1. 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, tc
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.
Spatial Coherence Length

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

3. The coherence time is the reciprocal of the bandwidth.
The coherence time is given by:
where Δν 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!

4. 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.

5. Irradiance of a sum of two waves
Same polarizations
Different polarizations
Same colors
Different 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!

6. The irradiance when combining a beam with a delayed replica of itself has “fringes.”
Okay, the irradiance is given by:
Suppose the two beams are E0 exp(iwt) and E0 exp[iw(t-t)], that is, a beam and itself delayed by some time t:
Fringes (in delay) - I t

7. Varying the delay on purpose
Simply moving a mirror can vary the delay of a beam by many wavelengths.
Input
beam
Mirror
E(t)
Output
beam
E(t–t)
Translation stage
Moving a mirror backward by a distance L yields a delay of:
Do not forget the factor of 2!
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.

8. We can also vary the delay using a mirror pair or corner cube.
Input
beam
E(t)
Mirror pairs involve two
reflections and displace
the return beam in space:
But out-of-plane tilt yields
a nonparallel return beam.
Mirrors
Output
beam
E(t–t)
Translation stage
Corner cubes involve three reflections and also displace the return
beam in space. Even better, they always yield a parallel return beam:
[Edmund
Scientific]
“Hollow corner cubes” avoid propagation through glass.

9. The Michelson Interferometer
Input
beam
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: L2
Output
beam
Mirror
Beam-
splitter L1
Delay
Mirror
“Dark fringe”
“Bright fringe”
Iout
where: DL = 2(L2 – L1)
Fringes (in delay):
DL = 2(L2 – L1)

10. Michelson-Morley experiment
19th-century physicists thought that light was a vibration of a medium, like sound. So they postulated the existence of “aether.”
Parallel and anti-parallel propagation
Michelson and Morley realized that the earth could not always be stationary with respect to the aether. And light would have a different phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether.
They observed no phase shift.
Goodbye aether.
Mirror
Perpendicular propagation
Beam-
splitter
Mirror
supposed velocity of earth through the aether

11. Huge Michelson Interferometers may someday detect gravity waves.
Gravity waves (emitted by all massive objects) ever so slightly warp space-time. Relativity predicts them, but they’ve never been detected.
Supernovae and colliding black holes emit gravity waves that may be detectable.
Gravity waves are “quadrupole” waves, which stretch space in one direction and shrink it in another. They should cause one arm of a Michelson interferometer to stretch and the other to shrink. L2
Mirror L1
Beam-
splitter
L1 and L2 = 4 km!
Mirror
Unfortunately, the relative distance (L1-L2 ~ cm) is less than the width of a nucleus! So such measurements are very very difficult!

12. The LIGO project CalTech LIGO Hanford LIGO
The building containing an arm
CalTech LIGO
A small fraction of one arm of the CalTech LIGO interferometer…
Hanford LIGO
The control center

13. The LIGO folks think big…
The longer the interferometer arms, the better the sensitivity.
So put one in space, of course.

14. Interference is easy when the light wave is a monochromatic plane wave
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.

15. Adding a non-monochro-matic wave to a delayed replica of itself
Constructive interference for all times (coherent) “Bright fringe”
Delay = 0:
Destructive interference for all times (coherent) “Dark fringe”)
Delay = ½ period (<< tc):
Incoherent addition No fringes.
Delay > tc:

16. Crossed Beams x q z Cross term is proportional to: Iout(x) x
Fringes (in position) x
Iout(x)
Fringe spacing:

17. Irradiance vs. position for crossed beams
Irradiance fringes occur where the beams overlap in space and time.
Figure courtesy of Hans Eichler, Laser Induced Dynamic Gratings

18. Big angle: small fringes. Small angle: big fringes.
The fringe spacing, L:
Large angle:
As the angle decreases to zero, the fringes become larger and larger, until finally, at q = 0, the intensity pattern becomes constant.
Small angle:

19. You can't see the spatial fringes unless the beam angle is very small!
The fringe spacing is:
L = 0.1 mm is about the minimum fringe spacing you can see:

20. 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.

21. The Michelson Interferometer (Misaligned)
Beam-
splitter
Input
beam
Mirror q z x
Suppose we misalign the mirrors,
so the beams cross at an angle
when they recombine at the beam
splitter. And we won't scan the delay.
If the input beam is a plane wave, the cross term becomes:
Fringes (in position) x
Iout(x)
Crossing beams maps delay onto position.

