Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse
generically leaves a two-level medium initially at rest in an excited state
constituted of static solitons and radiation. The soliton birth manifests as
sudden very large variations of the phase of the output pump pulse. This is
proved by building the IST solution of SRS on the semi-line, which shows
moreover that initial Stokes phase flips induce Raman spikes in the pump output
also for short pulse experiments.Comment: RevTex file, 4 page

Leon, J.

Mikhailov, A. V.

English

arXiv

Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse generically leaves a two-level medium initially at rest in an excited state constituted of static solitons and radiation. The soliton birth manifests as sudden very large variations of the phase of the output pump pulse. This is proved by building the IST solution of SRS on the semi-line, which shows moreover that initial Stokes phase flips induce Raman spikes in the pump output also for short pulse experiments

Leon, J

Mikhailov, AV

White Rose Research Online

promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in arXiv.  White Rose Research Online URL for this paper:  http://eprints.whiterose.ac.uk/76126/    Paper: Leon, J and Mikhailov, AV (1997) Raman Solitons and Raman spikes  http://arxiv.org/abs/solv-int/9703007v1   arXiv:solv-int/9703007v1  10 Mar 1997Raman Solitons and Raman spikesJ. Leon and A.V. Mikhailov+Physique Mathe´matique et The´orique, CNRS - UM2, 34095 MONTPELLIER (France)+ Landau Institute for Theoretical Physics, MOSCOW (Russia) andDept. Applied Math., Leeds University, LS2 9JT, LEEDS (U.K.)Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse generically leavesa two-level medium initially at rest in an excited state constituted of static solitons and radiation.The soliton birth manifests as sudden very large variations of the phase of the output pump pulse.This is proved by building the IST solution of SRS on the semi-line, which shows moreover thatinitial Stokes phase flips induce Raman spikes in the pump output also for short pulse experiments.Introduction. Stimulated Raman scattering (SRS) ofhigh energy Laser pulse in two-level medium has been in-tensively studied these last years (see e.g. the review [1]),more especially after the experiments by Dru¨hl, Wen-zel and Carlsten [2] showing that spikes of pump radi-ation (Raman spikes) occur spontaneously in the pumpdepletion zone, see also [3]. Consequently nonlinear Ra-man amplification was used as a means to observe atthe macroscopic level the fluctuation of the phase of theinitial Stokes vacuum [4].The SRS equations possess a Lax pair [5] and, as a con-sequence, theoretical and experimental works were partlydevoted to the search of the Raman soliton predicted in[5]. In particular it has been believed [6] that the spikeof pump radiation observed in [2] is a soliton which, inthe inverse spectral transform scheme (IST), would berelated to the discrete part of the spectrum. However ithas been proved that the observed Raman spike is nota soliton but merely a manifestation of the continuousspectrum [7].Then the question of the creation a genuine Ramansoliton is still an open problem (for 20 years now). Weshow here that the SRS equations, solved as a boundaryvalue problem on the semi-line, do induce the generationof solitons by pair, and that, after the passage of thepulses, the solitons are static in the medium. Althoughthe solitons are not seen in the output pump intensity(where only Raman spikes are seen), the signature ofthe soliton birth consists of very large fluctuations of thephase of the pump output, which remains to be experi-mentaly observed.Last we prove that the Raman spikes, observed in longpulse experiments (pulses with fwhm of 75 ns in [2] and15 ns in [3]), where damping is essential in the model,must occur also for short pulses (no damping), and givethe condition for their creation.The efficiency of the method of solution serves not onlyas a basis for seeking the Raman soliton, but also toevaluate exactly the result of any Raman amplification.Indeed the SRS equations are solved for arbitrary val-ues of the input (x = 0) pump and Stokes pulses on amedium with any initial (t = 0) state. The original