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.     

The relation between the boundary diffraction wave theory and physical optics

The physical optics surface integral is asymptotically reduced to a line integral along the contour of the diffracting edge. It is shown that the resultant integral can be separated into two sub-integrals which represent the reflected and transmitted diffracted fields. The integrands are transformed into the same forms with the potential function of the boundary diffraction wave theory.     

边界衍射波理论公式的准确性数值分析

利用数值分析的方法分析了由边界波理论公式所得的圆形光阑衍射场的光强分布.通过对边界波理论公式所得的光强分布曲线,对几何照明区与阴影区的光强分布曲线与面衍射积分公式所得的光强分布曲线进行比较,分析了边界波理论公式对衍射场描述的准确性.     

Conditions for the validity of the Debye integral representation of focused fields

A very simple sufficiency condition is obtained, under which the Debye diffraction integral may be expected to give a good approximation to the solution of a boundary value problem that is generally taken to represent a field in the region of focus. When the angular aperture of the focusing system is sufficiently small, the condition reduces to the requirement that the Fresnel number of the diffracting aperture, when viewed from the geometrical focus, is large compared to unity.     

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.     

Laser gain in femtosecond microstructured Nd:MgO:LiNbO3 crystals

Femtosecond laser writing has been used to create 1D and 2D diffracting microstructures in Nd³⁺:MgO co-doped LiNbO₃ crystals. The main characteristics of the diffracting device, such as its polarization dependence and diffraction efficiency have been investigated, showing that first order diffraction efficiencies in excess of 35% can be achieved. We have demonstrated that continuous wave laser oscillation from Nd³⁺ ions at 1.06 μm can be generated through the integrated diffraction gratings, thus leading to non-collinear intracavity laser propagation. The laser performance, in terms of laser thresholds and slope efficiencies, is reported and explained. The achievement of a controlled non-collinear propagation is an interesting feature in multi-frequency lasers, optical communication and optical switching. PACS 42.55.Rz; 42.70.Hj; 42.25.Fx     

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.     

Cornu's spiral in the Fresnel regime studied using ultrasound: a phase study

A simple experimental technique for measuring the phase and amplitude of diffracting ultrasound wave [A. Hitachi and M. Takata, Am. J. Phys. 78, 678 (2010)] has been applied to diffracting objects with straight edges as a demonstration of the Cornu spiral. Babinet's principle is studied observing the ultrasound field behind a slit and a complementary strip obstacle and is verified directly by comparing vectors (phasors) in the complex plane. The phase of the diffracted wave observed in the geometrical shadow of the straight screen has the form of a cylindrical wave originating at the edge of the straight screen as the boundary diffraction wave proposed by Young. In addition, the incident wave has a phase delay of π/4 behind the wave passing through on the center line of the slit, the plane of symmetry, has been observed as predicted by Huygens-Fresnel diffraction theory.     

Boundary diffraction waves at multiple Fresnel-Kirchhoff diffraction

The theory of boundary diffraction wave is generalized to the case of multiple Fresnel-Kirchhoff diffraction on several sequentially situated holes (screens). The generalized boundary wave is the sum of boundary diffraction waves with different multiplicities described by integrals whose maximal multiplicity is equal to the number of obstacles. It is shown that the proposed generalized boundary wave method provides a multiple time gain in calculations, as compared to the aperture integration method. Results of numerical calculations agree with experimental dependences.     

Rubinowicz transform of the MTPO surface integrals

The surface integral of the modified theory of physical optics is reduced to a line integral by using the Rubinowicz transform for the incident scattered fields by an arbitrary aperture in a black surface. The integral theorem of Kirchhoff is applied to the scattering geometry and the diffracted fields are expressed in terms of a line integral along the contour of the diffracting edge.     

