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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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Optical spatiotemporal Fourier synthesis: tutorial.

Murat Yessenov, Ayman F Abouraddy

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |September 22, 2025
    PubMed
    Summary
    This summary is machine-generated.

    This tutorial introduces spatiotemporal Fourier optics, combining spatial and temporal manipulation to create unique optical fields. It focuses on cylindrically symmetric fields, enabling propagation-invariant wave packets with tunable group velocity.

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    Area of Science:

    • Physical optics
    • Wave phenomena
    • Optical engineering

    Background:

    • Fourier optics is fundamental to understanding light manipulation.
    • Spatial Fourier optics enables imaging, microscopy, and holography.
    • Temporal Fourier optics allows for ultrafast pulse shaping.

    Purpose of the Study:

    • Introduce the emerging field of spatiotemporal Fourier optics.
    • Explain the joint manipulation of spatial and temporal spectra.
    • Focus on a specific class of nonseparable spatiotemporally structured fields.

    Main Methods:

    • Tutorial on spatiotemporal Fourier synthesis.
    • Focus on cylindrically symmetric fields (radial spatial frequency linked to wavelength).
    • Description of a system for preparing these structured optical fields.

    Main Results:

    • Spatiotemporally structured optical fields with unique propagation characteristics are produced.
    • Cylindrically symmetric fields form propagation-invariant wave packets.
    • These wave packets travel rigidly in linear media at a tunable group velocity.

    Conclusions:

    • Spatiotemporal Fourier optics offers novel ways to structure light.
    • This approach enables the creation of propagation-invariant wave packets.
    • The described system facilitates the preparation of specific spatiotemporally structured fields.