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Hypergeometric Vortices
Published in V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, Vortex Laser Beams, 2019
V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev
From Eq. (3.30) the complex amplitude of the HyG modes at γ=0 is seen to be proportional to the Bessel function of the first kind and of the integer and semi-integer orders. It is noteworthy that the above-discussed fields (both the HyG beams (3.13) and modes (3.25)) depend on the radial variable squared. This means that with increasing variable ρ the distance between the adjacent local maxima or minima will be decreased. An unlimited increase in the spatial frequency of the diffraction pattern is due to a (amplitude or phase) singularity of the complex amplitude at the initial plane (or waist) center. In practice, in order for such laser beams to be formed one should employ an annular diaphragm that would “block up” the center-coordinate singularity. In this case, however, the HyG beams can be generated only approximately.
Partial Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
In this section we study the various PDEs previously discussed in non-Cartesian coordinate systems such as polar coordinates. We will see that one of the additional concerns that this coordinate system introduces is the coordinate singularity at r = 0.
Fast transform spectral method for Poisson equation and radiative transfer equation in cylindrical coordinate system
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Peng-Hui Xiao, Ben-Wen Li, Zhong-Dong Qian
The first problem in solving Eq. (10) is the treatment of the coordinate singularity along the polar axis at the center r = 0. Most solvers of Poisson equation for Eq. (10) involve imposing extra pole conditions to approximate the solution in the vicinity of the origin accurately, which has been deliberated by Gottlieb and Orszag [33] and by Huang and Sloam [34]. Many references [33343536] discussed about various pole conditions. In this study, the radial interval is transformed into [–1, 1] to overcome this difficulty, and then the Gauss–Lobatto collocation points are adopted. Taking the total grid number as even to avoid the singularity at the origin, therefore no additional pole conditions are required.