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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
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
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
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
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
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
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
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?
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
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