Lecture 2: Propagation of lightwaves and pulses

Petr Kužel• Propagation of plane waves in linear homogeneous isotropic (LHI)

media:! refractive index! intensity! absorption

• Propagation of light pulses:! group velocity! group dispersion — chirp! superluminal effects (?)

Wave equation

0

0or

0

0

0

0

2

2

2

22

2

2

2

22

2

2

02

2

2

02

=∂

∂−∇

=∂∂−∇

=∂

∂εµ−∇

=∂∂εµ−∇

⇒

=∂∂−∧∇

=∂∂+∧∇

tcN

tcN

t

t

t

tHH

EE

HH

EE

DH

BE

( )( )EEE 2∇−⋅∇∇=∧∇∧∇

well-known identityrεε=ε 0

rrr iε ′′−ε′=ε

dielectric constant

00

1µε

=c

rcN ε=εµ= 20

2

refractive index

( ) ( )rrrrrr

rr

n

nn

ε′−ε ′′+ε′=κε′+ε ′′+ε′=

κ=ε ′′κ−=ε′

2222

22

21,

21

,2,κ−= inN

Propagation in the vacuum

01

01

2

2

22

2

2

22

=∂

∂−∇

=∂∂−∇

tc

tcHH

EE( )

( ) ck

e

e

ti

ti

ω=≡=

=

⋅−ω

⋅−ω

kHH

EE

rk

rk

with

0

0

• Back to Maxwell’s equations:

( )

0

10 000

0

=⋅−=⋅∇

=∧ωµ

=ωµ+∧−=∂∂+∧∇

EkE

HEkHEkBE

i

iit

kHEk ⊥⊥⊥ 00

( )01

00 EsH ∧η= −

( )01

0 EsB ∧= −c Ω≈εµ=η 377

0

00

kks =

Superpositions of plane waves• In the k-space: 3-dimensional

( ) ( ) ( )∫∫∫ ⋅−ω= kkErE rk det ti0,

( ) ( ) ( )∫∫∫ ⋅−ω= kkHrH rk det ti0,

• In the ω-space: 1-dimensional (propagation along z: kx = ky = 0)

( ) ( ) ( )∫ −= ωω ω dezt kzti0, EE

( ) ( ) ( )∫ −= ωω ω dezt kzti0, HH

• Pulse: central frequency ω0; central wave vector k0 = ω0/c.

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( )zticztizti

zkktiztizti

ecztdee

deeeztzt

00000

000000

10

01 ,,kk

kk

EE

EEE−−−−

−−−−−

−==

===

∫∫

ωωωω

ωωωω

ωω

ωω

Propagation in the matter

0

0

2

2

2

22

2

2

2

22

=∂

∂−∇

=∂∂−∇

tcN

tcN

HH

EE( )

( )N

ck

e

e

ti

ti

ω=≡=

=

⋅−ω

⋅−ω

kHH

EE

rk

rk

with

0

0

• Back to Maxwell’s equations:

( )

0

10 000

0

=⋅−=⋅∇

=∧ωµ

=ωµ+∧−=∂∂+∧∇

EkE

HEkHEkBE

i

iit

kHEk ⊥⊥⊥ 00

( )00

0 EsH ∧η

= N

( )00 EsB ∧=cN ε

µ=η=η 00

N

kks =

Propagation direction(vector k)

kkk ′′−′= i

• k' = n ω/c, k'' = κ ω/c• k' and k'' can have different different directions:

! k'∙r = const: constant phase planes! k''∙r = const: constant amplitude planes

• k'': boundary conditions of an absorbing sample

n1

N2 = n2 − iκ2

ki

kr

k'tx

y

z

α α

β'

k''t

Propagation direction(vector S)

• Energy flow (Poynting vector S)

∗∧=∧= HEHES Re21

( ) ( )

( )( ) rkrk EkEEkk

EEkEEkEkES

⋅′′−⋅′′−∗

∗∗∗∗∗∗

ωµ′

=⋅′′−′ωµ

=

=⋅−⋅ωµ

=∧∧ωµ

=

220

0

200

0

000

2Re

21

)(Re2

1Re2

1

eei

!"#

• Propagation along z (k // z)

( ) zcz enec

n α−ωκ−−

η=ωωµ= 2

00

220

10 2

2 EES

( )λπκ=ωκ=ωα 42

c

Optical pulses: dispersion• Dispersion relation (non-absorbing medium):

( )ωω= nc

k

• Optical pulse as a linear combination of eigenmodes:

( ) ( ) ( )( ) ( ) ( )( )∫∫ ωω== ω−ω−ω dedkekzt zktikztki00, EEE

• Mean frequency ω0 and wave vector k0: ∆ω ‹‹ ω0, ∆k ‹‹ k0:

( ) ( ) ( ) ( ) ( )2000

20

02

2

00

0 221 kkkkkk

dkdkk

dkdk g −β+−+ω=−

ω+−

ω+ω=ω v

( ) ( ) ( ) ( ) ( )200

10

20

02

2

00

0 221 ω−ωψ+ω−ω+=ω−ω

ω

+ω−ω

ω+=ω −

gkd

kdddkkk v

3gvβ−=ψ … group velocity dispersion (GVD)

