1. 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

2. Superposition of Ducks and ducklings

3. Constructive and destructive Superposition is just like adding two vectors, Waves of the same k vector, same frequency constructive destructive

4. 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 = 2I 0 Coherent constructive sum: the total intensity is I = 4I 0 Coherent destructive sum: the total intensity is I = 0 What happened to energy conservation???

5. Waves of the same k vector, same frequency Energy conservation If the energy is lost by destructive interference, it has to reappear somewhere else by 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

6. How to combine two identical beams? Waves of the same k vector, same frequency Energy conservation Mach Zehnder: Michelson

7. Random and Coherent source

8. 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

9. Two sine waves traveling in the same direction Waves of the same k vector, different frequencies

10. Two sine waves traveling in opposite directions “standing wave” Waves of the same k vector, different frequencies

11. Two sine waves with different frequencies: Beats Waves of the same k vector, different frequencies

12. How to measure a wavelength with highest accuracy? At z constant: a detector measures the difference frequency.

13. 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

14. k = 12,  = 0, dk = 1, d  = 2 k = 12,  = 7, dk = 1, d  = 2

15. 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