# Textbook Problem: Fiber Coupling Spheres

I am reading the book Fundamentals of Photonics. I try to solve problems at the end of chapter 1, but I got stuck on two problems.

1. Tiny glass ball are often used as lenses to couple light into and out of optical fibers. The fiber end is located at a distance $f$ from the sphere. For a sphere of radius $a=1$mm and refractive index $n=1.8$, determine $f$ such that a ray parallel to the optical axis at a distance $y=0.7$mm is focused onto the fiber, as illustrated in the picture.

Could you help me?

• Since y is given, you can determine the incidence angle of the red beam onto the glass surface (hint: arcsin should help). Using Snell law, find the angle of the beam traveling inside the ball, use the property of equilateral triangle to find the beam incidence angle at the second interface. Use Snell law again and find where the beam crosses the axis. – gigacyan Mar 16 '15 at 7:48
• Thank you for your help. Since I am not an English native speaker, I never see the work equilateral triangle before. Now I googled it and understand your comment. Thank again – TBBT Mar 16 '15 at 9:05
• I am also not a native speaker and I now realize that I used the wrong word. I meant the isosceles triangle, the one that has only two sides equal. Sorry if I confused you. – gigacyan Mar 16 '15 at 12:27

You can look for geometric optics: Paraxial Approximation, but the case when angles aren't small (it's strange a little);
In optical engineering we use it for calculations:

It seems that right answer is 0.156 mm