Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
1
Fundamentals of Modern Optics
Winter Term 2013/2014
Prof. Thomas Pertsch
Abbe School of Photonics
Friedrich-Schiller-Universität Jena
This script is based on the lecture series “Theoretische Optik” by Prof. Falk
Lederer at the FSU Jena and was adapted to English by Prof. Stefan Skupin
for the international education program of the Abbe School of Photonics.
Table of content
0. Introduction ............................................................................................... 4 1. (Ray optics - geometrical optics, covered by lecture Introduction to
Optical Modeling) ................................................................................... 16 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.6 1.6.1 1.6.2 1.6.3 Introduction ....................................................................................................... 16 Postulates ......................................................................................................... 16 Simple rules for propagation of light ................................................................. 17 Simple optical components............................................................................... 17 Ray tracing in inhomogeneous media (graded-index - GRIN optics) .............. 21 Ray equation..................................................................................................... 21 The eikonal equation ........................................................................................ 23 Matrix optics...................................................................................................... 24 The ray-transfer-matrix ..................................................................................... 24 Matrices of optical elements ............................................................................. 24 Cascaded elements .......................................................................................... 25 2. Optical fields in dispersive and isotropic media ...................................... 26 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.5.1 Maxwell’s equations...................................................................................... 26 Adaption to optics ............................................................................................. 26 Temporal dependence of the fields .................................................................. 29 Maxwell’s equations in Fourier domain ............................................................ 30 From Maxwell’s equations to the wave equation ............................................. 30 Decoupling of the vectorial wave equation ....................................................... 31 Optical properties of matter .............................................................................. 32 Basics ............................................................................................................... 33 Dielectric polarization and susceptibility ........................................................... 36 Conductive current and conductivity ................................................................ 37 The generalized complex dielectric function .................................................... 39 Material models in time domain ........................................................................ 43 The Poynting vector and energy balance ......................................................... 44 Time averaged Poynting vector ........................................................................ 44 Time averaged energy balance ........................................................................ 46 Normal modes in homogeneous isotropic media ............................................. 48 Transversal waves ............................................................................................ 49 Longitudinal waves ........................................................................................... 50 Plane wave solutions in different frequency regimes ....................................... 51 Time averaged Poynting vector of plane waves .............................................. 56 Beams and pulses - analogy of diffraction and dispersion............................... 56 Diffraction of monochromatic beams in homogeneous isotropic media .......... 58 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
2.5.2 2.5.3 2.5.4 2.5.5 2.6 2
Propagation of Gaussian beams ...................................................................... 70 Gaussian optics ................................................................................................ 77 Gaussian modes in a resonator ....................................................................... 81 Pulse propagation ............................................................................................. 86 (The Kramers-Kronig relation, covered by lecture Structure of Matter) ........... 99 3. Diffraction theory .................................................................................. 103 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.4 Interaction with plane masks ..........................................................................103 Propagation using different approximations ...................................................104 The general case - small aperture.................................................................. 104 Fresnel approximation (paraxial approximation) ............................................ 104 Paraxial Fraunhofer approximation (far field approximation) ......................... 105 Non-paraxial Fraunhofer approximation ......................................................... 107 Fraunhofer diffraction at plane masks (paraxial) ............................................107 Fraunhofer diffraction pattern ......................................................................... 107 Remarks on Fresnel diffraction ......................................................................112 4. Fourier optics - optical filtering.............................................................. 113 4.1 4.1.1 4.1.2 4.2 4.2.1 4.2.2 4.2.3 Imaging of arbitrary optical field with thin lens ...............................................113 Transfer function of a thin lens ....................................................................... 113 Optical imaging ............................................................................................... 114 Optical filtering and image processing ...........................................................116 The 4f-setup.................................................................................................... 116 Examples of aperture functions ...................................................................... 118 Optical resolution ............................................................................................ 119 5. The polarization of electromagnetic waves .......................................... 122 5.1 5.2 5.3 Introduction .....................................................................................................122 Polarization of normal modes in isotropic media............................................122 Polarization states ..........................................................................................123 6. Principles of optics in crystals............................................................... 125 6.1 6.2 6.3 6.4 6.4.1 6.4.2 6.4.3 6.4.4 Susceptibility and dielectric tensor .................................................................125 The optical classification of crystals ...............................................................127 The index ellipsoid ..........................................................................................128 Normal modes in anisotropic media ...............................................................129 Normal modes propagating in principal directions ......................................... 130 Normal modes for arbitrary propagation direction .......................................... 131 Normal surfaces of normal modes ................................................................. 135 Special case: uniaxial crystals ........................................................................ 137 7. Optical fields in isotropic, dispersive and piecewise homogeneous media
............................................................................................................. 140 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2 7.2.1 7.2.2 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.4 7.4.1 Basics .............................................................................................................140 Definition of the problem................................................................................. 140 Decoupling of the vectorial wave equation ..................................................... 141 Interfaces and symmetries ............................................................................. 142 Transition conditions....................................................................................... 142 Fields in a layer system  matrix method .....................................................143 Fields in one homogeneous layer .................................................................. 143 The fields in a system of layers ...................................................................... 145 Reflection – transmission problem for layer systems .....................................147 General layer systems .................................................................................... 147 Single interface ............................................................................................... 153 Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals ............. 160 Fabry-Perot-resonators .................................................................................. 167 Guided waves in layer systems ......................................................................173 Field structure of guided waves ...................................................................... 173 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
7.4.2 7.4.3 7.4.4 7.4.5 3
Dispersion relation for guided waves ............................................................. 174 Guided waves at interface - surface polariton ................................................ 176 Guided waves in a layer – film waveguide ..................................................... 178 how to excite guided waves............................................................................ 182 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
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)
Origin of light
 atomic system  determines properties of light (e.g. statistics, frequency,
line width)
 optical system  other properties of light (e.g. intensity, duration, …)
 invention of laser in 1958  very important development
Schawlow and Townes, Phys. Rev. (1958).
 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, ...
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Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
C)
Light can modify matter
 light induces physical, chemical and biological processes
 used for lithography, material processing, or modification of biological
objects (bio-photonics)
Fiber laser: Limpert, Tünnermann, IAP Jena, ~10kW CW (world record)
B)
Propagation of light through matter
 light-matter interaction
Hole “drilled” with a fs laser at Institute of Applied Physics, FSU Jena.
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)
Two-dimensional photonic crystal membrane.
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D)
Optics in our daily life
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E)
Optics in telecommunications
 transmitting data (Terabit/s in one fiber) over transatlantic distances
A small story describing the importance of light for everyday life, where all
things which rely on optics are marked in red.
1000 m telecommunication fiber is installed every second.
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Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
F)
Optics in medicine, life sciences
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Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
G)
Optical 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_2014-02-07s.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_2014-02-07s.docx
I)
Relativistic optics
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Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
K)
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What is light?
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
L)
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Schematic of optics
8
 electromagnetic wave (c= 3*10 m/s)
 amplitude and phase  complex description
 polarization, coherence
quantum optics
Spectrum of Electromagnetic Radiation
Region
Wavelength
(nanometers)
Wavelength
(centimeters)
Frequency
(Hz)
Energy
(eV)
Radio
> 108
> 10
< 3 x 109
< 10-5
Microwave
108 - 105
10 - 0.01
5
Infrared
10 - 700
Visible
700 - 400
3 x 109 - 3 x 1012
-5
0.01 - 7 x 10
12
3 x 10
- 4.3 x 10
14
7 x 10-5 - 4 x 10-5 4.3 x 1014 - 7.5 x 1014
-5
-7
14
7.5 x 10
- 3 x 10
17
10-5 - 0.01
electromagnetic optics
wave optics
0.01 - 2
2-3
3 - 103
Ultraviolet
400 - 1
4 x 10 - 10
X-Rays
1 - 0.01
10-7 - 10-9
3 x 1017 - 3 x 1019
103 - 105
Gamma Rays
< 0.01
< 10-9
> 3 x 1019
> 105
geometrical optics
 geometrical optics
  << size of objects  daily experiences
 optical instruments, optical imaging
 intensity, direction, coherence, phase, polarization, photons
 wave optics
   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
 no quantum optics  advanced lecture
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
M)
Literature
 Fundamental
1. Saleh, Teich, 'Fundamenals of Photonics', Wiley, 1992
2. Mansuripur, 'Classical Optics and its Applications', Cambridge, 2002
3. Hecht, 'Optik', Oldenbourg, 2001
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 (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
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D) Fermat’s principle
B
  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.
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
 The angle of refraction 2 depends on the angle of incidence 1 and is
given by Snell’s law:
n1 sin 1  n2 sin 2
 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.
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
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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
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C) Spherical interface (paraxial)
 paraxial imaging
2 
n1
n n y
1  2 1 (*)
n2
n2 R
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 1  n2 sin 2
for paraxial rays: n11  n22
 external reflection ( n1  n2 ): ray refracted away from the interface
 internal reflection ( n1  n2 ): ray refracted towards the interface
 total internal reflection (TIR) for:
2 
n

 sin 1  sin TIR  2
n1
2
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 lense (paraxial)
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 two spherical interfaces (R1, R2, ) apply (*) two times and assume
y=const ( small)
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variation of the path: r ( s)  r ( s)
B
B
A
A
L   nds   nds
n  grad n  r
ds 
2  1 
1
y
1
1 
with focal length:   n  1   
f
f
 R1 R2 
1 1 1
  (imaging formula)
z1 z2 f
z
m   2 (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)

 dr  d r 
 dr 
B
L   n r  s   ds
A
 dr 
2
 2dr  d r   d r  
2
 dr 
2
dr d r

 ds
ds ds
 dr d r 
 ds 1  
  ds
 ds ds 
dr d r
 ds 
ds ds
dr d r 

L    grad n  r  n 
 ds
ds ds 
A
B
d  dr  

   grad n   n    rds
ds
 ds  
A
B
integration by parts and A,B fix
L  0 for arbitrary variation
grad n 
A
computation:

 ds 1  2
B
  n(r )ds  0
2
2
d  dr 
 n  ray equation
ds  ds 
Possible solutions:
A) trajectory
x(z) , y(z) and ds  dz 1   dx dz    dy dz 
2
 solve for x(z) , y(z)
 paraxial rays  (ds  dz )
2
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d 
dx  dn
n  x, y , z   
dz 
dz  dx
 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 ()  two algebraic equation  2 x 2 matrix
Advantage: we can trace a ray through an optical system of many elements
by multiplication of matrices.
B)
homogeneous media
 straight lines
C) graded-index layer n(y) - paraxial, SELFOC
1.6.1 The ray-transfer-matrix
dy
 1 and dz  ds
dz
 1

n 2 ( y )  n02 1   2 y 2   n( y )  n0 1   2 y 2  for a  1
 2

d 
dy  d 
dy 
d2y
d2y
1 dn( y )
n
y

n
y

n
y








2
2




ds 
ds  dz 
dz 
dz
dz
n  y  dy
y ( z )  y0 cos z 
for n(y)-n0<<1:
d2y
  2 y
dz 2
0
sin z


( z ) 
dy
  y0 sin z  0 cos z
dz
in paraxial approximation:
y2  Ay1  B1
 y   A B   y1 
A B
M 
  2  




C D 
 2  C D   1 
2  Cy1  D1
A=0: same 1  same y2  focusing
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)
gradS  r    n  r 
2
24
1.6 Matrix optics
d 
dy  dn
n  x, y , z   
dz 
dz  dy
paraxial 
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
D=0: same y1  same 2  collimation
1.6.2 Matrices of optical elements
A)
free space
2
Remark: it is possible to derive Fermat’s principle from eikonal equation
 geometrical optics: Fermat’s or eikonal equation
S  rB   S  rA    grad S  r  ds   n  r ds
B
B
A
A
eikonal  optical path length  phase of the wave
1 d 
M

0 1 
B)
refraction on planar interface
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2. Optical fields in dispersive and isotropic media
0 
1
M

0 n1 n2 
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 developed in this lecture.
C) refraction on spherical interface
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
electromagnet 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.
1
0 

M

   n2  n1  n2 R n1 n2 
D) thin lens
 1
M
 1 f
E)
0
1 
reflection on planar mirror
Maxwell’s equations (macroscopic)
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
     C D      M  C D 
  1


 N 1  
M=MN….M2M
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 there 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
to influence of many point charges on the electromagnetic field in a
homogeneously distributed response of the solid state on the excitation by
the 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 electro-dynamic theory. Here we will
not redo this derivation. We will rather start directly from the averaged
Maxwell's equations equation.
rot E(r, t )  
B(r, t )
t
rot H (r, t )  jmakr (r, t ) 
 electric field
div D(r, t )  ext (r, t )
D(r, t )
t
E(r, t )
div B(r, t )  0
[V/m]
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[Vs/m2] or [tesla]
 magnetic flux density
or magnetic induction
 dielectric flux density
B(r, t )
D(r, t )
[As/m ]
 magnetic field
H (r, t )
[A/m]
 external charge density
27
2
3
ext (r , t ) [As/m ]
 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.
28
 free charge density
 ext (r, t ) [As/m3]
 macroscopic current density
jmakr (r, t )  jcond (r, t )  jconv (r, t ) [A/m2]
 conductive current density
jcond (r, t )  f  E
 convective current density
jconv (r, t )  ext (r, t ) v (r, t )
 In optics, we generally have no free charges which change at speeds
comparable to the frequency of light:
ext (r, t )  0  jconv (r, t )  0
D(r, t )  0E(r, t )  P(r, t )
H (r, t ) 
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 B(r, t )  M(r, t )
0
 dielectric polarization
P(r, t )
[As/m2],
 magnetic polarization
M(r, t )
[Vs/m2] (magnetization)
 electric constant (vacuum permittivity)
1
0 
 8.854  1012 As/Vm
0c 2
 magnetic constant (vacuum permeability)
 0  4  10 7 Vs/Am
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 describing these quantities.
 In optics, we generally deal with non-magnetizable media  M(r, t )  0
(exceptions are metamaterials with 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 the equations.
 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 ) 
 0
t
t
rot E(r, t )  0
0 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 polarization on the electric
field P(E) and the dependence of the (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 and current density are zero, and we can
solve Maxwell’s equations directly (most simple material model).
Remark:
 We can define a bound charge density
b (r , t )  div P (r, t )
 and a bound current density
jb (r, t ) 
P(r, t )
t
 This essentially means that we can describe the same physics in two
different ways (see generalized complex dielectric function below).
Complex field formalism:
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29
 Maxwell’s equations are also valid for complex fields and are easier to
solve
 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)].
 convention in this lecture
real physical field:
E r (r , t )
complex mathematical representation:
E(r, t )
 They are related by
Er (r, t )  12  E(r, t )  E (r, t )   Re  E(r, t ) 
Remark: 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  go
back to real quantities before! (relevant for, e.g., calculation of
Poynting vector, see 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(i t )  phase is fixed  coherent, infinite wave train e.g.:
E(r, t )  E(r )exp(it )
 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)
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
E(r, t )   E(r, )exp(i t )d 


E(r, ) 
1
 E(r, t )exp(it )dt
2 
Remark: The position of the sign in the exponent and the factor 1 / 2
can be defined differently in different textbooks.
2.1.3 Maxwell’s equations in Fourier domain
We want to plug the Fourier decompositions of our fields into Maxwell’s
equations in order to get a more simple description. For this purpose, we
need to know how a time derivative transforms into Fourier space.
Here we used integration by parts:


1
1
 

dt  E  r, t   exp  it   i

 dt E  r, t  exp  it   i E(r, )
2   t
2 

 rule:

FT

  i
t
Now we can write Maxwell’s equations in Fourier domain:
rot E(r, )  i0 H (r, )
0 div E(r, )   div P (r, )
rot H (r, )  j (r, )  iP(r, )  i0E(r, ) div H (r, )  0
2.1.4 From Maxwell’s equations to the wave equation
Maxwell's equations provide the basis to derive all possible mathematical
solutions. However we are interested in radiation fields which can be
described more easily by an adapted equation, which is the 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 a second time on rot E(r, t )   and substitute
rot H
rotrot E(r, t )  0rot
P(r, t )
E(r, t ) 
H(r, t )

 0  j(r, t ) 
 0
t
t 
t
t 
And find the wave equation for the electric field
rotrot E(r, t ) 
1 2
j(r, t )
 2P(r, t )
(
,
t
)




E
r
0
0
c 2 t 2
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, in particular:
div  0E(r, t )  P(r, t )  0
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31
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
0
Remarks:
 analog procedure possible for H, i.e., we can derive a wave equation
for the magnetic field
 generally, the wave equation for E is more convenient, because we
have P(E) given from the material model
 however, for inhomogeneous media H can be the better choice for the
numerical solution of the partial differential equation since it form a
hermitian operator
 analog procedure possible for H  E
 generally, wave equation for E is more convenient, because P(E)
given
 for inhomogeneous media H can be better choice
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and propagation only in the x-z-plane. Then all spatial derivatives along the
y-direction disappear and the operators in the wave equation simplify.
 generally all field components are coupled
 for translational invariance in e.g. the y-direction and propagation only in
the x-z-plane   / y  0
 x

rot rot E  grad div E   E  

 z
2
rotrot E(r, )  2 E(r, )  i0 j(r, )  02 P(r, )
c
E  E   E

 0 
 
E   E y  ,
 0 
 
with Nabla operator 


 Ez z    (2) E x 
 

0
    (2) E y 
  (2) 
Ex
Ez
   E z 
x  z 
Ex
x

(2)
Ex 
 
E   0 
 Ez 
 
  x 
2
2
(2)
  0  , and Laplace   2  2
x z


  z 
 gives two decoupled wave equations
 (2) E (r, ) 
and
div 0E(r, )  P(r, )   0

 decomposition of electric field
B) Frequency domain derivation
We can do the same procedure in the Fourier domain and find
32
 (2) E (r, ) 
2
E  (r, )  i 0 j (r, )   02 P  (r, )
c2
2
E (r, )  grad (2) div (2) E  i 0 j (r, )   02 P  (r, )
c2
 properties
 magnetic field:
H (r, )  
i
rot E(r, )
 0
 transferring the results from the Fourier domain to the time domain
 for stationary fields: take solution and multiply by e -it .
 for non-stationary fields and linear media  inverse Fourier
transformation

E(r, t )   E(r, )exp(i t )d 

2.1.5 Decoupling of the vectorial wave equation
 For arbitrary isotropic media generally all 3 field components are coupled.
 For problems with translational invariance in at least one direction, as e.g.
for homogeneous infinite media, layers or interfaces, there can be
decoupling of the components. Let’s assume invariance in the y-direction
 propagation of perpendicularly polarized fields E and E can be
treated separately
 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, leading eventually to a current
density we will use the same model but without restoring force. In the
literature, this simple approach is often called the Drude-Lorentz model
(named after Paul Drude and Hendrik Antoon Lorentz).
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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.
rotrot E(r, ) 
E(r, ) 
D(r, )  0E(r, )  P(r, )
The general response function and the respective susceptibility given above
simplifies for certain properties of the medium:
Different types of media
A)
 non-dispersive   ij (r, )  ij (r )  instantaneous: Rij (r , t )   ij (r )(t )
(Attention: This is unphysical!)
  ij (r , )   is a scalar constant
frequency domain
Pi (r, t )  0   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
P(r, )  0 E(r, )

P(r, t )  0E(r, t ) (unphysical!)
D(r, )  0 E(r, )

