Kubické nelinearity - Kerrov Kubické nelinearity - Kerrov javjav

Samomudulacia fázy, Samomudulacia fázy, samofokusáciasamofokusácia

KerrKerr effect effect:: The refractive index depends upon the intensity of the light propagating through the material.

0

t

BE

jt

DH

0 B D

HB

0 PED

0 EP 0

32

302010 EEEP

334 EPNL

102n

n

cc 0

00

0

1

c

EEPNL

233

Nonlinear medium Elmag. wave Response

tirki eerAE Re

E

PEPP

0

0

NLLins PPEEEP

.302010Re

32

PPP s

.Re

EEPNL

233

000 n

12n InInn

nn 23

0

3

2

1

nnn s Re

InnIn 2

Optický Kerrov jav

,63 32

3

0

IEE

PNL

2)(

2wE

I

Physical machanismPhysical machanism Veľkosť Veľkosť nnnelnelČas odozvy Čas odozvy

(s)(s)

Electronic movementElectronic movement 1010-15-15. 10. 10-13-13 1010-14-14.10.10-15-15

Molecules reorientationMolecules reorientation 1010-13-13. 10. 10-11-11 ~ 10~ 10-11-11

Libration and rotationesLibration and rotationes of moleculesof molecules

~ 10~ 10-13-13 ~ 10~ 10-13-13

Electrostriction, moleculesElectrostriction, molecules redistributiones redistributiones

~ 10~ 10-11-11 ~ 10~ 10-6-6

Iné mechanizmy: Sprievodné tepelné javy, fotorefrakcia.

0

t

BE

0

t

DH

nelPEP

)1(0 HB

0

0 B

0 D

PED

Dt

Ht

EEBt

E

2

2

00)()(

EEE

)()(

Ptt

E

cE

2

2

02

21

ncc r 0001 0 E

PED

..)(2

1),( )(

11111 zkezEetzE zkti

iii

Elektricky homogénne, izotropné (ε,μ – konštanty) navyše úplne priehľadné čo sa tíka magnetických vlastností (μr = 1). Prostredie je lineárne (platí Ohmov zákon) a nie sú v ňom volné náboje (ρ = 0).

..)(2

1),( )(

22222 zkezEetzE zkti

kkk

..)(2

1),( )(

33333 zkezEetzE zkti

jjj

,

..)(2

1),( )(

44444 zkezEetzE zkti

lll

)(12

2

12

21111 )(),(),( zkti

iiii ezEz

etzEz

tzE

Ptt

E

cE

2

2

02

21

)()( 12

2

1 zEdz

dkzE

dz

dimi

)(11

21

111 ).(2)(),( zktiiiiiii ezE

dz

dikzEketzE

),(),(1

11

2

2

2

2

tzEc

tzEtc i

ii

Pt

ezEec

ezEdz

dikzEke zkti

iiizkti

iiiii

2

2

0)(

112

2)(

112

11111 )().(2)(

ii kei 2 222 ck ii

Petk

iezE

dz

di

i

zktii

12

20)(

1 2).( 11

Petk

iezE

dz

di

i

zktii

12

20)(

1 2).( 11

1,, 21302010

32 EEEP

zkkktiljkljki

i

zktii ezEzEzEeeee

tk

iezE

dz

d )()(432432

312

200)(

143243211 )()()(

2).(

ljkieff eeee 4323

1

)()()()( zEzEzEizE

dz

d nef 200

nn ef2

E*(z) plus complex conjugate E(z) is constant

)()( zieEzE We expect solution in this form

zEz w

2

0 )0()( zEi weEzE2

)(

..),(2

zkeEetrE zEkti

Self-phase modulation

zkkktief

kzti ezEzEzEk

iezE

dz

d )()(*00)( )()()(2

).(

..,,),(2

,, zketyxEetrE ztyxEkti

Continuum generation and self-focusing

ρ zWeR

22

I

t

A0

τ0

20

2 2tetF

t2

t1 t

= / t

t

ω-ωlaser

t1

t2

Stokesovposun

Anti Stokesovposun

t

| E(t )|2

t2t1

nel | E(t) |2 z

tt2t1

| E(t) |2 n(t) (t)

..,,),(2

,, zketyxEetrE ztyxEkti

z

t

tE

2

z

t

tEt

2

21

2220max

ttt

tF

t

tFAz

zωχnτc

Αωz

τ

ΑσzΔω ef

00

20

0

20

max

2

F (t ) = exp[-t 2 / 2τ02]

2

2),(

2

1),(

dtetrErE ti

0 100 200 300100200300Δω=ω-ω laser (cm

-1)

δωM

Δωmax

SMF

Laser

t

ω-ω l aser

t 1 t 2

Stokesov posun

Anti Stokesov posun

t φ

z

t

tE

2

0

max 4

NM

0

2

M

0

22 2

ccM

zA

c

n

t

tF

t

tFA

c

nz

tt

20

0

2022

20

20max

212

t1 = - τ0 , t2 = τ0

zAc

nN 2

020max

42

zAc

nz 2

020

max 2)(

-101

-2 -1 0 1 2

1,0

1,5

Anti-Stokesovačasť

Stokesovačasť

-101

-2-101

2

1,0

1,5

t /τ0

r/W0

λ(t, r)/ λ0

Anti-Stokesovačasť

Stokesovačasť

r/W0 t /τ0

λ(t, r)/ λ0

a b

zωχnτc

ΑrRωzΔω ef

00

20

max

)(

2

20

2

20

20

202

0

22exp

.21

1,

t

W

r

c

tzAntr

-L/2

-L/2 L/2

2/4

nl(0)Lnl>>z0

L>> z0

Nelineárne médium

π/πLnl

-z0

z00 L/2

Nelineárne médium

L2

( / )2

L

nl( 0)

L >> L2

0

1

pL

SMFnl SMF 2ne /0w0

2

,

02

4)0(

vpkp

p

)0(,výstupnl

(p0 / pk,vp)2 << 1

nlLz0

L2 ~ 1.1z0

2

,

0

2

2

vlkp

pL

pk,v = 02/2n2

02

)0(z

z

L

znl

pk,vp = 02/πn2

(/)2 1 pk,vl / pk,vp 1.5

Nelinearne medium

n

ρ

..,,),(2

,, zketyxEetrE ztyxEkti

zWeR

22

Self-focusing

b) e)

546.1

L = 4.6 cm

h)

763.5

L = 4.6 cm

763.51

0.5mm

L = 4.6 cm

b)

763.51

L = 4.6 cm

579.066

L = 4.6 cm

9.5546.1

1

2

L = 3.4 cm i = 3

e)

546.14.0

L = 8.6 cm i = 3

738.46.9

1

2

3

L = 4.6 cm i = 2

4.0546.1

L = 8.6 cm i = 3

491.614.3

L = 8.6 cm i = 4

h)

435.8

4.2

L = 8.6 cm i = 5

nm811.5912.3 706.7 579.1 546.1 491.6 435.8 404.7 404.7 404.7 365.0

L = 8.6 cm

L = 4.6 cm

L = 3.4 cm

1 mm

0