Asymmetric Mathieu beams
Examples of asymmetric Mathieu-Gauss beams. Different rows correspond to different degrees of asymmetry (low asymmetry to high asymmetry, from up to down, respectively). Different columns correspond to different ellipticities (low ellipticity to high ellipticity, from left to right, respectively).
Optical lattices are structures of light generated by interfering optical laser beams, that resemble a crystal lattice. This interference creates a standing wave pattern that can be useful for the study and applications of light-matter interactions. A technique to generate such structures is using nondiffracting beams, i.e. beams whose intensity shape remains invariant under propagation, allowing a key feature in controlling light and optical manipulation of matter. However, obtaining nondiffracting beams can constitute a challenging problem because these beams must satisfy several restrictive mathematical conditions. In fact, an important and still open question is to determine what kind of transverse shapes imposed on a laser beam can be invariant propagated.
To date, there are several techniques focused on symmetry relations, numerical relaxation procedures or even in genetic algorithms to obtain novel and complex quasi-nondiffracting patterns. However, truly invariant and useful patterns are even by far more difficult to obtain, being the mathematical efforts to generate these nondiffracting beams formidable.
Researchers from Tecnologico de Monterrey at Mexico have been able to generate a novel kind of nondiffracting structure, the asymmetric Mathieu beam and they have corroborated experimentally their theoretical prediction by generating and propagating these beams, demonstrating their invariant behavior. These novel nondiffracting beams have three control parameters: the degree of ellipticity, the degree of asymmetry and the location of the respective symmetry breaking. Moreover, these beams can be implemented in an experimental setup by an easy modification to the standard and more well-known nondiffracting Mathieu beams. The generation of asymmetric Mathieu beams can be interesting for studying and developing new technologies in manipulation of Bose-Einstein condensates , applications of optical lattices for quantum computing, trapping of in-vivo and colloidal particles, and for soliton routing, just to name a few. This work has been published in Chinese Optics Letters, Vol. 16, Issue 12, 2018 (Arturo Barcelo-Chong et al., Asymmetric Mathieu beams).
Associate professor Servando Lopez-Aguayo, leader of the research project, believes that these results could shed light on novel techniques to generate even more complex invariant structures. As a holy grail, the Mexican researchers are looking to create truly and useful nondiffracting beams on demand, being also interested in diffraction management to achieve different kind of propagation dynamics.
Further work will be focused on applications of these nondiffracting modes, for example, in nonlinear optics for self-trapped beams in routing and steering and in the generation of dynamic optical asymmetric lattices.
非对称Mathieu光束
非对称Mathieu-高斯光束。每行光束对应于不同的非对称度（从上到下为低非对称到高非对称）。每列光束对应于不同的椭圆率（从左到右为低椭圆率到高椭圆率）。
光学晶格是由激光束相互干涉形成的光学结构，类似于晶体学中的晶格结构。这种干涉能够产生驻波图案，可用于光与物质之间相互作用的研究和应用。产生这种结构的一种方法是使用非衍射光束，即强度形状在传播中保持不变的光束，这种光束具有可控制光和进行物质光学操控的特点。然而，如何获得非衍射光束是一个非常有挑战的问题，因为这种光束必须满足几个限制条件。实际上，其中一个十分重要且尚未未决的问题是确定什么横截面形状的激光束能够在传播过程中形状保持不变。
目前，已经有利用对称关系、数值弛豫过程以及遗传算法的技术，以获得复杂的新型准非衍射模式。然而，由于在计算非衍射光束时所遇到的数学问题，要获得真正不变和有用的非衍射模式仍然十分困难。
来自墨西哥蒙特雷科技大学的学者们已经能够产生一种新型非衍射结构、非对称的Mathieu光束，他们通过产生和传播这些光束，证实了其理论预测。这种新型非衍射光束有三个控制参数：椭圆率、非对称度以及对称性缺失的位置。此外，这些光束可在实验装置中通过对标准非衍射Mathieu光束的简单修改来实现。非对称Mathieu光束可用于研究和开发操控玻色爱因斯坦凝聚的新技术，以及应用于量子计算的光学晶格、活体胶体粒子捕获以及孤子路由等领域。这项研究工作发表在Chinese Optics Letters 2018年第16卷第12期（Arturo Barcelo-Chong et al., Asymmetric Mathieu beams）。
研究项目负责人Servando Lopez-Aguayo副教授认为，这些研究结果有利于产生更复杂的不变结构的新技术。墨西哥蒙特雷科技大学的学者正在寻找按需产生非衍射光束的方法，同时也开始着手研究对衍射光束的操控，以实现不同类型的传播动力学。
下一步的工作将集中在这些非衍射模的应用中，例如，在非线性光学中自陷光束的传输与偏转以及动态光学中非对称晶格的生成。