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

2. Complex number Im
Z = a + i b
a = Re (z)
b = Im (z)
Z* = a - i b b z q Re a
Add and subtract exponentials and trigonometric functions
Plane waves, spherical waves

3. 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???

4. 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
A beam splitter is an element with a complex reflection coefficient
and a complex transmission coefficient
Incident intensity
Reflected intensity
Transmitted intensity
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
A beam splitter is an element with a complex reflection coefficient
and a complex transmission coefficient
Energy conservation:

6. Waves of the same k vector, same frequency
Energy conservation
Mach Zehnder:
Michelson

7. Waves of the same k vector, same frequency
Energy conservation
Antiresonant ring