A comparison of the vectorial nonparaxial approach with Fresnel and Fraunhofer approximations

Based on the vectorial Rayleigh diffraction integrals, a nonparaxial propagation equation of vectorial plane waves diffracted at a circular aperture is derived. The nonparaxial far-field expression, Fresnel and Fraunhofer diffraction formulae are given and treated as special cases of our general expression. The theoretical formulation permits us to study and compare the transversal and axial intensity distributions of diffracted plane waves both analytically and numerically. Illustrative numerical examples are given. It is shown that the vectorial nonparaxial approach has to be used if the aperture size is comparable with or less than the wavelength, and the knowledge of both transversal and axial intensity distributions is required to provide a comprehensive comparison of the paraxial and nonparaxial results.     

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.     

The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture

Based on the vectorial Rayleigh-Sommerfeld integrals, the general propagation integral expressions for on-axis nonparaxial vectorial spherical wave diffracted at a circular aperture are derived. The results are strict integral formulae for the light field on the axis, valid for either strong or weak focusing, both far and near zones, and for the systems in which the size of the aperture is comparable to or smaller than the wavelength. Thus, it has the advantage for general application. For convergent spherical waves, the numerical calculation results are compared with those obtained by using the integral formulae of scalar paraxial approximation and scalar nonparaxial approximation, confirming the consistence in the situation of scalar approximation.     

Propagation of vectorial nonparaxial Gaussian beams through an annular aperture

Starting from the vectorial Rayleigh diffraction integrals, the nonparaxial propagation of vectorial Gaussian beams through an annular aperture is studied. The analytical propagation expressions are derived, which permit us to treat the on-axis field and far field of vectorial nonparaxial Gaussian beams diffracted at the annular aperture, the nonparaxial diffraction at a circular aperture and a circular disc as our special cases in a unified way. The validity of our treatment is confirmed by direct numerical integration of the Rayleigh formulae. It is shown that the f-parameter and annular obscuration affect the beam nonparaxiality in the case of diffraction at the annular aperture.     

Focusing and diffraction by an optical lens and a small circular aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, analytical expressions for the transversal and axial field distribution of plane waves propagating through a thin lens followed by a small circular aperture are derived and used to study the focusing and diffraction properties of plane waves. Some special cases of our general result are discussed, and illustrative numerical calculation results are given. It is found that the vectorial nonparaxial approach should be applied if the aperture dimension is comparable with the wavelength or the focusing is strong.     

Nonparaxial propagation of a vectorial apertured off-axis Lorentz beam

Based on the vectorial Rayleigh--Sommerfeld integral formula and the complex Gaussian expansion of the hard-edge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis Lorentz beam passing through a rectangular aperture is derived. The unapertured case, the far field expression and the scalar paraxial result are also presented as special cases of the general formulae, respectively. Some numerical examples are also given to show the propagation characteristics of a nonparaxial vectorial off-axis Lorentz beam through a rectangular aperture. It is indicated that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behaviour.     

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.     

光阑对平面光波傍轴性的影响

 根据单色光傍轴度的定义,分别研究了圆形光阑和方形光阑对平面光波傍轴性的影响。推导出平面光波经圆形光阑和方形光阑衍射后其傍轴度的解析表达式。平面光波经圆形光阑或方形光阑衍射后,其傍轴性下降,傍轴度小于1,傍轴度下降量取决于光阑尺寸与入射平面光波波长的比值。数值计算结果表明：随光阑尺寸与入射平面光波波长的比值从零开始增大,平面衍射光波的傍轴度从零开始快速增大而后趋于一饱和值,但光阑的形状不影响平面衍射光波的傍轴性。     

平面波经微小圆孔衍射的传输特性

采用横截面上光强的精确表述,分析了单色平面波微小圆孔衍射后,其衍射光束的传输特性.发现二阶矩定义下的衍射光束束宽满足一般的双曲线传输规律.数值计算表明,在微小圆孔半径小于0.17个波长时,衍射光束的光束传输因子小于1.     

Anomalous spectral behavior of vectorial apertured ultrashort pulsed beams beyond the paraxial approximation

Based on the vectorial Rayleigh diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a circular hard-aperture is derived. The effect of f-parameter (f = 1/k₀w₀) on the spectral anomalies near phase singularities of the vectorial nonparaxial ultrashort pulsed beams is studied. It is shown that the spectral switch near the phase singularity of diffracted vectorial nonparaxial ultrashort pulsed beam still exists beyond paraxial regime, but disappears when the f-parameter is larger than a certain value.     

Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams from a hard-edged aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a rectangular hard-aperture is derived. The results of the far-field and paraxial cases can be regarded as special cases of the general expression. Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams passing through a hard-edged aperture is analyzed in detail. It is shown that the spectral switch near the phase singularity, which is predicated under the condition of the scalar paraxial theory, disappears with the increasing effect of the vectorial and nonparaxial nature.     

