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When a part of an optical wave-front experiences a sharp change in its phase, Fresnel diffraction becomes appreciable. Sharp change in phase occurs as a wave-front strikes with a phase step. The intensity distributions of diffraction patterns of the phase step is formulated by applying Fresnel–Kirchhoff integral. For while the incident light on the step is coherent, the Fresnel–Kirchhoff integral can be solved by using familiar Fresnel integrals. But, when the incident light is partially coherent, one can not express the diffraction integral as the Fresnel integrals and the problem is summarized in solving some unusual integrals. In this report, we propose Fourier transform method for solving the Fresnel–Kirchhoff integral. In this regard we use discrete Fourier transform method and calculate Fresnel diffraction from the 1D phase step by FFT-based algorithms. This method does not have any restriction on the coherence and profile shape of the incident light. We show that the method have appropriate solutions for coherent and partially coherent lights. For the case of the coherent light illumination of the step, the obtained results are in good agreement with the calculated results by using the Fresnel integrals in reported literatures.

The possibility of X-ray phase contrast imaging using already suggested three-block interferometer consisting of bi-level Fresnel zone plates is considered. The interferometer operates in the amplitude-division mode and does not impose strong requirements to spatial and temporal coherences of an initial radiation. The use of the Fresnel zone plates as the interferometer blocks allows one to obtain an optically magnified image of an object and to condense the radiation incident on the tested object.

Aims. We seek to study the Fresnel diffraction of external occulters that differ from a single mask in a plane. Such occulters have been used in previous space missions and are planned for the future ESA Proba 3 ASPIICS coronagraph. Methods. We studied the shading efficiency of double on-axis disks and generalized results to a 3D occulter. We used standard Fourier optics in an analytical approach. We show that the Fresnel diffraction of two and three disks on axis can be expressed using a Babinet-like approach. Results are obtained in the form of convolution integrals that can be written as Bessel-Hankel integrals; these are difficult to compute numerically for large Fresnel numbers found in solar coronagraphy. Results. We show that the shading efficiency of two disks is well characterized by the intensity of the residual Arago spot, a quantity that is easier to compute and therefore allows an interesting parametric study. Very simple conditions are derived for optimal sizes and positions of two disks to produce the darkest structure around the Arago spot. These conditions are inspired from empirical experiments performed in the sixties. A differential equation is established to give the optimal envelope for a multiple-disk occulter. The solution takes the form of a simple law, the approximation of which is a conical occulter, a shape already used in the SOHO Mission. Conclusions. In addition to quantifying expected results, the present study highlights unfortunate configurations of disks and spurious diffractions that may increase the stray light. Particular attention is paid to the possible issues of the future occulter spacecraft of ASPIICS.

The X-ray interferometric Fresnel holography was proposed and theoretically investigated. It was shown that under definite conditions the recorded interference pattern formed at the output surface ofthe analyzer crystal (the third block of interferometer) is the X-ray interferometric hologram (generally the Fresnel one) of the object under investigation. Further reconstruction of an image is performed with the help of visible light or a numerical method. As an example the recording of a Fresnel hologram of the simplest case of one-dimensional object, a narrow slit, and the reconstruction of an image by means of visible light were considered. This method may be used in X-ray microscopy and realized using synchrotron sources ofX-ray radiation, as well as X-ray free electron lasers.

The iterative Fresnel integrals method (IFIM) has been applied for the simulation and generation of the complete near-field Fresnel diffraction images created by N -apertures for the first time. The simulation can be performed in any PC using a MATLAB program developed by the authors. Necessary formalism was derived for the general N -slit problem, and a simulation algorithm was devised for this application. An interesting combination of interference effects with Fresnel diffraction was observed in the simulated images. Transition to the expected Fraunhofer diffraction pattern from Fresnel diffraction for N -apertures is also observed in the simulations under the appropriate conditions. Principal maxima of the expected Fraunhofer diffraction were observed at their expected positions, as well as the expected minima and the secondary maxima. The program can serve as a useful tool to study the complex phenomenon of Fresnel diffraction from N -apertures, and in addition, to study the near-field Fresnel diffraction from amplitude diffraction gratings.