D(r, t )  0E(r, t )
  E(r, ) 
Pi (r, )   0  ij (r , ) E j (r, )
j
 Obviously, things look friendlier in frequency domain. Using the wave
equation from before and assuming that there are no currents ( j  0)
we find
time domain description
   1 
Maxwell: div D  0  div E(r, )  0 for ()  0
 In frequency domain: we introduce the susceptibility
E(r , )  medium (susceptibility)  P (r , )

linear, homogenous, isotropic, non-dispersive media (most simple but
very unphysical case)
 homogenous  ij (r , )  ij ()
 isotropic   ij (r, )  (r, )ij
t
1
 ij ()exp(it )d 
2 
2
E(r, )  graddivE(r, )   02 P (r, )
c2
 and for auxiliary fields
 description in both time and frequency domain possible
 In time domain: we introduce the response function
E(r, t )  medium (response function)  P(r, t )
Rij (t ) 
2
E(r, )   02 P (r, )
c2
or
For the polarization P(E) (for j(E) very similar):
 response function and susceptibility are linked via Fourier transform
(convolution theorem)
34
B)
2
 2

E
(
r
,

)

0

E
(
r
,
t
)

E(r, t )  0

c2
c 2 t 2
 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  ()
P(r, )  0() E(r, )
D(r, )  0() E(r, )
div D(r, )  0  div E(r, )  0 for ()  0
  E(r, ) 
2
   E(r, )  0
c2
Helmholtz equation
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35
 This description is sufficient for many materials.
C) linear, inhomogeneous, isotropic, dispersive media  (r, )
P(r, )  0 (r, ) E(r, ),
D(r, )  0 (r, ) E(r, ).
div D(r, )  0
div D(r, )  0(r, )div E(r, )  0E(r, )grad (r, )  0,
 div E(r, )  
 E(r, ) 
grad (r, )
E(r, ).
(r, )
 grad (r, )

2
  r,  E(r, )  grad 
E(r, ) 

r

c2
(
,
)


 All field components couple.
D) linear, homogeneous, anisotropic, dispersive media  ij ()
Pi (r, )  0  ij () E j (r, )
j
Di (r, )  0  ij () E j (r, ).
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36
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 I) and II) separately, and join the results
later.
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 )  02s(r, t )  E(r, t )
2
t
t
m
 resonance frequency (electronic transition)  0
 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  ).
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.
 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 )  Np(r, t )  Nqs(r, t )
Hence, the governing equation for the polarization P(r, t ) reads as
2

q2 N
2
P
(
r
,
)

P
(
r
,
)
P
(
r
,
t
)

E(r, t )   0 fE(r, t )
t
g
t


0
t 2
t
m
with oscillator strength f 
1 e2 N
, for q=-e (electrons)
0 m
This equation is easy to solve in Fourier domain:
2 P(r , )  ig P(r , )  02 P(r, )  0 f E(r , )
 P(r, ) 
0 f
E(r, )
   2   ig
2
0
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with P(r, )  0()E(r, )  () 

2
0
37
f
    ig 
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, )  0   2
 E(r, )  0   E(r, )
2
j   0 j     ig j  




fj
      2

2
j   0 j     ig j  


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38
e
2

s(r, t )  g s(r, t )   E(r, t ),
2
t
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 )  02pE(r, t )
t
m
with plasma frequency 2p  f 
1 e2 N
0 m
Again we solve this equation in Fourier domain:
i j (r, )  g j (r, )  0 2p E(r, )
     is the complex, frequency dependent susceptibility
D(r , )  0E(r , )   0   E(r , )   0    E(r , )
    is the complex frequency dependent dielectric function
Example: (plotted for eta and kappa with         i    )
2
 j (r, ) 
0 2p
g  i
E(r, )     E(r, ).
Here we introduced the complex frequency dependent conductivity
   
 0 2p
g  i
 i
 0 2p
2  ig 
.
Remarks on plasma frequency
We consider a cloud of electrons and positive ions described by the total
charge density  in their self-consistent field E . Then we find according to
Maxwell:
 0divE(r, t )  (r, 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  02pE(r, t )
t
Now we apply divergence operator and plug in from above (red terms):
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 0  0 . This
corresponds to the case of zero restoring force.
div

j  gdivj  02pdivE(r, t )  2p(r, t )
t
With the continuity equation for the charge density (from Maxwell's equations)

  divj  0,
t
We can substitute the divergence of the current density and find:
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
39
2

  g   2p
2
t
t
40
Some special cases for materials in the IR und VIS:
A) Dielectrics (insulators) near phonon resonance (IR)
2

  g   2p  0,
2
t
t
 harmonic oscillator equation
If we are interested in dielectrics (insulators) near phonon resonance in the
infrared spectral range we can simplify the dielectric function as follows:
Hence, the plasma frequency p 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, ) and P(r, ) into the wave equation (in Fourier domain)
rotrot E(r, ) 
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2
E(r , )   02 P (r, )  i 0 j(r, )
c2


fj
f
()  1    2
 2
2
2





i



g



  ig
j 
0
j 
 0j

f
   2
,



 0 2   ig
  0 0 ()  i 0     E(r , )
Hence we can collect all terms proportional to E (r , ) and write
( ) 
2
rotrot E(r, )  2 ()E(r, )
c

2
0

 2 
2
0
 2   g 2 2
2
gf 
 2   g 2 2
2
,
. Lorentz curve
 width of resonance peak  g
i
()  1  () 
    ()  i()
0
 static dielectric constant in the limit   0 :
So, in general we have
0    
f
02
 so called longitudinal frequency L : (  L )  0


fj
2p
()  1    2
,


2
2  ig 
j   0 j     ig j  


 ()  0 : absorption and dispersion appear always together
Example: single resonance
because (from before)
    i

2
0
f
 resonance frequency  0
Here, we introduced the generalized complex dielectric function


fj
      2
,
2
j   0 j     ig j  


 ( )  ( )  i ( )  ( )  i ( )
( )   

2 
i
rotrot E(r, )  2 1  () 
    E(r, )
0
c 

 0 2p
2  ig 
.
() contains contributions from vacuum, phonons (lattice vibrations), bound
and free electrons.
  0
The contribution of vacuum and electronic transitions show almost no
frequency dependence (dispersion) in this regime and can be expressed as a
constant  . Let us study the real and the imaginary part of the resulting ()
separately:
   vacuum and electronic transitions
2
0  0 j
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12
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42
2 .2
ε'
ε′
ε′′
8
2
4
1 .8
ε0
0
ω0
ε∞
ω
ωL
1 .6
V IS
-4
 near resonance we find ()  0 (damping without absorption if  ''  0 )
 normal dispersion
 () /   0,
 anomalous dispersion
 () /   0
1 .4
0
4
6
8
ω in 10 1 5 s -1
The generalization of this approach in the transparent spectral range leads to
so-called Sellmeier formula:
Example: sharp resonance for undamped oscillator g  0
()  1  
12
ε
j
8
f j 02 j

2
0j
 2 
,
 describes many media very well (dispersion of absorption is neglected)
 oscillator strengths and resonance frequencies are fit parameters
4
ε0
0
ω0
ε∞
ω
ωL
C) Metals in visible spectral range
If we want to describe metals in visible spectral range we find
-4
()  1 
-8
 relation between resonance frequency 0 and longitudinal frequency
L (Lyddane-Sachs-Teller relation)
(L )   
f
 0 , f   0    02 (from above)
2
2



 0 L
 L  0
0
.

B) Dielectric media in visible spectral range
Dielectric media in visible spectral range can be described by a so-called
double resonance model (phonon in IR and electronic transition in UV).
()   
2

2
0p
fp
    ig p 
2

fe
,
   2   ige
2
0e
0 p    0e
   contribution of vacuum and other (far away) resonances
()  1 
2p
.
  ig 
2
2p
2  g
,
2
p  
() 
g 2p
  2  g 2 
.
Metals show a large negative real part of the dielectric function ()
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44
Examples
ε′
ε′′
20
A) dielectric media
RP (t ) 
ωP 2 -g 2
0
V IS
5
10
15
20
25
ω in 10 15 s -1
1 
1 
f
   exp  it  d  
exp  it  d ,
2
2




0    ig 
2
2
 Using the residual theorem we can find:
f
 g 
 exp   t  sin t
R (t )   
 2 
0

-2 0
t0
with
  02 
t0
g2
4
f
 g

 exp  2 (t  t)  sin (t  t) E(r, t)dt
 
t
P(r, t ) 
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, )  0()E(r, )
R j (t ) 
P(r, )  0()E(r, ).
The response function (or Green's function) R(t ) is then given by

1 
R(t ) 
()exp  it  d . ()   R (t ) exp  it  dt



2
To prove this, we can use the convolution theorem


P (r, t )   P (r, )exp -it  d    0  ()E(r, )exp -it  d 



1 
  0  ()  E(r, t )exp  it   dt  exp -it  d 

2 
Now we switch the order of integration, and identify the response function R
(red terms):
 1

 0 
()exp -i(t  t )  d E(r, t )dt 
 2  

 0  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 (t   t0 )  P (r, t )   0 R (t  t0 )e  Green's function.
For instantaneous (or non-dispersive) media we find:
R (t )  (t )
B) metals
 P  r , t    0 E  r , t 
(unphysical!)
2
1 
1   0 p
exp
exp  it  d ,






i
t
d




2  
2   g  i
 Using again the residual theorem we can find:
 p2
 exp   gt 
R(t )   g
0


t  0

t  0 
t
j(r, t )  0p2  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  2  / 0  1014 s
 pulse duration
 Tp
in general Tp  T0
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 duration of measurement
 Tm
45
in general Tm  T0
Hence, in general the detector does not recognize the fast oscillations of the
optical field  e  i0t (optical cycles) and delivers a time averaged value. For
the situation described above, the electro-magnetic fields factorize in slowly
varying envelopes and fast carrier oscillations:
→
×
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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 ),
 (r, t ')  H (r )
E
H
1
S(r, t )   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 )
1 
E(r, t )exp  i0t   c.c.  Er (r, t )
2
For such pulses, the momentary Poynting vector reads:
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  2i t   E
  (r, t )  H
  (r, t )exp  2i t  
 E
(r, t )  H
0
0 
4
1
 (r, t )  H
  (r, t )   1  E


  E
 2  (r, t )  H (r, t )  cos  20t 
2
1
  (r, t )  sin  2 t  .
  E  (r, t )  H
0

2
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  20t  ,  sin  20t  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 20t and ~ sin 20t 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:
1 1
S(r, t ) 
2 Tm

t Tm /2
 (r, t ')  H
  (r, t ')  dt '
 E

t Tm /2
 measurement destroys phase information
2.3.2 Time averaged energy balance
Let us motivate a little bit further the concept of the Poynting vector. It
appears naturally in the Poynting theorem, the 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 Hr
resp. Er: (note that we use real fields):
H r  rotEr  0 H r 
0Er 

Hr  0
t


Er  Er  rotH r  Er  ( jr  Pr )
t
t
Next, we subtract the two equations and get
H r  rotEr  Er  rotH r  0Er 



Er   0 H r  H r  Er  ( jr  Pr ).
t
t
t
This equation can be simplified by using the following vector identity:
div  Er  H r   H r  rotEr  Er  rotH r
Finally, with Er 

1 2
Er 
Er we find Poynting's theorem
t
2 t
1  2 1  2
 

0 Er   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 Poynting vector
(energy flux), we can identify the vacuum energy density u  12  0Er 2  12  0 H r 2 .
The right-hand-side of Poynting's theorem contains the so-called source
terms.
1
1
where u  0Er 2  0Hr 2  vacuum energy density
2
2
In the case of stationary fields and isotropic media (simple but important)
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47
1
Er (r, t )  E(r )exp  i0t   c.c.
2
1
H r (r, t )   H(r )exp  i0t   c.c.
2
Time averaging of the left hand side of Poynting’s theorem (*) yields:
1  2
1 
1
 0 Er (r , t )   0 H r2 (r , t )  div  Er (r , t )  H r (r, t )   div   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
would be gain of energy. In particular near resonances we have      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   S  ndA
A



  jr (r, t )  Pr (r, t )  Er (r, t )
t



1
(0 )E(r )e i0t  i00(0 )E(r )e i0t c.c. E(r)e i0t  c.c.
4 
Now we use our generalized dielectric function:



  0  
1 
 i00    0   i
 E(r) exp  i0t   c.c. E(r) exp  i0t   c.c.
00 
4 


1
 i00   0   1 E(r)E(r)  c.c.
4
Again, all fast oscillating terms  exp  2i0t  cancel due to the time average.
Finally, splitting   0  into real and imaginary part yields
1
1
 i00   0   1  i  0   E(r)E(r)  c.c.   00  0  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   00  0  E(r )E (r ).
2
This shows that a nonzero imaginary part of epsilon (      0 ) causes a
drain of energy flux. In particular, we always have      0 , otherwise there
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 solutions to the wave equation.
Because it is convenient to use the generalized complex dielectric function
()  1  () 
i
    ()  i()
0
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. We will see later that it is possible to construct general
solutions from the normal modes. The wave equation in Fourier domain reads
rotrot E(r, ) 
2
()E(r, )
c2
According to Maxwell the solutions have to fulfill additionally
0 1  ()  div E(r, )  0
In general, this additional condition implies that the electric field is free of
divergence:
1  ()  0  div E(r, )  0 (normal case)
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49
Let us for a moment assume that we already know that we can find plane
wave solutions of the form:
E(r , )  E()exp  ikr  ,
k  unknown complex wave-vector
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Hence, we have to solve
 2 2

 k  c 2 ()  E()  0 and k  E()  0.


which leads to the following dispersion relation
Then, the divergence condition implies that those waves are transversal
k  E()  transverse wave
The corresponding stationary field in time domain is given by
E(r, t )  E exp i  kr  t  
 monochromatic plane wave  normal mode
This is a monochromatic plane wave, the simplest solution we can expect, a
so-called normal mode.
If we split the complex wave vector into real and imaginary part k  k'  ik'',
we can define:
k ' r  const.
 planes of constant phase
k''r  const.
 planes of constant amplitude
In the following we will call the solutions
A) if those planes are identical
 homogeneous waves
B) if those planes are perpendicular  evanescent waves
C) otherwise
 inhomogeneous waves
We will see that in dielectrics      0  we can find a second, exotic type of
wave solutions: At   L  (L )  0, so-called longitudinal waves k  E()
appear.
50
k 2  k 2  kx2  ky2  kz2 
2
()
c2
We see that the so-called wave-number k ()  c () is a function of the
frequency. We can conclude that transversal plane waves are solutions to
Maxwell's equations in homogeneous, isotropic media, only if the dispersion
relation k () is fulfilled.
In general, k = k   ik  is complex. Therefore it is sometimes useful to
introduce the complex refractive index (if k   k  ):
k () 



()  nˆ ()   n()  i().
c
c
c
E(r, )  E() exp  ikr  ,
With the knowledge of the electric field we can compute the magnetic field if
desired:
H(r, )  
i
1
k  E()  exp  ikr 
rot E(r, ) 
0
0 
 H(r, )  H()exp  ikr  ,
with H() 
1
k  E() 
0 
2.4.2 Longitudinal waves
2.4.1 Transversal waves
As pointed out above, for   L the electric field becomes free of divergence:
 0() div E(r, )  0
 div E(r, )  0
Then, the wave equation reduces to the Helmholtz equation:
E(r, ) 
2
( )E(r, )  0.
c2
Hence, we have three scalar equations for E(r, ) (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, )  E()exp  ikr 
Immediately we see that the wave is transversal:
0  divE(r, )  ik  E(r, )  k  E().
Let us now have a look at the rather exotic case of longitudinal waves. Those
waves can only exist for ()  0 in dielectrics at the longitudinal frequency
  L . In this case, we cannot conclude that div E(r, )  0 , and the wave
equation reads (the l.h.s. vanishes because ()  0 ):
 rotrot E(r, L )  0
If we try our plane wave ansatz and assume k real, it is usefull to split the
electric field into transversal and longitudinal components with respect to the
wave vector:
E(r , )  E() exp  ikr   E  () exp  ikr   E () exp  ikr  ,
with, k  E () and k  E ()
With rot E()exp  ikr    ik  E()exp  ikr  we get from the wave equation:
k  k  E(r, L )   0
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51
Now we plug in the electric field (longitudinal and transversal):
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()  ()   
k  k   E  E   exp  ikr   0,
k  k  E  exp  ikr   k  k  E  exp  ikr   0,
f
02  2
 We can invert the dispersion relation k () 
 k 2 E   0  E(r, L )  E (L )exp  ikr 
52

()  (k ) :
c
ω
2.4.3 Plane wave solutions in different frequency regimes
2
dictates the (complex) wavenumber only. Thus, different solutions for the
complex wave vector k = k   ik  are possible. In addition, the generalized
dielectric function is complex. In this chapter we will discuss possible
scenarios and resulting plane wave solutions.
ω=
ω0 +
2
c
ε0
k
f
ε∞
ω0
A) Positive real valued epsilon       '     0
This is the regime favorable for optics. We have transparency, and the
frequency is far from resonances. The dispersion relation gives
k 2  k'2  k''2  2ik ' k'' 
c
k
ε∞
ω=
The dispersion relation for plane wave solutions k 2  k 2  kx2  ky2  kz2  c2 ()
2
2
()  2 n 2 ()
2
c
c

k ' k''  0
k
Example 2: free electrons
 for plasma and metal
 Again the imaginary part of () is neglected
There are two possibilities to fulfill this condition, either k''  0 or k'  k'' .
()  ()  1 
A.1) real valued wave-vector k''  0
 In this case the wave vector is real and we find the dispersion relation

 2
k ()  n()  
n()
c
cn 
2p
2
 We again invert the dispersion relation k () 

()  (k ) :
c
ω
 Because k''  0 these waves are homogeneous (trivial, because the
amplitude is constant).
Example 1: single resonance in dielectric material
 for lattice vibrations (phonons)
ω = ck
ε′
ε′′
ωp
k
A.2) complex valued wave-vector k'  k''
ω
 The second possibility to fulfill the dispersion relation leads to a complex
wave-vector and so-called evanescent waves. We find
k 2  k'2  k''2 
 Now the imaginary part of () is neglected
 This means that
2
() and therefore k''2  k'2  k 2
c2
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53
 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 i) ( 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, )  E() exp i k'   r  exp  k''()r 
 The planes defined by the equation k''()r = const. are the so-called
planes of constant amplitude, those defined by k'()r = const. are the
planes of constant phase. Because of k'  k'' these planes are
perpendicular to each other.
 The factor exp  k''( )r  leads to exponential growth of evanescent waves
in homogeneous space. Therefore, evanescent waves are not normal
modes of homogeneous space and can only exist at interfaces.
B) Negative real valued epsilon ()  ()  0
 k'  k''  evanescent waves
k 2  k'2  k''2  
k''2 
2
()
c2
2
()  k'2 .
c2
As above, these evanescent waves exist only at interfaces (like for
()  ()  0 ). The interesting point is that here we find evanescent waves
for all values of k'2 . In particular, case i) ( k'  0 ) is included. Hence, we can
conclude that for ()  ()  0 we find only evanescent waves!
C) Complex valued epsilon ()
This is the general case, which is in particular relevant near resonances.
From our (optical) point of view only weak absorption is interesting.
Therefore, in the following we will always assume ( )  ( ) . As we can
see in the following sketch, we can have ()  0, ()  0, or
()  0, ()  0.
This situation (negative but real () can occur near resonances in dielectrics
( 0    L ) or below the plasma frequency (   p ) in metals. Then the
ε′
ε′′
dispersion relation gives
k 2  k'2  k''2  2ik'  k'' 
54
B.2) k'  k''  0
 k''2  0 and k '2  k 2

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2
()  0
c2
ε′
ε′′
1
2
ω
ω
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

k''2 
2
()
c2
 E(r , )  exp  k''r   strong damping
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



()  nˆ ()   n()  i() ,
c
c
c


k'  n(),
k''  ().
c
c
k () 
The dispersion relation in terms of the complex refractive index gives
k2  k 2 
2
2
2
( )  2  n ( )  i( ) 
2
c
c
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55
Here we have
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Summary of normal modes
()  ()  i()  n ()   ()  2in() (),
2
and therefore 
2
()  n 2 ()  2 ()
()  2n() ()
n 2 () 
 
2
sgn   1    /   1 ,

2 
 2 () 
 
2
sgn   1    /   1 .

