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Asymmetric vortex laser beams
Published in V. V. Kotlyar, A. A. Kovalev, Accelerating and Vortex Laser Beams, 2019
In 1987, Gori [119] considered Bessel–Gauss beams (BG-beams). The complex amplitude of such beams is described by the product of the Gaussian function by the nth order Bessel function of the first kind and the phase function describing the angular harmonic. The complex amplitude of the BG-beam satisfies the paraxial propagation equation. These beams have a radially symmetric intensity distribution and have an orbital angular momentum (OAM). Radial symmetry is preserved when the beam propagates. But BG-beams are not free-space modes, since with propagation not only scale changes but also the light energy is redistributed between different rings in the intensity distribution in the beam cross section. The BG-beams have been summarized in several papers [120–122]. For example, in [122] the Helmholtz–Gauss beams a special case of which are the Bessel–Gauss beams are considered [119]. The BG beams have finite energy, but Bessel modes that do not possess finite energy are also known [1]. The Bessel modes are solutions to the Helmholtz equation [114] and, when propagated in a homogeneous space, retain their intensity, and therefore are also called diffracion-free Bessel beams [1]. The linear combination of the Bessel modes with arbitrary coefficients is also a solution of the Helmholtz equation. In [115], an algorithm was proposed for calculating the phase optical element which forms the Bessel diffraction-free beams with a given mode composition. In [116] it was proposed to consider the Mathieu beam as an alternative to the Bessel beam. In [117] it was shown that a linear combination of the even and odd Mathieu beams with complex coefficients has an OAM. But the Mathieu beams themselves do not possess OAM. It is interesting [110] that a linear combination of two Hermite–Gauss modes with complex coefficients that do not possess an OAM, possesses an OAM. The periodic Mathieu function can be decomposed into a Fourier series [114]. For example, the Mathieu even functions are expanded in cosines from the polar angle in a cylindrical coordinate system, and the odd functions in sine terms. Therefore, the Mathieu diffraction-free beam can be represented as a linear combination of the Bessel modes. Such beams are considered in [118].
Dynamic Stress Analysis of a Shallow Unlined Elliptical Tunnel under the Action of SH Waves
Published in Journal of Earthquake Engineering, 2022
Fuqing Chu, Hui Qi, Jing Guo, Guohui Wu
The dynamic stress concentration of an elliptical cavity under the action of SH waves was completed by Gross in 1973 using the Mathieu function in the elliptical coordinates (Gross 1973). At the same time, Datta used the method of matched asymptotic expansions to solve the SH wave diffraction problem of elliptic elastic cylinder inclusions (Datta 1974). These studies are conducted in full space, but for the study of shallow buried tunnels, the addition of horizontal boundaries is necessary. For the elastic wave scattering problem in half-space tunnels, the existing analytical solutions focus on circular structures, and less research has been done on elliptical structures. In fact, the multi-lane highway tunnel with a width greater than height and the railway tunnel with a height greater than width have been widely used (Yang, Zhang, and Yang 2015). Therefore, it is essential to study the seismic performance of these tunnels. Elliptic sections, as general shapes of geometries, can be effectively approximated by varying their axial ratios to various geometries. In addition, the Mathieu function is an effective method for solving elliptic boundary value problems. With the maturity of numerical computation and computing power, it has been widely used in physics, electromagnetic, and microwave technology (Alhargan 1996; Cojocaru 2008; Gutierrezvega et al. 2003; Mathieu 1868; Sebak 1991, 1994). However, the progress of the application of Mathieu function in elastic wave scattering is relatively slow, which has been started for a long time. In addition to the above-mentioned research on the elliptical cavity in the full space (Gross 1973), as far back as 1974, the solutions of the two-dimensional semi-elliptical alluvial valley and semi-elliptical canyon under SH waves were obtained by using the wave function expansion method in elliptical coordinate and Mathieu functions (Wong and Trifunac 1974a, 1974b). In 2013, the scattering and ground motion of steady SH plane waves by two kinds of semi-elliptic cylindrical protrusions in half-space according to Mathieu function were given (Lee and Amornwongpaibun 2013a, 2013b). Subsequently, ground move of steady SH wave by the semi-elliptical shallow hill which contains a concentric elliptical unlined tunnel inside were gave (Amornwongpaibun, Luo, and Lee 2016).