Homework on Reflection, Refraction and Optical Fibers

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Reflection
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Refraction
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Optical Fibers
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Reflection

Expand Your Thinking

Many of these questions have no "right" answer but are inquiry-style questions to encourage deeper thinking about the subject. Some instructors may therefore choose to use them as discussion questions rather than as homework.

1. List 5 examples of situations you have encountered outside of the classroom that exemplify the law of reflection. You do not have to restrict your answers to reflection of light but can include reflection of any object, such as a pool ball banking off the side of a pool table.

2. Two famous British scientists, Isaac Newton and Robert Hooke, debated the nature of light in the 1600s. Newton claimed light was composed of tiny "corpuscles", or particles. Hooke claimed light was a continuous wave. Think about the behavior of light when it is reflected by (a) by a smooth surface, and (b) by a rough surface. In each case, is the behavior of light more like particles, more like a wave, or explained equally well by either theory? Justify your answer.

3. We see objects when light reflected by them reaches our eyes. Do you think this reflection by most objects is total reflection or diffuse reflection? Explain.

4. Laser light is generally not visible as it travels through air. (If you have access to a laser or to a laser pointer, verify this for yourself.) Yet if you shine a laser through chalk dust, the beam is visible. Explain why this occurs.

5. If a laser beam is sent across a classroom, only students in the direct line of the beam would be able to see that the laser is shining. (Do NOT try to verify this - you should NEVER look into a laser!) But if the beam strikes a wall, the entire class will be able to see the spot made by the beam on the wall. Explain why this occurs.

6.

A scientist looking into a mirror sees an image of herself but only down to the hem of her labcoat. A scientist looking into a flat mirror hung perfectly perpendicular to the floor cannot see her feet but can only see down to the hem of her labcoat. Will she be able to see her feet if she backs away from the mirror? What if she moves toward the mirror? A drawing of light rays may help you explain your answer.

7. A stream of tennis balls striking a metal plate will exhibit total reflection, while the same stream of tennis balls reflecting off of an old, cracked sidewalk will exhibit diffuse reflection. What characteristic(s) of a surface distinguish(es) whether tennis balls exhibit diffuse or total reflection when striking that surface?

8. A stream of tennis balls striking a concrete wall will exhibit total reflection while a laser beam of light striking the same wall will be scattered in all directions. What characteristic(s) of an object distinguish(es) whether that object exhibits diffuse or total reflection when striking a given surface?

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9. Light strikes a mirror, making an angle of 25° to the surface. What angle will the reflected light make with the surface?

10. Light strikes a mirror, making an angle of 20° to the surface. What is the angle of reflection?

11. Light leaving a mirror makes an angle of 42° with respect to the normal to the surface. What was the angle of incidence?

Refraction

Expand Your Thinking

Many of these questions have no "right" answer but are inquiry-style questions to encourage deeper thinking about the subject. Some instructors may therefore choose to use them as discussion questions rather than as homework.

12. What are the physical limits on the index of refraction? (i.e., what values of n are physically impossible to achieve?) Explain your answer.

13. According to the theory of relativity, information cannot move between two points any faster than c, the speed of light in vacuum. Yet a shadow can move much faster than c. (a) Explain how a shadow can move faster than c, the speed of the light that causes the shadow. (b) How does the shadow moving faster than c not violate the limit on information transfer, when a shadow could conceivably carry information?

14. Two famous British scientists, Isaac Newton and Robert Hooke, debated the nature of light in the 1600s. Newton claimed light was composed of tiny "corpuscles", or particles. Hooke claimed light was a continuous wave. Think about the behavior of light when it travels from one medium to another. In particular, consider (a) the slowing of light, and (b) the refraction, or bending, of light. In each case, is the behavior of light more like particles, more like a wave, or explained equally well by either theory? Justify your answer.

15. The frequency f of a beam of light is related to the speed v and the wavelength λ of the light as follows: v = λf. How will this frequency change when the light moves from a denser medium to a rarer medium?

16. Two famous British scientists, Isaac Newton and Robert Hooke, debated the nature of light in the 1600s. Newton claimed light was composed of tiny "corpuscles", or particles. Hooke claimed light was a continuous wave. Think about the behavior of light when it approaches total internal reflection. For angles of incidence slightly less than the critical angle, light is partially reflected and partially transmitted. Is the behavior of light in this situation more like particles, more like a wave, or explained equally well by either theory? Justify your answer.

