Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
1
Fundamentals of Modern Optics
Winter Term 2014/2015
Prof. Thomas Pertsch
Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Table of content
0. Introduction ............................................................................................... 4
1. Ray optics - geometrical optics (covered by lecture Introduction to Optical
Modeling) .............................................................................................. 15
1.1
1.2
1.3
1.4
1.5
Introduction ......................................................................................................... 15
Postulates ........................................................................................................... 15
Simple rules for propagation of light ................................................................... 16
Simple optical components ................................................................................. 16
Ray tracing in inhomogeneous media (graded-index - GRIN optics) .................. 20
1.5.1 Ray equation ............................................................................................ 20
1.5.2 The eikonal equation ................................................................................ 22
1.6 Matrix optics........................................................................................................ 23
1.6.1 The ray-transfer-matrix ............................................................................. 23
1.6.2 Matrices of optical elements ..................................................................... 23
1.6.3 Cascaded elements.................................................................................. 24
2. Optical fields in dispersive and isotropic media ....................................... 25
2.1 Maxwell’s equations ........................................................................................ 25
2.1.1 Adaption to optics ..................................................................................... 25
2.1.2 Temporal dependence of the fields .......................................................... 29
2.1.3 Maxwell’s equations in Fourier domain .................................................... 30
2.1.4 From Maxwell’s equations to the wave equation ...................................... 30
2.1.5 Decoupling of the vectorial wave equation ............................................... 32
2.2 Optical properties of matter................................................................................. 33
2.2.1 Basics....................................................................................................... 33
2.2.2 Dielectric polarization and susceptibility ................................................... 36
2.2.3 Conductive current and conductivity......................................................... 38
2.2.4 The generalized complex dielectric function............................................. 39
2.2.5 Material models in time domain ................................................................ 43
2.3 The Poynting vector and energy balance ........................................................... 45
2.3.1 Time averaged Poynting vector ................................................................ 45
2.3.2 Time averaged energy balance ................................................................ 46
2.4 Normal modes in homogeneous isotropic media ................................................ 49
2.4.1 Transverse waves .................................................................................... 50
2.4.2 Longitudinal waves ................................................................................... 51
2.4.3 Plane wave solutions in different frequency regimes ............................... 52
2.4.4 Time averaged Poynting vector of plane waves ....................................... 58
2.5 The Kramers-Kronig relation ............................................................................... 59
2.6 Beams and pulses - analogy of diffraction and dispersion .................................. 62
2.7 Diffraction of monochromatic beams in homogeneous isotropic media .............. 64
2.7.1 Arbitrarily narrow beams (general case)................................................... 65
2.7.2 Fresnel- (paraxial) approximation ............................................................. 72
2.7.3 The paraxial wave equation ..................................................................... 76
2.8 Propagation of Gaussian beams......................................................................... 78
2.8.1 Propagation in paraxial approximation ..................................................... 79
2.8.2 Propagation of Gauss beams with q-parameter formalism....................... 84
2.8.3 Gaussian optics ........................................................................................ 84
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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2.8.4 Gaussian modes in a resonator ............................................................... 87
2.9 Dispersion of pulses in homogeneous isotropic media ....................................... 93
2.9.1 Pulses with finite transverse width (pulsed beams) .................................. 93
2.9.2 Infinite transverse extension - pulse propagation ................................... 100
2.9.3 Example 1: Gaussian pulse without chirp............................................... 101
2.9.4 Example 2: Chirped Gaussian pulse ...................................................... 104
3. Diffraction theory .................................................................................. 108
3.1 Interaction with plane masks............................................................................. 108
3.2 Propagation using different approximations ...................................................... 109
3.2.1 The general case - small aperture .......................................................... 109
3.2.2 Fresnel approximation (paraxial approximation) .................................... 109
3.2.3 Paraxial Fraunhofer approximation (far field approximation) .................. 110
3.2.4 Non-paraxial Fraunhofer approximation ................................................. 112
3.3 Fraunhofer diffraction at plane masks (paraxial) ............................................... 112
3.4 Remarks on Fresnel diffraction ......................................................................... 117
4. Fourier optics - optical filtering .............................................................. 119
4.1 Imaging of arbitrary optical field with thin lens .................................................. 119
4.1.1 Transfer function of a thin lens ............................................................... 119
4.1.2 Optical imaging using the 2f-setup ......................................................... 120
4.2 Optical filtering and image processing .............................................................. 122
4.2.1 The 4f-setup ........................................................................................... 122
4.2.2 Examples of aperture functions .............................................................. 125
4.2.3 Optical resolution ................................................................................... 126
5. The polarization of electromagnetic waves ........................................... 129
5.1 Introduction ....................................................................................................... 129
5.2 Polarization of normal modes in isotropic media............................................... 129
5.3 Polarization states ............................................................................................ 130
6. Principles of optics in crystals ............................................................... 132
6.1
6.2
6.3
6.4
Susceptibility and dielectric tensor .................................................................... 132
The optical classification of crystals .................................................................. 134
The index ellipsoid ............................................................................................ 135
Normal modes in anisotropic media.................................................................. 136
6.4.1 Normal modes propagating in principal directions .................................. 137
6.4.2 Normal modes for arbitrary propagation direction .................................. 138
6.4.3 Normal surfaces of normal modes ......................................................... 143
6.4.4 Special case: uniaxial crystals ................................................................ 145
7. Optical fields in isotropic, dispersive and piecewise homogeneous media
........................................................................................................... 148
7.1 Basics ............................................................................................................... 148
7.1.1 Definition of the problem ........................................................................ 148
7.1.2 Decoupling of the vectorial wave equation ............................................. 149
7.1.3 Interfaces and symmetries ..................................................................... 150
7.1.4 Transition conditions .............................................................................. 151
7.2 Fields in a layer system  matrix method ........................................................ 151
7.2.1 Fields in one homogeneous layer .......................................................... 151
7.2.2 The fields in a system of layers .............................................................. 153
7.3 Reflection – transmission problem for layer systems ........................................ 155
7.3.1 General layer systems............................................................................ 155
7.3.2 Single interface ...................................................................................... 162
7.3.3 Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals ...... 169
7.3.4 Fabry-Perot-resonators .......................................................................... 176
7.4 Guided waves in layer systems ........................................................................ 182
7.4.1 Field structure of guided waves .............................................................. 182
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
7.4.2
7.4.3
7.4.4
7.4.5
3
Dispersion relation for guided waves ..................................................... 184
Guided waves at interface - surface polariton ........................................ 186
Guided waves in a layer – film waveguide ............................................. 188
How to excite guided waves ................................................................... 192
This script originates from the lecture series “Theoretische Optik” given by Falk Lederer at
the FSU Jena for many years between 1990 and 2012. Later the script was adapted by
Stefan Skupin and Thomas Pertsch for the international education program of the Abbe
School of Photonics.
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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0. Introduction
• 'optique' (Greek)  lore of light  'what is light'?
• Is light a wave or a particle (photon)?
D.J. Lovell, Optical Anecdotes
• Light is the origin and requirement for life  photosynthesis
• 90% of information we get is visual
A)
What is light?
• electromagnetic wave propagating with the speed of c= 3 × 108 m / s
• wave = evolution of
− amplitude and phase  complex description
− polarization
 vectorial field description
− coherence
 statistical description
Spectrum of Electromagnetic Radiation
Region
Wavelength
Wavelength
[nm]
[m] (nm=10-9m)
Frequency
[Hz] (THz=1012Hz)
Energy
[eV]
Radio
> 108
> 10-1
< 3 x 109
< 10-5
Microwave
108 - 105
10-1 – 10-4
3 x 109 - 3 x 1012
10-5 - 0.01
Infrared
105 - 700
10-4 - 7 x 10-7
3 x 1012 - 4.3 x 1014
0.01 - 2
Visible
700 - 400
Ultraviolet
400 - 1
4 x 10-7 - 10-9
7.5 x 1014 - 3 x 1017
3 - 103
X-Rays
1 - 0.01
10-9 - 10-11
3 x 1017 - 3 x 1019
103 - 105
Gamma Rays
< 0.01
< 10-11
> 3 x 1019
> 105
7 x 10-7 - 4 x 10-7 4.3 x 1014 - 7.5 x 1014
2-3
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
B)
Origin of light
• atomic system  determines properties of light (e.g. statistics, frequency,
line width)
• optical system  Y other properties of light (e.g. intensity, duration, …)
• invention of laser in 1958  very important development
Schawlow and Townes, Phys. Rev. (1958).
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
• laser  artificial light source with new and unmatched properties (e.g.
coherent, directed, focused, monochromatic)
• applications of laser: fiber-communication, DVD, surgery, microscopy,
material processing, ...
Fiber laser: Limpert, Tünnermann, IAP Jena, ~10kW CW (world record)
C)
Propagation of light through matter
• light-matter interaction (G: Licht-Materie-Wechselwirkung)
effect
governed by
dispersion
↓
frequency
spectrum
diffraction
↓
spatial
frequency
absorption
↓
center of
frequency
spectrum
scattering
↓
wavelength
• matter is the medium of propagation  the properties of the medium
(natural or artificial) determine the propagation of light
• light is the means to study the matter (spectroscopy)  measurement
methods (interferometer)
• design media with desired properties: glasses, polymers, semiconductors,
compounded media (effective media, photonic crystals, meta-materials)
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Two-dimensional photonic crystal membrane.
D)
Light can modify matter
• light induces physical, chemical and biological processes
• used for lithography, material processing, or modification of biological
objects (bio-photonics)
Hole “drilled” with a fs laser at Institute of Applied Physics, FSU Jena.
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
E)
Optical telecommunication
• transmitting data (Terabit/s in one fiber) over transatlantic distances
1000 m telecommunication fiber is installed every second.
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
F)
Optics in medicine and life sciences
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
G)
Light sensors and light sources
• new light sources to reduce energy consumption
• new projection techniques
Deutscher Zukunftspreis 2008 - IOF Jena + OSRAM
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
H)
Micro- and nano-optics
• ultra small camera
Insect inspired camera system develop at Fraunhofer Institute IOF Jena
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
I)
Relativistic optics
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
J)
13
Schematic of optics
quantum optics
electromagnetic optics
wave optics
geometrical optics
• geometrical optics
⋅ λ << size of objects → daily experience
⋅ optical instruments, optical imaging
⋅ intensity, direction, coherence, phase, polarization, photons
G: Intensität, Richtung, Kohärenz, Phase, Polarisation, Photon
• wave optics
⋅ λ ≈ size of objects → interference, diffraction, dispersion, coherence
⋅ laser, holography, resolution, pulse propagation
⋅ intensity, direction, coherence, phase, polarization, photons
• electromagnetic optics
⋅ reflection, transmission, guided waves, resonators
⋅ laser, integrated optics, photonic crystals, Bragg mirrors ...
⋅ intensity, direction, coherence, phase, polarization, photons
• quantum optics
⋅ small number of photons, fluctuations, light-matter interaction
⋅ intensity, direction, coherence, phase, polarization, photons
• in this lecture
⋅ electromagnetic optics and wave optics
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
⋅ no quantum optics  subject of advanced lectures
K)
Literature
• Fundamental
1. Saleh, Teich, 'Fundamenals of Photonics', Wiley (1992)
in German: "Grundlagen der Photonik" Wiley (2008)
2. Hecht, 'Optic', Addison-Wesley (2001)
in German: "Optik", Oldenbourg (2005)
3. Mansuripur, 'Classical Optics and its Applications', Cambridge (2002)
4. Menzel, 'Photonics', Springer (2000)
5. Lipson, Lipson, Tannhäuser, 'Optik'; Springer (1997)
6. Born, Wolf, 'Principles of Optics', Pergamon
7. Sommerfeld, 'Optik'
• Advanced
1. W. Silvast, 'Laser Fundamentals',
2. Agrawal, 'Fiber-Optic Communication Systems', Wiley
3. Band, 'Light and Matter', Wiley, 2006
4. Karthe, Müller, 'Integrierte Optik', Teubner
5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Academic
6. Yariv, 'Optical Electronics in modern Communications', Oxford
7. Snyder, Love, 'Optical Waveguide Theory', Chapman&Hall
8. Römer, 'Theoretical Optics', Wiley,2005.
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1. Ray optics - geometrical optics (covered by
lecture Introduction to Optical Modeling)
The topic of “Ray optics – geometrical optics” is not covered in the course
“Fundamentals of modern optics”. This topic will be covered rather by the
course “Introduction to optical modeling”. The following part of the script
which is devoted to this topic is just included in the script for consistency.
1.1 Introduction
− Ray optics or geometrical optics is the simplest theory for doing optics.
− In this theory, propagation of light in various optical media can be
described by simple geometrical rules.
− Ray optics is based on a very rough approximation ( λ → 0 , no wave
phenomena), but we can explain almost all daily life experiences
involving light (shadows, mirrors, etc.).
− In particular, we can describe optical imaging with ray optics approach.
− In isotropic media, the direction of rays corresponds to the direction of
energy flow.
What is covered in this chapter?
− It gives fundamental postulates of the theory.
− It derives simple rules for propagation of light (rays).
− It introduces simple optical components.
− It introduces light propagation in inhomogeneous media (graded-index
(GRIN) optics).
− It introduces paraxial matrix optics.
1.2 Postulates
A)
B)
Light propagates as rays. Those rays are emitted by light-sources and
are observable by optical detectors.
The optical medium is characterized by a function n(r), the so-called
refractive index (n(r) ≥ 1 - meta-materials n(r) <0)
n=
c
cn
cn – speed of light in the medium
C) optical path length ∼ delay
i) homogeneous media
nl
ii) inhomogeneous media
B
∫ n(r)ds
A
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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D) Fermat’s principle
B
d ∫ n(r )ds =
0
A
Rays of light choose the optical path with the shortest delay.
1.3 Simple rules for propagation of light
A)
Homogeneous media
− n = const.  minimum delay = minimum distance
− Rays of light propagate on straight lines.
B) Reflection by a mirror (metal, dielectric coating)
− The reflected ray lies in the plane of incidence.
− The angle of reflection equals the angle of incidence.
C) Reflection and refraction by an interface
− Incident ray  reflected ray plus refracted ray
− The reflected ray obeys b).
− The refracted ray lies in the plane of incidence.
− The angle of refraction θ2 depends on the angle of incidence θ1 and is
given by Snell’s law:
n1 sin=
θ1 n2 sin θ2
− no information about amplitude ratio.
1.4 Simple optical components
A)
Mirror
i) Planar mirror
− Rays originating from P1 are reflected and seem to originate from P2.
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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ii) Parabolic mirror
− Parallel rays converge in the focal point (focal length f).
− Applications: Telescope, collimator
iii)
Elliptic mirror
− Rays originating from focal point P1 converge in the second focal point
P2
iv)
Spherical mirror
− Neither imaging like elliptical mirror nor focusing like parabolic mirror
− parallel rays cross the optical axis at different points
− connecting line of intersections of rays  caustic
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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− parallel, paraxial rays converge to the focal point f = (-R)/2
− convention: R < 0 - concave mirror; R > 0 - convex mirror.
− for paraxial rays the spherical mirror acts as a focusing as well as an
imaging optical element. paraxial rays emitted in point P1 are reflected
and converge in point P2
1 1
2
+ ≈
(imaging formula)
z1 z2 (− R)
paraxial imaging: imaging formula and magnification
m = -z2 /z1 (proof given in exercises)
B)
Planar interface
Snell’s law: n1 sin=
θ1 n2 sin θ2
for paraxial rays: n1θ1 = n2θ2
− external reflection ( n1 < n2 ): ray refracted away from the interface
− internal reflection ( n1 > n2 ): ray refracted towards the interface
− total internal reflection (TIR) for:
π
n
θ2 = → sin θ1= sin θTIR= 2
n1
2
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
C) Spherical interface (paraxial)
− paraxial imaging
θ2 ≈
n1
n −n y
θ1 − 2 1 (*)
n2
n2 R
n1 n2 n2 − n1
+ ≈
(imaging formula)
z1 z2
R
m= −
n1 z2
(magnification)
n2 z1
(Proof: exercise)
− if paraxiality is violated  aberration
− rays coming from one point of the object do not intersect in one point
of the image (caustic)
D) Spherical thin lens (paraxial)
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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− two spherical interfaces (R1, R2, ∆) apply (*) two times and assume
y=const (∆ small)
θ2 ≈ θ1 −
1
y
1
1 
with focal length: =−
( n 1)  − 
f
f
 R1 R2 
1 1 1
+ ≈ (imaging formula)
z1 z2 f
m= −
z2
(magnification)
z1
(compare to spherical mirror)
1.5 Ray tracing in inhomogeneous media (graded-index
- GRIN optics)
− n(r ) - continuous function, fabricated by, e.g., doping
− curved trajectories  graded-index layer can act as, e.g., a lens
1.5.1 Ray equation
Starting point: we minimize the optical path or the delay (Fermat)
B
d ∫ n(r )ds =
0
A
computation:
B
L = ∫ n r ( s )  ds
A
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
variation of the path: r ( s ) + δr ( s )
B
B
A
A
dL = ∫ dnds + ∫ ndds
=
dn grad n ⋅ dr
( dr + d dr )
dds
=
( dr )
=
2
2
−
( dr )
2
+ 2dr ⋅ d dr + ( d dr ) −
2
( dr )
2
dr d dr
⋅
− ds
ds ds
 dr d dr 
≈ ds 1 + ⋅
 − ds
 ds ds 
dr d dr
= ds ⋅
ds ds
≈ ds 1 + 2
=
dL
B
dr d dr 
 ds
ds 

∫  grad n ⋅ dr + n ds ⋅
A
=
B
d  dr  

integration by parts and A,B fix
∫  grad n − ds  n ds   ⋅ drds
A
δL =
0 for arbitrary variation
grad n =
d  dr 
 n  ray equation
ds  ds 
Possible solutions:
A) trajectory
dz 1 + ( dx dz ) + ( dy dz )
x(z) , y(z) and ds =
2
− solve for x(z) , y(z)
− paraxial rays  (ds ≈ dz )
2
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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d 
dx  dn
,
,
n
x
y
z
≈
(
)
dz 
dz  dx
d 
dy  dn
n ( x, y , z )  ≈

dz 
dz  dy
B)
homogeneous media
− straight lines
C) graded-index layer n(y) - paraxial, SELFOC
paraxial 
dy
 1 and dz ≈ ds
dz
 1

n 2 (=
y ) n02 (1 − α 2 y 2 ) ⇒ n( y ) ≈ n0 1 − α 2 y 2  for a  1
 2

d 
dy  d 
dy 
d2y
d2y
1 dn( y )
n
y
≈
n
y
≈
n
y
⇒
=
(
)
(
)
(
)
ds 
ds  dz 
dz 
dz 2
dz 2 n ( y ) dy
d2y
for n(y)-n0<<1:
= −α 2 y
2
dz
( z ) y0 cos αz +
y=
θ0
sin αz
α

θ( z ) =
dy
= − y0α sin αz + θ0 cos αz
dz
1.5.2 The eikonal equation
− bridge between geometrical optics and wave
− eikonal S(r) = constant  planes perpendicular to rays
− from S(r) we can determine direction of rays ∼ grad S(r) (like potential)
2
gradS ( r )  = n ( r )
2
Remark: it is possible to derive Fermat’s principle from eikonal equation
− geometrical optics: Fermat’s or eikonal equation
S ( rB ) −=
S ( rA )
∫
B
A
grad =
S ( r ) ds
B
∫ n ( r )ds
A
eikonal  optical path length ∼ phase of the wave
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
1.6 Matrix optics
− technique for paraxial ray tracing through optical systems
− propagation in a single plane only
− rays are characterized by the distance to the optical axis (y) and their
inclination (θ)  two algebraic equation  2 x 2 matrix
Advantage: we can trace a ray through an optical system of many elements
by multiplication of matrices.
1.6.1 The ray-transfer-matrix
in paraxial approximation:
y2= Ay1 + Bθ1
 y   A B   y1 
A B
=
=
→
M
  2 
C D 
θ2  C D   θ1 



θ=
Cy1 + Dθ1
2
A=0: same θ1  same y2  focusing
D=0: same y1  same θ2  collimation
1.6.2 Matrices of optical elements
A)
free space
1 d 
M=

0 1 
B)
refraction on planar interface
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
0 
1
M=

0 n1 n2 
C) refraction on spherical interface
1
0 

M=

 − ( n2 − n1 ) n2 R n1 n2 
D) thin lens
 1
M=
 −1 f
E)
0
1 
reflection on planar mirror
1 0 
M=

0 1 
F)
reflection on spherical mirror (compare to lens)
 1 0
M=

2 R 1
1.6.3 Cascaded elements
 y N +1   A B   y1 
A B
=
 θ  C D   θ=
 → M C D 
  1


 N +1  
M=MN….M2M
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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
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2. Optical fields in dispersive and isotropic media
2.1 Maxwell’s equations
Our general starting point is the set of Maxwell’s equations. They are the
basis of the electromagnetic approach to optics which is developed in this
lecture.
2.1.1 Adaption to optics
The notation of Maxwell’s equations is different for different disciplines of
science and engineering which rely on these equations to describe
electromagnetic phenomena at different frequency ranges. Even though
Maxwell's equations are valid for all frequencies, the physics of light matter
interaction is different for different frequencies. Since light matter interaction
must be included in the Maxwell's equations to solve them consistently,
different ways have been established how to write down Maxwell's equations
for different frequency ranges. Here we follow a notation which was
established for a convenient notation at frequencies close to visible light.
Maxwell’s equations (macroscopic)
In a rigorous way the electromagnetic theory is developed starting from the
properties of electromagnetic fields in vacuum. In vacuum one could write
down Maxwell's equations in their so-called pure microscopic form, which
includes the interaction with any kind of matter based on the consideration of
point charges. Obviously this is inadequate for the description of light in
condensed matter, since the number of point charges which would need to be
taken into account to describe a macroscopic object, would exceed all
imaginable computational resources.
To solve this problem one uses an averaging procedure, which summarizes
the influence of many point charges on the electromagnetic field in a
homogeneously distributed response of the solid state on the excitation by
light. In turn, also the electromagnetic fields are averaged over some
adequate volume. For optics this procedure is justified, since any kind of
available experimental detector could not resolve the very fine spatial details
of the fields in between the point charges, e.g. ions or electrons, which are
lost by this averaging.
These averaged electromagnetic equations have been rigorously derived in a
number of fundamental text books on electrodynamic theory. Here we will not
redo this derivation. We will rather start directly from the averaged Maxwell's
equations.
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∂B(r, t )
rot E(r, t ) =
−
div D(r, t ) =
rext (r, t )
∂t
∂D(r, t )
rot H (r, t ) =
jmakr (r, t ) +
div B(r, t ) =
0
∂t
− electric field (G: elektrisches Feld)
E(r, t )
[V/m]
− magnetic flux density
[Vs/m2] or [tesla]
B(r, t )
or magnetic induction
(G: magnetische Flussdichte oder magnetische Induktion)
− electric flux density
[As/m2]
D(r, t )
or electric displacement field
(G: elektrische Flussdichte oder dielektrische Verschiebung)
− magnetic field (G: magnetisches Feld) H (r, t )
[A/m]
− external charge density
ρext (ρ, t )
[As/m3]
− macroscopic current density
jmakr (r, t )
[A/m2]
Auxiliary fields
The "cost" of the introduction of macroscopic Maxwell's equations is the
occurrence of two additional fields, the dielectric flux density D(r, t ) and the
magnetic field H (r, t ) . These two fields are related to the electric field E(r, t )
and magnetic flux density B(r, t ) by two other new fields.
D(r, t ) =
ε0E(r, t ) + P (r, t )
=
H (r, t )
1
[B(r, t ) − M(r, t )]
µ0
− dielectric polarization
(G: dielektrische Polarisation)
− magnetic polarization
or magnetization
(G: Magnetisierung)
P( r , t )
[As/m2],
M (r , t )
[Vs/m2]
− electric constant
ε0 ≈ 8.854 × 10−12 As/Vm
or vacuum permittivity
(G: Vakuumpermittivität)
− magnetic constant
µ 0 = 4π × 10−7 Vs/Am
or vacuum permeability
(G: Vakuumpermeabilität)
electric constant and magnetic constant are connected by the speed of
light in vacuum c as
1
ε0 = 2
µ 0c
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Light matter interaction
In order to solve this set of equations, i.e. Maxwell's equations and auxiliary
field equations, one needs to connect the dielectric flux density D(r, t ) and
the magnetic field H (r, t ) to the electric field E(r, t ) and the magnetic flux
density B(r, t ) . This is achieved by modeling the material properties by
introducing the material equations.
• The effect of the medium gives rise to polarization P(r, t ) = f [ E] and
magnetization M (r, t ) = f [ B ] .  In order to solve Maxwell’s equations we
need material models which describe these quantities.
• In optics at visible wavelength, we generally deal with non-magnetizable
media. Hence we can assume M (r, t ) = 0 . Exceptions to this general
property are metamaterials wihich might possess some artificial effective
magnetization properties resulting in M (r, t ) ≠ 0 .
Furthermore we need to introduce sources of the fields into our model. This is
achieved by the so-called source terms which are inhomogeneities and hence
they define unique solutions of Maxwell's equations.
− free charge density (G: Dichte freier Ladungsträger)
ρext (ρ, t ) [As/m3]
− macroscopic current density (G: makroskopische Stromdichte)
consisting oft wo contributions
jmakr
=
(r, t ) jcond (r, t ) + jconv (r, t ) [A/m2]
⋅ conductive current density (G: Konduktionsstromdichte)
jcond (r, t ) = f [ E]
⋅ convective current density (G: Konvektionsstromdichte)
jconv (ρρ
, t ) = ρext ( , t ) v (ρ, t )
− In optics, we generally have no free charges which change at speeds
comparable to the frequency of light:
∂ρext (ρ, t )
≈ 0 → jconv (ρ, t ) =
0
∂t
• With the above simplifications, we can formulate Maxwell’s equations in the
context of optics:
∂H (r, t )
∂t
∂P(r, t )
∂E(r, t )
=
rot H (r, t ) j(r, t ) +
+ ε0
∂t
∂t
rot E(r, t ) = −µ 0
ε0 div E(r, t ) = − div P(r, t )
div H (r, t ) = 0
− In optics, the medium (or more precisely the mathematical material
model) determines the dependence of the induced polarization on the
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28
electric field P( E) and the dependence of the induced (conductive)
current density on the electric field j( E) .
− Once we have specified these relations, we can solve Maxwell’s
equations consistently.
Example:
− In vacuum, both polarization P and current density j are zero (most
simple material model). Hence we can solve Maxwell’s equations
directly.
Remarks:
− Even though the Maxwell's equations, in the way they have been
written above are derived to describe light, i.e. electromagnetic fields
at optical frequencies, they are simultaneously valid for other frequency ranges as well. Furthermore, since Maxwell's equations are linear
equations as long as the material does not introduce any nonlinearity
the principle of superposition holds. Hence we can decompose the
comprehensive electromagnetic fields into components of different
frequency ranges. In turn this means that we do not have to take care
of any slow electromagnetic phenomena, e.g. electrostatics or radio
wave, in our formulation of Maxwell's equations since they can be split
off from our optical problem and can be treated separately.
− We can define a bound charge density (G: Dichte gebundener
Ladungsträger) as the source of spatially changing polarization P .
ρb (ρ, t ) =
−div P (ρ, t )
− Analogously we can define a bound current density (G: Stromdichte
gebundener Ladungsträger) as the source of the temporal variation of
the polarization P .
jb (r, t ) =
∂P(r, t )
∂t
− This essentially means that we can describe the same physics in two
different ways since currents are in principle moving charges (see
generalized complex dielectric function below).
Complex field formalism (G: komplexer Feld-Formalismus)
− Maxwell’s equations are also valid for complex fields and are easier to
solve this way.
− This fact can be exploited to simplify calculations, because it is easier
to deal with complex exponential functions (exp(ix)) than with
trigonometric functions cos( x) and sin( x) .
− Hence we use the following convention in this lecture to distinguish
between the two types of fields.
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
real physical field:
Er (r, t )
complex mathematical representation:
E(r, t )
29
− Here we define their relation as
Er (r, t ) =
1
2
 E(r, t ) + E∗ (r, t )  = Re [ E(r, t )]
However, this relation can be defined differently in different textbooks.
− This means in general: For calculations we use the complex fields
[E(r, t )] and for physical results we go back to real fields by simply
omitting the imaginary part. This works because Maxwell’s equations
are linear and no multiplications of fields occur.
− Therefore, be careful when multiplications of fields are required in the
description of the material response or the field dynamics. In this case
you would have to go back to real quantities before you compute these
multiplication. This becomes relevant for, e.g., calculation of the Poynting vector, as can be seen in a chapter below.
2.1.2 Temporal dependence of the fields
When it comes to time dependence of the electromagnetic field, we can
distinguish two different types of light:
A) monochromatic light (stationary fields)
− harmonic dependence on temporal coordinate
− − exp(−iω t )  phase is fixed  coherent, infinite wave train e.g.:
=
E(r, t ) E(r )exp(−iωt )
− Monochromatic light approximates very well the typical output of a
continuous wave (CW) laser. Once we know the frequency we have to
compute the spatial dependence of the (stationary) fields only.
B) polychromatic light (non-stationary fields)
− finite wave train
− With the help of Fourier transformation we can decompose the fields
into infinite wave trains and use all the results from case A) (see next
section)
E(r=
,t)
∞
∫ E(r, ω)exp(−iωt )d ω
−∞
∞
1
=
E(r, ω)
∫ E(r, t )exp(iωt )dt
2p −∞
Remark: The position of the sign in the exponent and the factor 1 / 2π
can be defined differently in different textbooks.
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2.1.3 Maxwell’s equations in Fourier domain
In order to solve Maxwell's equations more easily we would like to introduce a
Fourier decompositions of the fields in the Maxwell's equations.
For this purpose, we need to find out how a time derivative of a dynamic
variable can be calculated in Fourier space.
A simple rule for the transformation a time derivative into Fourier space ca be
obtained using integration by parts:
+∞
+∞
1
1
 ∂

dt  E ( r, t )  exp ( iωt ) = −iω
∫
∫ dt E ( r, t ) exp ( iωt ) = −iω E(r, ω)
2p −∞  ∂t
2p −∞

Thus, a time derivative in real space transforms into a simple product with
−iω in Fourier space.
 rule:
∂
FT

→ − iω
∂t
Now we can write Maxwell’s equations in Fourier domain:
rot E(r, ω) =iωµ 0 H (r, ω)
ε0 div E(r, ω) =− div P(r, ω)
rot H (r, ω=
) j (r, ω) − iωP(r, ω) − iωε0E(r, ω) div H (r, ω=
) 0
2.1.4 From Maxwell’s equations to the wave equation
Maxwell's equations provide the basis to derive all possible mathematical
solutions of electromagnetic problems. However very often we are interested
just in the radiation fields which can be described more easily by an adapted
equation, which is the so-called wave equation. From Maxwell’s equations it
is straight forward to derive the wave equation by using the two curl
equations.
A) Time domain derivation
We start from applying the curl operator ( rot ) a second time on rot E(r, t ) = 
and substitute rot H with the other Maxwell equation.
rotrot E(r, t ) = −µ 0rot
∂H (r, t )
∂
∂P(r, t )
∂E(r, t ) 
= −µ 0  j(r, t ) +
+ ε0
∂t
∂t 
∂t
∂t 
We find the wave equation for the electric field as
1 ∂2
∂j(r, t )
∂ 2 P (r, t )
rotrot E(r, t ) + 2 2 E(r, t ) =
−µ 0
− µ0
c ∂t
∂t
∂t 2
The blue terms require knowledge of the material model. Additionally, we
have to make sure that all other Maxwell’s equations are fulfilled. This holds
in particular for the divergence of the electric field:
div [ ε0E(r, t ) + P (r, t )] =
0
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Once we have solved the wave equation, we know the electric field. From
that we can easily compute the magnetic field:
∂H(r, t )
1
= − rot E(r, t )
∂t
µ0
Remarks:
− An analog procedure is possible also for the magnetic field H , i.e., we
can derive a wave equation for the magnetic field as well.
− Normally, the wave equation for the electric field E is more
convenient, because the material model defines P(E) .
− However, for inhomogeneous media H can sometimes be the better
choice for the numerical solution of the partial differential equation
since it forms a hermitian operator.
− analog procedure possible for H instead of E
− generally, wave equation for E is more convenient, because P(E)
given
− for inhomogeneous media H can sometimes be better choice
B) Frequency domain derivation
We can do the same procedure to derive the wave equation also directly in
the Fourier domain and find
rotrot E(r, ω) −
ω2
E(r, ω)= iωµ 0 j(r, ω) + µ 0ω2 P (r, ω)
2
c
and
div ε0E(r, ω) + P(r, ω)  =0
− magnetic field:
H (r, ω) =−
i
rot E(r, ω)
ωµ 0
− transferring the results from the Fourier domain to the time domain
⋅ for stationary fields: take solution and multiply by e-iωt .
⋅ for non-stationary fields and linear media  inverse Fourier
transformation
E(r=
,t)
∞
∫ E(r, ω)exp(−iωt )d ω
−∞
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2.1.5 Decoupling of the vectorial wave equation
So far we have seen that for the general problem of electromagnetic waves
all 3 vectorial field components of the electric or the magnetic field are
coupled. Hence we have to solve a vectorial wave equation for the general
problem. However, it would be desirable to express problems also by scalar
equation since they are much easier to solve. For problems with translational
invariance in at least one direction, as e.g. for homogeneous infinite media,
layers or interfaces, this can be achieved since the vectorial components of
the fields can be decoupled.
Let’s assume invariance in the y-direction and propagation only in the x-zplane. Then all spatial derivatives along the y-direction disappear ( ∂ / ∂y =0 )
and the operators in the wave equation simplify.
 ∂∂x

