Schlieren Photography
Knife edge or filters makes
image sensitive to
n
x
x is  to edge.
n
x
Darker regions deflect
some light onto the blade.
Brighter regions deflect
light that would normally
hit the blade, past it.
Alcohol
drop into
water
Hydrogen gas
from nozzle
into air
Vortex
from blade
5 joule
spark
under
water
Ice cube
floating
in water
Optical rays vs. waves
Rays are description of light energy flow (Poyting
vector S), ignoring wave interference.
Applies best when l is _____ (k is ___)
Example: Snell’s law can predict “wave guiding”
based just on n at interfaces.
What if there is no interface? n varies continuously
here:
What continuous equation do rays obey?
Assume
1) n can change with position, but not with
direction of k or polarization of E.
2) significant changes in n must take place
over many wavelengths:
dn
dx
n
l
One type of graded index (GRIN) rod has an index
n(r) that increases outward from the center of the
rod
A plane wave enters from the left. Draw surfaces
of constant phase (how the planes and their
spacings morph) inside the rod based on the
speed of the waves, i.e. based on n  r  …what
parts get ahead of others?
a)
b)
I got it mostly right
I tried but got it mostly wrong
Now draw rays, which are perpendicular to the surfaces
of constant phase.
The effect of this rod is to _____ the light beam
a) converge
b) diverge
Surfaces of constant phase
We can write plane waves as
E  r , t   Eo e
i  k ( r )r t 
 Eo e
i  kvac R  r  t 
R(r )  n(r )uˆk  r
 
k vac R r
tells us the phase due to changes in position
 
k vac R r  constant are planes of constant phase.
What constant do we choose to get
surfaces separated by l r ?

What equation do rays obey?
spherical wave in spherically symmetric n(r)
E  r , t   Eo  r  e
i  kvac R  r  t 
R(r ) 
To conserve power of wave:
Eo  r  
n o c
I 
Eo
2
The waves propagate perpendicular to surfaces of
constant phase (contour surfaces)
k vac R  r   constant
2
Surfaces of constant phase
E  r , t   Eo  r  e
i  kvac R  r  t 
Find R(r )  R( z ,  ) for a cylindrical wave
a)
b)
I got it mostly right
I got it mostly wrong, but tried
Find Eo  r   f (  ) for a cylindrical wave
that conserves power
a)
b)
I got it mostly right
I got it mostly wrong, but tried
A few key steps to the eikonal equation
i  kvac R  r  t 
Put E  r , t   Eo  r  e
find equation for R  r  :

 Eo  r  e
2
ikvac R  r 
k
2
vac
into wave equation to
2
(
1


)

E
2 E 
0
2
2
c
t
n (r )Eo  r  e
2
ikvac R  r 
Approximation on l,k: _________
R  r   n  r 
Eikonal equation
0
dn
n
dx l
Eikonal Equation
R  r   n  r  R (phase) changes most in space where n is large
Surfaces of constant phase are
farther apart where
• n is smaller
• the wave moves faster
• l is larger
Rays are paths perpendicular
to contours (surfaces ) of R  r 
Comparing n-gradient effect to
Snell’s law for layered glass
If n is largest at the top of this layered glass structure,
what will the rays do?
a)
b)
turn downward
turn upward
Mirages
Rays leave the mountain top in all directions. On a hot
day, which ray paths can go to her eyes?
Another form of Eikonal equation
d
 nsˆ   n
R  r   n  r  sˆ
ds
s tells us where along a ray path we are.
ŝ gives us the direction of the ray.
If n changes reasonably slowly, the ray will turn
toward the direction of ____________
n!
increasing
Fermat’s principle
Light rays traveling from point A to point B travel by the
path(s) that…
minimize or maximize the time.
Sunset’s inverted mirage
Suppose the sun is actually below the horizon. You could
possibly still see it if n _________ with height.
a) increased b) decreased
Surface heating, inversions
Atmosphere can have complicated function of n(h)
“Green flash” at sunset
Highest frequencies appear last. Can you explain this?
Sunset blue flash, Christmas Eve, Paranal Observatory, Chile
Malibu CA, 24 min after sunset Feb 15 2014