1. Coherence and Interference Coherence Temporal coherence Spatial coherence Interference Parallel polarizations interfere; perpendicular polarizations don't. The Michelson Interferometer Fringes in delay Measure of Temporal Coherence The Fourier Transform Spectrometer The Misaligned Michelson Interferometer Fringes in position Measure of Spatial Coherence Opals use interference between tiny structures to yield bright colors.

2. 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: 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

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

4. 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 is very large (the entire visible spectrum). Lasers can have coherence times as long as about a second, which is amazing; that's >10 14 cycles!

5. The spatial coherence depends on the emitter size and its distance away. The van Cittert-Zernike Theorem states that the spatial coherence area A c 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.

6. Irradiance of a sum of two waves Different colors Different polarizations Same colors Same polarizations Interference only occurs when the waves have the same color and polarization. We also discussed incoherence, and that’s what this lecture is about!

7. The irradiance when combining a beam with a delayed replica of itself has fringes. Suppose the two beams are E 0 exp(i  t) and E 0 exp[i  t-  )], that is, a beam and itself delayed by some time  : Okay, the irradiance is given by: Fringes (in delay) - I 

8. Varying 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: Do 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–  ) Mirror Output beam

9. We can also vary the delay using a mirror pair or corner cube. Mirror pairs involve two reflections and displace the return beam in space: But out-of-plane tilt yields a nonparallel return beam. Corner cubes involve three reflections and also displace the return beam 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–  ) Mirrors Output beam [Edmund Scientific]

10. The Michelson Interferometer Beam- splitter Input beam Delay Mirror Fringes (in delay):  L = 2(L 2 – L 1 ) 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: I out L1L1 where:  L = 2(L 2 – L 1 ) L2L2 Output beam “Bright fringe” “Dark fringe”

11. The Michelson Interferometer Beam- splitter Input beam Delay Mirror The most obvious application of the Michelson Interferometer is to measure the wavelength of monochromatic light.  L = 2(L 2 – L 1 ) I out L1L1 L2L2 Output beam

12. Huge Michelson Interferometers may someday detect gravity waves. Beam- splitter Mirror L1L1 L2L2 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. Unfortunately, the relative distance ( L 1 -L 2 ~ 10 -16 cm) is less than the width of a nucleus! So such measurements are very very difficult! L 1 and L 2 = 4 km!

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

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

15. 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.

16. 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:

17. Suppose the input beam is not monochromatic (but is perfectly spatially coherent):  I out = 2I + c  Re{E(t+2L 1 /c) E*(t+2L 2 /c)} Now, I out will vary rapidly in time, and most detectors will simply integrate over a relatively long time, T : The Michelson Interferometer is a Fourier Transform Spectrometer The Field Autocorrelation! Beam- splitter Delay Mirror L1L1 L2L2 Recall that the Fourier Transform of the Field Autocorrelation is the spectrum!! Changing variables: t' = t + 2L 1 /c and letting  = 2(L 2 - L 1 )/c and T 

18. Fourier Transform Spectrometer Interferogram The Michelson interferometer output—the interferogram—Fourier transforms to the spectrum. The spectral phase plays no role! (The temporal phase does, however.) Integrated irradiance 0 Delay Michelson interferometer integrated irradiance 2/2/ 1/  Frequency Intensity  Spectrum  A Fourier Transform Spectrometer's detected light energy vs. delay is called an interferogram.

19. Fourier Transform Spectrometer Data Interferogram This interferogram is very narrow, so the spectrum is very broad. Fourier Transform Spectrometers are most commonly used in the infrared where the fringes in delay are most easily generated. As a result, they are often called FTIR's. Actual interferogram from a Fourier Transform Spectrometer

20. Fourier Transform Spectrometers Maximum path difference: 1 m Minimum resolution: 0.005 /cm Spectral range: 2.2 to 18  m Accuracy: 10 -3 /cm to 10 -4 /cm Dynamic range: 19 bits (5 x 10 5 ) A compact commercial FT spectrometer from Nicolet Fourier-transform spectrometers are now available for wave- lengths even in the UV! Strangely, they’re still called FTIR’s.

21. Crossed Beams  z x Cross term is proportional to: Fringes (in position) x I out (x) Fringe spacing:

22. Irradiance vs. position for crossed beams Irradiance fringes occur where the beams overlap in space and time.

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

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

25. 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.

26. The Michelson Interferometer and Spatial Fringes 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: Crossing beams maps delay onto position. Beam- splitter Input beam Mirror  z x Fringes (in position) x I out (x)

27. Michelson-Morley experiment 19 th -century physicists thought that light was a vibration of a medium, like sound. So they postulated the existence of a medium whose vibrations were light: aether. Michelson and Morley realized that the earth could not always be stationary with respect to the aether. And light would have a different path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether. Mirror Supposed velocity of earth through the aether Parallel and anti-parallel propagation Perpendicular propagation Beam- splitter Mirror

28. Michelson-Morley Experiment: Details If light requires a medium, then its velocity depends on the velocity of the medium. Velocity vectors add. Parallel velocities Anti-parallel velocities

29. Perpendicular velocity to mirror Perpendicular velocity after mirror Michelson-Morley Experiment: Details In the other arm of the interferometer, the total velocity must be perpendicular, so light must propagate at an angle.

30. Michelson-Morley Experiment: Details Perpendicular propagation Parallel and anti-parallel propagation The delays for the two arms depend differently on the velocity of the aether! If v is the earth’s velocity around the sun, 3 x 10 4 m/s, and L = 1 m, then: Let c be the speed of light, and v be the velocity of the aether.

31. Michelson-Morley Experiment: Results Michelson and Morley's results from A. A. Michelson, Studies in Optics Interference fringes showed no change as the interferometer was rotated. The Michelson interferometer was (and may still be) the most sensitive measure of distance (or time) ever invented and should’ve revealed a fringe shift as it was rotated with respect to the aether velocity. Their apparatus

32. Fresnel's Biprism A prism with an apex angle of about 179° refracts the left half of the beam to the right and the right half of the beam to the left. Fringe pattern observed by interfering two beams created by Fresnel's biprism