Interference

See Chapter 9 of Hecht

A single point createswaves with concentriccircles of light and dark bands.

This picture does not show an interference pattern. It is simply the concentric waves of two points sources drawn in the same plane. Contrastthis with the image to the right

This is the interference pattern of two waves.The two waves add or subtract to form the light and dark regions of the interference pattern

Waves are not simple two dimensional objects. When they interfere with each other, peaks andvalleys are formed. Many interference patternslook like two-dimensional systems of light and darkbands because they are being viewed from above. In this picture the system tilted so it can be viewedfrom the side.

E1

E2

Consider two waves intersecting at some location

222022

111011

cos

cos

trkEE

trkEE

At the point where the waves intersect the totalelectric field will be:

2220211101 coscos trkEtrkEET

Almost always the intensity of the radiation is detected

TAEcI 2

0

TA = Time Average

For convenience, neglect constants and say:

TAEI 2

1221

212

22

12 2

III

EEEEEITATATATA

Interference may, or may not, arise depending in the nature of I12 Getting interference is not easy.

TA

TA

trktrk

trktrkEE

trktrkEEI

222222

1111110201

222111020112

sinsincoscos

sinsincoscos2

coscos2

0 if 0 020112 EEI

The two beams must have some common polarization

Temporal Coherence

Waves should have the same frequency for interference. We will need:

0

coscos1

coscos0 2121

T

TAttdt

Ttt

In[8]:= Plot@Cos@100* tD* Cos@100.1* tD,8t, 0, 1<D

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Out[8]= … Graphics … In[10]:= Plot@Cos@100* tD* Cos@100.1* tD,8t, 0, 100<D

20 40 60 80 100

-1

-0.5

0.5

1

Out[10]= … Graphics …

T

Ttdt

T

T

2

sincoscos

1

0

This term decays away in a time called the coherence time:

2cohT

Coherence Length:

fcccTL cohcoh 22

This is the length for which wavetrains stay in phase.

It is extremely difficult to maintain coherence for two beams UNLESS they come from the same laser.

Consider two different, but similar lasers. At =850 nm one can “lock” laser to an atomic Cs transition.

Possible to have two lasers with:

sfT

kmkHzsmfcL

HzfkHzf

coh

coh

10/1

3100/103

1053.31008

14

Long, but not infinite. What is worse – “Mode Hops”Every s or so a laser will randomly shift its phase. Two laserswill do this independently and interference shifts.

Back to Interference CriteriaAssume two light beams from the same source.

cos

cos

sinsin

coscos

2

12

sinsincoscos

sinsincoscos2

coscos2

0201

22110201

2211

22110201

2222

11110201

2211020112

EE

rkrkEE

rkrk

rkrkEE

trktrk

trktrkEE

trktrkEEI

TA

TA

Depends on the phase difference of 2 waves

Note that:

cos2

22

cos

2221

202

222

201

211

020112

IIIII

EEIEEI

EEI

TATA

Constructive interference:

,...4,2,0for 2 2221 IIIII

Destructive interference:

,...5,3,for 2 2221 IIIII

Spatial CoherenceSource 1

Source 2

Observationpoint

Rays coming from extended source will have different phases. Hard to get interference between sources 1 and 2. Phase of light all mixed up.

Source 1

Source 2 Observationpoint

For point sources, arriving light has well defined phase.

2 Pinhole Sources

Grimaldi’s experiment of 1665Observation screen – no interference observed. No “fringes”

Sun

Sun is an extended source – no spatial coherence. No interference

2 pinholes or slits

YoungThomas Young, 1805. Used the sun, but an additional pinhole creates a (small) source with spatial coherence

Sun

1 pinhole2 pinholes

Observationscreen

Difference in path lengths is:

r1

r2 ddrr sin21

But Ly /

Constructive interference when

mrr 21

where m is an integer

L

dyI

L

ydI

r r kI I

r r k I I I I I I

20

20

2 1 20

2 1 0 0 0 2 1

cos 4

2

2cos 4

2cos 4

cos 2 2

Peaks and troughs on observing screen. Correctly predicts location of peaks of troughs,But not envelope (next week).

Waves versus particles (or so we think)

http://micro.magnet.fsu.edu/primer/java/doubleslit/

You can change wavelength of laser and the distancebetween the two slits.

Cool demonstration of double slit on the web. See:

Light & Matter, Waves & Particles – de Broglie

Wave and interference effects can be seen with matter too! Quantum wave properties:

photonsfor //, hchfphfE

De Broglie said, why not matter too?!?

ph /

Interference properties seen with electrons, neutrons, atoms, and now even molecules like C60 and C70!

Interferometers or the Double Slit: Interference seen even when only one particle is in system. Particle (be it electron, photon, atom, etc) goes through both slits at once.

Observed interference of C60 and C70

See results of Prof. Anton Zeilinger and his group

http://www.quantum.univie.ac.at/research/

Standing Waves

Consider two counter-propagating waves from a single laser.Say too that they have equal amplitudes.

