Interference See Chapter 9 of Hecht

This is the interference pattern of two waves.The two waves add or subtract to form the light and dark regions of the interference pattern This picture does not show an interference pattern. It is simply the concentric waves of two points sources drawn in the same plane. Contrast this with the image to the right A single point creates waves with concentric circles of light and dark bands.

Waves are not simple two dimensional objects. When they interfere with each other, peaks and valleys are formed. Many interference patterns look like two-dimensional systems of light and dark bands because they are being viewed from above. In this picture the system tilted so it can be viewed from the side.

Consider two waves intersecting at some location At the point where the waves intersect the total electric field will be: Almost always the intensity of the radiation is detected TA = Time Average For convenience, neglect constants and say:

Interference may, or may not, arise depending in the nature of I12 Getting interference is not easy. The two beams must have some common polarization

Temporal Coherence Waves should have the same frequency for interference. We will need:

This term decays away in a time called the coherence time: Coherence Length: This is the length for which wavetrains stay in phase.

It is extremely difficult to maintain coherence for two beams UNLESS they come from the same laser. Consider two different, but similar lasers. At l=850 nm one can “lock” laser to an atomic Cs transition. Possible to have two lasers with: Long, but not infinite. What is worse – “Mode Hops” Every ms or so a laser will randomly shift its phase. Two lasers will do this independently and interference shifts.

Back to Interference Criteria Assume two light beams from the same source. Depends on the phase difference of 2 waves

Note that: Constructive interference: Destructive interference:

Spatial Coherence Source 1 Observation point Source 2 Rays coming from extended source will have different phases. Hard to get interference between sources 1 and 2. Phase of light all mixed up. Source 1 Observation point Source 2 For point sources, arriving light has well defined phase.

2 Pinhole Sources Grimaldi’s experiment of 1665 Observation screen – no interference observed. No “fringes” Sun 2 pinholes or slits Sun is an extended source – no spatial coherence. No interference

Young Thomas Young, 1805. Used the sun, but an additional pinhole creates a (small) source with spatial coherence Sun 2 pinholes 1 pinhole Observation screen

Difference in path lengths is: But r1 Constructive interference when where m is an integer Peaks and troughs on observing screen. Correctly predicts location of peaks of troughs, But not envelope (next week).

Waves versus particles (or so we think)

Cool demonstration of double slit on the web. See: http://micro.magnet.fsu.edu/primer/java/doubleslit/ You can change wavelength of laser and the distance between the two slits.

Light & Matter, Waves & Particles – de Broglie Wave and interference effects can be seen with matter too! Quantum wave properties: De Broglie said, why not matter too?!?

Interference properties seen with electrons, neutrons, atoms, and now even molecules like C60 and C70! Interferometers or the Double Slit: Interference seen even when only one particle is in system. Particle (be it electron, photon, atom, etc) goes through both slits at once.

Observed interference of C60 and C70 See results of Prof. Anton Zeilinger and his group http://www.quantum.univie.ac.at/research/

Standing Waves Consider two counter-propagating waves from a single laser. Say too that they have equal amplitudes.

Standing Waves Consider two counter-propagating waves from a single laser. Say too that they have equal amplitudes.

Two traveling waves produces a standing wave.

Microwave Ovens Standing microwaves Peaks and troughs => Hot spots and cold spots => Nodes and anti-nodes Spinning dish hopefully brings all parts of food into contact with nodes Demo with marshmallows and a microwave

Beamsplitters In order to create multiple beams from a single laser one needs to use a “beamsplitter” R*I0 r*E0 E0 t*E0 Laser I0 T*I0 Usually (but not always) us 50-50 beamsplitter; half the light transmitted, half reflected. Example- half silvered mirror.

Polarizing Beamsplitters Some beamsplitters separated light according to polarization Laser

Mach-Zender Interferometer

Mach-Zender Interferometer Say that the length for the top path is L1 and L2 for the bottom. At detectors D1 and D2 If other factors would change the acquired phase for the two paths it would affect counts at D1 and D2.

L1 Mirror 2 BS2 L2 L2 Mirror 1 BS1 L1 Atoms or neutrons: Say interferometer is in a gravitational field. The arms of length L1 are parallel to the ground, while when particles are in the arms of length L2 they climb up against gravity. Interferometer pivoted about bottom arm by angle a.

Changed phase between two interferometer paths by insertion of Al in one path, or rotating interferometer in Earth’s gravitational field.

Atom Interferometry – Overlapping Na atoms from Bose-Einstein Condensate Atom Laser: Results from Wolfgang Ketterle’s group, MIT. Ketterle shares 2001 Noble Prize in Physics

Michelson Interferometer

L2 L1 I0 For equal arm lengths => L1 = L2 No light out of that beamsplitter port for all wavelengths – White Light Fringe

Michelson-Morley’s Search for the Aether

Homework 7, problem 1 Mirror 2 L Mirror 1 Incident light L Beamsplitter Calculate time for light to traverse each arm, according to ether theory. Take v to be Earth’s orbital velocity. Ether wind of speed v

Gravitational Radiation Detection Laser Interferometric Gravitational Wave Observatory LIGO

3 k m ( ± 1 s ) CALTECH Pasadena MIT Boston HANFORD Washington LIVINGSTON Louisiana

Hanford Observatory 4 km 2 km

Livingston Observatory 4 km

LIGO Interferometers Power Recycled Michelson Interferometer With Fabry-Perot Cavities end test mass Light bounces back and forth along arms about 30 times Light is “recycled” about 50 times input test mass Laser beam splitter signal 4 km Fabry-Perot arm cavity

Vibration Isolation Systems

Core Optics

Core Optics Suspension and Control

Core Optics Installation and Alignment

Washington 2k Pre-stabilized Laser Custom-built 10 W Nd:YAG Laser Stabilization cavities for frequency and beam shape

Fabry-Perot Interferometer Mirror 1 Mirror 2 Er Et E0 L Cavity of length L, incident light of amplitude E0 and wavelength l, k=2p/l

Real Mirrors have losses Er Et E0 It I0 A is the loss coefficient. For a very good mirror A~10-4 to 10-3

Free Spectral Range When 2kL changes by 2p we get another resonance Where m is some integer Free Spectral Range (FSR) = c/2L L=2cm => FSR=7.5GHz Cavity resonances function as “fence posts” or references

Fabry-Perot Interferometer as an Optical Spectrum Analyzer What’s going on with a laser??? An integral number of half-wavelengths fit into a laser cavity

gain frequency=(c/2L)n, n an integrer The laser medium will have some gain profile, as a function of frequency.

Overlap of gain profile and possible longitudinal modes

Resulting observed laser modes at various frequencies.

Fabry-Perot mirror mounted to a piezo-electric crystal Vary cavity length, and scan frequencies that resonate Mirror 1 Mirror 2 AC Voltage PZT L

The Speckle Effect