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A formula was derived in paper 1 of the this series for investigating the Fresnel diffraction field of a slit aperture when the mutual coherence function of the illumination contains a quadratic phase term. That formula is applied to study the intensity distribution in the Fresnel diffraction field of a slit aperture illuminated by a quasi-monochromatic incoherent slit source. The phase term has a big effect on the features of Fresnel diffraction.

An analytic expression is derived for the intensity distribution in the Fresnel diffraction “image” of a transparent object with arbitrary phase shift. Comparisons are made with the Cornu spiral treatment of slits and fine wires. The effects of wavelength variation and finite source width are indicated and appear in a brief mathematical appendix; methods of evaluating Fresnel integrals for plotting purposes are briefly described.

A general approximation is developed which permits the accurate computation of radiation patterns from circular apertures and reflectors by single integration both at wide angles and for large Fresnel numbers. Both the far-field and small angle Fresnel region approximations are obtained as special cases. The approximation is applicable to regions where the ratio of the aperture diameter to the distance to the point of observation is small and where this distance is many wavelengths. A comparison is made with the values determined by precise numerical integration for a certain range of parameters and shows good agreement for both the amplitudes and phases of the integrals within the expected range of validity of the approximation.

The intensity distribution in the Fresnel diffraction pattern of a slit aperture illuminated with partially coherent light is obtained, as a function of the coherence condition across the aperture, by evaluating the formula developed in a previous paper. It is shown that the Fresnel diffraction pattern is affected by the coherence condition of illumination across the aperture and its form.

A formulation is developed for investigating the effect of temporal coherence of the illumination on the intensity distribution in the Fresnel diffraction field of a slit aperture. The formula derived is applied to actually obtain the intensity distributions both along the axis normal to the slit aperture and in the Fresnel diffraction patterns for various receiving planes parallel to the aperture plane. It is found that the temporal coherence is less effective on the intensity distribution in the Fresnel diffraction field. The theoretical results for Fresnel diffraction patterns of a slit aperture illuminated by partially, temporally coherent light are experimentally confirmed.

Diffraction Patterns inside Elliptical Shadows Due to a Point Source of Light.\char22{}When a circular disc with its plane originally tangent to the light wave is rotated about an axis in its plane, the Arago spot changes to a figure with four cusps which move out along the axes of the elliptical shadow, two approaching the foci as limits and the other two going outside the shadow. These changes are shown by a set of photographs obtained with a disc 1.6 cm. diameter placed 2 meters from a pinhole 0.3 mm. in diameter and 5 meters from the photographic plate. The diffraction pattern was found to depend only on the ellipticity of the shadow whether produced by an ellipse or by an inclined disc. Each quadrant of the diffraction pattern was found to be associated with the quadrant of the shadow adjacent to it but on the opposite side of the major axis. Careful measurements of the photographs proves that in each case the diffraction pattern is the evolute of the geometrical shadow. The effect is as though each element of the edge of the shadow contributed a spot along its normal, the result being a caustic curve of diffraction.

The Fresnel zone plate is studied by the boundary-diffraction-wave theory (BDWT). Expressions are given for the fields in the transverse planes and along the axis; in particular, the wave amplitudes at the foci are considered in some detail. The BDWT provides a clear physical picture of the problem. The wave amplitude at any point is given by the sum of a finite number of rays determined by the number of zones in the plate. The foci are the points where the singly diffracted rays are in phase.

The intensity distribution in the Fresnel diffraction patterns of a circular aperture illuminated with the partially coherent light is examined experimentally. The experimental results show excellent quantitative agreement with the theoretical ones previously obtained and apparently indicates a gradual loss in contrast of the diffraction fringe structure as predicted from the theoretical results.

