Wigner Distribution Function Approach Research Articles

Wigner distribution function approach to analyze MIMO communication within a waveguide

Multiple-input-multiple-output (MIMO) communication is a technology to create high capacity wireless links. The main aim of this paper is to provide a foundation to mathematically model wireless chip to chip communication within complex enclosures. This paper mainly concentrates on modelling wave propagation between transmit and receive antennas through a phase space approach which exploits the relationship between the field-field correlation function (CF) and the Wigner distribution function (WDF). A reliable model of wireless chip-to-chip (C2C) communication helps mitigate the information bottleneck caused due to the wired connections between chips, thus, help improve the efficiency of electronic devices of the future. Placing complex sources such as printed circuit board (PCB) inside a cavity or enclosure results in multi-path interference and hence makes the prediction of signal propagation more difficult. Thus, the CFs can be propagated based on a ray transport approach that predicts the average radiated density, but not the significant fluctuations that occur about it. Hence, the WDF approach can be extended to problems in finite cavities that incorporates reflections as well. Phase space propagators based on classical multi-reflection ray dynamics can be obtained by considering the high-frequency asymptotics.

Analysis of optical systems with extended depth of field using the Wigner distribution function

We describe the use of a Wigner distribution function approach for exploring the problem of extending the depth of field in a hybrid imaging system. The Wigner distribution function, in connection with the phase-space curve that formulates a joint phase-space description of an optical field, is employed as a tool to display and characterize the evolving behavior of the amplitude point spread function as a wave propagating along the optical axis. It provides a comprehensive exhibition of the characteristics for the hybrid imaging system in extending the depth of field from both wave optics and geometrical optics. We use it to analyze several well-known optical designs in extending the depth of field from a new viewpoint. The relationships between this approach and the earlier ambiguity function approach are also briefly investigated.

Application of the Wigner distribution function to complex-argument Hermite– and Laguerre–Gaussian beams beyond the paraxial approximation

The Wigner distribution function (WDF) is applied to study the propagation of complex-argument Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation. The analytical expressions for their intensity distributions in free-space propagation are derived, which are expressed in terms of Hermite polynomials for nonparaxial complex-argument HG beams and in terms of the sum of finite Hermite polynomials for nonparaxial complex-argument LG beams. A detailed comparison of the WDF approach, series expansion method and paraxial expressions is made, which shows that in the paraxial regime the WDF approach and series expansion method deliver consistent results with that of paraxial expressions. Beyond the paraxial approximation, the WDF approach offers convergent results, whereas the series expansion method has a limited applicable range, within which it gives consistent results with that of WDF approach but beyond which it gives unrealistic and divergent results.

Optical systems under broad-band illumination: Analysis based on a wigner distribution function approach

The irradiance produced by an optical imaging system suffering from aberrations under broad-band illumination is analysed in the phase-space domain of the Wigner distribution function associated with a modified one-dimensional pupil aperture. To this end, a coordinate transformation is employed in such a way that the varying irradiance along different paths in the image space can be derived from different slices of the Wigner distribution function, which involve both the defocus distance and the dispersion properties of the optical system. The method is illustrated by comparing the axial irradiance distribution for different primary aberrations. Also, the irradiance along parabolic paths is obtained in order to analyse the coma effect.

Wigner distribution function approach to dissipative problems in quantum mechanics with emphasis on decoherence and measurement theory

We first review the usefulness of the Wigner distribution functions (WDF), associated with Lindblad and pre-master equations, for analyzing a host of problems in Quantum Optics where dissipation plays a major role, an arena where weak coupling and long-time approximations are valid. However, we also show their limitations for the discussion of decoherence, which is generally a short-time phenomenon with decay rates typically much smaller than typical dissipative decay rates. We discuss two approaches to the problem both of which use a quantum Langevin equation (QLE) as a starting-point: (a) use of a reduced WDF but in the context of an exact master equation (b) use of a WDF for the complete system corresponding to entanglement at all times.

