Optics L3

Shengwang Du

2015, the Year of Light

! 3.1 Basic Laws of EM Theory

!FE = q

!E

!FM = q

!v ×!B

!F = q

!E + q!v ×

!B

A charge in the EM field !E

q

What is the EM field? It can be sensed by an electric charge.

! 3.1.1 Faraday Induction Law

“Convert magnetism into electricity”, Michael Faraday, 1822

!E ⋅d!l = − d

dtC"∫!B ⋅d!S

A∫∫ = −∂!B∂t⋅d!S

A∫∫

!E ⋅d!l =

C"∫ (∇×!E) ⋅d

!S

A∫∫

∇×!E = −∂

!B∂t

!E

!B

Stokes’ theorem

! 3.1.3 Gauss’s Law - Electric

!E ⋅d!S

A"∫∫ =Qε0=1ε0

ρ dVV∫∫∫

!E ⋅d!S

A"∫∫ = (∇⋅!E)dV

V∫∫∫

∇⋅!E = ρ

ε0

Stokes theorm

! 3.1.3 Gauss’s Law - Magnetic

!B ⋅d!S

A"∫∫ = 0

∇⋅!B = 0

There is no magnetic charge (monopole)!

! 3.1.4 Ampere’s Law

“Convert electricity into magnetism”, James Maxwell

!B ⋅d!l = µ0C!∫

"J +ε0

∂!E∂t

$

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'

()⋅d!S

A∫∫

∇×!B = µ0

!J +ε0

∂!E∂t

$

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'

()

! Maxwell Equations - General

∇×!E = −∂

!B∂t

∇×!B = µ0

!J +ε0

∂!E∂t

$

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'

()

∇⋅!E = ρ

ε0

∇⋅!B = 0

Question: Do the above equations apply for vacuum or/and medium?

! Maxwell Equations in a Medium

∇×!E = −∂

!B∂t

∇×!B = µ0

!J +ε0

∂!E∂t

$

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'

()

∇⋅!E = ρ

ε0∇⋅!B = 0

ρ = ρ f + ρb

!J =!J f +!Jb

!P!M

Polarization

Magnetization

ρb = −∇⋅!PBounded charge density

Bounded current density !Jb =∇×

!M +

∂!P∂t

! Maxwell Equations in a Medium

∇×!E = −∂

!B∂t

∇×!H =!J f +

∂!D∂t

∇⋅!D = ρ f

∇⋅!B = 0

!D = ε0

!E +!P

!H =

!Bµ0

−!M

!P!M

Polarization

Magnetization

ρb = −∇⋅!PBounded charge density

Bounded current density !Jb =∇×

!M +

∂!P∂t

! Maxwell Equations in a Linear Medium

!D = ε0

!E +!P = ε

!E

!H =

!Bµ0

−!M =

!Bµ

!P!M

Polarization

Magnetization

ρb = −∇⋅!PBounded charge density

Bounded current density !Jb =∇×

!M +

∂!P∂t

∇×!E = −∂

!B∂t

∇×!B = µ

!J f +ε

∂!E∂t

$

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'

()

∇⋅!E =

ρ f

ε∇⋅!B = 0

!

Maxwell Equations (Vacuum, No Source)

∇×!E = −∂

!B∂t

∇×!B = µ0ε0

∂!E∂t

∇⋅!E = 0

∇⋅!B = 0

ρ = 0!J = 0

∇× (∇×) =∇(∇⋅) ∇2!E = µ0ε0

∂2!E

∂t2

∇2!B = µ0ε0

∂2!B

∂t2

! 3.2 Electromagnetic Waves

∇2!E = µ0ε0

∂2!E

∂t2=1c2∂2!E

∂t2

∇2!B = µ0ε0

∂2!B

∂t2=1c2∂2!B

∂t2

Recall: Ch2, Wave Motion ∇2Ψ =1v∂2Ψ∂t2

c = 1ε0µ0

≈ 3×108m / s

Maxwell: Light is EM wave!

! 3.2.1 Transverse Wave

0 =∇⋅!E =∇⋅ iEoxe

i(!k ⋅!r−ωt+φx ) + jEoye

i(!k ⋅!r−ωt+φy ) + kEoze

i(!k ⋅!r−ωt+φz )$

%&'

!E = iEoxe

i(!k ⋅!r−ωt+φx ) + jEoye

i(!k ⋅!r−ωt+φy ) + kEoze

i(!k ⋅!r−ωt+φz )

0 = Eox∂∂xei(!k ⋅!r−ωt+φx ) +Eoy

∂∂yei(!k ⋅!r−ωt+φy ) +Eoz

∂∂zei(!k ⋅!r−ωt+φz )

!k ⋅ !r = kx x + ky y + kz z

0 = ikxEoxei(!k ⋅!r−ωt+φx ) + ikyEoye

i(!k ⋅!r−ωt+φy ) + ikzEoze

i(!k ⋅!r−ωt+φz ) = i

!k ⋅!E

!k ⋅!E = 0 Transverse wave

!k ⊥!E

!k ⋅!B = 0

!k ⊥!B

! EM Wave: Transverse Wave

∇×!E = −∂

!B∂t

!E = iEoxe

i(kz−ωt+φEx ) + jEoyei(kz−ωt+φEy )

!B = iBoxe

i(kz−ωt+φBx ) + jBoyei(kz−ωt+φBy )

∇⇔ i!k ∂

∂t⇔−iω

!k ×!E =ω

!B

Recall: Ch2, Wave Motion

c = ωk

!B = 1

ω

!k ×!E = 1

ccω

!k ×!E = 1

cs ×!E

k = ωc

s = cω

!k =!kk

E =CB

! EM Wave: Transverse Wave

!k ⋅!E = 0

!k ⋅!B = 0

!B = 1

ω

!k ×!E = 1

cs ×!E

E =CB