1. Chapter 25
Wave Optics

2. Wave vs. Geometric Optics
Light is a wave (sometimes)
Has wavelength
And frequency
Travels with speed c
Geometric Optics – good when length scales are much larger than light wavelength
Wave Optics – good when length scales are on the same order as light wavelength

3. Interference Resultant Wave = add the amplitudes of the two waves
Waves interference
in phase – their maxima occur at the same time at a given point in space
out of phase – their maxima do not occur at the same time at a given point in space
Resultant Wave = add the amplitudes of the two waves
Section 25.1

4. Interference Special cases of phase differences
constructive interference = perfectly in phase = 0° phase difference
destructive interference = completely out of phase = 180° phase difference = π phase difference = λ/2 difference
Amplitude of wave
(E or B) fields is what
you are adding
Section 25.1

5. Conditions for Interference
The waves must have the same frequency
The waves must also have a fixed phase relationship
Means that phase difference between the waves remains constant (in time or space)
These waves are called coherent
We will only discuss coherent waves
Section 25.1

6. Coherence, cont.
With slightly different frequencies, the interference changes from constructive to destructive and back
NOT COHERENT
Section 25.1

7. Coherence Example Start with beam and split it into two coherent beams
One beam travels straight
Other beam takes a less travelled path (#yolo)
Recombine them and discuss conditions of
Constructive interference
Destructive interference

8. Intensity Example I = utotal x c = ½ εo c Eo2
Electric Fields add Linearly
Intensity does not add Linearly

9. Thin-Film Interference
Why all these pretty colors?
Section 25.3

10. Thin-Film Interference
Assume a thin soap film rests on a flat glass surface
incoming beam (1) split into two coherent beams
reflects part of the incoming light (2)
allows the rest to be transmitted through (refraction) (3)
reflects again at glass interface and exists soap film (refraction) (3)
Section 25.3

11. Thin-Film Interference
The two outgoing rays recombine (2) and (3) upon exiting film
Whether or not they are in phase / out of phase depends on:
Extra distance travelled by (3)
Phase changes at interfaces (air/soap, soap/glass)

12. Thin-Film Interference
Index of refraction of the film – speed of light in film
Speed of light inside the film
Wavelength changes inside film
The frequency does not change
Section 25.3

13. Thin-Film Interference
Extra distance travelled by (3) = 2d
Ignoring phase changes at interfaces
Constructive condition:
Section 25.3

14. Thin-Film Interference
Extra distance travelled by (3) = 2d
Ignoring phase changes at interfaces
Destructive condition:
Section 25.3

15. Phase Change at Interfaces
When light wave reflects from surface it may feel like changing phase
Why?
cuz…#yolo

16. Phase Change at Interfaces
There is a phase change whenever the reflection is from a larger n (than medium you are in)
No phase change if reflection is from a smaller n (than medium you are in)
The phase change is 180 degrees a.k.a. π radians a.k.a. λ/2 difference

17. Thin-Film Interference
Now we don’t ignore phase changes at interfaces
Beam (2) undergoes phase change of λ/2 at AS interface
Beam (3) undergoes phase change of λ/2 at SG interface
Lets determine condition of constructive interference:
Section 25.3

18. Thin-Film Interference
Lets determine condition of con(des)structive interference for B:
Constructive
Destructive

19. Thin-Film Interference
Lets determine condition of con(des)structive interference for C:
Constructive
Destructive

20. Thin-Film Example
Oil (n=1.51) is spread over puddle of water (n=1.33). In a region where the film looks blue (λ=450 nm) what is the minimum thickness of the film?

21. Light Through (n)-Slits
Light passes through
a slit – single slit (SS)
or two – double slit (DS)
or many – diffraction grating (DG)
If width of the slit ~ wavelength of the light things get weird
Section 25.4

22. Single-Slit Interference
Diffraction is the bending or spreading of a wave when it passes through a narrow opening
Like water, light is a wave
This will happen to light
Section 25.4

23. Huygens’ Principle
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

24. Double-Slit Interference
Light passes through two very narrow slits
Each slit acts as a point source of new waves
The double slit experiment showed that light acts like a wave
Thomas Young – 1800
Rays from each slit travel
different paths
When they meet, the phase diff.
will determine how they interfere
Section 25.5

25. Double Slit Analysis Assume W is very large
Slits are separated by a distance d
Difference in length between the paths of the two rays is
Section 25.5

26. 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 mean θ goes CW
Bright fringe = high light intensity = constructive interference
Dark fringe = zero light intensity = destructive interference
Section 25.5

27. Double-Slit Intensity Pattern
Bright fringe = high light intensity = constructive interference
Dark fringe = zero light intensity = destructive interference
Section 25.5

28. Double-Slit Intensity Pattern
Notation:
d is the distance between the slits
W is the distance between the slits and the screen
h is actual physical distance on the detector
θ is the angle with respect to center
Section 25.5

29. Double-Slit Example
Assume a DS setup with d=0.016 cm, W=2.4 m. The distance between the bright fringes is 0.77cm. What is the wavelength of the light used?
Section 25.5

30. Single-Slit Analysis Similar to DS Use Huygen’s principle
Points across the slit act as wave sources
These different waves interfere at the screen
Condition for destructive interference is:
no simple formula for bright fringes occur
Section 25.6

31. Single-Slit Analysis
Dark fringe = zero light intensity = destructive interference
Section 25.6

32. Diffraction Grating 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
Condition for constructive interference is:
Section 25.7

33. Diffraction Grating
Bright fringe = high light intensity = constructive interference
Section 25.7

34. Diffraction Grating
The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit
Intensity pattern depends on the number of slits
Larger the number of slits, the narrower the peaks
Section 25.7

35. 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

36. DG Example
A DG had 8000 lines uniformly spaced over 3cm. It is illuminated by mercury lamp which produces wavelength of 546 nm. What is the angle at which the third-order maximum appears?

37. 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

38. Telescope Example Look at a star through a telescope
Diffraction at the opening produces a circular diffraction spot
Oh wait… there are actually two stars
The two waves are incoherent -> do not interfere
Each source produces its own different pattern
Section 25.8

39. 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

40. Rayleigh Criterion, cont.
You can tell there are two sources if their angular separation is greater than the angular spread of a single diffraction spot
This 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

41. Rayleigh Scattering
When the wavelength is larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves
The light that scatters most is one with wavelength about same size as object
Blue light is scattered more than red
Called Rayleigh scattering
Section 25.9

42. Why is the sky blu?
Light from sun is scattered by molecules in atmosphere
The molecules are much smaller than the wavelength of visible light
They scatter blue light more than red (smaller wavelength)
This gives the atmosphere its blue color
Wow.
Section 25.9

43. Nature of Light
Interference and diffraction show convincingly that light has wave properties
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