1

Multiple filamentation of intense laser beams

Gadi FibichTel Aviv University

Boaz Ilan, University of Colorado at Boulder Audrius Dubietis and Gintaras Tamosauskas,

Vilnius University, Lithuania Arie Zigler and Shmuel Eisenmann, Hebrew

University, Israel

2

Laser propagation in Kerr medium

3

Vector NL Helmholtz model (cw)

EEEEEE

E

EEEEEEE

nnP

Pn

Pnkk

NL

NL

NL

**200

2

00

3212

00

2

02

0

14

1

,,

Kerr Mechanism

electrostriction0

non-resonant electrons

0.5

molecular orientation

3

4

Simplifying assumptions

Beam remains linearly polarized E = (E1(x,y,z), 0, 0) Slowly varying envelope E1 = (x,y,z) exp(i k0 z) Paraxial approximation

5

2D cubic NLS

Initial value problem in z Competition: self-focusing nonlinearity versus diffraction Solutions can become singular (collapse) in finite

propagation distance z = Zcr if their input power P is above a critical power Pcr (Kelley, 1965), i.e.,

),(),,0(

,0),,(

0

2

yxyxz

yxzzi yyxx

PP crdxdy 02

6

Multiple filamentation (MF)

If P>10Pcr, a single input beam can break into several long and narrow filaments

Complete breakup of cylindrical symmetry

7

Breakup of cylindrical symmetry

Assume input beam is cylindrically-symmetric

Since NLS preserves cylindrical symmetry

But, cylindrical symmetry does break down in MF Which mechanism leads to breakup of cylindrical

symmetry in MF?

yxrryx22

00),(),(

),(),,()(),(00

rzyxzryx

8

Standard explanation (Bespalov and Talanov, 1966)

Physical input beam has noise e.g. Noise breaks up cylindrical symmetry Plane waves solutions

of the NLS are linearly unstable (MI) Conclusion: MF is caused by noise in input beam

)],(1)[exp(),( 20 yxnoisecyx r

)exp( 2 zia a

9

Weakness of MI analysis

Unlike plane waves, laser beams have finite power as well as transverse dependence

Numerical support?Not possible in the sixties

10

G. Fibich, and B. Ilan Optics Letters, 2001

andPhysica D, 2001

11

Testing the Bespalov-Talanov model

Solve the NLS

Input beam is Gaussian with 10% noise and P = 15Pcr

)],(1.01)[exp(),( 20 yxrandcyx r

0),,(2 yxzzi

Z=0 Z=0.026

Blowup while approaching a symmetric profile Even at P=15Pcr, noise does not lead to MF in the

NLS

13

Model for noise-induced MF

,01

),,( 2

2

yxzzi

)],(1)[exp(),( 20 yxnoisecyx r

NLS with saturating nonlinearity (accounts for plasma defocusing)

Initial condition: cylindrically-symmetric Gaussian profile + noise

14

Typical simulation (P=15Pcr)

Ring/crater is unstable

15

MF pattern is random No control over number and location of

filaments Disadvantage in applications (e.g., eye

surgery, remote sensing)

Noise-induced MF

16

Can we have a deterministic MF?

17

Vectorial effects and MF

NLS is a scalar model for linearly-polarized beamsMore comprehensive model – vector nonlinear

Helmholtz equations for E = (E1, E2, E3) Linear polarization state E = (E1,0,0) at z=0 leads

to breakup of cylindrical symmetry Preferred direction Can this lead to a deterministic MF?

18

Linear polarization - analysis

vector Helmholtz equations for E = (E1,E2,E3)

EEEEEE

E

EEE

nnP

Pn

Pnkk

NL

NL

NL

**200

2

00

2

00

2

02

0

14

1

19

Derivation of scalar model

Small nonparaxiality parameter

Linearly-polarized input beam E = (E1,0,0) at z=0

12

1

000

rrk

f

20

NLS with vectorial effects

Can reduce vector Helmholtz equations to a scalar perturbed NLS for = |E1|:

*22*222

2

2

121

161

41

xxxx

zz

z

xx

i

f

f

21

Vectorial effects and MF

Vectorial effects lead to a deterministic breakup of cylindrical symmetry with a preferred direction

Can it lead to a deterministic MF?

22

Simulations

Cylindrically-symmetric linearly-polarized Gaussian beams 0 = c exp(-r2)

f = 0.05, = 0.5 No noise!

23

P = 4Pcr

Splitting along x-axis

Ring/crater is unstable

24

P = 10Pcr

Splitting along y-axis

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P = 20Pcr

more than two filaments

26

What about circular polarization?

27

G. Fibich, and B. Ilan Physical Review Letters, 2002

andPhysical Review E, 2003

28

Circular polarization and MF

Circular polarization has no preferred direction If input beam is cylindrically-symmetric, it will

remain so during propagation (i.e., no MF) Can small deviations from circular polarization

lead to MF?

