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Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2013-01-22.docx 1

Fundamentals of Modern Optics Winter Term 2012/2013

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 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) ............................................................ 16

1.1 Introduction ....................................................................................................... 16 1.2 Postulates ......................................................................................................... 16 1.3 Simple rules for propagation of light ................................................................. 17 1.4 Simple optical components............................................................................... 17 1.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) .............. 21

Ray equation ..................................................................................................... 21 1.5.1 The eikonal equation ........................................................................................ 23 1.5.2

1.6 Matrix optics ...................................................................................................... 24 The ray-transfer-matrix ..................................................................................... 24 1.6.1 Matrices of optical elements ............................................................................. 24 1.6.2 Cascaded elements .......................................................................................... 25 1.6.3

2. Optical fields in dispersive and isotropic media ...................................... 26 2.1 Maxwell’s equations ...................................................................................... 26

Adaption to optics ............................................................................................. 26 2.1.1 Temporal dependence of the fields .................................................................. 28 2.1.2 Maxwell’s equations in Fourier domain ............................................................ 29 2.1.3 From Maxwell’s equations to the wave equation ............................................. 29 2.1.4 Decoupling of the vectorial wave equation ....................................................... 30 2.1.5

2.2 Optical properties of matter .............................................................................. 31 Basics ............................................................................................................... 31 2.2.1 Dielectric polarization and susceptibility ........................................................... 34 2.2.2 Conductive current and conductivity ................................................................ 36 2.2.3 The generalized complex dielectric function .................................................... 37 2.2.4 Material models in time domain ........................................................................ 41 2.2.5

2.3 The Poynting vector and energy balance ......................................................... 42 Time averaged Poynting vector ........................................................................ 42 2.3.1 Time averaged energy balance ........................................................................ 44 2.3.2

2.4 Normal modes in homogeneous isotropic media ............................................. 46 Transversal waves ............................................................................................ 47 2.4.1 Longitudinal waves ........................................................................................... 48 2.4.2 Plane wave solutions in different frequency regimes ....................................... 49 2.4.3 Time averaged Poynting vector of plane waves .............................................. 55 2.4.4

2.5 Beams and pulses - analogy of diffraction and dispersion ............................... 55 Diffraction of monochromatic beams in homogeneous isotropic media .......... 57 2.5.1 Propagation of Gaussian beams ...................................................................... 68 2.5.2 Gaussian optics ................................................................................................ 74 2.5.3


Gaussian modes in a resonator ....................................................................... 78 2.5.4 Pulse propagation ............................................................................................. 83 2.5.5

2.6 (The Kramers-Kronig relation) .......................................................................... 96 3. Diffraction theory .................................................................................. 100

3.1 Interaction with plane masks .......................................................................... 100 3.2 Propagation using different approximations ................................................... 101

The general case - small aperture .................................................................. 101 3.2.1 Fresnel approximation (paraxial approximation) ............................................ 101 3.2.2 Paraxial Fraunhofer approximation (far field approximation) ......................... 102 3.2.3 Non-paraxial Fraunhofer approximation ......................................................... 104 3.2.4

3.3 Fraunhofer diffraction at plane masks (paraxial) ............................................ 104 Fraunhofer diffraction pattern ......................................................................... 104 3.3.1

3.4 Remarks on Fresnel diffraction ...................................................................... 109 4. Fourier optics - optical filtering .............................................................. 110

4.1 Imaging of arbitrary optical field with thin lens ............................................... 110 Transfer function of a thin lens ....................................................................... 110 4.1.1 Optical imaging ............................................................................................... 111 4.1.2

4.2 Optical filtering and image processing ........................................................... 113 The 4f-setup .................................................................................................... 113 4.2.1 Examples of aperture functions ...................................................................... 115 4.2.2 Optical resolution ............................................................................................ 116 4.2.3

5. The polarization of electromagnetic waves .......................................... 119 5.1 Introduction ..................................................................................................... 119 5.2 Polarization of normal modes in isotropic media ............................................ 119 5.3 Polarization states .......................................................................................... 120

6. Principles of optics in crystals ............................................................... 122 6.1 Susceptibility and dielectric tensor ................................................................. 122 6.2 The optical classification of crystals ............................................................... 124 6.3 The index ellipsoid .......................................................................................... 125 6.4 Normal modes in anisotropic media ............................................................... 126

Normal modes propagating in principal directions ......................................... 127 6.4.1 Normal modes for arbitrary propagation direction .......................................... 128 6.4.2 Normal surfaces of normal modes ................................................................. 132 6.4.3 Special case: uniaxial crystals ........................................................................ 134 6.4.4

7. Optical fields in isotropic, dispersive and piecewise homogeneous media ............................................................................................................. 137 7.1 Basics ............................................................................................................. 137

Definition of the problem ................................................................................. 137 7.1.1 Decoupling of the vectorial wave equation ..................................................... 137 7.1.2 Interfaces and symmetries ............................................................................. 138 7.1.3 Transition conditions ....................................................................................... 139 7.1.4

7.2 Fields in a layer system matrix method ..................................................... 139 Fields in one homogeneous layer .................................................................. 139 7.2.1 The fields in a system of layers ...................................................................... 142 7.2.2

7.3 Reflection – transmission problem for layer systems ..................................... 143 General layer systems .................................................................................... 143 7.3.1 Single interface ............................................................................................... 150 7.3.2 Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals ............. 157 7.3.3 Fabry-Perot-resonators .................................................................................. 163 7.3.4

7.4 Guided waves in layer systems ...................................................................... 169 Field structure of guided waves ...................................................................... 169 7.4.1 Dispersion relation for guided waves ............................................................. 170 7.4.2 Guided waves at interface - surface polariton ................................................ 172 7.4.3


Guided waves in a layer – film waveguide ..................................................... 174 7.4.4 how to excite guided waves............................................................................ 178 7.4.5

8. Statistical optics - coherence theory ..................................................... 181 8.1 Basics ............................................................................................................. 181 8.2 Statistical properties of light............................................................................ 183 8.3 Interference of partially coherent light ............................................................ 185


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, ...


Fiber laser: Limpert, Tünnermann, IAP Jena, ~10kW CW (world record)

B) Propagation of light through matter light-matter interaction

dispersion diffraction absorption scattering ↓ ↓ ↓ ↓ frequency spatial center of wavelength spectrum frequency frequency spectrum

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.


C) Light can modify matter light induces physical, chemical and biological processes used for lithography, material processing, or modification of biological

objects (bio-photonics)

Hole “drilled” with a fs laser at Institute of Applied Physics, FSU Jena.


D) Optics in our daily life

A small story describing the importance of light for everyday life, where all

things which rely on optics are marked in red.


E) Optics in telecommunications transmitting data (Terabit/s in one fiber) over transatlantic distances

1000 m telecommunication fiber is installed every second.


F) Optics in medicine, life sciences


G) Optical sensors and light sources new light sources to reduce energy consumption

new projection techniques

Deutscher Zukunftspreis 2008 - IOF Jena + OSRAM


H) Micro- and nano-optics ultra small camera

Insect inspired camera system develop at Fraunhofer Institute IOF Jena


I) Relativistic optics


K) What is light? electromagnetic wave (c= 3*108 m/s) amplitude and phase complex description polarization, coherence

Spectrum of Electromagnetic Radiation

Region Wavelength(nanometers)

Wavelength (centimeters)

Frequency (Hz)

Energy (eV)

Radio > 1088 > 10 < 3 x 109 < 10-5

Microwave 108 - 105 10 - 0.01 3 x 109 - 3 x 1012 10-5 - 0.01

Infrared 105 - 700 0.01 - 7 x 10-5 3 x 1012 - 4.3 x 1014 0.01 - 2

Visible 700 - 400 7 x 10-5 - 4 x 10-5 4.3 x 1014 - 7.5 x 1014 2 - 3

Ultraviolet 400 - 1 4 x 10-5 - 10-7 7.5 x 1014 - 3 x 1017 3 - 103

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


L) Schematic of 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

geometrical optics

wave optics

electromagnetic optics

quantum optics


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.


1. (Ray optics - geometrical optics) 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) Light propagates as rays. Those rays are emitted by light-sources and

are observable by optical detectors. B) The optical medium is characterized by a function n(r), the so-called

refractive index (n(r) 1 - meta-materials n(r) <0)

cnc

n

cn – speed of light in the medium

C) optical path length delay i) homogeneous media

nl ii) inhomogeneous media

( )B

A

n ds r

D) Fermat’s principle


( ) 0B

A

n ds r

Rays of light choose the optical path with the shortest delay.

1.3 Simple rules for propagation of light A) Homogeneous media

n = const. minimum delay = minimum distance Rays of light propagate on straight lines.

B) Reflection by a mirror (metal, dielectric coating) The reflected ray lies in the plane of incidence. The angle of reflection equals the angle of incidence.

C) Reflection and refraction by an interface Incident ray reflected ray plus refracted ray The reflected ray obeys b). The refracted ray lies in the plane of incidence.

The angle of refraction 2 depends on the angle of incidence 1 and is

given by Snell’s law: 1 1 2 2sin sinn n

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.


ii) Parabolic mirror Parallel rays converge in the focal point (focal length f). Applications: Telescope, collimator

iii) Elliptic mirror Rays originating from focal point P1 converge in the second focal point

P2

iv) Spherical mirror Neither imaging like elliptical mirror nor focusing like parabolic mirror parallel rays cross the optical axis at different points connecting line of intersections of rays caustic


parallel, paraxial rays converge to the focal point f = (-R)/2 convention: R < 0 - concave mirror; R > 0 - convex mirror. for paraxial rays the spherical mirror acts as a focusing as well as an

imaging optical element. paraxial rays emitted in point P1 are reflected and converge in point P2

1 1 2( )1 2z z R

(imaging formula)

paraxial imaging: imaging formula and magnification m = -z2 /z1 (proof given in exercises) B) Planar interface Snell’s law: 1 1 2 2sin sinn n

for paraxial rays: 1 1 2 2n n external reflection ( 1 2n n ): ray refracted away from the interface internal reflection ( 1 2n n ): ray refracted towards the interface total internal reflection (TIR) for:

2 2

2

1

sin sin nn

1 TIR


C) Spherical interface (paraxial)

paraxial imaging

1 2 12 1

2 2

n n n yn n R

(*)

1 2 2 1

1 2

n n n nz z R

(imaging formula)

1 2

2 1

n zmn z

(magnification)

(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)


two spherical interfaces (R1, R2, ) apply (*) two times and assume

y=const ( small)

2 1yf

with focal length: 1 2

1 1 11nf R R

1 2

1 1 1z z f (imaging formula) 2

1

zmz

(magnification)

(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

Ray equation 1.5.1Starting point: we minimize the optical path or the delay (Fermat)

( ) 0B

A

n ds r

computation:

B

A

L n s ds r


variation of the path: ( ) ( )s s r r

2 2

2 2 2

grad

2

1 2

1

B B

A A

L nds n ds

n n

ds d d d

d d d d d

d dds dsds ds

d dds dsds ds

d ddsds ds

r

r r r

r r r r r

r r

r r

r r

grad

grad

B

AB

A

d dL n n dsds ds

d dn n dsds ds

r rr

r r integration by parts and A,B fix

0L for arbitrary variation

grad d dn nds ds

r ray equation

Possible solutions: A) trajectory

x(z) , y(z) and 2 21ds dz dx dz dy dz

solve for x(z) , y(z) paraxial rays (ds dz )


, ,

, ,

d dx dnn x y zdz dz dx

d dy dnn x y zdz dz dy

B) homogeneous media straight lines

C) graded-index layer n(y) - paraxial, SELFOC

paraxial 1dydz and dz ds

22 2 20

220

1( ) 1 ( ) 12

n y n y n y yn

for 1a

2 2

2 2

1 ( )d dy d dy d y d y dn yn y n y n yds ds dz dz dz dz n y dy

for n(y)-n0<<1: 2

22

d y ydz

00

0 0

( ) cos sin

( ) sin cos

y z z z

dy

y

z y z zdz

The eikonal equation 1.5.2 bridge between geometrical optics and wave eikonal S(r) = constant planes perpendicular to rays from S(r) we can determine direction of rays grad S(r) (like potential)

2 2S n grad r r

Remark: it is possible to derive Fermat’s principle from eikonal equation geometrical optics: Fermat’s or eikonal equation

gradB A

B B

A AS S S ds n ds r r r r

eikonal optical path length phase of the wave


1.6 Matrix optics technique for paraxial ray tracing through optical systems propagation in a single plane only rays are characterized by the distance to the optical axis (y) and their

inclination () two algebraic equation 2 x 2 matrix Advantage: we can trace a ray through an optical system of many elements by multiplication of matrices.

The ray-transfer-matrix 1.6.1

in paraxial approximation:

2 1 1

2 1 1

y Ay B

Cy D

2 1

2 1

y yA B A BC D C D

M

A=0: same 1 same y2 focusing D=0: same 1y same 2 collimation

Matrices of optical elements 1.6.2A) free space

10 1

d

M

B) refraction on planar interface


1 2

1 00 n n

M

C) refraction on spherical interface

2 1 2 1 2

1 0n n n R n n

M

D) thin lens

1 01 1f

M

E) reflection on planar mirror

1 00 1

M

F) reflection on spherical mirror (compare to lens)

1 02 1R

M

Cascaded elements 1.6.3

1 1

1 1

N

N

y yA B A BC D C D

M M=MN….M2M


2. Optical fields in dispersive and isotropic media 2.1 Maxwell’s equations Our general starting point is the set of Maxwell’s equations. They are the basis of the electromagnetic approach to optics developed in this lecture.

Adaption to optics 2.1.1

Maxwell’s equations (macroscopic)

( , )rot ( , ) div ( , ) ( , )

( , )rot ( , ) ( , ) div ( , ) 0

tt t tt

tt t tt

B rE r D r r

D rH r j r B r

ext

makr

electric field ( , )tE r [V/m] magnetic flux density ( , )tB r [Vs/m2] or [tesla]

or magnetic induction dielectric flux density ( , )tD r [As/m2] magnetic field ( , )tH r [A/m] external charge density ( , )t rext [As/m3] macroscopic current density ( , )tj rmakr [A/m2]

Auxiliary fields

0

0

( , ) ( , ) ( , )

1( , ) ( , ) ( , )

t t t

t t t

D r E r P r

H r B r M r

dielectric polarization ( , )tP r [As/m2], magnetic polarization ( , )tM r [Vs/m2] (magnetization) electric constant (vacuum permittivity)

120 2

0

1 8.854 10c

As/Vm

magnetic constant (vacuum permeability)

70 4 10 Vs/Am


Light matter interaction The effect of the medium gives rise to polarization ( , )t fP r E and

magnetization ( , )t fM r B . In order to solve Maxwell’s equations we need material models describing those quantities.

