Magnetic fieldH
Electric fieldE
Dire
ctio
n of
pro
paga
tion
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
these two waves are out of phase
difference = 180 deg
Superposition
Add amplitudes for waves that are in phase
Superposition
Subtract amplitudes for waves that are out of phase by 180 deg
Superposition
A1 = A2 but the waves are out of phase by 180 deg.
Total destructive interference
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: many meters•low-coherent laser: 10 nm•sun: 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?
2 25 10 2 5 10 cos ??? 135
125 100cos(???) 135
100cos(???) 10
cos(???) 0.1
??? 84.26deg
coherentI
InterferenceInterference
Young’s Double Slit
single light source
pea
k
peak
peak
peak
peak
valley
valley
valley valle
y
valle
y
valle
y
Young’s Double Slit
single light source
pea
k
peak
peak
peak
peak
valley
valley
valley valle
y
valle
y
valle
y
screen
Young’s Double Slit Calculation
sin
tan
this is the distance
22 this is the distance converted to phase
d
ay
sd y ay
da s s
d ay
s
d
Slit separation = a
y
s
Young’s Double Slit Calculation
s
ayAA
dAA
AAAA
EEIcoherent
2cos22
2cos22
cos2
22
22
21212
22
1
221 Substitute in
the expression for phase difference
Young’s double slit
• Maxima occur whenever
, 0, 1, 2...m s
y ma
y – position on screenm – counter – wavelengths – distance from aperture to screena – slit separation
Young’s double slit
interference pattern for
monochromatic light
y
m=0
m=-1
m=-2
m=-3
m= 3
m= 2
m= 1
, 0, 1, 2...m s
y ma
Young’s double-slit
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
interferenceS
S’mirror
S
S’1
S’2
interference
Fresnel’s double prism
two thin prisms
Michelson Interferometer
Deformable Mirrors
Michelson Interferometer to Characterize Actuator Deflection of a
MEMS DM.
Applications of Applications of InterferenceInterference
Retinal Interference PatternsPotential 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?
n1n2
Reflected wave is shifted in phase by 180º (1/2 cycle)
reflected wave
incident wave
Thin Film Interference
n1n2
n2 < n1
Reflected wave continues with no change in phase
reflected wave
incident wave
What happens to a reflected wave when n2 < n1?
Reflectance of an AR Coating
550400 700
1
2
3
4
refle
ctan
ce (
%)
no ARCwith ARC
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?
5501 1 99.64 nm4 41.38destc
tn
Step 2
• What is the amplitude of reflectance at the surfaces?
1
1.38 10.16
1.38 1c air
c air
n nr
n n
2
1.5 1.380.0417
1.5 1.38g c
g c
n nr
n n
Step 3
2 2 21 2 1 2 1 2 1 2
1 2
2 21 2 1 2
2 cos
180 since they are out of phase
2
coherent
coherent
I E E A A A A p p
p p
I A A A A
• For 550 nm light….
Step 4
• For 400 nm light, what is the phase difference?
2 99.640.687 waves
4001.38
0.687 2 4.32 radians
waves
phase
Step 5
• For 400 nm light
2 2 21 2 1 2 1 2 1 2
2 21 2 1 2
2 cos
2 cos 4.32
coherent
coherent
I E E A A A A p p
I A A A A
Newton’s Rings
Summary
• If the phase changes are common to both surfaces (eg ARC), then
2
, 0,1,2...2const
mt m
n
2
12 , 0,1,2...
2dest
mt m
n
Summary
• If the phase changes are not common to both surfaces (eg soap bubble, or oil), then
2
, 0,1,2...2dest
mt m
n
2
12 , 0,1,2...
2const
mt m
n
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 ResolutionDiffraction 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
???
Airy Disc
circular aperture
Airy Disk
1.22
a
angle subtended at the nodal point
wavelength of the light
pupil diametera
angle subtended at the nodal point
wavelength of the light
pupil diameter
1.22
a
a
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
pupil diameter (mm)
dist
ance
from
pea
k to
1st m
inim
um
(min
utes
of a
rc 5
00 n
m li
ght)
Point Spread Function vs. Pupil Size
1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm
Perfect Eye
Rayleigh resolution
limit
Unresolved point sources
Resolved
Rayleigh Resolution Limit
At the Rayleigh resolution limit, the two points are separated by the angle…
min
min
angle subtended at the nodal point
wavelength of the light
pupil diameter
1.22
a
a
This is the same as the distance between the max and the first minimum for one Airy disk!!!
min
min
angle subtended at the nodal point
wavelength of the light
pupil diameter
1.22
a
a
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
pupil diameter (mm)
min
imum
ang
le o
f res
olut
ion
(min
utes
of a
rc 5
00 n
m li
ght)
Minutes of arc
20/20 20/105
arc
min
2.5
arc
min
1 arcmin
convolution
6 mm
3 mm
1 mm
20/20 E
DH
20/20 E
First light AO image of binary star k-Peg on the 3.5-m telescope at the Starfire Optical Range
September, 1997.
uncorrected corrected
arc of seconds 064.05.3
1090022.122.1 9
min
a
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 5 mm 6 mm 7 mm
Perfect Eye
Typical Eye
2.5.7: Image quality as a function of pupil size
opt
ica
l qu
alit
y(a
rb. u
nits
)
Best overall quality ~ 2 - 3 mm
0 2 4 6 8pupil size (mm)
PolarizationPolarization
Direction of Polarization
vertical horizontal diagonal
Any Polarization can be Expressed as a Sum of a Vertical and a Horizontal Component
A
Acos
Asi
n
diagonal polarization
(horizontal component)
(ver
tical
com
pone
nt)
2 2 2 2 2, cos , sinx yI A I A I A
y
x
Unpolarized Light
Most light is unpolarized. •sun •incandescent lamp•candlelight
EE
circular polarization elliptical polarization
Circular and Elliptical PolarizationCircular and Elliptical Polarization
linear polarization
E
unpolarized light
E
random polarization
E
Generating Polarized Light
Polarizing Filters
unpolarized light in
polarized light out
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
Es
Ep
Es
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.
Polarization by Reflectionre
flect
an
ce (
%)
angle (deg)
5
10
15
20
30 60 900
n=1.5
14.8 %
Rs Rp
56.3
Rs is the reflectance of the Es component.
Rp is the reflectance of the Ep component.
At 90º, both Rs and Rp are 100 %
Brewster’s angle
arctanB
n
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
Calcite
Haidinger’s Brush
Depolarized Parallel Polarized Randomly Polarized
Glaucoma Suspect
Disc Hyperpigmentation
Courtesy of Steve Burns and Ann Elsner, Schepens Eye Research Institute, Boston, MA
GDX Laser Diagnostic Technologies
Thick NFL Thin NFL
linear polarizationstrong
elliptical polarizationlinear polarization
weak elliptical polarization
GDX Image: AR left eye