Short Laser Pulse Propagation in Nonlinear Media. From Nonlinear Schrödinger Equation to Short Pulse Equation - PDF

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Short Laser Pulse Propagation in Nonlinear Media. From Nonlinear Schrödinger Equation to Short Pulse Equation A. T. Grecu, D. Grecu Department of Theoretical Physics National Institute of Physics and Nuclear

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Short Laser Pulse Propagation in Nonlinear Media. From Nonlinear Schrödinger Equation to Short Pulse Equation A. T. Grecu, D. Grecu Department of Theoretical Physics National Institute of Physics and Nuclear Engineering Horia Hulubei Bucharest, RO , Romania, EU Abstract The propagation of light pulses in weakly nonlinear dielectric media is discussed in two opposite limits.firstly, when the width of the pulse is large enough, the relevant equation is the well known cubic nonlinear Schrödinger NLS equation. It is a generic equation describing the propagation of quasi-monochromatic waves in weakly nonlinear media, irrespective of the physical problem under study. The second case corresponds to a short pulse containing only a few oscillations of the carrier wave. Its evolution is described by the short pulse equation SPE which is derived in more restrictive conditions. Both NLSE and SPE are completely integrable although through different inverse scattering transform methods. An interesting equivalence between SPE and sine-gordon equation SGE is noted which was used to find solutions of SPE starting from well known solutions of SGE. 1 Introduction A fundamental problem in nonlinear optics is the study of the propagation of a light laser pulse through a nonlinear dielectric medium. From Maxwell s equations, without free electric charges nor current densities and with constants magnetic permeability µ = µ 0 = ε 0 c 2 1, the basic equation describing the electric field evolution writes 2 E 1 2 E c 2 t 2 E = 1 2 P ε 0 c 2 t 2, 1 where P is the polarization vector of the medium which can be separated into a linear and a nonlinear part, P = PL + P NL. We consider an isotropic medium and E and P linearly polarized E = ee, P = ep, with e the polarization vector. In general the expression of P is of the form 1 P = χ 1 t τeτ dτ+ ε 0 + χ 3 t τ 1, t τ 2, t τ 3 Eτ 1 Eτ 2 Eτ 3 dτ 1 dτ 2 dτ 3, 2 where according to the causality principle, the susceptibilities χ i are vanishing for t τ i i = 1, 3. In this expression the term with quadratic dependence on the electrical field was omitted, as it vanishes due to the inversion symmetry, and higher order terms were neglected. A medium, thus approximated, is known as the nonlinear Kerr medium and it will be used through-out the present paper. Moreover we shall assume that the electrical field depends only on one space coordinate, let it be z, Ez, t, and then 1 becomes 2 E z E c 2 t c 2 t 2 χ 1 t τez, τ dτ = 1 ε 0 c 2 2 t 2 P NLE A general property of this equation, in the case of a weak nonlinearity, χ 3 E 3 χ 1 E, is easily obtainable if one looks for solutions of the following form where E 1 E 0 and E 0 z, t is a plane wave Ez, t = E 0 z, t + E 1 z, t, 4 E 0 z, t = A exp [ikz ωt] + cc. 5 Then, considering the linear equation in the left-hand side of 3 we get kω = ˆχ c 1 ωω, 5 where ˆχ 1 ω is the Fourier transform of χ 1 t ˆχ 1 ω = The linear equation satisfied by E 1 z, t is 2 E 1 z 2 1 c 2 2 E 1 t 2 1 c 2 2 t 2 χ 1 te iωt dt. Using the expression 5 for E 0, the right-hand side of 6 becomes 9ω2 c 2 3ω2 c 2 ˆχ3 ω, ω, ωa 3 e i3kz ωt 9ω2 c 2 χ 1 t τe 1 τ dτ = 1 ε 0 c 2 2 t 2 P NLE 0. 