Linear canonical transformation

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Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are Quadratic Phase Systems (QPS). The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform (LCT), a unitary, additive, three-parameter class of linear integral transform.

The LCT generalizes the Fractional Fourier transform, the Fresnel transform among others.

[edit] Definition

\mathcal{L}_{\alpha\beta\gamma}\{f(t)\}(t') = \sqrt{\beta} e^{-i \pi/4} \int_{-\infty}^\infty e^{-i \pi(\alpha t^2 - 2\beta tt' +\gamma t'^2)} f(t) dt

[edit] See also

Other time-frequency transforms:

[edit] References

  • S.A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Amer. 60, 1168–1177 (1970).
  • M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 8, 1772–1783, (1971).
  • B.M. Hennelly and J.T. Sheridan, "Fast Numerical Algorithm for the Linear Canonical Transform", J. Opt. Soc. Am. A 22, 5, 928–937 (2005).
  • H.M. Ozaktas, A. Koç, I. Sari, and M.A. Kutay, "Efficient computation of quadratic-phase integrals in optics", Opt. Let. 31, 35–37, (2006).
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