In optics, we generally deal with non-magnetizable media ( , ) 0t M r (exceptions are metamaterials with ( , ) 0t M r ).

In Maxwell’s equations we also have so-called source terms: free charge density

( , )t rext [As/m3]

macroscopic current density ( , ) ( , ) ( , )t t t j r j r j rmakr cond conv [A/m2]

conductive current density ( , )t fj r Econd

convective current density ( , ) ( , ) ( , )t t t j r r v rconv ext

In optics, we generally have no free charges which change at speeds corresponding to the frequency of light:

( , ) 0 ( , ) 0t t r j rext conv

With the above simplifications, we can formulate Maxwell’s equations in the

context of optics:

0 0

0

( , )rot ( , ) div ( , )

( , )rot

div

( , ) div ( , )

( ,

( , )) 0

)

( ,

t

tt

tt tt

tt ttt

H rE r E r

E rH r H

r

r

P

P rj r

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 those 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


( , ) ( , )t t r div P rb

and a bound current density

( , )( , ) ttt

P rj rb

This essentially means that we can describe the same physics in two different ways (see generalized complex dielectric function below).

Complex field formalism: 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: r ( , )tE r

complex mathematical representation: ( , )tE r

12( , ) ( , ) ( , ) Re ( , )t t t t E r E r E r E rr

This means in general: For calculation we use the complex fields [ ( , )tE r ] 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).

Temporal dependence of the fields 2.1.2When 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( )t i phase is fixed coherent, infinite wave train e.g.:

( , ) ( )exp( )t t E r E r i

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 A) (see next section)

( , ) ( , )exp( )

1( , ) ( , )exp( )2

t t d

t t dt

E r E r

E r E r

i

i

Maxwell’s equations in Fourier domain 2.1.3We 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 1exp , exp ( , )2 2

dt i t i dt t i tt it

E r E r E r

rule: FT it

Now we can write Maxwell’s equations in Fourier domain:

0 0

0

rot ( , ) ( , ) div ( , ) div ( , )rot ( , ) ( , ) ( , ) ( , ) div ( , ) 0

ii it

E r H r E r P rH r j r P r E r H r

From Maxwell’s equations to the wave equation 2.1.4From 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 ( , )t rot E r

00 0( , ) ( , ) ( , ) ( , )( , )tt t

t t ttt t

H r P rrotrot E rr Erot j r

And find the wave equation for the electric field

2

2

2

0 22 01 ( , ) ( , )( , ) ( , ) t t

t tt t

c t

E rrotrot E rr j r P

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:

0 ( , ) ( , ) 0t tdiv E r P r

Once we know the electric field, we can easily compute the magnetic field:

0

( , ) 1 ( , )t tt

H r rot E r


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 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

B) Frequency domain derivation We can do the same procedure in the Fourier domain and find

2

20 02( , ) ( , ) ( , ) ( , )

c

rotrot E r E r j r P ri

and

0 ( , ) ( , ) 0 div E r P r

magnetic field:

0

( , ) ( , )

H r rot E ri

transferring the results from the Fourier domain to the time domain for stationary fields: take solution and multiply by -i te . for non-stationary fields and linear media inverse Fourier

transformation

( , ) ( , )exp( )t t dE r E r

i

Decoupling of the vectorial wave equation 2.1.5 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 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 / 0y


(2)

(2)

(2)

0

x z

x z

E Exx x z

y

E E zz x z

E

E

E

rot rot E grad div E E

decomposition of electric field

E E E

0

, 00

x

y

z

EE

E

E E

with Nabla operator (2) 0x

z

, and Laplace 2 2

(2)2 2x z

gives two uncoupled wave equations

2(2) 2

0 02

2(2) (2) (2) 2

0 02

( , ) ( , ) ( , )

( , ) ( , ) ( , )

c

c

E E r j r P r

E E r grad div E j r P r

i

i

properties 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 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).

Basics 2.2.1We 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. We use the simplest approach, the so-called Drude-Lorentz model for free or bound charge carriers (electrons):

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 2E etc.), we can use linear response theory.

For the polarization ( )P E (for ( )j E very similar):

description in both time and frequency domain possible in time domain we introduce the response function

( , )tE r medium (response function) ( , )tP r

0( , ) ( , ) ( , )

t

i ij jj

P t R t t E t dt

r r r

with R̂ being a 2nd rank tensor , ,i x y z and summing over , ,j x y z

in frequency domain we introduce the susceptibility ( , )E r medium (susceptibility) ( , )P r

0( , ) ( , ) ( , )i ij jj

P E r r r

response function and susceptibility are linked via Fourier transform (convolution theorem)

( ) ( )exp(12

)ij ijR t t d

i

Obviously, things look friendlier in frequency domain. Using the wave equation from before we find (assumption of no current)

22

02

22

02

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

c

c

rotrot E r E r P r

E r E r graddivE r P r

and for auxiliary fields

0( , ) ( , ) ( , ) D r E 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) linear, homogenous, isotropic, non-dispersive media (most simple but

very unphysical case) homogenous ( , ) ( )ij ij r

isotropic ( , ) ( , )ij ij r r

non-dispersive ( , ) ( )ij ij r r instantaneous: ( , ) ( ) ( )ij ijR t t r r

( , )ij r is a scalar constant

0 0( , ) ( , ) ( , ) ( , )t t P r E r P r E r (unphysical!)

0 0( , ) ( , ) ( , ) ( , ) 1t t D r E r D r E r

Maxwell: 0divD ( , ) 0 div E r

2

2( , ) ( , ) 0c

E r E r 2

2 2( , ) ( , ) 0t tc t

E r E r

approximation only for a certain frequency range, because all media are dispersive

based on an unphysical material model B) linear, homogeneous, isotropic, dispersive media ( )

0

0

( , ) ( ) ( , ),( , ) ( ) ( , )

( , ) 0 ( , ) 0 ( ) 0.

P r E rD r E r

div D r div E r for

2

2( , ) ( , ) 0c

E r E r Helmholtz equation

This description is sufficient for many materials. C) linear, inhomogeneous, isotropic, dispersive media ( , ) r

0

0

( , ) ( , ) ( , ),( , ) ( , ) ( , ).

P r r E rD r r E r

0 0

div ( , ) 0div ( , ) ( , ) ( , ) ( , ) ( , ) 0,

( , )div ( , ) ( , ).( , )

D rD r r div E r E r grad r

grad rE r E rr


2

2

( , )( , ) , ( , ) ( , )( , )c

grad rE r r E r grad E rr

All field components couple. D) linear, homogeneous, anisotropic, dispersive media ( )ij

0

0

( , ) ( ) ( , )

( , ) ( ) ( , ).

i ij jj

i ij jj

P E

D E

r r

r r see chapter on crystal optics

This is the worst case for a linear response of the media. Before we start writing down the actual material model equations, let us summarize what we want to do:

What physics do we want to consider?

I) Bound electrons and lattice vibrations 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 ( 310 mass of electron) the resulting oscillations 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 ). Understanding the electronic transitions of bound electrons requires quantum theoretical treatment, which allows an accurate computation of the transition frequencies. However, a (phenomenological) classical treatment of the oscillation of bound electrons is possible and useful.

II) 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.

Dielectric polarization and susceptibility 2.2.2Let us first focus on bound charges (ions, electrons). In the so-called Drude model, the electric field ( , )tE r gives rise to a displacement ( , )ts r of charged particles from their equilibrium positions. In the easiest approach this can be modeled by a driven harmonic oscillator:


2

202 ( , ) ( , ) ( , ) ( , )qt g t t t

t t ms r s r s r E r

resonance frequency (electronic transition) 0 damping g charge q mass m

The induced electric dipole moment due to the displacement is given by ( , ) ( , ),t q tp r s r

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

( , ) ( , ) ( , )tqN t Nt P r p r s r Hence, the governing equation for the polarization ( , )tP r reads

2

20 02

2

( , ) ( , ) ( , ) ( , ) ( , )q Nm

t g t t f tt t

P r P r P r t E r E r

with oscillator strength 2

0

1 e Nfm

, for q=-e (electrons)

This equation is easy to solve in Fourier domain:

2 20 0( , ) ( , ) ( , ) ( , )g f P r P r P r E ri

0

2 20

( , ) ( , ).fg

P r E r

i

In general we have several 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

0 02 20

( , ) ( , ) ( , )j

j j j

fg

P r E r E ri

2 20

.j

j j j

fg

i

is the complex, frequency dependent susceptibility

0 0 0( , ) ( , ) ( , ) ( , ) D r E r E r E r


is the complex frequency dependent dielectric function

Example: (here is plotted by eta and kappa 2i )

Conductive current and conductivity 2.2.3Let 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.

2

2 ( , ) ( , ) ( , ),et g t tt t m

s r s r E r

The resulting induced current density is given by

( , ) ( , )Net tt

j r s r

and the governing dynamic equation reads as

2

20( , ) ( , ) ( , ) ( , )e Nt g t t t

m

j r j r E r E rt p

with plasma frequency 2

2

0

1 e Nfm

p

Again we solve this equation in Fourier domain:

20( , ) ( , ) ( , )g j r j r E rpi

2

0( , ) ( , ) ( , ).g

j r E r E rp

i

Here we introduced the complex frequency dependent conductivity


2

2 20 0 .

g g

p piii

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:

0 ( , ) ( , )t t divE r r 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):

20 ( , )g t

tj j E r

p

Now we apply divergence operator and plug in from above (red terms):

2 20 ( , ) ( , )tg t

t

ddiv j div ivE r rj p p

With the continuity equation for the charge density (from Maxwell's equations)

,t

divj 0

We can substitute the divergence of the current density and find:

22

2 gt t

p

2

2

2 0,gt t

p

harmonic oscillator equation

Hence, the plasma frequency p is the eigen-frequency of such a charge density.

The generalized complex dielectric function 2.2.4In 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)

22

0 02

20 0 0

( , ) ( , ) ( , ) ( , )

( ) ( , )

c

rotrot E r E r P r j r

E r

i

i

Hence we can collect all terms proportional to ( , )E r and write


0

2

2 1 ( )( , ) ( , )c

rotrot E r E ri

2

2 ( ()( , ) , )c

rotrot E r E r

Here, we introduced the generalized complex dielectric function

0

( ) 1 () ( )( )

ii

So, in general we have

2

20

22( ) 1 ,j

j j jg gf

p

ii

because (from before)

2 20

j

j j j

fg

i,

202 .

g

pii

( ) contains contributions from vacuum, phonons (lattice vibrations), bound and free electrons.

Some special cases for materials in the IR und VIS:

A) Dielectrics (insulators) near phonon resonance (IR) If we are interested in dielectrics (insulators) near phonon resonance in the infrared spectral range we can simplify the dielectric function as follows:

2 2 2 20

2 20

0

(

,

1) j

j j j

f fg

fg

g

ii

i

00

0

j

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

( ) ( ) ( ) ( ) ( ) i i

2 20

22 2 2 20

( ) ,f

g


22 2 2 2

0

( ) .gf

g

Lorentz curve

resonance frequency 0 width of resonance peak g

static dielectric constant in the limit 0 : 020

f

so called longitudinal frequency L : ( ) 0 L ( ) 0 : absorption and dispersion appear always together

Example: single resonance

near resonance we find ( ) 0 (damping without absorption if '' 0 ) normal dispersion ( ) / 0, anomalous dispersion ( ) / 0

Example: sharp resonance for undamped oscillator 0g

relation between resonance frequency 0 and longitudinal frequency

L (Lyddane-Sachs-Teller relation)

2 20

( ) 0f

LL

, 0 20f (from above)

-4

0

4

8

12

ε′ε′′

ω0 ωLω

ε∞

ε0

-8

-4

0

4

8

12

ω0 ωL ωε∞

ε0

ε


0

0 .

L

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 2 20 0

( ) ,f f

g g

p e

p p e ei i 0 0p e

contribution of vacuum and other (far away) resonances

The generalization of this approach in the transparent spectral range leads to so-called Sellmeier formula:

2

20

20( ) 1 ,j

j

j

j

f

describes many media very well (dispersion of absorption is neglected) oscillator strengths and resonance frequencies are fit parameters

C) Metals in visible spectral range If we want to describe metals in visible spectral range we find

2

2( ) 1 .g

p

i p

2 2

2 2 2 2( ) 1 , ( ) .

gg g

p p

Metals show a large negative real part of the dielectric function ( )

0 2 4 6 81.4

1 .6

1 .8

2

2.2

V IS

ε'

ω in 1015s -1


Material models in time domain 2.2.5Let us now transform our results of the material models back to time domain. In Fourier domain we found for homogeneous and isotropic media:

0

0

( )( )

( , ) ( , )( , ) ( , ).

D r E rP r E r

The response function (or Green's function) ( )R t is then given by

1( ) ( )exp .2

R t t d

i ( ) ( )expR t t dt

i

To prove this, we can use the convolution theorem

0

0

( , ) ( , )exp ( , )exp

1 (

( )

( ) , )exp exp2

t t d t d

t t dt t d

P r P r E r

E r

-i -i

i -i

Now we switch the order of integration, and identify the response function R (red terms):

0

0

(1 ( )exp ( )2

( )

, )

( , )

t t d t dt

t dtR t t

E r

E r

-i

For a “delta” excitation in the electric field we find the response or Greens function as the polarization: 0( , ) ( )t t t E r e 0 0( , ) ( )t R t t P r e Green's function.

