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Journal of the Optical Society of America A

Fractional Talbot effect: analysis in phase space

Markus E. Testorf, Jorge Ojeda‐Castañeda

Published Jan 1, 1996• 28 citations
Publisher

Abstract

The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.

DOI
10.1364/josaa.13.000119

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The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.
This thesis develops a phase-space generalization of the Talbot effect using the modular-variable formalism of quantum mechanics. Starting from the standard Fresnel-diffraction description of self-imaging behind periodic gratings, the work reframes the effect as the action of symplectic transformations on a discrete phase-space lattice, visualized through Wigner functions. The central result is that free propagation, responsible for the ordinary integer Talbot effect, is just one instance of a much larger symmetry: every element of the group SL(2,Z) maps modular eigenstates onto modular eigenstates, generating its own Talbot-like recurrence. Closed-form expressions for the associated quantum phases, written in terms of quadratic Gauss sums, are derived and shown to reduce to the classical Fresnel results in the appropriate limit. The extension to rational transformations, SL(2,Q), reproduces the fractional Talbot effect and yields consistent analytical formulas for rational shears, though a discrete path-integral construction reveals a structural obstruction that remains unresolved and is proposed as a direction for future work. The thesis also explores a first, preliminary application to quantum information processing: an encoding of qubits and qutrits in modular eigenstates based on torsion points of the modular torus, together with generalized Pauli gates and a phase-readout scheme based on SL(2,Z) interference. This application is left incomplete, since the displacement-operator formalism used cannot by itself generate the superposition required for universal quantum computation.
Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.
According to a scalar theory, diffraction affects the electric field amplitude as well as its angular spectrum, providing a natural space–frequency phase space on which the Wigner distribution can be defined. The phase space is equipped with an Euclidean structure, so that the effect of diffraction is a rotation of the Wigner distribution associated with the optical field. The rotation is shown to split into two specific rotations. Wigner distributions associated with transverse modes of a resonator are invariant in such rotations, and a complete theory of optical resonators is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouÿ phase.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
We propose a laser field configuration that acts as a quasiperiodic atom optical diffraction grating. Exact analytical results for the atomic center-of-mass wave function in the Fresnel region after the grating reveal a quasiperiodic density pattern, a semiclassical focusing effect, and a quasiperiodic self-imaging of the atomic wave function analogous to a Talbot effect. The high-contrast quasiperiodic pattern that forms at the focal plane survives both transverse and longitudinal velocity averaging. The quasi-Talbot effect should be observable using atoms from Bose-Einstein condensates.