Let us consider figure 3. A ray is launched from outside (air) at an angle , from the axis of the fiber.
The question is, under what conditions the ray is ultimately guided inside the core due to total internal reflections at the core cladding boundary?
2.
Let the ray makes an angle with the axis of the fiber inside the core, and let the ray make an angle with core
-cladding interface. Let be the angle of refraction in the cladding.
If < critical angle the ray is refracted in cladding. The ray which goes to cladding is lost and is not useful for communication. The ray which is confined to the core is useful for optical communication.
3.
Now as we increase the launching angle , the angle also increases.
, from the axis of the fiber. 
with the axis of the fiber inside the core, and let the ray make an angle 
with core
be the angle of refraction in the cladding. 
< critical angle the ray is refracted in cladding. The ray which goes to cladding is lost and is not useful for communication. The ray which is confined to the core is useful for optical communication. 
, the angle 
also increases. 
