Abstract
Solitons are of fundamental importance in photonics due to applications in optical data transmission and also as a tool for investigating novel phenomena ranging from light generation at new frequencies and wavetrapping to rogue waves. Solitons are also moving scatterers: they generate refractive index perturbations moving at the speed of light. Here we found that such perturbations scatter light in an unusual way: they amplify light by the mixing of positive and negative frequencies, as we describe using a first Born approximation and numerical simulations. The simplest scenario in which these effects may be observed is within the initial stages of optical soliton propagation: a steep shock front develops that may efficiently scatter a second, weaker probe pulse into relatively intense positive and negative frequency modes with amplification at the expense of the soliton. Our results show a novel alloptical amplification scheme that relies on soliton induced scattering.
Introduction
If compared to the welldeveloped field of traditional light scattering in which the medium is at rest, little attention has been devoted to the physics of scattering from a moving medium, in particular from a relativistically moving medium. Here we consider the remarkable ability of solitons to generate a copropagating refractive index inhomogeneity that propagates at relativistic speeds. The basics of scattering from a timechanging boundary were discussed in detail by Mendonça and coworkers (see e.g.^{1} and references therein). Examples of such “time refraction” have been predicted and observed from a moving plasma front^{1,2,3} and in waveguide structures^{4,5,6}. Recently, the nonlinear Kerr effect, i.e. the local increase of the medium refractive index induced by an intense laser pulse^{7}, was proposed to induce a moving refractive index inhomogeneity within a dispersive medium such as an optical fibre^{8}. The laser pulse induced relativistic inhomogeneity (RI) was then described in terms of a flowing medium in which the analogue of an event horizon may form^{8} and applications such a optical transistors have been proposed^{9}.
Intense laser pulses are also known to scatter from the selfinduced travelling RI: this selfscattering process leads to the resonant transfer of energy from the laser pulse to a significantly blueshifted peak, often referred to as resonant radiation (RR) or “optical Cherenkov” radiation^{10,11,12,13,14}. A recent discovery highlighted an additional scattered mode, further blueshifted with respect to the RR, identified as a mode excited on the negative frequency branch of the medium dispersion relation and therefore named “negative resonant radiation” (NRR)^{15}.
By applying a first Born approximation analysis, supported by numerical simulations, we show that the scattering of light by a soliton induced RI reveals a novel alloptical amplification mechanism. A schematic representation of the specific process we are considering is shown in Fig. 1: an incoming laser pulse, IN, interacts with a copropagating inhomogeneity, or scatterer. The momentum conservation law that governs the scattering process predicts that light may resonantly scatter into two output modes, RR and NRR (see SI for details). In the laboratory reference frame, both of these modes will have positive frequencies while in the reference frame comoving with the scatterer, RR is positive and NRR is negative valued. The key point in the following is precisely the presence of these two modes: mixing between the RR and NRR modes leads to a novel amplification process.
Results
First born approximation
In scattering theory, the first Born approximation allows us to estimate the elements of the scattering matrix S, which connects asymptotic input and output states. Kolesik et al. have shown that the positive RR mode is captured in great detail by the Born scattering approach^{16}. Here we extend this theory in order to also include the possibility of scattering to the NRR mode and find that the amplitude of each scattering channel is proportional to the amplitude of the Fourier component of the RI, (see SI): and, accounting also for the negative frequency branch, we similarly obtain: where ν is the velocity of the travelling RI. These relations (1) and (2) state that energy may be transferred to two modes, RR and NRR, and in order to do so the spectrum of the scattering potential must have nonzero Fourier components at frequencies equal to the relative distances between ω_{IN} and the resonant frequencies. In other words, the RI must present a sufficiently steep gradient so as to effectively excite the desired output modes. We now observe that we may derive a photon number balance equation by generalizing the ManleyRowe relation, adopted e.g. in nonlinear optics^{7}, to the case of a moving scatterer^{17,18}. We find that: where RR^{2} and NRR^{2} are the photon numbers of the RR and NRR modes normalized to the input photon number, IN^{2}. The negative sign in front of the NRR^{2} photon number is a direct consequence of the fact that the NRRmode has negative frequency in the comoving reference frame (see SI). So the difference between the normalized number of photons has to be equal to the photon number in the input mode. As a consequence, the total output photon number, RR^{2} + NRR^{2} > 1, i.e. we have amplification. The scattering process mediated by the travelling RI will amplify photons as a result of the coupling between the positive and negative frequency modes.
Numerical simulations
We verified our predictions by numerically simulating the one dimensional propagation of a scalar field propagating along the z direction^{19} in a transparent and isotropic dielectric medium (diamond and fused silica) that is arranged to scatter from a copropagating inhomogeneity (see Methods). We consider three separate cases: (i) the underlying physics are first exemplified by taking a “toymodel” situation in which the RI is simply a variation in the linear refractive index; (ii) the RI is generated by an actual solitonlike laser pulse through the nonlinear Kerr effect and light from the soliton itself is selfscattered; (iii) the RI is generated by a solitonlike pulse and a second, very weak probe pulse is scattered by the soliton.
