The variational formulation of the theory of non-stationary propagation of a femtosecond laser radiation
aa r X i v : . [ phy s i c s . op ti c s ] D ec The variational formulation of the theory ofnon-stationary propagation of a femtosecond laserradiation
Andrey D.BulyginAugust 12, 2018
Abstract
In this paper, an inverse variational problem is solved for the nonlocalnonlinear Schrdinger equation used in modeling filamentation in variousnonlinear media. The corresponding integral relations are found whichgeneralize the conservation laws for the non-conservative case. key words : nonlinear Schrodinger equation,variational formulation,virial the-orem,conservation laws,filamentation
The phenomenon of self-focusing and filamentation of high-power laser radi-ation (HFLR) is most often described on the basis of the so-called nonlinearSchrodinger equation (NSE) [1]. The variational formulation of this equation,in application to the problem of filamentation (with a specific type of equa-tions for the medium) is widely known in the literature solely for stationaryapproximation. Within this stationary approximation, only a very limited setof rigorous analytical results is known [1]. This property of the global collapsein the Kerr medium, Townes’ solution (which is also not analytic [2]) and theBespalov-Talanov instability [1]. One way or another, at the moment there areactually not exist known rigorous properties of non-stationary (NES), whichwould allow, for example, to evaluate the correctness of the numerical solu-tion. In this paper we propose a procedure for constructing the Lagrangianformulation of these equations (and hence also the Hamiltonian formulation) .The constructing the Hamiltonian formulation allows not only to automaticallywrite out the integrals of motion for non stationary models, but also providesa tool that allows the canonical implementation of the program of stochastic The transition from the Lagrangian to the Hamiltonian formulation in the case of anondegenerate theory is realized by means of the Legendre transformation, otherwise, i.e. inthe theory with constraints, the transition is carried out on the basis of the Dirac-Bergmanalgorithm [5]
The non stationary model of filamentation takes into account the effects ofdispersive spreading of the laser pulse, the effect of plasma accumulation overtime, cubic nonlinearity and higher-order nonlinearities [1]. In the so-calledconcomitant coordinate system, the equation for the complex envelope of theintensity of the light field is written in the Fourier representation in the form[11]: T u ≡ i ∂∂z U + ( ∂ µ ∂ µ + ǫ ker [ I ] + ǫ pl [ I ]) U + iα I U = 0 (1)Here ǫ ker [ I ] functional of the intensity, which we in accordance with theworks [11] , we choose in the form: ǫ ker ( t ) = n ker ( I + ǫ in [ I ])where ǫ in = Z ∞−∞ θ ( t − τ ) R ( t − τ ) I ( τ ) dτ. Here, the molecular response in the Kerr effect is modeled by the Green’sfunction of a damped oscillator: R ( t ) = exp ( − γ R t ) sin (Ω R t ). Equation for ǫ pl caused by the formation of a plasma under the action of laser radiation has theform: ∂ǫ pl ∂t = ( ǫ pl − ǫ pl ) ψ pl ( I ) + ψ pl ( I, ǫ pl )Here the ionization coefficient ψ pl is a nonlinear function of the intensity andcan be calculated, for example, according to the Popov-Perelomov-Terentyevmodel [14]); ǫ pl - the dimensionless concentration of molecules in the medium; ψ pl - The term by means of which takes into account such mechanisms as cascadeionization and recombination .The set of equations (1) can be represented in the form T u ≡ i ∂∂z U + ( ∂ µ ∂ µ + ǫ ker [ I ] + ǫ pl [ I ]) U + iα I U = 0 T ǫ pl ≡ ∂ǫ pl ∂t − ( ǫ pl − ǫ pl ) ψ pl ( I ) + ψ pl ( I, ǫ pl ) = 0 T ǫ in ≡ ∂ ǫ in ∂t + γ R ∂ǫ in ∂t + Ω R ǫ in = I (2)2his system of equations (2), generally speaking, is not Lagrangian, sincethe necessary condition for it is not satisfied δδǫ T u = δδu T ǫ Although, for the inertial part of the Kerr nonlinearity, the following condi-tion is satisfied: δδǫ in T u = δδu T ǫ in nevertheless the equation itself T ǫ in It is not Lagrangian, since