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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 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.

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...

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 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.

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.

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.

The Fresnel zone of optical gratings—the region behind the grating where secondary interference between the light coming through the different openings produces real images of the grating—can be understood by comparison with other types of wave phenomena. In the Fresnel zone of light diffracted by an optical grating, different types of amplitude modulation of the wave front are observable. Gratings of stationary ultrasonic waves, being phase gratings to the light passing through them, produce similar, though not identical Fresnel patterns. Between the planes in which the secondary interference patterns are true images of the gratings, the modulation changes from amplitude to phase modulation. This change can be demonstrated by an ultrasonic wave analog. Patterns formed by the diffraction of ultrasonic waves by an amplitude grating are given. Similar patterns in surface waves are given for both the phase grating and the amplitude grating analog.

The diffraction patterns of the central aperture and the first 10 rings of the classical Fresnel zone plate are calculated in the principal focal plane for off-axis distances as large as the radius of the 10th ring. The zone-plate pattern is found by summation; near the axis it is compared with closed-form approximations. The integrated flux within circles as large as the 10th ring is also calculated.

THERE is a point in connection with the ordinary expositions of Fresnel's theory of double refraction to which, on account of its frequent occurrence, it is perhaps worth while to direct attention. It is found in Aldis's “Tract on Double Refraction,” p. 7, in Preston's “Theory of Light,” third edition, p. 328, and in Basset's “Treatise on Physical Optics,” p. 115.

The method of Chandrasekhar is used to compute the parameters that characterize the scattered light leaving the top of a Rayleigh atmosphere. The atmosphere of this model lies above a smooth water surface, which reflects light according to Fresnel’s law. The combined atmosphere and water is called the Fresnel model. The results for the Fresnel model are compared with corresponding results for a second model, which is called the Lambert model, since its ground reflects light according to Lambert’s law. The reflectance at the ground is less than 0.1, if either the solar zenith angle θ0 0.6. The relative difference between the outward fluxes from the tops of the atmospheres of the two models is less than 0.07 if θ0 0.5, but the difference becomes large at small τ1. The maximum degree of polarization and the neutral-point positions at the top of the atmosphere of the Fresnel model are quite different from those of the Lambert model, when the total optical thickness is less than 0.5. The neutral points for the Fresnel model appear outside of the vertical plane through the sun for restricted ranges of both the total normal optical thickness and the solar zenith angle.

The effects of contrast in the Fresnel zone plate on the amplitude in the focal point due to a normally incident plane wave are obtained by direct evaluation of the Fresnel–Kirchhoff diffraction integral. The change in focal-point amplitude due to smoothing the ordinary, discontinuous, “square-wave” transmission function into a continuous, “sine-wave” function is also calculated. Finally, an expression is derived for the amplitude in the point image due to a plane wave impinging on the zone plate at a small angle of incidence.