Download Reflect/Refract

Optics
Reflection and Refraction
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 in three dimensions
Incident Wave
Reflected Wave
Total Wave
Ei  E0 sin  kx  t 
Er  E0 sin  kx  t 
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 E  E sin  k x  k y  t 
i
0i
x
y
2
2
  ck  c k x  k y
Er  E0 r sin  k x x  k y y   t 
•What will reflected wave look like?
•Assume it is reflected at x = 0
  c k x2  k y2
•It will have the same angular frequency
  
•Otherwise it won’t match in time
k y  k y
•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
c k x2  k y2  c k x2  k y2
kx2  k y2  kx2  k y2
kx  kx
k x  k x
Law of Reflection
•Since the frequency of all waves are the same, the total k ki = kr
for the incident and reflected wave must be the same.
•To match the wave at the boundary, ky must be the same before and after
ki sini = kr sinr
ki sini
sini = sinr
kr sinr
i r
i = r
Mirror
y
x
Geometric Optics and the Ray Approximation
•The wave calculations we have done assume
 i = r
the mirror is infinitely large
•If the wavelength is sufficiently tiny compared
to objects, this might be a good approximation
i r
•For the next week, we will always make
this approximation
Mirror
•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!
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
½
½
•The nodes will be separated by ½
•Then you get c from c = f
•Biggest error comes from
133Cs
measuring the distance
•Since this is the best way to
measure distance, we can use this to define the meter
•Speed of light is now defined as 2.99792458108 m/s
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
c  3.00 108 m/s
•Inside materials, however, the speed of light can be different
•Materials contain atoms, made of nuclei and electrons
Indices of Refraction
•The electric field from EM waves push on the electrons
Air (STP)
1.0003
•The electrons must move in response
Water
1.333
•This generally slows the wave down
c
Ethyl alcohol 1.361
v
•n is called the index of refraction
n
Glycerin
1.473
•The amount of slowdown can depend
Fused Quartz 1.434
on the frequency of the light
Glass
1.5 -ish
Cubic zirconia 2.20
Diamond
2.419
Refraction: Snell’s Law
ck
•The relationship between the angular frequency 

and the wave number k changes inside a medium
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, the
k1sin1
frequencies all match
•Once again, the y-component
1 r
index
n
1
of k must match
ck1 ck2
index n2

n2 k1  n1k2
y
n1
n2
k1 sin 1  k2 sin 2
x
n1n2 k1 sin 1  n2 n1k2 sin 2
n1 sin 1  n2 sin 2
Snell’s Law
2
k2sin2
c
f 
n
1 = r
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 as
a function of wavelength in vacuum
•Called dispersion
•This means that different types of light bend
by different amounts in any given material
•For most materials, the index of refraction
is 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, you
can get a double rainbow
Total Internal Reflection
•If sin 2 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
n1 sin c  n2
sin  c 
n2
n1
n1 sin 1  n2 sin 2
1
n1
2
n2
Optical Fibers
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 100 Tb/s
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’
i = r
P
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
Q
i r
X
i
Q’
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 the
slow glass very much
•Green path is bad too – it minimizes time
in glass, but makes path much longer
•Red path – a compromise – is best
•To minimize, set derivative = 0
P
n1 sin 1  n2 sin 2
1
s1
1 L – x
d1
x
s1 s2
n1s1 n2 s2 1 
2
2
2
t 


 n1 x  d1  n2  L  x   d 22 

v1 v2
c
c
c 


n
L

x
dt 1  n1 x

2
  1  n sin   n sin  
0


1
1
2
2
2
2 
dx c  x 2  d12
c
 L  x   d2 

d2
2
s2
2
Q
Light always
takes the
quickest
path