Wave Superposition What happens when two or more waves overlap in a certain region

of space at the same time?

To find the resulting wave according to the principle of superposition we should sum the fields (𝑬𝑬 & 𝑩𝑩) of the individual waves.

We should not sum the energy density (𝒖𝒖 ∝ 𝑬𝑬𝟐𝟐, 𝑩𝑩

𝟐𝟐) nor the

power density (𝑺𝑺 ∝ 𝑬𝑬 × 𝑩𝑩) of the individual waves.

After we have found the resulting wave, we can then calculate the resulting energy density and power density.

Note: Starting in the next slide, I will be using the Greek letter 𝝍𝝍 as a placeholder for a Cartesian component of the electric (𝑬𝑬) or magnetic (𝑩𝑩) fields.

1

just add them up !!

𝜓𝜓1 𝑟𝑟, 𝑡𝑡 𝜓𝜓2 𝑟𝑟, 𝑡𝑡

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓2 𝑟𝑟, 𝑡𝑡 + 𝜓𝜓2 𝑟𝑟, 𝑡𝑡

&

Given two waves

2

A) Two Planes Waves, Same Frequency

𝜓𝜓1 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 − 𝜔𝜔 𝑡𝑡

𝜓𝜓2 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2 − 𝜔𝜔 𝑡𝑡

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓1 𝑟𝑟, 𝑡𝑡 + 𝜓𝜓2 𝑟𝑟, 𝑡𝑡

= 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 − 𝜔𝜔 𝑡𝑡 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2 − 𝜔𝜔 𝑡𝑡

≡ 𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 − 𝜔𝜔 𝑡𝑡

𝛼𝛼1 ≡ 𝑘𝑘1. 𝑟𝑟 + 𝜀𝜀1

𝛼𝛼2 ≡ 𝑘𝑘2. 𝑟𝑟 + 𝜀𝜀2

𝜓𝜓0 = ? ?𝛼𝛼 = ? ?

to be determined3

𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 − 𝜔𝜔 𝑡𝑡 = 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 − 𝜔𝜔 𝑡𝑡 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2 − 𝜔𝜔 𝑡𝑡

𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 + 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼 𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 =

+ 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 + 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡

= 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 + 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡

𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 = 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2

𝜓𝜓0 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼 = 𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼24

𝑡𝑡𝑡𝑡𝑠𝑠 𝛼𝛼 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼2𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2

𝜓𝜓02 = 𝜓𝜓0,12 + 𝜓𝜓0,2

2 + 2 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2 − 𝛼𝛼1

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 − 𝜔𝜔 𝑡𝑡

= 𝜓𝜓0,1 − 𝜓𝜓0,22 + 4 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐2

𝛼𝛼2 − 𝛼𝛼12

Phase:

Amplitude:

Full Wave:5

Graphical Representation & Phasor:

𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼1

𝜓𝜓0,1𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼1

𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼2

𝜓𝜓0,2𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝛼𝛼2

𝛼𝛼2

𝛼𝛼1

𝛼𝛼

𝜓𝜓0𝛼𝛼2 − 𝛼𝛼1

6

Examples

7

A.1) Two Waves Propagating in a Collinear Direction

𝛼𝛼1 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘 𝑘𝑘1

𝑘𝑘2

8

Collinear Direction: Phase

𝛼𝛼1 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

𝑡𝑡𝑡𝑡𝑠𝑠 𝛼𝛼 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝛼𝛼 ≡ 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀

𝑡𝑡𝑡𝑡𝑠𝑠 𝜀𝜀 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀2𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2 9

Collinear Direction: Amplitude

𝛼𝛼1 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

𝛼𝛼2 − 𝛼𝛼1 = 𝜀𝜀2 − 𝜀𝜀1

𝜓𝜓02 = 𝜓𝜓0,12 + 𝜓𝜓0,2

2 + 2 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2 − 𝜀𝜀1

= 𝜓𝜓0,1 − 𝜓𝜓0,22 + 4 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐2

𝜀𝜀2 − 𝜀𝜀12

10

Collinear Direction: Summary

𝛼𝛼1 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀

𝑡𝑡𝑡𝑡𝑠𝑠 𝜀𝜀 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀2𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2

