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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2020-02-02s.docx 1

Fundamentals of Modern Optics Winter Term 2019/2020

Prof. Thomas Pertsch Abbe School of Photonics, Friedrich-Schiller-Universität Jena

Table of content 1. Introduction ............................................................................................................. 4 2. Optical fields in dispersive and isotropic media .................................................... 15

2.1 Maxwell’s equations .............................................................................................................. 15 Adaption to optics ...................................................................................................... 15 Temporal dependence of the fields .......................................................................... 19 Maxwell’s equations in Fourier domain ................................................................... 20 From Maxwell’s equations to the wave equation .................................................... 20 Decoupling of the vectorial wave equation .............................................................. 22

2.2 Optical properties of matter .................................................................................................. 23 Basics .......................................................................................................................... 23 Types of considered light-matter interactions (covered by lecture Structure of

Matter) ....................................................................................................................... 26 Dielectric polarization and susceptibility (covered by lecture Structure of Matter)

.................................................................................................................................... 27 Conductive current and conductivity (covered by lecture Structure of Matter) ... 28 Generalized complex dielectric function .................................................................. 30 Material models in time domain ............................................................................... 34

2.3 Poynting vector and energy balance ..................................................................................... 36 Time averaged Poynting vector ................................................................................ 36 Time averaged energy balance ................................................................................. 37

2.4 Kramers-Kronig relation (covered by lecture Structure of Matter) ..................................... 40 3. Solution of wave equations for homogeneous isotropic media ............................ 44

3.1 Normal modes in homogeneous isotropic media ................................................................ 44 Transverse waves (epsilon unequal zero) ................................................................. 45 Longitudinal waves (epsilon equal zero)................................................................... 47 Plane wave solutions in different frequency regimes .............................................. 47 Time averaged Poynting vector of plane waves ....................................................... 54

3.2 Beams and pulses as well as the analogy of diffraction and dispersion ............................. 54 3.3 Diffraction of monochromatic beams in homogeneous isotropic media ........................... 56

Arbitrarily narrow beams (general case) .................................................................. 57 Fresnel- (paraxial) approximation ............................................................................. 64 Paraxial wave equation .............................................................................................. 69

3.4 Propagation of Gaussian beams ............................................................................................ 70 Propagation in paraxial approximation..................................................................... 71 Propagation of Gaussian beams with q-parameter formalism ............................... 76 Gaussian optics .......................................................................................................... 77 Gaussian modes in a resonator ................................................................................. 80

3.5 Dispersion of pulses in homogeneous isotropic media ....................................................... 86 Pulses with finite transverse width (pulsed beams) ................................................ 86 Pulses with infinite transverse extension (pulse propagation) ............................... 94 Analogy of diffraction and dispersion ....................................................................... 94 Propagation of a Gaussian pulse without chirp ....................................................... 95


Propagation of a chirped Gaussian pulse ................................................................. 99 4. Diffraction theory ................................................................................................ 103

4.1 Interaction with plane masks .............................................................................................. 103 4.2 Propagation using different approximations ...................................................................... 104

General case - small aperture ................................................................................. 104 Fresnel approximation (paraxial approximation) ................................................... 105 Paraxial Fraunhofer approximation (far field approximation) ............................... 105 Non-paraxial Fraunhofer approximation ................................................................ 107

4.3 Paraxial Fraunhofer diffraction at plane masks .................................................................. 107 4.4 Remarks on Fresnel diffraction............................................................................................ 112

5. Fourier optics - optical filtering ........................................................................... 114 5.1 Imaging of arbitrary optical fields with a thin lens ............................................................. 114

Transfer function of a thin lens ............................................................................... 114 Optical imaging using the 2f-setup ......................................................................... 115

5.2 Optical filtering and image processing ................................................................................ 117 4f-setup .................................................................................................................... 117 Examples of aperture functions .............................................................................. 119 Optical resolution ..................................................................................................... 121

6. The polarization of electromagnetic waves ......................................................... 123 6.1 Polarization of normal modes in isotropic media............................................................... 123 6.2 Polarization states ................................................................................................................ 124

7. Principles of optics in crystals .............................................................................. 126 7.1 Susceptibility and dielectric tensor ..................................................................................... 126 7.2 Optical classification of crystals ........................................................................................... 128 7.3 Index ellipsoid ....................................................................................................................... 129 7.4 Normal modes in anisotropic media ................................................................................... 130

Normal modes propagating in principal directions ................................................ 131 Normal modes for arbitrary propagation direction ............................................... 132 Normal surfaces of normal modes .......................................................................... 137 Special case: uniaxial crystals .................................................................................. 139

8. Optical fields in isotropic, dispersive and piecewise homogeneous media ........ 142 8.1 Basics..................................................................................................................................... 142

Definition of the problem ........................................................................................ 142 Decoupling of the vectorial wave equation ............................................................ 143 Interfaces and symmetries ...................................................................................... 144 Transition conditions ............................................................................................... 144

8.2 Fields in a layer system matrix method ......................................................................... 145 Fields in one homogeneous layer ........................................................................... 145 Fields in a system of layers ...................................................................................... 147

8.3 Reflection – transmission problem for layer systems ........................................................ 149 General layer systems .............................................................................................. 149 Single interface......................................................................................................... 156 Periodic multi-layer systems – Bragg mirrors – 1D photonic crystals ................... 164 Fabry-Perot-resonators ........................................................................................... 171

8.4 Guided waves in layer systems ............................................................................................ 177 Field structure of guided waves .............................................................................. 177 Dispersion relation for guided waves ..................................................................... 179 Guided waves at interface - surface polariton ....................................................... 182 Guided waves in a layer – film waveguide .............................................................. 184 Excitation of guided waves ...................................................................................... 187


9. Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) .......................................................................................................... 190

9.1 Introduction .......................................................................................................................... 190 9.2 Postulates ............................................................................................................................. 190 9.3 Simple rules for propagation of light .................................................................................. 191 9.4 Simple optical components ................................................................................................. 191 9.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) ................................. 195

Ray equation ............................................................................................................ 195 Eikonal equation ...................................................................................................... 197

9.6 Matrix optics ......................................................................................................................... 198 Ray-transfer-matrix .................................................................................................. 198 Matrices of optical elements ................................................................................... 198 Cascaded elements .................................................................................................. 199

This script originates from the lecture series “Theoretische Optik” given by Falk LEDERER in the physics program at the Friedrich Schiller University Jena (Germany) for many years between 1990 and 2012. Later the script was adapted by Stefan SKUPIN and Thomas PERTSCH for the international education program in photonics at the Abbe School of Photonics, Friedrich Schiller University Jena (Germany).


1. Introduction • 'optique' (Greek) lore of light 'what is light'? • Is light a wave or a particle (photon)?

D.J. Lovell, Optical Anecdotes

• Light is one of the requirements for life photosynthesis • 90% of information we get is visual

A) What is light? • electromagnetic wave propagating with the speed of 83 10 /c m s= × • wave = evolution of

− amplitude and phase complex description − polarization vectorial field description − coherence statistical description

Spectrum of Electromagnetic Radiation

Region Wavelength [nm]

Wavelength [m] (nm=10-9m)

Frequency [Hz] (THz=1012Hz)

Energy [eV]

Radio > 108 > 10-1 < 3 x 109 < 10-5

Microwave 108 - 105 10-1 – 10-4 3 x 109 - 3 x 1012 10-5 - 0.01

Infrared 105 - 700 10-4 - 7 x 10-7 3 x 1012 - 4.3 x 1014 0.01 - 2

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

Ultraviolet 400 - 1 4 x 10-7 - 10-9 7.5 x 1014 - 3 x 1017 3 - 103

X-Rays 1 - 0.01 10-9 - 10-11 3 x 1017 - 3 x 1019 103 - 105

Gamma Rays < 0.01 < 10-11 > 3 x 1019 > 105


B) Origin of light • atomic system determines original properties of light (e.g. statistics, frequency,

line width) • laser control of emission properties of matter at the origin of emission

− laser: artificial light source with new and unmatched properties (e.g. coherent, directed, focused, monochromatic)

− invention in 1958: very important development • optical system modifies properties of light (e.g. intensity, duration, …)


Schawlow and Townes, Phys. Rev. (1958).

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

C) Propagation of light through matter • light-matter interaction (G: Licht-Materie-Wechselwirkung)

effect dispersion diffraction absorption scattering ↓ ↓ ↓ ↓ governed by 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, metamaterials)

Two-dimensional photonic crystal membrane.

D) Light can modify matter • light induces physical, chemical and biological processes • used for lithography, material processing, or modification of biological objects (bio-

photonics)

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


E) Optical telecommunication • transmitting data (Terabit/s in one fiber) over transatlantic distances

1000 m telecommunication fiber is installed every second.


F) Optics in medicine and life sciences


G) Light 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


J) Schematic of optics

• geometrical optics

⋅ λ << size of objects daily experience ⋅ optical instruments, optical imaging ⋅ intensity, direction, coherence, phase, polarization, photons

(G: Intensität, Richtung, Kohärenz, Phase, Polarisation, Photon)

• wave optics ⋅ λ ≈ size of objects interference, diffraction, dispersion, coherence ⋅ laser, holography, resolution, pulse propagation ⋅ intensity, direction, coherence, phase, polarization, photons

• electromagnetic optics

⋅ reflection, transmission, guided waves, resonators ⋅ laser, integrated optics, photonic crystals, Bragg mirrors ... ⋅ intensity, direction, coherence, phase, polarization, photons

• quantum optics

⋅ small number of photons, fluctuations, light-matter interaction ⋅ quantum cryptography quantum computation, teleportation … ⋅ intensity, direction, coherence, phase, polarization, photons

geometrical optics

wave optics

electromagnetic optics

quantum optics


• in this lecture ⋅ electromagnetic optics and wave optics ⋅ no quantum optics subject of advanced lectures

K) Literature • Fundamental

1. Saleh, Teich, 'Fundamenals of Photonics', Wiley (1992) in German: "Grundlagen der Photonik" Wiley (2008)

2. Hecht, 'Optic', Addison-Wesley (2001) in German: "Optik", Oldenbourg (2005)

3. Mansuripur, 'Classical Optics and its Applications', Cambridge (2002) 4. Menzel, 'Photonics', Springer (2000) 5. Lipson, Lipson, Tannhäuser, 'Optik'; Springer (1997) 6. Born, Wolf, 'Principles of Optics', Pergamon 7. Sommerfeld, 'Optik'

• Advanced 1. W. Silvast, 'Laser Fundamentals', 2. Agrawal, 'Fiber-Optic Communication Systems', Wiley 3. Band, 'Light and Matter', Wiley, 2006 4. Karthe, Müller, 'Integrierte Optik', Teubner 5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Academic 6. Yariv, 'Optical Electronics in modern Communications', Oxford 7. Snyder, Love, 'Optical Waveguide Theory', Chapman&Hall 8. Römer, 'Theoretical Optics', Wiley,2005.


2. Optical fields in dispersive and isotropic media

2.1 Maxwell’s equations Our general starting point is the set of Maxwell’s equations. They are the basis of the electromagnetic approach to optics, which is developed in this lecture.

Adaption to optics The notation of Maxwell’s equations is different for different disciplines of science and engineering, which rely on these equations to describe electromagnetic phenomena at different frequency ranges. Even though Maxwell's equations are valid for all frequencies, the physics of light matter interaction is different for different frequencies. Since light matter interaction must be included in the Maxwell's equations to solve them consistently, different ways have been established how to write down Maxwell's equations for different frequency ranges. Here we follow a notation, which was established for a convenient notation at frequencies close to visible light.

Maxwell’s equations (macroscopic) In a rigorous way the electromagnetic theory is developed starting from the properties of electromagnetic fields in vacuum. In vacuum one could write down Maxwell's equations in their so-called pure microscopic form, which includes the interaction with any kind of matter based on the consideration of point charges. Obviously this is inadequate for the description of light in condensed matter, since the number of point charges, which would need to be taken into account to describe a macroscopic object, would exceed all imaginable computational resources. To solve this problem one uses an averaging procedure, which summarizes the influence of many point charges on the electromagnetic field in a homogeneously distributed response of the solid state on the excitation by light. In turn, also the electromagnetic fields are averaged over some adequate volume. For optics this procedure is justified, since any kind of available experimental detector could not resolve the very fine spatial details of the fields in between the point charges, e.g. ions or electrons, which are lost by this averaging. These averaged electromagnetic equations have been rigorously derived in a number of fundamental text books on electrodynamic theory. Here we will not redo this derivation. We will rather start directly from the averaged Maxwell's equations.

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

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

tt t tt

tt t tt

∂= − = ρ

∂∂

= + =∂

B rE r D r r

D rH r j r B r

ext

makr


− electric field (G1: elektrisches Feld) ( , )tE r [V/m]

− magnetic flux density ( , )tB r [Vs/m2] or [tesla] or magnetic induction (G: magnetische Flussdichte oder magnetische Induktion)

− electric flux density ( , )tD r [As/m2] or electric displacement field (G: elektrische Flussdichte oder dielektrische Verschiebung)

− magnetic field (G: magnetisches Feld) ( , )tH r [A/m]

− external charge density ( , )tρ rext [As/m3]

− macroscopic current density ( , )tj rmakr [A/m2]

Auxiliary fields The "cost" of the introduction of macroscopic Maxwell's equations is the occurrence of two additional fields, the dielectric flux density ( , )tD r and the magnetic field

( , )tH r . These two fields are related to the electric field ( , )tE r and magnetic flux density ( , )tB r by two other new fields, ( , )tP r and ( , )tM r as

[ ]0

0

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

t t t

t t t

= ε +

= −µ

D r E r P r

H r B r M r

− dielectric polarization (G: dielektrische Polarisation) ( , )tP r [As/m2],

− magnetic polarization ( , )tM r [Vs/m2] or magnetization (G: Magnetisierung)

− electric constant 120 8.854 10−ε ≈ × As/Vm

or vacuum permittivity (G: Vakuumpermittivität)

− magnetic constant 70 4 10−µ = π× Vs/Am

or vacuum permeability (G: Vakuumpermeabilität) Here the electric constant and the magnetic constant are connected by the speed of light in vacuum c as

0 20

1c

ε =µ

1 Marked by "G:", the German translation of important terms will be given.


Light matter interaction In order to solve this set of equations, i.e. Maxwell's equations and auxiliary field equations, one needs to connect the dielectric flux density ( , )tD r and the magnetic field ( , )tH r to the electric field ( , )tE r and the magnetic flux density ( , )tB r . This is achieved by modeling the material properties by introducing the material equations. • The effect of the medium gives rise to polarization [ ]( , )t f=P r E and

magnetization [ ]( , )t f=M r B . In order to solve Maxwell’s equations we need material models, which describe these quantities.

• In optics at visible wavelength, we generally deal with non-magnetizable media. Hence we can assume ( , ) 0t =M r . Exceptions to this general property are metamaterials, which might possess some artificial effective magnetization properties resulting in ( , ) 0t ≠M r .

Furthermore we need to introduce sources of the fields into our model. This is achieved by the so-called source terms, which are inhomogeneities and hence they define the unique solutions of Maxwell's equations.

− free charge density (G: Dichte freier Ladungsträger)

( , )tρ rext [As/m3]

− macroscopic current density (G: makroskopische Stromdichte) consisting of two contributions

( , ) ( , ) ( , )t t t= +j r j r j rmakr cond conv [A/m2]

⋅ conductive current density (G: Konduktionsstromdichte)

[ ]( , )t f=j r Econd

⋅ convective current density (G: Konvektionsstromdichte)

( , ) ( , ) ( , )t t t= ρj r r v rconv ext

− In optics, we generally have no free charges, which change their local density in time at rates comparable to the oscillation frequency of the electro-magnetic fields of light (several Terahertz):

( , ) 0 ( , ) 0t tt

∂ρ≈ → =

∂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

∂= −µ ε =

∂∂

= + ε

−

∂+

∂∂=

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 induced polarization on the electric field

( )P E and the dependence of the induced (conductive) current density on the electric field ( )j E .

− Once we have specified these relations, we can solve Maxwell’s equations consistently.

Example: − In vacuum, both polarization P and current density j are zero (simplest

material model). Hence we can solve Maxwell’s equations directly. Remarks:

− for linear response of matter: principle of superposition applies electromagnetic effects at different frequencies coexist without influencing each other

− Even though the Maxwell's equations, in the way they have been written above, are derived to describe light, i.e. electromagnetic fields at optical frequencies, they are simultaneously valid for other frequency ranges as well. Furthermore, since Maxwell's equations are linear equations as long as the material does not introduce any nonlinearity the principle of superposition holds. Hence we can decompose the comprehensive electromagnetic fields into components of different frequency ranges. In turn this means that we do not have to take care of any slow electromagnetic phenomena, e.g. electrostatics or radio wave, in our formulation of Maxwell's equations since they can be split off from our optical problem and can be treated separately.

− We can define a bound charge density (G: Dichte gebundener Ladungsträger) as the source of spatially changing polarization P .

( , ) ( , )t tρ = −r div P rb

− Analogously we can define a bound current density (G: Stromdichte gebundener Ladungsträger) as the source of the temporal variation of the polarization P .

( , )( , ) ttt

∂=

∂P rj rb

− This essentially means that we can describe the same physics in two different ways since currents are in principle moving charges. Thus we can use either

( , )tj rb or ( , ) /t t∂ ∂P r (see generalized complex dielectric function below).

Complex field formalism (G: komplexer Feld-Formalismus) − Maxwell’s equations are also valid for complex fields and are easier to solve

this way.


− This fact can be exploited to simplify calculations, because it is easier to deal with complex exponential functions (exp( ))ix than with trigonometric functions cos( )x and sin( )x .

− Hence we use the following convention in this lecture to distinguish between the two types of fields.

real physical field: r ( , )tE r

complex mathematical representation: ( , )tE r

− In this lecture we define their relation as

[ ]12( , ) ( , ) ( , ) Re ( , )t t t t∗ = + = E r E r E r E rr

However, this relation can be defined differently in different textbooks. − This means in general: For calculations 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 in the description of the material response or the field dynamics. In this case you would have to go back to real quantities before you compute these multiplication. This becomes relevant for, e.g., calculation of the Poynting vector, as can be seen in a chapter below.

Temporal dependence of the fields When it comes to time dependence of the electromagnetic field, we can distinguish two different types of light:

A) monochromatic light (stationary fields) − harmonic dependence on temporal coordinate − exp( )t− ω i phase is fixed coherent, infinite wave train e.g.:

( , ) ( )exp( )t t= − ωE 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 all the results from case A) (see also next section). Based on this concept arbitrary temporal dynamics of the electromagnetic fields can be described by decomposing it into monochromatic light fields (so-


called Fourier decomposition), which later can be recomposed to obtain again the entire dynamics.

( , ) ( , )exp( )

1( , ) ( , )exp( )2

t t d

t t dt

∞

−∞

∞

−∞

= ω − ω ω

ω = ωπ

∫

∫

E r E r

E r E r

i

i

Remark: The position of the sign in the exponent and the factor 1/ 2π can be defined differently in different textbooks. The bar and the ω in ( , )ωE r indicate the frequency domain fields.

Maxwell’s equations in Fourier domain In order to solve Maxwell's equations more easily we would like to introduce a Fourier decomposition of the fields directly in the Maxwell's equations. The complexity of Maxwell's equations arises from the mixing of different derivatives with respect to space and time. The Fourier decomposition of the fields allows us to perform some or all of these derivatives analytically and hence, to considerably simplify the remaining equations. For this purpose, we need to find out how a derivative of a dynamic variable can be calculated in Fourier space. Here we will do this for the temporal derivatives but later in the course we will apply this concept also to spatial derivatives. A simple rule for the transformation of a time derivative into Fourier space can be obtained using integration by parts:

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

dt i t i dt t i tt it

+∞ +∞

−∞ −∞

ω = − ω ω = − ω ω π∂

π ∂∫ ∫E r E r E r

Thus, a time derivative in real space transforms into a simple product with i− ω in Fourier space.

rule: FT it

∂→ − ω

∂ under the condition

2( , )t t+∞

−∞∂ < ∞∫ E r

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

0 0

0

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

ii i

ω

−

ω = µ ω ε ω = − ω

ω = ω ω ε ω =− ωω ω

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 Maxwell's equations provide the basis to derive all possible mathematical solutions of electromagnetic problems. However very often we are interested just in the radiation fields, which can be described more easily by an adapted equation, which is the so-called wave equation. From Maxwell’s equations it is straight forward to derive the wave equation by using the two curl equations.


A) Time domain derivation We start from applying the curl operator ( rot ) a second time on rot ( , )t =E r and substitute rot H with the other curl equation from Maxwell's equations.

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

t t ttt t

∂ ∂=

∂ ∂ + + ε ∂ ∂ −µ = −µ

∂ ∂rot rot E H r P r E rrot j rr

We find the wave equation for the electric field as

2

2

2

0 0 221( , ) ( , ) ( , ) ( , )t

tct

tt

tt∂

+ =∂ ∂

−µ − µ∂ ∂ ∂

j r P rE r E rrot rot

The blue terms on the right hand side require knowledge of the material model. Additionally, we have to make sure that all other Maxwell’s equations are fulfilled. This holds in particular for the divergence of the electric field:

[ ]0 ( , ) ( , ) 0t tdiv E r P rε + =

Once we have solved the wave equation, we know the electric field. From that we can easily compute the magnetic field:

0

( , ) 1 ( , )t tt

∂= −

∂ µH r rot E r

Remarks: − An analog procedure is possible also for the magnetic field H , i.e., we can

derive a wave equation for the magnetic field as well. − Normally, the wave equation for the electric field E is more convenient,

because the material model defines ( )P E . − However, for inhomogeneous media the wave equation for H can some-

times be the better choice for the numerical solution of the partial differential equation since it forms a hermitian operator.

− analog procedure possible for H instead of E − generally, wave equation for E is more convenient, because ( )P E given

− for inhomogeneous media H can sometimes be a better choice

B) Frequency domain derivation We can do the same procedure to derive the wave equation also directly in the Fourier domain and find

22

0 0

2

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

ω − ω = µω

ω ωω + µ ωrot rot E r E r j r P ri

and

0 ( , ) ( , ) 0 ε ω + ω = div E r P r


− magnetic field:

0

( , ) ( , )ω = − ωµω

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 ω . ⋅ for non-stationary fields (and linear media) inverse Fourier

transformation

( , ) ( , )exp( )t t dE r E r∞

−∞

= ω − ω ω∫ i

Decoupling of the vectorial wave equation So far we have seen that for the general problem of electromagnetic waves all 3 vectorial field components of the electric or the magnetic field are coupled. Hence we have to solve a vectorial wave equation for the general problem. However, it would be desirable to express problems also by a scalar equation since they are much easier to solve. For problems with translational invariance in at least one direction, as e.g. for homogeneous infinite media, layers or interfaces, this can be achieved since the vectorial components of the fields can be decoupled. Let’s assume invariance in the y-direction and propagation only in the x-z-plane. Then all spatial derivatives along the y-direction disappear ( / 0y∂ ∂ = ) and the operators in the wave equation simplify.

( )

( )

(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

∆

− ∆ = ∆

∆

The decoupling becomes visible when the three components of the general vectorial field are decomposed in the following way. • 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

∂ ∂∆ = +

∂ ∂

Hence we obtain two wave equations for the ⊥E and E fields. • gives two decoupled wave equations


2(2) 2

0 02

2(2) (2) (2) 2

0 02

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

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

c

c

⊥ ⊥ ⊥⊥

ωω + ω = − ωµ ω − µ ω ω

ωω + ω − = − ωµ ω − µ ω ω

E r E r j r P r

E r E r grad div E j r P r

∆

∆

i

i

These two wave equations are independent as long as the material response, which is expressed by j and P , does not couple the respective field components by some anisotropic response.

Properties − propagation of perpendicularly polarized fields ⊥E and E can be treated

separately

− propagation of ⊥E is described by scalar equation − similarly the other two E-field components can be described by a scalar

equation for ⊥H − alternative notations used in some books: ⊥ 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 the same. More precisely, we will use a driven harmonic oscillator model to describe the motion of bound charges giving rise to a polarization of the medium. For free charges we will use the same model but without restoring force, leading eventually to a current density. In the literature, this simple approach is often called the Drude-Lorentz model (named after Paul Drude and Hendrik Antoon Lorentz).

Basics We are looking for ( )P E and ( )j E . In general, this leads to a many body problem in solid state theory, which is rather complex. However, in many cases phenomeno-logical models are sufficient to describe the necessary phenomena. As already pointed out above, we use the simplest approach, the so-called Drude-Lorentz model for free or bound charge carriers (electrons).

− assume an ensemble of non-coupling, driven, and damped harmonic oscillators

− free charge carriers: metals and excited semiconductors (intraband transitions)

− bound charge carriers: dielectric media and semiconductors (interband transitions)


− 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 dependence etc.), we can use linear response theory.

Basics of 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

(G: Responsfunktion) ( , )tE r acts on medium (response function) medium reacts by being

polarized ( , )tP r

0( , ) ( , ) ( , )t

i ij jj

P t R t t E t dt−∞

′ ′ ′= ε −∑ ∫r r r (convolution integral)

with R in the convolution integral being a 2nd rank tensor , ,i x y z= and summing over , ,j x y z=

− In frequency domain: we introduce the susceptibility (G: Suszeptibilität) as the transfer function

( , )ωE r medium (transfer function / susceptibility) ( , )ωP r

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

P Eω ω= χ ωε ∑r r r

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

( ) ( )exp(12

)ij ijR t t d∞

−∞π= χ ω − ω ω∫ i

− Obviously, things look friendlier in frequency domain. Using the wave equation from before and assuming that there are no currents ( 0)=j we find

22

02

22

02

( , ) ( , ) ( , )

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

c

c

ωω − ω = µ ω ω

ω∆ ω + ω − ω = −µ ω ω

rotrot E r E r P r

E r E r graddivE r P r

or

− and for auxiliary fields

0( , ) ( , ) ( , )ω = ε ω + ωD r E r P r


The general response function and the respective susceptibility given above simplifies for certain properties of the medium:

Simplification of the wave equation for different types of media A) linear, homogenous, isotropic, non-dispersive media (most simple but very

unphysical case) − homogenous ( , ) ( )ij ijχ ω = χ ωr

− isotropic ( , ) ( , )ij ijχ ω = χ ω δr r

− non-dispersive ( , ) ( )ij ijχ ω = χr r instantaneous: ( , ) ( ) ( )ij ijR t t= χ δr r

(Attention: This is unphysical!) ( , )ijχ ωr χ is a scalar constant

frequency domain time domain description

0 0( , ) ( , ) ( , ) ( , )t tω = ε ω ↔χ χ= εP r E r P r E r (unphysical!)

0 0( , ) ( , ) ( , ) ( , )t tω = ε ε ω ↔ = ε εD r E r D r E r

with 1ε = + χ

Maxwell: 0=div D ( , ) 0ω =div E r for ( ) 0ε ω ≠

2

2( , ) ( , ) 0cω

ω + ε ω =E r E r∆ 2

2 2( , ) ( , ) 0t tc tε ∂

− =∂

E r E r∆

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

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

0

0

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

( , ) ( ) ( , )

( , ) 0 ( , ) 0 ( ) 0

ω = ε χ ω ω ε ω = + χ ωω = ε ε ω ω

ω = ω = ε ω ≠

P r E rD r E r

div D r div E r for

( )2

2( , ) ( , ) 0cω

ω + ε ω ω =E r E r∆ Helmholtz equation

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

0

0

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

ω = ε χ ω ω

ω = ε ε ω ω

P r r E rD r r E r

div ( , ) 0ω =D r


0 0div ( , ) ( , ) ( , ) ( , ) ( , ) 0ω = ε ε ω ω + ε ω ε ω =D r r div E r E r grad r ,

( , )div ( , ) ( , ).

