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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
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Laser propagation in Kerr medium
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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
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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
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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
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Multiple filamentation (MF)
If P>10Pcr, a single input beam can break into several long and narrow filaments
Complete breakup of cylindrical symmetry
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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
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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
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Weakness of MI analysis
Unlike plane waves, laser beams have finite power as well as transverse dependence
Numerical support?Not possible in the sixties
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G. Fibich, and B. Ilan Optics Letters, 2001
andPhysica D, 2001
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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
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Z=0 Z=0.026
Blowup while approaching a symmetric profile Even at P=15Pcr, noise does not lead to MF in the
NLS
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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
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Typical simulation (P=15Pcr)
Ring/crater is unstable
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MF pattern is random No control over number and location of
filaments Disadvantage in applications (e.g., eye
surgery, remote sensing)
Noise-induced MF
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Can we have a deterministic MF?
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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?
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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
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Derivation of scalar model
Small nonparaxiality parameter
Linearly-polarized input beam E = (E1,0,0) at z=0
12
1
000
rrk
f
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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
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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?
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Simulations
Cylindrically-symmetric linearly-polarized Gaussian beams 0 = c exp(-r2)
f = 0.05, = 0.5 No noise!
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P = 4Pcr
Splitting along x-axis
Ring/crater is unstable
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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?
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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 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 -
+ -+
= = <<
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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
+ -+ -
+ +
- -
= =
é ù+ + + + =ê ú+ -+ ê ú+ ë ûé ù+ + + + =ê ú+-- ê ú+ ë û
DD
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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
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Derivation of scalar model
Small nonparaxiality parameter
Nearly circularly-polarized input beam at z=0
12
1
000
rrk
f
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
+ + +
+ + + +
- -
++ + =- -+ -+ ++ +
é ù- + + +ê ú+ + + +ê ú+ ë û++ + =+- +
D
D DÑ ÑD
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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
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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 polarization
If MF pattern is caused by vectorial effects, it should be deterministic and rotate with direction of input beam polarization
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A.Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, Optics Letters, 2004
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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 0 = c exp(-x2/a2-y2/b2)
Deterministic breakup of cylindrical symmetry with a preferred direction
Can it lead to deterministic MF?
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Possible MF patterns
Solution preserves the symmetries x -x and y -y
x
y
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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
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e = 1.09 central filament filament pair along
minor axis filament pair along
major axis
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e = 2.2 central filament quadruple of filaments very weak filament pair
along major axis
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All 4 filament types observed numerically
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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
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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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G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler,
Optics Letters, 2004
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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)
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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)
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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)
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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 air Ellipticity can be ``stronger’’ than noise
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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
|~),,(|