Optics i 18 Coherence Interference

8/18/2019 Optics i 18 Coherence Interference

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Coherence and Interference

Coherence

Temporal coherenceSpatial coherence

InterferenceParallel polarizations

interfere; perpendicularpolarizations don't.

The Michelson Interferometer Fringes in delay

Measure of Temporal CoherenceThe Fourier Transform Spectrometer

The Misaligned Michelson Interferometer Fringes in position

Measure of Spatial Coherence

Opals use interference

et!een tiny

structures to yield

right colors.


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The Temporal Coherence Time and

the Spatial Coherence Length

The temporal coherence time is the time the !a"e#fronts remain e$uallyspaced. That is% the field remains sinusoidal !ith one !a"elength&

The spatial coherence length is the distance o"er !hich the eam !a"e#

fronts remain flat&

Since there are

t!o trans"erse

dimensions% !e

can define a

coherence area.

Temporal

Coherence

Time% τc

Spatial

Coherence

ength


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Spatial

and

TemporalCoherence

(eams can ecoherent or

only partially

coherent

)indeed% e"en

incoherent*in oth space

and time.

Spatial and

TemporalCoherence&

Temporal

Coherence;

SpatialIncoherence

Spatial

Coherence;

TemporalIncoherence

Spatial and

Temporal

Incoherence


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The coherence time is the reciprocal of

the bandwidth.

The coherence time is gi"en y&

!here ∆ν is the light and!idth )the !idth of the spectrum*.

Sunlight is temporally "ery incoherent ecause its and!idth is

"ery large )the entire "isile spectrum*.

asers can ha"e coherence times as long as aout a second%

!hich is amazing; that's +,-, cycles/

1/c

vτ = ∆


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The spatial coherence depends on the

emitter size and its distance away.

The "an Cittert#0erni1e Theorem states that the spatial

coherence area Ac is gi"en y&

!here d is the diameter of the light source and D is the distance a!ay.

(asically% !a"e#fronts smooth

out as they propagate a!ay

from the source.

Starlight is spatially "ery coherent ecause stars are "ery far a!ay.

2 2

2c

D

d

λ

π Α =


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Irradiance of a sum of two waves

{ }2

*

2

1

1Rec E E

I I I

ε

= + +

×% %

2ifferent

colors

2ifferent polarizations

Same

colors

Same polarizations

1 2 I I I = +

1 2 I I I = + 1 2 I I I = +

Interference only occurs !hen the !a"es ha"e the same color and

polarization.

3e also discussed incoherence% and that4s !hat this lecture is aout/


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The irradiance when combining a beam with

a delayed replica of itself has fringes.

Suppose the t!o eams are E 0 exp(iωt ) and E 0 exp[iω(t -τ )]% that is%a eam and itself delayed y some time τ &

O1ay% the irradiance is gi"en y&

{ }*1 1 2 2Re I I c E E I ε = + × +% %

{ }*0 0 02 Re exp[ ] exp[ ( )] I I c E i t E i t ε ω ω τ = + × − −% %

{ }2

0 02 Re exp[ ] I c E iε ωτ = +%

20 02 cos[ ] I c E ε ωτ = +

%

0 02 2 cos[ ] I I I ωτ = +

Fringes (in delay

#

I

τ


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!arying the delay on purpose

Simply moving a mirror can vary the delay of a beam by many

wavelengths.

Since light travels 300 µm per ps, 300 µm of mirror displacement

yields a delay of 2 ps. Such delays can come about naturally, too.

Moving a mirror backward by a distance L yields a delay of:

τ = 2 L /cDo not forget the factor of 2!Light must travel the extra distanceto the mirror—and back!

Translation stage

Input

beam E(t)

E(t– τ )

Mirror

Output

beam


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"e can also vary the delay using

a mirror pair or corner cube.

Mirror pairs involve tworeflections and displacethe return beam in space:But out-of-plane tilt yieldsa nonparallel return beam.

Corner cubes involve three reflections and also displace the returnbeam in space. Even better, they always yield a parallel return beam:

“Hollow corner cubes” avoid propagation through glass.

