We investigated binary lenses with $1/r^n$ potentials in the asymmetric case with two lenses with different indexes n and m. These kinds of potentials have been widely used in several contexts, ranging from galaxies with halos described by different power laws to lensing by wormholes or exotic matter. In this paper, we present a complete atlas of critical curves and caustics for mixed binaries, starting from the equal-strength case, and then exploring unequal-strength systems. We also calculate the transitions between all different topology regimes. Finally we find some useful analytic approximations for the wide binary case and for the extreme unequal-strength case.

中文翻译：

---

混合二元系统引力透镜的焦散

我们调查了 $\mathrm{1个}/{\mathrm{[R}}^{ñ}$两个折射率为n和m的透镜在不对称情况下的电势。这些势能已在多种情况下得到广泛使用，从通过不同幂定律描述的具有光晕的星系到虫洞或外来物质的透镜化。在本文中，我们从等强度情况开始，然后探讨不等强度系统，给出了完整的混合二进制临界曲线和苛性图集。我们还计算了所有不同拓扑方案之间的过渡。最后，我们找到了一些有用的解析近似，适用于宽二元情况和极不等强度情况。