33. More about InterferenceMichelson Interferometer revisited

The field autocorrelationLIGO: the world’s largest interferometer

Other interferometersMach-Zehnder

Sagnac

Young’s double slit experimentwave-particle duality

Irradiance of a sum of two waves

2

*2

1

1Rec E E

I I I

Different wavelengths

Different polarizations

Same wavelength

Same polarizations

1 2I I I

1 2I I I 1 2I I I

Interference only occurs when two waves have the same wavelength and the same polarization.

A wave interferes with a time-delayed replica of itself only if the time delay is less than the coherence time.C?

Suppose the input beam is not monochromatic, so we write it as a general function E(t):(but suppose that it is a plane wave):

2 Re ( ') *( ' '

U IT c E t E t dt

The Michelson Interferometer produces the field autocorrelation

Beam-splitter

Inputbeam

DelayMirror Iout = 2I + c Re{E(t+L1 /c) E*(t+L2 /c)}

Changing variables: t' = t + L1 /c and letting = (L2 - L1)/c and T

/ 2 / 2

1 2

/ 2 / 2

( ) 2 Re ( / ) *( / )

T T

Out

T T

U I t dt U IT c E t L c E t L c dt

Measuring U as a function of gives the autocorrelation of E(t).

But Iout will vary rapidly as a function of time (fringes oscillate with the period of the light wave), and most detectors will simply integrate (i.e., average) over a relatively long time, T :

Reminder: the Autocorrelation Theorem

2

2

( ) *( { ( )}

(

F E t E t dt F E

E

For a light wave E(t), the Fourier transform of the autocorrelation:

The Fourier transform of a light-wave’s field autocorrelation is its spectrum!

A Michelson interferometer therefore can be used to measure the spectrum of a light wave. It is known as a Fourier Transform spectrometer.

For a monochromatic input wave, we saw that: 2 1 cos outI I And the Fourier transform of that is simply a delta function at frequency (ignore negative frequencies and the constant term), which is indeed the spectrum of the monochromatic input wave.

Simplest example:

Fourier Transform Spectrometer DataA Fourier Transform Spectrometer's detected light energy vs. delay is called an interferogram.

Fourier Transform Spectrometers find their most common use in the infrared where the fringes in delay are most easily generated. As a result, they are often called FT-IR spectrometers.

Interferogram

This interferogramis very narrow, so the spectrum is very broad.

“Fingerprint spectroscopy” for identification of unknown materials

Each absorption peak can be associated with a specific molecular vibrational motion.

Example:

Commercial FTIR instruments are widely used in chemical analysis

reminder: wavenumbers = (wavelength)-1, units of 1/cm

= 10 m = 2.5 m

Huge Michelson interferometers may someday detect gravity waves.

Gravitational waves (emitted by all moving objects) ever so slightly warp space-time. Colliding black holes and colliding neutron stars emit gravitational waves. But they’re tiny!

Gravitational waves are “quadrupole” waves, which stretch space in one direction and shrink it in another. They will cause one arm of a Michelson interferometer to stretch and the other to shrink.

The relative distance change (L1L2 ~ 10-16 cm) is less than the width of an atomic nucleus! So such measurements are very difficult!

Beam-splitter

Mirror

Mirror

L1

L2

L1 and L2 = 4 km!

have detected!

The LIGO project

Louisiana LIGO site

pipe containing one of the arms

Washington LIGO site

The Mach-Zehnder interferometer

The Mach-Zehnder interferometer requires two beam splitters.

It is often operated with something of interest in one arm, to measure phase changes.

Ludwig Mach1868 - 1951

Ludwig Zehnder1854 - 1949

A Mach-Zehnder interferogram

a plasma in an experimental fusion chamber

shift in fringes can be used to determine the plasma density

A Mach-Zehnder modulatorThis type of interferometer can also be built in a waveguide configuration. When combined with a voltage-driven phase modulator (like as in the Pockels effect), it offers a way to modulate light.

This is the most common design for digital modulators used in lightwave communications systems.

The Sagnac interferometer

The Sagnac interferometer can be used to sense rotation.

The two beams automatically take the same path around the interferometer, but in opposite directions. The paths can have different lengths, however, if the device is rotating.

