Ch 27 interference & wave nature of light online

Chapter 27

Interference and the Wave Nature of Light

AP Learning Objectives

Physical optics Interference and diffraction

Students should understand the interference and diffraction of waves, so they can:– Apply the principles of interference to coherent sources in

order to: Describe the conditions under which the waves reaching

an observation point from two or more sources will all interfere constructively, or under which the waves from two sources will interfere destructively.

Determine locations of interference maxima or minima for two sources or determine the frequencies or wavelengths that can lead to constructive or destructive interference at a certain point.

Relate the amplitude produced by two or more sources that interfere constructively to the amplitude and intensity produced by a single source.



Students should understand the interference and diffraction of waves, so they can:– Apply the principles of interference and diffraction to waves that pass through

a single or double slit or through a diffraction grating, so they can: Sketch or identify the intensity pattern that results when monochromatic

waves pass through a single slit and fall on a distant screen, and describe how this pattern will change if the slit width or the wavelength of the waves is changed.

Calculate, for a single-slit pattern, the angles or the positions on a distant screen where the intensity is zero.

Sketch or identify the intensity pattern that results when monochromatic waves pass through a double slit, and identify which features of the pattern result from single-slit diffraction and which from two-slit interference.

Calculate, for a two-slit interference pattern, the angles or the positions on a distant screen at which intensity maxima or minima occur.

Describe or identify the interference pattern formed by a diffraction grating, calculate the location of intensity maxima, and explain qualitatively why a multiple-slit grating is better than a two-slit grating for making accurate determinations of wavelength.



Students should understand the interference and diffraction of waves, so they can:– Apply the principles of interference to light reflected by

thin films, so they can: State under what conditions a phase reversal occurs

when light is reflected from the interface between two media of different indices of refraction.

Determine whether rays of monochromatic light reflected perpendicularly from two such interfaces will interfere constructively or destructively, and thereby account for Newton’s rings and similar phenomena, and explain how glass may be coated to minimize reflection of visible light.

Table Of Contents

1. The Principle of Linear Superposition

2. Young’s Double-Slit Experiment

3. Thin-Film Interference

4. The Michelson Interferometer (AP?)

5. Diffraction

6. Resolving Power (AP?)

7. The Diffraction Grating

8. Compact Discs, Digital Video Discs, and the Use of

Interference (AP?)

9. X-Ray Diffraction (AP?)

Chapter 27: Interference and the Wave Nature of

Light

Section 1:

The Principle of Linear Superposition

When two or more light waves pass through a given point, their electricfields combine according to the principle of superposition.

The waves emitted by the sources start out in phase and arrive at point P in phase, leading to constructive interference.

,3,2,1,0 12 mm


The waves emitted by the sources start out in phase and arrive at point P out of phase, leading to destructive interference.

,3,2,1,0 21

12 mm


If constructive or destructive interference is to continue occurring at a point, the sources of the waves must be coherent sources.

Two sources are coherent if the waves they emit maintain a constantphase relation.


27.1.1. In a shallow pool of water, there are two needle-like dippers that move up and down at the same constant frequency. The water waves move outward from each source as shown in the drawing. The wave crests, represented by solid lines, have an amplitude of 0.4 cm. What is the displacement of the water, relative to the undisturbed water level, at the point labeled P?

a) +0.8 cm

b) +0.4 cm

c) zero cm

d) 0.4 cm

e) 0.8 cm

27.1.2. In a shallow pool of water, there are two needle-like dippers that move up and down at the same constant frequency. The water waves move outward from each source as shown in the drawing. The wave crests, represented by solid lines, have an amplitude of 0.4 cm. What is the displacement of the water, relative to the undisturbed water level, at the point labeled P?

a) +0.8 cm

b) +0.4 cm

c) zero cm

d) 0.4 cm

e) 0.8 cm

27.1.3. Two wave pulses are sent down a stretched out rope. Pulse A is traveling toward the right with an amplitude of +2 mm. Pulse B is traveling toward pulse A from the right to the left with an amplitude of 4 mm. When the two pulses meet and completely overlap, what will be the maximum displacement of the rope relative to its undisurbed position?

a) zero mm

b) + 6 mm

c) + 2 mm

d) 2 mm

e) 4 mm

27.1.4. Complete the following sentence: In order for light to be considered completely coherent,

a) the phase difference of light at any two points must be constant.

b) it must originate from the same source.

c) its intensity at every point must be constant.

d) it must follow the same path.

e) it must be traveling at its vacuum speed.

