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)

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!

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

Michelson Interferometer

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

Frequencies, phases? Length of a pulse?

Beam diagnostic interferogram for light emitted by electron beam at Brookhaven

This light has coherence length of 1-2 mm

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.

Io

Io

I

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

Many-frequency case

2 1 cosI I dt t

Interferogram of gaussian pulse.

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

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

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

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

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!

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

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

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

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)

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

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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.

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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)