1. You play a song on two speakers (not stereo)
You play a song on two speakers (not stereo). How far back can you place the 2nd speaker and still hear interference effects?
(Ignore 1/r intensity drop off)

2. A simpler problem, clearly defined:
You play the same song on two speakers. With a microphone you average the intensity over the entire song.
How does Iavg(x) depend on the separation, and on the song?

3. A simpler problem, clearly defined:
How does Iavg(x) depend on the separation, and on the song?
Only on the power spectrum Iavg(w) of the song! No phase information matters.
Iavg(x) will be greatest at x=0, will oscillate with period of lavg, and the oscillations die out in a distance (v/Dw)

4. What if I play identical random noise on both speakers?
Iavg(x) still comes from Iavg(w) : will be greatest at x=0, will oscillate with period of lavg, and the oscillations die out in a distance (v/Dw).
Noise might be random, but can be interfered with itself over a distance a distance (v/Dw).

5. What if I play two different songs?
Iavg(x) will be constant =Iavg1 + Iavg2 …same as if I played independent noise on each speaker.

6. Temporal coherence and interference of light
How thick can a piece of glass be to still see interference fringes?
…it depends on the coherence length of the light we use!

7. Coherence time and coherence. length
“Longitudinal” coherence time tc, or length lc = ctc : time (distance) interval over which we can reasonably predict the phase of a wave at another time (or distance backward/forward in the wave), from a knowledge of the present phase
OR: time (distance) shift for an amplitude-splitting interference experiment, over which we can expect to see sharp fringes

8. What’s similar about these waves?
What’s different?

9. Michelson Interferometer
What can we learn from I(t), the interferogram?
Frequencies, phases? Length of a pulse?

10. Beam diagnostic interferogram for light emitted by electron beam at Brookhaven
This light has coherence length of 1-2 mm

11. Intensity measurements
Let Io be the intensity in each arm of the interferometer.
If t<< tc , we get typical interference, so at a bright fringe we should get ____ Io.
At a dark fringe we should get ____ Io.
If we move one arm so that t >> tc, there’s no interference (no fringes), and we should measure _____ Io. Why?

12. Io I

13. Single frequency case Time averaged intensity in one arm
Averaged intensity combined at detector
Fringes keep going as t increases! So tc is infinite for single frequency

14. Many-frequency case
Interferogram of gaussian pulse.

15. Many-frequency case I(t) g(t)
a dimensionless complex function to represent the oscillations in
g(t)
, the intensity in one arm

16. Suppose we have a short pulse, and put a thick piece of glass in the beam before the interferometer. The ___
a) wiggles shift
b) wiggles narrow
c) envelope shifts
d) envelope broadens
e) pattern stays the same

17. Why a long, dispersed pulse will have the same I(t) as the original short one.

18. print transparencies

19. Suppose we put a thick piece of absorbing colored glass that absorbs the outer parts of the spectrum The ___
a) envelope shifts
b) envelope narrows
c) envelope broadens
d) pattern stays the same

20. Suppose we put the thick piece of glass in one arm of the interferometer. What will happen?
This is a different theory from what we’re developing today

21. Summary
What can we learn about a beam of light from Michaelson interferometry?
Only things related to the power spectrum! No phase info.
For estimates use this!
We could also measure with a grating and detector, and get all the info from that.

22. If we FT-1 E(w), we get E(t)
If we FT-1 I(w), we get …..
… g (t), something that gives us the coherence time of the beam E(t)!
FT of

23. Suppose with filters we take sunlight and form I(w) as a rectangular function centered at wo.
The form of the wiggles g(t) of the interferogram will be _____
sinc
gaussian
rectangular I w

24. If the width of the rectangle is wo /10
The coherence time will be about
10 wo
1 /(10wo)
10 /wo
100 wo I wo w
How many oscillations will g(t) make before it dies down to about ½ or so of its peak amplitude?

25. time t (fs)
E(t) is shown with time increments of femtoseconds (10-15 sec).
The approx. frequency w=2p/T of the light is ______x1012 rad/sec
a) 5 b) c) 30 fs

26. time t (fs)
t (fs)
How many typical periods does it take for this light to get out of phase with previous part of the beam?

27. time t (fs)
Sketch what the interferogram I(t) would look like, in femtoseconds of delay t. Mark the coherence time and the average period of light.

28. Actual unnormalized interferogram shape (half of it)
Actual unnormalized interferogram shape (half of it). We know I(w) is “boxy” because of the ringing in g(t)!
delay t (fs)