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    The first Rayleigh-Sommerfeld diffraction formula is exactly computed using a 3D convolution. This enables a precise 3D angular spectrum method for analyzing electromagnetic fields without approximations.

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    Area of Science:

    • Optics and electromagnetism
    • Computational physics

    Background:

    • The first Rayleigh-Sommerfeld diffraction formula is fundamental in optics.
    • Exact analytical solutions for diffraction are computationally intensive.
    • Previous methods often involved approximations or were limited in scope.

    Purpose of the Study:

    • To present an exact, computationally tractable method for solving the first Rayleigh-Sommerfeld diffraction formula.
    • To introduce the 3D angular spectrum (3D-AS) method for electromagnetic field propagation.
    • To validate the numerical implementation of the 3D-AS method.

    Main Methods:

    • Representing the diffraction formula as a 3D convolution in the spatial domain.
    • Applying a 3D Fourier transform to convert the spatial domain representation to reciprocal space.
    • Numerically implementing the 3D-AS method, neglecting evanescent waves.

    Main Results:

    • The first Rayleigh-Sommerfeld diffraction formula is exactly formulated as a 3D convolution.
    • The 3D angular spectrum (3D-AS) method allows for approximate-free conversion of the diffracted field to reciprocal space.
    • Numerical implementation, neglecting evanescent waves, shows good agreement with theoretical predictions.

    Conclusions:

    • The 3D-AS method provides an accurate and efficient approach for analyzing electromagnetic diffraction.
    • The method is readily implementable numerically, offering a practical tool for optical simulations.
    • Further research can explore the inclusion of evanescent waves for even higher fidelity.