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。     

小孔阵列衍射特性与应用

以单色标量波衍射理论为基础,研究了均匀平面波从不同角度入射小孔阵列的衍射特性。运用单孔衍射理论,同时考虑相邻小孔间衍射光强的相互影响,建立了小孔阵列衍射的理论模型和光强分布的数值积分式,小孔为硬边小孔。利用Matlab对500 nm波长的平面波入射微小方孔阵列衍射图样进行了计算机仿真,得到了不同几何参量下平面波从不同角度入射时的衍射图样的一维和二维光强分布图,并将仿真结果用于微型数字式太阳敏感器的光学系统中的结构参量设计和图像处理中的参量确定。太阳敏感器的成像实验结果表明,小孔阵列衍射光强分布图的仿真结果正确、太阳敏感器光学系统参量设计合理。小孔阵列衍射理论为太阳敏感器的光学系统设计和图像处理提供了可靠的理论基础。     

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.     

Theory of Atom Optics: Feynman''s Path Integral Approach

The present theory of atom optics is established mainly on the Schr?dinger equations or the matrix mechanics equation. The authors present a new theoretical formulation of atom optics: Feynman’s path integral theory. Its advantage is that one can describe the diffraction and interference of atoms passing through slits (or grating), apertures, and standing wave laser field in Earth’s gravitational field by using a type of wave function and calculation is simple. For this reason, we derive the wave functions of particles in the following configurations: single slit (and slit with the van der Waals interaction), double slit, N slit, rectangular aperture, circular aperture, the Mach-Zehnder-type interferometer, the interferometer with the Raman beams, the Sagnac effect, the Aharonov-Casher effect, the Kapitza-Dirac diffraction effect, and the Aharonov-Bohm effect. The authors give a wave function of the state of particles on the screen in abovementioned configurations. Our formulas show good agreement with present experimental measurements.     

Angular criterion to distinguish between Fraunhofer and Fresnel diffraction

The distinction between Fresnel and Fraunhofer diffraction is an essential condition for the accurate analysis of diffracting structures. In this paper we propose a criterion based on the angle subtended by the first minimum of the diffraction pattern from the centre of the diffracting aperture. The determination of the minimum of the diffraction pattern is the crucial point to assure the accuracy of the criterion. Therefore, the applicability of adequate thresholds for detection is discussed. The criterion is also generalized by expressing it in terms of the number of Fresnel zones delimited by the aperture. Simulations are reported to illustrate the feasibility of the criterion.     

Theory of Atom Optics: Feynman’s Path Integral Approach

The present theory of atom optics is established mainly on the Schrödinger equations or the matrix mechanics equation. The authors present a new theoretical formulation of atom optics: Feynman’s path integral theory. Its advantage is that one can describe the diffraction and interference of atoms passing through slits (or grating), apertures, and standing wave laser field in Earth’s gravitational field by using a type of wave function and calculation is simple. For this reason, we derive the wave functions of particles in the following configurations: single slit (and slit with the van der Waals interaction), double slit, N slit, rectangular aperture, circular aperture, the Mach-Zehnder-type interferometer, the interferometer with the Raman beams, the Sagnac effect, the Aharonov-Casher effect, the Kapitza-Dirac diffraction effect, and the Aharonov-Bohm effect. The authors give a wave function of the state of particles on the screen in abovementioned configurations. Our formulas show good agreement with present experimental measurements.     

Boundary diffraction wave theory of junctions between two surfaces with different face impedances

The line integral of the boundary diffraction wave theory is derived for the diffraction process of waves by a junction between two surfaces with different face impedances. The exact solution of Maliuzhinets is used with this aim. The resultant integral is applied to the diffraction of waves by a circular junction between two impedance surfaces. The results are examined numerically.     

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 rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture

The rigorous electromagnetic theory of the diffraction of vector beams by an aperture is proposed and numerically evaluated by the diffraction of vector Gaussian beams by a circular aperture. The results are compared with those, by using the vector Rayleigh–Sommerfeld diffraction integrals and angular-spectrum expression theory, showing a good consistency. The numerical calculation shows that the result calculated by the rigorous theory is much more precise than those calculated by the integral method in diffraction near field, and a highly consistency reaches by the three methods in diffraction far field. A further extension of the theory is discussed.