Which is easiest to model in optics?
Glass
For light (l>>d_atom), amorphous materials are more
uniform than crystals, because of directional
differences in crystals
Uses of low-symmetry crystals
piezoelectricity (shape change with voltage)
electrical control of index of refraction
nonlinear optics: frequency sum/difference
polarizers
Light in a low-symmetry crystal
With these restoring forces, if we pull on an electron
along (1,1,1) direction: which way does it move?
Index change with E polarization
Resonances and n may
be different depending
on E and k directions
From what you know
about springs and the
resonance positions
shown, the “stiffer”
forces are along the
______ direction
a) x b) y
Light in a low-symmetry crystal
P (response)
  xx
 Px 

 
P



y
o
yx

 
P 

 z
  zx
 Px 
 x
 

P


0
y
o
 

P 
0
 z

nx  1   x
is generally not parallel to E (driven)
 xy  xz   Ex 
 
 yy  yz   E y 
 zy  zz   Ez 
0
y
0
0  Ex 
 
0  E y 

 z 
E
 z 
ny  1   y
Can find crystal
principal directions for
x, y, z so  is diagonal
nz  1   z
Light in a vacuum review
E  0
From the equation above we conclude that in a vaccum
_____:
a) E and B are perpendicular
b) E and k are perpendicular
c) S and E are perpendicular
The following are always true (definition and basic MaxEqn)
S  E
B
o
Bo 
k  Eo

Which two common features are not required?
1. S ‖ k 2. BE 3. Ek
4. Bk
a)
b)
c)
d)
e)
1,2
1,3
2,3
2,4
3,4
Light in a low-symmetry crystal

P
E 
=
o
o
 P  0
can occur in low symmetry crystals
k   o E  P   0
From the equation above we conclude_____:
a)
b)
c)
d)
E and P are parallel
E and P are perpendicular
If P (not ‖) E then E (not ) k
If P  E then E  k
Calcite
Unpolarized light entering a calcite crystal
B direction?
What does Snell’s law mean in low symmetry crystals?
It gives the direction of k, not S!
bucket brigade analogy
People on each yard line move identically (same phase)
to pass buckets to the next yard line, but diagonally.
Phase change (where they are in a cycle) travels to right
like ___; water travels diagonally like _____.
We get k direction from Snell's law…
…but it’s transcendental!
We get k direction from Snell's law…
…but it’s transcendental!
ni sin i  nt ( E , kt ) sin t  nt ( E , t ) sin t
To find k, we need n which depends on directions
k, E
Wave equation
2
2

E

P    P 
2
 E    
 

2
2
0
t
t

u
2
x
u
2
y
2
z
u
1
 2 2  2 2  2
2
2
n
n  nx
n  ny
n  nz
 
 

General case:
k is not along a crystal principal axis:
k  k x xˆ  k y yˆ  k z zˆ  k  u x xˆ  u y yˆ  u z zˆ   kuˆ
To find n for each k direction we must solve:

2
x
u y2
uz2
1
 2 2  2 2  2
2
2
n
n  nx
n  ny
n  nz
u
 
 

We get two solutions for n, for two perp. directions (polarizations) of E!
Special case:
k is along a crystal principal axis. Then the equation above is not
valid.
Example: k is along x axis Then we simply use n y , nz for the two
polarizations possible with that k. And Snell’s law is simple.
Suppose for a crystal that  x  8,  y  3,  z  2 . In
this crystal, light is propagating in a direction halfway
between the x and y axes. Write the equation needed
to find the two n’s.
a) I got it right
b) I tried

2
x
u y2
uz2
1
 2 2  2 2  2
2
2
n
n  nx
n  ny
n  nz
u
 
 

 x  8,  y  3,  z  2
If E is in the (1,1,0) direction, then P is in the ____ direction
a)
b)
c)
d)
e)
(0,0,1)
(8,3,0)
(3,8,0)
(3,8,2)
(8,3,2)
The crystal x and y are the dashed axes, and
 x  8,  y  3,  z  2
From the values above and
the shape of the wavelets
(grow at different speeds in
the different directions), we
can tell the line labeled
“optic” is the ____ (also
called the “slow”) axis.
a) x
b) y
(Huygen’s view of how elliptical wavelets give k different
from S)