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Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference, so we add wave intensities (not amplitudes)
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Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Jan 18, 2018

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Verity Robinson

Coherence time and coherence. length “Longitudinal” coherence time t c, or length l c = ct c : 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
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Page 1: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Can we see fringes coming from the interference of two lamps?

…from the interference of two lasers?

If the light is phase incoherent, no interference, so we add wave intensities (not amplitudes)

Page 2: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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!

Page 3: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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

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Page 4: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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

Page 5: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Michelson Interferometer

What can we learn from I(t), the interferogram?

Frequencies, phases? Length of a pulse?

Page 6: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Beam diagnostic interferogram for light emitted by electron beam at Brookhaven

This light has coherence length of 1-2 mm

Page 7: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Intensity measurements

Let Io be the intensity in each arm of the interferometer.

If we move one arm so that t >> tc, there’s no interference (no fringes), and we should measure _____ Io. Why?

If t<< tc , we get typical interference, so at a bright fringe we should get ____ Io.

At a dark fringe we should get ____ Io.

Page 8: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Io

Io

I

Page 9: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Single frequency case

Time averaged intensity in one arm

, ( ) exp exp

exp exp

o oI t I c E i t i t

i t i t

t t t

t

exp exp2o

o o ocI t E i t E i t I

Averaged intensity combined at detector

2 1 cosoI It t

Fringes keep going as t increases! So tc is infinite for single frequency

Page 10: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Many-frequency case

2 1 cosI I dt t

Interferogram of gaussian pulse.

Page 11: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Many-frequency case

, the intensity in one arm

2 1 cosI I dt t

oI I d

It

gt

2 1 ReoI It g t

1 ( )i

o

d I eFT I

II d

t

g t

a dimensionless complex function to represent the oscillations in I t

Page 12: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Suppose we have a short pulse, and put a thick piece of glass in the beam before the interferometer. The ___

a) wiggles shiftb) wiggles narrowc) envelope shiftsd) envelope broadens e) pattern stays the same

Page 13: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Suppose we put a thick piece of absorbing colored glass that absorbs the outer parts of the spectrum The ___

a) envelope shiftsb) envelope narrowsc) envelope broadensd) pattern stays the same

Page 14: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Suppose we put the thick piece of glass in one arm of the interferometer. What will happen?

, exp exp

exp exp

o oI t c E d i t i t i

i t i t i

t t

t

This is a different theory from what we’re developing today

Page 15: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

Summary

What can we learn about a beam of light from Michaelson interferometry?

2 1 cos 2 1 ReoI I d It t g t

Only things related to the power spectrum! No phase info.

c dt g t t

1

ct

We could also measure with a grating and detector, and get all the info from that.

I

For estimates use this!

Page 16: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

1 ( )

o

FT II

g t

If we FT-1 E(), we get E(t)

If we FT-1 I(), we get …..

… g (t), something that gives us the coherence time of the beam E(t)!

I g t

E t

FT of

I t

I t

Page 17: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

I

Suppose with filters we take sunlight and form I() as a rectangular function centered at o.

The form of the wiggles gt of the interferogram will be _____a) sincb) gaussianc) rectangular

Page 18: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

I

If the width of the rectangle is o /10

The coherence time will be about

a) 10 o

b) 1 /10oc) 10 /o

d) 100 o

How many oscillations will gt make before it dies down to about ½ or so of its peak amplitude?

o

Page 19: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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

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time t (fs)

Page 20: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

How many typical periods does it take for this light to get out of phase with previous part of the beam?

t (fs)0 500 1000 1500 2000 2500 3000

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time t (fs)

Page 21: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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

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time t (fs)

Page 22: Can we see fringes coming from the interference of two lamps? …from the interference of two lasers? If the light is phase incoherent, no interference,

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Actual unnormalized interferogram shape (half of it). We know I() is “boxy” because of the ringing in gt!

delay t (fs)