2 
Two important limiting cases of quasi-homogeneous plane waves:
Region 1) ,   0,   , (dielectric media)
n( )  ( ),
( ) 
1 ( )
2 ( )
In this regime propagation dominates ( n()  () ), and we have weak
absorption:
k'2  k''2 
k' 
2
(),
c2
2k'  k'' 


n() 
(),
c
c
k'' 
2
().
c2

1  ()
() 
c
2 c ()
a)
b)
c)
d)
undamped homogeneous waves and evanescent waves
evanescent waves
weakly damped quasi-homogeneous waves
strongly damped quasi-homogeneous waves
2.4.4 Time averaged Poynting vector of plane waves
S(r, t ) 
 

 

E(r , )  E()exp  ikr   E()exp i  n   ek r    exp      ek r   .
c
c



 
Region 2)   0,   0,    , (metals and dielectric media in socalled Reststrahl domain)
n() 
1 ()
,
2 ()
() 
() ,
In this regime damping dominates ( n()  () ), we find a very small
refractive index. Interestingly, propagation (nonzero n) is only possible due to
absorption (see time averaged Poynting vector below).

t Tm /2
  (r, t )  dt ,
 E (r, t )  H

t Tm /2
For plane waves we find:
E(r , t )  E exp  ikr  it   E exp  ik r  k r  it 
 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
1 1
2 Tm
H (r , t ) 
1
k  E (r , t )
 0
assuming a stationary case E(t )  E()
 S(r, t ) 
1 k
1

2
 
 2
exp  2k r  E  n 0 n k' exp  2   nk" r   E .
2 0
2 0
 c

2.5 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
56
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57
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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58
A pulse is a continuous superposition of stationary plane waves (normal
modes) with different frequencies
1
2
3
...
A) beam  finite transverse width  diffraction

E(r , t )   E()exp i  k  r   t   d .
k

C) pulsed beams  finite transverse width and finite duration 
diffraction and dispersion
2w
A pulsed beam is a continuous superposition of stationary plane waves
(normal modes) with different frequency and different propagation direction

plane wave
E(r , t )   E(k , ) exp i  k  r  t   d 3kd 
beam

A beam is a continuous superposition of stationary plane waves (normal
modes) with different propagation directions
k1
k2
k3
k5
k4
…

E(r , t )   E(k )exp i  kr   t   d 3k

B) pulse  finite duration  dispersion

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 dispersion can be treated in a very similar way. Treating diffraction in the
framework of wave-optical theory (or even Maxwell) 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   const. For
technical convenience and because it is sufficient for many important
problems, we will make the following assumptions and approximations:
 ()  ()  0,  optical transparent regime  normal modes are
stationary homogeneous and evanescent plane waves
 scalar approximation
E(r , )  E y (r , )e y  E y (r , )  u (r , ).
2Tp
stationary wave
2.5.1 Diffraction of monochromatic beams in homogeneous
isotropic media
pulse
 exact for one-dimensional beams and linear polarization
 approximation in two-dimensional case
In homogeneous isotropic media we have to solve the Helmholtz equation
 E(r, ) 
2
   E(r, )  0.
c2
In scalar approximation and for fixed frequency  it reads
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u (r, ) 
59
u (r, )  k 2   u (r, )  0.
In the last step we inserted the dispersion relation (wave number k). In the
following we often omit the fixed frequency  .
Arbitrarily narrow beams  the general case
Our naming convention is in the following: kx  , k y  , kz  .
Then, the dispersion relation reads:
k 2     2  2   2
Thus, to solve our problem we need only a two-dimensional Fourier
transform, with respect to transverse directions to the “propagation direction
z”:

Let us consider the following fundamental problem. We want to compute from
a given field distribution in the plane z  0 the complete field in the half-space
z  0 ( z is our “propagation direction”)
u (r )   U (, ; z )exp i  x   y   d d .

In analogy to the frequency we call spatial frequencies. If we plug
this expression into the scalar Helmholtz equation
 u (r )  k 2    u (r )  0
This way we can transfer the Helmholtz equation in two spatial dimensions
into Fourier space
 d2
2
2
2
 2  k     U (, ; z )  0,
 dz

 d2
2
 2   U (, ; z )  0.
dz


The governing equation is the scalar Helmholtz equation
 u (r , )  k 2    u (r , )  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
This equation is easily solved and yields the general solution
U (, ; z )  U1 (, )exp i (, ) z   U 2 (, )exp -i (, ) z  ,
depending on  (, )  k 2 ()   2  2 .
We can identify two types of solutions:
A)
 2  0,  2  2  k 2 , i.e., k real  homogeneous waves
γ

u (r, )   U (k , )exp ik ()r  d 3k
k

which can be interpreted as a superposition of normal modes with different
propagation directions and wavenumbers k () (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
dispersion relation:
k 2  k 2  kx2  k y2  kz2 
2
  
c2
60
 only two components of k are independent, e.g., k x , k y .
2
   u (r, )  0,
c2
scalar Helmholtz equation
2.5.1.1
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α
β
B)
 2  0,  2  2  k 2 , i.e., k complex, because   kz imaginary.
Then, we have k = k   ik  , with k  = e x   e y and k  = e z .
 k'  k''  evanescent waves
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61
γ
α
α ²+ β²>k²
k
β
We see immediately that in the half-space z  0 the solution  exp  iz 
grows exponentially. Because this does not make sense, we have to set
U 2 (, )  0 . In fact, we will see later that U 2 (, ) corresponds to backward
running waves, i.e., light propagating in the opposite direction. We therefore
find the solution:
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We can formulate a general scheme to describe diffraction:
1. initial field: u0 ( x, y )
2.
initial spectrum: U 0 (, ) by Fourier transform
3.
propagation: by multiplication with exp i  ,   z 
4.
new spectrum: U (, ; z )  U 0 (, )exp i  ,   z 
5.
new field distribution: u ( x, y, z ) by Fourier back transform
This scheme allows for two interpretations:
1)
 U 0 (, )exp i (, ) z 
Furthermore the following boundary condition holds: U ( , ;0)  U 0 ( , ). In
spatial space, we can find the optical field for z  0 by inverse Fourier
transform:

u (r )   U (, ; z )exp i  x   y   d d .


u (r )   U 0 (, )exp i  ,   z  exp i  x   y   d d .
The resulting field distribution is the Fourier transform of the propagated
spectrum

u (r )   U (, ; z ) exp i  x   y   d d .

U (, ; z )  U1 (, )exp i (, ) z 
 U (, ;0)exp i (, ) z 
2)
The resulting field distribution is a superposition of homogeneous and
evanescent plane waves ('plane-wave spectrum') which obey the
dispersion relation.



u (r )   U 0 (, )exp i x   y    ,   z  d d .

Let us now discuss the complex transfer function H (, ; z )  exp i  ,   z  ,
which describes the beam propagation in Fourier space. For z = const. (finite
propagation distance) it looks like:

For homogeneous waves (real  ) 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 in propagation direction for different
spatial frequencies , .
The initial spatial frequency spectrum or angular spectrum at z  0
 1 
U 0 ( , )   
 2 
2


62
u0 ( x, y ) exp  i  x  y   dxdy,
amplitude
phase

(boundary condition) follows from u0 ( x, y )  u ( x, y , 0).
Obviously, H (, ; z )  exp i  ,   z  acts differently on homogeneous and
As mentioned above the wave-vector components ,  are the so-called
spatial frequencies. Another common terminology is “direction cosine” for the
quantities  / k ,  / k , because of the direct link to the angle of the respective
pane wave with the optical z -axis.
evanescent waves:
A) homogeneous waves  2  2  k 2



exp i  ,   z   1, arg exp i  ,   z   0
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1

Upon propagation the homogeneous waves are multiplied by the
phase factor
0 .7 5
exp i k 2  2   2 z 


0 .5
B) evanescent waves  2  2  k 2

64
0 .2 5


exp i  ,   z   exp    2  2  k 2 z  , arg exp i  ,   z   0


0
-1 0
−π 0 π
10
a
α
2
-0 .2 5

Upon propagation the evanescent waves are multiplied by an
amplitude factor <1
 All spatial frequencies (-  ) are excited.
 Important spectral information is contained in the interval   2 / a .
 Largest important spectral frequency for a structure with width a is
  2 / 2 .
 Evanescent waves appear for   k .
exp    2  2  k 2 z   1


This means that their contribution gets damped with increasing
propagation distance z .
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 ( , )
 0 for 2  2  k 2 we get evanescent waves.
Example:
Let us consider the following one-dimensional initial condition which
corresponds to an aperture of a slit:

1
u0 ( x)  
0
General result
We have seen in the example above that evanescent waves appear for
structures < wavelength in the initial condition. Information about those small
structures gets lost for z   .
u0(x)
a
for x 
2.
otherwise
 To represent the relevant information by homogeneous waves the

2
2
k 
n  a
following condition must be fulfilled:
n
a

Conclusion
-a/2
a 
sin   
 2   sinc  a  
U 0 ()  FT u0 ( x) 


a 
2 
 
2 
a/2
x
In homogeneous media, only information about structural details with
x , y   / 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 ( , )  U ( , ; z )  H ( , ; z )U 0 ( , )  u ( x, y , z )
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 get the same phase shift 
U (, ; z )  U 0 (, )exp i  ,   z   U 0 (, )exp iCz 
 u ( x, y, z )  exp iCz  u0 ( x, y )
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 U 0 (, )  0 only for   ,    k 2   2  2  C   2  2  02 .
66
|H|
arg H
1
kz
It is straightforward to see that the relevant spatial frequencies lie on a
circular ring in the   ,   plane. For constant spectral amplitude on this ring
Fourier back-transform yields: u0 ( x, y )  J 0 (r ) (see exercises)
k
β
α
α
Amplitude
k
β
Phase
The propagation of the spectrum in Fresnel approximation works in complete
analogue to the general case, we just use the modified transfer function to
describe the propagation:
U F (, ; z )  H F (, ; z )U 0 (, )
For a coarse initial field distribution u0 ( x, y , z ) the angular spectrum U 0 ( , ) is
Bessel-beam (profile)
Bessel-beam
nonzero for 2  2  k 2 only. Then, only paraxial plane waves are relevant
for transmitting information.
Description in real space
2.5.1.2
Fresnel- (paraxial) approximation
The beam propagation formalism developed in the previous chapter can be
simplified for the important special case of narrowband angular spectrum
U 0 (, )  0
for
Of course it is possible to formulate beam propagation in Fresnel (paraxial)
approximation in position space as well:

uF ( x, y , z )   U F (, ; z )exp i  x   y   d d 

 2  2  k 2

  H F (, ; z )U 0 (, )exp i  x   y   d d 
In this situation the beam consists of plane waves with only small inclination
with respect to the optical (z) axis (paraxial (Fresnel) approximation). Then,
we can simplify the expression for  by a Taylor expansion:
  2  2 
 2  2
   k 2   2  2  k  1 
k
2
2k 
2k

The resulting expression for the transfer function in Fresnel approximation
reads:
   
H  exp  i (, ) z   exp  ikz  exp  i
z   H F (, ; z )
2k


2
2
In particular, in Fresnel approximation we find no evanescent waves, since
their existence was already excluded to justify the paraxial approximation.
Hence, all structural details x , y must be always much larger than the
wavelength x , y  10 / n   / n .


  hF ( x  x, y  y; z )u0 ( x, y)dxdy

with the spatial response function from the convolution theorem
 1 
hF ( x, y; z )   
 2 
2


H F (, ; z )exp i  x  y   d d 


  2  2
 1 
   exp  ikz   exp  i

2k
 2 

2

z  exp i  x  y   d d .

This Fourier integral can be solved and we find:

  x 2  y 2   
ik
 ik
 k

hF ( x, y; z )  exp  ikz  
exp i  x 2  y 2     
exp ikz 1 
,
2z
2z2 
 2z

 2z



The response function is a spherical wave in paraxial approximation.
To sum up, in position space paraxial beam propagation is given by:
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uF ( x, y, z )  

ik
2
2 
 k
exp  ikz   u0 ( x, y)exp i  x  x    y  y    dxdy.



2z
 2z
Of course, the two descriptions in position space and in the spatial Fourier
domain are completely equivalent.
67
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Fourier transformation back to position space leads to the so-called paraxial
wave equation:
i
 
V (, ; z )exp i  x  y   d d 
z  
1 

  2  2 V (, ; z )exp i  x  y  d d
2k  
Remark on the validity of the scalar approximation
i
ˆ , ,  ei xy z  d d 
E(r, )   E



2  
1  2
v( x, y, z )    2  2   V (, ; z )exp i  x  y   d d 
2k  x y  
z
divE(r, )  0  Eˆ x  Eˆ y  Eˆ z  0
i
A) One-dimensional beams
 translational invariance in y-direction: =0
 and linear polarization in y-direction: Eˆ  U
scalar approximation is exact since divergence condition is strictly
fulfilled
B) Two-dimensional beams
 Finite beam which is localized in the x,y-plane: ,   0
 and linear polarization, w.l.o.g. in y-direction: Eˆ x  0 , Eˆ y  U
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 inhomogeneous media. We start with
scalar Helmholtz equation (for inhomogeneous media already an
approximation assuming weak spatial fluctuations in (r, ) ):
divergence condition: Eˆ y  Eˆ z  0


Eˆ z  , ,    Eˆ y  , ,   
Eˆ y  , ,   0
2

k   2  2
In paraxial approximation (     k ) the scalar approximation is
automatically justified.
2.5.1.3
2
2
In paraxial approximation the propagated spectrum is given by
U F (, ; z )  H F (, ; z )U 0 (, )
  2  2
 U F (, ; z )  exp  ikz V (, ; z )  V (, ; z )  exp  i
2k

i

1
V (, ; z )    2  2 V (, ; z )
2k
z
2
(r, ).
c2
We use the ansatz u ( x, y, z )  v( x, y, z ) exp  ik0 z  with k0  k being the
average wavenumber. With the SVEA condition
k 0 v  v / z
2

v ( x, y, z )  2ik0 v ( x, y, z )   (2)v( x, y, z )   k 2 (r, )  k02  v ( x, y, z )  0,
2

z

z



0

z U 0 (, )

i
Let us introduce the slowly varying spectrum V (, ; z ) :
Differentiation of V with respect to z gives:
u ( x, y, z )  k 2 (r, )u ( x, y, z )  0 with k 2 (r, ) 
we can simplify the scalar Helmholtz equation as follows:
The paraxial wave equation
  2  2
 exp  ikz  exp  i
2k


1
v( x, y, z )   (2)v( x, y, z )  0 paraxial wave equation
2k
z
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  .
y
2
68

z V0 (, ).

 k 2 (r, )  k02 
1 (2)

 v ( x, y , z )  
v ( x, y , z ) 
 v ( x, y , z )  0
z
2 k0
2 k0


This is the paraxial wave equation for inhomogeneous media (weak index
contrast).
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Beam propagation scheme
Relation between transfer and response function
h ( x, y; z ) 

1
H (, ; z ) exp i  x  y   d d 
2  
(2)
2.5.2 Propagation of Gaussian beams
Transfer functions for homogeneous space
H (, ; z )  exp i  ,   z   exp i k 2   2  2 z  exact solution


  2  2
H F (, ; z )  exp ikz  exp i
 2k

z  Fresnel approximation

The propagation of Gaussian beams is an important special case. First of all,
the transversal fundamental mode of many lasers has Gaussian shape.
Second, in paraxial approximation it is possible to compute the Gaussian
beam evolution analytically.
u
0

with k  k ()  k0 n()  n()
c
y
x
Fundamental Gaussian beam in focus
The general form of a Gaussian beam is elliptic, with curved phase.
  x2 y 2 
u0 ( x, y )  v0 ( x, y )  A0 exp    2  2   exp i( x, y )  .


  wx wy  
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We can say that U 0 ( , )  0 for  2  2   16 / w02 , because exp  4   0.02
Here, we will restrict ourselves to rotational symmetry  wx2  wy2  w02 and
(initially) 'flat' phase ( x, y )  0 , which corresponds to a beam in the focus.
The Gaussian beam in the focal plane (flat phase) is characterized by
amplitude A and width w0 : u0 ( x 2  y 2  w02 )  A0 exp  1  A0 / e . In practice,
For paraxial approximation we need k 2    2  2 
 k 2  16 / w02
the so-called 'full width at half maximum' (FWHM) is often used instead of w0 .
(
u0 x 2 + y 2
)
2
2.5.2.1

U 0 (, )  V0 (, ) 

 2 
 x2  y2 
A0  exp  
 exp  i(x  y )  dxdy

w02 


1
2
  2  2  A0 2
  2  2 
A0 2
w0 exp  
w
exp


,
0
2 
4
ws2 
 4 / w0  4

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
U
0

 10 n
n
  2  2 
V (, ; z )  U 0 (, )exp  i
z
2k


2
2


  2  2
A
 
exp
 0 w02 exp   w02
 i
4
4 
2k



z .

