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ENE 451Fundamental of Optical Engineering
Lecture 6 extra
This can occur when 2 or more waves overlap each other in space.
Assume that 2 waves have the same frequency.
The interference results in an addition of wave amplitudes.
Interference
( , )
n
,
where = real amplitude
= phase
n i t zi r tn n
n
A r t e e
A
Two wave interference: the resulting field becomes
Interference
1 2
1 2
1 2
1 2 1 2where , , , and are real.
i t zi iAe A e e
A A
Power density, PA
Interference
1 2 1 2
1 2 1 2
*
1 2 1 2
2 21 2 1 2
2 21 2 1 2 1 22 cos
A
i i i i
i i
A
P
Ae A e Ae A e
A A A A e e
P A A A A
Interference
-15 -10 -5 0 5 10 150
0.5
1
1.5
phase shift (delta phi)
Pow
er d
ensi
ty (P
a)
Normalized power density vs. Phase shift
If lights behave without interference, we would expect where E1 and E2 are orthogonal.
If there is an interference, it is clearly seen that the term is the term for “interference”.
Interference
2 21 2AP A A
1 2 1 22 cosA A
PA is max for 1 - 2 = 0, 2, …, 2N PA is min for 1 - 2 = , 3, …, (2N+1)
Therefore,
Interference
22 21 2 1 2 1 2max
22 21 2 1 2 1 2min
2
2
A
A
P A A A A A A
P A A A A A A
Two waves N waves
Plane wave interference Diffraction grating
Young’s double slit experiment Interference filters
Reflection from film
Electro-optic modulators
Interferometers
Examples of interference