Macroscopic dielectric theory...Macroscopic dielectric theory And its integral along the path in the...
Transcript of Macroscopic dielectric theory...Macroscopic dielectric theory And its integral along the path in the...
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Macroscopic dielectric theory
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�∇× �E = −1
c
∂ �B
∂t�∇ · �E = 4πρ
�∇× �B =4π
c�J +
1
c
∂ �E
∂t�∇ · �B = 0
Macroscopic dielectric theory
Maxwellʼs equations
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�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 4πρ
�∇× �H =4π
c
��J + �Jext
�+
1
c
∂ �D
∂t
�∇ · �B = 0
�D = �E + 4π �P�H = �B + 4π �M
�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 0
�∇× �H =4π
c
�J +1
c
∂ �D
∂t
�∇ · �B = 0
Macroscopic dielectric theory
In a medium it is convenient to explicitly introduce induced charges and currents
With no external charges and currents
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�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 4πρ
�∇× �H =4π
c
��J + �Jext
�+
1
c
∂ �D
∂t
�∇ · �B = 0
�D = �E + 4π �P�H = �B + 4π �M
�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 0
�∇× �H =4π
c
�J +1
c
∂ �D
∂t
�∇ · �B = 0
Macroscopic dielectric theory
In a medium it is convenient to explicitly introduce induced charges and currents
With no external charges and currents
Electric polarization
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�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 4πρ
�∇× �H =4π
c
��J + �Jext
�+
1
c
∂ �D
∂t
�∇ · �B = 0
�D = �E + 4π �P�H = �B + 4π �M
�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 0
�∇× �H =4π
c
�J +1
c
∂ �D
∂t
�∇ · �B = 0
Macroscopic dielectric theory
In a medium it is convenient to explicitly introduce induced charges and currents
With no external charges and currents
Electric polarization
Magnetic polarization
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�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 4πρ
�∇× �H =4π
c
��J + �Jext
�+
1
c
∂ �D
∂t
�∇ · �B = 0
�D = �E + 4π �P�H = �B + 4π �M
�∇× �E = −1
c
∂ �B
∂t
�∇ · �D = 0
�∇× �H =4π
c
�J +1
c
∂ �D
∂t
�∇ · �B = 0
Macroscopic dielectric theory
In a medium it is convenient to explicitly introduce induced charges and currents
With no external charges and currents
Electric polarization
Magnetic polarization
Induced current density
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B =H
�P , �M and �J
�P = α�E�J = σE�M = χ�B
Macroscopic dielectric theory
If we assume the response is linear, we can write:
where α,σ e χ are the polarizability, conductivity and susceptibility tensors, respectively.
represent the response of the medium to the electromagnetic field
For optical frequencies we can neglect susceptibility (χ=0) so that
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�∇× �B =4π
c
�σ �E + α
∂ �E
∂t
�+
1
c
∂ �E
∂t
�Jpol = α∂ �E
∂t
�Johm = σ �E
Macroscopic dielectric theory
Conductivity (σ) and polarizability (α) both contain the material response giving rise to two induced currents, a polarization current:
and a ohmic current
By introducing into the fourth of Maxwellʼs equations we get:
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�∂W
∂t
�
abs
= �J · �E = σE2
�∂W
∂t
�
abs
= �Johm · �E = σE2
Macroscopic dielectric theory
If we consider solutions of Maxwellʼs equations in the form sin(ωt) we see immediately that Johm is in phase with E while Jpol is out phase by π/2. Polarization current does not give rise to absorption and is responsible of dispersion (refraction index). Ohmic current, in turn produce an absorpion:
The absorbed power per unit volume is given by:
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�D = � �E
� = 1 + 4πα
�E = �E0e−iωt
� = �1 + i�2 = � + i4πσ
ω
Macroscopic dielectric theory
As with simple a.c. circuits, absorption and dispersion can be treated with a single complex quantity.If we consider a field in the form
we get
in which we have introduced the complex dielectric function:
We introduce the dielectric tensor ε
�∇× �B =4π
c�J +
1
c
∂ �D
∂t=
1
c
��∂ �E
∂t− 4π �E
�=
�
c
∂ �E
∂t
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�E = �E0e−iωt
∂ �E
∂t= −iω �E0e
−iωt = −iω �E
�∇× �B =4π
cσ �E +
1
c(4πα + 1)
∂ �E
∂t=
=4π
cσ �E +
�
c
∂ �E
∂t
=
�4π
cσ + (−iω)
�
c
�i
ω
∂ �E
∂t=
=
��
c+ i
4πσ
c
�=
�
c
∂ �E
∂t
Macroscopic dielectric theory
�∇× �B =4π
c
�σ �E + α
∂ �E
∂t
�+
1
c
∂ �E
∂t
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∇2 �E − �
c2
∂2 �E
∂t2= 0
�E = �E0ei(�q·�r−ωt) + c.c.
