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Journal of the Optical Society of America A

Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems

Martin J. Bastiaans

Published Dec 1, 2000• 31 citations
Publisher

Abstract

A measure for the twist of Gaussian light is expressed in terms of the second-order moments of the Wigner distribution function. The propagation law for these second-order moments between the input plane and the output plane of a first-order optical system is used to express the twist in one plane in terms of moments in the other plane. Although in general the twist in one plane is determined not only by the twist in the other plane but also by other combinations of the moments, several special cases exist for which a direct relationship between the twists can be formulated. Three such cases, for which zero twist is preserved, are considered: (i) propagation between conjugate planes, (ii) adaptation of the signal to the system, and (iii) the case of symplectic Gaussian light.

DOI
10.1364/josaa.17.002475

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A measure for the twist of Gaussian light is expressed in terms of the second-order moments of the Wigner distribution function. The propagation law for these second-order moments between the input plane and the output plane of a first-order optical system is used to express the twist in one plane in terms of moments in the other plane. Although in general the twist in one plane is determined not only by the twist in the other plane but also by other combinations of the moments, several special cases exist for which a direct relationship between the twists can be formulated. Three such cases, for which zero twist is preserved, are considered: (i) propagation between conjugate planes, (ii) adaptation of the signal to the system, and (iii) the case of symplectic Gaussian light.
The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.
For one-mode light described by the Wigner function of a generic Gaussian form with five real parameters, the photon distribution function is obtained explicitly in terms of the Hermite polynomial of two variables. The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. The first and the second statistical moments of the photon distribution function are calculated. The generating function for the photon distribution is discussed. Various special cases, including the shifted thermal states, correlated and squeezed states, coherent states, etc., are considered.
Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.
The Wigner distribution function (WDF) is a quasi-probability distribution that provides a phase-space representation of quantum states. It offers insights into the state dynamics and contains statistical information of it, such as the marginal probabilities. This thesis investigates a protocol for estimating the WDF of quantum states using symplectic transformations, which are linear canonical transformations that preserve phase-space structure and can be implemented experimentally using the technique of Gaussian optics. These transformations are studied because the time evolution of the WDF can be described by means of these transformations in cases where the Hamiltonian is a second-order (quadratic) polynomial in the phase-space variables. Furthermore, it was found that the Fourier transform of the marginal probability (measured at the laboratory) is related to the characteristic function (Fourier transform of the Wigner function). This relationship is given by Equation 2.46, which states that, with each intensity data obtained by applying a given symplectic transformation, the value of the characteristic function (Fourier transform of the WDF) at a given point of the phase space can be obtained. Based on this, an optical setup was implemented to perform symplectic transformations, such as free-space propagation and passing through a lens. The study focuses on the WDF of a one-dimensional electric field of two Gaussian beams, separated by a distance 2d, where d = 1.388 ± 0.004mm. The two beams are passed through the transformation system (propagation-lens-propagation) and then the marginal probability (intensity) is measured with a CCD camera. After this, the Fourier transform was computationally applied to the intensity data and compared with the theoretical plots of the characteristic function (Equation 2.2). In doing so, it was found that it is necessary to multiply the data obtained by an additional scaling factor to improve them to better fit the theoretical results. Finally, the protocol to be followed to reconstruct the Wigner function of this state with the obtained data was described.
The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of , in the monomial basis, are shown to be essentially the SU (2) Clebsch–Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n -mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.