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Fractional Talbot effect in phase space: A compact summation formula.

K Banaszek, K Wodkiewicz, W P Schleich

|April 21, 2009

Summary

We present a phase space description of the fractional Talbot effect. This reveals that fractional Talbot images are generated by spatially displaced Wigner functions of the source field.

Area of Science:

Optics
Quantum mechanics
Diffraction phenomena

Background:

The fractional Talbot effect is a complex diffraction phenomenon observed with periodic gratings.
Understanding its behavior in phase space offers new insights into wave propagation.

Purpose of the Study:

To develop a phase space description for the fractional Talbot effect.
To derive a compact formula for the Wigner function at Talbot distances.

Main Methods:

Utilizing the phase space formalism.
Applying Fresnel diffraction principles to a one-dimensional periodic grating.

Main Results:

A compact summation formula for the Wigner function at rational Talbot distances was derived.

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The formula demonstrates that fractional Talbot images in phase space arise from a finite sum of displaced Wigner functions.

Conclusions:

The phase space approach provides a clear and concise representation of the fractional Talbot effect.
This formalism simplifies the understanding of image formation in fractional Talbot propagation.