ОПТИЧЕСКАЯ СПЕКТРОСКОПИЯ
ГЕТЕРОСТРУКТУР
Лекция 1
Maxwell’s Equations, Notations and Definitions
(in cgs units)
Maxwell’s microscopic equations:
∇ × E micro = −
∇ × H micro
1 ∂H micro
c ∂t
E micro = E micro (r, t )
1 ∂E micro 4π
=
+
j micro
c ∂t
c
∇ ⋅ E micro = 4πρ micro
∇ ⋅ H micro = 0
∂ρ micro
+ ∇ ⋅ j micro = 0
∂t
The Lorentz force:
The microscopic electric field
strength:
The microscopic magnetic field
strength :
H micro = H micro (r, t )
The microscopic electric charge
density:
ρ micro = ∑ qα δ (rα (t ) − r )
α
The microscopic electric current
density:
j micro = ∑ v α (r )qα δ (rα (t ) − r )
α
1
⎧
⎫
Fα = qα ⎨E micro (rα , t ) + vα × H micro (rα , t )⎬
c
⎩
⎭
From microscopic to macroscopic
(averaging procedure)
The macroscopic electric charge density:
ρ (r, t ) + ρ ext (r, t ) ≡ ⟨ ρ micro ⟩
external charge density
The macroscopic electric current density:
j(r, t ) + jext (r, t ) ≡ ⟨ jmicro ⟩
external current density
The macroscopic electric field strength:
E(r, t ) ≡ ⟨ E micro ⟩
The magnetic induction:
B(r, t ) ≡ ⟨ H micro ⟩
Maxwell’s macroscopic equations:
microscopic:
∇ × E micro
1 ∂H micro
=−
c ∂t
∇ × H micro =
1 ∂E micro 4π
+
j micro
c ∂t
c
macroscopic:
averaging
1 ∂B
c ∂t
1 ∂E 4π
∇×B =
( j + jext )
+
c ∂t
c
∇×E = −
∇ ⋅ E micro = 4πρ micro
∇ ⋅ E = 4π ( ρ + ρ ext )
∇ ⋅ H micro = 0
∇⋅B = 0
∂ρ micro
+ ∇ ⋅ j micro = 0
∂t
∂ρ
+∇⋅j = 0
∂t
∂ρ ext
+ ∇ ⋅ jext = 0
∂t
ρ ≡ −∇ ⋅ P
Polarization and magnetization
1 ∂B
c ∂t
1 ∂E 4π
∇×B =
+
( j + jext )
c ∂t
c
∇ ⋅ E = 4π ( ρ + ρ ext )
∇×E = −
∇⋅B = 0
∂ρ
+∇⋅j = 0
∂t
∂ρ ext
+ ∇ ⋅ jext = 0
∂t
polarization
j≡
∂P
+ c∇ × M
∂t
magnetization
ρ ≡ −∇ ⋅ P
D ≡ E + 4π P
∇⋅B = 0
∂ρ ext
+ ∇ ⋅ jext = 0
∂t
Constitutive equations:
displacement
H ≡ B − 4π M
Constitutive equation:
j = j(E, B )
1 ∂B
c ∂t
1 ∂D 4π
∇×H =
j ext
+
c ∂t
c
∇ ⋅ D = 4πρ ext
∇×E = −
magnetic field
strength
D=ε E
P=βE
permittivity
electric
susceptibility
B=μH
M=χH
permeability
magnetic
susceptibility
On the physical meaning of the P and M fields.
