Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches

The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the theory of the boundary diffraction wave. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals. As expected, our results are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory.     

Structure of boundary diffraction wave revisited

The physically appealing boundary diffraction wave theory which suggests that diffraction patterns arise due to interference of an undisturbed (geometrical) wave and the boundary diffraction wave generated by edge of the diffracting aperture, simplifies the solution of diffraction problems by reducing the Fresnel–Kirchhoff surface integral into a line integral over the illuminated boundary of the diffracting aperture. The present work reports experimental investigations carried out on the structure of the boundary diffraction wave. It has been shown that the boundary diffraction wave is continuous behind the diffracting aperture and apparently there does not exist any discontinuity at the geometrically light to shadow transition boundary, as is required by the theory. PACS 42.25.-p; 42.25.Fx; 42.25.Hz     

微小孔衍射——近场光学理论

推导了圆孔衍射公式，该公式不受孔径大小和到屏距离的限制，可以作为近场光学的理论。它满足麦克斯韦方程标量形式和基尔霍夫边界条件，包括传播波和衰减波。数值计算表明，此结果优于用Ｂｅｔｈｅ模型所得到的近场理论的结果     

双半透镜干涉场的研究

基于波前分析和基尔霍夫衍射积分式,导出了在会聚球面波照射下圆孔或半圆孔菲涅耳衍射场沿轴的复振幅分布,特别关注其相位分布,给出了像点两侧对称点之间的相位差公式,结果表明过像点的相位变化是连续的.进而将这些结果应用于双半透镜装置,即梅斯林干涉实验,分别两种情况(即焦距相等而物距不等和物距相等而焦距不等)给出了分别由单个半透镜产生的两个衍射场之间的相位差公式和相应的干涉强度公式.结果表明,这两个衍射场沿轴的有效相位差未必是0,亦未必是π,还可能出现锯齿型的反复变化.本研究为正确地分析双半透镜产生的两个衍射场之间的干涉场提供了一个可靠的理论途径.     

Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness

A 2D theoretical model of the diffraction of a plane electromagnetic wave by a slot in a perfectly conducting screen is constructed based on the partial domain method. The Tikhonov regularization is used to solve a system of algebraic equations for the slot-mode amplitudes. This makes it possible to extend the domain of applicability of the theory to conducting screens of arbitrary thickness and to significantly increase the accuracy of solution in the cases when the slot width and the screen thickness are comparable to the wavelength of the diffracted radiation. The absence of a continuous passage to the limit of an infinitesimal screen thickness from the case of an arbitrarily small finite thickness is demonstrated. The boundary conditions for the energy-flux vector are considered. A concept of the energy potential that is convenient for the computer calculations of the energy-flux lines of 2D diffraction fields is introduced.     

Diffraction of stochastic electromagnetic fields by a hole in a thin film with real optical properties

The classical Kirchhoff theory of diffraction is extended to the case of real optical properties of a screen and its finite thickness. A spectral power density of diffracted electromagnetic fields by a hole in a thin film with real optical properties was calculated. The problem was solved by use of the vector Green theorems and related Green function of the boundary value problem. A spectral and spatial selectivity of the considered system was demonstrated. Diffracted patterns were calculated for the coherent and incoherent incident fields in case of holes array in a screen of perfect conductivity.     

Zur Beugung an öffnungen in nichtebenen Schirmen

By means of an approximate method formerly published by this author, diffraction by an aperture in a screen, which is not plane, is discussed. For the special case of a circular aperture in a funnel-shaped screen, the diffraction field on the axis behind the screen is computed for an incident scalar plane wave (different boundary conditions) and for an incident electromagnetic plane wave propagating in the direction of the axis.     

The calculation research of classical diffraction formulas in convolution form

Based on the angular spectrum diffraction theory and the sampling theorem, the sampling conditions for calculation the Kirchhoff formula, the Rayleigh-Sommerfeld formula, the angular spectrum diffraction formula and the Fresnel diffraction formula in convolution form were studied. The results indicate that the diffraction calculation result is correct if the angular spectrum corresponding to the main energy of diffraction field can be fully transmitted by means of transfer functions when the samplings of initial wave fields satisfy the sampling condition. Compared with previous work, the sampling condition in this paper is less restricted. The diffraction calculation results are in agreement with experimental results.     

