Localised structures of light in dissipative nonlinear lattice models

最近，人们对利用光学设备进行通信和计算的可能性产生了浓厚的科学兴趣。虽然光纤技术使电子邮件能够跨大洲快速传输（例如，从英国到大西洋下的美国），但瓶颈是将电子邮件的击键转换为数字信号、光脉冲并再转换回来所需的电子元件。虽然计算机处理信息的速度很快——你的计算机的千兆赫时钟速度意味着每秒数十亿次操作——光脉冲以光速传播，原则上允许每秒数百万甚至数十亿次操作。原则上，如果我们可以使用光学而不是电子来进行计算和数据处理，我们就可以克服当前的瓶颈。这种全光学计算，或“光控光”一直是技术专家的梦想。目前最有前途的能够承载“光学计算机芯片”的材料是所谓的光子晶体，其中信息可以通过局部光团（也称为光孤子）进行处理和传输。现有的技术可以在这种晶体内部制造微小的线条和缺陷，形成网络型结构，用于光携带的信息的传输、存储和处理。本研究旨在利用数学方法了解光子芯片的精确性质如何影响其中孤子的动力学。需要解决的一个问题是，在这样的系统中，为了克服自然损耗而增加的少量增益如何影响孤子。此外，将光学位从晶体中的一个位置移动到另一个位置需要什么条件。当试图在计算机上模拟这些过程时，我们还将寻求解决是否需要捕捉晶格的精确细节，或者是否可以对晶体性质进行更广泛的描述。这些都是基本问题，我们将首先用一维晶体的简化情况来解决，然后是二维的情况，这是最实际的构型。这项研究的最终目的是为使用光子晶体作为光学计算介质的可行性提供基础信息。

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.