6 ˆχ3 ω, ω, ωa 3 e i3kz ωt ˆχ3 ω, ω, ω A 2 Ae ikz ωt 3ω2 c 2 ˆχ3 ω, ω, ω A 2 A e ikz ωt, where ˆχ 3 ω 1, ω 2, ω 2 is the Fourier transform of the cubic susceptibility ˆχ 3 ω 1, ω 2, ω 2 = χ 3 t 1, t 2, t 3 e iω 1t 1 +ω 2 t 2 +ω 3 t 3 dt 1 dt 2 dt 3. This expression suggests us to look for solutions of 6 of the form [1] Introducing 7 in 6 we find Ez, t = F 1 ze iωt + F 1 e iωt + F e ze 3iωt + F 3 e 3ωt 7 But d 2 F 3 dz 2 d 2 F 1 dz ˆχ1 3ω c 2 9ω 2 F 3 = 9ω2 c 2 ˆχ3 ω, ω, ωa 3 e 3ikωz ˆχ1 ω c 2 ω 2 F 1 = 3ω2 c 2 ˆχ3 ω, ω, ω A 2 Ae ikωz 9 k3ω = 3ω c 1 + ˆχ 1 3ω 3kω = 3ω c 1 + ˆχ 1 ω 9 and consequently F 3 is bounded no resonance with the right-hand side exists. On the other hand because of the presence of the exponential term exp ikωz in the right-hand side of 9, a solution of the inhomogeneous equation is F 1 z = i 3ω c ˆχ 3 ω, ω, ω 1 + ˆχ 1 ω z A 2 Ae ikωz 10 and the asymptotic expansion 4 will fail at distances z ˆχ 3 A 2 1. Such terms are called secular terms, and their appearance invalidates the simple expansion 4, so better asymptotic 264 methods must be applied we shall use in the next section the asymptotic method of the multiple scales. Momentarily let us try to eliminate these such terms by allowing the amplitude A constant up to this point to slowly variate along z and consider a supplemental term in the right-hand side of 6 of the form 2ik da dz eikz ωt 2ik da dz e ikz ωt. 11 We also choose the dependence Az in such a way as to suppress the resonance term. Writing ˆχ 3 ω, ω, ω = ˆχ 3 ω we get which integrated gives da dz = i3ω ˆχ 3 ω 2c 1 + ˆχ 1 ω A 2 A 12 A = A 0 exp and thus a new wave number may be introduced i 3ω ˆχ 3 ω 2c 1 + ˆχ 1 ω A 2 z c ω kω; A 2 = 1 + ˆχ 1 ω ˆχ 3 ω 1 + ˆχ 1 ω A Using the relation between the wave number k and the refractive index, k = n ω c, the equation allow us to define a nonlinear refractive index nω; A 2 = n 0 ω + n 2 ω A 2, n 0 ω = 1 + ˆχ 1 ω, 2n 0 n 2 = 3ˆχ 3 ω. 15 These general considerations show that even for a weak nonlinearity the propagation of a finite amplitude wave train in the medium is associated with a nonlinear refractive index and a nonlinear dispersion relation. We have to stress the importance of the frequency dependence of the linear susceptibility ˆχ 1 ω. Indeed, if ˆχ 1 doesn t depend on ω, one can see from 9 that k3ω = 3kω and the right-hand side of 7 is also resonant so the wave train would have a more complicated evolution. Usually ˆχω is a complex quantity, ˆχ ω = ˆχ ω + iˆχ ω for ω = ω r + iω i with ω i 0 to ensure the exponential decay of χt when t χt e ωit, and the real and imaginary part of ˆχω are related through the Kramers-Krönig relation causality condition ˆχ ω = 1 π P ˆχ ω ω ω dω, ˆχ ω = 1 π P ˆχ ω ω ω dω. One assumes that the corresponding imaginary part of n 0 ω is small and neglects altogether any imaginary part of n 2 ω. 2 Nonlinear Schrödinger Equation At first let us consider the one-dimensional propagation of a quasi-monochromatic wave ux, t in a weak nonlinear medium Lu = N u, 16 where L x, t is a linear differential operator with constant coefficients and N u a nonlinear operator of u and its derivatives. We assume that the linear problem Lu = 0 has plane wave solutions u = A expiθ + cc., θ = kx ωkt. Here ωk is given by the dispersion relation, solution of the algebraic equation lik, iω = 0, where lik, iω represents the action of L on expiθ, namely L x, expiθ = expiθlik, iω. 18 t We assume as well, that for any integer n 2 link, inω 0 no resonance condition. The quasi-monochromatic wave is written as ux, t = Ak, t exp [ikw ωt] dk + cc., 19 where Ak, t is different than zero on a small compact support around a wave vector k 0 Ak 0 iff k k 0 εk 0 The dispersion relation ωk can be expanded in a Taylor series around the point k 0 and the expression 19 writes ωk = ωk 0 + εk 0 ν = ω 0 + n=1 ε n k n 0 ν n ω n, ν 1 ω n = 1 n! d n ωk dk n 20 k=k0 +1 ux, t = εk 0 exp [ik 0 x ω 0 t] dνak 0 + εk 0 ν 1 exp { i [ k 0 εx ω 1 εtν k0ω 2 2 ε 2 tν 2 k0ω 3 3 ε 3 tν ]} + cc. 21 This discussion allows us to introduce in a natural way the so-called slow variables The solution can also be written as x 1 = εx, t 1 = εt, t 2 = ε 2 t, ux, t = exp [ik 0 x ω 0 t] Ux 1, t 1, t 2,..., where Ux 1, t 1, t 2,... depends only on the slow variables and therefore can be expanded in a power series of ε Ux 1, t 1, t 2,... = n=1 ε n Ψ n x 1, t 1, t 2,.... This