For instantaneous (or non-dispersive) media we find:

0( ) ( ) , ,R t t t t P r E r

5 10 15 20 25

-20

0

20 ε′ε′′

ω in 10 15s -1

V IS

ωP2-g2


Examples A) dielectric media

2 20

1 1( ) exp exp ,2 2P

fR t t d t dg

i ii

Using the residual theorem we can find:

exp sin 0

( ) 20 0

f g t t tR t

t

with 2

20 4

g

( , ) exp ( ) sin ( ) ( , )2

tf gt t t t t t dt

P r E r

B) metals

2

01 1( ) exp exp ,2 2jR t t d t d

g

pi ii

Using again the residual theorem we can find:

exp 0( )

0 0

gt tR t g

t

2p

0( , ) exp ( ) ( , )t

t g t t t dt

j r E r2p

2.3 The Poynting vector and energy balance Time averaged Poynting vector 2.3.1

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 ( , ) ( , ) ( , )t t t S r E r H rr r 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 140 02 / 10T s

pulse duration Tp in general 0T Tp


duration of measurement Tm in general 0T Tm Hence, in general the detector does not recognize the fast oscillations of the optical field 0i te (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:

01 ( , )exp . . ( , )2

t t c c t E r E rri

For such pulses, the momentary Poynting vector reads:

0 0

0

( , ) ( , ) ( , )

1 ( , ) ( , ) ( , ) ( , )4

1 ( , ) ( , )exp 2 ( , ) ( ,

1 ( , ) ( , ) cos 22

1 ( , ) ( , )

1 ( , ) ( , )2

)exp 2

i

4

s2

t t t

t t t t

t t t t t

t t t t t

t t

t

S r E r H r

E r H r E r H r

E r H r E r H

E r H r

E r H r

E H

r

r r

r r

i i

0n 2 .t

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 0 0cos 2 , sin 2t t changing over time scales of the optical cycle T0. Now, a measurement of the Poynting vector over a time interval mT leads to a time average of ( , )tS r

/2

/2

1( , ) ( , ') 't T

t Tt t dt

T

S r S rm

mm The fast oscillating terms 0~ cos 2 t and 0~ sin 2 t 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:

× →


/2

/2

1( , 1 ( , ') ')) ( , '2

t T

t Ttt dt t

T

E r H rS r m

mm 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):

( , ') ( ), ( , ') ( )t t E r E r H r H r

1( , ) ( ) ( ) .2

t S r E r H r

This is the definition for the optical intensity ( , )I t S r . We see that an intensity measurement destroys information on the phase.

( , )I t S r measurement destroys phase information

Time averaged energy balance 2.3.2Let 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):

0

0

0

( )

t

t t

r r

r r

r

r

r

r r r r

rotE H

E rot jEH

H

E P

H

E

Next, we subtract the two equations and get

0 0 ( ).

t t t

r r r r r r r r r r rH rotE E rotH E E H H E j P

This equation can be simplified by using the following vector identity:

r r r r r rdiv E H H rotE E rotH

Finally, with 212t t

r r rE E E we find Poynting's theorem

2 20 0

1 12 2t t t

r r r r r r rE H div E H E j P (*)

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

2 21 1

0 02 2u r rE H . The right-hand-side of Poynting's theorem contains the so-called source terms.


where 2 20 0

1 12 2

u r rE H vacuum energy density

In the case of stationary fields and isotropic media (simple but important)

0

0

1( , ) ( )exp . .21( , ) ( )exp . .2

t t c c

t t c c

E r E r

H r H r

r

r

i

i

Time averaging of the left hand side of Poynting’s theorem (*) yields:

2 20 0

1 1 1( , ) ( , ) ( , ) ( , ) ( ) ( )2 2 2

( , ) .

t t t tt t

t

E r H r div E r H r div E r H r

div S r

r r r r

Note that the time derivative removes stationary terms in 2 ( , )tE rr and 2 ( , )tH rr . Time averaging of the right hand side of Poynting’s theorem yields (source terms):

0 000 0 00

( , ) ( , )

1 . . . .

( , )

( ) ) ( )4

) (i t i i tt

tt

e

t

c c e c

t

ce

E r

E(r

P r

E rr )

j r

E(

rr r

i

Now we use our generalized dielectric function:

00

0 0

0

0 0 0 0

0 0

1 exp . . exp . .4

1 . .4

1

t c c t c c

c c

E(r) E(r)

E(r)E(r)

ii i i

i

Again, all fast oscillating terms 0exp 2 t i cancel due to the time average.

Finally, splitting 0 into real and imaginary part yields

0 0 0 0 0 0 01 11 . . ( ) ( ).4 2

c c E(r)E(r) E r E ri i

Hence, the divergence of the time averaged Poynting vector is related to the imaginary part of the generalized dielectric function:

0 0 01 ( ) ( ).2

div S E r E r


This shows that a nonzero imaginary part of epsilon ( 0 ) causes a drain of energy flux. In particular, we always have 0 , otherwise there 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.

V A

dV dA div S S n

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

0

( ) 1 ( ) ( ) ( ) i

i

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

2

2( , ) ( ) ( , )c

rotrot E r E r

According to Maxwell the solutions have to fulfill additionally 0 1 ( ) ( , ) 0 div E r

In general, this additional condition implies that the electric field is free of divergence:


1 ( ) 0 ( , ) 0 div E r (normal case)

Let us for a moment assume that we already know that we can find plane wave solutions of the form: ( , ) ( )exp , E r E kri

k unknown complex wave-vector Then, the divergence condition implies that those waves are transversal

( )k E transverse wave

The corresponding stationary field in time domain is given by

( , ) expt t E r E kr i

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' k'',iwe can define:

planes of constant phase ' .k r const planes of constant amplitude .k''r const

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 ( ) 0, L L so-called longitudinal waves ( )k E appear.

Transversal waves 2.4.1As pointed out above, for L the electric field becomes free of divergence:

0 ( )div ( , ) 0 E r div ( , ) 0 E r

Then, the wave equation reduces to the Helmholtz equation:

2

2( , ) ( ) ( , ) 0.c

E r E r

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: ( , ) ( )exp E r E kri

Immediately we see that the wave is transversal:


0 ( , ) ( , ) divE r k E ri ( ).k E

Hence, we have to solve

2

22 ( ) ( ) 0

c

k E and ( ) 0. k E

which leads to the following dispersion relation

2

2 2 2 2 22 ( )k k k k

c

k x y z

We see that the so-called wave-number ( ) ( )ck 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 ki is complex. Therefore it is sometimes useful to introduce the complex refractive index (if k k ):

ˆ( ) ( ) ( ) ( ) ( ) .k n n

c c c

i

( , ) ( )exp , E r E kri

With the knowledge of the electric field we can compute the magnetic field if desired:

0 0

0

1( , ) ( , ) ( ) exp

1 ( , ) ( )exp , ( ) ( )

H r rot E r k E kr

H r H kr H k Ewith

ii

i

Longitudinal waves 2.4.2Let 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 ( , ) 0 div E r , and the wave equation reads (the l.h.s. vanishes because ( ) 0 ):

( , ) 0 rotrot E r L

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: ( , ) ( )exp ( )exp ( )exp , E r E kr E kr E kri i i

with, ( ) k E and ( )k E

With ( )exp ( )exp rot E kr k E kri i i we get from the wave equation:


( , ) 0 k k E r L

Now we plug in the electric field (longitudinal and transversal):

exp 0,

exp exp 0,

k

k k E E kr

k k E kr k kE r

i

i i

2 0k E ( , ) ( )exp E r E krL L i

Plane wave solutions in different frequency regimes 2.4.3The dispersion relation for plane wave solutions 2

22 2 2 2 2 ( )

ck k k k k x y z

dictates the (complex) wavenumber only. Thus, different solutions for the complex wave vector k = k ki are possible. In addition, the generalized dielectric function is complex. In this chapter we will discuss possible scenarios and resulting plane wave solutions.

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

2 2

2 2 2 22 22 ' ( ) ( ) ' 0k n

c c k' k'' k k'' k k''i

There are two possibilities to fulfill this condition, either 0k'' or k' k'' .

A.1) real valued wave-vector 0k'' In this case the wave vector is real and we find the dispersion relation

2( ) ( ) ( )k n nc c

n

Because 0k'' these waves are homogeneous (trivial, because the amplitude is constant).

Example 1: single resonance in dielectric material for lattice vibrations (phonons)


Now the imaginary part of ( ) is neglected

2 20

( ) ( ) f

We can invert the dispersion relation ( ) ( )kc

( )k :

Example 2: free electrons

for plasma and metal Again the imaginary part of ( ) is neglected

2

2( ) ( ) 1

p

We again invert the dispersion relation ( ) ( )kc

( )k :

��

��

ε′ε′′

ω

ckω

ε∞

=

ω

k

0

ckω

ε=

0ω

20

fωε ∞

+


A.2) complex valued wave-vector k' k'' The second possibility to fulfill the dispersion relation leads to a complex

wave-vector and so-called evanescent waves. We find

2

2 2 22 ( )k

c

k' k'' and therefore 2 2 2k k'' k'

This means that

2 0k'' and 2 2k'k

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 2 2k'k , whereas the homogeneous waves of i) ( 0k'' ) obey 2 2k'k . If we plug our findings into the plane wave ansatz we get: for the evanescent waves:

ex( , ) ( ex p) p E r E k''r rk (' )i

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 This situation (negative but real ( ) can occur near resonances in dielectrics ( 0 L ) or below the plasma frequency ( p ) in metals. Then the dispersion relation gives

2

2 2 222 ( ) 0k

c k' k'' k' k''i

ckω =

ω

k

pω


As in the previous case A), the imaginary term has to vanish and ' 0 k k'' . Again this can be achieved by two possibilities.

B.1) 0k'

22

2 ( )c k''

( , ) exp E r k''r strong damping

B.2) 0 k' k'' k' k'' evanescent waves

22 2 2

2 ( )kc k' k''

22 2

2 ( ) .c k'' k'

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 2.k' In particular, case i) ( 0k' ) 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.

��

ε′ε′′

ω


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

ˆ( ) ( ) ( ) ( ) ( ) ,k n n

c c c

i

( ), ( ).n

c c

k' k''

The dispersion relation in terms of the complex refractive index gives

2 222 2

2 2( ) ( ) ( )k nc c

k i

Here we have

2 2( ) ( ) ( ) ( ) ( ) 2 ( ) ( ),n n i i

and therefore 2 2( ) ( ) ( )

( ) 2 ( ) ( )n

n

22 ( ) sgn 1 / 1 ,2

n

22 ( ) sgn 1 / 1 .2

Two important limiting cases of quasi-homogeneous plane waves:

Region 1) , 0, , (dielectric media)

1 ( )( ) ( ), ( )2 ( )

n

In this regime propagation dominates ( ( ) ( )n ), and we have weak absorption:

��

��

ε′ε′′

ω

1 2


2 2

2 22 2( ), 2 ( ).

c c k' k'' k' k''

1 ( )( ) ( ), ( )2 ( )

nc c c c

k' k''

, k' k'' k' k''

k' and k'' almost parallel homogeneous waves in homogeneous, isotropic media, next to resonances, we find damped, homogeneous plane waves, kk' k e with ke being the unit vector along k

( , ) ( )exp ( )exp exp .nc c

k kE r E kr E e r e ri i

Region 2) 0, 0, , (metals and dielectric media in so-called Reststrahl domain)

1 ( )( ) , ( ) ( ) ,2 ( )

n

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).

Summary of normal modes

a) undamped homogeneous waves and evanescent waves b) evanescent waves c) weakly damped quasi-homogeneous waves d) strongly damped quasi-homogeneous waves


Time averaged Poynting vector of plane waves 2.4.4

/2

/2

1 1( , ) ( , ) ( , ) ,2

t T

t Tt t t dt

T

S r E r H r m

mm

For plane waves we find:

0

( , ) exp exp1( , ) ( , )

t t t

t t

E r E kr E k r k r

H r k E r

i i i i

assuming a stationary case ( ) ( )t E E

2 20

0 0

1 1( , ) exp 2 exp 2 .2 2

ntc

S r r E n n rk Ek k' k"

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 (dispersion of pulses) will be treated in temporal Fourier domain (frequency domain), spatial effects (diffraction of beams) 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)

A) beam finite transverse width diffraction

plane wave beam

2w

k


A beam is a continuous superposition of stationary plane waves (normal modes) with different propagation directions

3( , ) ( )exp dt kt

k kr E rE i

B) pulse finite duration dispersion

stationary wave pulse

A pulse is a continuous superposition of stationary plane waves (normal modes) with different frequencies

( , ) ( )exp .dt t

E r E k ri

C) pulsed beams finite transverse width and finite duration diffraction and dispersion A pulsed beam is a continuous superposition of stationary plane waves (normal modes) with different frequency and different propagation direction

3( , ) ( )exp, d kt dt

k kE r E ri

p2T

k1 k2 k3 k4 k5

…

1 2 3

...


Diffraction of monochromatic beams in homogeneous 2.5.1isotropic media

Let us have a look at the propagation of monochromatic beams first. In this situation, we have to deal with diffraction only. We will see later that pulses and 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

y y y( , ) ( , ) ( , ) ( , ).E E u E r r e r r exact for one-dimensional beams and linear polarization approximation in two-dimensional case

In homogeneous isotropic media we have to solve the Helmholtz equation

2

2( , ) ( , ) 0.c

E r E r

In scalar approximation and for fixed frequency it reads

2

2

2

( , ) ( , ) 0,

( , ) ( , ) 0.

u u

u

c

uk

r r

r r

scalar Helmholtz equation

In the last step we inserted the dispersion relation (wave number k). In the following we often omit the fixed frequency .

