1. Chapter 25
RAY OPTICS
Lecture 12 April 28, 2005

2. Electromagnetic waves are transverse^ y

3. Poynting Vector and Intensity
Points in the direction of the wave
the magnitude is the rate of energy transfer per unit area carried by the wave S
Average{ [cos(kx-wt)]2 } = 1/2

4. Traveling Waves

5. Traveling Waves

6. Standing Waves between two flat mirrors

7. Radiation pressure
The theory of relativity: anything moving at the speed of light will carry momentum p=E/c
Light can push stuff!
My total energy output per unit time is constant
My energy output per unit time and area drops as the distance2 R
Near Earth:
P~ N/m2

8. Polarization
When E points in one direction the wave is linearly polarized
Points out of the screen ● E E
There are materials that absorb waves when E points in one direction
There are also other polarizations for which E changes direction
… but not in the other

9. I=I1 cos2 q Polarizers at angles reduce the intensity I=I0 I=I1=I0/2
Selects one polarization
Only the projection onto the transmission axis gets through
I=I1 cos2 q

10. Crossed polarizers transmit approximately what fraction of an electromagnetic wave?0%
25%
50%
75%
100%

11. YOUNG’S INTERFERENCE EXPT.
What is light?
I believe light is a stream of fast moving particles. This explains why and how light reflects and refracts.
Newton
I can also understand how and why light reflects and refracts if I assume it is a wave.
HUYGENS’ PRINCIPLE
Huygens
If light is a wave it can should be able to go around small obstacles…and it does!
YOUNG’S INTERFERENCE EXPT.
My equations predicted that light is a high frequency electromagnetic wave in1865.
T. Young
Maxwell

12. So, is light a wave or a particle?
Einstein
In many cases light behaves like a wave, but sometimes (when quantum effects are important) it behaves like a particle.
So, is light a wave or a particle?
Since it sometimes behaves like one and sometimes like the other it is neither.
Instead of trying to force it into some label convenient to us we should find out its properties.
Fermat

13. The ray approximation
Light behaves as a ray. In uniform media it travels in a straight line
When light propagates its wave nature is hidden if
We never look at distances of the order of l (or smaller)
All obstacles have typical sizes much larger than l
The wave nature of light is not important for d >> l

14. For smaller distances (d ~ l) the wave nature begins to show up
The ray approximation
For smaller distances (d ~ l) the wave nature begins to show up
For d << l the wave nature is central in understanding light’s behavior
When looking at features smaller than l the interference of light waves shows up

15. This is the shortest path
The shortest time principle – FERMAT’S PRINCIPLE
When light behaves as a ray and travels from point A to point B it
follows the path that gets it to B in the shortest time possible
In a uniform medium where the light speed is c1 …
… for constant speed the shortest path takes the least amount of time
In uniform media light rays travel in straight lines
This is the shortest path A
This path is longer B

16. Speed of light in the medium
When light behaves as a ray and travels from point A to point B it
follows the path that gets it to B in the shortest time possible
Let us look at a reflected ray L A B
x is such that it takes the least amount of time to go from A to B h
Speed of light in the medium x
mirror
L - x

17. II. The path of a light ray is reversible.
I. The law of reflection
q1 = q1
II. The path of a light ray is reversible.
III. The path of a light ray in vacuum defines what is meant by “a straight line”.

18. 25.17
The reflecting surfaces of two intersecting flat mirrors are at an angle of θ. If a light ray strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle of β=180º-2

19. Exercise 25.17
This looks like an application to the reflection formula and a bit of geometry
a-q
b =(p-q-a)+(a-q)=p-2q
p-q-a b q a a

20. When light behaves as a ray and travels from point A to point B it
follows the path that gets it to B in the shortest time possible
Look at a ray going form one medium with v1 to another with v2 v1 v2 A B h x
L - x

21. Index of Refraction v1 v2 Define the index of refraction: AB
Then under refraction,

22. Snell’s law of refraction
n2 sinq2 = n1 sin q1 n1 n2

23. longer than equal to shorter than perpendicular to parallel to
Given two slabs of transparent material of equal thickness, Fermat’s principle means that the part of a ray passing through the medium with the higher index of refraction is ______ the part passing through the lower index medium.
longer than
equal to
shorter than
perpendicular to
parallel to

24. 25.13
When the light in the figure passes through the glass block, it is shifted laterally by the distance d. If n = 1.76, find the value of d.

