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Related Experiment Videos

Zernike-Bessel representation and its application to Hankel transforms.

Charles Cerjan1

  • 1Lawrence Livermore National Laboratory, CA 94550, USA. cerjan1@llnl.gov

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|May 12, 2007
PubMed
Summary

This study leverages the Zernike polynomial basis and Fourier-Bessel expansion to precisely calculate Hankel transforms for radial functions. The method offers exact results for simple functions and high accuracy for complex ones, enhancing numerical analysis.

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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:

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  • 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.