Translation stage

Input

beam E(t)

E(t– τ )

MirrorsOutput

beam

[EdmundScientific]


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The #ichelson Interferometer

Beam-splitter

Input

beam

Delay

Mirror

Mirror

Fringes (in delay$

[ ] [ ]{ }

[ ]{ }

{ }

*

1 2 0 1 0 2

2

2 1 1 2 0 0

Re exp ( 2 ) exp ( 2 )

2 Re exp 2 ( ) ( / 2)

2 1 cos( )

out I I I c E i t kz kL E i t kz kL

I I I ik L L I I I c E

I k L

ε ω ω

ε

= + + − − − − −

= + + − ≡ = =

= + ∆

since

∆ L = 2(L2 – L1)

The Michelson Interferometer splits a

eam into t!o and then recominesthem at the same eam splitter.

Suppose the input eam is a plane

!a"e&

I out

L1

!here& ∆ L = 2( L2 – L1)

L2 Outputbeam

5(right fringe652ar1 fringe6


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The #ichelson

Interferometer

Beam-splitter

Input

beam

Delay

Mirror

Mirror

The most o"ious application of

the Michelson Interferometer is

to measure the !a"elength of

monochromatic light.

∆ L = 2(L2 – L1)

I out

L1

L2Outputbeam

{ } { }2 1 cos( ) 2 1 cos(2 / )out I I k L I Lπ λ = + ∆ = + ∆


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%uge #ichelson Interferometers

may someday detect gravity

waves.

Beam-splitter

Mirror

Mirror

L1

L2

7ra"ity !a"es )emitted y all massi"e o8ects* e"er so slightly !arp

space#time. 9elati"ity predicts them% ut they4"e ne"er een detected.

Superno"ae and colliding lac1 holes emit gra"ity !a"es that may e

detectale.

7ra"ity !a"es are 5$uadrupole6

!a"es% !hich stretch space in

one direction and shrin1 it in

another. They should cause

one arm of a Michelsoninterferometer to stretch and

the other to shrin1.

:nfortunately% the relati"e distance ) L1-L2 ~ 10-16 cm* is less than the

!idth of a nucleus/ So such measurements are "ery "ery difficult/

L1 and L2 1m/


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The LI&' proect

< small fraction ofone arm of the

CalTech I7O

interferometer=

The uilding

containing an arm

The control center

CalTech I7O

>anford I7O


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The LI&' fol)s

thin) big*

The longer the interferometer

arms% the etter the

sensiti"ity.

So put one in space%of course.


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Interference is easy when the light wave is a

monochromatic plane wave. "hat if it+s not,

For perfect sine !a"es% the t!o eams are either in phase orthey4re not. 3hat aout a eam !ith a short coherence time????

The eams could e in phase some of the time and out of phase

at other times% "arying rapidly.

9ememer that most optical measurements ta1e a long time% so

these "ariations !ill get a"eraged.


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-dding a

non

monochro

matic

wave to a

delayedreplica of

itself

2elay @ period

)AA τc*&

2elay + τc&

Constructi"e

interference for

all times

)coherent*5(right fringe6

2estructi"e

interference forall times

)coherent*

52ar1 fringe6*

Incoherent

addition Bo

fringes.

2elay -&


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Suppose the input eam is not monochromatic)ut is perfectly spatially coherent*&

⇒ I out = 2 I + c ε Re{ E (t+2L1 /c) E*(t+2L2 /c)}

Bo!% I out !ill "ary rapidly in time% and most detectors !ill simply

integrate o"er a relati"ely long time% T &/ 2 / 2

1 2

/ 2 / 2

( ) 2 Re ( 2 / ) *( 2 / )

T T

out

T T

U I t dt U IT c E t L c E t L c dt ε

− −

µ ⇒ µ + + +∫ ∫

The #ichelson Interferometer is a

Fourier Transform Spectrometer

The Field -utocorrelation/

Beam-splitter

Delay

Mirror

L1

L2

2 Re ( ') *( ' 'U IT c E t E t dt ε τ

∞

−∞

µ + − )∫ 9ecall that the Fourier Transform of the Field


18/32

Fourier Transform Spectrometer Interferogram

The Michelson interferometer outputthe interferogramFourier

transforms to the spectrum.

The spectral phase plays no role/ )The temporal phase does% ho!e"er.*

I n t e g r a t e d

i r r a d i a n c e

- 2elay

Michelson interferometer

integrated irradiance

2π/ω0

1/∆ω

Fre$uency

I n t e n s i t y

ω0

Spectrum

∆ω

< Fourier Transform Spectrometer's detected light energy "s. delay

is called an interferogram.


