Equations of Fermat's principle/Snell's law in a gradient lens (sugar aquarium)

Question

I'm trying to animate a beam of light inside a substance with gradient index of refraction as a function of y. A good example of that is a syrup inside an aquarium like here https://www.youtube.com/watch?v=BV3aRiL64Ak What equations will i need if i also want to animate the beam released from different angle than pi/2 to the aquarium?

Answer 1

HINT :

In the Figure we see the path of least time of a particle from point $\:\mathrm{A}_{0}\:$ to point $\:\mathrm{A}_{4}\:$ through 4 regions of variable speed, increasing towards positive $\:y$. This would be the light path with decreasing refraction index. Every intermediate path $\:\mathrm{A}_{j}\mathrm{A}_{j+2}\,(j=0,1,2)\:$ is a path of least time between points $\:\mathrm{A}_{j}\:$ and $\:\mathrm{A}_{j+2}\:$. So by Snell's Law⁽¹⁾ \begin{equation} \dfrac{v_{1}}{\sin\theta_1}=\dfrac{v_{2}}{\sin\theta_2}=\dfrac{v_{3}}{\sin\theta_3}=\dfrac{v_{4}}{\sin\theta_4}=\textrm{constant} \tag{01} \end{equation}

Now, if instead of the discrete regions we have a continuum with speed $\:v(y)\:$ being a continuous smooth increasing function of $\:y\:$, then in place of the piece-wise rectilinear path we would have a continuous smooth curve and \begin{equation} \dfrac{v(y)}{\sin[\theta(y)]}=v(y)\sqrt{1+\cot^{2}\theta}=v(y)\sqrt{1+\left(\dfrac{\mathrm{d} y}{\mathrm{d} x}\right)^{2}}=v(y)\sqrt{1+y'^{\,2}}=C_{1}=\textrm{constant} \tag{02} \end{equation}

Now \begin{equation} v(y)=\dfrac{C_{2}}{\textrm{n}(y)}=\dfrac{\textrm{constant}}{\textrm{n}(y)} \tag{03} \end{equation} where $\:\textrm{n}(y)\:$ the variable refraction index.

So, to find the path $\:y(x)\:$ for given function $\:\textrm{n}(y)\:$ we must solve the following differential equation \begin{equation} \dfrac{\sqrt{1+y'^{\,2}}}{\textrm{n}(y)}=\textrm{constant} \tag{04} \end{equation}

In case that $\:v(y)=\sqrt{2g\,y}\:$ equation (04) is expressed as \begin{equation} \sqrt{y\left(1+y'^{\,2}\right)}=\textrm{constant} \tag{05} \end{equation}

This is the equation of brachistochrone.⁽²⁾ That is the brachistochrone is the light path in a region with variable refraction index \begin{equation} \textrm{n}(y)=\dfrac{\textrm{constant}}{\sqrt{y}} \tag{06} \end{equation}

The path is valid inversely from point $\:\mathrm{A}_{4}\:$ to point $\:\mathrm{A}_{0}\:$ as in Figure below. This is the path through a medium with increasing refraction index.

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^{(1) About Snell's Law : Why one should follow Snell's law for shortest time?.}

^{(2) About Brachistochone : What is the position as a function of time for a mass falling down a cycloid curve?.}