1

Multiple filamentation of intense laser beams

Gadi FibichTel Aviv University

Boaz Ilan, University of Colorado at BoulderAudrius Dubietis and Gintaras Tamosauskas, Vilnius University, LithuaniaArie 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

,,

⋅+⋅+

=

⋅∇−=⋅∇

=−=+⋅∇∇−∆

γγ

εε

ε

Kerr Mechanism γ

electrostriction 0

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 = ψ(x,y,z) exp(i k0 z)

Paraxial approximation

5

2D cubic NLS

Initial value problem in zCompetition: self-focusing nonlinearity versus diffractionSolutions 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

ψψ

ψψψψ

==

∂+∂=∆=+∆+

PP crdxdy ≥= ∫ψ 02

6

Multiple filamentation (MF)

If P>10Pcr, a single input beam can break into several long and narrow filamentsComplete 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 MFWhich mechanism leads to breakup of cylindrical symmetry in MF?

yxrryx22

00),(),( +==ψψ

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

rzyxzryx ψψψψ =⇒=

8

Standard explanation (Bespalov and Talanov, 1966)

Physical input beam has noisee.g. Noise breaks up cylindrical symmetryPlane waves solutions

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

)],(1)[exp(),( 20 yxnoisecyx r +−=ψ

)exp( 2 zia a=ψ

9

Weakness of MI analysis

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

0),,(2

=+∆+ ψψψψ yxzzi

)],(1.01)[exp(),( 20 yxrandcyx r ⋅+−=ψ

0 0.02910

2

103

104

z

max

x,y |A

1(x,y

,z)|2

Gaussian Gaussian+noiseZ=0 Z=0.026

−1 1−1

1

x

y

A

−1 1−1

1

x

y

B

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

13

Model for noise-induced MF

,01

),,( 2

2

=+

+∆+ ψψε

ψψψ yxzzi

NLS with saturating nonlinearity (accounts for plasma defocusing)

Initial condition: cylindrically-symmetric Gaussian profile + noise

)],(1)[exp(),( 20 yxnoisecyx r +−=ψ

14

Typical simulation(P=15Pcr)

Ring/crater is unstable

15

Noise-induced MF

MF pattern is random No control over number and location of filaments Disadvantage in applications (e.g., eye surgery, remote sensing)

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 symmetryPreferred 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

⋅+⋅+

=

⋅∇−=⋅∇

−=+⋅∇∇−∆

γγ

εε

ε

19

Derivation of scalar model

Small nonparaxiality parameter

Linearly-polarized input beam

E = (E1,0,0) at z=0

12

1

000

<<==rrk

fπλ

20

NLS with vectorial effects

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

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞⎜⎝

⎛ +++

++++

−

−

=+∆+

ψψψψψψψ

ψ

ψψ

γγψ

γγ

ψψ

*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 ψ0 = c exp(-r2) f = 0.05, γ= 0.5 No noise!

23

P = 4Pcr

Ring/crater is unstable

Splitting along x-axis

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P = 10Pcr

Splitting along y-axis

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P = 20Pcr

more than two filaments

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What about circular polarization?

27

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

andPhysical Review E, 2003

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Circular polarization and MF

Circular polarization has no preferred directionIf 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 -

+ -+

= = < <

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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

+ -+ -

+ +

- -

= =

ש +י + + + ת= +ך -+ ת +ך כ�ש +י + + + ת= ך +-- ת +ך כ�

D

D

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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

⋅+⋅+

=

⋅∇−=⋅∇

−=+⋅∇∇−∆

γγ

εε

ε

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Derivation of scalar model

Small nonparaxiality parameter

Nearly circularly-polarized input beam

at z=0

12

1

000

<<==rrk

fπλ

1EE

e -

+

= < <

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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

+ + +

+ + + +

- -

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

ש -י + + ת+ ך + + + ת+ +ך כ�++ + =+- +

D

D Dׁ ׁ

D

33

Circular polarization and MF

New model is isotropic to O(f2) Neglected symmetry-breaking terms are O(εf2)Conclusion – small deviations from circular polarization unlikely to lead to MF

34

Back to Linear Polarization

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Testing the vectorial explanation

Vectorial effects breaks up cylindrical symmetry while inducing a preferred direction of input beam polarizationIf MF pattern is caused by vectorial effects, it should be deterministic and rotate with direction of input beam polarization

x

y

→n = (1,0,0)

x

yθ

→n = (cos(θ),sin(θ),0)

−1 0 1

−1

0

1

x

y

−1 0 1

−1

0

1

x

y

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?

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Ellipticity and MF

Use elliptic input beams ψ0 = c exp(-x2/a2-y2/b2)

Deterministic breakup of cylindrical symmetry with a preferred directionCan it lead to deterministic MF?

40

Possible MF patterns

Solution preserves the symmetries x -x and y -y

y

x

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Simulations with elliptic input beams

NLS with NL saturation

elliptic initial conditions ψ0 = c exp(-x2/a2-y2/b2)P = 66PcrNo noise!

,0005.01

),,( 2

2

=+

+∆+ ψψ

ψψψ yxzzi

e = 1.09central filamentfilament pair along minor axisfilament pair along major axis

e = 2.2central filamentquadruple of filamentsvery weak filament pair along major axis

44

All 4 filament types observed numerically

45

Experiments

Ultrashort (170 fs) laser pulsesInput 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

47

P=7Pcr

Additional filament pair along major axis

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P=18Pcr

Additional filament pair along minor axis

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P=23 Pcr

Additional quadruple of filaments

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All 4 filament types observed experimentally

51

Rotation Experiment

5Pcr 7Pcr 10Pcr 14Pcr

MF pattern rotates with orientation of ellipticity

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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 ellipticityAdvantage: 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

57

Experimental setup

Control astigmatism through lens rotation

58

MF pattern after 5 meters in air

Strong central filamentFilament pair along minor axisCentral and lower filaments are stable Despite high noise level, MF pattern is quite stableEllipticity dominates noise

average over 1000 shots

typical

59

typical

Average over 1000 shots

60

Control of MF - position

ϕ = 00 : one direction (input beam ellipticity)ϕ = 200 : all directions (input beam ellipticity +rotation lens)

61

Control of MF – number of filaments

ϕ = 00 2-3 filamentsϕ = 200 single filament

62

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 airEllipticity can be ``stronger’’ than noise

64

Theory needed

Currently, no theory for this high powerstrongly nonlinear regime (P>> Pcr)In contrast, fairly developed theory when P = O(Pcr)Why? the ``Townes profile’’ attractor

ZcrzaszL

rRzL

yxz →⎟⎠⎞

⎜⎝⎛

)()(1|~),,(|ψ