Here, we report two kinds of absolute optical instruments that can make stigmatic images for geometric optics in two-dimensional space. One is called the duplex Mikaelian lens, which is made by splicing two half-Mikaelian lenses with different periods. The other is an exponential conformal transformer of a duplex Mikaelian lens with a rational number of the ratio of different periods of its two half-Mikaelian lenses, which we call a duplex Maxwell’s fish-eye lens. Duplex Mikaelian lenses have continuous translation symmetry with an arbitrary real number, while duplex Maxwell’s fish-eye lenses have continuous rotation symmetry from zero to 2π. Hence, each duplex Maxwell’s fish-eye lens corresponds to a duplex Mikaelian lens. We further demonstrate the caustic effect of geometric optics in duplex Mikaelian lenses and duplex Maxwell’s fish-eye lenses. In addition, we investigate the Talbot effect of wave optics in the duplex Mikaelian lens based on numeric calculations. Our findings, based on splicing and exponential conformal mapping, enlarge the family of absolute optical instruments.

中文翻译：

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双面Mikaelian和双面Maxwell的鱼眼镜头

在这里，我们报告了两种绝对光学仪器，它们可以为二维空间中的几何光学产生像散图像。一种叫做双面米凯尔透镜，是通过将两个不同周期的半米凯尔透镜拼接而成的。另一个是双星米氏透镜的指数保形变换器，其两个半米氏透镜的不同周期之比的合理数量，我们称之为双星麦克斯韦鱼眼镜头。复式Mikaelian镜头有一个任意实数连续平移对称性，而复式麦克斯韦鱼眼透镜具有连续旋转对称性从零到2 π。因此，每个双麦克斯韦的鱼眼镜头都对应一个双麦角透镜。我们进一步论证了几何光学在双联Mikaelian透镜和双联Maxwell鱼眼镜头中的苛性作用。此外，我们基于数值计算研究了双光镜的米卡利安透镜中波光学的塔尔博特效应。基于拼接和指数共形映射，我们的发现扩大了绝对光学仪器的范围。