# Light Rays Passing Through Liquid Crystals

## Disciplines

Ordinary Differential Equations and Applied Dynamics

## Abstract (300 words maximum)

As light passes through a liquid crystal, it splits into two different rays, the ordinary ray and the extraordinary ray. The effective refractive index is a functional composed of those two rays. An attempt is made to find an explicit equation for the path of light by finding the minimum distance to pass through a given field. Using the Euler-Lagrange equation in order to minimize the effective refractive index, the resulting function is a highly nonlinear second-order ordinary differential equation. If solved, this would give the equation of light passing through a given liquid crystal according to Fermat's principle of least time. The equation was examined for three different types of fields through which the light passes. A numerical solution was evaluated by computing a metric and subsequent Christoffel symbols, then applying the Euler-Lagrange equation to find a system of geodesics and graphing numerically to visualize the path of light. The Ricci curvature is computed to find how non-flat the associated geometry is, and the curvature of the path is computed to find when maximum ray bending occurs. Finally, the numerical process is repeated in three dimensions and the Gaussian curvature is computed for various fields.

## Academic department under which the project should be listed

CSM - Mathematics

## Primary Investigator (PI) Name

Eric Stachura

Light Rays Passing Through Liquid Crystals

As light passes through a liquid crystal, it splits into two different rays, the ordinary ray and the extraordinary ray. The effective refractive index is a functional composed of those two rays. An attempt is made to find an explicit equation for the path of light by finding the minimum distance to pass through a given field. Using the Euler-Lagrange equation in order to minimize the effective refractive index, the resulting function is a highly nonlinear second-order ordinary differential equation. If solved, this would give the equation of light passing through a given liquid crystal according to Fermat's principle of least time. The equation was examined for three different types of fields through which the light passes. A numerical solution was evaluated by computing a metric and subsequent Christoffel symbols, then applying the Euler-Lagrange equation to find a system of geodesics and graphing numerically to visualize the path of light. The Ricci curvature is computed to find how non-flat the associated geometry is, and the curvature of the path is computed to find when maximum ray bending occurs. Finally, the numerical process is repeated in three dimensions and the Gaussian curvature is computed for various fields.