Activity: The Critical Angle and Fiber Optics (Equipment-Based)

In today's activity, you will measure the critical angle and index of refraction for a plastic prism and then use a simulation to investigate the effects of total internal reflection for fiber optics. The readings about these topics are found in the module on Reflection, Refraction, and Fiber Optics.

Equipment needed: Each group needs a laser, a plastic semi-circular prism, a device to vertically spread the laser beam mounted at an appropriate height (such as a cylindrical lens attached to a washer and mounted on a magnetic optical carrier), two pieces of blank paper, and a protractor.

Preparatory Questions:
The following questions should have been answered before the start of the activity. Your instructor may ask you to discuss your answers with group members or as a class before continuing.

Snell's Law states that n₁ sin q₁ = n₂ sin q₂. What does each symbol in the equation represent?

n_i

represents the index of refraction of medium i; q_i represents the angle between the path of light and the normal to the surface in medium i.

Consider the situation when q₂ = 90^o. q₁ is then called the critical angle q_c. For any angles q₁ > q_c, total internal reflection occurs. Which index of refraction, n₁ or n₂ must be larger for this to occur? Explain.

For total internal reflection (TIR) to occur, light must bend away from the normal. Another way to say this is that q₂ > q₁ . Sin q₂ will then be greater than sin q₁, since sine increases with angle for angles less than 90^o. If the angle in medium 2 is larger, Snell's Law can only be satisfied if medium 2 has a smaller index of refraction. Thus n₂ < n₁ if TIR occurs as light moves from medium 1 to medium 2. Total internal reflection can only occur when light moves from a denser medium (higher n) to a rarer medium (lower n).

In the experiment you will perform, the two media will be plastic and air. You will measure the index of refraction for the plastic. Do you expect to see total internal reflection when the light travels into the plastic from the air, when it travels into the air from the plastic, or both? Explain.

In the experiment, air has the smaller index of refraction, so TIR will occur as the light travels from the plastic into the air.

What would the critical angle for the plastic be if its index were 1.54?

n₁ sin q₁ = n₂ sin q₂.
n₁ sin q_c = n₂ sin 90^o = n₂, so
q_c = sin^-¹(n₂/n₁) = sin^-¹(1.00/1.54) = 40.5^o.

If you were using total internal reflection to trap light inside a fiber, would the core (inner layer) index have to be less than or greater than the cladding (outer layer) index of refraction? Explain.

Since TIR occurs when light moves into a rarer medium, the index of the core would have to be greater than the index of the cladding.

Measuring the Critical Angle of a Plastic Prism for a Given Wavelength

LASERS ARE NOT TOYS. DO NOT LOOK INTO THE LASER OR POINT YOUR LASER WHERE IT SHINES IN SOMEONE'S EYES. WHEN YOU ARE NOT USING THE LASER, TURN IT OFF.

Set up the equipment on top of a blank page of paper, as shown to the right. The laser light should shine through a device that spreads the light so you can see the path of the laser on the paper. The light should then strike the curved side of the plastic "prism", travel through the prism, and exit near the middle of the flat side of the prism. Some light will also be reflected off the flat side and exit the curved side.
View from Top (not to scale)
Laser shines through device, then through semicircular prism. Some light is reflected, the rest transmitted. Paper is underneath it all.

1. Rotate the prism, observing the behavior of the light beam exiting the flat face (refracted beam) and the behavior of the beam reflected from the flat face. Start with the flat side perpendicular to (but facing away from) the laser beam, and rotate it counterclockwise, as depicted in the diagram above. Continue rotating it until the flat side is parallel to the laser beam. Describe what happens to the refracted beam and reflected beam as you rotate the prism. How do you know when the critical angle has been reached?
When the flat side of the prism is perpendicular to the beam, all the light escapes. As the prism is rotated, the intensity of the refracted beam gradually decreases, and the intensity of the reflected beam increases. The refracted beam rotates toward the flat side of the prism; its intensity goes to zero as the beam becomes parallel with the prism's flat side. Through all this the angle between reflected and incident rays steadily increases.

2. Continue rotating the prism and determine whether you can achieve total internal reflection when the light enters the prism, when light leaves the prism, both, or neither? When you observe light entering the prism, the flat side should be facing the laser. (Light striking the curved face does not significantly refract - the angle of incidence is generally close to 0)
TIR is only possible when light leaves the prism, as it must travel from a denser medium into a rarer medium. Hopefully, students predicted this result in the preparatory questions.

