Phys 322 Chapter 12 Lecture 35

Basics of coherence theory Coherence review

Conditions for interference 1) 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= ctc. Spatial coherence: longitudinal and transverse The correlation of the phase of a light wave between different locations. The coherence time is the reciprocal of the bandwidth.

The coherence time is given by:

 c 1/v

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

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

Lasers can have coherence times as long as about a second, which is amazing; that's >1014 cycles! The Temporal Coherence Time and the Spatial Coherence Length

The temporal coherence time is the time the wave-fronts remain equally spaced. That is, the field remains sinusoidal with one wavelength:

Temporal Coherence

Time, c

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

Since there are Spatial two transverse Coherence Length dimensions, we can define a coherence area. Spatial and Spatial and Temporal Temporal Coherence Coherence:

Temporal Coherence; Spatial Beams can be Incoherence coherent or only partially coherent (indeed, Spatial even incoherent) Coherence; in both space and Temporal time. Incoherence

Spatial and Temporal 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: D22  c  d 2 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. Irradiance of a sum of two waves

Same polarizations Different polarizations

III 1 2 Same I  II12 cEE Re * colors 1 2

I  II I  II Different 12 12 colors

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

Okay, the irradiance is given by: * I  Ic1122 Re EE I 

Suppose the two beams are E0 exp(it) and E0 exp[it-)], that is, a beam and itself delayed by some time : IIc2 Re E exp[ itE ] * exp[  it ( )] 00 0 2 2Reexp[]Ic00  E i   2 2cos[]IcE00  Fringes (in delay)  I II22cos[]00 I 

-  Varying the delay on purpose

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

Input Mirror beam E(t) Output beam E(t–) Translation stage

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

Do not forget the factor of 2!   2 L /c Light must travel the extra distance to the mirror—and back!

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. The Michelson Interferometer Input beam

The Michelson Interferometer splits a L beam into two and then recombines 2 Output Mirror beam them at the same beam splitter. Beam- L splitter 1 Suppose the input beam is a plane Delay wave: Mirror

* Iout  I12 I  cRe E 0 exp i ( t  kz 2 kL  1 ) E 0 exp  i ( t  kz 2 kL 2 ) 2 II 2 I Re exp 2 ikLL (21  ) since II  12  I  ( c 0 / 2) E 0

 2IkL 1 cos( ) “Dark fringe” “Bright fringe”

Iout where: L = 2(L2 – L1)

Fringes (in delay):  L = 2(L2 –L1) 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. Constructive Adding a interference for non- Delay = 0: all times (coherent) monochro- “Bright fringe” matic

wave to a Destructive Delay = interference for delayed ½ period all times (coherent) replica of (<< c): “Dark fringe”) itself

Incoherent

Delay > c: addition No fringes. Crossed Beams x   k kkcos zkˆ sin xˆ   kk cos zkˆ sin xˆ z  rxxyyzzˆˆˆ    krkcos zk  sin x   k krk cos zk  sin x    *  IIc200 Re E exp[ itkrE (  )] 0  exp[  itkr ( )] Cross term is proportional to: * ReEitkzkxE00 exp (  cos sin exp   itkzkx (  cos sin 

2  ReEikx0 exp 2 sin Fringes (in position)  2 Iout(x)  Ekx0 cos(2 sin ) 

Fringe spacing: 2/(2sin) k  x 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.

The fringe spacing, : Large angle:

2/(2sin)k   /(2sin )

As the angle decreases to Small angle: zero, the fringes become larger and larger, until finally, at  = 0, the intensity pattern becomes constant. You can't see the spatial fringes unless the beam angle is very small!

The fringe spacing is:

   /(2sin )

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

  sin /(2 ) 0.5mm / 200   1/ 400 rad 0.15 Spatial fringes and spatial coherence

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

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. Coherence (chapter 12) Completely incoherent waves: no interference fringes Completely coherent waves: interference fringes best pronounced Add glass plate Laser Laser

Add glass plate Lamp Lamp

temporal coherence Coherence

I = I1 +I2 + I12 I  E tE t 12 1 2 T cross-correlation



I   E tE t   12 1 2 T Temporal coherence length is reflected in cross-correlation Visibility

I  I Visibility: V  max min Imax  Imin

2 I1 I2 V   12 I1  I2

Complex degree of coherence:   E t  E*t 1 2 T 12  2 2 E1 E2 T T

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

| | = 0.062 |12| = 0.703 |12| = 0.132 12 Spatial coherence: extended source

For double slit:

 a b  V  sinc   l  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: D22  c  d 2 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. The Michelson stellar interferometer

1. Case of double star (equal intensities)  angular distance interference disappears when: h  0 between the stars 2

 angular diameter 2. Single star, interference disappears when: h  1.22 0 of the star  Betelgeuse

-Orion, the first star for which diameter was established in 1920  h  1.22 0 h = 121”,  = 570 nm  0

  22.6108 rad  0.047" From known distance: diameter = 240 million miles (280 times larger than sun)