Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. II

In this second contribution the Fraunhofer diffraction patterns from apertures of the form of an equilateral triangle, a square, a regular pentagon, a hexagon and an octagon are compared with the calculated maps of the intensity distribution. For each shape of the aperture the first few maxima and minima of the intensity are tabulated and the formulae for the wave function are given characterizing the diffraction in the directions of the slowest and the steepest decrease of intensity.     

Fraunhofer Diffraction from Apertures Bounded by Regular Fractals

Abstract

Some properties of fields diffracted in the Fraunhofer region by apertures bounded by regular fractals are investigated. A recursion relation describing such apertures is introduced and the associated relation in the Fourier transform domain is described. For a triadic Koch aperture whose edge has the fractal dimension of D_s = 1·262, the recursion relation is numerically evaluated. Self-similar structures of intensity distributions in the Fraunhofer region are verified for the present objects. The relationship of the fractal dimension D _s of the fractal edge with the power-law decay of the Fraunhofer diffraction intensities is also verified.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.     

Algorithms for Fresnel Diffraction at Rectangular and Circular Apertures

This paper summarizes the theory of Fresnel diffraction by plane rectangular and circular apertures with a view toward numerical computations. Approximations found in the earlier literature, and now obsolete, have been eliminated and replaced by algorithms suitable for use on a personal computer.     

Optical Diffraction in Close Proximity to Plane Apertures. II. Comparison of Half-Plane Diffraction Theories

The accuracy and physical significance of the classical Rayleigh-Sommerfeld and Kirchhoff diffraction integrals are assessed in the context of Sommerfeld’s rigorous theory of half-plane diffraction and Maxwell’s equations. It is shown that the Rayleigh-Sommerfeld integrals are in satisfactory agreement with Sommerfeld’s theory in most of the positive near zone, except at sub-wavelength distances from the screen. On account of the bidirectional nature of diffraction by metallic screens the Rayleigh-Sommerfeld integrals themselves cannot be used for irradiance calculations, but must first be resolved into their forward and reverse components and it is found that Kirchhoff’s integral is the appropriate measure of the forward irradiance. Because of the inadequate boundary conditions assumed in their derivation the Rayleigh-Sommerfeld and Kirchhoff integrals do not correctly describe the flow of energy through the aperture.     

Fraunhofer Diffraction Patterns of Titled Planar Objects

Attention is drawn to the changes in the Fraunhofer diffraction patterns encountered when the planar objects become titled with respect to the optical axis. The differences between the far-field images observed in the well-known untilted case and the one investigated are shown and discussed using examples of familiar objects, such as square or rectangular apertures and linear diffraction gratings.     

基于夫琅禾费衍射的刀口直线度检测

现行的刀口直线度测量方法中,由于人眼分辨率和标准光隙带来的不确定度较大,故设计以基于夫琅禾费衍射原理的刀口直线度检测装置以提高检测精度.文章针对目前测量的现状,不仅设计了衍射检测装置并且进行了能力验证分析.     

The Far-field Diffraction Properties of AperturesI. Circular Apertures with Besinc Correlated Illumination

Counterfeiting of valuable papers and identification documents made of paper or plastics, such as banknotes, credit cards, passports, etc., is presently inhibited through the use of inks of many colours, intricate engraved designs on special papers which sometimes contain watermarks or embedded coloured paper platelets or metal threads. These methods have not always offered adequate protection. It is proposed to use optical multilayer coatings to validate such documents. One property of such coatings is that their spectral characteristics vary with the angle of incidence of the light. For most applications this variation is a decided disadvantage and efforts are often made to minimize it. But by this property optical multilayers differ essentially from all known types of dyes, inks, pigments and paints used in printing, photography and colour-copying machines. This angular variation can be made use of to enable the man in the street to distinguish at a glance an authentic document bearing such a coating from even the best counterfeit lacking it. With the same coating it is possible also to provide ever more discriminating tests of the authenticity of the document for the shop or bank, and for the crime laboratory where more sophisticated equipment can be used. The calculated and actual performance of a number of coatings designed for this purpose are described. These include some that permit positive identification not dependent on colour vision.     

