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We present numerical calculations of the Fresnel diffraction for periodic structures in an optical system with two apertures. In such a system, measurements are affected by the relation between the spatial frequency of the sample and the geometrical parameters involved (i.e., aperture diameters, radiometer-sample distance, in-plane rotation, and translation of the sample). This numerical calculation of the Fresnel diffraction enables us to establish criteria to choose the right geometrical parameters of the system to ensure invariance of the measurements when the sample is rotated or shifted. We use the theory of partial coherence to calculate the Fresnel diffraction through two successive apertures. By using the point spread function of the system, as in the theory of partial coherence, we avoid complicated statistical processes that are commonly used in this theory. We show some numerical results that verify our proposal.

A modified Fresnel zone plate that can produce an approximate Gaussian focal spot is proposed for the focusing and imaging of soft x rays and extreme ultraviolet radiation. The selection conditions for the positions and the widths of the concentric open rings are analytically presented. The focal spot size can be much smaller than the width of the narrowest open ring, and the sidelobes and the higher orders can be effectively suppressed. Through numerical experiments, we confirm that a Gaussian focal spot with a beam width of 7.7 nm can be produced by a modified Fresnel zone plate with a minimum structure size of 30 nm.

In a new type of illumination system for reflective liquid-crystal displays, the frontlight, unwanted shadows appear in certain viewing directions. It will be shown that for an accurate description of these shadows the geometrical optics approach is not satisfactory and that Fresnel diffraction has to be taken into account. A model for the diffraction effects was developed, and the predicted results correspond well to the measurements. In addition, some remarkable effects were explained from the theory of Fresnel zones.

We recognize that one can adapt any dispersion-compensated broadband optical Fourier transformer to achieve wavelength compensation in the Fresnel diffraction region just by inserting a diffractive lens at the input plane and vice versa. This unification procedure is employed in a second stage in the design of a novel hybrid (diffractive-refractive) optical setup that provides, in a sequential way, nearly wavelength-independent Fresnel diffraction patterns in the irradiance of the object transmittance.

The Arago point is a bright spot that appears at the center portion of the shadow of an obstacle such as a ball or a disk as the result of diffraction. Using this Arago point, we can judge the sphericity of a ball and the circularity of a disk, but it is difficult to distinguish the ball or the disk. To solve this problem, it is also necessary to observe the z coordinate component. As a method for doing this, we rotated the disk and observed its diffraction pattern as a function of the rotation angle. By employing the white-light laser as the light source and a three charge coupled device (3-CCD) camera as the detector, we have observed novel phenomena, for instance the deformation of the Arago point and the distortion of the circular fringes. Also, the color (wavelength) dependence of the diffraction angle could be directly understood in a visible sense.

Taking advantage of the slow optical response of low-temperature rare earth ion-doped crystal, we explore the similarity between causality and Fresnel diffraction by a straight edge. We relate these features to spectral resolution and selectivity. The concept is illustrated with experimental data.

The concept of the Fresnel number is generalized to make it possible to describe the regular polygon and slit illuminated by a homogeneous plane wave. The generalization is based on nonlinear regression of the axial intensity distribution curve. A useful analytical expression for the Fresnel number is presented. A simple experiment to show the different Fresnel numbers in one observation plane is illustrated.

By extension of the transitional operator method developed by Wunsche, the Rayleigh–Sommerfeld and Kirchhoff solutions to the diffraction of a converging spherical (or cylindrical) wave are expressed in terms of a series of derivatives of the field estimate that follows from the Fresnel approximation. This result allows a systematic assessment of the error associated with the paraxial wave model for focused fields and offers simple corrections to this model. In particular, for simple diffracting masks, the Fresnel approximation leads to estimates of the field that have a relative error near focus that is of the order of one on the square of the f-number. The number of significant digits in the field estimate is shown to be doubled by retaining just the first of the series of corrections derived here.

The influence of aperture stop position on the focusing properties of a lens is investigated. Expressions are obtained for the optical coordinates in terms of the system geometry. If the aperture stop is placed in the front focal plane of the lens, the effective Fresnel number is infinite. For other positions, the effective Fresnel number can be made either positive or negative, corresponding to focal shift either toward or away from the lens and a resulting coordinate rescaling.

An analytic formula for the light field amplitude (and therefore also intensity) behind a circular aperture illuminated by spherical or plane waves in the Fresnel limit is given, and the essential difference between Fresnel diffraction and Fraunhofer diffraction is discussed.

