The key objective of the project is to build a quantitative predictive capability for the focusing nonlinear electromagnetic waves. A standard mathematical model for describing this type of problems is the nonlinear Schroedinger equation (NLS). By now, this equation is relatively well studied. There are, however, indications that the NLS model may be oversimplified. An alternative is provided by a more comprehensive nonlinear Helmholtz equation (NLH), from which the NLS is, in fact, derived by employing the so-called paraxial approximation and neglecting the important phenomenon of backscattering. In contradistinction to the NLS, relatively little is known about the solvability of the NLH and uniqueness of its solutions. Moreover, this equation presents a considerable challenge for the numerical approach as well. Nonlinearity is a major hurdle, as it implies that the impinging and (back)scattered waves cannot be separated. Another key difficulty is the small magnitude of backscattering compared to that of the forward propagating wave. In the course of the project, the PI and his colleagues will develop, implement, and test an efficient numerical procedure for integrating the NLH. It will involve major modifications and improvements to the previously proposed methodology that has already proven successful and, in fact, unparalled in the literature. The methodology employs a high-order finite-difference approximation. Its central element is a special two-way nonlocal artificial boundary condition that makes the outer boundary transparent for all the outgoing waves and at the same time is capable of accurately prescribing the given impinging signal. It is expected that with the help of this methodology, a valuable new insight will be gained into a number of key outstanding questions in nonlinear optics, in particular, whether the nonparaxiality and backscattering may arrest the collapse (blow-up) of focusing nonlinear waves, and whether the NLH is capable of sustaining the so-called narrow spatial solitons, with the width on the order of only several wavelengths.
In the course of the project, a numerical methodology will be built to simulate the propagation of intense laser light through a variety of media and materials. This methodology has a solid mathematical foundation, and is expected to help address a number of challenging issues in the theoretical nonlinear optics. In addition to its potential theoretical merits, the methodology will be useful from the standpoint of applications as well. Indeed, the propagation of laser beams in materials is typically accompanied by the phenomena of nonlinear self-focusing and backscattering. The capability to quantitatively analyze and predict these key phenomena is extremely important for many of applications in modern science and engineering. The latter range from remote atmosphere sensing (when an earth-based powerful laser sends pulses to the sky, and backscattered radiation accounts for a substantial part of the detected signal), to laser surgery (propagation of laser beams in tissues), to transmitting information along optical fibers. There are other possible applications that involve, e.g., interactions between the co-propagating or counter- propagating laser beams. They may provide a vehicle for designing the so-called all- optical switches for the next generation of opto-electronic circuits. The proposed numerical methodology will yield a powerful tool for the accurate and robust analysis of the foregoing applications, along with many others.