=
rot rot E grad
div E − ∆ E 
=
∂
 ∂z
(
(
)
+ ∂∂Ez z   ∆ (2) E x 
 

0
 −  ∆ (2) E y 
  (2) 
∂Ex
∂Ez
+

∂x
∂z 
 ∆ E z 
∂Ex
∂x
)
The decoupling becomes visible when the three components of the general
vectorial field are decomposed in the following way.
• decomposition of electric field
 0 
 
 E⊥ =
E
E y ,
=
E E⊥ + E=

 0 
 
with Nabla operator ∇
(2)
 Ex 
 
 0 
 Ez 
 
 ∂ ∂x 
∂2
∂2


(2)
, and Laplace ∆ =
+ 2
=
2
 0 
∂
x
∂z
 ∂ ∂z 


Hence we obtain two wave equations for the E⊥ and E fields.
• gives two decoupled wave equations
∆ (2) E⊥ (r, ω) +
ω2
E⊥ (r, ω) =−iωµ 0 j⊥ (r, ω) − µ 0ω2 P ⊥ (r, ω)
2
c
ω2
∆ E (r, ω) + 2 E (r, ω) − grad (2) div (2) E =−iωµ 0 j (r, ω) − µ 0ω2 P  (r, ω)
c
(2)
These two wave equations are independent as long as the material response,
which is expressed by j and P , does not couple the respective field
components by some anisotropic response.
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33
Properties
− propagation of perpendicularly polarized fields E⊥ and E can be
treated separately
− propagation of E⊥ is described by scalar equation
− similarly the other field components can be described by a scalar
equation for H ⊥
− alternative notations:
⊥  s  TE (transversal electric)
 p  TM (transversal magnetic)

2.2 Optical properties of matter
In this chapter we will derive a simple material model for the polarization and
the current density. The basic idea is to write down an equation of motion for
a single exemplary charged particle and assume that all other particles of the
same type behave similarly. More precisely, we will use a driven harmonic
oscillator model to describe the motion of bound charges giving rise to a
polarization of the medium. For free charges we will use the same model but
without restoring force, leading eventually to a current density. In the
literature, this simple approach is often called the Drude-Lorentz model
(named after Paul Drude and Hendrik Antoon Lorentz).
2.2.1 Basics
We are looking for P( E) and j( E) . In general, this leads to a many body
problem in solid state theory which is rather complex. However, in many
cases phenomenological models are sufficient to describe the necessary
phenomena. As already pointed out above, we use the simplest approach,
the so-called Drude-Lorentz model for free or bound charge carriers
(electrons).
− assume an ensemble of non-coupling, driven, and damped harmonic
oscillators
− free charge carriers: metals and excited semiconductors (intraband)
− bound charge carriers: dielectric media and semiconductors
(interband)
− The Drude-Lorentz model creates a link between cause (electric field)
and effect (induced polarization or current). Because the resulting
relations P( E) and j( E) are linear (no E2 etc.), we can use linear
response theory.
For the polarization P(E) (for j(E) very similar):
− description in both time and frequency domain possible
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34
− In time domain: we introduce the response function
(G: Responsfunktion)
E(r, t )  medium (response function)  P(r, t )
t
Pi (r, t ) =
ε0 ∑ ∫ Rij (r, t − t ′) E j (r, t ′)dt ′
j −∞
with R̂ being a 2nd rank tensor
i = x, y, z and summing over j = x, y, z
− In frequency domain: we introduce the susceptibility
(G: Suszeptibilität)
E(r, ω)  medium (susceptibility)  P (r, ω)
Pi (r=
, ω) ε0 ∑ χij (r, ω) E j (r, ω)
j
− response function and susceptibility are linked via Fourier transform
(convolution theorem)
∞
1
Rij (=
t)
∫ χij (ω)exp(−iωt )d ω
2p −∞
− Obviously, things look friendlier in frequency domain. Using the wave
equation from before and assuming that there are no currents ( j = 0)
we find
ω2
rotrot E(r, ω) − 2 E(r, ω) =µ 0ω2 P (r, ω)
c
or
ω2
∆E(r, ω) + 2 E(r, ω) − graddivE(r, ω) =−µ 0ω2 P (r, ω)
c
− and for auxiliary fields
D(r, ω) =ε0E(r, ω) + P(r, ω)
The general response function and the respective susceptibility given above
simplifies for certain properties of the medium:
Simplification of the wave equation for different types of media
A) linear, homogenous, isotropic, non-dispersive media (most simple but
very unphysical case)
− homogenous  χij (r, ω) =χij (ω)
− isotropic  χij (r, ω) =χ(r, ω)δij
− non-dispersive  χij (r, ω) =χij (r ) → instantaneous: Rij (r, t ) =
χij (r )δ(t )
(Attention: This is unphysical!)
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35
 χij (r, ω) → χ is a scalar constant
frequency domain
time domain description
P(r, ω) =ε0χ E(r, ω)
↔
P(r, t ) = ε0χE(r, t ) (unphysical!)
D(r, ω) =ε0ε E(r, ω)
↔
D(r, t ) =ε0εE(r, t ) with ε = 1 + χ
Maxwell: div D = 0 → div E(r, ω) =0 for ε(ω) ≠ 0
ω2
ε ∂2
0
→ ∆ E(r, ω) + 2 ε E(r, ω) =0 → ∆ E(r, t ) − 2 2 E(r, t ) =
c
c ∂t
B)
− approximation is valid only for a certain frequency range, because all
media are dispersive
− based on an unphysical material model
linear, homogeneous, isotropic, dispersive media  χ(ω)
P(r, ω) =ε0χ(ω) E(r, ω)
D(r, ω) =ε0ε(ω) E(r, ω)
div D(=
r, ω) 0 ≠ div E(=
r, ω) 0 for ε(ω) ≠ 0
ω2
 ∆ E(r, ω) + 2 ε ( ω) E(r, ω) =0
c
Helmholtz equation
− This description is sufficient for many materials.
C) linear, inhomogeneous, isotropic, dispersive media  χ(r, ω)
P(r, ω) =ε0 χ(r, ω) E(r, ω),
D(r, ω) =ε0 ε(r, ω) E(r, ω).
div D(r, ω) =0
div D(r, ω) =ε0ε(r, ω)div E(r, ω) + ε0E(r, ω)grad ε(r, ω) =0,
→ div E(r, ω) =−
grad ε(r, ω)
E(r, ω).
ε(r, ω)
 grad ε(r, ω)

ω2
∆ E(r, ω) + 2 ε ( r, ω) E(r, ω) =−grad 
E(r, ω) 
c
 ε(r, ω)

− All field components couple.
D) linear, homogeneous, anisotropic, dispersive media → χij (ω)
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Pi (r, ω) =ε0 ∑ χij (ω) E j (r, ω)
j
Di (r, ω) =ε0 ∑ εij (ω) E j (r, ω).
36
 see chapter on crystal optics
j
− This is the worst case for a medium with linear response.
Before we start writing down the actual material model equations, let us
summarize what we want to do:
What kind of light-matter interaction do we want to consider?
I) Interaction of light with bound electrons and the lattice
The contributions of bound electrons and lattice vibrations in dielectrics and
semiconductors give rise to the polarization P . The lattice vibrations
(phonons) are the ionic part of the material model. Because of the large mass
of the ions (103 × mass of electron) the resulting oscillation frequencies will be
small. Generally speaking, phonons are responsible for thermal properties of
the medium. However, some phonon modes may contribute to optical properties, but they have small dispersion (weak dependence on frequency ω ).
Fully understanding the electronic transitions of bound electrons requires
quantum theoretical treatment, which allows an accurate computation of the
transition frequencies. However, a (phenomenological) classical treatment of
the oscillation of bound electrons is possible and useful.
II) Interaction of light with free electrons
The contribution of free electrons in metals and excited semiconductors gives
rise to a current density j . We assume a so-called (interaction-)free electron
gas, where the electron charges are neutralized by the background ions. Only
collisions with ions and related damping of the electron motion will be
considered.
We will look at the contributions from bound electrons / lattice vibrations and
free electrons separately. Later we will join the results in a generalized
material model which holds for many common optical materials..
2.2.2 Dielectric polarization and susceptibility
Let us first focus on bound charges (ions, electrons). In the so-called Drude
model, the electric field E(r, t ) gives rise to a displacement s(r, t ) of charged
particles from their equilibrium positions. In the easiest approach this can be
modeled by a driven harmonic oscillator:
∂2
∂
q
s(r, t ) + g s(r, t ) + ω02s(r, t ) =
E(r, t )
2
∂t
∂t
m
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37
− resonance frequency (electronic transition)  ω0
− damping  g
− charge  q
− mass  m
The induced electric dipole moment due to the displacement of charged
particles is given by
p(r, t ) = qs(r, t ),
We further assume that all bound charges of the same type behave identical,
i.e., we treat an ensemble of non-coupled, driven, and damped harmonic
oscillators. Then, the dipole density (polarization) is given by
=
P(r, t ) N=
p(r, t ) Nqs(r, t )
Hence, the governing equation for the polarization P(r, t ) reads as
∂2
∂
q2 N
2
P(r, t ) + g P(r, t ) + ω0 P(r, t ) =
E(r, t ) = ε0 fE(r, t )
∂t 2
∂t
m
1 e2 N
with oscillator strength f =
, for q=-e (electrons)
e0 m
This equation is easy to solve in Fourier domain:
−ω2 P(r, ω) − ig ωP(r, ω) + ω02 P(r, ω) =ε0 f E(r, ω)
=
→ P(r, ω)
ε0 f
E(r, ω)
( ω02 − ω2 ) − igω
with P(r, ω) =ε0χ(ω)E(r, ω)  χ(ω) =
(ω
2
0
f
− ω ) − ig ω
2
In general we have several different types of oscillators in a medium, i.e.,
several different resonance frequencies. Nevertheless, since in a good
approximation they do not influence each other, all these different oscillators
contribute individually to the polarization. Hence the model can be constructed by simply summing up all contributions.
− several resonance frequencies
fj


P(r, ω) =ε0 ∑  2
 E(r, ω) =ε0χ ( ω) E(r, ω)
2
ω
−
ω
−
i
g
ω
(
)
j 
j 
 0j



fj
χ ( ω) =∑  2

2
j  ( ω0 j − ω ) − ig j ω 


− χ ( ω) is the complex, frequency dependent susceptibility
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
38
D(r, ω) =ε0E(r, ω) + ε0χ ( ω) E(r, ω) =ε0ε ( ω) E(r, ω)
− ε ( ω) is the complex frequency dependent dielectric function
2
Example: (plotted for eta and kappa with ε ( ω) = η ( ω) + iκ ( ω)  )
2.2.3 Conductive current and conductivity
Let us now describe the response of a free electron gas with positively
charged background (no interaction). Again we use the model of a driven
harmonic oscillator, but this time with resonance frequency ω0 =0 . This
corresponds to the case of zero restoring force.
∂2
∂
e
s
(
r
,
t
)
+
g
s
(
r
,
t
)
=
−
E(r, t ),
∂t 2
∂t
m
The resulting induced current density is given by
j(r, t ) = − Ne
∂
s(r, t )
∂t
and the governing dynamic equation reads as
∂
e2 N
j(r, t ) + gj(r, t ) =
E(r, t ) =e0ω2pE(r, t )
∂t
m
1 e2 N
with plasma frequency ω = f =
e0 m
2
p
Again we solve this equation in Fourier domain:
−iω j (r, ω) + g j (r, ω) =ε0 ω2p E(r, ω)
 j (r, ω) =
ε0 ω2p
g − iω
E(r, ω) =σ ( ω) E(r, ω).
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
39
Here we introduced the complex frequency dependent conductivity
ε0 ω2p
ε0 ωω2p
.
σ ( ω) =
=−i 2
g − iω
−ω − ig ω
Remarks on plasma frequency
We consider a cloud of electrons and positive ions described by the total
charge density ρ in their self-consistent field E . Then we find according to
Maxwell:
ε0divE(ρρ
ρ( , t )
,t) =
For cold electrons, and because the total charge is zero, we can use our
damped oscillator model from before to describe the current density (only
electrons move):
∂
j + gj =
ε0ω2pE(r, t )
∂t
Now we apply divergence operator and plug in from above (red terms):
div
∂
j + gdivj =
ε0ω2pdivE(ρρ
, t ) =ω2pρ( , t )
∂t
With the continuity equation for the charge density (from Maxwell's equations)
∂
ρ + divj = 0,
∂t
We can substitute the divergence of the current density and find:
∂2
∂
− 2 ρ − g ρ = ω2pρ
∂t
∂t
∂2
∂
ρ + g ρ + ω2p =0  harmonic oscillator equation
2
∂t
∂t
Hence, the plasma frequency ωp is the eigen-frequency of such a charge
density.
2.2.4 The generalized complex dielectric function
In the sections above we have derived expressions for both polarization
(bound charges) and conductive current density (free charges). Let us now
plug our j (r, ω) and P(r, ω) into the wave equation (in Fourier domain)
rotrot E(r, ω) −
ω2
E(r, ω) =µ 0ω2 P (r, ω) + iωµ 0 j(r, ω)
2
c
= µ 0ε0ω2c(ω) + iωµ 0σ ( ω)  E(r, ω)
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
40
Hence we can collect all terms proportional to E(r, ω) and write

ω2 
i
rotrot E(r, ω) = 2 1 + c(ω) +
σ ( ω)  E(r, ω)
c 
ωε 0

ω2
rotrot
=
E(r, ω)
ε(ω)E(r, ω)
c2
Here, we introduced the generalized complex dielectric function
ε(ω) =1 + χ(ω) +
i
σ ( ω) =ε′(ω) + iε′′(ω)
ωε 0
So, in general we have


fj
ω2p
,
ε(ω) =1 + ∑  2
+
2
−ω2 − ig ω
j  ( ω0 j − ω ) − ig j ω 


because (from before)


fj
χ ( ω) =∑  2
,
2
ω
−
ω
−
g
ω
i
) j 
j ( 0 j

ε0 ωω2p
σ ( ω) =−i 2
.
−ω − ig ω
ε(ω) contains contributions from vacuum, phonons (lattice vibrations), bound
and free electrons.
Some special cases for materials in the infrared and visible spectral
range:
A) Dielectrics (insulators) in the infrared (IR) spectral range near
phonon resonance
If we are interested in dielectrics (insulators) near phonon resonance in the
infrared spectral range we can simplify the dielectric function as follows:


fj
f
ε(ω) 1 + ∑  2
with ω0 0 ω0 j and ω  ω0
+ 2
2
2
−
i
ω
−
ω
g
ω
(
)
j  ( ω0 j − ω ) − ig j ω 
0


 ε(ω)= ε∞ +
f
( ω02 − ω2 ) − igω
The contribution from electronic transitions shows almost no frequency
dependence (dispersion) in this frequency range far away from the electronic
resonances. hence it can be expressed together with the vacuum contribution
as a constant ε∞ .
Let us study the real and the imaginary part of the resulting ε(ω) separately:
ε∞  vacuum and electronic transitions
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
 ε( ω) =ℜε( ω) + i ℑε( ω) =ε′( ω) + i ε′′( ω)
ε′( ω) =ε∞ +
ε′′( ω) =
(ω
2
0
(ω
2
0
f
(ω
2
0
)
− ω2 )
2 2
−ω
2
+g ω
gf ω
)
2 2
−ω
2
2
2
+g ω
,
.  Lorentz curve
properties:
− resonance frequency: ω0
− width of resonance peak: g
− static dielectric constant in the limit ω → 0 : ε0 =ε∞ +
f
ω02
− so called longitudinal frequency ωL : ε′(ω = ωL ) = 0
− ε′′( ω) ≠ 0 : absorption and dispersion appear always together
− near resonance we find ε′( ω) < 0 (damping, i.e. decay of field, without
absorption if ε '' ≈ 0 )
∂ε′(ω) / ∂ω > 0
− frequency range with normal dispersion:
− frequency range with anomalous dispersion: ∂ε′(ω) / ∂ω < 0
Simplified example: sharp resonance for undamped oscillator g → 0
41
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42
− relation between resonance frequency ω0 and longitudinal frequency
ωL (Lyddane-Sachs-Teller relation)
ε′(ωL ) =ε∞ +
f
=0 , f =
2
2
ω
−
ω
( 0 L)
→ ωL =ω0
ε0
.
ε∞
(ε
0
− ε∞ ) ω02 (from above)
B) Dielectrics in the visible (VIS) spectral range
Dielectric media in visible (VIS) spectral range can be described by a socalled double resonance model, where a phonon resonance exists in the
infrared (IR) and an electronic transition exists in the ultraviolet (UV).
ε(ω) =ε∞ +
(ω
2
0p
fp
− ω2 ) − ig pω
+
fε
, with ω0 p 00
ω ω0 e
2
2
ω
−
ω
−
i
g
ω
( 0εε
)
ε∞  contribution of vacuum and other (far away) resonances
The generalization of this approach in the transparent spectral range leads to
the so-called Sellmeier formula.
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
ε′(ω) − 1 = ∑
j
f j ω02 j
(ω
2
0j
− ω2 )
43
,
− with j being the number of resonances taken into account
− describes many media very well (dispersion of absorption is neglected)
− oscillator strengths and resonance frequencies are often fit parameters
to match experimental data
C) Metals in the visible spectral range
If we want to describe metals in the visible spectral range we find
ω2p
ε(ω) = 1 − 2
with ωp >> ω
ω + ig ω
ε′( ω) = 1 −
ω2p
ω2 + g 2
,
ε′′( ω) =
g ω2p
ω ( ω2 + g 2 )
.
Metals show a large negative real part of the dielectric function ε′(ω) which
gives rise to decay of the fields. Eventually this results in reflection of light at
metallic surfaces.
2.2.5 Material models in time domain
Let us now transform our results of the material models back to time domain.
In Fourier domain we found for homogeneous and isotropic media:
D(r, ω) =ε0ε(ω)E(r, ω)
P(r, ω) =ε0χ(ω)E(r, ω).
The response function (or Green's function) R (t ) in the time domain is then
given by
R
(t )
=
1 ∞
d ω χ(ω)
χ(ω)exp ( −iωt )=
2p ∫ −∞
∫
∞
−∞
R(t )exp ( iωt ) dt
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
44
To prove this, we can use the convolution theorem
∞
∞
−∞
−∞
P(r, t ) =ω
e0 ∫ χ(ω)E(r, ω)exp (−iωt ) d ω
∫ P(r, )exp (−iωt ) d ω =
∞
=
e0 ∫ χ(ω)
−∞
1 ∞
E(r, t ′)exp ( iωt ′ ) dt ′ exp (−iωt ) d ω
2p ∫ −∞
Now we switch the order of integration, and identify the response function R
(red terms):
P(r, t ) = e0 ∫
1 ∞
χ(ω)exp (−iω(t − t ′) ) d ω E(r, t ′)dt ′
−∞ 2p ∫ −∞
((((((((((
∞
R ( t −t ′ )
∞
= e0 ∫ R(t − t ′)E(r, t ′)dt ′
−∞
For a “delta” excitation in the electric field we find the response or Greens
function as the polarization:
E(r, t ′′) =e δ(t ′′ − t0 )  P (r, t ) =
ε0 R (t − t0 )ε  Green's function
Examples
A) instantaneous media (unphysical simplification)
− For instantaneous (or non-dispersive) media, which cannot not really
exist in nature, we would find:
→ P ( r, t ) = ε0χE ( r, t )
R (t ) = χδ(t )
(unphysical!)
B) dielectrics
RP=
(t )
f
1 ∞
1 ∞
χ
ω
−
i
ω
=
ω
t
d
exp
exp ( −iωt ) d ω,
(
)
(
)
∫ −∞
2pp
2 ∫ −∞ ω02 − ω2 − ig ω
− Using the residual theorem we find:
f
 g 
 exp  − t  sin Ωt
R(t ) =  Ω
 2 
0

t≥0
with
Ω=
t<0
g2
ω −
4
2
0
t
f
 g

exp
=
− (t − t ′)  sin [ Ω(t − t ′) ] E(r, t ′)dt ′
P(r, t )
∫

Ω −∞
 2

C) metals
2
1 ∞
1 ∞ e 0 ωp
R=
dω
exp ( −iωt ) d ω,
j (t )
∫ −∞σ ( ω) exp ( −iωt )=
2pp
2 ∫ −∞ g − iω
− Using again the residual theorem we find:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
 ωp2
 exp ( − gt )
R(t ) =  g
0

2
p
j(r, t ) =e0ω
45

t ≥ 0

t < 0 
t
∫ exp [ − g (t − t′)] E(r, t′)dt′
−∞
2.3 The Poynting vector and energy balance
2.3.1 Time averaged Poynting vector
The energy flux of the electromagnetic field is given by the Poynting vector S .
In practice, we always measure the energy flux through a surface (detector),
S ⋅ n , where n is the normal vector of surface. To be more precise, the
Poynting vector =
S(r, t ) Er (r, t ) × H r (r, t ) gives the momentary energy flux.
Note that we have to use the real electric and magnetic fields, because a
product of fields occurs.
In optics we have to consider the following time scales:
− optical cycle:
T0 = 2π / ω0 ≤ 10−14 s
− pulse duration:
Tp
in general Tp >> T0
− duration of measurement: Tm
in general Tm >> T0
Hence, in general the detector does not recognize the fast oscillations of the
optical field − e − iω0t (optical cycles) and only delivers a time averaged value.
For the situation described above, the electro-magnetic fields factorize in
slowly varying envelopes and fast carrier oscillations:
1 
E(r, t )exp ( −iω0t ) + c.c. =Er (r, t )
2
For such pulses, the momentary Poynting vector reads:
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46
=
S(r, t ) Er (r, t ) × H r (r, t )
1 
 ∗ (r, t ) + E
 ∗ (r, t ) × H
 (r, t ) 
E(r, t ) × H

4
1 
 (r, t )exp ( −2iω t ) + E
 ∗ (r, t ) × H
 ∗ (r, t )exp ( 2iω t ) 
(r, t ) × H
+ E
0
0 
4
1
 (r, t ) × H
 ∗ (r, t )  + 1 ℜ E
 (r, t ) × H
 (r, t )  cos ( 2ω t )
= ℜ E
0



2
2
(((
((( (((((((
(((
=
sloω
fast
1
 ∗ (r, t ) × H
 ∗ (r, t )  sin ( 2ω t ) .
+ ℑ E
0

2
(((((
(((((((

fast
We find that the momentary Poynting vector has some slow contributions
which change over time scales of the pulse envelope Tp, and some fast
contributions 
cos ( 2ω0t ) , sin ( 2ω0t ) changing over time scales of the optical
cycle T0. Now, a measurement of the Poynting vector over a time interval Tm
leads to a time average of S(r, t ) .
S(r, t ) =
1
Tm
∫
t +Tm /2
S(r, t ')dt '
t −Tm /2
The fast oscillating terms ~ cos 2ω0t and ~ sin 2ω0t cancel by the integration
since the pulse envelope does not change much over one optical cycle.
Hence we get only a contribution from the slow term.
S(r, t ) =
1 1
2 Tm
∫
t +Tm /2
 (r, t ') × H
 ∗ (r, t ')  dt '
ℜ E

t −Tm /2
Let us now have a look at the special (but important) case of stationary
(monochromatic) fields. Then, the pulse envelope does not depend on time at
all (infinitely long pulses).
 (r, t ') E=
 (r, t ') H (r )
=
E
(r ),
H
1
S(r, t ) =
ℜ E(r ) × H ∗ (r )  .
2
This is the definition for the optical intensity I = S(r, t ) . We see that an
intensity measurement destroys information on the phase.
I = S(r, t )  measurement destroys phase information
2.3.2 Time averaged energy balance
Let us motivate a little bit further the concept of the Poynting vector. Some
interesting insight on the energy flow of light and hence also on the transport
of information can be obtained from the Poynting theorem, which is the
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
47
equation for the energy balance of the electromagnetic field. The Poynting
theorem can be derived directly from Maxwell’s equations. We multiply the
two curl equations by H r resp. Er (note that we use real fields):
∂
Hr =
0
∂t
∂
∂
Er ⋅ rotH r − ε0Er ⋅ E=
Er ⋅ ( jr + Pr )
r
∂t
∂t
H r ⋅ rotEr + µ 0 H r ⋅
Next, we subtract the two equations and get
H r ⋅ rotEr − Er ⋅ rotH r + ε0Er ⋅
∂
∂
∂
Er + µ 0 H r ⋅ H r =
−Er ⋅ ( jr + Pr ).
∂t
∂t
∂t
This equation can be simplified by using the following vector identity:
=
H r ⋅ rotEr − Er ⋅ rotH
div ( Er × H r )
r
Finally, with the substitution Er ⋅ ∂Er ∂t =
1
2
∂E2r ∂t we find Poynting's theorem
1 ∂ 2 1 ∂ 2
∂ 

ε0 Er + µ 0 H r + div ( Er × H r ) =−Er ⋅  jr + Pr  (*)
2 ∂t
2 ∂t
∂t 

This equation has the general form of a balance equation. Here it represents
the energy balance. Apart from the appearance of the divergence of the
Poynting vector (energy flux), we can identify the vacuum energy density
u = 12 ε0Er 2 + 12 µ 0 H r 2 . The right-hand-side of the Poynting's theorem contains
the so-called source terms.
1
1
where u = ε0Er 2 + µ 0 H r 2  vacuum energy density
2
2
In the case of stationary fields and isotropic media (simple but important)
1
E(r )exp ( −iω0t ) + c.c.
2
1
 H (r )exp ( −iω0t ) + c.c.
=
H r (r, t )
2
=
Er (r, t )
Time averaging of the left hand side of Poynting’s theorem (*) yields:
1 ∂ 2
1 ∂
1
ε0 Er (r, t ) + µ 0 H r2 (r, t ) + div [ Er (r, t ) × H r (r, t )=
div ℜ E(r ) × H ∗ (r ) 
]
2 ∂t
2 ∂t
2
{
}
= div S(r, t ) .
Note that the time derivative removes stationary terms in Er2 (r, t ) and H r2 (r, t ) .
Time averaging of the right hand side of Poynting’s theorem yields (source
terms):
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48
∂


−  jr (r, t ) + Pr (r, t )  Er (r, t )
∂t


1
=
− σ(ω0 )E(r )e − iω0t − iω0e0c(ω0 )E(r )e − iω0t c.c. E(r)e − iω0t + c.c.
4
Now we use our generalized dielectric function:
=−
=
1
4



σ ( ω0 ) 
 −iω0e0  c ( ω0 ) + i
 E(r) exp ( −iω0t ) + c.c. E(r) exp ( −iω0t ) + c.c.
ω0e0 



1
iω0e0 e ( ω0 ) − 1 E(r)E(r)∗ + c.c.
4
Again, all fast oscillating terms  exp ( ±2iω0t ) cancel due to the time average.
Finally, splitting ε ( ω0 ) into real and imaginary part yields
1
1
= iω0ε0 ε′ ( ω0 ) − 1 + iε′′ ( ω0 )  E(r)E(r)∗ + c.c. = − ω0ε0ε′′ ( ω0 ) E(r )E∗ (r ).
4
2
Hence, the divergence of the time averaged Poynting vector is related to the
imaginary part of the generalized dielectric function:
1
∗
 div S = − ω0ε0ε′′ ( ω0 ) E(r )E (r ).
2
This shows that a nonzero imaginary part of epsilon ( ε′′ ( ω) ≠ 0 ) causes a
drain of energy flux. In particular, we always have ε′′ ( ω) > 0 , otherwise there
would be gain of energy. In particular near resonances we have ε′′ ( ω) ≠ 0 and
therefore absorption.
Further insight into the meaning of div S gives the so-called divergence
theorem. If the energy of the electro-magnetic field is flowing through some
volume, and we wish to know how much energy flows out of a certain region
within that volume, then we need to add up the sources inside the region and
subtract the sinks. The energy flux is represented by the (time averaged)
Poynting vector, and the Poynting vector's divergence at a given point
describes the strength of the source or sink there. So, integrating the
Poynting vector's divergence over the interior of the region equals the integral
of the Poynting vector over the region's boundary.
∫ div
V
S=
dV
∫
A
S ⋅ ndA
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49
2.4 Normal modes in homogeneous isotropic media
Using the linear material models which we discussed in the previous chapters
we can now look for self-consistent solutions to the wave equation including
the material response.
It is convenient to use the generalized complex dielectric function to derive
the solution of the wave equation
ε(ω) =1 + χ(ω) +
i
σ ( ω) =ε′(ω) + iε′′(ω)
ωε0
We will do our analysis in Fourier domain. In particular, we will focus on the
most simple solution to the wave equation in Fourier domain, the so-called
normal modes. These normal modes are the stationary solutions of the wave
equation. Hence they usually correspond to waves which are infinitely extended in space and time. However as we will see later, by taking into account
the principle of superposition we can construct the general solutions from
these normal modes, by superimposing multiple normal modes to construct
also the transient waves.
We start from the wave equation in Fourier domain, which reads as
ω2
rotrot E(r, ω=
)
ε( ω)E(r, ω)
c2
According to Maxwell the solutions have to fulfill additionally the divergence
equation:
ε0 [1 + χ(ω) ] div E(r, ω) =0
In general, this additional condition implies that the electric field is free of
divergence:
for 1 + χ(ω) ≠ 0  div E(r, ω) =0 (normal case)
Let us for a moment assume that we already know that we can find plane
wave solutions of the following form in the frequency domain:
E(r, ω) = E(ω)exp ( ikr ) ,
k = unknown complex wave-vector
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50
The corresponding stationary field in time domain is given by:
=
E(r, t ) E exp i ( kr − ωt ) 
 monochromatic plane wave  normal mode
This is a monochromatic plane wave, the simplest solution we can expect, a
so-called normal mode.
Then, the divergence condition implies that these waves are transverse
k ⊥ E(ω)  transverse wave
Here transverse means that the electric field and the wave-vector are
oriented perpendicular to each other.
If we split the complex wave vector into real and imaginary part k = k' + ik'',
we can define:
o planes of constant phase
k ′r = const
o planes of constant amplitude k ′′r = const
In the following we will call the solutions
A) homogeneous waves  if these planes are identical
B) evanescent waves
 if these planes are perpendicular
C) inhomogeneous waves  otherwise
We will see that in dielectrics ( σ ( ω) =0 ) we can find a second, exotic type of
wave solutions: At ω = ωL → ε(ωL ) = 0, so-called longitudinal waves k  E(ω)
appear.
2.4.1 Transverse waves
Let us have a closer look at the transverse nature of the fields first. As
pointed out above, for ω ≠ ωL the electric field becomes free of divergence:
ε0ε(ω)div E(r, ω) =0  div E(r, ω) =0
Then, the wave equation reduces to the Helmholtz equation:
ω2
∆E(r, ω) + 2 ε( ω)E(r, ω) =0.
c
Hence, we have three scalar equations for E(r, ω) (from Helmholtz), and
together with the divergence condition we are left with two independent field
components. We will now construct solutions using the plane wave ansatz:
E(r, ω) = E(ω)exp ( ikr )
Immediately we see that the wave is transversal:
0= divE(ω)= ik ⋅ E(r, ω) → k ⊥ E(ω).
Hence, we have to solve
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 2 ω2

 −k + c 2 ε(ω)  E(ω) =0 and k ⋅ E(ω) =0.