We present what is to our knowledge the first experimental observation of controlled intensity f luctuations of Fresnel holograms of 8 microm x 8 microm pixel size by using a function for calculation of optical phase errors from the pixel area of an aperture of synthetic Fresnel holograms derived without approximation.

The amplitude transfer function for coherent imaging in the Fresnel approximation is derived. It is shown that this transfer function can solve problems in Fresnel diffraction that the more usual Fraunhofer approximation cannot. Because of its simple form, it allows for the formal solution of the aperture specification problem in diffractive optics by giving the complex optics transmission function in terms of a given object plane amplitude distribution and a desired image plane amplitude distribution. Two examples of this use are presented. Finally, the generalization of this transfer function to imaging in partially coherent and incoherent light follows easily.

The theory of Fresnel diffraction images is applied to Bragg-grating formation in a germanium-doped silica fiber. Fresnel diffraction images arise from the near-field diffraction at a periodic mask. The diffraction images are calculated as a function of the propagation distance for several mask configurations. The average of the diffraction-image intensities is calculated for a single longitudinal repetitive interval, and it is shown that the period of the resulting average intensity field is twice that of the original mask period. In some cases the periodic mask can be predicted for a desired average intensity field by calculation of the magnitude of its Fourier coefficients.

A dynamical theory is given for calculating the performances of stratified Fresnel linear zone plates (SFLZP’s). The Born expansion extended to the Fresnel diffraction is used to prove a fundamental theorem giving the diffraction efficiency and the diffraction pattern of transmission or reflection SFLZP’s. The method is valuable as long as the wavelength is smaller than the characteristic parameter of the zone plate. It allows us to evaluate the performances of reflection multilayer SFLZP’s, i.e., the so-called Bragg–Fresnel optics recently developed for x-ray optics.

The Wigner distribution function is used to describe general properties of the irradiance distribution at the Fresnel diffraction planes of one-dimensional phase gratings. We report a remarkably simple, analytical expression for the irradiance distribution at one quarter of the Talbot length of any phase grating. As illustrative examples, we consider binary and triangular phase profiles.

Laser beams that contain phase singularities can be generated with computer-generated holograms, which in the simplest case have the form of spiral Fresnel zone plates.

A generalization of the Fresnel approximation in diffraction theory is proposed. The phase term in the diffraction integral is approximated by a paraboloidal variation, not by a binomial expansion but rather by a matching at the critical points in asymptotic evaluation of the integral. The method provides a correction to the optical coordinates of the Fresnel diffraction theory that extends its region of validity. It is applied to diffraction by a circular aperture of a plane wave or focused beam, including effects caused by a large numerical aperture, finite Fresnel number, off-axis illumination, and the presence of aberrations. The method may also be used with other geometries: It is readily applied to cylindrical focusing.

A diffraction-free, high-efficiency light beam has been generated using a specific Fresnel zone plate and a suitable convex lens. The principle and experimental results are given.

A generalization of the Fresnel approximation in diffraction theory is proposed. The phase term in the diffraction integral is approximated by a parabolic variation, not by a binomial expansion but rather by matching up at the critical points for asymptotic evaluation of the integral. The method provides a correction to the optical coordinates of the Fresnel diffraction theory that extends its region of validity. It is applied to diffraction of an inclined plane wave by a half-plane.

We comment on the nature of an unexpected real image obtained in Fresnel holography in terms of pinhole imaging.

Results published previously [ J. Opt. Soc. Am.57, 610 ( 1967)] concerning the focusing properties of Fresnel zone plates were obtained by implicitly invoking the Fraunhofer approximation. A comparison of these results with those derived from Fresnel diffraction theory shows that the focusing properties of Fresnel zone plates predicted by Fraunhofer diffraction theory are incorrect except for resolution defined by the Rayleigh criterion.

A Fourier-series representation of a Fresnel zone plate is set up and used, with a paraxial approximation, to predict features of the image-forming and Fourier-transforming properties of the plate. These properties are compared and contrasted with those of a lens, and examples of a Fraunhofer diffraction pattern from a plate and a lens are illustrated.

The author presents a method of calculating the radial intensity distribution of the Fresnel diffraction pattern of a circular aperture. The method uses the concept of counting whole or partial Fresnel zones and has an accuracy of about five per cent of the maximum intensity, although it can be refined to improve this figure somewhat. The method can also be applied to calculations of the Fresnel diffraction patterns of more complicated apertures.

