Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Sekou Rowe - Howard University
Co-Author(s): Harena Yemane, Howard University, Washington, DC
The far field refractor problem in geometric optics is an inverse problem which deals with finding a refracting surface (lens) that is capable of reshaping a light beam from a given point source shining through an aperture cone Ω with prescribed illumination intensity, so that it covers a prescribed region Ω^⋇ in the far field with a predetermined intensity distribution. Our objective is to obtain a numerical approximation to this surface. The existence and uniqueness up to dilations of such surfaces has been established by using analytical methods which involve optimal mass transportation techniques and approximation by semi ellipsoids under some geometric conditions on Ω and Ω^⋇. A numerical algorithm to approximate such a surface is also demonstrated in the work of R. De Leo, C. E. Gutiérrez and H. Mawi in . In our work we exhibit an alternative method to numerically approximate the solution to the far field refractor problem by using MATLAB. Unlike the method discussed in the work of De Leo et al  our algorithm uses a direct application of the analytical existence proof which was given in . Reference:  R. De Leo, C. E. Gutiérrez and H. Mawi. On the Numerical Solution of the Far Field Refractor Problem. Nonlinear Analysis, Vol. 157, (2017).  C. E. Gutiérrez and H. Mawi, The refractor problem with loss of energy, Nonlinear Anal. Vol 82 (2013).
Funder Acknowledgement(s): This study was supported, in part, by a grant from NSF HBCU-UP program awarded for Henok Mawi, PhD, Assistant Professor, Department of Mathematics, Howard University, Washington, DC.
Faculty Advisor: Henok Mawi, firstname.lastname@example.org
Role: I worked to develop the MATLAB code that finds an approximate numerical solution for the far field refractor problem. I started by breaking the problem up into different aspects that I could more easily solve and translate into MATLAB code. I then worked to create a basic code that found all the preliminary information required to solve the problem and put it into a format that MATLAB could understand. I then added on code that used the theory to get a code that solved the problem. Finally, I added on code to make it a general solution.