3. Rotate the prism back until you have achieved the critical angle for the light traveling from the prism into air at the flat face. Use a sharp pencil (pen is ok, but sharp pencil is best) to CAREFULLY mark the following points:

The point at which light enters the front face of the prism (point A)

The point at which the light hits the back face of the prism (point B)

Another point along the back face of the prism (point C)

The point at which light exits the curved face of the prism (point D).

Remove the prism, label the points, and CAREFULLY draw the following lines, using a ruler and making sure the lines are longer than your protractor: Prism with points and lines identified

A line connecting points A and B, representing the incident ray

A line connecting points B and C, representing the back face of the prism (and the refracted ray)

A line connecting points B and D, representing the reflected ray

This sheet represents raw data, so make sure the names of all group members are on it.
The sheet should look like the image to the left, (without the laser light or the outline of the prism included for reference). The drawing should exhibit expected symmetry between incident and reflected rays.

4. Measure the angle between the incident ray and the reflected ray, estimating angles down to the nearest tenth of a degree. Be sure to line up your protractor as CAREFULLY as you can. A small error in measurement can lead to bad results in today's experiment!
Exact values will, of course, depend upon the material you use. Most plastic and glass will give results in the vicinity of the 81 degrees we obtained.

5. Calculate the critical angle from your angle in the previous question. How are they related?
The critical angle will be half of the angle measured above. The measured angle is the sum of the angle of incidence and the angle of reflection, which just equals twice the angle of incidence, according to the law of reflection.

6. How does this critical angle compare to the one you predicted for plastic in the Preparatory Questions?
Hopefully close, if the material used has an index near that given in the preparatory questions. Student values tend to vary between 1.45 and 1.65 when the prism has an index of 1.54.

7. Use your measured value of the critical angle to determine the index of refraction of your prism. Discuss your result in light of any discrepancies found in the previous question.
n₁ = n₂ /sin q_c Errors arise from many sources, including the width of the pencil lead, the accuracy of protractor use, and any carelessness when drawing points and lines. The ideal response will include a discussion of both the amount of error in the measurements and the possibility that the student's prism has an index of refraction other than the one used in the Preparatory Questions.

Applications to Optical Fibers

Go to this simulation of optical fibers by the Universitat Oldenburg, and answer the following questions. A few notes about the simulation:

The simulation is located at the bottom of the web page.
After you change parameters, you must click on the Trace button to see the effect of the new settings.
The angle denoted by i in this simulation (the angle of entry) is denoted by q₀ in the on-line readings for this course.
Assume that the fiber is in air, so n₀ is 1.0 throughout this activity.
Predict the answers to questions 8-9 before checking with the simulation.

8.	Will the angle of entry i affect whether or not the light is totally internally reflected as it enters the fiber? If so, how? If not, why not? Yes. the angle at which light enters the fiber determines its angle of incidence on the fiber wall. If the angle of entry is too big, the angle of incidence on the fiber wall will be less than the critical angle, causing light to escape.
9.	Based on what you know about total internal reflection, what range of values of n₂ (relative to the core index n₁) will keep the light trapped in the fiber? (You may assume the angle of incidence on the edge of the fiber is larger than the critical angle) Try out ns from your range and check. The cladding index n₂ must be less than the core index n₁ if TIR is to occur.
10.	Some of the n₂ values you expected to trap light in the fiber might have let light escape in the simulation due to the angle at which light strikes the edge of the fiber. In these situations would you need to increase or decrease the angle at which light strikes the top edge of the fiber (as measured to the normal) to trap the light? If light escapes, the angle of incidence at the top edge is less than the critical angle, so you need to increase the angle of incidence on the top edge.
11.	Does this desired increase or decrease in the angle at the edge of the fiber correspond to an increase or decrease of the entry angle i at which the light enters the fiber? Use the simulation to verify your answer. To increase the angle of incidence at the top edge, one must decrease the angle of entry i.
12.	Set the indices to n₁=1.54 and n₂ = 1.50, and find the largest value of i that results in total internal reflection by trying different values of i. Compare your result to the cut-off angle given by the simulation. To three decimal places, the maximum angle of entry is 20.4 degrees, in agreement with the cut-off angle calculated by the simulation.
13.	If you were designing a fiber, do you think you would prefer for your light to enter at an angle close to (but still less than) the cut-off angle, or at a much smaller angle close to zero degrees? What potential benefits do you envision this having? The primary advantage of light entering at an entry angle of close to zero is if all the light did so. If all of the light entering a fiber enters in a narrow cone, it travels essentially the same path and does not spread out, or disperse. (Such "modal dispersion" is the topic for another module.) A more basic way to look at it is that light entering along the axis of the fiber takes a more direct route and spends less time bouncing around than light that enters near the cut-off angle. Or, that light entering near the cut-off is more likely to exceed the critical angle at an edge should the fiber bend. Students might argue that light entering along the axis travels a shorter distance and so arrives more quickly. You could point out that such an advantage could be negated at the first bend. Additionally, the time it takes a single information "packet" to travel a fiber is much less important than the delay between information packets, or bits. The delay is dictated by dispersion (spreading), not by the transit time. This is an "extension" question and should probably not be graded for "correctness." It can lead to a discussion of modes and dispersion, it can be discussed in terms of path length traveled by the light, it can serve as a springboard to a discussion of acceptable bend in fibers, or it could be ignored.

