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Electric field
E
Magnetic field
H
The light wave is comprised
of an electric field and a
magnetic field.
The magnetic field, H is
always perpendicular to the
electric field.
Phase
these two waves are in phase
Phase
1/2 l difference = 180 deg
these two waves are out of phase
A2
A1
A1+A2
Superposition
Add amplitudes for
waves that are in phase
A2
A1-A2
A1
Superposition
Subtract amplitudes for
waves that are out of phase
by 180 deg
Superposition
A2
A1
A1-A2=0
Total destructive interference
A1 = A2 but the waves are out of phase by
180 deg.
Mutual Coherence
Two waves are said to be mutually coherent
when the phase difference between the two
waves does not change over time. (i.e. the crest of
the first wave is always a fixed distance from the
crest of the second wave)
When the phase difference between two waves
varies over time, the waves are said to be
mutually incoherent.
Mutual Coherence
• Coherent sources are generally derived
from the same source. That way, both
waves have the same wavelength and
the same random fluctuations in phase*.
*The wavetrain from any source (including a laser) is not
constant but undergoes random changes in phase
Coherence Length
The distance over which a wave can interfere with itself
*
*
or ….
The average length of a wavetrain
coherence length for..
•laser:
•low-coherent laser:
•sun:
many meters
10 nm
2 mm
Examples
1.What is the intensity of two mutually coherent waves,
one with amplitude 5 and another with amplitude 13 and
a phase difference between the two of
a) 90 degrees?
b) 180 degrees?
2. What is the intensity of two mutually incoherent
waves, one with amplitude 5 and another with amplitude
13?
Consider this example…
If two mutually coherent waves of
amplitude 5 and 10 respectively have a
combined intensity of 135, what is the
phase difference between them?
I coherent = 52 + 102 + 2  5 10  cos  ???  = 135
125 + 100 cos(???) = 135
100 cos(???) = 10
cos(???) = 0.1
??? = 84.26 deg
Interference
Young’s Double Slit
single light source
Young’s Double Slit
I
screen
single light source
Young’s Double Slit Calculation
d
Slit separation = a
d
sin  =

a
y
tan  =  
s
d y

=

a
s
d
l
2 =


y
s
d =
ay
this is the distance
s
2 ay
this is the distance converted to phase
ls
Young’s Double Slit Calculation
I coherent = E1 + E2 
2
= A1 + A2 + 2  A1 A2 cos1 -  2 
2
2
 d