27.1.5. Why is no interference pattern observed when light from two sources of differing wavelength interfere?

a) The intensities of the two waves will be necessarily different.

b) The light from the two different sources is not likely to be coherent.

c) If the two light sources are close enough to each other, they will produce an interference pattern.


Light

Section 2:

Young’s Double-Slit Experiment

In Young’s experiment, two slits acts as coherent sourcesof light.

Light waves from these slits interfere constructively anddestructively on the screen.

Young’s Double Slit Experiment

The waves coming from the slits interfere constructively ordestructively, depending on the difference in distances betweenthe slits and the screen.


sind

Bright fringes of a double-slit

Dark fringes of a double-slit

,3,2,1,0 sin md

m

,3,2,1,0 sin 21 m

dm


Example 1 Young’s Double-Slit Experiment

Red light (664 nm) is used in Young’s experiment with slits separatedby 0.000120 m. The screen is located a distance 2.75 m from the slits.Find the distance on the screen between the central bright fringe andthe third-order bright fringe.

dm

1sin

tanLy

m101.20

m106643sin

4

91

951.0

951.0tanm 75.2 m 0456.0

Conceptual Example 2 White Light and Young’s Experiment

The figure shows a photograph that illustrates the kind of interferencefringes that can result when white light is used in Young’s experiment.Why does Young’s experiment separate white light into its constituent colors? In any group of colored fringes, such as the two singled out, why is red farther out from the central fringe than green is? Why isthe central fringe white?

27.2.1. You are sitting in a closed room with no windows. The only light in the room originates from two identical bare, incandescent light bulbs. One is located on the wall to your left; and the other is located on the wall to your right. Bored, you look up at the ceiling and realize there is no interference pattern. Why is there no interference pattern?

a) The two light sources are not polarized.

b) The two light sources are not coherent.

c) The two light sources are in phase.

d) The interference pattern is too small to observe with the naked eye.

e) Interference of light is never observed, but the diffraction of light can easily be observed.

27.2.2. In a Young’s double slit experiment, green light is incident of the two slits; and the resulting interference pattern is observed a screen. Which one of the following changes would cause the fringes to be spaced further apart?

a) Move the screen closer to the slits.

b) Move the light source closer to the slits.

c) Increase the distance between the slits.

d) Use orange light instead of green light.

e) Use blue light instead of green light.

27.2.3. What happens to the locations of the maxima for double slit interference when the size of the slits is reduced?

a) Reducing the size of the slits has no effect on the locations of the maxima.

b) The distances between the maxima increase as the widths are reduced.

c) The distances between the maxima decrease as the widths are reduced.

d) Reducing the slit size only increases the number of maxima, but the locations of the initial maxima are not changed.

e) Reducing the slit size only decreases the number of maxima, but the locations of the initial maxima are not changed.

27.2.4. What happens to the width of the maxima for double slit interference when the size of the slits is reduced?

a) Reducing the size of the slits has no effect on the size of the maxima.

b) The widths of the maxima increase as the slit size is reduced.

c) The widths of the maxima decrease as the slit size is reduced.

d) Reducing the slit size only increases the number of maxima, but the widths of the initial maxima are not changed.

e) Reducing the slit size only decreases the number of maxima, but the widths of the initial maxima are not changed.