prob-lem is reduced to a Riccati equation (11) and linear in-tegral equations (12). In many interesting cases the Ric-cati equation (11) can be explicitly solved (see (15)) andimportant physical information can be obtained withoutsolving equations (12), such as the output (14).The SRS equations. For a medium of length L, a de-tuning ∆ω and a phase missmatch ∆k, in the dimen-sionless variables x = Z/L, t = (cT − ηZ)/L, whereZ and T are the physical variables, the SRS equationsfor short pulses (infinite dephasing time) in the slowlyvarying envelope approximation read [9]∂xaL= qaS e−i∆ωt e2ikx,∂xaS= −q¯aL ei∆ωt e−2ikx, (1)∂tq= −g∫dkaLa∗S ei∆ωt e−2ikx.The phase missmatch is represented there by the dimen-sionless parameter k = 12L(∆k −ηc∆ω). The envelopesaL (frequency ωL) and aS (frequency ωS) are scaled tothe peak intensity Im of the input laser field and thedimensionless coupling constant is given byg =L216Nα′0ǫ0ηmc2ωSωVI2m, (2)where α′0 is the differential polarizability at equilibrium,c/η is the light velocity in the medium of transition fre-quency ωV , and N is the density of oscillators of mass m.We have considered in (1) the cooperative interaction ofall k-components with the medium and then the input(x = 0) values of the light waves are also function of ksharply distributed around k = 0aL(k, 0, t) = JL(k, t), aS(k, 0, t) = JS(k, t). (3)Finally the electromagnetic field is obtained asE =Im2aLei(kLZ−ωLT ) +Im2√ωSωLaSei(kSZ−ωST ) + c.c.The system (1) can be solved on the infinte line forarbitrary asymptotic boundary values [8] and it has beenproved in [7] that this allows the interpretation of theexperiments of [2]. We will demonstrate that it can besolved completely on the semi-line x ∈ [0,∞), which ac-tually furnishes the solution at any point x = L andhence solves the finite interval case as a free end prob-lem.1Lax pair. The time evolution in (1) results as thecompatibility condition Ut − Vx + [U, V ] = 0 for the Laxpair (x ≥ 0, t ≥ 0)ϕx = Uϕ+ ikϕσ3, (4)ϕt = V ϕ+ ϕe−ikσ3xΩeikσ3x, (5)where the dispersion relation Ω is x-independent andU =(−ik q−q¯ ik),V =g4i∫dλλ− k(|aL|2 − |aS |2 2aLa¯Se−2iλx2a¯LaSe2iλx |aS |2 − |aL|2). (6)V (k) is discontinuous, therefore only the limits V ± =V (k± i0) are defined on the real k-axis. Such is also thecase for Ω, and Ω± will have to be determined from theboundary conditions on the solution ϕ.Scattering problem. Following standard methods inspectral theory, see e.g. [10], we define the solutions ϕ+and ϕ− according to(ϕ+11ϕ+21)=(10)−(−∫ x0dξqϕ+21∫ x0 dξq¯ϕ+11e2ik(x−ξ))(ϕ+12ϕ+22)=(01)−(∫∞xdξqϕ+22e−2ik(x−ξ)∫ x0dξq¯ϕ+12)(ϕ−11ϕ−21)=(10)+( ∫ x0dξqϕ−21∫∞xdξq¯ϕ−11e2ik(x−ξ))(ϕ−12ϕ−22)=(01)+(∫ x0 dξqϕ−22e−2ik(x−ξ)−∫ x0 dξq¯ϕ−12). (7)These solutions obey the reduction ϕ¯+1 (k¯) = iσ2ϕ−2 (k),the Riemann-Hilbert relationsϕ−1 = ϕ+1 − e2ikxρ∗ϕ−2 ,ϕ+2 = ϕ−2 + e−2ikxρϕ+1 , (8)and the boundsx = 0 : ϕ+ =(1 ρ0 1), ϕ− =(1 0−ρ∗ 1),x→∞ : ϕ+ →(1/τ 0e2ikxρ∗/τ∗ τ),ϕ− →(τ∗ −e−2ikxρ/τ0 1/τ∗). (9)which define the reflection coefficient ρ and the trans-mission coefficient τ (ρ∗(k) stands for ρ¯(k¯)). Note thatdet{ϕ(x)} = det{ϕ(∞)} = 1, hence for real k |τ |2 =1 + |ρ|2.The column vectors ϕ+1 and ϕ−2 are entire functions inthe k-plane, with good behaviors as k →∞ in the upperhalf-plane for ϕ+1 (and the lower one for ϕ−2 ). The vectorϕ+2 is meromorphic in the upper half-plane with a finitenumber N of simple poles kn (and ϕ−1 in the lower onewith poles k¯n). Consequently τ and ρ have meromorphicextensions in the upper half-plane where they possess theN simple poles kn (the bound states locations).Solution of SRS. From the boundary data (3) we havereadily (set ∆ω = 0 for simplicity because it can be scaledoff in aS)(aLaSe2ikx)= JL ϕ+1 + JS ϕ−2 e2ikx. (10)After careful computations of the boundary values of (5),using separately ϕ+ and ϕ−, we obtain both limits Ω+and Ω− of Ω and the following time evolution of the scat-tering coefficient ρ(k, t):ρt = −ρ2 C+[m∗]− 2ρ C+[φ]− C+[m]. (11)C+[f ] =1π∫dλλ− (k + i0)f(λ),m =iπ2gJLJ∗S , φ =iπ4g(|JL|2 − |JS |2).The reconstruction proceeds through the solution ofthe Riemann-Hilbert problem (8) which readsϕ+1 (k) =(10)+12iπ∫C−dλλ− kρ∗(λ)ϕ−2 (λ)e2iλx,ϕ−2 (k) =(01)+12iπ∫C+dλλ− kρ(λ)ϕ+1 (λ)e−2iλx, (12)where C+ (resp. C−) is a contour from −∞ + i0 to+∞+ i0 (resp. from −∞− i0 to +∞− i0) passing overall poles kn of ρ(k) (resp. under k¯n). Remember thenotation ρ∗(λ) = ρ¯(λ¯).Since m and φ are continuous bounded functions ofk, the coefficients of the Riccati equation (11) are ana-lytic functions in the upper half k-plane, where preciselyρ is meromorphic at time t = 0. Consequently ρ remainsmeromorphic at later times, which allows to prove thatthe reconstructed fields do obey the system of equations(1), or in short that the reconstructed ϕ does obey (9).Indeed from (12), the Cauchy theorem leads to∀x < 0 : ϕ+1 =(10), ϕ−2 =(01). (13)The formulae (11) and (12) with (10) furnish the solu-tion of (1), in particular the output field values at x = L(whatever be q for x > L). In the limit L → ∞, and by(9), the solution becomes explicit, and for instance theoutput pump readsx→∞ : aL →1τJL −ρτJS . (14)2Evolution of laser pulses. We are interested here inthe physically relevant case when the medium is initiallyin the groud state, that is ρ(k, 0) = 0. Next we choosea Stokes wave as a portion of the pump wave, both withLoretzian lineshape, namely, for κ > 0,|JL|2 = |A(t)|2κπ(k2 + κ2), JS = e−γ−iθ(t)JLwhere A(t) is the pulse shape of duration tm (t > tm ⇒A(t) = 0). The parameter γ (real positive constant) mea-sures the ratio pump/Stokes inputs. Considering the casewhen the input Stokes wave experiences one phase flip atsome time t0, we choose the phase θ(t) = 0 for t < t0 andθ(t) = π for t > t0.It follows from contour integration that the Riccatiequation (11) can be rewritten asρt = −i2ge−γk + iκ|A|2[ρ2e−iθ − 2ρ sinhγ − eiθ].Taking into account the phase flip, the general solutionwith zero initial datum readst ≤ t0 : ρ =sinh δcosh(δ − γ), (15)t ≥ t0 : ρ =ρ0 cosh(δ − δ0 + γ)− sinh(δ − δ0)cosh(δ − δ0 − γ)− ρ0 sinh(δ − δ0),with ρ0 = ρ(k, t0), δ0 = δ(k, t0) and the definitionsδ(k, t) =iT (t)k + iκ, T (t) =14g(1 + e−2γ)∫ t0dt′|A|2. (16)Soliton generation and signature. Let us consider firstthe case with no phase flip, that is t0 > tm. The spectraltransform ρ, first function in (15), has an infinite set ofmoving poles kn(t), n ∈ Z, given bykn = −iκ+T (t)(n+ 12 )π − iγ, (17)which are associated with solitons as soon as they lie inthe upper half plane (in the lower half plane they are theresonances).As t evolves, and for a given linewidth κ, these polesmove from the point −iκ and they may cross the realaxis (and generate solitons) if T (t) is large enough, whichmeans enough energy in the input pulses. Moreover, sincekn = −k¯−n−1, solitons are created by pair. After thepassage of the pulse, (t > tm), m and φ vanish in (11),T (t) is constant, and hence the whole solution becomest-independent. Consequently the laser pulses leave in themedium a finite number of static bi-solitons.The question of their observation is not straightfor-ward. Indeed, taking for simplicity the case of an infi-nite medium, the pump output is given in (14) wherethe coefficients ρ and τ are understood for real valuesof k (the essential missmatch). The main point here isthat both 1/τ and ρ/τ are holomorphic functions ion theupper half-plane of k, and hence an integrated intensitylike∫dk|aL|2 will not show the presence of the poles kn.Moreover, if at t = ts a resonance crosses the real axis atks to become a bound state (soliton birth), the outputintensity at that particular value ks will simply show afull depletion (1/τ(ks) = 0 and |ρ(ks)/τ(ks)| = 1).However, the phase of the transmission coefficient τ(k)diverges for all real k precisely at t = ts. Indeed the traceformula (with no pole on the real axis i.e. just before polecrossing) reads for k ∈ R [11]τ(k)|τ(k)|=∏ k − k¯nk − knexp[−i2π∫ln(1 + |ρ(λ)|2)λ− k],where the integral over λ is understood as the Cauchyprincipal value, and where the product concerns only thebound states if any. It is then clear that, for every k 6= ks,the phase of τ(k) diverges as t → ts because ρ(λ) has apole in λ = ks. The observation of true Raman solitonsshould go then with a measure of the phase of the pumpoutput.We remark here that, in the sharp line limit κ → 0and with absense of a flip in the boundary condition, theinfinite set of poles kn lie in the upper half-plane and|kn| is proportional to T (t) (17), which implies that thesolution becomes self-similar (the