From fiber optics to optical clocks to observing the fastest phenomena, ultra short pulses are of significant use and have a very non-intuitive behavior at times. The Mathematics Society, St. Stephen’s College, invited Dr. Farnum, Kean University, to campus on the 15th of January 2014 to host the talk, ‘Ultra-short Optical Pulses: Mathematical Models, Applications, and Asymptotics’.
Non-linear waves are characterized by a class of partial differential equations (PDEs), and are widely used to model a variety of physical phenomena, including water waves, tsunamis, fiber optic communications systems, mode-locked lasers, and Bose-Einstein condensates. The solutions of interest are typically localized structures, sometimes called solitons, or solitary waves, in contrast to waves of infinite extent. In general, such non-linear wave equations can be solved only by numerical methods, which may be computationally expensive, and may provide only limited intuition about the behavior of physical system being modeled.
Dr. Farnum talked about the NLS (Non-Linear Schrodinger) short optical pulse propagation and its complicated nature introduced by the non-linear nature of the pulses. Beginning the talk by introducing how these waves are generated and gradually moving up to the equation defining these waves, the topic was explained beautifully and intuitively by the speaker. Due to the non-linear and partial nature of the equation, the computation required to solve these equations was very abstruse and time consuming.
In the talk, Dr. Farnum concentrated on methods to model the propagation of ultra-short optical pulses, which have applications in laser science, communications technology, and measurement of ultra-fast physics, on the order of a femto-second and less. He introduced the standard model for optical pulse propagation, which is based on the Non-linear Schrodinger Equation, described the short-comings of this standard model, and proposed an improved model based on the Schafer Wayne Short Pulse Equation. Then, he went on to describe a method for reducing the partial differential equation to a low dimensional system of ordinary differential equations. This reduction greatly reduces computational time, and provides insight into the stability of solutions, and helps engineers to build lasers to generate shorter and shorter optical pulses.
In this way Dr. Farnum also introduced his own research work, which lead to an intuitive idea of the nature of these pulses. Though the talk pertained to complex concepts, it was simplified and beautifully presented.