Fractional Fourier transforms in two dimensions.

Area of Science:

• Mathematical Physics
• Signal Processing
• Quantum Optics

Background:

• The Fourier transform (FT) is fundamental in signal processing and physics.
• Fractional calculus extends integral transforms like the FT.
• Understanding the structure of fractional transforms is key to advanced applications.

Purpose of the Study:

• To analyze the fractionalization of the Fourier transform (FT).
• To explore the mathematical structure of the fractional Fourier transform (FrFT) in N dimensions.
• To clarify the relationship between FrFTs and special functions like Hermite-Gaussian and Laguerre-Gaussian beams.

Main Methods:

• Analysis of the repeated application of the fractional Fourier transform (FrFT).
• Characterization of the solution manifold using group theory (U(1), U(2)).
• Parameterization of the N-dimensional manifold as a fiber bundle over T2 and S2.
• Spectral analysis of FrFT eigenvalues and eigenfunctions.

Main Results:

• Repeated FrFT application recovers the FT.
• The 2D FrFT solution manifold is U(2), richer than the 1D U(1) case.
• Eigenvalues depend on T2 coordinates, eigenfunctions on S2 coordinates.
• Special FrFTs yield Hermite-Gaussian and Laguerre-Gaussian beams; generic FrFTs yield SU(2)-coherent states.

Conclusions:

• The N-dimensional FrFT structure is clarified.
• FrFTs are intrinsically linked to specific parameterizations of a fiber bundle.
• The spectral properties reveal connections to important beam types in optics.
• First-order systems in Sp(4, R) generate integral transforms equivalent to FrFTs.