In general, diffraction relations which were developed for optical purposes have been found to be useful in the prediction of electromagnetic fields due to radiating apertures but, because many radio-optical measurements must be made so close to the aperture that the mathematical approximations in the Fresnel field relation often are invalid, new relations, free from these approximations, must be developed. The optical, Fraunhofer relation has been found to predict the distant field with good accuracy and for this reason the Fresnel field is calculated from the distant field equation by the Fourier transform method. The calculation is found to result in an integral which reduces to Fresnel's integral when the standard optical approximations are made in it. The integral has not been evaluated.

Diffraction Patterns inside Shadows due to Point Sources of Light.---(1) Conic section shadows. In confirmation and continuation of the results in the preceding article, the patterns are found to depend only on the shape of the shadow, the Arago spot being obtained at the center of the circular shadow even when this was cast by a spiral edge of large pitch ground to the form of a truncated cone, when the point source was placed at the apex. Patterns inside the shadows of hyperbolic and parabolic plates were also obtained and are reproduced. As in the case of the ellipse, the predominant figure in each case is the evolute of the edge of the shadow. (2) This general relation between diffraction caustic and shadow is found to hold even in the case of the shadow of the involute of a circle, when the diffraction figure was identical with the generating circle which was, of course, the evolute of the edge of the shadow. A series of photographs of elliptical shadows show that the diffraction caustics are not continuous curves. The changes of detail and of color in the patterns with change of ellipticity of the shadow are described at some length.

The self-imaging phenomenon of a square wave binary amplitude grating is studied. The intensity distributions in the Fresnel diffraction image planes lying at distances z = (M -1/4)d 2/u from the grating illuminated by a plane wave are calculated. It is found that these images are very similar to the object structure. As a consequence, some potential applications are proposed.

Most university optics courses include the theory of Fresnel diffraction and practical class experiments to demonstrate diffraction from straight edges and slits. While students find these relatively easy to set up, normally using a laser with a beam expander (Eaton and Wiseman 1976), they often find their interpretation using a phase-amplitude diagram considerably more difficult. As an aid to the visualisation of the calculation and as a quantitative method for the derivation of the amplitude of the Fresnel diffraction pattern from a slit, the authors have found 'N' gauge model railway analogue very useful.

From Fresnel’s wave theory of light Poisson deduced in 1818 that a spot of light should appear at the center of the shadow of a circular obstruction. Poisson’s predicted spot had been observed by Maraldi one hundred years earlier. Arago performed Poissson’s thought experiment and rediscovered Maraldi’s long forgotten spot. This phenomenon is one historically imprtant confirmation of Fresnel’s wave theory of light and it is known as the Poisson-Arago bright spot.

The overestimation of power density by the Fraunhofer (far-field) approximation to the more exact Fresnel (near-field) diffraction integral is determined as a function of Fresnel number at arbitrary on-axis field points for rectangular and circular aperture illuminations of uniform amplitude and phase. For square and circular apertures, the overestimation is 2.8 percent and 1.3 percent, respectively, at a Fresnel number of 1/8 (corresponding to the far field boundary distance D-sq/ lambda) and 518 percent and 147 percent, respectively, at a Fresnel number of 1. Keywords: Antenna aperture; Aperture distribution of uniform intensity and phase circular aperture; Constructive interference; Destructive interference; Diffraction; Fresnel kirchhoff integral; Fresnel number; Optical aperture; Rectangular aperture.

In this study based on an original program of aberration calculations and automated optical lens design we try to increase ray tracing of a Fresnel system so that the original program not only contains the initial functions but also satisfies the calculations of a Fresnel or mixed system (both general and Fresnel lenses).

A description is given of the analysis and experiment on Fresnel diffraction by a semitransparent straight edge object under acoustically coherence-controllable illumination. The ringing pattern of Fresnel diffraction is heavily dependent on the phase of the optical wave transmitted by a semitransparent film as well as on the Raman-Nath acoustooptic interaction parameter v, associated with acoustic pressure, and the acoustic wavelength Λ. The ringing is appreciably deformed and suppressed on both sides near the geometrical shadow edge but remains almost unchanged at a distance of ±Λ from the edge even when the parameter v is steadily increasing.