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Fourier Transform Spectrometer 0ata

Interferogram

This interferogram

is "ery narro!% so

the spectrumis "ery road.

Fourier Transform Spectrometers are most commonly used in the

infrared !here the fringes in delay are most easily generated.


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Fourier Transform Spectrometers

MaDimum path difference& , m

Minimum resolution& -.--E cm

Spectral range& G.G to ,H µm


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[ ] [ ]{ }

[ ]{ }

*

0 0

2

0

2

0

Re exp ( cos sin exp ( cos sin

Re exp 2 sin

cos(2 sin )

E i t kz kx E i t kz kx

E ikx

E kx

ω θ θ ω θ θ

θ

θ

− − − − +

µ −

µ

Crossed 1eams

θ

k +

k −r

z

x

ˆˆcos sink k z k xθ θ + = +

ˆˆcos sink k z k xθ θ − = −

r

cos sink r k z k xθ θ +⇒ × = +r

cos sink r k z k xθ θ − × = −r r

{ }*0 0 02 Re exp[ ( )] exp[ ( )] I I c E i t k r E i t k r ε ω ω + −= + − × − − ×r rr r

Cross term is proportional to&

Fringes (in position

x

I out (x)

ˆ ˆ ˆr xx zz = + +

2 /(2 sin )k π θ Λ =Fringe spacing&


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Irradiance vs. position for crossed beams

Irradiance fringes occur !here the eams o"erlap in space and time.


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1ig angle$ small fringes.

Small angle$ big fringes.

2 /(2 sin )

/(2sin )

k π θ

λ θ

Λ =

=

The fringe spacing% Λ&


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The fringe spacing is&

Λ -., mm is aout the minimum fringe spacing you can see&

2ou can3t see the spatial fringes unless

the beam angle is very small/

sin /(2 )

0.5 / 200

1/ 00 !"# 0.15

! !

θ θ λ

θ µ µ

≈ = Λ

⇒ ≈

≈ = o

/(2sin )λ θ Λ =


25/32

Spatial fringes

and spatial

coherence

Interference is incoherent )nofringes* far off the aDis% !here

"ery different regions of the

!a"e interfere.

Interference is coherent )sharp

fringes* along the center line%!here same regions of the

!a"e interfere.

Suppose that a eam is temporally%

ut not spatially% coherent.


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The #ichelson

Interferometer

and Spatial FringesSuppose !e misalign the mirrorsso the eams cross at an angle!hen they recomine at the eamsplitter.


27/32

#ichelson#orley e4periment

,Jth

#century physicists thought that light !as a "iration of amedium% li1e sound. So they postulated the eDistence of a medium

!hose "irations !ere light& aether .

Michelson and Morley

realized that the earth could

not al!ays e stationary

!ith respect to the aether.


28/32

#ichelson#orley 54periment$ 0etails

If light re$uires a medium% then its "elocity depends on the "elocity ofthe medium. Kelocity "ectors add.

Parallel

"elocities


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Perpendicular

"elocity to mirror

Perpendicular

"elocity after mirror


In the other arm of the interferometer% the total "elocity must e

perpendicular% so light must propagate at an angle.

$"i#$t r

$%&t$&r r

$tot%" r

2 2$ $$ "i#$ %tot%" r t &t$&= −

$"i#$t r

$%&t$&r r

$tot%" r


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Perpendicular

propagation

$%&t$&r rParallel and

anti#parallel

propagation

2 2

$ $

2 1

[1 $ / ]

L Lt

c c

L

c c

∆ = +− +

= −

P

2 2

2 2

2$

2 1

1 $ /

Lt

c

L

c c

⊥∆ =−

=

−

The delays for the t!o arms depend

differently on the "elocity of the aether/

If $ is the earth4s "elocity around thesun% D ,- ms% and L , m% then&

et c e the speed of light% and $ e the "elocity of the aether.

1%~ 10 st t −⊥

∆ − ∆P


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#ichelson#orley

54periment$ 6esults

Michelson and Morley's results

from


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Fresnel3s 1iprism

< prism !ith an apeD

angle of aout ,LJ

refracts the left half of the

eam to the right and the

right half of the eam tothe left.

Fringe patternoser"ed y interfering

t!o eams created y

Fresnel's iprism

Optics i 18 Coherence Interference

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