Ultra-slow light pulses in a nonlinear metamaterial

We find the analytical expression for the threshold intensity necessary to launch ultra-slow light pulses in a metamaterial with simultaneous cubic electric and magnetic nonlinearity. The role played respectively by the permittivity, the permeability, the electric cubic nonlinearity, the magnetic cubic nonlinearity and the pulse duration is clearly identified and discussed.


Introduction
Temporal solitons, i.e. guided light pulses which propagate without dispersion due to the balancing between the group-velocity dispersion (GVD) and the self-phase modulation induced by a Kerr nonlinearity, play a fundamental role in optical communications systems [1,2]. In the past few years, metamaterials, i.e. artificial composites assembled in such a way that they show both an effective electric and magnetic response although the constituent materials are nonmagnetic, have been the subject of intense theoretical and experimental investigations due to their vast range potential applications from super-resolution [3] to cloaking [4]. The study of optical solitons in metamaterials is a new and exciting field of research which has already produced some important theoretical results: we cite for example the possibility to excite bright and dark gap solitons in a metamaterial cavity [5], a new nonlinear Schrödinger equations (NLSE) for metamaterials with only cubic electric nonlinearity [6], a generalized system of two, coupled, spatio-temporal NLSEs for metamaterials with simultaneous cubic electric and magnetic nonlinearity [7].
The aim of this paper is to arrive to an analytical expression for the threshold intensity needed to launch ultra-slow optical soliton in a metamaterial and to clearly put into evidence the role played respectively by the permittivity, the permeability, the electric cubic nonlinearity, the magnetic cubic nonlinearity and the pulse duration. "Slow-light" has recently received a great deal of attention in telecommunications for its numerous applications ranging from all-optical storage to all-optical switching [8]. Before going into the mathematical analysis of the problem we would like to discuss briefly the linear properties of a metamaterial which we describe by a Drude model [3] for both permittivity and permeability: ( ) The material is characterized by an opaque region (no propagative modes allowed) between ω 8 0. / ep mp = ω ω mp <ω<ω ep and two transparent regions (propagative modes allowed) respectively for ω<ω mp and ω>ω ep , note also that V g and n reach values close to zero near the edges of the propagative regions. The two band edges are defined respectively by the conditions ω=ω ep and ω=ω mp . It is indeed near the band edges of the propagative regions that ultra-slow GV temporal solitons make their appearance. In Fig.1 . The zones at the edges of the propagative region where low GV is achieved are also characterized by a strong dispersion that, as we will see later, substantially increases the threshold intensity necessary to launch the fundamental soliton. We also note that in the case of an impedance matched metamaterial, i.e. ep mp ω ω = , the opaque region disappears and, as a consequence, the GV remains substantially close to c over all the spectrum.
The paper is organized as follows: in Section 2 we derive the basic equations and arrive to a system of two NLSEs that couple together the electric and magnetic field, in Section 3 we discuss the analytical solutions and in particular we focus on the ultra-slow soliton solutions near the band edges of the metamaterial, in Section 4 we go to the conclusions.

Basic Equations
Now, let us start our analysis with the Maxwell equations in (1+1)dimension for linearly where Ẽ e H are the complex envelopes respectively for the electric and magnetic fields, ε and μ are respectively the relative permittivity and the permeability of the material, permittivity and permeability which we suppose real quantities, are respectively the cubic electric and magnetic nonlinearity which we suppose for simplicity nondispersive. In the derivation of the approximated expressions for t D ∂ ∂ / and we have neglected: a) the third order time derivative over the field envelopes; b) the first order time Note that by neglecting the third order spatio-temporal derivatives we are implicitly assuming that the electromagnetic pulse is slowly varying both in time and space. While in standard positive index materials (n>1, μ=1) the condition that the pulse is slowly varying in time generally implies that the pulse is also slowly varying in space, in our case the situation is not so simple because of the high dispersion present in the propagative regions near the electric or the magnetic plasma frequency of the metamaterial (see Fig.1), i.e. where slow GV solitons are present. In general the condition of slowly varying envelope in space is broken anytime the pulse extension in the material Δz=V g T 0 (T 0 is the temporal pulse duration) becomes comparable with the wavelength of the pulse λ=2πc/(nω 0 ), in other words Eq.(2) retains its validity when . It is clear that this condition put a lower limit on how short in time a low GV soliton can be. We will come later to a quantitative analysis of this condition. , and finally by neglecting the terms which are of the same order as the third order spatio-temporal derivatives over the envelopes and the first order derivatives over the nonlinear terms, Eq.
(2) can be put in the following form: In arriving to Eq.(3.1) we have also used the following equalities: (3. where T 0 is the pulse duration, L D is the so-called second-order dispersion length [1,2] and Z is the impedance of the medium which obviously is always positive. In the new variables, Eqs. (3) become: where ) sgn( 2 β stands for the sign of the GVD parameter.

