SUPERPOSITION OF WAVES

Waves of different k vector, same frequency

Counter-propagating waves

Intersecting waves

Waves mixing (AOM)

Co-propagating, random phase

Waves of the same k vector,same frequency

Waves of the same k vector direction, different frequencies

Beat note

Creation of an arbitrary

Group velocity

Superposition of Ducks and ducklings

Constructive and destructive Superposition is just like adding twovectors ,

Waves of the same k vector, same frequency

constructive destructive

Waves of the same k vector, same frequency

Example of a laser:

Constructive interference: adding two co-propagating beams of amplitude

The intensity in each beam is

Incoherent sum: the total intensity is I = 2I0

Coherent constructive sum: the total intensity is I = 4I0

Coherent destructive sum: the total intensity is I = 0

What happened to energy conservation???

Waves of the same k vector, same frequency Energy conservation

If the energy is lost by destructive interference, it has to reappear somewhere elseby constructive interference

and a complex transmission coefficient

Incident intensity

Energy conservation:

Reflected intensity

Transmitted intensity

A beam splitter is an element with a complex reflection coefficient

How to combine two identical beams?

Waves of the same k vector, same frequency Energy conservation

Mach Zehnder:

Michelson

Random and Coherent source

Interference of two circular waves – Wavelength (decreasing bottom to top) and wave centers distance (increasing to the right)

Waves of different k vector, same frequency

Two sine waves traveling in the same direction

Waves of the same k vector, different frequencies

Two sine waves traveling in opposite directions “standing wave”

Waves of the same k vector, different frequencies

Two sine waves with different frequencies: Beats

Waves of the same k vector, different frequencies

How to measure a wavelength with highest accuracy?

At z constant: a detector measures the difference frequency.

cos A + cos B = 2cos((A+B)/2)cos((A-B)/2).

cos((k+dk)x - (+d)t) + cos((k-dk)x - (-d)t) = 2cos(kx - t)cos((dk)x - (d)t)

Another look at superposition

k = 12, = 2, dk = 1, d = 0

k = 12, = 0, dk = 1, d = 2

                                                

k = 12, = 7, dk = 1, d = 2

                                                

Superposition of waves of different frequencies,leading to an arbitrary temporal profile.

Waves of the same k vector, different frequencies

Beat note

Creation of an arbitrary

Group velocity