Chapter 25 RAY OPTICS Lecture 12 April 28, 2005

Electromagnetic waves are transverse ^ y

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

Traveling Waves

Traveling Waves

Standing Waves between two flat mirrors

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~0.00001 N/m2

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

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

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

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

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

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

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

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

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

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”.

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

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

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

Index of Refraction v1 v2 Define the index of refraction: A B Then under refraction,

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

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

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.

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

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

Find the critical angle for a ray of light in glass Air: n2 = 1 Glass: n1 = 1.5 --> qc = 42o Fiber Optics clad, n2 < n1 Two Rt Angle Prisms No Loss of Light; use in optical instruments core, n1

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.

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.

Why is the sky blue?

why are sunsets red?

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

longer wavelength less scattered more scattered Iscattered=λ-4

Rainbows 400 Primary 420 Secondary 520 Colors Reversed

Examples

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

24.22

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

24.26

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.

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

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

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

24.21

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