procedure, applied to the nonlinear equation 16, will allow us to obtain the correct asymptotic behavior of its solutions, and it is well known in mathematics under the name of multiple scale method [2]. Generally, one considers ux, t = εφ 1 + ε 2 Φ The derivatives /t, /x have to be generalized to include the slow variables 22, namely x x + ε, x 1 Then the linear operator L x, t becomes L x + ε x 1, t t + ε t + ε ε t 1 t 2 + ε t 1 t 2 and can be expanded in a Taylor series about x, t L L x, + ε t ε2 L 11 2 x 2 1 L 1 + L 2 x 1 t L 12 + L 22 x 1 t 1 t Here the indexes 1 and 2 refer to partial differentiation of L with respect to /x and /t respectively L is viewed as a polynomial in /x and /t and differentiated accordingly with respect to these variables. Assuming that the nonlinear operator N u starts with cubic terms Kerr nonlinearity, the substitution of 23 and 24 into 16, in different orders of ε, will lead to Oε : LΦ 1 = 0 Oε 2 : LΦ 2 = Oε 3 : LΦ 3 = 1 2 L 1 L 1 + L 2 x 1 t 1 + L 2 x 1 t 1 L L 12 2 x 1 t 1 + L 22 2 t 2 1 Φ 1 Φ 2 + 2L 2 t 2 Φ 1 + cubic nonlinear terms In the first equation 25 we recover the linear equation with plane wave solutions. 25 Φ 1 = Ax 1, t 1, t 2,... expiθ + cc., with the phase θx, t = kx ωkt, ωk - the solution of the algebraic equation 17 and the amplitude A depending only on the slow variables. Substituting Φ 1 into the second equation 25 we obtain LΦ 2 A A = i l ω l k expiθ + cc. 26 t 1 x 1 We use the fact that Φ 1 depends on the fast variables x, t only through the argument of the exponential and we replace /x by ik, respectively /t by iω; then L 1 e iθ i k lik, iω = il k L 2 e iθ i ω lik, iω = il ω. The expiθ term in the right-hand side of 26 is a secular term and to keep our perturbation calculus valid we impose that Ax 1, t 1, t 2,... evolves according to the equation 27 l ω A t 1 l k A x 1 = The total differentiation of the dispersion relation 17 with respect to k gives dω l k + l ω = 0 dk dω dω 2 d 2 ω l kk + 2 l kω + l ωω + dk dk dk 2 l ω = Using the first equation 29 in 28 we see that Ax 1, t 1, t 2,... depends on x 1, t 1 only through the combination dω ξ = x 1 t 1 30 dk 267 and hence, on the first order slow space-time scales, the wave travels at group velocity dω dk. Since the right-hand side of 26 vanishes, Φ 2 has also the form of a plane wave with an amplitude depending only on the slow variables Φ 2 = Bx 1, t 1, t 2,...e iθ + cc. Introducing the expressions of Φ 1 and Φ 2 in the third equation 25, in order to remove the secular behavior one has to assume that Bx 1, t 1, t 2,... depends on x 1, t 1 through the variable ξ and Aξ, t 2,... satisfies the following relation il ω A t [ l kk + 2 dω l kω + dk which, using the second equation 29, writes i A t dω 2 2 A l ωω] dk ξ 2 + β A 2 A = 0 d 2 ω 2 A dk 2 ξ 2 + γ A 2 A = Here we considered only the simplest form of a cubic nonlinearity u 2 u which does not generate higher harmonics and as before we used the relation 27 to replace the derivative L 11, L 12, L 22 with the corresponding derivatives of lik, iω. The equation 31 is the well-known cubic nonlinear Schrödinger equation, a generic equation describing the propagation of quasi-monochromatic waves in weakly nonlinear media. The literature existent on this topic is very vast and we ll mention just a few references where more informations and applications can be found [1, 3 11]. It represents a completely integrable system with multisoliton solution that can be determined through the Inverse Scattering Transform IST [4 7,9]. The solution depends on the relative sign of α = 2 ω/k 2 and γ. If their product is positivefocusing case bright soliton solutions vanishing at ξ exist. We give below the expression of the one-soliton solution Aξ, t 2 = 2η exp { i [ 2vξ 4v 2 η 2 ]} t 2 + ϕ 0, 32 cosh [2ηξ + 4vt 2 ξ 0 ] where η is the amplitude and v is the soliton velocity. If αγ 0 defocussing Kerr medium darksoliton solution manifest having a finite limit when ξ. 3 Short Pulse Equation SPE The key