( , )ε ω

ω = − ωε ω

grad rE r E rr

( )2

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

( , )c ω ε ω

ω + ε ω ω = − ω ε ω

grad rE r r E r grad E rr

∆

− All field components couple. − This equation is also valid for case B) with grad ( , ) 0ε ω =r .

D) linear, homogeneous, anisotropic, dispersive media ( )ij→ χ ω

0

0

( , ) ( ) ( , )

( , ) ( ) ( , ).

i ij jj

i ij jj

P E

D E

ω = ε χ ω ω

ω = ε ε ω ω

∑

∑

r r

r r

− This is the worst case for a medium with linear response. − See chapter on crystal optics.

A similar classification of material responses could be made based on the conductivity ( , )σ ωr with the current density ( , ) ( , ) ( , )ω = σ ω ωj r r E r .

Before we start writing down the actual material model equations, let us define, which types of light matter interactions we would like to consider.

Types of considered light-matter interactions (covered by lecture Structure of Matter)

I) Interaction of light with bound electrons and the lattice The contributions of bound electrons and lattice vibrations in dielectrics and semiconductors give rise to the polarization P . The lattice vibrations (phonons) are the ionic part of the material model. Because of the large mass of the ions ( 310 ×mass of electron) the resulting oscillation frequencies will be small. Generally speaking, phonons are responsible for thermal properties of the medium. However, some phonon modes may contribute to optical properties, but they have small dispersion (weak dependence on frequency ω ). Fully understanding the electronic transitions of bound electrons requires quantum theoretical treatment, which allows an accurate computation of the transition frequencies. However, a (phenomenological) classical treatment of the oscillation of bound electrons is possible and useful.

II) Interaction of light with free electrons The contribution of free electrons in metals and excited semiconductors gives rise to a current density j. We assume a so-called (interaction-)free electron gas, where the


electron charges are neutralized by the background ions. Only collisions with ions and related damping of the electron motion will be considered. We will look at the contributions from bound electrons / lattice vibrations and free electrons separately. Later we will join the results in a generalized material model which holds for many common optical materials..

Dielectric polarization and susceptibility (covered by lecture Structure of Matter)

Let us first focus on bound charges (ions, electrons). In the Drude-Lorentz 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∂ ∂

+ ω∂ ∂

+ =

− resonance frequency (electronic transition) 0ω

− damping g

− charge q − mass m

The induced electric dipole moment due to the displacement of charged particles 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 as

2

20 02

2

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

t g t t f tt t

∂ ∂+ ω = ε

∂ ∂P r P r P r t E r E r+ =

with oscillator strength 2

0

1 e Nfm

=ε

, for q=-e (electrons)

This equation is easy to solve in Fourier domain:

2 20 0( , ) ( , ) ( , ) ( , )g f−ω ω − ω ω + ω ω = ε ωP r P r P r E ri

→ ( )0

2 20

( , ) ( , )g

fω −

εω = ω

− ωωP r E r

i


with 0( , ) ( , )ω = ε χ ω ωP r ( )E r ( )2 20

( ) fg

χ ω =ω − ω − ωi

In general we have several different types of oscillators in a medium, i.e., several different resonance frequencies. Nevertheless, since in a good approximation they do not influence each other, all these different oscillators contribute individually to the polarization. Hence the model can be constructed by simply summing up all contributions.

− several resonance frequencies

( ) ( )0 02 20

( , ) ( , ) ( , )j

j j j

fg

ω = ε ω = ε χ ω ω ω − ω − ω

∑P r E r E ri

( ) ( )2 20

j

j j j

fg

χ ω = ω − ω − ω

∑ i

− ( )χ ω is the complex, frequency dependent susceptibility

( ) ( )0 0 0( , ) ( , ) ( , ) ( , )ω = ε ω + ε χ ω ω = ε ε ω ωD r E r E r E r − ( )ε ω is the complex frequency dependent dielectric function

Example: (plotted for eta and kappa with ( ) ( ) ( ) 2iε ω = η ω + κ ω )

Conductive current and conductivity (covered by lecture Structure of Matter)

Let us now describe the response of a free electron gas with positively charged background (no interaction). Again we use the model of a driven harmonic oscillator,


but this time with resonance frequency 0 0ω = . 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∂

= ε ω∂

j r j r E r E rt p+ =

with plasma frequency 2

2

0

1 e Nfm

ω = =εp

Again we solve this equation in Fourier domain:

20( , ) ( , ) ( , )g− ω ω + ω = ε ω ωj r j r E rpi

( )2

0( , ) ( , ) ( , ).gε ω

ω = ω = σ ω ω− ω

j r E r E rp

i

Here we introduced the complex frequency dependent conductivity

( ) 2

2 20 0 .

g gε ω ε ω

σ ω = = −− ω −

ωω ω−

p piii

Remarks on plasma frequency We consider a cloud of electrons and positive ions described by the total charge density ρ in their self-consistent field E . Then we find according to Maxwell:

0 ( , ) ( , )t tε = ρdivE 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∂

ε ω∂ p+ =

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

2 20 ( , ) ( , )tg t

t∂

ε ω = ω ρ∂

ddiv j div ivE r rj p p+ =

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

,t

∂ρ +

∂divj 0=


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

22

2 gt t

∂ ∂− ρ − ρ = ω ρ

∂ ∂ p

2

2p2 0g

t t∂ ∂

ρ + ρ + ω =∂ ∂

harmonic oscillator equation

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

Generalized complex dielectric function In the sections above we have derived expressions for both polarization (bound charges) and conductive current density (free charges). Let us now plug our ( , )ωj r and ( , )ωP r into the wave equation (in Fourier domain)

( )

22

0 02

20 0 0

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

( ) ( , )

cω

ω − ω = µ ω ω + ωµ ω

= µ ε ω χ ω + ωµ σ ω ω

rotrot E r E r P r j r

E r

i

i

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

( )0

2

2 1 ( )( , ) ( , )c

+ χ ω + σ ω ω

=ε

ωω ωrotrot E r E ri

2

2 ( ()( , ) , )cω

ε ωω = ωrotrot E r E r

Here, we introduced the generalized complex dielectric function

( )0

( ) 1 () ( )( )σχ ′ ′′ε ω = + + = ε ωω + ωωω

εε

ii

So, in general we have

( )2 20

2

2( ) 1 j

j j j gf

g

ω − ω − ωε ω = + +

ω−ω − ω

∑ p

ii,

because (from before)

( ) ( )2 20

j

j j j

fg

χ ω = ω − ω − ω

∑ i, ( )

202 .

gε ωω

σ ω = −−ω − ω

pii

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


Some special cases for materials in the infrared and visible spectral range:

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

( ) ( )2 20

2 20

1( ) j

j j j

f fgg

+ ω − ω − ω

ε ω = +ω ω − ω −∑ ii

with 00 jω ω and 0ω ω

Close to a single isolated phonon resonance it can be approximated to

( )2 20

( ) fg∞ε ω = +

ω − ω − ωε

i

This function has the following parameters: − resonance frequency: 0ω

− resonance strength: f

− width of resonance peak: g

− epsilon from vacuum and electronic transitions: ∞ε

The contribution from electronic transitions shows almost no frequency dependence (dispersion) in this frequency range far away from the electronic resonances. Hence it can be expressed together with the vacuum contribution as a constant ∞ε .

∞ε vacuum and electronic transitions

Let us study the real and the imaginary part of the resulting ( )ε ω separately:

( ) ( ) ( ) ( ) ( )′ ′′ε ω = ℜε ω + ℑε ω = ε ω + ε ωi i

( )

( )

2 20

22 2 2 20

( ) ,f

g∞

ω − ω′ε ω = ε +

ω − ω + ω

( )22 2 2 2

0

( ) .gf

g

ω′′ε ω =ω − ω + ω

Lorentz curve


Properties:

− so called longitudinal frequency ωL : ( ) 0′ε ω = ω =L

− ( ) 0′′ε ω ≠ : absorption and dispersion appear always together

− frequency range with normal dispersion: ( ) / 0′∂ε ω ∂ω >

− frequency range with anomalous dispersion: ( ) / 0′∂ε ω ∂ω <

− static dielectric constant in the limit 0ω → : 0 20/f∞ε = ε + ω

− near resonance we find ( ) 0′ε ω < (damping, i.e. decay of field, without absorption if '' 0ε ≈ )

Simplified example: sharp resonance for undamped oscillator 0g →

− relation between resonance frequency 0ω and longitudinal frequency ωL

(Lyddane-Sachs-Teller relation)

( )2 20

( ) 0f∞′ε ω = ε + =

ω − ωLL

, ( )0 20f ∞= ε − ε ω (from above)

0

0∞

εω = ω

εL


B) Dielectrics in the visible (VIS) spectral range Dielectric media in visible (VIS) spectral range can be described by a so-called double resonance model, where a phonon resonance exists in the infrared (IR) at 0pω and an electronic transition exists in the ultraviolet (UV) at 0eω .

( ) ( )2 2 2 20 0

( ) ,f f

g g∞ε ω = ε + +ω − ω − ω ω − ω − ω

phonon electron

phonon phonon electron electroni i

with 0phonon 0electronω ω ω

∞ε contribution of vacuum and other (far away) high frequency resonances

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

( )20

2 20

( ) 1 j

j

j

j

f′ε ω −

ω=

− ωω∑

− with j being the number of resonances taken into account − describes many media very well (dispersion of absorption is neglected) − oscillator strengths and resonance frequencies are often fit parameters to

match experimental data

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

2

2( ) 1g

ωε ω = −

ω + ωp

i with ω >> ωp

where pω is the so-called plasma frequency of the metal. Hence the real and imaginary part of ( )ε ω of a metal is


( )2 2

2 2 2 2( ) 1 , ( ) .

gg g

ω ω′ ′′ε ω = − ε ω =

ω + ω ω +p p

Metals show a large negative real part of the dielectric function ( )′ε ω , which gives rise to decay of the fields. Eventually this results in reflection of light at metallic surfaces.

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

0

0

( )( )

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

ω = ε ωε ω

χω = ε ωω

D r E rP r E r

The response function (or Green's function) ( )R t in the time domain is then given by

( )1( ) ( )exp2

R t t d∞

−∞= χ ω − ω ω

π ∫ i ( )( ) ( )expR t t dt∞

−∞χ ω = ω∫ i

and the Polarization is determined by the convolution integral of the driving electric field with the response function as

− with the convolution integral to determine the polarization

0( , ) ( ) ( , )t R t t t dt∞

−∞′ ′ ′= ε −∫P r E r

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

∞ ∞

−∞ −∞

∞ ∞

−∞ −∞

= ω ω ω = ε ω ωχ ω

χ ω

ω

′ ′ ′= ε ω ω ωπ

∫ ∫

∫ ∫

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

( ) )

R t t

t t dt t dt

t dt tR t

∞

−∞

′−

∞

−∞

∞

−∞

′χ ω ω − ωπ

′−

′ ′= ε

′ ′= ε

∫∫

∫

P r E r

E r

-i

For a “delta” excitation in the electric field we find the response to be the response function (Greens function - GF) itself, which determines the polarization:

0( , ) ( )t t t= δ −E r e 0 0( , ) ( )t R t t= ε −P r e response function

Examples A) instantaneous media (unphysical simplification)

− For instantaneous (or non-dispersive) media, which cannot not really exist in nature, we would find:

( ) ( )0( ) ( ) , ,R t t t t= χδ → = ε χP r E r (unphysical!)

B) dielectrics

( ) ( ) ( )2 20

1 1( ) exp exp ,2 2

fR t t d t dg

∞ ∞

−∞ −∞= χ ω − ω ω = − ω ω

π π ω − ω − ω∫ ∫P i ii

− Using the residual theorem we find:

exp sin 0

( ) 20 0

f g t t tR t

t

− Ω ≥ = Ω <

with 2

20 4

gΩ = ω −

[ ]( , ) exp ( ) sin ( ) ( , )2

tf gt t t t t t dt−∞

′ ′ ′ ′= − − Ω − Ω ∫P r E r

C) metals

( ) ( ) ( )2

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

g∞ ∞

−∞ −∞

ε ω= σ ω − ω ω = − ω ω

π π − ω∫ ∫ pi ii

− Using again the residual theorem we find:

( )exp 0( )

0 0

gt tR t g

t

ω− ≥ =

<

2p

[ ]0( , ) exp ( ) ( , )

t

t g t t t dt−∞

′ ′ ′= ε ω − −∫j r E r2p


2.3 Poynting vector and energy balance

Time averaged Poynting vector The energy flux of the electromagnetic field is given by the Poynting vector S (named after its discoverer John Henry Poynting 1884). 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 −= π ω ≤ s

− pulse duration: Tpulse in general 0T T>>pulse

− duration of measurement: Tm in general 0T T>>m

Hence, in general the detector does not recognize the fast oscillations of the optical field 0i te ω−

(optical cycles) and only delivers a time averaged value. For the situation described above, the electro-magnetic fields factorize in slowly varying envelopes and fast carrier oscillations:

( )01 ( , )exp . . ( , )2

t t c c t − ω + = E r E r

ri

The tilde in ( , )tE r will be used throughout the script to indicates that this is the slowly varying part of the field. For such pulses, the momentary Poynting vector reads:

( ) ( )

( )

0

0

0

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

1 ( , ) ( , )e

1 ( , ) ( , ) co

xp 2 ( , ) ( , )ex

1 ( , )

p

( , )2

s

4

2

2

2

t t t

t t t t

t t t t t t

tt t t t

∗

∗

∗

∗

∗

ℜ × ω ℜ

= ×

= × + ×

×

+ × − ω + × ω

= +

S r E r H r

E r H r E r H r

E r H r E r H r

E r H rE r H r

sl w faso t

r r

i i

( )01 ( , ) ( , ) s2

.in 2t t t∗ ∗ ℑ × ω + E r H r

fast


We find that the momentary Poynting vector has some slow contributions which change over time scales of the pulse envelope Tp, and some fast contributions

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

/2

/2

1( , ) ( , ') 't T

t Tt t dt

T+

−= ∫S r S rm

mm

The fast oscillating terms 0~ cos2 tω and 0~ sin 2 tω 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+

−

∗ ℜ= × ∫ 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 ∗ = ℜ × 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 of the electromagnetic fields.

( , )I t= S r measurement destroys phase information

Time averaged energy balance Let us motivate a little bit further the concept of the Poynting vector. Some interesting insight on the energy flow of light and hence also on the transport of information can be obtained from the Poynting theorem, which is the 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 rH or rE (note that we use real fields):

0

0

0

( )

t

t t

∂⋅ + µ ⋅ =

∂∂ ∂

⋅ − ε ⋅ = ⋅ +∂ ∂

r r

r

r r

r r r rr r

rotE H

rotH jEE

H

E P

H

E

Next, we subtract the two equations and get

0 0 ( ).t t t

∂ ∂ ∂⋅ − ⋅ + ε ⋅ + µ ⋅ = − ⋅ +

∂ ∂ ∂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 rH rotE E rotH div E H

Finally, with the substitution 21r r r2t t⋅ ∂ ∂ = ∂ ∂E E E we find Poynting's theorem

( )2 20 0

change of Poynting vectorchange of vacuum energy density

1 12 2t t t

∂ ∂ ∂ ε + µ + × = − ⋅ + ∂ ∂ ∂ 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 divergence of the Poynting vector (energy flux), we can identify the vacuum energy density

2 21 1

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

where 2 20 0

1 12 2

u = ε + µr rE H vacuum energy density

In the case of stationary fields and isotropic media, which is a simple but important case,

( )

( )

0

0

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

t t c c

t t c c

= − ω +

= − ω +

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

∗∂ ∂ ε + µ + × = ℜ × ∂ ∂

=

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 derivatives remove stationary terms in 2 ( , )tE rr and 2 ( , )tH rr . Time averaging of the right hand side of Poynting’s theorem yields (source terms):

0000 0 0 0

( , )

(

( , )

1 . . . .

( , )

( ) ) (4

) )i t i t i t

t t

c

tt

c e c cee − ω− ω − ω

∂∂

− +

= − +− ω ε χ ω +σ ω

E r

E

j r

E(r

P r

( )E r r

rrr

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∗

= − − ω ε − ω + − ω +

σ ωχ ω +

ω ε

ε ω

= ω ε + −

E(r) E(r)

E(r)E(r)

ii i i

i

with 0

( ) 1 ( ) ( )iε ω = + χ ω + σ ω

ωε

Again, all fast oscillating terms ( )0exp 2 t± ω i cancel due to the time average. Finally, splitting ( )0ε ω into real and imaginary part yields

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

c c∗ ∗′ ′′ ′′ = ω ε ε ω − + ε ω + = − ω ε ε ω 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

∗′′= − ωε εωdiv S E r E r

This shows that a nonzero imaginary part of epsilon ( ( ) 0′′ε ω ≠ ) causes a drain of energy flux. In particular, we always have ( ) 0′′ε ω > , otherwise there would be gain of energy, which we will not consider in this course as it would require active media. In particular near resonances we have ( ) 0′′ε ω ≠ 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

Here A is the surface of the volume V .


2.4 Kramers-Kronig relation (covered by lecture Structure of Matter) In the previous sections we have assumed a very simple model for the description of the material's response to the excitation by the electromagnetic field. This model was based on quite strong assumptions, like a single charge which is attached to a rigid lattice etc. Hence, one could imagine that more complex matter could give rise to arbitrarily complex response functions if adequate models would be used for its description. However we can show from basic laws of physics, that several properties are common to all possible response functions, as long as a linear response to the excitation is assumed. These fundamental properties of the response function are formulated mathema-tically by the Kramers-Kronig relation. It is a general relation between ( )′ε ω (dispersion) and ( )′′ε ω (absorption). This means in practice that we can compute

( )′ε ω from ( )′′ε ω and vice versa. For example, if we have access to the absorption spectrum of a medium, we can calculate the dispersion. The Kramers-Kronig relation follows from reality and causality of the response function R of a linear system. That the response function is real valued is a direct consequence from Maxwell's equations which are real valued as well. Causality is also a very fundamental property, since the polarization must not depend on some future electric field. As we have seen in the previous sections, in time-domain the polarization and the electric field are related as:

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

tt R t t t dt t R t d

∞

−∞′ ′ ′= ε − ↔ = ε τ − τ τ∫ ∫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∞ ∞

ωτ ωτ

−∞ −∞

τ = ω χ ω = ω χ ωπ π∫ ∫ ( ) ( )∗χ ω = χ −ω

Causality of the response function implies:

( ) ( ) ( )R yτ = θ τ τ with ( )

1 for 01 for 020 for 0

τ >θ τ = τ =

τ <

Heaviside distribution


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

( ) ( ) ( )2 e Pi t idt t∞

ω

−∞

πθ ω = θ = + πδ ωω∫ defined as integral only

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

( ) ( )0 0( )d f f∞

−∞

ωδ ω − ω ω = ω∫ Dirac delta distribution

and the expression P(i/ )ω involving a Cauchy principal value:

0

P ( ) lim ( ) ( )i i id f d f d f∞ −α ∞

α→−∞ −∞ α

ω ω = ω ω + ω ω ω ω ω

∫ ∫ ∫ 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 (suscepti-bility) we expect a convolution:

( ) ( ) ( ) ( )

( ) ( )

e ei iyRd d

d y

∞ ∞ωτ ωτ

−∞ −∞

∞

−∞

θ τ τ

θ ω − ω

χ ω = τ = τ

=

τ

ω ω

∫ ∫

∫( ) ( )1 1P

2 2i

θ ω = + δ ωπ ω

( ) ( ) ( )1 P ( )2 2

y yd

i∞

−∞

ω ωχ ω = ω + ∗

π ω − ω∫

In order to derive the Kramers-Kronig relation we can use a small trick (this trick saves us using complex integration in the derivation). Because of the Heaviside function, we can choose the function ( )y τ for τ < 0 arbitrarily without altering the susceptibility! In particular, we can choose:

a) ( ) ( )y y−τ = τ even function

b) ( ) ( )y y−τ = − τ odd function

a) ( ) ( )y y−τ = τ

In this case ( ) ( )y y−τ = τ is a real valued and even function. We can exploit this property and show that

( ) ( ) ( ) ( )1 1e e2 2

i id y dy yy∞ ∞

ωτ − ωτ ∗

−∞ −∞

τ τ = τ τωπ π

ω== ∫ ∫ is real as well

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

( ) ( ) ( )1 P2 2

i y yd

∞∗

−∞

ω ωχ ω = − ω +

π ω − ω∫


Here P∫ is a so called principal value integral (G: Hauptwertintegral). Now we have expressions for ( ) ( )*,χ ω χ ω and can compute real and imaginary part of the susceptibility:

( ) ( ) ( ) ( ) ( ) ( ) ( )* 1 1P P2 2 2 2

i y y i y yd d y

∞ ∞

−∞ −∞

χ ω + χ ωω ω ω ω

ω + − ω +π ω − ω π ω − ω

= = ω∫ ∫

( ) ( ) ( )1 Pi

dy∗

∞

−∞

ωχ ω = = ω

π ω −−

ωχ ω ∫

Plugging the last two equations together we find the first Kramers-Kronig relation:

( ) ( )1 P d∞

−∞

ℜ χ ωℑχ ω = ω

π ω ω−

−∫ 1. K-K relation

Knowledge of the real part of the susceptibility (dispersion) allows us to compute the imaginary part (absorption).

b) ( ) ( )y y−τ = − τ

The second K-K relation can be found by a similar procedure when we assume that ( ) ( )y y−τ = − τ is a real odd function. We can show that in this case

( ) ( ) ( ) ( )1 1e e2 2

i id y dy y y∞ ∞

ωτ − ωτ

−

∗

−∞ ∞

= τ τ = − τ τωπ

ω= −π∫ ∫ is purely imaginary

With equation (*) we then find that

( ) ( ) ( )1 P2 2

i y yd

∞∗

−∞

ω ωχ ω = ω −

π ω − ω∫ (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

∞ ∞

−∞ −∞

∗χ ω − χ ωω ω ω ω

= ω + − ω + =π ω − ω π ω ω

ω−∫ ∫

( ) ( ) ( )* 1 Pi y

d∞

−∞

χ ωω

= = ωπ ω

χ− ω

+ ω ∫

and finally obtain

→ ( ) ( )1 P d∞

−∞

ℑ−

χ ωℜχ ω = ω

π ω ω∫ 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 ω : K-K relation for ε :

− ( ) ( )∗χ ω = χ −ω → ( ) ( )′ ′χ ω = χ −ω ( ) ( )′′ ′′χ ω = −χ −ω and

[ ]( ) ( ) 1 ( ) 1 ( )′ ′′χ ω = ε ω − = ε ω − + ε ωi

[ ]2 20

2 20

2 ( )( ) 1 P ,

( ) 12( ) P .

d

d

∞

∞

′′ωε ω′ε ω − = ωπ ω − ω

′ε ω −′′ε ω = − ω ω

π ω − ω

∫

∫

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

Example:

0( ) ( )′′ε ω δ ω − ω 02 20

( ) 1 ω′ε ω −ω − ω

Drude-Lorentz model


3. Solution of wave equations for homogeneous isotropic media

3.1 Normal modes in homogeneous isotropic media Using the linear material models, which we discussed in the previous chapters, we can now look for self-consistent solutions to the wave equation including the material response. It is convenient to use the generalized complex dielectric function to derive the solution of the wave equation

( )0

( ) 1 ( ) ( ) ( )′ ′′ε ω = + χ ω + σ ω = ε ω + ε ωωεi

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. These normal modes are the stationary solutions of the wave equation. Hence they usually correspond to waves, which possess a trivial spatio-temporal dynamics and which are infinitely extended in space and time. However as we will see later, by taking into account the principle of superposition we can construct the general solu-tions from these normal modes, by superimposing multiple normal modes to con-struct also the transient waves. In this context one should understand that by super-imposing infinitely many infinitely extended normal modes it is possible to form all sorts of localized, i.e. finitely extended, waves. We start from the wave equation in Fourier domain, which reads as

2

2( , ) ( ) ( , )cω

ω = ε ω ωrot rot E r E r

According to Maxwell the solutions have to fulfill additionally the divergence equation of the D field, which for homogeneous media can be transformed to:

[ ]0 1 ( ) ( , ) 0ε + χ ω ω =div E r

In general, the additional condition 1 ( ) 0+ χ ω ≠ implies that the electric field itself is free of divergence:

for the normal case of 1 ( ) 0+ χ ω ≠ ( , ) 0ω =div E r

Since we already know from the history of research in this field that the plane waves are the normal mode solutions, which we are looking for, we will use them as a 'smart' guess, i.e. an ansatz. Thus, here in the lecture we will restrict ourselves to showing the validity of this ansatz, meaning that we are showing that this ansatz is correctly fulfilling the Maxwell's equantions. We will start from plane wave solutions of the following form in the frequency domain:


( )( , ) ( , )exp ,ω = ωE r E k kri

k = unknown complex wavevector

The corresponding stationary field in time domain is given by:

( )( , ) ( , )expt t= ω ω E r E k kri −

monochromatic plane wave normal mode

This is a monochromatic plane wave, the simplest solution we can expect, a so-called normal mode. We insert this ansatz into Maxwell's equations to see if Maxwell's equations are fulfilled by this ansatz. Starting from the divergence equation for the E field (derived from the divergence equation for the D field in homogeneous media), we see that for ( ) 0ε ω ≠ the plane wave ansatz fulfills the divergence condition only if the so-called transversality condition ( )ωk E⊥ is fulfilled. Accordingly, these waves are called transverse waves.

0 ( , ) ( , )= ω = ⋅ ωdivE r k E ri ( )ωk E⊥ transverse wave

Here transverse means that the electric field and the wavevector are oriented perpendicular to each other. If we split the complex wavevector into real and imaginary part +k k' k'',i= we can define:

o planes of constant phase const′ =k r o planes of constant amplitude const′′ =k r

In the following we will call the solutions A) homogeneous waves if these two planes are identical (parallel) B) evanescent waves if these two planes are perpendicular C) inhomogeneous waves otherwise We will see that in dielectrics ( )( )0σ ω = we can find another, exotic type of wave solutions, since at the frequency ω = ωL with ( ) 0,ε ω =L so-called longitudinal waves

( )ωk E appear. Before we have a closer look at the detailed properties of these individual types of solutions, let us consider the two principal forms of plane waves, which are the transverse and longitudinal waves.

