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Related Concept Videos

Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and Faraday.
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Partial Differential Equations01:21

Partial Differential Equations

A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...
Poisson's And Laplace's Equation01:25

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.

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Related Experiment Video

Updated: Jun 22, 2026

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
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Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics.

Jethro H Greene, Allen Taflove

    Optics Express
    |June 17, 2009
    PubMed
    Summary

    This study introduces a novel numerical technique for modeling electromagnetic wave propagation in complex materials. The method accurately simulates vector spatial solitons, enabling the study of sub-wavelength interactions.

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    Published on: August 13, 2019

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    Published on: February 23, 2018

    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy
    09:43

    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy

    Published on: August 13, 2019

    Area of Science:

    • Computational Electromagnetics
    • Nonlinear Optics
    • Materials Science

    Background:

    • Modeling electromagnetic wave propagation in nonlinear materials is crucial for advanced optical technologies.
    • Existing methods often simplify the electric field to a single vector component, limiting accuracy.
    • Dispersive and nonlinear material properties significantly affect wave behavior.

    Purpose of the Study:

    • To present a full-vector numerical technique for modeling electromagnetic wave propagation in dispersive nonlinear materials.
    • To incorporate multiple polarization effects, including linear Lorentz, nonlinear Kerr, and nonlinear Raman.
    • To demonstrate the technique's capability by modeling vector spatial solitons.

    Main Methods:

    • Utilized the auxiliary differential equation finite-difference time-domain (ADE-FDTD) method.
    • Developed a full-vector Maxwell's equations solver.
    • Incorporated multiple-pole linear Lorentz, nonlinear Kerr, and nonlinear Raman polarization models.

    Main Results:

    • Successfully modeled electromagnetic wave propagation with two orthogonal electric field components.
    • Demonstrated the simulation of a spatial soliton with two vector components.
    • The numerical technique is novel in its ability to handle multi-component vector fields in such materials.

    Conclusions:

    • The presented ADE-FDTD method is the first to model electromagnetic wave propagation with two or three orthogonal vector components in dispersive nonlinear materials.
    • This technique opens possibilities for simulating sub-wavelength interactions of vector spatial solitons.
    • The method enhances the accuracy and scope of electromagnetic wave modeling in complex media.