Repetition lecture for Electromagnetic Optics ... · Different types of media •Linear: If P(r,t)...
Transcript of Repetition lecture for Electromagnetic Optics ... · Different types of media •Linear: If P(r,t)...
Repetition lecture for
Electromagnetic Optics, Polarization Optics, and
layered media
Cord Arnold / Anne L’Huillier
Electromagnetic optics
Maxwell Equations in a source free medium
MHB
PEDrrr
rrr
00
0
µµ +=
+=∈Flux
densities
Polarization
densities
Electric vacuum permitivity
Magnetic permeability
( )E
ED
EP
r
rr
rr
=∈
+=∈
=∈
χ
χ
10
0
Linear, homogeneous, and isotropic
Susceptibility
Electric permitivity
χ
χ
+=∈∈
=
∈∈
=+
1
1
0
0
n
Dielectric function
Refractive index
2
0
2
22
22 1
,0c
n
ct
EE =∈==
∂∂
∈−∇ µµr
r
Different types of media
• Linear: If P(r,t) is linearly related to Ԑ(r,t).
• Nondispersive: The response is instantaneous. The
polarization P(r,t) does not depend on
earlier times.
• Homogeneous: The relation between P and Ԑ is no
function of space.
• Isotropic: The relation between P and Ԑ is
independent of the direction of Ԑ.
• Spatially nondispersive: The relation between P and Ԑ is local.
Transverse electromagnetic (TEM) plane wave
E is orthogonal to H. Both
are orthogonal to the
direction of propagation k.
( ) ( ) ( ){ }( ) ( ) ( ){ }tjrHtrH
tjrEtrE
ω
ω
expRe,
expRe,
0
0
rrrr
rrrr
=
=Monochromatic EM-waves
( ) nncc
kEkE00
22 2,0
λπωω
ω ====+∇⇒rr
0000
0000
,,
,,
EkHHEk
HkEEHkrrrrrr
rrrrrr
⊥⇒−=×
⊥⇒∈=×
ωµ
ω
2
000
2
0
00
0
0
2
1
2
1
2
1Enc
EHEIS
H
E
∈==×==
∈==
η
µη
rrr
Impedance
Intensity
Vectorial spherical wave
Transmission bands for common materials in optics
'''12
'''
0 χχα
β
χχχ
jkjk
j
++=−=
+=
( )zUU α−= exp2
0
2
Absorption => attenuation
Implications of dispersion
The resonant medium
Damping
Driving force
Restoring force
The resonant medium
( )νννν
νχνχ
∆+−=
j22
0
2
00
OffsetResonance frequency
Width of the resonance
Multi resonance media
Dispersive media
Group velocity
Phase
velocity
( )( ) ( ) N
c
nn
cvg
0
000
00
'=
−=
λλλλ
Group index
Dispersive media
( )
( )
( )00
0
02
0
3
0
02
0
3
0
''
''
''2
λλ
λλ
λπλ
λ
ν
ω
nc
D
nc
D
nc
D
−=
=
=0,0
0,0
><
<>
λω
λω
DD
DD Normal dispersion
Anomalous dispersion
zD
zD
λκτ
νντ
σσ
σσ
=
=
Polarization of Light
One component in x-direction
zx
y
Components in x- and y-direction
zx
y
Components in x- and y-direction
zx
y
Circular polarization
zx
y
Elliptic polarization
zx
y
Visualization of the polarization state
Jones vectors
( )
xy
x
y
x
yyx
yxyxyx
y
x
A
A
a
aR
AAI
jaAA
AJ
ϕϕϕη
ϕ
−===+
=
=
=
,,2
exp,
22
,,,
r
Wave plates
λ/4 plate λ/2 plate
Reflection and refraction at a boundary
TE – Transverse Electric – orthogonal – (s)enkrecht
TM – Transverse Magnetic – parallel – (p)arallel
Boundary conditions
Fresnel equations (nonabsorbing media)
Overview
TE
TM
External Internal
Power reflectance
Application <-> polarization by reflection
Application <-> polarization filters in photography
Source: http://www.bhphotovideo.com
Isotropic <-> anisotropic
The refractive index ellipsoid
Biaxial: All refractive indices are different n1≠
n2≠n3
Uniaxial: Two refractive indices are identical,
n1= n2=no (ordinary refractive index)
and n3= ne (extraordinary refractive
index)
ne>no => positive uniaxial
ne<no => negative uniaxial
The z-axis of a uniaxial crystal is the
optic axis.
Isotropic: The index ellepsoid is a sphere.
Double refraction
Wave retarders
Uniaxial crystals: n1=n2=no, n3=ne
Optic axis
onon
en
θθ
4
λ2
λ4
3λ λ
Example:
λ/2-plate
( ) πππ
2,,2
=−=Γ dkk oe
Variable attenuator
Faraday (optical) isolator
Dielectric layered media
A photonic crystal is a periodic arrangement
of a dielectric material that exhibits strong
interaction with light
Variation of refractive index on the scale of the wavelength λ
Photonic Crystals; J.D. Joannopoulos
Definition
How to analyze a multilayer system
Scattering MatrixWave-Transfer Matrix
Si � Mi MΣ � S Σ
Analysis of multilayer optical system
Conversion
Cascaded system - Airy formulas
ϕ
ϕ
ϕ
ϕ
ϕ
j
j
j
j
err
etrtrr
ndkerr
ettt
2
2321
2
2123121213
02
2321
231213
1
,1
−
−
−
−
−+=
=−
=
Fabry-Perot Etalon (Mirror)
FinesseMaximum transmission
Distance between
resonances
Width of the
resonance
20
λπϕπϕ mdmndkm =⇔==⇔=
Dielectric Bragg grating
( ) πϕϕ mdndnk =+=+ 2211021
Condition
The accumulated phase shift must add to π, λ/2
Off-axis high-reflection mirrors
End of the course!