Transverse waves (epsilon unequal zero) Let us have a closer look at the transverse nature of the fields first. As pointed out above, for ω ≠ ωL the electric field becomes free of divergence:

0 ( )div ( , ) 0ε ε ω ω =E r div ( , ) 0ω =E r

Then, the wave equation reduces to the Helmholtz equation:


2

2( , ) ( ) ( , ) 0.cω

ω + ε ω ω =E r E r∆

Hence, we have three scalar equations for ( , )ωE 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:

( )( , ) ( )expω = ωE r E kri

Immediately we see that the wave is transversal:

0 ( , ) ( , )= ω = ⋅ ωdivE r k E ri → ( ).ωk E⊥

Hence, we have to solve

2

22 ( ) ( ) 0

c ω− + ε ω ω =

k E and ( ) 0.⋅ ω =k E

If ( ) 0ω ≠E the first term must be zero. This leads to the following dispersion relation

2

2 2 2 2 22 ( )k k k k

cω

= = + + = ε ωk x y z with k as the complex wavenumber

We see that the so-called wavenumber ( ) ( )ck ωω = ε ω is a function of the frequency. We can conclude that transverse plane waves are solutions to Maxwell's equations in homogeneous, isotropic media, only if the dispersion relation for ( )k ω is fulfilled. In general, ′ ′′+k = k ki is a complex valued vector. For cases where ′ ′′k k the vectorial character is not that important and it is sometimes useful to introduce the scalar complex refractive index n as an alternative notation:

[ ]ˆ( ) ( ) ( ) ( ) ( )k n n

c c cω ω ω

ω = ε ω = ω = ω + κ ωi

Remark: Instead of assuming that ˆ( )n ω and ( )ε ω have the same meaning, one should clearly distinguish between the two. While ( )ε ω is a property of the medium, ˆ( )n ω is a property of a particular type of the electromagnetic field in the medium, i.e.

a property of the infinitely extended monochromatic plane wave

( )( , ) ( )expω = ωE r E kri .

Hence, the coincidence of the complex refractive index ˆ( )n ω with ( )ε ω holds only for homogeneous media since ˆ( )n ω cannot be a local property while ( )ε ω is. From the knowledge of the electric field we can compute the magnetic field if desired:

( )0 0

1( , ) ( , ) ( ) exp ω = − ω = ω ωµ ωµH r rot E r k E kri

i×

( )( , ) ( )expω = ωH r H kri with 0

1( ) ( ) ω = ω ωµH k E×


and hence ( ) ( )ω ⊥ ωH E .

Longitudinal waves (epsilon equal zero) Let us now have a look at the rather exotic case of longitudinal waves. These waves can only exist for ( ) 0ε ω = in dielectrics, which occurs at the longitudinal frequency ω = ωL . In this case, we cannot conclude from MWQE that ( , ) 0ω =div E r , and hence the wave equation reads (the l.h.s. vanishes because ( ) 0ε ω = ):

( , ) 0ω =rot rot E r L

As for the transversal waves we try the plane wave ansatz and assume k to be real.

( )( , ) ( )expω = ωE r E kri

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

( , ) 0 ω = k k E r L× ×

Now we decompose the electric field into transversal and longitudinal components with respect to the wave vector:

( ) ( ) ( )( , ) ( )exp ( )exp ( )exp⊥ω = ω = ω ωE r E kr E kr E kri i i+

with ( )⊥ ωE k⊥ and ( )ωE k

This decomposed field is inserted into the wave equation:

( ) ( )

( ) ( )0

exp 0

exp exp 0

⊥

⊥

=

= + =

k k E E kr

k k E kr k krk E

i

i i

×

×

× +

× × ×

Since the cross product of k with the longitudinal field ( )ωE is trivially zero the remaining wave equation is:

2 0k ⊥ =E

Hence the transversal field ⊥E must vanish and the only remaining field component is the longitudinal field ( )ωE :

( )( , ) ( )expω = ωE r E krL L i

Plane wave solutions in different frequency regimes The dispersion relation ( )2 2 2 2 2 2 2 ( )k k k k c= = + + = ω ε ωk x y z for plane wave solutions determines the (complex, scalar) wavenumber k only. Thus, different solutions for the complex wavevector ′ ′′+k = k ki are possible. In addition, the generalized dielectric function ( )ε ω is complex. In this chapter we will discuss possible scenarios and resulting plane wave solutions.


A) Positive real valued epsilon ( ) ( )' 0ε ω = ε ω >

This is the favorable regime for optics. We have transparency, and the frequency of light is far from resonances of the medium. The dispersion relation gives

2 2

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

c cω ω′= − + ⋅ = ε ω = ω ⇒ ⋅ =k' k'' k k'' k k''i

Since k' can't be zero because ( )'ε ω is positive and real. There are only two possibilities to fulfill this condition, either 0=k'' or ⊥k' k'' .

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

2( ) ( ) ( )k n n

c cω ω π

ω = ω = = ωλn

Because 0=k'' these waves are homogeneous, i.e. planes of constant phase are parallel to the planes of constant amplitude. This is trivial, because the amplitude is constant everywhere.

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

− If we neglect the imaginary part of ( )ε ω , which mathematically corresponds

to an undamped resonance, we can further simplify the material equation to

2 20

( ) ( ) f∞′ε ω = ε ω = +

ω − ω

− We can invert the dispersion relation

( ) ( )kcω

ω = ε ω ( )kω


Example 2: free electrons − for plasma and metal − Again the imaginary part of ( )ε ω is neglected

2

2( ) ( ) 1ω

′ε ω = ε ω = −ωp

− We again invert the dispersion relation

( ) ( )kcω

ω = ε ω ( )kω

A.2) Complex valued wavevector ⊥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

cω

= − = ε ωk' k'' and therefore 2 2 2k= −k'' k'

This means that

2 0≠k'' 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'k being even larger than the wave number 2k , whereas the homogeneous waves of case A.1 ( 0=k'' ) obey 2 2k='k . If we plug our findings into the plane wave ansatz we get for the evanescent waves:

( ) ( )ex( , ) ( ex p) pω = ω ω ω −E r E k''r rk (' )i

Here we see that the planes defined by the equation ωk''( )r = const. are the s planes of constant amplitude, those defined by ωk'( )r = const. are the planes of constant phase. Because of ⊥k' k'' these planes are perpendicular to each other. The factor ( )exp − ωk''( )r leads to exponential growth of evanescent waves in homogeneous space in the direction of - ''( )ωk . Therefore, evanescent waves can't be physically justified normal modes of homogeneous space and can only exist in inhomogeneous space, where the exponential growth is truncated at a finite value, e.g. at an interface.

B) Negative real valued epsilon ( ) ( ) 0′ε ω = ε ω <

This situation (negative but real ( )ε ω ) can occur in dielectric materials near resonances ( 0ω < ω < ωL ) or in metals below the plasma frequency (ω < ωp ). Then the dispersion relation gives

2

2 2 222 ( ) 0k

cω ′= − + ⋅ = ε ω <k' k'' k' k''i

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) Vanishing real component of wavevector 0=k'

2

22 ( )

cω ′= ε ωk'' ( )( , ) expω −E r k''r strong damping


B.2) Complex-valued wavevector 0⋅ =k' k'' ⊥k' k'' evanescent waves

22 2 2

2 ( )kcω ′= − = − ε ωk' k''

22 2

2 ( ) .cω ′= ε ω +k'' k'

As above, these evanescent waves exist only at interfaces (like for ( ) ( ) 0′ε ω = ε ω > ). The interesting point is that here we find evanescent waves for all values of 2.k' Moreover, we see that case B.1 ( 0=k' ) is actually included in case B.2. Hence, we can conclude that for ( ) ( ) 0′ε ω = ε ω < we find only evanescent waves.

C) Complex valued epsilon ( )ε ω

This is the general case, which is relevant particularly near resonances. From our (optical) point of view only weak absorption is interesting because otherwise the information carried by light would be lost quickly. Therefore, in the following we will always assume ( ) ( )′′ ′ε ω << ε ω . As we can see in the following sketch, we can have

( ) 0, ( ) 0,′ ′′ε ω > ε ω > or ( ) 0, ( ) 0.′ ′′ε ω < 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

[ ] [ ]2 2 2

2 22 22 2 2ˆ( ) ( ) ( ) ( ) ( )i k n n

c c cω ω ω′ ′′+ = ω = ε ω = ω = ω + κ ωk k i

Since k' and k'' are almost parallel they can be represented approximately by scalars:

( ), ( )nc cω ω

= ω = κ ωk' k''

The dispersion relation in terms of the complex refractive index gives


[ ]

2 222 2

2 2( ) ( ) ( )k nc cω ω

= = ε ω = ω + κ ωk i

Here we have

2 2( ) ( ) ( ) ( ) ( ) 2 ( ) ( ),n n′ ′′ε ω = ε ω + ε ω = ω − κ ω + ω κ ωi i

and therefore 2 2( ) ( ) ( )

( ) 2 ( ) ( )n

n′ε ω = ω − κ ω′′ε ω = ω κ ω

( ) ( )22 ( ) sgn 1 / 1 ,2

n′ε ′ ′′ ′ω = ε + ε ε +

( ) ( )22 ( ) sgn 1 / 1 .2′ε ′ ′′ ′κ ω = ε + ε ε −

There are two important limiting cases of such quasi-homogeneous plane waves:

C.1) , 0,′ ′′ε ε > ′′ε << ,′ε (dielectric media)

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

n′′ε ω′ω ≈ ε ω κ ω ≈′ε ω

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

2 2

2 22 2( ), 2 ( ),

c cω ω′ ′′− = ε ω ⋅ = ε ωk' k'' k' k''

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

nc c c c

′′ω ω ω ω ε ω′= ω ≈ ε ω = κ ω ≈′ε ω

k' k''

We see that

⋅ ≈k' k'' k' k''

Hence, as we had assumed for case C already in the beginning, k' and k'' must be almost parallel, which indicates that under these conditions the plane waves are quasi-homogeneous wave. Thus, in homogeneous, isotropic media, next to resonances, we find damped, quasi-homogeneous plane waves, with ′ ′′ kk k e where ke is the unit vector along k

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

ω ω ω = ω = ω ω − κ ω k kE r E kr E e r e ri i

C.2) 0, 0,′ ′′ε < ε > ′′ε << ,′ε (metals and dielectric media in so-called Reststrahl domain)

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

n′′ε ω ′ω ≈ κ ω ≈ ε ω′ε ω


In this regime damping dominates ( ( ) ( )n ω κ ω ) and we find a very small refractive index. Interestingly, propagation (nonzero n) is only possible due to absorption (see time averaged Poynting vector below).

Summary of normal modes Here we summarize our previous findings about the properties of normal modes for different parameters of the material which they exist in. We do this at the example of dielectric media for frequencies close to one resonance. Here we can identify different frequency intervals, in which we find the typical behavior of the material's response.

There are the following frequency ranges, which give rise to the typical properties of normal modes: a) Frequency far below or far above resonance where ( ) 0′ε ω > , ( ) 0′′ε ω ≈

Typical normal modes: − undamped homogeneous waves case A.1 − evanescent waves case A.2

b) Frequency above resonance where ( ) 0′ε ω < , ( ) 0′′ε ω ≈ Typical normal modes: − evanescent waves cases B.1 and B.2

c) Frequency close to and below resonance where ( ) 0′ε ω > , ( ) 0′′ε ω > Typical normal modes: − weakly damped quasi-homogeneous waves case C.1

d) Frequency close to and above resonance where ( ) 0′ε ω > , ( ) 0′′ε ω > Typical normal modes: − strongly damped quasi-homogeneous waves case C.2

Optical systems work mainly in regime a) since here you find light propagating undamped over long distances through space. Furthermore one sometimes exploits


also regime b) when one would like to have reflection of light at surfaces, e.g. at a metallic mirror.

Time averaged Poynting vector of plane waves To understand the implication of the discussed properties of plane waves on experi-mentally measurable observables, we should look at their energy density flow by calculating their Poynting vector, which was defined as

/2

/2

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

t T

t Tt t t dt

T+ ∗

− ′ ′ ′= ℜ × ∫S r E r H r

m

mm

For plane waves we define the electric and magnetic field as

( ) ( )

0

( , ) exp exp1( , ) ( , )

t t t

t t

′ ′′= − ω = − − ω

= ×ωµ

E r E kr E k r k r

H r k E r

i i i i

Assuming a stationary case ( ) ( )exp( i t)t = ω − ωE E we find for the Poynting vector

[ ] ( )2 20

0 0

1 1( , ) exp 2 exp 22 2

ntc

ε ω = − = − ωµ µ

′κ′′S r r Ek k e e r E k' k" .

Here k'e is the unit vector along ′k and k''e is the unit vector along ′′k .

3.2 Beams and pulses as well as the analogy of diffraction and dispersion

Based on our understanding of the normal mode solutions of the wave equation, in this chapter we will analyze the propagation of light. In particular, we will answer the question how an arbitrary beam (spatial) or pulse (temporal) will change during propagation in isotropic, homogeneous, dispersive media. Relevant (linear) physical effects are diffraction and dispersion. Both phenomena can be understood very easily in the Fourier domain. Temporal effects, i.e. the dispersion of pulses, will be treated in temporal Fourier domain (temporal frequency domain). Spatial effects, i.e. the diffraction of beams, will be treated in the spatial Fourier domain (spatial frequency domain). We will see that: • Pulses with finite spatial width (i.e. pulsed beams) can be described as

superpositions of normal modes in the frequency- and spatial frequency domain. • Spatio-temporally localized optical excitations delocalize during propagation

because of the different phase evolution of the excited normal modes 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 (normal mode) beam

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

( ) 3( , ) ( )exp dt kt

∞

−∞= ∫ k kr E rE i − ω

B) pulse finite duration dispersion

stationary wave (normal mode) pulse

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

k

k1 k2 k3 k4 k5

…

ω


( )( , ) ( )exp .dt t∞

−∞= ⋅ ω ω∫E r E k ri ω−

C) pulsed beam 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∞

−∞= ⋅ ω ω∫ k kE r E ri ω−

3.3 Diffraction of monochromatic beams in homogeneous isotropic media

Let us have a look at the propagation of monochromatic beams first. In this situation, we have to deal with diffraction only. We will see later that pulses and their disper-sion can be treated in a very similar way. Treating diffraction in the framework of wave-optical theory (or even Maxwell's equations) allows us to describe rigorously many important optical systems and effects, i.e., optical imaging and its resolution limit as well as optical filtering, microscopy, gratings, ... In this chapter, we assume stationary (monochromatic) fields and therefore

constω = . For technical convenience and because it is sufficient for many important problems, we will make the following assumptions and approximations: • ( ) ( ) 0,′ε ω = ε ω > optical transparent regime normal modes are stationary

homogeneous and evanescent plane waves • scalar approximation

y y y( , ) ( , ) ( , ) ( , ).E E uω → ω → ω → ωE r r e r r

− exact for one-dimensional beams and linear polarization (see also section 2.1.5)

− approximation in two-dimensional case In homogeneous isotropic media we have to solve the Helmholtz equation

( )2

2( , ) ( , ) 0.cω

ω + ε ω ω =E r E r∆

For the scalar approximation and for fixed frequency ω it reads

ω1 ω2 ω3

...


( )

( )

2

2

2

( , ) ( , ) 0,

( , ) ( , ) 0.

u u

uc

uk

ω + ω =

ω + ω

ωε ω

ω =

r r

r r

∆

∆ scalar Helmholtz equation

In the last step we inserted the dispersion relation (wave number ( )k ω ). In the following we will often even omit the argument ω in our notation since the frequency ω is fixed anyway.

Arbitrarily narrow beams (general case) Let us consider the following fundamental problem: We want to compute from a given field distribution ( , ,0)u x y in the plane 0z = the complete field ( , , )u x y z in the half-space 0z > , where z is our “propagation direction”.

The governing equation to describe this z-dependence of the field distribution ind x and y is the scalar Helmholtz equation

( )2( , ) ( , ) 0u k uω + ω ω =r r∆ .

To solve this equation and to calculate the dynamics of the fields, we can switch again to the Fourier domain. However, this time we switch to the spatial Fourier domain. Let's first test what happens if we take the spatial Fourier transform over all three spatial coordinates

[ ] 3( , ) ( , )exp ( )u U d k∞

−∞ω = ω ω∫r k k ri .

This Fourier transform can be interpreted as a superposition of normal modes with different propagation directions and wavenumbers ( )k ω (here the absolute value of the wavevector k ). Naively, we could expect that we just constructed a general solution of our problem, but the solution is not correct because the normal modes have to obey the dispersion relation


( )2

2 2 2 2 22k k k k

cω

= = + + = ε ωk x y z .

Thus, out of the three vector components of k only two can be chosen freely, e.g. xk and yk . From now on we will use the following naming convention for the three wavevector components in Cartesian coordinates:

, ,k k k= α = β = γx y z .

Then, the dispersion relation reads:

2 2 2 2( )k ω = α + β + γ .

Hence, our previous attempt of a Fourier transform of all three spatial coordinates , ,x y z is overdetermined. This is because the dispersion relation is not fulfilled by all

combinations of , ,k k kx y z if the temporal frequency ω is fixed as well. Thus, to solve our problem we need only a two-dimensional Fourier transform, with respect to the transverse coordinates x and y , when z is the direction of propagation:

( )( ) ( , )exp; .u U x y d dz∞

−∞= α β α + β α β ∫∫r i

It will also be a convention for this section, that we will indicate the fields in the Fourier space by capital letters and the fields in real spatial space by small letters, instead of using u for the Fourier space. Furthermore, please be aware that

( , ; )U zα β is a mixed representation. It is in Fourier space with respect to x and y and thus depends in α and β . But it is still in real spatial space with respect to z. In analogy to the frequency ω we call α, β spatial frequencies. Now we insert this expression into the scalar Helmholtz equation

( )2( ) ( ) 0u k u+ ω =r r∆ .

This way we can transfer the Helmholtz equation in two spatial dimensions into Fourier space

22 2 2

2

22

2

( , ; ) 0,

( , ; ) 0.

d k U zdz

d U zdz

+ − α − β α β =

+ γ α β =

This turns the scalar Helmholtz equation from a partial differential equation into an ordinary differential equation, which can be easily solved and yields the general solution

[ ] [ ]1 2( , ; ) ( , )exp ( , ) ( , )exp ( , )U z U z U zα β = α β γ α β + α β γ α βi -i , (*)


with the arguments of the exponential terms depending on

2 2 2( , ) ( )kγ α β = ω − α − β .

Depending on α and β we can identify two types of solutions:

A) Homogeneous waves

2 0,γ ≥ 2 2 2kα + β ≤ k real homogeneous waves

B) Evanescent waves

2 0,γ < 2 2 2kα + β > k complex, because kγ = z is imaginary. Then, we have ′ ′′+k = k ki , with x y′ α + βk = e e and z′′ γk = e .

⊥k' k'' evanescent waves

We see immediately that in the half-space 0z > the solution (*) is ∼ ( )exp z− γi grows exponentially in z-direction. Because this does not make sense, this component of the solution must vanish and hence 2 ( , ) 0U α β = . In fact, we will see later that

2 ( , )U α β corresponds to backward running waves, i.e., light propagating in the opposite direction. We therefore reduce the general solution (*) to

[ ]1( , ; ) ( , )exp ( , )U z U zα β = α β γ α βi .

Furthermore the following boundary (initial) condition holds

1 0( , ) ( , ;0) ( , )U U Uα β = α β = α β ,

which determines uniquely the entire solution for all other positions 0z ≠ as well

k


[ ]

[ ]0

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

( , )exp ( , ) .

U z U z

U z

α β = α β γ α β

α β γ α β

i

i

To determine the solution in the original spatial space, we can find the optical field for 0z > by inverse Fourier transform:

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

∞

−∞= α β α + β α β ∫∫r i

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

∞

−∞γ α β= α β α + β α β ∫∫r ii

For homogeneous waves (real γ ) the red term above causes a certain phase shift for the respective plane wave during propagation. Furthermroe, for evanescent fields (imaginary γ ) the red term causes a decay of the fields along the propagation direction. Hence, we can formulate the following result: Diffraction is due to different phase shifts and amplitudes of the different excited normal modes when these modes are traveling in propagation direction. The indi-vidual phase shifts and amplitude reductions are determined according to different spatial frequencies ,α β of the normal modes.

The initial spatial frequency spectrum or angular spectrum at 0z = forms the initial condition of the initial value problem and follows from 0 ( , ) ( , ,0)u x y u x y= by Fourier transform:

( )2

0 01( , ) ( , )exp

2U u x y x y dxdy

∞

−∞

α β = − α + β π ∫∫ i .

As mentioned above the wave-vector components ,α β are the so-called spatial frequencies. Another common terminology is “direction cosine” for the quantities

/ ,kα / kβ , because of the direct link to the angle of the respective plane wave. For example / cos xkα = θ gives the angle of the plane wave's propagation direction with the x -axis.

Scheme for calculation of beam diffraction We can formulate a general scheme to describe the diffraction of beams: 1. Start from initial field 0 ( , )u x y

2. Calculate initial angular spectrum 0 ( , )U α β by Fourier transform

3. Describe propagation by multiplication with ( )exp , zγ α β i

4. Obtain new spectrum ( )0( , ; ) ( , )exp ,U z U zα β = α β γ α β i

5. Calculate new field distribution ( , , )u x y z by Fourier back transform


This scheme allows for two interpretations: 1) The resulting field distribution is the Fourier transform of the propagated spec-

trum

( )( ) exp .( , ; )u x y d dU z∞

−∞= α + β α β α β∫∫r i

2) The resulting field distribution is a superposition, i.e. interference, of homogene-ous and evanescent plane waves ('plane-wave spectrum'), which obey the dis-persion relation

( ) ( ) 0 x( ,) e p ,u d U zd∞

−∞α β γ α β= α β∫∫r

amplitude of phase factor which isinterference ofthe excited accumulated by theeigenstates to formeigenstates eigenstates durithe field pattern after

propagation

i [ ] exp x yα + β

shape of eigenstates(plan waves)

ngpropagation

i .

Remark: The excited eigenmodes do not exchange energy/information, they just superimpose (interfere).

The complex transfer function for homogeneous space To understand the diffraction of beams let us now discuss the complex transfer function of homogeneous space, which from the above formula can be extracted as

2 2 2

( , ; ) exp[ ( , ) ],

exp ( ) .

H z z

i k z

α β = γ α β

= ω − α − β

i

It determines the beam propagation in the Fourier space. For a finite propagation distance z = const. it looks like:

amplitude phase

Obviously, ( )( , ; ) exp ,H z zα β = γ α β i acts differently on homogeneous and

evanescent waves:


A) Homogeneous waves for 2 2 2kα + β <

( ) ( )( )exp , 1, arg exp , 0z zγ α β = γ α β ≠ i i

Upon propagation the homogeneous waves are multiplied by the phase factor

2 2 2exp k z − α − β i .

Each homogeneous wave keeps its amplitude. Homogeneous waves transport energy/information.

B) Evanescent waves for 2 2 2kα + β ≥

( ) ( )( )2 2 2exp , exp , arg exp , 0z k z z γ α β = − α + β − γ α β = i i

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

2 2 2exp 1k z − α + β − ≤ .

This means that their contribution gets damped with increasing propagation distance z .

Each evanescent wave keeps its phase. Evanescent waves do not transport energy/information.

Now the question is: When do we excite evanescent waves? Obviously, the answer lies in the initial condition, i.e. the boundary condition at 0z = . Whenever 0 ( , )u x y yields an angular spectrum with components 0 ( , ) 0U α β ≠ for 2 2 2kα + β > we excite evanescent waves.

Example: Slit Let us consider the following one-dimensional initial condition, which corresponds to the aperture of a slit:

01

( ) .20

axu x

≤=

for

otherwise

We assume that the slit is infinitely extended in the y-direction. Hence, we only study the dynamics in the x-z-plane.

-a/2 a/2 x

u0(x)


[ ]0 0

sin2( ) FT ( ) sinc

22

aaU u x

a

α α = = α α

− All spatial frequencies α (-∞ → ∞) are excited.

− Important spectral information is contained in the interval 2 / aα ≤ π . Largest important spectral frequency for a structure with width a is

2 / aα = π .

− Evanescent waves appear for kα > .

− To represent the relevant information by homogeneous waves the following

condition must be fulfilled: 2 2k naπ π

< =λ

anλ

>

General result We have seen in the example above that evanescent waves appear for structures < wavelength in the initial condition. Information about these small structures gets lost for z >> λ . In homogeneous media, only information about structural details having length scales of , /x y n∆ ∆ > λ are transmitted over macroscopic distances. Homogeneous media act like a low-pass filter for light.

Summary of the beam propagation scheme

1

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

→ α β → α β = α β α β →FT FT

with the transfer function ( )( , ; ) exp ,H z zα β = γ α β i

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

U0(α)


components have to get the same phase shift during the propagation. This condition assures that the superposition, i.e. interference, of the excited normal modes will always be the same. Then the propagation simplifies to a phase shift in real space:

( ) [ ]0 0( , ; ) ( , )exp , ( , )expU z U z U Czα β = α β γ α β ≡ α β i i ,

[ ] 0( , , ) exp ( , )u x y z Cz u x y= i .

Since in general ( , ) constγ α β ≠ the excitation 0 ( , )u x y must have a shape such that its Fourier transform has only components where the transfer function is of equivalent value

0 ( , ) 0U α β ≠ only for ( ) 2 2 2, k Cγ α β = − α − β = .

It is straightforward to see that the excited spatial frequencies must lie on a circular ring in the ( ),α β plane

2 2 20α + β = ρ .

To find out, what field distributions 0 ( , )u x y have a circular Fourier transform, we can just explore the result of the inverse Fourier transform of a circular angular spectrum distribution. For constant spectral amplitude on this ring the Fourier back-transform yields (see exercises):

0 0( , ) ( )u x y J r= ρ .

This corresponds to a so-called Bessel beam.

Bessel-beam (profile) Bessel-beam

Fresnel- (paraxial) approximation The beam propagation formalism developed in the previous chapter can be simplified for the important special case of a narrowband angular spectrum

0 ( , ) 0U α β ≠ only for 2 2 2kα + β .


In this situation the beam consists of plane waves having only small inclination with respect to the optical z -axis (paraxial (Fresnel) approximation). Then, we can simplify the expression for ( , )γ α β by a Taylor expansion to:

2 2 2 2

2 2 22( , ) 1

2 2k k k

k k α + β α + β

γ α β = − α − β ≈ − = −

Then the resulting expression for the transfer function in Fresnel approximation reads:

( ) ( )2 2

exp ( , ) exp exp ( , ; )2

H z kz z H zk

α + β= γ α β ≈ − = α β

Fi i i

Amplitude Phase

We can see that the absolute value of ( , ; )H zα βF is always one. Hence it does not account for the physics of evanescent waves. However, we must remember that the derivation of ( , ; )H zα βF as an approximation of ( , ; )H zα β required the assumption that the spatial frequency spectrum is narrow (paraxial waves). Thus, already at the beginning we had excluded the excitation of evanescent waves to justify the paraxial approximation. The assumption of a narrow frequency spectrum corresponds to the requirement that all structural details ,x y∆ ∆ of the field distribution in the excitation plane (at

0z = ) must be much larger than the wavelength:

, 10x yn nλ λ

∆ ∆ >

This requirement applies also to the phase of the excitation. Hence it is not sufficient that only the structural details of the intensity have a large scale. The underlying phase of the excitation field must fulfill this condition as well. This includes the condition that the phase of the beam should not have a strong inclination to the principal propagation direction, which in our notation is the positive z-direction. Also backward propagation is not allowed. The propagation of the spectrum in Fresnel approximation works in complete analogy to the general case. We just use the modified transfer function to describe the propagation:


0( , ; ) ( , ; ) ( , )U z H z Uα β = α β α βF F

Summary of Fresnel approximation For a coarse initial field distribution 0 ( , , )u x y z the angular spectrum 0 ( , )U α β is nonzero for 2 2 2kα + β only. Then, only paraxial plane waves are relevant for transmitting information and the transfer function of homogeneous space can be approximated by ( , ; )H zα βF .

