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    This study introduces hyperbolic deflecting lenses (HDLs) for precise light manipulation. These novel lenses offer new possibilities for advanced optical systems and virtual image generation.

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    Area of Science:

    • Optics and Photonics
    • Geometric Optics

    Background:

    • Deflecting lenses are crucial for manipulating light fields and creating virtual images.
    • Extending optical concepts to new domains is essential for technological advancement.

    Purpose of the Study:

    • To introduce and define general profiles for hyperbolic deflecting lenses (HDLs).
    • To investigate the geometric optical properties and deflection mechanisms of HDLs.
    • To explore the relationship between HDLs and hyperbolic absolute instruments.

    Main Methods:

    • Leveraging the correspondence between phase space and momentum space.
    • Analyzing the geometric optical properties of hyperbolic deflecting lenses.
    • Investigating the deflection mechanism within hyperbolic optical systems.

    Main Results:

    • General profiles for hyperbolic deflecting lenses (HDLs) are introduced.
    • The geometric optical properties and deflection mechanisms of HDLs are analyzed.
    • Inside-out HDLs are shown to correspond to hyperbolic absolute instruments for homogeneous regions.

    Conclusions:

    • This work establishes a foundation for developing deflecting lenses in hyperbolic optical systems.
    • Hyperbolic deflecting lenses offer a new paradigm for light manipulation.
    • The findings pave the way for future innovations in radially hyperbolic optical designs.