利用保角变换实现环形光栅的Talbot效应
利用保角变换实现环形光栅的Talbot效应
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DOI：
10.7498/aps.69.20191340
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
朱永元
中科院分区：
文献类型：
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作者：
杨哲宁;乐阳阳;洪煦昊;赵瑞智;陆蓉儿;冯霞;许亚光;袁旭东;张超;秦亦强;朱永元
Talbot效应是一种近场自成像效应, 通常只有周期光栅可以产生Talbot效应, 而环形光栅无法产生.在以点光源为入射光的圆周期结构中，发现光栅在一定的传播距离内不存在自成像效应。本文将保角变换与塔尔博特效应相结合，在物理空间中设计了一种特殊的介质，使得圆光栅在其中具有塔尔博特效应，计算了保角变换产生的折射率分布，得到了相应的自成像半径表达式。采用Lumerical乘积进行仿真验证，总结了该方法的适用条件。我们分别对有无设计介质的圆光栅进行了模拟。两种模拟中的光场分布彼此不同。在第二种情况下的光场与平面光栅的光场比第一种模拟具有更多的相似性。在第二种情况下，我们可以计算出一定的塔尔博特半径，在计算出的塔尔博特半径处的光场分布与圆光栅处的光场分布非常相似。但对于第一种情况，我们无法计算出某个塔尔博特半径，只能通过比较各距离处的光场与光栅结构，得到自成像精度最高的圆环半径。我们发现在第二种情况下所用的圆光栅的小周期使光场在塔尔博特半径处分叉。因此，我们对周期比入射波长大的圆光栅进行了第三次模拟。自成像结果与光栅结构很好地匹配。但是，这种方法存在一些局限性。根据保角变换，中心附近的折射率趋于无穷大，所以我们要去掉中心附近的介质。同样，当半径足够大时，折射率可以小于1，因此塔尔博特效应应该发生在该半径内。最后，我们证明了变换光学可以引入到圆光栅的自成像中，从而极大地拓展了塔尔博特效应的应用范围。
The Talbot effect is a near-field diffraction effect that occurs in periodic structures. In a circular periodic structure with a point source as incident light, it has been found that there is no self-imaging effect of the grating at a certain propagation distance. In this paper, we combine the conformal transformation with the Talbot effect and work out a special medium in the physical space, which allows the circular grating to have a Talbot effect within it. The refractive index distribution generated by conformal transformation is calculated and the corresponding self-imaging radius expression is obtained. Lumerical product is used for simulation verification, and the applicable condition of the method is summarized. We separately carry out the simulations of a circular grating with and without the designed medium. Light field distributions in the two simulations differ from each other. The light field in the second situation shares more similarities with the light field of a plane grating than the first simulation. What is more, in the second situation, we can work out a certain Talbot radius, and the light field distribution at the calculated Talbot radius is quite similar to that at the circular grating. But for the first situation, we cannot calculate a certain Talbot radius and can obtain only the radius of the ring with highest self-imaging accuracy by comparing light field at each distance with the grating structure. We find that the small period of the circular grating we used in the second situation makes the light field at Talbot radius furcate. So we carry out a third simulation of a circular grating with a large period compared with the incident wavelength. The self-imaging result matches the grating structure quite well. However, there are some limits in this method. According to the conformal transformation, the refractive index near the center tends to be infinite, so we have to remove the medium near the center. Also, when the radius is big enough, refractive index there can be smaller than 1, so the Talbot effect should happen within this radius. In conclusion, we show that the transformation optics can be introduced into the self-imaging of circular gratings, and thus greatly expanding the range of applications for the Talbot effect.