Specificity of the mathematical modeling of light fields in a sensing element for the fiber-based evanescent-wave mid-IR spectroscopy

Background and Objectives: The fiber-based evanescent-wave mid-IR spectroscopy is a prospective tool for the real-time remote chemical analysis of various substances, which have their vibrational spectra in the mid-IR spectral range. Chalcogenide fibers transparent in the mid-IR are considered as the most suitable sensing elements of the fiber-based mid-IR spectroscopic sensors. Earlier, to describe light fields in a chalcogenide fiber embedded into an absorbing medium, a ray optics approach based on the approximation of weak absorption of the medium was used. However, this approach is not applicable for the mid-IR spectroscopy of liquids since the absorption coefficients of liquids in the mid-IR can be of the order of 10$^3$ cm$^{-1}$. As the difference of refractive indices of a chalcogenide glass (2.4-3.4) and a liquid (1.28-1.35) is large, the weakly guiding approximation widely used to design the fiber-optic information networks is not applicable for the sensing elements modeling. Development of a reliable mathematical model of light fields in the chalcogenide sensing elements is an urgent problem. In this paper, a detailed analysis of such a mathematical model based on the electromagnetic theory of optical fibers is presented. Materials and Methods: A multimode single-index chalcogenide fiber embedded into an absorbing liquid is considered as a sensing element of a fiber-based spectroscopic sensor. For this sensing element, a model of a infinite cylindrical waveguide with a uniform core and an infinite uniform cladding with a complex-valued refractive index is proposed. To describe light fields in the sensing element, a mathematical model based on solution of a boundary value problem for Helmholtz equations in a rigorous electrodynamic formulation is developed. For classification 62 of the boundary value problem solutions, a complex plane of a fiber mode parameter in the cladding is used. Eigenwaves obtained by solution of the boundary value problem that satisfies the condition of exponential decay in the waveguide cross-section at infinity are identified as evanescent modes of the waveguide. The power of the modes is decreasing along the waveguide due to the external absorption. In computer modeling of the evanescent modes, an eigenvalue equation written for the modes parameters is solved numerically. As an absorbing liquid, pure acetone is chosen. The absorption coefficient of acetone, obtained experimentally, is used to evaluate the imaginary part of its refractive index. Results: Specificity of the boundary value problem formulation in application to the light fields in sensing elements of the fiber-based spectroscopic sensors has been revealed. A mathematical model of evanescent modes of a chalcogenide sensing element has been elaborated by using the rigorous electrodynamic approach. This model has been applied to calculate the longitudinal and transverse power flows of the HE v m evanescent modes in the cross-section of a chalcogenide fiber immersed into the pure acetone. It was demonstrated that in the rigorous mathematical model, the transverse power flows of evanescent modes are specifically nonzero. With the given parameters of the chalcogenide sensing element and the absorbing medium, the density of the transverse components of the power flow at a specified peak wavelength of an absorption band of acetone was 3-4 orders of magnitude lower than the longitudinal component density. Conclusion: The mathematical model of light fields in chalcogenide sensing elements for the mid-IR spectroscopy has been developed with account of the large difference in the refractive indices of the chalcogenide fiber core and the external medium having large absorption coefficients. Applicability of the mathematical model based on the rigorous electrodynamic approach was confirmed previously in our works where the results of computer modeling fit the experimental data obtained in the chalcogenide fiber based spectroscopic measurements.