1. Chapter 24
Wave Optics

2. Wave Optics
The wave nature of light is needed to explain various phenomena such as interference, diffraction, polarization, etc.

3. Interference

4. Interference
Light waves interfere with each other much like mechanical waves do
All interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine
For sustained interference between two sources of light to be observed, there are two conditions which must be met:
1) The sources must be coherent, i.e. they must maintain a constant phase with respect to each other
2) The waves must have identical wavelengths

5. Producing Coherent Sources
Old method: light from a monochromatic source is allowed to pass through a narrow slit
The light from the single slit is allowed to fall on a screen containing two narrow slits; the first slit is needed to insure the light comes from a tiny region of the source which is coherent
Currently, it is much more common to use a laser as a coherent source
The laser produces an intense, coherent, monochromatic beam, which can be used to illuminate multiple slits directly

6. Young’s Double Slit Experiment
Light is incident on a screen with a narrow slit, So
The light waves emerging from this slit arrive at a second screen that contains two narrow, parallel slits, S1 and S2
The narrow slits, S1 and S2 act as sources of waves
The waves emerging from the slits originate from the same wave front and therefore are always in phase
Thomas Young
(1773 – 1829)

7. Young’s Double Slit Experiment
The light from the two slits form a visible pattern on a screen, which consists of a series of bright and dark parallel bands called fringes
Constructive interference occurs where a bright fringe appears
Destructive interference results in a dark fringe
Thomas Young
(1773 – 1829)

8. Interference Patterns
Constructive interference occurs at the center point
The two waves travel the same distance, therefore they arrive in phase
The upper wave has to travel farther than the lower wave
The upper wave travels one wavelength farther
Therefore, the waves arrive in phase and a bright fringe occurs

9. Interference Patterns
The upper wave travels one-half of a wavelength farther than the lower wave
The trough of the bottom wave overlaps the crest of the upper wave
A dark fringe occurs
This is destructive interference

10. Interference Equations
The path difference, δ, is found from the tan triangle: δ = r2 – r1 = d sin θ
This assumes the paths are parallel
Although they are not exactly parallel, but this is a very good approximation since L is much greater than d

11. Interference Equations
For a bright fringe, produced by constructive interference, the path difference must be either zero or some integral multiple of the wavelength:
δ = d sin θbright = m λ; m = 0, ±1, ±2, …
m is called the order number
When m = 0, it is the zeroth order maximum and when m = ±1, it is called the first order maximum, etc.
Within the assumptions L >> d (θ is small) and d >> λ, the positions of the fringes can be measured vertically from the zeroth order maximum
y = L tan θ  L sin θ ; sin θ  y / L

12. Interference Equations
δ = d sin θbright = m λ; m = 0, ±1, ±2, …
sin θbright = m λ / d
sin θ  y / L
y = m λ L / d

13. Interference Equations
When destructive interference occurs, a dark fringe is observed
This needs a path difference of an odd half wavelength
δ = d sin θdark = (m + ½) λ; m = 0, ±1, ±2, …
Thus, for bright fringes
And for dark fringes

14. Uses for Young’s Double Slit Experiment
Young’s Double Slit Experiment gave the wave model of light a great deal of credibility
This experiment provides a method for measuring wavelength of light

15. Chapter 24 Problem 9
Monochromatic light falls on a screen 1.75 m from two slits separated by 2.10 mm. The first- and second-order bright fringes are separated by mm. What is the wavelength of the light?

16. Phase Changes Due To Reflection
An electromagnetic wave undergoes a phase change of 180° upon reflection from a medium of higher index of refraction than the one in which it was traveling (similar to a reflected pulse on a string

17. Phase Changes Due To Reflection
There is no phase change when the wave is reflected from a boundary leading to a medium of lower index of refraction (similar to a pulse in a string reflecting from a free support)

18. Interference in Thin Films

19. Interference in Thin Films
Interference effects are commonly observed in thin films (e.g., soap bubbles, oil on water, etc.)
The interference is due to the interaction of the waves reflected from both surfaces of the film
Recall: the wavelength of light λn in a medium with index of refraction n is λn = λ / n where λ is the wavelength of light in vacuum