Diffraction of an ultrashort pulse by an aperture

The scalar problem of diffraction of an infinitely short pulse by a plane screen is solved within Kirchhoff’s approximation. The response of an infinitely small aperture is calculated, and the explicit solution is found for the case of a circular aperture.     

Boundary diffraction wave theory of resistive surfaces with edge discontinuities

The line integral of the boundary diffraction wave theory is derived by considering the exact diffracted fields of a resistive half-plane. The line integral is generalized for arbitrary resistive surface with edge discontinuity. The method is applied to the diffraction problem of waves by a convex resistive spherical reflector and the resultant field expressions are investigated numerically.     

The method of the boundary diffraction wave for impedance surfaces

The line integral of the boundary diffraction wave theory is extended for the diffraction process of waves by the impedance surfaces with edge discontinuities. With this aim, the exact diffraction field expression of Maliuzhinets is transformed into a line integral. The method is applied to the scattering problems of waves by a spherical reflector with edge discontinuity and the diffracted fields are evaluated asymptotically. The resultant expressions of the waves are examined numerically.     

Geometrical theory of diffraction

Geometrical theory of diffraction (GTD) is an alternative model of diffraction propounded first by Thomas Young in 1802. GTD has a long history of nearly 150 years over which many eminent people enriched this model which has now become an accepted tool in the calculation of diffraction patterns. In the conventional Helmholtz-Kirchhoff theory the diffracted field is obtained by computing the net effect of the waves emitted by all points within the area of the aperture. But GTD reduces this problem to one of computing the net effect of waves from a few points on the boundary of the aperture or obstacle, thus simplifying considerably the labour involved in computations. Also the theory can easily be modified to include polarization effects. This has been done specifically by Keller (1962) who exploited the Sommerfeld solution of diffraction of electromagnetic waves at a half plane, making the theory more versatile than the Kirchhoff scalar wave theory. Interestingly the geometry of difffracted rays is predictable from a generalized Fermat principle. According to this the total path chosen by light from the source to the point of observation via the diffracting boundary is an extremum. Historically it should be stated that many of the salient features of GTD were established by a school led by Raman which was active from 1919–1945. Later when Keller (1962) revived GTD independently, he and others who followed him rediscovered many of the results of the Raman school. We have stressed wherever necessary the contributions of the Indian School. We have also discussed certain geometries where GTD can be effectively used. We get some new and interesting results, which can be easily understood on GTD, but are difficult to interpret on the conventional theory of diffraction.     

四孔径衍射屏在近场形成的光强分布及拓扑荷

利用基尔霍夫衍射理论计算模拟了四圆孔径衍射屏在菲涅耳深区形成的衍射光场的强度、零值线和相位的分布,发现衍射光场亮斑关于中心呈对称分布,在距离衍射屏较近的观察面上,光强值为零的点组成光强零值线段,该线段上光强等值线的离心率都接近或等于1,其两侧的光强值变化非常剧烈。复振幅的实部和虚部零值线多为封闭的曲线,零值线交叉点的个数为偶数,并且正负相位奇异点的个数相等。特殊相位奇异点周围的相位不仅呈对称分布,而且该点的拓扑荷的值近似为零。随着光波的传播,在不同的观察面上光强零值线段逐渐变短,最终趋于一点。     