22. Optical interferometry :
Optical interferometer = instrument that generates interference fringe patterns resulting from optical path differences
it divides initial beam into two or more parts that travel different optical paths and then brought together again to produce interference pattern
it is divided into two main classes (depending on how initial beam is separated):
a. Wavefront division interferometers : portions of same wavefront of a coherent beam of light are sampled (e.g. Young’s double slit, Lloyd’s mirror, Fresnel’s biprism)
b. Amplitude-division interferometers : uses beam splitter (semireflecting film, prisms) to divide initial beam into two parts (amplitude is shared); e.g. Michelson interferometer uses interference of 2 beams, Fabry-Perot interferometer uses multiple beams

23. Michelson Interferometer:
Arrangement for Michelson interferometer
Beam (1) from extended source S split by beam splitter BS (which has a thin, semitransparent front surface metallic or dielectric film, deposited on glass)  amplitude splitting
Reflected beam (2) and transmitted beam (3) have equal amplitudes, are then reflected (at normal incidence) by mirrors M2 and M1, respectively, and their directions are then reversed
Returning to BS, beam (1) is transmitted and beam (3) is reflected by semitransparent film, so that they come together again and leave interferometer as beam (4)

24. Michelson Interferometer:
One of the mirrors has tilting adjustment screws that allow the surface of M1 to be made perpendicular to M2
One of the mirrors is movable along direction of beam so that difference between optical paths of beams (2) & (3) can be varied
Beam (3) traverses BS 3X, beam (2) traverses BS 1X, thus, a plate C is inserted parallel to BS in path of beam (2) to compensate for this
This will ensure optical paths of two beams can be made precisely equal (especially when white light is used)
White light fringes seen from a Michelson interferometer when optical path difference is made zero

25. Actual interferometer has 2 optical axes perpendicular to one another
Michelson Interferometer:
Actual interferometer has 2 optical axes perpendicular to one another
Equivalent optics for Michelson interferometer
To derive the optical path difference, we shall use an equivalent optical system having single optical axis
We work with virtual images of source S and mirror M1 via reflection in BS mirror
Take S, M1, and beams (1) & (3) to be rotated anti-clockwise by 90° about point of intersection of beams with BS mirror
New position: source plane S’, virtual image M1’
Light from point Q on source plane S’ is reflected from both mirrors M2 and M1’ (parallel) at optical path difference d

26. (angle  = inclination of beams relative to optical axis)
Michelson Interferometer:
Now two reflected beams appear to come from two virtual images Q1’ and Q2’ of object Q
Separation of images S1’ and S2’ = 2 X mirror separation  distance between Q1’ and Q2’ = 2d
Optical path difference between two beams emerging from interferometer is:
(angle  = inclination of beams relative to optical axis)
For normal beam,  = 0 and p = 2d
If  = m for constructive interference, the beams will interfere constructively again at every /2 translation of one of the mirrors
This optical system is now equivalent to the case of interference due to a plane parallel air film illuminated by extended source
Virtual fringes of equal inclination is seen by looking into BS along ray (4)

27. where phase difference is Net optical path difference is
Michelson Interferometer:
Assuming equal amplitudes of the two interfering beams, irradiance of fringe system of concentric circles is given by:
where phase difference is
Net optical path difference is
Relative phase shift =  because beam (2) undergoes 2 external reflection but beam (3) undergoes only one
For dark fringes: or
If d is such that centre fringe is dark (normal rays), then, its order is
(neighbouring dark fringes decrease in order outwards, as cos  decreases from max value of 1)
(12-1)
(12-2)

28. Putting (12-3) into (12-1), we get
Michelson Interferometer:
In order to express order such that it increases in number outwards instead, we introduce p where:
Putting (12-3) into (12-1), we get
where central fringe is now of order p = 0 and neighbouring fringes increase in order, outward
(12-3)
(12-4)
Eqns (12-1) & (12-4) indicate:
as d varies, a particular point in fringe pattern ( = constant) corresponds to gradually changing values of order of m or p
m = mmax = 100
p = mmax  m = 0
m = 99
p = 1
m = 98
p = 2
m = 97
p = 3

29.  we can measure  when d is known
Michelson Interferometer:
From eqn (12-1) :
(differentiate)
This means that fringes are more widely spaced when optical path differences (denominator) are small
If mirror translates d, number m of fringes passing a point near or at centre of pattern is:
 we can measure  when d is known
(12-5)
E.g. Fringes observed due to monochromatic light in Michelson interferometer. When movable mirror translates 0.73 mm, a shift of 300 fringes is seen. What is the wavelength of the light? What displacement of the fringe system takes place when a slide of glass of index 1.51 and mm thick is inserted in one arm of interferometer? (Assume light beam to be normal to glass surface.)
Glass inserted, one arm is extended by path difference of d = ng t  nair t, thus,

30. Michelson Interferometer (continued)
Fringes of equal thickness

31. Michelson Interferometer (continued)
Fringes of equal Inclination
Fringes of equal thickness
(Path differences increases outward from centre)
White light fringes