17. You have a glass beaker full of an unknown liquid. How might you determine the liquid's composition using only a laser, a protractor, a ruler, a pencil, and a reference guide containing optical properties of various liquids?

18. Why couldn't you use a square piece of glass to measure the critical angle of the glass? A diagram will help explain your answer.

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You may want to refer to the Table of indices of refraction.

19. Light travels from a medium with n = 1.25 into a medium of n = 1.34, at an angle of 27° from the normal to the interface of the two media. (a) Will the speed of the light increase, decrease, or remain the same? (b) Will the wavelength of the light increase, decrease, or remain the same? (c) Will the light bend toward the normal, away from the normal, or not at all?

20. Light travels from a medium with n = 1.63 into a medium of n = 1.42, along the normal to the interface of the two media. (a) Will the speed of the light increase, decrease, or remain the same? (b) Will the wavelength of the light increase, decrease, or remain the same? (c) Will the light bend toward the normal, away from the normal, or not at all?

21.

Light incident on a crown glass prism makes an angle of 23.0° with the normal to the surface, as shown to the left. The prism is surrounded by air. (a) What is the angle of refraction inside the glass? (b) What is the angle of reflection at the surface?

22. Laser light travels through air and enters a crown glass tube. The path of the laser in air is not visible, but the glass scatters enough light to determine the path of the light in the tube. The angle between the light and the surface of the tube where the light enters is found to be 54° inside the glass. What was the angle between the light and the surface of the tube in the air outside the tube?

23. Light traveling through the crown glass tube of the previous question eventually encounters the side edge of the tube, which is perpendicular to the surface at which the light entered the tube. The angle of incidence on this side edge is 54°. Will the light escape the glass?

24. The crown glass tube of the previous problems is submerged in ethyl alcohol, and the angle at which the light enters the glass is adjusted to maintain a 54° angle of incidence on the side edge of the tube. (a) Will light striking the side of the tube with an angle of incidence of 54° escape? (b) What is the lowest value of n the medium outside the tube could have that would allow light with that 54° incident angle to escape? NOTE: This is a rather contrived experiment, since changing the medium outside of the tube also changes the relationship between the angle at which the light enters the tube and the angle at which it strikes the side edge of the tube. This complexity will be dealt with in the next section on Optical Fibers.

25. What interface between two materials from the Table will result in a critical angle of 62.2°? Indicate which material the light should leave and which it should enter.

26. Cubic zirconia has an index of refraction of 2.15. (a) Without doing any calculations, explain whether the critical angle for a cubic zerconia-air interface will be greater than or less than the critical angle for a diamond-air interface. (Diamond has an index of 2.42.) (b) Calculate the critical angles for the two interfaces and check your answer to (a).

Optical Fibers

Expand Your Thinking

Many of these questions have no "right" answer but are inquiry-style questions to encourage deeper thinking about the subject. Some instructors may therefore choose to use them as discussion questions rather than as homework.

27. Sometimes numerical aperture is expressed as just the sine of the cut-off angle: NA = sin θ_0max. How is this different from the definition given in your on-line reading? When might the two expressions be equivalent? How typical do you think such a condition is?

28. List three advantages of using a cladding other than air when making optical fibers.

29. You are talking about the exciting things you are learning in this course to a friend with a poor science background. He asks, "Why can't you just send light down an evacuated tube with a glass coating? After all, light travels fastest in vacuum, so this setup would seem the best." Write down an explanation of why a glass-lined evacuated tube would not guide light, using arguments and language your friend would understand.

30. When having lunch with a friend, you mention how intriguing optical fibers can be. In particular, you describe the photographs on the on-line reading page "What a Difference a Cladding Makes". Your friend wants to know why submerging the plastic fiber core in water causes light to escape, when the same beam at the same angle was trapped before the core was submerged. How do you explain this phenomenon to your friend?

Demonstrate Your Mastery

You may want to refer to the Table of indices of refraction.

31. What is the numerical aperture of a fiber with a core index of 1.62 and a cladding index of 1.55?

32. A fiber has a numerical aperture of 0.358. (a) What is the cut-off angle when light enters the fiber from air? (b) What is the cut-off angle for light entering the fiber from water?

33. A given fiber has a cut-off angle (when light enters the fiber from air) of 32°. (a) What is the numerical aperture of the fiber? (b) If the core of the fiber has an index of refraction of 1.56, what is the index of the cladding?