2.5.1.1 Arbitrarily narrow beams the general case

Let us consider the following fundamental problem. We want to compute from a given field distribution in the plane 0z the complete field in the half-space

0z ( z is our “propagation direction”)


2( , ) ( , ) 0u k u r r

We take the Fourier transform: 3( , ) ( , )exp ( )u U d k

r k k ri , 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:

2

2 2 2 2 22k k k k

c

k x y z

only two components of k are independent, e.g., , .k kx y

Our naming convention is in the following: , , .k k k x y z

Then, the dispersion relation reads:

Thus, to solve our problem we need only a two-dimensional Fourier transform, with respect to transverse directions to the “propagation direction z”:

( ) ( , )exp; .u U x y d dz

r i

In analogy to the frequency we call spatial frequencies. If we plug this expression into the scalar Helmholtz equation

2( ) ( ) 0u k u r r

we find in Fourier space

2 2 2 2k


22 2 2

2

22

2

( , ; ) 0,

( , ; ) 0.

d k U zdz

d U zdz

This equation is easily solved and yields the general solution 1 2( , ; ) ( , )exp ( , ) ( , )exp ( , ) ,U z U z U z i -i

depending on 2 2 2( , ) ( )k . We can identify two types of solutions: A) 2 0, 2 2 2k , i.e., k real homogeneous waves

B) 2 0, 2 2 2k , i.e., k complex, because k z imaginary. Then, we have k = k ki , with x y k = e e and z k = e .

k' k'' evanescent waves

We see immediately that in the half-space 0z the solution exp z i grows exponentially. Because this does not make sense, we have to set

2 ( , ) 0U . In fact, we will see later that 2 ( , )U corresponds to backward running waves, i.e., light propagating in the opposite direction. We therefore find the solution:

k

γ

α

β

k

α

β

k²² ²α + β >

γ


1

0

( , ; ) ( , )exp ( , )

( , ;0)exp ( , )

( , )exp ( , )

U z U z

U z

U z

i

i

i

Furthermore the following boundary condition holds: 0( , ;0) ( , ).U U In spatial space, we can find the optical field for 0z by inverse Fourier transform:

( ) ( , )exp; .u U x y d dz

r i

0( ) ( , )exp xp, e .u U x y d dz

r ii

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 0z

2

0 01( , ) ( , )exp ,

2U u x y x y dxdy

i

(boundary condition) follows from 0 ( , ) ( , ,0).u x y u x y 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. We can formulate a general scheme to describe diffraction:

0 ( , )u x y 0 ( , )U propagation = multiplication by exp , z i

new spectrum 0( , ; ) ( , )exp ,U z U z i

new field distribution ( , , ).u x y z This scheme allows for two interpretations:

1) The resulting field distribution is the Fourier transform of the propagated spectrum

( ) exp .( , ; )u x y d dU z

r i


2) The resulting field distribution is a superposition of homogeneous and evanescent plane waves ('plane-wave spectrum') which obey the dispersion relation.

0 ( , )exp) .,(u dU x y z d

r i

Let us now discuss the complex function ( , ; ) exp ,H z z i , which describes the beam propagation in Fourier space. For z = const. (finite propagation distance) it looks like:

amplitude phase

Obviously, ( , ; ) exp ,H z z i acts differently on homogeneous and evanescent waves:

A) homogeneous waves 2 2 2k

exp , 1, arg exp , 0z z i i

Upon propagation the homogeneous waves are multiplied by the phase factor

2 2 2exp k z i

B) evanescent waves 2 2 2k

2 2 2exp , exp , arg exp , 0z k z z i i

Upon propagation the evanescent waves are multiplied by an amplitude factor <1

2 2 2exp 1k z


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 0 ( , )u x y yields an angular spectrum 0 ( , )U 0 for 2 2 2k we get evanescent waves. Example: Let us consider the following one-dimensional initial condition which corresponds to an aperture of a slit:

0

1( ) .2

0

axu x

for

otherwise

0 0

sin2( ) FT ( ) sinc

22

aaU u x

a

All spatial frequencies (- ) are excited. Relevant spectral information is contained in the interval 2 / a .

Largest relevant spectral frequency for a structure with width a is 2 / 2 .

Evanescent waves appear for k .

To represent the relevant information by homogeneous waves the

following condition must be fulfilled: 2 2k na

a

n

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 .

��

-10 0 10

-0 .25

0

0.25

0.5

0 .75

1

2

a αππ−

-a/2 a/2 x


Conclusion: 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:

1

0 0 0( , ) ( , ) ( , ; ) ( , ; ) ( , ) ( , , )u x y U U z H z U u x y z

FT FT

Remark: diffraction free beams With our understanding of diffraction it is straight forward to construct so-called 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 same phase shift

0 0

0

( , ; ) ( , )exp , ( , )exp

( , , ) exp ( , )

U z U z U Cz

u x y z Cz u x y

i i

i

2 2 2 2 2 20 0( , ) 0 , .U k C only for

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: 0 0( , ) ( )u x y J r (see exercises)

Bessel-beam (profile) Bessel-beam

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

2 2 2

0 ( , ) 0U k for


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 Taylor expansion:

2 2 2 2

2 2 221

2 2k k k

k k

The resulting expression for the propagation function in Fresnel approximation reads:

2 2

exp ( , ) exp exp ( , ; )2

H z kz z H zk

Fi i i

In particular, in Fresnel approximation we find no evanescent waves! Structural details , 10 / /x y n n must be always much larger than the wavelength.

Amplitude Phase

The propagation of the spectrum in Fresnel approximation works in complete analogue to the general case, we just use the modified propagation function: 0( , ; ) ( , ; ) ( , )U z H z U F F

For a coarse initial field distribution 0 ( , , )u x y z the angular spectrum 0 ( , )U is nonzero for 2 2 2k only. Then, only paraxial plane waves are relevant for transmitting information. Of course it is possible to formulate beam propagation in Fresnel (paraxial) approximation in position space as well:

0

0

( , , ) ( , ; )exp

( , ; ) ( , )exp

( , ; ) ( , )

u x y z U z x y d d

H z U x y d d

h x x y y z u x y dx dy

F F

F

F

i

i

with the spatial response function from the convolution theorem

|H|

βα

k

1

β

arg H

kz

kα


2

2 2 2

1( , ; ) ( , ; )exp2

1 exp exp exp .2 2

h x y z H z x y d d

kz z x y d dk

F F i

i i i

This Fourier integral can be solved and we find:

2 22 2

2( , ; ) exp exp exp ,2 2

122

k k kh x y z kz x y kzz z

x yz z

F

i ii i i

The response function is a spherical wave in paraxial approximation. To sum up, in position space paraxial beam propagation is given by:

2 20( , , ) exp ( , )exp .

2 2k ku x y z kz u x y x x y y dx dyz z

F

ii i

Of course, the two descriptions in position space and spatial Fourier domain are completely equivalent. Remark on the validity of the scalar approximation:

ˆ( , ) , , ei x y z d d E r E

ˆ ˆ ˆ( , ) 0 0x y zE E E divE r

A) One-dimensional beams translational invariance in y-direction: =0

and linear polarization in y-direction: ˆyE 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: ˆ 0xE , ˆyE U

divergence condition: ˆ ˆ 0y zE E

2 2 2

ˆ ˆ ˆ, , , , , , 0z y yE E Ek

In paraxial approximation ( 2 2 2k ) the scalar approximation is automatically justified.


2.5.1.3 The paraxial wave equation In paraxial approximation the propagated spectrum is given by

0

2 2

0

( , ; ) ( , ; ) ( , )

exp exp ( , )2

z

U z H z U

kz Uk

F F

i i

Let us introduce the slowly varying spectrum ( , ; )V z :

( , ; ) exp ( , ; )U z kz V z F i 2 2

0( , ; ) exp ( , ).2

V z z Vk

i

Differentiation of V with respect to z gives:

2 21( , ; ) ( , ; )

2V z V z

z k

i

Fourier transformation back to position space leads to the so-called paraxial wave equation:

2 21 ( , ; )exp2

( , ; )expV z xz

V z x y

y

d

d

k

d

d

i

i

i

2 2

2 2

1 ( , ; )ex) p( ,2

, V zx y

x y d dz

x y zk

v

i i

(2)1( , , ) ( , , ) 02

v x y z v x y zz k

i paraxial wave equation

The slowly varying envelope ( , , )v x y z (Fourier transform of the slowly varying spectrum) relates to the scalar field as ( , , ) ( , , )exp .u x y z v x y z kzF i

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 ):

22 2

2( , , ) ( , ) ( , , ) 0 with ( , ) ( , ). u x y z k u x y z kc

r rr

We use the ansatz 0( , , ) ( , ex, ) pu x y z v x y z k z i with 0k k being the average wavenumber. With the SVEA condition

0 /k v v z


we can simplify the scalar Helmholtz equation as follows: 2

(2) 2 20 02

0

( , , ) 2 ( , , ) ( , , ) ( , ) ( , , ) 0,v x y z k v x y z v x y z k k v x y zz z

r

i

2 2

(2) 0

0 0

1 ( , )( , , ) ( , , ) ( , , ) 02 2

k kv x y z v x y z v x y zz k k

ri

This is the paraxial wave equation for inhomogeneous media (weak index contrast).

Beam propagation scheme

Relation between transfer and response function

2

1 e( xp(2

;, )) ,)

( ; x y d dh x y H zz

i

Transfer functions for homogeneous space

2 2 2( , ; ) exp , expH z z i k z i exact solution

2 2

( , ; ) exp exp2

H z kz i zk

F i Fresnel approximation

with 0( ) ( ) ( )k k k n nc


Propagation of Gaussian beams 2.5.2The 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.

Fundamental Gaussian beam in focus

The general form of a Gaussian beam is elliptic, with curved phase.

2 2

0 0 0 2 2( , ) ( , ) exp exp ( , )x yu x y v x y A x yw w

x y

i .

y

u0

x


Here, we will restrict ourselves to rotational symmetry 2 2 2w w w x y 0 and (initially) 'flat' phase ( , ) 0x y , 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 : 2 2 2

0 0 0( ) exp 1 /u x y w A A e 0 In practice, the so-called 'full width at half maximum' (FWHM) is often used instead of w0 .

( )

222 2 FWHM0 2

0

1exp22w

wu x yæ ö÷ç ÷+ = -ç ÷ç ÷çè ø

2FWHM

20

ln 22w

w- =- 2 2 2

FWHM 0 02ln 2w 1.386ww = »

2.5.2.1 Propagation in paraxial approximation Let us now compute the propagation of a Gaussian beam starting from the focus in paraxial approximation: 1) Field at 0z :

2 2

0 0 0 2( , ) ( , ) exp .x yu x y v x y Aw

0 2) Angular spectrum at 0z :

2 2

0 0 02 2

2 2 2 220

2 220

4 /

1( , ) ( , ) exp exp ( )2

exp exp ,4 4

x yU V A x y dxdyw

A Aw ww w

0 s

0

0 0

i

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

2w w s 0

Angular spectrum in the focal plane

C) Check if paraxial approximation is fulfilled:

β

U0

α


We can say that 0 ( , ) 0U for 2 2 216 / w 0 , because exp 4 0.02

For paraxial approximation we need 2 2 2k 2 216 /k w 0

2 2

22

16 2 ,2

wn n

n

0

paraxial approximation works for 10 10wn 0 n

D) Propagation of the angular spectrum:

( , ; ) ( , ; ) expU z V z kz i

2 2

0

2 2 2 22 20

( , ; ) ( , )exp2

exp exp .4 4 2

V z U zk

A w w zk

0 0

i

i

E) Fourier back-transformation to position space

2 2 20

2 2

0 2 2

2

0

2 2

0

2

20

2

( , , ) exp exp4

1 exp2 ( )11

1 exp .

4

11 /

2

/

Av x y z w x y d d

xkw

kw

z

w zk

z z

zz

yAw

x yAw z

0

0 0

0

0

0i

i

i

i

ii

With the Rayleigh length 0z which determines the propagation of a Gaussian beam:

220

00 .

2wz wk

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 0z


Some books use the “diffraction length” 02L zB , a measure for the “focus depth” of the Gaussian beam. E.g.: 10w 0 n 600 .L B n

From our computation above we know that the Gaussian beam evolves like:

2 2

0 20 0

1( , , ) exp .1 / /1zz z

x yv x y z Aw z

0i i

For practical use, we can write this expression in terms of z-dependent amplitude, width, etc.:

2 22 200

0 2 2 22 20 0 0

1 /( , , ) exp exp

1 / 1 / 1 /

zx y z zz x yv x y z A

z z w z z w z z

0 0

ii

2 2 2

2 20

2

2

0

20

0

(1( , , ) e )1 1

xp exp exp ,2

1

x y kv

x yzz zzz z z

z

x y z Aw

0

ii

Here we used that 202 /w z k0 . The (x,y)-independent phase ( )z is given by

0tan /z z , the so-called Gouy phase shift. In conclusion, we see that the propagation of a Gaussian beam is given by a z-dependent amplitude, width, phase curvature and phase shift:

2

2 22 2

( , , ) exp exp exp2

( ) ( )( ) ( )

A z zw z R z

x yx y kv x y z

i i

Discussion: The amplitude is given as:

0

0 02 22

1 1( ) ,11 zz

Lz

A z A A

B Hence, we get for the Intensity profile I~A²:


The on-axis intensity (x=y=0) evolves like

The beam width evolves like:

2 2

0

2( ) 1 1 ,z zw z w wz L

0 0

B

The radius of the phase curvature is given by


2 2

0( ) 1 12

z LR z z zz z

B

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

0

2tan z zz L

B

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

2 2

( , , ) ( ) .2 ( )x yx y z k z z

R z

const


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 0( )q z z z i 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”,

2 20

2 20

02 2 2 2

0 0 0 0

1 1 1 1( ) 1 1z z

z z

z zq z z z z z z z z z

i ii

we can observe that real and imaginary part are directly linked to radius of phase curvature and beam width:

2

1 1 .( ) ( ) ( )q z R z w z

ni because

220

0 02kwz w

n

Thus, the q-parameter describes beam propagation for all ! Example: propagation in homogeneous space by z d

A) initial conditions: 2

1 1(0) (0) (0)q R w

ni

B) propagation (by definition of q parameter) ( ) (0)q d q d C) q-parameter at z d determines new width and radius of curvature

2

1 1 1( ) (0) ( ) ( )q d q d R d w d

ni

Gaussian optics 2.5.3We 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 0 0,R w (i.e. 0q ) ,n nR w (i.e. nq ) after passing through n optical elements

z


We will evaluate the q-parameter at certain propagation distances, i.e., we will have values at discrete positions: ( )iq z iq . 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 BC D

M

B) propagation through multiple elements:

1 1ˆ ˆ ˆ ˆ.. .

A BC D

M M M MN N

C) matrix connects distances to the optical axis y and inclination angles before and after the element

2 1

2 1

ˆ .y y

M


Link to Gaussian beams Let us consider the distance to the intersection of the ray with the optical axis:

11

1

yz

1

2 1 1 12

12

1

1

11 1

yA By Ay Bz y

Az BCz DCy D C D

The distances 1 2,z z 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:

1 0 11

1 0 1

A q BqC q D

propagation through N elements:

0

0n

Aq BqCq D

with the matrix 1 1ˆ ˆ ˆ ˆ.. .

