Behavior of a laser beam in a spherically symmetric medium [closed]

Question

A laser beam propagates through a spherically symmetric medium. The refractive index varies with the distance from the centre of symmetry $r$ according to the equation: $$ \mu=\mu_0\frac{r}{r_0} $$ where $\mu_0=1$ and $r_0=0.3 m$ and $r_0\leq r\leq\infty$ and the beam's trajectory lies in a plane containing the centre of symmetry $C$. At a distance $r_1=1 m$, the beam makes an angle $\phi=30^\circ$ with $\vec{r_1}$(see figure). Find the minimum distance that the beam reaches from $C$

What I first did was apply $\mu(r) \sin(\theta)=\text{constant}$ and calculate it for the initial case of $[r_1,\phi]$ and for the final case, when $\theta=90^\circ$, giving an answer $r_f=2 m$ which seemed physically reasonable provided I didn't reach a point where $\theta$ becomes a critical angle.

However, the answer key reports it as $r_f=\frac{1}{\sqrt{2}} m$. Any other explanations I saw looked like they were forced to fit this answer. And if the inner regions are rarer anyway, how can the ray progress inside at all?

Please help

Answer 1

There are two problems in your solution.

First is that you simply misapplied your equation and multiplied by 2 instead of dividing.

But the second, and rather more significant, is that as the ray propagates inside the curved layered medium, even if the index of refraction is constant, $\theta$, being measured from the local radial direction, also changes. It's easy to see if you draw a chord from the starting point to any other point. In the middle of this chord the incidence angle is zero, so this point would be your solution if $\mu$ were constant and you chose the appropriate ending point.

So you should instead derive the equation that would work in terms of the local incidence angle, rather than using the angle relative to the radial direction at the starting point.

Just for reference I give the final result. Its derivation is left as an exercise for the reader (I do have the derivation, so I can confirm the equation is correct).

$$r\mu(r)\sin(\theta_{\mathrm{local}}(r))=\mathrm{const}.$$

You can easily check that applying it will yield the answer from the keys: $r_{\mathrm f}=m/\sqrt2.$