Macroscopic dielectric theory
By introducing the complex dielectric constant into Maxwellʼs equations we get
As well known a class of solutions of the wave equations are plane waves:
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n2 = �
�1 = n2 − κ2
�2 = 2nκ
Macroscopic dielectric theory
whose dispersion law
suggest a complex refraction
so that
and therefore
ω2 =c2
�q2
n = n + iκ
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�E = �E0ei(�q·�r−ω t) + c.c. = �E0e
−iω(t− nxc ) + c.c. = �E0e
−ωκxc e−iω(t−nx
c ) + c.c.
Macroscopic dielectric theory
A plane wave propagating along the x axis can be written in terms of the complex dielectric index:
x
E
c
ωκWednesday, July 10, 2013
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vp =ωk=λT
vg =∂ω∂k
Macroscopic dielectric theory
Phase and group velocity
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vp =ωk=λT
vg =∂ω∂k
Macroscopic dielectric theory
Phase and group velocity
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W = W 0e−ηx
η =2ωκ
c=
ω�2
nc=
4πσ
nc
Macroscopic dielectric theory
The absorption coefficient η is defined by the equation:
which describes the attenuation of the average energy flux, which is proportional to the square of the field , so:
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R = r∗r =(n− 1)2 + κ2
(n + 1)2 + κ2
Macroscopic dielectric theory
The complex reflection coefficient is given by the generalization of Frenel law:
So the reflectivity is:
If κ>>n, R≈1 and thw wave is rapidly attenuated in the medium. This happens, for example in metals at low frequency (high σ)
r =n− 1
n + 1=
(n− 1) + iκ
(n + 1) + iκ
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�P (t) =
� t
−∞G (t− t�) �E (t�) dt�
Macroscopic dielectric theory
The real and imaginary part of the dielectric function and in general of any function describing the linear response to a physical stimulus are not independent of each other. This is a consequence of the causality principle, which states that the response cannot occur in time preceding the stimulus.
Let us focus on polarization (response) and the electric field (stimulus).The following dependence will hold:
i.e. the polarization at a given time t depends on the history of the electric field weighted by the Green function G.
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�P (t) =
� t
−∞G (t− t�) �E0e
−iωt� (t�) dt� = �E0e−iωt
� ∞
0
G (τ) eiωτdτ
α (ω) =
� ∞
0
G (τ) eiωτdτ
Macroscopic dielectric theory
For a sinusoidal field (and a substitution of a variable) we ca write:
so we the complex response function (polarizability in this case) can always be expressed as:
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P
+∞�
−∞
α (ω)
ω − ω0dω = iπα (ω0)
� (ω) = 1 +1
iπP
+∞�
−∞
� (ω�)− 1
ω� − ωdω�
Macroscopic dielectric theory
And its integral along the path in the figure is zero, i.e. by applying Cauchyʼs theorem:
ω0
Im ω
Re ω
where P indicates the principal part of the integral.For the dielectric function we obtain the dispersion relation:
α(ω )If decreases to zero away from the real axis, the function α(ω)/(ω−ω0) has a single pole in ω0.