A BC D

M M M MN N

This works for all ABCD matrices given in chapter 1.6!!! Here: we will check two important examples: i) propagation in free space by z d : propagation (by definition of q parameter) ( ) (0)q d q d

1ˆ0 1

d

M

1 0 1 01 0

1 0 1 0 1A q B q dq q dC q D

ii) thin lens with focal length f


What does a thin lens to a Gaussian beam 2 2

2exp x yw

0

in paraxial

approximation?

no change of the width, but phase curvature fR : 2 2

exp2 fR

x yk

i

How can we see that? Trick:

We start from the focus which is produced by the lens with 2

0f

fn

wz z

and fw is focal width. The radius of curvature evolves as:

2

( ) 1 fzR z z z

z

for fz z

We can invert the propagation from the focal position to the lens at the distance of the focal length f and obtain fR f

0f 0f 1 0ˆ1 1f

M

double concave double convex lens lens defocusing focusing

1 0 1 01

1 0 1 0

20 0

021 0 0

1

1 1 1 n

n

A q B qqC q D q f

q f wqq q f w

i ifor

Be careful: Gaussian optics describes the evolution of beam width and phase curvature only! Changes of amplitude and reflection are not included!


Gaussian modes in a resonator 2.5.4In 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 Transversal fundamental modes (rotational symmetry)

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!):

1,2z is the position of mirror '1','2'; z=0 is the position of the focus!

d is the distance between the two mirrors 2 1z z d

because 20( ) zR z zz

radius of wave front <0 for z <0

from Chapter 1: beam hits concave mirror radius ( 1,2) 0.iR i beam hits convex mirror radius ( 1,2) 0.iR i

Examples: A) 1 2( ), ( ) 0R z R z ; 1 20, 0R R ; 1 20, 0z z


B) 1 2( ) 0, ( ) 0R z R z ; 1 2, 0R R ; 1 20, 0z z

According to our reasoning above, the conditions for stability are:

1 1 2 2( ), ( )R R z R R z

1 1 2 21 2

2 20 0, .R z R z

zzz

z

In both expressions we find the Rayleigh length z0, which we eliminate: 1 1 1 2 2 2( ) ( )z R z z R z

with 2 1z z d we find 21

1 2

.2

d R dz

R R d

Now we can choose 1 2, ,R R d and compute modes in the resonator. However, we have to make sure that those modes exist. In our computations above we have eliminated the Rayleigh length z0, a real and positive quantity. Hence, we have to check that so-called stability condition 2

0 0z > is fulfilled!

2 2 1 2 1 20 1 1 1 2

1 2

02

d R d R d R R dz R z z

R R d

The denominator 21 2 2R R d is always positive we need to fulfill

d0 z2z1

d

0z1 z2


1 2 1 2 0d R d R d R R d

If we introduce the so-called resonator parameters

1 21 2

1 , 1d dg gR R

We can re-express the stability condition as

1 2 1 21 2

21

1

2 1 2 1 2

2 1 2 1 2

1 0.

1R d R d g g Rd d

g g g g R

g g R RR R d

dR

R

This inequality is fulfilled for

1 20 1g g or 1 2

0 1 1 1d dR R

This final form of the stability condition can be visualized: The range of stability of a resonator lies between coordinate axes and hyperbolas:

Examples for a stable and an unstable resonator: A) 1 2 1 2 1 2 1 2, 0; , ; 0 1, 0 1; 0 1R R R d R d g g g g stable


B) 1 2 1 2 1 2 1 2, 0; , ; 0, 0 1; 0R R R d R d g g g g unstable

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 ( 1,2) 0.iR i beam hits convex mirror radius ( 1,2) 0.iR i 1,2z is the distance of mirror '1','2' to the focus!

d is the distance between the two mirrors 2 1z z d Examples: A) 1( ) 0R z , 2( ) 0R z 1 20, 0R R ; 1 20, 0z z

B) 1 2( ) 0, ( ) 0R z R z ; 1 2, 0R R ; 1 20, 0z z

d0 z2z1


Then the conditions for stability are: 1 1 2 2( ), ( )R R z R R z With analog calculation as above we find with for the resonator

parameters

1 21 2

1 , 1d dg gR R

the same stability condition

21 2 1 2 1 21 0,g g g g R R

1 20 1.g g

2.5.4.2 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:

2 2 2 2

02( , , ) exp exp exp ( ) .

( ) ( ) 2 ( )w x y k x yv x y z z

w z w z R zA

G i i

higher order modes: ( ,x y -dependency of phase is the same)

0,

2

,

2

( , , )( )

exp exp exp ( 1) ( ) .2 ( )

2 2( ) ( )l m l ml mx yA G G

w z w z

l m

wu x y zw z

k x y kz zR z

i i i

2

( ) ( )exp2l lG H

The functions Gl are given by the so-called Hermite polynomials: ( )lH ( 0 11, 2H H and 1 12 2 ).l l lH H lH

d

0z1 z2


Pulse propagation 2.5.5

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 is fix and therefore the wavenumber ( )k is 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 0T of

1310 s (100 fs) 100 10T s (100 ps).

Let us compute the spectrum of the (Gaussian) pulse:

2

0 20

20 2

02 20 0

( ) exp exp

4( ) exp 24 /

tf t tT

F TT T

s s

i

spectral width: 10 1 13 14 10 4 10s s s

(visible) center frequency of visible light: 150 2 4 10 s-1

optical cycle: 02 / 1.6T s fs

Hence, we have the following order of magnitudes:


0 0 0 s

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:

0 0

22

0 0 02

1( ) ...2

k kk k

The following terminology is commonly used in the literature:

000 0

0

1,nkk k

v c

Ph

phase velocity

0

1 kv

g

group velocity

0

2

2

kD

group velocity dispersion (GVD)

We reduce the complicated dispersion relation to three parameters: phase velocity Phv group velocity gv

group velocity dispersion (GVD) D

Discussion: A) group velocity and group index group velocity is the velocity of the center of the pulse (see below)

0 0

0 01 1( ) ( ) ( )k nk n n

c v c

g

0ω

sω

ω

k


0

0

0 00 0

( )( ) ( )

( )

c c nv vn nnn

g PHg g

0

0 0 0( ) ( ) group indexnn n

g

normal dispersion: / 0 ,n n n v v g g PH

anomalous dispersion: / 0 ,n n n v v g g PH

B) group velocity dispersion (GVD) (or simply dispersion) GVD changes pulse shape upon propagation (see below)

0

2

2

1

g

kD Dv

2

1 1

0 0

0 0

vD

v v

vD

vD

g

g g

g

g

Alternatively in telecommunications one often uses

2

1 2D cDv

g

Let us now discuss the propagation of pulsed beams. We start with the scalar Helmholtz equation, with the full dispersion (no Taylor expansion yet):

2

2( , ) ( ) ( , ) 0u uc

r r

In contrast to monochromatic beam propagation, we now have for each one Fourier component of the optical field:

dispersion relation: 2

22( ) ( )k

c

Hence, we need to consider the propagation of the Fourier spectrum (Fourier transform in space and time):

0( , , ; ) ( , , )exp , ,U z U z i


with 2 2 2, , ( )k

The initial spectrum at 0z is 0( , , )U

0 031( , , ) ( , , )exp

2U u x y t x y t dxdydt

i

Let us further assume Fresnel (paraxial) approximation is justified 2 2 2( )k

2 2

0( , , ; ) ( , exp exp2

, )U z U k z zk

i i

We see that propagation of pulsed beams in Fresnel approximation in Fourier space is described by the following propagation function (transfer function):

2 2

exp exp( , ,2

; ) k z zk

H z

FP i i

Now let us consider the Taylor expansion of ( )k from above. If the pulse is not too short, we can replace

0 0

22

0 0 02

1( ) ..2

k kk k

Moreover, in the second term

2 2

exp2

zk

i (which is already small due to

paraxiality) we can use 0 0( ) ( )k k k (breaks down for 0 15T fs). By doing so, we get the so-called parabolic approximation:

0

20

2

0

2 2

2 20

0

0

1( , , ; ) exp exp

exp exp2 2

1 1 1exp exp2 2

H z k z zv

D zk

z

k z z Dv k

FPg

g

i i

i i

i i

with 0

0( , , ; ) exp ( , , ; )H z k z H z FPFP i

In the last line of the above equation we have introduced a new variant of the propagation function. ( , , ; )H z FP is the propagation function for the slowly varying envelope ( , , , )v x y z t :


0 0

0 00

( , , , ) exp ( , , ) ( , , ; )

exp

exp ( , , ) ( , , ; )

exp

u x y z t k z U H z

x y t d d d

k z t U H z

x y t d d d

FP

FP

i

i

i

i

slowly varying envelope

0 0( , , , ) ( , , , )expu x y z t v x y z t k z t i

0( , , , ) ( , , ) ( , , ; )expv x y z t U H z x y t d d d

FP i

In order to complete the formalism, we also need the initial spectrum of the slowly varying envelope 0 0 0( , , ) ( , , ) expu x y t v x y t t i

0 0 031( , , ) ( , , )exp ( , , )

2U v x y t x y t dxdydt V

i

Thus, the propagation of the slowly varying envelope is given by:

0 ( , , ; )( , , , ) ( , , ) expv x y z t V xH z y t d d d

FP i

|u()|

0

|u()|

v(t)

t

u(t)

t


now: ( , , ; ) exp ( , , ; )H z z H zv

FP FPg

i

0( , , , ) ( ( ,, , ) ex, ; p)v x y z t V x y t dz v dH z d

gFP i

The last line above involves the propagation function ( , , ; )H z FP , the

propagation function for the slowly varying envelope in the co-moving frame of the pulse:

ztv

g

This frame is called co-moving because ( , , ; )H z FP 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:

2 22

00

( , , , ) ( , , )exp2

exp

zv x y z V Dk

x y d d d

i

i

The optical field u reads in the co-moving frame:

… 00 0 0 0( , , , ) ( , , , )exp ( , , , )expu x y z v x y z k z t v x y z k z z

v

g

i i

Finally, let us derive the propagation equation for the slowly varying envelope in the co-moving frame:

2 22

00

( , , ; ) ( , , )exp2zV z V D

k

i

2 22

0

( , , ; ) 1 ( , , ; )2

V z D V zz k

i

As before in the case of monochromatic beams, we use Fourier back-transformation to get the differential equation in time-position domain

2(2)

20

( , , , ) 1( , , , ) ( , , , ) 02 2

v x y z D v x y z v x y zz k

i

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)


2 2 2(2) 0 0

20 0

1 ( )( , , , ) ( , , , ) ( , , , ) ( , , , ) 02 2 2D k kv x y z v x y z v x y z v x y z

z k k

r i

with 0 0k k r

for 0D 'simple' diffraction beam propagation

2.5.5.2 Infinite transverse extension - pulse propagation Diffraction plays no role for sufficiently small propagation lengths Bz L . For broad beams, the diffraction length BL 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: 0 0 0( ) ( )expu t v t t i

2) initial spectrum: 0 0( ) ( )V U

3) propagation of the spectrum: 0

2( ; ) ( )ex

2pV z V Dz

i

4) back-transformation to leads to the following evolution of the slowly varying envelope in co-moving frame:

2

0( , ) ( )exp exp2Dv z V z d

i i

Description in time domain A) in time domain it is possible to describe pulse propagation by means of a

response function:

FT-1 of 2

( ; ) exp2DH z z

P i

22( ; ) exp2

h zDz Dz

P ii

and the evolution reads 0( , ) ( ; ) ( )v z h z v d

P

B) the evolution equation for slowly varying envelope in co-moving frame

reads 2

2

( , ) ( , ) 02

v z D v zz

i

Analogy of diffraction and dispersion DIFFRACTION DISPERSION


(2),( , (2

,,) ) 01k

x y x yv z v zz

i 2

2( , ) ( , ) 02

v z v zz

D

i

( , ) ( , )x y

0

1k

D but 0D can vary

Examples of pulse propagation

2.5.5.3 Example 1: Gaussian pulse without chirp use analogies to spatial diffraction

1. Initial pulse shape pulse without chirp corresponds to Gaussian pulse in the waist with flat phase

2 2

0 0 0 0 02 20 0

( ) exp exp ( ) exptu t A t v AT T

i

2. Initial pulse spectrum 2 2

0 00 0( ) exp

42T TV A

spectral width: 2 204 /s T

Use results from propagation of Gaussian beams:

0z describes Gaussian pulse 2

2 00 0

1 02 2

Tkz wD

hence anomalious GVD is equivalent to 'normal' diffraction

τ

0u

02T0u

e


Dispersion length: 02DL z

3. Evolution of the amplitude

2 2

00 2( , ) exp exp exp ( )

( ) ( ) 2 ( )Tv z A z

T z T z D R z

ii

with

0

2

0 0240

1( ) , ( ) 11 z

z

zA z A T z Tz

2 ( ) ( ) const.A z T z =

'Phase curvature' is not fitting to the description of pulses introduction of

new parameter Chirp

remember: 2 2

2 2 ( , , )( )kx y z

x y R z

monochromatic fields:

( )( )

arbitrary time dependence of phase

( ) ( )

and

2

2

( ) ( ) 0

chirp

parabolic approximation 'chirp' constant dimensionless chirp parameter (often just chirp)

2 2

02

( )2

TC

variable frequency of the pulse in time integration leads to:

0 02 2

2

0 0

( ) ( ) 2 , ( )T

CT

C


0C up-chirp 0C down-chirp

phase curvature ( )R z Chirp ( )C z Complete phase:

2 2

0 0 20

( ) ( )2 ( )

z z C zv DR z v T

g g

2

0 0( )2 ( ) ( )

T zC zDR z R z

with

2 2

0( ) z zR zz

2

20

02 2

0 0

( )1 z

z

z z zC zz z z

( ) ( ) 00 0

1(0) 0, sgn , ( )2

zC C z z C zz

= =- ¥ =- with 2

00 2

TzD

Attention: Chirp depends on sign of 0z and hence of .D Complete field:

2 2

00 0 02 2

0

( , ) exp exp ( ) exp ( ) exp( ) ( )Tu z A C z z k z t

T z T z T

i i i

Dynamics of a pulse is equivalent to that of a beam.

important parameter dispersion parameter 2

00 2

TzD

1) 0z z : no effect

2) 0z z : similar to beam diffraction

3) 0z z : asymptotic dependence

00

2( ) ( ) / / .

DT z T T z z T z consz t

z T 0 0

0

τ τ


2

( )D

T zzT

0

0D is only important if initial pulse is chirped, since otherwise quadratic dependence is observed.

