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Properties of Fourier Transform II

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The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
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Parseval's Theorem for Fourier transform

Parseval's Theorem for Fourier transform

Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
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Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
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Updated: Jun 22, 2026

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

Improved AWG Fourier optics model.

I Molina-Fernández, J Wangüemert-Pérez

|June 2, 2009

Summary

This study introduces an improved Fourier Optics model for calculating Arrayed Waveguide Grating (AWG) transmission. The new model enhances accuracy by applying consistent approximations to input and output sections, resolving prior inconsistencies.

Area of Science:

Photonics and Optical Engineering
Integrated Optics

Background:

Arrayed Waveguide Gratings (AWGs) are crucial components in wavelength-division multiplexing (WDM) systems.
Existing Fourier Optics models for AWGs exhibit reciprocity-related inconsistencies due to differing approximations for input/output sections.

Purpose of the Study:

To develop an improved Fourier Optics model for calculating AWG transmission characteristics.
To address and resolve reciprocity inconsistencies found in previous AWG modeling approaches.

Main Methods:

The improved model utilizes identical approximations for both input and output sections of the AWG.
Mathematical expressions summarizing the model are derived for clarity and interpretability.
Quasi-analytical expressions are developed under the Gaussian approximation for mode field profiles.

Related Experiment Videos

Last Updated: Jun 22, 2026

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

Main Results:

The enhanced model accurately calculates transmission characteristics between arbitrary input/output ports (IOPs).
Reciprocity inconsistencies present in prior models are eliminated.
Compact and interpretable mathematical expressions are obtained.

Conclusions:

The presented Fourier Optics model offers a more accurate and consistent method for analyzing AWG performance.
The model's simplicity and derived expressions facilitate easier design and analysis of AWG devices.
This work contributes to the advancement of photonic device modeling and simulation.