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Optical Event Horizons from the Collision of a Soliton and Its Own Dispersive Wave S Optical event horizons from the collision of a soliton and its own dispersive wave S. Wang, Arnaud Mussot, Matteo Conforti, A. Bendahmane, X. Zeng, Alexandre Kudlinski To cite this version: S. Wang, Arnaud Mussot, Matteo Conforti, A. Bendahmane, X. Zeng, et al.. Optical event horizons from the collision of a soliton and its own dispersive wave. Physical Review A, American Physical Society, 2015, 92 (2), 10.1103/PhysRevA.92.023837. hal-02388991 HAL Id: hal-02388991 https://hal.archives-ouvertes.fr/hal-02388991 Submitted on 2 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 92, 023837 (2015) Optical event horizons from the collision of a soliton and its own dispersive wave S. F. Wang,1,2 A. Mussot,1 M. Conforti,1 A. Bendahmane,1 X. L. Zeng,2 and A. Kudlinski1,* 1Laboratoire PhLAM, UMR CNRS 8523, IRCICA, USR CNRS 3380, Universite´ Lille 1, 59655 Villeneuve d’Ascq, France 2Key Laboratory of Specialty Fiber Optics and Optical Access Network, Shanghai University, 200072 Shanghai, China (Received 22 May 2015; published 20 August 2015) We observe experimentally the spectral signature of the collision between a soliton and the dispersive wave initially emitted from the soliton itself. This collision, interpreted in terms of an optical event horizon, is controlled by the use of an axially varying fiber which allows us to shape both the soliton and dispersive wave trajectories so that they both collide at a precise location within the fiber. The interaction of the dispersive wave with the soliton generates a reflected wave with a conversion efficiency which can be controlled by the input pump power. These experimental results are confirmed by numerical solution of the generalized nonlinear Schrodinger¨ equation and by the analytical calculation of the conversion efficiency. DOI: 10.1103/PhysRevA.92.023837 PACS number(s): 42.65.Tg I. INTRODUCTION enhances the deceleration of the soliton [17], reshapes the DW trajectory in the time domain, and also modifies the PM Event horizons can be mimicked in optical fibers by the condition [Eq. (1)] so that the DW collides with the soliton in nonlinear interaction of a weak linear radiation (usually termed the vicinity of its group velocity matching (GVM) wavelength. the probe wave) with an intense copropagating soliton [1–3]. This allows us to choose the fiber length at which the FEH is So-called fiber event horizons (FEHs) occur when the probe observed, so that the entire process occurs within a few meters. wave, traveling at a different group velocity with respect to The paper is organized as follows. In Sec. II we review the the soliton, is unable to pass through it during their collision. physics of a fiber event horizon in the framework of the theory The probe wave is therefore reflected onto the soliton, which of soliton-DW mixing that gives an analytical estimation of acts as a nonlinear barrier, altering its group velocity. In the the conversion efficiency. In Sec. III we describe the principle spectral domain, this results in the frequency conversion of the of the experiment, whose results are reported in Sec. IV and probe wave at ω into a reflected wave (RW) at ω = ω that P R discussed in Sec. V. We draw our conclusions in Sec. VI. satisfies the phase-matching (PM) condition [4–6] = D(ω) D(ωP ), (1) II. FIBER EVENT HORIZONS AND SOLITON–DISPERSIVE WAVE INTERACTION THEORY where D(ω) = β(ω) − β(ωS ) − β1 × (ω − ωS ) denotes the wave number in a reference frame moving with the soliton, FEHs can be rigorously described by means of the theory = β(ω) is the fiber propagation constant and β1 ∂ωβ(ωS )is of the mixing between a soliton and a DW [5]. We consider the group velocity at the soliton frequency ωS . The analogy the nonlinear Schrodinger¨ equation (NLSE) with third-order between event horizons and the nonlinear reflection of a weak dispersion (TOD): probe onto a soliton has attracted much interest over the past few years and opens innovative perspectives in the control β2 β3 i∂ u − ∂2u − i ∂3u + γ |u|2u = 0. (2) of light [7–10], in quantum physics [11,12], as well as in z 2 t 6 t superfluidity [13]. The few nonlinear optics experiments on Raman effect and dispersion terms with an order higher than 3 the subject were performed by causing the collision of an are neglected in order to keep the analysis as simple as possible intense soliton with a weak probe carefully adjusted at the