Fourier back-transformation to position space


 w2 i  
A0 2
w0   exp     2  2   0 
z   exp i  x  y   d d 
4
 4 2k  




1
x2  y 2
 A0
exp   2

2
2iz
 w0 1  2iz / (kw0 )  
1 2
kw0
v ( x, y , z ) 
 A0


x2  y 2
exp   2
.
z
 w0 1  iz / z0  
1 i
z0
1
With the Rayleigh length z0 which determines the propagation of a Gaussian
beam:
z0 
β
α
Angular spectrum in the focal plane
C) Check if paraxial approximation is fulfilled:
2
U ( , ; z )  V ( , ; z ) exp ikz 
E)
Angular spectrum at z  0 :
2
2
D) Propagation of the angular spectrum:
Propagation in paraxial approximation
 x2  y 2 
u0 ( x, y )  v0 ( x, y )  A0 exp  
.
w02 

 2    
    ,
 2   n   n 
 n
  
16
 paraxial approximation works for w0  10
Let us now compute the propagation of a Gaussian beam starting from the
focus in paraxial approximation:
1) Field at z  0 :
2)
w02 
æ w2
ö÷ 1
÷
= exp ççç- FWHM
çè 2w 02 ÷÷ø 2
w2
2
= 2 ln 2w 02 » 1.386w 02
- FWHM
= - ln 2  wFWHM
2
2w 0
72
kw02  2

w0 .
2
n
Note that we use the slowly varying envelope v !
Conclusion:
 Gaussian beam keeps its shape, but amplitude, width, and phase
change upon propagation
 Two important parameters: propagation length z and Rayleigh length
z0
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Some books use the “diffraction length” LB  2 z0 , a measure for the “focus
depth” of the Gaussian beam. E.g.: w0  10 n  LB  600 n .
From our computation above we know that the Gaussian beam evolves like:
v ( x, y, z )  A0


1
x2  y 2
exp   2
.
1  iz / z0
 w0 1  iz / z0  
For practical use, we can write this expression in terms of z-dependent
amplitude, width, etc.:
1 i
v( x, y, z )  A0
z
z0



2
2

x2  y 2


  x  y  z / z0 

exp
exp
i
 2
 2
2
2 
2 
1   z / z0 
w 1   z / z0   
 w0 1   z / z0   

 0 
v( x, y, z )  A0


 k
x2  y 2


i
2 
1
exp  2   z    exp  2
2



w
1
  
 0

 z
  z0   
1  


 z0 
x
 k x  y
 x2  y 2 
exp i
v ( x, y , z )  A( z ) exp   2

 2 R ( z )
 w ( z) 
2
A( z )  A0
1
 z
1  
 z0 
2
 A0
1
 2z 
1  
 LB 
Hence, we get for the Intensity profile I~A²:
2
,
The on-axis intensity (x=y=0) evolves like
 y2  

  z 2   exp i( z )  ,
z 1   0   
  z   
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 amplitude is given as:
 at different axial distances: (a) z  0 ; (b) z  z0 ; (c) z  2 z0 .
2
Here we used that w02  2 z0 / k . The (x,y)-independent phase ( z ) is given by
tan    z / z0 , the so-called Gouy phase shift.
Discussion:
The normalized beam intensity I / I 0 as a function of the radial distance
2
  exp i( z )

The normalized beam intensity I / I 0 at points on the beam axis (   0 )
as a function of z .
The beam width evolves like:
2
2
 z
 2z 
w( z )  w0 1     w0 1    ,
z
 LB 
 0
74
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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
 

x2  y 2 
 ( x, y , z )   k  z 
  ( z )   const.
2
R
(
z
)

 

The beam radius W ( z ) has its minimum value W0 at the waist ( z  0 ),
reaches
2W0 at z   z0 , and increases linearly with z for large z .
The radius of the phase curvature is given by
  z 2 
  L 2 
R ( z )  z 1   0    z 1   B  
  2 z  
  z  
Wavefronts of a Gaussian beam.
2.5.2.2
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


i 2 0 2 
q ( z ) z  iz0 z 2  z02
z  z0 z 1 
The radius of curvature R( z ) 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 (z=0) 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

z02
z2

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


 i 2n .
w ( z )
q( z ) R ( z )
z
2z
tan     
z0
LB
because z0 
kw02  2

w0
2
n
Thus, the q-parameter describes beam propagation for all z !
Example: propagation in homogeneous space by z  d
1
1


 i 2n
A) initial conditions:
q(0) R(0)
w (0)
B)
Phase retardation of the Gaussian beam ( z ) relative to a uniform plane
wave at the points of the beam axis.
propagation (by definition of q parameter) q(d )  q(0)  d
C) q-parameter at z  d determines new width and radius of curvature
1
1
1



 i 2n
q(d ) q(0)  d R(d )
w (d )
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2.5.3 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).
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 
before and after the element
 y2  ˆ  y1 
    M   .
 2
 1
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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":
y1
B
Az  B
y
Ay  B1
y
1

 1
z1  1  z2  2  1
2 Cy1  D1 C y1  D Cz1  D
1
1
A
The distances z1, z2 are connected by matrix elements, but not by normal
matrix vector multiplication.
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:
A1q0  B1
C1q0  D1
 propagation through N elements:
qn 
80
 no change of the width
Link to Gaussian beams
q1 
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Aq0  B
Cq0  D
 k  x2  y2  
 but change of phase curvature R f :  exp i

2
Rf


How can we see that?
Trick:
We start from the focus which is produced by the lens with z0  z f 
w2f
n
and w f is the focal width. 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
f 0
ˆ  1
M
 1 f

0
1 
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 :
 propagation (by definition of q -parameter) q(d )  q(0)  d
ˆ  1 d 
M
0 1 


q1 
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?
double concave
lens
 defocusing
q1 
double convex
lens
 focusing
A1q0  B1
q0

C1q0  D1  q0 f  1
i
1  q0 f  1 1


 n
q1
q0
 f w02
for
q0  i
w02
n
Be careful: Gaussian optics describes the evolution of the beam's width and
phase curvature only!
 Changes of amplitude and reflection are not included!
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2.5.4 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).
2.5.4.1
d
0
z2
z1
Transversal fundamental modes (rotational symmetry)
B)
R( z1 )  0, R( z2 )  0 ; R1, R2  0 ; z1  0, z2  0
z1
z2
0
Wave fronts 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 is the distance between the two mirrors 
z2  z1  d
2
0
z
 radius of wave front <0 for z <0
z
 from Chapter 1: beam hits concave mirror  radius Ri (i  1,2)  0.
 because R ( z )  z 
 beam hits convex mirror  radius Ri (i  1,2)  0.
Examples:
A) R( z1 ), R( z2 )  0 ; R1  0, R2  0 ; z1  0, z2  0
d
According to our reasoning above, the conditions for stability are:
R1  R( z1 ),

R2   R( z2 )
R1  z1 
z02
,
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!
 z02  R1 z1  z12  
d  R1  d  R2  d  R1  R2  d 
 R1  R2  2d 
2
0
The denominator  R1  R2  2d  is always positive we need to fulfill
2
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  d  R1  d  R2  d  R1  R2  d   0
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A)
R1 , R2  0; R1  d , R2  d ;  0  g1  1, 0  g 2  1;  0  g1 g 2  1  stable
B)
R1 , R2  0; R1  d , R2  d ;  g1  0, 0  g 2  1;  g1 g 2  0  unstable
If we introduce the so-called resonator parameters

d 
g1  1   ,
 R1 

d 
g 2  1  
 R2 
We can re-express the stability condition as
 d  R1  d  R2  d  R1  R2  d   dg1 g 2 R1 R2
1  g1 g 2  R1R2
d
 g1 g 2 1  g1 g 2  R1R2   0.
2
This inequality is fulfilled for

d 
d 
0  g1g2  1 or 0  1  1    1
 R1  R2 
This final form of the stability condition can be visualized: The range of
stability of a resonator lies between coordinate axes and hyperbolas:
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!
z2  z1  d
 d is the distance between the two mirrors 
Examples:
A) R( z1 )  0 , R( z2 )  0 R1  0, R2  0 ; z1  0, z2  0
Resonator stability diagram. A spherical-mirror resonator is stable if the
parameters g1  1  d / R1 and g 2  1  d / R2 lie in the unshaded regions
bounded by the lines g1  0 and g 2  0 , and the hyperbola g 2  1 / g1 .
R 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 g1  g 2 .
d
0
z1
B) R( z1 )  0, R( z2 )  0 ; R1, R2  0 ; z1  0, z2  0
Examples for a stable and an unstable resonator:
z2
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z1
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z2
0
Several low-order Hermite-Gaussian fuctions: (a) G0 (u ) ; (b) G1 (u ) ; (c)
d
G2 (u ) ; and (d) G3 (u ) .
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  
R
R
1
2 


the same stability condition
g1g2 1  g1g2  R1R2   0,
2
2.5.4.2
0  g1g2  1.
Intensity distributions of several low-order Hermite-Gaussian beams in the
transverse plane. The order (l , m) is indicated in each case.
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:
vG ( x, y, z )  A
 x y
w0
exp   2
w( z )
 w ( z)
2
2

 kx y 
 exp i 2 R( z )  exp i( z ).



2
2
higher order modes: ( x, y -dependence of phase is the same)
ul ,m ( x, y, z )  Al ,m

w0
x  
y 
Gl  2
Gm  2


w( z )  w( z )   w( z ) 
 k x2  y2 
exp i
 exp  ikz  exp i(l  m  1)( z ) .
 2 R( z ) 
  
Gl ()  H l ()exp   
 2
2
The functions Gl are given by the so-called Hermite polynomials:
Hl () ( H0  1, H1  2 and Hl 1  2Hl  2lHl 1 ).
2.5.5 Pulse propagation
2.5.5.1
Pulses with finite transverse width (pulsed beams)
In the previous chapters we have treated propagation of monochromatic
beams, where the frequency  was fix and therefore the wavenumber k ()
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 pulses. In this situation, we have typical envelope length T0 of 1013 s
(100 fs)  T0  1010 s (100 ps).
Let us compute the spectrum of the (Gaussian) pulse:
 t2 
f (t )  exp  i0t  exp   2 
 T0 
    0 2 
4
F ()  exp  
  s2  2  sT0  2
2
T0
4 / T0 

 spectral width: 4  1010 s 1  s  4  1013 s 1
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 group velocity is the velocity of the center of the pulse (see below)
 center frequency of visible light: 0  2   4  1015 s-1
 optical cycle: Ts  2 / 0  1.6 fs
k () 
Hence, we have the following order of magnitudes:
vg 
s  0    0    0
In this situation it can be helpful to replace the complicated frequency
dependence of the wave number (dispersion relation) by a Taylor expansion
at   0 . In general, a parabolic (or cubic) approximation will be sufficient:

n 
1 k
1
  n(0 )  0
n()  

c
vg  0 c 
 0 
c

n 
 n(0 )  0

 0 

ng (0 )  n(0 )  0
k

c
n(0 )
 vPH
ng (0 )
ng (0 )
n
 group index
 0
normal dispersion: n /   0  ng  n, vg  vPH
anomalous dispersion: n /   0  ng  n, vg  vPH
B)
ωs
group velocity dispersion (GVD) (or simply dispersion)
 GVD changes pulse shape upon propagation (see below)
D  D 
ω
ω0
k ()  k  0  
k
1  2k
   0  
 0
2 2
   0 
D
2
 ...
The following terminology is commonly used in the literature:
n  0 
1
k
 phase velocity
k  0   k0 , 
 0 
vPh 0
c
D 
 2k
2
 group velocity dispersion (GVD)
0
We reduce the complicated dispersion relation to three parameters:
 phase velocity vPh
 group velocity vg
 group velocity dispersion (GVD) D
Discussion:
A)
group velocity and group index

0
 1

  vg



 1
1 vg
  2
  vg 
vg 
 D 0
0
1 k
 group velocity

vg  0
 2k
2
 D0
vg

vg

0
0
Alternatively in telecommunication one often uses
D 
2
 1
    2 cD
  vg 

Let us now discuss the propagation of pulsed beams. We start with the scalar
Helmholtz equation, with the full dispersion (no Taylor expansion yet):
 u (r, ) 
2
() u (r, )  0
c2
In contrast to monochromatic beam propagation, we now have for each 
one Fourier component of the optical field:
dispersion relation: k 2 () 
2
()
c2
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Hence, we need to consider the propagation of the Fourier spectrum (Fourier
transform in space and time):
U (,, ; z)  U0 (,, )exp i  ,,  z 
with   , ,   k 2 ()   2  2
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H FP (, , ; z )  exp ik0 z  H FP (, , ; z )
In the last line of the above equation we have introduced a new variant of the
propagation function. H FP ( , ,  ; 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 (, , ) H FP (, , ; z ) 
The initial spectrum at z  0 is U0 (, , )
U 0 (, , ) 
1
90

 exp i  x  y   t   d d d 

u0 ( x, y, t )exp  i  x  y   t   dxdydt
3
 2   

u ( x, y, z, t )  exp i  k0 z  0t    U 0 (, , ) H FP (, , ; z ) 
Let us further assume Fresnel (paraxial) approximation is justified
 k 2 ()  2  2 
  2  2
U (, ,  ; z )  U 0 (, , ) exp ik    z  exp  i
2k   


z

We see that propagation of pulsed beams in Fresnel approximation in Fourier
space is described by the following propagation function (transfer function):



 exp i x   y   t  d d  d 


 slowly varying envelope
u ( x, y, z, t )  v( x, y, z , t )exp i  k0 z  0t  
  2  2 
H FP (, , ; z )  exp ik   z  exp  i
z
2k    

u(t)
v(t)
Now let us consider the Taylor expansion of k () from above. If the pulse is
not too short, we can replace
k ()  k  0  
k
1  2k
   0  
 0
2 2
   0 
t
2
 ..
0
   
Moreover, in the second term exp  i
z  (which is already small due to
2k   

paraxiality) we can use k ()  k (0 )  k0 (breaks down for T0  15 fs). By
doing so, we get the so-called parabolic approximation:
2
2
 1

H FP (, , ; z )  exp ik0 z  exp i    0  z  
v

 g
  2  2 
2
 D

 exp i    0  z  exp  i
z
2 k0
 2




 1

2
1 1 2
 exp ik0 z  exp i z   D  
  2   

v

2 k0

 g 2
 
with     0
t
|u()|
|u()|

0




v( x, y, z , t )   U 0 (, , ) H FP (, , ; z )exp i x  y   t  d d d 



In order to complete the formalism, we also need the initial spectrum of the
slowly varying envelope
u0 ( x, y , t )  v0 ( x, y , t ) exp  i0 t 
 V0 (, , ) 
1
 2  
3

v ( x, y, t )exp  i  x  y  t   dxdydt
 0
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Thus, the propagation of the slowly varying envelope is given by:



v( x, y, z, t )   V0 (, , ) H FP (, , ; z ) exp i x   y   t  d d d 



The next step is to introduce a co-moving reference frame with
 
H FP (, , ; z )  exp i
 vg

z  H FP (, , ; z )

i
i

 k 2 (r )  k02 
1 (2)
D 2

 v ( x, y, z, )   0
v ( x, y, z, ) 
v ( x, y, z, ) 
 v ( x, y, z, )  0
2
z
2 
2 k0
 2 k0



The last line above involves the propagation function H FP ( , ,  ; z ) , the
propagation function for the slowly varying envelope in the co-moving frame
of the pulse:
t 
z
vg
This frame is called co-moving because H FP ( , , ; z ) is now purely quadratic
in  , i.e., the pulse does not “move” anymore. In contrast, the linear  dependence in Fourier space had given a shift in the time domain. Thus, the
slowly varying envelope in the co-moving frame evolves as:
 z

2
 2  2  
v ( x, y, z, )   V0 (, , )exp i  D  
 

k0  
 2


 exp i x  y     d d  d 
with k0  k0  r 
 for D  0 'simple' diffraction  beam propagation
2.5.5.2
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     0 , corresponding to the assumption that we have a single
plane wave propagating in z-direction.
Description in frequency domain
1)
initial condition:
u0 (t )  v0 (t )exp  i0t 
2)
initial spectrum:
V0 ( )  U 0 ( )
3)
4)
The optical field u reads in the co-moving frame:
 z
2
   
V (, , ; z )  V0 (, , )exp i  D  

k0  
 2
2
i
2
2
1
V (, , ; z )
 2  2  
   D 
V (, , ; z )
2
z
k0 
As before in the case of monochromatic beams, we use Fourier backtransformation to get the differential equation in time-position domain
D 2

propagation of the spectrum: V (; z )  V0 ()exp i z  
2 

back-transformation to  leads to the following evolution of the slowly
varying envelope in the co-moving frame:

D 2

v ( z, )   V0 ()exp i z   exp  i    d 

2 

 


… u ( x, y, z , )  v ( x, y, z , )exp i  k0 z  0t    v ( x, y, z , )exp i  k0 z  0 z  0   
vg
 
 
Finally, let us derive the propagation equation for the slowly varying envelope
in the co-moving frame:
1 (2)
v ( x, y, z , ) D  2

 v ( x, y, z, )  0
v ( x, y, z , ) 
2
z
2 
2 k0
This is the scalar paraxial equation for propagation of so-called pulsed
beams. By using the slowly varying envelope approximation, it is possible to
generalize it for inhomogeneous media (weak index contrast)
v( x, y, z , t )   V0 (, , ) H FP (, , ; z ) exp i x  y    t  z vg   d d  d 




92
Description in time domain
A)
in time domain it is possible to describe pulse propagation by means of a
response function:

D 2
2
2 

exp  i
FT-1 of H P ( ; z )  exp i z    h P (; z ) 

2 
iDz

 2 Dz 
and the evolution reads

v ( z , )   h P (  ; z )v0 ()d 

Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
B)
the evolution equation for slowly varying envelope in the co-moving
93
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2. Initial pulse spectrum
v ( z , ) D 

v ( z, )  0
z
2 2
2
frame reads
i
V0 ()  A0
spectral width: 2s  4 / T02
Analogy of diffraction and dispersion
DIFFRACTION
i
DISPERSION

1

D 2
v( x, y, z )   (2)v( x, y, z )  0  i v ( z, ) 
v ( z, )  0
z
2k
2 2
z
(, )  
( x, y )

1
k0






Use results from propagation of Gaussian beams:
1 T2
k 
z0   w02   z0   0  0
2 D
2 
z0 describes Gaussian pulse
hence anomalious GVD is equivalent to 'normal' diffraction
Dispersion length: LD  2 z0
3. Evolution of the amplitude
 D but D  0 can vary
Examples of pulse propagation
2.5.5.3
 2T 2 
T0
0
exp  



4
2 


 2 
 i
T0
2 
exp  
exp

 2 D R ( z )  exp i( z )
2
T ( z)
 T ( z) 


v ( z , )  A0
with
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 with flat
phase
uV0
A( z )  A0 4
1
1
 
z
z0
,
2
 z
T ( z )  T0 1   
 z0 
2
A 2 ( z )T ( z ) = const.
'Phase curvature' is not fitting to the description of pulses  introduction of
new parameter Chirp
 2
2 
k
remember:  2  2   ( x, y, z ) 

x

y
R
( z)


monochromatic fields:
Vu00
ee
2T0
( )  


( )


 arbitrary time dependence of phase
τ
 t2 
u0 (t )  A0 exp   2  exp  i0t 
 T0 
 2 
 v0 ()  A0 exp   2 
 T0 

()
 2 ( ) ()
 () and 

 0  chirp

2

 parabolic approximation  'chirp' constant  dimensionless chirp
parameter (often just chirp)
C
T02  2  ( )
2 2
 variable frequency of the pulse in time
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95
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
integration leads to:

 ()

 ()  0  2C 2 ,

T0
  ()  0   C
C  0  up-chirp
leading
front
T ( z )  T0

T02
2
T ( z) 
C  0  down-chirp
2D
z
 T ( z ) / z  T0 / z0 
 const.
z0
T0
2D
z
T0
trailing
tail
τ
τ
phase curvature R( z )  Chirp C ( z )
Complete phase:
Gaussian pulse spreading as a function of distance z . For large
distances, the width increases at a rate 2 D / T0 , which is inversely


z 
2
z 
2
 ()  0     
 0      C ( z ) 2




vg  2 DR ( z )
vg 
T0


 C ( z) 
T02
z
 0
2 DR( z )
R( z )
proportional to the initial width T0 .
D  0 is only important if initial pulse is chirped, since otherwise quadratic
dependence is observed.
with
2.5.5.4
zz
z
z 2  z02
 C ( z)   2 0 2  
R( z) 
z  z0
z
z0 1 

z2
z02

1
z
T2
 C (0) = 0, C ( z0 ) = - sgn ( z0 ), C ( z  ¥) = - 0 with z0   0
2
z
2D
Attention: Chirp depends on sign of z0 and hence of D.
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
Complete field:
u ( z , )  A0

 2 
T0
2 
exp  
exp  iC ( z ) 2  exp i( z )  exp i  k0 z  0t  
2
T ( z)
T0 
 T ( z) 

Dynamics of a pulse is equivalent to that of a beam.
T2
important parameter  dispersion parameter z0   0
2D
1) z  z0 : no effect
2)
z  z0 : similar to beam diffraction
3)
z  z0 : asymptotic dependence
 2 (1  iC0 ) 
v0 ()  A0 exp  

T02


C0 – initial chirp
2. Input pulse spectrum
 2T 2 (1  iC ) 
0
0
V0 ()  A0 exp  

2
4(1
C
)

0


2
 spectral width: s 
4(1  C02 )
T02
 spectral width of chirped pulse is larger than that of unchirped pulse
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97
  4 / T  only for transform limited pulses
2
s
2
0
Aim: calculation of pulse width and chirp in dependence on z for given initial
conditions
Gaussian beam  q -parameter  similar for Gaussian pulse
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2) Evolution of q parameter
q( d )  q(0)  d 
1
2D 
T2 
 2 C ( d )  i 2 0  .
q(d ) T0 
T (d ) 
q( z )  q(0)  z.
Remember beams:
q( d ) 
1
1
2

i 2 .
q( z ) R ( z )
kw ( z )