32. Applications of Michelson Interferometer :
(1)Adapted to measure thin film (similar to that in preceding lecture)
(2)Adapted to determine refractive index of a gas
Insert evacuated cell with plane, parallel windows in path of beam (3), fill it with gas at desired pressure and temperature for which ngas is to be measured. Fringe system formed is monitored as gas is gradually pumped out. m of net fringe shift related to change in optical path during replacement of gas by vacuum is counted. If actual length of cell = L, change in optical path is
from index is:
(3)To determine wavelength difference between 2 closely spaced components of a spectral “line”,  and ’ (e.g. Na lines: & nm)
Wavelengths  and ’ form thier own fringe systems according to
and
(12-6)
(12-7)
(12-8)

33. For a fixed path difference d, m = m’’
Applications of Michelson Interferometer (determine wavelength difference):
We see two sets of fringe patterns viewing near centre of circular system, i.e., cos   1
For a fixed path difference d, m = m’’
When two fringe systems coincide (in-sync), resultant pattern appears sharp
coincide
out-of-sync
When one fringe system is midway (out-of-sync) with other fringe system, resultant pattern appears blur or wash-out
Say at 1st coincidence, the orders (m & m’) of the two systems corresponding to  and ’ are related through:
where N is an interger
if optical path difference here is d1, then from (12-8), we have
(12-9)
(12-10)

34. Applications of Michelson Interferometer (determine wavelength difference):
Distortion of fringes of equal thickness produced by candle flame when inserted in one arm of Michelson interferometer.
Variations in temperature produce variations in optical path length by changing the refractive index of the air.
Now at 2nd coincidence, occurring at optical path difference d2, we have: or
Subtract (12-10) from (12-11), and writing mirror movement as d = d2  d1, we get
Since  and ’ are very close, wavelength difference  may then be approximated by
Related to application (2)
(12-11)
(12-12)
(12-13)

35. Energy levels for sodium - note Na doublets 588. 99 & 589
Energy levels for sodium - note Na doublets & nm – related to application (3): 1 2 3 4 5
Energy (eV) 6s 5s 4s 3s
5p1/2
4p1/2
3p1/2
5p3/2
4p3/2
3p3/2 5d 4d
3d5/2,3/2 5f 4f
589.6
589.0
616.1
615.4
515.3
514.9
497.9
498.3
568.2
568.8
Some of the energy levels and transitions for Na. Lines between levels represent transitions (labelled by corresponding wavelength in nm)
(Grotian Diagram)

36. Variations on the Michelson Interferometer
Twyman-Green Interferometer

37. Twyman-Green interferometer Testing of flat surfaces
The basic use of a Twyman-Green interferometer is for measuring surface height variations. The Twyman-Green and Fizeau give the same interferograms for testing surface flatness; the main advantages of the Twyman-Green are more versatility and it is a non-contact test, so there is less chance of scratching the surface under test, while the main disadvantage is that more high-quality optical components are required.
Fringes of equal thickness
All rays are parallel
cosq=1 => 2d=ml

38. Twyman-Green interferometer(continued) Testing spherical surface and prism

39. Twyman-Green Interferometer

40. BS & M3 are half-silvered mirror (semi-transparent)
Mach-Zehnder Interferometer - variation of Michelson Interferometer
Parallel beam divided into two at beam splitter BS
Each divided beam totally reflected by mirrors M1 & M2
Then beams are brought together again by M3
Path lengths of beam (1) and (2) around rectangular system and through glass of BS and M3 are identical
BS & M3 are half-silvered mirror (semi-transparent)
M1 & M2 are mirrors
Say we want to measure the geometry of air flow around an object in a wind tunnel (detected as local variations of pressure and refractive index)
Put windowed test chamber into path (1) and another identical chamber in path 2 to maintain equality of optical paths. The model + streamline flow of air are introduced in test chamber. Air-flow pattern will be revealed by fringe pattern.

41. Mach-Zehnder interferometer Testing transparent medium
Optical path: S=Ln
L: geometrical path through medium
Optical path difference: D=S2-S1
=L(n-1)
Bright fringe: L(n-1) = Nl
for N=0,1,2,3…
Therefore: N=L(n-1)/l

42. Mach-Zehnder interferometer (continued)
Example:
Optical path difference:
L=length of medium
If refractive index varies in y-axis only:
Fringe separation depends on the gradient of refractive index n
Linear: n(y)=no+n’y
=> y=Nl/n’L
(2) exponential: n(y)=no-n1exp(-ay)

43. Mach-Zehnder interferometer (continued) Reference fringe
Interpreting the interferogram can be confusing because ±Df give the same fringe pattern e.g. figure (a). IT is ambiguous whether the fringe order number N increases or decreases as we proceed inward.
This problem can be resolved by adding a known reference fringe pattern (b).
In (c ) the fringe order N increases upwards while that of (d) increases downwards.

44. Mach-Zehnder interferometer (continued) Assigning fringe order number
Without reference fringe. In this case, it is assumed that the order is known. The background is assigned N=0 and the subsequent bright fringe N=1, etc…
With reference fringe. Here the order N increases upwards. A line parallel to the reference fringe at the position of interest is drawn. The first intersection is assigned N=1, etc,,,