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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...
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Related Experiment Video

Updated: Aug 26, 2025

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Concise and explicit expressions for typical spatial-structured light beams beyond the paraxial approximation.

Zhiwei Cui, Ju Wang, Wanqi Ma

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    |October 10, 2022
    PubMed
    Summary
    This summary is machine-generated.

    This study presents concise, non-paraxial analytical expressions for structured light beams, including Gaussian, Bessel, and Airy beams. These new formulas enhance understanding of light beam structures for advanced optical applications.

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    Area of Science:

    • Optics and Photonics
    • Electromagnetism
    • Wave Phenomena

    Background:

    • Structured light beams with complex amplitude, phase, and polarization are crucial for advanced optical applications.
    • Understanding the vectorial nature of these beams is key to unlocking their full potential.

    Purpose of the Study:

    • To derive explicit, non-paraxial analytical expressions for the electric field components of various structured light beams.
    • To provide more concise and accurate mathematical descriptions compared to existing literature.

    Main Methods:

    • Utilizing vectorial Rayleigh-Sommerfeld diffraction integrals.
    • Developing analytical expressions for fundamental Gaussian, Hermite-Gaussian, Laguerre-Gaussian, Bessel/Bessel-Gaussian, and Airy beams.

    Main Results:

    • New, concise analytical expressions for electric field components of structured light beams beyond the paraxial approximation.
    • Demonstration and analysis of electric field component distributions for these beams using the derived expressions.

    Conclusions:

    • The derived analytical expressions offer a more accurate and simplified description of structured light beams.
    • These findings facilitate a deeper understanding and further exploration of structured light for diverse applications.