Description in real space It is also possible to formulate beam propagation in Fresnel (paraxial) approximation in position space:

( )

( )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

∞

−∞

∞

−∞

∞

−∞

= α β α + β α β

= α β α β α + β α β

′ ′ ′ ′ ′ ′= − −

∫∫∫∫∫∫

F F

F

F

i

i

The spatial response function F( , ; )h x y z follows from the convolution theorem and is the Fourier transform of ( , ; )H zα βF :


( )

( ) ( )

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

∞

−∞

∞

−∞

= α β α + β α β π

α + β = − α + β α β π

∫∫

∫∫

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

= − + = − +

+

π π F

i ii i i

The response function corresponds to a spherical wave in paraxial approximation. Similar to Huygens principle, where from each point in the object plane a spherical wave is emitted towards the image plane, here paraxial approximations of spherical waves are emitted. To sum up, in position space paraxial beam propagation is given by:

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

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

∞

−∞

′ ′ ′ ′ ′ ′= − − + − π ∫∫F

ii i

Of course, the two descriptions in position space and in the spatial Fourier domain are completely equivalent.

The correspondence between real and frequency space

Relation between transfer and response function:


( )21 e( xp

(2;, )) ,

)( ; x y d dh x y H zz

∞

−∞α= α + β α ββ π ∫∫ i

Transfer functions for homogeneous space

( ) 2 2 2( , ; ) exp , expH z z i k z α β = γ α β = − α − β i exact solution

[ ]2 2

( , ; ) exp exp2

H z kz i zk

α + βα β = −

F i Fresnel approximation

with 0( ) ( ) ( )k k k n ncω

= ω = ω = ω

Remark on the validity of the scalar approximation In the previous description of the propagation of arbitrary beams we have used the scalar approximation of the vectorial fields. It is interesting to see, to what extend this approximation stays in correspondence to the conditions, which where necessary to derive the Fresnel approximation.

( ) ( )0

ˆ( , ) , , ei x y z d dα +β +γω = α β ω α β∫∫E r E

From the divergence condition we can derive conditions on the vectorial field components as

ˆ ˆ ˆ( , ) 0 0x y zE E Eω = → α + β + γ =divE r .

A) One-dimensional beams − translational invariance in y-direction: 0β = , 0α ≠ , 0γ ≠

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

scalar approximation is exact since divergence condition is strictly fulfilled B) Two-dimensional beams

− Finite beam which is localized in the x,y-plane: , 0α β ≠

− and linear polarization, w.l.o.g. in y-direction: ˆ 0xE = , ˆyE U=

divergence condition: ˆ ˆ 0y zE Eβ + γ =

( ) ( ) ( )2 2 2

0

ˆ ˆ ˆ, , , , , , 0z y yE E Ek

≈

β βα β ω = − α β ω = − α β ω ≈

γ − α − β

In paraxial approximation, where 2 2 2kα + β , the scalar approximation is automatically justified.


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

α + β− α β

α β = α β α β

=

F F

i i

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

( )( , ; ) exp ( , ; )U z kz V zα αβ β=F i 2 2

0( , ; ) exp ( , ).2

V z z Vk

α + βα β = − α β

i

Differentiation of V with respect to z gives:

( )2 21( , ; ) ( , ; )

2V z V z

z k∂

α β = α + β α β∂

i

Fourier transformation with respect to α and β back to position space in x and y leads to the so-called paraxial wave equation:

( )

( ) ( )2 21 ( , ; )exp2

( , ; )expV z xz

V z x y

y

d

d

k

d

d∞

−

∞

∞

∞

−

∂∂

= α

α β α + β α

β α + β α β

β

+

α β∫∫

∫∫i

i

i

( )2 2

2 21 ( , ; )ex) p( ,

2, V z

x yx y d d

zx y z

kv

∞

−∞

∂ ∂− + ∂ ∂

∂= α β α + β α β ∂ ∫∫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 kz=F 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 wave propagation in inhomogeneous media. We will include inhomogeneous media in this derivation even though the current chapter of this lecture is devoted to homogeneous media. We start from the scalar Helmholtz equation. However, we should mention that the extrapolation of our previous discussion towards inhomogeneous media requires another approximation. This approximation assumes small spatial fluctuations of

( , )ε ωr since otherwise the grad-div-term in the general wave equation would not


disappear. This is equivalent to having a weak index contrast between different spatial positions.

2( , , ) ( , ) ( , , ) 0u x y z k u x y z∆ + ω =r with 2

22( , ) ( , )k

cω

ω = ε ωrr

We make the ansatz of a slowly varying envelope

( )( , , ) ( , exp, )u x y z v x y z kz= i with k k=

where k is the spatially averaged wavenumber. This is the mean wavenumber in the particular volume of material, in which wave propagation is considered. Hence it is the average of the spatially varying material properties within the volume of interest.

2(2) 2 2

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

∂ ∂ + + ∆ + ω − = ∂ ∂r i

With the SVEA condition

/vkv z∂ ∂

the first 2nd order z derivative can be neglected and using this approximation, we can simplify the scalar Helmholtz equation as follows:

2 2

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

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

∂ ω −+ ∆ + = ∂

r

i

This is the paraxial wave equation for inhomogeneous media having a weak index contrast.

3.4 Propagation of Gaussian beams The propagation of Gaussian beams is an important special case. Optical beams of Gaussian shape are import because many beams in reality have a shape, which is at least close to the shape of a Gauss function. In some cases they even have exactly the Gaussian shape as e.g. the transversal fundamental modes of many lasers. The second reason why Gaussian beams are important, is the fact that in paraxial approximation it is possible to compute the evolution of Gaussian beams analytically.

Fundamental Gaussian beam in focus


The general form of a Gaussian beam at one plane ( constz = ) is elliptic, with a curved phase front

[ ]2 2

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

= = − + ϕ x y

i .

Here, we will restrict ourselves to excitation conditions with rotational symmetry ( 2 2 2w w w= =x y 0 ) and (initially) 'flat' phase ( , ) 0x yϕ = . Later we will see that this corresponds to the focus of a Gaussian beam. The Gaussian beam in such a focal plane with flat phase is characterized by amplitude A and width w0 , which is defined as

( )2 2 20 0 0( ) exp 1 /u x y w A A e+ = = − =0 .

In practice, the so-called 'full width at half maximum' (FWHM) of the intensity is often used instead of w0 . The FWHM of the intensity is connected to w0 of the field by

22 2 20 FWHM2 20 0

1exp22w

u x y wA

2FWHM

20

ln 22w

w 2 2 2

FWHM 0 02ln 2w 1.386ww

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

∞

−∞

+α β = α β = − − α + β

π

α + β α + β= − = − π π

∫∫

0 s

0

0 0

i

− We see that the angular spectrum has a Gaussian profile as well and that the widths in position space and Fourier space are linked by

2w w⋅ =s 0 .


Angular spectrum in the focal plane

− We should check if the paraxial approximation is fulfilled:

We can say that 0 ( , ) 0U α β ≈ for ( )2 2 216 / wα + β ≥ 0 , because ( )exp 4 0.02− ≈ .

For paraxial approximation we need ( )2 2 2k α + β

2 216 /k w 0

2 2

22

16 2 ,2

wn n

n

λ λ = ≈ π π λ

0

paraxial approximation works for 10 10wnλ

= λ0 n

3) Propagation of the angular spectrum:

2 2

0

2 2 2 22 20

( , ; ) ( , )exp2

exp exp .4 4 2

V z U zk

A w w zk

α + βα β = α β −

α + β α + β

= − − π 0 0

i

i

4) Fourier back-transformation to position space, when still using the slowly varying envelope v

( ) ( )

( )

( )

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

∞

−∞

= − α + β α + β α β π

+= −

+ +

+= − + +

+

∫ ∫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 π

= =λn

Note that we use the slowly varying envelope v ! Conclusion:

− A Gaussian beam keeps its shape, but its amplitude, width, and phase change upon propagation.

− There are two important parameters to characterize the beam's evolution: propagation length z and Rayleigh length 0z .

Some books use the “diffraction length” 02L z=B , which is a measure for the depth of focus” of the Gaussian beam. E.g.: 10w λ0 n 600 .L λB 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, and shape of the phase front:

( ) ( )( )

( )

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 20

2 2 2

2

0

0

0 2

1( , , ) exp exp ex

1

p (2 )1 1

x y x ykv x y z zz zzz z z

w

z

A

+ +− ϕ + +

=

+

0

ii

Here we used that 202 /w z k=0 . And the (x,y)-independent phase ( )zϕ is defined by

0tan /z zϕ = − , 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 + +

= −

ϕ

i i

The normalized beam intensity 0/I I as a function of the radial distance

2 2 1/2( )x yρ = + at different axial distances: (a) 0z = ; (b) 0z z= ; (c) 02z z= .

Discussion of beam parameters

Amplitude The amplitude evolves along z like:

0 02 2

0

1 1( ) ,211

A z A Azz

Lz

= =

++ B

Hence, we get for the Intensity profile 2I A :

The normalized beam intensity 0/I I at points on the beam axis

( 2 2 1/2( ) 0x yρ = + = ) as a function of the propagation distance z .


Width The beam width evolves along z like:

2 2

0

2( ) 1 1z zw z w wz L

= + = +

0 0

B

The beam radius ( )W z has its minimum value 0W at the waist ( 0z = ), reaches

02W at 0z z= ± , and increases linearly with z for large z .

Phase curvature The phase front of a Gaussian beam can be characterized by its radius of the phase curvature, which is given by

2 2

0( ) 1 12

z LR z z zz z

= + = +

B

The radius of curvature ( )R z of the wavefronts of a Gaussian beam. The dashed line is the radius of curvature of a spherical wave.

The flat phase in the focus ( 0z = ) corresponds to an infinite radius of curvature. The strongest curvature (minimum radius) appears at the Rayleigh distance 0z from the focus.

Gouy phase The (x,y)-independent Gouy phase is given by

0

2tan z zz L

ϕ = − = −B


The so-called Gouy phase is the retardation ( )zϕ of the phase of a Gaussian beam relative to a uniform plane wave at the points of the beam axis.

The Gouy phase is not important for many applications because it is ‘flat’. However, in resonators and in the context of nonlinear optics it can play an important role (i.e., harmonic generation in a focused beam geometries). Overall 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 +

Φ = + + ϕ =

const

Wavefronts of a Gaussian beam.

Propagation of Gaussian beams with q-parameter formalism In the previous chapter we derived the expressions for Gaussian beam propagation, i.e., we know how amplitude, width, and phase change with the propagation variable z . Furthermore, we have discovered that these the entire beam evolution depends essentially on the two parameters z and 0z . Therefore, one could combine these two real-valued parameters into a single complex-valued parameter, which we call the q-parameter:

0( )q z z z= −i q-parameter.

This complex beam parameter allows an even simpler computation of the evolution of a Gaussian beam than the expression of the last section. 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 see that real and imaginary part are directly linked to the radius of phase curva-ture and the beam width:


21 1( ) ( ) ( )q z R z w z

λ= +

πni because

220

0 02kwz wπ

= =λn

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

1) initial conditions: 21 1(0) (0) (0)q R w

λ= +

πni

2) propagation (by the definition of the q parameter) ( ) (0)q d q d= + 3) q-parameter at z d= determines new width and radius of curvature

21 1 1( ) (0) ( ) ( )q d q d R d w d

λ= +

+ π

ni

Gaussian optics We have seen in the previous chapter that the complex q-parameter formalism makes a simple description of beam propagation possible. The question is whether it is possible to treat also optical elements (lens, mirror, etc.) using this formalism. Hence by investigating beam propagation through such elements, we will go somehow beyond the general scope of this chapter, which was originally devoted to the investigation of homogeneous space only. Aim: for given 0 0,R w (i.e. 0q ) pass through n optical elements calculate ,n nR w

(i.e. nq )

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 the same ABCD-Matrices as for geometrical ray optics! This is remarkable because here we are doing wave-optics (but with Gaussian beams).

A short reminder of geometrical ray 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 Θ before and after the element

2 1

2 1

ˆ .y y

= Θ Θ M

Link to Gaussian beams Let us consider the distance to the intersection of the ray with the optical axis, as it was defined in chapter 9.6.1 on "The ray-transfer-matrix":

1 1 1tan( )z y= Θ Using the small angle approximation

1 1tan( )Θ ≈ Θ we can define this distance as:

11

1

yz =Θ

1

2 1 1 12

12

1

1

11 1

yA By Ay Bz y

Az BCz DCy D C D

++ Θ Θ

= = = =Θ + Θ +

++

Θ

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 . Then the propagation of q -parameters through an optical element is given by:

1 0 11

1 0 1

A q BqC q D

+=

+

Accordingly the propagation through N elements can be described by


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 8.6 for ray optics! Here: we will check two important examples: i) Propagation in free space over a length z d= : propagation (by definition of q -parameter) ( ) (0)q d q d= +

1ˆ0 1

d

M = 1 1 1 11, , 0, 1A B d C D= = = =

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 do to a Gaussian beam ( )2 2 20exp ( ) /x y w− + in paraxial

approximation? − no change of the width

− but change of 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

π= =

λ and

fw is the focal width. Hence the q-parameter is

20

1 n

f

iq w

λ=

π

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= − .

Now we compare this result to the description using the ABCD matrix of a thin lens from chapter 8.6:

0f < 0f > 1 0ˆ1 1f

−

Mthin lens =

double concave double convex

lens lens defocusing focusing

1 0 1 01

1 0 1 02

0 002

1 0 0

1

1 1 1 n

n

A q B qqC q D q f

q f wqq q f w

+= =

+ − +

− + λ π= = + = −

− π λi ifor

We see that we get the same result. However, be careful when using the q-parameter formalism. Gaussian optics des-cribes the evolution of the beam's width and phase curvature only. Changes of ampli-tude and reflection are not included!

Gaussian modes in a resonator In this chapter we will use our knowledge about paraxial Gaussian beam propagation to derive stability conditions for resonators. An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light (see He-Ne laser experiment in the experimental optics course at the end of the 1st semester).


3.4.4.1 Transversal fundamental modes (rotational symmetry)

Wave fronts, i.e. planes of constant phase, of a Gaussian beam.

The general idea to get a stable light configuration in a resonator is that mirrors and wave fronts (planes of constant phase) coincide. Then, radiation patterns are reproduced on every round-trip of the light through the resonator. Those patterns are the so-called modes of the resonator. In paraxial approximation and for 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, which is different to script used in the experimental optics course (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 8: beam hits concave mirror radius ( 1,2) 0iR i = < beam hits convex mirror radius ( 1,2) 0iR 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< >


The question, which arises now, is if there is a Gaussian beam, which fits into every mirror configuration or if some restrictions apply on 1R , 2R , and d . 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 ( )2

11 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 calculations above we have eliminated the Rayleigh length z0, being a real and positive quantity. Hence, we have to check that the 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+ + + +

= − = − >+ +

Since the denominator ( )21 2 2R R d+ + is always positive we need to fulfill

( )( )( )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) 1 2 0g g > and 1 21 0g g− >

or for

(2) 1 2 0g g < and 1 21 0g g− < .

Since the second case cannot be fulfilled for any real valued 1g and 2g , the following condition for the placement of the mirrors and their radii remains, to determine a stable cavity

1 20 1g g< < which is equivalent to 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 the coordinate axes and hyperbolas:

Resonator stability diagram. A spherical-mirror resonator is stable if the parameters 1 11 /g d R= + and 2 21 /g d R= + lie in the unshaded regions bounded by the lines 1 0g = and 2 0g = , and the hyperbola 2 11 /g g= . R is negative for a concave mirror and positive for a convex mirror. Various special configurations are indicated by letters. All symmetrical resonators lie along the line

1 2g g= .

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 the script for the HeNe experiment in the experimental optics course (and Wikipedia): In the experiment's scrip (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> >


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< <

3.4.4.2 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 the same wave fronts but different inten-sity 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

+ += − ϕ

G i i

And for higher order modes this expression generalizes to

( ) [ ]

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

× =

++ ϕ

+i i i

While the ,x y -dependence of phase is the same as for the fundamental mode, the amplitude's ,x y -dependence will be more complex:

2

( ) ( )exp2l lG H

ξξ = ξ −

.

Here, the functions ( )lG ξ are the Hermite-Gaussian functions and ( )lH ξ are the so-called Hermite polynomials with

0 11, 2H H= = ξ and 1 12 2l l lH H lH+ −= ξ − .


Several low-order Hermite-Gaussian functions: (a) 0 ( )G u ; (b) 1( )G u ; (c) 2 ( )G u ; and (d) 3( )G u .

Intensity distributions of several low-order Hermite-Gaussian beams in the transverse plane. The order ( , )l m is indicated in each case.

3.5 Dispersion of pulses in homogeneous isotropic media

Pulses with finite transverse width (pulsed beams) In the previous chapters we have treated the propagation of monochromatic beams, where the frequency ω was fixed and therefore the wavenumber ( )k ω was constant as well. This is the typical situation when we deal with continuous wave (cw) lasers. However, for many applications (spectroscopy, nonlinear optics, telecommunication, material processing) we need to consider the propagation of pulsed laser light. In this situation, we have a typical envelope length 0T of

13 10010 s(100fs) 10 s(100ps)T− −≤ ≤ .

Let us compute the spectrum of a Gaussian pulse:

( )

( )

2

0 20

20 2

02 20 0

( ) exp exp

4( ) exp 24 /

tf t tT

F TT T

= − ω −

ω − ω

ω − → ω = → ω =

s s

i

spectral width: 10 1 13 14 10 4 10s s− −⋅ ≤ ω ≤ ⋅s


− center frequency of visible light: 150 2 4 10ω = πν ⋅ s-1

optical cycle: 1502 / 1.6 10 s 1.6fsT −= π ω ≈ ⋅ =s

Hence, we have the following order of magnitudes:

0 0 0ω << ω → ω − ω = ω << ωs .

Thus out of the very broad range of electromagnetic frequencies we excite only a very small part. Accordingly, our models to describe the frequency-dependent res-ponse of the material, through which the light propagates, needs to cover only the narrow frequency range of the excitation as well. In this situation it can be helpful to replace also the complicated frequency depen-dence (dispersion relation) of the wave vector 2 2( ) ( ) ( )cω = ω ε ωk or the wave number ( ) ( ) ( )k cω = ω ε ω by a Taylor expansion at the central frequency 0ω = ω , which can be used to approximates the complex frequency dependence in a narrow frequency range quite well.

In most cases, a parabolic (or cubic) approximation of the frequency dependence in the dispersion relation will be sufficient:

( ) ( ) ( )0 0

22

0 0 021( )2

k kk kω ω

∂ ∂ω ≈ ω + ω − ω + ω − ω

∂ω ∂ω

The three expansion coefficients and their physical significance The following terminology for the individual expansion coefficients is commonly used in the literature. It associates the physics, which is inherited in the dispersion relation, with the three parameters of the Taylor expansion.


A) Phase velocity Phv

( ) ( )000 0

0

1,nkk k

v cω

ω = → = =ωPh

The phase velocity is the velocity of the phase fronts for the light at the central frequency 0ω = ω .

B) Group velocity gv

0

1kvω

∂=

∂ω g

The group velocity is the velocity of the center of the pulse (see detailed discussion below).

With

( ) ( )k ncω

ω = ω

we get

0 0

0 01 1 ( )k nnv cω ω

∂ ∂= = ω + ω

∂ω ∂ω g

0

0

0 00 0

( )( ) ( )

( )

c c nv vn nnn

ω

ω= = =

ω ω ∂ω + ω ∂ω

g PHg g

where g 0( )n ω is the so-called group index

0

0 0 0( ) ( ) nn nω

∂ω = ω + ω

∂ωg .

For normal dispersion: / 0n n n v v∂ ∂ω > → > → <g g PH

For anomalous dispersion: / 0n n n v v∂ ∂ω < → < → >g g PH

C) Group velocity dispersion (GVD) or simply dispersion Dω

0

2

2k Dω

ω

∂=

∂ω

The GVD changes the pulse shape upon propagation (see detailed discussion below).

0

2

21

g

kD Dvω

ω

∂ ∂= = = ∂ω ∂ω


21 1

0 0

0 0

vD

v v

vD

vD

∂∂= = − ∂ω ∂ω

∂→ > <

∂ω∂

→ < >∂ω

g

g g

g

g

Alternatively in telecommunication one often uses the derivative with respect to the wavelength as

21 2D cDvλ ω

−

∂ π= = ∂λ λ g

.

The wave equation for pulsed beams 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 ucω

+ ε ω =ω ωr r∆

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

dispersion relation : 2

22( ) ( )k

cω

= εω ω

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

( )0( , , ; ) ( , , )exp , ,U z U zα β = α β γω ω β ωα i

with ( ) 2 2 2, , ( )kγ α β = − αω − βω .

The initial spectrum at 0z = is 0 ( , , )U α β ω

( )

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

2U u x y t x y t dxdydt

∞

−∞α β ω = − α + β − ω π ∫∫∫ i .

Let us further assume that the Fresnel (paraxial) approximation is justified (narrow spatial frequency spectrum with 2 2 2( )k ω >> α + β )

( ) ( )2 2

0( , , ; ) ( , exp exp2

, )U z U k z zk

α + βωα β ω ≈ α − ω

β

ω

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

( , , ) kH z z zk

α + β= − α β ω ω

ω F i i

For a narrow excited spectrum (long pulses with respect to optical cycle), let us now consider the Taylor expansion of ( )k ω with respect to ω from above. If the pulse is not too short, we can replace the wave number ( )k ω by

( ) ( ) ( )0 0

22

0 0 021( )2

k kk kω ω

∂ ∂ω ≈ ω + ω − ω + ω − ω

∂ω ∂ω.

Moreover, in the second term 2 2exp[ ( ) / 2 ( )]z k− α + β ωi of the transfer function (which is already small due to paraxiality) we can approximate the frequency depen-dence of the wave number by 0 0( ) ( )k k kω ≈ ω = . This approximation assumes a non-dispersive diffraction term. This is sufficiently accurate to describe the diffraction of pulsed beams, which are not too short. For visible (VIS) or near infrared (NIR) light this approximation this approximation can be applied to describe the diffraction of pulses with a pulse length 0 15fsT ≥ . For shorter pulses the frequency spectrum would become very wide and dispersion and diffraction of the pulses would not be independent from each other (space-time coupling by non-factorizable operators). By introducing this approximation, we obtain the so-called parabolic approximation of the transfer function:

[ ] ( )

( )

0

2

0

20

0

2

1( , , ; ) exp exp

exp exp2 2

H z k z

zk

zv

D z

α β ω ≈ ×ω −

α + β × −

ω

ω − ω

FPg

i i

i i

The resulting propagation integral in parabolic approximation for the spatio-tempo-rally varying scalar field ( , , , )u x y z t is then

( )

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

exp .

u x y z t U H z

x y t d d d

∞

−∞= α β ω α β ω ×

× α + β − ω α β ω

∫∫∫ FP

i

In analogy to the already discussed slowly varying envelope, we can introduce the slowly varying envelope in space and time as

( )0 0( , , , ) ( , , , )expu x y z t v x y z t k z t = − ω i .

To obtain a transfer function for the slowly varying amplitude we can pull out the rapid phase oscillations along z from the transfer function as:

[ ]0( , , ; ) exp ( , , ; )H z k z H zα β ω ≈ α β ωFPFP i ,


where we have also introduced the frequency difference ω from the center frequency 0ω as

0ω = ω − ω .

The new transfer function ( , , ; )H zα β ωFP is then

( )2 2

0

21 1 1( , , ; ) exp2 2

H z z Dv k

α β ω = + − α + β

ωωFP

g

i .

It acts as the propagation function for the slowly varying envelope ( , , , )v x y z t as

( )

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

exp

v x y z t U H z

x y t d d d

∞

−∞= α β ω + ω α β ω

× α + β − ω α β ω

∫∫∫ FP

i

Remark: For the rapidly varying full field this would read as

( )

( )0 00( , , , ) exp ( , , ) ( , , ; )

exp .

u x y z t k z t U H z

x y t d d d

∞

−∞ = − α β ω α β ω ×

× α + β − α β ω

ω

ω

∫∫∫ FPi

i

Illustration of the slowly varying envelope in the spectral domain

In order to complete the formalism, we also need to define the initial spectrum of the slowly varying envelope

( )0 0 0( , , ) ( , , )expu x y t v x y t t= − ωi

|u(ω)|

ω ω0

|u(ω)|

ω

v(t)

t

u(t)

t


( )

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

2V v x y t x y t dxdydt

∞

−∞α β ω = − α + β − ω π ∫∫∫ i

Then the propagation of the slowly varying envelope is given by:

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

−∞α = α β ω α + β − ω α β ω β ω∫∫∫ FP i

Co-moving reference frame The next step is to introduce a co-moving reference frame by pulling out of the transfer function a term describing the propagation of the center of the pulse as

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

α β ω = α β ω

ωFP FP

g

i .

Then the new transfer function fort he slowly varying envelope in the co-moving reference frame is

2 2

2

0

( , , ; ) exp2zH z D

k α + β

α β ω = ω −

FP i .

Based on this transfer function the evolution of the slowly varying envelope would be described as

( )( )0 ( , ,( , , , ) ( , , ) exp; )v x y z t V x y tH z v d d dz∞

−∞α β ω = α β ω α + β − ω − α β ω ∫∫∫

gFP i .

In this formula the co-moving reference frame of the pulse is visible by the coupling of spatial and temporal coordinates in the term g/t z v− . In the temporal domain, this can be nicely expressed by introducing the so-called co-moving reference time τ as

zt

vτ = −

g

.

This frame is called co-moving because the pulse does not “move” anymore and it appears as it would rest in space and time. This is seen from the fact that

( , , ; )H zα β ω

FP is now purely quadratic in ω . In contrast, the linear ω -dependence in Fourier space had given a shift in the time domain. Eventually, 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

∞

−∞

α + β= α β ω ω − ×

× α + β − ω α β ω

τ

τ

∫∫∫ i

i

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


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

v

ω τ = τ − ω = τ − − ω τ g

i i .

Remark: The correspondence of the product with gexp[ ( / ) ]i v zω in the frequency domain to the shift of the coordinate by g( / )t z v− in the time domain is determined by the so-called shifting theorem of the Fourier transform.

Propagation equation in real space Finally, let us derive the propagation equation for the slowly varying envelope in the co-moving frame. We start from the transfer function

2 22

00

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

k α + β

α β ω = α β ω ω −

i .

Then we take the spatial derivative of the transfer function along the propagation direction z

2 22

0

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

V z D V zz k

∂ α β ω α + β= − ω − α β ω ∂

i .

This is an ordinary differential equation describing the spatial evolution of the spatio-temporal frequency spectrum along z for the slowly varying envelope in the co-moving reference frame. As before in the case of monochromatic beams, we use the Fourier back-trans-formation to get the differential equation in the time-position domain

2

(2)2

0

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

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

∂ τ ∂− τ + ∆ τ =

∂ ∂τ

i .