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� (−ω) = �∗ (ω)
�1(ω) = 1 +2
πP
∞�
0
ω��2(ω�)
ω�2 − ω2dω�
�2(ω) = −2ω
πP
∞�
0
�1(ω�)− 1
ω�2 − ω2dω�
Macroscopic dielectric theory
From:
we immediately obtain that
we can therefore rewrite the integral and write the Kramers-Kronig dispersion relations between the real and imaginary part of the dielectric function:
α (ω) =
� ∞
0
G (τ) eiωτdτ
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r (ω) =(n− 1) + iκ
(n + 1) + iκ= |r (ω)| eiθ
R(ω) = |r(ω)|2 =
����(n− 1) + iκ
(n + 1) + iκ
����2
ln (r (ω)) = ln (|r (ω)|) + iθ (ω) =1
2ln (R (ω)) + iθ (ω)
θ (ω) = −2ω
πP
∞�
0
ln (|r (ω�)|)ω�2 − ω2
dω� = −ω
πP
∞�
0
ln (R (ω�))
ω�2 − ω2dω�
Macroscopic dielectric theory
Dispersion relation for reflectivity
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r (ω) =(n− 1) + iκ
(n + 1) + iκ= |r (ω)| eiθ
R(ω) = |r(ω)|2 =
����(n− 1) + iκ
(n + 1) + iκ
����2
ln (r (ω)) = ln (|r (ω)|) + iθ (ω) =1
2ln (R (ω)) + iθ (ω)
θ (ω) = −2ω
πP
∞�
0
ln (|r (ω�)|)ω�2 − ω2
dω� = −ω
πP
∞�
0
ln (R (ω�))
ω�2 − ω2dω�
Macroscopic dielectric theory
Dispersion relation for reflectivity
We measure:
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r (ω) =(n− 1) + iκ
(n + 1) + iκ= |r (ω)| eiθ
R(ω) = |r(ω)|2 =
����(n− 1) + iκ
(n + 1) + iκ
����2
ln (r (ω)) = ln (|r (ω)|) + iθ (ω) =1
2ln (R (ω)) + iθ (ω)
θ (ω) = −2ω
πP
∞�
0
ln (|r (ω�)|)ω�2 − ω2
dω� = −ω
πP
∞�
0
ln (R (ω�))
ω�2 − ω2dω�
Macroscopic dielectric theory
Dispersion relation for reflectivity
We measure:
If we consider the natural logarithm of r(ω):
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�1 (0) = 1 +2
πP
∞�
0
�2 (ω�)
ω�dω�
Macroscopic dielectric theory
Static sum rule. For ω=0 we have:
so, if the static dielectric constant is ≠1, the imaginary part wil be ≠0 in some part of the spectrum, i.e the medium must absorb radiation. Because of the 1/ω factor in the integrand, the static dielectric function will be greater if the absorption occurs at low frequency.
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�2 (ω) = Aδ (ω − ω0)
�1 (0) = 1 +2
π
A
ω0
�1 (ω) = 1 +ω2
0 [�1 (0)− 1]
ω20 − ω2
�2 (ω) =π
2ω0 [�1 (0)− 1] δ (ω − ω0)
Macroscopic dielectric theory
Absorption localized at ω= ω0
therefore:
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Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
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md2y
dt2+ mγ
dy
dt+ ω2
0my = eE0e−iωt
Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
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md2y
dt2+ mγ
dy
dt+ ω2
0my = eE0e−iωt
Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
whose solution is
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md2y
dt2+ mγ
dy
dt+ ω2
0my = eE0e−iωt
y =eE0e−iωt
m [(ω20 − ω2)− iγω]
Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
whose solution is
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md2y
dt2+ mγ
dy
dt+ ω2
0my = eE0e−iωt
y =eE0e−iωt
m [(ω20 − ω2)− iγω]
Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
whose solution is
We obtain the dipole moment per unit volume by multiplying y by the charge e and by the oscillator density N. The dielectric function is therefore:
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md2y
dt2+ mγ
dy
dt+ ω2
0my = eE0e−iωt
y =eE0e−iωt
m [(ω20 − ω2)− iγω]
� (ω) = 1 + 4πα = 1 + 4πP
E= 1 +
4πe2N
m
1
(ω20 − ω2)− iγω
Macroscopic dielectric theory
Drude-Lorentz model.We describe the medium as an ensemble of harmonic oscillators whose resonance frequency and damping are ω0 and γ, respectively:
whose solution is
We obtain the dipole moment per unit volume by multiplying y by the charge e and by the oscillator density N. The dielectric function is therefore:
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�1 = 1 +4πe2N
m
ω20 − ω2
(ω20 − ω2)2 + (γω)2
�2 =4πe2N
m
γω
(ω20 − ω2)2 + (γω)2
5
4
3
2
1
0
-1
14121086420Photon Energy (eV)
!1
!2
Macroscopic dielectric theory
So:
ℏω0= 4eV
ℏγ= 1eV
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5
4
3
2
1
0
-1
1211109876543210Photon energy (eV)
eps1 eps2
�2 (ω) = −2ω
πP
∞�
0
�1 (ω�)− 1
ω�2 − ω2dω�
�2 (ω) = − 1
πP
∞�
0
�d�1 (ω�)
dω�
�· ln
�ω� + ω
ω� − ω
�dω�
Macroscopic dielectric theory
Integration by parts
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5
4
3
2
1
0
-1
1211109876543210Photon energy (eV)
eps1 eps2
�2 (ω) = −2ω
πP
∞�
0
�1 (ω�)− 1
ω�2 − ω2dω�
�2 (ω) = − 1
πP
∞�
0
�d�1 (ω�)
dω�
�· ln
�ω� + ω
ω� − ω
�dω�
Macroscopic dielectric theory
Integration by parts
General behaviour!
