Localised structures of light in dissipative nonlinear lattice models

最近，人们对使用光学设备进行通信和计算的可能性产生了很大的科学兴趣。虽然光纤技术可以实现电子邮件跨大陆的快速传输(例如，从英国到大西洋下的美国)，但瓶颈是将电子邮件的按键转换为数字信号、转换为光脉冲并返回所需的电子部件。虽然计算机处理信息的速度很快--计算机的GHz时钟速度意味着每秒可以进行数十亿次运算--但光脉冲的传播速度与光速一样，原则上每秒可以进行数十亿甚至数十亿次运算。那么，原则上，如果我们能够使用光学而不是电子学来进行计算和数据处理，我们就可以克服目前的瓶颈。一段时间以来，这种全光计算，或称“光控制光”，一直是技术人员的梦想。目前，最有希望容纳“光学计算机芯片”的材料是所谓的光子晶体，在这种晶体中，信息可以通过局域光团进行处理和传输，也称为光孤子。存在使这种晶体内的微小线条和缺陷形成网络型结构的技术，用于传输、存储和处理由光携带的信息。这项研究的目的是用数学来理解光子芯片的精确性质是如何影响其中孤子的动力学的。其中一个要解决的问题是，在这样的系统中，为了克服自然损耗而增加的少量增益是如何影响孤子的。此外，将光学比特从晶体中的一个位置移动到另一个位置需要什么条件。我们还将寻求解决的问题是，当试图在计算机上模拟这些过程时，是否需要捕捉晶格的精确细节，或者是否可以对晶体的性质进行更广泛的描述。这些都是基本问题，我们应该首先用一维晶体的简化情况来解决，然后用最实际的二维情况来解决。最终，这项研究的目的是提供有关将光子晶体用作光学计算介质的可行性的基本信息。

Recently there has been a lot of scientific interest in the possibility of using optical devices for communication and computation. While optical fibre technology enables rapid transmission of email across continents (e.g.~ from the UK to the USA under the Atlantic) the bottle neck is the electrical components required to turn the key strokes of your email into digital signals, into optical pulses, and back again. Although computers process information quickly --the GHz clock speed of your computer implies billions of operations persecond -- optical pulses travel at the speed of light which allowsin principle for millions of billions or even billions of billions ofoperations per second. In principle then, we could overcome the currentbottleneck if we could use optics rather than electronics toperform computation and data processing. This all optical computing,or `light controlling light' has remained a dream of technologists forsome time.Currently the most promissing materials which are able to host the 'optical computer chips' are so-called photonic crystals, where information can be processed and transmitted with localized blobs of light,also called optical solitons. Techniques exist to make tiny lines and defects inside such crystals to form a network type structure for transmission, storage and processing of information carried by light. This research aims tounderstand using mathematics how the precise nature of photonic chipsaffects dynamics of solitons in them. One of the issues tobe addressed is how the solitons are affected bysmall amounts of gain added to overcome the natural loss in suchsystems. Also, what conditions are required to move the optical bitsfrom one site in the crystal to another. We shall also seek toaddress whether, when trying to simulate these processes on acomputer, one needs to capture the precise details of the lattice orwhether one can get away with a much broader scale description of thecrystal's properties. These are fundamental questions which we shallfirst address using simplified situations of a one dimensional crystal, followed by the two-dimensional case, which is the most practical configuration. Ultimately the research is aimed at providing fundamental information on the feasibility of using photonic crystals as an optical computing medium.