which leads to the following dispersion relation
ω2
k = k = k + k + k = 2 ε(ω)
c
2
2
2
x
2
y
2
z
) ωc ε(ω) is a function of the
We see that the so-called wave-number k (ω=
frequency. We can conclude that transverse plane waves are solutions to
Maxwell's equations in homogeneous, isotropic media, only if the dispersion
relation k ( ω) is fulfilled.
In general, k = k ′ + ik ′′ is complex. Alternatively if k ′  k ′′ , it is sometimes
useful to introduce the complex refractive index n̂ :
k (ω=
)
ω
ω
ω
ε(ω=
)
nˆ (ω=
)
[ n(ω) + ik(ω)]
c
c
c
However, instead of assuming that nˆ (ω) and ε(ω) are just the same, one
should clearly distinguish between the two. While ε(ω) is a property of the
medium, nˆ (ω) is a property of a particular type of the electromagnetic field in
the medium, i.e. a property of the infinitely extended monochromatic plane
wave.
E(r, ω) = E(ω)exp ( ikr )
Hence, the coincidence of the complex refractive index nˆ (ω) with ε(ω)
holds only for homogeneous media since nˆ (ω) cannot not be a local property
while ε(ω) could.
With the knowledge of the electric field we can compute the magnetic field if
desired:
H (r, ω) =−
1
i
k × E(ω)  exp ( ikr )
rot E(r, ω) =
ωµ 0
ωµ 0 
→ H (r, ω)= H (ω)exp ( ikr ) ,
ωith H (ω)=
1
k × E(ω) 
ωµ 0 
2.4.2 Longitudinal waves
Let us now have a look at the rather exotic case of longitudinal waves. These
waves can only exist for ε( ω) =0 in dielectrics at the longitudinal frequency
ω = ωL . In this case, we cannot conclude that div E(r, ω) =0 , and hence the
wave equation reads (the l.h.s. vanishes because ε( ω) =0 ):
0
 rotrot E(r, ωL ) =
As for the transversal waves we try the plane wave ansatz and assume k to
be real.
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E(r, ω) = E(ω)exp ( ikr )
With rot E(ω)exp ( ikr )  = ik × E(ω)exp ( ikr ) we get from the wave equation:
k × k × E(r, ωL )  =
0
Now we decompose the electric field into transversal and longitudinal components with respect to the wave vector:
E(r, ω) = E(ω)exp ( ikr ) = E⊥ (ω)exp ( ikr ) + E (ω)exp ( ikr )
with E⊥ (ω) ⊥ k and E (ω)  k
This decomposed field is inserted into the wave equation:
k × k × ( E⊥ + E )  exp ( ikr ) = 0
k × k × E⊥  exp ( ikr ) + k × k × E  exp ( ikr ) =
0

((

=0
Since the cross product of k with the longitudinal field E (ω) is trivially zero
the remaining wave equation is:
k 2E⊥ = 0
Hence the transversal field E⊥ must vanish and the only remaining field
component is the longitudinal field E (ω) :
 E(r, ωL ) = E (ωL )exp ( ikr )
2.4.3 Plane wave solutions in different frequency regimes
(
)
The dispersion relation k 2 = k 2 = kx2 + ky2 + kz2 = ω2 c 2 ε(ω) for plane wave
solutions dictates the (complex) wavenumber k only. Thus, different solutions
for the complex wave vector k = k ′ + ik ′′ are possible. In addition, the
generalized dielectric function ε(ω) is complex. In this chapter we will discuss
possible scenarios and resulting plane wave solutions.
A) Positive real valued epsilon ε ( ω) =ε ' ( ω) > 0
This is the favorable regime for optics. We have transparency, and the frequency of light is far from resonances of the medium. The dispersion relation
gives
ω2
ω2 2
k = k' − k'' + 2ik '⋅ k'' = 2 ε′(ω) = 2 n (ω)
c
c
2
2
2
⇒
k '⋅ k'' = 0
There are two possibilities to fulfill this condition, either k'' = 0 or k' ⊥ k'' .
A.1) Real valued wave-vector k'' = 0
• In this case the wave vector is real and we find the dispersion relation
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k (ω)=
53
ω
ω 2π
n(ω)=
=
n(ω)
c
cn λ
• Because k'' = 0 these waves are homogeneous, i.e. planes of constant
phase are parallel to the planes of constant amplitude. This is trivial,
because the amplitude is constant.
Example 1: single resonance in dielectric material
− for lattice vibrations (phonons)
− Now the imaginary part of ε(ω) is neglected, which mathematically
corresponds to an undamped resonance
ε(ω) =ε′(ω) =∞ +
f
ω − ω2
2
0
)
− We can invert the dispersion relation k (ω=
ω
ε(ω)  ω(k ) :
c
Example 2: free electrons
− for plasma and metal
− Again the imaginary part of ε(ω) is neglected
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ε(ω) =ε′(ω) =1 −
54
ω2p
ω2
)
− We again invert the dispersion relation k (ω=
ω
ε(ω) → ω(k ) :
c
A.2) Complex valued wave-vector k' ⊥ k''
• The second possibility to fulfill the dispersion relation leads to a complex
wave-vector and so-called evanescent waves. We find
ω2
2
k'2 − k 2
k = k' − k''= 2 ε(ω) and therefore k''=
c
2
2
2
• This means that
 k''2 ≠ 0 and k '2 > k 2
• We will discuss the importance of evanescent waves in the next chapter,
where we will study the propagation of arbitrary initial field distributions.
What is interesting to note here is that evanescent waves can have
arbitrary large k '2 > k 2 , whereas the homogeneous waves of case A.1) (
k'' = 0 ) obey k '2 = k 2 . If we plug our findings into the plane wave ansatz we
get: for the evanescent waves:
{
}
E(r, ω) = E(ω) exp i k' ( ω) r  exp ( −k''(ω)r )
• The planes defined by the equation k''(ω)r = const. are the so-called planes
of constant amplitude, those defined by k'(ω)r = const. are the planes of
constant phase. Because of k' ⊥ k'' these planes are perpendicular to each
other.
• The factor exp ( −k''(ω)r ) leads to exponential growth of evanescent waves
in homogeneous space. Therefore, evanescent waves can't be physically
justified normal modes of homogeneous space and can only exist in
inhomogeneous space, where the exponential growth is suppressed, e.g.
at interfaces.
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B) Negative real valued epsilon ε(ω) =ε′(ω) < 0
This situation (negative but real ε(ω) can occur near resonances in dielectrics
( ω0 < ω < ωL ) or below the plasma frequency ( ω < ωp ) in metals. Then the
dispersion relation gives
ω2
k = k' − k'' + 2ik' ⋅ k''= 2 ε′(ω) < 0
c
2
2
2
As in the previous case A), the imaginary term has to vanish and k '⋅ k'' =
0.
Again this can be achieved by two possibilities.
B.1) k' = 0
ω2
 k'' = 2 ε′(ω)  E(r, ω) − exp ( −k''r )  strong damping
c
2
B.2) k' ⋅ k'' =
0
 k' ⊥ k''  evanescent waves
ω2
k =k' − k'' =− 2 ε′(ω)
c
2
2
2
ω2
k'' = 2 ε′(ω) + k'2 .
c
2
As above, these evanescent waves exist only at interfaces (like for
ε(ω) =ε′(ω) > 0 ). The interesting point is that here we find evanescent waves
for all values of k'2 . In particular, case B.1) ( k' = 0 ) is included. Hence, we
can conclude that for ε(ω) =ε′(ω) < 0 we find only evanescent waves!
C) Complex valued epsilon ε(ω)
This is the general case, which is relevant particularly near resonances. From
our (optical) point of view only weak absorption is interesting. Therefore, in
the following we will always assume ε′′( ω) << ε′( ω) . As we can see in the
following sketch, we can have ε′(ω) > 0, ε′′(ω) > 0, or ε′(ω) < 0, ε′′(ω) > 0.
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Let us further consider only the important special case of quasi-homogeneous
plane waves, i.e., k' and k'' are almost parallel. Then, it is convenient to use
the complex refractive index
ω2
ω2 2
ω2
2
[k′ + ik′′] = k (ω)= 2 ε(ω)= 2 nˆ (ω)= 2 [ n(ω) + ik(ω)]
c
c
c
2
2
Since k' and k'' are almost parallel:
ω
n(ω),
c
 κ' =
κ'' =
ω
κ(ω)
c
The dispersion relation in terms of the complex refractive index gives
ω2
ω2
2
k = k = 2 ε( ω)= 2 [ n( ω) + ik ( ω)]
c
c
2
2
Here we have
ε(ω) =ε′(ω) + iε′′(ω) =n 2 (ω) − κ 2 (ω) + 2in(ω) κ(ω),
ε′(ω=
) n 2 (ω) − κ 2 (ω)
and therefore 
ε′′(ω=
) 2n(ω) κ(ω)
2
n=
( ω)
ε′ 
2
sgn ( ε′) 1 + ( ε′′ / ε′) + 1 ,

2 
2
κ=
( ω)
ε′ 
2
sgn ( ε′) 1 + ( ε′′ / ε′) − 1 .

2 
Two important limiting cases of quasi-homogeneous plane waves:
C.1) ε′, ε′′ > 0, ε′′ << ε′, (dielectric media)
n( ω) ≈ ε′( ω),
κ( ω) ≈
1 ε′′( ω)
2 ε′( ω)
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In this regime propagation dominates ( n(ω)  κ(ω) ), and we have weak
absorption:
ω2
k' − k'' = 2 ε′(ω),
c
2
κ' =
ω2
2k' ⋅ k''= 2 ε′′(ω).
c
2
ω
ω
n(ω) ≈
ε′(ω),
c
c
κ'' =
ω
1 ω ε′′(ω)
κ(ω) ≈
c
2 c ε′(ω)
− k' ⋅ k'' ≈ k' k'' ,
− k' and k'' almost parallel  homogeneous waves
 in homogeneous, isotropic media, next to resonances, we find damped,
homogeneous plane waves, k'  k ′′  ek with e k being the unit vector along k
 ω

 ω

E(r, ω)= E(ω)exp ( iκr )= E(ω)exp i  n ( ω)( eκκ
r )   exp  − κ ( ω)( e r )  .

 c

 c
C.2) ε′ < 0, ε′′ > 0, ε′′ << ε′ , (metals and dielectric media in so-called
Reststrahl domain)
n( ω) ≈
1 ε′′( ω)
,
2 ε′( ω)
κ( ω) ≈
ε′( ω) ,
In this regime damping dominates ( n(ω)  κ(ω) ) and we find a very small
refractive index. Interestingly, propagation (nonzero n) is only possible due to
absorption (see time averaged Poynting vector below).
Summary of normal modes
Here we summarize our previous findings about the properties of normal
modes for different parameters of the material which they exist in. We do this
at the example of a dielectric media for frequencies close to one resonance.
Here we can identify different frequency intervals in which we find the typical
behavior of the material's reponse.
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There are the following frequency ranges, which give rise to the typical
properties of normal modes:
a) Frequency far below or far above resonance where ε′(ω) > 0 , ε′′(ω) ≈ 0
Typical normal modes:
− undamped homogeneous waves
− evanescent waves
b) Frequency above resonance where ε′(ω) < 0 , ε′′(ω) ≈ 0
Typical normal modes:
− evanescent waves
c) Frequency close to and below resonance where ε′(ω) > 0 , ε′′(ω) > 0
Typical normal modes:
− weakly damped quasi-homogeneous waves
d) Frequency close to and above resonance where ε′(ω) > 0 , ε′′(ω) > 0
Typical normal modes:
− strongly damped quasi-homogeneous waves
Optical systems work mainly in regime a) since here you find light propagating undamped over long distances through space. Furthermore one
sometimes exploits also regime b) when one would like to have reflection of
light at surfaces (e.g. at a metallic mirror).
2.4.4 Time averaged Poynting vector of plane waves
S(r, t ) =
1 1
2 Tm
∫
t +Tm /2
 ∗ (r, t ′)  dt ′,
ℜ E (r, t ′) × H

t −Tm /2
For plane waves we find:
E(r, t ) E exp ( ikr=
=
− iωt ) E exp ( ik ′r − k ′′r − iωt )
H=
(r, t )
1
k × E(r, t )
ωµ 0
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assuming a stationary case E(t )= E(ω)exp(− i ω t)
 S(r, t ) =
1 κ′
1
e
2
 ω
 2
exp [ −2κ ′′r ] E =
n 0 eκ' exp  −2 κ ( eκ" r )  E
2 ωµ 0
2 µ0
 c

with e k' being the unit vector along k ′ and e k'' being the unit vector along k ′′ .
2.5 The Kramers-Kronig relation
In the previous sections we have assumed a very simple model for the description of the material's response to the excitation by the electromagnetic
field. This model was based on quite strong assumptions, like a single charge
which is attached to a rigid lattice etc. Hence, one could imagine that more
complex matter could give rise to arbitrarily complex response functions if
adequate models would be used for its description. However we can show
from basic laws of physics, that several properties are common to all possible
response functions, as long as a linear response to the excitation is assumed.
These fundamental properties of the response function are formulated
mathematically by the Kramers-Kronig relation. It is a general relation
between ε′(ω) (dispersion) and ε′′(ω) (absorption). This means in practice that
we can compute ε′(ω) from ε′′(ω) and vice versa. For example, if we have
access to the absorption spectrum of a medium, we can calculate the dispersion.
The Kramers-Kronig relation follows from reality and causality of the response
function R of a linear system. That the response function is real valued is a
direct consequence from Maxwell's equations which are real valued as well.
Causality is also a very fundamental property, since the polarization must not
depend on some future electric field. As we have seen in the previous
sections, in time-domain the polarization and the electric field are related as:
t
∞
−∞
0
Pr (r, t ) = ε0 ∫ R (t − t ′)Er (r, t ′)dt ′ ↔ Pr (r, t ) = ε0 ∫ R(t)Er (r, t − t)d t
Reality of the response function implies:
∞
∞
1
1
e-iωτ
=
R ( τ)
d ω χ ( ω)=
d ω χ* ( ω) eiωτ → χ ( ω) = χ∗ ( −ω)
∫
∫
2π -∞
2π -∞
Causality of the response function implies:
1 for τ > 0
1

θ( τ)  =
for τ 0  Heaviside distribution
R ( τ ) =θ ( ττ
) y ( ) with =
2

0 for τ < 0
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In the following, we will make use of the Fourier transform of Heaviside
distribution:
∞
2πθ (=
ω)
∫ dtθ ( t ) e=
iωt
P
−∞
i
+ πd ( ω)  defined as integral only
ω
In Fourier space, the Heaviside distribution consists of the Dirac delta distribution
∞
f ( ω0 )  Dirac delta distribution
∫ d ωd(ω − ω0 ) f ( ω=)
−∞
and the expression P(i/ω) involving a Cauchy principal value:
∞
 −α

i
i
i
P ∫ d ω f (ω=
) lim  ∫ d ω f (ω) + ∫ d ω f (ω) 
α→0
ω
ω
ω
−∞
α
 −∞
  Cauchy principle value
As we have seen above, causality implies that the response function has to
contain a multiplicative Heaviside function. Hence, in Fourier space (susceptibility) we expect a convolution:
∞
∞
χ ( ω=
)
∫ d τR ( τ ) e
−∞

∞
) y( )e
∫ d τθ ( ττ
iωτ
=
iωτ
−∞
∞
=
∫ d ω θ ( ω − ω) y ( ω)
θ=
( ω)
1 i 1
P + δ ( ω)
2π ω 2
−∞
∞
i y ( ω) y ( ω)
1
=
χ ( ω)
P ∫ dω
+
(∗)
2π −∞
ω−ω
2
In order to derive the Kramers-Kronig relation we can use a small trick (this
trick saves us using complex integration in the derivation). Because of the
Heaviside function, we can choose the function y ( τ ) for t < 0 arbitrarily
without altering the susceptibility! In particular, we can choose:
a)
a) y ( −τ=
) y ( τ)
even function
b) y ( −τ ) = − y ( τ )
odd function
y ( −τ=
) y ( τ)
In this case y ( −τ=
) y ( τ ) is a real valued and even function. We can exploit
this property and show that
∞
∞
1
1
ωτ
 y ( ω) =
d ττ
y ( ) ei=
d ττ
y ( ) e −iωτ = y ∗ ( ω) is real as well
∫
∫
2π −∞
2π −∞
Hence, we can conclude from equation (*) above that
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61
∞
i y ( ω) y ( ω)
1
 χ ( ω) =− P ∫ d ω
+
ω−ω
2π −∞
2
∗
Here P ∫ is a so called principal value integral (G: Hauptwertintegral).
Now we have expressions for χ ( ω) , χ* ( ω) and can compute real and
imaginary part of the susceptibility:
∞
i y ( ω) y ( ω) 1 ∞
i y ( ω) y ( ω)
1
P ∫ dω
P ∫ dω
χ ( ω) + χ ( ω=
+
−
+
= y ( ω)
)
2π −∞
2
2π −∞
2
ω−ω
ω−ω
*
∞
i y ( ω)
1
χ ( ω) − χ ( ω=
) = P ∫ d ω
π −∞
ω−ω
∗
Plugging the last two equations together we find the first Kramers-Kronig
relation:
ℜ χ ( ω)
1
ℑχ ( ω) − P ∫ d ω
=
π −∞
ω−ω
∞
1. K-K relation
Knowledge of the real part of the susceptibility (dispersion) allows us to
compute the imaginary part (absorption).
y ( −τ ) = − y ( τ )
b)
The second K-K relation can be found by a similar procedure when we
assume that y ( −τ ) = − y ( τ ) is a real odd function. We can show that in this
case
∞
∞
1
1
 y ( ω) = ∫ d ττ
−
y ( ) eiωτ =
d ττ
y ( ) e −iωτ = − y ∗ ( ω) is purely imaginary
∫
2π −∞
2π −∞
With equation (*) we then find that
∞
i y ( ω) y ( ω)
1
 χ=
(see (*))
−
P ∫ dω
( ω)
ω−ω
2π −∞
2
∗
and
Again we can then compute real and imaginary part of the susceptibility
∞
i y ( ω) y ( ω) 1 ∞
i y ( ω) y ( ω)
1
χ ( ω) − χ ( ω) =
+
−
+
= y ( ω)
P ∫ dω
P ∫ dω
ω−ω
ω−ω
2π −∞
2
2π −∞
2
∗
∞
i y ( ω)
1
χ ( ω) + χ ( ω=
) = P ∫ d ω
π −∞
ω−ω
*
and finally obtain
∞
ℑ χ ( ω)
1
=
→ ℜχ
( ω) P ∫ d ω
π −∞
ω−ω
2. K-K relation
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62
The second Kramers-Kronig relation allows us to compute the real part of the
susceptibility (dispersion) when we know its imaginary part (absorption).
The Kramers-Kronig relation can also be rewritten in terms of the dielectric
function, where one applies also the symmetry relation for ω :
K-K relation for ε :
− χ ( ω) = χ∗ ( −ω) → χ′( ω) = χ′( −ω) χ′′( ω) = −χ′′( −ω) and
χ( ω) =ε( ω) − 1 =[ ε′( ω) − 1] + iε′′( ω)
2 ∞ ωε′′(ω)
d ω,
P
π ∫ 0 ω2 − ω2
∞ [ ε′(ω) − 1]
2
ε′′(ω) =− ω P ∫
d ω.
0 ω2 − ω2
π
− dispersion and absorption are linked, e.g., we can measure absorption
and compute dispersion
Example:
=
ε′(ω) − 1
ε′′(ω) − δ(ω − ω0 )  ε′(ω) − 1 −
ω0
 Drude-Lorentz model
ω02 − ω2
2.6 Beams and pulses - analogy of diffraction and
dispersion
In this chapter we will analyze the propagation of light. In particular, we will
answer the question how an arbitrary beam (spatial) or pulse (temporal) will
change during propagation in isotropic, homogeneous, dispersive media.
Relevant (linear) physical effects are diffraction and dispersion. Both phenomena can be understood very easily in the Fourier domain. Temporal effects,
i.e. the dispersion of pulses, will be treated in temporal Fourier domain
(temporal frequency domain). Spatial effects, i.e. the diffraction of beams, will
be treated in the spatial Fourier domain (spatial frequency domain). We will
see that:
• Pulses with finite spatial width (i.e. pulsed beams) are superposition of
normal modes (in frequency- and spatial frequency domain).
• Spatio-temporally localized optical excitations delocalize during
propagation because of different phase evolution for different frequencies
and spatial frequencies (different propagation directions of normal modes).
Let us have a look at the different possibilities (beam, pulse, pulsed beam)
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63
A) beam  finite transverse width  diffraction
k
plane wave (normal mode)
beam
A beam is a continuous superposition of stationary plane waves (normal
modes) with different wave vectors (propagation directions).
k1
k2
k3
k5
k4
…
∞
E(r, t ) = ∫ E(k )exp i ( kr − ω t )  d 3k
−∞
B) pulse  finite duration  dispersion
ω
stationary wave (normal mode)
pulse
A pulse is a continuous superposition of stationary plane waves (normal
modes) with different frequencies.
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ω1
ω2
64
ω3
...
=
E(r, t )
∫
∞
E(ω)exp i ( k ⋅ r − ω t )  d ω.
−∞
C) pulsed beams  finite transverse width and finite duration 
diffraction and dispersion
A pulsed beam is a continuous superposition of stationary plane waves
(normal modes) with different frequency and different propagation direction
=
E(r, t )
∫
∞
E(k , ω)exp i ( k ⋅ r − ωt )  d 3kd ω
−∞
2.7 Diffraction of monochromatic beams in
homogeneous isotropic media
Let us have a look at the propagation of monochromatic beams first. In this
situation, we have to deal with diffraction only. We will see later that pulses
and their dispersion can be treated in a very similar way. Treating diffraction
in the framework of wave-optical theory (or even Maxwell's equations) allows
us to treat rigorously many important optical systems and effects, i.e., optical
imaging and resolution, filtering, microscopy, gratings, ...
In this chapter, we assume stationary fields and therefore ω =const . For
technical convenience and because it is sufficient for many important
problems, we will make the following assumptions and approximations:
• ε( ω) =ε′( ω) > 0,  optical transparent regime  normal modes are
stationary homogeneous and evanescent plane waves
• scalar approximation
E(r, ω) → Ey (r, ω)e y → Ey (r, ω) → u (r, ω).
− exact for one-dimensional beams and linear polarization
− approximation in two-dimensional case
In homogeneous isotropic media we have to solve the Helmholtz equation
ω2
∆ E(r, ω) + 2 ε ( ω) E(r, ω) =0.
c
In scalar approximation and for fixed frequency ω it reads
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ω2
∆u (r, ω) + 2 ε ( ω) u (r, ω) =0,
c
∆u (r, ω) + k 2 ( ω) u (r, ω) = 0.
65
scalar Helmholtz equation
In the last step we inserted the dispersion relation (wave number k (ω) ). In the
following we often even omit the argument of the fixed frequency ω .
2.7.1 Arbitrarily narrow beams (general case)
Let us consider the following fundamental problem. We want to compute from
a given field distribution u ( x, y,0) in the plane z = 0 the complete field
u ( x, y, z ) in the half-space z > 0 , where z is our “propagation direction”.
The governing equation is the scalar Helmholtz equation
∆u (r, ω) + k 2 ( ω) u (r, ω) =0
To solve this equation and to calculate the dynamics of the fields, we can
switch again to the Fourier domain.
We take the Fourier transform
u=
(r, ω)
∫
∞
U (k , ω)exp [ik (ω)r ] d 3k
−∞
which can be interpreted as a superposition of normal modes with different
propagation directions and wavenumbers k (ω) (here the absolute value of
the wave-vector k ). Naively, we could expect that we just constructed a
general solution to our problem, but the solution is not correct because of the
dispersion relation:
ω2
k = k = k + k + k = 2 ε ( ω)
c
2
2
2
x
2
y
2
z
→ only two components of k are independent, e.g., kx , k y .
α, k y =
β, kz =
γ.
Our naming convention is in the following: kx =
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66
Then, the dispersion relation reads:
k 2 ( ω) = α 2 + β2 + γ 2
Thus, to solve our problem we need only a two-dimensional Fourier transform, with respect to the transverse coordinates x and y , when z is the
direction of propagation:
=
u (r )
∫∫
∞
U (α, β; z )exp i ( αx + βy )  d αd β.
−∞
In analogy to the frequency ω we call α, β spatial frequencies.
Now we plug this expression into the scalar Helmholtz equation
∆u (r ) + k 2 ( ω) u (r ) =
0
This way we can transfer the Helmholtz equation in two spatial dimensions
into Fourier space
 d2
2
2
2
k
+
−
α
−
β
 2
U (α, β; z ) = 0,
 dz

 d2
2
 2 + γ U (α, β; z ) =0.
 dz

This equation is easily solved and yields the general solution
U (α, β; z=
) U1 (α, β)exp [iγ (α, β) z ] + U 2 (α, β)exp [-iγ (α, β) z ],
depending on γ (α=
, β)
k 2 (ω) − α 2 − β2 .
We can identify two types of solutions:
A) Homogeneous waves
γ 2 ≥ 0,  α 2 + β2 ≤ k 2 , i.e., k real  homogeneous waves
k
B) Evanescent waves
γ 2 < 0,  α2 + β2 > k 2 , i.e., k complex, because γ =kz imaginary. Then,
we have k = k ′ + ik ′′ , with k ′ = αe x + βe y and k ′′ = γe z .
 k' ⊥ k''  evanescent waves
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67
We see immediately that in the half-space z > 0 the solution ∼ exp ( −iγz )
grows exponentially. Because this does not make sense, this component of
the solution must vanish U 2 (α, β) =0 . In fact, we will see later that U 2 (α, β)
corresponds to backward running waves, i.e., light propagating in the
opposite direction. We therefore find the solution:
U (α, β; z=
) U1 (α, β)exp [iγ (α, β) z ]
Furthermore the following boundary condition holds
U1 (α, β)U (α, β;0)
= U 0 (α, β)
which determines uniquely the entire solution for z ≠ 0 as
U (α, β; z )= U (α, β;0)exp [iγ (α, β) z ]
 U 0 (α, β)exp [iγ (α, β) z ]
In spatial space, we can find the optical field for z > 0 by inverse Fourier
transform:
=
u (r )
u (r )
=
∫∫
∞
∫∫
∞
U (α, β; z )exp i ( αx + βy )  d αd β.
−∞
U 0 (α, β)exp iγ ( α, β ) z  exp i ( αx + βy )  d αd β.
−∞
For homogeneous waves (real γ ) the red term above causes a certain phase
shift for the respective plane wave during propagation. Hence, we can
formulate the following result:
Diffraction is due to different phase shifts of the different normal modes when
these modes are traveling in propagation direction. The individual phase
shifts are determined according to different spatial frequencies α, β of the
normal modes.
The initial spatial frequency spectrum or angular spectrum at z = 0 forms the
initial condition of the initial value problem and follows from u0 ( x, y ) = u ( x, y,0)
by Fourier transform:
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 1 
=
U 0 ( α, β)  
 2p 
2
∫∫
∞
68
u0 ( x, y ) exp  −i ( αx + βy ) dxdy ,
−∞
As mentioned above the wave-vector components α, β are the so-called
spatial frequencies. Another common terminology is “direction cosine” for the
quantities α / k , β / k , because of the direct link to the angle of the respective
pane wave. For example α / k = cos θ x gives the angle of the plane wave's
propagation direction with the x -axis.
Scheme for calculation of beam diffraction
We can formulate a general scheme to describe the diffraction of beams:
1. initial field: u0 ( x, y )
2.
initial spectrum: U 0 (α, β) by Fourier transform
3.
propagation: by multiplication with exp iγ ( α, β ) z 
4.
5.
) U 0 (α, β)exp iγ ( α, β ) z 
new spectrum: U (α, β; z=
new field distribution: u ( x, y, z ) by Fourier back transform
This scheme allows for two interpretations:
1) The resulting field distribution is the Fourier transform of the propagated
spectrum
=
u (r )
2)
∫∫
∞
U (α, β; z ) exp i ( αx + βy )  d αd β.
−∞
The resulting field distribution is a superposition, i.e. interference, of
homogeneous and evanescent plane waves ('plane-wave spectrum')
which obey the dispersion relation
=
u (r )
∞
d αd β
∫∫
)))))
−∞
U 0 ( α, β ) exp {iγ ( α, β ) z} exp {i[ αx + βy ]}
))))) ))))))))) ))))))))))
αmplitude of phαse fαctor which is
interference of
the excited
αccumulαted βy the
eiγenstαtes to form
eiγenstαtes durinγ
the field pαttern αfter eiγenstαtes
propαγαtion
propαγαtion
shαpe of eiγenstαtes
(plαn wαves)
To understand the diffraction of beams let us now discuss the complex
transfer function H (α, β=
; z ) exp[iγ (α, β) z ] , which describes the beam propagation in Fourier space. For z = const. (finite propagation distance) it looks
like:
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amplitude
69
phase
=
; z ) exp iγ ( α, β ) z  acts differently on homogeneous and
Obviously, H (α, β
evanescent waves:
A) Homogeneous waves for α 2 + β2 ≤ k 2

(
)
exp i
=
ga
( , β ) z  1, arg exp iga
( , β ) z  ≠ 0
 Upon propagation the homogeneous waves are multiplied by the
phase factor
exp i k 2 − α 2 − β2 z 


 Each homogeneous wave keeps its amplitude.
B) Evanescent waves for α 2 + β2 > k 2

(
)
exp iga
z  exp  − a 2 + β2 − k 2 z  , arg exp iga
 0
( ,β)=
( , β ) z=


 Upon propagation the evanescent waves are multiplied by an
amplitude factor <1
exp  − α 2 + β2 − k 2 z  < 1


This means that their contribution gets damped with increasing
propagation distance z .
 Each evanescent wave keeps its phase.
Now the question is: When do we get evanescent waves? Obviously, the
answer lies in the boundary condition: Whenever u0 ( x, y ) yields an angular
spectrum U 0 (α, β) ≠ 0 for α2 + β2 > k 2 we get evanescent waves.
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70
Example: Slit
Let us consider the following one-dimensional initial condition which corresponds to an aperture of a slit:

1
u0 ( x ) = 
0
a
for x ≤
2.
otherwise
u0(x)
-a/2
a/2
x
a 
sin  a 
 2=
 sinc  a a 
U=
0 (a ) FT [u0 ( x ) ] 


2 
a 

 a
2 
U0(α)
− All spatial frequencies (-∞ → ∞) are excited.
− Important spectral information is contained in the interval a = 2π / a .
 Largest important spectral frequency for a structure with width a is
α = 2π / 2 .
− Evanescent waves appear for α > k .
− To represent the relevant information by homogeneous waves the
λ
2π
2π
<k = n → a>
following condition must be fulfilled:
n
λ
a
General result
We have seen in the example above that evanescent waves appear for
structures < wavelength in the initial condition. Information about these small
structures gets lost for z >> λ .
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71
Conclusion
In homogeneous media, only information about structural details having
length scales of ∆x , ∆y > λ / n are transmitted over macroscopic distances.
 Homogeneous media act like a low-pass filter for light.
Summary of beam propagation scheme
FT−1
FT
u0 ( x, y ) → U 0 (α, β) → U (α, β; z=
) H (α, β; z )U 0 (α, β) → u ( x, y, z )
=
; z ) exp iγ ( α, β ) z 
with the transfer function H (α, β
Remark: diffraction free beams
With our understanding of diffraction it is straight forward to construct socalled diffraction free beams, i.e., beams that do not change their amplitude
distribution during propagation. Translated to Fourier space this means that
all spatial frequency components have to get the same phase shift during the
propagation
U (α, β; z=
) U 0 (α, β)exp iγ ( α, β ) z  ≡ U 0 (α, β)exp [iCz ]
 u ( x, y, z ) = exp [iCz ] u0 ( x, y )
Since in general γ (α, β) ≠ const the excitation u0 ( x, y ) must have a shape such
that its Fourier transform has only components where the transfer function is
of equivalent value
β)
U 0 (α, β) ≠ 0 only for γ ( α,=
k 2 − α2 − =
β2 C
It is straightforward to see that the excited spatial frequencies must lie on a
circular ring in the ( α, β ) plane.
α 2 + β2 = ρ02
For constant spectral amplitude on this ring the Fourier back-transform yields
(see exercises):
u0 ( x, y=
) J 0 (rr )
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Bessel-beam (profile)
72
Bessel-beam
2.7.2 Fresnel- (paraxial) approximation
The beam propagation formalism developed in the previous chapter can be
simplified for the important special case of a narrowband angular spectrum
for
U 0 (α, β) ≠ 0
α 2 + β2 0 k 2
In this situation the beam consists of plane waves having only small
inclination with respect to the optical z -axis (paraxial (Fresnel) approximation). Then, we can simplify the expression for γ (α, β) by a Taylor
expansion to:
γ (α, β)=
 α 2 + β2 
α 2 + β2
k − α − β ≈ k 1 −
= k −
2k 2 
2k

2
2
2
The resulting expression for the transfer function in Fresnel approximation
reads:
 α 2 + β2
H
= exp ( iγ (α, β) z ) ≈ exp ( ikz ) exp  −i
2k

Amplitude
Phase

z=
 H F (α, β; z )

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73
We can see that this H F (α, β; z ) is always real valued. Hence it does not
account for the physics of evanescent waves. However, we must remember
that the derivation of H F (α, β; z ) as an approximation of H (α, β; z ) required the
assumption that the spatial frequency spectrum is narrow (paraxial waves).
Thus, already at the beginning we had excluded the excitation of evanescent
waves to justify the paraxial approximation.
The assumption of a narrow frequency spectrum corresponds to the requirement that all structural details ∆x , ∆y of the field distribution in the excitation
plane (at z = 0 ) must be much larger than the wavelength:
∆x , ∆y > 10λ / n  λ / n
This requirement applies also to the phase of the excitation. Hence it is not
sufficient that only the structural details of the intensity have a large scale.
The underlying phase of the excitation field must fulfill this condition as well.
This includes the condition that the phase of the beam should not have a
strong inclination to the principal propagation direction.
The propagation of the spectrum in Fresnel approximation works in complete
analogy to the general case. We just use the modified transfer function to
describe the propagation:
U F (α, β; z=
) H F (α, β; z )U 0 (α, β)
Summary of Fresnel approximation
For a coarse initial field distribution u0 ( x, y, z ) the angular spectrum U 0 ( α, β) is
nonzero for α 2 + β2  k 2 only. Then, only paraxial plane waves are relevant
for transmitting information and the transfer function of homogeneous space
can be approximated by H F (α, β; z ) .
Description in real space
It is also possible to formulate beam propagation in Fresnel (paraxial)
approximation in position space:
, y, z )
uF ( x=
=
=
∫∫
∫∫
∫∫
∞
U F (α, β; z )exp i ( αx + β y )  d αd β
−∞
∞
−∞
∞
H F (α, β; z )U 0 (α, β)exp i ( αx + β y )  d αd β
h ( x − x′, y − y′; z )u0 ( x′, y′)dx′dy′
−∞ F
The spatial response function hF ( x, y; z ) follows from the convolution theorem
and is the Fourier transform of H F (α, β; z ) :
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 1 
=
hF ( x, y; z )  
 2p 
2
2
∫∫
∞
−∞
74
H F (α, β; z )exp i ( αx + βy )  d αd β
∞
 α 2 + β2
 1 
  exp ( ikz ) ∫∫ −∞ exp  −i
2k
 2p 


z  exp i ( αx + βy )  d αd β.

