Nonparaxial Gaussian beam
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RESEARCH ARTICLE
Nonparaxial Gaussian beam N. V. Selina1
Received: 21 July 2019 / Accepted: 13 July 2020 / Published online: 4 August 2020 Ó The Optical Society of India 2020
Abstract A new analytical solution of the nonparaxial Helmholtz equation for the Gaussian beam has been obtained. It is shown that the beam retains the Gaussian distribution of the amplitude at propagation in space. The scale transformation of the beam has been determined. The Kogelnik–Li law applies to a nonparaxial Gaussian beam. Keywords Non-paraxial Gaussian beam Law KogelnikLi
Introduction The laser beam is approximated by the Gaussian beam model. Laser physics has many applications. Therefore, the study of the Gaussian beam model is very important. The well-known Gaussian beam formula is only valid in the case of its paraxiality [1]. The term ‘‘paraxial’’ is applicable when the beam mainly propagates along the optical axis. There are several papers that talk about paraxiality in a quantitative sense [2]. Depreciating the solution to the Helmholtz equation can be represented in the form of an expansion in plane-wave. This method is used in article [3], where the numerical calculation of the Gaussian beam transformations is given. The vector model of the Gaussian beam is developed on the same basis [4, 5]. The wave function of the beam in such a model is expressed in terms of a certain integral. The paraxial equation refers to differential equations of parabolic type. It is an approximation. The exact wave & N. V. Selina [email protected] 1
Kuban State Technological University, Krasnodar, Russia
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equation is of elliptic type. This is the nonparaxial Helmholtz equation. The nonparaxial beam model is used when the divergence or focus angle is large. The problem of solving the nonparaxial Helmholtz equation has been of interest to researchers for a long time. The mathematical solution of the elliptic wave equation is interesting in itself. This solution is necessary when calculating the parameters of laser systems used in the study of optical elements and their systems. Modern technologies make it possible to create optical devices with a diverse set of properties and capabilities. The sharp focusing of the laser beam or the use of diode lasers with a large beam divergence angle requires an accurate calculation of the spatial transformations of the laser beam. Therefore, the calculation of Gaussian beam divergence can be useful in theoretical and experimental studies of such processes. A vector model of a Gaussian beam is considered in [4–6]. The final solution is decomposed into flat waves. As a result, either a series or a Fourier integral is obtained for coordinates that are transverse to the beam axis. For an azimuthal variable, integration leads to Bessel functions; the radial coordinate integral has no analytical representation in elementary or special functions. Only numerical calculations are possible. The vector model of a Gaussian beam was also studied using the diffraction integral [6]. In the approximation of the di
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