A type of computer-generated hologram (CGH) is introduced which works in the Fresnel diffraction mode and generates the desired wave front on-axis. Such holograms rely on the use of the detour-phase method suitably modified for Fresnel diffraction and on the sampling theorem; therefore, they are of the cell-oriented type. The concept of a carrier or reference wave is not needed to introduce them; the Huygens-Fresnel principle provides a complete description of their diffracted waves. Consequently, no orders in the usual sense of grating orders or zone plates or natural holograms can be defined. Only orders in the sense of the sampling theorem exist for these computer-generated holograms, and the useful order is the central one; it is also the brightest and does not overlap with other orders. These questions are discussed and illustrated by experimental examples in connection with the problem of diffraction efficiency. Among the examples, the CGH of a single point on-axis is considered; it is in general not a zone plate.

The "spot of Arago" has been a controversial topic since its inception in 1818 when Poisson predicted its existence, which violated common sense, 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 wavefront aberrations of the incident beam and can therefore be used to advantage as a beam sample for wavefront analysis of annular beams. The implementation of this wavefront 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 will be presented along with a discussion of the capabilities and limitations of this particular beam sample for performing various wavefront sensing functions.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A study of Fresnel diffraction effects is presented for structures of interest for X-Ray Microlithography. This analysis uses the full optical constants of the mask absorber pattern. Therefore, our calculations take into account the fact that the photons experience a phase shift as they go through the mask's absorbing layer in addition to simple absorption. Results are presented which show the consequences of adding the phase effects to the Fresnel analysis. These results show that phase effects cannot be disregarded in modeling Fresnel intensity profiles on resists.

The scattering of a scalar plane wave by a totally reflecting sphere (hardcore potential) at high frequencies is treated by a modified Watson transformation. The behavior of the solution both in the near and far regions of space is discussed, as well as the accuracy and domain of applicability of the WKB approximation and classical diffraction theory. It is shown that different transformations are required in the forward and backward half-spaces, and corresponding integral representations for the primary wave are derived. The transformations are rigorously proved and the convergence of the residue series is discussed. In the shadow region, the physical interpretation of the complex angular momentum poles in terms of surface waves is in agreement with Keller's geometrical theory of diffraction. In the lit region, sufficiently far from the shadow boundary, the WKB expansion for the wave function is confirmed up to the second order. On the surface of the sphere, Kirchhoff's approximation is accurate, except in the penumbra region, where the behavior is described by Fock's function. The diffraction effects in the neighborhood of the shadow boundary are investigated and the corrections to classical diffraction theory are obtained. The shift of the shadow boundary is evaluated. The expression for the wave function in the Fresnel-Lommel region is derived and applied to the discussion of the Poisson spot and the behavior near the axis. The total scattering amplitude is evaluated for all angles, including the neighborhood of the forward and backward directions. The corrections to the forward diffraction peak and the transition to the region of geometrical reflection are discussed. The modified Watson transformation is also applied directly to the scattering amplitude. The connection between representations valid in different regions is established.

The object of this memoir was to remove from the molecular theory, difficulties in which the recent investigations of Mr Green appear to involve it. The question at issue is the ratio of the densities of the ether within and without a refracting medium. The usual mechanical hypotheses would appear to lead to the conclusion that the former density is the greater; whilst from the molecular hypothesis the latter appears to be the truth.

The Fresnel diffraction phenomenon referred to as Poisson's spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson's spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot's intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson's spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.

The x-ray focusing properties of linear Fresnel zone plates have been derived by solving the Helmholtz equation for the field propagating through the zones. We consider the imaging of a point object into the first diffraction order of a volume zone plate having its zones parallel to the optical axis. For plane wave illumination, the focal spot size is limited by the same material-dependent but wavelength-independent value that affects waveguide focusing. In marked contrast, for the one-to-one imaging condition, corresponding to specular reflection of the x rays from the zone boundaries, the image is found to have a minimal spot size approximately equal to the outermost zone width. Unlike x-ray waveguides, zone plates therefore do not appear to possess a fundamental limit to the smallest spot size to which they can focus.

