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Related Concept Videos

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.
Discrete Fourier Transform01:15

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
Properties of Fourier Transform I01:21

Properties of Fourier Transform I

The application of Fourier Transform properties in radio broadcasting is multifaceted, enabling significant advancements in the way signals are transmitted and received. Key areas where these properties are utilized include simultaneous multi-channel transmission, audio clip speed adjustments, live broadcast delays for different time zones, audio frequency adjustments, and signal demodulation.
In radio broadcasting, multiple audio signals often need to be transmitted simultaneously. The Fourier...
Aliasing01:18

Aliasing

Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
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Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
Properties of Fourier series II01:21

Properties of Fourier series II

Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...

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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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A Multimodal Wide-Field Fourier-Transform Raman Microscope

Published on: December 30, 2025

Matched coordinates and adaptive spatial resolution in the Fourier modal method.

Thomas Weiss1, Gérard Granet, Nikolay A Gippius

  • 14th Physics Institute, University of Stuttgart, Stuttgart, Germany. t.weiss@physik.uni-stuttgart.de

Optics Express
|May 13, 2009
PubMed
Summary

Researchers developed a new Fourier modal method for complex 2D shapes. This approach uses covariant Maxwell

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Area of Science:

  • Computational electromagnetics
  • Nanophotonics simulation

Background:

  • Fourier modal method (FMM) has seen improvements for faster convergence.
  • Applying FMM to arbitrary 2D shapes remains challenging.

Purpose of the Study:

  • Generalize the Fourier modal method for complex 2D geometries.
  • Improve the applicability of FMM to non-trivial planar structures.

Main Methods:

  • Utilized a covariant formulation of Maxwell's equations.
  • Employed a matched coordinate system aligned with structure interfaces.
  • Combined with adaptive spatial resolution techniques.

Main Results:

  • Successfully generalized the Fourier modal method for arbitrary 2D shapes.
  • Demonstrated a scheme easily combinable with adaptive spatial resolution.
  • Discussed a symmetric application of Fourier factorization.

Conclusions:

  • The covariant formulation and matched coordinate system enable FMM for complex geometries.
  • This generalization enhances the efficiency and applicability of FMM in nanophotonics.