Group velocity (first order effects)• Propagation of a pulse

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )$!$"#$!$"#

)(

1)(

1

))((0

)()(0

000

0000000,

B

gA

tzikg

zkti

kkztizktizkktkkizkti

tzetze

dkekedkeekzt gg

vv v

vv

−=−==

===

−−−ω

∞

∞−

−−−ω∞

∞−

−−−−ω ∫∫EE

EEE

• (A): field oscillation with the central frequency

wave front velocity:

• (B): propagation of envelope without deformation

group velocity

)(0

0

ω=ω=

nc

kv

0ω

ω=dkd

gv

( )ωω+

=⇒ωω=ddnn

cdkdnc

k gv

Optical pulses: energy flow• Poynting vector

czgg etzntz /22

10

)(2

)( ωκ−−η

=− vv ES

• Energy conservation law( )

( )

∗

=ωω+

∗

∗∗∗ωκ

⋅∂

∂ε

ωω++εκω=

=

⋅

∂∂ε′+⋅

∂∂

ωε′ω+ε′′ω=

∂∂

11

022

110

11

11

11/2

2

21

2

EEEE

EEEEEE

tddnnn

ttdd

tU

e

gcnddnnn

cz

$$ !$$ "#v

tU

et

nc

n

etzz

n

cz

g

czg

∂∂

−=

⋅

∂∂

η+⋅

ηκω−=

=−∂∂

η=⋅∇

ωκ−∗∗

ωκ−

/21

1

011

0

/221

0)(

2

EEEE

ES

v

v

0

00

00,1

εµ=η

µε=c

Group velocity dispersion GVD(second order effects)

• Main effect: pulse broadening in time

• This treatment is necessary for

! Ultrashort pulse (sub-50 fs) propagation

! Ultrashort pulse (ps or sub-ps) generation in lasers (multiplepasses through dispersive media)

! Short pulse (sub-ns) propagation in fibers (extremely longdistances)

( ) ( ) ( )2000 2

kkkkk g −β+−+ω=ω v

( ) ( ) ( )200

10 2

ω−ωψ+ω−ω+=ω −gkk v

Intuitive treatment• Spectral components coming from two

ends of the spectrum propagate withdifferent velocities

ω∆ω

=∆dd g

gv

v

• GVD is defined as

ω−=

ω

=ω

=ψdd

dd

dkd g

gg

vvv 22

2 11

• Spreading due to a thickness L of the dispersive material:

τ

ψ=ω∆ψ=ω∆ω

=∆≈τ∆0

22 2ln4 LLddLL g

gg

g

vv

vv

for a gaussian pulse shape

ω0

∆ω

vg(ω1) vg(ω2)

ω

z

Exact treatment

τ

ψ≈τ∆0

2ln4 L

(intuitive treatment)

• Superposition of the spectral components

( ) ( ) ( ) ( ) ( )∫∫∞

∞−

ω−ωβ−ω−ω−ω∞

∞−

ω−ω ωω=ωω= deeedezt gg ziztizktizkti 20

3000 )()2()()(

0)(

0, vvEEE

• One gets for a gaussian pulse for a crystal length L after all the integrations:

( ) ( )( ) ( )

( )1

12

2

00

11, +δ

δ−−α−

−ω

δ+=

izt

zktig

ei

eAztv

E 20

2ln42withτ

ψ=αψ=δ LL

• Spreading due to a thickness L of the dispersive material:

( )2

20

02

02ln411

τ

ψ+τ=δ+τ=τ LL

z

E

(A)

(B)

(C)

z = 0 z = z0 z = 2z0 z = 3z0

( ) ( )( )2

2

0 214

,L

Lztt

tz g

ψα+ψα−

+ω=∂Φ∂=ω

vMean frequency varies in time:

Chirp (group delay dispersion): LGDD ψ=ω∂Φ∂−= 2

2

Pulse spreading: example

20 40 60 80 100Bandwidth-limited pulse length (fs)

120

100

80

60

40

20

Puls

e le

ngth

afte

r the

pla

te (f

s)

Thickness:10 mm6 mm3 mm

Dispersion of a BK7 plate

“Superluminal” effects• What can be superluminal?

! Phase velocity?

! Group velocity?

! Velocity of an information?

• Definition of the speed of an information transfer

Information speed

Amplification process

The noise can only slow down the information transfer

Phase velocity

k, v

k cosθ, v/cosθ

θ

k = 0, v = ∞

• Group velocity can exceed the vacuum speed of the light:

! region of anomalous dispersion" dielectric resonance" gain medium

ωω+

=

ddnn

cgv

Group velocity

0 1 2 3 4Frequency (arb. u.)

10

8

6

4

2

0re

fract

ive

inde

x

10

8

6

4

2

0

abso

rptio

n in

dex

• group velocity

• group velocity can besuperluminal

• penetration depth =fraction of λ

ωω+

=

ddnn

cgv

0 1 2 3 4Frequency (arb. u.)

1.5

1

0.5

0

-0.5

-1

grou

p ve

loci

ty n

orm

aliz

ed to

c

3

2

1

0

pene

tratio

n de

pth

norm

aliz

ed to

λ

Dielectricresonance

Gain-assisted superluminal lightpropagationL. J. Wang, A. Kuzmich & A. DogariuNEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA

Gain-assistedanomalous dispersion:

Measured pulse advancementcorresponds to a negative group

velocity

Conclusion:

• pulse maximum andleading / trailing edgeof the pulse areadvanced

• there is almost (!!) nochange in the pulseshape

• the very first non-zerosignal can never (!!)be advanced