1
1
2 DC ( z )
, w2 ( z )  T 2 ( z ),

D
R( z )
T02
2
2
2
2 2
2
2


 2 Dd 1  C0   C0T0  iT0   T0 T (d ) C (d )T (d )  iT0 
2 D 1  C02 
2 D C 2 (d )T 4 (d )  T04 
1
2 DC ( z )
2D

i 2
q( z )
T02
T ( z)
2 4
 2 Dd 1  C02   C0T02 
  C ( d )T0 T ( d )
a) real part 
2
2
4
C ( d )T ( d )  T04 
1  C0 
T0  pulse width at z  0 , which is not necessarily in the 'focus' or waist
b) imaginary part
hence at z=0:
1
2D
 2 C0  i
q(0) T0
C0  C (0).
T02T 2 ( d )
C ( d )T 2 ( d )  iT02 
2 D C ( d )T 4 ( d )  T04  
2
4) Set two equations equal
1
2D 
T2 
 2 C ( z )  i 2 0  (*)
q ( z ) T0 
T ( z) 

T02  C0  i
1
 2 Dd 1  C02   C0T02  iT02 
d 
2

2 D 1  C0 
2 D 1  C02  
3) Inversion of general equation (*) for q(d)
Use analogy: however it is limited to homogeneous space
k
T02T 2 ( d )
1

1  C02  C 2 (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 ) ).
Idea:
a) q ( z )  q (0)  z and
b)
1
1
2D
 2 C0  i insert into 
q( z )
q(0) T0
1
2D 
T2 
 2 C ( z )  i 2 0  .
q( z ) T0 
T ( z) 
Important case: Where is the pulse compressed to the smallest length?
given: C0 , T0 , C ( d )  0 (focus)
unknown: d , T (d )

 d 
generally: 2 equations for C0 , T0 , z, C ( z ), T ( z )  3 values predetermined
1 T02C0
1
C0
  sgn( D)
L
2 D 1  C02 
2

1
 C02  D
here: z=d
b) T 2 ( d ) 
1) Determination of q parameter at input
T02  C0  i
2 D 1  C02 

a) real part must be zero  2 Dd 1  C02  C0T02   0
set a) and b) equal  T(z), C(z)
q(0) 
98
T02
1  C02 
Resulting properties:
1) A pulse can be compressed when the product of initial chirp and
dispersion is negative  C0 D  0.
2)
The eventual compression increases with initial chirp.
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99
Physical interpretation
If e.g. C0  0 and D  0  vg /   0  'red' is faster than 'blue'
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
t


0
100
Pr (r, t )   0  R (t  t )Er (r, t )dt   Pr (r, t )   0  R ()Er (r, t  ) d 
Reality of the response function implies:
R   


1
1

-i
*
i
 d     e  2  d     e        
2 

Causality of the response function implies:
1 for   0
1

R         y    with       for   0 Heaviside
2
0 for   0
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)
First the 'red tail' of the pulse catches up with the 'blue front' until
C ( z )  0 (waist), i.e. the pulse is compressed, no chirp remains.
2)
Then C ( z )  0 and red is in front. Subsequently the 'red front' is faster
than the 'blue tail', i.e. the pulse gets wider.
C ( z)  

z
z0 1 
z2
z02

T2
z0   0
2D
2.6 (The Kramers-Kronig relation, covered by lecture
Structure of Matter)
The topic of “Kramers-Kronig relation” is not covered in the course
“Fundamentals of modern optics”. This topic will be covered rather by the
course “Structure of matter”. The following part of the script which is devoted
to this topic is just included in the script for consistency.
It is possible to derive a very general relation between () (dispersion) and
() (absorption). This means in practice that we can compute () from
() and vice versa. For example, if we have access to the absorption
spectrum of a medium, we can calculate the dispersion relation.
The Kramers-Kronig relation follows from reality and causality of response
function R . That the response function is real valued is a direct consequence
from Maxwells equations which are real valued as well, and causality is also a
very fundamental property: The polarization must not depend on some future
electric field. As we have seen in the previous chapter, in time-domain the
polarization and the electric field are related as
In the following, we will make use of the Fourier transform of Heaviside
distribution:
2   

 dt  t  e
it
P

i
     defined in integral only

In Fourier space, the Heaviside distribution consists of the Dirac delta
distribution

 d (  0 ) f    f  0 
 Dirac delta distribution

and an expression involving a Cauchy principal value:

P  d

 

i
i
i
f ()  lim   d  f ()   d  f () 
0




 

 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:

   




i
 d R    e 
 d     y    e



i
   
1 i 1
P    
2  2
 d       y  

   

i y   y  
1

P  d
()

2 
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    for < 0 arbitrarily
without altering the susceptibility! In particular, we can choose:
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
a)
a) y     y   
even function
b) y      y   
odd function
101
         
y     y   
In this case y     y    is a real valued and even function. We can exploit
this property and show that


1
1
 i

i
 y   
 d  y    e  2  d  y    e  y   is real as well
2 

Hence, we can conclude from equation (*) above that
      
i y   y  
1

P  d

2 
2

Now we have expressions for     ,     and can compute real and
imaginary part of the susceptibility:
*

i y   y   1 
i y   y  
1
     *    



 y  
P  d
P  d


2  
2
2  
2

i y  
1
           P  d 
 


Plugging the last two equations together we find the first Kramers-Kronig
relation:
   
1
      P  d 
 


1. K-K relation
Knowledge of the real part of the susceptibility (dispersion) allows us to
compute the imaginary part (absorption).
b)
y      y   
The second K-K relation can be found in a similar procedure when we
assume that y      y    is a real odd function. We can show that in this
case
 y   


1
1
 i

i
 d  y    e   2  d  y    e   y   is purely imaginary
2 

With equation (*) we then find that
     

i y    y  
1

P  d
(see (*))

2 
2
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and
Again we can then compute real and imaginary part of the susceptibility
102

i y   y   1 
i y   y  
1
P  d
P  d



 y  
2 
2
2 
2


    *     

i y  
1
P  d
 

and finally obtain
    
   
1
P  d
 


2. K-K relation
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  :
K-K relation for  :
         ()  ( ) ()  ( ) and
( )  ( )  1   ( )  1  i( )
()  1 
2  ()
P
d ,
  0 2  2
  ()  1
2
()    P 
d .
0 2  2

 dispersion and absorption are linked, e.g., we can measure absorption
and compute dispersion
example:
()  (  0 )  ()  1 
0
 Drude model
  2
2
0
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3.2 Propagation using different approximations
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
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
H  , ; zB   exp  i (, ) zB  where  2  k 2 ()   2  2 .
Then we have no constraints with respect to spatial frequencies ,  . We get
homogeneous and evanescent waves and can treat arbitrary small structures
in the aperture by:

u ( x, y, z )   U  (, ) H  , ; zB  exp i  x   y   d d 

where U  (, )  FT u ( x, y ) 
t ( x, y ) with t ( x, y)  0 for x , y  a (aperture)
Derivation of the response function
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  , ; z  z A U  (, ; z A )exp i  x  y   d d 
We start from the Weyl-representation of a spherical wave:

1
i
1
exp  ikr  
exp i  x  y  z   d d 


2  
r
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

2
 1
2

exp  ikr      exp i  x  y  z   d d   FT 1  H    2  h
z  r
 
and therefore

h  x, y , z   
or

u ( x, y, z )    h  x  x, y  y, z  z A  u ( x, y, z A )dxdy

with h 
1
 2 
2
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.
1  1

exp  ikr   with r  x 2  y 2  z 2 .

2 z  r

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, zB  u ( x, y, z A ) dxdy

3.2.2 Fresnel approximation (paraxial approximation)
From the previous chapter we know that we can apply the Fresnel
approximation if 2  2  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  , ; zB   exp  ikzB  exp  i B   2  2  
 2k

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hF  x, y, zB   
105
 k

i
exp  ikzB  exp i
x2  y 2 

 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
2  2  k 2
and the additional condition for the so-called Fresnel number N F
N F  0.1 with
NF 
a a
 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  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:
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 k 
2
2 
exp i
 x  x    y  y   

2
z


B
 k

 x 2  2 xx  x2  y 2  2 yy  y2  
 exp i
2
z


B
 k



 k
k
2
2
2
2 
 exp i
 x  y   exp -i z  xx  yy exp i 2 z  x  y  
2
z
B
B
B






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
 x2  y2  
 2N F

2 zB
zB
 k

 x2  y2    1
for N F  0.1  exp i
 2 zB

This means that we can neglect the quadratic phase term in x',y' and we get
for the far field:
uFR ( x, y, zB )  
 k

i
exp  ikzB  exp i
( x2  y 2 )
 zB
2
z


B

ky  
  kx
 u ( x, y)exp -i  x  y   dxdy

zB  
  zB

uF ( x, y, zB )   H F (, ; zB )U  (, )exp i  x  y   d d 


  hF ( x  x, y  y; zB )u ( x, y)dxdy


 k 
i
2
2 

exp  ikzB   u ( x, y)exp i
 x  x   y  y   dxdy


 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:
uF ( x, y, zB )  
a
 k 
i
2
2 
exp  ikzB   u ( x, y)exp i
 x  x   y  y   dxdy
a

z
2
 zB
B


Now, let us have a closer look at the exponential expression:
106
 i
 2 
 zB
2
exp  ikzB U  (k
 k

x
y
, k )exp i
( x2  y 2 )
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
  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
frequency    kx / zB ;   ky / zB  with spectral amplitude U  (kx / zB , ky / zB )
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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u ( x, y, z A )  A exp i k x x  k y y  k z z A  
need to Fourier transform the field at the aperture. In order to apply this
approximation we have to check that:
A)
 2  2  k 2  smallest features x, y    narrow angular
spectrum (paraxiality)
B)
NF  azB  1  largest feature a determines zB 
2
Example: x, y  10, a  100,   1m 
a2

Form the previous chapter we know that the diffraction pattern in the far field
in paraxial Fraunhofer approximation is given as:
 far field
zB  104   1 cm
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 2  2  k 2
contribute to the far field distribution.
Evanescent waves decay for kzB  1  zB   .
with N F 
uFR ( x, y , zB )  i
U  (
 2 
kx
x  y  zB
2
2
2
2
,
The Fourier transform gives the spectrum:
x  y 2  zB 2
2
ky
x  y  zB
2
2
2
Let us calculate the field for inclined incidence of the excitation.
The field behind the mask is given by:
zB
2
2
x
y
,k )
zB zB
The diffraction pattern is proportional to the spectrum behind the mask at
x
y
  k ,  k .
zB
zB
U  (k
 x  y  zB
2
  zB 
U  (k
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)
a a 1 z0

 z B  zB
2
1
2
I ( x , y , zB )  u ( x , y , zB ) 
3.2.4 Non-paraxial Fraunhofer approximation
N F  0.1
108
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 diffraction patterns induced
by plane masks in (paraxial) Fraunhofer approximation.

x
y
,k )
zB zB

  x

 y
 
 
k
z
exp
i
 k x  x  i  k  k y  y dxdy


z A   t ( x , y )exp  i  k
2
 2 

 zB
 

  zB
A
 x

y
 A exp  ik z z A  T  k  k x , k  k y 
z
z
B
 B

Hence, the intensity distribution of the diffraction pattern is given as:
I ( x , y , zB ) 
3.3.1 Fraunhofer diffraction pattern
1
  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.
An inclination of the illuminating plane wave just shifts the pattern (in paraxial
approximation).
Examples
A) Rectangular aperture illuminated by normal plane wave
Let us consider an incident plane wave, with a wave vector perpendicular
( k x  k y  0) to the mask t
u ( x, y , z A )  A exp  ik z z A 
or, more general, inclined with a certain angle
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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C) One-dimensional periodic structure 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 period slit aperture
with: period b and a size of each slit 2a :
Fraunhofer diffraction pattern from a rectangular aperture. The central lobe
of the pattern has half-angular widths  x   / Dx and  y   / Dy .
B) Circular aperture (pinhole) illuminated by normal plane wave
N 1
t ( x) for x a
t ( x)  t1 ( x  nb) with t1 ( x)   s
elsewhere
n 0
0
1 for x 2  y 2a 2
t ( x, y )  
0 elsewhere

 J ka x 2  y 2
1 zB
I ( x , y , zB )  
 ka x 2  y 2
z
 B
Then, we can express the mask function t as:
The Fourier transform of the mask then reads
  
2
Airy disk


 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

The Fraunhofer diffraction pattern from a circular aperture produces the
Airy pattern with the radius of the central disk subtending an angle
  1.22  / D .
The first zero of the Bessel function (size of Airy disk):
ka
 0.61
with
  1.22  
zB
zB
a
So-called angle of aperture:  
2  x 2  y 2
2 1.22

zB
a
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
N 1
sin  N 2 
perform the summation:  exp  in  
sin  2 
n 0
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 we have
 x
 x 
TS  k   sinc  k a 
 zB 
 zB 
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and therefore

2
 x  sin N k2 zxB b
I  sinc  k a 
2 k x
 zB  sin 2 zB b
2
50
I
I0


111

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The width of a maximum in the far-field diffraction patter x N (smallest
length scale in the pattern) is determined by N * b , the total size of the
mask (largest length scale of the mask).
N =7
These observations are consistent with the general property of the Fouriertransform: small scales in position space give rise to a broad angular
spectrum.
40
3.4 Remarks on Fresnel diffraction
30
Fresnel number  N F 
20
10
a a
 zB
 NF  10
( a large, zB small, zB  1/ 30 z0 )  shadow
 NF  0.1
( zB  3z0 )  Fraunhofer  FT of aperture
 10  N F  0.1 ( 1 / 30 z0  zB  3 z0 )  Fresnel diffraction
xS
0
-4 2 -3 5 -2 8 -2 1 -1 4
-7
0
7
= xp
xN
14
21
28
35
42
k b x
N
2 zB π
We find three important parameters for the diffraction pattern of a
grating:
 width of diffraction pattern  first zero of slit function TS
k
z
xS
a    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).
 maxima of diffraction pattern  maxima of grid function

sin 2 N k2
sin
2

These are the so-called diffraction orders, which are determined
exclusively by the grating period.
 width of maximum  zeros of grid function
z
k xN
b    xN  B
Nb
2 zB
x
zB
k x
2 zB
z
k xP
b  n  xp  n B
b
2 zB
N
112
b
b


Fresnel diffraction from a slit of width D  2a . (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
N F  10,1,0.5 and 0.1 . The dashed area represents the geometrical
shadow of the slit. The dashed lines at | x | ( / D )d represent the width
of the Fraunhofer pattern in the far field. Where the dashed lines coincide
with the edges of the geometrical shadow, the Fresnel number
N F  a 2 / d  0.5 .
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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. Here, we will use
again paraxial approximation, which is in general sufficient for optical
systems.
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.
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  in Fourier domain
 interaction with lens or filter  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 

f
f
 2f




plane wave
And therefore the transfer function for a thin lens is given as (see chapter
2.6.3):


k
tL  x, y   exp  i
x2  y 2 

 2f

In Fourier domain we find consequently
TL  ,    i
f
 2 
2
 f

exp i   2  2  
 2k

114
4.1.2 Optical imaging
4. Fourier optics - optical filtering
spherical wave
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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 light
propagation recipe.
A)
Spectrum in object plane
U 0 (, )  FT u0 ( x, y ) 
B)
Propagation from object to lens
(lens positioned at distance f)
U  (, ; f )  H F  , ; f U 0 (, )
 i

U  (, ; f )  exp  ikf  exp     2  2  f  U 0 (, )
 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 )
U  (, ; f )  TL  ,    U  (, ; f )
 i
f

2
2 
 f
exp  ikf   exp i              



 2k
 2 
2
 i

 exp    2  2  f  U 0 (, )d d 
 2k

D) Propagation from lens to image plane
U (, ;2 f )  H F  , ; f U  (, ; f )
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U (, ;2 f )  i
115
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116
f

2
2 
 f
exp  2ikf   exp i              



2
k

 2 
2
 i

 i

 exp    2  2  f  exp     2  2  f  U 0 (, )d d 
k
2
2
k




 i

 i

Quadratic terms    2  2  f  and     2  2  f  in the exponent
2
k
2
k




f 
2
2
cancel with quadratic terms from i
         .
2k 
U (, ;2 f )  i
f

f


exp  2ikf   U 0 (, )exp  i       d d 

 k

 2 
2
4.2 Optical filtering and image processing
 i
f
f 
 f
exp  2ikf  u0   ,   
k
k


 2 
4.2.1 The 4f-setup
2
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
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:
u ( x, y,2 f )  FT 1 U (, ;2 f ) 
 i

f
f
 f
exp  2ikf    u0   , 
2
k
k

2

 


  exp i  x  y   d d 

With the coordinate transformation
x  
f
,
k
y  
f

k

d  
2
dx,
f
d  
2
dy
f
we get:

 k

1
 u ( x, y,2 f )  i exp  2ikf    u0  x, y  exp  i  xx  yy   dxdy
f
f



 2 
u ( x, y, 2 f )  i
2
k k 
exp  2ikf U 0  x, y 
f
f 
 f
The image in the second focal plane is the Fourier transform of the optical
field in the object plane.  like far field in Fraunhofer approximation, but
zB  f . This finding allows us to perform an optical Fourier transform, and in
the Fourier plane it is possible to manipulate the spectrum.
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 Fourier plane is the FT of the optical field in 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 manipulation.
Our final goal is the transmission function H A (, ;4 f ) of the complete
imaging system:

u (  x,  y , 4 f )    H A  , ;4 f U 0 (, ) exp i  x   y   d d 

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Note: We will see in the following calculation that the second lens does a
Fourier transform  exp -i x  y   . 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).
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 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 )    hA ( x  x, y  y)u0 ( x, y) dxdy

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 )  i
 2 
f
2
k k

exp  2ikf U   x, y;2 f 
f
 f

Now we can make the link to the initial spectrum in the object plane U 0 :

 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:

As usual, the response function is given as:
hA ( x, y ) 

H A  , ;4 f  exp i x   y  d d 
 2   
1
2
f
f
From above we have H A  , ;4 f  ~ p( , )
k k

f
f
hA ( x, y ) ~   p ( , )exp i x   y  d d 
k
k

f

We introduce the coordinate transform  x  ,
k

y
f
k




 k

 k
k 
hA ( x, y ) ~   p ( x, y )exp i  xx  yy   d xd y ~ P   x,  y 
f
f
f 





k

k
u ( x,  y,4 f ) ~   P   x  x  ,  y  y   u0 ( x, y)dxdy
f
f

 
The response-function is proportional to the Fourier transform of the
transmission mask.
k
k
x,   y
f
f
4.2.2 Examples of aperture functions
Then we can write:

f
f
u ( x, y,4 f ) ~  U 0  ,   p( , )exp  i  x  y   d d 
k k

Example 1: The ideal image (infinite aperture)
Be careful, we use paraxial approximation  limited angular spectrum
By passing to mirrored coordinates x   x, y   y we get

f
f
u ( x,  y,4 f ) ~   p ( , )U 0  ,   exp i  x  y   d d 
k k

Hence we can identify the transmission function of the system
f
f
H A  , ;4 f  ~ p( , )
k k
Summary
 Fourier amplitudes get multiplied by transmission mask
 transmission mask  transmission function
118
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).
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p  1  P    x  x     y  y    u( x,  y,4 f ) ~ u0 ( x, y)  mirrored original
Example 2: Finite aperture
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Translated to position space, the smallest transmitted structural information is
2
2 f

given by: rmin 
max nD
A more precise definition of the optical resolution can be derived the following
way:
1 for x 2  y 2   D / 2 2
p ( x, y )  
0 elsewhere

k

k
u ( x,  y,4 f ) ~   P   x  x  ,  y  y   u0 ( x, y)dxdy
f
f

 
 kD  x  x 2   y  y 2 
 J1 2 f

 u ( x, y)dxdy,
~ 
0
2
2
kD

 x  x    y  y 
2f
One point of the object x0 , y0 gives an Airy disk (pixel) in the image:
Spatial filtering. The transparencies in the object and Fourier planes have
complex amplitude transmittances f ( x, y ) and p ( x, y ) . A plane wave
traveling in the z 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 g ( x, y ) is a
2
2 
  kD
 J1  2 f  x0  x    y0  y   


2
2
 2kDf  x0  x    y0  y 



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:
f ( x, y ) . The system has a transfer function
H (vx , v y )  p(fvx , fv y ) .
filtered version of
      D / 2
1 for f  2 
k
H A  , ;4 f  ~ 
0 elsewhere
f
k
2
2
Further examples for 4f filtering:
 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.
k

2
2
2
 f   f 
         D / 2           D / 2 
k  k 
 f

2
2
2
With 2  a 2  2 we can define an upper limit for the angular frequencies max
which are transmitted (bandwidth of the system)
2n D
k2  D 
   max  f 2
f2 2
2
2
max

kD
rmin  1.22
2f
Hence we find:
Transmission function:
2
rmin 
1.22f
nD
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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
are plane waves E(r, t )  E exp i k   r  t  .