𝜓𝜓02 = 𝜓𝜓0,12 + 𝜓𝜓0,2

2 + 2 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2 − 𝜀𝜀1

= 𝜓𝜓0,1 − 𝜓𝜓0,22 + 4 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐2

𝜀𝜀2 − 𝜀𝜀12

𝛼𝛼 ≡ 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀

11

𝛼𝛼1 = −𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

− 𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

A.2) Two Plane Waves Propagating in Opposite Direction

𝑘𝑘1

𝑘𝑘2

12

Opposite Direction: Phase

𝛼𝛼1 = − 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

− 𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

𝑡𝑡𝑡𝑡𝑠𝑠 𝛼𝛼 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 − 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

𝛼𝛼 ≡ 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀

𝑡𝑡𝑡𝑡𝑠𝑠 𝜀𝜀 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀2𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2 13

𝛼𝛼2 − 𝛼𝛼1 = 𝑘𝑘2 − 𝑘𝑘1 . 𝑟𝑟 + 𝜀𝜀2 − 𝜀𝜀1 = 2 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2 − 𝜀𝜀1

𝜓𝜓02 = 𝜓𝜓0,12 + 𝜓𝜓0,2

2 + 2 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2 − 𝜀𝜀1

Opposite Direction: Amplitude

𝛼𝛼1 = −𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

− 𝑘𝑘1 = 𝑘𝑘2 = 𝑘𝑘

= 𝜓𝜓0,1 − 𝜓𝜓0,22 + 4 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑘𝑘. 𝑟𝑟 +

𝜀𝜀2 − 𝜀𝜀1214

Opposite Direction: Summary

𝛼𝛼1 = − 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀1 𝛼𝛼2 = 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2

− 𝑘𝑘1= 𝑘𝑘2 = 𝑘𝑘

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀

𝜓𝜓02 = 𝜓𝜓0,12 + 𝜓𝜓0,2

2 + 2 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀2 − 𝜀𝜀1

= 𝜓𝜓0,1 − 𝜓𝜓0,22 + 4 𝜓𝜓0,1 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑘𝑘. 𝑟𝑟 +

𝜀𝜀2 − 𝜀𝜀12

𝛼𝛼 ≡ 𝑘𝑘. 𝑟𝑟 + 𝜀𝜀 𝑡𝑡𝑡𝑡𝑠𝑠 𝜀𝜀 =𝜓𝜓0,1 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑠𝑠𝑠𝑠 𝜀𝜀2𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀1 + 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜀𝜀2

15

𝜓𝜓2 𝑟𝑟, 𝑡𝑡 = r𝜓𝜓1 𝑟𝑟, 𝑡𝑡 = 1

16

𝜓𝜓1 𝑟𝑟, 𝑡𝑡𝜓𝜓2 𝑟𝑟, 𝑡𝑡

Opposite DirectionDifferent Amplitudes

17

Opposite DirectionSame Amplitude

𝜓𝜓0,1 = 𝜓𝜓0,2

𝜓𝜓0 = 2 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘. 𝑟𝑟 +𝜀𝜀2 − 𝜀𝜀1

2

𝜓𝜓1 𝑟𝑟, 𝑡𝑡𝜓𝜓2 𝑟𝑟, 𝑡𝑡

18

A.3) If the waves have the same frequency 𝜔𝜔1 = 𝜔𝜔2 then 𝑘𝑘1 = 𝑘𝑘2,

and:

𝑘𝑘1

𝑘𝑘2 𝑘𝑘2,∥

𝑘𝑘1,⊥

𝑘𝑘2,⊥

𝑘𝑘1,∥

ii) counter propagating

i) co-propagating

𝑘𝑘2,∥𝑘𝑘2,⊥

𝑘𝑘1,⊥𝑘𝑘1,∥

19

B) Two Planes Waves, Different Frequencies

𝜓𝜓1 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑1 = 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘1. 𝑟𝑟 − 𝜔𝜔1 𝑡𝑡 + 𝜀𝜀1

𝜓𝜓2 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑2 = 𝜓𝜓0,2 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘2. 𝑟𝑟 − 𝜔𝜔2 𝑡𝑡 + 𝜀𝜀2