1
1
3
PV ≡ ∫ rρ d r = − ∫ r (∇ ⋅ P) d 3r
VV
VV
1 1
1 1
⎧ ∂P
⎫ 3
3
×
=
×
MV ≡
r
j
d
r
r
c
M
+
∇
×
⎨
⎬d r
∫
∫
V 2c V
V 2c V
⎩ ∂t
⎭
(the problem 1)
(the problem 2)
1
PV = ∫ P d 3r
VV
∂P 3
1 1
1
3
MV =
r
×
d
r
+
M
d
r
∫
∫
V 2c V
∂t
VV
∂P
c∇ × M >>
∂t
1
MV = ∫ M d 3r
VV
∂P
>> c∇ × M
∂t
MV =
∂P 3
1 1
r
×
d r
∫
∂t
V 2c V
Fig.1.1
Maxwell’s macroscopic equations
in cgs units:
1 ∂B(r, t )
c ∂t
1 ∂D(r, t ) 4π
∇ × H(r , t ) =
+
jext (r, t )
c ∂t
c
∇ × E(r, t ) = −
∂ρ ext (r, t )
+ ∇i j ext (r, t ) = 0
∂t
∂ρ (r, t )
+ ∇i j ( r , t ) = 0
∂t
∇iD(r, t ) = 4πρ ext (r, t )
∇i B ( r , t ) = 0
,
D(r, t ) ≡ E(r, t ) + 4π P(r, t )
PV ≡
1
3
r
ρ
d
r
∫
VV
H(r, t ) ≡ B(r, t ) − 4π M(r, t )
MV ≡
1 1
3
r
×
j
d
r
∫
V 2c V
D = L̂ε E
B = L̂μ H
The inhomogeneous wave equations in the cgs units take the form:
1 ∂ 2E(r, t )
4π ∂
∇ × ∇ × E(r, t ) + 2
=
−
j tot (r, t );
2
c
∂t
c ∂t
1 ∂ 2B(r, t ) 4π
∇ × ∇ × B (r , t ) + 2
=
∇ × j tot (r, t );
2
c
∂t
c
j tot (r, t ) =
PV ≡
∂ P( r , t )
+ c∇ × M(r, t ) + j ext (r, t )
∂t
1
3
r
ρ
d
r
∫
VV
MV ≡
1 1
3
r
×
j
d
r
∫
V 2c V
Maxwell’s Equations, Notations and Definitions
(in SI units)
Maxwell’s microscopic equations:
∂ B (r , t )
∂t
∂ D(r , t )
∇ × H(r , t ) =
+ j ext (r, t )
∂t
∇ × E(r, t ) = −
∇iD(r, t ) = ρ ext (r, t )
H(r, t ) = μ0−1B(r, t ) − M(r, t )
MV ≡
1 1
3
r
×
j
d
r
∫
V 2c V
The electric constant:
ε 0 = 8.85 ⋅ 10−12 [As / Vm]
∇i B ( r , t ) = 0
H ( r , t ) = μ B (r , t ) − M (r , t )
−1
0
∂
ρ ext (r, t ) + ∇i j ext (r, t ) = 0
∂t
j (r , t ) =
D(r, t ) = ε 0 E(r, t ) + P(r, t )
∂ P( r , t )
+ ∇ × M (r , t )
∂t
The Lorentz force:
The magnetic constant:
μ0 = 4 ⋅ 10−7 [Vs / Am]
ε 0 ⋅ μ0 = 1 / c 2
∂
ρ ( r , t ) + ∇i j ( r , t ) = 0
∂t
F(rα , t ) = qα ⋅ [ E(rα , t ) + vα × B(rα , t )]
MV ≡
⎧∂ P 1
⎫ 3
11
11
11
1 1
∂P 3
3
3
j
d
r
r
M
d
r
r
d
r
M
d
r
r
×
=
×
+
∇
×
=
×
+
⎨
⎬
∫
∫
∫
∫
V 2V
V 2 V ⎩ ∂ t μ0
V 2V
V μ0 V
∂t
⎭
1 ∂ 2E
∂
∇ × ∇ × E + 2 2 = − μ0
j tot ;
c ∂t
∂t
MV =
1 1
∂P 3
1
r×
d r + ∫ M d 3r
∫
V 2c V
∂t
VV
1 ∂ 2B
∇ × ∇ × B + 2 2 = μ0∇ × j tot ;
c ∂t
1 ∂2H
1
∇×∇× H + 2
=
−
c ∂ t2
μ0
j tot =
⎡ 1 ∂ 2M
⎤
+
∇
×
∇
×
M
⎢ c2 ∂ t 2
⎥ + ∇ × j tot ;
⎣
⎦
∂P 1
+ ∇ × M + j ext ;
∂ t μ0
Maxwell’s macroscopic equations
in cgs units:
1 ∂B(r, t )
∇ × E(r, t ) = −
c ∂t
∇ × B(r , t ) =
,
1 ∂D(r, t ) 4π
+
jext (r, t )
c
c ∂t
∇ ⋅ D(r, t ) = 4πρ ext (r, t )
∇ ⋅ B(r , t ) = 0
D(r, t ) ≡ E(r, t ) + 4π P(r, t )
D = L̂ε E
Spatial and frequency dispersion
t
Di (r, t ) = ∫ dt ' ∫ d 3r ' εˆij (r, r ' ; t , t ' ) E j (r ' , t ' )
−∞
If a medium is spatially homogeneous and uniform in time
εˆij (r, r ' ; t , t ' ) = εˆij (r − r ' ; t − t ' )
