Collapses and nonlinear laser beam combining. Pavel Lushnikov

Transcription

1 Collapses and nonlinear laser beam combining Pavel Lushnikov Department of Mathematics and Statistics, University of New Mexico, USA Support: NSF DMS , NSF DMS , NSF PHY , NSF DMS , DOE

2 Collaborators: Sergey Dyachenko (Brown U.), Denis Silantyev (UNM) and Natalia Vladimirova (UNM) Support: NSF DMS , NSF DMS , NSF PHY , NSF DMS , DOE

3 - Explosive instability (blow-up) formation of singularity in a finite time - Collapse blow-up with the contraction of the spatial extent of solution to zero

4 Self-focusing (collapse) of laser beam Laser beam Nonlinear medium Singularity point - 2D Nonlinear Schrödinger Equation - amplitude of light

5 Self-focusing of optical bullet Laser beam Nonlinear medium Singularity point - 3D Nonlinear Schrödinger Equation (NLSE)

6 Nonlinear Schrödinger Equation (NLSE) Conserved Integrals: - Hamiltonian: -optical power (in optics) or number of particles (in quantum mechanics) or wave action in oceanology

7 Mean square width: Virial theorem 1 : Singularity formation: 1 S.N.Vlasov, V.A Petrishchev, and V.I. Talanov (1971). V.E. Zakharov JETP, (1972)

8 Critical collapse of 2D Nonlinear Schrödinger Equation: Self-similar solution near singularity Soliton solution of NLSE: LogLog law 1 : 1 G. Fraiman (1985); M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem (1987); A. Dyachenko, A. Newell, A. Pushkarev and V.E. Zakharov (1992); F. Merle and P. Raphael (2006).

9 Strong vs. weak collapse of NLSE D=2: Strong critical collapse (mass critical) as above D=3: Weak supercritical collapse Self-similar variable Number of particles in collapsing region

10 Collapses at the National Ignition Facility

11 National Ignition Facility Target Chamber

12 Target

13 Thermonuclear burn D+T= 4 He (3.5 Mev)+n (14.1 Mev) Required temperature: 10 KeV=100 millions Celsius D+ 3 He = 4 He (3.7 Mev)+p (14.7 Mev) Required temperature: 100 KeV=1000 millions Celsius

14 Goal: propagation of laser light in plasma with minimal distortion Difficulties: collapses from self-focusing of light and Langmuir wave collapses Strong beam spray No spray Laser propagation in plasma

15 Main laser-plasma interaction effects at The National Ignition Facility 1. Stimulated Brillouin Scattering (SBS) Electromagnetic wave (from laser) k 1 Scattered electromagnetic wave Light k 2 Ion acoustic wave A particular version of SBS: cross-beam energy transfer (CBET) with both electromagnetic waves corresponding to two laser beams 2. Stimulated Raman Scattering (SRS) Electromagnetic wave (from laser) k 1 Light Scattered electromagnetic wave k 2 Electron plasma wave (Langmuir wave)

16 Collapse from Forward Stimulated Brillouin Scattering - amplitude of light - low frequency plasma density fluctuation - Landau damping - speed of sound P/ P cr =10 5

17 Stimulated Brillouin Scattering (SBS) is strongly affected by Z number both for forward SBS 1,2 (at the nonlinear stage results in multiple collapses 3,4 ) and backward SBS 5. E.g. gold plasma is subject to strong SBS for ~10 times smaller laser intensities than low Z plasma 1 P.M. Lushnikov and H.A. Rose. Physical Review Letters, 92, (2004). 2 P.M. Lushnikov and H.A. Rose. Plasma Physics and Controlled Fusion, 48, 1501 (2006). 3 P.M. Lushnikov and N. Vladimirova. Optics Letters, 35, (2010). 4 Y. Chung and P.M. Lushnikov, Physical Review E, 84, (2011). 5 A.O. Korotkevich, P.M. Lushnikov, and H.A. Rose, Phys. of Plasmas, 22, (2015).