34. Measuring the cut-off angle for light entering a particular fiber from air yields a value for numerical aperture of 0.567. What is the numerical aperture of this fiber when light enters the fiber from water?

35. A given fiber has a numerical aperture of 0.652. Will light entering the fiber (from air) at an angle of entry of 63° be trapped in the fiber?

36. Light entering at 37.9 degrees from the normal is trapped in this water-clad fiber. Light entering at an angle of 52.3 degrees from the normal escapes this water-clad fiber. The on-line reading page "What a Difference a Cladding Makes" shows photographs of light being trapped inside a water-clad fiber (redisplayed on the left) and of light escaping the same water-clad fiber (redisplayed on the right). These redisplayed photos also indicate the angle of entry for each case: 37.9° when light is trapped, and 52.3° when light escapes. (Test them with a protractor to confirm!) (a) Based on these given values and the images, what possible values could the cut-off angle for this (water-clad) fiber take? (b) One value in that possible range should be 45°. If the cut-off angle is indeed 45.0°, what is the index of refraction of the plastic fiber core? (The index of the water cladding is 1.33.) (c) Repeat step (b) for the minimum and maximum possible values of the cut-off angle you indicated in part (a). How different are these limits from the value calculated in (b)?

37. Light entering this air-clad fiber nearly perpendicular to the normal is still trapped. The on-line reading page "What a Difference a Cladding Makes" shows photographs of light being trapped inside an air-clad fiber even for angles of entry approaching 90°. One of these photos is redisplayed to the left. (a) Show that no cut-off angle can be defined (and thus all light remains trapped) when an air-clad (n₂ = 1.00) fiber has a core index of 1.60, provided the light enter the plastic from air as well (i.e., that n₀ = 1.00). (b) What is the minimum value of the core index that will trap all light when air is used as a cladding? If you have solved part (c) of the previous problem, comment on the consistency between this minimum core index and the lower bound on your calculated values of n for the plastic.

Put It All Together

38. A step-index fiber has a core index of refraction of n₁ = 1.40. Light of vacuum wavelength 633 nm enters the fiber from air (n₀=1.00) at an angle θ₀ as shown in the picture below.

Light enters the left side of a fiber at an angle of theta_0. n_0 outside the fiber is 1.00, n_1 of the core is 1.40, and the cladding index n_2 is unknown.

a.	What is the speed of light inside the fiber's core? Show your work.
b.	What is the wavelength of light inside the fiber's core? Show your work.
c.	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.
d.	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.
e.	The critical angle for the core-cladding interface is found to be 59.0°. What is the index of refraction of the cladding? Show your work.
f.	In your sketch of the light in the fiber, label the angle that should be compared to the critical angle (at the n₁/n₂ interface) when determining whether the light will remain trapped in the fiber. Call this angle θ₁. What do we call θ₀ when θ₁ equals the critical angle?
g.	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?
h.	What is the numerical aperture of this fiber? Show your work.
i.	Will light entering the fiber at an angle θ₀ = 30° be trapped in the fiber? How do you know this? (Hint: Consider the conditions on θ₀.)
j.	Use Snell's Law to determine the angle of refraction at the entrance to the fiber if the light enters the fiber at an angle of θ₀ = 30°.
k.	Use geometry to determine the angle of incidence at which light strikes the upper edge of the core.
l.	Compare your answer to the previous question to the critical angle for the core-cladding interface. Will light be trapped inside the fiber? Is this consistent with the conclusion you drew earlier?

9.	Light strikes a mirror, making an angle of 25° to the surface. What angle will the reflected light make with the surface?
10.	Light strikes a mirror, making an angle of 20° to the surface. What is the angle of reflection?
11.	Light leaving a mirror makes an angle of 42° with respect to the normal to the surface. What was the angle of incidence?