Optical systems under broad-band illumination: analysis based on a Wigner distribution function approach

Optical systems under broad-band illumination: analysis based on a Wigner distribution function approach

Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach

The on-axis image irradiance obtained from a radially symmetric optical system under broadband illumination is analysed in the phase-space domain of the Wigner distribution function associated with a modified one-dimensional pupil aperture. To this end, a coordinate transformation is employed in such a way that the varying on-axis irradiance can be derived from different slices of a two-dimensional Wigner distribution function, which involve both the defocus distance and the dispersion properties of the optical system. The method is illustrated by comparing the imaging performance of four different pupil apertures, when the system is illuminated with a light beam having a gaussian spectral content.

Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach

We propose a new method for the computation of the tristimuli values that correspond to the impulse response along the optical axis provided by an imaging optical system working under polychromatic illumination. We show that all the monochromatic irradiance distributions needed for this calculation can be obtained from the Wigner distribution function associated with a certain version of the pupil function of the system. The use of this single phase-space representation allows us to obtain the above merit function for aberrated systems with longitudinal chromatic aberration and primary spherical aberration. Some numerical examples are given to verify the accuracy of our proposal.

Time dependent, self-consistent simulations of field emission from silicon using the Wigner distribution function

Field emission current calculations from semiconductors typically rely on the assumptions that the emitted current is negligible and that the band bending at the surface may be calculated via Poisson’s equation with a Fermi–Dirac momentum distribution of electrons [the zero emitted current approximation (ZECA)]. This approach cannot take into account complications due to quantum confinement near the surface, and problems associated with scattering for large applied field. We have developed and applied a time-dependent, self-consistently calculated Wigner distribution function (WDF) approach for dealing with the semiconductor field emission problem, and have applied it to the case of silicon for the cases of 300 and 900 K. In particular, effects of self-consistency and scattering on the density and potential profiles were examined, comparing the results to the ZECA approach to show when quantum and scattering effects become significant. At low fields, the WDF approach may be coupled with the transmission coefficient approach to yield reasonable current estimates. At high fields, it is shown that current saturation effects set in, in which the current no longer exponentially rises with applied field. Finally, the time dependence of the current and particle densities after a sudden shift in the applied field are examined.

Numerical simulation of field emission and tunneling: A comparison of the Wigner function and transmission coefficient approaches

Quantum transport through one-dimensional potential barriers is usually analyzed using either the transmission coefficient (TC) or the Wigner distribution function (WDF) approach. Fast, accurate, and efficient numerical algorithms are developed for each and are compared for (a) calculating current-field relationships for field-emission potentials with silicon parameters (and current-voltage relationships for resonant tunneling diodes), (b) their ability to accommodate scattering, self-consistency, and time dependence, and for (c) the behavior of their ‘‘particle trajectory’’ interpretations. In making the comparisons, the concern will be on the ability of each method to be incorporated into a larger ensemble-particle Monte Carlo simulation; it is argued that, in this regard, the WDF approach has significant advantages. Since the TC calculations rely on the Airy function approach, a detailed comparison of this method is made with the widely used Wentzel–Kramers–Brillouin and Fowler–Nordheim approaches for the general problem of field emission from a material into the vacuum.

Numerical simulation of field emission from silicon

A common method for estimating current from semiconductor field emission is to assume that the potential inside the semiconductor is the same as that which would occur if no current were emitted [zero emitted current approximation (ZECA)], and then to use the calculated density at the surface in a standard Fowler–Nordheim analysis familiar from metallic field emission calculations. At high fields, it is obvious that this method fails, as the effect of current is no longer negligible in the calculation of the band bending effects. In this work, we shall present methods for calculating the band bending effects at arbitrary fields in the ZECA model; a comparison of the Wentzel–Kramers–Brillouin and Fowler–Nordheim approaches to field emission for silicon will be compared to the airy function approach of evaluating the transmission coefficient (TC) in the ZECA model; and finally, the limits of ZECA for field emission from silicon will be analyzed by comparing the TC calculations to Wigner distribution function (WDF) simulations performed with full band bending effects and scattering. As anticipated, the ZECA model, whether calculated by the TC or WDF approach, diverges from the full band bending WDF model for high fields. A discussion of this behavior is given in light of the density and current profiles.