Optics and Optical Design 5: Electromagnetic Optics ...

Optics and Optical Design

Chapter 5: Electromagnetic Optics

Lectures 9 & 10

Cord Arnold / Anne L’Huillier

Electromagnetic waves in dielectric media

EM‐optics compared to simpler theories

Electromagnetic spectrum

Electromagnetic optics describes all kinds of EM‐waves in all possible spectral ranges in possible kinds of media (vacuum, dielectric, conductive, etc.).

Example: THz imaging

The THz Network, www.thznetwork.net www.dailymail.co.uk

Particle ‐ wave

Wikipedia

X-ray imaging (shadow-graphy)

http://www.scienceiscool.org/

X-ray diffraction

X-ray image from the hand of Albert von Koelliker, taken in 1896.

Maxwell Equations in vacuum

Contributions from:‐Charles‐Augustin de Coulomb‐Hans Christian Örsted‐Carl Friedrich Gauss‐Jean‐Baptiste Biot‐André‐Marie Ampére‐Michael Faraday

‐ Unified by James Clerk Maxwell in 1861 as set of twenty equations.

‐ The current form, termed Maxwell Equations, was compressed by using vector notation by Oliver Heavyside in 1884.

Maxwell Equations in a source free medium

Boundary conditions

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.

Linear, nondispersive, homogeneous, isotropic, source‐free media

Anisotropic, linear, nondispersive media

The susceptibility tensor χ can have up to nine independent elements χji.

Dispersive media

Monochromatic electromagnetic waves

Introduce monochromatic fields

All fields and flux densities can be written in their monochromatic versions 

accordingly.

Transverse electromagnetic (TEM) plane wave

E is orthogonal to H. Both are orthogonal to the direction of propagation k.

Vectorial spherical wave

Example: focusing of vectorial waves

Vectorial solutions of the Helmholtz Equation

Absorption and dispersion

Transmission bands for common materials in optics

Implications of dispersion

Refractive index for different isotropic materials and crystals

The resonant medium

The resonant medium

Multi resonance media

Sellmeier Equation for the refractive index far from resonance

Kramers‐Kronig Relations 

The Kramers-Kronig relations relate mathematically the real and imaginary parts of the susceptibility to each other. Knowing one determines the other and vice versa.

Causal response function

Noncausal odd function

Signum function

Causal response function ththtth oo signum

0for real is and 0for 0 tthtth (causal funtion)

dtthtjthtdtthtjH sincosexp

Frequency space imaginary part of a causal response function

Frequency space real part of a causal response function

The real and imaginary parts are related because they originate from the same function and they contain the same information!

oo HHH SIGNUM

Imaginarypart

Realpart

The Drude Model for conductive media

ω<ωp – The effective permittivity is negative, β(ω) is imaginary. Light cannot propagate. => Perfect mirror.

ω>ωp – The effective permittivity is positive. Light can propagate. The refractive index is below 1.

ω=ωp – β(ω)=0. Light cannot propagate. But one can resonantly excite plasma waves. Plasmons!

Pulse propagation in dispersive media

Dispersive media

The field moves in respect to the envelope due to the difference of phase and group velocity

The pulse spreads due to group velocity dispersion (GVD)

Temporal and spectral representation of laser pulses and the time‐bandwidth product

Frequency0

∆

Frequency0

∆

Fourier transform

44.02 FWHM

Time-bandwidth product (Gaussian pulse)

Time

FWHM

Time

FWHM

Ele

ctric

fiel

d (a

.u.)

Ele

ctric

fiel

d (a

.u.)

Spe

ctra

l pow

er (a

.u.)

Spe

ctra

l pow

er (a

.u.)

tjtAtU 0exp Carrier frequency

Pulse envelope (spectrally broad)

Pulsed plane wave

Laser pulses in dispersive media

zjtzAzAtzA

zjzAzA

exp,0FF,~F,

exp,0~,~

11

Spectral plane wave propagator

2000

0

0

!2''

!1' n

c

Wave number expansion around a carrier ω0:

'/1 gvGroup velocity ’’ Group velocity dispersion

Plane wave propagation

Each frequency component evolves with a different wave

number

00

2

2

'',1'

gv

Group velocity and group velocity

dispersion (GVD) result from dispersion.

ms''

2

ms' Inverse of a speed Inverse of an acceleration

Group velocity and group index

0000

000

00

0000

0

020

0

0

20

0

00

0

20

00

20

20

0

0

0

0

','

'1

'

2

22

2,

nnNNc

nncv

nnc

nnc

n

cc

c

g

Depends on the change of the refractive index in respect to the wavelength

Group index

The speed of a pulse is determined by the rate of change of the refractive index

Refractive index for a typical material

Group velocity dispersion (GVD)

020

30

00

20

0

200

02

2

''2

''

22''

nc

D

cc

Refractive index for fused silica

GVD is proportional n’’(0), that is the curvature of n(0).

00

020

2

020

30 '',''

nc

Dnc

D

GVD for fused silica

zDzD 00 DD

Estimation for dispersive pulse broadening

Pulse broadening in dispersive media

Dispersive media

n>1

N>1=> vg<vp

Anomalousdispersion

Normaldispersion

Anomalousdispersion

N>1=> vg<vp

n<1