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)
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.
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 )  E exp i(kz  t )
Because the field is transversal we have two free complex field components
 Ex exp  ix  


E   Ey exp  iy  


0


with Ex , y and x ,y being real
Then the real electric field vector is given as
 Ex cos  t  kz  x  


Er (r, t )   Ey cos  t  kz  y  


0


Here, only the relative phase is interesting    y  x
Conclusion:
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Normal modes in isotropic, dispersive media are in general elliptically
polarized; Ex , Ey and phase difference     y   x  are free parameters
5.3 Polarization states
Let us have a look at different possible parameter settings:
   n (or Ex  0 or E y  0 )
A) linear polarization
B)
circular polarization

Ex  Ey  E ,    / 2
   / 2  counterclockwise rotation
Ey
Ex
   / 2  clockwise rotation
Ey
Ex
These pictures are for an observer looking contrary to the propagation
direction.
C) elliptic polarization  Ex  Ey  0,   n
0
 counterclockwise
    2
 clockwise
Example:
Remark
A linearly polarized wave can be written as a superposition of two counterrotating circularly polarized waves. Example: Let's observe the temporal
evolution at a fixed position kz  0 with    / 2 .
 cos  t  
 cos  t  
 cos  t  






E  sin  t    E   sin  t    2 E  0 
 0 


 0 
0






124
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6. Principles of optics in crystals
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 In complete analogy we find for the D field:
3
Di (r, )  0 ij () E j (r , )
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 direction of 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, )  0   E(r, )
Pi (r, )   0 
j 1
ij ()

D(r, )  0  ()E(r, )
As for the polarization we find:
 DE
We introduce the following notation:
 χˆ    ij

ε    
 susceptibility tensor

 dielectric tensor
ij
 σˆ   εˆ    ij   inverse dielectric tensor
1
3

ij
() D j (r, )  0 Ei (r, )
j 1
The following properties of the dielectric and inverse dielectric tensor are
important:
 ij , ij are real in the transparent region (omit  ), we have no losses
(see our assumptions above)
 The tensors are symmetric (hermitian), only 6 components are
independent
ij   ji , ij   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 ij , it means that we are looking for directions
3
In the following we will write E  E , because we assume monochromatic
light and the frequency  is just a parameter. Now (anisotropic):
3
j 1
where D  E , i.e., our principal axes:
D(r, )  0   E(r, )
E j (r, )
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 ij 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 ),
but the field is influenced by the symmetries of the crystal (see next
subchapter 6.2).
126
0 Ei  ij D j  Di
j 1
This is a so-called eigenvalue problem, with eigenvalues  . If we want to
solve for the eigenvalues we get
det  ij  I ij   0, with I ij  ij
This leads to a third order equation in  , hence we expect three solutions
(roots)  (  ) . The corresponding eigenvectors can be computed from
3
 D
ij
()
j
  (  ) Di(  ) .
j 1
The eigenvectors are orthogonal:
Di( ) Di(  )  0 for  (  )   ( )
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127
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:
ij  i ij ,
ij  i ij 
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128
 orthorhombic, monoclinic, triclinic
1
ij
i
1  
0
0 






0
0 
 ij  
2 
 0
3   
0
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.
1     2    3  
6.3 The index ellipsoid
The index ellipsoid offers a simple geometrical interpretation of the inverse
1
dielectric tensor σˆ  εˆ 
The defining equation for the index ellipsoid is
3
 x x
ij i
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....)
j
1
i , j 1
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:
3
 D D
ij
i
i , j 1
3
j
 0 Ei Di  2 wel
i 1
In the principal coordinate system the defining equation of the index ellipsoid
reads:
1    2    3    Di   0     Ei
B)
Cubic crystals behave like gas, amorphous solids, liquids, and have no
anisotropy.
uniaxial
 two crystallographic equivalent directions
 trigonal (quartz, lithium niobate), tetragonal, hexagonal
1    2    3  
C) biaxial
 no crystallographic equivalent directions
1 x12  2 x22  3 x32 
x12 x22 x32
  1
1  2 3
Here the elements of the dielectric tensor can be related to refractive indexes
ni  i
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→
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130
6.4.1 Normal modes propagating in principal directions
k
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.
→
k
n3
n1
Dx
n3
n2
Dy
Dx
Summary:
 anisotropic media  tensor instead of scalar
 in principal system: ni  i
 The index ellipsoid is degenerate for special cases:
 isotropic crystal:
sphere
 uniaxial crystal: rotational symmetric with respect to z-axis and n1  n2
n1
n2
Dy
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 , D y  0
6.4 Normal modes in anisotropic media
and
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    r  t 
In general we have E  D , but here k  D  0  k  E  0 , and the two
polarization directions x,y are decoupled:
 a solution where the spatial and temporal evolution of the phase are
connected by a dispersion relation   (k ) or k  k ()
D1 , 1  D1 exp i  k1 z  t    D1 exp i1 ( z ) exp -it  with k12 


Before – isotropic media
In isotropic media the normal modes are monochromatic plane waves


E(r, t )  E exp i k   r  t 
with the dispersion relation
k 2    k 2 () 
2
()
c2
with ()  0 and real as well as k  E  k  D  0 . The normal modes are
elliptically polarized, and the polarization is conserved during propagation.
Now – anisotropic media
What are the normal modes in anisotropic media?
Di   0i Ei
2
1 ()
c2
D2 , 2  D2 exp i  k2 z  t    D2 exp i2 ( z ) exp -it  with k22 
2
2 ()
c2
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 polarization in the direction of a principal axis (x or y) only the phase
changes during propagation, thus we found our normal modes:




D(a )  D1 exp i  k ar  t   e1  k a2 
D( b )  D2 exp i  k br  t   e 2
2 2
na  k 12  normal mode a
c2
 k 2b 
2 2
nbk  k 22  normal mode b
c2
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 For light propagation in principle direction we find two perpendicular
linearly polarized normal modes with E  D .
1
S    E  H 
2
hence k is not parallel to S because S  E
6.4.2 Normal modes for arbitrary propagation direction
6.4.2.1
132
Geometrical construction
Before we will do the derivation and actually calculate normal modes and
dispersion relation, let us have a look at the results visualized in the index
ellipsoid. It is actually possible to construct the normal modes geometrically:
 For a specific crystal and a given frequency  we take (in principal axis
system) the i and construct the index ellipsoid.
 We then fix the propagation direction  k / k  u .
 We draw a plane through the origin of index ellipsoid which is
perpendicular to k .
 If the index ellipse is a circle, the direction of this particular k-vector defines
the optical axis of the crystal.
6.4.2.2
Mathematical derivation of dispersion relation
Let us now derive the dispersion relation for normal modes of the form


D(r , t )  D exp i k    r  t 
E(r, t )  E exp i k    r  t 
In the isotropic case we found the dispersion relation
k 2 ()  k 2 () 
 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 na , nb of the normal modes for the propagation direction u  k / k



ka  na , k b  nb
c
c
 The directions of the principal axes of the index ellipse are the polarization
direction of the normal modes D( a ) and D( b ) .
 The electric field vectors of the normal modes E( a ) and E( b ) follow from
(a )
i
E
D(a)
 i ,
 0 i
( b)
i
E
D( b)
 i
 0 i
 Thus, D( a, b )  E(a, b ) , and E(a , b ) are not perpendicular to k.
 This has a direct consequence on the pointing vector:
2
()
c2
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    r  t  , but the wavenumber depends on the direction u


of propagation
k  k  , 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   (k1 , k2 , k3 ) or   ( k , u1 , u2 , u3 ) or k  k (, u1 , u2 , u3 ) .
We start from Maxwell's equations for the plane wave Ansatz:
k D  0
k  E   0 H
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k H  0
133
k  H  D
Now we follow the usual derivation of the wave equation:
 k   k  E   
2 1
2 1
2


D
k
k

E

k
E
D



c 2 0
c 2 0
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 k1 
 u1 
 u1 





With  k2   k ()  u2   n()  u2  we can write
k 
u  c
u 
 3
 3
 3
k 2ui2
ui2

1

i  2 2 
i  i   1
1  2 
 k  c 2 i 
 n 


 Here k  E does not vanish as in the isotropic case!
 In the principal coordinate system and with Di   0i Ei we find
 ki k j E j  k 2 Ei 
j
2
i Ei
c2

 2
2
 2 i  k  Ei  ki k j E j
c
j


Remark: for isotropic media the r.h.s. of this equation would vanish ( k  E = 0 ).
Now, we have the following problem to solve:
 2 1  k22  k32
c
k2 k1


k3k1

2
k1k2
 2  k12  k32
2
c2
 E  0
 1   
k 2 k3
  E2    0 
   
2
  k12  k22   E3   0 
c2 3
k1k3
k3 k 2
The straightforward way to solve this problem is using det ..  0 , which gives
the dispersion relation   (k ) for given ki / k . However, there is a more
easy way to obtain the dispersion relation:
trick:
 2
2
 2 i  k  Ei   ki k j E j
j
c

Ei  

2
c2
ki
i  k 2

kjEj
j
j
j
 
i

u12  n 2   2  n 2  3  n 2  u22  n 2  1  n 2  3  n 2  u32  n 2  1  n 2   2  n 2 
n
ki2
1
2 
2


k
i

c 2 

i
2
 1  n 2   2  n 2  3 
 The resulting equation is quadratic in (n 2 ) 2 since the n6 -term cancels.
Hence, we get two (positive) solutions na , nb , and therefore


ka  na , k b  nb for the two orthogonally polarized normal modes D( a )
c
c
and D( b ) .
In particular, for the propagation in direction of the principal axis
( u3  1, u1  u2  0 , see 6.4) we find:
  n 2  1  n 2   2  n 2   n 2  1  n 2   2  n 2  3 

k jE j.
j
  n 2  1  n 2   2  3  0
 na2  1 ,
nb2  2
Finally, we can compute the fields of the normal modes. We know:
 2
2
 2 i  k  Ei   ki k j E j
j
c

Because div E   j k j E j  0 we can divide and get the (implicit) dispersion
relation:

final form of DR
 Explicit calculation (multiplication by all denominators):
Now we multiply this equation by ki , perform a summation over the index ' i '
and rename i  j on l.h.s:
k E
ui2
1
i  n2     n2
i

Result:
 For given  i    and direction  u1 , u2  , (because u12  u22  u32  1 ) we can
compute the refractive index n  , u1 , u2  seen by the normal mode.
j
ki2
2
  k2
c2 i
134
and hence
Ei  

2
c2
ki
i  k 2

kjEj
j
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 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:
where the sum does not depend on the index i . Hence the last term of the
equation must be constant
 ( k j E j  const. )
j
k1

1  k 2
c2
2
:
k2

2  k 2
c2
2
:
k3

3  k 2
c2
2
1k1

1  k 2
c2
2
:
 2 k2

2  k 2
c2
2
:
 3 k3

3  k 2
c2
2
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 D
(a )
(b )
 0?
isotrop
(be careful: E E
(a)
(b )
 0)
kku
2

 2 2 

i
2
 ka  c 2 i  kb  c 2 i 







2
2
2
c
kk
iui
iui
2

k
 2 2 a b 2  ka2 

b
  kb  ka   i  2 2 
2  
i  2

 k a  c 2 i 
 kb  c 2 i  





D(a ) D(b ) ~ 
uniaxial
biaxial
and with Di  0i Ei
 D1 : D2 : D3 
optical axis
optical axis
Knowing that the last term is a constant we can calculate the relation of the
individual field components from the first part of the equation
 E1 : E2 : E3 
136
2 2
a b i i
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 2
2 
ka ,b  2 i  2 i  ui2

2
k u
c
c
iui2
 1 
 
1 

 2 2 
c 2 i  2 2 
  i
i  2
 ka ,b  c 2 i 
 ka ,b  c 2 i 
 ka ,b  c 2 i 






2
2
a ,b i
2
6.4.3 Normal surfaces of normal modes
In addition to the index ellipsoid there is a second geometrical interpretation,
the so-called normal surfaces:
isotrop: sphere
uniaxial: 2 points with na  nb in the poles  connecting line defines the
optical axis (for 1   2  or , 3  e the z-axis is the optical axis)
biaxial: 4 points with na  nb  connecting lines define two optical axes
How to read the figure:
 fix propagation direction ( u1 , u2 )  intersection with surfaces
 distances from origin to intersections with surfaces correspond to
refractive indices of normal modes
 definition of optical axis  na  nb
Summary: there are two geometrical constructions:
A) index ellipsoid (visualization of dielectric tensor)
 fix propagation direction  index ellipse  half lengths of principal
axes give na , nb (refractive indices of the normal modes)
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 na , nb
 optical axis connects points with na  nb
Conclusion:
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u
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) .
2
1
 u22 
e

1
u32
,

or nb 2
kb2 
138
2 2
nb  u1 , u2 , u3 
c2
Hence for a given direction ui one gets the two refractive indexes na , nb .
6.4.4 Special case: uniaxial crystals
Let us now treat the special (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, e.g., x,y-direction and 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 rotational symmetric
around the z-axis, and we have
The geometrical interpretation as normal surfaces is straightforward and can
be done, w.l.o.g., in the k2 , k3 or y, z plane ( u1  0 ). We have with ki2  k02 n 2 ui2
A)
B)
ordinary wave
extraordinary wave
ka2  k12  k22  k32  k02or
2
2
1  k1  k2  1 k32

1
e
or k02
k02
1   2  or , 3   e
which we call ordinary and extraordinary refractive indices.
Then, we expect two normal modes:
A) ordinary wave
 na independent of propagation direction
B)
 nb depends on propagation direction
extraordinary wave
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 kvector.
 The extraordinary wave D( e ) 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
i n2     n2
i

For uniaxial crystals this leads to
u12
u22
u32
1


 2
2
2
2
 n  or   n  or   n  e  n
n 2  n 2  e   n 2  or   u12  u22   n 2  n 2  or  u32   n 2  e   n 2  or 
2
A)
2
ordinary wave: independent of direction
na2  or  k 2   n 2  k 2
a
a
0 or
c2
2
B)
extraordinary wave (derivation is your exercise): dependent on direction
What about the fields? We know from before that
 D1 : D2 : D3 
or k1
 k
k
: 2 or 2
: 2 e 3
2


or  k 2
or  k 2
e  k 2
c2
c2
c2
For the extraordinary wave all denominators are finite, and in particular k1  0
implies D1( e )  0 , hence D( e) is polarized in the y-z plane. Then, D( or )  D( e )
implies that D ( or ) is polarized in x-direction.
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7. Optical fields in isotropic, dispersive and
piecewise homogeneous media
7.1 Basics
7.1.1 Definition of the problem
In summary, we find for the polarizations of the fields:
A) ordinary: D perpendicular to optical axis and k ,
D  k, D  E
B)
extraordinary: D perpendicular to k and in the plane k -optical axis
D  k , D  E , because D1   0 0r E1 , D3  0 e E3
If we introduce an angle  , as in the figures below, to describe the
propagation direction, a simple computation of nb2 () for the extraordinary
wave is possible (exercise):
u2  sin ,
nb 2    
u3  cos 
eor
or sin 2   e cos 2 
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 interfaces)
 system of layers
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  “guided” waves  no 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
The following classification for uniaxial crystals is commonly used
 or   e  negative uniaxial
 or   e  positive uniaxial
 orthogonality of normal modes of hom. space: no interaction in
homogeneous space
 inhomogeneity breaks this orthogonality  modes interact and exchange
energy
 however, locally the concept of eigenmodes is still very useful and we will
see that the interaction is limited to a small number of modes
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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
rotrot E(r, ) 
2
E(r, )  i0 j(r, )  02 P(r, )
c2
In general, for isotropic media all field components are coupled due to the
rotrot 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
ETM
 Ex 
  0  ,
E 
 z
H TM
 0  0
  H y    H 
 0  0

  
7.1.3 Interfaces and symmetries
Up to now we treated plane waves of the form
E(r, t )  E exp i  kr   t  
Here, homogeneous space implies  exp  ikr 
and monochromaticity leads to  exp  it 
Now, we will break homogenity in x-direction by considering an interface in yz – plane which is infinite in y and z

Then, we can split the electric field as E  E   E with
 0 
 
E   E y  ,
 0 
 
 Ex 
  x 
  (2) 
2
2
E   0  ,    0  ,  (2)  2  2
x
z
  z 
 Ez 


 
E is polarized perpendicular to the plane of propagation, E is polarized
parallel to this plane.
Common notations are:
perpendicular: 
 s  TE (transversal electric)

 p  TM (transversal magnetic)
parallel:
Both components are decoupled and can be treated independently:
 (2) E 

E  (r, )  i 0 j (r, )   02 P  (r, )
c2
2
2
 (2) E  2 E (r, )  grad (2) div (2) E  i 0 j (r, )   02 P  (r, )
c
From
H(r, )  
i
rot E(r, )
0
we can conclude that the corresponding magnetic fields are
 0  0
ETE   Ey    E  ,
 0  0
   
 Hx 
H TE   0 
H 
 z
142
x
medium I
y
interface
medium II
z
k
W.l.o.g. we can assume k   k x , 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 homogenity 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  t    ETM  x  exp i  kz z  t  
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) fields
TE: E  E y and H z continuous
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TM: Ez and H  H y continuous
B)
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  f   
 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; ); H ( x, z; ) and in the following we will omit  .
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This makes sense: The wave equation for the y-component of the electric
field is of second order, so we need to specify the field and its first derivative
as initial condition at x=0.
TM-polarization
analog for transversal components H y  H , Ez :
 2

2
 2  kfx  kz ,   H ( x)  0
 x

Ez ( x) 

i
H ( x)
0f x
Again, we succeed describing everything in transversal components.
We have the following problem to solve:


E ( x), H ( x) at x  d for
 calculate fields (E, H) and derivatives
x
x
given values at x=0
 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
TE-polarization
E, H  F
generalized field 1
We have to solve the wave equation (no y-dependence because of
translational invariance):
i 0 H z ,  i0 Ez  G
generalized field 2
 2
 2 2 


f  ETE ( x, z )  0
 2
z 2 c 2 
 x
HTE ( x, z )  HTE ( x) exp  ikz z 
  2 2
2
 2  2 f  kz  ETE ( x)  0
c
 x

i
rotETE ( x, z )
with: H TE ( x, z )  
0
Now let us extract the equations for transversal fields Ey  E , H z :
 2

