Linear canonical transformation

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.

Definition

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

See also

Other time-frequency transforms:

• Fractional Fourier transform
• continuous Fourier transform
• chirplet transform

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).