This is the scalar paraxial equation for propagation of so-called pulsed beams.

Comment: Extension to inhomogeneous media By using the slowly varying envelope approximation, it is possible to generalize the scalar paraxial equation also for inhomogeneous media, when a weak index contrast is assumed.

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 ∂ ∂ −

τ − τ + ∆ τ + τ = ∂ ∂τ

r i

with the averaged wave number ( )0 0k k≈ r

For 0D = the equation would be reduced to simple diffraction, as in the beam propagation scheme, which was derived earlier.


Pulses with infinite transverse extension (pulse propagation) Diffraction plays no role for sufficiently small propagation lengths 0z z . For broad beams, the Raleigh length 0z can be rather large and we can assume 0α = β ≈ , corresponding to the assumption that we have a single plane wave propagating in z-direction. Later in the lecture series we will see that this case is also valid for mode propagation in waveguides, as e.g. optical telecommunication fibers.

Description in frequency domain

1) initial condition: ( )0 0 0( ) ( )expu t v t t= − ωi

2) initial spectrum: 0 0( ) ( )V Uω = ω

3) propagation of the spectrum: 02( ; ) ( )ex

2pV z V Dz ω ω= ω i

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

[ ]20( , ) ( )exp exp

2Dv z V z d

∞

−∞

τ = ω ω − ωτ ω ∫ 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 ω = ω

P i 22( ; ) exp

2h z

Dz Dz τ

τ = − − π P i

i

and the evolution is described by the convolution integral

0( , ) ( ; ) ( )v z h z v d∞

−∞′ ′ ′τ = τ − τ τ τ∫ P

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

reads

2

2( , ) ( , ) 0

2v z D v z

z∂ τ ∂

− τ =∂ ∂τ

i

Analogy of diffraction and dispersion BEAM DIFFRACTION PULSE DISPERSION approximation: Fresnel/paraxial Taylor expansion of ( )k ω up to

quadratic term (+ co-moving frame) restriction: monochromacy plane wave

(2),( , (2

,,) ) 01k

x y x yv z v zz

+ ∆∂

=∂

i 2

2( , ) ( , ) 02

v z v zz

D ∂τ

∂=

∂−

∂ττ i


( , )x y τ

∇ ∂∂τ

0

1k

D− but 0D can vary

( , )α β ω

2 2

( , ; ) exp2

H z zk

α + βα β = −

F i 2( ; ) exp

2DH z z ω = ω

P i

In the following we will study two typical examples of pulse propagation.

Propagation of a Gaussian pulse without chirp use analogies to spatial diffraction

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

( )2

0 0 020

( ) exp exptu t A tT

= − − ω

i

2

0 0 20

( ) expv AT

ττ = −

2. Initial pulse spectrum 2 2

0 00 0( ) exp

42T TV A

ωω = − π

spectral width: 2 204 /s Tω =

Use results from propagation of Gaussian beams:

0z was used to describe Gaussian beam as 20 02

kz w =

V0

V0 e


Now the Gaussian pulse is described by 2

00

1 02

TzD

= −

Hence anomalious GVD is equivalent to 'normal' diffraction.

Dispersion length: 02DL z=

3. Evolution of the pulse

[ ]2 2

00 2( , ) exp exp exp ( )

( ) ( ) 2 ( )Tv z A z

T z T z D R z τ τ

τ = − − ϕ

ii

with the following parameters

Amplitude

( )0

2

0 0240

1( ) , ( ) 11 z

z

zA z A T z Tz

= = +

+

2( ) ( ) const.A z T z

Phase profile 'Phase curvature' is not fitting to the description of pulses since it can't be

measured instantaneously introduction of new parameter Chirp

Remember: The phase (x, y, )zΦ of a Gaussian beam has the following shape:

2 2

2 2 ( , , )( )kx y z

x y R z

Φ =

∂ ∂+

∂ ∂

For monochromatic fields the temporal dynamics of the phase is:

( )Φ τ = −ωτ ( )∂Φ τ

− = ω∂τ

arbitrary time dependence of phase

( ) ( )∂Φ τ

− = ω τ∂τ

and 2

2( ) ( ) 0∂ Φ τ ∂ω τ

− = ≠∂τ ∂τ

chirp

The chirp of a pulse describes the variation of the temporal frequency of the electric field in the pulse. parabolic approximation 'chirp' constant dimensionless chirp parameter (often just chirp)

2 2

02

( )2

TC ∂ Φ τ= −

∂τ

integration leads to:


0 02 2

2

0 0

( ) ( ) 2 , ( )T

CT

C∂Φ τ− = ω τ = ω + − Φ τ = ω +

∂ττ τ

τ

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

τ τΦ τ = −ω τ + − −ω τ + −

≡

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 on .D

leading front

trailing tail


Evolution of the chirp parameter C(z)

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 τ τ

τ = − − ϕ − ω

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

0

( )T z Tzz

≈ 0 2

( )D

T zzT

≈0

0

2( ) / / .

DT z z T z const

T= = =0

0

C(z)

z0


Gaussian pulse spreading as a function of distance z . For large distances, the width increases at a rate 02 /D T , which is inversely proportional to the initial width 0T (for D>0).

0D is only important if initial pulse is chirped, since otherwise the same quadratic dependence is observed, independent from the sign of D .

Propagation of a 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 (Nobel Prize 2018)

1. Input pulse shape

0 0 20

02 (1 )( ) exp Cv A

T τ +

τ = −

i C0 – initial chirp

2. Input pulse spectrum

2 2

0 00 0 2

0

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

C ω −

ω = − +

i

spectral width: 2

2 02

0

4(1 )CT+

ω =s

spectral width of chirped pulse is larger than that of unchirped pulse of the same length

( )2 204 / Tω =s only for transform limited pulses

Aim: calculation of pulse width and chirp in dependence on z for given initial conditions

(R/B) (R) (B)

(R) (B)


Gaussian beam q -parameter similar to Gaussian pulse

Use analogy: ( ) (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 (*)

Important: 0T is the pulse width at 0z = , which is not necessarily in the 'focus' or waist. Initial q-parameter at 0z = :

[ ]020

1 2(0)

D Cq T

= −i with 0 (0)C C=

Propagation through homogeneous space over distance z

a) ( ) (0)q z q z= + with [ ]020

1 2(0)

D Cq T

= −i

b) invert to calculate 1( )q z

c) determine ( )T z , ( )C z from 2

02 2

0

1 2 ( ) .( ) ( )

D TC zq z T T z

= −

i

Generally: 2 equations (real and imaginary part) with the 5 parameters 0 0, , , ( ), ( )C T z C z T z 3 values must be predetermined

Now lets go through the individual steps of the calculation for a propagation distance z=d here: z d=

1) Determination of q parameter at input

( )( )

200

20

(0)2 1

CTqD C

+=

+

i


2) Evolution of q parameter

( )( )

( )( )2 2 220 0 0 000

2 20 0

2 1( ) (0)

2 1 2 1Dd C C T TCTq d q d d

D C D C+ + ++

= + = + =+ +

ii

3) Inversion of general equation (*) for q(d)

2

02 2

0

1 2 ( )( ) ( )

D TC dq d T T d

= −

i

2 2 2 2

0 02 4 4

0

( ) ( ) ( )( )

2 ( ) ( )T T d C d T d T

q dD C d T d T

+ = +

i

4) Set two equations equal

( )

( )2 2 2 2 20 0 0

2 20 0

2 4 420

00

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

))

Dd C C T T T d C d T dT TD C d T d TD 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 ( 0 0, , ( )C T C d ), we can determine the other 2 unknown parameters ( , ( )d T d ).

Important case: Where is the pulse compressed to its smallest length? given: 0 0,C T & in the focus: ( ) 0C d =

unknown: , ( )d T d

a) From C(d)=0 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) From C(d)=0 imaginary part (***) becomes

( )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→ < This follows from (a), since only the distance d is positive.

2) The possible compression increases with initial chirp.

Physical interpretation If e.g. 0 0C < and 0 / 0D v> → ∂ ∂ω < →g 'red' is faster than 'blue'

Compression of a chirped pulse in a medium with normal dispersion. The low

frequency (marked R for red) occurs after the high frequency (marked B for blue) in the initial pulse (down-chirp), but it catches up since it travels faster. Upon further propagation, the pulse spreads again as the R component arrives earlier than the

B component (up-chirp).

1) First the 'red tail' of the pulse catches up with the 'blue front' until ( ) 0C z = (waist), i.e. the pulse is compressed. At this propagation distance the pulse has no remaining chirp.

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

= − = −+


4. Diffraction theory

4.1 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. 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 transmission 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 sequential processes: i) propagation from light source to aperture

not important, generally plane wave (no diffraction) ii) multiply field distribution of illuminating wave by transmission 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∞

+−∞

= α β − α β α + β α β ∫ ∫ 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−=π

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 exactly or in a paraxial approximation (Fresnel). In the following, we will see that a further approximation for very large zB is possible, the so-called Fraunhofer approximation.


4.2 Propagation using different approximations

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

( ) ( ), ; exp ( , )H z zα β = γ α βB Bi where 2 2 2 2( ) .kγ = ω − α − β

Then we have no constraints with respect to the spatial frequencies ,α β . 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∞

+−∞= α β α β α + β α β ∫∫ B i

where [ ]( , ) ( , )U u x y+ +α β =FT

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

( ) ( )1 1exp exp2

kr x y z d dr

∞

−∞

= α + β + γ α β π γ∫ ∫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). By taking the derivative of the above equation, we get

[ ] [ ]11 1 1exp(i ) exp i( x y z) FT 22 2

kr d d H hz r

∞−

−∞

∂ = − α + β + γ α β = − = − π ∂ π π ∫

and therefore

( ) ( )1 1, , exp2

h x y z krz r

∂ = − π ∂ i with 2 2 2 .r x y z= + +

Remark: The derivative operator in the response function is a signature, that by a pure definition of the initial field, the problem of wave propagation without Fresnel approximation is not well stated. For such a problem, which is described by an elliptic partial differential equation one needs to define the field at the entire surrounding boundary or one needs to define its derivative. From the point of view of physics, this corresponds to defining the direction of field propagation and not just the value of the field in a single plane. The resulting expression in position space for the propagation of monochromatic beams is also called 'Rayleigh-formula':

( , , z ) ( , , ) ( , , )A B B Au x y z h x x y y z u x y z dx dy∞

+−∞′ ′ ′ ′ ′ ′+ = − −∫∫


Fresnel approximation (paraxial approximation) From the previous chapter we know that we can apply the Fresnel approximation if

2 2 2kα + β << which is valid for a limited angular spectrum, which corresponds to a large size of the structures inside the aperture. Then

( ) ( ) ( )2 2, ; exp exp2zH z kzk

α β = − α + β B

F B Bi i

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

z z

= − + λ F B B

B B

ii i

Paraxial Fraunhofer approximation (far field approximation) A further simplification of the beam propagation is possible for many diffraction problems. Let us assume a narrow angular spectrum

2 2 2kα + β <<

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

0.1N F with a aNz

=λF

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 ( , )exp2

u x y z h x x y y z u x y dx dy

kkz u x y x x y y dx dyz z

∞

+−∞

∞

+−∞

′ ′ ′ ′ ′ ′= − −

′ ′ ′ ′ ′ ′= − − + − λ

∫∫

∫∫

B B

BB B

F F

ii i

In this situation it is easier to treat the beam propagation in position space, by solving the above convolution integral, 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 in the above integral we need to integrate only over the aperture and not over an infinite plane:

( ) ( ) ( )2 2( , , ) exp ( , )exp2

a

a

ku x y z kz u x y x x y y dx dyz z+−

′ ′ ′ ′ ′ ′= − − + − λ ∫∫B B

B BF

ii i


Now, let us have a closer look at the integral:

( ) ( )

2 2 2 2

2

2

2 2

2

2

( , )exp2

( , )e 2xp2

( , )exp exp ex

2

p2 2

a

a

a

a

a

a

ku x y x x y y dx dyz

x xx x y yy y

x y

ku x y dx dyz

k kx ky ku x y x yz z z

yz

x

−

−

−

+

+

+

′ ′ ′ ′ ′ ′− + −

′ ′ ′ ′ = ′ ′ ′ ′− + + − +

′ ′

′ ′ = + + ′ ′+

∫∫

∫∫

∫∫

B

B

B B B B

i

i

i -i i dx dy

′ ′

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

′ ′ + < = π B B

F

Hence,

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 field far from the object, i.e. the so-called 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

∞

+−∞

+

= − + λ

′ ′ ′ ′ ′ ′× +

π = − + λ

∫∫

BB

B B

FR BB

BB B BB

ii i

-i

i i i

This is the far-field in paraxial Fraunhofer approximation. From the fact that only a single plane wave contributes to every point in space, one can easily calculate the local Poynting vector and intensity at every point. Thus, also 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 +λ

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 zα = β =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 2kα + β << smallest features ,x y∆ ∆ >> λ narrow angular spectrum

(paraxiality)

B) 2 1azN λ= <<

BF largest feature a determines 2az λ>>B far field

Example: 4, 10 , 100 , 1 10 1x y a z∆ ∆ = λ = λ λ = µ → >> λ ≈m B cm

Non-paraxial Fraunhofer approximation The concept that the angular components of the input spectrum separate in the far field due to diffraction works also beyond the paraxial approximation.

If we have arbitrary angular frequencies in our spectrum, all 2 2 2kα + β ≤ contribute to the far field distribution. Evanescent waves decay for 1kz z>> → >> λB B .

0.1N F with 0F

B B for Gaussian beams

1a a zNz z

= = λ π

( )

( )2 2 2 2 2 2

2 2 2

2 2 2 2 2 2

22( , , )

( ; )e, xp

zx y z x y z

u

x y

x y z

kx k zx y

zz y

kx

yUz

+

+ + + +

+ ++ + +

π= −

+

λ

×

FR non-paraxialB

B B

B

B

A B

B

i

i

4.3 Paraxial Fraunhofer diffraction at plane masks Let us now plug things together and investigate examples of diffraction patterns induced by plane masks in (paraxial) Fraunhofer approximation.


Let us consider an incident plane wave with a wavevector, which is inclined with respect to the optical axis z

( )( , , ) expA x y z Au x y z A k x k y k z− = + + i

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

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

Hence, the diffraction pattern is proportional to the spectrum of the field behind the mask at

.

This spectrum is calculated by the Fourier transform of the field as:

( )

( )

( )

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

+

∞

−∞

′ ′ ′ ′ ′ ′= − − − −

π

= − −

∫ ∫

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

− −

λ B

B BB

This is the absolute square of the Fourier transform of the aperture function. In paraxial approximation an inclination of the illuminating plane wave just shifts the pattern transversely.

( )

22

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

z zz +λ

B BB BB

,x yk kz z

α = β =B B


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

Fraunhofer diffraction pattern from a rectangular aperture. The central lobe of the pattern has half-angular widths /x xDθ = λ and /y yDθ = λ .

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 Fraunhofer diffraction pattern from a circular aperture produces the Airy pattern with the radius of the central disk subtending an angle 1.22 / Dθ ≈ λ .

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


0.611.22kaz z a

ρ λρ = π → =

B B

with 2 2 2x yρ = +

So-called angle of aperture: B

2 1.22z aρ λ

Θ = = (in small angle approximation)

C) One-dimensional periodic structure (grating) illuminated by normal plane wave For periodic arrangements of slits we can gain deeper insight in the structure of the diffraction pattern. Let us assume a periodic slit aperture with: period b and a size of each slit 2a :

Then, we can express the mask function t as:

1

10

( ) ( )N

n

t x t x nb−

=

= −∑ with 1( )

( )0st x x a

t x

=

for elsewhere

Here S( )t x ist he transmission function of a single elementary slit and N ist he number of combined elementary slits. The Fourier transform of the mask is then given as

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 by using the following formula for the definition of the sinc-funtion:

( ) ( )

( )1

2

0 2

sinexp

sin

N

n

Nn

− δ

δ=

− δ =∑ 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 with s ( ) 1t x = we have

1B B

sincx xT k k az z

=

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:

− Global width of diffraction pattern first zero of slit function TS

xk az

= πS

B

2zxa

λ= 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).

− Position of local maxima of diffraction pattern maxima of grid function

( )( )

B

B

22

22

sinmax

sin

k xz

k xz

N b

b

2k x b n

z= πP

B

zx nb

λ= B

p


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

− Width of local maxima zero-points of grid function

2k xN b

z= πN

B

zx

Nbλ

= BN

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

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

4.4 Remarks on Fresnel diffraction Fresnel number a aN

z=

λFB

diffraction length from Gaussian beams 2

0az π

=λ

− 10NF ( a large, zλ B small, 01/ 30z z<B ) shadow

− 0.1NF ( B 03z z> ) Fraunhofer FT of aperture

− 10 0.1N F ( 0 01 / 30 3z z z< <B ) Fresnel diffraction

Fresnel diffraction from a slit of width 2D a= . (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffraction


beam. (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers F 10,1,0.5N = and 0.1. The dashed area represents the geometrical shadow of the slit. The dashed lines at | | ( / )x D d= λ represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number

2F / 0.5N a d= λ = .


5. Fourier optics - optical filtering From previous chapters we know how to propagate the optical field through homogeneous space, and we also know the transfer function of a thin lens. Thus, we have all tools at hand to describe optical imaging. While detailed designs of high resolution optical systems have to consider non-paraxial effects, usually the paraxial approximation is sufficient to obtain a principle understanding of optical systems. Hence we use the paraxial approximation here. Many imaging systems exploit the appearance of the Fourier transform of the original object in the so-called Fourier plane of the system in order to manipulate the angular spectrum of the object in this plane. Accordingly this field of optical science is called Fourier optics. In the following we will see that with the right setup of our imaging system we can generate the Fourier transform of the object on a much shorter distance than by far field diffraction in the Fraunhofer approximation. The general idea of Fourier optics is the following: 1) An imaging system generates the Fourier transform of the object in the Fourier

plane. 2) A spatial filter (e.g. an aperture) in the Fourier plane manipulates the field. 3) Another imaging system performs the Fourier back-transform and hence results

in a manipulated image. Mathematical concept: • propagation in free space calculated in Fourier domain • interaction with lens or filter calculated in position space

5.1 Imaging of arbitrary optical fields with a thin lens

Transfer function of a thin lens A thin lens changes only the phase of the optical field, since due to its infinitesimal thickness, no diffraction occurs. By definition, it transforms a spherical wave into a plane wave. If we write down this definition in paraxial approximation we get

2 2L

plane wavespherical wave in paraxial approximation

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

i k iikf i x y t x y ikff f f

− + = − λ λ

And therefore the response function for a thin lens is given as (see chapter 9.6):

( ) ( )2 2, exp2kt x y x yf

= − +

L i (by product)

By Fourier transforming the response function we find consequently the transfer function in the Fourier domain as


( )( )

( )2 22, exp

22f fT

kλ α β = − α + β π

L i i (by convolution)

Optical imaging using the 2f-setup Let us now consider optical imaging. We place our object in the first focus of a thin lens, with a field distribution 0 ( , )u x y , and follow the usual recipe for light propagation.

A) Spectrum in object plane

[ ]0 0( , ) FT ( , )U u x yα β =

B) Propagation from object to lens (lens positioned at distance f)

F 0( , ; ) (, ,; ) )(U f UH f− α βα β = α β

( ) 02 2exp exp ( )

2( , ; ) ( , )iikf f

kU f U−

− α + βα β =

α β

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

L

02 2

2 22

( , )

exp ( ) ( )(2

( , ; ) * ( , ; )

( ,

exp

)

( )

exp ( )2

) 2

Tf fi i

kikf

i fk

U f U f

U d d

+ −

∞

−∞

α β

λ ′ ′ − α − α + β − β

′ ′− α + β

α β = α β

=

′ ′ ′ ′⋅ α

β

π

α β

∫ ∫

RED: transfer function of lens; BLUE: transfer function of free space D) Propagation from lens to image plane


( ),( , ;2 ) ( , ; );U f U fH f +α β = α βα βF

( )( ) ( ) ( )

( ) ( )

2 22

2 20

2 2

( , ;2 ) exp exp22

exp (

2

exp2

, )2

f fU f kfk

fk

f U d dk

∞

−∞

′ ′

λ ′ ′α β = − α − α + β − β ⋅

− α + β

π

′ ′ ′ ′⋅ − α + β α β α β

∫∫i i i

i i

Quadratic terms from the free space transfer function ( )2 2

2f

k′ ′ − α + β

i and

( )2 2

2f

k − α + β

i in the exponent cancel with quadratic terms from

( ) ( )2 2

2fk

′ ′α − α + β − β i and only the mixed terms remain.

( )( ) ( )

( )( )

02

02

( , ;2 ) exp 2 ( , )exp2

exp 2 ,2

f fU f kf U d dk

f f fkf uk k

∞

−∞

λ ′ ′ ′ ′ ′ ′α β = − α β − αα + ββ α β π

λ = − − α − β π

∫∫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

−

∞

−∞

= α β

λ = − − α − β α + β α β π ∫ ∫i i i

With the coordinate transformation

2 2, ,f fx y d dx d dy

k k f fπ π′ ′ ′ ′= − α = − β → α = − β = −

λ λ

we get:

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

∞

−∞

′ ′ ′ ′ ′ ′= − − + λ ∫ ∫i i i


( ) ( )

2

02

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

ff

π= −

λ

i i

The image in the second focal plane is the Fourier transform of the optical field in the object plane. This is simmilar to the far field in Fraunhofer approximation, but for z f↔B . This finding allows us to perform an optical Fourier transform over shorter distances and with adjustable size in the Fourier plane. And in the Fourier plane it is possible to manipulate the spectrum.

5.2 Optical filtering and image processing

4f-setup For image manipulation (filtering) it would be advantageous if we could perform a Fourier back-transform by means of an optical imaging setup as well. It turns out that this leads to the so-called 4f-setup:

The filtering (manipulation) happens in the focal plane (Fourier plane after 2f) by applying a transmission mask ( , )p x y . In order to retrieve the filtered image, i.e. transforming it back to real space, we use a second lens: We know that the image in the Fourier plane is the FT of the optical field in the object plane.


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

=

We have to compute the imaging with the second lens after the manipulation of the spectrum in the Fourier plane. Our final goal is to derive the transmission function ( , ;4 )H fα βA of the complete imaging system:

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

∞

−∞

− − = α β α + β α β β α ∫ ∫ A i

Note: We will see in the following calculation that the second lens does a Fourier transform ( )exp x yα + β -i . In order to obtain a proper back transform we have to pass to mirrored coordinates ,x x y y→ − → − . Furthermore, we will see that 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+

π = − λ

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

′ ′α = β =

Then we can write:

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

k k

∞

−∞

α β α β − α + β α β ∫ ∫ 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

∞

−∞

− − α β α + βα α β β∫ ∫ i

Hence we can identify the transmission function of the system

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

α β α βA

Summary • Fourier amplitudes get multiplied by transmission mask • transmission mask transfer function • coordinates of image mirrored coordinates of object In position space we can formulate propagation through a 4f-system by using the response function ( , )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 the Fourier transform of the transfer function:

( )

( ) [ ] 21( , ) , ;4 exp

2h x y H f x y d d

∞

−∞

= α β α + β α βπ ∫ ∫A A i

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

α β α βA

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

∞

−∞

α β α + β α β∫ ∫A i

We introduce the coordinate transform ,f fx yk k

= α = β

( , ) ~ ( , )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

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


The 4-f system performs a Fourier transform followed by an inverse Fourier transform, so that the image is a perfect replica of the object (perfect only within the Fresnel approximation).

1p = ( ) (y)P xδ δ 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

Spatial filtering. The transparencies in the object and Fourier planes have complex amplitude transmittances ( , )f x y and ( , )p x y . A plane wave traveling in the z direction is modulated by the object transparency, Fourier transformed by the first lens, multiplied by the transmittance of the mask in the Fourier plane and inverse Fourier transformed by the second lens. As a result, the complex amplitude in the image plane ( , )g x y is a filtered version of the original field ( , )f x y in the object plane. The system has a transfer function ( , ) ( , )x y x yH v v p fv fv= λ λ .

Transmission function:

( ) ( ) ( ) ( )2 2 21 / 2, ;4 ~

0

f fk k DH f α β + ≤α β

for

elsewhereA

− finite aperture truncates large angular frequencies (low pass)


− determines optical resolution

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

( ) ( ) ( )22 2

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

α + β ≤ → α + β ≤

With 2 2 2ρ = α + β we can define an upper limit for the angular frequencies ρmax which are transmitted (bandwidth of the system)

22

22 2

k Df

ρ =

max 2

2n D

fπ

ρ =λmax

Translated to position space, the smallest transmitted structural information is given by:

min2 2 fr

nDπ λ

∆ ≈ =ρmax

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 by the following requirement: 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∆ = π

Hence we find: min1.22 fr

nDλ

∆ =


Further examples for 4f filtering

Examples of object, mask, and filtered image for three spatial filters: (a) low-pass filter; (b) high-pass filter; (c) vertical pass filter. Black means the transmittance is zero and white means the transmittance is unity.


6. The polarization of electromagnetic waves 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 pola-rized 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= ω − ω 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, which are the 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:

− The ellipse can degrade to a line: linear polarization. − The ellipse can become a circle: circular polarization.

6.1 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= ℜ − ωE r Er i

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

( )( )

exp

exp

0

E

E

ϕ

= ϕ

Ex x

y y

i

i with ,Ex y and ,ϕx y being real

Then the real electric field vector is given as

( )( )

cos

( , ) cos

0

E t kz

t E t kz

ω − − ϕ

= ω − − ϕ

E rx x

r y y

Usually, only the relative phase δ = ϕ − ϕy x is of interest.

Conclusion Normal modes in isotropic, dispersive media are in general elliptically polarized. The field amplitudes ,E Ex y and the phase difference ( )δ = ϕ − ϕy x are free parameters


6.2 Polarization states Let us have a look at different possible parameter settings:

A) linear polarization nδ = π or 0E =x or 0yE =

B) circular polarization , / 2E E E= = δ = ±πx y

/ 2δ = +π counterclockwise rotation

/ 2δ = −π clockwise rotation

These pictures are for an observer looking contrary to the propagation direction.

C) elliptic polarization 0,E E n≠ ≠ δ ≠ πx y

0 < δ < π counterclockwise

2π < δ < π clockwise

Ex

Ey

Ex

Ey


Examples

Remark on the correspondence between linear and circular polarization basis 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 / 2δ = ±π .

( )( )

( )( )

( )cos cos cossin sin 2 0

0 0 0

t t tE t E t E

ω ω ω ω + − ω =


7. Principles of optics in crystals In this chapter we will treat light propagation in anisotropic media, which will be a much more complicated than for the isotropic media before. Like in the isotropic case we will seek for the normal modes, and in order to keep things simple we assume homogeneous anisotropic media.

7.1 Susceptibility and dielectric tensor before: isotropy (optical properties independent of direction) now: anisotropy (optical properties depend on direction) The common reason for anisotropy in many optical media (in particular crystals) is that the polarization P depends on the direction of the electric field vector. The underlying reason is that in crystals the atoms have a periodic distribution with different symmetries in different directions. Prominent examples for anisotropic materials and their applications are:

− Lithium niobate electro-optical modulators, − Quartz polarizer, − liquid crystals displays.