= 2  A + 2  A cos
 2 
 l

 2ay 
2
2
= 2  A + 2  A cos

 ls 
2
2
Substitute in
the expression
for phase
difference
Young’s double slit
• Maxima occur whenever
ml s
y=
, m = 0, 1, 2...
a
y – position on screen
m – counter
l – wavelength
s – distance from aperture to screen
a – slit separation
y
Young’s double
slit
interference
pattern for
monochromatic
light
ml s
y=
, m = 0, 1, 2...
a
m= 3
m= 2
m= 1
m=0
m=-1
m=-2
m=-3
Young’s doubleslit interference
pattern for
white light
Example
• Given an aperture with a 0.1 mm slit
spacing, a wavelength of 500 nm, and a
screen held at a distance of 2 m. What
is the separation between maxima?
• What is the separation for 400 nm light?
Lloyd’s mirror
interference
S
mirror
S’
Fresnel’s double prism
S’1
interference
S
S’2
two thin prisms
Michelson Interferometer
Deformable Mirrors
Michelson Interferometer to Characterize
Actuator Deflection of a MEMS DM.
Applications of
Interference
Retinal Interference Patterns
Potential Acuity Meter
cataract
The laser beams bypass the cataract and generate scatter-free,
high resolution interference fringes on the retina to test retinal
function prior to cataract removal.
Thin Film Interference
What happens to a reflected wave when n2 > n1?
n2
n1
incident wave
reflected wave
Reflected wave is shifted in phase by 180º (1/2 cycle)
Thin Film Interference
What happens to a reflected wave when n2 < n1?
n2
n1
incident wave
reflected wave
n2 < n1
Reflected wave continues with no change in phase
Reflectance of an AR Coating
reflectance (%)
no ARC
with ARC
4
3
2
1
400
550
l
700
Why do ARCs Appear Purplish?
• green reflection is eliminated
• some reddish and bluish reflectance
remains (see graph)
• mixture of red and blue has purplish hue
• reflected color will change with angle
since effective thickness of coating
changes
Thin Film Problem
• What is the reflectance of a glass
(n=1.5) surface with a MgFl2
coating (n=1.38) optimized for 550
nm light for
1. 550 nm light?
2. 400 nm light?
Step 1
• What is the thickness of the coating?
tdest
l 1 550
1
=
=
= 99.64 nm
4n
4 1.38
c
Step 2
• What is the amplitude of reflectance at
the surfaces?
nc - nair 1.38 - 1
r1 =
=
= 0.16
nc + nair 1.38 + 1
ng - nc
1.5 - 1.38
r2 =
=
= 0.0417
ng + nc 1.5 + 1.38
Step 3
• For 550 nm light….
I coherent =  E1 + E2  = A + A2 + 2  A1 A2 cos  p1 - p2 
2
2
1
2
p1 - p2 = 180 since they are out of phase
I coherent = A + A2 - 2  A1 A2 =
2
1
2
Step 4
• For 400 nm light, what is the phase
difference?
2  99.64
waves =
= 0.687 waves
400
1.38
 phase = 0.687  2   = 4.32 radians
Step 5
• For 400 nm light
I coherent =  E1 + E2  = A + A2 + 2  A1 A2 cos  p1 - p2 
2
2
1
2
I coherent = A + A2 - 2  A1 A2 cos  4.32  =
2
1
2
Newton’s Rings
Summary
• If the phase changes are common to
both surfaces (eg ARC), then
tdest
m+ 1  l

2
=
,
tconst
2
n2
m = 0,1, 2...
m l
=
, m = 0,1, 2...
2 n2
Summary
• If the phase changes are not common
to both surfaces (eg soap bubble, or
oil), then
m l
tdest =
tconst
2 n2
, m = 0,1, 2...
m+ 1  l

2
=
,
2
n2
m = 0,1, 2...
Fringes of Equal Thickness Problem
• Two flat microscope slides, 10 cm long,
are touching at one end and are
separated by three microns on the
other. How many dark interference
bands will appear on the slide if you
look at the reflection for 450 nm light?
Diffraction and Resolution
Diffraction
“Any deviation of light rays from a
rectilinear path which cannot be
interpreted as reflection or refraction”
Sommerfeld, ~ 1894
Huygen’s Principle
Huygens' principle applied to both plane and spherical waves. Each point
on the wave front AA can be thought of as a radiator of a spherical wave
that expands out with velocity c, traveling a distance ct after time t. A
secondary wave front BB is formed from the addition of all the wave
amplitudes from the wave front AA.
Fresnel Diffraction
Fraunhofer Diffraction
• Also called far-field diffraction
• Occurs when the screen is held far from
the aperture.
• Occurs at the focal point of a lens
Diffraction and Interference
• diffraction causes light to bend
perpendicular to the direction of the
diffracting edge
• interference due to the size of the
aperture causes the diffracted light to
have peaks and valleys
rectangular aperture
square aperture
???
circular aperture
Airy Disc
Airy Disk
1.22  l
=
a
  angle subtended at the nodal point
l  wavelength of the light
a  pupil diameter

