Application of the Wigner distribution function in optics

This contribution presents a review of the Wigner distribution function and of some of its applications to optical problems. The Wigner distribution function describes a signal in space and (spatial) frequency simultaneously and can be considered as the local 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. It thus presents a link between Fourier optics and geometrical optics. The concept of the Wigner distribution function is not restricted to deterministic signals; it can be applied to stochastic signals, as well, thus presenting a link between partial coherence and radiometry. Some interesting properties of partially coherent light can thus be derived easily by means of the Wigner distribution function. Properties of the Wigner distribution function, for deterministic as well as for stochastic signals (i.e., for completely coherent as well as for partially coherent light, respectively), and its propagation through linear systems are considered; the corresponding description of signals and systems can directly be interpreted in geometric-optical terms. Some examples are included to show how the Wigner distribution function can be applied to problems that arise in the field of optics

Error Reduction In Two-Dimensional Pulse-Area Modulation, With Application To Computer-Generated Transparencies

The paper deals with the analysis of computer-generated half-tone transparencies that are realised as a regular array of area-modulated unit-height pulses and with the help of which we want to generate [via low-pass filtering] band-limited space functions by optical means. The mathematical basis for such transparencies is, of course, the well-known sampling theorem [1], which says that a band-limited func-tion y(x), say, [with x a two-dimensional spatial column vector] can be generated by properly low-pass filtering a regular array of Dirac functions whose weights are proportional to the required sample values yey(Xm) [with X the sampling matrix and m=(mi,m2)t an integer-valued column vector; the superscript t denotes transposition]

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals : critical sampling and rational oversampling

Gabor's expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is introduced and its relation to sampling of the sliding-window spectrum is shown. It is shown how Gabor's expansion coefficients can be found as samples of the sliding-window spectrum, where - at least in the case of critical sampling - the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is bi-orthonormal to the corresponding set of window functions. The Zak transform is introduced and its intimate relationship to Gabor's signal expansion is demonstrated. It is shown how the Zak transform can be helpful in determining the window function that corresponds to a given elementary signal and how it can be used to find Gabor's expansion coefficients. The continuous-time as well as the discrete-time case are considered, and, by sampling the continuous frequency variable that still occurs in the discrete-time case, the discrete Zak transform and the discrete Gabor transform are introduced. It is shown how the discrete transforms enable us to determine Gabor's expansion coefficients via a fast computer algorithm, analogous to the well-known fast Fourier transform algorithm. Not only Gabor's critical sampling is considered, but also the case of oversampling by a rational factor. An arrangement is described which is able to generate Gabor's expansion coefficients of a rastered, one-dimensional signal by coherent-optical means

Loss of coherence in double-slit diffraction experiments

By using optical models based on the theory of partially coherent light, and the quantum decoherence model proposed by Joos and Zeh [Z. Phys. B 59, 223 (1985)], we explore incoherence and decoherence in interference phenomena. The problem chosen to study is that of the double-slit diffraction experiment, a paradigmatic example in quantum mechanics. In particular, we perform an analysis of the experiment carried out with cold neutrons by Zeilinger et al. [Rev. Mod. Phys. 60, 1067 (1988).

Analysis and design of a leek-celery intercropping system using mechanistic and descriptive models

Intercropping leek (Allium porrum L.) and celery (Apium graveolens L.) was recognized as an option to reduce growth and reproductive potential of weeds while maintaining yield and product quality of both crops on a high level. To optimise the intercropping system for yield, quality and weed suppression a combined use of mechanistic and descriptive models, together with experimental work, was applied. An eco-physiological model was used to improve understanding of interplant competition based on physiological, morphological and phenological processes. The model was parameterised based on characteristics of the plants in monocultures and its performance was evaluated for the crop mixtures using experimental data from different growing seasons. After validation the model was used to simulate biomass production and quality of leek, celery and seed production of Common Groundsel (Senecio vulgaris L.) for a wide range of crop densities and times of weed emergence. In a second step, the results of the simulations where summarized using a descriptive hyperbolic yield-density model, which then allowed evaluation of the intercropping system in terms of productivity, product quality, and the ability to suppress weeds. The paper will explain this combined modelling approach and how it was used to design and optimise the leek-celery intercropping system. Moreover, this study shows that functional biodiversity, as represented by the intercropping system, can contribute to the improvement of the economical potential while increasing the sustainability of highly developed agricultural production systems