self-similar solution ofSRS is analized in [12]).Stokes phase flip and Raman spike. As shown in [7],the Raman spike observed in [2] and largely studied later,occurs at time tr for which ρ(k, tr) = 0. Indeed, then|τ(k, tr)| = 1 and from (14) the pump (intensity) is fullyrepleted.The second expression in (15) gives that ρ vanishes attr forcoth(δ0)− coth(δr − δ0) = 2 tanh γ, (18)which relates t0 (instant of phase flip) and tr (instant ofRaman spike) to the pulse area through (16).To get a simple insight in that formula, we may con-sider the dominant term k = 0 (resonance) for which δ0and δr − δ0 are real real positive. Then a Raman spikewill be allowed if (at k = 0)coth(δm − δ0) > coth(δ0)− 2 tanhγ > 0,where δm = δ(k, tm), and tm is the pulse duration. Thisis obtained without damping, which suggests the searchof Raman spikes in short pulse experiments (short withrespect to the medium dephasing time).We note finally that the occurence of a phase flip altersthe denominator of ρ, which can be used to change thediscrete spectrum and hence to modify the soliton partof the solution.3Conclusion. The following results have been ob-tained:1 - The proof that genuine Raman solitons do occur inSRS experiments and cannot be detected in the energyof the output but rather in its phase divergence.2 - The spontaneous generation of solitons out of vacuumin an integrable system.3 - The complete solution by IST of the boundary valueproblem for SRS on the semi-line. This solution relies onthe analytical properties of the nonlinear Fourier trans-form ρ(k, t) which are conserved in the time evolution.4 - The proof that Raman spikes are generic as they oc-cur as well for short duration pulses, as suspected in theexperimental work [3].5 - The proof that in the sharp line limit and with theproportional pump/Stokes input on a medium initiallyat rest, the solution is self-similar.Comments. 1 - The IST solution of SRS on the semi-line has been considered in [13] for the sharp line case.Following the same procedure in our inhomogeneouslybroadened case, and in particular neglecting some termsin the spectral data evolution, would lead to an evolutionfor ρ with non-analytic coefficients, in contradiction withthe necessary meromorphy of ρ.2 - IST on the semi-line for the nonlinear Schro¨dingerequation has been considered in [14], and the problem isconsiderably more difficult mainly because of the neces-sity of the knowledge of additional constraints in x = 0(e.g. the time derivative of the field).3 - Boundary conditions for nonlinear evolution equationsare known to be a source for soliton generation [15] [14],but this is the first instance of an integrable system whichshows explicitely the spontaneous generation of solitons,as motion of poles from the lower to the upper half-plane,from resonances to bound states.4 - At the moment we are not able to extract, from theexplicit form of the spectral data, the explicit form ofthe potential (the medium excitation field) because on thesemi-line no pure soliton exist (would ρ vanish on the realaxis that it would vanish everywhere and no bound stateswould be allowed). However the method provides the ex-plicit form of the physically relevant quantities which arethe light pulses output values.5 - The manifestation of the soliton birth as sudden largevariation of the phase of the output was quite unsus-pected and should lead to reconsider the measures of theoutput in SRS experiments.Aknowledgements. This work has been done aspart of the programm Dynamique nonline´aire et chainesmole´culaires - UM2. A.V.M. is grateful to the Labora-toire de Physique Mathe´matique et The´orique for hospi-tality and to the CNRS for support.[1] M.G. Raymer, I.A. Walmsley, Prog. in Optics, 28, 216(1991)[2] K. Dru¨hl, R.G. Wenzel, J.L. Carlsten, Phys. Rev. Lett.,51, 1171 (1983)[3] D.E. Gakhovich, A.S. Grabchikov, V.A. Orlovich, OpticsComm., 102, 485 (1993).[4] J.C. Englund, C.M. Bowden, Phys. Rev. Lett., 57, 2661(1986); M.G. Raymer, Z.W. Li, I.A. Walmsley Phys. Rev.Lett., 63, 1586 (1989); D.C. MacPherson, R.C. Swanson,J.L. Carlsten, Phys. Rev. A 40, 6745 (1989)[5] F.Y.F. 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Raman Solitons and Raman spikes

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