Hebri Davud, Rasouli Saifollah
J Opt Soc Am A Opt Image Sci Vis. 2019 Feb 1;36(2):253-263. doi: 10.1364/JOSAA.36.000253.
In this work, the diffraction-based discrimination of two-dimensional (2D) orthogonal separable and nonseparable periodic structures and prediction of the reduced Talbot distances for 2D orthogonal nonseparable periodic structures are presented. 2D orthogonal periodic structures are defined and classified into separable (multiplicative or additive) and nonseparable categories with the aid of a spatial spectrum lattice. For both the separable and nonseparable cases, the spatial spectra or far-field impulses are 2D orthogonal lattices. We prove that for a 2D orthogonal separable structure, in addition to the DC impulse, there are other impulses on the coordinate axes. As a result, if all the spectrum impulses of a structure on the coordinate axes, except for the DC impulse, vanish, we conclude that the structure is nonseparable. In the second part of this work, using a unified formulation, the near-field diffraction of the 2D orthogonal separable and nonseparable periodic structures is investigated. In general, the Talbot distance equals the least common multiple of the individual Talbot distances in the orthogonal directions, say, = . For the 2D orthogonal nonseparable periodic structures having Fourier coefficients only with odd indices, we have found surprising results. It is shown that for this kind of structure, the Talbot distance strongly depends on the number theoretic properties of the structure. Depending on the ratio of the structure's periods in the orthogonal directions, , the Talbot distance reduces to 2, 4, or 8. In addition, for the 2D orthogonal nonseparable sinusoidal grating, we show that, regardless of the value of , self-images are formed at distances smaller than the conventional Talbot distances attributed to and that we name the reduced Talbot (RT) distances. Halfway between two adjacent RT distances, the formation of negative self-images with a complementary amplitude of the self-images is predicted. Halfway between two adjacent self-image and negative-elf-image, subimages are formed. As another interesting result, we show that the intensity patterns of the subimages are 2D multiplicatively separable with halved periods in both directions. Finally, we show that 2D almost periodic structures with impulses on zone-plate-like concentric circles have self-images under plane wave illumination.
在这项工作中，我们提出了基于衍射对二维（2D）正交可分离和不可分离周期结构的辨别方法，并对二维正交不可分离周期结构的约化塔尔博特距离进行了预测。借助空间频谱晶格定义了二维正交周期结构，并将其分为可分离（乘法或加法）和不可分离两类。对于可分离和不可分离两种情况，空间频谱或远场脉冲都是二维正交晶格。我们证明，对于二维正交可分离结构，除了直流脉冲外，坐标轴上还有其他脉冲。因此，如果一个结构在坐标轴上的所有频谱脉冲（直流脉冲除外）都消失，我们就得出该结构是不可分离的结论。在这项工作的第二部分，我们使用统一的公式研究了二维正交可分离和不可分离周期结构的近场衍射。一般来说，塔尔博特距离等于正交方向上各个塔尔博特距离的最小公倍数，即 = 。对于仅具有奇数索引傅里叶系数的二维正交不可分离周期结构，我们发现了惊人的结果。结果表明，对于这种结构，塔尔博特距离强烈依赖于该结构的数论性质。根据结构在正交方向上的周期比 ，塔尔博特距离会减小到 2、 4或 8。此外，对于二维正交不可分离正弦光栅，我们表明，无论 的值如何，都会在小于归因于 和 的传统塔尔博特距离的距离处形成自成像，我们将其称为约化塔尔博特（RT）距离。在两个相邻的RT距离中间，预测会形成具有与自成像互补幅度的负自成像。在两个相邻的自成像和负自成像中间，会形成子图像。作为另一个有趣的结果，我们表明子图像的强度图案在两个方向上都是二维乘法可分离的，周期减半。最后，我们表明在平面波照明下，在类似波带片的同心圆上具有脉冲的二维准周期结构会形成自成像。