非傍轴衍射光束横截面上的光强及其能量传输

运用角谱表示的矢量衍射理论,以平面波微小圆孔的非傍轴衍射为例,给出了垂直于光束传输方向的横截面上不同定义的三种光强（传统光强I、功率密度Jz、时间平均能流密度矢量的z分量〈Sz〉）的计算公式,并进行了详细的数值计算和比较研究.指出对微小孔衍射的能量传输,必须考虑光场的矢量特性.应用能流密度矢量S和功率流密度矢量J,分别计算了微小孔非傍轴衍射的透射系数,得到了一些新的结论.     

非傍轴矢量高斯光束的光强表示

 基于瑞利-索末菲衍射积分，未使用任何近似，对非傍轴矢量光束的两种光强表示式，即传统光强公式和时间平均坡印廷矢量的z分量进行了研究。对非傍轴矢量高斯光束详细数值计算结果的比较表明，两种表示式之间的差异，即两者的相对误差与束腰宽度及传输距离和波长的比值有关。对非傍轴矢量高斯光束，若传输距离与波长的比值为10，束腰宽度与波长的比值大于等于0.8，则最大相对误差不到1.5%。因此，传统光强公式是可用的。     

Far-field divergence of a vectorial plane wave diffracted by a circular aperture from the vectorial structure

Based on the vectorial structure of an electromagnetic wave, the analytical and concise expressions for the TE and TM terms of a vectorial plane wave diffracted by a circular aperture are derived in the far-field. The expressions of the energy flux distributions of the TE term, the TM term and the diffracted plane wave are also presented. The ratios of the power of the TE and TM terms to that of the diffracted plane wave are examined in the far-field. In addition, the far-field divergence angles of the TE term, the TM term and the diffracted plane wave, which are related to the energy flux distribution, are investigated. The different energy flux distributions of the TE and TM terms result in the discrepancy of their divergence angles. The influences of the linearly polarized angle and the radius of the circular aperture on the far-field divergence angles of the TE term, the TM term and the diffracted plane wave are discussed in detail. This research may promote the recognition of the optical propagation through a circular aperture.vspace1mm     

非傍轴矢量高斯光束单缝衍射的严格理论

提出非傍轴矢量光束衍射严格的电磁场理论,并以非傍轴矢量高斯光束的单缝衍射为例加以说明.与基于矢量瑞利-索末菲衍射积分推出的解析结果做了数值计算比较,表明了二者的一致性.对理论的进一步推广做了简要讨论. 关键词： 严格的电磁场衍射理论 矢量瑞利-索末菲衍射积分 非傍轴矢量高斯光束的单缝衍射     

Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field

Based on the vector angular spectrum method and the stationary phase method and the fact that a circular aperture function can be expanded into a finite sum of complex Gaussian functions, the analytical vectorial structure of a four-petal Gaussian beam (FPGB) diffracted by a circular aperture is derived in the far field. The energy flux distributions and the diffraction effect introduced by the aperture are studied and illustrated graphically. Moreover, the influence of the f-parameter and the truncation parameter on the non-paraxiality is demonstrated in detail. In addition, the approximate formulas obtained in this paper can degenerate into un-apertured case when the truncation parameter tends to infinity. This work is beneficial to strengthen the understanding of vectorial properties of the FPGB diffracted by a circular aperture.     

非傍轴矢量余弦-高斯光束两种光强表示的比较研究

从矢量瑞利-索末菲衍射积分公式出发,以非傍轴矢量余弦－高斯(CoG)光束为例,对非傍轴矢量光束的两种光强表示式,即传统光强公式和时间平均坡印廷矢量的z分量进行了比较研究.对非傍轴矢量CoG光束轴上和横向光强分布详细的数值计算和比较结果表明,两种光强表示式之间的相对误差η与w0/λ、z/λ和偏心参量b有关,其中w0,λ和z分别为束腰宽度,波长和传输距离.当偏心参量b较小,且束腰宽度与波长相比不很小时,例如,b≤0.8,w0/λ≥0.8,z/λ=10时,二者间的最大相对误差ηmax＜2%,传统光强公式可以使用.     

Phase singularities of nonparaxial cosh-Gaussian vortex beams diffracted by a rectangular aperture

Analytical expression for the propagation of nonparaxial cosh-Gaussian (ChG) beams diffracted by a rectangular aperture is derived based on the vector Rayleigh-Sommerfeld diffraction integrals and expansion of the aperture window function into a finite sum of complex Gaussian functions, and used to study the phase singularities of nonparaxial diffracted ChG vortex beams. The pair creation, annihilation, motion of phase singularities in the diffracted field and the dependence of position and number of phase singularities on the aperture and beam parameters, as well as on the beam nonparaxiality are illustrated by numerical examples.     

非傍轴矢量光束二阶矩的发散特性

基于非傍轴矢量光束横截面上光强的精确表示和矢量二阶矩理论,应用严格的角谱表示的矢量衍射理论,对任意非傍轴矢量衍射光束的传输进行了系统的研究.给出了光束的束宽、横向发散角的积分表达式,论证了非傍轴矢量光束二阶矩的发散特性.对非傍轴矢量光束的束宽、发散角的定义及一些相关的问题进行了分析和讨论. 关键词： 矢量衍射理论 光束传输 矢量二阶矩 发散特性     

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.