this is anequation with friction, namely, an equation of the type of a damped oscillator.Thus, the non-Lagrangianity of the system (2) is due to two circumstances.First, the equations for nonlinear dielectric permittivity contain the first timederivatives, this applies to both the inertial Kerr part and the plasma part.Secondly, the term in the equation for the light field, which causes interactionwith the medium, does not have a corresponding partner in the equation for themedium, i.e. ∂ ǫ T intu = ∂ u T intǫ .The solution of the first problem is presented in a general form in the paper[8]. Thus, considering the intensity of the laser field as the driving force in theright-hand side of the equation for the harmonic damped oscillator, to whichthe inertial Kerr nonlinearity is modeled, we can, respectively, write the actionfor the inertial term in the form: S in = n ker Z e γ R t (( ˙ ǫ in p ǫ − ǫ in ˙ p ǫ − ( p ǫ + γ R ǫ in p ǫ + ǫ in Ω R )) / ǫ in I ( t )) d~xdz. Where p ǫ = ∂ǫ/∂t .We will notice that to receive the initial system of the equations it is nec-essary to increase together full Lagranzhian systems by this integrating mul-tiplier (because of presence of the member of interaction) that corresponds tomultiplication of all equations by this multiplier, with only that remark thatto compensate the excess composed arising in the equations for the field frommembers in the form of γ R t∂ t U ∂ t U ∗ of it is necessary to add to action the mem-ber in shape γ R ( ∂ t U U ∗ − ∂ t U ∗ U ) e γ R t , what leading to actually to physicallyequivalent system of the equations. However given compensating composed ispurely imaginary that represents some complexity for physicalThe situation with the equation for the plasma is somewhat different, to findsuch an integrating factor that would ensure the Lagrangianity of the equationfor the plasma itself and compatibility with the equation for the light field, itseems to us possible, only by approximate perturbative methods. However, thepractical value of this approach in this case is extremely small. n this situation,it seems reasonable to introduce an auxiliary field. From physical considerations,one can guess which auxiliary field should be introduced in order to ensure theLagrangianity of the complete system of equations. The remark can serve asa remark that the dynamics of electrons, in general, should be described by3n equation for a complex field having the meaning of the probability densityamplitude. Thus, it is natural to try to change from the real function of theplasma density to the complex value χ , which has the meaning of the amplitudeof the density of the number of particles participating in the energy exchangewith the light field, which satisfies the Schrodinger equation and is related tothe electron density ǫ pl , as χχ ∗ = ǫ − ǫ pl . Then it is easy to verify that thesystem of equations (2) is equivalent to the following system of equations: T (0) u + iα I U ≡ i ∂∂z U + ( ∂ µ ∂ µ + ǫ ker [ I ] + ǫ pl [ I ]) U + iα I U = 0 T (0) χ + iα χ χ ≡ i ∂χ∂τ m + χI + iα χ χ = 0 T (0) ǫ in + γ R ∂ǫ in ∂t ≡ ∂ ǫ in ∂t + γ R ∂ǫ in ∂t + Ω R ǫ in = I (3)For a given system of equations, it is no longer difficult to write down anaction: S c = S ku + S χ + S in + S kin + iS f + iS in . (4)Here we have selected a part of the action, so that the variation of this partgives a derivative with respect to the evolution variable z for the field U and thenonlinearity of the Kerr type: S ku = Z ( i ( ∂∂z U ) U ∗ − i ( ∂∂z U ∗ ) U + ǫ k I / e γ R t d~xdz. The term in action responsible for the kinetic part has, respectively, the form: S d = Z ∂∂ µ U ∂∂ µ U ∗ e γ R t d~xdz. Part of the action, the variation of which gives a derivative for the field χ onthe variable t: S χ = Z ( i ( ∂∂t χ ) χ ∗ − i ( ∂∂t χ ∗ ) χ ) e γ R t d~xdz The term in action determining the phase interaction of the fields χ and U: S int = Z ( I ( χχ ∗ ) ) e γ R t d~xdz. Finally, the imaginary part of the action, which determines the exchange ofquantity of matter: S f = Z αe γ R t d~xdz. and : S in = Z γ R ( ∂ t U