25. Exercise 25.13
This looks like an application to the refraction formula and a bit of geometry q1
q1 -q2 h L q2 d

26. Total internal reflection
When light goes from medium 1 to medium 2 with n1 > n2
Larger than 1
If we increase q1 the right hand side grows
Eventually, when sinq1 = n2/n1 we get sinq2=1
If we increase q1 beyond that the wave in medium 2 disappears
The ray suffers
total internal reflection
θc is when sinθ2=1 i.e. n1/n2 sin θc =1

27. Find the critical angle for a ray of light in glass
Air: n2 = 1 Glass: n1 = > qc = 42o
Fiber Optics
clad, n2 < n1
Two Rt Angle Prisms
No Loss of Light; use in
optical instruments
core, n1

28. Constructive interference creates the new wave front.
Huygen’s principle (1678)
Each point on a wave front is a source of secondary spherical wavelets.
Constructive interference creates the new wave front.

29. Endoscope "Foreign Body" in the Stomach
Swallowed Quarter Here is a quarter which a young
man swallowed and which is lying
in the stomach. These are easily
removed with a wire snare or
device for grasping a coin.

30. Why is the sky blue?

31. why are sunsets red?

32. Dispersion
If n depends on l we get dispersion
blue bent more than red

33. longer wavelength
less scattered
more scattered
Iscattered=λ-4

34. Rainbows
400
Primary
420
Secondary
520
Colors Reversed

35. Examples

36. 24.22
At what distance from a 100W electromagnetic wave point source does Emax=15V/m

37. 24.22

38. 24.26
A possible means of space flight is to place an absorbing sheet into orbit around the Earth and then use the light from the Sun to push this “solar sail.” Suppose a sail of area 6105 m2 and mass 6000kg is placed in orbit facing the Sun.
a) what is the force exerted on the sail?
b) What is the sail’s acceleration?
c) How long does it take the sail to reach the Moon, 3.84108 m away? Ignore all gravitational effects, assume that the acceleration calculated in part b) remains constant, and assume a solar intensity of 1340 W/m2

39. 24.26

40. 24.35
An important news announcement is transmitted by radio waves to people sitting next to their radios, 100 km from the station, and by sound waves to people sitting across the news room, 3M from the newscaster. Who receives the news first? Explain. Take the speed of sound in air to be 343 m/s.

41. Exercise 24.35
the sound and radio waves start at the same time
sound covers a distance d in a time d/v
radio waves cover a distance L in a time L/c
Light wins

42. 24.41
In the figure, suppose that the transmission axes of the left and right polarizing disks are perpendicular to each other.
Also, let the center disk be rotated on a the common axis with angular speed . Show that if unpolarized light is incident on the left disk with an intensity Imax, the intensity of the beam emerging from the right disk is
This means that the intensity of the emerging beam is modulated at a rate four times the rate of rotation of the center disk. Hint: Use the trig. identities
cos2=(1+ cos2)/2 and sin2=(1- cos2)/2

43. Exercise 24.41 q p/2-q p/2-q Imax
c is so large that the polarizers appear frozen to a “bit” of light
... At time t the rotation angle will be q=wt
the intensity is decreased by (cos q)2 q
p/2-q
p/2-q
Imax
Polarization after the 3rd polarizer
Polarization after the 2nd polarizer
Polarization after the 1st polarizer

44. 24.21

45. p=E/c I=I0 cos2 traveling waves ^ dampled SHO driven and damped SHO
positive is from L to S
positive v will use (+), negative v (-)
p=E/c ^
dampled SHO
I=I0 cos2
driven and damped SHO