3( , , ), 1E EE E E Ee -

+ -+

= = <<

29

Standard model (Close et al., 66)

Neglects E3 while keeping the coupling to the weaker E- component

( )( )

0 0

2 2

22

exp( ) , exp( )

1 (1 2 ) 011 (1 2 ) 0

1

| | | |

| || |z

z

i z i z

i

i

k kE E

gg

gg

y y

y yy yy

yyy y y

+ -+ -

+ +

- -

= =

é ù+ + + + =ê ú+ -+ ê ú+ ë ûé ù+ + + + =ê ú+-- ê ú+ ë û

DD

30

Circular polarization - analysis

vector Helmholtz equations for E = (E+,E-,E3)

EEEEEE

E

EEE

nnP

Pn

Pnkk

NL

NL

NL

**200

2

00

2

00

2

02

0

14

1

31

Derivation of scalar model

Small nonparaxiality parameter

Nearly circularly-polarized input beam at z=0

12

1

000

rrk

f

1EE

e -

+= <<

32

NLS for nearly circularly-polarized beams

Can reduce vector Helmholtz equations to the new model:

Isotropic to O(f2)

( ) ( )

( )

22 2

22 2 2 2* *

2

1 1 21 1 4

42(1 )

1 2 01

| | | |

| | ( ) | | ( )

| |

z zz

z

i

i

f

f

gg g

ggg

y yy y yy y

y y y yy y y y

yy y y

+ + +

+ + + +

- -

++ + =- -+ -+ ++ +

é ù- + + +ê ú+ + + +ê ú+ ë û++ + =+- +

D

D DÑ ÑD

33

Circular polarization and MF

New model is isotropic to O(f2) Neglected symmetry-breaking terms are O(f2)Conclusion – small deviations from circular

polarization unlikely to lead to MF

34

Back to Linear Polarization

35

Testing the vectorial explanation

Vectorial effects breaks up cylindrical symmetry while inducing a preferred direction of input beam polarization

If MF pattern is caused by vectorial effects, it should be deterministic and rotate with direction of input beam polarization

36

A.Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, Optics Letters, 2004

37

First experimental test of vectorial explanation for MF

Observe a deterministic MF patternMF pattern does not rotate with direction of

input beam polarizationMF not caused by vectorial effects Possible explanation: collapse is arrested by

plasma defocusing when vectorial effects are still too small to cause MF

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So, how can we have a deterministic MF?

39

Ellipticity and MF

Use elliptic input beams 0 = c exp(-x2/a2-y2/b2)

Deterministic breakup of cylindrical symmetry with a preferred direction

Can it lead to deterministic MF?

40

Possible MF patterns

Solution preserves the symmetries x -x and y -y

x

y

41

Simulations with elliptic input beams

NLS with NL saturation

elliptic initial conditions 0 = c exp(-x2/a2-y2/b2) P = 66Pcr No noise!

,0005.01

),,( 2

2

yxzzi

e = 1.09 central filament filament pair along

minor axis filament pair along

major axis

e = 2.2 central filament quadruple of filaments very weak filament pair

along major axis

44

All 4 filament types observed numerically

45

Experiments

Ultrashort (170 fs) laser pulses Input beam ellipticity is b/a = 2.2 ``Clean’’ input beamMeasure MF pattern after propagation of

3.1cm in water

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P=4.8Pcr

Single filament

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Additional filament pair along major axis

P=7Pcr

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Additional filament pair along minor axis

P=18Pcr

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Additional quadruple of filaments

P=23 Pcr

50

All 4 filament types observed experimentally

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Rotation Experiment

MF pattern rotates with orientation of ellipticity

5Pcr 7Pcr 10Pcr 14Pcr

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2nd Rotation Experiment

MF pattern does not rotate with direction of input beam polarization

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Dynamics in z

54

G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler,

Optics Letters, 2004

55

Control of MF in atmospheric propagation

Standard approach: produce a clean(er) input beam

New approach: Rather than fight noise, simply add large ellipticity

Advantage: easier to implement, especially at power levels needed for atmospheric propagation (>10GW)

56

Ellipticity-induced MF in air

Input power 65.5GW (~20Pcr) Noisy, elliptic input beam

Typical Average over 100 shots

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Experimental setup

Control astigmatism through lens rotation

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typical

average over 1000 shots

MF pattern after 5 meters in air

Strong central filament Filament pair along minor

axis Central and lower filaments

are stable Despite high noise level, MF

pattern is quite stable Ellipticity dominates noise

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typical

Average over 1000 shots

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Control of MF - position

= 00 : one direction (input beam ellipticity) = 200 : all directions (input beam ellipticity

+rotation lens)

61

Control of MF – number of filaments

= 00 2-3 filaments = 200 single filament

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Ellipticity-induced MF is generic

cw in sodium vapor (Grantham et al., 91) 170fs in water (Dubietis, Tamosauskas, Fibich,

Ilan, 04) 200fs in air (Fibich, Eisenmann, Ilan, Zigler, 04) 130fs in air (Mechain et al., 04)Quadratic nonlinearity (Carrasco et al., 03)

63

Summary - MF

Input beam ellipticity can lead to deterministic MF

Observed in simulations Observed for clean input beams in waterObserved for noisy input beams in air Ellipticity can be ``stronger’’ than noise

64

Theory needed

Currently, no theory for this high power strongly nonlinear regime (P>> Pcr)

In contrast, fairly developed theory when P = O(Pcr)

Why? the ``Townes profile’’ attractor

Z crzaszLr

RzL

yxz

)()(1

|~),,(|