12.	What are the physical limits on the index of refraction? (i.e., what values of n are physically impossible to achieve?) Explain your answer.
13.	According to the theory of relativity, information cannot move between two points any faster than c, the speed of light in vacuum. Yet a shadow can move much faster than c. (a) Explain how a shadow can move faster than c, the speed of the light that causes the shadow. (b) How does the shadow moving faster than c not violate the limit on information transfer, when a shadow could conceivably carry information?
14.	Two famous British scientists, Isaac Newton and Robert Hooke, debated the nature of light in the 1600s. Newton claimed light was composed of tiny "corpuscles", or particles. Hooke claimed light was a continuous wave. Think about the behavior of light when it travels from one medium to another. In particular, consider (a) the slowing of light, and (b) the refraction, or bending, of light. In each case, is the behavior of light more like particles, more like a wave, or explained equally well by either theory? Justify your answer.
15.	The frequency f of a beam of light is related to the speed v and the wavelength λ of the light as follows: v = λf. How will this frequency change when the light moves from a denser medium to a rarer medium?
16.	Two famous British scientists, Isaac Newton and Robert Hooke, debated the nature of light in the 1600s. Newton claimed light was composed of tiny "corpuscles", or particles. Hooke claimed light was a continuous wave. Think about the behavior of light when it approaches total internal reflection. For angles of incidence slightly less than the critical angle, light is partially reflected and partially transmitted. Is the behavior of light in this situation more like particles, more like a wave, or explained equally well by either theory? Justify your answer.
17.	You have a glass beaker full of an unknown liquid. How might you determine the liquid's composition using only a laser, a protractor, a ruler, a pencil, and a reference guide containing optical properties of various liquids?
18.	Why couldn't you use a square piece of glass to measure the critical angle of the glass? A diagram will help explain your answer.

27.	Sometimes numerical aperture is expressed as just the sine of the cut-off angle: NA = sin θ_0max. How is this different from the definition given in your on-line reading? When might the two expressions be equivalent? How typical do you think such a condition is?
28.	List three advantages of using a cladding other than air when making optical fibers.
29.	You are talking about the exciting things you are learning in this course to a friend with a poor science background. He asks, "Why can't you just send light down an evacuated tube with a glass coating? After all, light travels fastest in vacuum, so this setup would seem the best." Write down an explanation of why a glass-lined evacuated tube would not guide light, using arguments and language your friend would understand.
30.	When having lunch with a friend, you mention how intriguing optical fibers can be. In particular, you describe the photographs on the on-line reading page "What a Difference a Cladding Makes". Your friend wants to know why submerging the plastic fiber core in water causes light to escape, when the same beam at the same angle was trapped before the core was submerged. How do you explain this phenomenon to your friend?

31.	What is the numerical aperture of a fiber with a core index of 1.62 and a cladding index of 1.55?
32.	A fiber has a numerical aperture of 0.358. (a) What is the cut-off angle when light enters the fiber from air? (b) What is the cut-off angle for light entering the fiber from water?
33.	A given fiber has a cut-off angle (when light enters the fiber from air) of 32°. (a) What is the numerical aperture of the fiber? (b) If the core of the fiber has an index of refraction of 1.56, what is the index of the cladding?
34.	Measuring the cut-off angle for light entering a particular fiber from air yields a value for numerical aperture of 0.567. What is the numerical aperture of this fiber when light enters the fiber from water?
35.	A given fiber has a numerical aperture of 0.652. Will light entering the fiber (from air) at an angle of entry of 63° be trapped in the fiber?
36.	The on-line reading page "What a Difference a Cladding Makes" shows photographs of light being trapped inside a water-clad fiber (redisplayed on the left) and of light escaping the same water-clad fiber (redisplayed on the right). These redisplayed photos also indicate the angle of entry for each case: 37.9° when light is trapped, and 52.3° when light escapes. (Test them with a protractor to confirm!) (a) Based on these given values and the images, what possible values could the cut-off angle for this (water-clad) fiber take? (b) One value in that possible range should be 45°. If the cut-off angle is indeed 45.0°, what is the index of refraction of the plastic fiber core? (The index of the water cladding is 1.33.) (c) Repeat step (b) for the minimum and maximum possible values of the cut-off angle you indicated in part (a). How different are these limits from the value calculated in (b)?
37.	The on-line reading page "What a Difference a Cladding Makes" shows photographs of light being trapped inside an air-clad fiber even for angles of entry approaching 90°. One of these photos is redisplayed to the left. (a) Show that no cut-off angle can be defined (and thus all light remains trapped) when an air-clad (n₂ = 1.00) fiber has a core index of 1.60, provided the light enter the plastic from air as well (i.e., that n₀ = 1.00). (b) What is the minimum value of the core index that will trap all light when air is used as a cladding? If you have solved part (c) of the previous problem, comment on the consistency between this minimum core index and the lower bound on your calculated values of n for the plastic.