Multiple filamentation of intense laser beams Gadi Fibich

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Multiple filamentation of intense laser beams Gadi Fibich Tel Aviv University l l l Multiple filamentation of intense laser beams Gadi Fibich Tel Aviv University l l l 1 Boaz Ilan, University of Colorado at Boulder Audrius Dubietis and Gintaras Tamosauskas, Vilnius University, Lithuania Arie Zigler and Shmuel Eisenmann, Hebrew University, Israel

Laser propagation in Kerr medium 2 Laser propagation in Kerr medium 2

Vector NL Helmholtz model (cw) Kerr Mechanism electrostriction 0 non-resonant electrons 3 0. 5 Vector NL Helmholtz model (cw) Kerr Mechanism electrostriction 0 non-resonant electrons 3 0. 5 molecular orientation 3

Simplifying assumptions l l l 4 Beam remains linearly polarized E = (E 1(x, Simplifying assumptions l l l 4 Beam remains linearly polarized E = (E 1(x, y, z), 0, 0) Slowly varying envelope E 1 = (x, y, z) exp(i k 0 z) Paraxial approximation

2 D cubic NLS l l l 5 Initial value problem in z Competition: 2 D cubic NLS l l l 5 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. ,

Multiple filamentation (MF) l l 6 If P>10 Pcr, a single input beam can Multiple filamentation (MF) l l 6 If P>10 Pcr, a single input beam can break into several long and narrow filaments Complete breakup of cylindrical symmetry

Breakup of cylindrical symmetry l Assume input beam is cylindrically-symmetric l Since NLS preserves Breakup of cylindrical symmetry l Assume input beam is cylindrically-symmetric l Since NLS preserves cylindrical symmetry l But, cylindrical symmetry does break down in MF Which mechanism leads to breakup of cylindrical symmetry in MF? l 7

Standard explanation (Bespalov and Talanov, 1966) l Physical input beam has noise e. g. Standard explanation (Bespalov and Talanov, 1966) l Physical input beam has noise e. g. Noise breaks up cylindrical symmetry Plane waves solutions l of the NLS are linearly unstable (MI) Conclusion: MF is caused by noise in input beam l l 8

Weakness of MI analysis l l l 9 Unlike plane waves, laser beams have Weakness of MI analysis l l l 9 Unlike plane waves, laser beams have finite power as well as transverse dependence Numerical support? Not possible in the sixties

G. Fibich, and B. Ilan Optics Letters, 2001 and Physica D, 2001 10 G. Fibich, and B. Ilan Optics Letters, 2001 and Physica D, 2001 10

Testing the Bespalov-Talanov model l l 11 Solve the NLS Input beam is Gaussian Testing the Bespalov-Talanov model l l 11 Solve the NLS Input beam is Gaussian with 10% noise and P = 15 Pcr

Z=0 l l 12 Z=0. 026 Blowup while approaching a symmetric profile Even at Z=0 l l 12 Z=0. 026 Blowup while approaching a symmetric profile Even at P=15 Pcr, noise does not lead to MF in the NLS

Model for noise-induced MF NLS with saturating nonlinearity (accounts for plasma defocusing) Initial condition: Model for noise-induced MF NLS with saturating nonlinearity (accounts for plasma defocusing) Initial condition: cylindrically-symmetric Gaussian profile + noise 13

Typical simulation (P=15 Pcr) Ring/crater is unstable 14 Typical simulation (P=15 Pcr) Ring/crater is unstable 14

Noise-induced MF l l l 15 MF pattern is random No control over number Noise-induced MF l l l 15 MF pattern is random No control over number and location of filaments Disadvantage in applications (e. g. , eye surgery, remote sensing)

Can we have a deterministic MF? 16 Can we have a deterministic MF? 16

Vectorial effects and MF l l l 17 NLS is a scalar model for Vectorial effects and MF l l l 17 NLS is a scalar model for linearly-polarized beams More comprehensive model – vector nonlinear Helmholtz equations for E = (E 1, E 2, E 3) Linear polarization state E = (E 1, 0, 0) at z=0 leads to breakup of cylindrical symmetry Preferred direction Can this lead to a deterministic MF?

