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Related Concept Videos

Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Beams with Symmetric Loadings01:15

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The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI...
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Beams with Unsymmetric Loadings01:17

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Analyzing a supported beam under unsymmetrical loadings is essential in structural engineering to understand how beams respond to varied force distributions. This analysis involves calculating the deflection and identifying points where the slope of the beam is zero, which are crucial for ensuring structural stability and functionality.
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Distribution of Stresses in a Narrow Rectangular Beam01:11

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In studying beam stress distribution, examining an elemental section is essential. To determine the average shearing stress on this face, the calculated shear is divided by the surface area. Importantly, shearing stresses on the beam's transverse and horizontal planes mirror each other, indicating a consistent stress distribution along the upper region of the beam. Notably, shearing stresses are absent at the beam's upper and lower surfaces due to the absence of applied forces in these...
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has...
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Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
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Related Experiment Video

Updated: May 5, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Cylindrical quasi-Gaussian beams.

F G Mitri

    Optics Letters
    |December 11, 2013
    PubMed
    Summary

    This study introduces an exact solution for cylindrical quasi-Gaussian beams using the complex-source-point method. The findings are useful for designing optical systems and understanding light-matter interactions.

    Area of Science:

    • Optics and Electromagnetism
    • Mathematical Physics

    Background:

    • Gaussian beams are fundamental in optics but often require approximations for cylindrical geometries.
    • Exact solutions are needed for precise modeling in advanced optical applications.

    Purpose of the Study:

    • To introduce an exact analytical solution for cylindrical quasi-Gaussian beams.
    • To derive electromagnetic field components without approximations.

    Main Methods:

    • Utilizing the complex-source-point method in cylindrical coordinates.
    • Applying Maxwell's equations and Lorenz's gauge condition.
    • Deriving Cartesian components of the electromagnetic field from vector potentials.

    Main Results:

    • An exact solution for cylindrical quasi-Gaussian beams satisfying Helmholtz and Maxwell's equations.

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  • Demonstration of the theory for tightly focused and quasi-collimated beams.
  • Derivation of electromagnetic field components without approximations.

    • Conclusions:

    • The developed method provides an exact representation of cylindrical quasi-Gaussian beams.
    • The results are applicable to beam-forming design in microscopy, optical tweezers, and scattering studies.