This Fourier integral can be solved and we find:

 ( x 2 + y 2 )  
ik
 ik
 k 2
2 
hF ( x, y; z ) =
exp ( ikz ) −
exp i ( x + y )   =
−
exp ikz 1 +
 ,
2
pp
2
z
2
z
2
z
2
z





 

The response function corresponds to a spherical wave in paraxial approximation. Similar to Huygens principle, where from each point in the object
plane a spherical wave is emitted towards the image plane, here paraxial
approximations of spherical waves are emitted.
To sum up, in position space paraxial beam propagation is given by:
∞
ik
2
2 
 k
uF ( x, y, z ) =
exp ( ikz ) ∫∫ u0 ( x′, y′)exp i ( x − x′ ) + ( y − y′ )   dx′dy′.
−

−∞
2pz
 2z 
Of course, the two descriptions in position space and in the spatial Fourier
domain are completely equivalent.
The correspondence between real and frequency space
Relation between transfer and response function:
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=
h ( x, y; z )
75
∞
1
H (α, β; z ) exp i ( αx + βy )  d αd β
2 ∫∫ −∞
(2p)
Transfer functions for homogeneous space
H (α, β
; z ) exp iγ ( α, β=
=
) z  exp i k 2 − α 2 − β2 z  exact solution
 α 2 + β2
H F (α, β; z ) =exp [ikz ] exp i
 2k
with k = k (ω) = k0 n(ω) =

z  Fresnel approximation

ω
n(ω)
c
Remark on the validity of the scalar approximation
In the previous description of the propagation of arbitrary beams we have
used the scalar approximation of the vectorial fields. It is interesting to see, to
what extend this approximation stays in correspondence to the conditions
which where necessary to derive the Fresnel approximation.
E(r,=
ω)
ˆ
∫∫ E ( α, β, ω)e
i( αx +βy +γz )
d αd β
divE(r, ω) = 0 → αEˆ x + βEˆ y + γEˆ z = 0
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A) One-dimensional beams
− translational invariance in y-direction: β =0
− and linear polarization in y-direction: Eˆ y = U , Eˆ x = 0
 scalar approximation is exact since divergence condition is strictly
fulfilled
B) Two-dimensional beams
− Finite beam which is localized in the x,y-plane: α, β ≠ 0
− and linear polarization, w.l.o.g. in y-direction: Eˆ x = 0 , Eˆ y = U
 divergence condition: βEˆ y + γEˆ z =0
β
β
Eˆ z ( α, β, ω) = − Eˆ y ( α, β, ω) = −
Eˆ y ( α, β, ω) ≈ 0
γ
k 2 − α 2 − β2
In paraxial approximation ( α 2 + β2  k 2 ) the scalar approximation is
automatically justified.
2.7.3 The paraxial wave equation
In paraxial approximation the propagated spectrum is given by
U F (α, β; z=
) H F (α, β; z )U 0 (α, β)
 α 2 + β2
= exp ( ikz ) exp  −i
2k


z U 0 (α, β)

Let us introduce the slowly varying spectrum V ( α, β; z ) :
 α 2 + β2 
 U F (α, β; z ) = exp ( ikz )V (α, β; z )  V (α, β;=
z ) exp  −i
z V0 (α, β).
2k


Differentiation of V with respect to z gives:
i
∂
1 2
α + β2 )V (α, β; z )
V (α, β; =
z)
(
∂z
2k
Fourier transformation back to position space leads to the so-called paraxial
wave equation:
i
∂ ∞
V (α, β; z )exp i ( αx + βy )  d αd β
∂z ∫∫ −∞
1 ∞ 2
=
α + β2 )V (α, β; z )exp i ( αx + βy )  d αd β
(
∫∫
−
∞
2k
∂
∂2  ∞
1  ∂2
 i v( x, y, z ) −  2 + 2  ∫∫ V (α, β; z )exp i ( αx + βy )  d αd β
=
∂z
2k  ∂x
∂y  −∞
76
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i
77
∂
1
v( x, y, z ) + ∆ (2)v( x, y, z ) =
0 paraxial wave equation
∂z
2k
The slowly varying envelope v( x, y, z ) (Fourier transform of the slowly varying
spectrum) relates to the scalar field as uF ( x, y, z ) = v( x, y, z )exp ( ikz ) .
Extension of the wave equation to weakly inhomogeneous media
(slowly varying envelope approximation - SVEA)
There is an alternative, more general way to derive the paraxial wave equation, the so-called slowly varying envelope approximation. This approximation
even allows us to treat wave propagation in inhomogeneous media. We will
include inhomogeneous media in this derivation even though the current
chapter of this lecture is devoted to inhomogeneous media.
We start from the scalar Helmholtz equation. However, we should mention
that the extrapolation of our previous discussion towards inhomogeneous
media requires another approximation. This approximation assumes small
spatial fluctuations of ε(r, ω) . This is equivalent to having a weak index
contrast between different spatial positions.
0 with k 2 (r, ω)=
∆u ( x, y, z ) + k 2 (r, ω)u ( x, y, z ) =
ω2
ε(r, ω)
c2
We make the ansatz
( )
 with k = k
u ( x, y, z ) = v( x, y, z ) exp ikz
where k is the spatially averaged wavenumber. This is the mean wave
number in the volume, in which wave propagation is considered. Hence it is
the average of the spatially varying material properties of the medium.
With the SVEA condition
  ∂v / ∂z
kv
we can simplify the scalar Helmholtz equation as follows:
∂2
∂
v( x, y, z ) + 2ik v( x, y, z ) + ∆ (2)v( x, y, z ) +  k 2 (r, ω) − k 2  v( x, y, z ) =0,
2
∂z (((
∂z
(

≈0
 k 2 (r, ω) − k 2 
1 (2)
∂
0
 i v ( x, y , z ) +  ∆ v ( x, y , z ) + 
 v ( x, y , z ) =

∂z
2k
2
k


This is the paraxial wave equation for inhomogeneous media having a weak
index contrast.
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78
2.8 Propagation of Gaussian beams
The propagation of Gaussian beams is an important special case. Optical
beams of Gaussian shape are import because many beams in reality have a
shape which is at least close to the shape of a Gauss function. In some cases
they even have exactly the Gaussian shape as e.g. the transversal fundamental modes of many lasers. The second reason why Gaussian beams are
important, is the fact that in paraxial approximation it is possible to compute
the evolution of Gaussian beams analytically.
Fundamental Gaussian beam in focus
The general form of a Gaussian beam at one plane ( z = const ) is elliptic, with
curved phase
  x2 y 2 
u0 ( x, y=
) v0 ( x, y=
) A0 exp  −  2 + 2   exp [iϕ( x, y ) ] .


  wx wy  
2
2
w=
w02 ) and
Here, we will restrict ourselves to rotational symmetry ( w=
x
y
0 . Later we will see that this corresponds to the
(initially) 'flat' phase ϕ( x, y ) =
focus of a Gaussian beam. The Gaussian beam in such a focal plane with flat
phase is characterized by amplitude A and width w0 , which is defined as
u0 ( x 2 + y 2= w02 )= A0 exp ( −1)= A0 / e .
In practice, the so-called 'full width at half maximum' (FWHM) of the intensity
is often used instead of w0 . The FWHM of the intensity is connected to w0 of
the field by
(
2
u0 x + y
-
2
)
2
æ w2
ö÷ 1
÷
= exp ççç- FWHM
çè 2w 2 ÷÷ø 2
0
2
wFWHM
2
= 2ln 2w 02 » 1.386w 02
= - ln 2  wFWHM
2
2w 0
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79
2.8.1 Propagation in paraxial approximation
Let us now compute the propagation of a Gaussian beam starting from the
focus in paraxial approximation:
1) Field at z = 0 :
 x2 + y2 
u=
v=
A0 exp  −
.
0 ( x, y )
0 ( x, y )
w02 

2)
Angular spectrum at z = 0 :
U 0 (α, β=
) V0 (α, β=
)
=
1
( 2p )
 x2 + y 2 
A0 ∫∫ exp  −
 exp [ −i(αx + βy ) ] dxdy
−∞
w02 

∞
2
 α 2 + β2  A0 2
 α 2 + β2 
A0 2
=
w0 exp  −
w
exp
−
,
0
2 
2
w
w
4pp
4
4
/
0 
s



We see that the angular spectrum has a Gaussian profile as well and
that the width in position space and Fourier space are linked by
ws ⋅ w0 =
2.
Angular spectrum in the focal plane
Check if paraxial approximation is fulfilled:
We can say that U 0 ( α, β) ≈ 0 for ( α2 + β2 ) ≥ 16 / w02 , because
(
exp ( −4 ) ≈ 0.02 For paraxial approximation we need k 2 2 α 2 + β2
 k 2 2 16 / w02

3)
2
2
16
 2λ   λ 
w 2 =
  ≈  ,
2
 2π   πn   n 
 n
 λ 
2
0
λ
 paraxial approximation works for w0  10 = 10λ n
n
Propagation of the angular spectrum:
V (α, β; z=
) U 0 (α, β) HF (α, β;z)
)
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4)
80
Fourier back-transformation to position space, when still using the slowly
varying envelope
With the Rayleigh length
beam:
which determines the propagation of a Gaussian
Note that we use the slowly varying envelope !
Conclusion:
− Gaussian beam keeps its shape, but amplitude, width, and phase
change upon propagation
− Two important parameters: propagation length and Rayleigh length
Some books use the “diffraction length”
depth” of the Gaussian beam. E.g.:
, a measure for the “focus

From our computation above we know that the Gaussian beam evolves like:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
81
For practical use, we can write this expression in terms of z-dependent
amplitude, width, and shape of the phase front:
Here we used that
. The (x,y)-independent phase
, the so-called Gouy phase shift.
is given by
In conclusion, we see that the propagation of a Gaussian beam is given by a
z-dependent amplitude, width, phase curvature and phase shift:
The normalized beam intensity
as a function of the radial distance
at different axial distances: (a)
; (b)
; (c)
.
Discussion
The amplitude evolves along z like:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Hence, we get for the Intensity profile
82
:
The normalized beam intensity
at points on the beam axis (
) as a function of the propagation distance .
The beam width evolves along
The beam radius
reaches
at
like:
has its minimum value
at the waist (
, and increases linearly with for large
The radius of the phase curvature is given by
.
),
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83
The radius of curvature
of the wavefronts of a Gaussian beam. The
dashed line is the radius of curvature of a spherical wave.
The flat phase in the focus (
) corresponds to an infinite radius of curvature. The strongest curvature (minimum radius) appears at the Rayleigh
distance from the focus.
The (x,y)-independent Gouy phase is given by
The so-called Gouy phase is the retardation
of the phase of a Gaussian beam relative to a uniform plane wave at the points of the beam axis.
The Gouy phase is not important for many applications because it is ‘flat’.
However, in resonators and in the context of nonlinear optics it can play an
important role (i.e., harmonic generation in focused geometries).
The wave fronts (planes of constant phase) of a Gaussian beam are given by
Wavefronts of a Gaussian beam.
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84
2.8.2 Propagation of Gauss beams with q-parameter formalism
In the previous chapter we gave the expressions for Gaussian beam
propagation, i.e., we know how amplitude, width, and phase change with the
propagation variable z . However, the complex beam parameter
q ( z )= z − iz0
q-parameter
allows an even simpler computation of the evolution of a Gaussian beam. In
fact, if we take the inverse of the “q-parameter”,
1
1
z
z
1
=
=2
+i 2 0 2 =
2
q ( z ) z − iz0 z + z0
z + z0 z 1 +
(
z02
2
z
)
+i
(
1
z0 1 +
z2
z02
)
we can observe that real and imaginary part are directly linked to radius of
phase curvature and beam width:
1
1
λn
kw02 π 2
=
+i 2
because=
z0 =
w0
q( z ) R( z )
πw ( z )
2
λn
Thus, the q-parameter describes beam propagation for all z !
Example: propagation in homogeneous space by z = d
1
1
λ
+ i 2n
A) initial conditions: =
q (0) R(0)
πw (0)
( d ) q(0) + d
B) propagation (by definition of q parameter) q=
C) q-parameter at z = d determines new width and radius of curvature
1
1
1
λ
=

+ i 2n
q (d ) q(0) + d R(d )
πw (d )
2.8.3 Gaussian optics
We have seen in the previous chapter that the complex q-parameter
formalism makes a simple description of beam propagation possible. The
question is whether it is possible to treat optical elements (lens, mirror, etc.)
as well.
Aim: for given R0 , w0 (i.e. q0 )  Rn , wn (i.e. qn ) after passing through n
optical elements
We will evaluate the q-parameter at certain propagation distances, i.e., we
will have values at discrete positions: q( zi ) → qi .
Surprising property: We can use ABCD-Matrices from ray optics!
This is remarkable because here we are doing wave-optics (but with
Gaussian beams).
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85
How did it work in geometrical optics?
A) propagation through one optical element:
ˆ =  A B .
M
C D 


B)
propagation through multiple elements:
ˆ =M
ˆ M
ˆ ..M
ˆ =  A B .
M
N
N −1
1
C D 


C) matrix connects distances to the optical axis y and inclination angles Θ
before and after the element
 y2  ˆ  y1 
 Θ  = M  Θ .
 2
 1
Link to Gaussian beams
Let us consider the distance to the intersection of the ray with the optical axis,
as it was defined in chapter 1.6.1 on "The ray-transfer-matrix". Using the
small angle approximation ( tan(Θ1 ) ≈ Θ1 = y1 z1 ) we can define this distance
as:
y1
+B
y2 Ay1 + BΘ1
Az1 + B
Θ1
y1
=
=
z2 =
=
z1 =
Θ 2 Cy1 + DΘ1 C y1 + D Cz1 + D
Θ1
Θ1
A
The distances z1 , z2 are connected by matrix elements, but not by normal
matrix vector multiplication.
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86
It turns out that we can pass to Gaussian optics by replacing z by the
complex beam parameter q . The propagation of q -parameters through an
optical element is given by:
q1 =
A1q0 + B1
C1q0 + D1
 propagation through N elements:
qn =
Aq0 + B
Cq0 + D
ˆ =M
ˆ M
ˆ ..M
ˆ =  A B .
with the matrix M
N
N −1
1
C D 


 This works for all ABCD matrices given in chapter 1.6 for ray optics!!!
Here: we will check two important examples:
i) propagation in free space by z = d :
( d ) q(0) + d
 propagation (by definition of q -parameter) q=
ˆ = 1 d  =
A1 1,=
B1 d ,=
C1 0,=
D1 1
M
0 1 


q=
1
ii)
A1q0 + B1 q0 + d
=
= q0 + d
C1q0 + D1
0 +1
thin lens with focal length f
What does a thin lens do to a Gaussian beam exp ( −( x 2 + y 2 ) / w02 ) in
paraxial approximation?
− no change of the width
 k ( x2 + y 2 ) 

− but change of phase curvature R f : × exp i
Rf
 2

How can we see that?
Trick:
We start from the focus which is produced by the lens with z=
z=
0
f
and w f is the focal width. Hence the q-parameter is
πw2f
λn
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87
λ
1
= i n2
q0
πw f
The radius of curvature evolves as:
  z f 2 
R( z ) =+
z 1    ≈ z for z  z f
  z  
We can invert the propagation from the focal position to the lens at the
distance of the focal length f and obtain R f = − f
f <0
double concave
lens
 defocusing
=
q1
f >0
 1
ˆ
M
thin lens = 
 −1 f
0
1 
double convex
lens
 focusing
A1q0 + B1
q0
=
C1q0 + D1 − q0 f + 1
1 − q0 f + 1 1
iλ
=
=
+ n2
for
q1
q0
− f πw0
πw02
q0 =
−i
λn
Be careful: Gaussian optics describes the evolution of the beam's width and
phase curvature only!
 Changes of amplitude and reflection are not included!
2.8.4 Gaussian modes in a resonator
In this chapter we will use our knowledge about paraxial Gaussian beam
propagation to derive stability conditions for resonators. An optical cavity or
optical resonator is an arrangement of mirrors that forms a standing wave
cavity resonator for light waves. Optical cavities are a major component of
lasers, surrounding the gain medium and providing feedback of the laser light
(see He-Ne laser experiment in Labworks).
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
2.8.4.1
88
Transversal fundamental modes (rotational symmetry)
Wave fronts, i.e. planes of constant phase, of a Gaussian beam.
The general idea to get a stable light configuration in a resonator is that
mirrors and wave fronts (planes of constant phase) coincide. Then, radiation
patterns are reproduced on every round-trip of the light through the resonator.
Those patterns are the so-called modes of the resonator.
In paraxial approximation and Gaussian beams this condition is easily
fulfilled: The radii of mirror and wave front have to be identical!
In this lecture we use the following conventions (different to Labworks script,
see remark below!):
− z1,2 is the position of mirror '1','2'; z=0 is the position of the focus!
d
− d is the distance between the two mirrors  z2 − z1 =
z02
 radius of wave front <0 for z <0
− because R( z )= z +
z
− from Chapter 1: beam hits concave mirror  radius R=
i (i 1,2) < 0.
− beam hits convex mirror  radius R=
i (i 1,2) > 0.
Examples:
A) R ( z1 ), R( z2 ) > 0 ; R1 > 0, R2 < 0 ; z1 > 0, z2 > 0
B)
R( z1 ) < 0, R( z2 ) > 0 ; R1 , R2 < 0 ; z1 < 0, z2 > 0
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89
According to our reasoning above, the conditions for stability are:
R1 = R( z1 ),
R2 = − R( z2 )
2
 R1 =z1 + z0 ,
z1
− R2 =z2 +
z02
.
z2
In both expressions we find the Rayleigh length z0, which we eliminate:
z1 ( R1 − z1 ) =
− z2 ( R2 + z2 )
with z2= z1 + d we find z1 = −
d ( R2 + d )
.
R1 + R2 + 2d
Now we can choose R1 , R2 , d and compute modes in the resonator. However,
we have to make sure that those modes exist. In our calculations above we
have eliminated the Rayleigh length z0, a real and positive quantity. Hence,
we have to check that the so-called stability condition z02 > 0 is fulfilled!
d ( R1 + d )( R2 + d )( R1 + R2 + d )
R1 z1 − z12 =
−
>0
 z02 =
2
( R1 + R2 + 2d )
The denominator ( R1 + R2 + 2d ) is always positive we need to fulfill
2
 −d ( R1 + d )( R2 + d )( R1 + R2 + d ) > 0
If we introduce the so-called resonator parameters

d 
g1 =
1 +  ,
 R1 

d 
g2 =
1 + 
 R2 
We can re-express the stability condition as
−d ( R1 + d )( R2 + d )( R1 + R2 + d ) = dg1 g 2 R1R2
(1 − g1 g 2 ) R1R2
d
=
g1 g 2 (1 − g1 g 2 )( R1R2 ) > 0.
2
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90
This inequality is fulfilled for
(1)
and
(2)
and
or for
which results in the following condition for the placement and radii of the
mirrors for a stable cavity
which is equivalent to
This final form of the stability condition can be visualized: The range of
stability of a resonator lies between coordinate axes and hyperbolas:
Resonator stability diagram. A spherical-mirror resonator is stable if the
parameters
and
lie in the unshaded regions
bounded by the lines
and
, and the hyperbola
.
is negative for a concave mirror and positive for a convex mirror.
Various special configurations are indicated by letters. All symmetrical
resonators lie along the line
.
Examples for a stable and an unstable resonator:
A)
stable
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B)
unstable
R1 , R2 < 0; R1 < d , R2 > d ; 
g1 ≤ 0, 0 ≤ g 2 ≤ 1; g1 g 2 ≤ 0
Remark connection to He-Ne-Labwork script (and Wikipedia):
In Labworks (he_ne_laser.pdf) a slightly different convention is used:
− Direction of z-axis reversed for the two mirrors
− beam hits concave mirror  radius Ri (i = 1,2) > 0.
− beam hits convex mirror
 radius Ri (i = 1,2) < 0.
− z1,2 is the distance of mirror '1','2' to the focus!
− d is the distance between the two mirrors  z2 + z1 = d
Examples:
A) R( z1 ) < 0 , R( z2 ) > 0 R1 < 0, R2 > 0 ; z1 < 0, z2 > 0
B) R ( z1 ) > 0, R( z2 ) > 0 ; R1 , R2 > 0 ; z1 > 0, z2 > 0
91
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92
Then the conditions for stability are:
=
R1 R=
( z1 ),
R2 R( z2 )
With analog calculation as above we find with for the resonator
parameters


d 
d 
=
g1 =
1 −  , g 2 1 − 
 R1 
 R2 
the same stability condition
g1 g 2 (1 − g1 g 2 )( R1R2 ) > 0,
2
2.8.4.2
0 < g1 g 2 < 1.
Higher order resonator modes
For the derivation of the above stability condition we needed the wave fronts
only. Hence, there may exist other modes with same wave fronts but different
intensity distribution. For the fundamental mode we have:
 x2 + y 2 
 k x2 + y 2 
w
vG ( x, y, z ) =
exp
A 0 exp  − 2

i 2 R ( z )  exp [iϕ( z ) ].
(
)
w( z )
w
z




higher order modes: ( x, y -dependence of phase is the same)
ul ,m ( x, y, z ) = Al ,m

w0
x  
y 
Gl  2
G
×
2
m
w( z )  w( z )   w( z ) 
 k x2 + y 2 
exp i
 exp ( ikz ) exp [i(l + m + 1)ϕ( z ) ].
 2 R( z ) 
 x2 
Gl (x=
) H l (x)exp  − 
 2
The functions Gl are given by the so-called Hermite polynomials:
H l (ξ) ( H 0 = 1, H1 = 2ξ and H l +1 =2ξH l − 2lH l −1 ).
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Several low-order Hermite-Gaussian fuctions: (a)
; and (d)
.
; (b)
93
; (c)
Intensity distributions of several low-order Hermite-Gaussian beams in the
transverse plane. The order
is indicated in each case.
2.9 Dispersion of pulses in homogeneous isotropic
media
2.9.1 Pulses with finite transverse width (pulsed beams)
In the previous chapters we have treated the propagation of monochromatic
beams, where the frequency
was fix and therefore the wavenumber
was constant as well. This is the typical situation when we deal with continuous wave (cw) lasers.
However, for many applications (spectroscopy, nonlinear optics, telecommunication, material processing) we need to consider the propagation of
of
pulses. In this situation, we have a typical envelope length
.
Let us compute the spectrum of the (Gaussian) pulse:
 spectral width:
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94
− center frequency of visible light: ω0 = 2πν  4 ⋅ 1015 s-1
 optical cycle:
Ts = 2π / ω0 ≈ 1.6 fs
Hence, we have the following order of magnitudes:
ωs << ω0 → ω − ω0 = ω << ω0
In this situation it can be helpful to replace the complicated frequency
dependence (dispersion relation) of the wave vector k (ω) or the wave
number k (ω) by a Taylor expansion at the central frequency ω = ω0 .
In most cases, a parabolic (or cubic) approximation of the frequency dependence in the dispersion relation will be sufficient:
1 ∂ 2k
∂k
k (ω) ≈ k ( ω0 ) +
( ω − ω0 ) +
2 ∂ω2
∂ω ω0
( ω − ω0 )
2
+ ...
ω0
The three expansion coefficients and their physical significance
The following terminology for the individual expansion coefficients is commonly used in the literature. It associates the physics, which is inherited in the
dispersion relation, with the three parameters of the Taylor expansion.
A) Phase velocity vPh
k ( ω0 ) = k0 , →
n ( ω0 )
1
k
= 0 =
vPh ω0
c
 velocity of the phase front for the light at the central frequency ω = ω0
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95
B) Group velocity vg
∂k
1
=
∂ω ω0 vg
 group velocity is the velocity of the center of the pulse (see below)
ω
∂n 
1 ∂k
1
=  n(ω0 ) + ω0
n(ω) → =

∂ω ω0 
c
vg ∂ω ω0 c 
ω)
k (=
c
c
n(ω0 )
= = vPH
ng (ω0 )

∂n  ng (ω0 )
 n(ω0 ) + ω0

∂ω ω0 

=
vg
ng (ω0 )= n(ω0 ) + ω0
∂n
 group index
∂ω ω0
normal dispersion: ∂n / ∂ω > 0  ng > n, vg < vPH
anomalous dispersion: ∂n / ∂ω < 0  ng < n, vg > vPH
C) Group velocity dispersion (GVD) or simply dispersion Dω
∂ 2k
∂ω2
= Dω
ω0
 GVD changes pulse shape upon propagation (see below)
=
D D=
ω
D=
∂ 2k
∂ 1
=

∂ω2 ω ∂ω  vg
0



∂ 1
1 ∂vg
  = − 2
∂ω  vg 
vg ∂ω
∂vg
<0
∂ω
∂v
→ D<0 g >0
∂ω
→ D >0
Alternatively in telecommunication one often uses
=
Dλ
∂ 1
2π
=
  − 2 cDω
∂λ  vg 
λ
Let us now discuss the propagation of pulsed beams. We start with the scalar
Helmholtz equation, with the full dispersion (no Taylor expansion yet):
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96
ω2
∆ u (r, ω) + 2 ε(ω) u (r, ω) = 0
c
In contrast to monochromatic beam propagation, we now have for each
frequency ω one Fourier component of the optical field:
)
dispersion relation: k 2 (ω=
ω2
ε(ω)
c2
Hence, we need to consider the propagation of the Fourier spectrum (Fourier
transform in space and time):
U (α, β, ω; z=
) U 0 (α, β, ω)exp iγ ( α, β, ω) z 
with γ ( α=
, β, ω)
k 2 (ω) − α 2 − β2
The initial spectrum at z = 0 is U 0 ( α, β, ω)
=
U 0 (α, β, ω)
1
∞
u0 ( x, y, t )exp  −i ( αx + βy − ω t )  dxdydt
3
( 2p ) ∫∫∫ −∞
Let us further assume that the Fresnel (paraxial) approximation is justified
(k 2 (ω) >> α 2 + β2 )
 α 2 + β2
U (α, β, ω ; z ) ≈ U 0 (α, β, ω) exp ik ( ω) z  exp  −i
2k ( ω )


z

We see that propagation of pulsed beams in Fresnel approximation in Fourier
space is described by the following propagation function (transfer function):
 α 2 + β2
=
H F (α, β, ω; z ) exp ik ( ω) z  exp  −i
2k ( ω )


z

Now let us consider the Taylor expansion of k ( ω) from above. If the pulse is
not too short, we can replace the wave number k (ω) by
k (ω) ≈ k ( ω0 ) +
∂k
1 ∂ 2k
ω
−
ω
+
(
0)
∂ω ω0
2 ∂ω2
( ω − ω0 )
2
+ ..
ω0
Moreover, in the second term exp[−i(α 2 + β2 ) z / {2k (ω)}] of the transfer
function (which is already small due to paraxiality) we can approximate the
frequency dependence of the wave number by k (ω) ≈ k (ω0 ) =k0 . This
approximation is sufficiently accurate to describe the diffraction of pulsed
beams which are not to short. But this approximation would break down for
T0  15 fs since for such short pulses the frequency spectrum would become
very wide. By introducing this approximation, we obtain the so-called parabolic approximation:
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97
 1

H FP (α, β, ω; z ) ≈ exp [ik0 z ] exp i ( ω − ω0 ) z  ×

 vg
 α 2 + β2
2
 D

× exp i ( ω − ω0 ) z  exp  −i
2 k0
 2



z


ω 1

2
1 1 2
2
= exp [ik0 z ] exp i z  + D ω −
α
+
β
(
) 
v
2 k0

 g 2
 
with ω = ω − ω0
Based on the last line of the above equation we can introduce a new variant
of the propagation function, where the frequency argument is replaced by the
frequency difference ω from the center frequency ω0
H FP (α, β, ω; z ) ≈ exp [ik0 z ] H FP (α, β, ω; z )
The new transfer function H FP (α, β, ω; z ) is the propagation function for the
slowly varying envelope v( x, y, z , t ) :
∞
u ( x, y, z , t ) exp [ik0 z ] ∫∫∫ U 0 (α, β, ω) H FP (α, β, ω; z ) ×
=
−∞
× exp i ( αx + βy − ω t )  d αd βd ω
∞
u (=
x, y, z , t ) exp i ( k0 z − ω0t )  ∫∫∫ U 0 (α, β, ω) H FP (α, β, ω; z ) ×
−∞
(
)
× exp i αx + βy − ω t  d αd βd ω


Illustration of the slowly varying envelope in the spectral domain
=
u ( x, y, z , t ) v( x, y, z , t )exp i ( k0 z − ω0t ) 
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
u(t)
98
v(t)
t
|u(ω)|
t
|u(ω)|
ω0
∫∫∫
v( x=
, y, z , t )
∞
ω
ω
(
)
U 0 (α, β, ω) H FP (α, β, ω; z )exp i αx + βy − ω t  d αd βd ω


−∞
In order to complete the formalism, we also need to define the initial spectrum
of the slowly varying envelope
=
u0 ( x, y, t ) v0 ( x, y, t )exp ( −iω0t )
 V0 (α, β, ω)
=
1
∞
v0 ( x, y, t )exp  −i ( αx + βy − ωt )  dxdydt
3
( 2p ) ∫∫∫ −∞
Thus, the propagation of the slowly varying envelope is given by:
v( x,=
y, z , t )
∫∫∫
∞
(
)
V (α, β, ω) H FP (α, β, ω; z ) exp i αx + β y − ω t  d αd β d ω


−∞ 0
Co-moving reference frame
The next step is to introduce a co-moving reference frame with
 ω
=
H FP (α, β, ω; z ) exp i
 vg
v( x,=
y, z , t )
∫∫∫
∞

z  H FP (α, β, ω; z )