We introduce and demonstrate a new interferometric method called longitudinal-differential (LD) interferometry, which measures the spatially resolved phase difference of the scattered field by an object relative to the illumination. This method is combined with a high-resolution interference microscope that allows recording three-dimensional field distributions in amplitude and phase. The method is applied to study the axial phase behavior of Arago spots, an effect observable in low-Fresnel-number systems behind objects with a size comparable to the wavelength. We directly observe the initial phase delay in the Arago spot and prove that the local phase velocity exceeds the speed of light in air. Such LD phase studies are applicable not only to the Arago spot but also to other kinds of light interactions with wavelength-scale objects, e. g., photonic nanojets. (C) 2012 Optical Society of America

Various formulations have been proposed for the Fresnel approximation for diffraction by a circular aperture. These formulations can describe the effects of convergent illumination, finite value of Fresnel number, and off-axis illumination. The retention of a further term, which is dependent on the coordinates of the focus and the observation point, is proposed. This results in a redefined axial optical coordinate, giving improved prediction of the diffracted field for off-axis points.

We describe how Fourier analysis in projective coordinates allows inversion in the Fresnel zone. Since the longitudinal resolution of 3D coherence imaging falls inversely in the square of range in both the Fresnel and Fraunhofer zones, extension to the Fresnel zone dramatically improves longitudinal resolution by removing far-field range constraints in Michelson rotational shear interferometry.

A mathematical model to describe the behavior of low-resolution Fresnel lenses encoded in any low-resolution device (e.g., a spatial light modulator) is developed. From this model the diffraction efficiency is calculated in terms of all the parameters that characterize these lenses.

This paper proposes two systems of Fresnel Zone Plates which could focus two parallel monochromatic and coherent beams (with different wavelengths), eliminating the wellknown chromatical aberrations of the classical construction.

A point source focused by a Fresnel zone plate may interfere constructively or destructively with the undiffracted background. Fresnel zone plates with unusually small dark centers empirically lead to constructive interference.

Investigations of the three-dimensional intensity distribution near the focus are extended here to systems with Fresnel numbers N < 0.5. Isophotes (contours of equal intensity) are presented that demonstrate the structure of the field deep within the region of Fresnel diffraction.

By direct numerical-integration comparisons, it is established that the Fresnel approximation for collimated propagation is quite good (within about 2% in amplitude and 0.02 rad in phase) in every case, including that with the limit of a high Fresnel number. Moreover, the Fresnel approximation begins to break down in phase for spherical-wave propagation for beams faster than about f/12. It has been discovered, however, that if one also invokes the paraxial approximation, that is, replaces the spherical wave by a quadratic phase front, then the Fresnel approximation becomes valid for expanding (or diverging) beams as well. This result is substantiated through the use of stationary-phase arguments.

The weighted-ordinate method of colorimetry is applied to Fresnel diffraction of white light by a circular aperture to determine the dominant wavelength, purity, and relative luminance of the center of the diffraction pattern. This chromaticity information is compared to experimental results and graphs are included to allow easy determination of these quantities for most combinations of aperture radius, source distance, and observation distance that would be of interest.

The behavior along the axis of the intensity arising from the diffraction of a uniform, converging spherical wave at a circular aperture is studied on the basis of the theory of the boundary-diffraction wave. The results are used to determine the location of the principal intensity maximum and to elucidate the dependence of the focal shift both on the Fresnel number and on the f number of the focusing geometry. Analytic as well as numerical results are obtained. Comparison with microwave experiments of Farnell [ Can. J. Phys.36, 935 ( 1958)] is also made.

The general theory of the diffraction and propagation of light is considered. Particular attention is paid to the case of uniformly illuminated rectangular or circular apertures, for which the derivative of the field given by the Fresnel approximation is evaluated exactly. The limitations of the Fresnel approximation are considered, and thus a useful, simple way of understanding its validity is produced. This treatment yields an estimate of the accuracy of the Fresnel approximation in the general case and a method by which a modified Fresnel treatment can be used over a substantially extended region.

