Basics of coherencetheory

Chapter 12

Phys 322Lecture 35

Conditions for interference1) For producing stable pattern, the two sources must have nearly

the same frequency.2) For clear pattern, the two sources must have similar amplitude.3) For producing interference pattern, coherent sources are

required.

Temporal coherence:Time interval in which the light resembles a sinusoidal wave. (~10 ns for natural light)Longitudinal coherence length: lc= ctc.Spatial coherence: longitudinal and transverseThe correlation of the phase of a light wave between different locations.

Coherence review

The coherence time is the reciprocal of the bandwidth.

The coherence time is given by:

where is the light bandwidth (the width of the spectrum).

Sunlight is temporally very incoherent because its bandwidth isvery large (the entire visible spectrum).

Lasers can have coherence times as long as about a second,which is amazing; that's >1014 cycles!

1/c v

The Temporal Coherence Time and the Spatial Coherence LengthThe temporal coherence time is the time the wave-fronts remain equally spaced. That is, the field remains sinusoidal with one wavelength:

The spatial coherence length is the distance over which the beam wave-fronts remain flat:

Since there are two transverse dimensions, we can define a coherence area.

Temporal Coherence

Time, c

Spatial Coherence

Length

Spatial and Temporal Coherence

Beams can be coherent or

only partially coherent (indeed, even incoherent)

in both space and time.

Spatial andTemporal

Coherence:

TemporalCoherence;

Spatial Incoherence

Spatial Coherence;

TemporalIncoherence

Spatial andTemporal

Incoherence

The spatial coherence depends on the emitter size and its distance away. The van Cittert-Zernike Theorem states that the spatial coherence area Ac is given by:

where d is the diameter of the light source and D is the distance away.

Basically, wave-fronts smooth out as they propagate away from the source.

Starlight is spatially very coherent because stars are very far away.

2 2

2cD

d

Irradiance of a sum of two waves

2

*2

1

1Rec E E

I I I

Different colors

Different polarizations

Same colors

Same polarizations

1 2I I I

1 2I I I 1 2I I I

Interference only occurs when the waves have the same color and polarization.

We also discussed incoherence, and that’s what this lecture is about!

The irradiance when combining a beam with a delayed replica of itself has fringes

Suppose the two beams are E0 exp(it) and E0 exp[it-)], that is, a beam and itself delayed by some time :

Okay, the irradiance is given by:

*1 1 2 2ReI I c E E I

*0 0 02 Re exp[ ] exp[ ( )]I I c E i t E i t

2

0 02 Re exp[ ]I c E i

20 02 cos[ ]I c E

0 02 2 cos[ ]I I I

Fringes (in delay)

-

I

Varying the delay on purposeSimply 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:

2 L /cDo not forget the factor of 2!Light must travel the extra distance to the mirror—and back!

Translation stage

Input beam E(t)

E(t–)

MirrorOutput beam

The Michelson Interferometer

Beam-splitter

Inputbeam

Delay

Mirror

Mirror

Fringes (in delay):

*1 2 0 1 0 2

22 1 1 2 0 0

Re exp ( 2 ) exp ( 2 )

2 Re exp 2 ( ) ( /2)

2 1 cos( )

outI 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

since

L = 2(L2 – L1)

The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter.

Suppose the input beam is a plane wave:

Iout

L1

where: L = 2(L2 – L1)

L2 Outputbeam

“Bright fringe”“Dark fringe”

Interference is easy when the light wave is a monochromatic plane wave. What if it’s not?

For perfect sine waves, the two beams are either in phase or they’re not. What about a beam with a short coherence time????

The beams could be in phase some of the time and out of phase at other times, varying rapidly.

Remember that most optical measurements take a long time, so these variations will get averaged.

Adding a non-

monochro-matic

wave to a delayed

replica of itself

Delay = ½ period

(<< c):

Delay > c:

Constructive interference for all times (coherent) “Bright fringe”

Destructive interference for all times (coherent) “Dark fringe”)

Incoherent addition No fringes.

Delay = 0:

*0 0

20

20

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

Crossed Beams

k

k

z

xˆˆcos sink k z k x

ˆˆcos sink k z k x

cos sink r k z k x

cos sink r k z k x

*0 0 02 Re exp[ ( )] exp[ ( )]I I c E i t k r E i t k r

Cross term is proportional to:

Fringes (in position)

x

Iout(x)

ˆ ˆ ˆr xx yy zz

2 /(2 sin )k Fringe spacing:

Irradiance vs. position for crossed beams

Irradiance fringes occur where the beams overlap in space and time.

Big angle: small fringes.Small angle: big fringes.

2 /(2 sin )/(2sin )

k

The fringe spacing, :

As the angle decreases to zero, the fringes become larger and larger, until finally, at = 0, the intensity pattern becomes constant.

Large angle:

Small angle:

The fringe spacing is:

= 0.1 mm is about the minimum fringe spacing you can see:

You can't see the spatial fringes unlessthe beam angle is very small!

sin /(2 )0.5 / 200

1/ 400 rad 0.15m m

/(2sin )

Spatial fringes and spatial coherence

Interference is incoherent (no fringes) far off the axis, where very different regions of the wave interfere.Interference is coherent (sharp fringes) along the center line, where same regions of the wave interfere.

Suppose that a beam is temporally, but not spatially, coherent.

Coherence (chapter 12)Completely incoherent waves: no interference fringesCompletely coherent waves: interference fringes best pronounced

Laser

Add glass plate

Laser

temporal coherence

LampAdd glass plateLamp

Coherence

I = I1 + I2 + I12

T

tEtEI 2112

cross-correlation

T

tEtEI 2112

Temporal coherence length is reflected in cross-correlation

Visibility

Visibility:minmax

minmax

IIII

V

1221

212

IIII

V

TT

T

EE

tEtE2

22

1

*21

12

Complex degree of coherence:

Coherent limit: |12| = 1Incoherent limit: |12| = 0Partial coherence: 0<|12|<1

Examples

|12| = 0.703 |12| = 0.132 |12| = 0.062

Spatial coherence: extended source

l

basincV

For double slit:

The spatial coherence depends on the emitter size and its distance away. The van Cittert-Zernike Theorem states that the spatial coherence area Ac is given by:

where d is the diameter of the light source and D is the distance away.

Basically, wave-fronts smooth out as they propagate away from the source.

Starlight is spatially very coherent because stars are very far away.

2 2

2cD

d

The Michelson stellar interferometer

1. Case of double star (equal intensities) interference disappears when:

2

0hangular distance between the stars

2. Single star, interference disappears when:022.1h

angular diameter of the star

Betelgeuse

-Orion, the first star for which diameter was established in 1920

022.1h h = 121”, 0 = 570 nm

"047.0106.22 8 rad

From known distance: diameter = 240 million miles(280 times larger than sun)