2.5.5.4 Example 2: Chirped Gaussian pulse Important because of:

short pulse lasers chirped pulses Chirp is introduced on purpose, for subsequent pulse compression analogy to curved phase focusing chirped pulse amplification (CPA) Petawatt lasers

1. Input pulse shape

0 0 20

02 (1 )( ) exp Cv A

T

i C0 – initial chirp

2. Input pulse spectrum

2 2

0 00 0 2

0

(1 )( ) exp4(1 )T CV A

C

i

spectral width: 22 0

20

4(1 )CT

s

spectral width of chirped pulse is larger than that of unchirped pulse

2 204 /T s only for transform limited pulses


Aim: calculation of pulse width and chirp in dependence on z for given initial conditions Gaussian beam q -parameter similar for Gaussian pulse Use analogy: however it is limited to homogeneous space ( ) (0) .q z q z

Remember beams:

2

1 1 2 .( ) ( ) ( )q z R z kw z

i

2 22

0

1 1 2 ( ), ( ) ( ),( )

DC zk w z T zD R z T

2 20

1 2 ( ) 2( ) ( )

DC z Dq z T T z

i

2

02 2

0

1 2 ( )( ) ( )

D TC zq z T T z

i

0T pulse width at 0z , which is not in the 'focus' or waist hence at z=0:

020

1 2(0)

D Cq T

i 0 (0).C C

Idea:

a) ( ) (0)q z q z and 020

1 2(0)

D Cq T

i insert into 1( )q z

b) 2

02 2

0

1 2 ( ) .( ) ( )

D TC zq z T T z

i

set a) and b) equal T(z), C(z) generally: 2 equations for 0 0, , , ( ), ( )C T z C z T z 3 values predetermined here: z=d

1) Determination of q parameter at input

200

20

(0)2 1

CTqD C

i

2) Evolution of q parameter

20 2 2 20

0 0 0 02 20 0

1( ) (0) 2 12 1 2 1

CTq d q d d Dd C C T TD C D C

ii


3) Inversion of equation for q(d)

2

02 2

0

1 2 ( ) .( ) ( )

D TC dq d T T d

i

2 2

2 2002 4 4

0

( )( ) ( ) ( )2 ( ) ( )

T T dq d C d T d TD C d T d T

i

4) Set two equations equal

2 2 2 2 20 0 0

2 20 0

2 40

4200

2 1 ( )

2 ( ) ( )1

(

2

) ( )T TDd C C

D C d T d TD

T T T d C d T d

C

i i

a) real part

2 2 2 40 0 0 0

2 2 4 40 0

2 1 ( ) ( )1 ( ) ( )

Dd C C T C d T T dC C d T d T

b) imaginary part 2 2

02 2 4 40 0

( )11 ( ) ( )

T T dC C d T d T

If we predetermine 3 parameters, we can determine the other 2 unknown parameters. Important case: Where is the pulse compressed to the smallest length? given: 0 0, , ( ) 0C T C d (focus) unknown: , ( )d T d

a) real part must be zero 2 20 0 02 1 0Dd C C T

0 0

2 20 0

201 s1

2 2n

1g ( )

1 DC Cd T D L

C CD

b) 2

2 020

( )1

TT dC

Resulting properties: 1) A pulse can be compressed when the product of initial chirp and

dispersion is negative 0 0.C D 2) The eventual compression increases with initial chirp. Physical interpretation: If e.g. 0 0C and 0 / 0D v g 'red' is faster than 'blue'


1) First the 'red tail' of the pulse catches up with the 'blue front' until

( ) 0C z (waist), i.e. the pulse is compressed, no chirp remains. 2) Then ( ) 0C z and red is in front. Subsequently the 'red front' is faster

than the 'blue tail', i.e. the pulse gets wider.

2

20

20

0

0

( )21 z

z

z TC z zDz

2.6 (The Kramers-Kronig relation) 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

0 0 0( , ) ( ) ( , ) ( , ) ( ) ( , )

tt R t t t dt t R t d

P r E r P r E rr r r r

Reality of the response function implies:

- *1 1e e2 2

i iR d d

Causality of the response function implies:


R y with

1 for 01 for 020 for 0

Heaviside

In the following, we will make use of the Fourier transform of Heaviside distribution:

2 e Pi t idt t

defined in integral only

In Fourier space, the Heaviside distribution consists of the Dirac delta distribution

0 0( ) Dirac delta distributiond f f

and an expression involving a Cauchy principal value:

0

P ( ) lim ( ) ( )i i id f d f d f

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:

e ei iyRd d

d y

1 1P2 2

i

1 P ( )2 2

y yd

i

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:

a) y y even function

b) y y odd function

a) y y

In this case y y is a real valued and even function. We can exploit this property and show that


1 1e e2 2

i id y dy yy

is real as well

Hence, we can conclude from equation (*) above that

1 P2 2

i y yd

Now we have expressions for *, and can compute real and imaginary part of the susceptibility:

* 1 1P P2 2 2 2

i y y i y yd d y

1 Pi

dy

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

1 1e e2 2

i id y dy y y

is purely imaginary

With equation (*) we then find that

1 P2 2

i y yd

(see (*)) and

Again we can then compute real and imaginary part of the susceptibility

1 1P P2 2 2 2

i y y i y yd yd

* 1 Pi y

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

2 20

2 20

2 ( )( ) 1 P ,

( ) 12( ) P .

d

d

dispersion and absorption are linked, e.g., we can measure absorption and compute dispersion

example:

0( ) ( ) 02 20

( ) 1

Drude model


3. Diffraction theory 3.1 Interaction with plane masks In this chapter we will use our knowledge on beam propagation to analyze diffraction effects. In particular, we will treat the interaction of light with thin and plane masks/apertures. Therefore we would like to understand how a given transversally localized field distribution propagates in a half-space. There are different approximations commonly used to describe light propagation behind an amplitude mask: A) If we use geometrical optics we get a simple shadow. B) We can use scalar diffraction theory with approximated interaction, i.e., a

so-called aperture is described by a complex transfer function ( , )t x y with ( , ) 0t x y for ,x y a (aperture)

Here we consider the description based on scalar diffraction theory. Then we can split our diffraction problem into three processes: i) propagation from light source to aperture

not important, generally plane wave (no diffraction) ii) multiply field distribution of illuminating wave by transfer function ( , , ) ( , ) ( , , )u x y z t x y u x y z A A

iii) propagation of modified field distribution behind the aperture through homogeneous space

( , , ) , ; ( , ; )expu x y z H z z U z x y d d

A A i

or

( , , ) , , ( , , )u x y z h x x y y z z u x y z dx dy

A A

with

12

1 FT2

h H

In the following we will use the notation z z z B A . According to our choice of the propagation function H , resp. h , we can compute this propagation either exact 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.


3.2 Propagation using different approximations The general case - small aperture 3.2.1

We know from before that for arbitrary fields (arbitrary wide angular spectrum) we have to use the general propagation function

, ; expH z z B Bi where 2 2 2 2( ) .k

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:

( , , ) ( , ) , ; expu x y z U H z x y d d

B i

where ( , ) ( , )U u x y FT

Derivation of the response function We start from the Weyl-representation of a spherical wave:

1 1exp exp2

kr x y z d dr

i

i i

Now we can compute the response function h , which we did not do in the previous chapter, where we computed only Fh (Fresnel approximation)). The following trick shows that

2112 exp exp FT 2kr x y z d d H hz r

i i

and therefore

1 1, , exp2

h x y z krz r

i with 2 2 2 .r x y z

The resulting expression in position space for the propagation of mono-chromatic beams is also called 'Rayleigh-formula':

( , , ) , , ( , , )u x y z h x x y y z u x y z dx dy

B A

Fresnel approximation (paraxial approximation) 3.2.2From the previous chapter we know that we can apply the Fresnel approximation if 2 2 2k which is valid for a limited angular spectrum, and therefore a large size of the structures inside the aperture. Then

2 2, ; exp exp2zH z kzk

B

F B Bi i


2 2, , exp exp2kh x y z kz x y

z z

F B B

B B

ii i

Paraxial Fraunhofer approximation (far field 3.2.3approximation)

A further simplification of the beam propagation is possible for many diffraction problems. Let us assume a narrow angular spectrum

2 2 2k

and the additional condition for the so-called Fresnel number NF

0.1N F with a aNz

F

B

where a is the (largest) size of the aperture (like the "beam width"). Obviously, a larger aperture needs a larger distance zB to fulfill 0.1NF . 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:

2 2

( , , ) ( , ; ) ( , )exp

( , ; ) ( , )

exp ( , )exp2

u x y z H z U x y d d

h x x y y z u x y dx dy

kkz u x y x x y y dx dyz z

B B

B

BB B

F F

F

i

ii i

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 ( , ) 0t x y for ,x y a (aperture)

( , ) 0u x y for ,x y a

This means that we need to integrate only over the aperture in the above integral:

2 2( , , ) exp ( , )exp2

a

a

ku x y z kz u x y x x y y dx dyz z

B B

B BF

ii i

Now, let us have a closer look at the exponential expression:


2 2

2 2

2 22

2

2

2

exp2

exp2

exp exp exp2 2

2 2x xx x y yy y

xx yy

k x x y yz

kz

k k kx yz z

x yz

B

B

B B B

i

i

i -i i

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

2

2 2 22k kax y Nz z

B B

F

for 0.1N F 2 2exp 12k x yz

B

i

This means that we can neglect the quadratic phase term in x',y' and we get for the far field:

2 2

22 2

( , , ) exp exp ( )2

( , )exp

2exp ( )exp ( ),

2

ku x y z kz x yz z

kx kyu x y x y dx dyz z

x kkz U x yykz zz z

k

BB

B B

FR BB

BB B BB

ii i

-i

i i i

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

2

21( , , ) ( , ; )x yI x y z U k k z

z zz

FR B AB BB

Interpretation: For a plane in the far field at z z B in each point ,x y only one angular frequency / ; /kx z ky z B B with spectral amplitude ( / , / )U kx z ky z B B 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


need to Fourier transform the field at the aperture. In order to apply this approximation we have to check that: A) 2 2 2k smallest features ,x y narrow angular

spectrum (paraxiality) B) 2 1a

zN BF largest feature a determines 2az B far field

Example: 4, 10 , 100 , 1 10 1x y a z m B cm

Non-paraxial Fraunhofer approximation 3.2.4The 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 2k contribute to the far field distribution. Evanescent waves decay for 1kz z B B .

0.1N F with 01a a zNz z

F

B B

2 2 2 2 2 2

2 2 2

2 2 2 2 2

2

2

2( , , )

( ; )exp,

u x y z

kx kyU

zx y z x y z

x y zx y z x z

z ky

B

B

F

B

B

A

B

R

B

B i

i

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.

Fraunhofer diffraction pattern 3.3.1

Let us consider an incident plane wave, with a wave vector perpendicular ( 0)x yk k to the mask t

( , , ) expA z Au x y z A k z i

or, more general, inclined with a certain angle


( , , ) expA x y z Au x y z A k x k y k z i

Form the previous chapter we know that the diffraction pattern in the far field in paraxial Fraunhofer approximation is given as:

22

21( , , ) ( , , ) ( , )x yI x y z u x y z U k k

z zz

B BB BB

The diffraction pattern is proportional to the spectrum behind the mask at

,x yk kz z

B B

.

Let us calculate the field for inclined incidence of the excitation. The field behind the mask is given by:

( , , ) ( , , ) ( , ) exp ( , )A A x y z Au x y z u x y z t x y A k x k y k z t x y i

The Fourier transform gives the spectrum:

2

( , )

exp ( , )exp2

exp ,

z A x y

z A x y

x yU k kz z

A x yk z t x y k k x k k y dx dyz z

x yA k z T k k k kz z

B B

B B

B B

i i i

i

Hence, the intensity distribution of the diffraction pattern is given as:

2

21( , , ) ,x y

x yI x y z T k k k kz zz

B

B BB

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

1 ,

( , )0

x a y bt x y

for elsewhere

2 2( , , ) sinc sincx yI x y z ka kbz z

BB B


B) Circular aperture (pinhole) illuminated by normal plane wave

2 2 21

( , )0

x y at x y

for elsewhere

22 2

1

2 2

J( , , )

kaz

kaz

x yI x y z

x y

B

B

B Airy disk

The first zero of the Bessel function (size of Airy disk):

0.611.22kaz z a

B B

with

2 2 2x y

So-called angle of aperture: 2 1.22z a

B

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 :


Then, we can express the mask function t as:

1

10

( ) ( )N

n

t x t x nb

with 1

( )( )

0st x x a

t x

for elsewhere

The Fourier transform of the mask then reads

1

10

( )expN

n

x xT k t x nb k x dxz z

B B

i

With the new variable x nb X we can simplify further:

1

0

1

0

( )exp exp

exp

aN

n a

N

n

x x xT k t X k X k nb dXz z z

x x xT k T k k nbz z z

SB B B

SB B B

i i

i

We see that the Fourier transform ST of the elementary slit St appears. The second factor has its origin in the periodic arrangement. With some math we can identify this second expression as a geometrical series and

perform the summation:

12

0 2

sinexp

sin

N

n

Nn

i

Thus we finally write:

2

2

sin

sin

k xz

k xz

N bx xT k T kz z b

B

B

SB B

For the particular case of a simple grating of slit apertures we have

sincx xT k k az z

S

B B

and therefore

222

22

sinsinc

sin

k xz

k xz

N bxI k az b

B

BB


We find three important parameters for the diffraction pattern of a

grating: width of diffraction pattern first zero of slit function TS

xk az

S

B

2zxa

B

S

The width of the total far-field diffraction pattern Sx (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

22

22

sin

sin

k xz

k xz

N b

bB

B

2k x b n

z P

B

zx nb

Bp

These are the so-called diffraction orders, which are determined exclusively by the grating period.

width of maximum zeros of grid function

2k xN b

z N

B

zx

Nb

BN

The width of a maximum in the far-field diffraction patter Nx (smallest length scale in the pattern) is determined by *N b , the total size of the mask (largest length scale of the mask).

-42 -35 -28 -21 -14 -7 0 7 1 4 2 1 2 8 3 5 4 2

0

1 0

2 0

3 0

4 0

5 0

B2

k b xN

z π

0

I

I7N=

Sx

Nxpx=


These observations are consistent with the general property of the Fourier-transform: small scales in position space give rise to a broad angular spectrum.