Focusing characteristics of cross like Fresnel zone plate is well analyzed. This element is composed of two perpendicular linear Fresnel zone plates. Focusing properties of the element is compared to a classic and square Fresnel zone plate. Analysis clarify that focusing of the elements is well corrected in comparison to square Fresnel zone plate. Resolution is noticeably increased, DC noise and diffraction efficiency are substantially reduced and also secondary maxima have been evidently suppressed. Theoretical results have been completely verified by experiment.

We present the study of the wave motion in the Talbot interferometer with an additional element such as a lens for all related audiences. Our solutions are in the analytic form. A general principle of the Talbot effect, which is the optical near-field effect, is the Fresnel diffraction. The Fresnel integral is rather complicated. We therefore introduce an alternative method which is based on the wave propagation through the transmission functions of the grating and the lens. Our method has been proved by a simple experimental setup.

Phase shifted linear Fresnel zone plates are presented for the first time. These diffractive elements are linear Fresnel zone plates in which their phase are laterally shifted. The impact of the phase shifting on their diffractive and focusing properties are studied. It is demonstrated that by shifting the phase, two parallel linear beams can be generated at the focal plane. Furthermore, as they are propagating, a non-diffractive line-shaped beam is generated at a given distance from the focus. Transverse intensity profile of the beam at different distances as well as its cross section of propagation along the optical axis clearly shows that the intensity profile of the beam is really kept unchanged as it is propagated. All results are completely verified by experiments.

Fresnel theory, with certain level of assumptions, explains satisfactorily observed diffraction patterns. It is a scalar theory and basically it deals special way of superposition of fields originated from several points of Huygens wave front. In the present paper a total new approach is proposed according to which the interacting electric fields are not originated from the secondary emitters of the same wave front, instead, they are originated from the successive Huygens wave fronts. The proposed approach is explained with the help of the circular aperture. This new view supports Fresnels theory without a set of assumptions.

It is shown that when a coherent beam of light illuminates a Fresnel's double mirror, Fresnel diffraction becomes appreciable. The subject fundamentally differs from interference. We calculate the diffracted field by applying Fresnel–Kirchhoff integral. We modified the common Fresnel's double mirror by imposing an initial height between the mirrors, as the height is chosen small enough so that the application of Fresnel's double mirror is maintained. We show by simulation and experiment that a phase singularity causes from the initial height and modifies the diffracted field from the Fresnel's double mirror. One can adjust the location of the line singularity by changing the angle between the mirrors. Also, the anomalous behavior of a polychromatic beam diffracted from the modified Fresnel's double mirror at the neighborhood of the line singularity, is investigated by simulation.

There are many 3D data structures for computing diffraction field, the description based on 3D point-cloud is the elemental method of other descriptions, the diffraction field of 3D scene can be thought as the superposition of the diffraction field of each sampled light point and other descriptions can be thought as grouping with point-cloud by different way. The paper focuses on devising a computation method of 3D Fresnel field which is based on Fresnel diffraction field propagation reversibility. The proposed method combines the reversibility of Fresnel propagation and the requirement of sampling spacing by Nyquist criterion, i.e. to find the best space positions to optimize the number of sampling points and the reconstruction for the 3D Fresnel field. The method can be applied to reduce the computational complexity and improve computational efficiency.

In this work, we propose and analyze a novel kind of binary Fresnel zone plate with single focus. It consists of a Fresnel zone plate whose zones have rough edges. We give analytical results for the intensity along the optical axis and demonstrate that lateral roughness of the zones produces the disappearance of secondary foci as a blurring of the edges. Besides, we corroborate its behavior by numerical simulations and experiments. This kind of Fresnel zone plate can be useful in a wide range of photonic applications, even for focusing with soft and hard X-rays or extreme ultraviolet radiation.