(i)  “Toymodel” RI. The RI is simulated as a linear propagating refractive index inhomogeneity that moves with a speed ν = 1.23 × 10^{8} m/s that is just slightly slower than the group velocity of an input probe pulse in the medium, ν_{g} = 1.25 × 10^{8} m/s. In this case, as a dielectric medium we choose diamond, which does not exhibit any resonances over a very broad bandwidth (from less than one Terahertz through to ultraviolet wavelengths). The probe pulse is very weak so that no nonlinear effects are excited and the physics are dominated solely by linear scattering from the RI. Figures 2(a)–(b) show the evolution along propagation of the envelope (plotted in the frame moving at the RI velocity) and of the spectrum, respectively. We launch an input Gaussian probe pulse (carrier wavelength 2 µm, pulse width 20 fs), together with a superGaussian shaped RI (see Methods), with amplitude δn_{0} = 0.08 and superGaussian order m = 40 corresponding to a rising time ~ 3 fs; σ = 50 fs. This ensures a wide RI Fourier spectrum , so as to meet both conditions (1) and (2). After ~ 0.2 mm the probe pulse reaches the moving RI [whose position is highlighted in Fig. 2(a) by the vertical black dotted lines] and slows down due to the RI increase of refractive index. Scattering occurs towards two blueshifted resonant modes, RR and NRR, which due to dispersion both travel slower than the RI. The medium dispersion relation is shown in Fig. 2(d) in (ω′, ω) coordinates where the comoving frequency ω′ = γ(ω − νk); k = ωn(ω)/c is the medium dispersion relation and , being c the speed of light in vacuum. In the comoving frame, the absolute value of scattered mode frequencies are determined by the input mode comoving frequency (horizontal solid and dashed lines, see SI): the corresponding laboratory reference frame frequencies can be seen to be in perfect agreement with the numerically observed values.
In order to verify the photon number amplification predicted by the ManleyRowe relation (3), we evaluated the photon number evolution, as shown in Fig. 2(c) for two different RI gradients, obtained by varying the order of the superGaussian function that describes the RI, m = 40 and m = 200 (these correspond to rise times from the background refractive index to the maximum of the RI of 3 fs and 0.7 fs, respectively). Dotted, solid and dashed lines correspond to the normalized photon numbers in the IN mode, total output photons RR^{2} + NRR^{2} and output photon difference RR^{2} − NRR^{2}, respectively. As can be seen, the difference in photon numbers is conserved while the sum of photon numbers is larger than 1 (amplification) and increases with increasing RI steepness. This simplified “toymodel” therefore gives a direct confirmation of the predictions based on the first Born approximation model.
(ii)  Solitoninduced RI. The RI is physically generated by an intense solitonlike laser pulse through the nonlinear Kerr effect, i.e. δn = n_{2}I, where I is the peak intensity and n_{2} the nonlinear Kerr coefficient (see Methods). The soliton may interact with the selfinduced refractive index variation and scatter into new output modes. As a Kerr medium we choose common fused silica glass, whose dispersion relation is well known^{20}. We launch a solitonlike mode in the anomalous dispersion region, with central wavelength 2 µm, pulse width 7 fs and input intensity 10 TW/cm^{2}. Figures 3(a)–(b) show the envelope and spectral evolution, respectively, along propagation over a distance z = 0.4 cm. The input pulse initially selfsteepens, forms a shock front and, at z ~ 0.3 mm, when conditions (1)–(2) are met, energy starts to resonantly transfer from the input mode towards two blueshifted modes, RR and NRR, both travelling slower with respect to the input soliton. The soliton frequency is redshifted due to a recoil effect^{10} and subsequently emerges with a slightly lower propagation velocity. In Figure 3(c)–(d) we compare the nonlinear propagation with a numerical simulation in which the nonlinear source term has been replaced with a RI in the linear polarization with the same amplitude, steepness and propagation velocity of the soliton induced perturbation of Figs. 3(a)–(b). We observe a nearly perfect agreement with the nonlinear case confirming that soliton shedding of RR and NRR modes is nothing more than a specific realization of the more general relativistic scattering process. The “toymodel” RI scattering is indeed able to capture all the essential features of the resonant energy transfer, and coupling between positive and negative frequency modes.