18 Stimulated Raman Scattering (SRS) Electromagnetic wave (from laser) Light k 1 k 1 - k 2, ω ω p Scattered Electromagnetic wave k 2 Electron plasma wave (Langmuir wave) In the kinetic regime: 0 25 <k λ D < 0 45 λ D Debye length Found that SRS can be reduced by the filamentation from the collapse of Langmuir waves

19 Collapse of Langmuir waves: Generalized Zakharov Eq. - amplitude of Langmuir wave - nonlinear Landau damping - nonlinear frequency shift - speed of ion-acoustic waves - density of low frequency fluctuations

20 Kinetic effects in Generalized Zakharov Eq: - nonlinear Landau damping - nonlinear frequency shift

21 Kinetic effects require to solve 3+3 Vlasov equation (3 velocity dimensions and 3 spatial dimensions) for the phase space distribution function f(r,v,t) t v E v f 0 E f dv x dv y dv z Scaled units: electron thermal units 1 P.M. Lushnikov, H.A. Rose, D.A. Silantyev, and N. Vladimirova. Physics of Plasmas, 21, (2014). 2 D.A. Silantyev, P.M. Lushnikov and H.A. Rose. Submitted to Physics of Plasmas, ArXiv: (2016). 3 D.A. Silantyev, P.M. Lushnikov and H.A. Rose. Submitted to Physics of Plasmas, ArXiv: (2016).

22 Another example of collapse: If quadratic optical nonlinearity is added to Kerr nonlinearity then NLSE is replaced by Davey-Stewartson (Benney-Roskes) equation 1,2 1 M.J. Ablowitz, G. Biondini, S. Blair, Phys. Lett. A 236 (1997) M.J. Ablowitz, G. Biondini, S. Blair, Phys. Rev. E 63 (2001) 605.

23 Collapse of Davey-Stewartson (Benney-Roskes) equation The Hamiltonian Virial theorem collapse 1 1 G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Physica D, 72, 61 (1994)

24 Critical NLSE collapse ground state soliton of NLSE Critical collapse in Davey-Stewartson Eq (DSE): ground state soliton of DSE 1 M.J. Ablowitz and H. Segur (1979); M.J. Ablowitz, I. Bakirtas and B. Ilan (2005).

25 Critical collapse of 2D Nonlinear Schrödinger Equation: Self-similar solution near singularity Soliton solution of NLSE: LogLog law 1 : 1 G. Fraiman (1985); M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem (1987); A. Dyachenko, A. Newell, A. Pushkarev and V.E. Zakharov (1992); F. Merle and P. Raphael (2006).

26 But simulations failed to confirm log-log law in a convincing way 1 although the exact proof of the existence of log-log scaling was given 2 Example of NLS simulations: L(t) depends on initial conditions 1 See e.g.: N.E. Kosmatov, V.F. Shvets and V.E. Zakharov (1991); G. D. Akrivis, V.A. Dougalis, O.A. Karakashian, and W.R. McKinney (2003). 2 F. Merle and P. Raphael (2006).

27 A little of history of 2D NLS collapse G.A. Askaryan: Self-focusing of laser beam R.Y. Chiao, E. Garmire and C.H. Townes: self-trapping of laser beam, Townes soliton as instability threshold V.I. Talanov: lens transform S.N.Vlasov, V.A Petrishchev, and V.I. Talanov: Virial theorem and exact proof of collapse formation M.I. Weinstein. Sharp estimates on collapse E.A. Kuznetsov and S.K. Turitsyn: conformal symmetry and Noether theorem G. Fraiman : almost log-log scaling of collapse M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem: log-log scaling of collapse A. Dyachenko, A. Newell, A. Pushkarev and V.E. Zakharov: collapse turbulence V.M. Malkin: collapse in terms of the excess of number of particles above critical F. Merle and P. Raphael: exact proof of existence of log-log scaling