The experimental study about the diffraction of high-density grating in deep Fresnel region is performed in this paper. A microscope-magnification method is advanced, and the diffraction intensity distributions of high-density grating in deep Fresnel region are measured. The quasi-Talbot image of grating is also obtained in experiment. The corresponding theoretic results are also provided for convenient comparison with the experimental ones. The coincidence of experimental and theoretical results verifies the reliability of microscope-magnification method. This method can be used in the measurement of the diffraction of sub-wavelength structure in near field and deep Fresnel region.

The intensity distribution in the vicinity of the focus of circular Fresnel zone plate formed by Bessel-Gauss beam is studied. Dark hollow beams are obtained near focus under the illumination of high order Bessel-Gauss beam, the width and the length of central hollow depends on the order of the beam and the number of plate zones. These beams with special intensity distribution in the field of laser processing and trapping particles have potential application value.

Diffraction of matter-waves is an important demonstration of the fact that objects in nature possess a mixture of particle-like and wave-like properties. Unlike in the case of light diffraction, matter-waves are subject to a vacuum-mediated interaction with diffraction obstacles. Here we present a detailed account of this effect through the calculation of the attractive Casimir-Polder potential between a dielectric sphere and an atomic beam. Furthermore, we use our calculated potential to make predictions about the diffraction patterns to be observed in an ongoing experiment where a beam of indium atoms is diffracted around a silicon dioxide sphere. The result is an amplification of the on-axis bright feature which is the matter-wave analogue of the well-known `Poisson spot' from optics. Our treatment confirms that the diffraction patterns resulting from our complete account of the sphere Casimir-Polder potential are indistinguishable from those found via a large-sphere non-retarded approximation in the discussed experiments, establishing the latter as an adequate model., 13 pages, 10 figures

A technique and investigation results are presented on the effect of the number of Fresnel zones on the diffraction efficiency and quality of the wave front formed by a diffractive optical element. Equations are presented for the microstructure reliefs of diffractive optical elements making (regardless of the number of Fresnel zones) the wavefront shape coinciding with the calculated one and ensuring 100% diffraction efficiency at a single wavelength and, in the case of two-layer structures, almost 100% diffraction efficiency in a wide spectral range.

We applied the iterative Fresnel integrals method for the numerical computation of Fresnel diffraction patterns from rectangular apertures tilted at an arbitrary angle to the optical axis. Detailed theoretical formalism is developed and discussed, and then is applied for the numerical computation and simulation of the actual diffraction patterns for an arbitrary optical configuration. The generated intensity distributions (images) show distortion and stretching in the direction of the tilt, but not in the other orthogonal direction. Significant decrease of the intensity is also predicted and observed, the decrease being proportionate with the tilt angle. The simulated images qualitatively resemble those published in the literature. In addition to single-axis tilts, simultaneous rotations (tilts) of the aperture in two orthogonal coordinate axes were also briefly considered and simulated.

In high frequency, due to the locality of the scattering, the importance of currents is localized near the scattering centers, which correspond the stationary phase points and the edge diffraction points, as is defined by the GTD. This localization would provide the reduction of unknowns in the method of moments (MoM), though the area of importance moves with the location of the source and the observer. If this concept is extended for the observer on the surface of scatterer, the reaction matrix is transformed into the sparse band matrix. Authors have proposed the use of Fresnel zone number as the threshold for the truncation of the scatterers or the reaction matrix. This talk reviews the concept and the advancement of the high frequency technique based upon the locality of scattering in terms of Fresnel zone number.

A physical interpretation of the focal shift is given based on geometrical optics. The results of an analytical study of the focal shift depending on the Fresnel numbers are presented. The effect of the aperture diameter and the width of the central zone of the phase Fresnel lens on the distribution of the light intensity along the axis upon illumination by a convergent light beam is analyzed.

