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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
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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 =
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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.
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#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.
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#ichelson#orley 54periment$ 0etails
If light re$uires a medium% then its "elocity depends on the "elocity ofthe medium. Kelocity "ectors add.
Parallel
"elocities
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Perpendicular
"elocity to mirror
Perpendicular
"elocity after mirror
#ichelson#orley 54periment$ 0etails
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
$tot%" r
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#ichelson#orley 54periment$ 0etails
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