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