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In short:

I want to know the necessary conditions in order for a ray to 'bend backwards' and complete a full circle.

In detail:

Suppose theres a medium with a radially symmetric index of refraction $n(\vec{r})=n(r)$. A ray is 'launched' in the $\hat{\theta}$ direction. I would like to know what basic requirements of this function $n(r)$ are such that this ray continuously 'bends' to complete a full circle.

My intuition says that $n(r)$ must have a maximum at some radius $r^\ast$ so that the ray always refracts toward the gradient in the index of refraction, i.e. towards the maximum $r^\ast$, i.e. total internal reflection $$\frac{d}{dr}n(r^\ast)=0$$ $$\frac{d^2}{dr^2}n(r^\ast)<0$$

In other words, I'd like to think of this a sort of waveguide whereby the core is the area around $r^\ast$, and the cladding is the rest of the medium.

Intuition aside, I would like to able to reach this conclusion via the Eikonal equation: $$\frac{d}{ds}(n(\vec{r})\hat{e})=\nabla n(\vec{r})$$

My attempt:

The path element: $$ds=\frac{\partial s}{\partial \theta}d\theta + \frac{\partial r}{\partial r}dr $$

Since we want the path to always be in the $\hat{\theta}$ direction: $dr=0$ $$ds=\frac{\partial s}{\partial \theta}d\theta =rd\theta$$ Plugging this into the Eikonal equation, we get for the radial component:

$$\frac{n(r)}{r}\frac{d}{d\theta}(\hat \theta )=\frac{dn(r)}{dr}\hat{r}$$ After evaluating the derivative on the LHS we get:

$$-\frac{n(r)}{r}\hat{r}=\frac{dn(r)}{dr}\hat{r}$$ We can solve the ODE by separation:

$$\frac{dn}{n}=-\frac{dr}{r}$$

Integrating: $$ln(n(r))=-ln(r)+Const$$ And finally:$$n(r)=\frac{Const}{r}$$

But this cannot be correct because this function does not have a maximum! The ray would refract inwards, circling in until it reaches the origin.

My question:

Is my intuition incorrect? Can I not think of this as a type of waveguide?

OR

Is my intuition correct, and my solution of the Eikonal equation incorrect?

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    $\begingroup$ I request we should not close this question. The OP has made a reasonable attempt, and there are some interesting issues to discuss comparing the ray (Eikonal) answer to this problem and the Maxwell equation solution. In particular the OP's assumption that there is needfully a maximum in the profile is wrong for an interesting reason. $\endgroup$ – Selene Routley Jan 29 '18 at 11:14
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    $\begingroup$ Yes you can think of this in terms of waveguides - see my answer here. There is not needfully a maximum in the profile and I suspect your Eikonal answer is correct - I need to check and hopefully I shall get to write an answer before the question gets closed (hopefully it won't be). $\endgroup$ – Selene Routley Jan 29 '18 at 11:24
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The problem which you are going to tackle via an ray optic approach looks to me like the more or less well known "Whispering gallery mode" which you can have in certain (dielectric microwave and optical)structures; you may also consider waveguide type ring resonators or resonating rings where you can have such kind of modes. In case of a standing wave pattern you have a superposition of clockwise and counter clockwise rotating "rays" but there are methods to excite only a wave in a specific desired direction e.g. clockwise only or counterclockwise only

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  • $\begingroup$ Seems rather brief... $\endgroup$ – QuIcKmAtHs Jan 29 '18 at 10:16

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