An analysis of Fresnel images of the amplitude binary diffraction grating with open ratio α/d < 0·5 is presented. Particular attention is paid to images appearing behind the self-imaging grating at distances z = 0 to d 2/λ. An attempt is made at a systematic examination of Fresnel images that have the form of binary gratings with ratio α′/d′ different from that of the original grating.

Using the Hopkins algorithm, expressions are derived for the intensity patterns, in both the Fresnel and far-field regions, associated with the diffraction of a plane-wave elliptical Gaussian beam truncated by an elliptical aperture. This is accomplished by evaluating the diffraction integral subject to the Fresnel approximation. Numerical results are presented that indicate how the truncation parameter affects the side-lobe level in the Fresnel and far-field regions.

A review and commentary is given of practical methods for computing the Fresnel diffraction patterns of circular and linear apertures. The general problem of approximating transcendental functions is also considered.

The rigorous expansion developed by the present authors for the velocity potential of a circular piston source [J. Acoust. Soc. Am. 74, 1044–1047 (1983)] is extended to include the space average pressure in the nearfield and the Fresnel diffraction of incident plane waves by an infinitely thin rigid disk. The present theoretical framework has the remarkable advantage that it includes neither approximations nor numerical integrations.

Simple equations for tracing rays through the Fresnel zone plate have been derived. The spot diagram constructed by means of the ray tracing shows an appearance of imagery of the Fresnel zone plate as a focusing element. The spot diagram can be obtained not only for the real images but also for the imaginary images. Some demonstrative examples of the spot diagram are given.

An expression is derived for the intensity distribution due to a partially coherent field at a distance satisfying the Fresnel approximation. The Collet-Wolf equivalence theorem, which provides conditions under which sources of different states of coherence generate the same intensity distribution in the far zone, is examined in the Fresnel zone.

The propagation problem of mutual intensity from a source plane to an object plane is discussed. The propagated mutual intensity remains unmodified in any object plane in the paraxial approximation region of Fresnel diffraction if the mutual intensity in the source plane is space-invariant and propagated to a region of interest around the propagation axis in which the Fresnel diffraction effect by the edge of an aperture in the source plane is almost negligible.

Analysis of the Fresnel image of the binary-amplitude diffraction grating with open ratio 0·5 is presented. Fresnel images situated at the distance z = Nd2/8λ (N = 1,2…) from the grating which was illuminated by a plane wave were calculated and investigated by experiment. Forms of intensity distributions in the second grating plane are presented, the second grating cooperating with the self-imaging grating.

The lateral shift of Fresnel images is explained. Derivations are presented for a one-dimensional amplitude grating of arbitrary transmittance profile.

A wave incident on a volume of random fluctuations of medium parameters will be scattered in all directions. Two kinds of effects on random volume scattering are considered: 1) multiple scattering effect, and 2) Fresnel diffraction effect. The cumulative forward scattering is known to be responsible for the scintillation phenomenon. When strong, such cumulative or multiple scattering can drive the signal statistics into the saturated regime in which the complex amplitudes behave as a complex Gaussian process. These cumulative scattering effects on signals of a bistatic radar are investigated. Additionally, to allow for the possibility that the size of the irregularity is comparable or larger than the first Fresnel zone, Fresnel diffraction instead of Fraunhofer diffraction is considered. A generalized formula for calculating average scattered power, taking both multiple scattering and Fresnel diffraction into account, is derived. It shows that the Fresnel diffraction pattern appearing around the forward direction tends to be smeared out by multiple scattering. Also, both multiple scattering and Fresnel diffraction effects weaken and broaden the forward-scattered beam, and reduce the backscattered power when compared with the single scattering and Fraunhofer diffraction cases respectively. Under certain conditions, the enhancement effect of the backward scattered power is confirmed.

The conditions determining the lateral localization of Fresnel images of a linear binary amplitude grating with respect to the optical axis are derived. The conditions represent the third basic parameter, besides the axial localization and the lateral magnification, characterizing the diffraction images formed by the self-imaging phenomenon.

Lommel functions were used to solve the Fresnel-Kirchhoff diffraction integral for the case of a spherical obstacle. Comparisons were made between Fresnel diffraction theory and Mie scattering theory. Fresnel theory is then compared to experimental data. Experiment and theory typically deviated from one another by less than 10 percent. A unique experimental setup using mercury spheres suspended in a viscous fluid significantly reduced optical noise. The major source of error was due to the Gaussian-shaped laser beam.