2
2
2
2
 2  kfx  kz ,   E ( x)  0 with kfx  kz ,   2 f    kz
c
 x

i 
H z ( x)  
E ( x)
0 x
and write down the problem to solve:
 2

2
 2  kfx  kz ,   F ( x)  0
 x

We use the ansatz from above:
ETE ( x, z )  ETE ( x)exp  ikz z  ,
144
G ( x)   f

F ( x)
x
with  fTE  1,  fTM 
1
f
We know the general solution of this system (harmonic oscillator):
F ( x)  C1 exp  ikfx x   C2 exp  ikfx x 
G ( x)   f

F ( x)  i f kfx C1 exp  ikfx x   C2 exp  ikfx x  
x
We have as initial conditions F(0), G(0) given, and can compute the constants
C1, C2:
F (0)  C1  C2
G (0)  i f kfx C1  C2 
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
F ( x)  cos  kfx x  F (0) 

1
i
C1   F (0) 
G (0) 
2
f kfx

146
1
sin  kfx x  G(0)
f kfx
G ( x)  f kfx sin  kfx x  F (0)  cos  kfx x  G (0)
We can write this formally as

1
i
C2   F (0) 
G (0) 
f kfx
2

The final solution of the initial value problem is therefore:
 F ( x) 
 F (0) 
ˆ

m
,
G ( x) 
G (0) 
where the 2 x 2-matrix m̂ describes propagation of the fields:
1
F ( x)  cos  kfx x  F (0) 
sin  kfx x  G (0)
 f kfx

 cos  kfx x 
ˆ  x  
m

kfx  f sin  kfx x 
G ( x)   f kfx sin  kfx x  F (0)  cos  kfx x  G (0)
1

sin  kfx x  
kfx  f

cos  kfx x  
By resubstituting we have the electromagnetic field in the layer 0  x  d .
 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
7.2.2 The fields in a system of layers
 A system of layers is characterized by i , d i
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 'GRIN' - GradedIndex-Profiles.
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, it just
requires matrix multiplication:
A) two layers
F
F
F
ˆ 2 (d 2 )    m
ˆ 2 (d 2 )m
ˆ 1 (d1 )  
m
G
 d1 d2
 G d1
 G 0
B)
N layers
N
F
F
ˆ F
ˆ i ( di )    M
 m
G
G
G
 d1 d2 .. dN  D i 1
 0
 0
N
ˆ  m
M
 ˆ i (di )
i 1
i
ˆ i have the same form, but different if , di , kfx

All matrices m
2
i    kz2 .
c2
Summary of matrix method:
 F (0) and G (0) given ( E , H z for TE, E z , H for TM)
 kz , if , i , di given  matrix elements
From above, we know the fields in one layer:
 multiplication of matrices (in the right order)  total matrix
 fields F ( D ) and G ( D )
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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 on the other side. We
have seen that after splitting in TE/TM polarization, continuous (transversal)
field components are sufficient to describe the whole field. What we will do
now is to link those field components with accessible fields, 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 incident ( k I ) , reflected ( k R ) and
transmitted ( k T ) fields:
 ksx 
k I   0  ,
k 
 z
with ksx 
  ksx 
k R   0  ,
 k 
 z 
2
s  kz2  ks2 ()  kz2 ,
c2
kcx 
 kcx 
k T   0 
k 
 z
2
c  kz2  kc2 ()  kz2 ,
c2
where s () and  c () are dielectric functions of substrate and cladding.
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148
sive, 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 I  ks sin R  I   R (reflection)
2. ks sin I  kc sin T  ns sin I  nc sin T (Snellius)
Let us now rewrite the fields in order to solve the reflection transmission
problem:
A) field in substrate
complex amplitudes FI , FR .
Fs ( x, z )  exp  ikz z   FI exp  iksx x   FR exp  iksx x  
Gs ( x, z )  isksx exp  ikz z   FI exp  iksx x   FR exp  iksx x  
B) field in layer system
 matrix method
Ff ( x, z )  exp  ikz z  F ( x)
Gf ( x, z )  exp  ikz z  G ( x)
and the amplitudes F ( x) and G ( x) are given by
F
F
ˆ
 G   M ( x)  G 
 x
 0
C) field in cladding
Fc ( x, z )  exp  ikz z  FT exp ikcx  x  D  
multi layer system
Gc ( x, z )  ic 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 I ), i , 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
As we have seen before, the k z component of the wave vector is conserved,
 k x gives the direction of the wave (forward or backward). The total length of
the wave vector on each layer is given by the dispersion relation for disper-
Field in cladding at x  D
x0
field in substrate at
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149
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:
FT

  M 11
 i k F   
 c cx T   M 21
FI  FR
M 12  


.
M 22   isksx  FI  FR  
We consider FI as known, and FR and FT as unknown and get:
  k M  c kcx M 11   i  M 21  sksxc kcx M 12  F
FR  s sx 22
I
sksx M 22   c kcx M 11   i  M 21  sksx  c kcx M 12 


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B) TM-polarization
1
TM  .

F  H  Hy,
In the case of TM polarization we have the problem that an analog calculation
to TE would lead to HR,T / HI , 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:
kz
N
FT 
2 s ksx ( M 11M 22  M 12 M 21 )
FI
( s ksx M 22   c kcx M 11 )  i ( M 21   s ksx  c kcx M 12 )
FT 
2 s ksx
FI
N
f
kx
f
ETM
Those are the general formulas for reflected and transmitted amplitudes. The
matrix elements depend on 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:
Ex
k
  sin    z
TM
E
k
k
TM
 E   Ex ,
kz
TE  1
With Maxwell we can link E x to H y :
i) reflected field
ERTE  RTE EITE
E
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
TE
22
sx
cx
cx
TE
11
TE
21
TE
11
TE
21
sx cx
sx cx
TE
12
TE
12
N TE
ETTE  TTE EITE
sx
M
result:
ERTM,T
TM
I
E

s H R ,T
, 
s,c H I
E TM  
k
1
Hy  
Hy
0
c0 
s / c relevant for transmission only
Hence we find the following for TM polarization:
with the reflection coefficient
with the transmission coefficient
k
1
1
kz H y 
 k  H   Ex 
0
0
ERTM  RTM EITM
ii) transmitted field
TTE 
Ex
As can be seen in the figure, we can express the amplitude of the ETM field in
terms of the Ex component:
A) TE-polarization
F  E  Ey ,
150
TE
22
2ksx
2k
 sx ,
TE
 kcx M   i  M 21
 ksx kcx M 12TE  N TE
RTM 
TE
11
->We get complex coefficients for reflection and transmission, which
determine the amplitude and phase of the reflected and transmitted
light.
  k M   k M   i   M  k k M 
 k M   k M   i   M  k k M 


c sx
TM
22
s cx
TM
11
s c
TM
21
sx cx
TM
12
c sx
TM
22
s cx
TM
11
s c
TM
21
sx cx
TM
12
N TM
TM
T
E
 TTM E
TM
I
with the transmission coefficient
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TTM 
2 sc ksx
 k
c sx
M
TM
22
 skcx M
TM
11
  i   M
s c
TM
21
 ksx kcx M
TM
12

151

2 sc ksx
N TM
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
TE
TE
M 22
 kcx M 11TE   i  M 21
 ksx kcx M 12TE
sx
TTE 
RTM 
M
7.3.1.2
 kcx M
TE
11
  i M
TE
21
 ksx kcx M
TE
12


 k
 k
c sx
1 
2
2
ksx EI  ksx ER 

20 
2
  kz2  kc2 ()  kz2
2 c
c
may be complex!
In contrast, in the cladding
The energy flux from the layer system into the cladding is
kcx 
S c ex 
1
2
  kcx  ET .
20
EI  ER 
2
TM
TM
M 22
 skcx M 11TM   i  sc M 21
 ksx kcx M 12TM 
TM
TM
M 22
 skcx M 11TM   i  sc M 21
 ksx kcx M 12TM 
2 sc ksx
.
N TM
2
1
S e x    E  H  e x
2
1
H 
 k   E 
0
we find
S ex 
1
1
2
2
  k e x  E 
  kx  E .
20
20
Since in an absorption free medium the energy flux is conserved, in an
absorption free layer the energy flux is also conserved.
2
s  kz2  ks2 ()  kz2 is supposed to be real,
c2
because we have our incident wave coming from there. The total energy flux
from the substrate to the layer system is given as
In the substrate, ksx 
  kcx 
2
ET
ksx
We now will compute the global reflectivity  and transmissivity  of a layer
system. Of course, we will decompose into TE and TM polarizations and
relate to the reflectivities TE,TM and transmissivities TE,TM . We know:
TM
ER  ETE
R  ER ,
ET  ETTE  ETTM
Reflectivity and transmissivity
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
With
S s ex 
152
Because we have energy conservation: S s e x  S c e x 
2ksx
NTE
c sx
TTM 
TE
22
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2
2
EI  ETE
 ETM

R
R
2
  kcx  TE 2
ET  ETM
T
ksx

2

2
2


  kcx 
  kcx 
2
2
2
2
  RTE 
TTE  EITE   RTM 
TTM  EITM .
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:
ETM
E
d
TE
I
E
 EI cos ,
TM
I
E
ETE
 EI sin .
2
Then, we can divide by the (arbitrary) amplitude EI and write


  kcx 
  kcx 
2
2
2
2
1   RTE 
TTE  cos 2    RTM 
TTM  sin 2 
ksx
ksx





k


2
2
2
2
cx
1  RTE cos 2   RTM sin 2  
TTE cos 2   TTM sin 2 
ksx


The red and blue terms can be identified as

1   
The global reflectivity and transmitivity are therefore given as

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153
As above, we can assume that ksx is always real, because otherwise we
have no incident wave. k cx is real for nc  ns sin I , but imaginary for
nc  ns sin I (total internal reflection).
  TE cos 2   TM sin 2 
  TE cos   TM sin 
2
2
with the reflectivities
2
TE,TM  RTE,TM ,
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TE,TM
  kcx 
2
TTE,TM .

ksx
for the two polarization states TE and TM.
The matrix for a single interface is the unit matrix
1 0
ˆ m
ˆ (d  0)  
M

0 1
7.3.2 Single interface
and it is easy to compute coefficients for reflection and transmission, and
reflectivity and transmissivity. Using the formulas from above we find:
7.3.2.1
A) TE-polarization
(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   0  ,
k 
 z
  ksx 
k R   0  ,
 k 
 z 
 kcx 
k T   0  .
k 
 z
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 


RTE 
 ksx  kcx   ns cos I 
 ksx  kcx  ns cos I 
TTE 
2ksx
2ns cos I
2ns cos I


2
2
2
 ksx  kcx  ns cos I  nc  ns sin I ns cos I  nc cos T
TE  RTE 
2
TE 
nc2  ns2 sin 2 I
n  n sin I
2
c
2
s
ksx  kcx
2
ksx  kcx
2
2

TTE 
2ksx
NTE
ns cos I  nc cos T
ns cos I  nc cos T
4k   kcx 
  kcx 
2
.
TTE  sx
2
ksx
ksx  kcx
 TE  TE  1
The continuous component of the wave vector, expressed in terms of the
angel of incidence, is


kz 
s sin I  ns sin I.
c
c
B) TM-Polarisation
RTM 
 ksx 
2
2
2
 2
2





 sin 2 I 
k
ni  ns2 sin 2 I
z
2 i
2 i
2 s
c
c
c
c

ns cos I ,
c
kcx 
 2

nc  ns2 sin 2 I  nc cos T ,
c
c
TTM 
2 sc ksx
.
N TM
1 0
ˆ m
ˆ (d  0)  
with: M

0 1
Then, the discontinuous component is given as
 kix 
 c ksx M 22  skcx M 11   i  sc M 21  ksx kcx M 12 
c ksx M 22  skcx M 11   i  sc M 21  ksx kcx M 12 



RTM 
 ksxc  kcxs   nsnc2 cos I  ns2
 ksxc  kcxs  nsnc2 cos I  ns2
TTM 
2ksx cs
2ns2 nc cos I
2ns cos I


,
2
 ksxc  kcxs  nsnc cos I  ns2 nc2  ns2 sin 2 I nc cos I  ns cos T
nc2  ns2 sin 2 I
n  n sin I
2
c
2
s
2

nc cos I  ns cos T
nc cos I  ns cos T
154
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TM  RTM 
2
TM 
ksx c  kcx s
ksx c  kcx s
155
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2
TE 
,
2
4k   kcx  sc
  kcx 
2
TTM  sx
2
ksx
ksx c  kcx s
ksx  ic
2
ksx  ic
2
1
TE 
4ksx  kcx 
ksx  kcx
2
156
 0.
Remark
 TM  TM  1
Remark
It may seem that we have a problem for I  0 . For I  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:
For metals in visible range (below the plasma frequency) we have

c  ns2 sin 2 I always
always TIR, because:   c   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


RTE  1  exp i TE 
 tan   tan
 TE
2
exp  i 
ksx  ic Z


 exp  2i 
ksx  ic Z  exp  i 

sin 2 I  sin 2 Itot
n 2 sin 2 I  nc2
c
 s

.
cos I
ksx
ns cos I
B) TM-polarization
7.3.2.2

Let us now consider the special case when all incident light is reflected from
the interface. This means that the reflectivity is unity.
TE 
ksx  kcx
2
ksx  kcx
2
TM 
ksxc  kcx s
2
ksxc  kcxs
2
 2
nc  ns2 sin 2 I we can compute the smallest angle of incidence
c
with TE,TM  1 :
 tan   tan
kcx  0  nc  ns sin Itot
nc
.
ns
For angles of incidence larger than this limit angle, I  Itot we have
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):
 TM

2

 c  s s
 tan TE ,
ksx c c
2
 TM   TE ,
As a consequence, incident linearly polarized light gets generally elliptically
polarized after TIR  Fresnel prism
Remark
The field in the cladding is evanescent  exp  ikxc x   exp  c x  .
 The averaged energy flux in the cladding normal to the interface
vanishes.
S
 2 2
2
ns sin I  nc2  ic  i k z2  2 c  imaginary
i
c
c
   kcx   0  TIR
exp  i 
ksx c  i cs Z


 exp  2i 
ksx c  ic s Z  exp  i 
In conclusion, we have seen that the phase shifts of the reflected light at TIR
is different for TE and TM polarization, and because  s   c
With kcx 
sin Itot 

RTM  1  exp i TM 
Total internal reflection (TIR) for S>C
7.3.2.3
x


1
2
 kE
20


x
1
2
  kx  E  0.



20
0
The Brewster angle
There exists another special angle with particular reflection properties. For
TM-polarization, for incident light at the Brewster angle  B we find RTM  0 :
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TM 
ksx c  kcx s
2
ksx c  kcx s
2
157
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158
 0,
 ksx c  kcx s
c2  s  sin 2 Bs   s2  c  sin 2 Bs 
sin 2 B 
sc  s  c 
s    
2
s
2
c


cos 2 B  1  sin 2 B  1 
c
 s  c 
total internal reflection
c
s

 s  c   s  c 
angle of incidence
With the last two lines we can write the final result for the Brewster angle:
tan B 
c
.
s
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.
total internal reflection
angle of incidence
7.3.2.4
sin B nc

tan B 
cos B ns


 ns sin B  nc cos B  nc sin   B 
2

At the same time the angle of the transmitted light is always
ns sin B  nc sin T  T 
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.

 B ,
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  s   c .
Let us start with an incident plane wave in TE polarization:
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EI ( x, z )  EI exp i  ksx x  kz z    EI ( x, z )  EI exp i  z  s x  
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Thus, the reflected beam appears shifted by d   (Goos-Hänchen Shift).
ER ( x, z )  EI exp i  z  s x   exp i    
with


ns sin I ,
c
s 
2 2
ns   2 .
c2
The reflected plane wave gets a phase shift, which depends on the angle of
incidence (here characterized by the transverse wave number ). We want to
treat beams, which we can write as a superposition of plane waves (Fourier
amplitude eI    ):
EI ( x, z )  d eI    exp i  z  s    x  
  0  
In the Fourier integral, we have to integrate over angular frequencies with
non-zero amplitudes eI    only ( eI   0     0 for      only)

EI ( x, z )  exp  i 0 z   d eI   0    exp i  z   s x  


ER ( x, z )  exp  i 0 z   d eI   0    exp i  z  s x   exp i   0     .

Let us make the following assumptions:

ns )
c
Then, it is justified to expand the phase angle into a Taylor series up to first
order:
  0       0  

     0   
 0
Then, the reflected beam at the interface at x  0 is given as:


ER (0, z )  exp i  0 z     0  

d eI   0    exp i  z     

We can identify the remaining integral as a shifted version of the incident
beam profile at x  0


ER (0, z )  exp i    0    0 EI  z    .
 TE
2
  2arctan
 0 = c ns sin I0 mean angular frequency
 all Fourier component undergo TIR (       Itot )
tan   tan

c

 c
ksx
s
 2  c2 nc2
2
We assume a mean angle of incidence I0 :
 small divergence of beam (narrow spectrum,  
Let us finally compute the shift d   . We know from before that the phase
shift for TIR is given as:

c2
2
n 
2
s
2
 2arctan
c
ksx
  ksx2  c2 
2
2
ksx 
c
1
2
2ksx
1 

c ksx
 
 2 
 c
 2
 2
c2

c ksx
ksx2
ksx2  c2
1 2
ksx
 
1
1
   d  2 xET tan I0 with xET 


and tan I0 
2
0
2
2

ksx  s
c
 0  c2 nc
 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 those multi-layer systems
become periodic. Periodic structures are important in physics (lattices,
crystals, atomic chains, waveguide arrays…), and we can gain insight in
general features of such systems by looking at optical properties of periodic
(dielectric) multi-layer systems, 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 resonators
(laser, interferometer).
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.
In order to keep things simple we will treat:
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 semi-infinite periodic multi-layer systems  x  0,  1 , d1  ,   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:
ER  EI  E0
and
iksx  EI  ER  
E0 iE0

2 2ksx
and
ER 
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). For our example of 2
periodically repeated layers, we have:
( x)  ( x  ) with the period   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; )  exp i  kx  kz ,  x  kz z  Ekx ( x)
If the Bloch wave is a solution to our problem, we can set the two expressions
equal:
Mˆ  exp iK   I EE 
looking for solutions which have the same amplitude after one period of the
medium, but we allow for a different phase  exp i  kx  kz ,  x  . Here kx is

the (yet) unknown Bloch vector. Because in this easy example we deal with a
one-dimensional problem, the Bloch vector is actually a scalar.
In the following, we will find a dispersion relation for the Bloch-waves
2
kx  kz ,  , in complete analogy to the DR for plane waves kx2  2     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 .
0
'
N
and with   exp  iK   we have to solve an eigenvalue problem.
Mˆ  ( K )I EE 
0
'
N
This eigenvalue problem determines the Bloch vector K and will finally give
our dispersion relation.
ˆ  I  0 to compute the
As usual, we use the solvability condition det M


dispersion relation expressed in   exp iK   . Hence we still need to
compute K afterwards.
  exp  iK    
with Ekx ( x   )  Ekx ( x ) being a periodic function. In other words, we are

E
E
 exp  iK    '  .
 E' 
  N 1
 E  N
ˆ m
ˆ  d2  m
ˆ  d1   M ij   kmik(2) mkj(1)
with M
E0 iE0

2 2ksx
E
 E .
x

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
ˆ E
M
 E' 
 E' 
  N 1
  N
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
i0 H z  G 
162
On the other hand, we know from our matrix method:
E
 E0
x 0
or
EI 
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 M11  M 22  
2
 