(
)
V (α, β, ω) H FP (α, β, ω; z ) exp i αx + βy − ω ( t − z vg )  d αd βd ω


−∞ 0

The last line above involves the propagation function H FP (α, β, ω; z ) , which is
the propagation function for the slowly varying envelope in the co-moving
frame of the pulse:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
t= t−
99
z
vg
This frame is called co-moving because H FP (α, β, ω; z ) is now purely quadratic
in ω , i.e., the pulse does not “move” anymore. In contrast, the linear ω dependence in Fourier space had given a shift in the time domain. Thus, the
slowly varying envelope in the co-moving frame evolves as:
v ( x, =
y, z , τ)
 z
2
α 2 + β2  
∫∫∫ −∞V0 (α, β, ω)exp i 2  Dω − k0  ×


∞
{
}
× exp i αx + βy − ω τ  d αd βd ω
The optical field u reads in the co-moving frame as:
 

ω
ω0t )  v ( x, y, z , t)exp i  k0 z − 0 z − ω0 t  
u ( x,=
y, z , t) v ( x, y, z , t)exp i ( k0 z −=


vg
 
 
Propagation equation in real space
Finally, let us derive the propagation equation for the slowly varying envelope
in the co-moving frame. We start from the transfer function
 z
2
α 2 + β2  

V (α, β, ω; z =
) V0 (α, β, ω)exp i  Dω −

k0  
 2
Then we take the spatial derivative of the transfer function along the propagation direction z
2
1
∂V (α, β, ω; z )
α 2 + β2  
i
= −  Dω −
V (α, β, ω; z )
2
k0 
∂z
As before in the case of monochromatic beams, we use Fourier back-transformation to get the differential equation in the time-position domain
∂v ( x, y, z , τ) D ∂ 2
1 (2)

i
−
v
(
x
,
y
,
z
,
τ
)
+
Dτ
v ( x, y, z , ) =0
∂z
2 ∂τ2
2 k0
This is the scalar paraxial equation for propagation of so-called pulsed
beams.
Comment: Extension to inhomogeneous media
By using the slowly varying envelope approximation, it is possible to generalize the scalar paraxial equation also for inhomogeneous media, when a weak
index contrast is assumed.
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
100
 k02 (r ) − k02 
∂
D ∂2
1 (2)
i v ( x, y, z , τ) −
v ( x, y, z , τ) +
Dτ
v ( x, y, z , ) + 
 v ( x, y, z , τ) =0
∂z
2 ∂τ2
2 k0
2
k
0


with k0 ≈ k0 ( r )
For D = 0 the equation would be reduced to simple diffraction, as in the beam
propagation scheme which was derived earlier.
2.9.2 Infinite transverse extension - pulse propagation
Diffraction plays no role for sufficiently small propagation lengths z  LB . For
broad beams, the diffraction length LB can be rather large and we can
assume α = β ≈ 0 , corresponding to the assumption that we have a single
plane wave propagating in z-direction. Later in the lecture series we will see
that this case is also valid for mode propagation in waveguides, as e.g.
optical telecommunication fibers.
Description in frequency domain
u0 (t ) v0 (t )exp ( −iω0t )
1) initial condition: =
2)
3)
4)
initial spectrum:
V0 (ω=
) U 0 (ω)
D 

propagation of the spectrum: V (ω; z=
) V0 (ω)exp i z ω2 
2 

back-transformation to τ leads to the following evolution of the slowly
varying envelope in the co-moving frame:
v (=
z , τ)
 D 2
V
ω
(
)exp
0
∫ −∞
iz 2 ω  exp [ −iωτ] d ω
∞
Description in time domain
A) In time domain it is possible to describe pulse propagation by means of a
response function:
 D 
FT of H P =
(ω; z ) exp iz ω2  
h P (τ; z )
=
 2 
-1

2
τ2 
exp  −i

−ipDz
 2 Dz 
and the evolution is described by the convolution integral
v (=
z , τ)
B)
∫
∞
−∞
′; z )v0 ( ′)d ′
h P (τ − τττ
The evolution equation for slowly varying envelope in the co-moving
frame reads
i
∂v ( z , τ) D ∂ 2
−
v ( z , τ) =0
∂z
2 ∂τ2
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Analogy of diffraction and dispersion
DIFFRACTION
∂
1
i v( x, y, z ) + ∆ (2)v( x, y, z ) = 0 
∂z
2k
(α, β) 
101
DISPERSION
∂
D ∂2
i v ( z , τ) −
v ( z , τ) = 0
∂z
2 ∂τ2
ω
( x, y )

τ
∇

∂
∂τ
1
k0

− D but D  0 can vary
In the following we will study two typical examples of pulse propagation.
2.9.3 Example 1: Gaussian pulse without chirp
use analogies to spatial diffraction
1. Initial pulse shape
pulse without chirp  corresponds to Gaussian pulse in the waist (focus) with
flat phase
V0
V0
e
 τ2 
 t2 
v0 (τ) A0 exp  − 2 
u0 (t ) A0 exp  − 2  exp ( −iω0t )  =
=
 T0 
 T0 
2. Initial pulse spectrum
 ω2T02 
T0
V0 (ω) A0
exp  −
=

4 
2 p

 spectral width: ω2s =4 / T02
Use results from propagation of Gaussian beams:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
102
1 T02
k 2
0
z0 describes Gaussian pulse z0 =  w0   z0 = −
2 D
2 
Hence anomalious GVD is equivalent to 'normal' diffraction.
Dispersion length: LD = 2 z0
3. Evolution of the amplitude
2
2
 ττ

 i

T0
exp  −
exp  −
 exp [iϕ( z ) ]
2
T ( z)
 T ( z) 
 2 D R( z ) 
=
v ( z , τ) A0
with
A=
( z ) A0 4
1
1+
( )
z
z0
2
,
 z
T=
( z ) T0 1 +  
 z0 
2
A2 ( z )T ( z ) = const.
'Phase curvature' is not fitting to the description of pulses  introduction of
new parameter Chirp
Remember: The phase of a Gaussian beam Φ (x, y, z ) has the following
shape:
 ∂2
∂2 
k
 2 + 2  Φ ( x, y , z ) =
∂y 
R( z )
 ∂x
For monochromatic fields the temporal dynamics of the phase is:
Φ ( τ) = −ωτ
→
−
∂Φ ( τ)
= ω
∂τ
 arbitrary time dependence of phase
∂ 2Φ (τ) ∂ω(τ)
∂Φ (τ)
=
≠ 0  chirp
−
=ω(τ) and −
∂τ2
∂τ
∂τ
The chirp of a pulse describes the variation of the temporal frequency of the
electric field in the pulse.
 parabolic approximation  'chirp' constant  dimensionless chirp
parameter (often just chirp)
C= −
T02 ∂ 2 Φ ( τ)
2 ∂τ2
integration leads to:
∂Φ (τ)
ττ2
−
=ω(τ) =ω0 + 2C 2 ,
− Φ (τ) =ω0 τ + C 2
∂τ
T0
T0
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
C > 0 → up-chirp
leading
front
C < 0 → down-chirp
trailing
tail
phase curvature R( z )  Chirp C ( z )
Complete phase:
2
2


z 
ττ
z 
≡ −ω0  τ +  − C ( z ) 2
Φ (τ) = −ω0  τ +  −

 2 DR( z )

v
vg 
T0
g



 C ( z) =
T02
z
= − 0
2 DR( z )
R( z )
with
zz
z
z 2 + z02
− 20 2=
−
 C ( z) =
R( z ) =
z + z0
z
z0 1 +
(
z2
z02
)
z0
1
T02
 C (0) = 0, C ( z0 ) = - sgn ( z0 ), C ( z ® ¥) = - with z0 = −
2
z
2D
Attention: Chirp depends on sign of z0 and hence on D.
Complete field:
u ( z , t) A0
=

 t2 
T0
t2 
exp  −
exp
(
)
exp [iϕ( z ) ] exp i ( k0 z − ω0t ) 
C
z
−
i

2
2
(
)
T ( z)
T
z
T


0 

Dynamics of a pulse is equivalent to that of a beam.
important parameter  dispersion parameter z0 = −
1)
z << z0 : no effect
2)
z  z0 : similar to beam diffraction
3)
z  z0 : asymptotic dependence
T ( z ) ≈ T0
T02
2D
2D
z
→ T ( z ) / z =T0 / z0 =
=const.
z0
T0
103
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
104
Gaussian pulse spreading as a function of distance . For large distances, the width increases at a rate
, which is inversely proportional to the initial width .
is only important if initial pulse is chirped, since otherwise the same
quadratic dependence is observed, independent from the sign of .
2.9.4 Example 2: Chirped Gaussian pulse
Important because of:
−
−
−
−
short pulse lasers → chirped pulses
Chirp is introduced on purpose, for subsequent pulse compression
analogy to curved phase  focusing
chirped pulse amplification (CPA)  Petawatt lasers
1. Input pulse shape
C0 – initial chirp
2. Input pulse spectrum
 spectral width:
 spectral width of chirped pulse is larger than that of unchirped pulse
only for transform limited pulses
Aim: calculation of pulse width and chirp in dependence on
conditions
for given initial
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105
Gaussian beam  q -parameter  similar to Gaussian pulse
Use analogy: however it is limited to homogeneous space
q=
( z ) q(0) + z.
Remember beams:
1
1
2
=
+i 2 .
q( z ) R ( z )
kw ( z )
k→−
1
1
2 DC ( z )
→
, w2 ( z ) → T 2 ( z ),
D
R( z )
T02
1
2 DC ( z )
2D

=
−i 2
2
q( z )
T0
T ( z)
1
2D 
T02 
=
C ( z ) − i 2  (*)
q ( z ) T02 
T ( z) 
Important: T0 is the pulse width at z = 0 , which is not necessarily in the 'focus'
or waist.
hence at z = 0 :
1
2D
=
[C0 − i] with C0 = C (0)
q(0) T02
Idea:
1
1
2D
( z ) q (0) + z and=
a) q=
insert
into

C
−
i
[
]
0
q( z )
q(0) T02
1
2D 
T02 
b)
C
(
z
)
.
=
−
i
q ( z ) T02 
T 2 ( z ) 
set a) and b) equal  T ( z ) , C ( z )
generally: 2 equations for C0 , T0 , z, C ( z ), T ( z )  3 values predetermined
here: z = d
1) Determination of q parameter at input
T02 ( C0 + i)
q(0) =
2 D (1 + C02 )
2) Evolution of q parameter
q( d )= q(0) + d=
T02 ( C0 + i)
1
 2 Dd (1 + C02 ) + C0T02 + iT02 
+
d
=
2
2 

2 D (1 + C0 )
2 D (1 + C0 )
3) Inversion of general equation (*) for q(d)
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
106
1
2D 
T02 
=
C (d ) − i 2
q (d ) T02 
T (d ) 
T02T 2 ( d )
C ( d )T 2 ( d ) + iT02 
2
4
4 
2 D C ( d )T ( d ) + T0 
q( d )
4) Set two equations equal
 2 Dd (1 + C02 ) + C0T02 + iT02  T02T 2 (d ) C (d )T 2 (d ) + iT02 

=


2
4
4
2
2 D C (d )T (d ) + T0 
2 D (1 + C0 )
 2 Dd (1 + C02 ) + C0T02 
C ( d )T02T 4 ( d )


= 2
a) real part
2
C ( d )T 4 ( d ) + T04 
+
C
1
( 0)
T02T 2 ( d )
1
= 2
b) imaginary part
2
+
1
C
( 0 ) C (d )T 4 (d ) + T04 
If we predetermine 3 parameters ( C0 , T0 , C (d ) ), we can determine the other 2
unknown parameters ( d , T (d ) ).
Important case: Where is the pulse compressed to the smallest length?
given: C0 , T0 & in the focus: C (d ) = 0
unknown: d , T (d )
(
)
0
a) real part must be zero  2 Dd 1 + C02 + C0T02  =
C0
1
1 T02C0
−
=
−
g
(
D
)
LD
n
s
 d=
2
2 D (1 + C02 )
2
+
C
1
( 0)
T02
b) T ( d ) =
(1 + C02 )
2
Resulting properties
1) A pulse can be compressed when the product of initial chirp and
dispersion is negative → C0 D < 0.
2)
The possible compression increases with initial chirp.
Physical interpretation
If e.g. C0 < 0 and D > 0 → ∂vg / ∂ω < 0 → 'red' is faster than 'blue'
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Compression of a chirped pulse in a medium with normal dispersion. The
low frequency (marked R for red) occurs after the high frequency (marked
B for blue) in the initial pulse, but it catches up since it travels faster. Upon
further propagation, the pulse spreads again as the R component arrives
earlier than the B component.
1)
2)
First the 'red tail' of the pulse catches up with the 'blue front' until
(waist), i.e. the pulse is compressed. At this propagation distance the
pulse has no remaining chirp.
Then
and red is in front. Subsequently the 'red front' is faster
than the 'blue tail', i.e. the pulse gets wider.
107
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108
3. Diffraction theory
3.1 Interaction with plane masks
In this chapter we will use our knowledge on beam propagation to analyze
diffraction effects. In particular, we will treat the interaction of light with thin
and plane masks/apertures. Therefore we would like to understand how a
given transversally localized field distribution propagates in a half-space.
There are different approximations commonly used to describe light propagation behind an amplitude mask:
A) If we use geometrical optics we get a simple shadow.
B) We can use scalar diffraction theory with approximated interaction, i.e., a
so-called aperture is described by a complex transfer function
t ( x, y ) with t ( x, y ) = 0 for x , y > a (aperture)
Here we consider the description based on scalar diffraction theory. Then we
can split our diffraction problem into three processes:
i) propagation from light source to aperture
→ not important, generally plane wave (no diffraction)
ii) multiply field distribution of illuminating wave by transfer function
u+ ( x, y, z A ) = t ( x, y )u− ( x, y, z A )
iii)
propagation of modified field distribution behind the aperture through
homogeneous space
u ( x, =
y, z )
∞
∫ ∫ H ( α, β; z − z )U
A
−∞
+
(α, β; z A )exp i ( αx + βy )  d αd β
or
u ( x, y , z =
)
∞
∫ ∫ h ( x − x′, y − y′, z − z ) u ( x′, y′, z
A
−∞
with h =
1
( 2π )
2
+
A
)dx′dy′
FT −1 [ H ]
In the following we will use the notation zB= z − z A . According to our choice of
the propagation function H , resp. h , we can compute this propagation either
exactly or in a paraxial approximation (Fresnel). In the following, we will see
that a further approximation for very large zB is possible, the so-called
Fraunhofer approximation.
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109
3.2 Propagation using different approximations
3.2.1 The general case - small aperture
We know from before that for arbitrary fields (arbitrary wide angular spectrum)
we have to use the general propagation function
2
H ( α, β;=
zB ) exp ( iγ (α, β) zB ) where γ=
k 2 (ω) − α 2 − β2 .
Then we have no constraints with respect to spatial frequencies α, β . We get
homogeneous and evanescent waves and can treat arbitrary small structures
in the aperture by:
u ( x=
, y, z )
∫∫
∞
U + (α, β) H ( α, β; zB ) exp i ( αx + βy )  d αd β
−∞
where U + (α, β) =FT [u+ ( x, y ) ]
Derivation of the response function
We start from the Weyl-representation of a spherical wave:
i
1
1
=
exp ( ikr )
∫ γ exp i ( αx + βy + γz ) d αdβ
r
2p ∫ −∞
∞
Now we can compute the response function h , which we did not do in the
previous chapter, where we computed only hF (Fresnel approximation)). The
following trick shows that
∞
∂ 1

−2p  exp ( ikr ) =
∂z  r

−1
∫ ∫ exp i ( αx + βy + γz ) d αdβ= FT [ H ]=
( 2p )
2
h
−∞
and therefore
h ( x, y , z ) = −
1 ∂ 1

exp ( ikr )  with r =

2p ∂z  r

x2 + y 2 + z 2 .
The resulting expression in position space for the propagation of monochromatic beams is also called 'Rayleigh-formula':
u ( x, y, z )=
∞
∫ ∫ h ( x − x′, y − y′, z ) u ( x′, y′, z
−∞
B
+
A
)dx′dy′
3.2.2 Fresnel approximation (paraxial approximation)
From the previous chapter we know that we can apply the Fresnel
approximation if α2 + β2 << k 2 which is valid for a limited angular spectrum,
and therefore a large size of the structures inside the aperture. Then
 z

=
H F ( α, β; zB ) exp ( ikzB ) exp  −i B ( α 2 + β2 ) 
 2k

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110
 k

i
exp ( ikzB ) exp i
hF ( x, y, zB ) =
x 2 + y 2 )
−
(
λ zB
 2 zB

3.2.3 Paraxial Fraunhofer approximation (far field
approximation)
A further simplification of the beam propagation is possible for many diffraction problems. Let us assume a narrow angular spectrum
α 2 + β2 << k 2
and the additional condition for the so-called Fresnel number N F
N F N 0.1 with N F =
a a
λ zB
where a is the (largest) size of the aperture (like the "beam width").
Obviously, a larger aperture needs a larger distance zB to fulfill N F N 0.1 .
Hence the approximation, which we derive in the following, is only valid in the
so-called 'far field', which means far away from the aperture.
To understand the influence of this new condition on the Fresnel number, we
have a look at beam propagation in paraxial approximation:
uF ( x, y, z=
B)
∫∫
∞
h ( x − x′, y − y′; zB )u+ ( x′, y′)dx′dy′
−∞ F
∞
 k 
i
2
2 
=
−
exp ( ikzB ) ∫∫ u+ ( x′, y′)exp i
x − x′ ) + ( y − y′ )   dx′dy′
(

−∞
λ zB
 2 zB 
In this situation it is easier to treat the beam propagation in position space,
because
u+ ( x, y ) = t ( x, y )u− ( x, y ) , and t ( x, y ) = 0 for x , y > a (aperture)
→ u+ ( x, y ) = 0 for x , y > a
This means that we need to integrate only over the aperture in the above
integral:
a
 k 
i
2
2 
−
uF ( x, y, zB ) =
exp ( ikzB ) ∫∫ u+ ( x′, y′)exp i
x − x′ ) + ( y − y′ )   dx′dy′
(

−a
λ zB
 2 zB 
Now, let us have a closer look at the integral:
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∫∫
111
 k 
2
2 
u+ ( x′, y′)exp i
x − x′ ) + ( y − y′ )   dx′dy′
(

−a
 2 zB 
a
a
 k

 x 2 − 2 xx′ + x′2 + y 2 − 2 yy′ + y′2   dx′dy′
= ∫∫ u+ ( x′, y′)exp i
−a
 2 zB

∫∫
 k

 kx

 k

 x 2 + y 2   exp −i [ x′ + y′] exp i
 x′2 + y′2   dx′dy′
u+ ( x′, y′)exp i
−a
 2 zB

 zB

 2 zB

a
So far, nothing happened, we just sorted the factors differently. But here
comes the trick:
Because of the integration range, we have x′, y′ < a and therefore
k
ka 2
2
2
 x′ + y′  <
→
=
2πN F
2 zB 
zB
 k

 x′2 + y′2   ≈ 1
→for N F N 0.1 → exp i
 2 zB

This means that we can neglect the quadratic phase term in x′, y′ and we get
for the field far from the object, the so-called far field:
 k

i
uFR ( x, y, zB ) =
exp ( ikzB ) exp i
( x2 + y 2 )
−
λ zB
 2 zB

  kx
∞
ky  
×∫∫ u+ ( x′, y′)exp −i  x′ + y′   dx′dy′
−∞
zB  
  zB
( 2p ) exp ikz U (k x , k y )exp i k ( x 2 + y 2 ) 
=
−i
( B) +


2
λ zB
zB
zB
 2 zB

This is the far-field in paraxial Fraunhofer approximation. Surprisingly, the
intensity distribution of the far field in position space is just given by the
Fourier transform of the field distribution at the aperture
I FR ( x, y, zB ) 
1
( λ zB )
2
x
y
U + (k , k ; z A )
zB zB
2
Interpretation
For a plane in the far field at z = zB in each point x, y only one angular
=
zB ; β ky / zB ) with spectral amplitude U + (kx / zB , ky / zB )
frequency
( α kx /=
contributes to the field distribution. This is in contrast to the previously
considered cases, where all angular frequencies contributed to the response
in a single position point.
In summary, we have shown that in (paraxial) Fraunhofer approximation the
propagated field, or diffraction pattern, is very simple to calculate. We just
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112
need to Fourier transform the field at the aperture. In order to apply this
approximation we have to check that:
A) α 2 + β2 << k 2  smallest features ∆x, ∆y >> λ  narrow angular
spectrum (paraxiality)
B)
N=
F
a2
λ zB
<< 1  largest feature a determines zB >> aλ  far field
2
Example: ∆x, ∆y = 10λ, a = 100λ, λ = 1µµ → zB >> 104 λ ≈ 1 cm
3.2.4 Non-paraxial Fraunhofer approximation
The concept that the angular components of the input spectrum separate in
the far field due to diffraction works also beyond the paraxial approximation.
If we have arbitrary angular frequencies in our spectrum, all α 2 + β2 ≤ k 2
contribute to the far field distribution.
Evanescent waves decay for kzB >> 1 → zB >> λ .
N F N 0.1
uFR non−paraxiaλ ( x, y, zB ) = −i
×U+ (
kx
2
a a 1 z0
=
λ zB π zB
with =
NF
2
( 2p )
2
zB
λ x 2 + y 2 + zB 2
x + y + zB
2
,
x 2 + y 2 + zB 2
ky
2
2
x + y + zB
2
(
; z A )exp ik x 2 + y 2 + zB 2
)
3.3 Fraunhofer diffraction at plane masks (paraxial)
Let us now plug things together and investigate examples of diffraction
patterns induced by plane masks in (paraxial) Fraunhofer approximation.
Let us consider an incident plane wave, with a wave vector perpendicular
(k=
k=
0) to the mask t
x
y
u− ( x, y, z A ) = A exp ( ik z z A )
or, more general, inclined with a certain angle
u− ( x=
, y, z A ) A exp i ( k x x + k y y + k z z A ) 
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113
For inclined incidence of the excitation the field behind the mask is given by:
u+ ( x=
, y, z A ) u− ( x, y, z A=
)t ( x, y ) A exp i ( k x x + k y y + k z z A )  t ( x, y )
Form the previous chapter we know that the diffraction pattern in the far field
in paraxial Fraunhofer approximation is given as:
2
I ( x, y, zB ) 
u ( x , y , zB )
1
( λ zB )
2
x
y
U + (k , k )
zB zB
2
Hence, the diffraction pattern is proportional to the spectrum of the field
behind the mask at
x
y
=
α k =
,β k .
zB
zB
This spectrum is calculated by the Fourier transform of the field as:
U + (k
x
y
,k )
zB zB
  x

 y
 
′
′
′
′ ′ ′
exp
(
,
)exp
k
z
t
x
y
k
k
x
k
k
−
−
−
−
i
i
i
(
)
 

z A ∫∫
x
y  y  dx dy
2
z
z
2
p
( )

 B
 
−∞
  B
∞
A
 x

y
= A exp ( ik z z A ) T  k − k x , k − k y 
zB
 zB

Hence, the intensity distribution of the diffraction pattern is given as:
I ( x , y , zB ) −
1
( λ zB )
2
 x

y
T  k − kx , k − k y 
zB
 zB

2
This is the absolute square of the Fourier transform of the aperture function.
In paraxial approximation an inclination of the illuminating plane wave just
shifts the pattern transversely.
Examples
A) Rectangular aperture illuminated by normal plane wave
1 for x a, y b
t ( x, y ) = 
0 elsewhere


x
y
I ( x, y, zB )  sinc 2  ka  sinc 2  kb 
 zB 
 zB 
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Fraunhofer diffraction pattern from a rectangular aperture. The central lobe
of the pattern has half-angular widths
and
.
B) Circular aperture (pinhole) illuminated by normal plane wave
 Airy disk
The Fraunhofer diffraction pattern from a circular aperture produces the
Airy pattern with the radius of the central disk subtending an angle
.
The first zero of the Bessel function (size of Airy disk):
with
So-called angle of aperture:
114
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115
C) One-dimensional periodic structure (grating) illuminated by normal
plane wave
For periodic arrangements of slits we can gain deeper insight in the
structure of the diffraction pattern. Let us assume a periodic slit aperture
with: period b and a size of each slit 2a :
Then, we can express the mask function t as:
t ( x)
=
ts ( x) for x a
with
t
(
x
)
=
(
)
t
x
nb
−

1
∑
1
elsewhere
n =0
0
N −1
The Fourier transform of the mask is given as
 x  N −1 ∞

x 
T  k  − ∑ ∫ t1 ( x′ − nb)exp  −ik x′  dx′
zB 
 zB  n=0 −∞

With the new variable x ′ − nb =
X ′ we can simplify further:
 x  N −1 a




x
x
T  k  − ∑ ∫ tS ( X ′)exp  −ik X ′  exp  −ik nb  dX ′
zB 
zB 
 zB  n = 0 − a


 x
 x  N −1


x
T  k  − TS  k  ∑ exp  −ik nb 
zB 
 zB 
 zB  n = 0

We see that the Fourier transform TS of the elementary slit tS appears.
The second factor has its origin in the periodic arrangement. With some
math we can identify this second expression as a geometrical series and
perform the summation by using the following formula for the definition of
the sinc-funtion:
sin ( N δ2 )
i
−
δ
=
n
exp
(
)
∑
sin ( δ2 )
n =0
N −1
Thus we finally write:
(
 x
 x  sin N k2 zxB b
T  k   TS  k 
k x
 zB 
 zB  sin 2 zB b
(
)
)
For the particular case of a simple grating of slit apertures with ts ( x) = 1
we have
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 x 
 x 
T1  k  = sinc  k a 
 zB 
 zB 
and therefore
(
2
 x  sin N k2 zxB b
I  sinc  k a 
2 k x
 zB  sin 2 zB b
2
(
)
)
We find three important parameters for the diffraction pattern of a
grating:
− Global width of diffraction pattern  first zero of slit function TS
k
λz
xS
a = π  xS = B
2a
zB
The width of the total far-field diffraction pattern xS (largest length
scale in the pattern) is determined by the size a of the individual slit
(smallest length scale in the mask).
− Position of local maxima of diffraction pattern  maxima of grid
function
(
 sin 2 N k2 zx b
B
max 
2 k x
 sin 2 z b
B

(
)
) 



λz
k xP
b = nπ  xp = n B
b
2 zB
These are the so-called diffraction orders, which are determined
exclusively by the grating period.
− Width of local maxima  zero-points of grid function
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117

The width of a maximum in the far-field diffraction patter
(smallest
length scale in the pattern) is determined by
which is the total
size of the mask (largest length scale of the mask).
These observations are consistent with the general property of the Fouriertransform: small scales in position space give rise to a broad angular
spectrum and vice versa.
3.4 Remarks on Fresnel diffraction
Fresnel number
diffraction length from Gaussian beams
−
(
−
(
−
(
large,
small,
)  Fraunhofer
)
)  shadow
 FT of aperture
 Fresnel diffraction
Fresnel diffraction from a slit of width
. (a) Shaded area is the
geometrical shadow of the aperture. The dashed line is the width of the
Fraunhofer diffraction beam. (b) Diffraction pattern at four axial positions
marked by the arrows in (a) and corresponding to the Fresnel numbers
and
. The dashed area represents the geometrical
shadow of the slit. The dashed lines at
represent the width
of the Fraunhofer pattern in the far field. Where the dashed lines coincide
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
with the edges of the geometrical shadow, the Fresnel number
N F= a 2 / λd= 0.5 .
118
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119
4. Fourier optics - optical filtering
From previous chapters we know how to propagate the optical field through
homogeneous space, and we also know the transfer function of a thin lens.
Thus, we have all tools at hand to describe optical imaging. While detailed
designs of high resolution optical systems have to consider non-paraxial
effects, usually the paraxial approximation is sufficient to obtain a principle
understanding of optical systems. Hence we use the paraxial approximation
here.
Many imaging systems exploit the appearance of the Fourier transform of the
original object in the so-called Fourier plane of the system in order to
manipulate the angular spectrum of the object in this plane. Accordingly this
field of optical science is called Fourier optics. We will see in the following
that with the right setup of our imaging system we can generate the Fourier
transform of the object on a much shorter distance than by far field diffraction
in the Fraunhofer approximation.
The general idea of Fourier optics is the following:
1) An imaging system generates the Fourier transform of the object in
Fourier plane.
2) A spatial filter (e.g. an aperture) in the Fourier plane manipulates the
field.
3) Another imaging system performs the Fourier back-transform and hence
results in a manipulated image.
Mathematical concept:
• propagation in free space  calculated in Fourier domain
• interaction with lens or filter  calculated in position space
4.1 Imaging of arbitrary optical field with thin lens
4.1.1 Transfer function of a thin lens
A thin lens changes only the phase of the optical field, since due to its
infinitesimal thickness, no diffraction occurs. By definition, it transforms a
spherical wave into a plane wave. If we write down this definition in paraxial
approximation we get
 k

i
i
− exp ( ikf ) exp i
x 2 + y 2 )  tL ( x, y ) =
− exp ( ikf )
(
λf
λf
 2f
((((

((((((((((((

sphericaλ wave
pλane wave
And therefore the response function for a thin lens is given as (see chapter
2.8.3):
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120


k
tL ( x, y ) =
exp  −i
x2 + y 2 )
(
 2f

By Fourier transforming the response function we find consequently the
transfer function in the Fourier domain as
TL ( α, β ) = −i
λf
 f

exp
i ( α 2 + β2 ) 
2

 2k

( 2p )
4.1.2 Optical imaging using the 2f-setup
Let us now consider optical imaging. We place our object in the first focus of
a thin lens, with a field distribution u0 ( x, y ) , and follow the usual recipe for light
propagation.
A)
Spectrum in object plane
U 0 (α, β) =FT [u0 ( x, y ) ]
B)
Propagation from object to lens
(lens positioned at distance f)
=
U − (α, β; f ) H F ( α, β; f )U 0 (α, β)
 i

U − (α, β; f ) = exp ( ikf ) exp  − ( α 2 + β2 ) f  U 0 (α, β)
 2k

C) Interaction with lens
(multiplication in position space or convolution in Fourier domain)
u + ( x, y , f ) = tL ( x, y ) u − ( x, y , f )
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121
, β; f ) TL ( α, β ) ∗ U − (α, β; f )
U + (α=
−i
λf
∞
2
2 
 f 
 ⋅
′
′
exp
exp
α
−
α
+
β
−
β
i
i
kf
(
)
(
)
(
)

2
∫∫


−∞
2
k

( 2p )
 i

⋅ exp  − ( α′2 + β′2 ) f  U 0 (α′, β′)d α′d β′
 2k

D) Propagation from lens to image plane
=
U (α, β;2 f ) H F ( α, β; f )U + (α, β; f )
U (α, β;2 f ) = −i
λf
∞
2
2 
 f
exp ( 2ikf ) ∫∫ exp i ( α − α′ ) + ( β − β′ )   ⋅

−∞
 2k 
( 2p )
2
 i

 i

⋅ exp  − ( α′2 + β′2 ) f  exp  − ( α 2 + β2 ) f  U 0 (α′, β′)d α′d β′
 2k


 2k
 i

 i

Quadratic terms  − ( α′2 + β′2 ) f  and  − ( α 2 + β2 ) f  in the exponent
 2k

 2k

f
2
2
cancel with quadratic terms from i ( α − α′ ) + ( β − β′ )  and only the mixed

2k 
terms remain.
U (α, β;2 f ) = −i
=−i
λf
∞
f


exp ( 2ikf ) ∫∫ U 0 (α′, β′)exp  −i ( αα′ + ββ′ )  d α′d β′
−∞
 k

( 2p )
2
λf
f 
 f
−
α
−
β
exp
2
,
i
kf
u
(
)
0

2
k 
 k
( 2p )
We see that the spectrum in the image plane is given by the optical field in
the object plane.
E) Fourier back transform in image plane
u ( x, =
y,2 f ) FT −1 [U (α, β;2 f ) ]
= −i
∞
λf
f
 f
exp ( 2ikf ) ∫ ∫ u0  − α, −
k
 k
( 2p )
−∞
2
With the coordinate transformation

β  exp i ( αx + βy )  d αd β

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122
we get:

The image in the second focal plane is the Fourier transform of the optical
field in the object plane. This is simmilar to the far field in Fraunhofer
approximation, but for
. This finding allows us to perform an optical
Fourier transform over shorter distances.
And in the Fourier plane it is possible to manipulate the spectrum.
4.2 Optical filtering and image processing
4.2.1 The 4f-setup
For image manipulation (filtering) it would be advantageous if we could
perform a Fourier back-transform by means of an optical imaging setup as
well. It turns out that this leads to the so-called 4f-setup:
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The filtering (manipulation) happens in the second focal plane (Fourier plane
after 2f) by applying a transmission mask p ( x, y ) . In order to retrieve the
filtered image we use a second lens:
We know that the image in the Fourier plane is the FT of the optical field in
the object plane.
k k 
u ( x, y,2 f ) = AU 0  x, y 
f 
 f
We have to compute the imaging with the second lens after the manipulation
of the spectrum in the Fourier plane.
Our final goal is the transmission function H A (α, β;4 f ) of the complete imaging system:
=
u ( − x, − y , 4 f )
∞
∫ ∫ H ( α, β;4 f )U
−∞
A
0
(α, β) exp i ( αx + βy )  d αd β
Note: We will see in the following calculation that the second lens does a
Fourier transform  exp i ( αx + βy )  . In order to obtain a proper back
transform we have to pass to mirrored coordinates x → − x, y → − y .
The transmission mask p ( x, y ) contains all constrains of the system (e.g. a
limited aperture) and optical filtering (which we can design).
A) Field behind transmission mask
k k 
u+ ( x, y,2 f ) = u ( x, y,2 f ) p ( x, y ) ~ AU 0  x, y  p ( x, y )
f 
 f
B) Second lens  Fourier back-transform of field distribution
u ( x, y,4 f ) =
( 2p )
−i
2
k k

exp ( 2ikf )U +  x, y;2 f 
f
λf
 f

Now we can make the link to the initial spectrum in the object plane U 0 :
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∞
 k

u ( x, y,4 f ) ~ ∫ ∫ u+ ( x′, y′,2 f )exp  −i ( xx′ + yy′ )  dx′dy′
f


−∞
∞
k
 k

k 
~ ∫ ∫U 0  x′, y′  p ( x′, y′)exp  −i ( xx′ + yy′ )  dx′dy′
f 
f
 f


−∞
Here we do not care about the amplitudes and just write ~:
To get the anticipated form we need to perform a coordinate transformation:
=
α
k
k
′, β
x=
y′
f
f
Then we can write:
∞
f
f
u ( x, y,4 f ) ~ ∫ ∫ U 0 ( α, β ) p ( α, β)exp  −i ( αx + βy )  d αd β
k k
−∞
By passing to mirrored coordinates x → − x, y → − y we get
∞
f
f
u (− x, − y,4 f ) ~ ∫ ∫ p ( α, β)U 0 ( α, β ) exp i ( αx + βy )  d αd β
k k
−∞
Hence we can identify the transmission function of the system
f
f
H A ( α, β;4 f ) ~ p ( α, β)
k k
Summary
• Fourier amplitudes get multiplied by transmission mask
• transmission mask  transfer function
• coordinates of image  mirrored coordinates of object
In position space we can formulate propagation through a 4f-system by using
the response function hA ( x, y )
u (− x, − y,4
=
f)
∞
∫ ∫h
−∞
A
( x − x′, y − y′)u0 ( x′, y′)dx′dy′
As usual, the response function is given as the Fourier transform of the
transfer function:
=
hA ( x, y )
1
∞
H A ( α, β;4 f ) exp {i[ αx + βy ]} d αd β
2
( 2p ) ∫ −∞∫
f
f
From above we have H A ( α, β;4 f ) ~ p ( α, β)
k k
∞
f
f
hA ( x, y ) ~ ∫ ∫ p ( α, β)exp {i[ αx + βy ]} d αd β
k
k
−∞
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We introduce the coordinate transform
The response-function is proportional to the Fourier transform of the
transmission mask.
4.2.2 Examples of aperture functions
Example 1: The ideal image (infinite aperture)
Be careful, we use paraxial approximation  limited angular spectrum
The 4-f system performs a Fourier transform followed by an inverse
Fourier transform, so that the image is a perfect replica of the object
(perfect only within the Fresnel approximation).
→
Example 2: Finite aperture
 mirrored original
125
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126
Spatial filtering. The transparencies in the object and Fourier planes have
complex amplitude transmittances
and
. A plane wave
traveling in the
direction is modulated by the object transparency,
Fourier transformed by the first lens, multiplied by the transmittance of the
mask in the Fourier plane and inverse Fourier transformed by the second
lens. As a result, the complex amplitude in the image plane
is a
filtered version of the original field
in the object plane. The system
has a transfer function
.
Transmission function:
− finite aperture truncates large angular frequencies (low pass)
− determines optical resolution
4.2.3 Optical resolution
A finite aperture acts as a low pass filter for angular frequencies.
With
we can define an upper limit for the angular frequencies
which are transmitted (bandwidth of the system)

Translated to position space, the smallest transmitted structural information is
given by:
A more precise definition of the optical resolution can be derived the following
way:
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∞
k

k
u (− x, − y,4 f ) ~ ∫ ∫ P  ( x′ − x ) , ( y′ − y )  u0 ( x′, y′)dx′dy′
f
f

−∞ 
∞
~∫∫
−∞
J1  2kDf

kD
2f
( x′ − x ) + ( y′ − y )
2
( x′ − x ) + ( y′ − y )
2
2
2


u0 ( x′, y′)dx′dy,′
One point of the object x0 , y0 gives an Airy disk (pixel) in the image:
2
2 
  kD
J
x
−
x
+
y
−
y
(
)
(
)


1
2
0
0
f
 

−

2
2
 2kDf ( x0 − x ) + ( y0 − y )



2
We can define the limit of optical resolution:
Two objects in the object plane can be independently resolved in the image
plane as long as the intensity maximum of one of the objects is not closer to
the other object than its first intensity minimum:
kD
Drmin = 1.22π
2f
Hence we find:
1.22λf
Drmin =
nD
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Further examples for 4f filtering
Examples of object, mask, and filtered image for three spatial filters: (a)
low-pass filter; (b) high-pass filter; (c) vertical pass filter. Black means the
transmittance is zero and white means the transmittance is unity.
128
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129
5. The polarization of electromagnetic waves
5.1 Introduction
We are interested in the temporal evolution of the electric field vector Er (r, t ) .
In the previous chapters we mostly used a scalar description, assuming
linearly polarized light. However, in general one has to consider the vectorial
nature, i.e. the polarization state, of the electric field vector.
We know that the normal modes of homogeneous isotropic dielectric media
=
E(r, t ) E exp i k ( ω) r − ωt  .
are plane waves
{
}
If we assume propagation in z direction (k-vector points in z-direction),
divE(r, t ) = 0 implies that we can have two nonzero transversal field
E ,E
components  x − and y − components x y
The orientation and shape of the area which the (real) electric field vector
covers is in general an ellipse. There are two special cases:
− line (linear polarization)
− circle (circular polarization)
5.2 Polarization of normal modes in isotropic media
0
k =  0 
k 
 
 propagation in z direction
The evolution of the real electric field vector is given as
Er (r, t ) = ℜ{E exp [i(kz − ωt )]}
Because the field is transversal we have two free complex field components
 Ex exp ( iϕx ) 


=
E  Ey exp ( iϕy ) 


0


with Ex ,y and ϕx ,y being real
Then the real electric field vector is given as
 Ex cos ( ωt − kz − ϕx ) 


E
=
 Ey cos ( ωt − kz − ϕy ) 
r (r , t )


0


Here, only the relative phase is interesting  δ = ϕy − ϕx
Conclusion:
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Normal modes in isotropic, dispersive media are in general elliptically
polarized. The field amplitudes Ex , Ey and the phase difference δ = ( ϕy − ϕx )
are free parameters
5.3 Polarization states
Let us have a look at different possible parameter settings:
A) linear polarization
 δ = nπ (or Ex = 0 or E y = 0 )
B)
circular polarization
 Ex = Ey = E , δ = ±π / 2
δ = +π / 2  counterclockwise rotation
Ey
Ex
δ = −π / 2  clockwise rotation
Ey
Ex
These pictures are for an observer looking contrary to the propagation
direction.
C) elliptic polarization  Ex ≠ Ey ≠ 0, δ ≠ nπ
0<δ<π
 counterclockwise
π < δ < 2π
 clockwise
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Examples
Remark
A linearly polarized wave can be written as a superposition of two counterrotating circularly polarized waves. Example: Let's observe the temporal
with
.
evolution at a fixed position
131
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6. Principles of optics in crystals
In this chapter we will treat light propagation in anisotropic media (the worst
case). Like in the isotropic case before we will seek for the normal modes,
and in order to keep things simple we assume homogeneous media.
6.1 Susceptibility and dielectric tensor
before:
isotropy (optical properties independent of direction)
now:
anisotropy (optical properties depend on direction)
The common reason for anisotropy in many optical media (in particular
crystals) is that the polarization P depends on the direction of the electric
field vector. The underlying reason is that in crystals the atoms have a
periodic distribution with different symmetries in different directions.
Prominent examples for anisotropic materials are:
− Lithium Niobat  electro-optical material
− Quartz
 polarizer
− liquid crystals  displays, NLO
In order to keep things as simple as possible we make the following
assumptions:
− one frequency- (monochromatic), one angular frequency (plane wave)
− no absorption
From previous chapters we know that in isotropic media the normal modes
are elliptically polarized, monochromatic plane waves. The question is how
the normal modes of an anisotropic medium look like  ???
Before (isotropic)
P(r, ω) =ε0χ ( ω) E(r, ω)
D(r, ω) =ε0ε ( ω) E(r, ω)
In the following we will write E  E , because we assume monochromatic
light and the frequency ω is just a parameter.
Now (anisotropic)
3
=
Pi (r, ω) e0 ∑
j =1
cij (ω)

E j (r, ω)
tensor components
The linear susceptibility tensor has 3x3=9 tensor components. Direct consequences of this relation between polarization P and electric field E are:
− P  E : the polarization is not necessarily parallel to the electric field
− The tensor elements χij depend on the structure of crystal. However,
we do not need to know the microscopic structure because of the
different length scales involved (optics: 5 ⋅ 10−7 m ; crystal: 5 ⋅ 10−10 m ),
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but the field is influenced by the symmetries of the crystal (see next
section).
− In complete analogy we find for the D field:
3
Di (r, ω) =ε0 ∑εij (ω) E j (r, ω)
j =1
D(r, ω) =ε0 ε (ω)E(r, ω)
As for the polarization we find:
− DE
We introduce the following notation:
− χˆ = ( χij )
 susceptibility tensor
− ε =
(ε )
− σˆ =
( εˆ )
 dielectric tensor
ij 
−1
=
(σ )
ij
 inverse dielectric tensor
3
∑σ
j =1
ij
(ω) D j (r, ω) =ε0 Ei (r, ω)
The following properties of the dielectric and inverse dielectric tensor are
important:
− σij , εij are real in the transparent region (omit ω ), we have no losses
(see our assumptions above)
− The tensors are symmetric (hermitian), only 6 components are
independent
εij =
ε ji , σij =
σ ji .
− It is known (see any book on linear algebra) that for such tensors a
transformation to principal axes by rotation is possible (matrix is
diagonalizable by orthogonal transformations).
− If we write down this for σij , it means that we are looking for directions
where D  E , i.e., our principal axes:
3
ε0 Ei =∑σij D j  λDi
j =1
This is a so-called eigenvalue problem, with eigenvalues λ . If we want to
solve for the eigenvalues we get
det σij − λI ij  = 0, with I ij = dij
This leads to a third order equation in λ , hence we expect three solutions
(roots) λ ( α ) . The corresponding eigenvectors can be computed from
3
∑σ D
j =1
ij
(α)
j
=
λ ( α ) Di( α ) .
The eigenvectors are orthogonal:
Di(β ) Di( α ) = 0 for λ ( α ) ≠ λ (β )
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The directions of the principal axes (defined by the eigenvectors) correspond
to the symmetry axes of the crystal.
The diagonalized dielectric and inverse dielectric tensors are linked:
The above reasoning shows that anisotropic media are characterized in
general by three independent dielectric functions (in the principal coordinate
system).
It is easier to do all calculations in the principal coordinate system (coordinate
system of the crystal) and back-transform the final results to the laboratory
system.
6.2 The optical classification of crystals
Let us now give a brief overview over crystal classes and their optical
properties:
A) isotropic
− three crystallographic equivalent orthogonal axes
− cubic crystals (diamond, Si....)

Cubic crystals behave like gas, amorphous solids, liquids, and have no
anisotropy.
B) uniaxial
− two crystallographic equivalent directions
− trigonal (quartz, lithium niobate), tetragonal, hexagonal
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135
C) biaxial
− no crystallographic equivalent directions
− orthorhombic, monoclinic, triclinic
6.3 The index ellipsoid
The index ellipsoid offers a simple geometrical interpretation of the inverse
dielectric tensor
. The defining equation for the index ellipsoid is
which describes a surface in three dimensional space.
Remark on the physics of the index ellipsoid:
The index ellipsoid defines a surface of constant electric energy density:
In the principal coordinate system the defining equation of the index ellipsoid
reads:
This equation can be interpreted as the defining equation of an ellipsoid
. From our discussion of the normal
having semi-principal axes of length
modes in isotropic media we know that
corresponded to the refractive
index of the normal modes. We will show in the following discussion that also
determine the
for anisotropic media, there will be special cases where the
phase velocity of normal modes. Hence the elements of the dielectric tensor
which define the semi-principal axes of the epsilon ellipsoid can be related to
refractive indexes
This is the reason why the ellipsoid, which represents graphically the epsilon
tensor, is called index ellipsoid.
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136
Graphical representation of the epsilon tensor of an anisotropic crystal by
the so-called index ellipsoid.
Summary:
• anisotropic media → tensor instead of scalar
 in principal system: ni= εi
• The index ellipsoid is degenerate for special cases:
− isotropic crystal: sphere
− uniaxial crystal: rotational symmetric with respect to z-axis and n1 = n2
6.4 Normal modes in anisotropic media
Let us now look for the normal modes in crystals. A normal mode is:
• a solution to the wave equation, which shows only a phase dynamics
during propagation while amplitude and polarization remain constant
 most simple solution  exp {i k ( ω) r  ωt }
• a solution where the spatial and temporal evolution of the phase are
connected by a dispersion relation ω = ω(k ) or k= k (ω)
Before – isotropic media
In isotropic media the normal modes are monochromatic plane waves
{
}
E
=
(r, t ) E exp i k ( ω) r − ωt 
with the dispersion relation
ω2
k ( ω=
) k (ω=) 2 ε(ω)
c
2
2
with ε(ω) > 0 and real as well as k ⋅ E = k ⋅ D = 0 . The normal modes are
elliptically polarized, and the polarization is conserved during propagation.
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137
Now – anisotropic media
What are the normal modes in anisotropic media?
6.4.1 Normal modes propagating in principal directions
Let us first calculate the normal modes for propagation in the direction of the
principal axes of the index ellipsoid, which is the simple case.
We assume without loss of generality that the principal axes are in x, y, z
direction and the light propagates in z direction ( k → kz ) . Then, the fields
are arbitrary in the x,y-plane
Dx , Dy ≠ 0
and
Di =ε0εi Ei
In general we have E  D , but here k ⋅ D = 0 ⇒ k ⋅ E = 0 , and the two
( x 1,=
y 2) are decoupled:
polarization directions =
ω2
D1 , e1  D1=
exp i ( k1 z - ωt )  D1 exp [iϕ1 ( z ) ] exp (-iωt ) with k = 2 ε1 (ω)
c
2
1
ω2
D2 , e 2  D2=
exp i ( k2 z - ωt )  D2 exp [iϕ2 ( z ) ] exp (-iωt ) with k = 2 ε 2 (ω)
c
2
2
We see that in contrast to isotropic media, normal modes can't be elliptically
polarized, since the polarization direction would change during propagation.
But, for linear polarization in the direction of a principal axis (x or y) only the
phase changes during propagation, thus we found our normal modes:
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D
D
(a)
( b)

{
}
ω2 2
→ k = 2 na = k12 → normal mode a
c
}
ω2 2
→ k = 2 nb = k 22 → normal mode b
c
= D1 exp i ( k ar − ωt )  e1
{
= D2 exp i ( k br − ωt )  e 2
138
2
a
2
b
For light propagation in principle direction we find two perpendicular
linearly polarized normal modes with E  D .
Remark on the indices in the index ellipsoid
The indices ni= εi in the index ellipsoid are connected to the indices na and
nb of the normal modes propagating along the principal axis. However please
be careful about the direction correspondence. For example, the two normal
modes propagating along the z direction have phase velocities determined
by the indices n1 = nx and n2 = n y determined by the direction of their electric
field rather than by the direction of their propagation.
6.4.2 Normal modes for arbitrary propagation direction
6.4.2.1
Geometrical construction
Before we will do the mathematical derivation and actually calculate normal
modes and dispersion relation, let us preview the results visualized in the
index ellipsoid. Actually, it is possible to construct the normal modes
geometrically. We start from the normal modes which we have determined for
propagation in the principal directions of the crystal and try to generalize to
arbitrary propagation directions:
• For a specific crystal and a given frequency ω we take the εi in the
principal axis system and construct the index ellipsoid.
• We then fix the propagation direction of the normal mode which we would
like to look at  k / k = u .
• We draw a plane through the origin of index ellipsoid which is
perpendicular to k .
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139
• The resulting intersection is an ellipse, the so-called index ellipse.
• The half-lengths of the principle axes of this ellipse equal the refractive
indices
of the normal modes for the propagation direction

and
• The directions of the principal axes of the index ellipse are the polarization
direction of the normal modes
and
.
and
follow from
• The electric field vectors of the normal modes
, and
are not perpendicular to
• Thus,
• This has a direct consequence on the pointing vector:
hence
is not parallel to
because
• If the index ellipse is a circle, the direction of this particular k-vector defines
the optical axis of the crystal.
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6.4.2.2
140
Mathematical derivation of dispersion relation
Let us now derive mathematically the dispersion relation for normal modes of
the form
{
}
D exp {i k ( ω ) r − ωt }
=
E(r, t ) E exp i k ( ω ) r − ωt 
=
D(r, t )
In the isotropic case we found the dispersion relation
ω2
k (ω)= k (ω)= 2 ε(ω)
c
2
2
where the absolute value of the k-vector is independent of its direction. The
fields of the normal modes are elliptically polarized.
In the anisotropic case the normal modes are again monochromatic plane
waves  exp {i k ( ω) r  ωt } , but the wavenumber depends on the direction u
of propagation, where u = k / k . Hence
k= k ( ω, u )
and the polarization of the normal modes is not elliptic.
In the following, we start again from Maxwell’s equations and plug in the
plane wave ansatz. We will use the following notation for the directional
dependence of k :
 k1 
 u1 


k =  k2  = k  u2  with u12 + u22 + u32 =
1
k 
u 
 3
 3
Our aim is to derive ω = ω(k1 , k2 , k3 ) or ω = ω(k , u1 , u2 , u3 ) or k= k (ω, u1 , u2 , u3 ) .
We start from Maxwell's equations for the plane wave Ansatz:
k D = 0
k × E = ωµ 0 H
k H = 0
k × H = −ωD
Now we follow the usual derivation of the wave equation:
ω2 1
ω2 1
2
D  −k ( k ⋅ E ) + k E =
D
− k × ( k × E )  =
c 2 ε0
c 2 ε0
− Here k ⋅ E does not vanish as it would have in the isotropic case!
− In the principal coordinate system and with Di =ε0εi Ei we find
ω2
−ki ∑k j E j + k 2 Ei =
εE
2 i i
c
j
 ω2
2
k
ε
−
 2 i
 Ei =−ki ∑k j E j
j
c

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141
Remark: for isotropic media the r.h.s. of this equation would vanish ( k ⋅ E = 0 ).
Now, we have the following problem to solve:
 ωc2 ε1 − k22 − k32

k2 k1


k3k1

k1k2
2
ω2
c2
 E  0
 1   
k 2 k3
  E2  = 0 
   
ω2
ε − k12 − k22   E3   0 
c2 3
k1k3
ε 2 − k12 − k32
k3 k 2
The general way to solve this problem is using det [..] = 0 , which gives the
dispersion relation ω = ω(k ) for given ki / k .
However, there is an easy way to show some general properties of the
dispersion relation. We start from the following trick:
 ω2
2
 2 εi − k  Ei =−ki ∑k j E j  Ei = −
j
c

(
ki
ω
ε − k2
c2 i
2
kE
∑
)
j
j
j
Now we multiply this equation by ki , perform a summation over the index ' i '
and rename i ↔ j on the l.h.s:
∑k E
j
j
j
= −∑
i
(
ki2
ω2
ε − k2
c2 i
)∑
j
k jE j.
Because=
div E ∑ j k j E j ≠ 0 we can divide and get the (implicit) dispersion
relation:
ki2
∑i  2 ω2  = 1
 k − c 2 εi 


 k1 
 u1 
 u1 
ω
With  k2  =
k (ω)  u2  =n(ω)  u2  we can write
k 
u  c
u 
3
 
 3
 3
k 2ui2
ui2
=
1→ ∑
1
∑i  2 ω2  =
εi 
i 
1 − 2 
 k − c 2 εi 
 n 


1
ui2
= 2 final form of DR
 ∑ 2
i 
 n − εi  n
Discussion of results
For given εi ( ω) and direction ( u1 , u2 ) we can compute the refractive index
1 , it is sufficient
n ( ω, u1 , u2 ) seen by the normal mode. Because u12 + u22 + u32 =
to fix two components ( u1 , u2 ) of u to determine the direction.
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A more explicit form of the dispersion relation can be obtained by multiplying
with denominators:
u12 ( n 2 − ε 2 )( n 2 − ε3 ) n 2 + u22 ( n 2 − ε1 )( n 2 − ε3 ) n 2 + u32 ( n 2 − ε1 )( n 2 − ε 2 ) n 2 =
(n
2
− ε1 )( n 2 − ε 2 )( n 2 − ε3 )
The resulting equation is quadratic in (n 2 ) 2 since the n6 -terms cancel. Hence,
we get two (positive) solutions na , nb and therefore k=
na (ω / c) and
a
k=
nb (ω / c) for the two orthogonally polarized normal modes D(a ) and D( b ) .
b
In particular, for the propagation in direction of the principal axis ( u3 = 1 and
u=
u=
0 , see 6.4.1) we find:
1
2
(
)(
( n − ε )( n
)
2
 n 2 − ε1 n 2 − ε 2 n=

2
1
2
(n
2
− ε1 )( n 2 − ε 2 )( n 2 − ε3 )
− ε 2 ) ε3 = 0
 na2 =
ε1 , nb2 =
ε2
Finally, we can derive some properties of the fields of the normal modes, i.e.
the eigenfunctions.
We start from the eigenvalue equation, which we had derived above
 ω2
2
 2 εi − k  Ei =−ki ∑k j E j .
j
c

For cases where the first factor of the l.h.s. is unequal zero (propagation
directions not parallel to the principal axes) we can divide by this term
Ei = −
(
ki
ω
ε − k2
c2 i
2
)∑
j
k jEj .
The sum does not depend on the index i . Hence the last term of the equation
must be constant

∑k E
j
j
j
= const.
Knowing that the last term is a constant we can derive a relation of the
individual field components from the first part of the equation
 E1 : E2 : E3 =
2
k1
:
ε1k1
:
ω
ε − k2
2 1
c
2
k2
:
ε 2 k2
:
ω
ε − k2
2 2
c
2
k3
ω
ε − k2
2 3
c
and with Di =ε0εi Ei
 D1 : D2 : D3 =
2
ω
ε − k2
2 1
c
2
ω
ε − k2
2 2
c
2
ε 3 k3
ω
ε − k2
2 3
c
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Please be aware that this relation can only be applied for propagation directions not parallel to the principal axes.
How are the normal modes polarized?
− The ratio between the field components is real
 phase difference 0 → linear polarization
How do we see the orthogonality D(a ) D(b ) = 0 ? (be careful: E(a ) E(b ) ≠ 0 )
(a)
D D
(b )
ka kb εi2ui2
~∑
ω2  2 ω2 
i  2
 ka − c 2 εi  kb − c 2 εi 







2
2
c
k a kb  2
εiui
εiui
2

=
ka ∑
− kb ∑
2
2
2
2
2
ω ( kb − k a )  i  2 ω 
ω 
i  2

 ka − c 2 εi 
 kb − c 2 ε i  





2
Since the two red terms vanish due to the dispersion relation, it follows that
D(a ) D(b ) = 0 . The vanishing of the red terms can be seen when rewriting the
dispersion relation:
 2 ω2
ω2  2
k
−
ε
+
 a ,b c 2 i c 2 εi  ui
ka2,bui2
ω2
εiui2


1= ∑
= ∑
= 1+ 2 ∑
c i  2 ω2 
 2 ω2 
ω2  i
i  2
 ka ,b − c 2 εi 
 ka ,b − c 2 εi 
 ka ,b − c 2 εi 






6.4.3 Normal surfaces of normal modes
In addition to the index ellipsoid, which is a graphical representation of the
material properties of crystals from which the properties of the normal modes
can be interpreted as shown above, we can derive a direct graphical
representation of the dispersion relation of normal modes in crystals. This
graphical representation of the dispersion relation is called normal surfaces:
If we plot the refractive indices (wave number or norm of the k-vector divided
by k0 ) of the normal modes in the ki -space (normal surfaces), we get a
centro-symmetric, two layer surface.
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optical axis
optical axis
uniaxial
biaxial
isotrop
Normal surfaces as the graphical representation of the dispersion relation
of normal modes in crystals.
isotrop: sphere
uniaxial: 2 points with
optical axis (for
in the poles  connecting line defines the
,
the z-axis is the optical axis)
biaxial: 4 points with
 connecting lines define two optical axes
How to read the figure:
− fix propagation direction (
)  intersection with surfaces
− distances from origin to intersections with surfaces correspond to
refractive indices of normal modes
− definition of optical axis 
Summary: there are two geometrical constructions:
A) index ellipsoid (visualization of dielectric tensor)
− fix propagation direction
index ellipse
half lengths of principal
(refractive indices of the normal modes)
axes give
B)
− optical axis  index ellipse is a circle
− for uniaxial crystals the optical axis coincides with one principal axis
normal surfaces (visualization of dispersion relation)
− fix propagation direction
intersection with surfaces
distances from origin give
− optical axis connects points with
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145
Conclusion
In anisotropic media and for a given propagation direction we find two normal
modes, which are linearly polarized monochromatic plane waves with two
different phase velocities c na , c nb and two orthogonal polarization directions
D ( a ) , D (b ) .
6.4.4 Special case: uniaxial crystals
Let us now investigate the special and simpler case of uniaxial crystals. In
biaxial crystals we do not find any other effects, just the description is more
complicated.
The main advantage of uniaxial crystals is that we have rotational symmetry
in one plane. Therefore all three-dimensional graphs (index-ellipsoid, normal
surfaces) can be reduced to two dimensions, and we can sketch them more
easily. As we have seen before, uniaxial crystals have trigonal, tetragonal, or
hexagonal symmetry.
Let us assume (without loss of generality) that the index ellipsoid is
rotationally symmetric around the z-axis, and we have
ε1 =ε 2 =εor , ε3 =εε ,
which are the ordinary and extraordinary dielectric constants.
Then, we expect two normal modes:
A) ordinary wave
 na independent of propagation direction
B)
extraordinary wave
 nb depends on propagation direction
The z-axis is, according to definition, the optical axis with na = nb .

(or)
The ordinary wave D is polarized perpendicular to the z-axis and the
k-vector and it does not interact with ee .
(e)
The extraordinary wave D is polarized perpendicular to the k-vector
(or)
and D .
Let us now derive the dispersion relation: From above we know the implicit
form

ui2
1
= 2
 ∑ 2
i 
 n − εi  n
For uniaxial crystals this leads to
u12
u22
u32
1
+ 2
+ 2
=
2
 n − εor   n − εor   n − εε  n 2
2
2
2
n 2  n 2 − εε   n 2 − εor  ( u12 + u22 ) + n 2  n 2 − εor  u=
3
 n − εε   n − εor 
2
A) ordinary wave: independent of direction
2
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146

B) extraordinary wave (derivation is your exercise): dependent on direction
Hence for a given direction
one gets the two refractive indexes
.
The geometrical interpretation as normal surfaces is straightforward and can
or y, z plane (
).
be done, w.l.o.g., in the ,
Normal surfaces for a uniaxial crystal.
The shape of the normal surfaces can be derived in the following way.
We have with
A) ordinary wave
B) extraordinary wave
What about the fields? We know from before that
For the extraordinary wave all denominators are finite, and in particular
, hence
is polarized in the y-z plane. Then,
implies
implies that
is polarized in x-direction.
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In summary, we find for the polarizations of the fields:
perpendicular to optical axis and ,
A) ordinary:
B)
extraordinary:
perpendicular to
and in the plane
-optical axis
, because
If we introduce an angle , as in the figures below, to describe the
propagation direction, a simple computation of
for the extraordinary
wave is possible (exercise):
The following classification for uniaxial crystals is commonly used
 negative uniaxial
 positive uniaxial
147
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148
7. Optical fields in isotropic, dispersive and
piecewise homogeneous media
7.1 Basics
7.1.1 Definition of the problem
Up to now, we always treated homogeneous media. However, in the context
of evanescent waves we already used the concept of an interface. This was
already a first step in the direction we now want to pursue. When we treated
interfaces so far we never considered effects of the interface, we just fixed
the incident field on an interface and described its further propagation in the
half-space.
In this chapter, we will go further and consider reflection and transmission
properties of the following physical systems:
− interface
− layer (2 subsequent interfaces)
− system of layers (arbitrary number of subsequent interfaces)
Aims
• We will study the interaction of monochromatic plane waves with arbitrary
multilayer systems  interferometers, dielectric mirrors, …
by superposition of such plane waves we can then describe interaction of
spatio-temporal varying fields with multilayer systems
• We will see a new effect, the “trapping” of light in systems of layers  new
types of normal modes in inhomogeneous space  “guided” waves
(propagation of confined light beams without diffraction)
Approach
• take Maxwell's transition condition for interfaces
• calculate field in inhomogeneous media  matrix method
• solve reflection-transmission problem for interface, layer, and system of
layers,
• apply the method to consider special cases like Fabry-Perot-interferometer,
1D photonic crystals, waveguide…
Background
• orthogonality of normal modes of homogeneous space
 no interaction of normal modes in homogeneous space
• inhomogeneity breaks this orthogonality
 modes interact and exchange energy
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• however, locally the concept of eigenmodes is still very useful and we will
see that the interaction at the inhomogeneities is limited to a small number
of modes
7.1.2 Decoupling of the vectorial wave equation
Before we will start treating a single interface, it is worth looking again at the
wave equation in homogeneous space in frequency domain
ω2
rotrot E(r, ω) − 2 E(r, ω)= iωµ 0 j(r, ω) + µ 0ω2 P (r, ω)
c
In general, for isotropic media all field components are coupled due to the
rot rot operator. However, for problems with translational invariance in at least
one direction (homogeneous infinite media, layers or interfaces) a
simplification is possible. Let us assume, e.g. translational invariance of the
system in y direction and propagation in the x-z-plane  ∂ / ∂y =0
 ∂∂x

rot rot E grad
=
div E − ∆ E 
=
∂
 ∂z
(
(
)
+ ∂∂Ez z   ∆ (2) E x 
 

 −  ∆ (2) E y 
0
  (2) 
∂Ex
∂Ez
∆ E z 
+
∂x
∂z 
 
∂Ex
∂x
)
Then, we can split the electric field as =
E E⊥ + E with
 0 
 
E⊥ =  E y  ,
 0 
 
 Ex 
 
E =  0  , ∇ (2) =
 Ez 
 
 ∂ ∂x 
2
2
 0  , ∆ (2) = ∂ + ∂


∂x 2 ∂z 2
 ∂ ∂z 


E⊥ is polarized perpendicular to the plane of propagation, E is polarized
parallel to this plane.
Common notations are:
perpendicular: ⊥
 s  TE (transversal electric)

parallel:
 p  TM (transversal magnetic)
Both components are decoupled and can be treated independently:
∆ (2) E⊥ (r, ω) +
ω2
E⊥ (r, ω) =−iωµ 0 j⊥ (r, ω) − µ 0ω2 P ⊥ (r, ω)
2
c
ω2
∆ E (r, ω) + 2 E (r, ω) − grad (2) div (2) E =−iωµ 0 j (r, ω) − µ 0ω2 P  (r, ω)
c
(2)
From
H (r, ω) =−
i
rot E(r, ω)
ωµ 0
we can conclude that the corresponding magnetic fields are
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 0 
=
ETE =
Ey 
 0 
 
