Propagation of doughnut-shaped super-Gaussian beams, convolution theorem and Hankel transform (original) (raw)
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An analytic formula for a certain type of a cylindrical beam, which might be called a γ beam, has been derived directly from the paraxial equation and independently using the method of the Hankel transform formulated in our previous work [T. Radożycki, Opt. Laser Technol. 147, 107670 (2022)]. The fundamental properties of this beam are analyzed and the parameters characterizing the beam shape are identified. The connection with Gaussian and elegant Laguerre-Gauss beams is demonstrated. In the plane perpendicular to the propagation axis, this beam is shown to display an expanding ring of high-energy concentration. At large radial distances the spatial profile exhibits a power-law falloff. The phase of the wave is also studied in this paper. Close to the symmetry axis it is shown to be typical of modes that exhibit vortex character, such as Gaussian beams of nth order, but at large distances it reveals a peculiar behavior distinct from Gaussian-type beams.
Proceedings of SPIE, 1999
Gaussian beams are an important fundamental concept of optics. They occur naturaly in the treatment of laser beams and resonators, and can also be used as a basis for study of general diffraction problems. However, a limitation of the usual formulation of Gaussian beams is that they are derived using a paraxial approximation. This breaks down when beam divergence is large. Moreover, for investigation of the polarisation properties of beams a vectorial treatment is necessary. Numerous authors have considered correction terms which extend the validity of the beams to higher divergence angles. An alternative is to derive a form of beam which is an exact solution of the wave equation for the scalar case or Maxwell's equations for the vectorialcase. This can be accomplished using the complex source point method. However, this leads to the presence of unphysical singularities. These can be avoided by introducing complex sink-source pairs. The resulting solution is a rigorous solution of Maxwell's equations which reduces to a conventional Gaussian beam for small angles of divergence. The resonance conditions for a resonator can be derived. Various different polarisations can also be considered, including plane polarised, transverse electric and transverse magnetic modes.
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An entropic formulation to describe the free propagation of Gaussian beams, in a similar way to the thermodynamic Ž . Ž . Ž . theory is developed. We consider two basic applications: 1 an extension to super-Gaussian-like SGL beams, and 2 the Ž . effect of lenses convergent and divergent on the propagation of Gaussian beams. We are interested in such applications because the SGL profiles are obtained through the convolution product using rectangle and Gaussian functions and so, they can be related to the Gaussian beams. The propagation in the Fresnel and Fraunhofer regions are studied, obtaining the laws for the optical entropy of the system. Also, we include some properties and a brief discussion about the condition under which the beam can be considered as an isolated system. For both applications, the evolution of the characteristic width is derived from the entropic postulates. q
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An explicit formula for a new type of beams, which in this work are called the "special" hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using the method of the Hankel transform formulated in our previous work. The fundamental properties of these beams are analyzed. The parameters that define the beam shape have been identified and related to those of the fundamental Gaussian beam. The analytical expressions for the SHBG beams include an additional parameter γ, which allows the beam's shape to be modified to some extent. In the plane perpendicular to the propagation direction, these beams exhibit the annular nature. Interestingly, initially (i.e. near the beam's spot) a single ring splits into a number of rings as one is moving along the beam. This is especially apparent for γ close to unity, as this effect then appears for values of z relatively small compared to the Rayleigh length i.e., where the energy concentration in the beam is still high. The phase of the wave, whose behavior is in certain aspects typical of modes having the vortex character, is also studied in this paper.