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Methods of Obtaining Topography01:25

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Topography involves measuring and mapping land elevations, natural features, and artificial structures to create accurate representations of the terrain. Topographic surveying relies on traditional and modern methods, each with distinct advantages and limitations.Traditional Surveying Methods:Transit stadia surveys and plane table surveys were widely used traditional surveying methods. These techniques relied on instruments like theodolites and stadia rods for measuring distances and angles,...
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Continuous -time Fourier Transform01:11

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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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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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Discrete-time Fourier transform01:26

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The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
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Properties of Fourier Transform II01:24

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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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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.
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Fourier ptychographic topography.

Hao Wang, Jiabei Zhu, Jangwoon Sung

    Optics Express
    |May 8, 2023
    PubMed
    Summary
    This summary is machine-generated.

    Fourier ptychographic topography (FPT) offers a new method for high-throughput surface characterization. This technique achieves nanoscale accuracy with a wide field-of-view and high resolution, overcoming previous limitations in topography measurement.

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    Area of Science:

    • Optics and Photonics
    • Materials Science
    • Metrology

    Background:

    • Accurate topography measurement is critical for semiconductor metrology and surface characterization.
    • Existing methods face a trade-off between field-of-view (FOV) and spatial resolution.
    • High-throughput and precise topography measurement remains a significant challenge.

    Purpose of the Study:

    • To introduce a novel topography technique, Fourier ptychographic topography (FPT), for enhanced surface characterization.
    • To demonstrate FPT's capability for simultaneous wide FOV and high spatial resolution.
    • To achieve nanoscale height reconstruction accuracy for reflective surfaces.

    Main Methods:

    • Developed a reflection-mode Fourier ptychographic microscopy technique (FPT).
    • Utilized a custom computational microscope with programmable LED arrays (brightfield and darkfield).
    • Employed a sequential Gauss-Newton-based phase retrieval algorithm with total variation regularization for topography reconstruction.

    Main Results:

    • Achieved a synthetic numerical aperture (NA) of 0.84, increasing native NA by 3×, across a 1.2 × 1.2 mm² FOV.
    • Obtained a diffraction-limited resolution of 750 nm.
    • Demonstrated nanoscale height reconstruction accuracy, outperforming standard optical profilometry on complex patterns.
    • Characterized low spatial (0.529 nm) and temporal (0.027 nm) noise.

    Conclusions:

    • Fourier ptychographic topography (FPT) effectively overcomes the FOV-resolution trade-off in topography measurement.
    • FPT provides high-throughput, accurate, and high-resolution surface characterization for reflective samples.
    • The technique shows promise for advanced metrology and inspection applications, especially for complex micro/nanostructures.