28 Working on NLSE and DSE collapse in parallel NLSE blow-up variables and lens transform where - adiabatically slow small parameter Looking for solution in the form

29 Looking for the DSE collapsing solution as For blow up variables Collapse and For Davey-Stewartson Eq. transforms into where - adiabatically slow small parameter and

30 Looking for solution in the form In adiabatic approximation of slow

31 Tail minimization principle: during collapse dynamics system dynamically select collapsing solution with minimal tail amplitude Then we look for V 0 with the minimal tail

32 NLSE: In adiabatic approximation of slow minimizing tails by shooting method: V

33 Approximation through ground state soliton R

34 NLSE: Full solution V match the envelope of V 0 of in the tail: V0 V - from numerics V 0 soliton with R ground state soliton with V V 0 to the left from

35 DSE: In adiabatic approximation of slow minimizing tails by Newton-conjugate-gradient method (J. Yang, 2009) combined with the correct choice of the asymptotic at infinity: 0 p or q

36 Approximation through DSE ground state soliton Ground state soliton solution of Davey-Stewartson Eq.

37 NLSE: Full solution V match the envelope of V 0 of in the tails along both spatial directions

38 NLSE: How to extract L(t) and from simulations: The analysis of Taylor series solution of at Then is found from the implicit equation for each given and DSE: qualitatively similar procedure

39 But simulations failed to confirm log-log law in a convincing way 1 although the exact proof of the existence of log-log scaling was given 2 Example of NLS simulations: L(t) depends on initial conditions 1 See e.g.: N.E. Kosmatov, V.F. Shvets and V.E. Zakharov (1991); G. D. Akrivis, V.A. Dougalis, O.A. Karakashian, and W.R. McKinney (2003). 2 F. Merle and P. Raphael (2006).

40 NLSE: Modifying the standard theory L(t) is not universal but b t ( b ) is universal:

41 DSE: universality of b t ( b )

42 Focus on NLSE collapse: Recall that we are looking for solution in the form - has the imaginary part because of slow dependence of on : Also we need to make sure that has only outgoing waves for In analogy with Gamov a-decay theory introduce nonself-adjoint problem where as can be determined from the balance of the norm of V

43 In other words, we look at as the Schrodinger equation with the effective potential U: and complex eigenvalue E: 2 turning points and of WKB:

44 Solution near V0 V - from numerics V 0 soliton with R ground state soliton with V V 0 to the left from

45 Oscillating tail is given by the linear combination of confluent hypergometric functions of the first and second kinds: Matching asymptotics and using WKB give Here of ground state soliton is determined by the asymptotic Asymptotics of complex solution

46 Introducing the number of particles to the left of the second turning point and balancing the flux of particles through that point New basic ODE system Here

47 Compare with old basic ODE system of the standard theory Asymptotic solution near collapse time t c : 1 G. Fraiman (1985); M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem (1987); A. Dyachenko, A. Newell, A. Pushkarev and V.E. Zakharov (1992); V. F. Malkin (1993).

48 Returning to previous Figure L(t) is not universal but b t ( b ) is universal:

49 Finding asymptotic of a new basic ODE system

50 Using from the inversion of previous expression and inverting that equation

51 Asymptotic of new basic ODE system

52 Simulations vs. analytic

53 Simulations vs next order analytic 0.35 L log-log Solid numerics Dashed - analytics t0 t

54 Simulations vs. analytic larger interval starting from the initial Gaussian L Solid numerics Dashed - analytics t0 t

55 In comparison, the standard log-log scaling dominates only for amplitudes above Googol 10 = 10 = Googolplex 1 P.M. Lushnikov, S.A. Dyachenko and N. Vladimirova. Physical Review A, v. 88, (2013).