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓1 𝑟𝑟, 𝑡𝑡 + 𝜓𝜓2 𝑟𝑟, 𝑡𝑡

20

𝜓𝜓1 𝑟𝑟, 𝑡𝑡 = 𝑡𝑡 + 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 + 𝐵𝐵

𝐴𝐴 ≡12𝑘𝑘1 + 𝑘𝑘2 . 𝑟𝑟 −

12𝜔𝜔1 + 𝜔𝜔2 𝑡𝑡 +

12𝜀𝜀1 + 𝜀𝜀2

𝐵𝐵 ≡12𝑘𝑘1 − 𝑘𝑘2 . 𝑟𝑟 −

12𝜔𝜔1 − 𝜔𝜔2 𝑡𝑡 +

12𝜀𝜀1 − 𝜀𝜀2

𝜓𝜓2 𝑟𝑟, 𝑡𝑡 = 𝑡𝑡 − 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 − 𝐵𝐵

𝑡𝑡 ≡12𝜓𝜓0,1 + 𝜓𝜓0,2

𝑏𝑏 ≡12𝜓𝜓0,1 − 𝜓𝜓0,2

21

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 𝜓𝜓1 𝑟𝑟, 𝑡𝑡 + 𝜓𝜓2 𝑟𝑟, 𝑡𝑡

= 𝑡𝑡 + 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 + 𝐵𝐵 + 𝑡𝑡 − 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 − 𝐵𝐵

= 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 + 𝐵𝐵 + 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 − 𝐵𝐵 + 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 + 𝐵𝐵 − 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 − 𝐵𝐵

= 2 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝐵𝐵 − 2 𝑏𝑏 𝑐𝑐𝑠𝑠𝑠𝑠 𝐴𝐴 𝑐𝑐𝑠𝑠𝑠𝑠 𝐵𝐵

22

Same Amplitude

𝑡𝑡 ≡12𝜓𝜓0,1 + 𝜓𝜓0,2 = 𝜓𝜓0,1

𝑏𝑏 ≡12𝜓𝜓0,1 − 𝜓𝜓0,2 = 0

𝜓𝜓0,1 = 𝜓𝜓0,2

𝜓𝜓 𝑟𝑟, 𝑡𝑡 = 2 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐 𝐵𝐵

= 2 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐12𝑘𝑘1 + 𝑘𝑘2 . 𝑟𝑟 −

12𝜔𝜔1 + 𝜔𝜔2 𝑡𝑡 +

12𝜀𝜀1 + 𝜀𝜀2

𝑐𝑐𝑐𝑐𝑐𝑐12𝑘𝑘1 − 𝑘𝑘2 . 𝑟𝑟 −

12𝜔𝜔1 − 𝜔𝜔2 𝑡𝑡 +

12𝜀𝜀1 − 𝜀𝜀2

23

𝑘𝑘 ≡12𝑘𝑘1 + 𝑘𝑘2 𝜔𝜔 ≡

12𝜔𝜔1 + 𝜔𝜔2 𝜀𝜀 ≡

12𝜀𝜀1 + 𝜀𝜀2

∆𝑘𝑘 ≡12𝑘𝑘1 − 𝑘𝑘2 ∆𝜔𝜔 ≡

12𝜔𝜔1 − 𝜔𝜔2 ∆𝜀𝜀 ≡

12𝜀𝜀1 − 𝜀𝜀2

𝜓𝜓 𝑟𝑟, 𝑡𝑡 =

= 2 𝜓𝜓0,1 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 . 𝑟𝑟 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀 𝑐𝑐𝑐𝑐𝑐𝑐 ∆𝑘𝑘 . 𝑟𝑟 − ∆𝜔𝜔 𝑡𝑡 + ∆𝜀𝜀