 Ex 
=
ETM  0  ,
E 
 z
150
 Hx 
=
H TE  0 
H 
 z
0
 E ,
 
0
 
 0 
=
H TM =
H y 
 0 


0
H 
 
0
 
7.1.3 Interfaces and symmetries
Up to now we treated plane waves of the form
=
E(r, t ) E exp i ( kr − ω t ) 
− homogeneous space implies:
− monochromaticity leads to:
exp ( ikr )
exp ( −iωt )
Now, we will break the homogenity in x-direction by considering an interface
in y-z – plane which is infinite in y and z
x
medium I
interface
medium II
y
z
k
W.l.o.g. we can assume
k = ( kx , 0, kz )
by choosing an appropriate coordinate system (plane of incidence is the x, z plane), and then the problem does not depend on the y -coordinate.
As pointed out before, we can split the fields in TE and TM polarization
E = ETE + ETM and treat them separately.
We still have homogeneity in z -direction, and therefore we expect solutions 
exp (ikz z ) . The wave vector component kz has to be continuous at the
interface (follows strictly from continuity of transverse field components, see
7.1.4.). Therefore, we can write for the electric field:
E( x, z , t ) ETE ( x ) exp i ( kz z − ωt )  + ETM ( x ) exp i ( kz z − ωt ) 
=
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7.1.4 Transition conditions
From Maxwell’s equations follow transition conditions for the field
components. Here we will use that Et, Ht (transverse components) are
continuous at an interface between two media. This implies for the:
A) Continuity of fields
TE: E = E y and H z continuous
TM: Ez and H = H y continuous
B) Continuity of wave vectors
homogeneous in z-direction  phase eikz z  k z continuous
7.2 Fields in a layer system  matrix method
We will now derive a quite powerful method to compute the electromagnetic
fields in a system of layers with different dielectric properties.
7.2.1 Fields in one homogeneous layer
Let us first compute the fields in one homogeneous layer of thickness d and
dielectric function εf ( ω)
• aim: for given fields at x = 0  calculate fields at x = d
• strategy:
− Do computation with transverse field components (because they are
continuous).
− The normal components can be calculated later.
We will assume monochromatic light (one Fourier component, E ( x, z; ω)
H ( x, z; ω) ) and in the following we will often omit ω in the notation.
TE-polarization
We have to solve the wave equation (no y -dependence because of
translational invariance):
 ∂2
∂ 2 ω2 
+
+ 2 εf  ETE ( x, z ) =
0
 2
2
∂
x
∂
z
c


We use the ansatz from above:
ETE ( x, z ) = ETE ( x)exp ( ikz z ) and HTE ( x, z ) = HTE ( x)exp ( ikz z )
 ∂ 2 ω2
2
 2 + 2 εf − kz  ETE ( x) =0
c
 ∂x

with: H TE ( x, z ) = −
i
rotETE ( x, z )
ωµ0
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Now let us extract the equations for transversal fields Ey = E , H z :
 ∂2

ω2
2
2
0 with kfx ( kz , ω)= 2 εf ( ω) − kz2
 2 + kfx ( kz , ω)  E ( x) =
c
 ∂x

H z ( x) = −
i ∂
E ( x)
ωµ0 ∂x
This makes sense since the wave equation for the y-component of the
electric field is a second order differential equation. Hence we need to specify
the field and its first derivative as initial condition at x = 0 to determine a
unique solution.
TM-polarization
analog for transversal components H y = H , Ez :
 ∂2

2
+
ω
k
k
,
0
)
(
 2
 H ( x) =
fx
z
∂
x


Ez ( x) =
i
∂
H ( x)
ωε0εf ∂x
Again, we succeed describing everything in transversal components.
Now we have the following problem to solve:
− calculate fields (E, H) and derivatives
∂
∂
E ( x), H ( x) at x = d for
∂x
∂x
given values at x = 0
− calculate the fields at x = d
− at the end: HTM  ETM  E = ETM + ETE
Because the equations for TE and TM have identical structure, we can treat
them simultaneously. We rename
E, H → F
generalized field 1
iωµ0 H z , − iωε0 Ez → G
generalized field 2
and write down the problem to solve:
 ∂2

2
+
ω
k
k
,
0
(
)
 2
 F ( x) =
fx
z
∂
x


G ( x) = α f
∂
F ( x)
∂x
with α fTE =1, α fTM =
1
εf
We know the general solution of this system (harmonic oscillator equation):
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153
F=
( x) C1 exp ( ikfx x ) + C2 exp ( −ikfx x )
∂
G ( x) =
αf
F ( x) =
iα f kfx C1 exp ( ikfx x ) − C2 exp ( −ikfx x ) 
∂x
We have as initial conditions F (0), G (0) given:
F (0)
= C1 + C2
G (0) =
iα f kfx [C1 − C2 ]
from which we can compute the constants C1 , C2 :
=
C1

1
i
F
(0)
−
G
(0)


2
α f kfx

=
C2

1
i
G (0) 
 F (0) +
2
α f kfx

The final solution of the initial value problem is therefore:
=
F ( x) cos ( kfx x ) F (0) +
1
sin ( kfx x ) G (0)
α f kfx
G ( x) = −α f kfx sin ( kfx x ) F (0) + cos ( kfx x ) G (0)
By resubstituting we have the electromagnetic field in the layer 0 ≤ x ≤ d .
7.2.2 The fields in a system of layers
In the previous subchapter we have seen how to compute the electromagnetic field in a single dielectric layer, dependent on the transverse field
components E y , H z (TE) and H y , Ez (TM) at x = 0 . We can generalize our
results to systems of dielectric layers, which are used in many optical
devices:
− Bragg mirrors
− chirped mirrors for dispersion compensation
− interferometer
− multi-layer waveguides
− Bragg waveguide
− metallic interfaces and layers
We can even go further and “discretize” an arbitrary inhomogeneous (in one
dimension) refractive index distribution. This is important for so-called 'GRIN'
- Graded-Index-Profiles.
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154
From above, we know the fields in one layer:
F ( x) cos ( kfx x ) F (0) +
=
1
sin ( kfx x ) G (0)
α f kfx
G ( x) = −α f kfx sin ( kfx x ) F (0) + cos ( kfx x ) G (0)
We can write this formally in matrix notation as
 F ( x) 
 F (0) 
ˆ ( x) 

=m
,
G
(
x
)
G
(0)




where the 2x2-matrix m̂ describes propagation of the fields:

 cos ( kfx x )
ˆ ( x) = 
m
−k α sin ( k x )
fx
 fx f
1

sin ( kfx x ) 
kfx αf

cos ( kfx x ) 
− To compute the field at the end of the layer we set x = d .
ˆ ( x) = 1.
− We assume no absorption in the layer  m
− A system of layers is characterized by εi , d i
If multiple layers are considered, the fields between them connect continuously since the field components used for the description of the fields are
the continuous tangential components.
Hence, we can directly write the formalism for a multilayer system since it just
requires matrix multiplications:
A) Two layers
F
F
F
ˆ
ˆ
ˆ
d
d
d
=
m
=
m
m
(
)
(
)
(
)
2
2 
2
2
1
1 
G


 d1 + d2
 G d1
 G 0
B) N layers
F
=

 G d + d +..+ d =D
1
2
N
N
F
F
ˆ
ˆ
ˆ
ˆ i (di )
=
m i (di )   M   with M = ∏m
∏
G
G
i =1
i =1
 0
 0
N
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i
ˆ i have the same form, but different α if , di , k=
All matrices m
fx
155
ω2
ε ω − kz2 .
2 i( )
c
Summary of matrix method
− F (0) and G (0) given ( E , H z for TE, E z , H for TM)
− kz , αif , εi , di given  matrix elements
− multiplication of matrices (in the right order)  total matrix
− fields F ( D) and G ( D)
7.3 Reflection – transmission problem for layer systems
7.3.1 General layer systems
7.3.1.1
Reflection- and transmission coefficients  generalized
Fresnel formulas
In the previous chapter, we have learned how to link the electromagnetic field
on one side of an arbitrary multilayer system with the field on the other side.
We have seen that after splitting in the TE/TM-polarizations, continuous
(transversal) field components are sufficient to describe the whole field. What
we will do now is to link those field components with the fields, which are
accessible in an experimental configuration, i.e. incident, reflected, and
transmitted fields. In particular, we want to solve the reflection-transmission
problem, which means that we have to compute reflected and transmitted
fields for a given angle of incidence, frequency, layer system and polarization.
We introduce the wave vectors of the incident ( kI ) , reflected ( k R ) and
transmitted ( k T ) fields:
 ksx 
k I =
kR
=
0  ,
k 
 z
with k=
sx
ω2
2
ε − k=
z
2 s
c
 −ksx 
 0 ,
kT
 =

 k 
 z 
ks2 (ω) − kz2 ,
k=
cx
 kcx 
 0 


k 
 z
ω2
2
ε − k=
z
2 c
c
kc2 (ω) − kz2 ,
where εs (ω) and εc (ω) are the dielectric functions of the substrate and
cladding and k z is the tangential component of the wave vector which is
continuous throughout the layer system.
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156
Multi layer system
As we have seen before, the k z component of the wave vector is conserved
and ± kx determines the direction of the wave (forward or backward). The
total length of the wave vector in each layer is given by the dispersion relation
for dispersive, isotropic, homogeneous media. As a consequence, the k x
component changes its value in each layer.
Remark on law of reflection and transmission (Snellius)
It is possible to derive Snellius law just from the fact that kz is a conserved
quantity:
1. ks sin ϕI =ks sin ϕ R  ϕI =ϕ R (reflection)
2. ks sin=
ϕI kc sin ϕT  ns sin=
ϕI nc sin ϕT (Snellius)
Let us now rewrite the fields in order to solve the reflection transmission
problem:
A) Field in substrate
The fields in the substrate can be expressed based on the complex
amplitudes of the incident FI and reflected field FR as:
=
Fs ( x, z ) exp ( ikz z )  FI exp ( iksx x ) + FR exp ( −iksx x ) 
Gs ( x, z ) =
iαsksx exp ( ikz z )  FI exp ( iksx x ) − FR exp ( −iksx x ) 
B) Field in layer system
The fields inside the layer system can be expressed as
Ff ( x, z ) = exp ( ikz z ) F ( x)
Gf ( x, z ) = exp ( ikz z ) G ( x)
where the amplitudes F ( x) and G ( x) are given by matrix method as
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157
F
ˆ ( x)  F 
=
M
G 
G 
 x
 0
C) field in cladding
The fields in the cladding can be expressed based on the complex amplitude
of the transmitted field FT as:
=
Fc ( x, z ) exp ( ikz z ) FT exp ikcx ( x − D ) 
Gc ( x, z ) =
iαc kcx exp ( ikz z ) FT exp ikcx ( x − D ) 
Note that in the cladding we consider a forward (transmitted) wave only.
Reflection transmission problem
We want to compute FR and FT for given FI , kz (  sin ϕI ), εi , d i . We know that
F and G are continuous at the interfaces, in particular at x = 0 and x = D .
We have:
F
F
ˆ
 G  = M ( D)  G  .
 D
 0
Field in cladding at x = D
field in substrate at x = 0
On the other hand, we have expressions for the fields at x = 0 and x = D from
our decomposition in incident, reflected and transmitted field from above.
Hence, we can write:
FI + FR
FT


  M 11 ( D) M 12 ( D)  
=
 iα k F   M ( D) M ( D)   iα k F − F  .
R )
22
  s sx ( I
 c cx T   21
We consider FI as known, and FR and FT as unknown and get:
FR =
( αsksx M 22 − αc kcx M 11 ) − i ( M 21 + αsksxαc kcx M 12 ) F
I
αsksx M 22 + α c kcx M 11 ) + i ( M 21 − αsksx α c kcx M 12 )
((((((((((
(((((((((
N
FT =
2α s ksx ( M 11M 22 − M 12 M 21 )
FI
(α s ksx M 22 + α c kcx M 11 ) + i ( M 21 − α s k sx α c kcx M 12 )
FT =
2α s ksx
FI
N
These are the general formulas for reflected and transmitted amplitudes.
Please remember that the matrix elements depend on the polarization
direction  M ijTE ≠ M ijTM .
Let us now transform back to the physical fields, and write the solution for the
results of the reflection transmission problem for TE and TM polarization:
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158
A) TE-polarization
F = E = Ey ,
αTE = 1
i) reflected field
ERTE = RTE EITE
with the reflection coefficient
RTE
( k M − k M ) − i( M + k k M )
=
k M + k M ) + i( M − k k M )
((((((((
(((((((((

sx
TE
22
sx
TE
22
cx
cx
TE
11
TE
21
TE
11
TE
21
sx cx
sx cx
TE
12
TE
12
N TE
ii) transmitted field
ETTE = TTE EITE
with the transmission coefficient
TTE
2ksx
2ksx
,
=
TE
TE
TE
TE
( ksx M 22 + kcx M11 ) + i( M 21 − ksx kcx M12 ) NTE
 We get complex coefficients for reflection and transmission, which
determine the amplitude and phase of the reflected and transmitted
light.
B) TM-polarization
F = H = Hy,
αTM =
1
.
ε
In the case of TM polarization we have the problem that an analog calculation
to TE would lead to H R,T / H I , i.e., relations between the magnetic field.
TM
However, we want ER,T
/ EITM . Therefore, we have to convert the H -field to the
E TM -field:
As can be seen in the figure, we can express the amplitude of the ETM field in
terms of the Ex component:
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159
Ex
k
= − sin ϕ = − z
TM
E
k
k
 E TM = − Ex ,
kz
With Maxwell we can link Ex to H y :
E= −
1
1
kz H y 
( k × H )  Ex =
ωε0ε
ωε0ε
result:
ERTM,T
εs H R ,T
, →
εs,c H I
=
TM
I
E
k
1
E TM =
−
Hy =
−
Hy
ωε0ε
cε0 ε
εs / εc relevant for transmission only
Hence we find the following for TM polarization:
ERTM = RTM EITM
with the reflection coefficient
RTM
( ε k M − ε k M ) − i( ε ε M + k k M )
=
ε k M + ε k M ) + i( ε ε M − k k M )
((((((((((
(((((((((
TM
22
c sx
TM
22
c sx
s cx
s cx
TM
11
s c
TM
11
s c
TM
21
TM
21
sx cx
sx cx
TM
12
TM
12
N TM
ETTM = TTM EITM
with the transmission coefficient
2 εsεc ksx
2 εsεc ksx
=
N TM
( εcksx M 22TM + εskcx M11TM ) + i( εsεc M 21TM − ksx kcx M12TM )
TTM
In summary, we have found different complex coefficients for reflection and
transmission for TE and TM polarization. The resulting generalized Fresnel
formulas for multilayer systems are
RTE
(k
=
(k
sx
sx
TTE =
RTM
TE
TE
M 22
− kcx M 11TE ) − i ( M 21
+ ksx kcx M 12TE
M
2ksx
N TE
(ε k
=
(ε k
TTM =
c sx
c sx
TE
22
+ kcx M
TE
11
) + i( M
TE
21
− ksx kcx M
TE
12
)
)
TM
TM
M 22
− εskcx M 11TM ) − i ( εsεc M 21
+ ksx kcx M 12TM )
TM
TM
M 22
+ εskcx M 11TM ) + i ( εsεc M 21
− ksx kcx M 12TM )
2 εsεc ksx
.
N TM
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7.3.1.2
Reflectivity and transmissivity
160
In the previous chapter we have computed the coefficients of reflection and
transmission, which relate the electric fields in TE and TM polarization of
incident, reflected and transmitted wave. However, in many situations it is
more important to know the relation of energy fluxes, the so called reflectivity
and transmissivity. In order to get information on these quantities we have to
compute the energy flux perpendicular to the interface:
− flux through a surface with x = const
1
ℜ ( E × H∗ ) ex
2
S e x=
With
=
H∗
1
k ∗ × E∗ )
(
ωµ 0
we find
1
1
2
2
S e x = ℜ ( k ∗e x ) E = ℜ ( kx ) E .
2ωµ 0
2ωµ 0
Since in an absorption free medium the energy flux is conserved, in an
absorption free layer the energy flux is also conserved.
In the substrate
k=
sx
ω2
2
ε − k=
z
2 s
c
ks2 (ω) − kz2
is supposed to be real-valued, because we have our incident wave coming
from there. The total energy flux from the substrate to the layer system is
given as
=
S s ex
1 
2
2
ksx EI − ksx ER 

2ωµ 0 
In contrast, in the cladding
k=
cx
ω2
2
ε − k=
z
2 c
c
kc2 (ω) − kz2
may be complex-valued. The energy flux from the layer system into the
cladding is
S=
e
c x
1
2
ℜ ( kcx ) ET .
2ωµ 0
Because we have energy conservation: S s e x = S c e x 
2
2
E
=
ER +
I
ℜ ( kcx )
2
ET
ksx
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161
Now we will compute the global reflectivity ρ and transmissivity τ of a layer
system. Of course, we will decompose into TE and TM polarizations and
relate to the reflectivities ρTE,TM and transmissivities τ TE,TM . We know:
TM
ER =
ETE
R + ER ,
2
2
2
ET =
ETTE + ETTM
+ ETM
+
EI = ETE
R
R
ℜ ( kcx ) TE 2
ET + ETTM
ksx
(
2
)
2
2


ℜ ( kcx )
ℜ ( kcx )
2
2
2
2
TTE  EITE +  RTM +
TTM  EITM .
=  RTE +
ksx
ksx




Here, we just substituted the reflected and transmitted field amplitudes by
incident amplitudes times Fresnel coefficients. Now, we decompose the
incident field as follows:
EI cos δ,
EITE =
EI sin δ.
EITM =
2
Then, we can divide by the (arbitrary) amplitude EI and write


ℜ ( kcx )
ℜ ( kcx )
2
2
2
2
=
1  RTE +
TTE  cos 2 δ +  RTM +
TTM  sin 2 δ
ksx
ksx




ℜ ( kcx )
2
2
2
2
=
1 RTE cos 2 δ + RTM sin 2 δ +
TTE cos 2 δ + TTM sin 2 δ
ksx
(
)
The red and blue terms can be identified as
(
1= ρ + τ
The global reflectivity and transmissivity are therefore given as
2
2
ρ = ρδ
+ ρδ
TE cos
TM sin
2
2
τ = τδ
+ τδ
TE cos
TM sin
with the reflectivities
2
=
ρTE,TM RTE,TM ,
=
τTE,TM
ℜ ( kcx )
2
TTE,TM .
ksx
for the two polarization states TE and TM.
)
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162
7.3.2 Single interface
7.3.2.1
(classical) Fresnel formulas
Let us now consider the important example of the most simple layer system,
namely the single interface. The relevant wave vectors are (as usual):
 ksx 
k I =
kR
=
0  ,
k 
 z
 −ksx 
 0 ,
kT
 =

 k 
 z 
 kcx 


 0 .
k 
 z
The continuous component of the wave vector, expressed in terms of the
angel of incidence, is
k=
z
ω
ω
εs sin ϕ=
ns sin ϕI.
I
c
c
Then, the discontinuous component is given as
 k=
ix
=
ksx
ω2
2
ε − k=
z
2 i
c
ω
ns cos ϕI ,
c
ω2
ω2
ω 2
ε − 2 εs sin 2 ϕ=
ni − ns2 sin 2 ϕI
I
2 i
c
c
c
=
kcx
ω 2
ω
ϕI
nc − ns2 sin 2=
nc cos ϕT ,
c
c
As above, we can assume that ksx is always real, because otherwise we
have no incident wave. kcx is real for nc > ns sin ϕI , but imaginary for
nc < ns sin ϕI (total internal reflection).
The matrix for a single interface is the unit matrix
1 0
ˆ= m
ˆ (d= 0)= 
M

0 1
and it is easy to compute coefficients for reflection and transmission, and
reflectivity and transmissivity. Using the formulas from above we find:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
A) TE-polarization
RTE =
( ksx M 22 − kcx M 11 ) − i ( M 21 + ksx kcx M 12 ) ,
ksx M 22 + kcx M 11 ) + i ( M 21 − ksx kcx M 12 )
((((((((
(((((((
TTE =
2ksx
N TE
RTE
=
ksx − kcx )
(=
( ksx + kcx )
TTE
=
2ksx
2ns cos ϕI
2ns cos ϕI
=
=
( ksx + kcx ) ns cos ϕI + nc2 − ns2 sin 2 ϕI ns cos ϕI + nc cos ϕT
ns cos ϕI − nc2 − ns2 sin 2 ϕI ns cos ϕI − nc cos ϕT
=
2
ns cos ϕI + nc − ns2 sin 2 ϕI ns cos ϕI + nc cos ϕT
2
ρTE
= RTE=
=
τTE
ksx − kcx
ksx + kcx
2
2
ℜ ( kcx )
4ksxℜ ( kcx )
2
=
TTE
.
2
ksx
ksx + kcx
 ρTE + τTE =1
B) TM-Polarisation
RTM =
( εc ksx M 22 − εskcx M 11 ) − i ( εsεc M 21 + ksx kcx M 12 )
εc ksx M 22 + εskcx M 11 ) + i ( εsεc M 21 − ksx kcx M 12 )
(((((((((
((((((((((

TTM =
2 εsεc ksx
.
N TM
1 0
ˆ= m
ˆ (d= 0)= 
with: M

0 1
=
RTM
ksx εc − kcx εs )
(=
( ksxεc + kcxεs )
nsnc2 cos ϕI − ns2 nc2 − ns2 sin 2 ϕI nc cos ϕI − ns cos ϕT
=
nsnc2 cos ϕI + ns2 nc2 − ns2 sin 2 ϕI nc cos ϕI + ns cos ϕT
2ksx εcεs
2ns2 nc cos ϕI
2ns cos ϕI
=
TTM =
=
,
( ksxεc + kcxεs ) nsnc2 cos ϕI + ns2 nc2 − ns2 sin 2 ϕI nc cos ϕI + ns cos ϕT
2
ρTM
= RTM=
=
τTM
ksx εc − kcx εs
ksx εc + kcx εs
2
2
,
ℜ ( kcx )
4ksxℜ ( kcx ) εsεc
2
=
TTM
2
ksx
ksx εc + kcx εs
 ρTM + τTM =1
163
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164
Remark
It may seem that we have a problem for ϕI =0 . For ϕI =0 , TE and TM
polarization should be equivalent, because the fields are always
polarized parallel to the interface. However, formally we have RTE = − RTM ,
TTE = TTM . The “strange” behavior of the coefficient of reflection can be
explained by the following figures:
7.3.2.2
Total internal reflection (TIR) for εs>εc
Let us now consider the special case when all incident light is reflected from
the interface. This means that the reflectivity is unity.
2
ρTE
With kcx =
with ρTE,TM
2
k − kcx
=sx
2
ksx + kcx
ρTM
k ε −k ε
=sx c cx s 2
ksx εc + kcx εs
ω 2
nc − ns2 sin 2 ϕI we can compute the smallest angle of incidence
c
=
1:
=
kcx 0 
=
nc ns sin ϕItot
nc
sin ϕItot =
.
ns
For angles of incidence larger than this limit angle, ϕI > ϕItot we have
kcx
ω 2 2
ω2
2
2
=i
ns sin ϕI − nc = iµc = i k z − 2 εc  iµaginary
c
c
0  TIR
→ ℜ ( kcx ) =
Obviously, we find the same angle of TIR for TE and TM polarization. The
energy fluxes are given as (here TE, same result for TM):
ksx − iµc
=
1
2
ksx + iµc
2
=
ρTE
=
τTE
4ksxℜ ( kcx )
=
0.
2
ksx + kcx
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165
Remark
For metals in visible range (below the plasma frequency) we have
ω
εc − ns2 sin 2 ϕI always
always TIR, because: ℜ ( εc ) < 0 → kcx=
c
imaginary.
In the case of TIR the modulus of the coefficient of reflection is one, but
the coefficient itself is complex  nontrivial phase shift for reflected light:
A) TE-polarization
exp ( ia )
k − iµc Z
RTE =⋅
1 exp iΘ TE = sx
=∗=
=exp ( 2ia )
ksx + iµc Z
exp ( −ia )
(
)
Θ TE
ns2 sin 2 ϕI − nc2
sin 2 ϕI − sin 2 ϕItot
µc
 tan a = tan
=−
=−
=−
.
2
ksx
ns cos ϕI
cos ϕI
B) TM-polarization
exp ( ia )
k e − iµces Z
=∗=
=exp ( 2ia )
RTM =⋅
1 exp iΘ TM = sx c
ksx ec + iµces Z
exp ( −ia )
)
(
 tan a = tan
Θ TM
=−
2
Θ
µ c es e s
= tan TE ,
ksx ec ec
2
In conclusion, we have seen that the phase shifts of the reflected light at TIR
is different for TE and TM polarization, and because ε s > εc
Θ TM > Θ TE ,
As a consequence, incident linearly polarized light gets generally elliptically
polarized after TIR  Fresnel prism
Remark
x ) exp ( −µc x ) .
The field in the cladding is evanescent − exp ( ikxc=
 The averaged energy flux in the cladding normal to the interface
vanishes.
S
7.3.2.3
x
(
1
2
=
ℜ kE
2ωµ 0
)
x
1
2
=
ℜ ( kx ) E =
0.
2ωµ 0 
The Brewster angle
=0
There exists another special angle with particular reflection properties. For
TM-polarization, for incident light at the Brewster angle ϕ B we find RTM = 0 :
2
=
ρTM
ksx εc − kcx εs
=
0,
2
ksx εc + kcx εs
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166
 ksx εc = kcx εs
εc2 ( εs − sin 2 ϕBεs ) =εs2 ( εc − sin 2 ϕBεs )
=
sin 2 ϕB
εsεc ( εs − εc )
=
εs ( εs2 − εc2 )
cos 2 ϕB = 1 − sin 2 ϕB = 1 −
εc
( εs + ε c )
εc
εs
=
( εs + ε c ) ( ε s + ε c )
With the last two lines we can write the final result for the Brewster angle:
tan ϕB =
εc
.
εs
The Brewster angle exists only for TM polarization, but for any ns  nc .
There is a simple physical interpretation, why there is no reflection at the
interface for the Brewster angle.
tan=
ϕB
sin ϕB nc
=
cos ϕB ns
π

ϕB nc cos=
ϕB nc sin  − ϕB 
 ns sin=
2

At the same time the angle of the transmitted light is always
ns sin ϕB= nc sin ϕT  ϕT=
π
− ϕB ,
2
Hence, at Brewster angle reflected and transmitted wave propagate in perpendicular directions. If we interpret the reflected light as an emission from
oscillating dipoles in the cladding, no reflected wave can occur for TM polarization (no radiation in the direction of dipole oscillation).
In summary, we have the following results for reflectivity and transmittivity at
a single interface with ε s > εc .
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167
total internal reflection
angle of incidence
total internal reflection
angle of incidence
7.3.2.4
The Goos-Hänchen-Shift
The Goos-Hänchen shift is a direct consequence of the nontrivial phase shift
of the reflected light at TIR. It appears when beams undergo total internal
reflection at an interface. The reflected beam appears to be shifted along the
interface. As a result it seems as if the beam penetrates the cladding and
reflection occurs at a plane parallel to the interface at a certain depth, the socalled penetration depth. For sake of simplicity we will treat here TE-polarization only.
Let us start with an incident plane wave in TE polarization:
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with
ω
α = ns sin ϕI , =
γs
c
ω2 2
n − α2.
2 s
c
This gives rise to a reflected plane wave as
=
ER ( x, z ) EI exp i ( αz − γs x )  exp iΘ ( α )  .
The reflected plane wave gets a phase shift Θ(α) , which depends on the
angle of incidence (here characterized by the transverse wave number α).
Now we want to treat beams, which we can write as a superposition of plane
waves (Fourier amplitude eI ( α ) ):
EI ( x, z )=
∫d αe ( α ) exp i( αz + γ ( α ) x )
I
s
We assume a mean angle of incidence ϕI0 :
α 0 = ωc ns sin ϕI0 mean angular frequency
α = α0 + ε
In the Fourier integral, we have to integrate over angular frequencies with
non-zero amplitudes eI ( α ) only ( eI ( α 0 + e ) ≠ 0 for −∆ ≤ ε ≤ ∆ only)
∆
EI (=
x, z ) exp ( iα 0 z ) ∫ d eeI ( α 0 + ee
) exp i ( z + γs x )
−∆
∆
ER (=
x, z ) exp ( iα 0 z ) ∫ d eeI ( α 0 + ee
) exp i ( z − γs x ) exp iΘ ( α0 + e ) .
−∆
Let us make the following assumptions:
− small divergence of beam (narrow spectrum, ∆ 
− all Fourier component undergo TIR ( Θ ( α ) > Θ Itot )
ω
ns )
c
Then, it is justified to expand the phase shift Θ(α) of the reflected wave into a
Taylor series up to first order:
Θ ( α0 + ε ) ≈ Θ ( α0 ) +
∂Θ
ε =Θ ( α 0 ) + Θ′ε
∂α α0
Then, the reflected beam at the interface at x = 0 is given as:
{
}∫
ER=
(0, z ) exp i α 0 z + Θ ( α 0 ) 
∆
d eeI ( α 0 + e ) exp i ( z + Θ′ ) e 
−∆
We can identify the remaining integral as a shifted version of the incident
beam profile at x = 0
{
}
ER=
(0, z ) exp i Θ ( α 0 ) − α 0Θ′ EI ( z + Θ′ ) .
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Thus, the reflected beam appears shifted by d = −Θ′ (Goos-Hänchen Shift).
Let us finally compute the shift d = −Θ′ . We know from before that the phase
shift for TIR is given as:
tan
Θ TE
2
=−
µc
µ
=− c
ksx
γs
a 2 − ωc2 nc2
µc
Θ = −2arctan
= −2arctan 2
ω
ksx
n2 − a2
c2 s
2
α ( ksx2 + µ c2 )
−2α
2α
µc
ksx −
∂Θ
µ c ksx
1
2µ c
2ksx
1 α
Θ′ =
=−2 ×
×
=−2
=−2
2
2
2
2
µ
∂α
µ c ksx
ksx
ksx + µ c
1 + 2c
ksx
Θ′ a =−d =−2 xET tan ϕI0 with x=
ET
0
1
=
µc
1
α 02 − ωc2 nc2
2
and tan ϕI0 =
aa
=
ksx γ s
 xET depth of penetration
7.3.3 Periodic multi-layer systems - Bragg-mirrors - 1D
photonic crystals
In the previous chapters we have learned how to treat (finite) arbitrary multilayer systems. Interesting effects occur when these multi-layer systems become periodic, so-called Bragg-mirrors. The reflectivity of such mirrors is
almost 100% in certain frequency ranges; the more layers the closer we get
to this ideal value. Bragg mirrors are important for building resonators (laser,
interferometer). Furthermore periodic structures are of general importance in
physics (lattices, crystals, atomic chains …). We can learn many things about
the general features of such periodic systems by looking at the optical
properties of periodic (dielectric) multi-layer systems
In our theoretical approach, we will assume these layer systems as infinite,
i.e. consisting of an infinite number of layers, and we treat them as so-called
one-dimensional photonic crystals. We will discuss effects like band gaps,
dispersion and diffraction in such periodic media, and gain understanding of
the basics of Bragg reflection and the physics of photonic crystals.
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In order to keep things simple we will treat:
− semi-infinite periodic multi-layer systems  x > 0, ( ε1 , d1 ) , ( ε 2 , d 2 ) 
− TE-polarization only
− monochromatic light
At the interface between substrate and Bragg-mirror ( x = 0 ) we have incident
and reflected electric field:
E=
ER + EI
0
and
∂E
′ iksx ( EI − ER )
= E=
0
∂x 0
Alternatively the incident and reflected field can be expressed by the field at
the interface and its derivative as
E
=
I
E0 iE0′
−
2 2ksx
E=
R
and
E0 iE0′
+
2 2ksx
In chapter 7.2, we developed a matrix formalism involving the generalized
fields F and G . Because here we treat TE polarization only, we can use
directly the electric field amplitude and its derivative with respect to x , because E = F and
iωµ 0 H=
G=
z
∂E
= E′ .
∂x
Let us now calculate the field in the multi-layer system. From before, we know
how to treat finite systems with the matrix method. Here, we want to treat an
infinite periodic medium (like a one-dimensional crystal).
As a particular example we will investigate an infinite system consisting of just
two periodically repeated layers. The two layers should consist of
homogeneous material with ε1 and ε 2 having a thickness of d1 and d 2 ,
respectively. Hence we have:
ε( x) = ε( x + Λ ) with the period Λ= d1 + d 2
For infinite periodic media, we can make use of the Bloch-theorem to find the
generalized normal modes (Bloch modes or Bloch waves). We seek for
solutions like:
}
{
=
E ( x, z; ω) exp i  kx ( kz , ω) x + kz z  Ekx ( x)
with
Ekx ( x + Λ ) = Ekx ( x)
In other words, Ekx ( x) is a periodic function and we are looking for solutions
E ( x, z; ω) which have the same amplitude after one period of the medium, but
{
}
we allow for a different phase  exp i  kx ( kz , ω) x  . Here kx is the (yet)
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unknown Bloch vector. Because in this easy example we deal with a onedimensional problem, the Bloch vector is actually a scalar.
In the following, we will find a dispersion relation for the Bloch-waves
ω2
2
kx ( kz , ω) , in complete analogy to the DR for plane waves kx= 2 ε ( ω) − kz2 in
c
homogeneous media. In order to make the difference to the homogeneous
case more obvious, we change the notation for the Bloch vector: k x  K .
According to the Bloch-theorem (our ansatz) we have a relation between E
and E ′ when we advance by one period of the multi-layer system (from
period N to period N + 1 ):
E
E
exp ( iK Λ )  '  .
 E '=

