optics and optical design 5: electromagnetic optics
TRANSCRIPT
Optics and Optical Design
Chapter 5: Electromagnetic Optics
Lectures 9 & 10
Cord Arnold / Anne L’Huillier
Electromagnetic waves in dielectric media
EM‐optics compared to simpler theories
Electromagnetic spectrum
Electromagnetic optics describes all kinds of EM‐waves in all possible spectral ranges in possible kinds of media (vacuum, dielectric, conductive, etc.).
Example: THz imaging
The THz Network, www.thznetwork.net www.dailymail.co.uk
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.
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.
Maxwell Equations in a source free medium
Boundary conditions
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.
Linear, nondispersive, homogeneous, isotropic, source‐free media
Anisotropic, linear, nondispersive media
The susceptibility tensor χ can have up to nine independent elements χji.
Dispersive media
Monochromatic electromagnetic waves
Introduce monochromatic fields
All fields and flux densities can be written in their monochromatic versions
accordingly.
Transverse electromagnetic (TEM) plane wave
E is orthogonal to H. Both are orthogonal to the direction of propagation k.
Vectorial spherical wave
Example: focusing of vectorial waves
Vectorial solutions of the Helmholtz Equation
Absorption and dispersion
Transmission bands for common materials in optics
Implications of dispersion
Refractive index for different isotropic materials and crystals
The resonant medium
The resonant medium
Multi resonance media
Sellmeier Equation for the refractive index far from resonance
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.
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
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
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!
Pulse propagation in dispersive media
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)
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
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
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
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
Pulse broadening in dispersive media
Dispersive media
n>1
N>1=> vg<vp
Anomalousdispersion
Normaldispersion
Anomalousdispersion
N>1=> vg<vp
n<1