparency regions, linearity between the change of the index of refraction and concentration is preserved to a high degree. This can be shown with help of the Kramers-Kronig relations which connect the integrated absorbance to the index of refraction change at lower wavenumbers than the corresponding band. This finding builds the foundation not only for refractive index sensing, but also for new interferometric approaches in IR spectroscopy, which allow measuring the complex index of refraction function.

Removing interference-based effects from infrared spectra - interference fringes re-revisited.

Substantial refractive index mismatches between substrate and layers lead to undulating baselines, which are known as interference fringes. These fringes can be attributed to multiple reflections inside the layers. For thin and plane parallel layers, these multiple reflections result in wave interference and electric field intensities which strongly depend on the location within the layer and wavenumber. In particular, the average electric field intensity is increased in spectral regions where the reflectance is reduced. Therefore, the most important precondition for the Beer-Lambert law to hold, absorption as the single reason for electric field intensity changes, is no longer valid and, since absorption is proportional to the electric field intensity, considerable deviations from the Beer-Lambert law result. Fringe removal is consequently synonymous with correcting deviations from the Beer-Lambert law in the spectra. Within this contribution, we introduce an appropriate formalism based on wave optics, which allows a particularly fast and simple correction of any interference based effects. We applied our approach for correcting transmittance spectra of Poly(methyl methacrylate) layers on silicon substrates. The interference effects were successfully removed and correct baselines, in good agreement with the calculated spectra, were obtained. Due to its sound theoretical foundation, our formalism can be used as benchmark to test the performance of other methods for interference fringe removal.

Quantitative Evaluation of Infrared Absorbance Spectra - Lorentz Profile versus Lorentz Oscillator.

We present the theoretical basis for a profound upgrade of the method of absorbance band fitting ("band deconvolution"), which requires only minute changes in the code of corresponding spectrometer software. This upgrade is based on a (re-)connection of the damped harmonic oscillator model ("Lorentz oscillator") and the Lorentz profile used for band fitting. Based on this reconnection, we provide a proper extension to multiple oscillators. As a result, band fitting allows directly obtaining all oscillator parameters with very good accuracy, at least for the not too strong oscillators present in organic and biological matter. Accordingly, this could be the initial spark to open the way to a long-awaited paradigm shift in infrared spectroscopy: Away from a mere oscillator position-based, towards an also intensity-based quantitative interpretation of spectra. As an extra, absorbance band fitting ("Poor Man's Dispersion Analysis"), allows to obtain the index of refraction function in one go.

Beer's Law - Why Absorbance Depends (Almost) Linearly on Concentration.

Beer's law assumes a strictly linear dependence of the absorbance from concentration. Usually, chemical interactions and instrumental imperfection are made responsible for experimental deviations from this linearity. In this contribution we show that even in the absence of such interactions and instrumental errors, absorbance should be only approximately proportional to concentration. This can be derived from the quadratic dependence of the complex refractive index, and, by that, of the molar attenuation coefficient, from the dielectric constant and its frequency dispersion. Following dispersion theory, it is the variation of the real and the imaginary part of the dielectric function that depends linearly on concentration in the absence of interactions between the oscillators. We show that this linear correlation translates into a linear dependence of the absorbance for low concentrations or molar oscillator strengths based on an approximation provided by Lorentz in 1906. Accordingly, Beer's law can be derived from dispersion theory.

Beer's Law-Why Integrated Absorbance Depends Linearly on Concentration.

As derived by Max Planck in 1903 from dispersion theory, Beer's law has a fundamental limitation. The concentration dependence of absorbance can deviate from linearity, even in the absence of any interactions or instrumental nonlinearities. Integrated absorbance, not peak absorbance, depends linearly on concentration. The numerical integration of the absorbance leads to maximum deviations from linearity of less than 0.1 %. This deviation is a consequence of a sum rule that was derived from the Kramers-Kronig relations at a time when the fundamental limitation of Beer's law was no longer mentioned in the literature. This sum rule also links concentration to (classical) oscillator strengths and thereby enables the use of dispersion analysis to determine the concentration directly from transmittance and reflectance measurements. Thus, concentration analysis of complex samples, such as layered and/or anisotropic materials, in which Beer's law cannot be applied, can be achieved using dispersion analysis.

