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Related Concept Videos

Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

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.
The first moment-area theorem determines the slope at any point on the beam. This theorem indicates that the change in slope between two points on a beam...
Deflection of a Beam01:19

Deflection of a Beam

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.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...
Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

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...
Method of Superposition01:20

Method of Superposition

The method of superposition is a crucial technique in structural engineering, used to analyze the effect of multiple loads on beams. This approach involves calculating the deflection and slope for each load on a beam separately, and then summing these effects to determine the overall impact. It is applicable only when the beam material remains within its elastic limit, ensuring that deformations are linearly elastic.
When applying the method of superposition, each type of load—whether...
Principal Stresses in a Beam01:11

Principal Stresses in a Beam

In prismatic beams subject to arbitrary transverse loading, It is essential to analyze the interaction between shear forces and bending moments in order to understand stress distribution and ensure structural integrity. The highest normal or bending stress occurs at the outer fibers of the beam, decreasing linearly to zero at the neutral axis. In contrast, shear stress peaks at the neutral axis and diminishes toward the outer surfaces.
Analyzing principal stresses is crucial, especially in...
Deformation of a Beam under Transverse Loading01:15

Deformation of a Beam under Transverse Loading

Understanding beam deflection, particularly for indeterminate beams with overhanging segments and multiple concentrated loads, is crucial for ensuring structural integrity and functionality. The process begins with constructing an accurate free-body diagram, which helps identify the forces and moments acting on the beam. This diagram is vital for visualizing how bending moments vary along the beam's length, influencing its curvature.
The insights from the bending moment diagram extend to...

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Related Experiment Video

Updated: May 31, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

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

Published on: August 12, 2013

Complex modes and instability of full-vectorial beam propagation methods.

Huan Xie1, Wangtao Lu, Ya Yan Lu

  • 1Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China.

Optics Letters
|July 5, 2011
PubMed
Summary

Full-vectorial beam propagation methods (FVBPMs) are analytically unstable for optical waveguides with complex modes. This instability persists regardless of computational adjustments, highlighting a fundamental limitation.

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Characterization of Anisotropic Leaky Mode Modulators for Holovideo
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Characterization of Anisotropic Leaky Mode Modulators for Holovideo

Published on: March 19, 2016

Related Experiment Videos

Last Updated: May 31, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

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

Published on: August 12, 2013

Characterization of Anisotropic Leaky Mode Modulators for Holovideo
09:36

Characterization of Anisotropic Leaky Mode Modulators for Holovideo

Published on: March 19, 2016

Area of Science:

  • Optics and Photonics
  • Computational Electromagnetics

Background:

  • Full-vectorial beam propagation methods (FVBPMs) are essential for modeling light propagation in optical waveguides.
  • High-index-contrast waveguides present unique challenges for accurate wave propagation modeling.

Purpose of the Study:

  • To investigate the analytical stability of paraxial and wide-angle FVBPMs.
  • To identify the limitations of current FVBPMs when applied to waveguides with complex modes.

Main Methods:

  • Analysis of paraxial and wide-angle FVBPMs based on diagonal Padé approximants.
  • Theoretical investigation of mode confinement in high-index-contrast waveguides.

Main Results:

  • Demonstrated analytical instability in paraxial and wide-angle FVBPMs for waveguides supporting complex modes.
  • Confirmed that computational domain size, resolution, and perfectly matched layers do not resolve the instability.
  • Identified high mode confinement around the waveguide core as the root cause.

Conclusions:

  • Current FVBPM formulations are analytically unstable for complex modes in optical waveguides.
  • The inherent instability necessitates the development of alternative or modified propagation methods.
  • Understanding mode confinement is critical for accurate FVBPM application.