24

𝑘𝑘 . 𝑟𝑟 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀 = 𝜑𝜑

∆𝑘𝑘 . 𝑟𝑟 − ∆𝜔𝜔 𝑡𝑡 + ∆𝜀𝜀 = 𝜑𝜑𝑚𝑚

𝑣𝑣𝑝𝑝 =𝜔𝜔𝑘𝑘

𝑣𝑣𝑔𝑔 =∆𝜔𝜔∆𝑘𝑘

=𝑑𝑑𝜔𝜔𝑑𝑑𝑘𝑘

phase velocity

group velocity25

Phase & Group Velocities

26

Phase & Group Velocities

27

𝑘𝑘 𝜔𝜔 = 𝑠𝑠 𝜔𝜔 𝑘𝑘𝑜𝑜 = 𝑠𝑠 𝜔𝜔𝜔𝜔𝑐𝑐

𝑑𝑑𝑘𝑘 𝜔𝜔𝑑𝑑𝜔𝜔

=1𝑐𝑐

𝑠𝑠 𝜔𝜔 + 𝜔𝜔𝑑𝑑𝑠𝑠 𝜔𝜔𝑑𝑑𝜔𝜔

𝑠𝑠𝑔𝑔 𝜔𝜔 ≡𝑐𝑐𝑣𝑣𝑔𝑔

= 𝑐𝑐𝑑𝑑𝑘𝑘𝑑𝑑𝜔𝜔

= 𝑠𝑠 𝜔𝜔 + 𝜔𝜔𝑑𝑑𝑠𝑠 𝜔𝜔𝑑𝑑𝜔𝜔

𝑑𝑑𝑠𝑠 𝜔𝜔𝑑𝑑𝜔𝜔

< 0anomalous dispersion

𝑑𝑑𝑠𝑠 𝜔𝜔𝑑𝑑𝜔𝜔

> 0

𝑑𝑑𝑠𝑠 𝜔𝜔𝑑𝑑𝜔𝜔 > 0

normal dispersion

normal dispersion

𝑠𝑠 𝜔𝜔 =𝑐𝑐𝑣𝑣𝑝𝑝

phase velocity

group velocity

28

Beat Frequency: ∆𝜔𝜔

∆𝜔𝜔 𝑇𝑇 = 2 𝜋𝜋 𝑇𝑇 =2 𝜋𝜋∆𝜔𝜔

=1∆𝜐𝜐

29

Fourier Analysis

in a nutshell …

30

Let’s start with a periodic function:

𝑓𝑓 𝑥𝑥 =𝑡𝑡02

+ �𝑚𝑚=1

𝑡𝑡𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 + �𝑚𝑚=1

𝑏𝑏𝑚𝑚 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥

𝑡𝑡𝑚𝑚 =2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

𝑏𝑏𝑚𝑚 =2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

𝑚𝑚 = 0, 1, 2, 3, …

𝑚𝑚 = 1, 2, 3, …

31

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥 =2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2 𝑡𝑡0

2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+2𝑃𝑃�𝑚𝑚=1

𝑡𝑡𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+2𝑃𝑃�𝑚𝑚=1

𝑏𝑏𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

𝑓𝑓 𝑥𝑥 =𝑡𝑡02

+ �𝑚𝑚=1

𝑡𝑡𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 + �𝑚𝑚=1

𝑏𝑏𝑚𝑚 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥

=2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2 𝑡𝑡0

2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+1𝑃𝑃�𝑚𝑚=1

𝑡𝑡𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 −𝑚𝑚𝑥 𝑥𝑥 + 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 + 𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+1𝑃𝑃�𝑚𝑚=1

𝑏𝑏𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 −𝑚𝑚𝑥 𝑥𝑥 + 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 + 𝑚𝑚𝑥 𝑥𝑥

= 𝑡𝑡𝑚𝑚𝑚 𝑚𝑚𝑥 = 0, 1, 2, 3, …

2𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 ��−𝑃𝑃2

�+𝑃𝑃2𝑑𝑑𝑥𝑥multiply by and integrate over

on both sides of the equation,

with 32

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥 =2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2 𝑡𝑡0

2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+2𝑃𝑃�𝑚𝑚=1

𝑡𝑡𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+2𝑃𝑃�𝑚𝑚=1

𝑏𝑏𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

𝑓𝑓 𝑥𝑥 =𝑡𝑡02

+ �𝑚𝑚=1

𝑡𝑡𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 + �𝑚𝑚=1

𝑏𝑏𝑚𝑚 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥

=2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2 𝑡𝑡0

2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+1𝑃𝑃�𝑚𝑚=1

𝑡𝑡𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 −𝑚𝑚𝑥 𝑥𝑥 + 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 + 𝑚𝑚𝑥 𝑥𝑥 𝑑𝑑𝑥𝑥

+1𝑃𝑃�𝑚𝑚=1

𝑏𝑏𝑚𝑚 ��−𝑃𝑃2

�+𝑃𝑃2𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 −𝑚𝑚𝑥 𝑥𝑥 − 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 + 𝑚𝑚𝑥 𝑥𝑥

= 𝑏𝑏𝑚𝑚𝑚

2𝑃𝑃𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚𝑥 𝑥𝑥 ��−𝑃𝑃2

�+𝑃𝑃2𝑑𝑑𝑥𝑥multiply by and integrate over

on both sides of the equation,

with 𝑚𝑚𝑥 = 1, 2, 3, … 33

𝑓𝑓 𝑥𝑥 =

+ �𝑚𝑚=1

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑚 𝑐𝑐𝑠𝑠𝑠𝑠2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥

+ �𝑚𝑚=1

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥 +

=1𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑚 +𝑡𝑡02

𝑡𝑡𝑚𝑚

𝑏𝑏𝑚𝑚

34

𝑓𝑓 𝑥𝑥 =

=1𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑚 + �

𝑚𝑚=1

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

𝑘𝑘 ≡2 𝜋𝜋𝑃𝑃

𝑚𝑚 ∆𝑘𝑘𝜋𝜋

=2𝑃𝑃∆𝑚𝑚

35

𝑓𝑓 𝑥𝑥 =

=1𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑑𝑑𝑥𝑥𝑚 + �

𝑚𝑚=1

2𝑃𝑃�

�−𝑃𝑃2

�+𝑃𝑃2𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐

2 𝜋𝜋𝑃𝑃

𝑚𝑚 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

= �0

∞𝑑𝑑𝑘𝑘𝜋𝜋

�−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

= �−∞

∞ 𝑑𝑑𝑘𝑘2 𝜋𝜋

�−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑥

Make the period goes to infinity: lim

𝑃𝑃 →∞

36

0

𝑓𝑓 𝑥𝑥 = �−∞

∞ 𝑑𝑑𝑘𝑘2 𝜋𝜋

�−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑚

0 = 𝑠𝑠 �−∞

∞ 𝑑𝑑𝑘𝑘2 𝜋𝜋

�−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑐𝑐𝑠𝑠𝑠𝑠 𝑘𝑘 𝑥𝑥𝑚 − 𝑥𝑥 𝑑𝑑𝑥𝑥𝑚

𝑓𝑓 𝑥𝑥 = �−∞

∞ 𝑑𝑑𝑘𝑘2 𝜋𝜋

�−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑥𝑥′−𝑥𝑥 𝑑𝑑𝑥𝑥𝑚

=1

2 𝜋𝜋�−∞

∞𝐹𝐹 𝑘𝑘 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑥𝑥 𝑑𝑑𝑘𝑘

𝐹𝐹 𝑘𝑘 ≡ �−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑒𝑒+ 𝑖𝑖 𝑘𝑘 𝑥𝑥′ 𝑑𝑑𝑥𝑥𝑚where

37

𝑓𝑓 𝑥𝑥 =1

2 𝜋𝜋�−∞

∞𝐹𝐹 𝑘𝑘 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑥𝑥 𝑑𝑑𝑘𝑘

𝐹𝐹 𝑘𝑘 = �−∞

+∞𝑓𝑓 𝑥𝑥𝑥 𝑒𝑒+ 𝑖𝑖 𝑘𝑘 𝑥𝑥′ 𝑑𝑑𝑥𝑥𝑚

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞𝐹𝐹 𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

𝐹𝐹 𝜔𝜔 = �−∞

+∞𝑓𝑓 𝑡𝑡𝑥 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡𝑚

space x and spatial frequency k

time t and angular frequency ω

38

𝑥𝑥, 𝑘𝑘

𝑡𝑡,𝜔𝜔

𝑓𝑓(𝑡𝑡) 𝐹𝐹 𝜔𝜔

𝑓𝑓(𝑡𝑡)

𝑡𝑡

𝐹𝐹 𝜔𝜔

𝜔𝜔

Big picture:

39

Fourier Transform Pairs with Mathematica

40

(Lateral) Spatial Confinement

and

Wave Divergence

∆𝑘𝑘 ≈1𝐿𝐿

41

𝐿𝐿2

Example: top hat function

𝑓𝑓 𝑥𝑥

𝑥𝑥

−𝐿𝐿2≤ 𝑥𝑥 ≤

+𝐿𝐿2

when

otherwise

𝑓𝑓 𝑥𝑥 =

𝐴𝐴

0

42

𝑘𝑘 𝐿𝐿2

−𝜋𝜋−2𝜋𝜋−3𝜋𝜋 +𝜋𝜋 +2𝜋𝜋 +3𝜋𝜋 +4𝜋𝜋

𝑘𝑘𝑚𝑚 𝐿𝐿2

= 𝑚𝑚 𝜋𝜋

𝑚𝑚 = ±1, ±2, ±3, …

at 𝐹𝐹 𝑘𝑘 = 0 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘 𝐿𝐿2

𝐹𝐹 𝑘𝑘 = 𝐴𝐴 𝐿𝐿 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘 𝐿𝐿2

𝑘𝑘+1 − 𝑘𝑘−1 =4 𝜋𝜋𝐿𝐿

𝐿𝐿 ∆𝑘𝑘 ≈ 4 𝜋𝜋

𝐹𝐹 𝑘𝑘 = �−𝐿𝐿2

+𝐿𝐿2𝐴𝐴 𝑒𝑒+ 𝑖𝑖 𝑘𝑘 𝑥𝑥′ 𝑑𝑑𝑥𝑥𝑚

43

𝑓𝑓(𝑥𝑥) =1

2 𝜋𝜋�−∞

∞𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑥𝑥 𝑑𝑑𝑘𝑘

𝐹𝐹 𝑘𝑘 = 𝐴𝐴 𝐿𝐿 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘 𝐿𝐿2

−𝐿𝐿2≤ 𝑥𝑥 ≤

+𝐿𝐿2

when

otherwise𝑓𝑓 𝑥𝑥 = �

𝐴𝐴

0

𝐹𝐹 𝑘𝑘

𝑘𝑘 𝐿𝐿2

𝐴𝐴 𝐿𝐿 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘 𝐿𝐿2

�𝐹𝐹 𝑘𝑘𝐴𝐴 𝐿𝐿

44

𝑓𝑓(𝑥𝑥) =1𝜋𝜋�0

∞cos(𝑘𝑘 𝑥𝑥) 𝑑𝑑𝑘𝑘𝐴𝐴 𝐿𝐿 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑘𝑘 𝐿𝐿2

even function

(Longitudinal) Pulse Duration

and

Spectral Width

∆𝜈𝜈 ≈1𝑇𝑇

45

𝑓𝑓 𝑡𝑡 =

−𝑇𝑇2≤ 𝑡𝑡 ≤

+𝑇𝑇2

when

otherwise

Example: truncated harmonic wave

𝐴𝐴 cos 𝜔𝜔𝑜𝑜 𝑡𝑡

0

46

𝜔𝜔𝑜𝑜 = 100

𝑇𝑇 = 2

𝑓𝑓 𝑡𝑡

𝑡𝑡

𝐴𝐴 = 1

𝐹𝐹 𝜔𝜔 =𝐴𝐴2𝑇𝑇 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 + 𝜔𝜔𝑜𝑜 + 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜

𝐹𝐹 𝜔𝜔 = �−𝑇𝑇2

+𝑇𝑇2𝐴𝐴 cos 𝜔𝜔𝑜𝑜 𝑡𝑡 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡𝑚

47

even function

𝜔𝜔𝑚𝑚 𝑇𝑇2

=−𝜔𝜔𝑜𝑜 𝑇𝑇

2+ 𝑚𝑚 𝜋𝜋

𝑚𝑚 = ±1, ±2, ±3, …

at 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑇𝑇2𝜔𝜔 + 𝜔𝜔𝑜𝑜 = 0

𝜔𝜔+1 − 𝜔𝜔−1 =4 𝜋𝜋𝑇𝑇

𝑇𝑇 ∆𝜔𝜔 ≈ 4 𝜋𝜋

𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜 = 0 𝜔𝜔𝑚𝑚 𝑇𝑇

2=𝜔𝜔𝑜𝑜 𝑇𝑇

2+ 𝑚𝑚 𝜋𝜋at

48

𝜔𝜔𝑜𝑜 = 100

𝑇𝑇 = 2

𝐴𝐴 = 1𝐹𝐹 𝜔𝜔

𝜔𝜔

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞𝐹𝐹 𝜔𝜔 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

𝐹𝐹 𝜔𝜔 =𝐴𝐴2𝑇𝑇 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 + 𝜔𝜔𝑜𝑜 + 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔𝐴𝐴

2𝑇𝑇 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 + 𝜔𝜔𝑜𝑜 + 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�0

∞𝐴𝐴 𝑇𝑇 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 + 𝜔𝜔𝑜𝑜 + 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

𝑓𝑓(𝑡𝑡) ≅1

2 𝜋𝜋�0

∞𝐴𝐴 𝑇𝑇𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐

𝑇𝑇2𝜔𝜔 − 𝜔𝜔𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

49

𝐹𝐹 𝜔𝜔

even function

Example: light emission and lifetime of an excited state

50

After emission

Do we get photons with just one single frequency?

or

Is the emitted light completely monochromatic?

