I've read about GRIN (Gradient Refractive Index) lens but nowhere have I come across a derivation of its focal length (in the paraxial limit, approximately). Here is a typical GRIN lens –
Let it's refractive index be given by $n(r) = n_0(1-\frac{kr^2}{2})$, where $r$ is the distance from the axis, and is small in the paraxial limit, $d$ is the thickness and $k$ is the gradient constant. I've seen that the focal length $f$ for such a setup is given by $\frac{1}{n_0\sqrt{k}sin(d\sqrt{k})}$. But they don't derive this result, for any reason whatsoever. So I tried to use the fact that rays would meet at the focus at the same time, and used the paraxial limit approximations and ended up with $$f=\frac{1}{n_0dk}$$ which means that it holds only for very small thickness of the lens. I believe that I need to calculate the optical path when a ray at a distance $r$ from the axis is passing through the lens with the help of Snell's law and Fermat's principle, which essentially states-
$$\delta \int n(r) \,ds = 0$$
where $ds=\sqrt{dx^2+dy^2}$. But I'm not really able to see how to set up the path in order to calculate the path integral in this case. Can someone help me out in this? Is there a better and easier way to derive this result? If so, it would be of great help if someone could show me the way.
Best Answer
The ray propagates perpendicular to the wavefront. The speed of the wavefront is $$v(r) = c/n = \frac{2c}{(1 - kr^2)}$$
In time $dt$, the wave propagates $$dx(r) = v(r)dt$$ and $$dx(r+dr) = (v(r) + \frac{dv}{dr}dr)dt$$ The wavefront tilts. $$d\theta \approx tan(d\theta) = \frac{\frac{dv}{dr}drdt}{dr}$$ or $$\frac{d\theta}{dt} = \frac{dv}{dr}$$ Finally, $$\frac{d\theta}{dx} = \frac{d\theta}{dt} \frac{dt}{dx} = \frac{dv/dr}{v(r)}$$
If you make a small angle assumption that the ray does not descend too far as it traverses the lens, you can use the lens thickness to figure out the tilt at any height.
$$\Delta \theta = \frac{d\theta}{dx} \Delta x$$
I have left evaluating the derivative for you, and the work of showing how well the rays come to a focus.