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arXiv:physics/9711004 · Optics Express

Fractional Talbot effect in phase space: A compact summation formula

Konrad Banaszek, Krzysztof Wodkiewicz, Wolfgang Peter Schleich

Published Mar 2, 1998• 26 citations
arXivHTMLPDF

Abstract

A phase space description of the fractional Talbot effect, occurring in a one--dimensional Fresnel diffraction from a periodic grating, is presented. Using the phase space formalism a compact summation formula for the Wigner function at rational multiples of the Talbot distance is derived. The summation formula shows that the fractional Talbot image in the phase space is generated by a finite sum of spatially displaced Wigner functions of the source field.

DOI
10.1364/oe.2.000169

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A phase space description of the fractional Talbot effect, occurring in a one--dimensional Fresnel diffraction from a periodic grating, is presented. Using the phase space formalism a compact summation formula for the Wigner function at rational multiples of the Talbot distance is derived. The summation formula shows that the fractional Talbot image in the phase space is generated by a finite sum of spatially displaced Wigner functions of the source field.
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.
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.
We theoretically study the Talbot effect resulting from Raman-induced grating based on periodic spatial modulation of the Raman gain and dispersion in the field of a standing pump wave. Features of integer and fractional Talbot effects are demonstrated for one- and two-dimensional (1D and 2D) Raman-induced gratings. It is shown that the intensity of diffraction images can increase due to Raman amplification in the grating. Glass-shaped diffraction patterns are demonstrated for 2D gratings. It is also shown that in the vicinity of the Talbot planes there are planes in which the diffraction patterns are spatially compressed and the intensity becomes greater. The results expand the possibility of using the Talbot effect in various applications.
Any quantum (Schrödinger) wavefunction on a sphere is necessarily periodic in time. The corresponding statement down one dimension, for a circular line instead, is the quantum version of the 'Talbot effect' for a diffraction grating in paraxial optics (which is fully analogous to quantum mechanics). In the circle case the 'revival' of any initial wavefunction at the period, or 'Talbot time', is accompanied by a kind of partial revival at any rational fraction of the period, increasing in complexity for less simple fractions. In particular, any piecewise constant initial wavefunction is again piecewise constant at such times. By contrast, in the sphere case, the simplest piecewise constant wave, constant on hemispheres is shown not to retain its piecewise constancy at rational fractions of the period, but instead, rather strikingly, to develop infinities at calculable locations. The calculation requires the uniform asymptotic form of the Legendre polynomials together with the Poisson sum formula leading to Gauss sums.
The quantum Talbot effect is observed from a domain-engineered nonlinear photonic crystal dispensing with a real grating. We deduce and experimentally verify the quantum self-imaging formula which is related to the crystal's structure parameter and working wavelengths. A two-photon Talbot carpet is captured to characterize the Fresnel diffraction dynamics of entangled photons wherein the quantum fractional Talbot effect is specified. The compact and stable quantum Talbot effect can be considered as the contactless diagnosis of domain's homogeneity and developed for new types of entangled photon source and quantum technologies such as quantum lithography with improved performance.