Optics: Derivation of $\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$

Question

I have been given this formula from optics here, with no background: $$\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$$

Where $n$ is the refractive index and $\hat{u}$ is a unit vector tangent to the path $s$ that light takes inside a medium.

Does anyone know if this formula has a name? I am looking specifically for a derivation of it. I have looked through the optics book by Hecht with no luck - I assume it comes from fermats principle of least time in some form.

Any help greatly appreciated.

Answer 1

This equation is called the ray equation and it can indeed be derived from Fermat's principle. I guess you can find more about its derivation in, e.g., Born and Wolf's Principles of optics or in Fundamentals of Photonics by Saleh and Teich.