A simple model of a gravitational lens from geometric optics.

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n (r) 2 = 1 + C 2 / r 2 , where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images. Editor's Note: This article describes a simple theoretical model for gravitational lensing. The authors analyze a graded index of refraction that reproduces the behavior for light passing near the event horizon of a black hole. The mathematical simplicity of the model permits exploration of the effects of gravitational lensing—including bending, reflection, and the formation of Einstein rings—using only integral calculus and Fermat's principle. The authors illustrate many interesting lensing phenomena with 2D and 3D graphics. The model described in this paper could be introduced as a "theoretical toy model" to complement classroom demonstrations of gravitational lensing such as a "logarithmic lens" or the stem of a wine glass, making gravitational lensing and its use in modern astrophysics accessible to introductory physics students. [ABSTRACT FROM AUTHOR]