Transcript Application: Anti-Reflective Coatings

Application: Anti-Reflective Coatings
• Your eyeglasses (possibly) and sophisticated multi-element
optical systems like telephoto lenses (definitely) rely on antireflective coatings to reduce extraneous images, loss of contrast,
and other image degradation due to unwanted reflections.
• Straightforward application of thin-film interference:
 Coating of SiO, MgF2 or other.
Hard, transparent, and easy to
apply in a thin uniform coating
(e.g. by vapor deposition).
 nair < ncoating < nglass – two phase
jumps
 Thickness /4 for best
transmission (destructive
interference for reflection) where
 is the wavelength in the film.
 Explicitly, film=air/n
 Single-layer coating optimized
for one wavelength, typically
~550 nm (yellow/green). Visible
color in white light is whiteyellow/green  purple
How is the energy of an incident wave shared
between reflected and transmitted waves at a
boundary?
• Derivable from Maxwell’s
equations and physical
boundary conditions on the
E and B fields.
x
y
E = cB
Polarized along x.
Propagating toward +z.
Normally incident on
boundary at z = z0
between region 1 (n1)
and region 2 (n2).
z
E x and By must be continuous across the bdy :
E1x  z  z0   E2x  z  z0 
B1y  z  z0   B2y  z  z0 
dBy
dE x
and
must be continuous across the bdy :
dz
dz
dE1x  z  z0  dE2x  z  z0 

dz
dz
dB1y  z  z0  dB2y  z  z0 

dz
dz
Apply these to a harmonic EM
wave crossing this bdy.



Region 1: E1  E0cos k 1 z  t  RE0cos k 1 z  t + 

Region 2 : E2  TE0cos k 1 z  t


(R and T are the reflection and transmission coefficients.)
Applying the bdy conditions to these fields (laborious),...
n -n
R= 2 1
n2 + n1
Square reflection coefficient to get relative reflected intensity :
n - n 
IR
= R2 = 2 1 2
I0
 n2 + n1 
2
Energy conservation demands R 2  T 2 = 1, so...
IT
4n1n2
=T 2 =
2
I0
 n2 + n1 
Analysis is similar, but more
complicated, for non-normal
incidence  Fresnel formulas.
Example:
- Air(n1=1)
Lucite(n2=1.5)
R2=.04
T2=.96
Michelson Interferometer
• Interferometer: sensitive
instrument that can measure
physical parameters that
change the phase of a wave.
–
Path length (distance), refractive index,
motion with respect to wave medium,…
• Michelson’s interferometer
produces interference fringes by
splitting a monochromatic beam,
sending the two parts along
different paths, and recombining
to form an interference pattern.
One path has a movable mirror
(the other has a fixed one).
Precise distance measurements
are made by moving the mirror
and counting the number m of
interference fringes that pass a
reference and determining the
distance as d = m/2.
Demonstration: Michelson
Interferometer
Closer Look
Diffraction
• More of the same! Interference in different
circumstances.
• News flash: Light does not really propagate like
a geometrical ray or simple particle.
• Demonstration: Laser and Diffraction Objects
– In coherent, monochromatic, light, razor blades cast
fuzzy shadows, pinholes produce interference
fringes, and solid spheres appear to have holes….
The “Poisson Spot”
The center of the shadow is the only
point that is equidistant from every
point on the circumference, so there
must be constructive interference.
Diffraction around knife edges, through slits and pinholes,
and in other circumstance were studied extensively by
Fresnel, both experimentally and by Huygens construction.
“Fresnel Diffraction” is the general case, with full detailed
analysis and diffraction patterns that depend on distance
from the obstacle.
“Fraunhofer Diffraction” refers
to the simplified case where the
light source and observation
point are both far from the
obstacle, so that plane waves
can be assumed and rays taken
to be parallel. This is the
approximation we use.
(Essentially the small-angle
approximation we used before.)
Multiple-Slit Interference
P
S1
d
S2
d
S3

Two slits were nice,
how about three?
• Assume coherent, monochromatic
waves. Waves through S1 interfere
constructively with those through S2
when
d sin  = n (n = 0, 1, 2,…)
• Waves from S2 interfere constructively
with those from S2 when
d sin  = n (n = 0, 1, 2,…)
• The maxima overlap, but the minima
are more complicated.
– When S1 and S2 cancel, S3 is left over.
Full cancellation occurs when S1+S2
cancels S3 or S2+S3 cancels S1.
– Full cancellation occurs when there is
a 120 (or 240 or…) phase spacing of
the waves from the sources.
Abbreviated derivation:
E1  E0 cos t -   E2  E0 cos t  E3  E0 cos t +  
d sin  

As
previously,

=





Add the fields :
E1  E3  2E0 cos t  cos 
E1  E2  E3  2E0 cos t  cos   E0 cos t   E0 cos t  2 cos  +1
Condition for total destructive interference is
1
2
2 cos   1  0  cos       120 
2
3
The pattern repeats and the minima occur at path differences of
 2 4
d sin   ,
,
,...
3 3 3
(Every one - third wavelength, whole wavelengths, which are maxima.)
Full intensity analysis
Principal Maxima
Observed intensity pattern
Secondary Maxima
Demonstration: Multiple Slits
We didn’t go through this exercise because we care about 3 slits!
N parallel slits makes a
Diffraction Grating
• Principal maxima:
d sin  = m 
(m = 0, 1, 2,… is the “order”)
• Minima:
d sin  = m /N
(m = 0, 1, 2,… as above,
with m = N, 2N,… are
excluded)
Diffraction Grating – Intensity Distribution
• Intensity (from superposition of
fields):
N2
Imax =
I0
<I> = N I0
• Width of maxima:
Principal Maxima
As N gets
bigger, the
maxima get
sharper
 = /(N d)
(sharper for big N)
• What does a diffraction grating
do with white light?
– Each wavelength gets its own
set of principal maxima at
characteristic angles.
– White light is dispersed into its
component wavelengths – a
spectrum like that from
dispersion, only better.
Secondary
maxima are
negligible for
a real grating
with N of
thousands.
• Anyone have a diffraction
grating?
– A CD/DVD is an inadvertent
reflection grating
– A real diffraction grating is a
plate of optical glass, ruled
with thousands of parallel
grooves, or a plastic cast of a
ruled glass grating. With light
passing through you have a
transmission grating and with
a coating of aluminum you
get a reflection grating.
Demonstration: Projected Spectra with Grating