We review three different methods of generating digital Fresnel holograms of 3-D real-existing objects illuminated by incoherent light. In the first method, a scanning hologram is generated by a unique scanning system in which Fresnel zone plates (FZP) are created by a homodyne rather than the common heterodyne interferometer. During the scanning, the FZP projected on the observed object is frozen rather than varied as previously. In each scanning period, the system produces an on-axis Fresnel hologram. The twin image problem is solved by a linear combination of at least three holograms taken with three FZPs with different phase values. The second hologram reviewed here is the digital incoherent modified Fresnel hologram. To calculate this hologram, multiple-viewpoint projections of the 3-D scene are acquired, and a Fresnel hologram of the 3-D scene is directly computed from these projections. This method enables to obtain digital holograms by using a simple digital camera, which operates under regular light conditions. The last digital hologram reviewed here is the Fresnel incoherent correlation hologram. In this motionless holographic technique, light is reflected from the 3-D scene, propagates through a diffractive optical element (DOE), and is recorded by a digital camera. Three holograms are recorded sequentially, each for a different phase factor of the DOE. The three holograms are superposed in the computer, such that the result is a complex-valued Fresnel hologram that does not contain a twin image.

Although Thomas Young is generally given credit for being the first to provide evidence against Newton’s corpuscular theory of light, it was Augustin Fresnel who first stated the modern theory of diffraction. We review the history surrounding Fresnel’s 1818 paper and the role of the Poisson spot in the associated controversy. We next discuss the boundary-diffraction-wave approach to calculating diffraction effects and show how it can reduce the complexity of calculating diffraction patterns. We briefly discuss a generalization of this approach that reduces the dimensionality of integrals needed to calculate the complete diffraction pattern of any order diffraction effect. We repeat earlier demonstrations of the conventional Poisson spot and discuss an experimental setup for demonstrating an analogous phenomenon that we call a “second-order Poisson spot.” Several features of the diffraction pattern can be explained simply by considering the path lengths of singly and doubly bent paths and distinguishing between first- and second-order diffraction effects related to such paths, respectively.

An indirect algorithm for Fresnel diffraction calculation at any distance is presented in this paper. Using this indirect algorithm, the Fresnel diffraction on a near plane can be realized in two steps, the first is to acquire the diffractive field on a far field and the second is to obtain the diffractive field on a near plane using an inverse Fresnel integral. Compared with conventional algorithms, this algorithm is valid at any range of distance. The sampling theorem, as the basis of this indirect algorithm, is discussed in detail. Theoretical calculations show good agreements with experimental results.

An objective lens composed of a spherical Fresnel zone plate mirror and aspheric mirrors is designed. The Fresnel zone plate with a spherical shape is analyzed, and the method to approximately replace the aspherical mirror with spherical Fresnel zone plate is deduced. The objective lens is designed with a single spherical Fresnel zone plate mirror and three aspherical mirrors. Under the condition of 100 × magnification, 2.5 Fresnel number and 120° field angle, the modulation transfer function can reach above 40% at 0.6 line pairs/mm on the magnification side, and the distortion is less than 2.2%. This method can provide a reference for the application of Fresnel zone plate in visible light imaging, and has a bright future with the continuous development of the fabrication technique of Fresnel devices.

We define the effective Fresnel number of a focused cylindrical spherical aberrated beam. Based on the effective Fresnel number, a simple analytical expression for evaluating the relative focal shift is presented. It is shown that the relative focal shift is determined by the effective Fresnel number, which is dependent on the spherical aberration and Fresnel number of the focused system. To illustrate the approach, a comparison between the results obtained by this new method and the numerical integral calculation is made. The results indicate that the relative focal shift can be simply and exactly evaluated using this new method. A special case is discussed, in which the effective Fresnel number turns out to be a pure imaginary number, and the corresponding relative focal shift is positive.

