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Optics and Optical Design
Chapter 5: Electromagnetic Optics
Lectures 9 & 10
Cord Arnold / Anne L’Huillier
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Electromagnetic waves in dielectric media
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EM‐optics compared to simpler theories
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Electromagnetic spectrum
Electromagnetic optics describes all kinds of EM‐waves in all possible spectral ranges in possible kinds of media (vacuum, dielectric, conductive, etc.).
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Example: THz imaging
The THz Network, www.thznetwork.net www.dailymail.co.uk
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Particle ‐ wave
Wikipedia
X-ray imaging (shadow-graphy)
http://www.scienceiscool.org/
X-ray diffraction
X-ray image from the hand of Albert von Koelliker, taken in 1896.
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Maxwell Equations in vacuum
Contributions from:‐Charles‐Augustin de Coulomb‐Hans Christian Örsted‐Carl Friedrich Gauss‐Jean‐Baptiste Biot‐André‐Marie Ampére‐Michael Faraday
‐ Unified by James Clerk Maxwell in 1861 as set of twenty equations.
‐ The current form, termed Maxwell Equations, was compressed by using vector notation by Oliver Heavyside in 1884.
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Maxwell Equations in a source free medium
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Boundary conditions
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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.
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Linear, nondispersive, homogeneous, isotropic, source‐free media
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Anisotropic, linear, nondispersive media
The susceptibility tensor χ can have up to nine independent elements χji.
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Dispersive media
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Monochromatic electromagnetic waves
Introduce monochromatic fields
All fields and flux densities can be written in their monochromatic versions
accordingly.
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Transverse electromagnetic (TEM) plane wave
E is orthogonal to H. Both are orthogonal to the direction of propagation k.
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Vectorial spherical wave
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Example: focusing of vectorial waves
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Vectorial solutions of the Helmholtz Equation
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Absorption and dispersion
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Transmission bands for common materials in optics
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Implications of dispersion
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Refractive index for different isotropic materials and crystals
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The resonant medium
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The resonant medium
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Multi resonance media
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Sellmeier Equation for the refractive index far from resonance
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Kramers‐Kronig Relations
The Kramers-Kronig relations relate mathematically the real and imaginary parts of the susceptibility to each other. Knowing one determines the other and vice versa.
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Causal response function
Noncausal odd function
Signum function
Causal response function ththtth oo signum
0for real is and 0for 0 tthtth (causal funtion)
dtthtjthtdtthtjH sincosexp
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Frequency space imaginary part of a causal response function
Frequency space real part of a causal response function
The real and imaginary parts are related because they originate from the same function and they contain the same information!
oo HHH SIGNUM
Imaginarypart
Realpart
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The Drude Model for conductive media
ω<ωp – The effective permittivity is negative, β(ω) is imaginary. Light cannot propagate. => Perfect mirror.
ω>ωp – The effective permittivity is positive. Light can propagate. The refractive index is below 1.
ω=ωp – β(ω)=0. Light cannot propagate. But one can resonantly excite plasma waves. Plasmons!
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Pulse propagation in dispersive media
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Dispersive media
The field moves in respect to the envelope due to the difference of phase and group velocity
The pulse spreads due to group velocity dispersion (GVD)
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Temporal and spectral representation of laser pulses and the time‐bandwidth product
Frequency0
∆
Frequency0
∆
Fourier transform
44.02 FWHM
Time-bandwidth product (Gaussian pulse)
Time
FWHM
Time
FWHM
Ele
ctric
fiel
d (a
.u.)
Ele
ctric
fiel
d (a
.u.)
Spe
ctra
l pow
er (a
.u.)
Spe
ctra
l pow
er (a
.u.)
tjtAtU 0exp Carrier frequency
Pulse envelope (spectrally broad)
Pulsed plane wave
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Laser pulses in dispersive media
zjtzAzAtzA
zjzAzA
exp,0FF,~F,
exp,0~,~
11
Spectral plane wave propagator
2000
0
0
!2''
!1' n
c
Wave number expansion around a carrier ω0:
'/1 gvGroup velocity ’’ Group velocity dispersion
Plane wave propagation
Each frequency component evolves with a different wave
number
00
2
2
'',1'
gv
Group velocity and group velocity
dispersion (GVD) result from dispersion.
ms''
2
ms' Inverse of a speed Inverse of an acceleration
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Group velocity and group index
0000
000
00
0000
0
020
0
0
20
0
00
0
20
00
20
20
0
0
0
0
','
'1
'
2
22
2,
nnNNc
nncv
nnc
nnc
n
cc
c
g
Depends on the change of the refractive index in respect to the wavelength
Group index
The speed of a pulse is determined by the rate of change of the refractive index
Refractive index for a typical material
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Group velocity dispersion (GVD)
020
30
00
20
0
200
02
2
''2
''
22''
nc
D
cc
Refractive index for fused silica
GVD is proportional n’’(0), that is the curvature of n(0).
00
020
2
020
30 '',''
nc
Dnc
D
GVD for fused silica
zDzD 00 DD
Estimation for dispersive pulse broadening
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Pulse broadening in dispersive media
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Dispersive media
n>1
N>1=> vg<vp
Anomalousdispersion
Normaldispersion
Anomalousdispersion
N>1=> vg<vp
n<1