assumption made in the derivation of the nonlinear Schrödinger equation in nonlinear optics as well as in any other contexts is that the pulse width is large in comparison to the oscillations of the carrier frequency. In most applications this assumption is well satisfied. But with the technological advance of creating shorter and shorter pulses e. g. the chirped pulse amplification technique [12] with durations in the femtosecond range, this assumption is no longer valid. Indeed if a pulse contains only a few oscillations of the carrier wave, the hypothesis of a slowly varying amplitude is meaningless, and new approaches are needed to describe the propagation of these ultra-short pulses in dielectric media. One possibility is to complete the NLS equation with additional higher order nonlinear terms quintic or derivative nonlinear terms [13 15]. Recently a new approach, based on the fact that the pulse is broad in the Fourier space, was developed by several authors [16 19], its merit being that it leads to a new completely integrable equation as the generic equation describing in certain simplifying conditions the propagation of short pulses in cubic nonlinear dielectric media. The generic equation describing the propagation of linearly polarized light of frequency far from any resonant frequencies of the medium, is given in 13. We ll assume that the Fourier transform of the linear susceptibility can be approximated by a polynomial in λ λ being the wavelength, related to the angular frequency ω through the usual relation ω = 2π/λ, namely [18] ˆχ 1 λ ˆχ 1 0 ˆχ 1 2 λ2, where ˆχ 1 0, ˆχ1 2 are constants. Then the Fourier transform of the linear part of 3 writes z x, E 0 2 u x χ1 0 c 2 ω 2 û 2π 2 χ 1 2 û. As for the nonlinear term, we expect that only the instantaneous contribution will affect the propagation of short pulses and thus we consider P NL = χ 3 u 3 with χ 3 constant. Thus the equation under study has the form 2 u x 2 = 1 2 u c 2 l t c 2 u + χ 3 2 u 3 2 t 2, 34 where c l = c/ 1 + ˆχ 1 0, c 2 = 1/ 2π ˆχ 1 2 and we ll take c l = 1 for simplicity. The expression 33 represents the opposite approximation to the one used in the previous section in deriving the NLS equation an expansion in power series of frequency was applied. As discussed in [18] these approximations are well satisfied for infrared light in silica fibers. In the multiple scales analysis of the equation 34 besides the usual slow variables x n = ε n x, 35 an ultra-fast variable T = t x 36 ε is introduced the order of magnitude of the small parameter ε is 1/λ. Then ux, t may be expanded in a power series ux, t = εa 0 T, x 1, x 2,... + ε 2 A 1 T, x 1, x 2, For x = 0 ux = 0, t = εa 0 t/ε + ε 2 A 1 t/ε +... and for functions A 0 t/ε, A 1 t/ε decaying at infinity, the function ux = 0, t vanishes fast enough to describe a real short pulse when ε 1. As t 1 ε T, x 1 ε T + ε +... x 1 it is easily seen that the terms of order ε 1 cancel each other, no terms of order ε 0 exist and in order ε we get 2 2 A 0 = T x 1 c 2 A 0 + χ 2 T 2 A This is the Short Pulse Equation that was sought. With an appropriate scaling and the redefinition of the variables, it may be written in the standard form u xt = u u 3 xx, 39 where the subscripts x, t represent partial derivation with respect to that variable. Soon after this equation was deduced and proven to describe better numerically the propagation of short pulses in weakly nonlinear dielectric media [18,19], it was shown that it is completely integrable [20] and it was solved using the inverse scattering transform method [20,21] Wadati, Konno, Ichikawa variant [23]. An interesting equivalence between SPE and sine-gordon equation SGE was found and various solutions of SGE were used to derive solutions for SPE [23 26]. The Hamiltonian structure and the short pulse equation hierarchy were discussed in [27, 28]. Recently a vector short pulse equation was studied in [29] and an integrable discretization was investigated in [30]. It is worth mentioning that the equation 39 appeared some time before in the differential geometry as one of Rabelo s equations describing pseudospherical surfaces [31 33]. 