The study of scattered fields from nonspherical scatterers is becoming an important subject particularly in the area of theoretical modeling of microwave backscatter from vegetation. The generalized Rayleigh-Gans approximation has been widely used in the calculation of scattered fields from scatterers where at least one of its dimensions is comparably smaller than the wavelength. In such calculations, far-field approximations are used, which are not accurate when the observation points are in the Fresnel and near-field regions of the scatterer. Hence, the effects of Fresnel phase and amplitude corrections for the scattered field of circular disks, needles, and cylinders have been examined and shown to be significant in the calculation of closely spaced scatterers. However, the effects of Fresnel corrections in the scattered fields from general ellipsoids and elliptic disks have not yet been studied. In this paper, the scattered fields from general ellipsoids and elliptic disks are formulated based on the generalized Rayleigh-Gans approximation for cases with and without Fresnel corrections. Theoretical analysis shows that the Fresnel effects are important particularly at larger frequencies and at the null locations in the plots of the backscattering cross section. These effects become more important as the ellipticity of the scatterer increases. Comparisons with measurement data demonstrate that the calculated results with Fresnel corrections provide a better match compared with those without.

We study the Fresnel diffraction of Gaussian beam truncated by one circular aperture, and give the general analytic expression of the Fresnel diffraction of truncated Gaussian beam denoted by Bessel functions. Then the characteristic of the axial diffraction fluctuation and the influence of the caliber of the circular aperture and the wave waist of Gaussian beam on the diffraction distributions are discussed, respectively. Through the numerical calculations, the characteristics of the transverse diffraction are presented and the relationship of the fluctuation of the transverse diffraction profile and the position of the axial point is shown. The physical origin of the fluctuation of Fresnel diffraction intensities of truncated Gaussian beam is expressed in terms of Fresnel half-zone theory. These phenomena and the conclusions are important for the measurement of the parameters of the beam and its applications.

In this work, Fresnel's mirror and Young's double-slit experiments are compared. Numerical calculations show that Fresnel's experiment and Young's experiment present significant differences between their interference patterns when the optical source is extended rather than point-like. Those differences agree with the analogous between the overlapping of two quasi-monochromatic beams on the temporal domain (the analogous of Young's experiment) and these same beams superimposed previous modulation with sawtooth-wave (the analogous of Fresnel's interferometer). The implemented algorithms allow evaluating the cases of fully spatially coherence, fully spatially incoherence and spatially partially coherence. Computer simulations are presented to show the validity of our proposal.

The paper is devoted to investigation of fluctuations of the position of the focal spot in Fresnel’s zone of a round aperture.

This appendix explains complex Fresnel integrals. Several forms of Fresnel integrals have been defined by various authors, and as a result, the definition often depends on the application. The complex Fresnel integral is related to the cosine and integral functions. Another function used in diffraction theory is the modified Fresnel integral. This book is intended as an advanced text for courses in antennas, with a focus on the mature but vital background field of aperture antennas. It is is useful for students of final year, MSc, PhD and Post‐Doctoral, as well as readers who are moving from academia into industry, beginning careers as wireless engineers, system designers, in R&D, or for practising engineers.

When the Fresnel–Kirchhoff (FK) diffraction integral is evaluated exactly (instead of using the Fresnel approximation), the well-known mathematical inconsistency in the FK boundary conditions leads to unacceptable results for the intensity of Poisson's spot. The Rayleigh–Sommerfeld (RS) integral has no inconsistencies and leads to an accurate description. The case for RS is bolstered by the observation that it is equivalent to Fourier propagation.

Recurring to the characteristic of Bessel function, we give the analytic expression of the Fresnel diffraction by a circular aperture, thus the diffractions on the propagation axis and along the boundary of the geometrical shadow are discussed conveniently. Since it is difficult to embody intuitively the physical meaning from this series expression of the Fresnel diffraction, after weighing the diffractions on the axis and along the boundary of the geometrical shadow, we propose a simple approximate expression of the circular diffraction, which is equivalent to the rigorous solution in the further propagation distance. It is important for the measurement of the parameter of the beam, such as the quantitative analysis of the relationship of the wave error and the divergence of the beam. In this paper, the relationship of the fluctuation of the transverse diffraction profile and the position of the axial point is discussed too.