27.2.5. Without changing the slit width in the double slit experiment, what effect on the interference pattern does reducing the height of the slit have? Assume that the height always remains somewhat larger than the wavelength of light incident on the slits.

a) There is no effect on the pattern.

b) The distances between the maxima will increase.

c) The widths of the maxima will increase.

d) The number of maxima will increase.

e) The height of the maxima will decrease, but there is otherwise no effect.


Light

Section 3:

Thin-Film Interference

Because of reflection and refraction,two light waves enter the eye when lightshines on a thin film of gasoline floating on a thick layer of water.

Because of the extra distance traveled, therecan be interference between the two waves.

nvacuum

film

Thin Film Interference

Phase Change as Boundaries

When light travels through a material with a smaller refractive index towards a material with a larger refractive index,

– reflection at the boundary occurs along with a phase change that is equivalent to one-half of a wavelength in the film.

When light travels from a larger towards a smaller refractive index,

– there is no phase change upon reflection.

Constructive Thin-film Interference

Occurs when:

Therefore,

ceinterferen veconstructi

for Condition

reflection toduechange phasenet

wavelength-Half2 waveby traveled

distance Extra

...,3,2,2

12 filmfilmfilmfilmt

...,3,2,1,0221

mm

tfilm

Destructive Thin-film Interference

Occurs when:

Therefore,

ceinterferen veconstructifor Condition

reflection toduechange phasenet

wavelength-Half2 waveby traveled

distance Extra

...,2

5,

2

3,

2

1

2

12 filmfilmfilmfilmt

...,3,2,1,02

mm

t film

Example 3 A Colored Thin Film of Gasoline

A thin film of gasoline floats on a puddle of water. Sunlight falls perpendicularly on the film and reflects into your eyes. The film hasa yellow hue because destructive interference eliminates the colorof blue (469 nm) from the reflected light. The refractive indices of theblue light in gasoline and water are 1.40 and 1.33. Determine the minimum non-zero thickness of the film.

film25

film23

film21

film21 ,,2 t

2filmm

t

nm 168

2

1.40nm 4691t

Conceptual Example 4 Multicolored Thin Films

Under natural conditions, thin films, like gasoline on water or likethe soap bubble in the figure, have a multicolored appearance that oftenchanges while you are watching them. Why are such films multicoloredand why do they change with time?

Newton’s Rings

If monochromatic light is incident on an accurate spherical surface which is placed on an optically flat plate

– Circular Fringes are created

Quick way to test the quality of a lens for camera manufacturers

27.3.1. A special system is set up in a lab that lets its user select any wavelength between 400 nm and 700 nm with constant intensity. This light is directed at a thin glass film (n = 1.53) with a thickness of 350 nm and that is surrounded by air. As one scans through these possible wavelengths, which wavelength of light reflected from the glass film will appear to be the brightest, if any?

a) 428 nm

b) 535 nm

c) 657 nm

d) 700 nm

e) Since the intensity of the light is constant, all wavelengths of light reflected from the glass will appear to be the same.

21

22

12

22

1

m

nt

m

t

mt

film

film

27.3.2. Blue light ( = 512 nm) is illuminating a thin film of plastic (nP = 1.60) that is on top of a glass sheet (nG = 1.45). Which of the following statements best describes the light that an observer sees coming from the thin film, if it has a uniform thickness of 0.200 m?

a) You would see alternating bright and dark bands.

b) You would see the spectrum of colors.

c) You would see the film as uniformly bright due to constructive interference.

d) You would see the film as uniformly dark due to destructive interference.

e) You cannot see any effects because the film is too thin.

27.3.3. Blue light ( = 512 nm) is illuminating a thin film of plastic (nP = 1.45) that is on top of a glass sheet (nG = 1.53). Which of the following statements best describes the light that an observer sees coming from the thin film, if it has a uniform thickness of 0.265 m?

a) You would see alternating bright and dark bands.

b) You would see the spectrum of colors.

c) You would see the film as uniformly bright due to constructive interference.

d) You would see the film as uniformly dark due to destructive interference.

e) You cannot see any effects because the film is too thin.