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Macroscopic dielectric theoryBy considering the complex refraction index, one can identify regions in which transmissivity, absorption or reflectivity are the main effects:
T A R
n,κ
2.0
1.5
1.0
0.5
0.0
121086420
Photon Energy (eV)
n !
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Macroscopic dielectric theoryBy considering the complex refraction index, one can identify regions in which transmissivity, absorption or reflectivity are the main effects:
T A R
n,κ
2.0
1.5
1.0
0.5
0.0
121086420
Photon Energy (eV)
n !
R
0.3
0.2
0.1
R
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� =
�1−
ω2p
ω2 + γ2
�+ i
γω2p
ω3 + γ2ω
ω2p =
4πNe2
m
Macroscopic dielectric theory
Optical properties of metals can be obtained from the Lorentz-Drude model by setting ω0=0 and by defining the plasma frequency:
The dielectric functions becomes:
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Macroscopic dielectric theory
Free electron metal wi th ℏωp=8eV and ℏγ=0.5 eV
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σ =ω�2
4π
σ0 = limω→0
ω�2
4π=
Ne2
mγ
Macroscopic dielectric theory
The conductivity, σ is given by:
which in the statc limit becomes:
which can be compared with the expression derived in the transport theory to find that γ=1/τ
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R ∼= 1− 2
�ω
2πσ= 1− 2
�2ω
ω2pτ
Macroscopic dielectric theory
At very low frequencies (ω<<1/τ) the reflectivity is:
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Macroscopic dielectric theory
Free electron metal wi th ℏωp=8eV and ℏγ=0.5 eV
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�1∼= 1−
ω2p
ω2
�2∼=
ω2pγ
ω3
�1(ω) = 1 +2
πP
∞�
0
ω��2(ω�)
ω�2 − ω2dω�
�1(ω)ω→∞ ∼= 1− 2
ω2π
∞�
0
ω��2(ω�)dω�
Macroscopic dielectric theory
At frequencies much higher than the plasma frequency we get:
which represent the behaviour of any material at high energy.If we now consider the KK relation:
At high energy we can neglect ω´2 in the integrand denominator because ϵ2 decreases very rapidly (as 1/ω3, superconvergence theorem) and therefore:
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�1∼= 1−
ω2p
ω2
�2∼=
ω2pγ
ω3
�1(ω) = 1 +2
πP
∞�
0
ω��2(ω�)
ω�2 − ω2dω�
�1(ω)ω→∞ ∼= 1− 2
ω2π
∞�
0
ω��2(ω�)dω�
Macroscopic dielectric theory
At frequencies much higher than the plasma frequency we get:
which represent the behaviour of any material at high energy.If we now consider the KK relation:
At high energy we can neglect ω´2 in the integrand denominator because ϵ2 decreases very rapidly (as 1/ω3, superconvergence theorem) and therefore:
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∞�
0
ω��2(ω�)dω� =
π
2ω2
p =2π2e2N
m
ωmax�
0
ω��2(ω�)dω� ∼=
π
2
4πe2
mNeff
Macroscopic dielectric theory
so we get the sum rule:
which relates he dielectric function to the total number of electrons contributing to it. We can define the effective number of electrons contributing to the dielectric function up to a frequency ωmax:
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�2(ω) = −2ω
πP
∞�
0
�1(ω�)− 1
ω�2 − ω2dω�
�2 (ω)ω→∞ =2
πω
∞�
0
(�1 (ω�)− 1) dω�
∞�
0
(�1 (ω)− 1) dω = 0
Macroscopic dielectric theory
The other KK relation
at high frequency becomes
and by comparison with the expression of the dielectric function at high energy we get:
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-5
-4
-3
-2
-1
0
1
2
3
4
5
!1
, !2
14121086420
Photon energy (eV)
1.0
0.8
0.6
0.4
0.2
0.0
Re
flectiv
ityMacroscopic dielectric theory
Free electron metal wi th ℏωp=8eV and ℏγ=0.5 eV
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1.0
0.8
0.6
0.4
0.2
0.0
Reflectivity
2015105
Photon energy (eV)
Macroscopic dielectric theory
Normal incidence Ag experimental Reflectivity
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-8
-6
-4
-2
0
2
4
6
!1
, !