It was recently shown that, when a converging spherical wave is focused in a diffraction-limited system of sufficiently low Fresnel numbers, the point of maximum intensity does not coincide with the geometrical focus but is located closer to the exit pupil. In the present paper both qualitative and quantitative arguments are presented that elucidate the modifications that the whole three-dimensional structure of the diffracted field undergoes as the Fresnel number is gradually decreased. Contours of equal intensity in the focal region are presented for systems of selected Fresnel numbers, which focus uniform waves.

The problem of two-dimensional diffraction of light by a slit is solved rigorously on the basis of Huyagens-Fresnel principle assuming the amplitude of a secondary wavelet emitted from an infinite strip of infinitesimal width on the slip aperture to be proportional to cos (θ/2)/√r, where θ is the angle of diffraction and r the distance from the strip. The solution of the problem where a plane-wave of light is normally incident on a slit in a perfectly black screen is expressed in terms of the Fresnel integral. With slight modification, the method is also applicable to the diffraction in a light-absorbing medium.

An analogy between the generalized radiance from a planar Lambertian source and the Fresnel diffraction amplitude from a plane aperture of the same size and shape as the source is exhibited.

A discussion of the effect on the primary monochromatic aberrations of moving the stop or the object for a system of axially symmetric curved Fresnel surfaces is presented. Mathematical expressions, which are generalizations of similar results for non-Fresnel systems, for the changes in the aberrations are derived.

A convenient expression describing the location of Fresnel diffraction patterns when a diffracted spherical wave is focused by a lens is given. Experimental results confirm the predicted positions of 26 axial extrema corresponding to integer numbers of Fresnel half-period zones. Experimental radial intensity profiles are presented for some of these positions.

A convolution formulation of Fresnel diffraction is presented. The diffracted amplitude is expressed as a convolution in either direct or reciprocal space. Approximations involve the use of parabolic wavefronts and the omission of the obliquity factor. The formulation is readily applied to many optical phenomena.A theory of Fresnel transforms is given. The Fresnel transforms are expressed as a convolution in either direct or reciprocal space. Two examples of their use are given.

The exact calculation of the field at a point due to a single Fresnel zone is carried out by using the Maggi–Rubinowicz contour integral. The result agrees with the Fresnel theorem in the limit for k very large, k being the propagation constant of the incident wave. The results obtained suggest a new interpretation of the physical meaning of the Maggi–Rubinowicz contour integral in diffraction theory as representing a contribution of elementary or Fresnel zones, in exactly the same manner that the Kirchhoff integral does when considered as an expression of Huygens’ principle.

A new kind of interferometer for testing any unit magnification optical system is described. The method utilizes diffraction beam splitting and involves the use of two identical Fresnel zone plates, one by the side of the other. The interference patterns obtained are easy to interpret because the fringes are the loci of equal optical path difference of the wavefront under test with respect to a reference sphere.

The consistency of the Fresnel zone construction is customarily checked by verifying that it correctly describes wave propagation from a point source in free space. The calculation, based on an approximate evaluation of the contribution of individual Fresnel zones to the total field, leads to the Fresnel theorem: the total field is just one-half that due to the first zone alone.We show here that if the Fresnel zones are defined on a plane passing through the midpoint of the line joining the source point to the field point, the field due to each Fresnel zone may be calculated exactly. When the Fresnel zones are defined on a plane perpendicular to the line between the source and field points, the contribution of the first zone is equal to the total field multiplied by the factor 1+ (1+π/kR′)−2. Here, k is the wavenumber and R′ is the distance separating source and field points. Thus, for this geometry, the Fresnel theorem holds only in the limit kR′≫1; for arbitrary kR′, the factor quoted must be used.A formula, valid for any Fresnel zone, and for arbitrary orientation of the plane on which the Fresnel zones are defined, is given in the text.