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.The exact origin of the form of Beer's law, as we employ it nowadays [1,2] AðṽÞ ¼ e * ðṽÞ � c � dwherein e * ðṽÞ is the molar attenuation coefficient, c the concentration and d the sample thickness, remains unclear. It seems that it was not employed in this form before about 1900, as it was not discussed nor provided in this form in Kayser's handbook of spectroscopy, [3] which was the reference work at this time. When Max Planck derived his particular kind of dispersion theory and employed it to compare the results with Beer's law in 1903, [4] he was most probably unaware of Equation (1), otherwise he probably would have derived it from dispersion theory. Recently, this derivation was carried out, first from dispersion theory [5] and, subsequently, from simple electromagnetic theory. [6] One of the main results of the derivation is that the molar attenuation coefficient is inversely proportional to the index of refraction, which is itself a function of concentration. Concerning the nature of this dependence, as was already stated in Ref.[5], a law comparable to Beer's law, but for the index of refraction function, can be derived. In ref. [5] the nature of this dependence was not investigated in detail. As we will show in the following, the use of the index of refraction variation instead of the absorbance to investigate concentration may have several advantages. To derive the concentration dependence of the complex index of refraction, it is possible to start from simple electromagnetic theory. Accordingly, the macroscopic polarizationP is related to the dipole momentp induced in an atom or a molecule by the following relation:Here, N is the number of dipole moments per unit volume (which is equal to the molar concentration multiplied by Avogadro's constant, N = N A ·c). Inherent to Equation (2) is the assumption that there are no interactions between the microscopic dipoles, i....
Plasmonic waveguides are crucial building blocks for integrated on-chip mid-infrared (mid-IR) sensors, which have recently attracted great interest as a sensing platform to target enhanced molecular sensing. However, while hosting a wide range of applications from spectroscopy to telecommunication, the mid-IR lacks suitable broadband solutions that provide monolithic integration with III-V materials. This work reports a novel concept based on hybrid semiconductor-metal surface plasmon polariton waveguides, which result in experimentally demonstrated low loss and broadband devices. Composed of a thin germanium slab on top of a gold layer, the waveguiding properties can be directly controlled by changing the geometrical parameters. The measured losses of our devices are as low as 6.73 dB/mm at 9.12 µm and remain <15 dB/mm in the mid-IR range of 5.6–11.2 µm. The octave-spanning capability of the waveguides makes them ideal candidates for combination with broadband mid-IR quantum cascade laser frequency combs and integrated spectroscopic sensors.
Novel laser light sources in the mid-infrared region enable new spectroscopy schemes beyond classical absorption spectroscopy. Herein, we introduce a refractive index sensor based on a Mach-Zehnder interferometer and an external-cavity quantum cascade laser that allows rapid acquisition of high-resolution spectra of liquid-phase samples, sensitive to relative refractive index changes down to 10−7. Dispersion spectra of three model proteins in deuterated solution were recorded at concentrations as low as 0.25 mg mL−1. Comparison with Kramers-Kronig-transformed Fourier transform infrared absorbance spectra revealed high conformance, and obtained figures of merit compare well with conventional high-end FTIR spectroscopy. Finally, we performed partial least squares-based multivariate analysis of a complex ternary protein mixture to showcase the potential of dispersion spectroscopy utilizing the developed sensor to tackle complex analytical problems. The results indicate that laser-based dispersion sensing can be successfully used for qualitative and quantitative analysis of proteins.
We report on a mid-infrared (mid-IR) photothermal spectrometer for liquid-phase samples for the detection of water in organic solvents, such as ethanol or chloroform, and in complex mixtures, such as jet fuel. The spectrometer is based on a Mach−Zehnder interferometer (MZI) employing a He-Ne laser, a mini-flow cell with two embedded channels placed in the interferometer's arms, and a tunable external cavity quantum cascade laser (EC-QCL) for selective analyte excitation in a collinear arrangement. In this study, the bending vibration of water in the spectral range 1565−1725 cm −1 is targeted. The interferometer is locked to its quadrature point (QP) for most stable and automated operation. It provides a linear response with respect to the water content in the studied solvents and photothermal analyte spectra, which are in good agreement with FTIR absorbance spectra. The method is calibrated and validated against coulometric Karl Fischer (KF) titration, showing comparable performance and sensitivity. Limits of detection (LODs) for water detection in the single-digit ppm range were obtained for chloroform and jet fuel due to their low background absorption, whereas lower sensitivity has been observed for water detection in ethanol due to pronounced background absorption from the solvent. In contrast to KF titration, which requires toxic reagents and produces waste, the developed method works reagent-free. It can be applied in an online format in the chemical industry as well as for fuel quality control, being industrial applications where traces of water need to be accurately determined, preferably in real-time. It thus holds great promise as a green alternative to the offline KF titration method, which is the current standard method for this application.
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