Homework A on Reflection, and Refraction (solutions)

A step-index fiber has a core index of refraction of n₁ = 1.55. Light of vacuum wavelength 633 nm enters the fiber from air (n₀=1) at an angle q₀ as shown in the picture below.

1.	What is the speed of light inside the fiber's core? Show your work.
	n = c / v gives a value for v of 1.935 x 10⁸ m/s.
2.	What is the wavelength of light inside the fiber's core? Show your work.
	n = l₀ / l gives a value for lof 408.4 nm.
3.	In which direction does the light bend as it enters the fiber (toward the top wall, toward the bottom wall, or neither)? Justify your answer. Sketch the approximate path that the light takes before it hits the inner wall (top or bottom) of the fiber, clearly showing the correct bending.
	The light will bend towards the bottom wall. When light enters a denser medium and slows down, it bends towards the normal.
4.	Should the index of refraction of the cladding be greater or less than the index of refraction of the core if the fiber is to guide the light through total internal reflection? Explain.
	The cladding should have a lower index than the core. Snell's Law says n₁ sin q₁ = n₂ sin q₂. Total internal reflection occurs when Snell's Law gives a nonsense (greater than 1) value for sin q₂. This occurs only when n₁ > n₂.
5.	The critical angle for the core-cladding interface is found to be 75.4^o. What is the index of refraction of the cladding? Show your work.
	sin q_c = n₂ / n₁ gives a value for n₂ of 1.5.
6.	Label the angle in your sketch of the light in the fiber that would correspond to the critical angle if appropriate conditions were met. Under what condition(s) will that angle equal the critical angle?
	The light will strike the inner wall at the critical angle when the entering beam is refracted to an angle of 90 degrees minus the critical angle; in this case, q1 = 14.6^o. Applying Snell's Law at the point of entry, sin q₀ = (1.55 / 1) * sin 14.6^o gives a value for q₀ of 23 degrees.
7.	When the light strikes the inner wall of the fiber, some (or all) of it may be reflected. Sketch the path of the reflected light in the fiber. What condition must be met by the reflected light?
	The reflected light will make the same angle* with the surface as the incoming light does.*
8.	What is one thing you would like to know about fiber optics and/or optical communication? Email your response to this question to Dr. DJ (wagnerdj@rpi.edu), and she will provide Dr. Persans a list to help in his preparation of his guest lecture.