20. Interference in Thin Films
Recall: an electromagnetic wave traveling from a medium of index of refraction n1 toward a medium of index of refraction n2 undergoes a 180° phase change on reflection when n2 > n1 and there is no phase change in the reflected wave if n2 < n1
Ray 1 undergoes a phase change of 180° with respect to the incident ray

21. Interference in Thin Films
Ray 2, which is reflected from the lower surface, undergoes no phase change with respect to the incident wave
Ray 2 also travels an additional distance of 2t before the waves recombine
For constructive interference, taking into account the 180° phase change and the difference in optical path length for the two rays:
2 n t = (m + ½) λ; m = 0, 1, 2 …

22. Interference in Thin Films
Ray 2, which is reflected from the lower surface, undergoes no phase change with respect to the incident wave
Ray 2 also travels an additional distance of 2t before the waves recombine
For destructive interference
2 n t = m λ; m = 0, 1, 2 …

23. Interference in Thin Films
Two factors influence thin film interference: possible phase reversals on reflection and differences in travel distance
The conditions are valid if the medium above the top surface is the same as the medium below the bottom surface
If the thin film is between two different media, one of lower index than the film and one of higher index, the conditions for constructive and destructive interference are reversed

24. Interference in Thin Films, Example
An example of different indices of refraction: silicon oxide thin film on silicon wafer
There are two phase changes

25. Newton’s Rings
Another method for viewing interference is to place a planoconvex lens on top of a flat glass surface
The air film between the glass surfaces varies in thickness from zero at the point of contact to some thickness t
A pattern of light and dark rings – Newton’s Rings – is observed
Newton’s Rings can be used to test optical lenses

26. Problem Solving for Thin Films
Identify the thin film causing the interference
Determine the indices of refraction in the film and the media on either side of it
Determine the number of phase reversals: zero, one or two
The interference is constructive if the path difference is an integral multiple of λ and destructive if the path difference is an odd half multiple of λ
The conditions are reversed if one of the waves undergoes a phase change on reflection

27. Problem Solving for Thin Films
Equation
1 phase reversal
0 or 2 phase reversals
2 n t = (m + ½) l
constructive
destructive
2 n t = m l

28. Chapter 24 Problem 24
Two rectangular optically flat plates (n = 1.52) are in contact along one end and are separated along the other end by a 2.00-mm- thick spacer. The top plate is illuminated by monochromatic light of wavelength nm. Calculate the number of dark parallel bands crossing the top plate (including the dark band at zero thickness along the edge of contact between the plates).

29. Diffraction
Huygen’s principle requires that the waves spread out after they pass through slits
This spreading out of light from its initial line of travel is called diffraction
In general, diffraction occurs when waves pass through small openings, around obstacles or by sharp edges

30. Diffraction
A single slit placed between a distant light source and a screen produces a diffraction pattern
It has a broad, intense central band
The central band is flanked by a series of narrower, less intense secondary bands called secondary maxima
The central band will also be flanked by a series of dark bands called minima
This result cannot be explained by geometric optics

31. Fraunhofer Diffraction
Joseph von Fraunhofer
1787 –1826
Fraunhofer Diffraction
Fraunhofer Diffraction occurs when the rays leave the diffracting object in parallel directions
The screen is very far from the slit and the lens is converging
A bright fringe is seen along the axis (θ = 0) with alternating bright and dark fringes on each side

32. Single Slit Diffraction
According to Huygen’s principle, each portion of the slit acts as a source of waves
The light from one portion of the slit can interfere with light from another portion
The resultant intensity on the screen depends on the direction θ
All the waves that originate at the slit are in phase

33. Single Slit Diffraction
Wave 1 travels farther than wave 3 by an amount equal to the path difference (a / 2) sin θ
If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results
In general, destructive interference occurs for a single slit of width a when
sin θdark = mλ / a; m = 1, 2, …

34. Single Slit Diffraction
The general features of the intensity distribution are shown
A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes
The points of constructive interference lie approximately halfway between the dark fringes

35. Chapter 24 Problem 36
A screen is placed 50.0 cm from a single slit, which is illuminated with light of wavelength 680 nm. If the distance between the first and third minima in the diffraction pattern is 3.00 mm, what is the width of the slit?

