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How can it be shown geometrically that paraxial rays meet at the focus of thin lens? (just like it's done for mirror)

I tried to model this using Snell's law in desmos but the rays do not seem to meet at a point.

Edit: After making changes the paraxial rays close to the principal axis are intersecting at the focus as per Lensmaker's formula (thanks to @AndyChen)

Link

Shub
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Your code has a mistake regarding the refracted ray. Its angle of the slope should be $$\beta=\alpha-\sin^{-1}{\big({\sin{\alpha} \over n}\big)}$$ (although you define $\beta$ as the refracted angle, I find this is the easiest way to fix your code). A minor note about the existence of focus is that only rays which are near the axis are under consideration.

Also, you can try to prove that the focus $f$ is $$f={R \over 2(n-1)}$$ where $R$ is the radius of the curvature of the lens and $n$ is the refractive index. You already know how to implement the Snell's law, and to prove the equation, you just need to take one more step and use the approximation $\theta \sim \sin{\theta} \sim \tan{\theta}$ when $\theta$ is small (this requirement is satisfied if rays are near the axis). You will find the paraxial rays closely meet at a single point, which is the focus of the thin less.

Andy Chen
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  • I made changes as per you said regarding the refracted angle but the graph still doesn't seem correct :( – Shub Oct 09 '22 at 16:36
  • To me, it looks great. Try to make lens thinner and rays closer to the axis. You can also change the refractive index, and you will see the focus $f={R \over 2(n-1)}$. – Andy Chen Oct 09 '22 at 16:42
  • Note that we require thin lens and near-axis rays to make the theory about focus work, if you select rays which are too far from the axis or lenses which are too thick, the approximation no longer holds and we cannot get expected results. – Andy Chen Oct 09 '22 at 16:47
  • Thanks a lot for going through the mess of the equations I wrote and fixing the graph :) This is one of the things that made me so confused. – Shub Oct 13 '22 at 09:23
  • I'm also glad my answer helped, and your code is actually quite clear with each step being well-separated. – Andy Chen Oct 15 '22 at 05:58