27.3.4. Which one of the following choices does not affect interference of light when the thickness of a thin film is much less than the wavelength of light?

a) path length difference

b) phase shifts upon reflection

c) the angle of incidence

27.3.5. Which one of the following statements provides the most convincing evidence that visible light is a form of electromagnetic radiation?

a) Two light sources can be coherent.

b) Light can be reflected from a surface.

c) Light can form a double-slit interference pattern.

d) Light can be diffracted through an aperture.

e) Light travels through vacuum at the same speed as X-rays.


Light

Section 4:

The Michelson Interferometer

(AP?)

A schematic drawing ofa Michelson interferometer.

Michelson Interferometer

27.4.1. The drawing shows the Michelson interferometer with the addition of a closed cylinder, the ends of which are transparent. The sources provides a beam of monochromatic light of wavelength . Initially, the cylinder, which has a length L, is evacuated. This apparatus may be used to determine the index of refraction for the gas in the cylinder. As gas is very slowly allowed to enter the cylinder, a student counts the number of fringes N that pass by a fixed point in the viewing telescope. Which of the following is the correct expression that determines the index of refraction for the gas?

a)

b)

c)

d)

e)

L

Nn

2

12

L

Nn

LN

n 2

2

LNn

2

Nn


Light

Section 5:

Diffraction

Diffraction

Diffraction is the bending of waves around obstacles or the edges of an opening.

Huygens’ principle– Every point on a wave front acts as

a source of tiny wavelets that move forward with the same speed as the wave;

– The wave front at a latter instant is the surface that is tangent to the wavelets.

The extent of the diffraction increases as the ratio of the wavelengthto the width of the opening increases.

Effect of /W on Diffraction

Single Slit Diffraction of Light

This top view shows five sources of Huygens’ wavelets.

Central Bright Band

These drawings show how destructiveinterference leads to the first dark fringeon either side of the central bright fringe.

Destructive Interference (Dark Band)

Dark fringes forsingle slitdiffraction

,3,2,1 sin mW

m

Equation for Dark Fringes (Single Slit)

Other types of “Single Slit” Diffraction

27.5.1. A human hair is placed directly in front of the opening of a laser pointer. The light has a wavelength of 532 nm. On a screen 2.0 m in front of the laser, a diffraction pattern is observed with minima spaced 0.0164 m apart. Determine the approximate thickness of the hair.

a) 40 m

b) 48 m

c) 53 m

d) 65 m

e) 79 m

27.5.2. A laser uniformly illuminates two narrow, identical slits and an interference pattern is observed on a screen. Now, imagine that one of the two slits is completely covered so that no light can pass through it. Which of the following statements best describes what is subsequently observable on the screen, if anything?

a) The width of the maxima and their spacing looks the same as before.

b) The maxima are spaced farther apart, but their width remains the same.

c) The maxima are spaced closer together and their width is smaller.

d) The maxima are spaced farther apart and their width increases.

e) Only a narrow band of light is observed on the screen.

27.5.3. Which of the following must be satisfied if interference is to occur for light passing though a single slit?

a) The light source must be a point source.

b) The light must be traveling with an angle of incidence of 0 toward the slit.

c) The distance from the slit to the observation screen must be greater than the width of the slit.

d) The width of the slit must be comparable to the wavelength of light.

e) The light must be comprised of a single wavelength.

27.5.4. In a dark room, Jennifer is conducting an experiment. The two sides of a slit-like opening are initially 5.0 cm apart. Jennifer shines green laser light on the opening. She then continually brings the two sides closer together, narrowing the slit-like opening. At what point will Jennifer observe an interference (diffraction) pattern on a screen behind the opening?

a) She’ll she interference as soon as the slit width becomes similar to the path difference between the Huygen’s wavelets.

b) She’ll be unable to observe interference unless the waves undergo edge effects at the opening.

c) Once the opening becomes comparable to an integer multiple of the wavelength, she’ll be able to see the interference.

d) She’ll be unable to observe interference by using green light. She should use white light.

