Dean C E, Marston P L
Appl Opt. 1991 Aug 20;30(24):3443-51. doi: 10.1364/AO.30.003443.
Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [P. L. Marston and E. H. Trinh, Nature London 312, 529-531 (1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U - U(c))(3) = - d(infinity)V(2), where U and V are horizontal and vertical scattering angles and U(c) is the cusp point direction. The far-field opening rate d(infinity) is calculated for the transverse cusp. It is shown that d(infinity) has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wave front and hence to d(infinity). Measurements of d(infinity) for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q(4) approximately 1.31, the critical value for generation of a hyperbolic-umbilic focal section, the predicted d(infinity) goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d(infinity) vanishes as q approaches critical values for lips and transition events.
人们观察到，扁球形水滴散射的光在彩虹区域附近会产生尖点焦散[P. L. 马斯顿和E. H. 丁，《自然》伦敦312, 529 - 531 (1984)]。研究了产生远场横向尖点的一般局部波前的主曲率。结果表明，该波前会产生一条横向尖点曲线(U - U(c))(3) = - d(infinity)V(2)，其中U和V分别是水平和垂直散射角，U(c)是尖点方向。计算了横向尖点的远场开口率d(infinity)。结果表明，d(infinity)与一般波前的参数有简单的依赖关系。我们定义水滴的纵横比q = D/H，其中H是高度，D是散射水滴的赤道宽度。采用广义光线追踪方法将q与出射波前的主曲率和形状参数联系起来，进而与d(infinity)联系起来。对从声悬浮水滴散射激光的d(infinity)测量似乎支持了该计算。当q趋近于q(4)≈1.31（产生双曲脐状焦斑的临界值）时，预测的d(infinity)趋近于无穷大。对发散的性质进行了数值研究，同时也研究了当q接近唇状和过渡事件的临界值时d(infinity)消失的速率。