51

𝑓𝑓 𝑡𝑡 ∝ 𝐴𝐴 𝑒𝑒 �−𝑡𝑡𝑇𝑇

𝑓𝑓 𝑡𝑡 = 𝐴𝐴 cos 𝜔𝜔𝑜𝑜 𝑡𝑡 𝑒𝑒 �− 𝑡𝑡𝑇𝑇

52

𝑓𝑓 𝑡𝑡

𝑡𝑡

𝜔𝜔𝑜𝑜 = 40𝑇𝑇 = 2

𝐴𝐴 = 1

𝐹𝐹 𝜔𝜔 = �−∞

∞𝐴𝐴 cos 𝜔𝜔𝑜𝑜 𝑡𝑡𝑥 𝑒𝑒

�− 𝑡𝑡𝑇𝑇 𝑒𝑒+ 𝑖𝑖 𝜔𝜔 𝑡𝑡′ 𝑑𝑑𝑡𝑡𝑚

53

𝐹𝐹 𝜔𝜔

𝜔𝜔

=𝐴𝐴 𝑇𝑇

1 + 𝑇𝑇2 𝜔𝜔 − 𝜔𝜔𝑜𝑜 2 +𝐴𝐴 𝑇𝑇

1 + 𝑇𝑇2 𝜔𝜔 + 𝜔𝜔𝑜𝑜 2

𝜔𝜔𝑜𝑜 = 40𝑇𝑇 = 2

𝐴𝐴 = 1

even function

54

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞ 𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 − 𝜔𝜔𝑜𝑜 2 +

𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 + 𝜔𝜔𝑜𝑜 2 𝑒𝑒− 𝑖𝑖 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

=1𝜋𝜋�0

∞ 𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 − 𝜔𝜔𝑜𝑜 2 +

𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 + 𝜔𝜔𝑜𝑜 2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

≅𝐴𝐴 𝑇𝑇𝜋𝜋�0

∞ 11 + 𝑇𝑇2 𝜔𝜔 − 𝜔𝜔𝑜𝑜 2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

𝑓𝑓(𝑡𝑡) =1

2 𝜋𝜋�−∞

∞ 𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 − 𝜔𝜔𝑜𝑜 2 +

𝐴𝐴 𝑇𝑇1 + 𝑇𝑇2 𝜔𝜔 + 𝜔𝜔𝑜𝑜 2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔 𝑡𝑡 𝑑𝑑𝜔𝜔

55

∆𝜈𝜈 ≅1𝑇𝑇

𝜈𝜈 =𝐸𝐸2 − 𝐸𝐸1

ℎ±

1𝑇𝑇

56

Coherence Time & Coherence Length

𝑇𝑇 ≅1∆𝜈𝜈

Coherence Time

𝑙𝑙 = 𝑐𝑐 𝑇𝑇 ≅𝑐𝑐∆𝜈𝜈

Coherence Length

𝑇𝑇

𝑇𝑇

∆𝑡𝑡 < 𝑇𝑇

∆𝑡𝑡 > 𝑇𝑇

Phase difference is constant

Phase difference varies randomly

coherent

𝑡𝑡

𝑡𝑡

incoherent

57

𝑙𝑙 ≅𝑐𝑐∆𝜈𝜈

=𝜆𝜆2

∆𝜆𝜆

Examples of coherence length

𝑐𝑐 = 𝜆𝜆 𝜈𝜈 ∆𝜈𝜈 =𝑐𝑐𝜆𝜆2

∆𝜆𝜆

White light

He-Cd laser

Fiber laser

𝜆𝜆 ≈ 550 𝑠𝑠𝑚𝑚 ∆𝜆𝜆 ≈ 300 𝑠𝑠𝑚𝑚 𝑙𝑙 ≈ 1 𝜇𝜇𝑚𝑚

𝜆𝜆 ≈ 442.5 𝑠𝑠𝑚𝑚 ∆𝜈𝜈 ≈ 163 MHz 𝑙𝑙 ≈ 1.8 𝑚𝑚

𝜆𝜆 ≈ 1.55 𝜇𝜇𝑚𝑚 ∆𝜈𝜈 ≈ 1.4 kHz 𝑙𝑙 ≈ 210 𝑘𝑘𝑚𝑚