269 In the followings we shall discuss briefly the equivalence between SPE and SGE, and using simple solutions of SGE we shall derive the corresponding solutions of SPE. According to [25,28] we introduce a new dependent variable r r 2 = 1 + u 2 x 40 and write the equation 39 as Multiplying 41 by u x, and using u xt = u1 + u 2 x u2 u xx. 41 rr t = u x u xt, rr x = u x u xx the SPE 41 may be easily expressed as a conservative law 1 r t = 2 u2 r Let us define the hodograph transformation x 42 Then dy = rdx u2 rdt, dτ = dt. 43 x r y, t τ u2 r y and in terms of the new variables the equations 40 and 41 write r 2 = 1 + r 2 u 2 y r τ = r 2 uu y. 44 We introduce a new dependent variable z = zy, τ through the relation u y = sin z. 45 Using this relation in the first equation 44 we get cos z = 1 r. Differentiating this equation with respect to τ we obtain successively sin zz τ = 1 r 2 r τ = uu y = u sin z, so that u = z t, 46 which introduced in 45 transforms it into the SGE z yτ = sin z. 47 But the hodograph transformation 43 is invertible, namely dx = 1 r dy 1 2 u2 dτ, dt = dτ 48 and xy, τ is determined by solving the following system of linear partial differential equations x y = 1 r = cos z, x τ = 1 2 u2 xy, τ = dy cos z + C, with C an integration constant. Then if zy, τ is a solution of the SGE 47, the corresponding solution of the SPE is given by in parametric form uy, τ = d zy, τ, 50 dτ with τ = t and xy, τ given by 49. The success of the analysis relies on whether one can perform the integration over the variable y in 49. Two solutions of SPE will be determined starting from simple solutions of SGE. As the first example we consider the one-kink solution of SGE 47 zy, τ = zy + τ = 4 arctan [expy + τ]. Using 50 we find 2 uy, τ = coshy + τ. 51 Also the integration in 49 is easily performed giving xy, τ = xy + τ = y + τ 2 tanhy + τ. 52 We used cos 4α = 1 8 sin 2 α cos 2 α, α = arctan ξ, ξ = expy + τ and sin α = ξ/ 1 + ξ 2, cos α = 1/ 1 + ξ 2 2 resulting cos 4α = 1. The solution ux, t is the one loop soliton moving from cosh 2 y+τ right to left with the velocity c = 1. From 51 and 52 we get for t = 0 u x = u y dx/dy = 2 sinh y cosh 2 y 2, so that the solution has two discontinuity points for y = ±y 0, cosh y 0 = 2, x±y see figure 1. It is a multivalued solution and therefore it is not convenient for applications in optical u x,t x Figure 1: One loop solution ux of SPE for t = 0. fibers. A convenient solution may be obtained if one starts from a breather solution of SGE m sin ψ zy, t = 4 arctan, 53 n cosh φ where n = 1 m 2, ψ = ny t, φ = my + t 54 and m is a real parameter, 0 m 1. The solution of SPE is now given by [23, 25] m sin ψ sinh φ + n cos ψ cosh φ uy, t = 4mn m 2 sin 2 ψ + n 2 cosh 2 φ m sin 2ψ n sinh 2φ xy, t = y + t + 2mn m 2 sin 2 ψ + n 2 cosh 2 φ, which is nonsingular and single-valued if m m cr = sinπ/8 [23], representing a true wave packet. A typical pulse is represented in figure 2 for m = 0.27 m cr at two different times u x,t u x,t x x Figure 2: The pulse solution ux, t of SPE for m = 0.27: left t = 4.5, right t = Conclusions In the present paper we discussed two opposite limits of a light pulse propagation in a weak nonlinear dielectric medium. In the first case when the pulse width is large enough, containing many oscillations of the carrier wave, the relevant equation describing its propagation is the well known NLSE. The conditions in which it is derived are quite general and this explains its presence in many areas of physics. In the opposite case, the pulse is so short that it contains just a few oscillations and its propagation is described by the SPE. Although the conditions are less general, the SPE represents an attractive result as it is also completely integrable. An interesting link betwe
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