In order to keep things as simple as possible we make the following assumptions: − monochromacy single frequency ω , − plane wave single angular frequency, − no absorption real valued ′ε = ε

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 and their dispersion relation in an anisotropic medium look like.

Before (isotropic)

( )0( , ) ( , )χω = ωε ωP r E r

( )0( , ) ( , )εω = ωε ωD r E r

In the following we will write E E , because we assume monochromatic light and the frequency ω is just a parameter.

Now (anisotropic)

tensor compone

3

01 nts

( ( ))( , ) ,iji jj

P E=

ω = ωχ ωε ∑r r

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 ijχ 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 section).

− In complete analogy we find for the D field:

3

01

0

( , ) ( ) ( , )

( , ) ( ) ( , )

i ij jj

D E=

ω = ε ε ω ω

ω = ε ε ω

∑r r

D r E r ω

As for the polarization we find: − D E

We introduce the following notation: − ( )ˆ ij= χχ susceptibility tensor

− ( )ij= εε dielectric tensor

− ( ) ( )1ˆ ˆ ij−= = σσ ε inverse dielectric tensor

3

01

( ) ( , ) ( , )ij j ij

D E=

σ ω ω = ε ω∑ r r

The following properties of the dielectric and inverse dielectric tensor are important:

− , ij ijσ ε are real in the transparent region (omit ω ), we have no losses (see

our assumptions above) − The tensors are symmetric (hermitian), only 6 components are independent

, ij ji ij jiε = ε σ = σ .

− 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 ijσ , 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=

ε = σ λ∑

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

det 0,ij ij ij ijI I σ − λ = = δ with

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


3

( ) ( ) ( )

1

.ij j ij

D Dα α α

=

σ = λ∑

The eigenvectors are orthogonal: 3

( ) ( )

10i i

iD Dβ α

=

=∑ for ( ) ( )α βλ ≠ λ

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

ε = ε δ σ = σ δ = δε

( )( )

( )( )

1

2

3

0 00 00 0

ij

ε ω ε = ε ω ε ω

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.

7.2 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 3ε ω = ε ω = ε ω ( )0i iD E= ε ε ω

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 3ε ω = ε ω ≠ ε ω

C) biaxial − no crystallographic equivalent directions − orthorhombic, monoclinic, triclinic

( ) ( ) ( )1 2 3ε ω ≠ ε ω ≠ ε ω

7.3 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=

σ =∑

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= =

σ = ε =∑ ∑ el

In the principal coordinate system the defining equation of the index ellipsoid simplifies to

2 2 2

1 1 2 2 3 3 1x x xσ + σ + σ = and can be expressed by the three diagonal dielectric tensor components as

2 2 21 2 3

1 2 3

1x x x+ + =

ε ε ε .


This equation can be interpreted as the defining equation of an ellipsoid having semi-principal axes of length iε . From our discussion of the normal modes in isotropic

media we know that ε corresponded to the refractive index of the normal modes. We will show in the following discussion that also for anisotropic media, there will be special cases where the iε determine the phase velocity of normal modes. Hence

the elements of the dielectric tensor which define the semi-principal axes of the epsilon ellipsoid can be related to refractive indexes

i in = ε

This is the reason why the ellipsoid, which represents graphically the epsilon tensor, is called index ellipsoid.

Graphical representation of the epsilon tensor of an anisotropic crystal by the so-called index ellipsoid.

• The index ellipsoid is degenerate for special cases: − isotropic crystal: sphere − uniaxial crystal: rotational symmetric with respect to z-axis and n n=1 2

7.4 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 tω ω k r i −

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


Before – isotropic media In isotropic media the normal modes are monochromatic plane waves

( ) ( , ) expt t= ω ω E r E k ri −

with the dispersion relation

( )2

2 22( ) ( )k

cω

ω = ω = ε ωk

with ( ) 0ε ω > and real as well as 0⋅ = ⋅ =k E k D . The normal modes are elliptically polarized, and the polarization is conserved during propagation.

Now – anisotropic media What are the normal modes in anisotropic media?

Normal modes propagating in principal directions Let us first calculate the normal modes for propagation in the direction of the principal axes of the index ellipsoid, which is the simple case.

We assume without loss of generality that the principal axes are in , ,x y z direction and the light propagates in z direction ( )kk z→ . Then, the fields

are arbitrary in the x,y-plane

, 0x yD D ≠

and

0i iD E= ε εi

In general we have E D , but here 0 0⋅ = ⇒ ⋅ =k D k E , and the two polarization directions are decoupled:

( )1, expx x xD D k z tε ω i − with 2

21 2 ( )xk

cω

= ε ω


( )2, expy y yD D k z tε ω i − with 2

22 2 ( )yk

cω

= ε ω

We see that in contrast to isotropic media, normal modes can't be elliptically polarized, since the polarization direction would change during propagation. But, for linear polarization in the direction of a principal axis (x or y) only the phase changes during propagation, thus we found our normal modes:

( )

( )

2( ) 2

2

2( ) 2

2

exp normal mode a

exp normal mode b

x x x

y y y

D tc

D tc

ω = ω → = ε →

ω = ω → = ε →

D k r e k

D k r e k

aa a

bb b

i

i

−

−

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

Remark on the indices in the index ellipsoid The indices i in = ε in the index ellipsoid are connected to the indices an and bn of the normal modes propagating along the principal axis. However please be careful about the direction correspondence. For example, the two normal modes propagating along the z direction have phase velocities determined by the indices

a xn n= and b yn n= determined by the direction of their electric field rather than by the direction of their propagation.

Normal modes for arbitrary propagation direction

7.4.2.1 Geometrical construction Before we will do the mathematical derivation and actually calculate normal modes and dispersion relation, let us preview the results visualized in the index ellipsoid. Actually, it is possible to construct the normal modes geometrically. We start from the normal modes which we have determined for propagation in the principal directions of the crystal and try to generalize to arbitrary propagation directions: • For a specific crystal and a given frequency ω we take the iε in the principal axis

system and construct the index ellipsoid. • We then fix the propagation direction of the normal mode which we would like to

look at / k =k u . • We draw a plane through the origin of index ellipsoid which is perpendicular to u .


• The resulting intersection is an ellipse, the so-called index ellipse. • The half-lengths of the principal axes of this ellipse equal the refractive indices

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

ak ncω

= a and k ncω

=b b

• The directions of the principal axes of the index ellipse are the polarization directions 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= =ε ε ε ε

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∗= ℜ ×S E H

hence k is not parallel to 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.


7.4.2.2 Mathematical derivation of dispersion relation Let us now derive mathematically the dispersion relation for normal modes of the form

( ) ( )

( , ) exp

( , ) exp

t t

t t

= ω

= ω

E r E k r

D r D k r

i

i

ω −

ω −

with E and D being connected in principal axes notation by 0i i iD E= ε ε . In the isotropic case we found the dispersion relation

2

2 22( ) ( ) ( )k

cω

ω = ω = ε ωk

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, but the wavenumber depends on the direction u of propagation, where / k=u k . Hence

( ),k k= ω u

and the polarization of the normal modes is not elliptic. In the following, we start again from Maxwell’s equations and plug in the plane wave ansatz. We will use the following notation for the directional dependence of k :

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 kω = ω or 1 2 3,( , , )u uk uω = ω or 1 2 3( , , , )k k u u u= ω .

We start from Maxwell's equations for the plane wave Ansatz:

00= = ωµk D k E H ×

0= = −ωk H k H D ×

Now we follow the usual derivation of the wave equation:

( )2

20

1cω

− = εk k E D× × ( )

22

20

1cω

− =ε

k k E k E D⋅ +

− Here k E⋅ does not vanish as it would have in the isotropic case since the transversality condition from the divergence equation ( ( , ) 0div t =D r

⊥D k ) applies only to the D field. − In the principal coordinate system and with 0i i iD E= ε ε we find

2

22i j j i i i

j

k k E k E Ecω

− = ε∑ +


2

22 i i j j

jik E

ck Ek

ωε − = −

∑

Remark: for isotropic media the r.h.s. of this equation would vanish (k E = 0⋅ ). Now, we have the following problem to solve:

2

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

ω

ω

ω

ε − − ε − − = ε − −

The general way to solve this problem is using [ ]det .. 0= , which gives the dispersion relation ( )kω = ω for given /ik k . But this results in a complicated mathematical problem from which it would be difficult to derive intuition on the properties of the normal modes. However, there is an easy way to show some general properties of the dispersion relation. We start from the following trick:

2

22 i i i j j

j

k E k k Ec

ωε − = −

∑

( )2

22

ii j j

jic

kE k Ekω

= −ε − ∑

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

( )2

2

2

2.i

j j j jj i jic

kk E k Ekω

= −ε −∑ ∑ ∑

Because div 0j j jk E∑E = ≠ we can divide and get the (implicit) dispersion relation:

2

22

2

1i

ii

k

kc

= ω− ε

∑

With 1 1 1

2 2 2

3 3 3

( ) ( )k u uk k u n u

ck u u

ω = ω = ω

we can write

2 2 2

22

22

1 11

i i

ii ii

k u u

k nc

= → =ε ω −− ε

∑ ∑

22

2 1i

ii nnu

= ε−∑ final form of DR


Discussion of results

For given ( )iε ω and direction ( )1 2,u u we can compute the refractive index

( )1 2, ,n u uω seen by the normal mode. Because 2 2 21 2 3 1u u u+ + = , it is sufficient to fix

two components ( )1 2,u u of u to determine the direction.

A more explicit form of the dispersion relation can be obtained by multiplying with 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

− ε − ε + − ε − ε + − ε − ε =

− ε − ε − ε

The resulting equation is quadratic in 2 2( )n since the 6n -terms cancel. Hence, we get two (positive) solutions ,n na b and therefore ( / )k n c= ωa a and ( / )k n c= ωb 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 1u = and

1 2 0u u= = , see 6.4.1) we find:

( )( ) ( )( )( )2 2 2 2 2 21 2 1 2 3n n n n n n− ε − ε = − ε − ε − ε

( )( )2 21 2 3 0n n− ε − ε ε =

2 21 2,n n= ε = ε a b

Finally, we can derive some properties of the fields of the normal modes, i.e. the eigenfunctions. We start from the eigenvalue equation, which still involves the eigenfunctions and which we had derived above

2

22 i i i j j

j

k E k k Ec

ωε − = −

∑ .

For cases where the first factor of the l.h.s. is unequal zero (propagation directions not parallel to the principal axes) we can divide by this term

( )2

22

ii j j

jic

kE k Ekω

= −ε − ∑ .

The sum does not depend on the index i . Hence the last term of the equation must be constant

.j jj

k E =∑ const

Knowing that the last term is a constant we can derive a relation of the individual field components from the first part of the equation


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

=ω ω ωε − ε − ε −

and with 0i i iD E= ε ε

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

ε ε ε=

ω ω ωε − ε − ε −

Please be aware that this relation can only be applied for propagation directions not parallel to the principal axes. How are the normal modes polarized?

− The ratio between the field components is real phase difference 0 linear polarization

How do we see the orthogonality ( ) ( ) 0=D D ba ? (be careful: ( ) ( ) 0≠E 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

ε ω ω− ε − ε

= − ω −

ε ε ω ω− ε − ε

∑

∑ ∑

D D ba

Since the two red terms vanish due to the dispersion relation, it follows that

( ) ( ) 0=D D ba . The vanishing of the red terms can be seen when rewriting the dispersion relation:

2 22 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 kc c c

ω ω− ε + ε ω ε = = = + ω ω ω− ε − ε − ε

∑ ∑ ∑

Normal surfaces of normal modes In addition to the index ellipsoid, which is a graphical representation of the material properties of crystals from which the properties of the normal modes can be interpreted as shown above, we can derive a direct graphical representation of the dispersion relation of normal modes in crystals. This graphical representation of the dispersion relation is called normal surfaces: If we plot the refractive indices (wave number or norm of the k-vector divided by 0k ) of the normal modes in the ik -space (normal surfaces), we get a centro-symmetric, two layer surface.


Normal surfaces as the graphical representation of the dispersion relation of normal modes in crystals.

isotrop: sphere

uniaxial: 2 points with n n=a b in the poles connecting line defines the optical axis

(for 1 2ε = ε = εor , 3ε = εe the z-axis is the optical axis)

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

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 n=a 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 n=a b

isotrop

biaxial

optical axis

uniaxial

optical axis


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 .

Special case: uniaxial crystals Let us now investigate the special and simpler case of uniaxial crystals. In biaxial crystals we do not find any other effects, just the description is more complicated. The main advantage of uniaxial crystals is that we have rotational symmetry in one plane. Therefore all three-dimensional graphs (index-ellipsoid, normal surfaces) can be reduced to two dimensions, and we can sketch them more easily. As we have seen before, uniaxial crystals have trigonal, tetragonal, or hexagonal symmetry. The following classification for uniaxial crystals is commonly used or eε > ε negative uniaxial

or eε < ε positive uniaxial

Let us assume (without loss of generality) that the index ellipsoid is rotationally symmetric around the z-axis, and we have

1 2 3,ε = ε = ε ε = εor e ,

which are the ordinary and extraordinary dielectric constants. Then, we expect two normal modes:

A) ordinary wave • na independent of propagation direction

• The ordinary wave (or)D is polarized perpendicular to the z-axis and the k-vector and it does not interact with eε .

B) extraordinary wave • nb depends on propagation direction

• The extraordinary wave (e)D is polarized perpendicular to the k-vector and (or)D . The z-axis is, according to definition, the optical axis with a bn n= .

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

2

22

1i

i i

unn

= − ε

∑

For uniaxial crystals this leads to

2 2 21 2 3

22 2 2

1u u unn n n

+ + = − ε − ε − ε or or e

,


which can be expanded to

( ) 2 22 2 2 2 2 2 2 2 2 21 2 3n n n u u n n u n n − ε − ε + + − ε = − ε − ε e or or e or .

A) ordinary wave independent of direction

2n = εa or 2

2 2 202k n k

cω

= = εa a or

B) extraordinary wave dependent on direction

( ) ( )

2 2 2 21 2 2 23

1 2 32 21 , , ,b bb

u u u k n u u un c

+ ω+ = =

ε ε

e or

Hence for a given direction u 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 2k , 3k or y, z plane ( 0u =1 ). The shape of the normal surfaces can be derived in the following way. We have with

2 2 2 2

0ik k n u= i A) ordinary wave

2 2 2 2 2

1 2 3 0k k k k k= + + = εa or B) extraordinary wave

( )2 2 21 2 3

2 20 0

1 1 1k k k

k k+

+ =ε εe or

Normal surfaces for a uniaxial crystal.


If we introduce an angle θ , as in the figures above, to describe the propagation direction, a simple computation of 2 ( )bn θ for the extraordinary wave is possible:

2 3sin , cosu u= θ = θ

( )2b 2 2sin cos

n ε εθ =

ε θ + ε θe or

or e

Determination of field components 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

ε ε ε=

ω ω ωε − ε − ε −

or or e

or or e

.

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

1 0eD = , hence ( )eD 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 2 0 2 3 0 3,or eD E D E= ε ε = ε ε


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

8.1 Basics

Definition of the problem Up to now, we always treated homogeneous media. However, in the context of eva-nescent waves we already used the concept of an interface. This was a first step in the direction, which we want to pursue now. When we treated interfaces so far we never considered effects of the interface, we just fixed the incident field on an interface and described its further propagation in the half-space. In this chapter, we will go further and consider reflection and transmission properties of the following physical systems:

− interface − layer (2 subsequent interfaces) − system of layers (arbitrary number of subsequent interfaces)

To do this in a simplified way, we will switch back to isotropic media. The principle effects, which we discuss in this chapter will, however, be valid for anisotropic media as well.

Aims • We will study the interaction of monochromatic plane waves with arbitrary

multilayer systems interferometers, dielectric mirrors, … • By superposition of such plane waves we can then describe interaction of spatio-

temporal varying fields with multilayer systems. • We will see a new effect, the “trapping” of light in systems of layers new types

of normal modes in inhomogeneous space “guided” waves (propagation of confined light beams without diffraction).

Approach • take Maxwell's transition condition for interfaces • calculate field in inhomogeneous media matrix method • solve reflection-transmission problem for interface, layer, and system of layers, • apply the method to consider special cases like Fabry-Perot-interferometer, 1D

photonic crystals, waveguide…

Background • Due to the orthogonality of normal modes of homogeneous space there is no

interaction of normal modes in homogeneous space.


• The inhomogeneity breaks this orthogonality and modes interact, i.e. exchange energy.

• However, locally the concept of eigenmodes is still very useful and we will see that for the considered inhomogeneity of planar surfaces the interaction at the inhomogeneities is limited to a small number of modes.

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

2

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

cω

ωω − ω = µ ω + µ ωωrotrot E r E r j r P ri

In general, for isotropic media all field components are coupled due to the rot rot operator. However, for problems with translational invariance in at least one direction (homogeneous infinite media, layers or interfaces) a simplification is possible. Let us assume, e.g. translational invariance of the system in y direction and propagation in the x-z-plane / 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 with

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. Common notations are:

perpendicular: ⊥ s TE (transversal electric) parallel: p TM (transversal magnetic)

Both components are decoupled and can be treated independently:

2(2) 2

0 02

2(2) (2) (2) 2

0 02

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

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

c

c

⊥ ⊥ ⊥⊥

ωω + ω = − ωµ ω − µ ω ω

ωω + ω − = − ωµ ω − µ ω ω

E r E r j r P r

E r E r grad div E j r P r

∆

∆

i

i

From


0

( , ) ( , )ω = − ωωµ

H r rot E ri

we can conclude that the corresponding magnetic fields are

0 0, 0

0 0

HE E

H

= = =

E Hx

TE y TE

z

0 00 ,

0 0

EH H

E

= = =

E Hx

TM TM y

z

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

( )( , ) expt t= − ω E r E kri

− homogeneous space implies: ( )exp kri − monochromaticity leads to: ( )exp t− ωi

Now, we will break the homogenity in x-direction by considering an interface in y-z – plane which is 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). Then the problem does not depend on the y -coordinate.

Transition conditions From Maxwell’s equations follow transition conditions for the field components. Here we will use that Et, Ht (transverse components) are continuous at an interface between two media since they are tangential to the interface. This implies for the:

x

z

k

medium I

medium II

interface y


A) Continuity of fields − TE: yE E= and zH continuous

− TM: zE and yH H= continuous

− As pointed out before, we can always split the fields in TE and TM polarization +E E ETE TM= and treat them separately.

B) Continuity of wave vectors − We have homogeneity in z -direction, and therefore we expect solutions to be

( )exp k z zi . The wave vector component kz has to be continuous at the interface (follows strictly from continuity of transverse field components above).

homogeneous in z-direction phase e zik z zk continuous

Therefore, we can write for the electric field:

( ) ( ) ( ) ( )( , , ) exp expTE TMx z t k z t k tx zx= − ω + − ω E E Ez zi i

8.2 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 Let us first compute the fields in one homogeneous layer of thickness d and with dielectric function ( )fε ω • aim: for given fields at 0x = calculate fields at x d= • strategy:

− Do computation with transverse field components (because they are continuous).

− The normal components can be calculated later. We will assume monochromatic light (one Fourier component, ( , ; )E x z ω ( , ; )H x z ω ) and in the following we will often omit the argument/parameter ω in the notation.

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

∂ ∂ ω+ + ε =

∂ ∂ E

We use the ansatz from above:

( )z( , ) ( )expTE TEx z x ik z=E E and ( )z( , ) ( )expTE TEx z x ik z=H H


2 2

2f z2 2 ( ) 0TEk x

x c ∂ ω

+ ε − = ∂

E

with: TE TE0

( , ) ( , )ix z x z= −ωµ

H rotE

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

( )2

2fx z2 , ( ) 0k k E x

x ∂

+ ω = ∂

with ( ) ( )2

2 2fx z f z2,k k k

cω

ω = ε ω −

z0

( ) ( )iH x E xx

∂= −

ωµ ∂

This makes sense since the wave equation for the y-component of the electric field is a second order differential equation. Hence we need to specify the field and its first derivative as initial condition at 0x = to determine a unique solution.

TM-polarization analog for transversal components y z,H H E= :

( )2

2fx z2 , ( ) 0k k H x

x ∂

+ ω = ∂

z0 f

( ) ( )iE x H xx

∂=

ωε ε ∂

Again, we succeed describing everything in transversal components. Now we have the following problem to solve for TE and TM:

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

∂ ∂∂ ∂

at x d= for

given values at 0x = − calculate also the other field components at x d= − at the end: HTM ETM E = ETM + ETE

Generalized transverse fields F & G 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 Gωµ − ωε → generalized field 2

and write down the generalized problem, which we have to solve:


( )2

2fx z2 , ( ) 0k k F x

x ∂

+ ω = ∂

,

f( ) ( )G x F xx

∂= α

∂ with fTE fTM

f

11,α = α =ε

.

We know the general solution of this system since the problem is similar to an harmonic oscillator equation:

( ) ( )

( ) ( )

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

= + −

∂= α = α − − ∂

We have as initial conditions (0), (0)F G given:

[ ]

1 2

f fx 1 2

(0)(0)

F C CG i k C C

= +

= α −

from which we can compute the constants 1 2,C C :

1

f fx

2f fx

1 (0) (0)2

1 (0) (0)2

iC F Gk

iC F Gk

= − α

= + α

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

= +α

= −α +

By resubstituting to the physical E and H fields, we have the electromagnetic field in the layer 0 x d≤ ≤ .

Fields in a system of layers In the previous subchapter we have seen how to compute the electromagnetic field in a single dielectric layer, dependent on the transverse field components yE , zH (TE) and yH , zE (TM) at 0x = . 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 Based on the analytic solutions we can even treat layer systems with layers of very different thicknesses, since the amount of computation is independent from the layer's thickness. We can even go further and “discretize” an arbitrary inhomogeneous (in one dimension) ε distribution. This is important for the modelling of so-called 'GRIN' – Graded-Index Profiles.

A continuously varying ε distribution can be described by its discretization as a system of multiple layers each having a homogeneous ε .

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

= +α

= −α +

We can write this formally in matrix notation as

( ) (0)

ˆ ( ) ,( ) (0)

F x Fx

G x G

=

m

where the 2x2-matrix m describes propagation of the fields:

( )( ) ( )

( ) ( )

1cos sinˆ

sin cos

k x k xkx

k k x k x

α= − α

m fx fxfx f

fx f fx fx

− To compute the fields 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 idε .

If multiple layers are considered, the fields between them connect continuously since the field components used for the description of the fields are the continuous tangential components.


Hence, we can directly write the formalism for a multilayer system since it just requires matrix multiplications:

A) Two layers

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

ˆˆ ( )N

i iid d d D

F F Fd

G G G=+ + + =

= =

∏m M

N

with 1

ˆ ˆ ( )N

i ii

d=

= ∏M m

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

2, , .i if i id k k

cω

α = ε ω − 2zfx

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

− , , ,if i ik dα εz given matrix elements

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

8.3 Reflection – transmission problem for layer systems

General layer systems

8.3.1.1 Reflection- and transmission coefficients generalized Fresnel formulas In the previous chapter, we have learned how to link the electromagnetic field on one side of an arbitrary multilayer system with the field on the other side. We have seen that after splitting in the TE/TM-polarizations, continuous (transversal) field components are sufficient to describe the whole field. What we will do now is to link those field components with the fields, which are accessible in an experimental configuration, i.e. incident, reflected, and transmitted fields. In particular, we want to solve the reflection-transmission problem, which means that we have to compute reflected and transmitted fields for a given angle of incidence, frequency, layer system and polarization.


We introduce the wave vectors of the incident ( )kI , reflected ( )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 cω ω

= ε − = ω − = ε − = ω −ssx z s z cx c z c z

where ( )ε ωs and ( )ε ωc are the dielectric functions of the substrate and cladding and zk is the tangential component of the wave vector, which is continuous through-out the layer system. As we have seen before, the zk component of the wave vector is conserved and k± x determines the direction of the wave (forward or backward). The total length of the wave vector in each layer is given by the dispersion relation for dispersive, isotropic, homogeneous media. As a consequence, the xk 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 kϕ = ϕ ϕ = ϕs I s R I R (reflection)

2. sin sin sin sink k n nϕ = ϕ ϕ = ϕs I c T s I c T (Snellius)

It should be noted that these formulas are valid independently from the specific layer system separating the two semiinfinite half-spaces, i.e. the substrate region and the cladding region. These formulas are derived just from the contin-uity of the transverse wavevector component kz .

εc

x

z

Multi layer system

εs


Let us now connect the propagating waves (incident, reflected, transmitted) to the local fields at the interfaces in order to solve the reflection transmission problem:

A) Field in substrate The fields in the substrate s ( , )F x z and s ( , )G x z can be expressed based on the complex amplitudes of the incident FI and reflected field FR as:

B) Field in layer system The fields inside the layer system can be expressed as

where the amplitudes ( )F x and ( )G x are given by matrix method as

C) Field in cladding The fields in the cladding c ( , )F x z and c ( , )G x z can be expressed based on the complex amplitude of the transmitted field FT as:

( ) ( )

( ) ( )( , ) exp exp

( , ) exp exp

F x z k z F k x D

G x z k k z F k x D

= − = α −

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. Hence, we exclude any reflection at inhomogeneities after the boundary from the layer system to the cladding.

Reflection-transmission problem The aim is to compute FR and FT for given FI , kz ( sin ), ,i idϕ ε I . We know that F and G are continuous at the interfaces, in particular at 0x = and x D= . We have:

0

ˆ ( ) .D

F FD

G G

=

M

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

( ) ( ) ( )( ) ( ) ( )

( , ) 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

= + − = α − −

s z I sx R sx

s s sx z I sx R sx

i i i

i i i i

( )( )

( , ) exp ( )

( , ) exp ( )

F x z k z F x

G x z k z G x

=

=f z

f z

i

i

0

ˆ ( )x

F Fx

G G

=

M


On the other hand, we have expressions for the fields at 0x = 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 D M D

k F Fk F M D M D+

= α −α

I RT

s sx I Rc cx T ii

We consider IF as known, and RF and TF as unknown and solve for RF and TF :

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

N

k M k M M k k MF F

k M k M M k k Mα − α − + α α

=α + α + − α α

s sx c cx s sx c cxR I

s sx c cx s sx c cx

i

i

11 22 12 21

22 11 21 12

2 ( )( ) ( )

s sxT I

s sx c cx s sx c cx

k M M M MF Fk M k M i M k k M

α −=

α + α + − α α

2 s sx

T IkF F

Nα

=

These are the general formulas for reflected and transmitted amplitudes. Please remember that the matrix elements depend on the polarization direction

.ij ijM M≠TE TM

Let us now transform back to the physical fields, and write the solution for the results of the reflection-transmission problem for TE and TM polarization:

A) TE-polarization

, 1F E E= = α =y TE

i) reflected field

E R E=TE TER TE I

with the reflection coefficient

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

N

k M k M M k k MR

k M k M M k k M− − +

=+ + −

TE

TE TE TE TEsx cx sx cx

TE TE TE TE TEsx cx sx cx

i

i

ii) transmitted field

E T E=TE TET TE I

with the transmission coefficient

( ) ( )22 11 21 12

2 2 ,k kTNk M k M M k k M

= =+ + −

sx sxTE TE TE TE TE

TEsx cx sx cxi

We get complex coefficients for reflection and transmission, which determine the amplitude and phase of the reflected and transmitted light.