Deviations from Beer's law on the microscale - nonadditivity of absorption cross sections.

Beer's law assumes a linear dependence of absorbance on concentration, accordingly the index of absorption and the molar attenuation coefficient are material properties and the absorption cross section, including absorbance itself, must be additive if chemical interactions are excluded. Under the "no interaction" condition, a linear dependence should also exist between macroscopic polarization and the number of induced dipole moments per unit volume. The latter linear dependence is the basis for dispersion theory. Invoking Maxwell's wave equation, Beer's law has been derived recently from dispersion theory. As a result, Beer's law is a limiting law. Accordingly, indices of absorption and molar attenuation coefficients including absorption cross sections are no material properties and the latter can per se not be additive. Indeed, as we show in this contribution, not even for particles very small compared to wavelength, where the scattering cross sections can be neglected, is this additivity a given, except for comparably large distances between the particles. We investigate the magnitude of these critical distances with the help of finite difference time domain calculations for amorphous SiO2 spheres in the infrared spectral range. Based on electric field maps, we conclude that the deviations scale with oscillator strengths and, correspondingly, with local electric fields and nearfield effects.

Employing Theories Far beyond Their Limits - Linear Dichroism Theory.

Using linear polarized light, it is possible in case of ordered structures, such as stretched polymers or single crystals, to determine the orientation of the transition moments of electronic and vibrational transitions. This not only helps to resolve overlapping bands, but also assigning the symmetry species of the transitions and to elucidate the structure. To perform spectral evaluation quantitatively, a sometimes "Linear Dichroism Theory" called approach is very often used. This approach links the relative orientation of the transition moment and polarization direction to the quantity absorbance. This linkage is highly questionable for several reasons. First of all, absorbance is a quantity that is by its definition not compatible with Maxwell's equations. Furthermore, absorbance seems not to be the quantity which is generally compatible with linear dichroism theory. In addition, linear dichroism theory disregards that it is not only the angle between transition moment and polarization direction, but also the angle between sample surface and transition moment, that influences band shape and intensity. Accordingly, the often invoked "magic angle" has never existed and the orientation distribution influences spectra to a much higher degree than if linear dichroism theory would hold strictly. A last point that is completely ignored by linear dichroism theory is the fact that partially oriented or randomly-oriented samples usually consist of ordered domains. It is their size relative to the wavelength of light that can also greatly influence a spectrum. All these findings can help to elucidate orientation to a much higher degree by optical methods than currently thought possible by the users of linear dichroism theory. Hence, it is the goal of this contribution to point out these shortcomings of linear dichroism theory to its users to stimulate efforts to overcome the long-lasting stagnation of this important field.

Removing interference-based effects from the infrared transflectance spectra of thin films on metallic substrates: a fast and wave optics conform solution.

A hybrid formalism combining elements from Kramers-Kronig based analyses and dispersion analysis was developed, which allows removing interference-based effects in the infrared spectra of layers on highly reflecting substrates. In order to enable a highly convenient application, the correction procedure is fully automatized and usually requires less than a minute with non-optimized software on a typical office PC. The formalism was tested with both synthetic and experimental spectra of poly(methyl methacrylate) on gold. The results confirmed the usefulness of the formalism: apparent peak ratios as well as the interference fringes in the original spectra were successfully corrected. Accordingly, the introduced formalism makes it possible to use inexpensive and robust highly reflecting substrates for routine infrared spectroscopic investigations of layers or films the thickness of which is limited by the imperative that reflectance absorbance must be smaller than about 1. For thicker films the formalism is still useful, but requires estimates for the optical constants.

The Electric Field Standing Wave Effect in Infrared Transmission Spectroscopy.

