Gradient Index Optics (GRIN)

In tracing GRIN Optics rays follow curved trajectories due to Fermat's principal, for which in a inhomogeneous refractive media where index depends as a function of position, , we obtain (Ref 1):

By defining a parametric vector , we can calculate components within the Cartesian Coordinate system, and then using calculus of variations we can obtain the following ray equation:

We can now simply integrate this equation twice to calculate the ray trajectories from an initial position as follows:

Finally we can calculate the trajectory using an ODE solver. This ODE solver utilizes a higher order symplectic energy loss minimization adaptive step methodology¹ to determine optimal step size, and minimize computation time for computationally rigorous GRIN systems.

References

1. B. E. Saleh and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons (2019).
2. Simon Tsaoussis, Hossein Alisafaee, "Ray tracing tool for arbitrary gradient index optical components," Proc. SPIE 11483, Novel Optical Systems, Methods, and Applications XXIII, 114830Y (21 August 2020); doi: 10.1117/12.2569760