Zernike-Bessel representation and its application to Hankel transforms

Area of Science:

• Optics and Photonics
• Mathematical Physics
• Numerical Analysis

Background:

• Hankel transforms are crucial in analyzing systems with radial symmetry, common in optics and physics.
• Existing methods for computing Hankel transforms can be computationally intensive or lack precision for certain functions.
• The Zernike polynomial basis and Fourier-Bessel expansion are established mathematical tools with known properties.

Purpose of the Study:

• To exploit the duality between Zernike polynomials and Fourier-Bessel expansions for efficient Hankel transform calculation.
• To develop a numerical method for evaluating Hankel transforms of truncated radial functions with high accuracy.
• To investigate the applicability and limitations of the Fourier-Bessel representation in Hankel transform pairs.

Main Methods:

• Utilizing the mathematical relationship between the Zernike polynomial basis and the Fourier-Bessel expansion.
• Applying this duality to formulate a computational approach for Hankel transforms.
• Testing the method on simple truncated radial functions and more complex cases.

Main Results:

• Achieved exact Hankel transform calculations for simple truncated radial functions.
• Demonstrated high numerical accuracy for the Hankel transforms of more complicated radial functions.
• Gained insights into the limitations of the Fourier-Bessel representation, particularly for infinite-range Hankel transforms.

Conclusions:

• The proposed method provides an accurate and efficient way to compute Hankel transforms using Zernike polynomials and Fourier-Bessel expansions.
• This approach offers a valuable tool for numerical analysis in fields requiring radial symmetry analysis.
• Understanding the limitations is key for applying this method to diverse Hankel transform pairs.