1. Chapter 25
Wave Optics

2. Wave Optics
The field of wave optics studies the properties of light that depend on its wave nature
Originally light was thought to be a particle and that model successfully explained the phenomena discussed in geometric options
Other experiments revealed properties of light that could only be explained with a wave theory
Maxwell’s theory of electromagnetism convinced physicists that light was a wave

3. Wave vs. Geometric Optics
The wavelength of light plays a key role in determining when geometric optics can or cannot be used
When discussing image characteristics over distances much greater than the wavelength, geometric optics is extremely accurate
When dealing with sizes comparable to or smaller than the wavelength, wave optics is required
Examples include interference effects and propagation through small openings
Even more experiments led to the quantum theory of light
Light has properties of both waves and particles

4. Interference One property unique to waves is interference
Interference of sound waves can be produced by two speakers
When the waves are in phase, their maxima occur at the same time at a given point in space
Section 25.1

5. Interference, cont.
The total wave displacement at the listener’s location is the sum of the displacements of the two individual waves
If two waves are in phase, the sum of their displacements is large
The wave interfere constructively
This produces a large amplitude and a large intensity
Section 25.1

6. Interference, final
The maximum of one wave can coincide with the minimum of the other wave
These waves are out of phase
The interference is destructive when the waves are out of phase
If the waves are 180° out of phase, the sum of the displacements of the two waves is zero
Section 25.1

7. Conditions for Interference
Two or more interfering waves travel through different regions of space over at least part of their propagation from source to destination
The waves are brought together at a common point
The waves must have the same frequency and must also have a fixed phase relationship
This means that over a given distance or time interval the phase difference between the waves remains constant
Such waves are called coherent
Section 25.1

8. Coherence
The eye cannot follow variations of every cycle of the wave, so it averages the light intensity
For waves to interfere constructively, they must stay in phase during the time the eye is averaging the intensity
For waves to interfere destructively, they must stay out of phase during the averaging time
Both of these possibilities involve the wave having precisely the same frequency
Section 25.1

9. Coherence, cont.
With slightly different frequencies, the interference changes from constructive to destructive and back
Over a large number of cycles, the waves average no interference
Section 25.1

10. Michelson Interferometer
The Michelson interferometer is based on the interference of reflected waves
Two reflecting mirrors are mounted at right angles
A third mirror is partially reflecting
Called a beam splitter
Section 25.2

11. Michelson Interferometer, cont.
The incident light hits the beam splitter and is divided into two waves
The waves reflect from the mirrors at the top and right and recombine at the beam splitter
The only difference between the two waves is that they travel different distances between their respective mirrors and the beam splitter
The path length difference is ΔL = 2L2 – 2L1
Section 25.2

12. Michelson Interferometer, final
The path length difference is related to the wavelength of the light
If N is an integer, the two waves are in phase and produce constructive interference
If N is a half-integer the waves will produce destructive interference
Section 25.2

13. Interference Conditions
For constructive interference, ΔL = m λ
For destructive interference, ΔL = (m + ½) λ
m is an integer in both cases
If the interference is constructive, the light intensity at the detector is large
Called a bright fringe
If the interference is destructive, the light intensity at the detector is zero
Called a dark fringe
Section 25.2

14. Measuring Length with a Michelson Interferometer
Use the light from a laser and adjust the mirror to give constructive interference
This corresponds to one of the bright fringes
The mirror is then moved, changing the path length
The intensity changes from high to zero and back to high every time the path length changes by one wavelength
If the mirror moves through N bright fringes, the distance d traveled by the mirror is
Section 25.2

15. Measuring Length, cont.
The accuracy of the measurement depends on the accuracy with which the wavelength is known
Many laboratories use helium-neon lasers to make very precise length measurements
A similar type of interference effect is used to read information from CDs and DVDs
Section 25.2

16. LIGO LIGO – Laser Interferometer Gravitational Wave Observatory
Designed to detect very small vibrations associated with gravitational waves that arrive at the Earth from distant galaxies
By using a long distance between the beam splitter and the mirrors, the LIGO interferometers are sensitive to very small percentage changes in that distance
Section 25.2

17. Thin-Film Interference
Assume a thin soap film rests on a flat glass surface
The upper surface of the soap film is similar to the beam splitter in the interferometer
It reflects part of the incoming light and allows the rest to be transmitted into the soap layer after refraction at the air-soap interface
Section 25.3