B) TM-polarization

1, .F H H= = α =εy TM

In the case of TM polarization we have the problem that an analog calculation to TE would lead to /H HR,T I , i.e., relations between the magnetic field. However, we

want /E ETM TMR,T I . Therefore, we have to convert the H -field to the ETM -field:

As can be seen in the figure, we can express the amplitude of the TME field in terms of the xE component:

sin

,

E kE k

kE Ek

= − ϕ = −

= −

x zTM

TMx

z

With Maxwell we can link xE to yH :

( )0

1×

ωε εE k H= −

0

1E k H=ωε εx z y

0 0

1kE H Hc

= − = −ωε ε ε ε

TMy y

result: , ,

,

,E HE H

ε=

ε

TMR T R TsTMI s c I

→ /ε εs c relevant for transmission only

Hence we find the following for TM polarization:

E R E=TM TMR TM I

with the reflection coefficient

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

N

k M k M M k k MR

k M k M M k k Mε − ε − ε ε +

=ε + ε + ε ε −

TM

TM TM TM TMc sx s cx s c sx cx

TM TM TM TM TMc sx s cx s c sx cx

i

i

E T E=TM TMT TM I

with the transmission coefficient


( ) ( )22 11 21 12

2 2k kT

Nk M k M M k k Mε ε ε ε

= =ε + ε + ε ε −

s c sx s c sxTM TM TM TM TM

TMc sx s cx s c sx cxi

In summary, we have found different complex coefficients for reflection and transmission for TE and TM polarization. The resulting generalized Fresnel formulas for multilayer systems are

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

k M k M M k k MR


=+ + −

TE TE TE TEsx cx sx cx

TE TE TE TE TEsx cx sx cx

i

i

2kTN

= sxTE

TE

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

k M k M M k k MR


=ε + ε + ε ε −

TM TM TM TMc sx s cx s c sx cx

TM TM TM TM TMc sx s cx s c sx cx

i

i

2

.k

TNε ε

= s c sxTM

TM

8.3.1.2 Reflectivity and transmissivity In the previous chapter we have computed the coefficients of reflection and transmission, which relate the electric fields in TE and TM polarization of incident, reflected and transmitted wave. However, in many situations it is more important to know the relation of energy fluxes, the so-called reflectivity and transmissivity. In order to get information on these quantities we have to compute the energy flux perpendicular to the interface:

− flux through a surface with x = const For a monochromatic plane wave follows

( )12

∗= ℜS e E H ex x×

With ( )0

1∗ ∗ ∗= ×ωµ

H k E

we find

( ) ( )2 2*

0 0

1 1 .2 2 xk∗= ℜ = ℜωµ ωµ

S e k e E Ex x

Since in an absorption free medium the energy flux is conserved, in an absorption free layer system the energy flux is also conserved. In the substrate

2

2 2 22 ( )k k k k

cω

= ε − = ω −ssx z s z

x z


is supposed to be real-valued, because we have our incident wave coming from there. The total energy flux from the substrate to the layer system is given as

2 2

0

12

k k = − ωµS e E Ex sx I sx Rs

In contrast, in the cladding

22 2 2

2 ( )k k k kcω

= ε − = ω −cx c z c z

may be complex-valued. The energy flux from the layer system into the cladding is

( ) 2

0

1 .2

k= ℜωµ

S e Ex cx Tc

Because we have energy conservation:

=S e S ex xs c

( )2 2 2kk

ℜ= +E E Ecx

I R Tsx

Now we will compute the global reflectivity ρ and transmissivity τ of a layer system. Of course, we will decompose into TE and TM polarizations and relate to the reflectivities ρTE,TM and transmissivities τTE,TM . We know:

( ) ( )

( ) ( )

2 22 2

22 22

2

2 2 .

,

kR T

k

k

kR T

k

k

= + = +

ℜ= + + +

= ℜ

+

ℜ

+ +

E E

E E E E E E

E E

E

E

E

TE TER T

cx TETE TE

TM TMR T

cx

TE

TMTM T

TM TE TMR R R T T T

cxI

s

I Ms

Isx

x

x

Here, we just substituted the reflected and transmitted field amplitudes by incident amplitudes times Fresnel coefficients. Now, we decompose the incident field as follows:

cos , sin .E E= δ = δE ETE TMI I I I

Then, we can divide by the (arbitrary) amplitude 2EI and write


( ) ( )

( ) ( ) ( )

2 2

2 22

2 22 2

2 22 22

1 cos

cos sic

si

in

n

os s1 n

R R

R R

k kT T

k k

kT T

kτ

ρ

= + δ + + δ

δ + δ

ℜ ℜ

ℜδ + δ

= +

cx cxTE TM

sx sx

cxTE TM

sx

TE TM

TE TM

The red and blue terms can be identified as 1 = ρ + τ

The global reflectivity and transmissivity are therefore given as

2 2

2 2

cos sincos sin

ρ = ρ δ + ρ δ

τ = τ δ + τ δTE TM

TE TM

with the reflectivities

( )2 2

, , , ,, .k

R Tk

ℜρ = τ = cxTE TM TE TM TE TM TE TM

sx

for the two polarization states TE and TM.

Single interface

8.3.2.1 (classical) Fresnel formulas Let us now consider the important example of the most simple layer system, namely the single interface. The relevant wave vectors are (as usual):

0 , 0 , 0 .k k k

k k k

− = = =

k k ksx sx cx

I R T

z z z

The continuous component of the wave vector, expressed in terms of the angel of incidence, is

.sin sink nc cω ω

= ε ϕ = ϕz s I s I

cladding

substrate


Then, the discontinuous component is given as

2 2 2

2 2 2 2 22 2 2 sin sini i i ik k n n

c c c cω ω ω ω

= ε − = ε − ε ϕ = − ϕx z s I s I

2 2 2cos , sin cos ,ck n k n n nc c cω ω ω

= ϕ = − ϕ = ϕsx s I cx c s I T

As above, we can assume that ksx is always real, because otherwise we have no incident wave. kcx is real for sinn n> ϕc s I , but imaginary for sinn n< ϕc s I (total internal reflection). The matrix for a single interface is the unit matrix

1 0ˆ ˆ ( 0)0 1

d = = =

M m

and it is easy to compute coefficients for reflection and transmission, and reflectivity and transmissivity. Using the formulas from above we find:

A) TE-polarization

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

N

k M k M M k k MR


=+ + −

TE

sx cx sx cxTE

sx cx sx cx

i

i, 2kT

N= sx

TETE

with: 1 0ˆ ˆ ( 0)0 1

d = = =

M m

( )( )

( )

2 2 2

2 2 2

2 2 2

cos sin cos coscos coscos sin

2 2 cos 2 coscos coscos sin

k k n n n n nRk k n nn n n

k n nTk k n nn n n

− ϕ − − ϕ ϕ − ϕ= = =

+ ϕ + ϕϕ + − ϕ

ϕ ϕ= = =

+ ϕ + ϕϕ + − ϕ

sx cx s I c s I s I c TTE

sx cx s I c Ts I c s I

sx s I s ITE

sx cx s I c Ts I c s I

( ) ( )

22

2

22

4.

k kR

k k

k k kT

k k k

−ρ = =

+

ℜ ℜτ = =

+

sx cxTE TE

sx cx

cx sx cxTE TE

sx sx cx

1ρ + τ =TE TE


B) TM-Polarisation

( ) ( )( ) ( )

22 11 21 12

22 11 21 12

N

k M k M M k k MR


=ε + ε + ε ε −

TM

c sx s cx s c sx cxTM

c sx s cx s c sx cx

i

i

2.

kT

Nε ε

= s c sxTM

TM

with: 1 0ˆ ˆ ( 0)0 1

d = = =

M m

( )( )

( )

2 2 2 2 2

2 2 2 2 2

2

2 2 2 2 2

cos sin cos coscos coscos sin

2 2 cos 2 cos ,cos coscos sin

k k n n n n n n nRk k n nn n n n n

k n n nTk k n nn n n n n

− ϕ − − ϕ ϕ − ϕ= = =

+ ϕε εε ε

ε ε

+ ϕϕ + − ϕ

ϕ ϕ= = =

+ ϕ +ϕε ϕϕ −ε +

sx cx s c I s c s I c I s TTM

sx cx c I s Ts c I s c s I

sx s c I s ITM

c s

c s

c s

csx cx c I s Tss c I s c s I

( ) ( )

22

2

22

,

4

k kR

k k

k k kT

k k k

ε − ερ = =

ε + ε

ℜ ℜ ε ετ = =

ε + ε

sx c cx sTM TM

sx c cx s

cx sx cx s cTM TM

sx sx c cx s

1ρ + τ = TM TM

Remark It may seem that we have a problem for 0ϕ =I . For 0ϕ =I , TE and TM

polarization should be equivalent, because the fields are always polarized parallel to the interface. However, formally we have R R= −TE TM , .T T=TE TM The “strange” behavior of the coefficient of reflection can be explained by the following figures:

8.3.2.2 Total internal reflection (TIR) for εs>εc Let us now consider the special case when all incident light is reflected from the interface. This means that the reflectivity is unity.

2

2

k kk k

−ρ =

+cxsx

E

sx cx

T 2

2

k kk k

ε − ερ =

ε + εsx c scx

csx c x

TM

s


With 2 2 2sin 0k n ncω

= − ϕ =cx c s I we can compute the smallest angle of incidence

with 1ρ =TE,TM :

0 sink n n= = ϕcx c s Itot

sin .nn

ϕ = cItot

s

For angles of incidence larger than this limit angle, ϕ > ϕI Itot we have

2

2 2 2 22sin zk n n k

c cω ω

= ϕ − = µ = − ε imaginarycx s I c c ci i i

→ ( ) 0kℜ =cx TIR

Obviously, we find the same angle of TIR for TE and TM polarization. The energy fluxes are given as (here TE, same result for TM):

( )2

2 2

41 0.

k kkk k k

ℜ− µρ = = τ = =

+ µ +sx cxsx c

TE TE

sx c sx cx

i

i

Remark For metals in visible range (below the plasma frequency) we have always TIR,

because: ( ) 0ℜ ε <c → 2 2sink ncω

= ε − ϕccx s I always imaginary.

In the case of TIR the modulus of the coefficient of reflection is one, but the coefficient itself is complex nontrivial phase shift for reflected light:

A) TE-polarization

( ) ( )( ) ( )

2 2 2 2 2

exp1 exp exp 2

exp

sin sin sintan tan .

2 cos cos

ZRk Z

nn

k

kn

∗

α−= ⋅ Θ = = = = α

+ µ − α

Θ ϕ − ϕ − ϕα = = − = − = −

µϕ

µ

ϕ

TE

TE

TEsx

sx c

c

s

c

s I c I Itot

s Ix I

iii i

i i

B) TM-polarization

( ) ( )

( ) ( )exp1 exp exp 2

exp

tan tan tan ,2 2

ZR

k

kk

Z ∗

α−= ⋅ Θ = = = = α

ε + µ ε −ε µ

α

Θ Θεα = = −

ε

µε

=ε

ε

TM

TM TE

TMsx c c s

c s

sx c

sx c c s

s

c

iii i

i i

In conclusion, we have seen that the phase shifts of the reflected light at TIR is different for TE and TM polarization, and because s cε > ε


,Θ > ΘTM TE

As a consequence, incident linearly polarized light gets generally elliptically polarized after TIR Fresnel prism

Remark

The field in the cladding is evanescent ( ) ( )exp exp .k x x= −µ xc ci

The averaged energy flux in the cladding normal to the interface vanishes.

( ) ( )2 2

0 0 0

1 1 0.2 2x

k=

= ℜ = ℜ =ωµ ωµ

S k E E

xx

8.3.2.3 The Brewster angle There exists another special angle with particular reflection properties. For TM-polarization, for incident light at the Brewster angle ϕB we find 0R =TM :

2

2 0,kk k

kε − ε

ρ = =ε + ε

TM

sx c cx s

sx c cx s

k kε = εsx c cx s

( ) ( )2 2 2 2sin sinε ε − ϕ ε = ε ε − ϕ εc s B s s c B s

( )

( ) ( )2

2 2sinε ε ε − ε ε

ϕ = =ε + εε ε − ε

s c s c cB

s cs s c

( ) ( )2 2cos 1 sin 1 ε εϕ = − ϕ = − =

ε + ε ε + εc s

B Bs c s c

With the last two lines we can write the final result for the Brewster angle:

tan .εϕ =

εc

Bs

The Brewster angle exists only for TM polarization, but for any .n ns c

There is a simple physical interpretation, why there is no reflection at the interface for the Brewster angle.


sintancos

sin cos sin2

nn

n n n

ϕϕ = =

ϕ

π ϕ = ϕ = − ϕ

B cB

B s

s B c B c B

At the same time the angle of the transmitted light is always

sin sin ,2

n n πϕ = ϕ ϕ = − ϕs B c T T B

Hence, at Brewster angle reflected and transmitted wave propagate in perpendicular directions. If we interpret the reflected light as an emission from oscillating dipoles in the cladding, no reflected wave can occur for TM polarization (no radiation in the direction of dipole oscillation). In summary, we have the following results for reflectivity and transmissivity at a single interface with s cε > ε .


8.3.2.4 The Goos-Hänchen-Shift The Goos-Hänchen shift is a direct consequence of the nontrivial phase shift of the reflected light at TIR. It appears when beams undergo total internal reflection at an interface. The reflected beam appears to be shifted along the interface. As a result it seems as if the beam penetrates the cladding and reflection occurs at a plane parallel to the interface at a certain depth, the so-called penetration depth. For sake of simplicity we will treat here TE-polarization only.

Let us start with an incident plane wave in TE polarization:

( ) ( )( , ) exp ( , ) expE x z E k x k z E x z E z x = + → = α + γ I I sx z I I si i

total internal reflection

angle of incidence

total internal reflection

angle of incidence


with

transverse wave number: S Isinncω

α = ϕ

longitudinal wave number: 2

2 2S S2 n

cω

γ = − α

This gives rise to a reflected plane wave as

( ) ( )( , ) exp expE x z E z x = α − γ Θ αR I si i .

The reflected plane wave gets a phase shift ( )Θ α , which depends on the angle of incidence (here characterized by the transverse wave number α). Now we want to treat beams, which we can write as a superposition of plane waves with Fourier amplitude I ( )e α :

( ) ( )( )( , ) expE x z d e z x = α α α + γ α ∫I I si

We assume a mean angle of incidence :

0α = sinc nω ϕs I0 mean angular frequency

0α = α + ε

In the Fourier integral, we have to integrate over angular frequencies with non-zero amplitudes ( )e αI only ( ( )0 0e α + ε ≠I for −∆ ≤ ε ≤ ∆ only)

[ ]

[ ] [ ]

I 0 I 0 S

R 0 I 0 S 0 0

( , ) exp( ) ( )exp ( )

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

E x z i z e i z x d

E x z i z e i z x R i d

∆

−∆

∆

−∆

= α α + ε ε + γ ε

= α α + ε ε − γ α + ε Θ α + ε ε

∫

∫

Let us make the following further assumptions:

− small divergence of beam (narrow spectrum, ncω

∆ s )

− all Fourier component undergo TIR ( ( ) ItotΘ α > Θ ) totα > α

0( ) 1R α + ε = in the equation above

Then, it is justified to expand the phase shift ( )Θ α of the reflected wave into a Taylor series up to first order:

( ) ( ) ( )

0

0 0 0α

∂Θ ′Θ α + ε ≈ Θ α + ε = Θ α + Θ ε∂α

Then, the reflected beam at the interface at 0x = is given as:

( ) ( ) ( )00 0(0, ) exp expE z z d e z∆

−∆ = α + Θ α ′ε α + ε + Θ ε ∫R I ii

ϕI0


We can identify the remaining integral as a shifted version of the incident beam profile at 0x =

( ) ( )0 0(0, ) exp .E z E z′ ′ = Θ α − α Θ + Θ R Ii

Thus, the reflected beam appears shifted by d ′= −Θ (Goos-Hänchen Shift).

Let us finally compute the shift d ′= −Θ . We know from before that the phase shift for TIR is given as:

tan2 k

Θ= −−

µ µ=

γTE c c

sx s

2

2

2

2

2 2

2 22arctan 2arctan cc

sx c

nk n

ω

ω

α −µΘ = − = −

− α

c

s

( )2 2

2 2 2 2

2

2 21 2 2 12 2 2

1

sx csx c

c sx c sx

c sx sx c c sx

sx

kk

k kk k k

k

α + µα − α− µ∂Θ µ µ α′Θ = = − × × = − = −

µ∂α + µ µ+

02 tand x

α′Θ = − = − ϕET I0 with

2

22 20

1 1

c

xnω

= =µ α −

ETc c

and tansx sk

α αϕ = =

γI0

xET depth of penetration

Periodic multi-layer systems – Bragg mirrors – 1D photonic crystals In the previous chapters we have learned how to treat (finite) arbitrary multi-layer systems. In addition, interesting effects occur when these multi-layer systems be-come semi-infinite periodic stacks of layers, so-called Bragg mirrors. The reflectivity of such mirrors is almost 100 % in certain frequency ranges; the more layers the closer we get to this ideal value. Bragg mirrors are important for building resonators (lasers, interferometers). Furthermore periodic structures are of general importance in physics (lattices, crystals, atomic chains …). We can learn many things about the general features of such periodic systems by looking at the optical properties of periodic (dielectric) multi-layer systems In our theoretical approach, we will assume these layer systems as infinite, i.e. consis-ting of an infinite number of layers, and we treat them as so-called one-dimensional


photonic crystals. We will discuss effects like band gaps, dispersion, and diffraction in such periodic media and gain understanding of the basics of Bragg reflection and the physics of photonic crystals. In order to keep things simple we will treat:

− semi-infinite periodic multi-layer systems with [ ]1 1 2 2 1 1 2 20,( , ),( , ),( , ),( , ),x d d d d> ε ε ε ε

− placed on a substrate [ ]S0,x < ε

− TE-polarization only − monochromatic light

At the interface between substrate and Bragg-mirror ( )0x = we have the incident and the reflected electric field:

0 R IE E E= + and ( )0 sx0

I RE E ik E Ex

∂ ′= = −∂

Alternatively the incident and reflected field can be expressed by the field at the interface and its derivative as

0 0I

sx2 2E iEE

k′

= − and .

In chapter 8.2, we developed a matrix formalism involving the generalized fields F and G . Because here we treat TE polarization only, we can use directly the electric field amplitude E and its derivative E′ with respect to x , because

and 0 zEi H G Ex

∂ ′ωµ = = =∂

.

Let us now calculate the field in the multi-layer system. From before, we know how to treat finite systems with the matrix method. Here, we want to treat an infinite periodic medium (like a one-dimensional crystal). As a particular example we will investigate an infinite system consisting of just two periodically repeated layers. The two layers should consist of homogeneous materials with 1ε and 2ε having a thickness of 1d and 2d , respectively. Hence we have:

( ) ( )x xε = ε + Λ with the period 1 2d dΛ = +

For infinite periodic media, we can make use of the so-called Bloch theorem to find the generalized normal modes (Bloch modes or Bloch waves). We seek for solutions like:

( ) ,( , ; ) exp ( )kE x z x k zk k E xω = + ωxzx zi

with

( ) ( )k kE x E x+ Λ =x x

0 0R

sx2 2E iEE

k′

= +

E F=


In other words, ( )kE xx

is a periodic function and we are looking for solutions ( , ; )E x z ω , which have the same periodicity. In detail, the solutions should have the

same amplitude after one period of the layer system, but they can have a different phase. This pahse difference cannot assume arbitrary phase values but must obey

( ) ,exp k k x ω x zi . Here kx is the Bloch vector, which is yet unknown. Because in this easy example we deal with a one-dimensional problem, the Bloch vector is actually a scalar. In the following, we will find a dispersion relation for the Bloch-waves ( ),k k ωx z , in complete analogy to the DR for plane waves 2

22 2( )x zc

k kω= ε ω − in homogeneous media. In order to make the difference to the homogeneous case more obvious, we change the notation for the Bloch vector to:

xk K .

According to the Bloch-theorem (our ansatz) we have a relation for E and E′ when we advance by one period of the multi-layer system (from period N to period 1N + ):

( )

( )' 'exp .E E

KE E

+ Λ Λ

= Λ

N 1 N

i

On the other hand, we know from our matrix method, that the fields at both sides of the same part of the layer system are connected by:

( )

' 'ˆE E

E E+ Λ Λ

=

M

N 1 N

with ( ) ( )2 1ˆ ˆ ˆd d=M m m (2) (1)

kij ik kjM m m= ∑ .

If the Bloch wave is a solution to our problem, we can set the two expressions equal:

ˆ ˆ exp( ) 0N

EiK

EΛµ

− Λ = ′

M I

.

And with ( )exp Kµ = Λi we can formulate the following eigenvalue problem:

'ˆ ( ) 0

EK

EΛ

µ =

M I−

N

.

This eigenvalue problem determines the Bloch vector K and will finally give our dispersion relation. As usual, we use the solvability condition ˆdet 0µ =M I− to compute the dispersion relation expressed for µ . Hence we still need to compute K afterwards from

, as

( ) ( ) ( ) 211 22 11 22exp 1.

2 2M M M M

K± ±

+ + µ = Λ = ± −

i

( )exp Kµ = Λi


Note that we assumed non-absorbing media 1 2( 0, 0)′′ ′′ε = ε = for which ˆdet 1=M . This explains why the off-diagonal elements of the matrix do not appear in the formula. Moreover, because of ˆdet 1=M we have 1+ −µ µ = . The corresponding eigenvectors (field and its derivative at x N= Λ ) can be computed from

( ) 'ˆ exp 0

EK

EΛ

Λ =

M I

N

i−

11 12

21 22

0N

M M EM M E

Λ

− µ = ′− µ

.

From the first row the following condition can be derived:

( )11 12 0M E M E′− µ + =

Since the investigated system is linear and invariant for the phase, the absolute amplitude and phased of the E -component of the eigenvector can be chosen arbitrarily. Here we take 1E = and get for the full eigenvector

( )'

11 12

1.

/E

EM ME Λ

Λ

= µ −

N

N

If field values of the Bloch mode, i.e. the function ( ) ( )k kE x E x+ Λ =x x

, inside the layers are desired, they can be computed by using the matrix formalism and the above eigenvector ( , )NE E Λ′ .

Physical properties of infinite multilayer systems We are interested in the reflection properties of an infinite Bragg mirror. Based on the electric field and derivative at the interface ( 0x = ), E0 and E’0, we can express the reflectivity of the Bragg mirror as

2

EE

ρ = R

I

with 0 0R

sx2 2E iEE

k′

= + and 0 0I

sx2 2E iEE

k′

= − from before

2'

0 0'

0 0

k E Ek E E

ρ = sx

sx

+i-i

With our knowledge of the eigenvector from above we can compute the reflectivity:

110 0

12

' ME EM

µ −=

2112'

0 0 12'

110 0

12

Mkk E E M

Mk E E kM

µ −

ρ = =µ −

sxsx

sxsx

+i+i-i -i

According to this formula two scenarios are possible:


A) total internal reflection ρ = 1 Hence µ has to be real, which results in the condition

( )11 22 12

M M+≥

with [for our example 1 2 1 2( , , , )d dε ε ]

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

211 1 1 2 2 1 1 2 2

1

122 1 1 2 2 1 1 2 2

2

cos cos sin sin

cos cos sin sin .

kM k d k d k d k dkkM k d k d k d k dk

= −

= −

xx x x x

x

xx x x x

x

This defines the so-called band gap, i.e. frequencies of excitation for which no propagating solutions exist.

B) propagating normal modes Hence µ must be complex, which results in the condition

( )11 22 12

M M+<

We can compute a more explicit version of the dispersion relation by starting from

( ) ( ) ( ) 211 22 11 22exp 1.

2 2M M M M

K+ +

µ = Λ = ± −

i

and splitting the exponential into

( ) ( ) ( ) ( ) ( ) 2exp cos sin cos cos 1K K K K Kµ = Λ = Λ + Λ = Λ ± Λ −i i .

By comparing the two expressions we get

( ) ( )11 22cos ,2

M MK k

+ω Λ =z .

Only if the Bloch vector K fulfills this DR the Bloch wave is a solution to Maxwell’s equations in infinite periodic media. This is in complete analogy to plane waves in homogeneous media with the DR ( )2

22 2.

ck kω= ε ω −x z

Interpretation

• For the case of total internal reflection (µ real, ( )11 22 12

M M+≥ ) the Bloch vector

K is complex, ( ) ( )( ) ( )( )exp exp expK K Kµ = Λ = ℜ Λ −ℑ Λi i .

Hence ( ) ,K k nℜ ω Λ = πz and


( ) ( ) ( ) ( ) 211 22 11 22, ln 1 1 .

2 2n M M M M

K k + + ℑ ω Λ = − − ± −

z

The ± accounts for exponentially damped and growing solution, as we usually expect in the case of complex wave vectors and evanescent waves.

• There is an infinite number of so-called band gaps or forbidden bands, because 1...n = ∞ . These band gaps are interesting for Bragg mirrors and Bragg waveguides.

The band gaps correspond to "forbidden" frequency ranges, where no propagating solution exists.

• The limits of the bands are given by

( ) ,K k nℜ ω Λ = πz and ( ) , 0K kℑ ω Λ =z

( ), /K k nω = π Λz

• Outside the band gaps, i.e., inside the bands, we find propagating solutions, which have different properties than the normal modes in homogeneous media (different dispersion relation). We can exploit the strong curvature, i.e. frequency dependence, of DR for, e.g., dispersion compensation or diffraction free propagation

Special case: normal incidence In general there is a complex interplay between the angle of incidence and frequency of light determining the reflection properties of multilayer systems. Therefore let us have a look at the simpler case of normal incidence ( )0k =z . In a graphical representation of the dispersion relation for 0k =z it is common to use the following dimensionless quantities

cGω

and KG

with the scaling constant 2G π

=Λ

Examples for normal incidence


It is common to use the reduced band structure, where the information for all possible Bloch vectors is mapped onto the Bloch vectors in the following interval

0.5 ( / ) 0.5k G− ≤ ≤ , which is the so-called Brillouin zone.

Because of the periodicity in the spatial frequency space eiKΛ it is sufficient to describe the dispersion relation in one period. Thus we need only K−π ≤ Λ ≤ π

0.5KG

≤ to describe the entire dispersion relation for all spatial frequencies.

