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First-order optical systems with unimodular eigenvalues.

Martin J Bastiaans1, Tatiana Alieva

  • 1Faculteit Elektrotechniek, Technische Universiteit Eindhoven, Eindhoven, Netherlands. m.j.bastiaans@tue.nl

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|July 13, 2006
PubMed
Summary

Lossless first-order optical systems with unimodular eigenvalues are similar to fractional Fourier transformers. This similarity transformation is performed by another lossless optical system, revealing new optical mode properties.

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Area of Science:

  • Optics
  • Quantum Optics
  • Mathematical Physics

Background:

  • First-order optical systems are fundamental in optics.
  • Fractional Fourier transformers (FFTs) have unique properties related to Hermite-Gauss functions.
  • Understanding the mathematical structure of optical systems is crucial for designing new devices.

Purpose of the Study:

  • To establish a similarity transformation between unimodular first-order optical systems and separable fractional Fourier transformers.
  • To formulate the eigenfunctions of unimodular first-order optical systems.
  • To explore practical optical realizations of these systems.

Main Methods:

  • Analysis of the ray transformation matrix of lossless first-order optical systems.
  • Diagonalization of real symplectic matrices with unimodular eigenvalues.

Related Experiment Videos

  • Utilizing Hermite-Gauss functions as a basis for eigenfunctions.
  • Investigating system decompositions for optical implementation.

    • Main Results:

    • Demonstrated similarity between unimodular systems and separable fractional Fourier transformers via a specific transformation.
    • Identified the transformation system as another lossless first-order optical system.
    • Formulated eigenfunctions of unimodular systems as Hermite-Gaussian-type modes.
    • Considered two system decompositions, detailing one for optical realization.

    Conclusions:

    • Unimodular first-order optical systems share a deep connection with fractional Fourier transformers.
    • The eigenfunctions of these systems are related to established mathematical functions, extending their applicability.
    • The findings pave the way for novel optical system designs and implementations.