The starting point of the article is the puzzling fact that one cannot recover the Fresnel coefficients by letting tend the width of a slab to infinity. Without using the so-called limiting absorption principle, we show by a convenient limit analysis that it is possible to define rigorously the field diffracted by a semi-infinite periodic medium., Comment: 14 pages, 13 figures, submitted to Optics Communications

An analytical expression based on an improved Rayleigh–Sommerfeld diffraction formula with evanescent term is derived for analyzing the axial light intensity distribution throughout the whole space behind a Fresnel zone plate. The effects of the number of Fresnel zones and the size of aperture on the axial intensity distribution are calculated for two kinds of Fresnel zone plate with larger and smaller aperture. The validity of the general formulae for calculating the focal lengths and the relative intensities of the foci of a Fresnel zone plate is analyzed.

In a diffracted wavefield occurs in patterns phase variations. The most familiar example is the Fraunhofer diffraction pattern through a single slit: the theory predicts that the side bands of the diffraction pattern alternate in phase. This paper deals with phase reversal phenomenon in diffraction patterns. Two interesting systems are recognized: the diffraction through a half plane and the diffraction through a single slit.

It is shown by using the Fresnel-Kirchhoff diffraction theory and the method of images that a scalar field confined by a 3-dimensional optical waveguide can be generated in free space by a suitable light source. In the method the boundaries of a waveguide are replaced by virtual sources. This allows one to present the wave guiding as the interference and diffraction of multiple light beams produced in free space by the guide equivalent source (Fresnel waveguide). Thus, the scalar optics of a 3-dimensional waveguide is presented as the free-space beam optics. The method is illustrated by construction of the Fresnel sources for the triangular, rectangular and hexagonal waveguides. The numerical examples demonstrate the diffraction-free and self-imaging propagation in the free-space of the eigenmodes of the Fresnel rectangular-waveguide.

The Fresnel interference pattern of a triple-gaussian-slit light interferometer is investigated both theoretically and experimentally. It is shown that the structure of Fresnel fringes of the interferometer is similar to the fractional Talbot images of diffraction gratings. A dependence of the Fresnel fringes on the phase difference between the three slits of the interferometer is studied. It is shown that a change of the the phase of light field in one of the slit of the interferometer leads not only to a lateral shift of the Fresnel fringes but (in contrast to Young’s interferometer) also changes of their form and amplitude.

The optical guiding of X-ray radiation by capillary waveguides is analysed by using the Fresnel-Kirchhoff diffraction theory and the method of images. The guiding of the continuous, short-pulse, coherent, partially coherent and incoherent waves is presented as the diffraction and interference of several X-ray beams produced in free space by the Fresnel source of the waveguide. Using the Fresnel source properties, the necessary conditions for influence of the diffraction and interference on the output characteristics of the capillary X-ray optics in the near- and far-field diffraction zones are derived. The experimental data and computer simulations presented confirm the wave optics behaviour of X-rays under such conditions.

A solid immersion Fresnel zone plate (SIFZP) is designed and its near-field imaging characteristics are analyzed using the vector angular spectrum method. Results show SIFZP can generate a high intensity and subwavelength focus.

Fractional Fourier transformation of an object can be approximated by the object's free-space Fresnel diffraction pattern under some restricted conditions and plane wave illumination according to Hua's method. A better approximation is achieved under least-squared conditions developed in this paper. Simulation results verify that our theoretical development works for any fractional order a compared with the previous approach.

Fringe wave current components of modified edge representation (MER) equivalent edge currents (EECs) are modified by Fresnel zone number. The dipole wave scattering from flat square plates are detailed with numerical examples and the accuracy of the fields predicted by this new fringe wave expressions is confirmed by comparing with MoM.