When band ratios in infrared absorbance spectra of films are compared (which had been converted from transmittance spectra), it can be noted that even after background correction and removal of interference fringes these band ratios change with the thickness of the films. The main goal of this work is to show that this effect is a consequence of an electric field standing wave based on the coherent superposition of light waves in the film. We further investigate how transmittance and reflectance, as well as absorbance and the (from absorbance) regained index of absorption, depend on the thickness of the film and how these parameters influence the positions of bands. We compare the results with those for the incoherent case and the case where a single pass of light through the film without reflection loss is assumed.

Employing Theories Far beyond Their Limits-The Case of the (Boguer-) Beer-Lambert Law.

For spectroscopists, the (Bouguer-)Beer-Lambert law is unquestionably an essential principle, since it is inseparably linked with one of the most important quantities in spectroscopy, the absorbance. In spite of its importance, a quantitative discussion of the legitimacy of relating the transmittance, the quantity that is usually measured, to the absorbance by assuming a logarithmic relation between both quantities cannot be found in literature. In this contribution, we quantitatively discuss, based on examples, the errors that can be introduced by disregarding the exact solution based on Maxwell's equations and show that these errors can easily exceed one order of magnitude. We also re-derive the Beer-Lambert law, thereby providing guidance as how to convert transmittance into absorbance properly.

Sophisticated Attenuated Total Reflection Correction Within Seconds for Unpolarized Incident Light at 45°.

The most common mid-infrared (MIR) attenuated total reflection (ATR) accessory has a nominal angle of incidence of 45° and does not have a polarizer. A spectrum recorded with such an accessory does not hold enough information for the sophisticated ATR correction of MIR spectra with strong peaks, which are often strongly affected by refractive index changes due to anomalous dispersion. Here we show that a 45° ATR spectrum recorded without a polarizer and the polarization angle for the same ATR Fourier transform infrared spectroscopy system provide enough information to determine the ATR s-polarized spectrum. Further analysis with an improved non-iterative Kramers-Kronig analysis immediately yields the complex refractive index function. The analysis is about two orders of magnitude faster than iterative formalism and runs within seconds on a typical office PC. The effectiveness of our advanced ATR correction formalism is showcased through its application to water, employing diamond, ZnSe, and Ge ATR crystals, along with two distinct ATR accessories. Additionally, the formalism is applied to octadecane spectra. Potential sources of errors such as incidence angle spread, dispersion of the polarization angle, and the influence of reflection at the air/ATR crystal interface are investigated by simulations.

Dispersion Analysis of Perpendicular Modes Using a Hybrid Two-Trace Two-Dimensional (2T2D) Smart Error Sum.

Using linear dichroism theory, one would assume that a z-cut of a uniaxial crystal is equivalent to an x-cut to determine the perpendicular component of the dielectric tensor and the corresponding oscillator parameters. However, Fresnel's equations show that the effect of interfaces in the form of the continuity relations of the different components of the electric field must be considered. A consequence of the continuity relations is that perpendicular modes increase less significantly in strength with increasing angle of incidence than expected. This is a consequence of the fact that it is the inverse of the perpendicular component of the dielectric function that increasingly becomes important with a growing angle of incidence. An inverse dielectric function, however, has typically much smaller values than the dielectric function. An additional consequence is that perpendicular modes are blueshifted and coupled in such a way that oscillator strength is transferred to the higher wavenumber mode. Thus, the spectral signatures of perpendicular modes are often weak and masked by the parallel modes when two modes overlap. Accordingly, to enable dispersion analysis, it is suggested to use a hybrid of the conventional residual sum of squares and the two-trace two-dimensional (2T2D) smart error sum, which can correct systematic multiplicable errors in the experimental spectrum. As demonstrated for fresnoite (Ba2TiSi2O8), this is an important step toward determining the perpendicular component of the dielectric tensor and the corresponding oscillator parameters using dispersion analysis, since asynchronous 2T2D correlation spectra are, in particular, sensitive to perpendicular modes.