Solution and focus property of the nonparaxial vector beams in the parabolic coordinates
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摘要: 为了求解柱坐标系下非傍轴矢量波动方程，得到光束的电场解析表达式，基于轴对称情况下沿角向偏振的电场，将非傍轴近似情况下的矢量波动方程进行了抛物线坐标的转化，利用分离变量法进行了相应求解，并给出了相应的数值计算。结果表明，非傍轴近似情况下，矢量波动方程的解能描述一种光束的电场，该场的解析表达式与合流超几何函数以及梅杰函数的解有关；光束的光强分布与第1类零阶贝塞尔模式光束类似；光束在近光轴处的光强表现为无限大并且沿边缘方向急剧衰减；在焦平面上沿着径向方向光强急剧减小。所得结果对于探究非傍轴近似情况下矢量光束的传输特性有一定的意义。Abstract: In order to solve the nonparaxial vector wave equation in the cylindrical coordinates and obtain electric field expression of the beams, based on the electric field along the azimuthal polarization under the axisymmetric circumstance, the vector wave equation under the nonparaxiality similar circumstances was transformed to the parabolic coordinates and was solved appropriately with the separation variables method. The corresponding numerical calculation was made. The results show that the new analytical solution of the nonparaxial vector wave equation is discussed to describe the propagation of a laser beam. The electric field of such a beam is found to be based on the solutions of the confluent hypergeometric function and the Meijer functions. The intensity distribution of beam is similar to the first-class zero-order Bessel beam mode. The intensity of the light beam near the optical axis is nearly infinite, and decays rapidly along the peripheral direction and decreases sharply along the radial direction in the focal plane. The acquired results are of certain significance for exploring the propagation properties of vector beams in case of nonparaxial approximation.
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