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.     

The vectorial angular-spectrum representation and Rayleigh–Sommerfeld diffraction formulae

The equivalence of the vectorial angular-spectrum representation and Rayleigh–Sommerfeld (RS) diffraction formulae is studied. Based on the angular-spectrum representation and the Weyl representation of a spherical wave, the vectorial RS diffraction formulae of the first and second kinds are derived in a simple way. Numercial results of diffracted divergent spherical waves are given to illustrate the application of the two vectorial RS diffraction formulae.     

Intensity distributions near focus of strongly converging spherical waves diffracted at a circular aperture

Based on the Rayleigh–Sommerfeld diffraction integral, the diffraction of converging spherical waves at a circular aperture is studied in a general case. The expression for the intensity near focus of strongly converging spherical diffracted waves is derived, which reduces to the well known result expressed in terms of Lommel functions for the case of weakly converging spherical diffracted waves. The intensity distributions at the geometrical focal plane and along the axis are given. Numerical comparative examples are presented to illustrate the more general applicability of our results.     

Nonparaxial diffraction of vectorial plane waves at a small aperture

By using the vectorial Rayleigh diffraction integrals, a nonparaxial propagation equation of vectorial plane waves diffracted at a small rectangular aperture is derived analytically and some special cases are discussed. Numerical calculation results are given to illustrate the applicability and validity of our theoretical formulae. It is shown that for the apertured case the ratio of the aperture width and wavelength affects the beam nonparaxiality. The nonparaxial approach presented in this paper has to be used for diffracted plane waves if the aperture width is comparable with or less than the wavelength.     

Propagation of vectorial Gaussian beams behind a circular aperture

Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum.     

圆孔的菲涅耳衍射

得到了圆孔菲涅耳衍射光强分布的表达式,此式由贝塞尔函数的数列(或Lommel函数)给出,因此很容易利用普及的数学软件算出结果.并且利用CCD照相机量度了衍射光强分布,与运算的结果作了比较,也与Burch在1985报导的结果作了比较.     

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

Focusing properties of twisted Gaussian Schell-model beams passing through a system: The aperture lens or the aperture and lens separated

Starting from the propagation law of partially coherent light, the focusing properties of twisted Gaussian Schell-model (TGSM) beams passing through an optical system consisting of an aperture lens or through a system consisting of a separated aperture and lens are studied. The propagation expressions for focused TGSM beams have been derived. Our main attention is focused on the effect of the coherence of partially coherent light, the twist parameter and the distance between the aperture and lens on the focusing properties of TGSM beams. Numerical calculation results are given to illustrate how the coherence of partially coherent light, the twist parameter and the distance between the aperture and lens affect the axial irradiance in the focused field.     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Optical tubular structures produced by diffraction of circular apertures

The coherent optical field diffracted through a circular aperture can produce optical tubular structures single-closed or twice-closed, by choosing the suitable experimental conditions. Both structures can be generated on the same optical set-up only by changing the experimental parameters involved on it, i.e. the aperture radius, the on-axis observation point, the wavelength and the on-axis source position. Such structures, generated in this way, have strong intensity gradients at their edges and their dimensions can be controlled in order to make them appropriate structures for optical trapping of particles, for instance.     

Focusing of flattened Gaussian beams by an annular lens

Focusing properties of flattened Gaussian beams (FGBs) passing through an annular lens is studied based on the Collins formula. It is found that the on-axis irradiance distributions of focused FGBs are unsymmetrical with respect to the geometrical focal plane even for large values of Fresnel number F_w associated with the beam, so that there exist focal shifts in general. Detailed numerical results show the dependence of focal shifts on the beam and system parameters. Focal shifts of FGBs by a lens without central obscuration and focal shifts of Gaussian beams by an annular lens can be treated as special cases of the obscure ratio =0 and beam order N=0, respectively. Furthermore, focal shifts of plane waves by an annular lens can also be treated as a special case of N=0 and F_w→∞.     

大孔径会聚光束的圆孔和圆屏衍射及其反演变换

实验中发现,经常使用的大孔径会聚光束的圆孔和圆屏衍射图样不同于一般的菲涅耳圆孔和圆屏衍射图样.通过对这种光束进行的实验性研究发现,这是一种由球差引起的畸变了的会聚球面波光束.解释了衍射图样的形成,证明了这种光束的衍射图样中存在反演变换关系.利用这种光束可以对微小的衍射物体进行反演放大.     

Near-field properties of diffraction through a circular subwavelength-size aperture

Analytical nonparaxial vectorial electric field expressions for both Gaussian beams and plane waves diffracted through a circular aperture are derived by using the vector plane angular spectrum method for the first time,which is suitable for the subwavelength aperture and the near-field region.The transverse properties of intensity distributions and their evolutions with the propagating distance,and the power transmission functions for diffracted fields containing the whole field,the evanescent field and the propagating field are investigated in detail,which is helpful for understanding the relationship between evanescent and propagating components in the near-field region and can be applied to apertured near-field scanning optical microscopy.     

Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism

Effects of third-order astigmatism on the focused structure of linearly and circularly polarized Laguerre-Gaussian beams have been investigated by using vectorial Debye-Wolf integral. The results have been presented for total intensity distribution and squares of the polarization components at the focal plane of a high numerical aperture system, for two values of the topological charge. Astigmatism results in the stretching of the intensity pattern as well as of the squares of the polarization components. A split is observed in the intensity pattern of a focused beam having double topological charge, and also in the pattern of the longitudinal polarization component of circularly polarized beam even with unit topological charge.     

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.     

An opposed-anvil-type apparatus with an optical window and a wide-angle aperture for neutron diffraction

We designed new anvil assemblies for acquiring high-quality neutron diffraction data and ruby fluorescence spectra inside a sample chamber. The conical aperture of Ni-binded WC anvils was expanded by a factor of two. A hybrid gasket made of TiZr- and Al-alloy was developed to prevent outward extrusion. A small and optically transparent window of moissanite was introduced to allow for the determination of pressure and hydrostaticity by measurement of ruby fluorescence spectra. High pressure-generation tests that make use of Bi electrical conductivity and ruby pressure markers revealed that pressure could be determined over 10 GPa. In situ synchrotron X-ray diffraction experiments were also carried out using NaCl as the pressure calibrants. The maximum pressure achieved was approximately 13 GPa. The neutron diffraction intensity from the newly generated anvil assemblies was 2.5–3.0 times greater than that using the standard toroidal anvil assemblies used previously.     

Angular spectrum analysis of near-field diffraction of a small aperture

The near-field diffraction of a nanometer aperture is studied in this paper according to the angular spectrum theory. The mathematic expressions of the near-field diffraction are presented, and the influences of the polarization of the electromagnetic field and the evanescent wave on the diffraction in near field are analyzed by means of vector and scalar diffraction theory, respectively. The transverse and longitudinal diffraction intensity distributions of a small aperture are numerically calculated, and the results show the near-field diffraction of a small aperture is influenced strongly by the vector property of the electromagnetic field and the evanescent component.     

Axial intensity distribution behind a Fresnel zone plate

An analytical expression based on an improved Rayleigh–Sommerfeld diffraction formula with evanescent term is derived for analyzing the axial light intensity distribution throughout the whole space behind a Fresnel zone plate. The effects of the number of Fresnel zones and the size of aperture on the axial intensity distribution are calculated for two kinds of Fresnel zone plate with larger and smaller aperture. The validity of the general formulae for calculating the focal lengths and the relative intensities of the foci of a Fresnel zone plate is analyzed.