Controlled experiments of the diffraction of sound by a curved surface

Controlled measurements of the sound field from a point source above a curved surface are described. The measurements were made in the frequency range between 0.3 and 10 kHz, in the case of a rigid boundary and a surface of finite impedance. Receiver positions include all of the area within, and above, the shadow zone and for various source heights. Particular attention is given to the region across the shadow boundary. The measurements are compared to diffraction theory expressed in terms of a residue series, or creeping wave solution. The calculation is extended by removing restrictive approximations and by carrying the computation to higher-order terms. A numerical algorithm allows the extension to the general case of a finite impedance. Above the shadow boundary, the sound field is calculated using geometrical theory that accounts for reflections from a curved surface. Deep within the shadow, theory and measurements agree to, typically, 0.5 dB. The same agreement is obtained between measurements and the geometrical theory well above the shadow boundary. In the vicinity of the shadow boundary, both theories agree to within 0.5 dB but differ from the measured results by 2 to 5 dB. Finally, the theory is compared to measurements obtained outdoors above a grass covered curved ground with no refraction and above flat ground with refraction.     

Cylindrical-sinc beam

Propagation of diffracted beams in free space has already been identified. One possible solution is derived from the Helmholtz wave equation and this solution is named as cylindrical-sinc beam. Therefore, cylindrical-sinc beam is a new beam type which can be obtained from Helmholtz equation. Diffraction properties of this new beam type were observed from an opaque aperture screen. Additionally, geometrical theory of diffraction is used to determine numerical values of diffracted fields. This new beam type which does not exist in the literature was observed by using these methods passing through an opaque aperture. The obtained expression was analyzed numerically. Simulation results of the beam depending on the length of aperture and distance to the observation point were added.     

圆孔“衍射波”的相位特性

在标量衍射理论的基础上,导出了圆孔衍射的“边界衍射波”的级数表达式,提出圆孔衍射产生的“衍射波”概念,对“边界波”和“衍射波”的相位特性作了讨论和比较.     

Diffraction of a plane electromagnetic wave by a slot in a conducting screen with a transverse dielectric layer

The problem of diffraction of a plane electromagnetic wave by a slot in a planar perfectly conducting arbitrary thick screen with an infinite planar dielectric layer passing through the slot transversely to the screen is solved rigorously. In each of the field existence domains (two domains on either side of the screen and the interior of the slot), the solution is represented as an expansion in piecewise harmonic or exponential modes that allow for reflection and refraction at the boundaries of the dielectric layer. It is found that a set of functions describing such modes is complete enough to construct a solution satisfying all boundary conditions of the diffraction problem. The procedures of solution construction for the case at hand and for the same diffraction structure without the dielectric layer are compared.     

SiO2波导布拉格器件中声表面波及声光相互作用分析

针对SiO2光波导声光布拉格器件,计算了SiO2非对称平板波导TE模式的横向场分布;给出了SiO2/ZnO/Air层状介质结构的性能方程、运动方程和麦克斯韦方程,推导出这种层状结构的特征方程,并结合所满足的边界条件,得到了各层介质的位移及电磁场分布;计算了声表面波所引起的光学相对介质隔离率张量的变化,最后讨论了声光衍射效率和光场与声场的重叠积分、声功率、声频率、声孔径和光波导参数之间的关系。结果表明,在低频范围内光场与声表面波场重叠良好;低阶模的重叠积分始终大于高阶模重叠积分,最低阶模与声表面波相互作用最强,所需声功率最小;当声功率一定时,增加声孔径可以提高衍射效率。     

Partially coherent Fresnel diffraction by a slit aperture. II. Axial intensity distributions

The general formula derived in a previous paper is applied to obtain the intensity distribution along the axis normal to a slit aperture in the Fresnel diffraction field when the spatial mutual coherence function of the illumination over the aperture takes three different forms of correlation. It is shown that the axial intensity distribution of Fresnel diffraction by the slit aperture is effectively influenced by the coherence condition across the diffracting aperture.     

Zur Theorie einer Klasse von Beugungsproblemen mittels singulärer Integralgleichungen. V Hochfrequenz-Beugung ebener elektromagnetischer Wellen an einer ideal leitenden Kreisblende

The high-frequency diffraction of a plane electromagnetic wave by a circular aperture in a infinite plane conducting screen is considered at normal incidence. The asymptotic solution of the singular integral equation for the Fourier-transform of the current density on the screen is found by means of elementary function-theoretic methods. Five terms of the high-frequency development of the transmission coefficient are given.