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Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

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,...

Region of Convergence of Laplace Tarnsform

Region of Convergence of Laplace Tarnsform

The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...

Gauss's Law: Spherical Symmetry

Gauss's Law: Spherical Symmetry

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...

Poisson's And Laplace's Equation

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

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Related Experiment Video

Updated: Sep 11, 2025

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Laguerre-Gaussian-to-Hermite-Gaussian mode conversion revisited.

Somaye Fathollazade, Saifollah Rasouli, Davud Hebri

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|August 12, 2025

Summary

This study generalizes optical astigmatism theory for Laguerre-Gaussian (LG) to Hermite-Gaussian (HG) mode conversion. Different astigmatic conditions lead to distinct conversion scenarios and control over output mode properties.

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Last Updated: Sep 11, 2025

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Area of Science:

Optics and Photonics
Quantum Optics
Laser Physics

Background:

Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) modes are fundamental solutions in paraxial optics.
Optical astigmatism is a common aberration that can induce mode conversion.
Understanding and controlling LG-to-HG mode conversion is crucial for various optical applications.

Purpose of the Study:

To generalize the theoretical framework for Laguerre-Gaussian-to-Hermite-Gaussian (LG-to-HG) mode conversion.
To identify distinct conversion scenarios based on astigmatism conditions.
To explore astigmatic phase profiles for controlled LG-to-HG mode conversion.

Main Methods:

Theoretical analysis of LG beam diffraction by astigmatic optical elements.
Investigating three types of astigmatic elements: cylindrical lenses, quadratic curved-line gratings, and elliptical zone plates.
Exploring two families of astigmatic phase profiles for independent control over mode stretching and orientation.

Main Results:

Identified three distinct LG-to-HG conversion scenarios based on astigmatism.
Showed that LG modes of different orders convert to corresponding HG modes at specific distances.
Demonstrated independent control over the stretching and orientation of converted HG modes using tailored astigmatic phase profiles.

Conclusions:

The generalized theoretical framework accurately predicts LG-to-HG mode conversion under astigmatism.
Astigmatic optical elements offer versatile methods for controlling light beam modes.
Theoretical predictions show good qualitative agreement with experimental observations.