Mathematics 229: Mathematical Modeling

Mathematics 229: Mathematical Modeling (Spring 2002)

Instructor

Description

The topic for Math 229 this year will be Nonlinear Fiber Optics; this is based in part on a remark by Dan Gauthier that “companies are desperate for people in this field”. The first half of the course will consist of prepatory lectures by me: first on the derivation of the PDE which describe the phenomena, including the physical background, and then on singular-perturbation techniques we will use to study wave propagation. The second half of the course will be project oriented: I will propose specific projects in this area for 2-or-3-person students teams to research, under my supervision. (I hope these projects will benefit Dan Gauthier in Physics, who proposed this area.) During the second half, class time will be divided approximately equally between additional supplementary lectures by me, student research, and student lectures presenting their research.

The grade will be based mostly on the projects; either oral and written presentations will be required. There will also be modest homework assignments to support the first half of the course.

Outline

Preliminary material
- Maxwell's equations in a continuous medium
- Linear fiber modes
- Theory of dispersive linear waves
Nonlinear propagation of light in a fiber
- Derivation of the nonlinear Schroedinger equation
- Spreading of optical pulses through nonlinearity
- Solitons and inverse scattering
- Nonlinearity and polarization
Energy-dissipating phenomena, and their use in signal amplification^*
- Stimulated Raman scattering
- Stimulated Brillouin scattering

^* Yes, that's right, a dissipative phenomenon is used to amplify the signal!

Prerequisites

No prior knowledge of the underlying physics will be required---I will cover this during the first part of the course. I will assume knowledge of the elementary theory of PDE, including Fourier transform; any one of the courses Math 114, 133, 211, or 232 will suffice as a prerequisite. (Multi-variable calculus and linear algebra are also required.)

Textbook

Nonlinear Fiber Optics, 3rd edition, by Govind Agrawal

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Last modified: 30 November 2001