Francisco F. Medina

Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment

Theoretical considerations, simulations and experiments have been carried out to distinguish between Fresnel and the Fraunhofer diffraction regarding the formation of interference patterns by a conventional Youngs double-pinhole arrangement with variable separation, which is illuminated by a coherent plane wave. We show that the optical path difference introduced by this setup fits the Fresnels phase difference between the pinholes. Consequently, it is possible to determine the number of Fresnel zones subtended by a circle centered on one of the pinholes and with radius equals to the pinhole separation. Then, we propose a criterion for distinguishing between Fresnel and Fraunhofer diffraction based on this number of Fresnel zones, which is applicable for diffracting apertures with any shape.

An experiment on the particle–wave nature of electrons

A primary electron beam of a transmission electron microscope is scattered into secondary beams by the planes of atoms of a single crystal. These secondary beams are focused to form a diffraction pattern on the final screen. This experiment is similar to the Thompson one which, independently by Davisson and Germer, demonstrated the de Broglie hypothesis of the existence of electron waves. Without changing the experimental apparatus, it is possible to realize an interference experiment with electrons coming from two spatially separated sources in analogy with the optical Young set-up. Both experiments are clear evidence of the electron wave-like behaviour. By varying the conditions of illumination, changes of the fringe visibility which reveal the spatial coherence properties of the electron beam, are displayed.

Optical tubular structures produced by diffraction of circular apertures

The coherent optical field diffracted through a circular aperture can produce optical tubular structures single-closed or twice-closed, by choosing the suitable experimental conditions. Both structures can be generated on the same optical set-up only by changing the experimental parameters involved on it, i.e. the aperture radius, the on-axis observation point, the wavelength and the on-axis source position. Such structures, generated in this way, have strong intensity gradients at their edges and their dimensions can be controlled in order to make them appropriate structures for optical trapping of particles, for instance.

Fresnel–Fraunhofer diffraction and spatial coherence

In this paper we discuss the influence of the spatial coherence properties of the optical field in diffraction phenomena. Then, we propose a criterion for distinguishing between Fresnel and Fraunhofer diffraction of optical fields in any state of spatial coherence.

Partially coherent imaging with Schell-model beams

Abstract The agreement between the general description of imaging with quasimonochromatic spatially partial coherent optical fields and the concept of the Schell-model beam is analysed. Radiator pairs of object Schell-model beams at the entrance window of the imaging system yield interferograms with different periods, visibilities and fringe orientation onto the entrance pupil, whose superposition determines the intensity distribution there. Furthermore, the structure of the optical field in the entrance pupil will be modified by the properties of the imaging system through their transfer to the exit pupil. Radiator pairs in the exit pupil generate interferograms onto the exit window, whose superposition yields the image intensity distribution. In the case of ideal imaging, the optical field in the exit window will also be constituted by image Schell-model beams, which are a scaled, inverted and proportional version of the object Schell-model beams, and the superposed interferograms in the exit window will be Youngs interferograms.

Discovering the puzzling behaviour of electrons with the Grimaldi–Young experiment

An experiment analogous to that devised by Grimaldi and subsequently repeated by Young to study the nature of light has been realized with electrons. Following the Grimaldi and Young line of thought, an original approach is presented to introduce undergraduate physics students to the wave behaviour of electrons. An electron microscope equipped with a low coherent source of electrons and a thin platinum wire, acting as an opaque obstacle, is used to reproduce the experimental conditions adopted by Grimaldi and Young with light. Electron interference fringes produced in the geometrical shadow of the obstacle are interpreted by assuming that electrons behave like a sound or a light wave. This hypothesis is confirmed by the modelling of the experimental electron interference patterns.

Angular criterion to distinguish between Fraunhofer and Fresnel diffraction

Summary The distinction between Fresnel and Fraunhofer diffraction is an essential condition for the accurate analysis of diffracting structures. In this paper we propose a criterion based on the angle subtended by the first minimum of the diffraction pattern from the centre of the diffracting aperture. The determination of the minimum of the diffraction pattern is the crucial point to assure the accuracy of the criterion. Therefore, the applicability of adequate thresholds for detection is discussed. The criterion is also generalized by expressing it in terms of the number of Fresnel zones delimited by the aperture. Simulations are reported to illustrate the feasibility of the criterion.

Spatial partial coherence in Young's interferograms

Youngs interferograms with high visibility reveals a high degree of spatial coherence of first order. But, spatially partial coherence of second order can be observed when it interferes itself through a compensated Michelsons interferometer attached at the exit of the Youngs slit pair. We show that the patterns at the exit of the Michelsons interferometer are Youngs interferograms with modulation fringes, which allow an estimation of the degree of the high order spatial coherence.

Resolution, focus depth in the Fresnel-Fraunhofer domains

Defocused imaging can be analyzed as Fresnel diffraction. The defocus parameter, that characterizes this imaging, is related to an effective Fresnel number, which is induced by the geometry of the imaging set-up. From this point of view, perfectly focused images result from Fraunhofer diffraction. Therefore, the same criteria for distinguishing between Fraunhofer and Fresnel diffraction can be applied to determine if the image is or not focused. As a consequence, new definitions of the focus depth can be deduced. In addition, resolution of two point-objects under different states of spatial coherence and focus conditions is discussed and some resolution criteria are deduced. Then, they are compared to the classical resolution criteria, which are considered as applicable to the extreme cases of fully spatial coherence and fully spatial incoherence. Some classical examples, such as, imaging of one point object and two near point objects, are discussed to illustrate the analysis.

Far field properties of spatial coherence beams

Spatial coherence of optical fields can be considered as beam structured if the size of the coherence patch varies slowly through the propagation of the optical field. As a consequence, the correlation of the optical field will be concentrated in a finite region around the direction of propagation. For properly describing it in the Fraunhofer-Fresnel domain, the marginal cross-spectral density is introduced. The superposition of spatial coherence beams in this domain is also analysed. It produces an interference field and a spatial coherence Moire. DEFINITION OF SPATIAL COHERENCE BEAMS Optical fields can take the form of beams that come as close as possible to spatially localised and non-diverging waves. As a consequence, the beam power is principally concentrated within a small region surrounding the beam axis [1]. Similarly, the spatial coherence of optical fields can be considered as beam structured if the coherence patches in two planes along the direction of propagation are comparable in size. We call these structures Spatial Coherence Beams. They are quite different from the conventional spatially coherent beams, which are beams conformed by spatially coherent optical fields [2]. However, the structure of spatial coherence beams at a specific plane is described by the corresponding crossspectral density [3]. Specifically, let us assume an optical field, whose cross-spectral density is given by