18. Thin-Film Interference, cont.
The transmitted ray is partially reflected at the bottom surface
The two outgoing rays meet the conditions for interference
They travel through different regions
One travels the extra distance through the soap film
They recombine when they leave the film
They are coherent because they originated from the same source and initial ray
Section 25.3

19. Thin-Film Interference, final
The index of refraction of the film also needs to be accounted for
From the speed of the wave inside the film
The wavelength changes as the light wave travels from a vacuum into the film
The frequency does not change
The number of extra wavelengths is
Section 25.3

20. Frequency of a Wave at an Interface
When a light wave passes from one medium to another, the waves must stay in phase at the interface
The frequency must be the same on both sides of the interface
Section 25.3

21. Phase Change and Reflection
When a light wave reflects from a surface it may be inverted
Inversion corresponds to a phase change of 180°
There is a phase change whenever the index of refraction on the incident side is less than the index of refraction of the opposite side
If the index of refraction is larger on the incident side the reflected ray in not inverted and there is no phase change
Section 25.3

22. Phase Change and Reflection, Diagram
Section 25.3

23. Phase Changes in a Thin Film
The total phase change in a thin film must be accounted for
The phase difference due to the extra distance traveled by the ray
Any phase change due to reflection
For a soap film on glass, nair < nfilm < nglass
There are phase changes for both reflections at the soap film interfaces
The reflections at both the top and bottom surfaces undergo a 180° phase change
Section 25.3

24. Thin-Film Interference, Case 1
Both waves reflected by a thin film undergo a phase change
The number of extra cycles traveled by the ray inside the film completely determines the nature of the interference
If the number of extra cycles, N, is an integer, there is constructive interference
If the number of extra cycles is a half-integer, there is destructive interference
Section 25.3

25. Case 1, cont. Equations are
These equations apply whenever nair < nfilm < n(substance below the film)
Section 25.3

26. Thin-Film Interference, Case 2
Assume the soap bubble is surrounded by air
There is a phase change at the top of the bubble
There is no phase change at the bottom of the bubble
Since only one wave undergoes a phase change, the interference conditions are
Section 25.3

27. Thin-Film Interference: White Light
Each color can interfere constructively, but at different angles
Blue will interfere constructively at a different angle than red
When you look at the soap film the white light illuminates the film over a range of angles
Section 25.3

28. Antireflection Coatings
Nearly any flat piece of glass may act like a partially reflecting mirror
To avoid reductions in intensity due to this reflection, antireflective coatings may be used
The coating makes a lens appear slightly dark in color when viewed in reflected light
Section 25.3

29. Antireflective Coatings, cont.
Many coatings are made from MgF2
nMgF2 = 1.38
There is a 180° phase change at both interfaces
Destructive interference occurs when
Section 25.3

30. Antireflective Coatings, final
The smallest possible value of d that gives destructive interference corresponds to m = 0
MgF2 is a popular material for antireflective coatings because it can be made into very uniform films with small thicknesses
Although an antireflective coating will work best only at one wavelength, it will give partially destructive interference at nearby wavelengths
To function over the entire range of visible wavelengths, the coatings are made using multiple layers that give perfect destructive interference at different wavelengths
Section 25.3

31. Thin-Film Interference
Assume a thin soap film rests on a flat glass surface
The upper surface of the soap film is similar to the beam splitter in the interferometer
It reflects part of the incoming light and allows the rest to be transmitted into the soap layer after refraction at the air-soap interface
Section 25.3

32. Phase Change and Reflection, Diagram
Section 25.3

33. Case 1, cont. Equations are
These equations apply whenever nair < nfilm < n(substance below the film)
Section 25.3

34. Thin-Film Interference, Case 2
Assume the soap bubble is surrounded by air
There is a phase change at the top of the bubble
There is no phase change at the bottom of the bubble
Since only one wave undergoes a phase change, the interference conditions are
Section 25.3

35. Thin-Film Interference
Section 25.3

36. Coherent Interference Intensity
I = utotal x c = ½ εo c Eo2
Electric Fields add linearly
Intensity units W/m2

37. Light Through a Single Slit
Light passes through a slit or opening and then illuminates a screen
As the width of the slit becomes closer to the wavelength of the light, the intensity pattern on the screen and additional maxima become noticeable
Section 25.4