  M 11  M 22  

  1.
2


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     1 .
The corresponding eigenvectors (field and its derivative at x  N  ) can be
computed from
Mˆ  exp iK   I EE 
0
'
N
M 12  E 
M 
  11
 0
M 22   
 M 21
  E  N 
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163
  M 11    E  M 12 E  0

If field values of the Bloch mode, i.e. the function Ekx ( x   )  Ekx ( x ) , inside
the layers are desired, they can be computed by using the matrix formalism
E
and the above eigenvector  '  .
 E  N
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
ER
EI
E0 iE0
E
iE 

and EI  0  0 from before
2 2ksx
2 2ksx
2
with ER 
k E + iE0'
   sx 0
ksx E0 -iE0'
2
With our knowledge of the eigenvector from above we can compute the
reflectivity:
E '0 
k E + iE0'
  M 11
E0    sx 0
ksx E0 -iE0'
M 12
2
  M 11
M 12

  M 11
ksx -i
M 12
2
ksx + i
According to this formula two scenarios are possible:
A) total internal reflection  = 1
Hence  has to be real which results in the condition

 M 11  M 22 
2
1
with [for our example (n1,n2,d1,d2)]
M 11  cos  k1x d1  cos  k2 x d 2  
164
B) propagating normal modes
Hence  must be complex which results in the condition
1


E
  ' 
 EN .
 E  N    M 11  / M 12 

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k2 x
sin  k1x d1  sin  k2 x d 2 
k1x
k
M 22  cos  k1x d1  cos  k2 x d 2   1x 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.
 M 11  M 22 
2
1
We can compute the explicit dispersion relation:
  exp  iK  
 M11  M 22  

2
  M 11  M 22  

  1.
2


2
  exp  iK    cos  K    i sin  K    cos  K   
 cos  K  kz ,    
cos  K  
2
1
 M 11  M 22 
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  c2     kz2 .
Interpretation
 For the case of total internal reflection (  real,
 M11  M 22 
2
 1 ) the Bloch
vector K is complex,   exp  iK    exp  i  K    exp    K    .
Hence  K  kz ,    n and
2



 M 22 
  M  M 22  
n M

 K  kz ,      ln   1  11
  11

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.
 There is an infinite number of so-called band gaps or forbidden bands,
because n  1... . Those band gaps are interesting for Bragg mirrors and
Bragg waveguides
 The limits of the bands are given by
 K  kz ,    n and  K  kz ,    0
 K  kz ,   n / 
 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
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Special case: normal incidence
1.2
1
0.3
/cG
0.4
0
0.7
Re k
Im k
0.6
0.6
0.1
Examples for normal incidence
0.4
0.8
0.2
2
K

and
with the scaling constant G 
cG
G

Re k
Im k
Re k
Im k
0.4
/cG
In general there is a complex interplay of angle of incidence and frequency of
light on the properties of multilayer systems. Therefore let us have a look at
the simpler case of normal incidence  kz  0  . Then, the band gaps correspond to "forbidden" frequency ranges. In a graphical representation of the
dispersion relation for kz  0 it is common to use the following dimensionless
quantities
0
0.25
0.5
0.75
k/G
1
1.25
0.2
n1=1.4, d1=0.5
n2=3.4, d2=0.5
0.5
0
0
0.25
k/G
0.4
dispersion relation.
Inside the band gap, we find damped solutions:
0.2
0.1
0.1
0
0
0.25
0.5
0.75
k/G
n1=1.4, d1=0.5
n2=3.4, d2=0.5
1
1.25
0
0.5
K
 0.5 to describe the
G
Because of e iK  we need only   K    
0.3
0
0.25
0.5
0.75
k/G
1
1.25
1
n =1.0, n =1.4,
1.5
0
n1=1.4, d1 = 17/24 
n2=3.4 , d2 =7/24 
0.9
1
1 6 d /d 1
0.8
0.7
Transmission
/cG
/cG
0.3
0.2
166
0.6
0.5
0.4
0.3
0.2
It is common to use the reduced band structure - Brillouin zone, where the
information for all possible Bloch vectors is mapped onto the Bloch vetors up
to (k / G )  0.5 .
0.1
0
field in a Bragg-mirror
0.3
0.4
0.5
/cG
0.6
0.7
Transmission
Let us quantify the damping. In our example (n1,n2,d1,d2) we have
 M11  M 22   cos   n d
2

c
1 1
 
 1  n2 n1   
 

 cos  n2 d 2      sin  n1d1  sin  n2 d 2 
 c
 2  n1 n2   c
 c

In the middle of the first band gap (optimum configuration for high reflection)



we have B n1d1  B n2 d 2  , with B being the Bragg frequency, and
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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expression for ( K ) and assume small index contrast n2  n1   n2  n1  we
find
( K )max  2
n2  n1
n2  n1
168
cladding
(do derivation as an exercise)
 Damping is proportional to index contrast n2  n1
  M 11  M 22 

The spectral width of the gap 
 1 is then
2


gap 
2B
( K )max

substrate
(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, a so-called FabryPerot-Resonator. In this case, one single layer, the so-called cavity, is
distinguished from the others and we are interested in the forward and
backward propagating fields inside. The other layers function as mirrors, and
may be periodic multilayer-systems 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. For simplicity, we will restrict ourselves to TEpolarization. The figure shows our setup with two mirrors at x=0 and x=D,
characterized by coefficients of reflection and transmission R0, T0, RD, TD.
 EI , ER and ET ,  amplitudes of incident, reflected and transmitted
external fields in substrate and cladding.
 E , E  amplitudes of forward and backward running internal fields
inside the cavity.
Using the known coefficients of reflection and transmission, we can link the
field amplitudes. Our aim is to eliminate E and E
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 )
And with E ( D)  E (0) exp  ikfx D  E (0) 
ET
exp  ikfx D 
TD
C) At the upper mirror inside the cavity:
E ( D)  RD E ( D) 
RD
ET
TD
and with E (0)  E ( D)exp  ikfx D 
E (0) 
RD
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
 EI 
RD
E
ET exp  ikfx D   T 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 in
coefficients of the mirrors (R0, T0, RD, TD), cavity properties and angle of
incidence (D, kfx 
2
 f  k z2 ) reads:
c2
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Note:  and  for a lossless mirror are the same for both sides of the mirror.
For lossy mirrors only  is the same,  is then side-dependent.
For discussing the effect of the phase shift  we rewrite



cos   cos2  sin 2  1  2sin 2
2
2
2
Plugging everything in we get
T0TD exp  ikfx D 
E
.
TTE  T 
EI 1  R0 RD exp  2ikfx D 
This is the general transmission function of a lossless Fabry-Perot resonator.
In general, the mirror coefficients are complex and the fields get certain
phase shifts ( R, T )0,D  ( R , T )0,D exp  i0R,,DT  . Obviously, only the phase shifts
1  R  1  R 
2

2
0
D
1  R0 RD  2 R0 RD 1  2sin 2 2 
2
2
induced by the coefficients of reflection R0 , RD are important for the
1
2


4 R0 RD
 1  R0 RD 
2


sin
 .
2
2
2
2
2
1  R0 1  RD
 1  R0 1  RD

transmissivity of the FP resonator TE  TTE .
2
For given R0 , RD , T0 , TD and R0 , RD , the general transmissivity of a lossless
Fabry-Perot resonator reads:
2
T0 TD
TTE  T 
2
2
1  R0 RD
2

2
2

2

2
 




ksx
2
2
1  R0 1  RD
kcx

2


 1   2

1  m 



4
m
sin 2 



2
2

2
2
1  m
 1  m 
1  m   4m sin 2  
2  

1


  1  F sin 2 
2


R  R0  RD
1
2
2

1

2
T0 TD
kcx
ksx 1  R0 2 RD 2  2 R0 RD cos 
T0 TD
2
m  R0  RD  0  D ,
a) asymmetric FP-resonator
k
2 k fx
2
1  RD
 sx 1  R0
k fx
kcx

b) symmetric FP-resonator (Airy-formula for transmissivity)
Depending on whether the two mirrors have identical properties we
distinguish between symmetric and asymmetric FP-resonators.
Because we assume no losses we can use energy conservation at each
mirror to eliminate T0,D :

 1   2

4  0 D
0 D


sin 2  .


2
 1  0  1  D  1  0  1  D 


Discussion
2
 
0  R0 , D  RD
kcx 2
T .
ksx
2

and with


 2 R0 RD cos  2k fx D  R0  RD 
  



Here we introduced the phase-shift  which the field acquires in one roundtrip in the cavity.

170
with
F
4m
1  m 
2
,
1

 k fx D  
2
The Airy-formula (see also Labworks script, where phase shifts  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   1 for  / 2  n .
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172
with  the FWHM and    the distance in rad between two resonances.
1

 
1
2
1  F sin  m    
2
2




For narrow resonances (small line width  ) we can write
1
2
2

1
2
   



1
F

F

  
  1  
2
2
2
F
  
 


Remarks and conclusions
 We can do an analog calculation for TM-polarization  R
TM
resp. 
TM
 Resonances of the cavity with  MAX  1 occur for  / 2  k fx DMAX    m with
kfx 
2 2
nF  nS2 sin 2 I

 DMAX

 m   
m   


kfx
2 nF2  nS2 sin 2 I

cavity
2
 transmission properties of a given resonator depend on I and .
 MIN 
 minimum transmission is given as
1
1 F
 The line width  (FWHM) is inversely proportional to the finesse  .
 The Airy-formula can be expressed in terms of the finesse
1
  2 2

2
  1  
 sin  .
2 
   
 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    m reduced transmission by factor 1/2
k


D  m   m 
2
2
2
2

 k  2 nf  

D
 kD   
 it is favorable to have large F  e.g.:
4m
100  F 
m

1  m 
4 1  m 
1  

2
m
m2
4
m2
 100  m  0.2  m  0.8.
 pulses and beams can be treated efficiently in Fourier domain:
 e.g. TE: E (, , )  T (, , ) E (, , )
T
TE
I
Fourier back transformation: ET ( x, y , t )  FT 1  ET (, , ) 
 beams, because they always contain a certain range of incident angles I ,
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:

distance between resonance


full width at half maximum of resonance 
m
 

F 
 2
1  m








nfD D
D
example:   5  107 m ,   30, nf D  4  103m 
  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  )   these intra-cavity fields can be
very high  important for nonlinear effects
 Lifetime of photons in cavity: Via the "uncertainty relation":
Tc  const.  1 it is possible to define a lifetime of photons inside the
cavity:
1

k  nF  
c
D
c 
n D D
1
 
 F
 Tc 
 .
nF D
c


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173
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.
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 ,i  calculate ER,ET
Inside each layer we have plane waves  E  kz ,  exp i  kx x  kz z  t   .
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  c 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 (vs.
cladding and substrate? And what about a single interface?
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 1. condition: kz2 
174
2
max s,c ()
c2
 oscillating solution (standing wave) in the core (layers, fiber core, film)
~ A sin(k fx x)  B cos(k fx x)
with kix 
2
i ()  kz2  0
c2
 2. condition: kz2  2 max i ()
c i
2
Note that this 2. condition is not obvious at this stage. It appears due to
transition conditions at the boundaries to substrate and cladding.
In summary, the z - component of the wave vector of guided waves has to
fulfill:


 
max  ns ,c   k z  max  ni 
i
c

c 
The field structure in substrate and cladding is given as:
EC ( x, z )  ET exp  ikz z  exp  c  x  D  
ES ( x, z )  ER exp  ikz z  exp  s x 
xD
x0
If we go back to our usual reflection transmission problem, we have here
reflected and transmitted (evanescent) field for zero incident field EI  0 .
We will see in the following how this can be exploited to derive the dispersion
relation for guided waves.
7.4.2 Dispersion relation for guided waves
As we have seen above, we have ET , ER  0 for EI  0, and this can be
exploited. The coeffiecients for reflection and transmission are
Guided waves have the following field structure:
 plane wave in propagation direction ~ exp(ik z z )
 evanescent waves in substrate and cladding
~ exp[  c ( x  D )]
cladding
~ exp( µs x )
substrate
with s,c  kz2 
2
s,c ()  0
c2
E 
T TE,TM   T 
 EI 
TE,TM
E 
, R TE,TM   R 
 EI 
TE,TM
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
2  02
   0 In the case of resonance we get action for infinitesimal cause.
Hence, we can get the dispersion relation of guided waves by looking for a
vanishing denominator in R, T This reasoning is a general principle in
physics:
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175
The poles of the response function (or Greens function) are the resonances
of the system.
We know the coefficients of reflection and transmission for a layer system
from before, and they have the same denominator:
R
FR  sksx M 22   c kcx M 11   i  M 21  sksx c kcx M 12 

FI  sksx M 22   c kcx M 11   i  M 21  sksx  c kcx M 12 
Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-02-07s.docx
 d 2 2

2
 2  2 ( x)  E ( x)  kz E ( x)
c
 dx


 d 2 2m

2m
 2  2 V ( x)  ( x)   2 E( x)


 dx

guided waves

discrete energy eigenvalues

E  Vext
kz2 

max s,c 
c2
2
e
The pole is then given as:
With (because we have evanescent waves in substrate and cladding)
ksx  is  i kz2 

s (),
c2
kcx  i c  i kz2 

c ()
c2
We can write the general dispersion relation in an arbitrary layer system
1

M 11TE,TM  ssM 12TE,TM 
M 21TE,TM  s s M 22TE,TM  0
 c c
 c c
1

In addition to the material dispersion in the layers
Here, as usual, we have TE  1,
ki2 () 
TM 
2
i ()  ki2x  kz2
c2
we get the waveguide dispersion relation kz  , geometry 
The waveguide dispersion relation gives a discrete set of solutions, so-called
waveguide modes. For given i , d i ,  we get kz   .
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  

F exp  µ s x 
x
Hence, for given F(0) we get G (0)   s F (0)
( F (0)  free parameter)
 1 
F
F
ˆ
ˆ
 G   M ( x)  G   M ( x)    F (0).
 x
 0
 s
Analogy of optics to the stationary Schrödinger equation in QM
Optics (e.g. TE-polarization)
eS
2
Quantum mechanics
V
VS
2
kz
w2
c2
  sksx M 22  c kcx M 11   i  M 21  sksx c kcx M12   0
2
176
ef
eC
VC
E
Tunnel effect
kz2 
2
Film
c2

E  VBarriere
e
V
VB
2
kz
2
w
c2
E
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 ist that on both sides of the interface we have
2
evanescent waves: kz2  2 c,s , because
c
s,c  kz2 
2
c,s  0
c2
The general dispersion relation we derived before reads
M 11TE,TM  ssM12 TE,TM 
1

M 21TE,TM  s s M 22 TE,TM  0
 c c
 c c
and with the matrix for a single interface:
ˆ  1 0
M
0 1


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we get the dispersion relation
A)
1
177
ss
0
 c c
TE-polarization (   1)
s   c  0,  no solution because c , s  0
B)
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kz   
178
 s     c
c s      c
TM polarization
s    0
c  0
s    c
TM-polarization (   1 / )
c s

0
c s
with  c , s  0  c  s  0,
 on of the media has to have negative  (dielectric near resonance or
metal)
Surface polaritons may have very small effective wavelengths in z-direction:
c  1, s    1  kz   
In dielectrics
0(T)    L
we can find surface-phonon-polaritons.
In metals
  p
we can find surface-plasmon-polaritons.
Remark
Surface polaritons occur in TM polarization only, similar to the phenomenon
k
k
of Brewster-angle (no reflection for cx  sx  0, )
c s
Let us now compute the explicit dispersion relation for surface polaritons.
W.o.l.g. we assume s ()  0 , and because s () is near a resonance it will
show a much stronger  dependence than  c .
 µc s 
2
  µs c 
2



2 
2
 2s   kz2  2 c2   c2 kz2  2 s   
c
c




k z   
 s     c
c  c  s   
There is a second condition for existence of surface polaritons: c  s    0
Conclusion:
2  s     c
2

  eff  
 s    c  eff
7.4.4 Guided waves in a layer – film waveguide
The prototype of a waveguide is the film waveguide, where the waveguide
2
consists of one guiding layer with 2  f     kz2    .
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M 11  ssM 12 
cos  kfx d  
1

M 21  s s M 22  0
 c c
 c c
ss
 k

sin  kfx d   f fx sin  kfx d   s s cos  kfx d   0
f kfx
 c c
 c c

1 s s
 k      c c 
sin  kfx d 
 c c
 tan  kfx d  
 f 2 fx2 s s
k



cos  kfx d 
f kfx  cscs
f fx
 s s
cc f kfx
kfx  s c 



 
s 
tan  kfx d   f 2 c

kfx
 c2s
 c s
f
Here: TE-Polarisation    1
tan  kfx d  
Such film waveguides are the basis of integrated optics. Typical parameters
are:
d  a few   d 
  103  101
Fabrication of film waveguides can be achieved by coating, diffusion or ion
implantation. The matrix of a single layer (film) is given as:

cos  kfx d 
ˆ TE,TM  m
ˆ TE,TM (d )  
M
  k sin  k d 
fx
 f fx
sin  kfx d  

cos  kfx d  
1
f kfx
From this matric we can compute the dispersion relation for guided modes:
kfx  s  c 
,
2
kfx
 cs
This waveguide dispersion relation is an implicit equation for kz . For given
frequency  and thickness d we get several solutions with index kz
 
Here is an example for fixed frequency  , the effective index neff  kz /  
c
versus the thickness d:
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In a symmetric waveguide the fundamental more (   0 ) has cut-off = 0!
If we plot the dispersion curves for each mode we get a graphical
representation of the dispersion relation:
f
f
7.4.5 how to excite guided waves
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. c  s )
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
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
s  k z2 
2
2
  0  no guiding  k z2  2 s
2 s
c
c
We can plug this cut-off condition in the DR: tan  kfx d  
kfx  s  c 
,
2
 cs
kfx
  s s  c
 


f  s d   f
 s c
tan 
 f  s
 f  s
c

 d co
TE

c
f   s
  d co 
TE
s  c


 v 
arctan
 f  s


c
 f  s


arctan


a  v
 max /2

s  c
a0
with parameter of asymmetry a: s  f
a
s   c
 we can define a cut-off frequency for k   when we keep d fix
z
 we can define a cut-off thickness for k  d  when we keep  fix
z
Then, inside the waveguide we have (without radiative modes):
E ( x, z )  a E ( x)exp  ikz z 

 E ( x,0)  a E ( x)

with : P 
a 
E ( x )

kz 
2
E ( x) dx
20  
kz
20 P


Ein ( x) E ( x)dx,

 mode  couples to the incident field Ein (0) with amplitude a .
 Gauss-beam couples very good to the fundamental mode
B) adaption of wave vector  coupling through the interface
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eC
kz ef
eS
f
We know that k z is continuous at interface. The condition for the existence of
guided modes is
kz 

c,s
c
 We got a problem!
There are two solutions:
i) coupling by prism
we bring a medium with  p  f
kz 
d  z   d   z 
  z   A sin  gz  mit
g
2
P  period
P
coupling works for m’th diffraction order:
k zv  k z  mg
(prism) near the waveguide

2
p  kpx 
 p  kz2  0 .
c
c2
x
z
ep
möglichst
weit hinten
evaneszent
kpx kz
kz
kzn
grating on waveguide (modulated thickness of layer d):
2

kz 
c,s  ks2,cx 
c,s
c2
c

f
c
g
kz
ef
eS
but dispersion relation for waves in bulk media dictates
kz 
eC
ef
 light can couple to the waveguide via optical tunneling: ATR ('attenuated
total reflection')
ii) coupling by grating


ns sin   mg.
c
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