27.5.5. Which one of the following statements best explains why the diffraction of sound is more apparent than the diffraction of light under most circumstances?

a) Sound waves are longitudinal, and light waves are transverse.

b) Sound requires a physical medium for propagation.

c) Light waves can be represented by rays while sound waves cannot.

d) The speed of sound in air is six orders of magnitude smaller than that of light.

e) The wavelength of light is considerably smaller than the wavelength of sound.

27.5.6. In a single slit experiment, what effect on the diffraction pattern would result as the slit width is decreased?

a) The width of the central band would increase.

b) The width of the central band would decrease.

c) The width of the central band would not change.

27.5.7. In a single slit experiment, what effect on the first two minima in the diffraction pattern would result as the slit width is decreased?

a) The width of the two minima would increase.

b) The width of the two minima would decrease.

c) The width of the two minima would not change.

27.5.8. In a single slit experiment, what effect on the central minimum in the diffraction pattern would result as the wavelength of the light is decreased?

a) The width of the central maximum would increase.

b) The width of the central maximum would decrease.

c) The width of the central maximum would not change.

27.5.9. Light of wavelength 600 nm is incident upon a single slit with width 4 × 104 m. The figure shows the pattern observed on a screen positioned 2 m from the slits. Determine the distance s.

a) 0.002 m

b) 0.003 m

c) 0.004 m

d) 0.006 m

e) 0.008 m

27.5.10. Light of 600.0 nm is incident upon a single slit. The resulting diffraction pattern is observed on a screen that is 0.50 m from the slit. The distance between the first and third minima of the diffraction pattern is 0.80 mm. Which range of values listed below contains the width of the slit?

a) 0.1 mm to 0.4 mm

b) 0.4 mm to 0.8 mm

c) 0.8 mm to 1.2 mm

d) 1.2 mm to 1.6 mm

e) 1.6 mm to 2.0 mm

27.5.11. Visible light of wavelength 589 nm is incident on a diffraction grating that has 3500 lines/cm. At what angle with respect to the central maximum is the fifth order maximum observed?

a) 17.9

b) 23.8

c) 35.7

d) 71.3

e) A fifth order maximum cannot be observed with this grating.


Light

Section 6:

Resolving Power

(AP?)

Three photographs of an automobile’s headlights, taken atprogressively greater distances.

Resolving Power

First minimum of a circular diffraction pattern

D

22.1sin

diameter of hole

Effect of Diffraction of Resolving Power

Resolving Two Objects

Rayleigh criterion

Two point objects are just resolved when the first dark fringe inthe diffraction pattern of one falls directly on the central bright fringe in the diffraction patter of the other.

D

22.1min

Conceptual Example 8 What You See is Not What You Get

The French postimpressionist artist Georges Seurat developed atechnique of painting in which dots of color are placed close togetheron the canvas. From sufficiently far away the individual dots are notdistinguishable, and the images in the picture take on a more normalappearance.

Why does the camera resolve the dots, while his eyes do not?

27.6.1. A spy satellite is at an altitude 650 km above the Earth’s surface. How large must the satellite’s camera lens be so that its resolution is 25 cm? Assume the average wavelength of light of 480 nm.

a) 1.8 m

b) 2.7 m

c) 0.55 m

d) 1.5 m

e) 0.85 m

27.6.2. A special microscope has been set up that allows the user to view a specimen using light from among the colors listed below. Which of these would you choose to use for the best resolution?

a) yellow

b) red

c) violet

d) blue

e) green

27.6.3. The Hubble Space Telescope in orbit above the Earth has a 2.4 m circular aperture. The telescope has equipment for detecting ultraviolet light. What is the minimum angular separation between two objects that the Hubble Space Telescope can resolve in ultraviolet light of wavelength 95 nm?

a) 4.8 × 108 rad

b) 7.0 × 108 rad

c) 1.9 × 107 rad

d) 1.5 × 107 rad

e) 3.3 × 109 rad

27.6.4. A spy satellite is in orbit at a distance of 1.0 × 106 m above the ground. It carries a telescope that can resolve the two rails of a railroad track that are 1.4 m apart using light of wavelength 600 nm. Which one of the following statements best describes the diameter of the lens in the telescope?

a) It is less than 0.14 m.

b) It is greater than 0.14 m and less than 0.23 m.

c) It is greater than 0.23 m and less than 0.35 m.

d) It is greater than 0.35 m and less than 0.52 m.

e) It is greater than 0.52 m.