Ultra-slow solitons
The fundamental soliton solutions of Eqs(4) can be easily found by noting that this system of two coupled NLSEs can be decoupled into a single NLSE by using the following transformations: . The transformations used to decouple the system of Eq.(4) u u Zu u = = 2 1 , 7 have an obvious physical meaning: the electric and the magnetic field must be proportional each other through the impedance Z of the medium, as one may expect. The single NLSE in the u variable takes the following form: The fundamental soliton solutions of Eq.(5) are well known [1,2].
and . Dark soliton:  of the pulse mus be T 0 >>2πc/(V g nω 0 )~100ps, which means at least ~1ns pulses. There is one last issue that deserves to be discussed in some detail and this is the effect of higher order dispersion terms. As we have already remarked in the manuscript, we are operating near the band edges of the structure, i.e. in regions of extremely high dispersion, as it is evident, for example, from Fig.1(b) where the GVD parameter β 2 is represented. On the other hand, in order to arrive to analytical solutions, we have neglected the third and higher order dispersion terms. Now, in conventional fibers, third and higher order dispersion terms are generally small and can be treated by perturbation approaches for a pulse duration T 0 >>1ps [1,2]; in our case, given the remarkably high dispersion present near the band edges, we may expect that much longer pulses than those of a conventional fiber are needed in order to reduce the effects of the third and high order dispersion. We would like to remind the reader that we have already set up a lower limit to the pulse duration, i.e. T 0 >>2πc/(V g nω 0 ), based on the requirement that the pulse must be slowly varying in space; in our case, for a soliton with a group velocity 10 c, this condition calls for pulses that are at least 1ns in duration. Let us here concentrate on the third order dispersion in particular. In order to consider the third order dispersion as a small perturbation we need to impose the condition that where δ 3 is the third order dispersion parameter: ) 6 /( Here we consider for simplicity the case that the pulse is tuned near the high frequency band edge and the electric nonlinearity is positive. It can be demonstrated by using a perturbation approach [2,13] that the fundamental bright soliton solution modified by the third order dispersion can be written in the following form: In other words, what the third order dispersion do luen the tual GV of t soliton, GV which takes the following form: es is to inf ce ac he bright 2 0 where clearly V g =1 β Similar results to those exposed above can be expected if ω 0 is tuned near ω mp , except that in this latter case bright solitons can be launched when ( ) 0 3 < χ μ .Here we have explicitly analyzed the case: ω <ω . In the opposite situation: ω >ω , we would have bright solitons at . A final note regarding the absorption of the metamaterial. In this work we have neglected the absorption of the metamaterial. In currently metam e absorption is still so high that it is premature to think about any practical soliton application. Nevertheless, in principle, there is nothing that prevents the possible availability of low-loss metamaterials in the near future.

Conclusions
In conclusion, we have performed an analytical study on the possibility to available aterials in the near infrared [12] th excite ultra-slow band-edges of a metamaterial. We have investigated the role played respectively solitons near the by the permittivity, the permeability, the electric cubic nonlinearity, the magnetic cubic nonlinearity and the pulse duration. We hope that our results may stimulate further research aiming at the study of this new class of materials in applications which involve "slow light", such as all-optical buffering and switching for example.    Fig.4: vs.