Mathematical Analysis of Novel Nonlinear Waves in Dissipative Optical Systems

这个项目从理论上研究光学中的耗散非线性波动系统，这些系统表现出新的和反直觉的性质，例如守恒行为和单向波传播。这种光学系统的例子包括某些支持沿激光腔边缘的单向光传播而不受散射和无序影响的先进激光器，这在光学计算机的设计中作为光学二极管是有希望的。其他例子包括某些耗散二次谐波产生系统，其特征是具有连续功率范围的孤子族，可用于信号处理设备。在这样的系统中表现出来的特性是耐人寻味的，它们将在这个项目中得到全面的研究。通过先进的数学技术和复杂的数值计算，将对这些光学系统的运行有新的见解，从而可以评估和优化它们在光学二极管和非线性数据处理方面的应用。该项目所涉及的问题是应用数学和光学的前沿课题。在数学上，这个项目冒险进入像保守系统一样行为的耗散系统的未知领域，这为非线性波的研究开辟了新的研究方向。在物理方面，这些研究将使人们更好地了解下一代激光设备，这将直接应用于光学和光子学的许多分支。在教育方面，将培训研究生，并监督本科生荣誉论文和本科生暑期研究，因此该项目将有助于培养STEM领域的年轻人才。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project theoretically investigates dissipative nonlinear wave systems in optics, which exhibit novel and counter-intuitive properties such as conservative-like behaviors and unidirectional wave propagation. Examples of such optical systems include certain advanced lasers supporting unidirectional light propagation along the edges of the laser cavity immune to scattering and disorder, which holds promise as an optical diode in the design of optical computers. Other examples include certain dissipative second-harmonic-generation systems featuring soliton families with continuous ranges of powers, which may be exploited for signal processing devices. Properties manifested in such systems are intriguing, and they will be investigated comprehensively in this project. Through advanced mathematical techniques and sophisticated numerical computations, new insight will be gained on the operation of these optical systems, so that their applications for optical diodes and nonlinear data processing can be assessed and optimized. The problems undertaken in this project are at the cutting edge of applied mathematics and optics. Mathematically, this project ventures into the unchartered territory of dissipative systems behaving like conservative systems, which opens new research directions in the study of nonlinear waves. On the physical side, these studies will lead to better understanding of next-generation laser devices, which will have direct applications to many branches of optics and photonics. Educationally, graduate students will be trained, and undergraduate honors thesis and undergraduate summer research will be supervised, so that this project will help train young talents in the STEM fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.