8.03 Lecture 17 - MIT OpenCourseWare · 2020-01-27 · 8.03 Lecture 17 *Review: Snell’s Law n 1...
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8.03 Lecture 17 *Review: Snell’s Law n 1 sin θ 1 = n 2 sin θ 2 Polarization: Another way to “add more dimensions”! If wee choose our coordinate system such that the wave is going in the z -direction then: E(z,t) = Re ψ 0 · e i(kz-ωt) where ψ = ψ 1 ˆ x + ψ 2 ˆ y. Can be understand as superposition of two EM waves!: ψ 1 = A 1 e iψ 1 ψ 2 = A 2 e iψ 2 Or sometimes we write it as E = Re Z · e i(kz-ωt) Where Z = ψ 1 ψ 2 (1.) If we add two waves with no phase difference: E 1 = E 0 cos(kz - ωt)ˆ x E 2 = E 0 cos(kz - ωt)ˆ y E = E 1 + E 2 Write it in matrix notation: E = Re (E 0 ˆ x + E 0 ˆ y)e i(kz-ωt) E = Re E 0 E 0 e i(kz-ωt)
Transcript of 8.03 Lecture 17 - MIT OpenCourseWare · 2020-01-27 · 8.03 Lecture 17 *Review: Snell’s Law n 1...
8.03 Lecture 17
*Review: Snell’s Lawn1 sin θ1 = n2 sin θ2
Polarization: Another way to “add more dimensions”! If wee choose our coordinate system suchthat the wave is going in the z-direction then:
~E(z, t) = Re[~ψ0 · ei(kz−ωt)
]where ~ψ = ψ1x+ ψ2y. Can be understand as superposition of two EM waves!:
ψ1 = A1eiψ1 ψ2 = A2e
iψ2
Or sometimes we write it asE = Re
[Z · ei(kz−ωt)
]Where
Z =(ψ1ψ2
)(1.) If we add two waves with no phase difference:
~E1 = E0 cos(kz − ωt)x~E2 = E0 cos(kz − ωt)y
~E = ~E1 + ~E2
Write it in matrix notation:
~E = Re[(E0x+ E0y)ei(kz−ωt)
]E = Re
[(E0E0
)ei(kz−ωt)
]
Z = E0
(11
)Linearly polarized! Other examples:
Z = E0
(10
)Z = E0
(01
)Z = E0
(cos θsin θ
)
(2.) If we add two waves together with the same amplitude but a phase difference of π/2
~E1 = E0 cos(kz − ωt)x~E2 = E0 sin(kz − ωt)y
= E0 cos(kz − ωt− π/2)y~E = Re
[(E0x− iE0y)ei(kz−ωt)
]E = Re
[E0
(1−i
)ei(kz−ωt)
]
Z = E0
(1−i
)
Clockwise “right-handed.” Circularly polarized!
Counter-clockwise:Z = E0
(1i
)
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(3.)We can also add two waves with different amplitude
~E1 = E02 cos(kz − ωt)x
~E2 = E0 sin(kz − ωt)y~E = ~E1 + ~E2
E = Re[E0
(1/2−i
)ei(kz−ωt)
]
“Elliptically polarized”:
Z = E0
(1/2i
)Z = E0
(A
iB
)Z = E0
(C
−iD
)
(4.) There is another way to produce elliptically polarized EM waves: phase difference:
∆φ 6= π
2 ,3π2 · · · otherwise, circularly polarized
Example:
~E1 = E0 cos(kz − ωt)x~E2 = E0 cos(kz − ωt+ ∆φ)y
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Elliptically polarized
In general: A ≥ |B|
Z =(ψ1ψ2
)= eiφ
(A cos θ − iB sin θA sin θ + iB cos θ
)(5.) “Unpolarized” light: EM waves produced independently by a large number of uncorrelatedemitters. Not:
Because that gives zero!
*Emitted at different time with slightly different frequency!
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Polarizer Example: grid of metal wires:
1. If the EM wave is in the y direction then it will induce movement of the electron in the ydirection (in the metal wires). EM wave is reflected like what we worked on before with metalplates.
2. EM wave in the x direction cannot induce movement of electrons in the x direction
In this case, the “Easy Axis” is x
P0 =(
1 00 0
)for polarizer with x easy axis:
Pπ/2 =(
0 00 1
)for polarizer with y easy axis:
In general:
Pθ =(
cos2 θ cos θ sin θcos θ sin θ sin2 θ
)Pπ/4 =
(1/2 1/21/2 1/2
)
Intensity ∝ 〈 ~E2〉. After passing through the polarizer the perpendicular component is eliminated~E0 ⇒ | ~Ef | = | ~E0| cos θ ⇒ If ∝ 〈 ~Ef 〉 ⇒ If = I0 cos2 θ
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8.03SC Physics III: Vibrations and WavesFall 2016
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