U ∗ − ∂ t U ∗ U ) e γ R t d~xdz. H c = R αd~x . Generally speaking, the introduction of an imaginary action is not avery good procedure , in that, for example, it can not automatically write outthe integrals of motion, nevertheless, such models with imaginary action arediscussed, for example, in work [12] or is used in describing scattering processeswith absorption of so-called optical model of the nucleus. For us, the immediatebenefit of such a record is that it immediately shows how unambiguously theionization coefficients are related to the plasma equations and the nonlinearabsorption coefficient in the NSE equation for the light field.Next, we turn to a number of special cases:a) S f = 0, S χ = 0, S int = 0, γ R = 0 - This situation is a medium with a cubicinertial nonlinearity in the absence of any dissipative mechanisms. In this case,the original system is a real and Lagrangian system.Conservation laws corresponding to external global symmetries can be ex-pressed by means of the energy-momentum tensor P νµ constructed in the stan-dard way in terms of the Lagrange function.In particular, the law of conservation of energy (corresponding to the sym-metry with respect to the shift in the evolutionary variable, which in this caseis z), written in a differential form, will take the following form: ∂h g ∂z = − ∂ j P zj Here h g ≡ P zz = ∂ µ U ∂ µ U ∗ + n ker ( I / p ǫ − Ω R ǫ in ) / ǫ in I ( t ))and p ǫ ( t ) = Z θ ( t − ζ ) exp ( − γ R ( t − ζ ))( − Ω R ǫ in ( ζ ) + I ( ζ )) dζ Accordingly, in the integral form we have: H g ≡ Z h g d~x = H d + H ker + H in = const Next we will include in our consideration dissipative mechanisms associatedwith the oscillatory friction γ R >
0. Then, after a series of transformations, onecan obtain: ∂H g ∂z = n ker Z ( γ R p ǫ ( Z ∞−∞ θ ( x − τ ) R ( x − τ ) s ( τ ) dτ )) d~x this situation also arises when considering the usual linear absorption in a medium ifone looks at the Schrodinger equation from the position of quantum mechanics, then thecorresponding evolution operator is not Hermitian (and therefore he can not be associatedwith a physical quantity), and evolution is not unitary, however, in optics, the considerationof such equations is standard s ν = ( ∂ ν U U ∗ − ∂ ν U ∗ U ) / (2 i ). This integral relation is a generalizationof the widely known integral relation for a medium with instantaneous cubicnonlinearity [13] (and as far as we know it is written out in explicit form for thefirst time), and can be similar to him (see for example [10]) be used to verifythe correctness of numerical calculations, but already for a medium with inertialcubic nonlinearity.b) We consider the case only with condition S f = 0. In this case, the actionis real, and the interaction between the radiation and the field χ reduce to afictitious phase rotation of the field χ . This situation corresponds to the stage ofnonstationary self-focusing in the medium by inertial cubic nonlinearity. Thenit is not difficult to write down the integrals of motion. So the integrals ofmotion corresponding to the internal symmetry - the phase rotation, have theform: χχ ∗ = const The law of conservation of the number of particles for plasma. Z ( U U ∗ ) d~x = const The law of conservation of the number of particles for a light field.This situation is of interest because in it for the Hamilton function H g ≡ H d + H ker + H in + H int + H χ the simple relation: H g ≡ Z h g d~x = const And in this case the difference in the expression for H from the previousone is given by the term H int + H χ ≡ H int ≡ − R ( I ( χχ ∗ ) ) d~x , which in thecase ( chi chi ∗ ) = const reduces to a linear combination of the integral of themotion H in an inertial cubic medium and the integration expressing the lawof conservation of the number of particles.c) Let us pass to the consideration of the general case. The integrands ofmotion for the number of particles in this case take the form:( χχ ∗ e R t α ρ ρdξ ) = const Z ( U U ∗ e R z R α I Idζd~x/ ( R Id~x ) ) d~x = const For H g ≡ H d + H ker + H in + H int + H χ from the equations of