3.4 Remarks on Fresnel diffraction Fresnel number a aN

zF

B

10NF (a large, z B small, 01/ 30z zB ) shadow 0.1NF ( 03 )z z B Fraunhofer FT of aperture 10 0.1N F ( 0 01 / 30 3z z z B ) Fresnel diffraction


4. Fourier optics - optical filtering From previous chapters we know how to propagate the optical field through homogeneous space, and we also know the transfer function of a thin lens. Thus, we have all tools at hand to describe optical imaging. 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 Transfer function of a thin lens 4.1.1

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

2 2exp exp , exp2kkf x y t x y kf

f f f

plane wavespherical wave

L

i ii i i

And therefore the transfer function for a thin lens is given as (see chapter 2.6.3):

2 2, exp2kt x y x yf

L i

In Fourier domain we find consequently

2 22, exp

22f fT

k

L i i


Optical imaging 4.1.2Let us now consider optical imaging. We place our object in the first focus of a thin lens, with a field distribution 0( , )u x y , and follow the usual light propagation recipe.

A) Spectrum in object plane 0 0( , ) FT ( , )U u x y

B) Propagation from object to lens (lens positioned at distance f) 0( , ; ) , ), ; (HU f f U F

20

2exp exp2

( , ; ) ( , )kf fk

U f U

ii

C) Interaction with lens (multiplication in position space or convolution in Fourier domain)

( , , ) , ( , , )u x y f t x y u x y f L

2

2 20

22

( , ; ) ( , ; )

exp

ex

,

exp22

p ( , )2

U T

f fk

f U f

kf

f U d dk

L

ii

i

i

D) Propagation from lens to image plane ,( , ;2 ) ( , ; );U f U fH f F


2 22

2 20

2 2

( , ;2 ) exp exp22

exp (

2

exp2

, )2

f fU f kfk

fk

f U d dk

i i i

i i

Quadratic terms 2 2

2f

k

i and 2 2

2f

k

i in the exponent

cancel with quadratic terms from 2 2

2fk i .

02

02

( , ;2 ) exp 2 ( , )exp2

exp 2 ,2

f fU f kf U d dk

f f fkf uk k

i i i

i i

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

1

02

( , ,2 ) FT ( , ;2 )

exp 2 , exp2

u x y f U f

f f fkf u x y d dk k

i i i

With the coordinate transformation

2 2, ,f fx y d dx d dyk k f f

we get:

01( , ,2 ) exp 2 , exp ku x y f kf u x y xx yy dx dyf f

i i i

2

0

2exp( , 2 ) ,2, k k ku x y f U x y

f ff

f

i i

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 z fB . This finding allows us to perform an optical Fourier transform, and in the Fourier plane it is possible to manipulate the spectrum.


4.2 Optical filtering and image processing The 4f-setup 4.2.1

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:

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.

0( , ,2 ) ,k ku x y f AU x yf f

We have to compute the imaging with the second lens after manipulation. Our final goal is the transmission function ( , ;4 )H f A of the complete imaging system:

0( , ,4 ) ( , ), exp;4H fu x y f U x y d d

A i


Note: We will see in the following calculation that the second lens does a Fourier transform exp x y -i . In order to obtain a proper back transform we have to pass to mirrored coordinates ,x x y y . The transmission mask ( , )p x y contains all constrains of the system (e.g. a limited aperture) and optical filtering (which we can design).

A) Field behind transmission mask

0( , ,2 ) ( , ,2 ) ( , ) ~ , ( , )k ku x y f u x y f p x y AU x y p x yf f

B) Second lens Fourier back-transform of field distribution

22

( , ,4 ) exp 2 , ;2k ku x y f kf U x y ff f f

i i

Now we can make the link to the initial spectrum in the object plane 0U :

0

( , ,4 ) ~ ( , ,2 )exp

~ , ( , )exp

ku x y f u x y f xx yy dx dyf

k k kU x y p x y xx yy dx dyf f f

i

i

Here we do not care about the amplitudes and just write ~: To get the anticipated form we need to perform a coordinate transformation:

,k kx yf f

Then we can write:

0( , ,4 ) ~ , ( , )expf fu x y f U p x y d d

k k

i

By passing to mirrored coordinates ,x x y y we get

0(( , ,4 ) , )~ , expf fp

k ku x y f U x y d d

i

Hence we can identify the transmission function of the system

, ;4 ~ ( , )f fH f pk k

A

Summary Fourier amplitudes get multiplied by transmission mask transmission mask transmission function


coordinates of image mirrored coordinates of object In position space we can formulate propagation through a 4f-system by using the response function ( , )h x yA

0( , ,4 ) ( , ) ( , )u x y f h x x y y u x y dx dy

A

As usual, the response function is given as:

21( , ) , ;4 exp

2h x y H f x y d d

A A i

From above we have , ;4 ~ ( , )f fH f pk k

A

,( , ) ~ ( )exph x y p x y d df fk k

A i

We introduce the coordinate transform ,f fx yk k

( , ) ~ ( , )exp ~ ,k k kh x y p x y xx yy d xd y P x yf f f

A i

0( , ,4 ) ~ , ( , )k ku x y f P x x y y u x y dx dyf f

The response-function is proportional to the Fourier transform of the transmission mask.

Examples of aperture functions 4.2.2

Example 1: The ideal image (infinite aperture) Be careful, we use paraxial approximation limited angular spectrum

1p P x x y y 0( , ,4 ) ~ ( , )u x y f u x y mirrored original


Example 2: Finite aperture

22 21 / 2( , )0

x y Dp x y

for elsewhere

Transmission function:

2 2 21 / 2, ;4 ~

0

f fk k DH f

for

elsewhereA

finite aperture truncates large angular frequencies (low pass) determines optical resolution

Optical resolution 4.2.3A finite aperture acts as a low pass filter for angular frequencies.

22 2

2 2 2/ 2 / 2f f kD Dk k f

With 2 2 2a we can define an upper limit for the angular frequencies max which are transmitted (bandwidth of the system)

22

22 2

k Df

max 2

2n D

f

max

Translated to position space, the smallest transmitted structural information is

given by: min2 2 fr

nD

max

A more precise definition of the optical resolution can be derived the following way:


0

2 21 2

02 22

( , ,4 ) ~ , ( , )

J~ ( , ) ,

kDf

kDf

k ku x y f P x x y y u x y dx dyf f

x x y yu x y dx dy

x x y y

One point of the object 0 0,x y gives an Airy disk (pixel) in the image:

22 2

1 2 0 0

2 22 0 0

J kDf

kDf

x x y y

x x y 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:

min 1.222kD r

f

Hence we find: min1.22 fr

nD

Further examples for 4f filtering:



5. The polarization of electromagnetic waves 5.1 Introduction We are interested in the temporal evolution of the electric field vector ( , )tE rr . 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 ( , ) expt t E r E k ri .

If we assume propagation in z direction (k-vector points in z-direction), ( , ) 0t divE r implies that we can have two nonzero transversal field

components x and y components ,E Eyx The orientation and shape of the area which the (real) electric field vector covers is in general an ellipse. There are two special cases:

line (linear polarization) circle (circular polarization)

5.2 Polarization of normal modes in isotropic media

00k

k propagation in z direction

The evolution of the real electric field vector is given as

( , ) exp ( )t kz t E r Er i

Because the field is transversal we have two free complex field components

exp

exp

0

E

E

Ex x

y y

i

i with ,Ex y and ,x y being real

Then the real electric field vector is given as

cos

( , ) cos

0

E t kz

t E t kz

E rx x

r y y

Here, only the relative phase is interesting y x

Conclusion:


Normal modes in isotropic, dispersive media are in general elliptically polarized; ,E Ex y and phase difference y x are free parameters

5.3 Polarization states Let us have a look at different possible parameter settings: A) linear polarization n (or 0E x or 0yE )

B) circular polarization , / 2E E E x y

/ 2 counterclockwise rotation

/ 2 clockwise rotation

These pictures are for an observer looking contrary to the propagation

direction. C) elliptic polarization ,E E n x y

0 counterclockwise 2 clockwise Example:

Ex

Ey

Ex

Ey


Remark A linearly polarized wave can be written as a superposition of two counter-rotating circularly polarized waves. Example: Let's observe the temporal evolution at a fixed position 0kz with / 2 .

cos cos cossin sin 2 0

0 0 0

t t tE t E t E


6. Principles of optics in crystals In this chapter we will treat light propagation in anisotropic media (the worst case). Like in the isotropic case before we will seek for the normal modes, and in order to keep things simple we assume homogeneous media.

6.1 Susceptibility and dielectric tensor before: isotropy (optical properties independent of direction) now: anisotropy (optical properties depend on direction) The common reason for anisotropy in many optical media (in particular crystals) is that the polarization P depends on 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):

0( , ) ( , ) P r E r

0( , ) ( , ) D r E r

In the following we will write E E , because we assume monochromatic light and the frequency is just a parameter. Now (anisotropic):

3

01

( , ) (( ) , )i jj

ijP E

r rTensorkomponenten

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 - 75 10 m ; crystal - 105 10 m ), but the field is influenced by the symmetries of the crystal (see next subchapter 6.2).


In complete analogy we find for the D field:

3

01

0

( , ) ( ) ( , )

( , ) ( ) ( , )

i ij jj

D E

r r

D r E r

As for the polarization we find: D E

We introduce the following notation: ˆ ij χ susceptibility tensor

ij ε dielectric tensor

1ˆ ˆ ij σ ε inverse dielectric tensor

3

01

( ) ( , ) ( , )ij j ij

D E

r r

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 where D E , i.e., our principal axes:

3

01

i ij j ij

E D D

This is a so-called eigenvalue problem, with eigenvalues . If we want to solve for the eigenvalues we get

det 0,ij ij ij ijI I with

This leads to a third order equation in , hence we expect three solutions (roots) ( ). The corresponding eigenvectors can be computed from

3

( ) ( ) ( )

1

.ij j ij

D D

The eigenvectors are orthogonal: ( ) ( ) 0i iD D for ( ) ( )


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:

1,ij i ij ij i ij iji

1

2

3

0 00 00 0

ij

The above reasoning shows that anisotropic media are characterized in general by three independent dielectric functions (in the principal coordinate system). It is easier to do all calculations in the principal coordinate system (coordinate system of the crystal) and back-transform the final results to the laboratory system.

6.2 The optical classification of crystals Let us now give a brief overview over crystal classes and their optical properties: A) isotropic

three crystallographic equivalent orthogonal axes cubic crystals (diamond, Si....)

1 2 3 0i iD E

Cubic crystals behave like gas, amorphous solids, liquids, and have no anisotropy.

B) uniaxial two crystallographic equivalent directions trigonal (quartz, lithium niobate), tetragonal, hexagonal

1 2 3

C) biaxial no crystallographic equivalent directions


orthorhombic, monoclinic, triclinic

1 2 3

6.3 The index ellipsoid The index ellipsoid offers a simple geometrical interpretation of the inverse dielectric tensor 1ˆ ˆ σ ε The defining equation for the index ellipsoid is

3

, 1

1ij i ji j

x x

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 3

0, 1 1

2ij i j i ii j i

D D E D w

el

In the principal coordinate system the defining equation of the index ellipsoid reads:

2 2 22 2 2 1 2 3

1 1 2 2 3 31 2 3

1x x xx x x

Here the elements of the dielectric tensor can be related to refractive indexes i in


Summary: anisotropic media tensor instead of scalar in principal system: i in

The index ellipsoid is degenerate for special cases: Cubic crystal: sphere Uniaxial crystal: rotational symmetric with respect to z-axis and n n1 2

6.4 Normal modes in anisotropic media Let us now look for the normal modes in crystals. A normal mode is: a solution to the wave equation, which shows only a phase dynamics

during propagation while amplitude and polarization remain constant most simple solution exp t k r i

a solution where the spatial and temporal evolution of the phase are connected by a dispersion relation ( ) k or ( ) k k

From before: In isotropic media the normal modes are monochromatic plane waves

( , ) expt t E r E k ri

2

2 22( ) ( )k

c

k

with ( ) 0 real and 0 k E k D . Normal modes are elliptically polarized, and the polarization is conserved during propagation. now: What are the normal modes in anisotropic media? simple for propagation in direction of principal axis of the index ellipsoid

D

D

k→

x

y

3n

2n1n


Normal modes propagating in principal directions 6.4.1Let us first calculate the normal modes for propagation in the direction of the principal axes.

We assume without loss of generality that the principal axes are in , ,x y z direction and the light propagates in z-direction ( )kk z . Then, the fields are arbitrary in the x,y-plane , 0x yD D

and 0i iD E i

In general we have E D , but here 0 0 k D k E , and the two polarization directions x,y are decoupled:

11 1 11 1, exp expexp ( )D DD k z zt t ii -i with 2

21 12 ( )k

c

22 2 22 2, exp expexp ( )D DD k z zt t ii -i with 2

22 22 ( )k

c

We see that, in contrast to isotropic media, elliptic polarization is not a normal mode, the polarization direction changes during propagation. But, for polarization in direction of a principal axis (x,y) only the phase changes during propagation, thus we found our normal modes:

2( ) 2 2 2

1 1 12

2( ) 2 2 2

2 2 22

exp normal mode a

exp normal mode b

D t nc

D t nc

D k r e k k

D k r e k k

aa a a

bb b bk

i

i

D

D

k→

x

y

3n

2n1n


For light propagation in principle direction we find two perpendicular linear polarized normal modes with E D .

Normal modes for arbitrary propagation direction 6.4.2

6.4.2.1 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 given frequency and crystal 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 and which is

perpendicular to k .