(iii)  Probe pulse scattering from a solitoninduced RI. Finally, we consider the case in which the soliton is accompanied by a second, delayed probe pulse. This second pulse is much weaker and therefore does not form a soliton or excite any nonlinear Kerr effects. Figure 4 shows the results for a 1.9 µm central wavelength probe pulse, pulse width 8 fs and input intensity 5 × 10^{9} W/cm^{2} (complete movie showing the full dynamics is also included). The probe pulse is scattered by the soliton induced RI shown in Fig. 3(a)–(b). The initial probe pulse delay, t_{0} = 15 fs, is adjusted such that it encounters the soliton when it forms the steepest shockfront and is then scattered relatively efficiently, simultaneously spectrally recoiling to a slightly redshifted wavelength and emitting two blueshifted RR and NRR modes. In Fig. 4(c) we show the overall photon number evolution for the probe pulse (black solid line) and for the soliton pulse (blue dashed curve). In order to highlight the crossscattering dynamics, the two curves have been normalized at each propagation distance with respect to two independent simulations for the probe and soliton pulses alone. We thus note that the overall probe pulse photon number increases by nearly 1%, followed by some weaker oscillations that are originating from a remnant of the probe pulse that is trapped and thus continuously interacts with the soliton [see Fig. 4(a), trapped light is visible for z > 1.5 mm, around t ~ 0 fs]. This clear photon number increase thus indicates that solitoninduced scattering in this regime, very differently from standard crossphase modulation, may lead to true amplification of the weaker pulse.
Discussion
Summarising, we have shown that a RI amplifies and scatters light to higher frequencies. Likewise, if the probe pulse were to be reduced to the level of quantum fluctuations, we may expect to see the RI excite the vacuum states. Bearing in mind that the motion of photons in the vicinity of a RI may also be described in terms of an effective curved spacetime metric^{21,22}, we expect that measurements of vacuum fluctuations excited by a RI would give an experimental direct window into the physics of the quantum vacuum in curved spacetimes^{23,24,25}.
These results therefore open the perspective for novel alloptical control schemes that may be implemented in a wide variety of geometries and applications together with a novel numerical and experimental approach for the investigation of fundamental phenomena.
Methods
Numerical model
The code adopted for the numerical results presented in the main text is a 1D Unidirectional Pulse Propagation Equation (UPPE) solver^{19}, where the spectral components E_{ω}(z) of the real field E(z, t), are obtained by Fourier transform and obey to the following model equation, in the pulse or inhomogeneity comoving frame: being the vacuum permittivity, c the speed of light in vacuum, and P_{ω} the spectral component at frequency ω of the realvalued polarization as obtained by Fourier transform.
Dispersion of the medium is described by k_{z}(ω) and the shift −ω/v indicates that Eq. (4) is solved in the local frame of the scatterer. Below we report the “linear” (i) and “nonlinear” (ii) implementation of the polarization source term.
(i)  “Linear source term”. For the numerical results shown in Fig. 2 and in Fig. 3(c)–(d), the polarization source term is implemented as a linear relativistic inhomogeneity (RI), travelling at velocity v along the z direction, i.e., where the background term, n_{0}, is the refractive index at the input carrier frequency and τ = t − z/v is the local time coordinate. The “artificial” RI (δn) has a superGaussian longitudinal profile of amplitude δn_{0} and thickness σ: δn(τ) = δn_{0} exp[−(τ − t_{0})^{m}/σ^{m}]; being m a positive even number identifying the Gaussian order and t_{0} the initial temporal position.
(ii)  “Nonlinear source term”. The numerical model is extended to include the nonlinear case presented in Fig. 3(a)–(b); that is the case in which the inhomogeneity is induced by an intense soliton through the nonlinear Kerr effect. The polarization source term is where is the complex valued analytical signal and gives the dimensionless refractive index variation.
The probe pulse scattering by the solitoninduced RI, is modelled by solving a second equation for the real field of the probe pulse, E_{2}(z, t), coupled to the equation for the soliton (4): The coupling is indeed given by the “crossphase modulation” term of the nonlinear polarization, i.e., , where is the complex valued analytical signal of the pump (soliton), as in the previous case.
We intentionally neglect all the other nonlinear terms, e.g. third harmonic generation and Raman scattering, in order to highlight the generation of the RR and NRR modes. Moreover, we recall that, in order to capture the coupling between positive and negative frequencies, the model equation (and especially the source term), must be implemented by calculating the ωfrequency component of the real polarization P(z, t). This is achieved for each propagation step by means of back and forth Fourier transforms allowing for an evaluation of the real electric field E(z, t) from its complex spectral components E_{ω}(z).
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Acknowledgements
D.F. and U.L. acknowledge financial support from the Engineering and Physical Sciences Research Council EPSRC, Grants EP/J00443X/1 and EP/J004200/1.
Author information
Affiliations
Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria, Via Valleggio 11, IT22100 Como, Italy
 E. Rubino
 , A. Lotti
 & S. L. Cacciatori
Centre de Physique Theorique CNRS, Ecole Polytechnique, F91128 Palaiseau, France
 A. Lotti
 & A. Couairon
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo 32, 20133 Milano, Italy
 F. Belgiorno
School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
 U. Leonhardt
School of Engineering and Physical Sciences, SUPA, HeriotWatt University, Edinburgh EH14 4AS, UK
 D. Faccio
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Contributions
D.F. and E.R. developed the ideas and performed the numerical simulations. A.L. and A.C. implemented the numerical codes, F.B. and S.C. provided the mathematical models. All authors contributed to the manuscript preparation.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to D. Faccio.
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