56 Evolution of the average output power of nearly diffraction limited fiber lasers (emitting either a continuous wave or ultrashort pulses) 1 IPG Photonics 1 C. Jauregui, J. Limpert, and A. Tunnermann, Nature Photonics 7, 861 (2013)

57 Schematic of a high-power double-clad fiber amplifier The signal is coupled in the fiber core, which contains the active material, whereas the pump is coupled in the fiber cladding. This structure allows the pump to be progressively absorbed by the active material in the core as the pump propagates along the fiber. This absorbed pumped energy is used to amplify the signal.

58 Advantages of fiber lasers - Alignment-free laser systems - High efficiency (50-80%) - Compact design - Maintenance-free operation Disadvantage of fiber lasers - Mode instabilities limiting average power

59 Commercially available IPG Photonics Fiber lasers up to 50kW 1 1

60 Overcoming power limitations: Laser beam combining Standard schemes 1 1

61 Coherent beam combining: Combine several laser beams such that the phase of each laser beam is controlled to ideally produce the combined beam with the coherent phase. Example 1 : five 500W laser beams into 1.9kW Gaussian beam with a good beam quality M 2 = 1.1. Difficulties in coherent beam combining: -Complicated adaptive optics scheme -Bad scaling with power due to nonlinearity 1 S. M. Redmond, D. J. Ripin, C. X. Yu, S. J. Augst, T. Y. Fan, P. A. Thielen, J. E. Rothenberg, and G. D. Goodno, Opt. Lett. 37, 2832 (2012).

62 New proposal 1,2 : Use nonlinearity to our advantage to achieve combining of multiple laser beams into a diffraction-limited beam by the strong selffocusing in a waveguide with the Kerr nonlinearity. 1 P.M. Lushnikov and N. Vladimirova, Optics Letters 39, (2014). 2 P.M. Lushnikov and N. Vladimirova, Optics Express 23, (2015).

63 Nonlinear laser beam combining (1) Simpler limit: combining of a few laser beams into a single diffraction-limited beam

64 Propagation and combining of 3 beams along z z Cross sections at different z

65 Propagation and combining of 7 beams along z z Cross sections at different z

66 Laser beam quality M 2 after exit of the collapsed beam from Kerr media based on least square fit of beam waste w on the propagation distance z 1 1 T. Sean Ross, Laser Beam Quality Metrics, SPIE Press, 204 pages (2013)

67 (2) Nonlinear laser beam combining of multiple beams Beams from fiber lasers Side-by-side combining Nonlinear propagation of multiple beams Output coherent beam Laser amplitudes and phases in fiber cross sections:

68 z-dependence of the maximum light amplitude at the crosssection vs. z: Regularization of collapse

69 Probability density function (PDF) of the collapse distance N=6N c N=5N c N=4N c

70 Inverse cascade before collapse

71 Physical units: -wavelength in vacuum - linear index of refraction - nonlinear Kerr index with - laser intensity Critical power: - ground state soliton

72 Optical fiber and laser intensity parameters for the nonlinear beam combining in fused silica: - Laser intensity for the continuous wave operations Optical fiber length Optical fiber diameter Combined beam power Requires to combine several hundreds of the commercially available fiber lasers. The proposed scheme does not high quality beams for combining.

73 Short pulse operations for the nonlinear beam combining in fused silica: Fused silica optical damage threshold: Fiber length and cross section can be scaled down by a factor ocompare with the continuous wave operations 1 P.M. Lushnikov and N. Vladimirova, Optics Letters 39, (2014). 2 P.M. Lushnikov and N. Vladimirova, Optics Express 23, (2015).

74 Conclusion -Suggest that optical collapses will be around for a very long time -Propose to achieve a nonlinear beam combining by propagating multiple laser beams in the waveguide with the Kerr nonlinearity. - Large fluctuations during propagation seed the collapse event resulting in the formation of near diffraction-limited beam. -Optical fiber length with optical fiber diameter is sufficient to achieve the combined beam power