Optical Diffraction in Close Proximity to Plane Apertures. IV. Test of a Pseudo-Vectorial Theory

Rayleigh’s pseudo-vectorial theory of the diffraction of polarized light by apertures which are small compared to the wavelength of light is analyzed with respect to its mathematical rigor and physical significance. It is found that the results published by Rayleigh and Bouwkamp for s-polarized incident do not obey the conditions assumed in their derivation and must therefore be dismissed. It is also found that the theory leads to paradoxical predictions concerning the polarization of the diffracted field, so that the pseudo-vectorial approach is intrinsically incapable of describing polarization effects.     

光散射粒度测量中采用Fraunhofer衍射理论或Mie理论的讨论

从理论和实验两方面出发,讨论了颗粒的折射率对光能分布及粒径分布反演结果的影响,同时给出了衍射理论的完整适用条件及基于衍射理论的粒度仪的最佳光靶尺寸。     

Fraunhofer Diffraction in Partially Space Coherent Light by Slit Aperture with Amplitude Filters

Fraunhofer diffraction patterns produced by a narrow rectangular aperture having a non-uniform illumination with partially coherent light are presented. Graphs of intensity distributions in the diffraction patterns for three cases of amplitude filters and four forms of the mutual coherence at the slit are given. An increase in the half-width at half-height (H.W.H.H.) and a decrease in the intensity at the centre are observed. Tolerances for the value of the correlation interval in the aperture based on the Strehl intensity criterion have been calculated for various amplitude filters.     

On the Fraunhofer (Far Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background

The diffracting objects considered are opaque and semi-transparent objects of known cross section which are situated in an otherwise transparent area and illuminated with a collimated quasi-monochromatic beam of light. The plane in which the resultant diffraction pattern is considered is at a sufficient distance from the object for the approximation of Fraunhofer (far field) diffraction to be made whilst still remaining in the near field of the surrounding transparent area. The resultant intensity distribution is investigated both theoretically and experimentally for objects which present both circular and rectangular cross sections to the beam.     

Problems of Photography at the Diffraction Limit

A number of recent publications show that the relation of various image properties to the line width and to the aperture of a diffraction-limited lens can be computed from the light distribution in the Airy disk. These properties include unit-line intensity and density profiles, profiles of line pairs, the relative “printing quality” of lines, changes of light intensity with line width, and the increase of line width with exposure. The finest line of given quality that can be produced by a diffraction-limited lens and the rate of increase of line width with exposure are directly proportional to the F-number of the lens.     

Study of the properties of non-integer order vortex beams at Fraunhofer zone

The intensity and phase distributions of an optical vortex beam with non-integer values of the topological charge are analyzed in Fraunhofer region. There are two annular rings with different size and shape in the intensity patterns. The petal-like bright spots appear on the larger annular rings with higher intensity, and the small rings with lower-intensity. There are ellipse-like dark spots in the middle of the annular rings. The intensity patterns for positive and negative topological charges of vortex beams are mirrored in the x axis, and the number of bright spots and dark spots is related to the topological charge of vortex beam. In phase patters, the new born phase vortex moves gradually to the origin along the y axis, and the positions of phase vortices within the central region move regularly with the increase of the non-integer topological charge of vortex beam.     

Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection

Diffraction and reflection of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit are considered in the context of ray theory. A closed-form solution for both diffraction and reflection is obtained, and the polarization character of the diffracted plus reflected electric field is obtained. It is found that the magnitude of the reflected electric field is multivalued for forward scattering. This is interpreted in terms of the variation of the spheroid's Gaussian curvature at the points where grazing ray incidence occurs.     

On the Phase Distributions in Images and Fraunhofer Diffraction Patterns Produced by a Thin,Non-aberrant Lens Illuminated by a Monochromatic Point Source of Light

The paraxial approximation is used to show that the phase distribution in the image plane for an object which itself introduces no phase distortions is that which would be obtained from a monochromatic point source placed in the Fraunhofer diffraction plane and vice versa. The idea of image formation as amplitude modulation of a carrier is suggested.     

Diffraction of low-energy electrons at a rippled surface

An analysis is made of a kinematic model of the scattering of low-energy electrons at a onedimensional periodic toothed structure simulating a rippled surface. Calculations of the intensity profiles of the diffraction peaks are made for periodic systems of identical symmetrical triangular teeth with a variable number of atoms on each generatrix of the tooth and various angles of inclination of the generatrices to the horizontal. The main effect observed as a result of scattering at a periodic toothed structure involves broadening or splitting of the diffraction peaks compared with the diffraction patterns from a smooth chain of atoms. Pis’ma Zh. Tekh. Fiz. 24, 64–69 (July 12, 1998)