Multiple interference fringes generated by a Fabry-Perot interferometer are used for realizing phase Fresnel lenses. The authors deal with a parallelism between the decomposition of a spherical wavefront in Fresnel zones and a multiple interference pattern obtained with a FP cavity with specially designed mirrors. In the first section, the authors show that the phase profile of a Fresnel lens can be described as the superposition of pseudo-sinusoidal functions whose amplitude and phase are given by a Fourier decomposition. Synthesis of the Fresnel lens profile can be carried out by multiple interference fringes formed by a FP cavity observed in reflection. The reflection coefficients of the mirrors of the cavity make the amplitude and the phase of multiple waves proportional to the Fourier terms. Experimental results are reported in the second section. Multiple fringe patterns are recorded on bleached holographic plates. The diffraction efficiency of Fresnel lenses thus obtained is 65%.

The author confirms the recent work of Southwell (1981) showing that the Fresnel approximation is valid for points within the geometrically defined beam even very close to the diffracting object. By providing a physical explanation for this fact, the paper goes on to derive qualitatively the shape of the entire region in which the Fresnel approximation is valid.

The Fresnel–Kirchhoff theory is used to derive formulas which permit rapid calculation by personal computer of the radial distribution of irradiance in a Fresnel diffraction pattern behind a circular aperture illuminated by a plane wave. An experiment to measure the radial distribution of irradiance is described. The experimental results are in good, but not complete, agreement with the theory.

The convolution formulation of Fresnel diffraction for cascaded aberration-free optical systems is presented. Limitations of the application of this description are discussed.

The results of an investigation of flux redistribution at the focal region of a planar Fresnel ring mirror are reported. A parabolic mirror of large aperture was used to provide a parallel beam of light which was directed at the Fresnel test object. A cotton thread grid was used as a mapping aid to provide a 25 x 25 matrix of spatial data points. (SPH)

The ‘‘spot of Arago’’ has been a controversial topic since its inception in 1818 when Poisson predicted its existence in an attempt to discredit Fresnel’s wave theory of light. Arago performed the experiment and found the surprising prediction was true, thus putting Fresnel’s theory on a firm technical foundation. In recent years, the spot of Arago, which exists as a bright spot at the center of the geometrical shadow of a circular obstruction, has caused substantial grief in various high‐energy laser applications and has come to be considered more of a nuisance than a curiosity. This paper suggests that the size and shape of the spot of Arago is characteristic of the wave‐front aberrations of the incident beam and can therefore be used to advantage as a beam sample for wave‐front analysis of annular beams. The implementation of this wave‐front sampling scheme would eliminate the requirement for a special beam‐sampling optical component and thus reduce to a minimum the deleterious effects upon the beam frequently accompanying the use of such components. Both experimental and numerical results are presented along with a discussion of the capabilities and limitations of this particular beam sample for diagnostic purposes.

The converging prolate spheroidal functions are introduced. They are shown to exhibit phase conjugacy properties under both finite and infinite Fresnel transformation. Their use in Fresnel optics is illustrated. In particular, the number of degrees of freedom of a Fresnel diffracted field is evaluated.

The qualitative features of the lowest order mode profile of an unstable resonator can be predicted by a simple technique based on Fresnel zone theory. The number of rings in the pattern and the occurrence of a local minimum or a local maximum on-axis can be predicted by computing the number of Fresnel zones subtended at the observation plane by the cavity aperture uniformly illuminated by a point source at the virtual focus. This technique has been verified by computer calculations of mode profiles both inside and external to the cavity for moderate Fresnel numbers. The results were also confirmed by experimentally measured mode profiles on an unstable cavity He-Xe laser. The predicted cyclic nature of the mode loss with cavity aperture diameter was also confirmed experimentally.

The diffracted wave field and the irradiance distribution are found for the case of Fresnel diffraction of a circular Gaussian wave diffracted by a Wood zone plate.

In the problem on the interaction of an acoustic wave with a rigid sphere coated with a thin compressible layer /1/, the non-stationary pressure distribution on the sphere is found. The method of numerical inversion of the Laplace integral transform is used, together with asymptotic relations that hold in the case of a sufficiently thin coating. It is shown that the behaviour of the pressure is qualitatively different in the cases of a rigid sphere and a sphere with a coating. In the pressuretime dependence, successive series of oscillations are discovered, which are not seen with a rigid sphere, see /2, 3/. The pressure rise corresponding to the instant of interaction of the enveloping wave (the “Poisson spot” /4/) is displaced in time and in some cases exceeds twice the incident wave amplitude.

In the case of a periodic object, the wave amplitude in every point of a Fresnel diffraction pattern may be obtained from the values of the wave amplitude at a finite number of points in the object...