OpticsReflection and Refraction

Reflection i 0 sin kx t E E

0 sinr kx t E E

Incident WaveReflected Wave

Total Wave

•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 in three dimensions

Waves Going at Angles•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 0 sini i x yk x k y t E E

2 2x yc k k ck

•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 ky value•Otherwise it won’t match at boundary

•kx must be negative• So it is going the other way

0 sinr r x yk x k y t E E

y yk k

2 2x yc k k

2 2 2 2x y x yc k k c k k

x xk k x xk k 2 2 2 2x y x yk k k k

Law of Reflection

•Since the frequency of all waves are the same, the total kfor the incident and reflected wave must be the same.•To match the wave at the boundary, ky must be the same before and after

Mirror

IncidentRefl

ected

ki k r

x

y

i r

ki sini kr sinr

ki = kr

ki sini = kr sinr

sini = sinr

i = r

Geometric Optics and the Ray Approximation

•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•Physical optics will come later

•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!

Mirror

i r

i = r

Measuring the Speed of Light

•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

½ ½

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 83.00 10 m/sc

•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

cv

n

Indices of RefractionAir (STP) 1.0003Water 1.333Ethyl alcohol 1.361Glycerin 1.473Fused Quartz 1.434Glass 1.5 -ishCubic zirconia 2.20Diamond 2.419

Refraction: Snell’s Law•The relationship between the angular frequency and the wave number k changes inside a medium

ck

n

•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, thefrequencies all match•Once again, the y-componentof k must match index n1

2

1 r

1 = r

1 2

1 2

ck ck

n n index n2

x

y

1 1 2 2sin sink k

k1sin1

k2sin2

2 1 1 2n k n k

1 2 1 1 2 1 2 2sin sinn n k n n k

1 1 2 2sin sinn n

cf

n

Snell’s Law

Dispersion•The speed of light in a material can depend on frequency

•Index of refraction n depends on frequency•Confusingly, its dependence is often given asa function of wavelength in vacuum•Called dispersion

•This means that different types of light bendby different amounts in any given material•For most materials, the index of refractionis higher for short wavelength

Red Refracts Rotten

Blue Bends Best

Prisms•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

Rainbows

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

Total Internal Reflection•If sin2 comes out bigger than one, then none of the light is refracted

•It is all reflected•This can only happen if it is going from a high index to low index material•The minimum incident angle where this happens is called the critical angle

1 1 2 2sin sinn n

2

1

n2

n1

1 2sin cn n

2

1

sin c

n

n

Optical Fibers

Protective Jacket

•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 100 Tb/s

Low n glass High n glass

Fermat’s Principle (1)•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’

P

Q

X

Q’

ri

i

i = 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 points

Fermat’s Principle (2)•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

P

Q

d1

x

L – x

d2

s1

s2

1 2

1 2

s st

v v 1 1 2 2n s n s

c c 22 2 2

1 1 2 2

1n x d n L x d

c

0dt

dx

21

2 2 2 21 2

1 n L xn x

c x d L x d

1 1 2 2

1sin sinn n

c

1

2

1 1 2 2sin sinn n

Light always takes the quickest

path

1

2