For a complex valued monochromatic plane wave
E j (r ' , t ' ) = E j (ω , k ) exp[i ( k ⋅ r '−ω t ' )]
one obtains
t
Di (r, t ) = ∫ dt ' ∫ d 3r ' εˆij (r − r ' ; t − t ' ) E j (ω , k ) exp[i ( k ⋅ r '−ω t ' )] =
−∞
t
= ∫ dt ' ∫ d 3r ' εˆij (r − r ' ; t − t ' ) E j (ω , k ) exp{− i[( k ⋅ (r − r ' ) − ω (t − t ' )]}exp[i ( k ⋅ r − ω t )] =
−∞
⎛∞
⎞
= ⎜ ∫ dτ ∫ d 3R εˆij ( R;τ ) exp{− i[( k ⋅ R − ωτ ]}⎟ E j (ω , k ) ⋅ exp[i ( k ⋅ r − ω t )] =
⎝0
⎠
= Di (ω , k ) ⋅ exp[i ( k ⋅ r − ω t )]
So,
Di (r, t ) = Di (ω , k ) ⋅ exp[i ( k ⋅ r − ω t )]
where
Di (ω , k ) = ε ij (ω , k ) E j (ω , k )
∞
with
ε ij (ω, k ) ≡ ∫ dτ ∫ d 3R εˆij (R;τ ) exp[− i ( k ⋅ R − ωτ )]
0
General properties of
ε ij (ω , k )
∞
ε (ω, k ) = ∫ dτ ∫ d 3R εˆij ( R;τ ) exp[i ( k * ⋅ R − ω *τ )] = ε ij ( −ω * ,−k * )
*
ij
0
According to the kinetic coefficient symmetry principle
εˆij (r, r ' ; t , t ' ) = εˆij (r ' , r; t , t ' )
which yields for the
ε ij (ω , k ) = ε ji (ω ,− k )
(ω, k ) space:
If, due to some symmetry properties (for instance, inversion symmetry),
ε ij (ω , k ) = ε ij (ω ,− k )
the medium is called non-gyrotropic.
On the other hand, from the kinetic coefficient symmetry,
ε ij (ω ,− k ) = ε ji (ω , k )
So, for non-gyrotropic media
ε ij (ω , k ) = ε ji (ω , k )
Linear response and Kramers-Kronig relations.
In the case of an isotropic medium:
ε ij (ω , k ) = ε (ω , k )δ ij
∞
ε (ω, k ) = ∫ dτ ∫ d 3R εˆ( R;τ ) exp[− i ( k ⋅ R − ωτ )]
0
At
ω = ω '+iω" :
∞
ε (ω, k ) = ∫ dτ ∫ d 3R εˆ( R;τ ) exp[− i ( k ⋅ R − ω 'τ )]⋅ exp( −ω" )
0
I (ω , k ) =
ε ( z, k ) − 1
∫C z − ω dz
z = ω '+iω"
Fig.1.2
ε (ω ' , k ) − 1
dω ' = 0
ω '−ω
−∞
+∞
I (ω, k ) = −iπ [ε (ω , k ) − 1] + P ∫
ε (ω, k ) − 1 = −
ε (ω ' , k ) − 1
dω ' = 0
π −∞ ω '−ω
i
+∞
P∫
ε (ω, k ) = ε ' (ω, k ) + iε " (ω, k )
ε " (ω ' , k )
ε ' (ω, k ) − 1 = P ∫
dω '
π −∞ ω '−ω
1
ε " (ω, k ) = −
+∞
ε ' (ω ' , k ) − 1
dω '
π −∞ ω '−ω
1
+∞
P∫
Taking into account the general relation
ε ij* (ω, k ) = ε ij ( −ω * ,−k * )
we have for an isotropic medium
ε * (ω , k ) = ε ( −ω * ,−k * )
On the other hand, from the kinetic coefficient symmetry principle,
ε (ω , k ) = ε (ω ,− k )
So, for the real frequency
ω
and real wave vector
k ,
the following relations are valid:
ε * (ω , k ) = ε ( −ω ,−k )
and
ε (ω , k ) = ε (ω ,− k )
ε * (ω , k ) = ε ( −ω , k )
which gives
ε ' (ω, k ) − iε " (ω, k ) = ε ' ( −ω, k ) + iε " ( −ω, k )
ε ' (ω, k ) = ε ' ( −ω, k )
ε " (ω, k ) = −ε " ( −ω, k )
ε '(ω , k ) − 1 =
ε "(ω , k ) = −
2
π
2ω
π
+∞
P∫
0
+∞
P∫
0
ω '⋅ ε "(ω ', k )
dω '
2
2
ω' −ω
ε '(ω ', k ) − 1
dω '
ω '2 − ω 2