Linear polarization - analysis l 18 vector Helmholtz equations for E = (E 1, Linear polarization - analysis l 18 vector Helmholtz equations for E = (E 1, E 2, E 3)

Derivation of scalar model l Small nonparaxiality parameter l Linearly-polarized input beam E = Derivation of scalar model l Small nonparaxiality parameter l Linearly-polarized input beam E = (E 1, 0, 0) at z=0 19

NLS with vectorial effects l 20 Can reduce vector Helmholtz equations to a scalar NLS with vectorial effects l 20 Can reduce vector Helmholtz equations to a scalar perturbed NLS for = |E 1|:

Vectorial effects and MF l l 21 Vectorial effects lead to a deterministic breakup Vectorial effects and MF l l 21 Vectorial effects lead to a deterministic breakup of cylindrical symmetry with a preferred direction Can it lead to a deterministic MF?

Simulations l l l 22 Cylindrically-symmetric linearly-polarized Gaussian beams 0 = c exp(-r 2) Simulations l l l 22 Cylindrically-symmetric linearly-polarized Gaussian beams 0 = c exp(-r 2) f = 0. 05, = 0. 5 No noise!

P = 4 Pcr Ring/crater is unstable 23 Splitting along x-axis P = 4 Pcr Ring/crater is unstable 23 Splitting along x-axis

P = 10 Pcr 24 Splitting along y-axis P = 10 Pcr 24 Splitting along y-axis

P = 20 Pcr 25 more than two filaments P = 20 Pcr 25 more than two filaments

What about circular polarization? 26 What about circular polarization? 26

G. Fibich, and B. Ilan Physical Review Letters, 2002 and Physical Review E, 2003 G. Fibich, and B. Ilan Physical Review Letters, 2002 and Physical Review E, 2003 27

Circular polarization and MF l l l 28 Circular polarization has no preferred direction Circular polarization and MF l l l 28 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?

Standard model (Close et al. , 66) l 29 Neglects E 3 while keeping Standard model (Close et al. , 66) l 29 Neglects E 3 while keeping the coupling to the weaker Ecomponent

Circular polarization - analysis l 30 vector Helmholtz equations for E = (E+, E-, Circular polarization - analysis l 30 vector Helmholtz equations for E = (E+, E-, E 3)

Derivation of scalar model l Small nonparaxiality parameter l Nearly circularly-polarized input beam at Derivation of scalar model l Small nonparaxiality parameter l Nearly circularly-polarized input beam at z=0 31

NLS for nearly circularly-polarized beams l 32 Can reduce vector Helmholtz equations to the NLS for nearly circularly-polarized beams l 32 Can reduce vector Helmholtz equations to the new model: l Isotropic to O(f 2)

Circular polarization and MF l l l 33 New model is isotropic to O(f Circular polarization and MF l l l 33 New model is isotropic to O(f 2) Neglected symmetry-breaking terms are O( f 2) Conclusion – small deviations from circular polarization unlikely to lead to MF

Back to Linear Polarization 34 Back to Linear Polarization 34

Testing the vectorial explanation l l 35 Vectorial effects breaks up cylindrical symmetry while Testing the vectorial explanation l l 35 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

A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, B. Optics Letters, 2004 36 A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, B. Optics Letters, 2004 36

First experimental test of vectorial explanation for MF l l 37 Observe a deterministic First experimental test of vectorial explanation for MF l l 37 Observe a deterministic MF pattern does not rotate with direction of input beam polarization MF not caused by vectorial effects Possible explanation: collapse is arrested by plasma defocusing when vectorial effects are still too small to cause MF

So, how can we have a deterministic MF? 38 So, how can we have a deterministic MF? 38

Ellipticity and MF l l l 39 Use elliptic input beams 0 = c Ellipticity and MF l l l 39 Use elliptic input beams 0 = c exp(-x 2/a 2 -y 2/b 2) Deterministic breakup of cylindrical symmetry with a preferred direction Can it lead to deterministic MF?

