1,721,182 research outputs found
Curvature radius of conic sections: A kinematic derivation
No full textConics are relevant in many undergraduate course on classical physics, from free fall to Rutherford scattering. A pedagogical derivation of the intrinsic expressions of the curvature radius of different types of conic sections is presented. Our proof is carried out without resorting to any coordinate systems, but rather on using only elementary kinematic concepts together with basics of vector calculus and the very definition of conics. As a byproduct application of the present analysis, a simple and compact deduction of the Newton 'inverse square law' for gravitation from the three Kepler laws is also presented
Huygens' cycloidal pendulum: An elementary derivation
No full textA pedagogical derivation of the Huygens cycloidal pendulum, suitable for undergraduates, is here presented. Our derivation rests only on simple algebraic and geometrical tricks, without the need of Calculus
Sharp-edge diffraction under Gaussian illumination: A paraxial revisitation of Miyamoto-Wolf's theory
No full textA “genuinely” paraxial version of Miyamoto-Wolf's theory aimed at dealing with sharp-edge diffraction under Gaussian beam illumination is presented. The theoretical analysis is carried out in such a way the Young-Maggi-Rubinowicz boundary diffraction wave theory can be extended to deal with Gaussian beams in an apparently straightforward way. The key for achieving such an extension is the introduction of suitable “complex angles” within the integral representations of the geometrical and boundary diffracted wave components of the total diffracted wavefield. Surprisingly enough, such a simple (although not rigorously justified) mathematical generalization seems to work well within the Gaussian realm. The resulting integrals provide meaningful quantities that, once suitably combined, give rise to predictions that are in perfect agreement with results already obtained in the past. An interesting and still open theoretical question about how to evaluate “Gaussian geometrical shadows for arbitrarily shaped apertures is also discussed
“Analytical continuation” of flattened Gaussian beams
No full textA purely analytical extension of flattened Gaussian beams [Opt. Commun. 107, 335 (1994)] to any values of beam order is here proposed. Due to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary ABCD optical systems can definitely be solved in closed form via a particular bivariate confluent hypergeometric function
Paraxial Sharp-Edge Diffraction of Vortex Beams by Elliptic Apertures
Full text linkA semi-analytical computational algorithm to model the wavefield generated by
paraxial diffraction of a class of Laguerre-Gauss beams by sharp-edge elliptic
apertures is here developed. Thanks to such a powerful computational tool, some
basic aspects of an intriguing and still unexplored singular optics scenario
can be studied, within a geometry as simple as possible, with arbitrarily high
accuracies
Paraxial Sharp-Edge Diffraction: A General Computational Approach
No full textA general reformulation of classical sharp-edge diffraction theory is
proposed within paraxial approximation. The, not so much known, Poincar\'e
vector potential construction is employed directly inside Fresnel's 2D integral
in order for it to be converted into a single 1D contour integral over the
aperture boundary. Differently from the recently developed paraxial
revisitation of BDW's theory, such approach can be applied to arbitrary
wavefield distributions impinging onto arbitrarily shaped sharp-edge planar
apertures. A couple of interesting examples of application of the proposed
method is presented
Correspondence between super-Gaussian and flattened Gaussian beams
No full textRelations connecting the parameters of a super-Gaussian with those of a flattened Gaussian beam are determined by minimizing the mean squared difference of the two profiles. Simplified analytical expressions are suggested and tested for values of the power parameter of the super-Gaussian function up to 20
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