36. Diffraction Grating
The diffraction grating consists of many equally spaced parallel slits (or grooves)
A typical grating contains several thousand lines per centimeter
The intensity of the pattern on the screen is the result of the combined effects of interference and diffraction

37. Diffraction Grating The condition for maxima is
d sin θbright = m λ; m = 0, 1, 2, …
The integer m is the order number of the diffraction pattern
The zeroth order maximum corresponds to m = 0
The first order maximum corresponds to m = 1

38. Diffraction Grating
Note the sharpness of the principle maxima and the broad range of the dark area in contrast to the broad, bright fringes characteristic of the two-slit interference pattern
If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

39. Polarization of Light
An unpolarized wave: each atom produces a wave with its own orientation of E, so all directions of the electric field vector are equally possible and lie in a plane perpendicular to the direction of propagation
A wave is said to be linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point
Polarization can be obtained from an unpolarized beam by selective absorption, reflection, or scattering

40. Polarization by Selective Absorption
The most common technique for polarizing light
Uses a material that transmits waves whose electric field vectors in the plane are parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction

41. Polarization by Selective Absorption
The intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2 θ, where Io is the intensity of the polarized wave incident on the analyzer
This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other
Étienne-Louis Malus
1775 – 1812

42. Chapter 24 Problem 60
Light of intensity I0 and polarized parallel to the transmission axis of a polarizer, is incident on an analyzer. (a) If the transmission axis of the analyzer makes an angle of 45° with the axis of the polarizer, what is the intensity of the transmitted light? (b) What should the angle between the transmission axes be to make I/I0 = 1/3?

43. Polarization by Reflection
When an unpolarized light beam is reflected from a surface, the reflected light can be completely polarized, partially polarized, or unpolarized
It depends on the angle of incidence
If the angle is 0° or 90°, the reflected beam is unpolarized
For angles between this, there is some degree of polarization
For one particular angle, the beam is completely polarized

44. Polarization by Reflection
The angle of incidence for which the reflected beam is completely polarized is called the polarizing (or Brewster’s) angle, θp
Brewster’s Law relates the polarizing angle to the index of refraction for the material
Sir David Brewster
1781 – 1868

45. Polarization by Scattering
When light is incident on a system of particles, the electrons in the medium can scatter – absorb and reradiate – part of the light (e.g., sunlight reaching an observer on the ground becomes polarized)
The horizontal part of the electric field vector in the incident wave causes the charges to vibrate horizontally
The vertical part of the vector simultaneously causes them to vibrate vertically
Horizontally and vertically polarized waves are emitted

46. Optical Activity
Certain materials display the property of optical activity
A substance is optically active if it rotates the plane of polarization of transmitted light
Optical activity occurs in a material because of an asymmetry in the shape of its constituent materials

47. Answers to Even Numbered Problems
Chapter 24:
Problem 4
2.40 µm

48. Answers to Even Numbered Problems
Chapter 24:
Problem 6
1.52 cm
(b) 2.13 cm

49. Answers to Even Numbered Problems
Chapter 24:
Problem 16
640 nm
(c) 360 nm (m = 1), 600 nm (m = 2)

50. Answers to Even Numbered Problems
Chapter 24:
Problem 20
233 nm

51. Answers to Even Numbered Problems
Chapter 24:
Problem 28
99.6 nm

52. Answers to Even Numbered Problems
Chapter 24:
Problem 52
0.336
0.164

53. Wave Optics
The particle nature of light was the basis for ray (geometric) optics
The wave nature of light is needed to explain various phenomena such as interference, diffraction, polarization, etc.

54. Uses for Young’s Double Slit Experiment
Young’s Double Slit Experiment provides a method for measuring wavelength of the light
This experiment gave the wave model of light a great deal of credibility
It is inconceivable that particles of light could cancel each other