dynamics in fullform, one can obtain: ∂∂z H g = − R ( iα I ( U Φ ∗ U − U ∗ Φ U ) + iα χ ( χ δG ∗ χ ( I ) δI ∂s ν ∂x ν − χ ∗ δG χ ( I ) δI ∂s ν ∂x ν ) + γ R p ǫ ∂G ǫ ( I ) ∂I ∂s ν ∂x ν ) d~x χ ( I ) is a well-known function of I in many cases, because for a numberof important applications ψ pl = 0. In this case, the right-hand side has the formof an explicit function of the field U . After simple transformations it is possibleto obtain: ∂H g ∂z = − Z α I ((2 h d + I δh ker δI ) − ∂ ν ∂ ν I ) − χχ ∗ ( α I + ¯ α Iρ ) d~x Where¯ α Iρ ≡ ∂ z Z t ( α ρ ) dξ = − Z t (( α I + ∂ ν s ν ) α Iρ + ( χχ ∗ ( ∂ z Z ξ ( α ρ ) dζ )) α ρρ ) dξ This ratio is, again, useful for the possibility of checking and correctingnumerical calculations already in the complete model of filamentation. Conclusion
In this paper, we propose a variational formulation of the system of equationsfor the nonlinear Schrodinger equation and the plasma density. To proceed tothe variational formulation, we needed to apply various methods, this is themethod of integrating factors for solving the inverse variational problem andthe method of auxiliary fields. So for the terms describing the effect of inertialcubic self-action, the transition to the variational formulation can be achievedby the method of integrating factors. However, the compatibility condition forthe equations for plasma and NSE can hardly be achieved by a similar method.The variational formulation in this case can be achieved by introducing an aux-iliary field Also in this paper we analyze equations for NLES integrals in variousof its formulations including a complete nonstationary model with nonlinear dis-sipation . Knowledge of these conservation laws is a valuable tool not only forchecking the correctness of numerical schemes widely used in numerical mod-eling of NLES, but also for developing fully conservative methods of numericalintegration of these equations.
References [1]
R.W. Boyd , S.G. Lukishova , Y.R. Shen , Self-focusing: Past andPresent ,
Springer Science, Business Media: LLC. (2009).[2]
K.D. Moll , L.A. Gaeta , G. Fibich , Phys.Rev.Lett. :20 ,203902(2003) It should be noted that in the literature, a more general equation is used to describea broadband extremely short laser pulse than the nonlinear Schrodinger equation, namely anonlinear equation of the generalized Kadomtsev-Petviashvili type [15,1]. As far as we know,its Lagrangian function is written out only for the case of local nonlinearity considered by usin the first section. The construction of the Lagrangian formulation, for this more generalform of the equations, in the case of nonlocal effects, such as inertial cubic nonlinearity andplasma formation, requires a separate study that goes beyond the scope of this work.
L.L.Tatarinova ,M.E.Garcia ,Phys. Rev. A. :2,021806 (2008)[4] Alekseenko V. N.
The integrals of the motion of nonlinear equa-tions of Schrodinger type // Differ. Uravn. 1976. Vol. 12, No. 6.P. 11211122(In Russ.)[5]
M. Henneaux and C. Teitelboim , Quantization of gauge systems
Princeton U.P., NJ (1992).[6]
Yu.E. Geints,A.D. Bulygin, A.A. Zemlyanov
Applied PhysicsB. :1,243255 (2012)[7]
Chekalin S. V., Kandidov V. P.
From self-focusing light beams tofemtosecond laser pulse filamentation Phys. Usp. 56 123140 (2013); DOI:10.3367/UFNe.0183.201302b.0133[8]
D.M. Gitman, V.G. Kupriyanov arXiv:hep-th/0605025v4 28 Feb 2007[9]
P.O. Kazinski, S.L.Lyakhovich, A.A. Sharapov
JHEP. :X 076(1-42) (2005)[10]
Kandidov,V P et al.
Nonlinear-optical transformation of a high-powerfemtosecond laser pulse in air //Quantum Electronics.2003.Vol. 33, No. 1.P. 69–75.[11]
A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramirez-Gongora, and M. Kolesik
Eur.Phys.J. Journal Special Topics :1,5-76 (2011)[12]
Menskii M. B.
Dissipation and decoherence in quantum systems Phys.Usp. 46 11631182 (2003); DOI: 10.1070/PU2003v046n11ABEH001680[13]
Vlasov S. N., Talanov V. I.
Samofokusirovka Voln (Self-focusing ofWaves), Nizhny Novgorod:Izd. IPF RAN.1997. 220 p.[14]
S.V. Popruzhenko,V.D. Mur, V.S. Popov,D. Bauer
Phys. Rev.Lett. :19, 193003 (2008)[15]