Download Electromagnetism of Continuous Media — Optics

Electromagnetism of Continuous
Media — Optics
Petr Kužel
Course for 3rd year of
University studies
(Université Paris-Nord)
Series of 14 lectures in Department
of Dielectrics, IOP
Topics of Interest
•
•
•
•
•
•
•
•
•
Layered systems and optical coatings:
Birefringence and polarization optics:
Dispersion and pulse propagation:
Electro- and magneto-optics:
Nonlinear optics:
Waveguides and fibers:
Inhomogeneous media:
Coherence:
Diffraction, gaussian beams:
in detail
in detail
yes
yes
basics
basics
basics
marginal
no
Electromagnetic Spectrum
380 nm
770 nm
visible spectrum
RADIO WAVES
OPTICAL
RANGE
γ-RAYS
MW
(radar)
X-RAYS
UV
vacuum near
wavelength
(m)
frequency:
10-16
10-14
10-12
3×1020 Hz
UHF
VHF
FM
AC
AM
IR
near
10-10
10-8
(1 Å) (1 nm)
10-6
(1 µm)
3×1017 Hz
6×1014 Hz
far
10-4
1 THz
10-2
(1 cm)
100
(1 m)
1 GHz
102
104
106
(1 km)
1 MHz
50 Hz
108
Notation conventions
×≡∧
curl E ≡ rot E ≡ ∇ × E ≡ ∇ ∧ E
div E ≡ ∇ ⋅ E
Maxwell Equations
∂B
=0
∂t
∂D
= j
∇∧ H −
∂t
∇⋅ D = ρ
∇⋅B = 0
∇∧E+
∂
B ⋅ dS
∂t ∫
∂
⋅
=
H
l
D ⋅ dS + J
d
∫
∂t ∫
∫ E ⋅ dl = −
∫ D ⋅ dS = Q
∫ B ⋅ dS = 0
D = εE = ε 0 E + P
B = µH = µ 0 H + M
E 2t − E1t = 0
H 2t − H1t = js
D2 n − D1n = σ
B2 n − B1n = 0
Wave equation
N 2 ∂2E
∇ E− 2
=0
2
c ∂t
2
N 2 ∂2H
∇ H− 2
=0
2
c ∂t
2
E = E0 ei (ωt − k ⋅r )
H = H 0 ei (ωt − k ⋅r )
Maxwell Equations
∂B
=0
∂t
∂D
= j
∇∧ H −
∂t
∇⋅ D = ρ
∇⋅B = 0
∇∧E+
Faraday (induction)
E 2t − E1t = 0
Ampère (induction)
H 2t − H1t = js
Coulomb (electric charges)
D2 n − D1n = σ
No magnetic charges
D = εE = ε 0 E + P
B = µH = µ 0 H + M
B2 n − B1n = 0
Discontinuity at the interface
• Ideal conductor (σ = ∞)
• Real conductor (σ < ∞)
E
E
H
H
E 2t − E1t = 0
E 2t − E1t = 0
H 2t − H1t = js
inhomogeneity
H 2t − H1t = 0
Time-resolved reflectivity
ref
lec
ted
pro
be
• Pump pulse:
high density of free carriers
(non-equilibrium state)
• Probe pulse:
monitoring of the
permittivity change
mp
u
p
ed
y
la
de
e
b
o
pr
Inhomogeneous
distribution
Properties of the permittivity
•
•
•
•
•
dispersive [ε = ε(ω)] — not dispersive (ε does not depend on ω)
absorbing (ε is complex) — not absorbing or transparent (ε is real)
inhomogeneous [ε = ε(r)] — homogeneous (ε does not depend on r)
anisotropic (ε is a second rank tensor) — isotropic (ε is a scalar)
nonlinear [ε = ε(E,H)] — linear (ε does not depend on the fields)
D(ω) = ε(ω) E (ω) = ε 0 E (ω) + P (ω)
P (ω) = ε 0 χ(ω) E (ω)
ε(ω) = ε 0 (1 + χ(ω))
Spectral decomposition
Real notation:
• Monochromatic wave
a (t ) = A cos(ωt + α ) = Re Aeiωt ,
• Polychromatic wave
a (t ) = ∑ Ak cos(ωk t + α k )
{
}
with A = A eiα
k
∞
a (t ) = ∫ A(ω) cos(ωt + α(ω)) dω
0
Complex notation:
• Monochromatic wave
a(t ) = Aeiωt
∞
• Polychromatic wave
a(t ) = ∫ A(ω)e
0
iωt
dω
(
or
a(t ) =
or
1
a(t ) = ∫ A(ω)eiωt dω
2 −∞
1
Aeiωt + A∗e −iωt
2
∞
)
Calculation of the field products
• If only mean values are needed (energy density and flow):
a(t )b(t )
in complex notation:
a (t )b(t )
T
1
1
= ∫ A cos(ωt + α ) B cos(ωt + β) = AB cos(α − β)
2
T0
1
= Re AB ∗ .
2
{ }
• If instantaneous values are needed (nonlinear optics):
a(t ) =
(
)
1
Aeiωω+ A∗e −iωω
2
b(t ) =
(
1
Beiωt + B ∗e −iωt
2
)
Maxwell equations for the
spectral components
∇ ∧ E = − iω B
∇ ∧ H = j + iωD
∇⋅ D = ρ
∇⋅B = 0
Introduction of the generalized permittivity:
Total current (free + polarization charges):
iσ 

jTOT = σE + iωD = iω  ε −  E = iωεˆ E
ω

We define a new D which involves the free charges:
∇ ∧ E = −iωB
∇ ∧ H = iωD
∇⋅ D = 0
∇⋅B = 0
Dˆ = εˆ E
Third Maxwell equation becomes:
1
∇ ⋅ Dˆ = ∇ ⋅ (ε − i σ ω)E = ρ + ∇ ⋅ j = 0
iω
Conservation of energy
All conservation laws have the general form:
∂ρ
+∇⋅ j = P
∂t
Work of the field per unit of time exerted on the charges (Lorentz force):
q(E + v ∧ B )⋅ v = qv ⋅ E → j ⋅ E
From Maxwell equations:
∂B
∂t
∂D
∇∧H = j+
∂t
∇∧E = −

/⋅ H 
∂D
∂B
E
H
⋅
+
⋅
+ ∇ ⋅ (E ∧ H ) = − j ⋅ E

∂t
∂t
/⋅ E 

∂U
+∇⋅S =−j⋅E
∂t
Conservation of energy: continued
E⋅
∂D
∂B
+H⋅
∂t
∂t
+ ∇ ⋅ (E ∧ H ) = − j ⋅ E
∂U
+∇⋅S =−j⋅E
∂t
Linear non-dispersive (and non-absorbing) media:
U = 12 (E ⋅ D + H ⋅ B )
Absorbing medium: for a monochromatic wave the energy change
corresponds to the heat:
Q=
( )
( )
ω
Im ε∗ EE ∗ = ω Im ε∗ E 2
2
Medium with small dispersion: quasi-monochromatic wave:
1  d (ωε )
 1  d (ωε ) 2

U = 
E ⋅ E ∗ + µ0 H ⋅ H ∗  = 
E + µ0 H 2 
4  dω
 2  dω