27.6.5. The headlights of a car are 1.6 m apart and produce light of wavelength 575 nm in vacuum. The pupil of the eye of the observer has a diameter of 4.0 mm and a refractive index of 1.4. What is the maximum distance from the observer that the two headlights can be distinguished?

a) 8.0 km

b) 9.1 km

c) 11 km

d) 13 km

e) 16 km


Light

Section 7:

The Diffraction Grating

An arrangement consisting of a large number of closely spaced,parallel slits is called a diffraction grating.

Diffraction Grating

The conditions shown here lead to the first- and second-order intensitymaxima in the diffraction pattern.

Explanation of Diffraction

The bright fringes produced bya diffraction grating are much narrower than those produced bya double slit.

Principal maxima of adiffraction grating

,3,2,1,0 sin md

m

distance betweenslits

Grating vs Double Slit

Example 9 Separating Colors With a Diffraction GratingA mixture of violet (410 nm) light and red (660 nm) light falls ontoa grating that contains 1.0x104 lines/cm. For each wavelength,find the angle that locates the first-order maximum.

24m101.0

m10410sinsin

6

91violet1

d

41m101.0

m10660sinsin

6

91red1

d

27.7.1. Two monochromatic beams of light, one red and one blue, are directed at the same spot on a diffraction grating. The resultant diffraction pattern is observed on a screen located a short distance behind the grating. Which of the following best describes the observed pattern?

a) The central maximum appears to be purple. The maxima on each side would alternate, first red, then blue.

b) The central maximum appears to be purple. The maxima on each side would alternate, first blue, then red.

c) The central maximum is red. The maxima on each side would alternate, first blue, then red.

d) The central maximum is blue. The maxima on each side would alternate, first red, then blue.

e) The central maximum is blue. The maxima on each side would alternate, first blue, then red.

27.7.2. Three monochromatic light beams are directed at a diffraction grating. The resulting pattern, shown in grayscale below, is observed on a screen 2 m from the grating. What is the correct order from top to bottom of the three light beams used?

a) green, red, blue

b) red, green, blue

c) blue, green, red

d) green, blue, red

e) blue, red, green

27.7.3. Diffraction occurs when light passes through a single slit. Rank the following three choices in decreasing order, according to the extent of the diffraction that occurs (largest diffraction first): A - blue light, narrow slit, B - red light, narrow slit, C - blue light, wide slit

Note: The blue light referred to in choices A and C is the same wavelength. Also, the narrow slit referred to in choices A and B is the same width.

a) A, B, C

b) B, A, C

c) C, A, B

d) A, C, B

e) B, C, A


Light

Section 8:Compact Discs, Digital Video Discs, and the Use of Interference(AP?)

Storing info in CD/DVD/Blueray

Pit thickness allows for

destructive interference

– Thin film interference

Light either reflects or doesn’t

– No interference or

destructive

Photoreceptor interprets as 1

or 0 for binary code.

Diffraction Gratings use

By using a diffraction grating,

3 beams reflect off the surface

into photoreceptors

Center beam reflects as

previous described.

Outer beams used to make

sure the center beam is

tracking the disc’s information

correctly


Light

Section 9:

X-Ray Diffraction

(AP?)

X-ray Diffraction

Space between atoms/ions in

crystals acts as “slits” of a

diffraction grating

Resulting diffraction pattern

can be used to back calculate

the crystal structure

NaCl

Actual Diffraction Patterns

DNA

Documents

Ch 27 interference & wave nature of light online