and to consider only the essential physical effects. Indeed, it fiber input in terms of power, wavelength detuning, and delay has been shown that the Raman deceleration does not play a [1,3,14–16]. However, it is still unclear whether this analogy fundamental role in the process [3]. with event horizons persists when the probe parameters Consider the solution as the sum of a fundamental soliton are not specifically controlled, as was the case in previous u and linearly dispersing waves g [5]: experiments. s iks z Here, we show experimentally that this process is actually u(z,t) = us (t)e + g(z,t), (3) very robust, as FEHs can be observed from the collision √ = =| | 2 = between a soliton and the dispersive wave (DW) emitted where us (t) P sech(t/T0), P β2 /(γT0 ), and ks directly from this soliton at the early stage of propagation. γP/2. By inserting Eq. (3)inNLSEEq.(2) and linearizing, at The soliton is excited by an ultrashort pulse and generates a first order we obtain an evolution equation for the perturbation phase-matched DW acting as the probe wave. The collision β β − 2 2 − 3 3 + between the soliton and this DW is controlled by using a i∂zg ∂t g i ∂t g photonic crystal fiber (PCF) with a longitudinally varying 2 6 β3 ∗ dispersion landscape. The varying dispersion along the fiber − i ∂3|u |eiks z + γ |u |2g e2iks z + 2γ |u |2g = 0. (4) 6 t s s s Consider now that DWs are the sum of the resonant radiation *Corresponding author: [email protected] emitted from the soliton and other linear waves (i.e., the probe) 1050-2947/2015/92(2)/023837(6) 023837-1 ©2015 American Physical Society S. F. WANG et al. PHYSICAL REVIEW A 92, 023837 (2015) 80 D(ω) TW 40 D(ωp) Collision RW 2ks - D(ωp) ks (ω) 0 D ωGVM Fiber length -40 DW emission Wavelength -80 Fiber diameter Time -40 -20 02040 ω FIG. 2. (Color online) Schematic illustration of the FEH exper- iment using an axially varying fiber. The gray region corresponds FIG. 1. (Color online) Graphical solution of Eqs. (6)–(8)fora to the shaping of the DW trajectory induced by the axial variation toy model with the following values of (nondimensional) parameters: of fiber dispersion. TW, transmitted wave through the soliton (same = =− = β2 1, β3 0.1, γ 1. wavelength as DW); RW, reflected wave on the soliton. generated elsewhere: and the wave number of the linear waves is fixed by the − = =| | 2 g(z,t) = weiD(ωp )z iωp t + ψ(z,t), (5) approximated dispersion relation (TOD 0) as k β2 /2 ( is the frequency shift from ωGV M ). where D(ω) is the linear dispersion relation that, when The reflection and transmission coefficients can be calcu- = 2 + 3 considering TOD only, reads D(ω) β2ω /2 β3ω /6. We lated analytically for this potential as [19] can recognize three source terms in the second line of Eq. (4) √ that can force the system efficiently if the following resonance cosh2 15 π conditions are fulfilled [5]: ρ() = 2 √ , (13) 2 + 2 15 sinh (πT0) cosh 2 π = 2 D(ω) ks, (6) sinh (πT0) τ() = √ . (14) 2 2 15 sinh (πT0) + cosh π D(ω) = 2ks − D(ωp), (7) 2 Interestingly, with our approximations, the conversion effi- = D(ω) D(ωp). (8) ciency ρ does not depend on β2 for a fundamental soliton. Equation (6) describes the condition for generating the Equations (13) and (14) have been obtained before in Refs. standard resonant radiation (RR) [18], whereas Eqs. (7) and (8) [3,16] by resorting to the concept of comoving-frequency describes two additional peaks generated by four-wave mixing conservation. Our calculations show that we can rigorously between the soliton and the probe wave. It is easy to see that describe FEHs in the frame of NLSE, without evoking any only Eq. (8) can describe interactions near the GVM point relativistic argument. Equations (13) and (14) indicate that a perfect horizon ωGV M (see Fig. 1). If we consider a probe whose frequency is near GVM frequency, we can study the behavior of the with 100% reflection is only realized with an incident wave = generated radiation by investigating the following reduced precisely at the GVM frequency ( 0). However, in this equation, where only the source term that is almost in phase case the probe wave and soliton propagate at exactly the same (resonant) with the radiation has been considered: velocity, and they would require infinite distance to interact. This suggests that an event horizon in a strict sense is unlikely β2 2 β3 3 2 to be observed in optical fibers where a fundamental soliton i∂zψ − ∂ ψ − i ∂ ψ + 2γ |us (t)| ψ = 0.
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