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 ,n na b of the normal modes for the propagation direction / ku k

ak nc

a , k nc

b b

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

( ) ( )

( ) ( )

0 0

,i ii i

D DE E

a ba b

i i

Thus, ( , ) ( , )D Ea b a b , and ( , )E a b are not perpendicular to .k This has a direct consequence on the pointing vector:


1

2 S E H

k S because S E

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

( , ) exp

( , ) exp

t t

t t

E r E k r

D r D k r

i

i

In the isotropic case we found the dispersion relation

2

2 22( ) ( ) ( )k

c

k

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 t k r i , but the wavenumber depends on the direction uof 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 :

1 1

2 2

3 3

k uk k uk u

k with 2 2 21 2 3 1u u u

Our aim is to derive 1 2 3( , , )k k k or 1 2 3,( , , )u uk u or 1 2 3( , , , )k k u u u . We start from Maxwell's equations for the plane wave Ansatz: 00 k D k E H

0 k H k H D


Now we follow the usual derivation of the wave equation:

2

20

1c

k k E D

22

20

1c

k k E k E D

Here k E does not vanish as in the isotropic case! In the principal coordinate system and with 0i i iD E we find

2

22i j j i i i

j

k k E k E Ec

2

22 i i j j

jik E

ck Ek

Note: for isotropic media the r.h.s. of this equation vanishes (k E = 0 ). Thus, we have the following problem to solve:

2

2

2

2

2

2

2 21 2 3 1 2 1 3 1

2 22 1 2 1 3 2 3 2

2 233 1 3 2 3 1 2

000

c

c

c

k k k k k k Ek k k k k k E

Ek k k k k k

The straightforward way to solve this problem is using det .. 0 , which gives the dispersion relation ( )k for given /ik k . However, there is a more easy way to obtain the dispersion relation:

trick: 2

22 i i i j j

j

k E k k Ec

2

22

ii j j

jic

kE k Ek

Now we multiply this equation by ik , perform a summation over the index ' i ' and rename i j on l.h.s:

2

2

2

2.i

j j j jj i jic

kk E k Ek

Because div 0j j jk EE we can divide and get the (implicit) dispersion relation:

2

22

2

1i

ii

k

kc


With 1 1 1

2 2 2

3 3 3

( ) ( )k u uk k u n u

ck u u

we can write

2 2 2

22

22

1 11

i i

ii ii

k u u

k nc

22

2 1i

ii nnu

final form of DR

Result: For given i and direction 1 2,u u , (because 2 2 2

1 2 3 1u u u ) we can compute the refractive index 1 2, ,n u u seen by the normal mode.

Explicit calculation (multiplication by all denominators):

2 2 2 2 2 2 2 2 2 2 2 21 2 3 2 1 3 3 1 2

2 2 21 2 3

u n n n u n n n u n n n

n n n

The resulting equation is quadratic in 2 2( )n since the 6n -term cancels. Hence, we get two (positive) solutions ,n na b , and therefore

,k n k nc c

a a b b for the two orthogonally polarized normal modes ( )D a

and ( )D b . In particular, for the propagation in direction of the principal axis ( 3 1 21, 0u u u , see 6.3) we find:

2 2 2 2 2 21 2 1 2 3n n n n n n

2 21 2 3 0n n

2 21 2,n n a b

Finally, we can compute the fields of the normal modes. We know:

2

22 i i i j j

j

k E k k Ec

and hence

2

22

ii j j

jic

kE k Ek

where the sum does not depend on the index i


( .j jj

k E const )

1 2 31 2 3 2 2 2

2 2 21 2 32 2 2

: : : :k k kE E Ek k k

c c c

and with 0i i iD E

1 1 2 2 3 31 2 3 2 2 2

2 2 21 2 32 2 2

: : : :k k kD D Dk k k

c c c

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 ( ) ( ) 0D D ba ? (be careful: ( ) ( ) 0E E ba )

2 2( ) ( )

2 22 2

2 2

22 2

2 2

2 2

2

2 22 2

2

2

~ a b i i

i i i

ia i b i

a ba b

i

i ia i b i

b a

k k u

k kc c

c k k k kk k

u u

k kc c

D D ba

Since the two red terms vanish due to the dispersion relation, it follows that ( ) ( ) 0D D ba . The vanishing of the red terms can be seen when rewriting the

dispersion relation: 2 2

2 2,2 2 2 2 2 2

,22 2 2

2 2 2, , ,2 2 2

1 1a b i i i

a b i i i

i i ia b i a b i a b i

k uk u c c u

ck k k

c c c

Normal surfaces of normal modes 6.4.3In addition to the index ellipsoid there is a second geometrical interpretation, the so-called normal surfaces:

If we plot the refractive indices (wave number or norm of the k-vector divided by 0k ) of the normal modes in the ik -space (normal surfaces), we get a centro-symmetric, two layer surface:


biaxial: 4 points with n na b connecting lines define two optical axes

uniaxial: 2 points with n na b in the poles connecting line defines the optical axis (for 1 2 or , 3 e the z-axis is the optical axis)

isotrop: sphere How to read the figure:

fix propagation direction ( 1 2,u u ) intersection with surfaces distances from origin to intersections with surfaces correspond to

refractive indices of normal modes definition of optical axis n na b

Summary: there are two geometrical constructions: A) index ellipsoid (visualization of dielectric tensor)

fix propagation direction index ellipse half lengths of principal axes give ,n na b (refractive indices of the normal modes)

optical axis index ellipse is a circle for uniaxial crystals the optical axis coincides with one principal axis

B) normal surfaces (visualization of dispersion relation) fix propagation direction intersection with surfaces distances from origin give ,n na b

optical axis connects points with n na b Conclusion: In anisotropic media and for a given propagation direction we find two normal modes, which are linearly polarized monochromatic plane waves with two different phase velocities c na , c nb and two orthogonal polarization directions

( ) ( ),D D ba .

biaxial uniaxial

isotrop

optical axis optical axis


Special case: uniaxial crystals 6.4.4Let 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

1 2 3, or e which we call ordinary and extraordinary refractive indices. Then, we expect two normal modes: A) ordinary wave n independent of propagation direction B) extraordinary wave n depends on propagation direction The z-axis is, according to definition, the optical axis with a bn n

The ordinary wave ( )orD is polarized perpendicular to the z-axis and the k-vector.

The extraordinary wave ( )eD is polarized perpendicular to the k-vector and ( )orD .

Let us now derive the dispersion relation: From above we know the implicit form

22

2 1i

ii nnu

For uniaxial crystals this leads to

2 2 21 2 3

22 2 2

1u u unn n n

or or e

2 22 2 2 2 2 2 2 2 2 21 2 3n n n u u n n u n n e or or e or

A) ordinary wave: independent of direction

2n a or

22 2 2

02k n kc

a a or

B) extraordinary wave (derivation is your exercise): dependent on direction

2 2 2 21 2 2 23

1 2 32 2

1 , , ,b bb

u u u k n u u un c

e or

Hence for a given direction iu one gets the two refractive indexes ,n na b .


The geometrical interpretation as normal surfaces is straightforward and can be done, w.l.o.g., in the k2, k3 or y, z plane ( 0u 1 ). We have with 2 2 2 2

0ik k n u i

A) ordinary wave 2 2 2 2 21 2 3 0k k k k k a or

B) extraordinary wave 2 2 2

1 2 32 20 0

1 1 1k k k

k k

e or

What about the fields? We know from before that

1 2 31 2 3 2 2 2

2 2 22 2 2

: : : :k k kD D Dk k k

c c c

or or e

or or e

For the extraordinary wave all denominators are finite, and in particular 1 0k implies ( )

1 0D e , hence ( )D e is polarized in the y-z plane. Then, ( ) ( )D Dor e implies that ( )D or is polarized in x-direction.

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 1 0 0 1 3 0 3,D E D E r e

If we introduce an angle , as in the figures below, to describe the propagation direction, a simple computation of 2 ( )bn for the extraordinary wave is possible (exercise): 2 3sin , cosu u

2b 2 2sin cos

n

e or

or e

The following classification for uniaxial crystals is commonly used or e negative uniaxial or e positive uniaxial


7. Optical fields in isotropic, dispersive and piecewise homogeneous media

7.1 Basics Definition of the problem 7.1.1

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

And here is what we will do: 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

And here is how we will do it: 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, consider special cases like Fabry-Perot-interferometer, 1D photonic

crystals, waveguide…

Decoupling of the vectorial wave equation 7.1.2Before we will start by treating a single interface, it is worth looking again at the wave equation in frequency domain

2

20 02( , ) ( , ) ( , ) ( , )

c

rotrot E r E r j r P ri

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 / 0y

(2)

(2)

(2)

0

x z

x z

E Exx x z

y

E E zz x z

E

E

E

rot rot E grad div E E

Then, we can split the electric field as E E E

2 2

(2) (2)2 2

0

, 0 , 0 ,0

x

y

z

E xE

x zzE

E E

E is polarized perpendicular to the plane of propagation, E is polarized parallel to this plane. Both components are decoupled and can be treated independently:

2(2) 2

0 02

2(2) (2) (2) 2

0 02

( , ) ( , ) ( , )

( , ) ( , ) ( , )

c

c

E E r j r P r

E E r grad div E j r P r

i

i

alternative notations are: s TE (transversal electric) p TM (transversal magnetic)

From

0

( , ) ( , )

H r rot E ri

we can conclude that the corresponding magnetic fields are

0 0

, 00 0

HE E

H

E Hx

TE y TE

z

0 0

0 ,0 0

EH H

E

E Hx

TM TM y

z

Interfaces and symmetries 7.1.3Up to now we treated plane waves of the form

( , ) expt t E r E kri

Here, homogeneous space implies exp kri


and monochromaticity leads to exp t i Now, we will break homogenity in x-direction by considering an interface in y-z – plane infinite in y and z

W.l.o.g. we can assume , 0,k kk x z by choosing an appropriate coordinate system (plane of incidence is the x,z plane), and then the problem does not depend on y - coordinate. Hence, we can split the fields in TE and TM polarization E E ETE TM and treat them separately. We still have homogenity in z - direction, and therefore we expect solutions exp k zzi . 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:

( , , ) exp expTE TMx z t k z t k tx zx E E Ez zi i

Transition conditions 7.1.4From 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= Ey and Hz continuous TM: Ez and H=Hy continuous B) wave vectors homogeneous in z- direction phase e zik z zk 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.

Fields in one homogeneous layer 7.2.1Let us first compute the fields in one homogeneous layer of thickness d and dielectric function f

x

z

k


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 .

TE-polarization We have to solve the wave equation (no y-dependence because of translational invariance):

2 2 2

f2 2 2 ( , ) 0TE x zx z c

E

We use the ansatz from above: z( , ) ( )expTE TEx z x ik zE E , z( , ) ( )expTE TEx z x ik zH H

2 2

2f z2 2 ( ) 0TEk x

x c

E

with: TE TE0

( , ) ( , )ix z x z

H rotE

Now let us extract the equations for transversal fields y z,E E H :

2

2fx z2 , ( ) 0k k E x

x

with 2

2 2fx z f z2,k k k

c

z0

( ) ( )iH x E xx

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 y z,H H E :

2

2fx z2 , ( ) 0k k H x

x

z0 f

( ) ( )iE x H xx

Again, we succeed describing everything in transversal components.


We have the following problem to solve:

calculate fields (E, H) and derivatives ( ), ( )E x H xx x

at x d for

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 ,E H F generalized field 1

0 z 0 z,i H i E G generalized field 2

and write down the problem to solve:

2

2fx z2 , ( ) 0k k F x

x

f( ) ( )G x F xx

with fTE fTMf

11,

We know the general solution of this system (harmonic oscillator):

1 fx 2 fx

f f fx 1 fx 2 fx

( ) exp exp

( ) ( ) exp exp

F x C ik x C ik x

G x F x i k C ik x C ik xx

We have as initial conditions F(0), G(0) given, and can compute the constants C1, C2:

1 2

f fx 1 2

(0)

(0)

F C C

G i k C C

1f fx

2f fx

1 (0) (0)2

1 (0) (0)2

iC F Gk

iC F Gk

The final solution of the initial value problem is therefore:


fx fxf fx

f fx fx fx

1( ) cos (0) sin (0)

( ) sin (0) cos (0)

F x k x F k x Gk

G x k k x F k x G

By resubstituting we have the electromagnetic field in the layer 0 x d .

The fields in a system of layers 7.2.2In the previous subchapter we have seen how to compute the electromagnetic field in a single dielectric layer, dependent on the transverse field components Ey, Hz, (TE) and Hy, 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 dimenion) refractive index distribution. This is important for 'GRIN' - GRaded-Index-Profiles.

From above, we know the fields in one layer:

fx fxf fx

f fx fx fx

1( ) cos (0) sin (0)

( ) sin (0) cos (0)

F x k x F k x Gk

G x k k x F k x G

We can write this formally as

( ) (0)

ˆ ,( ) (0)

F x FG x G

m


where the 2 x 2-matrix m̂ describes propagation of the fields:

1cos sinˆ

sin cos

k x k xkx

k k x k x

m fx fxfx f

fx f fx fx

To compute the field at the end of the layer we set .x d We assume no absorption in the layer ˆ 1x m .

A system of layers is characterized by ,i id Hence, we can directly write the formalism for a multilayer system, it just requires matrix multiplication: A) two layers:

1 2 1

2 2 2 2 1 10

ˆ ˆ ˆ( ) ( ) ( )d d d

F F Fd d d

G G G

m m m

B) N layers:

1 2

1.. 0 0

1

ˆˆ ( )

ˆ ˆ ( )

N

i iid d d D

N

i ii

F F Fd

G G G

d

m M

M m

N

All matrices ˆ im have the same form, but different 2

2, , .i if i id k k

c

2zfx

Summary of matrix method: (0)F and (0)G given ( , zE H for TE, ,zE H for TM)

, , ,if i ik d z given matrix elements

multiplication of matrices (in the right order) total matrix fields ( )F D and ( )G D

7.3 Reflection – transmission problem for layer systems General layer systems 7.3.1

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 ( )kI , reflected ( )kR and transmitted ( )kT fields:

0 , 0 , 0k k k

k k k

k k ksx sx cx

I R T

z z z

with 2 2

2 2 2 2 2 22 2( ) , ( ) ,k k k k k k k k

c c

ssx z s z cx c z c z

where ( ) s and ( ) c are dielectric functions of substrate and cladding.

As we have seen before, the kz 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 dispersive, isotropic, homogeneous media. As a consequence, the kx 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. sin sink k s I s R I R (reflection) 2. sin sin sin sink k n n s I c T s I c T (Snellius)

Let us now rewrite the fields in order to solve the reflection transmission problem:

multi layer system


A) field in substrate complex amplitudes , .F FI R

( , ) exp exp exp

( , ) exp exp exp

F x z k z F k x F k x

G x z k k z F k x F k x

s z I sx R sx

s s sx z I sx R sx

i i i

i i i i

B) field in layer system matrix method

( , ) exp ( )

( , ) exp ( )

F x z k z F x

G x z k z G x

f z

f z

i

i

and the ampliudes ( )F x and ( )G x are given by

0

ˆx

F FG G

M

C) field in cladding

( , ) exp exp

( , ) exp exp .

F x z k z F k x D

G x z k k z F k x D

c z T cx

c c cx z T cx

i i

i i i

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 id I . We know that F and G are continuous at the interfaces, in particular at x=0 and x=D. We have:

0

ˆ .D

F FG G

M

Field in cladding at x D field in substrate at 0x

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:

11 12

21 22

.F FF M M

k F Fk F M M

I RT

s sx I Rc cx T ii

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