1.	Rotate the prism, observing the behavior of the light beam exiting the flat face (refracted beam) and the behavior of the beam reflected from the flat face. Start with the flat side perpendicular to (but facing away from) the laser beam, and rotate it counterclockwise, as depicted in the diagram above. Continue rotating it until the flat side is parallel to the laser beam. Describe what happens to the refracted beam and reflected beam as you rotate the prism. How do you know when the critical angle has been reached? When the flat side of the prism is perpendicular to the beam, all the light escapes. As the prism is rotated, the intensity of the refracted beam gradually decreases, and the intensity of the reflected beam increases. The refracted beam rotates toward the flat side of the prism; its intensity goes to zero as the beam becomes parallel with the prism's flat side. Through all this the angle between reflected and incident rays steadily increases.
2.	Continue rotating the prism and determine whether you can achieve total internal reflection when the light enters the prism, when light leaves the prism, both, or neither? When you observe light entering the prism, the flat side should be facing the laser. (Light striking the curved face does not significantly refract - the angle of incidence is generally close to 0) TIR is only possible when light leaves the prism, as it must travel from a denser medium into a rarer medium. Hopefully, students predicted this result in the preparatory questions.
3.	Rotate the prism back until you have achieved the critical angle for the light traveling from the prism into air at the flat face. Use a sharp pencil (pen is ok, but sharp pencil is best) to CAREFULLY mark the following points: The point at which light enters the front face of the prism (point A) The point at which the light hits the back face of the prism (point B) Another point along the back face of the prism (point C) The point at which light exits the curved face of the prism (point D). Remove the prism, label the points, and CAREFULLY draw the following lines, using a ruler and making sure the lines are longer than your protractor: A line connecting points A and B, representing the incident ray A line connecting points B and C, representing the back face of the prism (and the refracted ray) A line connecting points B and D, representing the reflected ray This sheet represents raw data, so make sure the names of all group members are on it. The sheet should look like the image to the left, (without the laser light or the outline of the prism included for reference). The drawing should exhibit expected symmetry between incident and reflected rays.
4.	Measure the angle between the incident ray and the reflected ray, estimating angles down to the nearest tenth of a degree. Be sure to line up your protractor as CAREFULLY as you can. A small error in measurement can lead to bad results in today's experiment! Exact values will, of course, depend upon the material you use. Most plastic and glass will give results in the vicinity of the 81 degrees we obtained.
5.	Calculate the critical angle from your angle in the previous question. How are they related? The critical angle will be half of the angle measured above. The measured angle is the sum of the angle of incidence and the angle of reflection, which just equals twice the angle of incidence, according to the law of reflection.
6.	How does this critical angle compare to the one you predicted for plastic in the Preparatory Questions? Hopefully close, if the material used has an index near that given in the preparatory questions. Student values tend to vary between 1.45 and 1.65 when the prism has an index of 1.54.
7.	Use your measured value of the critical angle to determine the index of refraction of your prism. Discuss your result in light of any discrepancies found in the previous question. n₁ = n₂ /sin q_c Errors arise from many sources, including the width of the pencil lead, the accuracy of protractor use, and any carelessness when drawing points and lines. The ideal response will include a discussion of both the amount of error in the measurements and the possibility that the student's prism has an index of refraction other than the one used in the Preparatory Questions.