38. Single-Slit Diffraction
Water wave example of single-slit diffraction
All types of waves undergo single-slit diffraction
Water waves have a wavelength easily visible
Diffraction is the bending or spreading of a wave when it passes through an opening
Section 25.4

39. Huygens’ Principle
It is useful to draw the wave fronts and rays for the incident and diffracting waves
Huygen’s Principle can be stated as all points on a wave front can be thought of as new sources of spherical waves
Section 25.4

40. Double-Slit Interference
Light passes through two very narrow slits
When the two slits are both very narrow, each slit acts as a simple point source of new waves
The outgoing waves from each slit are like simple spherical waves
The double slit experiment showed conclusively that light is a wave
Experiment was first carried out by Thomas Young around 1800
Section 25.5

41. Young’s Double-Slit Experiment
Light is incident onto two slits and after passing through them strikes a screen
The incident light shown is a plane wave
This is easy to achieve with a laser
Could also be achieved with a lens
Section 25.5

42. Young’s Experiment, cont.
The experiment satisfies the general requirements for interference
The interfering waves travel through different regions of space as they travel through different slits
The waves come together at a common point on the screen where they interfere
The waves are coherent because they come from the same source
Interference will determine how the intensity of light on the screen varies with position
Section 25.5

43. Young’s Experiment, final
Assume the slits are very narrow
According the Huygen’s principle, each slit acts as a simple source with circular wave fronts as viewed from above
The light intensity on the screen alternates between bright and dark as you move along the screen
These areas correspond to regions of constructive interference and destructive interference
Section 25.5

44. Double Slit Analysis
Determine the path length between each slit and the screen
Assume W is very large
If the slits are separated by a distance d, then the difference in length between the paths of the two rays is
ΔL = d sin θ
Section 25.5

45. Double Slit Analysis, cont.
If ΔL is equal to an integral number of complete wavelengths, then the waves will be in phase when they strike the screen
The interference will be constructive
The light intensity will be large
If ΔL is equal to a half number of complete wavelengths, then the waves will not be in phase when they strike the screen
The interference will be destructive
The light intensity will be zero
Section 25.5

46. Conditions for Interference
For constructive interference, d sin θ = m λ
m = 0, ±1, ±2, …
Will observe a bright fringe
For destructive interference, d sin θ = (m + ½) λ
Will observe a dark fringe
Section 25.5

47. Double-Slit Intensity Pattern
The angle θ varies as you move along the screen
Each bright fringe corresponds to a different value of m
Negative values of m indicate that the path to those points on the screen from the lower slit is shorter than the path from the upper slit
Section 25.5

48. Spacing Between Slits Notation: For m = 1,
d is the distance between the slits
W is the distance between the slits and the screen
h is the separation between the slits
For m = 1,
Since the angle is very small, sin θ ~ θ and θ ~ λ/d
Between m = 0 and m = 1, h = W tan θ
Section 25.5

49. Approximations
Since the wavelength of light is small, the angles involved in the double-slit analysis are also small
For small angles, sin θ ~ θ and tan θ ~ θ
Using the approximations, h = W θ = W λ / d
Section 25.5

50. Interference with Monochromatic Light
The conditions for interference state the interfering waves must have the same frequency
This means they must have the same wavelength
Light with a single frequency is called monochromatic
Consists of one color
Light sources with a variety of wavelengths are generally not useful for double-slit interference experiments
The bright and dark fringes may overlap or the total pattern may be a “washed out” sum of bright and dark regions
No bright or dark fringes will be visible
Section 25.5

51. Single-Slit Interference
Slits may be narrow enough to exhibit diffraction but not so narrow that they can be treated as a single point source of waves
Assume the single slit has a width, w
Light is diffracted as it passes through the slit and then propagates to the screen
Section 25.6

52. Single-Slit Analysis
The key to the calculation of where the fringes occur is Huygen’s principle
All points across the slit act as wave sources
These different waves interfere at the screen
For analysis, divide the slit into two parts
Section 25.6

53. Single-Slit Fringe Locations
If one point in each part of the slit satisfies the conditions for destructive interference, the waves from all similar sets of points will also interfere destructively
Destructive interference will produce a dark fringe
Conditions for destructive interference are
w sin θ = ±m λ
m = 1, 2, 3, …
The negative sign will correspond to a fringe below the center of the screen
Section 25.6