Inside the band gap, we find damped solutions:

n1=1.4, d1=0.5Λ n2=3.4, d2=0.5Λ

0 0.25 0.5 0.75 1 1.25k/G

0

0.1

0.2

0.3

0.4ω

/cG

Re kIm k

0 0.25 0.5 0.75 1 1.25 1.5k/G

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω/c

G

Re kIm k

n1=1.4, d1 = 17/24 Λ n2=3.4 , d2 =7/24 Λ

n1=1.4, d1=0.5Λ n2=3.4, d2=0.5Λ

0 0.25 0.5 0.75 1 1.25k/G

0

0.1

0.2

0.3

0.4

ω/c

G

Re kIm k

0 0.25 0.5

k/G

0

0.2

0.4

0.6

0.8

1

1.2

ω/c

G


field in a Bragg-mirror Transmission

Let us quantify the damping. In our example (n1,n2,d1,d2) we have

( )11 22 2 11 1 2 2 1 1 2 2

1 2

1cos cos sin sin2 2

M M n nn d n d n d n dc c n n c c

+ ω ω ω ω = − +

In the middle of the first band gap (optimum configuration for high reflection) we

have 1 1 2 2 2n d n d

c cω ω π

= =B B , with Bω being the Bragg frequency, and therefore

( )11 22 2 1

1 2

1 12 2

M M n nn n

+ = − + < −

. If we plug this (for the first gap 1n = ) in our

expression for ( )Kℑ and assume a small index contrast ( )2 1 2 1n n n n− << + we find

2 1max

2 1

( ) 2 n nKn n

−Λℑ ≈

+ (do derivation as an exercise)

Damping is proportional to the index contrast of the subsequent layers 2 1n n−

The spectral width of the gap ( )11 22 1

2M M +

≥

is then

max2 ( )Kω

∆ω ≈ ΛℑπB

gap (do derivation as an exercise)

Spectral width is proportional to index contrast as well.

Fabry-Perot-resonators In this chapter we will treat a special multi-layer system, the so-called Fabry-Perot-resonator. To construct a Fabry-Perot-resonator, one can start from a highly reflecting periodic multi-layer system (Bragg reflector). If one changes just a single layer somewhere in the middle of the otherwise periodic layer system, a so-called cavity is formed, and we are interested in the forward and backward propagating fields of the entire layer system.

0.3 0.4 0.5 0.6 0.7ω/cG

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tran

smiss

ion

2 1n n−

n0=1.0, n

1=1.4, n

2=1.6, d

1/d

2=1


While the single layer, which is distinguished from the periodic stack, forms the cavity, the other layers function as mirrors, and may be periodic multilayer-systems or metal films. Fabry-Perot-resonators are very important in optics, as they appear as: • Fabry-Perot- interferometer • laser with plane mirrors Fabry-Perot-Resonator with active medium inside the

cavity • nonlinear optics high intensities inside the cavity nonlinear optical effects for

low intensity incident light: − bistability − modulational instability − pattern formation, solitons

Here, we want to compute the transmission properties of the resonator for arbitrary plane mirrors. This task could be achieved employing the matrix method which we developed in the previous sections. However we will take a different approach to achieve deeper physical insight into the cavity's behavior. For simplicity, we will restrict ourselves to TE-polarization. The figure shows our setup with two mirrors at 0x = and x D= , characterized by coefficients of reflection and transmission 0R , 0T , DR , and DT .

− ,E EI R and ,ET amplitudes of incident, reflected and transmitted external

fields in substrate and cladding − ,E E+ − amplitudes of internal fields running forward and backward inside

the cavity Using the known coefficients of reflection and transmission of the two mirrors, we can eliminate E+ and E− by connecting the field amplitudes:

A) At the lower mirror inside the cavity:

0 0 (0) (0)ET EE R − ++ =I

cladding

substrate


B) At the upper mirror outside the cavity:

( )E T E D+=T D

And with ( )(0)( ) expE D k DE++ = fxi ( )(0) expE k DT

E+ = −Tfx

D

i

C) At the upper mirror inside the cavity:

( ) ( )DE D R E D− +=

with T( )D

EE DT+ = from B)

and with ( )(0) ( )expE D DE k−− = fxi

( )e p(0) xR E DT

E k− = DT fx

D

i

D) we substitute (0)E+ and (0)E− in A)

( ) ( )0 0 exp expR ET E R E k D k DT T

+ = −D TI T fx fx

D D

i i

( ) ( ) 00

1 exp exp .E k D R R k D ET T

= − −I fx D fx TD

i i

Thus, the coefficient of transmission for the whole FP-resonator expressed by the coefficients of the mirrors ( 0R , 0T , DR , and DT ), the cavity properties and the angle

of incidence ( D , 2

22 f zk k

cω

= ε −fx ) reads:

( )

( )0

0

exp.

1 exp 2T T k DER

TE R k D

= =−

D fxTTE

I D fx

ii

This is the general transmission function of a lossless Fabry-Perot resonator. In general, the mirror coefficients are complex and the fields get certain phase shifts

( ),0, 0, ,( , ) ( , ) expR T R T= ϕR TD D 0 Di . Obviously, mainly the phase shifts induced by the

coefficients of reflection 0 ,R RD are important for the transmissivity of the FP resonator

2Tτ TE TE . For given , ,R R0 D ,T T0 D and 0 , Dϕ ϕR R , the general transmissivity of a lossless Fabry-Perot resonator reads:

2 22 2

2 20

2

1 2 cos(2 )

.

f Dx

T TT T

R R R R k D

k Tk

δ

= =+ − + +

τ =

ϕ ϕ

R R0 D

TE

0 D 0 D

cx

sx


Here we introduced the phase-shift δ , which the field acquires in one round-trip in the cavity.

Discussion Depending on whether the two mirrors have identical properties we distinguish between symmetric and asymmetric FP-resonators.

a) asymmetric FP-resonator

2 2

2 21 2 cosT Tk

k R R R Rτ =

+ − δ0 Dcx

sx 0 D 0 D

Because we assume no losses we can use energy conservation at each mirror to eliminate T0,D :

( ) ( )

( )( )

2 22 2

2 2

1 1

1 1

fx

fx

kkT T R Rk kk R Rk

= − −

= − −

sx0 D 0 D

cx

sx0 D

cx

Note: τ and ρ for a lossless mirror are the same for both sides of the mirror. For lossy mirrors only τ is the same, ρ is then side-dependent.

For discussing the effect of the phase shift δ we rewrite

2 2 2cos cos sin 1 2sin2 2 2δ δ δ

δ = − = −

Plugging everything in we get

( )( )( )

( )( )( ) ( )( )

2 2

2 22 2

2 2 22

12

2

1 2 2sin1

4sin .

2

1

1

1

1 1 1 1

R

R R

R R R

R R

R R R

R

R

R

δ

−

− −

−

τ =+ −

− − −

−

δ = +

−

0

0 D

0 D 0 D

00 D

D 0 D

D

and with

2 2

0 ,R Rρ = ρ =0 D D

( )

( )( ) ( )( )

12

0 20

0 0

1 4sin .

1 1 1 1 2

− − ρ ρ ρ ρ δ τ = + − ρ − ρ − ρ − ρ

D D

D D


b) symmetric FP-resonator (Airy-formula for transmissivity)

2 2

0 0, DR Rρ = = = ρ = ρ ϕ = ϕ = ϕR R Rm 0 D D

( )

( )( )( ) ( )

1

2 22

2 22 2

1

1 1 4 sin21 11 4 sin

2

−

=

− ρ − ρ ρ δ τ = = + δ − ρ − ρ − ρ + ρ

m m

m mm m

1

21 sin2

F−δ τ = +

Airy-formula

with ( )2

41

F ρ=

− ρm

m

and 2 fxk Dδ

= + ϕ

The Airy-formula (see also Labworks script, where phase shifts ϕ due to the mirrors are not considered) gives the transmissivity of a symmetric, lossless Fabry-Perot-resonator. Only for this case we can get the maximum transmissivity 1τ = for

/ 2 nδ = π .

Remarks and conclusions • We can do an analog calculation for TM-polarization RTM resp. ρTM

• Resonances of the cavity with 1τ =MAX occur for / 2 fxk D mδ = + ϕ = πMAX with

( )

2 2 2

2 2 2

2 sin

2 sin

2

k n n

mmDk n n

ϕπ

π= − ϕ

λ−π − ϕ λ

= =− ϕ

λ

cavity

fx F S I

MAXfx F S I

where Iϕ is the angle of incidence in the substrate


• transmission properties of a given resonator depend on ϕI and .λ

• minimum transmission is given as 1

1 Fτ =

+MIN

• it is favorable to have large F , e.g.:

( )

( )

2

2 2

41001

4 1 41 100 0.2 0.8.

F ρ= =

− ρ

− τρ = − τ ≈ ≈ τ = ρ =

τ τ

m

m

mm m m m

m m

• pulses and beams can be treated efficiently in Fourier domain: e.g. TE: ( , , ) ( , , ) ( , , )E T Eα β ω = α β ω α β ωT TE I Fourier back transformation: [ ]1( , , ) FT ( , , )E x y t E−= α β ωT T

• beams, because they always contain a certain range of incident angles ϕI , they produce interference rings in the farfield output (or image of lens, like in labworks).

• a quantity often used to characterize a resonator is the finesse:

∆

Φ = =ε

distance between resonancefull width at half maximum of resonance

with ε the FWHM and ∆ = π the distance in rad between two resonances. To calculate the FWHM ε we can start from:

1

2 11 sin2 2

F m−

ε + π ±

For narrow resonances (small line width ε ) we can write

12 11

2 2F

− ε + ≈

2 212

FF

ε = ε =

2 1F

ρ∆ πΦ = = = π

ε − ρm

m

• The line width ε (FWHM) is inversely proportional to the finesse Φ . • The Airy-formula can be expressed in terms of the finesse

12

221 sin .2

− Φ δ τ = + π

• A Fabry-Perot-resonator can be used as a spectroscope. Then, we can ask for its resolution (here: normal incidence). Resonances (maximum transmission) occur at:


kD m+ ϕ = π

reduced transmission by factor 1/2

2

2 2 22

k

D

kD D m m

k nD

n D D

→ + ϕ ± = π ± = π ±

π π∆ = ∆λ =

λ Φ

∆λ λ λ= λ

∆Φ

εΦΦ

π

λ

ε

f

f

example: 75 10−λ = ⋅ m , 330, 4 10n D −Φ = = ⋅ mf 122 10−∆λ = ⋅ m

• The field amplitudes (here forward field) inside the cavity are given as:

( )E T E D+=T D Because 2 1 / (1 )T − − ρ Φ these intra-cavity fields can be very

high important for nonlinear effects • Lifetime of photons in cavity: Via the "uncertainty relation": 1T∆ω = ≈const.c it is

possible to define a lifetime of photons inside the cavity:

1

1 .

ncc nTn D

kD

Dc

D

= ∆ω =

π∆ω = = =

π∆

ΦΦ

εΦ ∆ω π

F

Fc

F

8.4 Guided waves in layer systems Finally, we want to explore our layer systems as waveguides. For many applications it is interesting to have waves, which propagate without diffraction. This is crucial for integrated optics, where we want to guide light in very small (micrometric or smaller) dielectric layers (film, fiber), or optical communication technology, where some light encoded information is transported over long distances. Moreover, waveguides are important in nonlinear optics, due to the strong confinement of light over long propagation distances (since there is no diffractive spreading as in free space) nonlinear effects become important. Here, we will treat waveguiding in one dimension only because we restrict ourselves to layer systems, but the general concepts developed here can be transferred to other settings with two dimensional confinement, e.g. where the waveguide is a fiber.

Field structure of guided waves Let us first do some general consideration about the field structure of guided waves. We want to find guided waves in a layer system. In such systems, till now, we have solved the reflection and transmission problem: For given ,, , i ik d εEz I we calculated

,E ER T .


Inside each layer we have plane waves

( ) ( ), expk k x k z tα αω + − ω E z x zi .

The question is, how can we trap (or guide) waves within a finite layer system? A possible hint gives the effect of total internal reflection (TIR), where the transmitted field at the interface of the multi-layer system to the cladding is:

( ) ( )( , ) exp expx z k z x= −µE ET T z ci .

Obviously, in the case of TIR we have no energy flux in the cladding medium. If TIR is the key mechanism to guide light, we can have TIR on two sides, i.e. to the cladding and to the substrate. And maybe we can also have a similar mechanism at a single interface.

We will concentrate our discussion first on a system of layers, which we will call the core of the waveguide, and which we place between a semi-infinite substrate and cladding. The single interface, where the core is absent, will be considered at a later stage. According to our discussion the guided waves have the following field structure: • plane wave in propagation direction ~ exp( )zik z

• evanescent waves in substrate and cladding

c~ exp[ ( )]x D−µ − cladding

s~ exp( )µ x substrate

with 2

2, ,2 ( ) 0k

cω

µ = − ε ω >s c z s c

1. condition: 2

2,2 max ( )k

cω

> ε ωz s c

• oscillating solution (standing wave) in the core (layers, fiber core, film)

~ sin( ) cos( )fx fxA k x B k x+

with 2

22 ( ) 0i ik k

cω

= ε ω − >x z


2. condition: 2

22 max ( )ii

kcω

< ε ωz

Note that this 2. condition is not obvious at this stage. It appears due to transition conditions at the boundaries to substrate and cladding. The fact that the fields in the substrate and gladding are of evanescent nature requires that both fields are exponentially decreasing away from the interface to the core. The transition conditions at the interfaces impose continuity of the tangential components of the fields and their first derivative normal to the surface. This fact was shown during the derivation of the transmission reflection properties in a multilayer system. Since the field in the core must continuously connect the fields and derivatives of the fields in substrate and cladding, it must have some non-monotonic profile. This condition excludes evanescent field profiles in the core and leaves only oscillating radiation-type fields as a possible solution in the core of a waveguide. For waveguide cores consisting of multiple layers, the field has to be of radiating nature in at least one of these core layers. In summary, the z -component of the wave vector of guided waves has to fulfill:

,max maxs c z iin k n

c cω ω < <

In summary, the field structure in substrate and cladding is given as:

( ) ( )( ) ( )

( , ) exp exp

( , ) exp exp 0

x z k z x D x D

x z k z x x

= −µ − > = µ <R

E E

E EC T z c

S z s

i

i

and in the core as

f f( , ) exp( )exp( ) D x 0i i z xix z ik z ik x= ≥ ≥E E

Dispersion relation for guided waves If we compare the principal shapes of the fields for the guided wave to our usual reflection transmission problem, we see that for guided waves the reflected and transmitted (evanescent) field exist for zero incident field . In the following we will use this condition to derive the dispersion relation for guided waves. Thus we have for I 0E → . Please remember that is true even though there is no energy transport connected to these fields because of their evanescent nature. Consequently the coefficients for reflection and transmission are

with I 0E →

0→IE

, 0E E ≠T R , 0E E ≠T R

,E ET RE E

= =

TE,TM TE,TMTE,TM TE,TMT R

I I


Therefore we must have in the case of guided waves. In this sense, guided waves can be considered as resonances of the multi-layer system. To appreciate this concept, let us compare to a driven harmonic oscillator

, x = action, F = cause

In the case of resonance ( 0ω = ω ) we get an action for an infinitesimal cause. Hence, we can get the dispersion relation of guided waves by looking for a vanishing denominator in the expressions for This reasoning is a general principle in physics: The poles of the response function (or Greens function) are the resonances of the system. Furthermore, a resonance of a system is equivalent to an eigenmode of the system, which is localized in our case. We know the coefficients of reflection and transmission for a layer system from before, and they have the same denominator:

The pole is then given as:

Because we have evanescent waves in the substrate and the cladding the imaginary valued sxk and cxk can be expressed as real numbers with

Now we can write the general dispersion relation of guided modes in an arbitrary layer system as

Here, as usual, we have TE 1α = , TM 1α = ε for the two independent polarizations.

This waveguide dispersion relation gives a discrete set of solutions, which are the so-called waveguide modes. For given we get

We can see that the dispersion relation of guided modes depends on the material's dispersion in the individual layers as well as in the substrate and the cladding according to the dispersion relation of homogeneous space as

.

,R T → ∞

2 20

Fx =ω − ω

,R T

( ) ( )( ) ( )

22 11 21

22 11 21 12

12

k M k M Mk M k M M k k MFR

k k MFα − α −α + α + − α α

+ α α= = s sx c cx s sx c cx

s sx c cx s sx cI cx

R

i

i

( ) ( )22 11 21 12 0k M k M M k k Mα + α + − α α s sx c cx s sx c cxi

2 22 2

2 2( ), ( )k i i k k i i kc cω ω

= µ = − ε ω = µ = − ε ωsx s z s cx c z c

11 12 21 221 0M M M Mµ

+ µ +α

αµα

+α µ

TE,TM TE,TM TE,TM TE,TMs ss

c cs

c c

, ,i idε ω ( ).k ν ωz

22 2 2

2( ) ( )i i ik k kcω

ω = ε ω = +x z


In addition to the material's dispersion in the layers the dispersion relation of guided modes depends on the geometry of the waveguide's layer system

In the case of guided waves it is easy to compute the field (mode profile) inside the layer system: • take from dispersion relation

• in the substrate we have:

Hence, for given (0)F we get ( free parameter)

Analogy of optics to the stationary Schrödinger equation in QM Optics (e.g. TE-polarization) Quantum mechanics

guided waves discrete energy eigenvalues

Tunnel effect

( ),k ωz geometry

zk

( ) exp( ),sF x F µ x= ( ) ( ) exp sG x F µ xx

∂= α

∂

s(0) (0)G F= αµ (0)F →

0

1ˆ ˆ( ) ( ) (0).x

F Fx x F

G G

= = αµ M M

s

222

22 )( ( ) ( )d E x E xx

kxcd

=

+

ωε z ↔

2

2 222 ( ( ) ( )) 2m Ex xVd

dx

xm

ψ = ψ −

−

↔

2

2,2 maxk

cω

> εz s c ↔ E V< ext

22

2kcω

> εFilmz ↔ E V< Barriere


Guided waves at interface - surface polariton Let us first have a look at the most simple case where the guiding layer structure is just an interface.

Our condition for guided waves is that on both sides of the interface we have

evanescent waves: , because

The general dispersion relation we derived before reads

and with the matrix for a single interface:

we get the dispersion relation

A) TE-polarization (

→ no solution because

B) TM-polarization (

with

→ on of the media has to have negative (dielectric near resonance or metal)

22

,2kcω

> εz c s

22

s,c ,2 0kcω

µ = − ε >z c s

11 22 21 211 0M MM Mα µ

+ α µ + +α µ α µ

TE,TM TE,TTE,TM TE,TM Ms ss s

c c c c

110ˆ

0

=

M

1 0α µ+ =

α µs s

c c

1)α =

0,µ + µ =s c , 0µ µ >c s

1/ )α = ε

0µ µ+ =

ε εc s

c s

, 0 ,0ε ⋅ ε <µ µ >c s c s

ε


In dielectrics we can find surface-phonon-polaritons.

In metals we can find surface-plasmon-polaritons.

Remark Surface polaritons occur in TM polarization only, similar to the phenomenon of

Brewster-angle (no reflection for )

Let us now compute the explicit dispersion relation for surface polaritons. W.o.l.g. we assume , and because is near a resonance it will show a much stronger dependence than .

There is a second condition for existence of surface polaritons:

Conclusion:

TM polarization

( )ω < ω < ω0 T L

ω < ωp

0,k k− =

ε εcx sx

c s

s ( ) 0ε ω < s ( )ε ωω cε

( ) ( )

( ) ( )

( ) ( )( )

2 2

2 22 2 2 2 2

z c c z s2 2

s c

c s

c s s c

s

z

µ µ

k kc c

kc

ε = ε

ω ωε ω − ε = ε − ε ω

ε ω εωω =

ε + ε ω

( ) 0ε + ε ω <c s

( ) ( )( )

kc

ε ω εωω =

ε ω + εs c

zs c

( ) 0ε ω <s 0ε >c ( )ε ω > εs c


Surface polaritons may have very small effective wavelengths in z-direction:

Guided waves in a layer – film waveguide The prototype of a waveguide is the film waveguide, where the waveguide consists of

one guiding layer with .

Such film waveguides are the basis of integrated optics. Typical parameters are:

( ) ( ) ( )( )

2 21, 1 kεπ π

εω ε

ε ω= ε ω ≈ → ω

+= = → λ << λ

λ λε effeff

s c

s cc s z

( ) ( )2

22 f k

cω

ε ω > ωz


a few wavelengths

3 1f s 10 10− −∆ε = ε − ε ≈ −

Fabrication of film waveguides can be achieved by coating, diffusion or ion implantation. The matrix of a single layer (film) is given as:

From this matrix we can compute the dispersion relation for guided modes:

Here: TE-Polarisation

This waveguide dispersion relation is an implicit equation for . For given frequency and thickness d we get several solutions with index

Here is an example for fixed frequency , the effective index versus

the thickness d :

d ≈

( ) ( )( ) ( )

1cos sinˆ ˆ ( )sin cos

kk d k dd

k k d k dα

= = −α M mTE,TM TE,TM f fxfx fx

f fx fx fx

( ) ( ) ( ) ( )

( )( ) ( ) ( )

11 12 21 22

2 2

1 0

cos sin sin cos 0

1sin

tancos

M M M M

kk d k d k d k dk

kk dk d kk d k

k

α µ+ α µ + +

α µ α µ

α µ α α µ+ − + =

α α µ α µ

α µ+α α µ + α µα µ

= = =α α µ α − α α µ µ−α µ α

s ss s

c c c c

s s f fx s sfx fx fx fx

f fx c c c c

s s

f fx s s c cfx c cfx

f fx s sfx f fx c s c s

c c f fx

( ) 2

2

tan

k

k dk

µ µ+ α α α =µ µ−

α α α

fx s c

f c sfx

fx c s

c s f

( )1α =

( ) ( )2tan ,

kk d

kµ + µ

=− µ µ

fx s cfx

fx c s

kzω k νz

ω eff z /n kcω =


We can see in the figure that for large thickness d we have many modes. If we decrease d, more and more modes vanish at a certain cut-off thickness. Definition of cut-off: a guided mode vanishes cut-off (here w.o.l.g.

The idea of the cut-off is that a mode is not guided anymore. Guiding means evanescent fields in the substrate and cladding, so cut-off means

no guiding

We can plug this cut-off condition in the DR:

with parameter of asymmetry a:

we can define a cut-off frequency for when we keep d fix

we can define a cut-off thickness for when we keep fix

)ε < εc s

22

s s2 0zkcω

µ = − ε = 2

2s2zk

cω

= ε

( ) ( )2tan ,

kk d

kµ + µ

=− µ µ

fx s cfx

fx c s

f s s c s cf s

f s f s

tan dc

ε − ε ε − εω ε − ε ε − ε = = ε − ε ε − ε

( )TE s cco

f sf

arctans

cd v ε − ε ω = + π ε − εε − ε

( )TE

comax /2f s

arctancd a vπ

→ ω = + π

ε − ε

s c

s f

s c

0aa

ε ≈ ε →

ε ≈ ε → ∞

ε ≈ ε

( )zk ω

( )zk d ω


In a symmetric waveguide the fundamental more ( ) has cut-off = 0! If we plot the dispersion curves for each mode we get a graphical representation of the dispersion relation:

Excitation of guided waves Finally, we want to address the question how we can excite guided waves. In principle, there are two possibilities, we can adapt the field profile or the wave vector

A) adaption of field front face coupling

Then, inside the waveguide we have (without radiative modes):

mode ν couples to the incident field with amplitude .

Gauss-beam couples very good to the fundamental mode B) adaption of wave vector coupling through the interface

0ν =

( )kz

( )

2

0

0

( , ) ( )exp

( ,0) ( ) ( )

: ( )2

,2

( ) ( )

E x z a E x k z

E x a E x E x

kP

E x

E x dx

ka dxP

E x

ν ν νν

ν ν µν

∞ν

ν ν−∞

∞ν

ν −ν

ν∞

=

≈

=ωµ

=ωµ

∑

∑ ∫

∫

∫

z

z

inz

i

with

(0)Ein aν

f


We know that is continuous at interface. The condition for the existence of guided modes is

but dispersion relation for waves in bulk media dictates

We got a problem! There are two solutions:

i) coupling by prism we bring a medium with (prism) near the waveguide

.

light can couple to the waveguide via optical tunneling: ATR ('attenuated total reflection')

ii) coupling by grating

zk

,ck ω

> εcz s

, ,

22,2k k

c cω

<εω

ε= −z c s s cx c s

ε > εp f

z fkcω

< ε z pkcω

< ε 2

2px p z2 0k k

cω

= ε − >


grating on waveguide (modulated thickness of layer d):

coupling works for m’th diffraction order:

( ) ( )

( ) ( ) 2sin P period

d z d z

z A gz mit gP −

= + ς

πς = =

s sin .

zv zk k mg

n mgc

= +

ω= φ +


9. Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling)

The topic of “Ray optics – geometrical optics” is not covered in the course “Fundamentals of modern optics”. This topic will be covered rather by the course “Introduction to optical modeling”. The following part of the script, which is devoted to this topic, is just included in the script for consistency.

9.1 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 ( 0λ → , 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.

9.2 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 dsδ =∫ r

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

9.3 Simple rules for propagation of light A) Homogeneous media

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

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

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

− The angle of refraction θ2 depends on the angle of incidence θ1 and is given

by Snell’s law:

1 1 2 2sin sinn nθ = θ

− no information about amplitude ratio.

9.4 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 nθ = θ

for paraxial rays:

− 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 2π

θ = → 2

1

sin sin nn

θ = θ =1 TIR

1 1 2 2n nθ = θ


C) Spherical interface (paraxial)

− paraxial imaging

1 2 12 1

2 2

n n n yn n R

−θ ≈ θ − (*)

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 lens (paraxial)


− two spherical interfaces (R1, R2, ∆) apply (*) two times and assume y=const (∆

small)

2 1yf

θ ≈ θ − 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)

9.5 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 Starting point: we minimize the optical path or the delay (Fermat)

( ) 0B

A

n dsδ =∫ r

computation:

( )

B

A

L n s ds= ∫ r


variation of the path: ( ) ( )s s+ δr 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

δ = δ + δ

δ = ⋅ δ

δ = + δ −

= + ⋅ δ + δ −

δ≈ + ⋅ −

δ ≈ + ⋅ −

δ= ⋅

∫ ∫

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

δ δ = ⋅ δ + ⋅

= − ⋅ δ

∫

∫

r rr

r r integration by parts and A,B fix

0Lδ = 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 = − α ⇒ ≈ −

α for 1

( ) ( ) ( ) ( )2 2

2 21 ( )d dy d dy d y d y dn yn y n y n y

ds ds dz dz dz dz n y dy ≈ ≈ ⇒ =

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

22

d y ydz

= −α

00

0 0

( ) cos sin

( ) sin cos

y z z z

dy

y

z y z zdz

= α + αα

θ = = − α α + θ α

θ

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

( ) ( )2 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


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

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

Ray-transfer-matrix

in paraxial approximation:

2 1 1

2 1 1

y Ay B

Cy D

= + θ

θ = + θ

2 1

2 1

y yA B A BC D C D

= → = θ θ

M

A=0: same 1θ same y2 focusing

D=0: same 1y same 2θ collimation

Matrices of optical elements A) free space

B) refraction on planar interface

10 1

d =

M


C) refraction on spherical interface

D) thin lens

E) reflection on planar mirror

F) reflection on spherical mirror (compare to lens)

Cascaded elements

1 1

1 1

N

N

y yA B A BC D C D

+

+

= → = θ θ

M M=MN….M2M

1 2

1 00 n n

=

M

( )2 1 2 1 2

1 0n n n R n n

= − −

M

1 01 1f

= −

M

1 00 1

=

M

1 02 1R

=

M

fundamentals of modern optics - uni-jena.de · 2020. 2. 2. · script "fundamentals of modern...

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