Gradient Index Optics (GRIN)

Gradient Index Optics Setup and Theory in KostaCLOUD

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, n(r)n(\vec{r}), we obtain (Ref 1):

δn(r)ds=0\delta\int n(\vec{r}) \text{d}s = 0

By defining a parametric vector r(s)\vec{r}(s), we can calculate components within the Cartesian Coordinate system, and then using calculus of variations we can obtain the following ray equation:

dds(n(r)drds)=n(r)\frac{\text{d}}{\text{d}s}(n(\vec{r}) \frac{\text{d}\vec{r}}{\text{d}s}) = \nabla n(\vec{r})

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

r(s)=0s1n(r(t))0tn(r(t))dtdt\vec{r}(s) = \int_0^s\frac{1}{n(\vec{r}(t'))}\int_0^{t'} \nabla n(\vec{r}(t)) \text{d}t\text{d}t'

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


  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

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