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
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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?
39
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
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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
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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)
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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
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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