Light reflecting through a geometry

[Imagin_e]

I am discussing physics with a friend and we need someone to confirm a thing that we're not agreeing on.

We are discussing incident light that is passing through different geometries, and I want to know how the light behaves when it reflects inside a half sphere (of glass for example). Maybe one of you can recommend a good PDF/book, or can just show me an image of it.

Thanks!

 

[Drakkith]

A ray entering or leaving the half-sphere will refract at an angle according to Snell's law. If the ray is attempting to leave the half-sphere and the angle of incidence is too high, the light will undergo total internal reflection and will not exit, instead acting as if the glass-air boundary is a mirror and reflect at the same angle that it struck the boundary at.

Do you know how to use Snell's law?

 

[Cutter Ketch]

I'll assume what you meant, and you can clarify if I am wrong.

Suppose light is coming from a source which is far away compared to the radius of the sphere, and suppose the flat side of the half sphere faces the source. Light entering the flat surface is normal to that surface and so refraction does not cause the rays to bend at that surface. Inside the glass we still have parallel rays normal to the flat surface traveling toward the curved surface.

At the curved surface near the center of the lens the rays are still close to normal to the surface and pass through the glass-air interface into the air. Going away from the center the angle of incidence increases and refraction bends the rays more and more according to Snell's law. The center of the sphere acts as a lens focusing the rays in the air just past the spherical surface.

For the central portion of a lens made of typical glass the rays will focus at a distance of about twice the radius. Further out from the center the rays bend more and reach the axis closer than the focal point of the central rays. Even further from the center of the lens the angle of incidence exceeds the critical angle of 40-45 degrees in typical glass and instead of refracting through the glass air interface, the rays reflect.

The angle of reflection equals the angle of incidence and the rays near 45 deg of incidence pretty much take a sharp 90 degree turn across the sphere where they again encounter the surface at pretty much the same angle of incidence and reflect again to come out the flat surface traveling back toward the source with varying angles.

Continuing even further from the center, the angles of incidence get steeper and the net angle of the ray upon reflection becomes more and more obtuse. Now the ray will reflect more than twice as it rattles around the perimeter still to eventually emerge out the flat side.

 

[Cutter Ketch]

Ps: without guidance I assumed a configuration where many interesting things happen. Turn the lens around so that the nearly parallel rays are incident on the curved surface first and there is no internal reflection. The hemisphere acts as a pretty good fast lens depending on how much aberration you can tolerate. The outer rays focus much shorter than the inner ones, but much of the lens makes a tolerably good blur spot.