?Georges Sagnac

1869 - 1926

Sagnac Interferometer: gyroscope

20 0

20

cross term exp( ) exp( )

sin ( )

E jkd E jkd

I kd

2

20

(2 / ) 2 ( ) / 2 / cross term sin (2 / )

d R T R R c R c cI k c

AreaArea

Suppose that the beam splitter moves by a distance, d, in thetime, T, it takes light to circumnavigate the Sagnac interferometer.

As a result, one beam will travel more, and the other less distance.

If R = the interferometer radius, and = its angular velocity:

Optical gyroscope

Thus, the Sagnac Interferometer is sensitive to rotation, and the sensitivity depends on its area. And it need not be round!

The double slit experimentSuppose we illuminate an opaque screen with two narrow slits in it, and look at the light transmitted through:

If the slits are very narrow, then the diffraction pattern is easy to compute (in the Fraunhofer regime):

Aperture(x1) = sum of two delta functions

Ediffracted (x0) = Fourier transform{sum of two delta functions}

= sum of two complex exponentials (recall: beat patterns)

We can also understand this as a manifestation of interference between two light sources:

Young’s double slit

Thomas Young1773 - 1829

In 1801, Young discovered interference effects using this approach, and concluded that he had proven that light was a wave.

A dozen years later, he was the first to attempt a translation of the Rosetta Stone, from which we now know how to read ancient hieroglyphs.

Young’s double slit as a demonstration of the quantum nature of light

a stream of particles, or waves

a particle pattern a wave pattern

Questions:What if we send our stream of particles one at a time?What if we look to see which slit each particle went through?

Diffraction pattern formed by a stream of electrons hitting a

double slit, one at a time.

Louis de Broglie predicted that, if waves can behave like particles, then particles should also behave as waves. And their wavelength is related to their momentum, = h/p

Louis de Broglie, 1892-1987

A layman’s explanation of particle-wave duality:http://www.felderbooks.com/papers/quantum.html

• confirmed by Clinton Davisson and Lester Germer (Bell Labs) in 1927• Nobel prize for de Broglie, 1929

Young’s double slit works for particles too

But we can destroy the interference pattern, by looking at the slits…

1st electron 2nd electron 3rd electron

4th electron 5th electron

Eventually, a diffraction pattern

But then, what if we try to detect which slit each electron went through?

No more diffraction pattern.

Knowing that the electron went through

one particular slit changes its behavior.

Is light a particle or a wave? Can we be more careful about how we measure?

A Mach-Zender interferometer

Particle: signal should appear on D1 or D2, but not both (anti-correlated)Wave: interference leads to fringes, signals on both detectors

If we remove this beam splitter…

Particle: signal should appear on D1 or D2, but not both (anti-correlated)Wave: signals on both detectors but no fringes (correlated)

This is a large enough distance so that the information about whether the beam splitter was in or out could not reach the first beam splitter in time, due to the relativistic speed limit on information.

Wheeler’s “delayed choice” experimentFor each photon, we either remove the 2nd beam splitter, or we don’t.

We make the decision individually for each photon, after the photon has passed the input beam splitter.

Did the single photon interact with the first beam splitter as if it was a wave or as if it was a particle?

48 meters

V. Jacques et al., Science (March 2007)

beam splitter present

beam splitter absent

this is proof that the signal traveled by both paths, not just one

this is proof that the signal traveled by one or the other path, not both

Not shown here: the two signals are almost perfectly anti-correlated

An afterward: Schroedinger’s catA thought experiment proposed (but never actually carried out, so far as we know) by Erwin Schroedinger, one of the pioneers of quantum theory:

You set up an experiment where a particle has a 50% chance of decaying after an hour. The particle is in a box, and you don't look inside. You also put a cat in the box, with an apparatus that will kill the cat if the particle decays. After an hour, what is the state of the system?

An afterward: Schroedinger’s cat

Quantum mechanics says that these two statements are definitively different, and that the intuitive one is wrong.

Experiments have verified this claim (not with cats).

• Intuitive answer: The cat is either alive or dead, and you just don’t know which, until you look in the box.

• Non-intuitive answer: The cat is both alive and dead, until you look. By looking, you cause the system to ‘collapse’ into either one or the other of these states, with a 50% probability of each.