Standing Waves

Consider two counter-propagating waves from a single laser.Say too that they have equal amplitudes.

tkzEtkzEEtotal coscos 00

2/cos42cos22

2cos22

000

2221

kzIkzII

kzIIIII

2/cos4 20 kzII

Two traveling waves produces a standing wave.

In[2]:= Plot@Cos@xD̂2,8x, 0, 6Pi<D

2.5 5 7.5 10 12.5 15 17.5

0.2

0.4

0.6

0.8

1

Out[2]= … Graphics …

Microwave OvensStanding microwaves

Peaks and troughs => Hot spots and cold spots => Nodes and anti-nodes

Spinning dish hopefully brings all parts of food into contact with nodes

Demo with marshmallows and a microwave

BeamsplittersIn order to create multiple beams from a single laser one needs to use a “beamsplitter”

LaserE0

r*E0

t*E0

I0

R*I0

T*I0

Usually (but not always) us 50-50 beamsplitter; half the light transmitted, half reflected. Example- half silvered mirror.

Polarizing Beamsplitters

Some beamsplitters separated light according to polarization

Laser//EEE

//E

E

Mach-Zender Interferometer

Mach-Zender Interferometer

0E

0* Et

0* Er

Say that the length for the top path is L1 and L2 for the bottom.

21

21

02

02

2

001

ikLikL

ikLikL

eEreEtE

eErteErtE

At detectors D1 and D2

2sin

2cos

21201

21201

LLkII

LLkII

If other factors would change the acquired phase for the two paths it would affect counts at D1 and D2.

BS1

Mirror 1

Mirror 2

BS2L2

L2

L1

L1

Atoms or neutrons: Say interferometer is in a gravitational field. The arms of length L1 are parallel to the ground, while when particles are in the arms of length L2 they climb up against gravity.

Interferometer pivoted about bottom arm by angle .

Changed phase between two interferometer paths by insertion of Al in one path, or rotating interferometer in Earth’s gravitational field.

Atom Interferometry – Overlapping Na atoms from Bose-Einstein Condensate

Atom Laser: Results from Wolfgang Ketterle’s group, MIT.Ketterle shares 2001 Noble Prize in Physics

Michelson Interferometer

0E

L1

L2

I0

212

0 sin LLkIIout

For equal arm lengths => L1 = L2

No light out of that beamsplitter port for all wavelengths – White Light Fringe

Michelson-Morley’s Search for the Aether

Homework 7, problem 1Mirror 2

Mirror 1

Beamsplitter

Incident light

L

L

Ether wind of speed v

Calculate time for light to traverse each arm, according to ether theory. Take v to be Earth’s orbital velocity.

Gravitational Radiation Detection

Laser Interferometric Gravitational Wave ObservatoryLIGO

3030 km(±10 ms)

CALTECHPasadena

MITBoston

HANFORDWashington

LIVINGSTONLouisiana

Hanford Observatory

4 km

2 km

Livingston Observatory

4 km

LIGO Interferometers

Laser

end test mass

Light bounces back and forth along arms about 30 times

input test massLight is “recycled” about 50 times

signal

Power RecycledMichelsonInterferometer With Fabry-Perot Cavities

Power RecycledMichelsonInterferometer With Fabry-Perot Cavities

4 km Fabry-Perot arm cavity

beam splitter

Vibration Isolation Systems

Core Optics

Core Optics Suspension and Control

Core Optics Installation and Alignment

Washington 2k Pre-stabilized Laser

Custom-built10 W Nd:YAG

Laser

Stabilization cavities for frequency

and beam shape

Fabry-Perot Interferometer

Mirror 1 Mirror 2

LE0

ErEt

...

...4

1212102

121001

52121210

321210210

LikLik

r

LikLikikLt

etrrrtEetrtEErE

etrrrrtEetrrtEettEE

Cavity of length L, incident light of amplitude E0 and wavelength , k=2/

Mirror

E0

ErEt

Real Mirrors have losses

I0

Ir

It

1 ATR

A is the loss coefficient. For a very good mirror A~10-4 to 10-3

Free Spectral Range

When 2kL changes by 2 we get another resonance

mc

Lfm

L

24

22

2

Where m is some integer

L

cfm

L

cf

22

Free Spectral Range (FSR) = c/2LL=2cm => FSR=7.5GHz

Cavity resonances function as “fence posts” or references

Fabry-Perot Interferometer as an Optical Spectrum Analyzer

What’s going on with a laser???

An integral number of half-wavelengths fit into a laser cavity

integeran is where2

n

nL

cflaser

The laser medium will have some gain profile, as a function of frequency.

gain

frequency=(c/2L)n, n an integrer

Overlap of gain profile and possible longitudinal modes

Resulting observed laser modes at various frequencies.

Mirror 1 Mirror 2

L

PZT

Fabry-Perot mirror mounted to a piezo-electric crystalVary cavity length, and scan frequencies that resonate

AC Voltage

The Speckle Effect