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Related Concept Videos

Fast Fourier Transform01:10

Fast Fourier Transform

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...
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.
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
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...
Parseval's Theorem for Fourier transform01:15

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.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a signal's...
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...

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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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Published on: December 30, 2025

Optical implementation of iterative fractional Fourier transform algorithm.

Joonku Hahn, Hwi Kim, Byoungho Lee

    Optics Express
    |June 17, 2009
    PubMed
    Summary
    This summary is machine-generated.

    This study presents an optical method for iterative fractional Fourier transform, enabling 2D and 3D intensity distribution synthesis. The technique uses digital holography and spatial light modulators for complex optical field manipulation.

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    Published on: February 12, 2014

    Area of Science:

    • Optics and Photonics
    • Digital Holography
    • Fourier Optics

    Background:

    • Iterative algorithms are crucial for complex optical field manipulation.
    • Fractional Fourier Transform (FrFT) offers unique capabilities in optical signal processing.
    • Existing methods for optical FrFT implementation face limitations in flexibility and resolution.

    Purpose of the Study:

    • To propose and demonstrate an optical implementation of the iterative fractional Fourier transform algorithm.
    • To achieve precise control over complex optical fields for advanced applications.
    • To enable the synthesis of desired intensity distributions in optical domains.

    Main Methods:

    • Utilizing phase-shifting digital holography for accurate measurement of complex optical fields.
    • Employing a phase-type spatial light modulator for dynamic modulation of optical wavefronts.
    • Developing a novel iterative fractional Fourier transform system integrating these optical components.

    Main Results:

    • Successful demonstration of two-dimensional (2D) intensity distribution synthesis in the fractional Fourier domain.
    • Simultaneous achievement of three-dimensional (3D) intensity distribution synthesis.
    • Formation of desired intensity patterns at multiple focal planes within the optical system.

    Conclusions:

    • The proposed optical iterative fractional Fourier transform system provides a robust platform for complex optical field synthesis.
    • The integration of digital holography and spatial light modulators offers high flexibility and precision.
    • This technique holds potential for applications in optical information processing, imaging, and beam shaping.