Possible MF patterns l Solution preserves the symmetries x -x and y -y y Possible MF patterns l Solution preserves the symmetries x -x and y -y y x 40

Simulations with elliptic input beams l NLS with NL saturation l elliptic initial conditions Simulations with elliptic input beams l NLS with NL saturation l elliptic initial conditions 0 = c exp(-x 2/a 2 -y 2/b 2) P = 66 Pcr No noise! l l 41

l l 42 e = 1. 09 central filament pair along minor axis filament l l 42 e = 1. 09 central filament pair along minor axis filament pair along major axis

l l 43 e = 2. 2 central filament quadruple of filaments very weak l l 43 e = 2. 2 central filament quadruple of filaments very weak filament pair along major axis

All 4 filament types observed numerically 44 All 4 filament types observed numerically 44

Experiments l l 45 Ultrashort (170 fs) laser pulses Input beam ellipticity is b/a Experiments l l 45 Ultrashort (170 fs) laser pulses Input beam ellipticity is b/a = 2. 2 ``Clean’’ input beam Measure MF pattern after propagation of 3. 1 cm in water

P=4. 8 Pcr l 46 Single filament P=4. 8 Pcr l 46 Single filament

P=7 Pcr l 47 Additional filament pair along major axis P=7 Pcr l 47 Additional filament pair along major axis

P=18 Pcr l 48 Additional filament pair along minor axis P=18 Pcr l 48 Additional filament pair along minor axis

P=23 Pcr l 49 Additional quadruple of filaments P=23 Pcr l 49 Additional quadruple of filaments

All 4 filament types observed experimentally 50 All 4 filament types observed experimentally 50

Rotation Experiment 5 Pcr 51 7 Pcr 10 Pcr MF pattern rotates with orientation Rotation Experiment 5 Pcr 51 7 Pcr 10 Pcr MF pattern rotates with orientation of ellipticity 14 Pcr

2 nd Rotation Experiment MF pattern does not rotate with direction of input beam 2 nd Rotation Experiment MF pattern does not rotate with direction of input beam polarization 52

Dynamics in z 53 Dynamics in z 53

G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, Optics Letters, 2004 54 G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, Optics Letters, 2004 54

Control of MF in atmospheric propagation l l l 55 Standard approach: produce a Control of MF in atmospheric propagation l l l 55 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 (>10 GW)

Ellipticity-induced MF in air l l 56 Input power 65. 5 GW (~20 Pcr) Ellipticity-induced MF in air l l 56 Input power 65. 5 GW (~20 Pcr) Noisy, elliptic input beam Typical Average over 100 shots

Experimental setup l 57 Control astigmatism through lens rotation Experimental setup l 57 Control astigmatism through lens rotation

MF pattern after 5 meters in air l l l 58 Strong central filament MF pattern after 5 meters in air l l l 58 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 average over 1000 shots typical

Average over 1000 shots typical 59 Average over 1000 shots typical 59

Control of MF - position l l 60 = 00 : one direction (input Control of MF - position l l 60 = 00 : one direction (input beam ellipticity) = 200 : all directions (input beam ellipticity +rotation lens)

Control of MF – number of filaments l l 61 = 00 = 200 Control of MF – number of filaments l l 61 = 00 = 200 2 -3 filaments single filament

Ellipticity-induced MF is generic l l l 62 cw in sodium vapor (Grantham et Ellipticity-induced MF is generic l l l 62 cw in sodium vapor (Grantham et al. , 91) 170 fs in water (Dubietis, Tamosauskas, Fibich, Ilan, 04) 200 fs in air (Fibich, Eisenmann, Ilan, Zigler, 04) 130 fs in air (Mechain et al. , 04) Quadratic nonlinearity (Carrasco et al. , 03)

Summary - MF l l l 63 Input beam ellipticity can lead to deterministic Summary - MF l l l 63 Input beam ellipticity can lead to deterministic MF Observed in simulations Observed for clean input beams in water Observed for noisy input beams in air Ellipticity can be ``stronger’’ than noise

Theory needed l l l 64 Currently, no theory for this high power strongly Theory needed l l l 64 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




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