A vectorial diffraction model is presented for the focusing of a Gaussian laser beam with a wavelength of 800 nm by a parabolic metallic mirror with a diameter of 15 mm and a focal length of 150 mm. The model is based on a rigorous calculation of the reflected electromagnetic field using s- and p-polarization basis functions, complex Fresnel coefficients, and the Kirchhoff–Rayleigh surface integral. The reflective coating is characterized by a complex refractive index n = 0.145 + 4.5i, corresponding to silver in the near-infrared spectral range. The incident beam has a waist radius of 3 mm at the mirror’s vertex plane. The field distribution in the focal plane is numerically computed on a 300×300 grid over a ±30 μm region. Focus quality is evaluated using three criteria: total intensity, radial intensity distribution, and the full width at half maximum (FWHM) of the focal spot. A focal spot with FWHM ≈ 8.56 μm is obtained, in close agreement with the theoretical diffraction-limited estimate. The results demonstrate that accounting for the vectorial nature of the field and the dissipative properties of the metal enables accurate prediction of polarization distortions and energy losses in practical mirror-based focusing systems.
Keywords: vectorial diffraction model, parabolic metallic mirror, Gaussian laser beam, Fresnel coefficients, complex refractive index
The paper describes a multifactorial nonlinear regression model of revenue dynamics of the mining and metallurgical company Severstal, based on retrospective information for 2009-2021. Production volumes by type were used as independent variables: hot-rolled and cold-rolled sheet, galvanized sheet and sheet with other metal coating, rolled products, large diameter pipes, other pipe products and profiles. The criteria of multiple definition and Fisher, as well as the average absolute approximation error, were used as criteria for the adequacy of the model. A model competition was held to select the best regression dependence. As a result, a model is constructed containing inverse transformations of two independent variables in the right part.
Keywords: regression model, least squares method, adequacy criteria, mining and metallurgical company, revenue, model competition