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Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • The 2D nonseparable linear canonical transform (NS-LCT) is a unitary transform for paraxial scalar wave fields.
    • It models non-axially symmetric optical systems, unlike separable transforms.
    • Numerical approximation of 2D-NS-LCT lacks extensive literature.

    Purpose of the Study:

    • Develop a general sampling theorem for the 2D-NS-LCT.
    • Generalize existing 1D sampling theorems to the 2D case.
    • Determine conditions for a unitary discrete 2D-NS-LCT.

    Main Methods:

    • Formulation of a novel sampling theorem for the 2D-NS-LCT.
    • Analysis of sampling rates for discrete transform properties.
    • Extension of 1D sampling theory to 2D nonseparable transforms.

    Main Results:

    • A generalized sampling theorem for the 2D-NS-LCT is established.
    • Specific sampling rates are identified to ensure the unitarity of the discrete transform.
    • The work provides a foundation for numerical analysis of complex optical systems.

    Conclusions:

    • The developed sampling theorem is crucial for accurate numerical implementation of 2D-NS-LCT.
    • Ensuring unitarity of the discrete transform is vital for preserving physical properties.
    • This research bridges a gap in the numerical treatment of 2D-NS-LCTs.