2
2015105
Photon energy (eV)
!1
!2
Macroscopic dielectric theory
Experimental dielectric function of Ag:
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Macroscopic dielectric theory
E x p e r i m e n t a l Reflectivity of Al
Refl
ectiv
ity (%
)
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Macroscopic dielectric theoryReflection and refraction at an arbitrary angle
����k��� =
��� �k����� =
ω
cAll waves have the same frequency, ω, and
vφ =ω
k� =c
n⇒ k� =
����k���� =
ω
c(1− δ + iβ)The refracted wave has phase velocity
�E = �E0e−i(ωt−�k·�r)
�E � = �E �0e
−i(ωt−�k�·�r)
�E �� = �E ��0e−i(ωt− �k��·�r)
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�k · �r0 = �k� · �r0 = �k�� · �r0
kx = k�x = k��
x
φ = φ��;sin φ
sin φ� = n
|kz| = |k�z| = |k��
z |k sin φ = k� sin φ� = k�� sin φ��
Macroscopic dielectric theory
Kinematic boundary conditions:
at the boundary (z=0):
nothing occurs along x
so along z
therefore:
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n = (1− δ + iβ)
φc = arcsin (1− δ)
θc =√
2δ
Macroscopic dielectric theory
If we write the complex index of refraction as
and assume β→0 then n≃1-δ and we can have total external reflection for angles above the critical angle
or, in terms of the glancing incidence
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Macroscopic dielectric theory Dynamic boundary conditions for an s polarized wave:�E = �E0e
−i(ωt−�k·�r)
�E � = �E �0e
−i(ωt−�k�·�r)
�E �� = �E ��0e−i(ωt− �k��·�r)
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Macroscopic dielectric theory Dynamic boundary conditions for an s polarized wave:
E0 = E�0 = E
��0
H0 cos φ−H��0 cos φ = H
�0 cos φ
�
tangential electric and magnetic fields are continuos:
�E = �E0e−i(ωt−�k·�r)
�E � = �E �0e
−i(ωt−�k�·�r)
�E �� = �E ��0e−i(ωt− �k��·�r)
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Macroscopic dielectric theory Dynamic boundary conditions for an s polarized wave:
E0 = E�0 = E
��0
H0 cos φ−H��0 cos φ = H
�0 cos φ
�
tangential electric and magnetic fields are continuos:
�H (�r, t) = n
�k
k× �E (�r, t)
(E0 − E ��0 ) cos φ = nE �
0 cos φ�
since
we obtain
�E = �E0e−i(ωt−�k·�r)
�E � = �E �0e
−i(ωt−�k�·�r)
�E �� = �E ��0e−i(ωt− �k��·�r)
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E �0
E0=
2 cos φ
cos φ +�
n2 − sin2 φ
E ��0
E0=
cos φ−�
n2 − sin2 φ
cos φ +�
n2 − sin2 φ
Rs =
���cos φ−�
n2 − sin2 φ���2
���cos φ +�
n2 − sin2 φ���2
Macroscopic dielectric theory
For an s polarized wave:
so the reflectivity is
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E �0
E0=
2n cos φ
n2 cos φ +�
n2 − sin2 φ
E ��0
E0=
n2 cos φ−�
n2 − sin2 φ
n2 cos φ +�
n2 − sin2 φ
Rp =
���n2 cos φ−�
n2 − sin2 φ���2
���n2 cos φ +�
n2 − sin2 φ���2
Macroscopic dielectric theory
For a p polarized wave:
so the reflectivity is
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10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Refle
ctiv
ity
9080706050403020100Angle of incidence(degrees)
Rs RpBroken: n=1.5Continuos:n=1.01
Macroscopic dielectric theory
Reflectivity as a function of ϕ and n
Brewster angle
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θb = arctan�
n2
n1
�
Macroscopic dielectric theory
Brewsterʼs angle
When the reflected a n d r e f r a c t e d waves fo rm an angle of 90˚ i.e. when:
the reflected wave is 100% s polarized (if no absorption occurs!)
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0.01
2
3
4
5678
0.1
2
3
4
5678
1
Refle
ctiv
ity
9080706050403020100Angle of Incidence (Degrees)
R s R p
Ag
Macroscopic dielectric theory
Ag s- and p- polarized reflectivity at ℏω=20eV
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100
80
60
40
20
0
Phot
on E
nerg
y (e
V)
806040200Incidence angle (degrees)
1.00.80.60.40.20.0Reflectivity scale
Macroscopic dielectric theory
Experimental reflectivity of Ag for p-polarized light
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Macroscopic dielectric theory
Wednesday, July 10, 2013