标量衍射积分公式比较分析

Comparison on the Scalar Diffraction Integral Formulae

由于菲涅耳基尔霍夫衍射公式、瑞利索末菲衍射公式1和2共存于标量衍射积分公式体系,3个衍射积分公式的优劣难以断定。为了比较3个衍射积分公式,基于小衍射源和衍射远场的特点简化3个衍射积分公式,给出计算衍射远场平面上的标量光场总功率和衍射远场半球面上的矢量光场总功率的函数表达式,并以被积函数的收敛性判断3个衍射积分公式的适用范围,以衍射源总功率为标准,判断3个衍射积分公式的计算精度。分析表明,3个衍射积分公式均适用于计算傍轴标量衍射场,而只有瑞利索末菲衍射公式1才适用于计算非傍轴标量衍射场;3个衍射积分公式均适用于计算傍轴和非傍轴矢量衍射场。其中,对于垂直入射至小圆孔的平面波受限非傍轴衍射,基于瑞利索末菲衍射公式2计算的衍射远场观察半球面上总功率的相对计算误差的绝对值最小。

As the Fresnel-Kirchhoff diffraction formula and the first and second solution of Rayleigh-Sommerfeld diffraction formulae coexist in the scalar diffraction integral formulae system, it is difficult to judge which one is the best. For the sake of comparing three diffraction formulae, the specialties of small diffracted source and far field diffraction are employed to simplify three diffraction formulae, the expressions of the scalar far field total powers in observation plane and vector far field total powers in observation hemisphere are presented. Based on the convergence of integrand, the applicable scopes of three diffraction formulae are introduced. And using the total power of diffracted source as standard, the computational precisions of the diffraction formulae are clarified. The analysis results indicate that, for the hypothesis of scalar diffraction field, three diffraction formulae are suitable for computing the paraxial scalar diffraction beam, but only the first solution of Rayleigh-Sommerfeld diffraction formula is appropriate for computing the non-paraxial diffraction beam. For the nature of vector diffraction field, three diffraction formulae are applicable for computing the paraxial and non-paraxial vector diffraction beam. Hereinto, for the normal incident plane wave diffracted by small circular aperture, the absolute value of relative calculated error of the far field total power which is computed by the second solution of Rayleigh-Sommerfeld diffraction formula is the least.