Solutions of two optical problems
Solutions of two optical problems
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两个光学问题的解
DOI：
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发表时间：
1954
期刊：
影响因子：
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通讯作者：
A. Young
中科院分区：
文献类型：
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作者：
A. Fletcher;T. Murphy;A. Young
第一个问题是找出球内折射率的球对称分布（球外折射率是均匀的），使光线从给定的外部点a开始，落在球上，到达给定的外部点B的精确焦点a，其中AB必须经过球的中心O。吕尼贝格（1944）给出的解在这里是从阿贝尔型的积分方程推导出来的，与吕尼贝格的方法略有不同，然后进行发展，以提供数值结果。问题的一般效用系数如表1所示。其余的表是关于特殊情况的，特别是关于下列三种情况：A在无穷远处，入射光束由平行光线组成，b等于球半径的2.3倍、2.5倍和2.7倍。在这三种情况下，球中心的折射率分别是球外的1.150倍、1.137倍和1.126倍，折射率从中心到表面的变化如表3所示。第二个问题是找出一个圆柱形透镜的折射率的轴对称分布，在垂直于轴的平面上终止，使平行光束平行于轴传播，并正常地入射到平面上，正好在透镜轴上的一点上的焦点。再次从阿贝尔型积分方程中发现，折射率必须与sech （πr/2F）成正比，其中r表示到轴的距离，f表示焦点到平面的距离。
The first problem is to find the spherically symmetrical distribution of refractive index inside a sphere (the refractive index outside being uniform ) that brings rays starting from a given external pointA, and falling on the sphere, to an exact focus a t a given external point B, where AB necessarily passes through the centre O of the sphere. The solution given by Luneberg (1944) is here derived from an integral equation of Abel’s type, in a way slightly different from Luneberg’s, and is then developed so as to provide numerical results. Coefficients of general utility in the problem are given in table 1. The remaining tables relate to particular cases, and especially to the three cases in which, A being at infinity so that the incident beam consists of parallel rays, OB equals 2.3, 2.5 an d 2.7 times the radius of the sphere. In these three cases the refractive index at the centre of the sphere is respectively 1.150, 1.137 and 1.126 times that outside the sphere, and the variation of refractive index from centre to surface is indicated in final form in table 3. The second problem is to find the axially symmetrical distribution of refractive index in a cylindrical lens terminated by a plane face perpendicular to the axis that brings a parallel beam travelling parallel to the axis and incident normally on the plane face exactly to a focus at a point on the axis of the lens. It is found, again from an integral equation of Abel’s type, that the refractive index must be proportional to sech (πr/2F), where r denotes distance from the axis and Fdenotes the distance of the focus from the plane face.