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Reflection

Ch. 35

Reflection

- What happens when our wave hits a conductor?
- E-field vanishes in a conductor
- Let’s say the conductor is at x = 0
- Add a reflected wave going other direction
- In reality, all of this is occurring inthree dimensions

Incident WaveReflected WaveTotal Wave

- Up to now, we’ve only considered waves going in the x- or y-direction
- We can easily have waves going at angles as well

- What will reflected wave look like?
- Assume it is reflected at x = 0
- It will have the same angular frequency
- Otherwise it won’t match in time
- It will have the same kyvalue
- Otherwise it won’t match atboundary
- kx must be negative
- So it is going the other way

ki=kr

- Since the frequency of all waves are the same, the total k for the incident and reflected wave must be the same.
- To match the wave at the boundary, kymust be the same before and after

kisini

krsinr

kisini=krsinr

ki

kr

sini=sinr

i=r

Incident

Reflected

i

r

Mirror

y

x

Geometric Optics and the Ray Approximation

i=r

- The wave calculations we have done assumethe mirror is infinitely large
- If the wavelength is sufficiently tiny comparedto objects, this might be a good approximation
- For the next week, we will always makethis approximation
- It’s called geometric optics
- In geometric optics, light waves are represented by rays
- You can think of light as if it is made of little particles
- In fact, waves and particles act very similarly
- First hint of quantum mechanics!

i

r

Mirror

A light ray starts from a wall at an angle of 47 compared to the wall. It then strikes two mirrors at right angles compared to each other. At what angle does it hit the wall again?

A) 43 B) 45 C) 47 D) 49 E) 51

= 47

47

43

47

Mirror

47

43

43

- This works for any angle
- In 3D, you need three mirrors

Mirror

½

½

- Take a source which produces EM waves with a known frequency
- Hyperfine emission from 133Cs atom
- This frequency is extremely stable
- Better than any other method of measuring time
- Defined to be frequency f = 9.19263177 GHz
- Reflect waves off of mirror
- The nodes will be separated by ½
- Then you get c from c = f
- Biggest error comes frommeasuring the distance
- Since this is the best way tomeasure distance, we can use this to define the meter
- Speed of light is now defined as 2.99792458108 m/s

133Cs

The Speed of Light in Materials

- The speed of light in vacuum c is the same for all wavelengths of light, no matter the source or other nature of light

- Inside materials, however, the speed of light can be different
- Materials contain atoms, made of nuclei and electrons
- The electric field from EM waves push on the electrons
- The electrons must move in response
- This generally slows the wave down
- n is called the index of refraction
- The amount of slowdown can dependon the frequency of the light

Indices of Refraction

Air (STP) 1.0003

Water 1.333

Ethyl alcohol 1.361

Glycerin 1.473

Fused Quartz 1.434

Glass 1.5 -ish

Cubic zirconia 2.20

Diamond 2.419

k1sin1

1

r

2

k2sin2

- The relationship between the angular frequency and the wave number k changes inside a medium

- Now imagine light moving from one medium to another
- Some light will be reflected, but usually most is refracted
- The reflected light again must obey the law of reflection
- Once again, the frequencies all match
- Once again, the y-component of k must match

1=r

index n1

index n2

y

x

Snell’s Law

34

2

n4 = 1.33

2

n5 = 1.5

3

3

n6 = 1

4

4

n2 = 1.5

n3 = 2.4

n1 = 1

5

5

A light ray in air enters a region at an angle of 34. After going through a layer of glass, diamond, water, and glass, and back to air, what angle will it be at?

A) 34 B) Less than 34C) More than 34 D) This is too hard

6

- The speed of light in a material can depend on frequency
- Index of refraction n depends on frequency
- Confusingly, its dependence is often given as a function of wavelength in vacuum
- Called dispersion
- This means that different types of lightbend by different amounts in any givenmaterial
- For most materials, the index of refractionis higher for short wavelength

Red Refracts Rotten

Blue Bends Best

- Put a combination of many wavelengths (white light) into a triangular dispersive medium (like glass)

- Prisms are rarely used in research
- Diffraction gratings work better

- Lenses are a lot like prisms
- They focus colors unevenly
- Blurring called chromatic dispersion
- High quality cameras use a combination of lenses to cancel this effect

- A similar phenomenon occurs when light bounces off of the inside of a spherical rain drop
- This causes rainbows
- If it bounces twice, you canget a double rainbow

A trick question:

A light ray in diamond enters an air gap at an angle of 60, then returns to diamond. What angle will it be going at when it leaves out the bottom?

A) 60 B) Less than 60C) More than 60 D) None of the above

n1 = 2.4

60

2

2

n2 = 1

3

n3 = 2.4

- This is impossible!
- Light never makes it into region 2!
- It is totally reflected inside region 1
- This can only happen if you go from a high index to a low
- Critical angle such that this occurs:
- Set sin2 = 1

Protective Jacket

Low n glass

High n glass

- Light enters the high index of refraction glass
- It totally internally reflects – repeatedly
- Power can stay largely undiminished for many kilometers
- Used for many applications
- Especially high-speed communications – up to 40 Gb/s

P

Q

i

X

i

Q’

- Light normally goes in straight lines. Why?
- What’s the quickest path between two points P and Q?
- How about with mirrors? Go from P to Q but touch the mirror.
- How do we make PX + XQ as short as possible?
- Draw point Q’, reflected across from Q
- XQ = XQ’, so PX + XQ = PX + XQ’
- To minimize PX + XQ’, take a straight line from P to Q’

i = r

r

We can get: (1) light moves in straight lines, and (2) the law of reflection if we assume light always takes the quickest path between two poins

1

s1

d1

1

L – x

x

d2

2

2

s2

P

- What about refraction?
- What’s the best path from P to Q?
- Remember, light slows down in glass
- Purple path is bad idea – it doesn’t avoid theslow glass very much
- Green path is bad too – it minimizes timein glass, but makes path much longer
- Red path – a compromise – is best
- To minimize, set derivative = 0

Light always takes the quickest path

Q

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