Repetition lecture for

Electromagnetic Optics, Polarization Optics, and

layered media

Cord Arnold / Anne L’Huillier

Electromagnetic optics

Maxwell Equations in a source free medium

MHB

PEDrrr

rrr

00

0

µµ +=

+=∈Flux

densities

Polarization

densities

Electric vacuum permitivity

Magnetic permeability

( )E

ED

EP

r

rr

rr

=∈

+=∈

=∈

χ

χ

10

0

Linear, homogeneous, and isotropic

Susceptibility

Electric permitivity

χ

χ

+=∈∈

=

∈∈

=+

1

1

0

0

n

Dielectric function

Refractive index

2

0

2

22

22 1

,0c

n

ct

EE =∈==

∂∂

∈−∇ µµr

r

Different types of media

• Linear: If P(r,t) is linearly related to Ԑ(r,t).

• Nondispersive: The response is instantaneous. The

polarization P(r,t) does not depend on

earlier times.

• Homogeneous: The relation between P and Ԑ is no

function of space.

• Isotropic: The relation between P and Ԑ is

independent of the direction of Ԑ.

• Spatially nondispersive: The relation between P and Ԑ is local.

Transverse electromagnetic (TEM) plane wave

E is orthogonal to H. Both

are orthogonal to the

direction of propagation k.

( ) ( ) ( ){ }( ) ( ) ( ){ }tjrHtrH

tjrEtrE

ω

ω

expRe,

expRe,

0

0

rrrr

rrrr

=

=Monochromatic EM-waves

( ) nncc

kEkE00

22 2,0

λπωω

ω ====+∇⇒rr

0000

0000

,,

,,

EkHHEk

HkEEHkrrrrrr

rrrrrr

⊥⇒−=×

⊥⇒∈=×

ωµ

ω

2

000

2

0

00

0

0

2

1

2

1

2

1Enc

EHEIS

H

E

∈==×==

∈==

η

µη

rrr

Impedance

Intensity

Vectorial spherical wave

Transmission bands for common materials in optics

'''12

'''

0 χχα

β

χχχ

jkjk

j

++=−=

+=

( )zUU α−= exp2

0

2

Absorption => attenuation

Implications of dispersion

The resonant medium

Damping

Driving force

Restoring force

The resonant medium

( )νννν

νχνχ

∆+−=

j22

0

2

00

OffsetResonance frequency

Width of the resonance

Multi resonance media

Dispersive media

Group velocity

Phase

velocity

( )( ) ( ) N

c

nn

cvg

0

000

00

'=

−=

λλλλ

Group index

Dispersive media

( )

( )

( )00

0

02

0

3

0

02

0

3

0

''

''

''2

λλ

λλ

λπλ

λ

ν

ω

nc

D

nc

D

nc

D

−=

=

=0,0

0,0

><

<>

λω

λω

DD

DD Normal dispersion

Anomalous dispersion

zD

zD

λκτ

νντ

σσ

σσ

=

=

Polarization of Light

One component in x-direction

zx

y

Components in x- and y-direction

zx

y

Components in x- and y-direction

zx

y

Circular polarization

zx

y

Elliptic polarization

zx

y

Visualization of the polarization state

Jones vectors

( )

xy

x

y

x

yyx

yxyxyx

y

x

A

A

a

aR

AAI

jaAA

AJ

ϕϕϕη

ϕ

−===+

=

=

=

,,2

exp,

22

,,,

r

Wave plates

λ/4 plate λ/2 plate

Reflection and refraction at a boundary

TE – Transverse Electric – orthogonal – (s)enkrecht

TM – Transverse Magnetic – parallel – (p)arallel

Boundary conditions

Fresnel equations (nonabsorbing media)

Overview

TE

TM

External Internal

Power reflectance

Application <-> polarization by reflection

Application <-> polarization filters in photography

Source: http://www.bhphotovideo.com

Isotropic <-> anisotropic

The refractive index ellipsoid

Biaxial: All refractive indices are different n1≠

n2≠n3

Uniaxial: Two refractive indices are identical,

n1= n2=no (ordinary refractive index)

and n3= ne (extraordinary refractive

index)

ne>no => positive uniaxial

ne<no => negative uniaxial

The z-axis of a uniaxial crystal is the

optic axis.

Isotropic: The index ellepsoid is a sphere.

Double refraction

Wave retarders

Uniaxial crystals: n1=n2=no, n3=ne

Optic axis

onon

en

θθ

4

λ2

λ4

3λ λ

Example:

λ/2-plate

( ) πππ

2,,2

=−=Γ dkk oe

Variable attenuator

Faraday (optical) isolator

Dielectric layered media

A photonic crystal is a periodic arrangement

of a dielectric material that exhibits strong

interaction with light

Variation of refractive index on the scale of the wavelength λ

Photonic Crystals; J.D. Joannopoulos

Definition

How to analyze a multilayer system

Scattering MatrixWave-Transfer Matrix

Si � Mi MΣ � S Σ

Analysis of multilayer optical system

Conversion

Cascaded system - Airy formulas

ϕ

ϕ

ϕ

ϕ

ϕ

j

j

j

j

err

etrtrr

ndkerr

ettt

2

2321

2

2123121213

02

2321

231213

1

,1

−

−

−

−

−+=

=−

=

Fabry-Perot Etalon (Mirror)

FinesseMaximum transmission

Distance between

resonances

Width of the

resonance

20

λπϕπϕ mdmndkm =⇔==⇔=

Dielectric Bragg grating

( ) πϕϕ mdndnk =+=+ 2211021

Condition

The accumulated phase shift must add to π, λ/2

Off-axis high-reflection mirrors

End of the course!