54. Single-Slit Analysis – Bright Fringes
There is no simple formula for the angles at which the bright fringes occur
The intensity on the screen can be calculated by adding up all the Huygens waves
There is a central bright fringe with other bright fringes that are lower in intensity
The central fringe is called the central maximum
The central fringe is about 20 times more intense than the bright fringes on either side
Section 25.6

55. Single-Slit – Central Maximum
The width of the central bright fringe is approximately the angular separation of the first dark fringes on either side
The full angular width of the central bright fringe = 2 λ / w
If the slit is much wider than the wavelength, the light beam essentially passes straight through the slit with almost no effect from diffraction
Section 25.6

56. Double-Slit Interference with Wide Slits
When the slits of a double-slit experiment are not extremely narrow, the single-slit diffraction pattern produced by each sit is combined with the sinusoidal double-slit interference pattern
A full calculation of the intensity pattern is very complicated
Section 25.6

57. Diffraction Grating
An arrangement of many slits is called a diffraction grating
Assumptions
The slits are narrow
Each one produces a single outgoing wave
The screen is very far away
Section 25.7

58. Diffraction Grating, cont.
Since the screen is far away, the rays striking the screen are approximately parallel
All make an angle θ with the horizontal axis
If the slit-to-slit spacing is d, then the path length difference for the rays from two adjacent slits is
ΔL = d sin θ
If ΔL is equal to an integral number of wavelengths, constructive interference occurs
For a bright fringe, ΔL = d sin θ = m λ
m = 0, ±1, ±2, …
Section 25.7

59. Diffraction Grating, final
The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit
The overall intensity pattern depends on the number of slits
The larger the number of slits, the narrower the peaks
Section 25.7

60. Grating and Color Separation
A diffraction grating will produce an intensity pattern on the screen for each color
The different colors will have different angles and different places on the screen
Diffraction gratings are widely used to analyze the colors in a beam of light
Section 25.7

61. Diffraction and CDs
Light reflected from the arcs in a CD acts as sources of Huygens waves
The reflected waves exhibit constructive interference at certain angles
Light reflected from a CD has the colors “separated”
Section 25.7

62. Optical Resolution
For a circular opening of diameter D, the angle between the central bright maximum and the first minimum is
The circular geometry leads to the additional numerical factor of 1.22
Section 25.8

63. Telescope Example Assume you are looking at a star through a telescope
Diffraction at the opening produces a circular diffraction spot
Assume there are actually two stars
The two waves are incoherent and do not interfere
Each source produces its own different pattern
Section 25.8

64. Rayleigh Criterion
If the two sources are sufficiently far apart, they can be seen as two separate diffraction spots (A)
If the sources are too close together, their diffraction spots will overlap so much that they appear as a single spot (C)
Section 25.8

65. Rayleigh Criterion, cont.
Two sources will be resolved as two distinct sources of light if their angular separation is greater than the angular spread of a single diffraction spot
This result is called the Rayleigh criterion
For a circular opening, the Rayleigh criterion for the angular resolution is
Two objects will be resolved when viewed through an opening of diameter D if the light rays from the two objects are separated by an angle at least as large as θmin
Section 25.8

66. Scattering
When the wavelength is larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves
The amplitude of the scattered wave depends on the size of the scattering object compared to the wavelength
Blue light is scattered more than red
Called Rayleigh scattering
Section 25.9

67. Blue Sky
The light we see from the sky is sunlight scattered by the molecules in the atmosphere
The molecules are much smaller than the wavelength of visible light
They scatter blue light more strongly than red
This gives the atmosphere its blue color
Section 25.9

68. Scattering, Sky, and Sun Blue sky Sun near horizon
Although violet scatters more than blue, the sky appears blue
The Sun emits more strongly in blue than violet
Our eyes are more sensitive to blue
The sky appears blue even though the violet light is scattered more
Sun near horizon
There are more molecules to scatter the light
Most of the blue is scattered away, leaving the red
Section 25.9

69. Nature of Light
Interference and diffraction show convincingly that light has wave properties
Certain properties of light can only be explained with a particle theory of light
Color vision is one effect that can be correctly explained by the particle theory
Have strong evidence that light is both a particle and a wave
Called wave-particle duality
Quantum theory tries to reconcile these ideas
Section 25.10

70. Quantum Mechanics?! Single electrons fired through double slit
Interference? With…?
Quantum Computing