(minutes of arc 500 nm light)
distance from peak to 1st minimum
1.22  l
=
a
  angle subtended at the nodal point
2.5
l  wavelength of the light
2
a  pupil diameter
1.5
1
0.5
0
1
2
3
4
5
pupil diameter (mm)
6
7
8
Point Spread Function vs. Pupil Size
1 mm
2 mm
3 mm
4 mm
Perfect Eye
5 mm
6 mm
7 mm
Unresolved
point sources
Rayleigh
resolution
limit
Resolved
Rayleigh Resolution Limit
At the Rayleigh resolution limit, the two points
are separated by the angle…
1.22  l
 min =
a
 min  angle subtended at the nodal point
l  wavelength of the light
a  pupil diameter
This is the same as the distance between the
max and the first minimum for one Airy disk!!!
(minutes of arc 500 nm light)
minimum angle of resolution
 min
1.22  l
=
a
 min  angle subtended at the nodal point
2.5
l  wavelength of the light
2
a  pupil diameter
1.5
1
0.5
0
1
2
3
4
5
pupil diameter (mm)
6
7
8
Minutes of arc
20/10
5 arcmin
2.5 arcmin
20/20
1 arcmin
convolution
6 mm
20/20 E
3 mm
1 mm
DH
20/20 E
uncorrected
corrected
First light AO image of binary star k-Peg on the 3.5-m telescope at the
Starfire Optical Range
September, 1997.
 min
1.22  l 1.22  900 10-9
=
=
= 0.064 seconds of arc
a
3.5
About 1000 times better than the eye!
Keck telescope: 10 m reflector: about 4500 times better than the eye
Point Spread Function vs. Pupil Size
1 mm
2 mm
3 mm
4 mm
Perfect Eye
Typical Eye
5 mm
6 mm
7 mm
2.5.7: Image quality as a function of pupil size
optical quality
(arb. units)
Best overall quality ~ 2 - 3 mm
0
2
4
pupil size (mm)
6
8
Polarization
Direction of Polarization
vertical
horizontal
diagonal
Any Polarization can be Expressed as a Sum of a
Vertical and a Horizontal Component
A
Asin
(vertical component)
y
diagonal polarization

x
Acos
(horizontal component)
I = A , I x = A cos  , I y = A sin 
2
2
2
2
2
Unpolarized Light
Most light is unpolarized.
•sun
•incandescent lamp
•candlelight
Circular and Elliptical Polarization
=?
E
E
linear polarization
random polarization
E
circular polarization
E
unpolarized light
E
elliptical polarization
Generating Polarized Light
Polarizing Filters
polarized light out
unpolarized light in
Example
• Unpolarized light is incident on a
polaroid filter whose orientation is
vertical (90 degrees). It is followed by a
filter whose orientation is 180 degrees.
If 100 units of intensity are incident on
the pair of filters, how many units of light
will emerge?
Example
• If you add a 3rd filter oriented 45
degrees from the horizontal in between
the two original filters, how much light
emerges?
Polarization by Reflection
Ep
Es
B
Es is the component of the polarization
that is parallel to the reflecting surface.
Ep is the component of polarization that is
perpendicular to Es.
Es
Es
Ep
Polarization by Reflection
Rs is the reflectance of the Es component.
Rp is the reflectance of the Ep component.
20
reflectance (%)
Rs
n=1.5
Rp
14.8 %
At 90º, both Rs
and Rp are 100 %
15
10
5
56.3
0
30
60
angle (deg)
90
Brewster’s angle
 n 
 B = arctan  
n
Polarization by Scattering
Applications of Polarization
• Haidinger’s brushes
• Polarizing sunglasses
– reflections from flat surfaces (roads, water,
snow, carhoods) are horizontally polarized.
– These are suppressed by having glasses
that transmit only the vertically polarized
component of light
• Reducing specular reflections
LCD screen
Calcite
Haidinger’s Brush
Disc Hyperpigmentation
 Glaucoma Suspect
Depolarized
Parallel Polarized
Courtesy of Steve Burns and Ann Elsner,
Schepens Eye Research Institute, Boston, MA
Randomly Polarized
GDX Laser Diagnostic Technologies
linear polarization
linear polarization
strong
elliptical polarization
weak
elliptical polarization
Thick NFL
Thin NFL
GDX Image: AR left eye