 ( N +1)Λ
 E  NΛ
On the other hand, we know from our matrix method:
E
ˆ E
=
M
 E' 
 E' 
 ( N +1)Λ
  NΛ
ˆ =m
ˆ ( d2 ) m
ˆ ( d1 )  M ij = ∑ kmik(2) mkj(1)
with M
If the Bloch wave is a solution to our problem, we can set the two expressions
equal:
{Mˆ − exp (iK Λ ) I} EE 
'
=
0
NΛ
=
µ exp (iK Λ ) we have to solve the following eigenvalue problem:
and with
{Mˆ − µ( K )I} EE 
'
0
=
NΛ
This eigenvalue problem determines the Bloch vector K and will finally give
our dispersion relation.
ˆ − µI} =
As usual, we use the solvability condition det{M
0 to compute the dispersion relation expressed=
in µ exp (iK Λ ) . Hence we still need to compute
K afterwards.
=
µ ± exp ( i=
K±Λ )
{ }
( M 11 + M 22 ) ±
2
2
 ( M 11 + M 22 ) 

 − 1.
2


ˆ = 1 , which explains why the off-diagonal elements
Note that we used det M
{ }
ˆ =1
of the matrix do not appear in the formula. Moreover, because of det M
we have µ +µ − =1 .
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The corresponding eigenvectors (field and its derivative at x= N Λ ) can be
computed from
{Mˆ − exp (iK Λ ) I} EE 
'
=
0
NΛ
M 12  E 
M −µ
  11
=0
M 22 − µ  E ′  N Λ
 M 21
From the first row the following condition can be derived:
( M 11 − µ ) E + M 12 E′ = 0
Since the investigated system is linear and invariant for the phase, the
absolute amplitude and phased of the E -component of the eigenvector can
be chosen arbitrarily. Here we take E = 1 and get for the full eigenvector
1


E
=

 ENΛ .
 E' 
  NΛ ( µ − M 11 ) / M 12 
If field values of the Bloch mode, i.e. the function Ekx ( x + Λ ) = Ekx ( x ) , inside
the layers are desired, they can be computed by using the matrix formalism
and the above eigenvector ( E , E ′) N Λ .
Physical properties of infinite multilayer systems
We are interested in the reflection properties of an infinite Bragg mirror.
Reasoning in terms of the electric field and derivative at the interface ( x = 0 ),
E0 and E’0, we can express the reflectivity of the Bragg mirror as
E
ρ= R
EI
2
E0 iE0′
E0 iE0′
and E
from before
+
=
−
I
2 2ksx
2 2ksx
with E=
R
ksx E0 + iE0'
 ρ=
ksx E0 -iE0'
2
With our knowledge of the eigenvector from above we can compute the
reflectivity:
µ − M 11
2
ksx + i
'
ksx E0 + iE0
M 12
µ − M 11
=
=
E '0 =
E0  ρ
'
µ − M 11
ksx E0 −iE0
M 12
ksx −i
M 12
According to this formula two scenarios are possible:
A) total internal reflection r = 1
Hence µ has to be real which results in the condition
2
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
( M 11 + M 22 )
2
173
≥1
with [for our example (ε1 , ε 2 , d1 , d 2 ) ]
M 11 cos ( k1x d1 ) cos ( k2 x d 2 ) −
=
=
M 22 cos ( k1x d1 ) cos ( k2 x d 2 ) −
k2 x
sin ( k1x d1 ) sin ( k2 x d 2 )
k1x
k1x
sin ( k1x d1 ) sin ( k2 x d 2 ) .
k2 x
This defines the so-called band gap, i.e. frequencies of excitation for which no
propagating solutions exist.
B) propagating normal modes
Hence µ must be complex which results in the condition

( M 11 + M 22 )
2
<1
We can compute the explicit dispersion relation:
=
µ exp (=
iK Λ )
( M 11 + M 22 ) ±
2
2
 ( M 11 + M 22 ) 

 − 1.
2


=
µ exp ( iK=
Λ ) cos ( K Λ ) + i sin ( K=
Λ ) cos ( K Λ ) ±
{cos ( K Λ )}
2
−1
( M 11 + M 22 )
 cos { K ( kz , ω) Λ} =
2
In infinite periodic media, only if the Bloch vector K fulfills this DR the Bloch
wave is a solution to Maxwell’s equations. This is in complete analogy to
2
plane waves in homogeneous media with the DR kx2 = ωc2 ε ( ω) − kz2 .
Interpretation
• For the case of total internal reflection ( µ real,
( M 11 + M 22 )
2
≥ 1 ) the Bloch
=
µ exp ( i=
K Λ ) exp ( iℜ ( K ) Λ ) exp ( −ℑ ( K ) Λ ) .
vector K is complex,
Hence ℜ{ K ( kz , ω) Λ} = nπ and
2

 M +M

M
+
M


(
)
(
)
n


11
22
11
22

ℑ{ K ( kz , ω) Λ} =− ln ( −1) 
± 
− 1 .



2
2








The ± accounts for exponentially damped and growing solution, as we usually
expect in the case of complex wave vectors and evanescent waves.
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• There is an infinite number of so-called band gaps or forbidden bands,
because =
n 1...∞ . These band gaps are interesting for Bragg mirrors and
Bragg waveguides. The band gaps correspond to "forbidden" frequency
ranges, where no propagating solution exists.
• The limits of the bands are given by
ℜ{K ( kz , ω) Λ} = nπ and ℑ{K ( kz , ω) Λ} =0
 K ( kz , ω) = nπ / Λ
• Outside the band gaps, i.e., inside the bands, we find propagating solutions
which have different properties than the normal modes in homogeneous
media (different dispersion relation).
 We can exploit the strong curvature, i.e. frequency dependence, of DR
for, e.g., dispersion compensation or diffraction free propagation
Special case: normal incidence
In general there is a complex interplay between the angle of incidence and
frequency of light determining the reflection properties of multilayer systems.
Therefore let us have a look at the simpler case of normal incidence ( kz = 0 ) .
In a graphical representation of the dispersion relation for kz = 0 it is common
to use the following dimensionless quantities
2π
ω
K
and
with the scaling constant G =
Λ
cG
G
Examples for normal incidence
0.7
Re k
Im k
0.4
Re k
Im k
0.6
0.5
ω/cG
ω/cG
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
0
0.25
0.5
0.75
k/G
n1=1.4, d1=0.5Λ
n2=3.4, d2=0.5Λ
1
1.25
0
0
0.25
0.5
0.75
k/G
1
1.25
1.5
n1=1.4, d1 = 17/24 Λ
n2=3.4 , d2 =7/24 Λ
It is common to use the reduced band structure - Brillouin zone, where the
information for all possible Bloch vectors is mapped onto the Bloch vectors in
the following interval −0.5 ≤ (k / G ) ≤ 0.5 .
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1.2
Re k
Im k
0.4
175
1
0.3
ω/cG
ω/cG
0.8
0.2
0.6
0.1
0.4
0
0
0.25
0.5
0.75
k/G
1
1.25
0.2
n1=1.4, d1=0.5Λ
n2=3.4, d2=0.5Λ
0
Because of eiK Λ we need only −π ≤ K Λ ≤ π 
0
0.25
k/G
0.5
K
≤ 0.5 to describe the disperG
sion relation.
Inside the band gap, we find damped solutions:
1
n0=1.0, n1=1.4,
0.8
0.7
Transmission
1 6 d /d 1
0.9
0.6
0.5
0.4
0.3
0.2
0.1
0
field in a Bragg-mirror
0.3
0.4
0.5
ω/cG
0.6
0.7
Transmission
Let us quantify the damping. In our example (n1,n2,d1,d2) we have
( M 11 + M 22 ) cos  ω n d  cos  ω n d  − 1  n2 + n1  sin  ω n d  sin  ω n d 
=

  1 1 
2 2
2 2
 1 1

2
c

c
 2  n1 n2   c
 c

In the middle of the first band gap (optimum configuration for high reflection)
ωB
ωB
π
n1d1 =
n2 d 2
, with ωB being the Bragg frequency, and
we have =
c
c
2
( M 11 + M 22 ) = − 1  n2 + n1  < −1 . If we plug this (for n = 1 ) in our
therefore


2
2  n1 n2 
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176
expression for ℑ( K ) and assume a small index contrast n2 − n1 << ( n2 + n1 )
we find
Λℑ( K ) max ≈ 2
n2 − n1
(do derivation as an exercise)
n2 + n1
 Damping is proportional to index contrast of the subsequent layers n2 − n1
 ( M 11 + M 22 )

The spectral width of the gap 
≥ 1 is then
2


∆ωgaπ ≈
2ωB
Λℑ( K ) max
π
(do derivation as an exercise)
 Spectral width is proportional to index contrast as well.
7.3.4 Fabry-Perot-resonators
In this chapter we will treat a special multi-layer system, the so-called FabryPerot-resonator. To construct a Fabry-Perot-resonator, one can start from
highly reflecting periodic multi-layer system (Bragg reflector). If one changes
just a single layer somewhere in the middle of the otherwise periodic layer
system, a so-called cavity is formed, and we are interested in the forward and
backward propagating fields of the entire layer system.
While the single layer which is distinguished from the periodic stack forms the
cavity, the other layers function as mirrors, and may be periodic multilayersystems or metal films. Fabry-Perot-resonators are very important in optics,
as they appear as:
• Fabry-Perot- interferometer
• laser with plane mirrors  Fabry-Perot-Resonator with active medium
inside the cavity
• nonlinear optics  high intensities inside the cavity  nonlinear optical
effects for low intensity incident light:
− bistability
− modulational instability
− pattern formation, solitons
Here, we want to compute the transmission properties of the resonator for
arbitrary plane mirrors. This task could be achieved employing the matrix
method which we developed in the previous sections. However we will take a
different approach to achieve deeper physical insight into the cavity's behavior.
For simplicity, we will restrict ourselves to TE-polarization. The figure shows
our setup with two mirrors at x = 0 and x = D , characterized by coefficients of
reflection and transmission R0 , T0 , RD , and TD .
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177
cladding
substrate
− EI , ER and ET ,  amplitudes of incident, reflected and transmitted
external fields in substrate and cladding
− E+ , E−  amplitudes of internal fields running forward and backward
inside the cavity
Using the known coefficients of reflection and transmission of the two mirrors,
we can eliminate E+ and E− by connecting the field amplitudes:
A)
At the lower mirror inside the cavity:
T0 EI + R0 E− (0) =
E+ (0)
B)
At the upper mirror outside the cavity:
ET = TD E+ ( D)
E+ (0)
And with E+ ( D) = E+ (0) exp ( ikfx D ) =
ET
exp ( −ikfx D )
TD
C) At the upper mirror inside the cavity:
=
E− ( D) R=
D E+ ( D )
RD
ET
TD
and with E− (0) = E− ( D)exp ( ikfx D )
R
E− (0) = D ET exp ( ikfx D )
TD
D) we substitute E+ (0) and E− (0) in A)
T0 EI + R0 E− (0) =
E+ (0)
T0 EI + R0
 E=
I
RD
ET
ET exp ( ikfx D ) =
exp ( −ikfx D )
TD
TD
1
{exp ( −ikfx D ) − R0 RD exp (ikfx D )} ET .
T0TD
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Thus, the coefficient of transmission for the whole FP-resonator expressed by
the coefficients of the mirrors ( R0 , T0 , RD , and TD ), the cavity properties and
ω2
ε f − k z2 ) reads:
2
c
the angle of incidence ( D , kfx
=
T0TD exp ( ikfx D )
ET
=
.
EI 1 − R0 RD exp ( 2ikfx D )
T=
TE
This is the general transmission function of a lossless Fabry-Perot resonator.
In general, the mirror coefficients are complex and the fields get certain
=
( R, T )0,D ( R , T )0,D exp iϕ0R,,DT . Obviously, only the phase shifts
phase shifts
induced by the coefficients of reflection R0 , RD are important for the
2
transmissivity of the FP resonator τTE  TTE .
For given R0 , RD , T0 , TD and ϕR0 , ϕRD , the general transmissivity of a lossless
Fabry-Perot resonator reads:
(
2
2
2
TTE= T=
)
2
2
T0 TD
2
1 + R0 RD − 2 R0 RD cos(2k fx D + ϕR0 + ϕRD )
((((
δ
τ=
kcx 2
T .
ksx
Here we introduced the phase-shift δ , which the field acquires in one roundtrip in the cavity.
Discussion
Depending on whether the two mirrors have identical properties we
distinguish between symmetric and asymmetric FP-resonators.
a) asymmetric FP-resonator
2
2
T0 TD
k
τ = cx
ksx 1 + R0 2 RD 2 − 2 R0 RD cos δ
Because we assume no losses we can use energy conservation at each
mirror to eliminate T0,D :
ksx
2
2 k fx
2
2
T0 TD =
1 − R0
1 − RD
k fx
kcx
(
) (
(
)(
k
2
2
= sx 1 − R0 1 − RD
kcx
)
)
Note: τ and ρ for a lossless mirror are the same for both sides of the mirror.
For lossy mirrors only τ is the same, ρ is then side-dependent.
For discussing the effect of the phase shift δ we rewrite
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179
δδδ
cos δ = cos 2 − sin 2 = 1 − 2sin 2
2
2
2
Plugging everything in we get
(1 − R ) (1 − R )
0
2
D
2
τ=
2
2
1 + R0 RD − 2 R0 RD (1 − 2sin 2 δ2 )
−1
2


4 R0 RD
 (1 − R0 RD )
2δ
= 
+
sin  .
2
2
2
2
2
1 − R0 1 − RD
 1 − R0 1 − RD

(
) (
)(
)(
)
and with
2
=
ρ0 R0 , ρ=
RD
D
(
2
−1
)
 1 − ρρ 2

4 ρρ
0 D
δ


0 D
=
τ 
+
sin 2  .
2
 (1 − ρ0 ) (1 − ρD ) (1 − ρ0 ) (1 − ρD )


b) symmetric FP-resonator (Airy-formula for transmissivity)
2
2
ρm =R0 =RD =ρ0 =ρD ,
ϕR =ϕR0 =ϕRD


2
2


1 − ρm )
4ρδ
(
 (1 − ρm )
2 
=
τ
= 
+
sin 
2
2
2
2
2δ
−
ρ
−
ρ
1
1
(
)
(
)
 ((
m
m
(1 − ρm ) + 4ρm sin

2 

=1
δ

τ= 1 + F sin 2 
2

with
F=
4ρm
(1 − ρm )
2
−1
−1
δ
and= k fx D + ϕ
2
The Airy-formula (see also Labworks script, where phase shifts ϕ due to the
mirrors are not considered) gives the transmissivity of a symmetric, lossless
Fabry-Perot-resonator. Only for this case we can get the maximum
transmissivity τ =1 for δ / 2 =nπ .
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180
Remarks and conclusions
• We can do an analog calculation for TM-polarization  R TM resp. ρTM
• Resonances of the cavity with τ MAX =
1 occur for δ / 2= k fx DMAX + ϕ= mπ with
kfx =
2π 2
nF − nS2 sin 2 ϕI
λ
m − ϕπ )
(
mπ − ϕ λ
DMAX =
=
2 nF2 − nS2 sin 2 ϕI
kfx

λ
cavity
2
where ϕI is the angle of incidence in the substrate
• transmission properties of a given resonator depend on ϕI and λ.
1
• minimum transmission is given as τ MIN =
1+ F
• it is favorable to have large F , e.g.:
4ρm
100= F=
ρm
(1 − ρm )
4 (1 − τm )
4
=1 − τ 
≈
≈ 100 τ = 0.2 ρ
m
2
2
2
ττ
m
m
m
m
= 0.8.
• pulses and beams can be treated efficiently in Fourier domain:
 e.g. TE: ET (α, β, ω=
) TTE (α, β, ω) EI (α, β, ω)
Fourier back transformation: ET=
( x, y, t ) FT −1 [ ET (α, β, ω) ]
• beams, because they always contain a certain range of incident angles ϕI ,
they produce interference rings in the farfield output (or image of lens, like
in labworks).
• a quantity often used to characterize a resonator is the finesse:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
Φ
181
distancε bεtwεεn rεsonancε
∆
=
full width at half maximum of rεsonancε ε
with ε the FWHM and ∆ = π the distance in rad between two resonances.
To calculate the FWHM ε we can start from:
−1

ε 
1
2
1 + F sin  mπ ±   
2 
2


For narrow resonances (small line width ε ) we can write
−1
2
2

1
ε
 ε  
 F =
ε
1 + F    ≈
 1 =
2
2
2






F=
2
F
ρm
∆ π
=
F =π
ε 2
1 − ρm
• The line width ε (FWHM) is inversely proportional to the finesse Φ .
• The Airy-formula can be expressed in terms of the finesse
−1
  2Φ 2

2δ
τ= 1 + 
sin
 .

2 
  π 
• A Fabry-Perot-resonator can be used as a spectroscope. Then, we can ask
for its resolution (here: normal incidence).
Resonances (maximum transmission) occur at:
kD + ϕ= mπ
reduced transmission by factor 1/2
Dk
ε
π
D= mπ ± = mπ ±
2
2
2Φ
π
2π
=
Dk
Dλ
n=
2 f
λ
ΦD
→ kD + ϕ ±
ε
Dλ
λ
λ

 λ
=
λ
nfΦD ΦD
D
example: λ = 5 ⋅ 10−7 m , Φ = 30, nf D = 4 ⋅ 10−3 m 
∆λ= 2 ⋅ 10−12 m
• The field amplitudes (here forward field) inside the cavity are given as:
−2
ET = TD E+ ( D) Because T − 1 / (1 − ρ) − Φ these intra-cavity fields can be
very high  important for nonlinear effects
• Lifetime of photons in cavity: Via the "uncertainty relation":
=
∆ωTc const. ≈ 1 it is possible to define a lifetime of photons inside the
cavity:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
182
1
π
nΦ Dω
=
c
ΦD
c π
1
n ΦD D
Dω
=
 =
Tc = Φ
 .
nΦ ΦD
Dω
cπ
ε
D=
k
7.4 Guided waves in layer systems
Finally, we want to explore our layer systems as waveguides. For many applications it is interesting to have waves which propagate without diffraction.
This is crucial for integrated optics, where we want to guide light in very small
(micrometric or smaller) dielectric layers (film, fiber), or optical communication
technology where some light encoded information is transported over long
distances. Moreover, waveguides are important in nonlinear optics, due to
confinement and long propagation distances nonlinear effects become important. Here, we will treat wave-guiding in one dimension only because we
restrict ourselves to layer-systems, but general concepts developed can be
transferred to other settings, e.g. where the waveguide is a fiber.
7.4.1 Field structure of guided waves
Let us first do some general consideration about the field structure of guided
waves. We want to find guided waves in a layer system. In such systems, till
now, we have solved the reflection and transmission problem: For given
kz , EI , di ,εi we calculated ER , ET .
Inside each layer we have plane waves
− Eα ( kz , ω) exp i ( kαx x + kz z − ωt ) .
The question is, how can we trap (or guide) waves within a finite layer
system? A possible hint gives the effect of total internal reflection, where the
transmitted field is:
=
ET ( x, z ) ET exp ( ikz z ) exp ( −µc x )
Obviously, in the case of TIR we have no energy flux in the cladding medium.
If TIR is the key mechanism to guide light, can we have TIR on two sides, i.e.
in cladding and substrate? And what about a single interface?
We will concentrate our discussion first on a system of layers, which we will
call the core of the waveguide, and which we place between semi-infinite
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
183
substrate and cladding. The single interface, where the core is absent, will be
considered at a later stage.
According to our discussion the guided waves have the following field
structure:
• plane wave in propagation direction ~ exp(ik z z )
• evanescent waves in substrate and cladding
~ exp[−µ c ( x − D)]
cladding
~ exp( µs x)
substrate
ω2
k − 2 εs,c (ω) > 0
c
2
z
with µs=
,c
 1. condition: kz2 >
ω2
max {εs,c (ω)}
c2
• oscillating solution (standing wave) in the core (layers, fiber core, film)
~ A sin(k fx x) + B cos(k fx x)
ω2
ε (ω) − kz2 > 0
2 i
c
with k=
ix
2
 2. condition: kz2 < ω2 max {εi (ω)}
c i
Note that this 2. condition is not obvious at this stage. It appears due to
transition conditions at the boundaries to substrate and cladding. The fact that
the fields in the substrate and gladding are of evanescent nature requires that
both fields are exponentially growing towards the interface to the core. The
transition conditions at the interfaces impose continuity of the tangential
components of the fields and their first derivative normal to the surface (This
fact was shown during the derivation of the transmission reflection properties
in a multilayer system). Since the field in the core must continuously connect
the fields and derivatives of the fields in substrate and cladding, it must have
some non-monotonic profile. This condition excludes evanescent field profiles
in the core and leaves only oscillating radiation-type fields as a possible
solution.
In summary, the z -component of the wave vector of guided waves has to
fulfill:
ω

ω 
max  ns ,c  < k z < max  ni 
i
c

c 
The field structure in substrate and cladding is given as:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
=
EC ( x, z ) ET exp ( ikz z ) exp  −µc ( x − D ) 
ES ( x, z )= ER exp ( ikz z ) exp ( µs x )
184
x>D
x<0
7.4.2 Dispersion relation for guided waves
If we compare the principal shapes of the fields for the guide wave to our
usual reflection transmission problem, we see that for guided waves the
reflected and transmitted (evanescent) field exist for zero incident field
EI → 0 . In the following we will use this condition to derive the dispersion
relation for guided waves.
Thus we have ET , ER ≠ 0 for EI → 0 . Consequently the coefficients for
reflection and transmission are
TE,TM
 ET 
E 
=
T TE,TM =
, R TE,TM  R 

 EI 
 EI 
TE,TM
with EI → 0
and we find R, T → ∞ in the case of guided waves. In this sense, guided
waves are resonances of the system. Let us compare to a driven harmonic
oscillator
x=
F
, x = action, F = cause
ω − ω02
2
In the case of resonance ( ω = ω0 ) we get action for infinitesimal cause.
Hence, we can get the dispersion relation of guided waves by looking for a
vanishing denominator in the expressions for R, T This reasoning is a general
principle in physics:
The poles of the response function (or Greens function) are the resonances
of the system. Furthermore, a resonance of a system is equivalent to an
eigenmode of the system, which is somehow localized.
We know the coefficients of reflection and transmission for a layer system
from before, and they have the same denominator:
R
=
FR
=
FI
( αsksx M 22 − αc kcx M 11 ) − i ( M 21 + αsksxαc kcx M 12 )
( αsksx M 22 + αc kcx M 11 ) + i ( M 21 − αsksxαc kcx M 12 )
The pole is then given as:
→ ( αsksx M 22 + αc kcx M 11 ) + i ( M 21 − αsksx α c kcx M 12 )  0
Because we have evanescent waves in substrate and cladding the imaginary
valued ksx and kcx can be expressed as real numbers with
ksx
ω2
=
iµs =
i k − 2 εs (ω),
c
2
z
kcx
ω2
=
iµc =
i k − 2 εc (ω)
c
2
z
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185
We can write the general dispersion relation of guided modes in an arbitrary
layer system
M 11TE,TM + αsµs M 12TE,TM +
1
αµ
M 21TE,TM + s s M 22TE,TM  0
α cµ c
α cµ c
Here, as usual, we have α TE =
1, α TM =1 ε for the two independent
polarizations.
The waveguide dispersion relation gives a discrete set of solutions, so-called
waveguide modes. For given εi , d i , ω we get kzν ( ω) .
We can see that the dispersion relation of guided modes depends on the
material's dispersion in the individual layers as well as in substrate and
cladding according to the dispersion relation of homogeneous space as
ω2
k (ω)= 2 εi (ω)= ki2x + kz2 .
c
2
i
In addition to the material's dispersion in the layers the dispersion relation of
guided modes depends on the geometry of the waveguide's layer system
kz ( ω, geometry )
In the case of guided waves it is easy to compute the field (mode profile)
inside the layer system:
• take k z from dispersion relation
• in the substrate we have:
F ( x) = F exp( µs x),
G ( x) = α
∂
{F exp ( µ s x )}
∂x
Hence, for given F (0) we get G (0) = αµs F (0)
( F (0) → free parameter)
F
F
ˆ=
ˆ ( x)  1  F (0).
=
M
x
M
(
)
 αµ 
G
G
 x
 0
 s
Analogy of optics to the stationary Schrödinger equation in QM
Optics (e.g. TE-polarization)
Quantum mechanics
 d 2 ω2

2
 2 + 2 ε( x)  E ( x) = kz E ( x)
c
 dx

↔
 d 2 2m

2m
x) − 2 Eψ ( x)
 2 − 2 V ( x)  ψ ( =


 dx

guided waves
↔
discrete energy eigenvalues
ω2
k > 2 max {εs,c }
c
↔
E < Vext
2
z
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186
Tunnel effect
ω2
k > 2 εFilm
c
2
z
E < VBarriere
↔
7.4.3 Guided waves at interface - surface polariton
Let us first have a look at the most simple case where the guiding layer
structure is just an interface.
Our condition for guided waves is that on both sides of the interface we have
ω2
2
evanescent waves: kz > 2 εc,s , because
c
µs,c=
ω2
k − 2 εc,s > 0
c
2
z
The general dispersion relation we derived before reads
M 11TE,TM + αsµsM 12 TE,TM +
1
αµ
M 21TE,TM + s s M 22 TE,TM  0
α cµ c
α cµ c
and with the matrix for a single interface:
we get the dispersion relation
A)
1+
ˆ = 1 0
M
0 1


αsµs
=
0
α cµ c
TE-polarization ( α =1)
µs + µc =0, → no solution because µc , µs > 0
B)
TM-polarization ( α= 1 / ε)
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187
µ c µs
+
=
0
ε c εs
with µc , µs > 0  εc ⋅ εs < 0,
→ on of the media has to have negative ε (dielectric near resonance or
metal)
In dielectrics
ω0(T) < ω < ωL
we can find surface-phonon-polaritons.
In metals
ω < ωp
we can find surface-plasmon-polaritons.
Remark
Surface polaritons occur in TM polarization only, similar to the phenomenon
k
k
0, )
of Brewster-angle (no reflection for cx − sx =
ε c εs
Let us now compute the explicit dispersion relation for surface polaritons.
W.o.l.g. we assume εs (ω) < 0 , and because εs (ω) is near a resonance it will
show a much stronger ω dependence than εc .
( µcε s )
2
= ( µs εc )
2


ω2  2  2 ω2
ε 2s ( ω) kz2 − 2 εc2  =
ε c  kz − 2 εs ( ω) 
c
c




ω εs ( ω) ε c
k z ( ω) =
c ε c + εs ( ω)
There is a second condition for existence of surface polaritons: εc + εs ( ω) < 0
Conclusion:
ω εs ( ω ) ε c
kz ( ω) =
c εs ( ω ) + ε c
TM polarization
εs ( ω ) < 0
εc > 0
εs ( ω ) > ε c
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188
Surface polaritons may have very small effective wavelengths in z-direction:
εc = 1, εs ( ω) ≈ 1 → kz (=
ω)
2 π εs ( ω ) ε c
2π
=
→ λ eff << λ
λ εs ( ω) + εc λ eff
7.4.4 Guided waves in a layer – film waveguide
The prototype of a waveguide is the film waveguide, where the waveguide
ω2
consists of one guiding layer with 2 ε f ( ω) > kz2 ( ω) .
c
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189
Such film waveguides are the basis of integrated optics. Typical parameters
are:
a few wavelengths
Fabrication of film waveguides can be achieved by coating, diffusion or ion
implantation.
The matrix of a single layer (film) is given as:
From this matrix we can compute the dispersion relation for guided modes:
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
M 11 + αsµs M 12 +
cos ( kfx d ) +
190
1
αµ
M 21 + s s M 22  0
α cµ c
α cµ c
αsµs
α k
αµ
sin ( kfx d ) − f fx sin ( kfx d ) + s s cos ( kfx d ) =
0
αf kfx
α cµ c
α cµ c
αsµs
αf kfx ( αsµs + αcµc )
sin ( kfx d )
α cµ c
= tan
=
( kfx d ) α k =
2
αµ
cos ( kfx d )
αf2 kfx
− α cαsµcµs
f fx
− s s
αcµc αf kfx
1+
kfx  µs µc 
+


α f  α c αs 
tan ( kfx d ) =
2
kfx
µµ
− c2s
α c αs
αf
Here: TE-Polarisation ( α =1)
tan ( kfx d ) =
kfx ( µs + µc )
,
2
kfx
− µ c µs
This waveguide dispersion relation is an implicit equation for kz . For given
frequency ω and thickness d we get several solutions with index kzν
 ω
Here is an example for fixed frequency ω , the effective index neff = kz /  
c
versus the thickness d :
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191
We can see in the figure that for large thickness d we have many modes. If
we decrease d, more and more modes vanish at a certain cut-off thickness.
Definition of cut-off:
a guided mode vanishes  cut-off (here w.o.l.g.
The idea of the cut-off is that a mode is not guided anymore. Guiding means
evanescent fields in the substrate and cladding, so cut-off means
 no guiding
We can plug this cut-off condition in the DR:
with parameter of asymmetry a:
 we can define a cut-off frequency for
 we can define a cut-off thickness for
when we keep d fix
when we keep
fix
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192
In a symmetric waveguide the fundamental more ( ν =0 ) has cut-off = 0!
If we plot the dispersion curves for each mode we get a graphical
representation of the dispersion relation:
f
7.4.5 How to excite guided waves
Finally, we want to address the question how we can excite guided waves. In
principle, there are two possibilities, we can adapt the field profile or the wave
vector ( kz )
A)
adaption of field  front face coupling
Then, inside the waveguide we have (without radiative modes):
E ( x, z ) = ∑aν Eν ( x)exp ( ikzν z )
ν
 E ( x,0) ≈ ∑aν Eν ( x)
ν
with : Pν =
aν =
∫E ( x )
µ
kzν ∞
2
(
)
E
x
dx
ν
2ωµ 0 ∫ −∞
kzν
2ωµ 0 Pν
∫
∞
−∞
Ein ( x) Eν ( x)dx,
 mode ν couples to the incident field Ein (0) with amplitude aν .
 Gauss-beam couples very good to the fundamental mode
B) adaption of wave vector  coupling through the interface
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193
We know that k z is continuous at interface. The condition for the existence of
guided modes is
kz >
ω
εc,s
c
but dispersion relation for waves in bulk media dictates
=
kz
ω2
ω
ε − ks2,cx <
εc,s
2 c ,s
c
c
 We got a problem!
There are two solutions:
i) coupling by prism
we bring a medium with ε p > εf (prism) near the waveguide
ω
kz <
εf
c
ω
kz <
ε p  k=
px
c
ω2
ε − kz2 > 0 .
2 p
c
 light can couple to the waveguide via optical tunneling: ATR ('attenuated
total reflection')
ii) coupling by grating
Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2015-02-14s.docx
grating on waveguide (modulated thickness of layer d):
d ( z )= d + ς ( z )
=
ς ( z ) A sin=
( gz ) mit g
2π
P − period
P
coupling works for m’th diffraction order:
k zv= k z + mg
=
ω
ns sin φ + mg .
c
194