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

LCD screen

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